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Photonic bandstructure in semiconductor slab waveguides with strong two-dimensional texture Pacradouni, Vighen 2006

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PHOTONIC B A N D S T R U C T U R E IN SEMICONDUCTOR SLAB WAVEGUIDES WITH STRONG TWO-DIMENSIONAL T E X T U R E by V i g h e n Pacradouni B . S c , M c g i l l University, 1989 B . E n g . , M c g i l l University, 1991 M . S c , University of B r i t i s h Columbia, 1993  A THESIS S U B M I T T E D IN P A R T I A L F U L F I L L M E N T O F THE REQUIREMENTS FOR T H E D E G R E E OF Doctor of Philosophy in T H E F A C U L T Y OF G R A D U A T E STUDIES (Physics) T H E UNIVERSITY OF BRITISH C O L U M B I A A p r i l 2006 © V i g h e n Pacradouni, 2006  Abstract Fabrication techniques were developed and used to realize sub-100 nm-thick semiconductor membranes that contain two-dimensional (2D) square arrays of holes that completely penetrate the membrane. W i t h the pitch of the arrays on the order of 500 nm, broadband optical reflectivity spectra obtained from these "2D planar photonic crystal membranes" reveal resonant features w i t h energies and linewidths that disperse as the in-plane momentum (angle of incidence) of the exciting radiation is varied i n the near-infrared. A numerical solution of Maxwell's equations was developed to interpret these reflectivity spectra i n terms of a "photonic bandstructure" for electromagnetic modes localized i n the vicinity of the porous membrane. A complete understanding of the complex (energy and lifetime) dispersion and polarization properties of this interesting new class of optical materials was obtained. It is shown that this bandstructure can be largely explained by applying symmetry considerations to the renormalization of slab waveguide modes by the high index contrast 2D scattering potential.  ii  Table of Contents  Abstract  ii  Table of Contents  iii  List of Tables  vii  List of Figures  viii  Acknowledgements  xxiii  Dedication  xxiv  1 Introduction  1  2 Modelling  19  2.1  Introduction  19  2.2  Description of Finite Difference R e a l Space Approach  20  2.2.1  Scattering from a periodically textured layer  20  2.2.2  Discretization and Integration of Maxwell's Equations i n R e a l Space 27  2.2.3  Eigenmodes on a Discrete Mesh  34  2.2.4  Significance of Using Pendry Modes  39  2.2.5  Computation of Reflection and Transmission Coefficients  42  2.2.6  Buried Gratings  48  2.3  Results  51  2.3.1  Introduction  51  2.3.2  Near-Normal Incidence Reflectivity from Untextured Uniform Slab  51  iii  2.4  2.3.3  B o u n d Modes of Untextured Slab  52  2.3.4  Slab Waveguide w i t h I D G r a t i n g  55  2.3.5  Waveguide w i t h 2D weak 2nd order grating  62  2.3.6  2D Strong Texture  69  Convergence  75  2.4.1  Convergence of the Integration Scheme Alone  77  2.4.2  Overall Convergence Behaviour for Textured Structures  78  2.5  Comparison w i t h Green's Function Based M o d e l  87  2.6  Significance of Modification to Pendry's M e t h o d  92  2.7  Summary  96  3 Fabrication  98  3.1  Introduction and Overview  3.2  Processing Techniques and Equipment  102  3.2.1  Electron B e a m Lithography  102  3.2.2  Etching  106  3.2.3  Oxidation  Ill  3.2.4  O p t i c a l Microscopy  113  3.2.5  A t o m i c Force Microscope  114  3.3  3.4  98  Fabrication Procedure  115  3.3.1  S E M measurement of Layer Structure  115  3.3.2  Cleaving  115  3.3.3  Lithography  117  3.3.4  Etching  119  3.3.5  Oxidation  120  Fabrication Results  121  iv  4  Optical Experiments  127  4.1  Introduction  127  4.2  Experimental Apparatus  128  4.2.1  131  Fourier Transform Spectrometer  4.3  Alignment  132  4.4  A n g u l a r Resolution  135  4.5  Experimental Procedure  135  5 Results  137  5.1  Introduction  137  5.2  Resonant Reflectivity  137  5.3  Polarization Properties  139  5.4  Dispersion  140  5.5  Fitting with Model  143  5.6  E x t r a c t i o n of M o d e Frequency and L i n e w i d t h from Spectra  145  5.6.1  E x t r a c t i o n of Frequency and L i n e w i d t h from Experimental Spectra 145  5.6.2  E x t r a c t i o n of M o d e Parameters from M o d e l Spectra  146  5.7  Bandstructure  149  5.8  Linewidths  151  5.8.1  L i n e w i d t h of Lowest Energy B a n d  151  5.8.2  Linewidths of 4th and 5th Bands  154  5.9  Effect of F i l l i n g Fraction  155  6 Discussion  158  6.1  Introduction  158  6.2  Resonant Features i n Reflectivity  159  6.3  Polarization Properties and Basic Dispersion  160  6.4  Gaps and Symmetry Influences on Linewidths  165  v  6.5  Effect of F i l l i n g Fraction  166  6.6  Gaps A w a y from Zone-Centre  169  6.7  Extrinsic Influences on Linewidths  172  7 Summary and Conclusions  175  7.1  Summary  175  7.2  Conclusions  177  7.3  Future Directions  178  Bibliography  180  A Generalization of FDRS Method to an Arbitrary 2D Lattice  184  vi  List of Tables  3.1  N o m i n a l and measured thicknesses of layers i n sample A S U 721  116  3.2  E t c h conditions  120  3.3  Pitches and doses specified for e-beam lithography  123  3.4  Extent of oxidation and yield  124  3.5  Pitches and hole radii and corresponding air filling fractions estimated from A F M profiles from three gratings having nominally the same pitch but different electron beam exposure times  5.1  126  M o d e l , Nominal, and Measured sample structure parameters for grating 22 on sample A S U 721  145  vii  List of Figures  1.1  Schematic diagram of 3D photonic crystal concept. R is any one of the lattice vectors that define the crystal symmetry  1.2  2  Photonic bandstructure arising from 3D face-centred-cubic photonic lattice comprised of non-spherical "atoms" and dielectric contrast of — = 13.0 [1]  3  1.3  2D photonic crystal "waveguide" w i t h tight bend  5  1.4  Scheme invented by Yablonovitch [2] for fabricating a 3D photonic crystal which exhibits a complete gap. It consists of machining three sets of cylindrical airholes separated by 120° into a block of material w i t h sufficiently high dielectric contrast at a 35° angle on a 2D periodic grid. The resulting interconnecting network of voids forms a face-centred-cubic ( F C C ) lattice w i t h non-spherical "atoms"  1.5  7  Schematic diagram of an asymmetric 3-layer slab waveguide w i t h 2 D photonic crystal embedded i n core  9  1.6  Schematic diagram of an asymmetric 3-layer slab waveguide  1.7  Schematic dispersion diagram of the modes of an asymmetric 3-layer slab  10  waveguide. The solid line represents the light line for the core material. T h e dot-dashed line represents the line light-line for the superstrate, i.e., top semi-infinite layer (air). T h e dotted line represents the light line for the substrate, i.e., b o t t o m semi-infinite layer 1.8  10  Schematic diagram of 2D photonic crystal w i t h defect embedded i n slab waveguide  14  viii  2.1  Schematic diagram of a two-dimensionaUy periodically textured slab structure  2.2  21  Discrete in-plane (x — y) lattice on which E and H ' fields are computed on a series of planes separated by g/2 = 7. The dielectric constant is sampled on the lattice indicated by the black dots. T h e fields are calculated on the sublattice indicated by the crossed squares. T h e planes shown span a unit cell of the periodicity i n the in-plane direction. The light and dark grey shading of the planes indicates whether the E or H ' fields are computed there. T h e white hole i n the center schematically depicts dielectric texture comprised of a circular airhole  2.3  33  Scheme of real-space points on which the fields and dielectric constant are sampled on the unit cell. T h e dielectric constant is sampled at points represented by the black dots. T h e electric or magnetic fields are sampled only on those points w i t h the crossed squares  2.4  43  Scheme of reciprocal lattice vectors corresponding to in-plane wavevector of launched modes  44  2.5  2D grating embedded i n an arbitrary multilayer dielectric stack  49  2.6  Component scattering matrices for calculation of overall reflection (scattering) matrix from grating embedded i n an arbitrary multilayer dielectric stack  2.7  50  Waveguide consisting of 120 nm-thick slab of material w i t h dielectric constant e' = 12.25  2.8  52  Specular reflectivity spectrum for light incident at 10° (from the normal) on a 120 nm-thick slab of material of dielectric constant e' = 12.25. The solid curve depicts the spectrum for the s polarization; the dashed curve depicts the spectrum for the p polarization  ix  53  2.9  P l o t of mode frequency versus in-plane wavevector, (3, for the lowest order T E (solid curve) and T M (dotted curve) modes of a planar waveguide consisting of a 120 n m slab of material w i t h e' = 12.25, clad above and below by air (e' = 1). The dot-dashed lines represent the lightlines for these materials. The solid vertical line depicts the part of LO — /3 space that is explored i n the calculation of the spectra shown i n Figure 2.10 below. The dots at the intersection of the vertical line w i t h the dispersion curves indicate the appearance of a resonant feature i n the spectra  54  2.10 Specular reflectivity spectrum for evanescent field incident on 120 n m slab of material w i t h e' = 12.25, w i t h in-plane wavector of 20, 000 c m  - 1  . The  solid line is for s polarized light, the dashed line is for p polarized light. .  55  2.11 Schematic dispersion plot for single-mode slab waveguide w i t h shallow I D grating etched into its surface. T h e dashed lines represent the light lines for the low-index cladding layer (larger slope) and the high-index core layer (smaller slope). T h e solid lines represent the dispersion of Bloch states i n the presence of the grating  56  2.12 Slab waveguide w i t h "weak" I D texture:  the structure consists of a  120 nm-thick slab of e' — 12.25 material, clad above and below by air, w i t h a 10 n m deep I D square-toothed grating having a pitch A = 500 n m and an air filling fraction of 25% etched into the surface 2.13 Calculated specular reflectivity spectrum for evanescent w i t h in-plane wavevector /3 = (PQ/2)X  = (TT/A)X.  56 light incident  The solid curve is for s  polarized light, the dashed curve is for p polarized light. T h e corresponding waveguide structure is depicted i n Figure 2.12  x  57  2.14 (a) Calculated specular reflectivity spectrum for evanescent light incident w i t h in-plane wavevector, (3 = I.OIPGX,  i.e., slightly detuned from  the boundary between the 2nd and 3rd B r i l l o u i n zones, (b) Calculated specular reflectivity spectrum for light incident w i t h in-plane wavevector, (3 = O.OIPGX,  i.e., slightly detuned from the B r i l l o u i n zone-centre,  (c) Calculated 1st order diffraction coefficient, i.e., coefficient for scattering from a field component at (3 = O.OIPGX  to a field component at  (3 = 1.01/5GX. The solid curves are for s polarized light; the dashed curves are for p polarized light. T h e waveguide structure under study is depicted i n Figure 2.12 2.15 (a) Calculated specular reflectivity spectrum for evanescent light incident w i t h in-plane wavevector, (3 = PG , at the edge of the 1st B r i l l o u i n zone, X  (b) Calculated specular reflectivity spectrum for radiation incident w i t h zero in-plane wavevector, i.e., at the B r i l l o u i n zone-centre, (c) Calculated 1st order diffraction coefficient, i.e., coefficient for scattering from a field component at (3 = 0.01PG  X  to a field component at (3 = I.OIPQX.  The  solid curves are for 5 polarized light; the dashed curves are f o r p polarized light. The waveguide structure under study is depicted i n Figure 2.12. . . 2.16 Calculated near-normal incidence specular reflectivity spectrum for light incident w i t h in-plane wavevector, j3 = O.Ol/fcz, for s polarized light. The waveguide structure is identical to the one specified i n Figure 2.12 except that the grating has an air filling fraction of 75% rather than 25% 2.17 Slab waveguide w i t h "strong" I D texture: the structure consists of a 120 n m thick slab of e' = 12.25 material, clad above and below by air, w i t h a I D square-toothed grating that completely penetrates the slab. grating has a pitch, A = 500 nm, and an air filling fraction of 25%  The  2.18 Calculated near-normal incidence specular reflectivity spectrum for s polarized light incident w i t h in-plane wavevector, /3 = 0.01/?GX. T h e waveguide structure, depicted i n Figure 2.17, consists of a 120 n m thick slab of e' = 12.25 material, clad above and below by air, completely penetrated by I D square-toothed grating having a pitch, A = 500 nm, and an air filling  fraction of 25%  63  2.19 Slab waveguide consisting of 125 n m of e' = 12.25 material w i t h air above and below, having a square lattice of cylindrical airholes, 25 n m deep, embedded at its centre. T h e lattice has a pitch of A = 500 nm, and the radius, r, of the holes is r / A = 0.10  64  2.20 Schematic in-plane bandstructure of resonant modes of waveguide w i t h a square grating shown i n 1st 2D B r i l l o u i n zone 2.21 Symmetry directions for the square lattice  64 65  2.22 Calculated specular reflectivity spectrum for light incident on a 125 n m slab waveguide of e' = 12.25 material w i t h air above and below, and a square lattice of cylindrical airholes, 25 n m deep, embedded at its centre. T h e lattice has a pitch of A = 500 nm, and the radius, r, of the holes is r / A = 0.10. Solid lines are for the s polarization, while dashed lines depict the p polarization. T h e spectra were calculated for light incident w i t h in-plane wavevector fixed at P/PG = 0.01 i n the T — X direction, which is defined i n Figure 2.21  66  2.23 Expanded view of the spectrum depicted i n Figure 2.22: (a) lowest frequency cluster (b) intermediate frequency cluster (c) high frequency cluster  67  2.24 Calculated specular reflectivity spectrum from same structure as i n F i g ure 2.23. The spectra were calculated for light incident w i t h zero in-plane wavevector, P/PG = 0.00, corresponding to normal incidence  xii  68  2.25 Calculated specular reflectivity spectrum from same structure as i n F i g ure 2.23.  T h e spectra were calculated for light incident w i t h in-plane  wavevector fixed at P/PG = 0.01 i n the [x + y] direction (i.e., along the r — M symmetry direction) 2.26 Calculated specular reflectivity spectrum for light incident on a 125 n m slab waveguide of e' = 12.25 material w i t h air above and below, and a square lattice of cylindrical airholes, 25 n m deep embedded at its centre. T h e spectra were calculated for light incident w i t h in-plane wavevector fixed at P/PG — 0.001 i n the x direction ( X symmetry direction).  The  lattice has a pitch of A = 500 nm, and the radius, r, of the holes is: (a) r / A = 0.25 (b) r / A = 0.32 (c) r / A = 0.40 2.27 Strongly 2D textured t h i n asymmetric slab waveguide: the structure consists of a 80 nm-thick slab of e' = 12.25 material which forms the guiding core, clad above by air and below by 2 pm of material w i t h e' = 2.25. T h e semi-infinite half-plane below is composed of e' = 12.25 material. T h e core waveguide layer is penetrated by a 2D square array of holes on a pitch, A = 500 nm, and w i t h hole radii given by r / A = 0.5 2.28 Calculated near-normal incidence specular reflectivity spectrum for light incident w i t h in-plane wavevector, P = 0.03/?G, along the T — X direction for the structure depicted i n Figure 2.27 w i t h r / A = .2821. T h e solid line represents the s polarized spectrum, the dashed line depicts the p polarized spectrum 2.29 Calculated near-normal incidence specular reflectivity spectrum for light incident w i t h in-plane wavevector, P = 0.03/?G, along the T — M direction for the structure depicted i n Figure 2.27 w i t h r / A = .2821. T h e solid line represents the s polarized spectrum, the dashed line depicts the p polarized spectrum  xiii  2.30 Calculated specular reflectivity spectra for light incident on the structure depicted i n Figure 2.27 w i t h r / A = .2821. (a) N o r m a l incidence specular reflectivity, i.e.., light incident w i t h (3 = 0. (b) Reflectivity for s polarized evanescent light incident w i t h (3 = (3QX 2.31 Logarithmic plot of the absolute value of the error on the bound mode frequencies of an untextured slab waveguide as calculated by the F D R S method as a function of the number of mesh points i n each of the x and y directions included i n the calculation. T h e (artificial) unit cell for the F D R S method was set to a square of side A = 500 nm, and the in-plane wave vector of the incident evanescent light was set to (3- = 2ir/A. mc  The  structure used for the calculation consisted of a 120 n m slab of material w i t h e' = 12.25 clad above and below by air (e' = 1.0).  T h e different  curves (lines) are for increasing numbers, (specifically: 12, 22, 32, 42, 62, and 122), from top to b o t t o m on the graph, of mesh points i n the z direction 2.32 Slab waveguide w i t h "strong" I D texture: the structure consists of a slab of thickness d of e' = 12.25 material, clad above and below by air, w i t h a I D square-toothed grating having a pitch, A = 500 n m , and an air filling fraction of 25% that completely penetrates the slab 2.33 M o d e frequency (a) and w i d t h (b) vs.  the "number of elements"  (see  text) used i n F D R S calculation for the structure depicted i n Figure 2.32 w i t h d = 80 nm. T h e results are for the second-order lossy mode at the zone-centre. Plots w i t h round markers and solid lines depict results w i t h anti-aliasing spectral filtering of the e' — / / profile implemented; plots w i t h square markers and dashed lines depict results without filtering of the e' — / / profile  2.34 (a) M o d e frequency and (b) width, vs.  square of the x — y mesh size  normalized by the unit cell size (or grating pitch) used i n the F D R S calculation for the structure depicted i n Figure 2.32 w i t h d = 80 nm. The results are for the second-order lossy mode at the zone-centre. The insets provide a magnified view of the small mesh size regions of their respective graphs. The F D R S calculation was carried out w i t h anti-aliasing spectral filtering of the e' — u! profile implemented. T h e dashed lines represent the best fit line obtained from a linear least squares fit. In (b) the rightmost three points were ignored i n the fitting 2.35 Strongly 2D textured t h i n asymmetric slab waveguide: the structure consists of a 80 nm-thick slab of e' = 12.25 material which forms the guiding core, clad above by air and below by material w i t h e' = 2.25. The core waveguide layer is penetrated by a 2D square array of holes on a pitch, A = 500 nm, and w i t h hole radii given by r / A = 0.2821 2.36 M o d e frequency (a) and w i d t h (b) vs.  square of the x — y mesh size  normalized by the unit cell size (or grating pitch) used i n the F D R S calculation for the structure depicted i n Figure 2.35. T h e results are for the second-order lossy mode at the zone-centre.  T h e F D R S calculation  was carried out w i t h anti-aliasing spectral filtering of the e' — / / profile implemented. T h e mesh sizes correspond (from left to right in the graph) to 21 x 21, 19 x 19, 17 x 17, 15 x 15, 13 x 13, 11 x 11, and 9 x 9 elements used i n the calculation  xv  2.37 Near-normal incidence specular reflectivity spectra for light incident w i t h in-plane wavevector 3 = 0.02/?G along the Y — X direction calculated w i t h the F D R S model (lower plot) and the G F model (upper plot). T h e parameters used for the F D R S model are as described i n section 2.4.2 using 17 elements. T h e G F calculation included the nine reciprocal lattice vectors at 0, ±PG%, ±3ai),  ±PG% ± PGV- The solid lines represent the s polar-  ized spectra, and the dashed lines represent the p polarized spectra. T h e structure being modelled is the same as the one depicted i n Figure 2.27. The resonance widths and shape are i n excellent agreement. T h e mode frequencies agree to w i t h i n 1.5%  90  2.38 L o c a t i o n of the upper and lower edge of the 2nd order gap i n a I D textured symmetric waveguide as a function of guide and grating thickness, d (see Figure 2.32). The waveguide core is made of material w i t h e' = 12.25 and clad above and below by air (e' = 1.0).  T h e grating, which completely  penetrates the waveguide core, has an air filling fraction of 25%.  The  dashed lines are for the calculation w i t h the G F model, the solid lines w i t h the F D R S model which was verified to be converged to w i t h i n 1% for the upper band and 0.2% for the lower band (using 17 "elements" i n the calculations). The G F calculation included the three reciprocal lattice vectors at 0 and ±PG&  91  xvi  39 Calculated normal incidence specular reflectivity spectra for light incident w i t h zero in-plane wavevector.  T h e calculation is performed (a)  using Pendry's original approximations for k and (b) the approximations for k implemented i n this work. T h e solid and dashed lines, which overlap completely here, represent the s polarized and p polarized spectra respectively. T h e light is incident on a film of material having a dielectric constant of 12.25 textured w i t h a square array of 300 n m diameter cylindrical holes filled w i t h a material having a dielectric constant of 2.25, all on a substrate w i t h a dielectric constant of 2.25. T h e pitch of the grating is 500 n m .40 Calculated near-normal incidence specular reflectivity spectrum for light incident w i t h in-plane wavevector, f3 — 0.01/?G, along the F — X direction. T h e calculation is performed (a) using Pendry's original approximations for k and (b) the approximations for k implemented in this work. The solid lines represent the s polarized spectra, the dashed lines depict the p polarized spectra. T h e light is incident on a film of material having a dielectric constant of 12.25 textured w i t h a square array of 300 n m diameter cylindrical holes filled w i t h a material having a dielectric constant of 2.25, a l l on a substrate w i t h a dielectric constant of 2.25. T h e pitch of the grating is 500 n m .1  Scanning electron microscope ( S E M ) image of freestanding a i r / G a A s / a i r slab waveguide patterned w i t h 2D square lattice of airholes  .2  H i g h magnification ( S E M ) image of the same structure as i n Figure 3.1 .  .3  Schematic diagram of oxide-based high-index contrast waveguide w i t h strong 2D texture  xvii  3.4  (a) F i e l d emission scanning electron microscope column, corresponding to the column of the Hitachi S-4100 S E M : (1) Electron source (2) B e a m paths showing the effect of successive apertures i n the column, (3) Apertures (4) Condenser Lens (5) Deflection Coils (6) Objective lens (7) Specimen; (b) Magnetic Lens detail showing how the magnitude of the axial magnetic component B  z  varies i n the lens region; (c) Detail showing geometry of  beam aperture angle a at the sample  105  3.5  Schematic diagram of E C R plasma etcher  108  3.6  O x i d a t i o n Setup  112  3.7  Schematic diagram of A F M tip operation and position detection  114  3.8  A s grown sample layer structure for A S U 721  116  3.9  Top view of grating 22 on sample A S U 721 as imaged by the optical m i croscope. T h e size of the grating is 90 / m i x 90 //m. Some processing was performed on the image to enhance the contrast between the oxidized region, which extends out around the grating and is slightly darker compared to the unoxidized parts of the sample  3.10 4 iim x 4 pm A F M image of grating 22  122 125  3.11 Section through diameter of holes i n image i n Figure 3.10. T h e arrows indicate the leading and trailing edges of the depression i n the profile, which are used to obtain a measurement of the diameter of the holes at the surface 4.1  126  Schematic diagram of scattering geometry used i n specular reflectivity measurement.  T h e dimensions of the sample and pattern depicted are  the nominal values for grating 22 on sample A S U 721  128  4.2  Schematic diagram of experimental setup  129  4.3  Schematic diagram of sampleholder  131  4.4  Schematic diagram of F T I R spectrometer  133  xviii  5.1  Experimental (a) and calculated (b) specular reflectivity spectra on grating 22 at incidence angle of 9 = 5° and 0 = 0° for p-polarization of reflected light  5.2  138  Experimental (a) and calculated (b) specular reflectivity spectra on grating 22 at incidence angle of 9 = 5° and 0 = 0° for s-polarization of reflected light  5.3  139  Experimental (a) and calculated (b) specular reflectivity spectra on grating 22 at incidence angle of 9 = 5° and 4> = 45° for p-polarization of reflected light  5.4  140  Experimental (a) and calculated (b) specular reflectivity spectra on grating 22 at incidence angle of 9 = 5° and <f> = 45° for s-polarization of reflected light  5.5  141  Experimental specular reflectivity spectra for s polarization at various incident angles, 9, along the T — X direction (0 = 0). The plots have been offset from one another for the sake of clarity; the vertical scale on the reflectivity traces is linear  5.6  142  F i t to specular reflectivity spectrum obtained from grating 22 obtained w i t h light incident at 9 = 50° along the T — X direction. The solid line represents the experimental spectrum while the dashed line represents the "best fit" spectrum generated by a Fano lineshape function summed w i t h an A i r y function. The initial fit to the Fabry-Perot fringes was performed over a wider range of energies (6000 c m  - 1  to 11,500 c m ) not shown - 1  here. The fit using a Fano lineshape and A i r y function was performed using data points i n the range 6000 c m  xix  - 1  - 9000 c m  - 1  147  5.7  F i t to spectra generated by the G F model, (a) Specular reflectivity spect r u m for reference (b) 1st order scattering to 1st order scattering +3QX field component.  —QG  X  field component (c)  In (b) and (c) the symbols  correspond to the model values, and the smooth lines are the result of a fit w i t h a Fano lineshape function.  T h e spectra are for 10° angle of  incidence along the T — X direction 5.8  148  M o d e frequency versus magnitude of in-plane wavevector for experiment (solid lines and markers) and model (dotted lines and hollow markers) for s (circles) and p (squares) polarized radiation. T h e inset schematically depicts the bandstructure for the case of weak texture  5.9  150  Experimental (filled circles and solid lines) and model (filled squares and dotted lines) linewidths of the lowest frequency band appearing along the T - X direction i n Figure 5.8  152  5.10 Experimental (dots and solid lines) and model (open circles and dotted lines) linewidths of 5th lowest (a), and 4th lowest (b) bands i n Figure 5.8 vs.  in-plane wavevector along the T - X direction. The crossed circle  represents a zero linewidth that was inferred from the absence of any feature i n the spectrum  155  5.11 Experimental specular reflectivity spectra for s polarized light (solid lines) and p polarized lines (dotted lines) (a) grating 22 (b) grating 23 (c) grating 24 which have the same pitch but different air filling fractions of (a) 0.28 (b) 0.33 (c) 0.49. T h e spectra have been offset vertically from one another for the sake of clarity. T h e vertical scale on the reflectivity traces is linear. T h e dark, thick lines have been drawn i n by hand to serve as a guide to the eye 6.1  157  Schematic dispersion of resonant states attached to the core of a slab waveguide w i t h square photonic lattice  xx  160  6.2  Dominant Fourier field components of eigenmodes i n lowest four bands along the T — X  (9 = 0) direction depicted i n Figure 5.8: the solid  black arrows represent the propagation vectors associated w i t h the i n plane variation of the field components; dashed lines represents the electric fields and their x and y components; the grey arrows represent the incident in-plane wavevector, i.e., detuning from zone-centre 6.3  161  Dominant Fourier field components of eigenmodes i n the four bands along the V — M direction depicted i n Figure 5.8: the solid arrows represent the propagation vectors associated w i t h the in-plane variation of the field components; the grey arrows represent the incident in-plane wavevector, i.e., detuning from zone-centre; the dashed lines represents the electric fields and their x and y components  6.4  163  Calculated specular reflectivity spectra for s polarized (solid lines) and p polarized (dotted lines) for 9 = 2°, <j> = 0 w i t h (a) r / A = 0.29 and (b) r / A = 0.33. T h e other parameters i n the model are as the fitted results for grating 22. T h e curves have been shifted vertically for clarity  6.5  167  Schematic depiction of the i n plane electric field intensity distribution over the unit cell for a I D grating of the symmetric, lossy (left) and antisymmetric, non-lossy (right) band edge states for a grating w i t h (a) 25% (b) 50% (c) 75% air filling fraction. In (b) the lossy and lossless modes b o t h modes have the same fraction of their total field energy (area under the curve) i n the high-index region. The dielectric profile is indicated by the dashed line  169  xxi  6.6  Dominant Fourier field components of eigenmodes i n the 4 t h and 5th lowest energy bands along T — X direction i n Figure 5.8 i n the vicinity of their anti-crossing: the solid arrows represent the propagation vectors associated w i t h the in-plane variation of the field components; the grey arrows represent the incident in-plane wavevector, i.e., detuning from zone-centre; dashed lines represents the electric fields a n d their x a n d y components  6.7  171  Scattering amplitudes calculated using the F D R S Code for s polarized light incident at /3  inc  « 0.25/?G£ corresponding to 9 = 25° along the  r — X direction, where the anti-crossing occurs. T h e various plots depict the scattering amplitudes to s polarized light w i t h in-plane wavevector: ( ) Pine (b) Pinc-Pcx a  (f) f3  inc  (c) (3 ±(3 y inc  G  (d) i3  inc  + p x (e) f3 G  inc  +  p [-x±y] G  + P [x ±y}. The structure is the same as that for grating 22 w i t h G  the parameters as obtained from fitting w i t h the G F code A.l  173  Scheme of real-space points on which the fields and dielectric constant are sampled o n the unit cell. The dielectric constant is sampled at points represented by the black dots. The electric or magnetic fields are sampled only o n the points w i t h squares. A i and A represent the primitive lattice 2  vectors of the photonic crystal A.2  187  Scheme of reciprocal lattice vectors corresponding to in-plane wavevector of launched modes for photonic crystal w i t h oblique 2 D lattice  xxii  188  Acknowledgements I would like to thank my contemporaries i n the Photonic Nanostructures G r o u p at U B C , postdoctoral researchers, M a n o j Kanskar, P a u l Paddon, and Jennifer Watson, graduate students, A l e x Busch, Jody Mandeville, Francois Sfigakis, M u r r a y M c C u t c h e o n , and A l l a n Cowan, as well as U B C Advanced Materials and Process Engineering Laboratory research engineers, J i m M a c K e n z i e and A l Schmalz, for their instruction, insights, industry, and cooperation on various scientific and technical problems relevant to this dissertation. I would also like to thank Shane Johnson, Simon Watkins, and T o m Tiedje for growing the samples used throughout the course of the research described herein. I would like to gratefully acknowledge the N a t u r a l Sciences and Engineering Research C o u n c i l of Canada, the B C Science Council, and the Canadian Cable Labs Fund, for funding portions of my doctoral studies and research program. A s well, I would like to thank my research supervisory committee members, M a r k Halpern, L o m e Whitehead, Jochen Meyer, D a v i d Jones, and B i l l McCutcheon, for their helpful advice and questions over the years. I am also very grateful to B i l l M c C u t c h e o n and Jess Brewer for their detailed comments on the penultimate draft of this dissertation. Finally, I would like to thank, my research supervisor, Jeff Young, for all of his valuable advice, criticism, guidance, and mentoring, as well as for the energy, enthusiasm, and diligence he has devoted to the supervision of this dissertation.  xxiii  Dedication  To my loving parents and late grandmother.  xxiv  Chapter 1 Introduction  Photonic crystals are structures i n which the dielectric constant is given a periodic spatial variation, or "texture", as, for example, depicted schematically i n Figure 1.1. For electromagnetic radiation w i t h a wavelength on the order of the spatial period of the dielectric variation, this texturing can be used to modify and control the optical properties of the photonic crystal. The one-dimensional ( I D ) version of this concept, the dielectric stack, has existed for some time and forms the basis of applications like Distributed Bragg Reflector ( D B R ) mirrors and optical filters. However, it is fair to say that the field of photonic crystals was invented by Yablonovich and J o h n (independently) [3, 4]. They generalized the concept of dielectric texture to two dimensions (2D) and three dimensions (3D) and realized that this texturing could drastically modify the totality of allowed electromagnetic states.  Further, they recast the resulting electro-  magnetic ( E M ) Maxwell's equations for light i n a periodic dielectric into the formalism of existing condensed matter band theory for electrons i n a periodic atomic potential. Here, the periodic dielectric plays the role of a periodic potential that couples the freely propagating electromagnetic eigenmodes into photonic B l o c h states that can be used to describe the E M excitations of the photonic crystal i n analogy to the electronic B l o c h states formed i n "normal", i.e., atomic, crystals. Figure 1.2 depicts the photonic bandstructure i n a face-centred-cubic photonic crystal [1]. A t the B r i l l o u i n zone boundaries, the superposition of forward propagating and Bragg scattered radiation results i n standing wave eigenstates w i t h zero group velocity i n the direction perpendicular to the zone boundaries. For a definition of B r i l l o u i n zones, B l o c h states, and their importance i n solving problems involving potential functions w i t h 1  e(r)= e( r + R)  Figure 1.1: Schematic diagram of 3D photonic crystal concept. lattice vectors that define the crystal symmetry.  2  R is any one of the  X  W  K„  Figure 1.2: Photonic bandstructure arising from 3D face-centred-cubic photonic lattice comprised of non-spherical "atoms" and dielectric contrast of — = 13.0 [1].  discrete translation symmetry see [5, 6]. T h e periodic dielectric generally alters the dispersion and eigenstates everywhere in the B r i l l o u i n zone. T h e stronger the dielectric contrast, the stronger the scattering and the greater the modification to the photonic dispersion. One particular feature of this altered dispersion is that frequency gaps open up, at least along a given crystal symmetry direction, as depicted in Figure 1.2.  In other words, there exists a frequency band  w i t h i n which there are no classical propagating electromagnetic modes w i t h wavevectors along the given symmetry direction.  Furthermore, if all of the gaps i n the various  3  symmetry directions overlap partially, as they do here, then the region of complete overlap is said to form a complete photonic bandgap.  T h i s means that no classical  propagating radiation w i t h a frequency w i t h i n the bandgap is allowed inside the crystal, and consequently, that externally impinging radiation will decay exponentially w i t h distance into the crystal. The existence of a complete gap requires a crystal structure w i t h a high degree of symmetry, as well as a large dielectric contrast i n the texturing. The symmetry causes the energies of the modes not to vary greatly i n different directions, while the high dielectric contrast causes the gaps that open up to be quite large. The combination of these two effects, then, increases the likelihood for the gaps i n a l l of the different directions to overlap i n some frequency band. A material w i t h a full photonic bandgap is appealing as a platform for b o t h research as well as novel device applications. If a point defect is introduced into the otherwise infinite periodic lattice, then it is possible to localize light to the defect site and thereby trap it. Due to the full bandgap that exists w i t h i n the surrounding infinite photonic lattice for some range of frequencies, the light trapped at the defect cannot propagate away. T h e defect site can be thought of as a high quality cavity surrounded by perfect, omnidirectionally reflecting mirrors. If an electronic resonance, such as that occurring near the bandedge of an embedded quantum well or quantum dot, is tuned to the energy of the defect mode, then there is only a single photon mode that can interact w i t h the electronic resonance. T h e sub-micron size of the localized photons means that the coupling of the electronic transition and the photon can be very strong, on the order of an m e V , making these structures interesting for the study of strong coupling effects [7, 8, 9]. It has been suggested that near-thresholdless lasers [3] might be realized i n such systems, where there are effectively no continuum electromagnetic modes to compete w i t h the lasing mode below threshold. W h i l e the introduction of a point defect into a photonic crystal can create a trap for light, a line defect can localize light into a channel. Once again, any light having  4  ••mr © c m w• m m o • •)  • • •• • • •• # ®m •  m •  (ft \^  mm  © ®mm m m m m m mo © o m o o o o o J o © O " ~) o © o o o o o o 1  w  o o © 'j o o © o ©  Figure 1.3: 2D photonic crystal "waveguide" w i t h tight bend.  a frequency w i t h i n the photonic bandgap of the surrounding photonic crystal cannot escape.  If the defect channel is properly designed, the light can only propagate along  the channel, even when the channel undergoes a tight bend, as depicted i n Figure 1.3 for the case of a 2D photonic crystal. Such lossless guiding around tight bends cannot be achieved w i t h conventional index-guided waveguides, as the critical angle of total internal reflection is exceeded if the radius of curvature becomes too small. T h e ability of photonic crystal waveguides to guide light nimbly around corners on a photonic chip could be an essential tool i n the large scale integration of photonic circuits. Even i n the absence of a complete photonic bandgap, periodic dielectric texture confers the ability to drastically modify the dispersion of photons i n a medium, beyond and separate from that attainable using composition control i n untextured bulk material. T h i s has already found application i n dispersion compensation elements and compact delay components i n optical fibre communication systems [10]. Furthermore, enhancement of gain i n laser structures and of non-linear optical cross-sections, e.g., for sum  5  frequency generation, have been predicted for weakly dispersive photons possessing small group velocities [11, 12]. T h e properties of 2D and 3 D photonic crystals have been studied theoretically for pure dielectrics [1, 2, 13, 14, 15, 16, 17, 18]. T h e y have also been verified experimentally i n the microwave regime, where the lattice constants of the structures realized by assembling rods or machining slabs of polystyrene are on the order of centimeters. The bandstructure depicted i n Figure 1.2 actually represents a lattice formed by machining three sets of cylindrical airholes separated by 120° into a block of material w i t h a dielectric constant e' = e/e = 13.0 (where e is the permittivity of free space) at a 35° angle 0  0  on a 2D periodic grid (see Figure 1.4). T h e resulting interconnecting network of voids forms a face-centred-cubic ( F C C ) lattice w i t h non-spherical "atoms". Similar structures have found application i n the microwave region as high-Q niters and broadband antennae platforms [19]. Semiconductors, i n many ways, make ideal hosts for photonic crystals. T h e refractive index difference between air and several important group I V and III-V semiconductors, A n ~ 2.5, is sufficient to allow the formation of complete optical bandgaps using appropriate lattice geometries [1]. Furthermore, artificial semiconductor quantum wells, wires and dots can be formed i n a number of ways to achieve tunable electronic properties (resonant frequencies, densities of states) throughout the near-infrared part of the spect r u m . In fact, it was recognized from the outset that incorporating photonic crystals into semiconductor hosts, rich i n electronic resonances and ubiquitous i n optical communication systems, would have even greater scientific and technological implications in the fields of quantum electronics and opto-electronics. However, the texture on the order of 200 - 500 n m required to operate i n the near-infrared part of the spectrum has proven to be extremely challenging to realize i n three dimensions. Several strategies have been suggested and attempted to date. Cheng and Scherer [20] have used an ion beam to m i l l three sets of cylinders, as i n Yablonovitch's scheme  6  Figure 1.4: Scheme invented by Yablonovitch [2] for fabricating a 3D photonic crystal which exhibits a complete gap. It consists of machining three sets of cylindrical airholes separated by 120° into a block of material w i t h sufficiently high dielectric contrast at a 35° angle on a 2D periodic grid. The resulting interconnecting network of voids forms a face-centred-cubic ( F C C ) lattice w i t h non-spherical "atoms".  7  depicted i n Figure 1.4, into a G a A s / A l G a A s heterostructure through a 2D periodic mask of circular holes on a sub-micron pitch. Broadband transmission measurements d i d suggest that these structures exhibit an optical gap near normal incidence consistent w i t h model calculations. However, the limitations of ion milling preclude making the structures any deeper. A more straightforward but tedious approach involves layer by layer assembly of semiconductor rods made by forming a surface grating on a sacrificial liftoff layer [21]. T h e textured surface is then turned upside down and fused to a previouslylaid down set of rods rotated by 90°. The sacrificial layer is etched away, freeing the rods from the substrate and leaving the I D array of rods fused to the layer previously laid down. A l t h o u g h this technique addresses the challenge of achieving high-contrast dielectric texture normal to the semiconductor wafer surface, it does not seem practical for reproducibly fabricating photonic crystals on a large scale. Another approach involves self-assembly of monodispersed opal or polystyrene spheres w i t h lattice constants from ~ 250 - 500 n m followed by infiltration of the resulting voids w i t h a high-index material. However, this fabrication method produces crystals w i t h lattice imperfections that result i n poor stop-bands [22]. In summary, despite considerable effort, a good-quality bulk 3D photonic crystal useful i n the optical or near-infrared spectrum is still not available. In contrast, 2D patterning of semiconductors, via nanolithography and etching, and epitaxial growth of multilayer semiconductor waveguides are mature technologies.  It  may be possible, then, to access many of the potentially useful and interesting properties of photonic crystals simply by etching a 2D array of holes i n a planar semiconductor waveguide. Such a planar waveguide-based photonic crystal is depicted i n Figure 1.5. A n additional advantage to this approach is that it is compatible w i t h the vast majority of existing opto-electronics components like semiconductor lasers, modulators, and couplers. T h e tradeoff is that it is impossible to perfectly localize light i n this geometry because the structure cannot support true photonic bandgaps but, at best, pseudo-gaps.  8  airholes (En)  A  E  Q  (air)  Figure 1.5: Schematic diagram of an asymmetric 3-layer slab waveguide w i t h 2D photonic crystal embedded in core.  T h i s is because there are always solutions to Maxwell's equations corresponding to radiation propagating i n the direction perpendicular to the plane of the slab waveguide (see Figure 1.7 below). Figure 1.6 schematically depicts an untextured planar (or slab) waveguide structure along w i t h the intensity profile of one of its guided modes. The electromagnetic excitations of the slab waveguide structures can be labelled by their in-plane wavevector, /?, and frequency, cu. Figure 1.7 depicts a schematic dispersion diagram for a structure like the one shown i n Figure 1.6. Three relevant light lines are also depicted on the diagram, one for each of the semi-infinite cladding layers as well as one for the guiding core. These c represent the photon dispersion given by ui = Q for propagating radiation i n bulk \/e/ o e  material w i t h an index of refraction (=-y/e/e ) equal to that of their respective layers. 0  T h e three light lines on the diagram serve to delineate the boundaries between four regimes.  Below the light line of the core is a "forbidden" region where there are no  solutions to Maxwell's equations i n the absence of external sources.  The variation of  the fields i n the direction normal to the interfaces is necessarily exponentially growing or decaying. Between the core and substrate lightlines, there is a discrete set of guided modes bound to the guiding core. In the case of these guided modes, the fields i n the  g  0  (air)  2 Figure 1.6: Schematic diagram of an asymmetric 3-layer slab waveguide.  CO  Figure 1.7: Schematic dispersion diagram of the modes of an asymmetric 3-layer slab waveguide. T h e solid line represents the light line for the core material. The dot-dashed line represents the line light-line for the superstrate, i.e., top semi-infinite layer (air). T h e dotted line represents the light line for the substrate, i.e., b o t t o m semi-infinite layer.  10  cladding layers decay exponentially away from the core such that no energy is carried away i n the transverse direction. In contrast, above the lightline of the superstrate (air), there exists a continuum of radiation modes. These modes are plane waves that propagate throughout the structure and undergo multiple reflections at the boundaries. Between the superstrate and substrate lightlines, there exists another continuum of modes w i t h fields that decay away from the core i n the superstrate but radiate down into the substrate. These are referred to as substrate modes. Additionally, all of the electromagnetic modes can be separated into two orthogonal polarizations, transverse electric ( T E ) and transverse magnetic ( T M ) , based on whether they have their electric ( T E ) or magnetic ( T M ) field completely transverse to their direction of propagation, which lies i n the plane of the slab. In the case considered here, there is only one T E polarized guided mode i n the region of to — (3 space depicted i n Figure 1.7. T h e details of planar waveguide theory can be found i n reference [23]. Slab waveguides, themselves, can be classified into two types: "hard" guides.  "soft" guides and  "Soft" guides are comprised of a thicker (~ 1 fxm for near-infrared  wavelengths) core, and cladding layers having a relatively small difference i n refractive index from that of the core ( A n PS 0.1).  "Hard" guides or membranes are comprised  of a t h i n core (~ 0.1 ^ m for near-infrared wavelengths) w i t h cladding layers having a larger refractive index difference ( A n ~ 2) from that of the core.  "Soft" guides are  found i n the vast majority of opto-electronic applications, i n part because they can be realized as semiconductor heterostructures which can be fabricated relatively easily using epitaxial growth techniques employing composition control i n some suitably chosen alloy system (e.g., G a A s / A l G a A s ) . Since large refractive index contrast cannot be achieved between semiconductor alloys, "hard" guides require the t h i n guiding core to be made freestanding, and therefore, to have air for its cladding layers, or to be clad on one or b o t h sides by a thick oxide layer (n ~ 1.5). F r o m the point of view of embedding waveguides w i t h planar photonic crystals, there  11  are advantages and drawbacks to using each of these as hosts. In order to achieve strong scattering, it is necessary that the crystal penetrate a significant portion of the core. For "soft" guides, this requires etching deep (> 1 /mi) high-aspect ratio holes, which presents a considerable challenge. Moreover, the introduction of a material w i t h dielectric constant far lower than that of the cladding layers generally reduces the volume-averaged index of .the core layer enough to cut off guiding. Consequently, when planar photonic crystals are embedded i n "soft" guides, the holes are made as deep as possible but placed i n a cladding layer. T h i s is done at the expense of scattering strength. Nevertheless, such "soft" guide based crystals are appealing due to their compatibility w i t h the overwhelming majority of opto-electronic devices. A n early success i n this geometry was achieved by Wendt et al. [24], who etched a honeycomb array of holes ~ 1 /jm deep on a sub-micron pitch into a special slab waveguide i n a G a A s / A l G a A s / I n G A s heterostructure.  Photoluminescence excited i n the  vicinity of the crystal exhibited preferential propagation along certain of its symmetry directions. More experimental work i n this area has been done by Labilloy et al. [25]. They have obtained evidence of a photonic bandgap and mapped dispersion by exciting broadband photoluminescence i n quantum dots embedded w i t h i n a 2 D planar waveguide and measuring the transmission through a finite 2D photonic crystal interposed between the excited spot and a cleaved facet. They use the dropout i n the transmission spectrum to determine the extent of the photonic band gap along the symmetry direction of the crystal. They then use oscillations i n the spectra outside the gap, arising from the variation of the round-trip phase through the finite-sized grating as a function of the in-plane wavevector, to deduce the mode dispersion. In contrast to "soft" waveguide-based photonic crystals, "hard" waveguide-based crystals require only shallow etches to completely penetrate the t h i n waveguide core. A n additional advantage is that they can still support guided modes despite the introduction  12  of a large amount of low dielectric material into the core. O n the other hand, freestanding waveguides tend to be less robust.  Oxide-based structures avoid this problem; but  the presence of an oxide, its quality, and the ease and reliability w i t h which it can be incorporated pose other difficulties, especially when one attempts to make active devices. Nevertheless, these porous membranes represent the extreme, non-perturbative limit of waveguide-based photonic crystals. T h e introduction of 2D in-plane periodic dielectric texture has an effect on bound modes similar to the effect on plane waves caused by the introduction of 3D periodic texture i n bulk dielectrics. T h i s can be seen i n the weakly textured case by applying a perturbative method [26, 27]. T h e dispersion of the bound modes is modified and gaps are opened up at the boundaries of the 2D B r i l l o u i n zones, leading to a bandstructure for the slab modes. It is interesting to address to what extent 2D planar photonic crystal waveguides can be used to implement the novel ideas mentioned above for 3D photonic crystals. For instance, point or line defects can still be introduced to create guides or cavities, but the photon confinement is achieved due to the photonic crystal i n the in-plane directions, and due to the guiding properties of the slab structure i n the transverse direction. Figure 1.8 schematically exemplifies a two-dimensionally textured waveguide w i t h a point defect. However, i n any geometry that lacks translational invariance i n the transverse direction there w i l l always be some coupling to propagating modes outside of the waveguide. A s is evident from Figure 1.7, at any frequency there exist modes that propagate directly through the waveguide. Consequently, although the bandstructure for modes localized to the slab waveguide core may exhibit frequency gaps, there always exist other nonresonant states at those frequencies.  T h i s geometry does not support full photonic  bandgaps, but rather, at most, only pseudo-gaps. It is of considerable importance to understand and quantify the differences between having a true gap, as opposed to only a pseudo-gap (or in-plane gap). F r o m a practical  13  Figure 1.8: Schematic diagram of 2D photonic crystal w i t h defect embedded i n slab waveguide.  point of view this may be framed i n terms of two key questions: (a) To what extent is the total Density of States modified ? and (b) How high are the intrinsic Q's that can be achieved ? These questions have been addressed by others to some extent. It is known that planar photonic crystals do achieve a significant redistribution of the overall density of states i n the vicinity of the pseudo-gap or the discrete defect states w i t h i n the gap. Painter et al. [28] were able to demonstrate an optically pumped defectmode laser i n a 2D textured III-V semiconductor waveguide based photonic crystal. S. Fan et al. have demonstrated photon confinement by fabricating a silicon channel waveguide on an insulating substrate [29, 30]. A line of holes was then etched into the silicon channel, omitting the hole i n the centre, to form a linear, I D photonic crystal w i t h defect for optical modes confined to the silicon channel. A localized photon was formed at the defect w i t h an effective volume of < 0.1 pm . 3  The same group has also  demonstrated low losses through tight bends using photonic crystals embedded i n slab waveguides [31]. Referring again to Figure 1.7, one is able to characterize the bound modes by their dispersion.  It has also been shown that each such mode has a definite polarization.  14  Furthermore, one can see that the bound modes are discrete modes, i n the sense that for any /? there are no modes w i t h a neighbouring ui.  This is also the case for the  modes of a 3D photonic lattice as described i n Figure 1.2. T h e addition of 2D texture to the slab waveguide causes some of the bound modes to m i x w i t h radiative modes i n the continuum. W h i l e some renormalized modes remain purely bound, the bandstructure for discrete modes now describes localized states which are generally only quasi-bound, or leaky. T h i s leakiness can be described by an associated lifetime that has no analog i n the case of pure 3D (or pure 2D) photonic crystals. It should be mentioned that such leaky modes also exist i n the case of an untextured slab waveguide and are also associated w i t h lifetimes. In this case, however, they occur i n the continuum regime. Moreover, it should be emphasized that, i n b o t h cases, the lifetimes are not associated w i t h an absorptive process.  T h a t is, the lifetimes do not describe a loss of electromagnetic energy, but  rather the diminishment of its localization i n the waveguide core. T h e points identified above provide the main motivation for this thesis, which thoroughly characterizes the dispersion, polarization, and lifetime properties of discrete, slab modes of planar waveguides w i t h strong 2D dielectric texture. F r o m a theoretical point of view, the bound and quasi-bound modes i n I D textured waveguides have been studied extensively w i t h coupled mode theory ( C M T ) [32, 33, 34], which is a perturbative approach.  Most of this work has been done i n the context  of Distributed Feedback Lasers ( D F B ' s ) , which incorporate weak I D texture i n a slab waveguide geometry.  This weak texture i n D F B ' s results from particular conditions:  either the index-contrast w i t h i n the grating is small, or it is placed away from the waveguide core, or both. Coupled M o d e Theory solves the separate scalar equations for the T E and T M modes, considering only the lowest order couplings between the modes of the waveguide. In the case of 2 D texturing of a general planar waveguide, however, the true eigenmodes of the slab can never strictly be expressed as superpositions of solely T E or solely  15  T M modes, and the problem is inherently one of a vector nature. In this vein, a 2D vector coupled-mode formalism has been presented by Paddon and Young [27]. A l t h o u g h this formalism provides a transparent understanding of the non-trivial vector coupling which occurs i n 2D waveguides and insight into the polarization properties that arise from symmetry, it remains limited to weak texture. Experimentally, we and others have shown that angle-resolved specular reflectivity measurements can be used to map out the dispersion, lifetimes, and polarization properties of these states away from the zone-centre [35, 36, 37]. In a somewhat related vein, Yablonovitch has demonstrated that coupling to radiation modes resulting from the presence of a photonic crystal can be used to enhance light extraction from light emitting diodes [2]. Our group was the first to report experimental and model results which showed that photonic bandedge states manifest themselves as Fano-like resonances [38] i n normalincidence specular reflectivity spectra taken on a strongly 2D-textured waveguide [39]. T h i s occurred when the energy of the incident light matched that of the photonic eigenstates at zone-centre.  T h e experiments were performed on a "hard" waveguide-based  photonic crystal [39], which was fabricated by etching a periodic 2D array of holes completely through a 240 n m G a A s layer, then etching away an underlying sacrificial layer through the newly formed holes to create a freestanding,  "porous waveguide".  The  results were compared w i t h a finite-element model that was developed principally by the author. T h e model allows for the calculation of specular reflectivity and scattering spectra, whence we may obtain the dispersion, lifetimes, and polarization properties of the modes. Qualitative agreement as well as crude quantitative agreement was obtained between the model and experiment. These freestanding "porous waveguide" samples proved to be b o t h difficult to fabricate and quite frail. A s a result, a more robust structure and fabrication technique for "hard" waveguide-based photonic crystals was developed by the author i n conjunction  16  w i t h F . Sfigakis, a Master's student i n the group. In addition, the use of a new and much-improved optical scattering setup, developed by W . J . Mandeville, a doctoral student i n the group, allowed for easy variation of the angle of the incident and scattered light. A t the same time, a more efficient Green's function ( G F ) based numerical model specific to the t h i n "hard" waveguide geometry, was developed by A . Cowan (another fellow doctoral student) and J . F . Young [40]. T h i s much faster code allowed manual fitting of the model-generated bandstructures (dispersions and lifetimes) to those obtained from experiment. T h i s entire body of work, which provides for a clear interpretation of optical scattering data i n terms of the dispersion, lifetime, and polarization properties of 2D photonic crystals embedded i n thin, "hard" waveguides, is the subject of this thesis. T h e principal elements of this work include: (1) a numerical model for calculating specular reflectivity and scattering from an infinite 2D photonic crystal embedded i n an arbitrary slab waveguide layer structure, based on integration of Maxwell's equations through the textured layer on a discrete mesh, exploiting the discrete translational symmetry of the problem; (2) a fabrication technique for making structurally robust and robustly guiding t h i n semiconductor slab waveguides based on a wet oxidation technique; (3) angle-resolved specular reflectivity measurements that allow determination of mode dispersion and lifetimes, and which exhibit interesting polarization symmetries and anti-crossings away from high-symmetry points i n the B r i l l o u i n zone; and (4) a comparison of these measurements w i t h simulations based on a computationally efficient Green's function-based ( G F ) model. T h i s G F model is validated using the slower, "exact" model presented here, for the case of t h i n "hard" waveguides studied i n this work. Resulting from this body of work is the realization that, even i n this extreme non-perturbative limit of strongly textured t h i n waveguides, the photonic eigenstates i n the vicinity of the second order T E gap can be understood qualitatively, and i n some cases semi-quantitatively, as renormalized T E polarized slab modes. T h e organization of the remainder of this thesis is as follows.  17  Chapter 2 presents  the finite difference model developed by the author. Results from the model are used to introduce certain concepts related to bandstructure arising i n the slab waveguide geometry as well as the approach used i n this thesis for probing it. Chapter 3 presents the fabrication process used to realize the textured waveguide structures studied. Chapter 4 describes the experimental apparatus and procedure used to perform angle-resolved spectroscopy and the capabilities and limitations thereof. Chapter 5 presents the experimental data obtained, as well as a description of the fitting technique used to obtain dispersion and lifetime information. Chapter 6 provides an interpretation of the results and a qualitative explanation of the observed dispersion and polarization properties, using simple symmetry and kinematical arguments . Chapter 7 presents conclusions and some direction for future work.  18  Chapter 2 Modelling  2.1  Introduction  T w o different techniques were used to model the linear optical scattering properties of t h i n porous slab waveguides i n this work. T h e first technique, which is referred to as the Finite Difference Real Space approach ( F D R S ) , involves the integration of Maxwell's equations through the textured layer on a discrete mesh defined on the unit cell of the photonic lattice. In our first attempt at this finite-element approach, we tried to apply an algorithm developed by J . B . Pendry [18] for calculating reflection and transmission coefficients through pure 2D and 3D photonic crystals. not converge well i n our geometry.  However, this algorithm d i d  T h e specular reflectivity spectra d i d not contain  the correct number of resonant features expected from symmetry considerations, nor were the polarization selection rules obeyed.  T h e problem was traced to the use of  forward differences (rather than symmetric differences) i n the integration scheme, which introduced an artificial chirality into the calculation. Consequently, a new integration scheme that converges well i n our waveguide geometry was devised by the author i n collaboration w i t h P. Paddon. T h i s new scheme was then implemented by the author. Additionally, the model was extended (by the author) to allow for efficient modelling of buried gratings. T h i s new model is presented i n detail in this chapter. T h e second model used i n this work was developed by J . F Y o u n g and A . R Cowan. It is referred to as the Green's Function ( G F ) model. T h i s model is more computationally efficient than the F D R S approach, w i t h its limitation being that it is only valid for  19  a textured layer that is t h i n compared to the wavelength of light therein. A detailed description of this model can be found elsewhere [40, 41] and is not given here.  Due  to its computational efficiency, it was used extensively for fitting to the experimental results presented i n Chapter 4. T h e F D R S model was developed first, and the process of its development led to most of our basic understanding of the symmetry and polarization properties of the electromagnetic modes attached to 2D photonic crystals embedded i n slab waveguides. It also serves as a benchmark for more efficient codes, such as the G F code, which are based on approximations specific to a particular geometry. A comparison of the results from the G F and F D R S models is presented i n section 2.5. T h e remainder of this chapter provides a detailed description of the F D R S model. It also presents and discusses a number of spectra generated by the model for various relevant sample structures.  Additionally, it explores the convergence behaviour of the  algorithm. Finally, a discussion is presented regarding the necessity and significance of the modifications made to the original method of J . B . Pendry. A summary is given at the end of the chapter.  2.2 2.2.1  Description of Finite Difference R e a l Space A p p r o a c h Scattering from a periodically textured layer  We begin by considering the structure depicted i n Figure 2.1. The dielectric function, e(r) i n region II is periodic i n the plane of the slab:  e(p; z) = e(p + R*; z) where p is a vector i n the x — y plane and Rj is a 2D real-space lattice vector also i n the x — y plane. For simplicity, we assume that each material i n the structure is linear, isotropic, and homogenous, so that it can be described i n bulk form by a scalar dielectric  20  Figure 2.1: Schematic diagram of a two-dimensionally periodically textured slab structure.  21  constant. Therefore, the dielectric function may be written as a Fourier series: =  e'(p; z) 4 ^  where the G ;  > m  4, »  exp(tG,,  ra  • p)  (2.2)  are the reciprocal lattice vectors of the dielectric modulation ("photonic 27T  crystal") , e.g., G /  27T  = l~^  + ~^~$> ^  x  > m  m  o r  ^  a  S  (  l  u  a  r  e  lattice having a pitch A , where I  and m are integers. T h e harmonic solutions to Maxwell's equations i n the homogeneous regions, I and III, for a given frequency can be expanded i n terms of a basis set consisting of transverse polarized plane waves (in general, a different set for each region to allow for different e's i n regions I and III) where all of the fields have a spatial-temporal variation of the form exp ( i k • r) exp (—iut) subject to the dispersion relation: = Ik| = kl + hi + k>  e'l/m^  2  (2.3)  and where 'i/in  e  — i/in/eo 6  (2.4)  represents the dielectric constant i n region I or III, which does not vary spatially, and c is the speed of light i n vacuum. There are, i n general, two degenerate, orthogonal polarizations, labelled s and p according to whether the electric or magnetic field, respectively, lies completely i n the plane of the slab interfaces. W i t h the unit normal to the interfaces denoted by n , these plane wave modes can be written as: E< >(r) = , *E |n x k| a  n  exp(zk • r - itot)  X  0  HW(r) =  k  ^  ( f i  r  k )  |n x k|  — exp(ik • r - iut) = — k x E ^ ( r ) phi pjuj  (2.5)  (2.6)  and: H  ( p )  (r) = , *H n x k n  X  0  22  e x p ( i k • r - itot)  (2.7)  E ( p )  =  kx(nxk)go |n x k| eto  . _ r  i  u  t  )  =  l  k  x  H  ( ) P  ( r )  (  2  8  )  ecu  where e and u are the permittivity and permeability of the medium, respectively, and E  and HQ are normalization constants bearing units of electric and magnetic field  0  respectively. For the special case of k parallel to n (i.e., normal incidence), the fields of the polarized plane waves lie entirely i n the x — y plane, and hence the two independent polarizations may be defined w i t h respect to any two orthogonal in-plane directions. For convenience, & and y are chosen, so our basis and the modes i n this situation are defined as x or y polarized according to: E  (r) = E exp(ik • r - iut)x  ( x )  (2.9)  0  H^fr) = —  E ex.p(ik-T-iujt)y 0  =  — k x E^(r)  (2.10)  and H ( r ) = H exp(ik-r-icot)x  (2.11)  (y)  0  E ^ ( r ) = - ^ # e x p ( z k • r - iut)y = - ^ k x H ( r ) (y)  0  (2.12)  T h e situation we wish to consider is somewhat similar to that of a diffraction grating. We consider a plane wave of definite frequency, u>, and 3D wavevector, k = (3 +  k z, z  incident on the structure from the left, where {3 is the in-plane (x — y plane) component of the wavevector.  The incident plane wave scatters to a discrete set of plane waves  travelling backward i n region I as well as a discrete set of plane waves travelling forward i n region III. T h e in-plane wavevectors of the discrete set of scattered plane waves i n b o t h regions are given by (3 + G /  > m  . Physically, this arises from the fact that the periodic  dielectric texture modulates the incident field, generating polarization w i t h the same periodicity as the dielectric texture. T h i s polarization will, i n turn, generate an electric field having spatial Fourier components restricted to those described by the discrete set of reciprocal lattice vectors, G /  ) T n  , that define the dielectric texture.  23  Most generally, the scattering process can be described i n terms of the following discrete set of transverse polarized plane waves on either side of the textured structure:  (2.13) H}^'  ( r ) = -[(/3 + G  J / / J i )  I > m  ) + wffMz]  x E<"*>(r)  (2.14)  and  (2.15) E  g  f  =  + G«.«) + ^ £  7 / / / / )  2] x H^)(r)  (2.16)  where fi  7/777)  The quantity, wf^^ ^, 111  = ±[z't/n^/c  2  - |/3 + G , , | ] 2  r a  1 / 2  (2-17)  represents the ^-component of the wavevector for the par-  ticular plane wave, and equation 2.17 was obtained from the dispersion relation, equation 2.3. W h e n w\[l~ ^ ^ ,I  III  >  is real-valued, we have a plane wave travelling (+) right  (i.e., +z direction) or (—) left according to whether the positive or negative solution of the square root is selected. W h e n w^~'^  is imaginary, then we have an evanescent  Iiri  plane wave that is decaying towards the (+) right or towards the (—) left. For the sake of concision, we refer to the modes simply as right-going or left-going according to the above convention regardless of whether they are actually propagating or evanescent. We can then classify the plane waves as r i g h t - (+) or left-going (—) and order them w i t h respect to polarization and in-plane wave-vector i n a systematic manner. For the situation we are considering of a right-going (+) plane wave incident on the textured region from the left, we wish to compute the amplitudes of the scattered right-going (+) plane waves i n region III, which we denote by (ax)ifori = 1, 2, 3,... (Transmitted), where the subscript i, now labels b o t h the polarization and reciprocal lattice vector. W e 24  also wish to compute the amplitudes of the scattered left-going plane waves i n region I which we denote by (OR)J (Reflected). T h e scattering problem can thus be written i n m a t r i x form as: a  T  a  R  where a j  nc  = T  + +  a  (2.18)  i n c  = T-+a  (2.19)  i n c  contains the amplitudes of the right-going (+) plane wave modes i n region I  (incident). Note that for the experiment under consideration this last column m a t r i x is all zero except for the one element corresponding to the sole plane wave incident from the left. These column matrices are of dimension 2N x N where N x N is the number of reciprocal lattice vectors, i n 2 D , included i n the calculation and the factor 2 arises from the two independent polarizations (s and p). T h e matrix element T^ , +  therefore, represents the scattering coefficient for plane  wave i travelling toward the right (+) i n region I to plane wave j leaving the textured layer i n region III and travelling toward the right (+). T h e matrix element T^ , +  rep-  resents the scattering coefficient from plane wave i travelling toward the right (+) i n region I and reflected/diffracted back to left-going (—) plane wave j i n region I. The problem, then, is to calculate the matrices T  and T~ .  ++  +  T h i s is accomplished  i n the present model by integrating Maxwell's equations through the textured layer. However, i n order to do so, one must perform the inverse scattering calculation using the following mathematical manipulation.  a c  =  (T  a*  =  T- (T  in  + +  +  )- a  (2.20)  1  + +  r  )- a 1  r  (2.21)  Now, by specifying a single plane wave of unit amplitude travelling toward the right i n region III, integrating backwards, and projecting onto the forward and backward travelling plane wave basis i n region I, we obtain a column of each of the matrices 25  (T" "" ") and T~ (T )~ , 1  1  -1  +  ++  1  respectively. T h e matrices may thus be constructed i n their  entirety by launching each of the basis plane waves i n region I. T h e matrices T T~  +  +  and  +  are then obtained by matrix inversion and multiplication. If we also independently consider the situation where one of the left-going modes  is incident on the textured layer from region III and find the scattering amplitudes to diffracted/transmitted left-going modes i n region I, (br)u and diffracted/reflected rightgoing modes i n region III, (&R)J, we can define:  where b ;  n c  b  T  = T~b  b  R  = T ~b  m  (2.22)  c  (2.23)  +  mc  contains the amplitudes of the left-going (—) plane wave modes i n region III  (incident). T h e m a t r i x element  , therefore, represents the scattering coefficient from plane  wave % travelling toward the left (—) i n region III to plane wave j leaving the textured layer i n region I and travelling toward the left (—). T h e m a t r i x element  , represents  the scattering coefficient from plane wave i travelling toward the left (—) i n region III and reflected/diffracted back to right-going (+) plane wave j i n region III. Therefore, by a similar argument, we may obtain (T  )  _ 1  and T ~(T  )  +  _ 1  by spec-  ifying plane waves of unit amplitude travelling toward the left i n region I and integrating forward, and hence obtain T  and T ~. +  The four matrices, T , ++  T~ , +  T  , and  T~ +  completely specify the scattering properties of the textured layer. Since the dielectric function is assumed to have the same discrete translation symmetry i n the x and y directions for a l l z, Bloch's theorem may be exploited to render the computation tractable. Bloch's theorem requires that for any field component, F(r) =  F(p;z)  F(p + R;z)  = exp(i/3-R)F(p;  26  z)  (2.24)  Therefore, we need only compute the fields on the finite unit cell of the photonic crystal. T h i s makes it possible to discretize Maxwell's equations on a finite number of mesh points w i t h finite grid size so that they may be integrated forward or backward i n a finite number of computational steps.  2.2.2  Discretization and Integration of Maxwell's Equations in Real Space  In order to compute the reflection and transmission matrices defined i n the previous subsection, we need to be able to propagate the fields through the textured layer. T h a t is, given the fields at one edge of the textured layer, we must compute the fields at the other boundary. This can be accomplished by stepping through the layer i n the direction normal to the slab (z) by successively computing the fields on adjacent intermediate planes lying parallel to the slab and mutually separated by some finite (z) step size. Therefore, we need to be able to calculate the fields on one plane given the fields on an adjacent plane, allowing for a spatially varying scalar permittivity, e(r) =  e(p;z).  We begin by considering Maxwell's equations i n the absence of free charges and currents:  V x E(r,t)  (2.25)  dB(r,t)/dt  V x H(r,<) =  dB(r,t)/dt  (2.26)  V-D(r,t) = 0  (2.27)  V-B(r.t) = 0  (2.28)  and the constitutive equations:  D(r,t) = e(r)E(r,t) = e [l + Xe(r)]E(r, t) = e e'(r)E(r, t)  (2.29)  B(r, t) = "(r)H(r, t) = /x [l + X (r)]H(r, t) = /x /x'(r)H(r, t)  (2.30)  0  0  0  m  27  0  where Xe(r) and x ( r ) are the electric and magnetic susceptibilities respectively, which m  are properties of the media, and eo and (io are the permittivity and permeability of free space, respectively. Equations 2.25 and 2.26 may be Fourier transformed w i t h respect to both space and time to give: zk x E ( k , u) = iu;B(k, cu)  (2.31)  ik x H ( k , u) = - « j D ( k , u)  (2.32)  A s stated i n the previous section, the approach we take is to solve Maxwell's equations on a finite number of spatial mesh points. A s we shall, see this has the effect of approximating the spatial derivatives i n the equations 2.25-2.26 w i t h difference terms between neighbouring mesh points. Consequently, we now define a rectilinear mesh w i t h spacings a, b, and g i n the x, y, and z directions, respectively, and make the following approximations:  [exp(  2  ftjy ~ [expi  ,ik b.  1  [exp(  y  2  ik a.. x  exp(-  ;  ..  2  ^  = «*  exp( = ^ ) ] / i 6 ^ Ky  ;  ,ik q. z  exp(-  2 >  ~ik g., z  2  .. )\hg = A  (2.33) (2.34) (2.35)  K  Z  which are valid provided that: ka «  1  (2.36)  kb «  1  (2.37)  kg «  1  (2.38)  x  y  z  so that we define: K = K X + K y + K Z f« k. X  y  Z  (2.39)  Note that the greater the spatial frequency content i n the fields, the finer the mesh required for this approximation to remain valid. It is also important to emphasize that 28  we have not made any assumptions here about the orientation of the coordinate system w i t h respect to the photonic crystal. M a k i n g the substitutions, k = K , i n equations 2.31 and 2.32, carrying out the crossproducts and expanding component-by-component yields the following set of six equations:  exp{ik ^)H y  D  x  «  v  ibu  «  exp(ik ^-)H * igu  «  exp(ik ^-)H * iau '  x  x  D  z  y  x  «  v  «  z  «  exp(ik ^)H * iau  exp(-ik ^)H ^ (2.41) iau  y  exp(-ik ^)H * iau -- •  exp(ik ^-)H -± ibu •  x  x  x  y  exp(- -ik -)E  z  y  y  y  z  exp(-ik ^)H —A (2.42) ibu y  +  +  igu -- •  exp(ik ^)E ^— iau iau  a. ~ exp( -ik -)E ^— iau iau  exp(ik ^)E -= ibu ibu  x  z  x  y  exp(ik -)E ^ „ „ , . , iau  y  x  +  x  exp(- -ik ^)E ^ „igu „,., z  y  exp(ik ^)E  z  ibu • x  z  z  exp(ik ^)E ^ igu x  B  exp(-ik ^-)H * igu  z  +  ibu  ibu • z  B  exp(-ik ^-)H * (2.40) igu  z  z  exp(ik ^)E B  exp(ik ^-)H * igu  y  +  z  D  exp(-ik ^)H  z  x  x  y  igu  ^  (2.43)  ixp(-ik ?-)E + *  z  y  :  x  ^  z  a  a  (2.44)  ;  Bxp{-ik ^)E y  x  +  ibu  x  (2.45)  It is possible to transform the expression for the fields given above back to real-space by recognizing that the inverse Fourier transform of a field component times a phase shift is the translated real-space field. T h a t is: exp{±ik ^)F  &  Fi(xT^,y,z)  exp(±ik -)F  O  FfayT^z)  o  F|(x,y,2T|)  x  y  t  l  exp(±ik ^)F z  l  b  =  Fj(rTa)  =  *}(r =F C)  =  *i(>T7)  ( - ) 2  46  where i } denotes a component of the E or H field, and a , £, and 7 are defined as 7  29  follows. a, a = —x C = -v 2 2  g„ -z 2  7 =  (2.47)  The real space fields are then given by:  A,  H (r + C) - H (r z  - Q  z  r  H (r + 7) - H (r y  ibu H (r  (2.48)  iguj  + 7) ~ H (v  x  7)  y  x  H ( r + cx) - H {r - a )  - 7)  z  z  igui  iau>  H (r + a) - ff„(r - a )  ^(r + Q - ^(r - Q  y  (2.49) (2.50)  iau) Bx(T.  By(V  BJT  E (r + C) - ^ ( r - C ) z  E (r + 7) - £ „ ( r - 7)  +  y  ibu E (r x  igu  £,(r + a) - £ ( r - a)  + 7) - E ( r - 7) x  z  *0W E (r + a ) - ^ ( r - a ) iau y  E (T + Q - E (T - Q ibu)  |  x  x  (2.51) (2.52) (2.53)  The constitutive equations 2.29 and 2.30 are used to eliminate the D and B fields to yield:  E (v  H (r + C) - H (r z  z  x  H (r  + 7) - H {r  v  x  (2.54).  a)  z  iguje(r)  iaue(r)  H (r + a ) - i J „ ( r - a ) iaa;e(r)  #*(r + C) - # x ( r - Q i6a;e(r)  y  z  z  Ey{v + 7) - ^ , ( r - 7)  +  ibu>/j,(r) E (r x  + 7) - E (r x  igufj.(r) - 7)  ^ ( r + a) - £ ( r z  igujfi(r)  a)  mu>/z(r)  E (r + a ) - E „ ( r - a )  £ ( r + fl - E (v  y  HJT  7)  H (r + a ) - ff,(r -  - 7)  E ( r + C) - E (r - _ 0  Hy(T  -  v  iguje(r)  + 7) - H (v  x  z  H (r  ibue(r)  EJT  E (r  - Q  iawju(r)  g  '  x  ibup(v)  - Q  (2.55) (2.56) (2.57) (2.58) (2.59)  Finally, the z-component of the fields are eliminated, terms containing r ± 7 collected on the left-hand side, and the substitutions:  H ' = zH/' gtQui 30  (2.60)  c=l/Vio7^ e' = e/e H  =  1  (2-61) (2.62)  0  (2.63)  fi/iMj  are made, where c is the speed of light i n vacuum. T h e substitutions are made i n order to make the equations for the E and H fields look similar.  T h e result is the  following set of four equations:  E {v x  +  )  1  «  £; (r- ) + ^u'(r)F;(r) + ^ { x  7  H' (r + a + C)-H' (r e'(r + a)b  + a-C)  H'{v)-H>{T-2ct)  ^(r + 2a)-^(r)  x  x  +  e'(r — a) a  +  H' (r-a  + C)-H' (r-a-C) e'(r — a)b  x  x  ^  3  e'(r + a) a  ^ ( r + a + Q - g ; ( r - a + C)  H' {v + a - Q - H' (r - a - <)  e'(r + C)a  e'(r - C)a  y  y  H' (r) - H' (T - 2C)  H' (r + 2C) - J£(r)  e'(r-C)&  e'(r + C)6  g  x  x  c  }  (2-65)  2  £ ( r + a + C) - E (r + a - C)  £ * ( r - a + C) - #x(r - a - C)  fx'(r + a)b  fj,'(r — a)b  x  x  E (v)  - E (r - 2a)  y  y  /i'(r  —  |  ^(r + 2a)-^(r)  a)a  u'(r + a ) a  c F;(r- )-6'(r)^(r) + — { 2  ^ (  r  + 7)  «  7  |  E ; ( r + a + C ) - ^ ( r - a + C)  tf (r y  + a - C) - # (r - a - <) y  /x'(r + C)a  A»'(r-C)a  J % ( r ) - J % ( r - 2 Q _ i%(r + 2Q - i%(r) /x'(r-C)6  ^'(r + C)& 31  1  1  ;  Given the values of the E and H ' fields on a discrete mesh on two adjacent planes separated from one another by a distance 7 = g/2 i n the z direction, these equations provide for the calculation of the E and H ' fields on a discrete mesh on two more such planes translated i n the z direction by an amount g from the original planes. It applies when the spatial variation of the scalar dielectric constant is accurately described by the discrete mesh as determined by a, b, and g. In other words, this allows us to "integrate" the fields forward or backward through a given distance i n the z direction i n steps of g given the values of the E and H ' fields on one pair of planes.  If these planes are  aligned parallel w i t h the slab of textured material depicted i n Figure 2.1, that is, the coordinate system is aligned so that z points i n the direction perpendicular to the slab, then repeated application of these equations allows us to integrate Maxwell's equations through a textured layer. The situation is illustrated i n Figure 2.2 for the case of a square photonic crystal. Specifically, to integrate the fields forward, one begins w i t h E(r' — 7), and H'(r'), and uses equations 2.64 and 2.65 which manifestly gives the expression for E(r' + 7). One then calculates H'(r' + 27) from E(r' + 7) and H'(r') according to equations 2.66 and 2.67 making the substitution r = r' + 7. It is important to note that the order i n which the E or H ' fields must be calculated is crucial and depends on the direction of integration for a given initial ordering of the E and H ' planes. For example, i n the above case, it would be impossible to calculate the H ' field since the value of H'(r' — 7) is not known. In order to integrate the fields backward one begins w i t h the initially specified values E(r'), H'(r' + 7 ) , and uses equations 2.66 and 2.67 to calculate H'(r' — 7). T h e n one calculates E(r' — 27) from E(r') and H'(r' — 7) according to equations 2.64 and 2.65, making the substitution r = r' — 7. T h e above expressions for the fields may be recast as a transfer matrix, M,  32  that  E  H' E  H'  E  Figure 2.2: Discrete in-plane (x — y) lattice on which E and H' fields are computed on a series of planes separated by g/2 = 7. T h e dielectric constant is sampled on the lattice indicated by the black dots. T h e fields are calculated on the sublattice indicated by the crossed squares. The planes shown span a unit cell of the periodicity i n the in-plane direction. T h e light and dark grey shading of the planes indicates whether the E or H' fields are computed there. The white hole i n the center schematically depicts dielectric texture comprised of a circular airhole.  33  relates the fields on one pair of planes to the fields on a subsequent pair of planes.  y{QiX  E  z  + l)  'x(Ql,l> )  H  z  = M  (2.68) x(e ;  z + 7)  Ey(Q ;  z + 7)  'x(Q ,m  Z+ 2 )  'y(QN, ;z  + 2 )  E  E {Q ,N;Z-J) X  NtN  N  NtN  V(QN,N\ ~~I)  E  Z  H  N  H  N  Hy(g ;z)  7  7  NtN  where M is a AN  2  x AN  2  matrix, and  are the A ^ 2 D (in-plane) lattice vectors of 2  the discrete mesh on which the fields are evaluated. Note that, i n principle, the m a t r i x M may be dependent on z. Referring to Figure 2.2, the  are depicted by the crossed  squares, and taking the centre of the hole as the origin, they may be defined explicitly as:  Qij = ( ~ N  2.2.3  \)C + (« - 1)(C + OL) + (j - 1)(< - a )  i,j = 1...N  (2.69)  Eigenmodes on a Discrete M e s h  In order to compute the matrices T , ++  T~ , +  T~ +  a basis set of AN -vectors 2  is chosen  i n regions I and III. Each one of these is integrated through the textured layer and projected onto each of the basis vectors on the other side. More specifically, to construct the matrices T , ++  T~  the basis vectors that represent right-going plane waves i n region  +  III are integrated separately back through the textured layer using equations 2.64 - 2.67, and the resulting 4A^ -vector is projected onto each of the basis vectors i n region I. 2  Similarly, to construct the matrices T  , T~ +  the basis vectors that represent left-going  plane waves i n region I are integrated forward through the textured layer using equations  34  2.64 - 2.67 and the resulting 4iV -vector is projected onto each of the basis vectors i n 2  region III. W h i l e one might be tempted to use the basis vectors defined i n equations 2.5 to 2.8, which are polarized plane waves, these are not, i n fact, eigenvectors of the  discretized  version of Maxwell's equations applied to a uniform medium w i t h e' = ^j/ ni  Formally, i n order to construct a basis set of the eigenvectors of the discretized Maxwell's equations i n regions I and III one must use the set of eigenvectors of the m a t r i x M i n equation 2.68, defined by the equations 2.64 - 2.67, where the spatially constant value of e' or e' :  IH  is used i n equations 2.64 - 2.67. <—>  Unfortunately, the matrix M is not Hermitean. T h i s means that its right eigenvectors, defined by: Mv  ( i )  = A  ( i )  v  w  (2.70)  and its left eigenvectors, defined by: u  M  m  = A u w  ( i ) i f  (2.71)  where the superscript H denotes the Hermitean conjugate, are not the same, and neither set is orthogonal. T h a t is:  m (j)  u  ^  u  § i j  (2.73)  T h i s nominally presents a problem when performing the projection operations. However, it is t r i v i a l to show that  and, therefore, the projection of an arbitrary 47V -vector, a, e.g., a wavefield integrated 2  through the textured region, onto the basis of right eigenvectors is given by AN  2  a = J2vjV  ij)  35  (2.75)  where rjj = u 3L  (2.76)  {j)H  Thus, each right eigenvector, for which we seek an expansion coefficient, has a unique left eigenvector that can be used i n equation 2.76 to find that expansion coefficient. Note that right and left eigenvectors have no correlation w i t h right and left going plane waves. B o t h right and left going plane waves are right eigenvectors of the transfer m a t r i x when there is no dielectric variation, and together constitute a complete basis.  However,  because these right eigenvectors are not orthogonal, the corresponding left eigenvectors are needed to efficiently carry out the expansion of an arbitrary vector i n terms of the basis of plane waves. Pendry [18] has shown that, to a good approximation, the right eigenvectors of M i n a uniform medium are given by the 2D in-plane Discrete Fourier Transform ( D F T ) of s and p polarized plane waves described i n Fourier space as i n equations 2.13 to 2.16, where K, the approximation made for k i n deriving the transfer m a t r i x equations, is substituted for k. T h a t is, the right eigenvectors may be constructed as follows. First we define the two 3-vectors representing the Discrete Fourier Transform of the discretely sampled fields. EW">(G,, )  = n x  m  H'(., ,////i) ±  ( G i m )  =  i/m  K  H ' ^ / ^ ( Q , E(p.±^(  G l  m  x  «f££  g(., ,////i) ±  (2.77) ( G j m )  ( 2  ) = n x  , ) = K\^1 m  x H'<*±.'/'">(G  .  7 8 )  (2.79)  l > m  )  where the s or p superscript denotes the polarization of the plane wave, the  (2.80) I/III  superscripts denote the region i n which the plane wave is being defined (see figure 2.1), the ± subscripts denote whether it is right-going or left-going, and the I, m subscripts label the reciprocal lattice vector.  36  To construct the K  L M  .  ±  we begin w i t h  k = {G x  ltm  + /3)-x;  ky = {G  lim  + p)-y  (2-81)  and use equations 2.33 and 2.34 to obtain the x and y components of  We then  use the dispersion relation to solve for the two values of k corresponding to the rightz  and left-going waves, using e ^  / i 7  as appropriate for the particular region. Finally, for  each such value of k , we use equation 2.35 to obtain the corresponding two values of z  the z-components of  K ^.±. J  However, the dispersion relation that must be used is not precisely that for actual plane waves i n a homogeneous medium given b y equation 2.3. Rather, it is modified such that K is again substituted for k , as i n the definition of the polarized basis set of plane waves itself (equations 2.13 - 2.16). T h i s new dispersion relation is given by:  2 2 2 w V o e o C j / m = K K = - ^ [ l - cosk a] + —[1 - cosk b] + — [1 - cosk g] a o g H  x  y  z  (2.82)  Explicitly, the two ( ± ) solutions for k i n each region (I/III) and for each reciprocal z  lattice vector G ;  i,m,±j/ni  k  are given by:  > m  i c o s - K l - ^ V o ^ / m  =  2  +  -[i  9  2  - o s ( a [ G , , + p] • x)] + -[l - cos(&[G,, + p] • y)]} (2.83) 9  C  m  m  Notice that we have dropped the normalization constants present i n equations 2.13 2.16 since they are superfluous for vectors to be used as a basis set i n subsequent calculations i n which they cancel out.  Consequently, the units of electric field here are  the same as those for k . Finally, notice that equations 2.77 - 2.80 refer to the reduced magnetic field, H ' , defined b y equation 2.60, which is relevant to the transfer m a t r i x defined by equations 2.64 - 2.67 and has the same units as electric field. 37  The E and H ' fields are then grouped together and their z-components dropped to form the "right" 4-vectors: R ft S/ ,±,I/III) x  P  {Gi  (s/p,±,I/III)  E .V  (G;,m)  ~(s/p,±,I/III)  (2.84)  ,  ~{s/p,±,I/III)  H'  (G/, ) m  The real-space right eigenvectors are then obtained by projecting these onto the mesh points: R,I/III  E;  E {QIX,Z ) X  0  E.  y(Qi,i, o)  E  z  -i R  (s/p,±,I/III) (8/p,±,I/M).  —  e  (i[/3+G,, ]-  e  (i[/3+G ]-^. )  m  eiil  )  (s/p,±,I/III),  'x(QiX' o)  H  z  ~(s/p,±,I/III)  E {g ;z ) x  jtn  ,  g(s/p,±,/////) G / , )  0  m  j  x  (s/p,±,i/in)  E .y H'x{Qj,n'i  H  O)  Z  (r  l«!,mj  /im  n  s/p,±,I/III) (s/p,±,I/III)  ,  -i R E {QN,N'I X  O)  Z  Ey(QN,Ni H'X(QN,N'I H'y{QN,N'i  o)  z  (s/p,±,I/III),  O)  Z  o)  (Q, )  K'x  Z  J  J  (G,, ,«/p,±)  H  m  m  (8/p,±,I/III)  ( *,m) G  (2.85) The corresponding left eigenvectors are also shown by Pendry to be constructed by  38  forming the "left" 4-vectors from the "right" 4-vectors according to: -l L  g(s/p,±,/////)  E  (s/p,±,I/III) V  (  1  G/,m)  (s/p,±,I/III)  H'  (Q,m)  x  ~(s/p,±,I/III),  0  -1  0  0  1  0  0  0  0  0  0  0  0  ^(p/s,±,I/III)  X  (r  E.  (p/s,±,I/III)  l  G  it  l,m)  l/lll  c  t  —(p/s,±,I/III)  0 I/III  (2.86) (notice the swapping of s and p subscripts). T h e real space left eigenvectors may then be constructed i n the same manner as i n equation 2.85.  2.2.4  Significance of U s i n g P e n d r y M o d e s  If we consider a uniform medium so that D = e ; E and B = yUoH, assume harmonically u n  time-varying fields, and drop the time varying exponential, e~~  , from b o t h sides of  ZU}t  equations 2.25 and 2.26, they become : (2.87)  V x E ( r ) = zcj/i H(r) 0  V x H ( r ) = -iooe E(r)  (2.88)  nni  W i t h o u t making any approximations, we may again proceed as i n section 2.2.2, carry out the curls, and eliminate the ^-components of the fields to produce an equation, for example, for the E  y  field: (2.89)  For an s polarized plane wave it is a simple matter to show that the difference term i n the curly braces i n equation 2.89 is identically zero. T h i s is because: H^ \x,y,z)  =  H^e e y e *  (2.90)  H^ \x,y,z)  =  H^e * e y e >  (2.91)  s  x  s  ikxX  ik  39  x  ik  ik  y  y  ik  ik  so that  ^di  {  ~W  Hy{r)  2Hx{T)}  -  =  ^  ^  ^  ^  - vHi \ k  (2-92)  s)  but since from equations 2.5 - 2.8 we have H  it follows that  S ) X  x  V  oyox  kk  =  kk  z  (2.93)  x  z  (2.94)  y  = 0.  -—H (r)}  {——H (T)  =  oy  1  W h e n Maxwell's equations are discretized through the procedure described i n sect i o n 2.2.2, the analogous equation to equation 2.89 is obtained by rearranging equation 2.65 to yield the difference equation corresponding to the ^-derivative: ^(r + 7 ) - ^ ( r - 7 ) H (T Y  + a + C)-H (T-a  + C)  V  - ^ ' ( r ^ r )  _  H'(r + a - Q - H' {v - cc -  (2.95)  9  y  e'(r + C)a  fl  e'(r - C)a  H' (v) - H' (v - 2C) x  + j {  =  H' (r +  x  2Q-H' (r)^  x  e'(r-<)6  x  e>(r + C)b  +  1  1  '  For a plane wave i n a uniform medium the translated fields, e.g., H' (r + a — £ ) i n y  equation 2.95 may be rewritten i n terms of a phase factor multiplying an untranslated field. A s s u m i n g a square array of mesh points, i.e., a = b, we obtain:  £,(r + 7 ) - ^ ( r - 7 )  =  - ^ ' ( r ) i £ ( r ) + -f-{ [e * e ^H' (r) ik  a  - e~ * e ^  IK  ik  y  _  [ ik*<* -ikyC ^ e  e  _  H  +  (2.97)  e  a  ik  H' (r)] y  -i**« -i*i,C#/( )] e  r  [H' (v)-e-^H' {r)} x  + mr)-e  x  K(r)}}  2ikyC  (2-98)  T h e term i n curly braces can be further simplified to yield 4sm(k C)[H' (r) y  x  s i n ( ^ C ) - H' {r) sm{k a)} y  40  x  (2.99)  and making use of the fact that H' (r) = K K ; H' (r) = K K x  Z  x  y  Z  (2.100)  V  the term i n curly braces becomes:  4 sin(A;j C)^{ x sm(k () K  /  y  — K, sm(k a} y  (2.101)  x  which, using equations 2.33 - 2.34, reduces to , • 4sm(k C)K { > s  y  f  2sin(La)  Z  2sm(k ()  >N  y  Ct  . ., ,, ^ sin^a)} = 0  v  sm(k ()  CL  ,  n  (2.102)  Therefore, the terms i n curly braces i n equation 2.95, which represent the discrete analog to the double partial derivative terms i n curly braces i n equation 2.89, cancel to yield identically zero as was shown for the exact continuous case. Thus these "Pendry" modes w i l l propagate as eigenmodes of the discretized integration scheme and w i l l not m i x into one another as they propagate. It is important to note that had we used "true" plane waves, that is, H' (r) = k k ;H' (r) x  z  x  y  = kk z  (2.103)  y  then we would have obtained Asm(kyC)k {k sm(k () z  x  - A ^ s i n ^ a : ) } fa 0  y  (2.104)  which is only approximately equal to zero to the extent that k a, k b x  y  1. T h i s means  that these fields are not eigenstates of the discretized integration operation, and so they are "mixed" as they are propagated by equations 2.64 - 2.67. T h e same also holds true for a l l similar terms i n curly braces i n equations 2.64, 2.66, and 2.67 for the x component of the E field of an s polarized mode and for the x and y components of the H field of a p polarized mode, respectively.  41  2.2.5  Computation of Reflection and Transmission Coefficients  In what follows, the algorithm for the construction of the matrices ( T and hence T , ++  T~ , +  T  , T~ +  + +  )  _ 1  , T  _ +  (T  + +  )  _ 1  ,  is described i n detail for the case of a square photonic  lattice. T h e method can be generalized to accommodate other types of photonic lattices, and an outline for how this may be done is presented i n A p p e n d i x A . O f course, only a finite number of plane waves are included i n the calculation, w i t h there being an equal number of s and p polarized plane waves as well as an equal number of right-going and left-going plane waves. Thus ii N x N reciprocal lattice vectors are included i n two-dimensions, the size of the T matrices that result is 2A'' x 2A'' . 2  2  We note that equations 2.64-2.67 require evaluation of the fields at r, r ± (a ± £), r ± 2 £ , r ± 2a, as well as evaluation of e' and p! at r, r ± £, and r ± a.  T h e scheme  of real-space lattice points and reciprocal lattice vectors used for the square lattice is depicted i n Figures 2.3 and 2.4, respectively. Given / 3 , the value of k is calculated from the dispersion relation, for each recipinc  z  rocal lattice vector, Gi ,  using equation 2.82 w i t h the value of the dielectric constant  tm  on the right hand side of the structure, e' . in  A s previously mentioned, there will be  two solutions for k , each of which will be either purely real or purely imaginary for the z  case of lossless dielectrics (purely real e'). One solution is labelled as right-going or (+), and the other as left-going or (—) according to the sign of the real or imaginary part of k. z  For the case of real k corresponding to propagating waves, its sign corresponds to z  that of the group velocity; for imaginary k corresponding to evanescent fields, its sign z  determines whether the fields are growing or decaying i n the positive z direction. The right-going solution is selected and K calculated according to equations 2.33-2.35. Then, the right eigenvector for a particular polarization is calculated by using equations 2.77 to 2.80 and, dropping the z components, is used to form the 4-vectors as i n equation 2.84. These "plane waves" are then projected onto the N x N real-space mesh points on the unit cell depicted i n figure 2.3 according to  42  a  Figure 2.3: Scheme of real-space points on which the fields and dielectric constant are sampled on the unit cell. The dielectric constant is sampled at points represented by the black dots. T h e electric or magnetic fields are sampled only on those points w i t h the crossed squares.  43  y  Figure 2.4: Scheme of reciprocal lattice vectors corresponding to in-plane wavevector of launched modes  44  -1 / / / fi(s/p, + ,IH)  E.  R {Gm,n)  {s/p, + JII)  s  (r  [i(/3+G , )-ei,i]  e  ~(s/p,+,III)  H'  H' (Qi,i) x  H (Q v  l t l  )  MP,+,III)  H  S/P, + )  (G  x  (G  mj7l  m]7  n  )  j)  III  it  (s/p,+,I/III)  Ex  m  l  G  r  ^(s/p,+,I/III)  x  (n  ~(s/p,+,I/III)  H' x .  y(Sj,i)  H  _  H'  s/ ,+) P  III  \^xm,n)  (s/p +,////7) )  J/  m ) n  )  it  fi(s/p,+,I/III)  (G )  E-IX  E (QN,N)  (G  /  mtn  X  (s/p,+,I/III)  Ey  E {QN,N) V  {Gm,n)  H' ' { /P  H' (QN,N)  x  X  ~(s/p,+,I/III) H {QN,N) V  (G ,n,s/p,+) m  [i(/3+G,„) m  ~(s/p,+,I/III)  ,  J  (G , ) m  n  ,  J  (2.105) where the superscript J i 7 indicates that it is a plane wave basis mode i n region III. The subscripts  G  m<n  and q(= s, p), identify the particular Fourier component and polarization  launched, and the subscript (+) indicates the right-going mode (as distinct from the right eigenvector). T h e right eigenvectors are used regardless of whether the mode is rightor left- going. E q u a t i o n 2.68 is applied repeatedly to propagate the field backwards i n discrete steps  45  to the left hand side of the textured layer to yield the fields on each mesh point g  1  in  £r (e ) p  M  H' (Qi,i) prop  x  (Gm,n,g,+) III  Er (Qj,i) p  (2.106)  (G , ,<7,+) m  n  I HI  Er (e , ) p  N N  H' (QN,N) PT0P  X  where the superscript "prop" indicates that the fields are those that have been propagated back to the left hand side for the particular mode launched forward on the right hand side as indicated by the subscripts on the 4-vectors. A Discrete Fourier Transform is then performed to extract, i n turn, each Fourier component of the fields corresponding to each reciprocal lattice vector, G  r > s  , where the  subscripts on the 4-vectors continue to identify the particular launched mode.  III  in Er {Q d  Er (G ,s)  p  p  r  EV (Gr,s)  h  VOP  e x p [ - z ( G , + (3) • £ •,] r  s  H' ° (Gr,s) Pr  P  x  Hy ° (Gr, ) Pr  f° ^u)  E  =  P  S  46  P  (2.107)  The m a t r i x elements are then computed by projecting the resulting Fourier transformed 4-vector on the left hand side of equation 2.107 corresponding to each of the reciprocal lattice vectors, G  r ) S  , onto the plane wave basis for the half-space on left side  of the grating (region I). T h i s is done by taking the inner product with each of the corresponding left eigenvectors for that basis. Specifically, the elements of the matrix ( T  + +  )  are obtained by projection onto the  _ 1  right-going modes on the left hand side of the structure (region I): III  E  (Gr,s)  PrOP  {  T  +  +  )  (G ,s,u),(G ,n,q) T  m  H'  VL T  x  ( r,s) ( r,s) G  H'  (2.108)  H'r (G , ) p  r s  G  H' ° (Gr,s) PT  J  P  y  may then be obtained by matrix inversion. T h e elements of the product matrix  ++  T~ {T )~ +  ++  are obtained by projection onto the left-going modes i n region I.  1  ' ( r,s) 7)  ^(G )  By"' [r- (r +  + +  )- ] 1  ( G r  ,.  f U )  ,  rjS  V +  -l L H  \  -l / / / Er (G , ) p  r  E  s  (Gr,s)  PT0P  ( G T I  (G ,s)  X  Hence T~  G  r,s)  r  H'  (G ) r>s  y  H? P(G IO  r  (2.109)  may be obtained by m a t r i x multiplication of the above result w i t h T  Similarly, the matrices T  and T~ (T )~ +  ++  1  +  +  .  are obtained by launching left-going  modes on the left hand side (region I), projecting onto the real space lattice, integrating forward, Fourier analyzing the resulting wave field, and projecting onto the left eigenvectors on the right hand side(region III):  47  -,///) - 777)  ^ (G,, ) prop  s  G,) r  (G , ,u),(G ,„,g) r  s  m  H' V H'  {u -  777)  ^  s  (G , ) (Gr )  X  (G , ) r  (2.110)  s  ^; (G,, ) prop  r s  •—(u -,III)  rop  s  #I (G , ) prop  ]S  r  s  ^ (G,, ) ^ (Gr, ) prop  s  E [T  +  (T  )  1  ](G ,„u),(G r  r o > n  ,g) —  K+,777) l/ « , t , i J J U p  Pr0P  \  S  (u,+,III)  H'x  vG , ) r  iY>°P(G , )  s  r  (u,+,777),  V The  matrix T  (Gr,s)  H  J  ^  /  p r o p  s  (G , ) r  s  (2.111]  , is obtained by m a t r i x inversion, and T  +  is obtained by matrix  multiplication.  2.2.6  Buried Gratings  It is also desirable to compute the scattering from a 2D grating buried i n the middle of  a dielectric stack as depicted i n Figure 2.5. T h e extension of the model to allow  for  efficient calculation of scattering from such structures was performed by the author.  In  the description of the calculation that follows, the reader is referred to Figure 2.6.  The  calculation proceeds by first determining the matrices T  + +  , T~ , T ~, T +  +  for the  textured layer, taking its adjacent untextured layers as the semi-infinite layers i n the calculation described i n the preceding section. These matrices are then combined with their diagonal counterparts, R ,  R~ , R ~, and R  ++  +  , which describe the scattering off  +  the entire dielectric stack located to the right of the grating, i n a multiple scattering calculation to yield intermediate matrices C , ++  off  C~ , C ~, C +  +  , which describe scattering  the grating and dielectric stack to the right of the grating. These latter matrices are  then combined w i t h L , ++  L~ , L ~, L +  +  , which describe the scattering from the entire  48  49  R  T  ++  R Figure 2.6: Component scattering matrices for calculation of overall reflection (scattering) matrix from grating embedded i n an arbitrary multilayer dielectric stack.  dielectric stack to the left of the grating, i n a multiple scattering calculation to yield F ~ , the matrix that describes reflection and diffraction from the entire buried grating +  structure. T h e expression for C~  +  C~+ = T~  +  is given by:  + T—[I  and the expression for F ~  F~*  = L~  where, the P^  +  ±1  + L  - P; R- P+_ T ]- P- R- P+ T +  +1  +1  +  1  1  +  +1  ++  +1  (2.112)  is given by:  [I - (P-_ C- P;_ L -)]- P-_ C- P;_ L +  1  +  1  1  +  1  ++  1  (2.113)  are diagonal matrices accounting for the phase acquired or exponen-  tial growth or decay of the plane wave corresponding to each polarization and Fourier component i n the calculation as it "propagates" right (+) or left (—) i n the layers to the immediate left (g — 1) or right (g + 1) of the grating layer. 50  T h e untextured dielectric stacks specularly reflect and transmit but do not diffract light. Consequently, the matrices L  ± : t  , L  ±  T  , i?  ± = t  ,i?  ± : F  are diagonal, and their diagonal  elements can be efficiently calculated i n a multiple scattering calculation using Fresnel [42] reflection and transmission coefficients at each interface and plane wave propagation through each layer. T h i s multiple-scattering calculation avoids the need to integrate Maxwell's Equations through thick layers which becomes computationally expensive since the number of integration steps must increase linearly w i t h layer thickness.  Moreover, the multiple  scattering calculation yields a result which is exact for all of the basis functions included i n the calculation.  2.3 2.3.1  Results Introduction  In this section we illustrate how reflectivity calculations can be used to reveal the nature of the photonic eigenmodes of two-dimensionally strongly-textured high-index contrast semiconductor waveguides.  In order to introduce the various concepts involved,  we proceed first through an investigation of the untextured, weakly and strongly onedimensionally textured, and the weakly two-dimensionally textured cases.  2.3.2  Near-Normal Incidence Reflectivity from Untextured Uniform Slab  Figure 2.8 shows the specular reflectivity spectrum over a broad spectral range, for both the s and p polarizations, for light incident from air at an angle of 9 = 10° from the normal onto a 120 nm-thick piece of e' = 12.25 material. T h e structure is depicted i n Figure 2.7. T h e s and p spectra are somewhat different, as expected, w i t h both exhibiting fringes arising from the Fabry-Perot cavity formed w i t h i n the high-index material due to reflections from the interfaces. The m a x i m a i n the reflectivity correspond to resonant leaky states where the fields are localized to some extent to the high-index region. A 51  discussion of the number of Fourier components included i n the calculation, the number of in-plane mesh points, and intermediate planes (2-steps) that must be used to get well-converged results is deferred to section 2.4.  2.3.3  Bound Modes of Untextured Slab  The 120 nm-thick slab of material w i t h dielectric constant e' = 12.25 constitutes a slab 2  waveguide which supports bound electromagnetic modes. These modes have fields i n the cladding layers above and below the slab that decay exponentially away from the slab and thus carry away no energy. B o u n d modes have a dispersion curve falling between the light lines for the refractive index of the core and cladding layers, as depicted i n Figure 1.7. These modes are, i n a sense, bare modes that will be mixed by a 2D texture to form the renormalized eigenstates of the planar 2 D photonic crystal. One way to determine the frequency of such a mode for a given in-plane wavevector is by computing the specular reflectivity spectrum for light w i t h that particular wavevector. The specular reflectivity spectrum will then exhibit a pole when the energy and in-plane wavevector match the dispersion of one of these bound modes.  52  This corresponds to  0.8  Figure 2.8: Specular reflectivity spectrum for light incident at 10° (from the normal) on a 120 nm-thick slab of material of dielectric constant e' = 12.25. The solid curve depicts the spectrum for the s polarization; the dashed curve depicts the spectrum for the p polarization.  53  performing the specular reflectivity calculation w i t h an evanescent incident field. A n example of this is given below.  Figure 2.9 depicts dispersion curves for the  lowest order T E and T M bound modes supported by the slab structure of Figure 2.7. Figure 2.10 depicts the specular reflectivity spectrum obtained when an incident field having an in-plane propagation constant of 20,000 c m  - 1  impinges on the slab. T h i s  spectrum corresponds to probing ui — 3 space along the line depicted i n Figure 2.9. A feature (pole) appears i n each of the s, and p polarized spectra, when the frequency matches that of corresponding T E or T M bound mode, as indicated by the dots i n Figure 2.9.  0 0  40  10  50  Figure 2.9: P l o t of mode frequency versus in-plane wavevector, 3, for the lowest order T E (solid curve) and T M (dotted curve) modes of a planar waveguide consisting of a 120 n m slab of material w i t h e' = 12.25, clad above and below by air (e' = 1). T h e dot-dashed lines represent the lightlines for these materials. T h e solid vertical line depicts the part of u — 3 space that is explored i n the calculation of the spectra shown i n Figure 2.10 below. T h e dots at the intersection of the vertical line w i t h the dispersion curves indicate the appearance of a resonant feature i n the spectra.  54  'I  1 7  1 8  1 9  1 10  1 11  1 12  ^1 13  F R E Q U E N C Y co/2rcc [ 10 cm" ] 3  l  Figure 2.10: Specular reflectivity spectrum for evanescent field incident on 120 n m slab of material w i t h e' = 12.25, w i t h in-plane wavector of 20, 000 c m . T h e solid line is for s polarized light, the dashed line is for p polarized light. - 1  2.3.4  Slab Waveguide with ID Grating  Figure 2.11 schematically depicts the effect on the dispersion of the  electromagnetic  modes when a shallow I D grating is etched into the guide as shown i n Figure 2.12. Gaps open up at the B r i l l o u i n zone boundaries as the various orders (PG, 2PG) of the dielectric modulation couple together the bound modes which existed i n degenerate pairs at ± / ? G / 2 and ±PG fc» the untextured structure. It should be noted that one such dispersion plot r  exists for each of the T E and T M polarized modes, w i t h the T M dispersion plot shifted higher i n frequency.  Finally, we note that the periodicity of the dielectric allows us to  describe the mode dispersion i n a zone-folded scheme i n the first B r i l l o u i n zone. W h e n the specular reflectivity calculations are performed w i t h incident (evanescent) light having in-plane wavevector given by (PG/2)X,  two features are seen to occur i n  each polarization as shown i n Figure 2.13. These occur again as poles i n the reflectivity spectra indicating that the excitations remain completely bound to the guide. T h e eigenmodes i n the vicinity of the second order gap at in-plane wavevector PGX,  55  Figure 2.11: Schematic dispersion plot for single-mode slab waveguide w i t h shallow I D grating etched into its surface. The dashed lines represent the light lines for the low-index cladding layer (larger slope) and the high-index core layer (smaller slope). The solid lines represent the dispersion of Bloch states i n the presence of the grating.  e/=1.0 Figure 2.12: Slab waveguide w i t h "weak" I D texture: the structure consists of a 120 nm-thick slab of e' = 12.25 material, clad above and below by air, w i t h a 10 n m deep I D square-toothed grating having a pitch A = 500 n m and an air filling fraction of 25% etched into the surface.  5G  4000  — i — i I I 5000 6000 7000 8000 F R E Q U E N C Y co/2rc [ cm" ]  i 9000  1  Figure 2.13: Calculated specular reflectivity spectrum for evanescent light incident w i t h in-plane wavevector /3 ={J3Q/2)X = (-rr/A)x. The solid curve is for s polarized light, the dashed curve is for p polarized light. T h e corresponding waveguide structure is depicted i n Figure 2.12.  although primarily composed of superpositions of field components at ±/?G£> are nevertheless coupled via the ±(3QX component of the dielectric modulation, to a component at (3 — 0 which can radiate into the semi-infinite layer, thereby rendering them leaky (or lossy). Figure 2.14a depicts the specular reflectivity spectrum for evanescent light incident w i t h in-plane wavevector, /3 = 1.0ip x, G  i.e., slightly detuned from the boundary  between the 2nd and 3rd B r i l l o u i n zones, for b o t h the s and p polarizations. T h e B l o c h states again manifest themselves as resonant features i n the spectra, however, they are no longer true poles. T h e y have non-zero linewidths owing to their leaky nature. T h e zeroth order radiative component of the B l o c h states i n the vicinity of the 2nd order gap allows us to excite them w i t h a propagating field incident from above the grating.  Figure 2.14b depicts the specular reflectivity spectra for light incident from  above the grating at near-normal incidence for b o t h the s and p polarizations. Once again, two features per polarization occur i n the specular reflectivity spectra when the energy and in-plane wavevector match the dispersion of the modes. Here, the features occur as bipolar resonances i n the specular reflectivity spectra, which are bounded by unity as required by energy (flux) conservation for propagating incident and reflected 57  Figure 2.14: (a) Calculated specular reflectivity spectrum for evanescent light incident w i t h in-plane wavevector, (3 = 1.0ip x, i.e., slightly detuned from the boundary between the 2nd and 3rd B r i l l o u i n zones, (b) Calculated specular reflectivity spectrum for light incident w i t h in-plane wavevector, (3 = 0.0ip x, i.e., slightly detuned from the B r i l l o u i n zone-centre, (c) Calculated 1st order diffraction coefficient, i.e., coefficient for scattering from a field component at (3 = 0.0ip x to a field component at (3 = 1.0ip x. The solid curves are for s polarized light; the dashed curves are for p polarized light. T h e waveguide structure under study is depicted i n Figure 2.12. G  G  G  G  58  fields. T h i s is like h i t t i n g a diffraction grating near-normal incidence but w i t h a wavelength such that there are no propagating diffracted orders. T h e diffraction is only into these evanescent "bound" modes. Figure 2.14 (c) depicts the calculated 1st order scattering coefficient. T h i s is the coefficient for scattering from a field component at (3 = Ancident^ to a field component at (3 — Ancident^ + PG - In this case this corresponds to scattering from (3 = O.OIAJX to X  (3 = I . O I A J X . T h i s is the amount of the incoming radiation at near-normal incidence, w i t h in-plane wavevector (3 = 0 . 0 l A ? x , that is scattered by the grating to light having E , and H fields which are evanescent i n the direction normal to the slab (£) and which vary sinusoidally i n the x direction w i t h an in-plane wavevector of (3 = I.OIPGX.  It essentially  represents the coupling between the zeroth order, radiating component, and the field component resembling a bound mode of the untextured slab. T h i s scattering coefficient also displays resonant features when the energy and in-plane wavevector match the dispersion of the B l o c h states. It is important to note that, i n a l l of the spectra shown i n Figure 2.14, one of each pair of features is narrow compared to the other . In fact, precisely at normal incidence, the narrow feature vanishes completely from the normal-incidence specular reflectivity and first-order scattering spectra, but not from the evanescent specular reflectivity spectrum. T h i s occurs because symmetry requires an exact cancellation of the radiative component of that particular state at the precise centre of the B r i l l o u i n zone. Figure 2.15 shows the results of the reflectivity calculations carried out exactly at the zone-centre ((3 — 0). There are still two distinct B l o c h states at zone-centre, as evidenced by the two features i n the evanescent specular reflectivity spectrum. However, only one of them has a nonzero, zeroth order, radiative component which allows us to couple to it w i t h radiation from above the grating. Nevertheless, we can still couple to the other B l o c h state w i t h an evanescent field, phase matched to the field component of the B l o c h state at ~  59  Q x. G  Figure 2.15: (a) Calculated specular reflectivity spectrum for evanescent light incident w i t h in-plane wavevector, j3 — (3QX, at the edge of the 1st B r i l l o u i n zone, (b) Calculated specular reflectivity spectrum for radiation incident w i t h zero in-plane wavevector, i.e., at the B r i l l o u i n zone-centre, (c) Calculated 1st order diffraction coefficient, i.e., coefficient for scattering from a field component at f3 = O.OlfScx to a field component at (3 = I.OIPGX. The solid curves are for s polarized light; the dashed curves are for p polarized light. T h e waveguide structure under study is depicted i n Figure 2.12.  60  Figure 2.16 shows the near-normal incidence specular reflectivity spectrum for the same structure but w i t h the grating having an-air filling fraction of 75% rather than 25%.  Notice that the broader mode is now the lower-frequency mode. T h e frequency  (energy) ordering of the lossy, and non-lossy modes are dependent upon the duty cycle (air filling fraction) of the grating. T h e frequency ordering is basically due to the differing distributions of the real space field intensities of the eigenmodes i n the high and lowindex regions of the unit cell.  T h i s effect is discussed i n greater detail i n Chapter 6  (section 6.5) where a schematic depiction is given of the in-plane field intensity for the resonant eigenmodes at zone-centre.  i.o H  8050  8100 8150 8200 FREQUENCYCO/2TIC [cm"']  8250  Figure 2.16: Calculated near-normal incidence specular reflectivity spectrum for light incident w i t h in-plane wavevector, (3 = O.Olpcx, for s polarized light. T h e waveguide structure is identical to the one specified i n Figure 2.12 except that the grating has an air filling fraction of 75% rather than 25%.  It should be mentioned that, i n comparison to the photonic crystals discussed later, these structures are considered "weakly textured" because although the dielectric contrast between the teeth and trenches i n the gratings is large, the grating itself occupies  61  a small fraction (~ 20%) of the total core thickness, resulting in a relatively small interaction of the mode w i t h the grating. T h e w i d t h of the gap is approximately 1.5 percent of its centre frequency, which, nevertheless, is still larger than that corresponding to gratings used i n typical D F B ' s , which is on the order of 0.1%. We now consider the situation when the I D grating is made to completely penetrate the guiding layer. Such a structure is depicted i n Figure 2.17. Figure 2.18 shows the specular reflectivity spectrum for s polarized light incident from above the grating at 5° from normal. We consider only the spectrum for the s polarization. We notice that the gap between the modes has increased substantially, indicating stronger coupling between the ±(3Q Fourier components.  In fact, the gap is now ~ 20% of the centre frequency,  indicating a very strong influence of the texture on the photonic eigenstates.  Figure 2.17: Slab waveguide w i t h "strong" I D texture: the structure consists of a 120 n m thick slab of e' = 12.25 material, clad above and below by air, w i t h a I D square-toothed grating that completely penetrates the slab. T h e grating has a pitch, A = 500 nm, and an air filling fraction of 25%.  2.3.5  Waveguide with 2D weak 2nd order grating  Next, we consider the situation where a weak 2D square grating is etched into a waveguide. One example of such a structure is that where a 2D array of cylindrical airholes, 10 n m deep, is buried i n the core of our 125 n m waveguide of e' = 12.25 material. T h e structure, which is depicted schematically i n Figure 2.19, also illustrates the ability of the model to perform calculations incorporating buried gratings. G2  i.oH  |  0.8  H 0.6H  §  0.2 11 F R E Q U E N C Y GV2TC [10  3  cm" ] 1  Figure 2.18: Calculated near-normal incidence specular reflectivity spectrum for s polarized light incident w i t h in-plane wavevector, (3 = O.Ol/fex. The waveguide structure, depicted i n Figure 2.17, consists of a 120 n m thick slab of e' = 12.25 material, clad above and below by air, completely penetrated by I D square-toothed grating having a pitch, A = 500 nm, and an air filling fraction of 25%.  W i t h a 2D grating, i n the vicinity of the 2D B r i l l o u i n zone-centre, there are now four field components resembling bound modes of the waveguide w i t h in-plane propagation constants of ±PG%  a  n  d ^PGV,  that are coupled together via the ±2(5GX  and  ±2PGV  Fourier components of the dielectric modulation. Additionally, a l l four of these couple to a zeroth order component, which radiates out normally from the structure, as i n the I D case, via the ±BQX  and ±6GV components of the texture. Figure 2.20 schematically  depicts the expected 2D bandstructure i n the vicinity of the zone-centre as consisting of four band sheets shown i n the reduced 2D B r i l l o u i n zone. There are actually two sets of these clusters of four sheets, centred at two different frequencies, corresponding to those of the T E and T M modes of the untextured slab. T h e radiative, zeroth order component of the B l o c h state once again allows us to probe these modes w i t h radiation incident from above the grating i n a specular reflectivity calculation. Figure 2.22 depicts the specular reflectivity spectrum for light incident from above this structure w i t h in-plane wavevector along the T — X  symmetry direction.  The  symmetry directions for the square lattice are depicted i n Figure 2.21. Three clusters  63  Figure 2.19: Slab waveguide consisting of 125 n m of e' = 12.25 material w i t h air above and below, having a square lattice of cylindrical airholes, 25 n m deep, embedded at its centre. T h e lattice has a pitch of A = 500 nm, and the radius, r, of the holes is r / A = 0.10.  W  2  *  Figure 2.20: Schematic in-plane bandstructure of resonant modes of waveguide w i t h a square grating shown i n 1st 2D B r i l l o u i n zone.  64  of four bipolar features appear corresponding to Fano resonances [38] resulting from the four leaky modes described above. T h e precise Fano lineshape function is described i n Chapter 5 (section 5.6.1), and a physical explanation of the reason behind the bipolar nature of the Fano lineshape is given i n Chapter 6 (section 6.2). T h e cluster occurring at the lowest frequencies corresponds to the modes arising primarily from the combination of the four T E field components at  ±PGX  and  ±PGV-  The cluster of four modes occurring  at the highest frequencies corresponds to the modes arising from the combination of the four T M field components at  ±PQX  and  ±PGV-  There is an additional cluster of four  modes occurring at an intermediate frequency. These correspond to modes arising from the combination of T E field components at ±PG&  ± PGV-  Figure 2.21: Symmetry directions for the square lattice.  Figure 2.23 shows an expanded view i n the vicinity of each cluster. We note that i n the lowest frequency cluster (where the modes are comprised primarily of T E field components) three features occur i n the s polarization, while one occurs i n the p polarization. Conversely, i n the highest frequency cluster (where the modes are comprised primarily of T M field components) three features occur i n the p polarization while one occurs i n the s polarization. We now examine the properties of the modes occurring i n the lowest frequency cluster i n greater detail.  Figure 2.24 shows reflectivity spectra for light incident w i t h zero 65  8  9 FREQUENCY  10 (O/2TTC  11 [10 cm" ] 3  1  Figure 2.22: Calculated specular reflectivity spectrum for light incident on a 125 n m slab waveguide of e' = 12.25 material w i t h air above and below, and a square lattice of cylindrical airholes, 25 n m deep, embedded at its centre. The lattice has a pitch of A = 500 nm, and the radius, r, of the holes is r / A = 0.10. Solid lines are for the s polarization, while dashed lines depict the p polarization. The spectra were calculated for light incident w i t h in-plane wavevector fixed at P/PG = 0.01 i n the T — X direction, which is defined i n Figure 2.21.  66  i.<H £  1 1 1  0.8-  (a)  u  S  0.2-1 0.07700  7750  7800  7850  7900  7950  8000  F R E Q U E N C Y CO/2TIC [ cm" ] 1  1.0AIL +>•  FLE  H U  0.80.6-  (b)  0.4-  >  r  0.20.09600  9640  9680  9720  F R E Q U E N C Y CO/2TIC [ cm" ]  1.0 H £  0.8  -j  i—i >  H  0.6-|  S  0.2'  (c)  0.0' 11.50  —I— 11.55  11.60  —1— 11.65  11.70  F R E Q U E N C Y co/27ic [10 cm" ] 3  1  Figure 2.23: Expanded view of the spectrum depicted i n Figure 2.22: (a) lowest quency cluster (b) intermediate frequency cluster (c) high frequency cluster  67  in-plane wavevector (zone-centre), corresponding to normal incidence.  The s and p  polarized spectra, which now merely correspond to x and y polarization of the electric field, are identical, as expected from symmetry considerations, and each contains a lone feature. Thus, precisely at zone-centre, the narrow features vanish from the spectra, and the surviving, broad features, one i n each of the s and p (or rather, x and y) polarized spectra, become degenerate.  1.0H  0.8  i—i  > P  0.6 H  U W  w 0.4 H 0.27700  7750  7800  T 7850  7900  T 7950  8000  F R E Q U E N C Y co/2rcc [ cm"'] Figure 2.24: Calculated specular reflectivity spectrum from same structure as i n F i g ure 2.23. T h e spectra were calculated for light incident w i t h zero in-plane wavevector, P/PG = 0.00, corresponding to normal incidence.  Figure 2.25 depicts reflectivity spectra for light incident w i t h in-plane wavevector oriented along the direction  +  ( M symmetry direction). In this case, two resonances  appear i n each polarization (s and p). Figure 2.26 depicts three specular reflectivity spectra, i n the vicinity of the T E gap, for off-normal incident light along the X direction, for three different diameters of the airholes. T h e result illustrates the fact that the precise ordering of the broad and narrow features corresponding to the frequency ordering of the leaky and less leaky modes is dependent on the details of the unit cell of the square lattice. These polarization and symmetry properties are discussed further i n Chapter 6. 68  i.oH  j  0.20.0  J  1  1  7800  7820  1  1  7840  7860  1  7880  7900  FREQUENCYco/27tc [cm" ] 1  Figure 2.25: Calculated specular reflectivity spectrum from same structure as i n F i g ure 2.23. T h e spectra were calculated for light incident w i t h in-plane wavevector fixed at P/PG — 0.01 i n the [x + y] direction (i.e., along the T — M symmetry direction).  2.3.6  2D Strong Texture  We now investigate structures w i t h a "strong" 2 D grating. A s i n the I D case, this can be realized by having the dielectric texture extend all the way through the core of the waveguide. Figure 2.27 schematically depicts one such structure which is very similar to those investigated experimentally i n the present work. T h e waveguide structure consists of a 80 n m thick slab of e' = 12.25 material which forms the guiding core, clad above by air and below by 2  of material w i t h e' = 2.25. T h e semi-infinite half-plane below is  also composed of e' — 12.25 material. T h e core waveguide layer is penetrated by a 2 D square array of holes on a pitch, A = 500 nm, and w i t h hole radii given by r / A = 0.5. Due to the significant reduction of the average refractive index of the core layer, the guiding properties, i.e., the frequencies of the T E and T M modes having P = PQ, can no longer be safely approximated by those of the untextured slab. Based on a calculation using the air-filling-fraction-averaged refractive index of the core, the effective untextured structure is seen to support only T E guided modes, w i t h the T M modes being below the  69  (a)  7950 8000 8050 8100 FREQUENCY co/2nc [ cm"] 1  Figure 2.26: Calculated specular reflectivity spectrum for light incident on a 125 n m slab waveguide of e' = 12.25 material w i t h air above and below, and a square lattice of cylindrical airholes, 25 n m deep embedded at its centre. T h e spectra were calculated for light incident w i t h in-plane wavevector fixed at 3/(3G — 0.001 i n the x direction ( X symmetry direction). T h e lattice has a pitch of A = 500 n m , and the radius, r , of the holes is: (a) r / A = 0.25 (b) r / A = 0.32 (c) r / A = 0.40  70  Figure 2.27: Strongly 2D textured thin asymmetric slab waveguide: the structure consists of a 80 nm-thick slab of e' = 12.25 material which forms the guiding core, clad above by air and below by 2 fxm of material w i t h e' = 2.25. The semi-infinite half-plane below is composed of e' = 12.25 material. The core waveguide layer is penetrated by a 2D square array of holes on a pitch, A = 500 nm, and w i t h hole radii given by r / A = 0.5.  low frequency cutoff. Consequently, spectra are shown only in the vicinity of the second order T E gap. Figures 2.28 and 2.29 show reflectivity spectra for light incident on the structure depicted i n Figure 2.27 along the two symmetry directions. T h e specular reflectivity spectra for this structure exhibit the same symmetry and polarization properties as in the case of weak texture.  However, the range over which  the features occur is now much larger. T h i s is as expected owing to the much stronger scattering induced by the holes that now completely penetrate the waveguide core. Figure 2.30 (a) shows the specular reflectivity spectrum on the same structure at norm a l incidence. A s i n the case of weak texture, a lone feature appears, as the two narrow features vanish and the broad features overlap exactly. Finally, as i n the I D case, we confirm the presence of the lossless features at zone-centre by calculating the reflectivity for evanescent light incident at /3 — (3QX, which comprises one of the dominant field  71  8.5  9.0  9.5  10.0  F R E Q U E N C Y CO/2TIC [10  3  cm" ] 1  Figure 2.28: Calculated near-normal incidence specular reflectivity spectrum for light incident w i t h in-plane wavevector, j3 = 0.03(3 , along the F—X direction for the structure depicted i n Figure 2.27 w i t h r / A = .2821. The solid line represents the s polarized spectrum, the dashed line depicts the p polarized spectrum. G  -i 8.5  1  1  9.0  9.5  F R E Q U E N C Y W/2TCC [10  r* 10.0 3  cm" ] 1  Figure 2.29: Calculated near-normal incidence specular reflectivity spectrum for light incident w i t h in-plane wavevector, Q = 0.03/fc, along the T - M direction for the structure depicted i n Figure 2.27 w i t h r / A = .2821. The solid line represents the s polarized spectrum, the dashed line depicts the p polarized spectrum.  72  components i n a l l four modes at zone-centre. This spectrum is shown i n Figure 2.30 (b). T h e two lossless modes manifest themselves as poles while the two lossy modes appear as degenerate resonances w i t h finite linewidth. T h e nature of the various photonic eigenstates corresponding to each of the resonant features i n the specular reflectivity spectra is described i n Chapter 6.  73  - i 8.5  1  1  9.0  9.5  r~ 10.0  F R E Q U E N C Y (O/2KC [10  3  cm" ] 1  8.5 9.0 9.5 10.0 F R E Q U E N C Y c o / 2 7 i c [10 cm" ] 3  1  Figure 2.30: Calculated specular reflectivity spectra for light incident on the structure depicted i n Figure 2.27 w i t h r / A = .2821. (a) N o r m a l incidence specular reflectivity, i.e.., light incident w i t h f3 = 0. (b) Reflectivity for s polarized evanescent light incident w i t h P = P x. G  74  2.4  Convergence  We recall that i n the F D R S algorithm described above the structure being modelled is described by a finite, discrete, 3D mesh of the electric susceptibility and magnetic permeability constants. T h e components of the electric and magnetic fields are computed on a finite, discrete, 3D mesh, and a scattering matrix is computed for a finite, discrete set of plane wave like basis functions. There are, i n principle, several distinct issues w i t h respect to convergence having to do w i t h the choice of the number of mesh points, and the number of basis wave functions to include i n the calculation. Firstly, for any given basis state, the accuracy of the propagation algorithm is limited by the sizes of the in-plane and z direction meshes relative to the wavevector of the basis state, as expressed by equations 2.36-2.38. The second issue relates to the number of plane wave-like basis functions included in the calculation. Even for t h i n sinusoidal gratings, higher order scattering requires, i n principle, that an infinite number of plane waves be included. O f course, for any given structure, there w i l l only be significant scattering to a finite subset of these. T h e end result is that the number of plane waves to be included, either to capture the essential physics, or further, to give acceptable quantitative results w i l l depend on the structure. The t h i r d issue has to do w i t h the relationship between the number of mesh points used for the field and the number of basis functions used i n the calculation. We recall that the basis functions are essentially plane waves w i t h a particular in-plane wavevector; they represent the different discrete in-plane Fourier components of the fields. The Nyquist criterion requires that the number of mesh points i n each direction (x and y) be at least equal to the number of Fourier components (or basis functions) i n that direction. T h i s simply allows the basis functions to be represented on the real-space mesh without aliasing. However, if the number of in-plane mesh points exceeds the number of basis functions included i n the calculation, then the fields at the end of the integration may contain significant higher spatial frequency content than that included i n the plane wave  75  basis states onto which the fields are projected. This leads to a "scattering out of the basis" and quantitative errors.  Because the problem is cast as an inverse scattering  calculation requiring the transfer m a t r i x to be inverted to yield the scattering matrices, one flagrant manifestation of this inconsistency is that the computed specular reflectivity spectra contain values that exceed unity for propagating waves and thus violate the principle of flux conservation. Ideally, to insure self-consistency, it is preferable to perform the calculation w i t h the same number of mesh points as basis functions.  However, this would mean that the  basis functions w i t h the highest in-plane propagation vectors are not integrated very N 2TY A accurately because k / a / x y  x y  = (~2~~^~)( jy)  =  7 1  ^  Since these higher order field  components are relatively small, the significance of this error should be small. However, i n the waveguide structures of interest here, these higher order components sometimes become erroneously enhanced and thereby introduce false modes i n the reflectivity spectra. In some cases, this leads to spurious features appearing i n the spectral range i n the vicinity of the second order T E gap. One solution is to simply increase the number of basis functions and mesh points simultaneously such that the spurious features gradually move up and out of the range of interest. T h i s is computationally very expensive. However, it turns out that by having the number of mesh points exceed the number of basis functions by one or two, the spurious features are quickly moved up and out of the frequency range of interest. This comes at the cost of the violation of flux conservation. However, the excess reflected flux i n this regime on the structures studied is only i n the range of a couple of percent. N o such spurious features arise i n the spectra presented i n this section and so the calculations were carried out w i t h the same number of mesh points as basis functions. A fourth issue concerns the inherent pitfalls i n attempting to represent the continuous p e r m i t t i v i t y and permeability functions, often containing sharp transitions, on a discretely sampled mesh. We recall that, as a result of equations 2.64 to 2.67, the number  76  of mesh points used to describe the structure i n terms of its p e r m i t t i v i t y and permeability (e' — //-mesh) is related to that of the mesh points for the fields, as indicated by figure  2.2. This means that adjusting the e' — /i'-mesh necessarily requires adjusting  the field mesh as well. T h e remainder of this section describes the various tests performed to determine the convergence properties of the F D R S calculations and discusses the results w i t h a view to the issues raised above.  2.4.1  Convergence of the Integration Scheme Alone  A simple, basic test of the accuracy and convergence of the code was performed by using it to determine the frequency of the bound modes of an untextured slab from the appearance of the poles i n the evanescently probed specular reflectivity for large in-plane propagation vector. Such a test is attractive because an exact answer can be calculated independently by using Fresnel coefficients [42]. It should also be noted that since the problem possesses continuous translational symmetry i n the in-plane directions, the size of the in-plane unit cell for the calculation, may, i n principle, be chosen arbitrarily. Thus, the size of the in-plane mesh on which the fields are integrated may be varied continuously. Additionally, since there is no scattering to higher order in-plane Fourier components i n the case of the untextured waveguide, it is sufficient to include only one plane wave corresponding to the in-plane Fourier component of the field having the i n plane propagation constant for which it is desired to find the bound mode frequency i n the calculation. This further allows only one mesh-point per unit cell to be used i n the integration of the fields. A s a result the F D R S code becomes much less computationally expensive than i n the textured case. However, purely for convenience, i n the studies on the convergence properties of the untextured waveguide mode frequencies presented here, the F D R S parameters were set as follows. The in-plane unit cell was set to a square of side A = 500 nm, which  77  was typical of the actual textured structures studied and modelled i n the present work. O n l y one plane wave was used but the number of in-plane mesh points per unit cell and the number of z-mesh points were varied. T h e in-plane wave vector of the incident evanescent light was set to (3- = 2ir/A. mc  The untextured waveguide structure used i n  the calculation consisted of a 120 n m slab of material w i t h e' = 12.25 clad above and below by air (e' = 1.0). Figure 2.31 depicts a logarithmic plot of the absolute value of the error as a function of the number of mesh points that were included i n the calculation i n each of the x and y directions. T h e different curves (lines) from top to b o t t o m on the graph are for increasing numbers (specifically: 12, 22, 32, 42, 62, and 122) of mesh points i n the z direction. A s the number of z steps increases the curves approach a straight line, w i t h a slope of —2 on the logarithmic plot, indicating that for sufficiently fine z-mesh, the error decreases as the inverse square of the number of mesh points i n each of the x, and y directions. Since there is no scattering to higher order in-plane Fourier components i n the case of the untextured waveguide, it is sufficient to include i n the calculation only the one i n plane Fourier component having the in-plane propagation constant for which it is desired to find the bound mode frequency. The error here may arise only from the coarseness of the mesh on which Maxwell's equations are integrated. Therefore, the preceding analysis tests only the convergence of the integration scheme, itself.  2.4.2  Overall Convergence Behaviour for Textured Structures  In the case of a textured waveguide, the eigenmodes contain, i n principle, an infinite number of in-plane Fourier components of the fields. It is expected that only a finite subset of these are significant, and so the issue arises of how many need be included i n the calculation to yield quantitatively accurate results. A s i n the untextured case, the number of mesh points i n the in-plane (x — y), as well as i n the normal (z) direction may  78  1.0  1.1  1.2  1.3  1.4  LOG(NO. O F M E S H POINTS IN E A C H OF X A N D Y DIRECTIONS)  Figure 2.31: Logarithmic plot of the absolute value of the error on the bound mode frequencies of an untextured slab waveguide as calculated by the F D R S method as a function of the number of mesh points i n each of the x and y directions included i n the calculation. The (artificial) unit cell for the F D R S method was set to a square of side A = 500 nm, and the in-plane wave vector of the incident evanescent light was set to P- = 27r/A. The structure used for the calculation consisted of a 120 n m slab of material w i t h e' = 12.25 clad above and below by air (e' = 1.0). T h e different curves (lines) are for increasing numbers, (specifically: 12, 22, 32, 42, 62, and 122), from top to b o t t o m on the graph, of mesh points i n the z direction. mc  79  also be varied. Rather than exploring the entire parameter space, which is computationally expensive, particularly for two-dimensionally textured structures, the approach adopted i n the work described below is to vary these parameters i n a related fashion. T h i s is done as follows.  We define iV to be the number of reciprocal lattice vectors  (or Fourier components of the field) used per reciprocal-space direction, as shown i n figure 2.4. Thus, i n 2 D , there will be N x N reciprocal lattice vectors. We also set equal to N the number of field mesh points per real-space direction, so that there are N x N field mesh points i n 2 D . Finally, we set to N + 2 the number of pairs of planes on which the electric and magnetic field are alternately defined (see figure 2.3). T h e additional two pairs of planes account for the fact that one pair of planes is required i n each of the untextured regions, so that the number of planes i n the textured region remains N. Therefore, i n this scheme, the aspect ratio of the three-dimensional mesh or "element" remains unchanged as iV is varied. We note that for the 2D square lattice, the mesh sizes for the approximations defined i n equations 2.33-2.35 are given by:  a = b=V2  A N  d  (2.114)  (2.115)  where d is the thickness of the textured region. T h e F D R S calculation may then be performed several times for the same physical structure but for increasing values of this one parameter, N, called the "number of elements", w i t h respect to which the convergence properties of the mode frequencies and widths may be studied. Figure 2.33 shows the results of such a study for the strongly one-dimensionally textured slab waveguide depicted i n Figure 2.32 w i t h slab and grating thickness, d = 80 n m . T h e plots w i t h the square markers and dashed lines i n Figure 2.33a and Figure 2.33b represent the frequency and w i d t h respectively of the lossy mode at the zone-centre (P = 0) as a function of the number of elements used i n the F D R S calculation. These plots indicate that the convergence behaviour of this model w i t h respect to b o t h the  80  mode frequency and linewidth possesses a strong oscillatory component.  e '=12.25  ej =1.0  A=500 nm  G!  2  =1.0  Figure 2.32: Slab waveguide w i t h "strong" I D texture: the structure consists of a slab of thickness d of e' = 12.25 material, clad above and below by air, w i t h a I D square-toothed grating having a pitch, A = 500 nm, and an air filling fraction of 25% that completely penetrates the slab.  B y damping out this oscillatory component, the convergence properties of the model can be improved. This can be achieved by spectrally filtering the underlying c' — p! mesh such that it contains no higher spatial Fourier components than that permitted by the Nyquist criterion for the x — y mesh employed for the particular calculation as dictated by the number of elements used therein. In other words, an anti-aliasing filter must be applied to the underlying e' — p' profile of the ideal structure that is appropriate for the coarseness of the mesh on which the fields are to be sampled for the particular calculation. The filter used i n this work is a Kaiser window [43] w i t h -^Kaiser  s e  /?Kaiser  = 2.3 and  t equal to the number of elements used i n the F D R S calculation (in order to  optimally satisfy the Nyquist criterion). The plots w i t h the round markers and solid lines i n Figure 2.33a and Figure 2.33b represent the corresponding results of the F D R S calculations when they are carried out w i t h this spectral pre-filtering of the e' — / / profile implemented. Monotonic convergence w i t h respect to the number of elements is now exhibited. In order to obtain an estimate of the upper bound on the error of the F D R S calculation it is useful to re-plot the above convergence results w i t h spectral pre-filtering as 81  11.2*  11.1 -  ,".  ',  , ,  L ^ r T ; '.;  \.  'l  "  i:  i  "  *  M  "  "  •  a 11.0 o  (a)  10.910.8 10.7-  OH  10.6 300-  i i i  Ii  i i i I i i i i  I  i i i i I i i i i  Ii  i i i  i  i i i i  )  i i i i I i i i i  I  i i i i I  280260-  a 240X H Q  (b)  220200180| i i i i | II i i | i i i i | i i i i | i i i i | i i i i | i i II |  10  40 30 20 NO. OF ELEMENTS  50  Figure 2.33: M o d e frequency (a) and w i d t h (b) vs. the "number of elements" (see text) used i n F D R S calculation for the structure depicted i n Figure 2.32 w i t h d = 80 nm. T h e results are for the second-order lossy mode at the zone-centre. Plots w i t h round markers and solid lines depict results w i t h anti-aliasing spectral filtering of the e' — // profile implemented; plots w i t h square markers and dashed lines depict results without filtering of the e' — JJ! profile.  82  a function of the square of the mesh size normalized by the size of the unit cell (grating pitch). T h i s plot is shown i n Figure 2.34. The dashed lines represent the best fit line obtained from a least-squares fit. T h e y-intercept of the best fit line may be taken as an estimate of the final converged value, and, hence, an estimate of the error for a particular calculation may be obtained by subtracting it from the value of the calculation. In the case of the width, there was a regime at the larger mesh sizes (fewer elements used) where the convergence was not very linear w i t h respect to the square of the mesh size, but when the mesh size squared decreases below a certain point (~ 10 x 10~ on 3  the graph, corresponding to 15 elements), b o t h the linearity and rate of convergence improve. Therefore, the fitting was only carried out using the results for these smaller mesh sizes as indicated by the presence of the dashed line. For this structure it was found that a computation w i t h 15 elements was sufficient to produce results that are converged to w i t h i n 1% for the mode frequencies, and to w i t h i n 1.7% for the width. Similar convergence behaviour is exhibited i n the case of two-dimensional strong texture. Figure 2.36 depicts the convergence of the mode frequency and w i d t h for the second-order lossy mode at the zone-centre for the structure depicted i n Figure 2.35. T h i s structure is very similar to those used i n the experimental work described later i n this thesis. Here it is found that a computation w i t h 17 elements, requiring approximately two hours of runtime on a current P C (AMD Duron 1600 1.6 G H z , 1 G B R A M ) , produces results that are converged to w i t h i n 1% for the mode frequency and to w i t h i n 7% for the w i d t h . Finally, it is perhaps not surprising that the F D R S calculations converge roughly linearly w i t h the square of the mesh size since this is the leading error term i n the approximations i n equations 2.33 to 2.35.  83  11.1 -E  e  »^ ^  11.0-j X  o  N  -  10.9-i  u  (a)  N  10.8-j  N  s v  \ N N  w  10.7 \  X  I  I I I I '[• 1 I I I  I 1  V  I T T | I I I I | I I I T  |  •  296292-  •  (b)  •  X  H  *  •  /  '  i i i i I i i i  1  r | T  T i i |  m  i i i i | i  1  10 20 30 40 X-Y MESH SIZE SQR ( a / A f xlO" 3  Figure 2.34: (a) M o d e frequency and (b) w i d t h , vs. square of the x — y mesh size normalized by the unit cell size (or grating pitch) used i n the F D R S calculation for the structure depicted i n Figure 2.32 w i t h d = 80 n m . T h e results are for the second-order lossy mode at the zone-centre. T h e insets provide a magnified view of the small mesh size regions of their respective graphs. T h e F D R S calculation was carried out w i t h anti-aliasing spectral filtering of the e' — / / profile implemented. T h e dashed lines represent the best fit line obtained from a linear least squares fit. In (b) the rightmost three points were ignored i n the fitting.  84  A=500 nm e '=2.25 3  Figure 2.35: Strongly 2D textured t h i n asymmetric slab waveguide: the structure consists of a 80 nm-thick slab of e' = 12.25 material which forms the guiding core, clad above by air and below by material w i t h e' = 2.25. T h e core waveguide layer is penetrated by a 2D square array of holes on a pitch, A = 500 nm, and w i t h hole radii given by r / A = 0.2821.  85  • 10.20 d g  o 10.15^  (a)  KlO.lOd ® 10.05  i  110.00-3 9.95  9!  A n—i—i—i—r  -i—i—i—i—i—i—r  230 -j  220-1 S 210^  g  (b)  200^ 190^ T  T  5 10 15 20 X-YMESH SIZE SQR ( a / A f xlO"  3  Figure 2.36: M o d e frequency (a) and w i d t h (b) vs. square of the x — y mesh size normalized by the unit cell size (or grating pitch) used i n the F D R S calculation for the structure depicted i n Figure 2.35. The results are for the second-order lossy mode at the zone-centre. The F D R S calculation was carried out w i t h anti-aliasing spectral filtering of the e' — fj,' profile implemented. The mesh sizes correspond (from left to right i n the graph) to 21 x 21, 19 x 19, 17 x 17, 15 x 15, 13 x 13, 11 x 11, and 9 x 9 elements used i n the calculation.  86  2.5  Comparison with Green's Function Based Model  Due to r u n times of several hours required to produce well-converged, resolved spectra over a broad frequency range for comparison to those obtained experimentally, the F D R S model presented i n this thesis was used for validation of the more computationally efficient Green's Function ( G F ) based model [40, 41] which was then used for extensive fitting to the experimental spectra. In the G F model, the I D or 2D grating is treated as a modification to the linear susceptibility, A x ,  of some untextured slab waveguide structure.  are then arranged so that the polarization arising from A x  Maxwell's equations  is isolated as a driving  term on the right hand sides. In this formulation, the homogeneous solution is merely the known solution for the case of an untextured waveguide. Using a Green's function formalism, a particular solution may then be found for the spatial Fourier transform of the electric field. The solution takes the form of an integral of the polarization generated by A% times the Green's function. T h e polarization is then re-expressed i n terms of A x times the electric field to yield an integral equation. Due to the discrete translational symmetry of the problem, A x  may be expanded i n an infinite but discrete Fourier  series over the reciprocal lattice vectors of the grating. Also, due to Bloch's theorem, only the corresponding discrete set of spatial Fourier components of the field need be computed. Using the fact that A x is non-zero only i n the grating layer, and making the approximation that the fields do not vary significantly over the thickness of the grating (thin grating approximation), the integral equation is converted to a set of algebraic equations which couple the various Fourier components of the fields through the various Fourier components of A x -  A full, detailed description of the G F model is beyond the  scope of this thesis and is given elsewhere [40, 41]. However, from the brief description given above, it is important to note: 1) that the number of reciprocal lattice vectors included i n any actual calculation w i l l be finite and may, therefore, affect the accuracy of the results; and 2) that the accuracy of the results w i l l also be affected by the extent  87  to which the fields remain constant over the thickness of the grating. In the remainder of this section we demonstrate the accuracy of this Green's function-based reflectivity calculation, when applied to a t h i n slab, by comparing it w i t h results from the F D R S model. Figure 2.37 compares the near-normal (P = 0.02/3^ along the F — X direction) specular reflectivity spectra from a 2D textured slab, as calculated using the G F model (upper plot), and using the F D R S model (lower plot). T h e structure used i n the comparison is the same as that depicted i n Figure 2.35 and is intended to be a somewhat simplified version of that used i n the experimental work described i n this thesis. It consists of an 80 n m slab of material w i t h e' = 12.25 completely penetrated by a 2D square array of cylindrical holes on a 500 n m pitch and 25% air filling fraction, all sitting on a semiinfinite substrate w i t h e' = 2.25. T h e number of mesh points and plane waves used for the F D R S model are as described i n section 2.4.2 using 17 elements. T h e G F calculation included the nine reciprocal lattice vectors at 0,  ±PG%,  ±PGV,  ±PG%  ±  PGV-  T h e computation time required for the G F model is approximately 1/2000 of the time required for the F D R S approach. In Figure 2.37 it is clear that the spectra are i n excellent agreement, b o t h w i t h respect to the locations and widths of the resonances. T h e absolute frequencies of the modes differ by less than 1.5%, which is smaller than uncertainties arising from grating fabrication tolerances (pitch, filling fraction, etch profile), epitaxial growth tolerances (layer thickness and alloy composition), uncertainties i n literature values of refractive indices, as well as experimental uncertainties on the in-plane wavevector of the probing beam. Furthermore, the widths of the broad, p polarized features were extracted from the results of b o t h calculations and found to agree to w i t h i n 6.6%. Similar calculations on the same structure at precisely normal incidence (P = 0) using b o t h models yielded spectra, exhibiting, as expected, a single broad degenerate feature i n each of the s and p polarized spectra. Here, the agreement between the two models was w i t h i n 1.0% w i t h respect to the absolute frequency, and to within  88  6.5% w i t h respect to the width. To address the limitations imposed by the thin-grating approximation made i n the G F model we consider a symmetric planar waveguide consisting of a slab of varying thickness of e' = 12.25-material, w i t h air (e' = 1.0) above and below, completely penetrated by a I D grating having an air-filling fraction of 25%. T h e structures are similar to that depicted i n Figure 2.17. Figure 2.38 plots the location of the upper and lower edges of the 2nd order gap as calculated w i t h the G F model (dotted curve), and F D R S model (solid curve) for varying thickness of the grating (=slab). T h e F D R S model was verified to be converged to w i t h i n 1% for the upper band and 0.2% for the lower band (using 17 "elements" i n the calculations). T h e G F calculation included the three reciprocal —*  lattice vectors at 0 and ± / ? G X . It is clear that for t h i n gratings the two methods agree well. W h e n the waveguide-grating is 80 n m thick the two methods agree to w i t h i n 1.4% for the upper edge, and 0.27% for the lower edge. However, when the waveguide is 250 n m thick (a thickness greater then those that are usually of interest i n the context of studying photonic crystal membranes, which is the object of the present work) the two methods agree to w i t h i n 6.2% for the upper edge, and 0.8% for the lower edge.  The  percent difference between the two methods increases as the thickness of the grating and slab-waveguide increases. However, it is not possible, i n general, to specify a simple relationship between the G F model's accuracy and the grating thickness.  For a given structure, the accuracy  depends on the mode being studied, as demonstrated i n this example.  Furthermore,  different layer structures may result i n different errors for a fixed grating thickness. The F D R S model is, therefore, useful for benchmarking and verifying the accuracy of the G F model for each type of structure.  89  1  8000  1  1  1  1  9000  10000  '  1  1  11000  1  12000  FREQUENCY co/27ic [cm" ] Figure 2.37: Near-normal incidence specular reflectivity spectra for light incident w i t h in-plane wavevector P = 0.02/?G along the T — X direction calculated w i t h the F D R S model (lower plot) and the G F model (upper plot). T h e parameters used for the F D R S model are as described i n section 2.4.2 using 17 elements. T h e G F calculation included —*  the nine reciprocal lattice vectors at 0, ± / ? G ^ > ±PGV, ±PG% ± PGV- T h e solid lines represent the s polarized spectra, and the dashed lines represent the p polarized spectra. T h e structure being modelled is the same as the one depicted i n Figure 2.27. T h e resonance widths and shape are i n excellent agreement. T h e mode frequencies agree to w i t h i n 1.5%.  90  — I — I — I — I — I — I — I — I — I — I — I — I — I — I — I — I  100  150  200  250  Grating Thickness [nm] Figure 2.38: Location of the upper and lower edge of the 2nd order gap i n a I D textured symmetric waveguide as a function of guide and grating thickness, d (see Figure 2.32). T h e waveguide core is made of material w i t h e' = 12.25 and clad above and below by air (e' = 1.0). The grating, which completely penetrates the waveguide core, has an air filling fraction of 25%. T h e dashed lines are for the calculation w i t h the G F model, the solid lines w i t h the F D R S model which was verified to be converged to w i t h i n 1% for the upper band and 0.2% for the lower band (using 17 "elements" i n the calculations). T h e G F calculation included the three reciprocal lattice vectors at 0 and ±PG X  91  2.6 As  Significance of Modification to Pendry's Method  previously mentioned, this scheme for integration of Maxwell's equations on a discrete  mesh is a modified version of a similar scheme published by J . B Pendry [ 4 4 ] .  The  principal difference lies i n the approximations Pendry makes for the wavevector. Instead of equations 2 . 3 3 to 2 . 3 5 , Pendry defines two approximate wavevectors:  k  fa  [exp(ik a)  — l]/ia  =  K  k  fa  [exp(ik b)  - l]/ib  =  «  fa  [exp(ik g)  - l]/ig  =  x  y  k  z  x  y  z  X  J  ( 2 . 1 1 6 )  « f  and k  fa  — [exp(—ik a) — l]/ia  =  K  k  fa  -[exp(-ik b)  -l]/ib  =  K*  k  fa  ~[exp(-ik g)  - l]/ig  =  x  y  z  Substitution of  into  ( 2 . 1 1 6 )  x  y  z  and  ( 2 . 3 1 )  into  ( 2 . 1 1 7 )  X  ( 2 . 1 1 7 )  Fourier transformation to  ( 2 . 3 2 ) ,  real space, and elimination of the z components i n a manner similar to the derivation presented i n section 2 . 2 yields, for example, for the E  component of the fields:  x  E (r + g) = E (r) + ^ ^ ( ) ^ ( r ) + ^ { x  x  r  , H' (v + a - b) - H' (r + a) x  e'(r H( y  H' (r - b) -  x  + a)6  H' (r) x  e'(r)b  - s) - H (r)  T  x  H (r) - H (r + *  v  y  y  e'(r)a  e'(r + a)a  }  ( 2 . 1 1 8 )  Comparing this w i t h equation 2 . 6 4 derived i n section 2 . 2 :  £ * ( r + 7) = ^ ( r - ) + ^ ' ( r K ( r ) + ^ { 7  H (r x  c  a  z  + a + C)-H (r  + a-C)  x  e'(r + a)b H' {v)-H' (v-2a) y  y  e'(r  —  a) a  H' (r  |  t  -  a + Q -  H' (T  e'(v —  a)b  '  +  H>(r + 2a) -  e'(r 92  + a)  H (v) y  a  }  x  -  a - Q  ( 2 . 1 1 9 )  we see that b o t h contain the four terms i n braces corresponding to discrete, numerical derivatives arising from V x H i n Maxwell's Equations. However, Pendry's approximations for k lead to forward or backward difference terms for the derivatives rather than the centred difference terms that arise i n the approximations for k used i n the present work. This introduces an artificial chirality for the scattering to Fourier components i n the ±x  and ±y directions i n the plane, and this asymmetry prevents the model from pro-  ducing specular reflectivity spectra w i t h features that exhibit the expected degeneracies at points of high symmetry, such as zone-centre. Figures 2.39 and 2.40 compare the specular reflectivity spectra produced by the model using Pendry's original approximations for k to those from the model implemented i n this work. T h e specular reflectivity spectra are for light incident from air on a 80 n m film of material having a refractive index of 3.5, textured w i t h a square array of 300 n m diameter cylindrical holes filled w i t h a material having a refractive index of 1.5, all on a substrate of refractive index 1.5. Figure 2.39 is for the case of normal incidence, while Figure 2.40 is for the case of light incident w i t h in-plane wavevector of P/PG = 0.01 along the T — X direction. A t normal incidence, Pendry's original scheme does produce identical spectra for b o t h the s and p reflectivity. However, it produces three distinct features i n each of these spectra rather than one degenerate feature.  More generally, for the case of ra-  diation incident w i t h a wavevector P = 0.01/?G along the T — X symmetry direction, Pendry's original scheme yields three features i n each polarization rather than the combination required by the symmetry of the 2D square lattice.  93  3s+lp  10.4  10.6  10.8  11.0  11.2  11.4  11.6  11.8  12.0  12.2  Frequency <JD/2TIC [cm ] 1  Figure 2.39: Calculated normal incidence specular reflectivity spectra for light incident w i t h zero in-plane wavevector. T h e calculation is performed (a) using Pendry's original approximations for k and (b) the approximations for k implemented i n this work. The solid and dashed lines, which overlap completely here, represent the s polarized and p polarized spectra respectively. The light is incident on a film of material having a dielectric constant of 12.25 textured w i t h a square array of 300 n m diameter cylindrical holes filled w i t h a material having a dielectric constant of 2.25, all on a substrate w i t h a dielectric constant of 2.25. The pitch of the grating is 500 n m .  94  0.0 -I 10.4  1  1  1  1  1  1  1  1  10.6  10.8  11.0  11.2  11.4  11.6  11.8  12.0  1 12.2  Frequency CO/2TTC [cm ] 1  Figure 2.40: Calculated near-normal incidence specular reflectivity spectrum for light i n cident w i t h in-plane wavevector, (3 = O.OIPG, along the T — X direction. T h e calculation is performed (a) using Pendry's original approximations for k and (b) the approximations for k implemented i n this work. T h e solid lines represent the s polarized spectra, the dashed lines depict the p polarized spectra. The light is incident on a film of material having a dielectric constant of 12.25 textured w i t h a square array of 300 n m diameter cylindrical holes filled w i t h a material having a dielectric constant of 2.25, all on a substrate w i t h a dielectric constant of 2.25. T h e pitch of the grating is 500 n m .  95  2.7  Summary  T h i s chapter has described a numerical model for calculating optical scattering from an arbitrary slab waveguide structure w i t h an embedded 2D photonic crystal. T h e algor i t h m involves the projection of plane wave-like basis functions onto a real-space mesh and integration of each of these functions through the structure i n the direction perpendicular to the slab, using an approximate, discretized version of Maxwell's equations i n real space, followed by projection back onto the basis of plane wave functions, so as to form a transfer matrix. T h e plane wave-like basis functions are the actual eigenfunctions for the translation operation on the discrete mesh using the approximate, discretized Maxwell's equations i n a bulk, untextured medium. T h e model is capable of computing efficiently the scattering from structures w i t h buried photonic lattices by combining the real-space integration scheme for the textured layer w i t h Fresnel coefficients computed for the untextured layers, using a transfer matrix formalism. Scattering spectra that were generated using this numerical model for various planar waveguide structures w i t h b o t h strong and weak, I D and 2D photonic lattices were presented  interpreted,  and discussed i n terms of the dispersion, lifetimes, and polarization properties of the slab modes of their various respective structures.  Results of an investigation of the  convergence behaviour of the numerical algorithm were presented for the t h i n "hard" waveguide photonic crystals of interest i n this work which showed that convergence was greatly improved w i t h the application of anti-aliasing pre-filtering to the dielectric profile. T h e results also showed that by using a mesh size and set of basis functions that required a r u n time of only a couple of hours on a current P C (AMD Duron 1600 1.6 G H z , 1 G B R A M ) the mode frequencies were converged to w i t h i n approximately 1% and the widths to w i t h i n 7%. M o d e frequencies and widths were calculated using a more efficient Green's function ( G F ) based method specific to the t h i n "hard" waveguide photonic crystal geometry and found to agree w i t h the results from the method presented here to w i t h i n 1.5% for the frequencies and 7% for the widths. Results were presented  96  which indicated that the agreement between the two models decreased as the depth of the textured layers was increased. F i n a l l y an explanation was given of how the incorrect polarization properties and lack of degeneracies expected at high crystal symmetry points exhibited by spectra using Pendry's original approximations were attributable to the introduction of an artificial chirality into the equations, and how this was rectified i n the model presented i n this work.  97  Chapter 3 Fabrication  3.1  Introduction and Overview  One of the author's early attempts at realizing a robustly guiding waveguide w i t h strong 2D texture is depicted i n the micrograph shown i n Figure 3.1 taken using a scanning electron microscope ( S E M ) . A view taken at higher magnification is shown i n Figure 3.2. The structure consists of a 100 n m G a A s film sitting on a 2.0 fxm layer of Alo.6Gao.4As on a G a A s substrate. It was grown by M e t a l Organic V a p o u r Phase E p i t a x y by Simon W a t k i n s ' group at Simon Eraser University.  A 20 yum x 100 jivn array of holes of  diameter 350 n m and period 480 n m was etched through the top G a A s layer using aqueous citric acid/hydrogen peroxide solution and an etch-mask of P M M A made by electron beam lithography. After removal of the mask, the structure was dipped i n an aqueous hydrofluoric acid solution, which penetrated through the etched holes i n the G a A s layer to selectively etch away the underlying Alo.6Gao.4As film while leaving the G a A s layers intact. T h e structure, therefore, constitutes a "freestanding", a i r / G a A s / a i r waveguide w i t h high-index-contrast 2D texture embedded directly i n the guiding core. Figure 3.2 also serves to illustrate the extent to which these thin, diaphanous structures represent the extreme non-perturbative limit of textured slab waveguides. The problem w i t h this approach is the lack of robustness, which additionally places an upper limit on the size of the freestanding area that can be produced before the whole structure buckles and collapses. In an attempt to fabricate a batch of ten such structures, each having a length of 90 fim and widths varying from 10 fj,m to 100 fj,m, only one structure, having a w i d t h of 10 /um, survived.  98  Figure 3.1: Scanning electron microscope ( S E M ) image of freestanding a i r / G a A s / a i r slab waveguide patterned w i t h 2 D square lattice of airholes.  99  Figure 3.2: H i g h magnification ( S E M ) image of the same structure as i n Figure 3.1  100  GaAs/Al G a A s 3  7  \t t - 8 0 m  Figure 3.3: Schematic diagram of oxide-based high-index contrast waveguide w i t h strong 2D texture  W i t h the advent of wet-lateral oxidation techniques [45, 46, 47] for oxidizing A l G a A s to produce aluminum oxide, an alternative approach was developed. It consists of oxidizing, rather than etching away, an underlying A l ^ G a i - ^ A s layer through the 2D array of airholes that penetrate the waveguide core. T h e refractive index of aluminum oxide, n xide ~ 1.6, which forms the underlying cladding layer i n this scheme, is sufficiently low 0  to maintain robust guiding even i n the presence of airholes in the waveguide core. Such an oxide-based structure was fabricated by the author and is depicted schematically i n figure 3.3. The sample was grown by Molecular B e a m E p i t a x y by Shane Johnson at the Center for Solid State Electronics Research, A r i z o n a State University. The oxidation process requires A l G a i _ A s with high a l u m i n i u m composition in order to proceed. Therefore, x  x  Alo.98Gao.02As was grown as the cladding layer below the waveguide core.  Al.wGawAs  was grown to form the lower 35 n m of the core, adjacent to the layer to be oxidized. T h i s was done on the basis of reports i n the literature [48] that a better interface quality is achieved between this alloy and a l u m i n i u m oxide than w i t h pure G a A s . A s with the  101  "freestanding" sample, the underlying cladding layer was made thick enough so that the value of the electric field i n the evanescent tail of the bound mode is insignificant at the G a A s substrate, thereby making negligible any coupling to substrate modes. T h e remainder of the fabrication process was developed and carried out by the author. The holes were made by dry etching the waveguide through a 225 nm-thick P M M A mask created using electron-beam lithography. T h e P M M A mask was then removed. T h e oxide layer was formed by placing the etched sample i n an atmosphere comprised of nitrogen gas and water vapour at 425°C for 20 minutes. A t o m i c force microscopy was used to determine the uniformity of the pitch and the hole diameters. T h e following section describes the techniques and equipment used i n fabricating the oxide-based structure described above, while section 3.3 lays out the sequence of processing steps carried out. Finally, section 3.4 presents results from subsequent inspection of the fabricated structures using optical, scanning electron, and atomic force microscopy.  3.2 3.2.1  Processing Techniques and Equipment Electron Beam Lithography  Overview Electron beam lithography consists of selectively bombarding areas of an organic polymer mask w i t h an electron beam that can be swept i n a controlled fashion across a certain "field of view". T h e electron beam modifies the polymer i n one of two ways. In the case of a "positive" resist, the beam breaks bonds and creates smaller polymer chains, while in the case of a "negative" resist, it induces the formation of cross linked bonds. W h e n an exposed positive resist is dipped into a developer solution, the developer dissolves the smaller chains, thereby removing the portion of the resist subjected to the electron beam. Conversely, i n the case of a "negative" resist, the exposed area is impervious to the developer so that the pattern of resist that is dissolved is the negative of the pattern swept  102  by the electron beam. T h e remainder of this subsection provides a general description of the processing steps, techniques, and equipment used to create the polymer mask by electron beam lithography. A more detailed description of the specific recipes used to produce the samples used i n the optical experiments is deferred to section 3.3.  Resist Application A positive resist, Polymethylmethacrylate ( P M M A ) , was used i n the present work. A t h i n film of the resist was applied to the sample by a technique known as "spin coating". T h i s involves placing a few drops of the resist, dissolved i n highly volatile solvent (chlorobenzene), i n the centre of the sample and spinning it at 500 — 10,000 rotations per minute (rpm). T h e centrifugal force distributes the fluid over the sample until a uniform thickness of the resist layer is achieved i n steady state. T h e resulting thickness of the resist coating depends primarily on the spin rate (500-10,000 rpm). T h e sample is subsequently heated to just beyond the P M M A ' s glass temperature i n order to relieve stress i n the resist and to promote evaporation of the solvent.  Exposure T h e system used to create and deliver an electron beam to the polymer resist is comprised of a scanning electron microscope ( S E M ) interfaced to a computer running specialized nanolithography software. T h e microscope used i n this work is a Hitachi model S-4100. T h e m a i n components of the scanning electron microscope are an electron source, beam deflection system, electromagnetic lenses, and apertures.  These are shown i n  Figure 3.4. The top electromagnetic lens is used to provide a demagnified image of the source, approximately 1 n m i n diameter, which is then imaged onto the target by a second (objective) electromagnetic lens. T h e advantage of having a two-lens system, i n conjunction w i t h beam limiting apertures, lies i n its ability to deliver a monochromatic beam of electrons as well as to provide for flexibility i n the "working distance" between  103  the sample and the scan coils. The beam-limiting apertures also control the m a x i m u m angle, a , subtended by the beam i n order to limit aberrations and to minimize the beam spot size. T h e electron source is a cold cathode field emission tip. It consists of a small piece of crystalline tungsten etched to a point and welded onto a tungsten filament. A large negative voltage is applied to the tip w i t h respect to the walls of a small surrounding chamber. T h i s induces a large potential gradient (electric field) i n the vicinity of the pointed tip, which reduces the w i d t h of the potential barrier confining electrons to the tungsten, thus allowing them to quantum mechanically tunnel out. F i e l d emission tips are brighter than those based on thermionic emission, thereby allowing for shorter exposure times. However, adsorption of individual atoms of gas onto the tip leads to fluctuations i n current. Longer term downward drift of the emitted current occurs as several atomic layers b u i l d up on the tip, and eventually the current becomes unusably low and the tip must be "flashed" to clean it. This is done by briefly passing a large current through the tungsten filament and tip. A s a result, one must wait two to three hours after flashing for the current to become stable, after which it is usable for about four to eight hours before it has decayed by nearly an order of magnitude. Once extracted, the electron beam is accelerated towards the specimen by an accelerating potential difference, V . The electrons are then focussed onto the sample by acc  the pair of electromagnetic lenses, and the entire beam can be deflected i n the x and y directions by the deflection coils. D u r i n g lithography the signals that control the amount of current fed to the deflection coils are provided by a 80486 50 M H z I B M - A T computer running the " N P G S " nanolithography software (from J . C Nabity Lithography Systems) via a 16 bit Digitalto-Analog / Analog-to-Digital ( D / A - A / D ) interface card. T h e software allows the user to specify the required pattern of exposure using a computer aided design package, "Design C a d " , as well as the desired exposure for each  104  Figure 3.4: (a) F i e l d emission scanning electron microscope column, corresponding to the column of the Hitachi S-4100 S E M : (1) Electron source (2) B e a m paths showing the effect of successive apertures i n the column, (3) Apertures (4) Condenser Lens (5) Deflection Coils (6) Objective lens (7) Specimen; (b) Magnetic Lens detail showing how the magnitude of the axial magnetic component B varies i n the lens region; (c) Detail showing geometry of beam aperture angle a at the sample. z  105  pattern element.  The software then transforms the information to a series of (x, y)  points and dwell times.  In order to convert the required dose to a dwell time, the  software requires the beam current. T h i s is obtained by measuring the current entering a Faraday cup inserted into the beampath just before the sample (see Figure 3.4). The Faraday cup is a small conducting chamber w i t h a pinhole entrance for the electrons, and the current is measured using a sensitive picoammeter. Due to the gradual degradation of the emitted current described above, this measurement must be done immediately prior to writing each pattern. U p o n entering the t h i n resist layer, the high-energy electron beam (~ 20 — 30 keV) spreads slightly due mainly to inelastic, small-angle, forward scattering. After having traversed the resist layer, the electrons continue losing energy through inelastic processes but also experience large-angle, elastic scattering, which causes some electrons to re-enter the resist w i t h lower energy, where they multiply scatter and give up more energy to the polymer over an extended lateral area. The result is that the radial distribution of total energy deposited i n the resist is much broader than the waist of the focused electron beam incident on the surface, which is on the order of a nanometre. T h e distribution of deposited energy, therefore, has a tail that extends beyond the typical sub-micron lengthscale of the features to be written. T h i s leads to what is called the "proximity effect" whereby the energy deposited to create a certain pattern element contributes to the total absorbed energy dose of nearby pattern elements. In addition to increasing the m i n i m u m feature sizes attainable, scattering also renders the requisite doses patterndependent. In the present work, this means that the dose required to produce holes of a certain diameter depends on the pitch of the array of holes to be created.  3.2.2  Etching  After the array of holes is created i n the resist layer by electron beam lithography, an etching process is used to transfer it onto the core layer of the waveguide.  106  Conventional wet chemical etching techniques involve immersing the sample into a liquid reactive solution. However, wet-etching tends to be isotropic; this leads to undercutting of the mask which makes the m i n i m u m feature sizes considerably larger than those attainable i n the etch-mask. " D r y " etching techniques, which involve bombardment of the sample w i t h reactive ion species i n a rarefied atmosphere, allow for greater anisotropy and higher aspect ratios of the etched sidewalls, which, i n turn, allow for feature sizes virtually limited by the mask. A n electron-cyclotron resonance ( E C R ) plasma etcher was used i n this work.  It  consists of a cylindrical process chamber, approximately 12 inches i n diameter and 20 inches tall, w i t h multiple optical ports, gas inlet and outlet ports, and adjacent airlock. A system computer is interfaced to D / A - A / D boards which read out various transducers and remotely control valves and power supplies. A schematic diagram of the E C R etcher is shown i n figure 3.5. T h e reactive ions are obtained by igniting a plasma i n the reactor vessel at low pressure, i.e., 1-100 mTorr.  The ions are then driven down onto the sample by an  electric field induced normal to the sample surface. T h e electron cyclotron resonance ( E C R ) plasma etching system used i n this work creates a plasma by delivering microwave radiation resonant w i t h the cyclotron frequency of electrons i n the presence of an applied magnetic field. T h e magnetic confinement i n an E C R plasma enhances the degree of ionization, and thereby enables operation at low pressures.  The long ionic mean free  paths attainable allow for high directionality of the bombarding ions w i t h lower ionic kinetic energies.  T h e result is that anisotropic etching can be achieved without the  excessive damage induced by energetic ions. T h e ions are accelerated towards the sample by the D C component of an electric field created i n its vicinity by applying an 13.56 M H z R F potential between the reactor walls and an electrode placed just below the sample. T h e D C component of the electric  107  Microwave 2.45 GHz Gas inlet  /  Microwave window Magnet  Process Chamber  Gas inlet  5SI  Figure 3.5: Schematic diagram of E C R plasma etcher  108  field arises from the difference i n mobility between the negatively charged electrons and positively charged ions, which leave the plasma at different rates at the anode and cathode. T h e D C bias can be controlled by varying the amount of R F power delivered. However, the R F power required to achieve high directionality of the ions is still low enough to have little effect on the plasma ionization, which is essentially controlled by the amount of microwave power delivered. Thus, this system allows for almost independent control of these two important plasma etching parameters. T h e corrosive process gases are stored i n gas bottles housed i n exhausted gas cabinets just outside the clean room and delivered to the etcher through electro-polished stainless steel tubing. For safety reasons, gas sensors are placed inside the gas cabinets as well as i n the vicinity of the etcher, itself, inside the cleanroom. For a typical etch process, the sample is placed on a sample holder i n the airlock. T h e airlock is then pumped down, and a motorized load arm loads the sample holder onto the chuck i n the middle of the reaction vessel. D u r i n g etching, the sample sits on a liquid cooled chuck; the temperature of the chuck is measured by a thermocouple monitored by the system computer. A helium flow is maintained on the backside of the wafer to increase thermal conductivity and thereby improve temperature control during the etch process. The process vacuum system for evacuating the reaction chamber consists of a Pfeiffer corrosive service turbomolecular pump backed by a mechanical pump package. A  VAT  Series 64 throttling gate valve mounted between the process chamber and the turbo pump is used to control process pressures i n the reactor chamber. It is controlled by computer through a VAT PM5 pressure controller. Process chamber pressure is monitored by an MKS Baratron capacitance manometer. Process gas flow, itself, is regulated by MKS mass flow controllers under control of the system computer. T h e reactor is configured w i t h an N d - F e - B permanent magnet which is used to generate the electron cyclotron resonance zone. T h e magnet produces a m a x i m u m field of  109  approximately 4600 G at the top of the reactor vessel. Microwaves are transmitted from a magnetron to the E C R through a m i l spec # W R 284 waveguide w i t h m i l spec.  #CPR-284 G  flanges.  T h e output frequency of the  microwave power supply is 2.455 G H z . T h e output power is continuously variable and is normally controlled remotely by the system computer.  Impedance matching to the  plasma load is allowed for by partial insertion of three stubs mounted on three different translation stages into the hollow waveguide cavity. T h e R F source is A C coupled to the electrode through an autotuning impedance matching network that matches the 50 Q output impedance of the source to the i m pedance presented by the plasma.  T h e sample wafer is electrically isolated from the  chuck and the R F electrode so that its potential floats. T h e R F source is a  Plasma  Therm Type HFS 500E. It operates at a frequency of 13.56 M H z . T h e matching network is a Plasma  Therm AMNS-500E  modified and repaired by the U B C Physics &  Astronomy Electrical Shop. The gas mixture employed i n this work consisted of argon ( A r ) , chlorine ( C l ) , and 2  boron trichloride ( B C I 3 ) . T h i s combination was selected based on reports i n the literature that it provided good anisotropy, little damage to the substrate as well as low etch selectivity between G a A s and A l A s [49]. Low selectivity was particularly important i n the present work as the waveguide core to be etched was comprised of a G a A s layer as well as an Alo.3Gao.7As layer. One problem that tends to arise i n etching A l G a A s is that it has a high affinity for any residual O2 or H 0 2  i n the chamber and readily oxidizes to form A I 2 O 3 , which cannot be easily etched  by CI2. T h i s causes the formation of a step i n the etch profile since the Alo.3Gao.7As etches more slowly. T h e B C I 3 works as a scavenger for any residual 0 chamber.  Secondly, it readily etches any A 1 0 2  3  2  or H 0 i n the 2  as soon as it is formed, thus allowing  the chlorine ions to etch both layers evenly. The A r is used mainly as a carrier gas and to stabilize the plasma, as A r ion etching is purely a kinetic process w i t h a much lower  110  etch rate.  3.2.3  Oxidation  The process known as "wet lateral oxidation" of A l G a A s , discovered i n the early 1990's [45, 46, 47], simply entails exposing the face of a film of A l G a i _ A s w i t h x > .84 to x  x  a damp but otherwise inert atmosphere at high temperature.  T h e oxygen i n the water  reacts to form an oxide w i t h aluminum and gallium, while the arsenic and hydrogen combine to form arsine gas. The reaction proceeds from an exposed face at a constant rate which depends on the precise conditions but which increases w i t h the a l u m i n i u m composition of the alloy. The exact nature of the oxide formed is an outstanding issue i n the literature, and possibly dependent on oxidation temperature and other conditions. Potential candidates include: ( A l G a i _ ) 2 i n a similar form to 7 - A I 2 O 3 , crystalline phases such as a - A l 0 3 , x  x  /3-A\ 0 , 2  ( 5 - A l 0 , amorphous alumina A 1 0 , hydrates ( A l 0 ) - n H 0 , and hydroxides  3  2  Al (OH) . m  2  n  3  y  2  3  2  Nevertheless, the oxides so produced have refractive indices that lie be-  tween 1.5 and 1.75, which is sufficient to provide the low-index waveguide cladding layer required i n the present work. Figure 3.6 depicts the setup used for oxidation. It was designed and assembled by the author and Francois Sfigakis, a former Master's student. Stainless steel piping is used throughout as it is relatively inert (compared w i t h copper or brass) up to 500°C. A quartz oven tube is used i n order to support temperatures up to 525°C. The valves and other P y r e x glassware used away from the oven are capable of withstanding temperatures up to 200°C. Nitrogen ( N ) , which does not react w i t h G a A s or A l G a A s at a temperature 2  below 600°C, is used as the carrier gas. Nitrogen gas from the building supply is bubbled through the boiler ( B L ) , which consists of a sealed flask of de-ionized water maintained at 95°C, and then delivered to the quartz tube sitting inside a three-zone Lindberg furnace (model 5500 series)  111  112  maintained at 400°C - 425°C. T h e flow rate of the carrier gas, N2, is controlled by a pair of mass flow controllers, each comprised of a Vacuum General valve (model CV2 31S01) and a Tylan Corp. mass flow meter (model FM-360).  T w o parallel controllers were used i n order to increase the  m a x i m u m controlled gas throughput possible. T h e boiler is heated by the combination of hemispherical heating mantle and heating tape.  T h e temperature of the boiler is  measured using a calibrated thermocouple and maintained by manually varying the A C voltage applied to the heating elements by using a variable transformer. Heating tape is wrapped around the stainless steel tubing, bellows, and around the parts of the quartz tube lying outside the oven. T h e oven temperature is automatically regulated by a Lindberg Proportional-IntegralDerivative ( P I D ) programmable controller based on a temperature measurement fed back to it from a thermocouple i n contact w i t h the quartz tube i n the centre of the oven. T h e exhaust gases containing toxic arsine are bubbled through a flask of water, dubbed the bubbler ( B B ) , to partially dissolve and remove arsine, before being vented up the fumehood exhaust. T w o empty flasks act as water traps ( T P ) . One is placed between the oven and bubbler to prevent water from being sucked into the oven by the pressure differential as the oven cools.  The other trap is placed before the boiler to  prevent water from being sucked into the gas lines.  3.2.4  Optical Microscopy  Immediately after oxidation, the patterns were viewed under the optical microscope. T h e holes i n the patterns are not resolvable optically. O n the other hand, the presence of aluminum oxide can be detected from the colour difference between the region around a n d under the patterns and the rest of the sample. T h i s colour difference is due to the different optical lengths of the cavities formed by either the Alo.98Gao.02As or aluminum oxide sandwiched between the waveguide core and substrate.  113  Figure 3.7: Schematic diagram of A F M tip operation and position detection  The optical microscope was also used to detect patterns that were unusable due to particulate matter that had been deposited on the sample surface prior to or during processing, or due to non-uniformity of the resist thickness.  3.2.5  Atomic Force Microscope  A t o m i c force microscopy was used after fabrication to measure the uniformity, pitch, and air filling fraction of the 2D gratings. The atomic force microscope ( A F M ) used was a Digital Instruments  model MMAFM-2  operating i n "Tapping Mode".  A t o m i c force microscopy consists of scanning a pointed probe mounted on a cantilever across the sample surface. T h e tip position is measured w i t h the use of a laser beam which is reflected off the backside of the cantilever, near the tip, and onto an array of four photodetectors as depicted i n Figure 3.7. T h e vertical position of the tip is inferred from the difference signal between the top and b o t t o m pairs of photodetectors. In "Tapping Mode" operation, oscillation is induced near the cantilever's resonant frequency using a piezoelectric crystal. W h e n the oscillating tip comes i n close proximity to a surface, the tip begins to intermittently contact it, and the cantilever oscillation is  114  reduced due to energy loss caused by the tip contacting the surface. The tendency for reduction i n oscillation amplitude is used to identify and measure surface features.  The  cantilever oscillation amplitude is actually maintained constant by a feedback loop, and the surface features are inferred from the amplitude of the error signal. T h e advantage of "Tapping Mode" is that it inherently prevents the tip from sticking to the surface and causing damage during scanning. It should also be noted that "Tapping Mode" A F M was selected over S E M i n order to avoid damage to these very porous structures by the prolonged electron bombardment of small areas which is required for filling fraction measurements. Finally, it should be noted that b o t h the radius of curvature of the very end of the tip and its aspect ratio limit the resolution of the images obtained. It should also be emphasized that the image obtained cannot always simply be considered to be a convolution of the tip shape w i t h the topography of the sample since some portions of the sample may be inaccessible to the tip due to its non-zero w i d t h and non-vertical profile.  3.3 3.3.1  Fabrication Procedure S E M measurement of Layer Structure  T h e layer structure for the as-grown sample is depicted i n Figure 3.8.  B y viewing a  cleaved edge of the sample using the field emission S E M , it was possible to measure the thickness of the Alo.9sGao.02As layer as well as the combined thickness of the G a A s and Al .7Gao.3^4s layers. These measured thicknesses, along w i t h their nominal values, are 0  given i n Table 3.1.  3.3.2  Cleaving  A small, approximately 5 m m x 5 m m , piece of sample was cleaved off the quarter wafer by scribing the wafer at the desired cleave locations, placing it over a t h i n wire aligned 115  1. GaAs 2.  ^ 0 . 3 0 ^  a  0.70  ^  3. A l o.98 Ga o.o2 As  4. GaAs substrate and buffer layer  Figure 3.8: A s grown sample layer structure for A S U 721  Layer N o .  Material  N o m i n a l Thickness [nm]  1 2  GaAs Alo.7Gao.3As  45 35  90 ± 10  3  Alo.98Gao.02As  1600  1800 ± 8 0  4  GaAs  substrate  substrate  Measured Thickness [nm]  Table 3.1: N o m i n a l and measured thicknesses of layers i n sample A S U 721  116  w i t h the scribe, and applying pressure to one side of the sample.  3.3.3  Lithography  Resist Application and Pre-bake The sample was manually rinsed to remove particles from the surface. It was held w i t h tweezers while deionized (DI) water from a gun was sprayed on the surface . It was subsequently sprayed liberally w i t h acetone from a squirt bottle to remove the water, and then sprayed w i t h methanol to remove the acetone. Finally, the sample surface was blown w i t h dry nitrogen gas. After cleaning, the sample was placed on the chuck of the photo-resist spinner. T h e resist used was Polymethylmethacrylate ( P M M A ) 950K molecular weight. It is shipped by the supplier, Olin, i n 4% solution i n chlorobenzene.  Using a pipette and bulb, a  single drop of the resist solution was applied to the centre of the sample, and the sample was spun at 8000 r p m for 30 seconds to produce a uniform 225 nm-thick coating. After spin-on, the sample was removed from the spinner and placed on a hot-plate, pre-heated to 175°C, and left to bake overnight i n a fumehood.  Exposure The sample was then fixed to a ~ 7 m m stainless steel stub using a carbon based conductive glue w i t h the resist-coated side facing up. In order to provide greater electrical contact and mechanical support, two t h i n strips of aluminum foil were placed across two diagonally opposing corners of the sample. Small pieces of graphite were deposited on the resist near the edges of the sample by rubbing a 2 B pencil gently against them. These provided small features useful i n subsequent focussing of the electron beam on the resist surface. The stub holding the sample was mounted onto a holder and inserted into the S E M  117  m a i n chamber through the air-lock and locked into place on the t r a n s l a t i o n / r o t a t i o n / t i l t stage. T h e tip was then flashed and left for 2.5 hours to allow the adsorption on the tip to reach its stable configuration. T h e electron beam was then turned on. T h e electron beam and apertures were aligned and the astigmatism adjustment performed as per the procedures specified i n the S E M manual. T h e edge of the sample was then translated into the field of view, taking care not to expose any of the more central parts of the sample to the electron beam. T h e resist surface was moved into the focal plane of the electron beam so as to bring the graphite pieces lying on the surface into focus. T h e sample was then translated such that each of its edges was, i n turn, brought into the field of view. B y thus verifying that all of the edges remained i n focus, it was possible to ensure that the sample was sitting flat and was, in fact, oriented normal to the beam. Finally, for convenience, the sample was rotated to align its edges w i t h those of the field of view. Next, the movable Faraday cup was inserted into the beam path; the beam current was measured, and its value was entered into the nanolithography software. The sample was then translated to bring the desired exposure area under the beam, the x — y raster control switched to computer control, the Faraday cup blocking the beam removed, and a key depressed on the computer to command the software to start writing the pattern. Once the pattern was written, the sample was translated to bring another portion of it under the field of view, and the procedure i n this paragraph was repeated for each of the patterns defined i n the nanolithography software run file. After all the patterns were written, the electron beam was turned off, the sample was removed from the S E M and returned to the clean room for development and etching.  Development First, the sample was flushed w i t h dry nitrogen gas to remove the graphite particles. It was then immersed i n developer solution (from O C G Y Inc.) consisting principally of  118  methyl isobutyl ketone ( M I B K ) diluted i n isopropyl alcohol (IPA), M I B K : I P A 1:3, for 90 seconds. T h e development process was stopped by immersing the sample i n an I P A rinse for 60 seconds. T h i s was followed by immersion of the sample i n D I water for 30 seconds followed by drying w i t h nitrogen gas. T h e sample was then immersed i n a high-contrast developer solution consisting of ethoxy ethanol ( E E ) diluted in methanol, E E : m e t h a n o l 3:7, for 15 seconds to remove any t h i n film of P M M A remaining at the b o t t o m of the developed holes. T h i s final development process was stopped by immersing the sample i n methanol for 30 seconds. T h i s was followed, once again, by rinsing of the sample i n D I water for 30 seconds followed by drying w i t h nitrogen gas. After a brief inspection under the optical microscope, to confirm the appearance of 90 x 90 pm patterns, the sample was transferred to the etcher.  3.3.4  Etching  T h e sample holder that shipped w i t h the etcher is designed to hold 3"-diameter wafers. Consequently, a 3" silicon wafer was placed i n the sample holder, and the sample was fixed to the centre of the silicon wafer using a drop of conductive paint obtained from Astex-PlasmaQuest. T h e load-lock cover was closed and the load-lock pumped down. T h e sample was then inserted into the reaction chamber by the motorized load arm.  The steps and  conditions programmed into the etcher system control computer are given i n Table 3.2. After the chamber was pumped down to a base pressure of ~ 1 x 10~ Torr, the 7  process gases were allowed to flow for 30 seconds before the microwave source was activated to ignite the plasma. Simultaneously, the R F source was manually activated to induce the D C bias. After the etch, the chamber was purged w i t h nitrogen gas for 30 seconds i n order to help clear out the toxic process gases before the sample was transferred back to the air lock by the motorized load arm.  119  Process Pressure  10 m T o r r  A r flow rate  20.1 seem 1.6 seem 2.2 seem  CI2 flow rate BCI3 flow rate Microwave power R F Bias Chuck temperature Backside He pressure Pre-etch purge time E t c h time  100 W a t t s 100 Volts 5°C 5 Torr 45 seconds 160 seconds  Table 3.2: E t c h conditions The sample was then removed from the air lock, rinsed, and left i n a beaker of acetone for two hours i n order to remove the remaining P M M A mask. T h i s was immediately followed by rinsing i n methanol and D I water for 30 seconds, each followed by drying w i t h nitrogen gas.  3.3.5  Oxidation  W i t h reference to Figure 3.6, the sample was placed on a quartz boat that was introduced into the quartz tube and carefully pushed to the centre of the tube and oven using a rod. T h e system was then purged w i t h nitrogen gas for one hour by opening valves V 2 , V 4 , V 5 , and setting the three-way valve V 6 to vent to atmosphere. W i t h the nitrogen purge still on, the oven was turned on and the P I D controller was programmed to ramp the temperature up to 400°C i n 30 minutes. Meanwhile the boiler was filled w i t h fresh D I water, and the hemispherical heating mantle was turned on to slowly heat the water up to 95°C. The power to the heating tape was also turned on at this point. Once the oven had reached 400°C, valves V 2 and V 5 were closed, the mass flow controllers were turned on and set to provide a flow of 100 seem (standard cubic centimetres per minute), and valve V 3 was opened. Finally, the three-way valve, V 6 ,  120  was switched to deliver the moist nitrogen gas into the oven tube. After the desired oxidation time had elapsed (usually twenty to sixty minutes), valves V 6 and V 4 were closed, and valve V 5 was opened to deliver 100 seem of dry nitrogen gas while the oven was ramped down to 200°C i n two hours. The boiler, heating tape, and oven were turned off, and the sample and oven were allowed to cool overnight w i t h the controlled 100 seem nitrogen purge on. T h e sample was then removed from the oxidation oven.  3.4  Fabrication Results  Lithography, etching, and oxidation were performed, as described previously i n this chapter, on sample A S U 721 so as to produce twenty-four patterns arranged on a 0.5 m m grid i n 6 rows of 4 gratings each.  T h e doses and pitches programmed into the  lithography software are given i n Table 3.3. Note that two copies of each pitch-dose combination were produced for the sake of redundancy. T h e sample was oxidized for twenty minutes. Figure 3.9 shows a magnified view of one of these, grating 22, and its vicinity. The dark, square box is the grating itself. Surrounding it is a region that is slightly darker than the rest of the sample. T h i s region was oxidized outward from the grating, and we can see that it extends out approximately 10 / i m from the edges of the grating. Table 3.4 summarizes the yield and extent of oxidation around each of the gratings. After the optical spectroscopy measurements were completed, atomic force microscopy was carried out on gratings 20, 22, 23, 24 to verify pattern uniformity as well as the air filling fraction of the holes produced. A 4 / m i x 4 (im image of grating 22 is shown i n Figure 3.10. T h e lateral scales were calibrated by comparing the value obtained from an A F M measurement of the pitch of a grating to that obtained from a diffraction measurement using a helium-neon laser, and from observation i n the S E M . T h e image shows good uniformity i n b o t h the hole size and pitch over its field of view. T h i s was also the  121  90  L i m  Figure 3.9: Top view of grating 22 on sample A S U 721 as imaged by the optical microscope. T h e size of the grating is 90 fxm x 90 ptm. Some processing was performed on the image to enhance the contrast between the oxidized region, which extends out around the grating and is slightly darker compared to the unoxidized parts of the sample.  122  Grating  gl g2 g3 4 g  g5 g6 g7 g8 g9  gio gll gl2 gl3 gl4 gl5 gl6 gl7 gl8 gl9 g20 g21 g22 g23 g24  Dose [fC] 35 50 65 25 35 45 30 45 60 45 60 75 30 50 65 25 35 45 30 45 60 45 60 75  P i t c h nm] 550 550 550 500 500 500 525 525 525 575 575 575 550 550 550 500 500 500 525 525 525 575 575 575  Table 3.3: Pitches and doses specified for e-beam lithography  123  Grating  Extent of oxidation out from edge [/im]  gl g2 g3 4  -  g5 g6  12 12 6 12 14 8  -  -  g  g7 g8 g9  gio gll gl2 gl3 gl4  -  15 3 9 13 12 9 12 9 10 15 10 11 10  gl5 gl6 gl7 gl8 gl9 g20 g21 g22 g23 g24  Condition bad bad bad bad top left corner bad 2 bad patches good good good good good good spot on left edge good good good good good good good spot on left edge good good good  Table 3.4: Extent of oxidation and yield  124  Figure 3.10: 4 pm x 4 //m A F M image of grating 22.  case i n images taken from other regions of this and other gratings. Figure 3.11 shows a slice of the above image for grating 22 taken through the diameters of the holes. The profile and radius of the A F M tip preclude drawing any conclusions regarding the profile and depth of the holes. However, the diameter of the holes at the surface may be inferred from the measured profile by taking the distance between the leading and trailing edges as indicated by the arrows i n Figure 3.11. Table 3.5 gives the measured hole diameters and pitches of the gratings as well as their air filling fractions, calculated w i t h the assumption of cylindrical holes.  125  o  m  CM  1  1.00  '  t  2.00  3.00  4.00  JJM Figure 3.11: Section through diameter of holes i n image i n Figure 3.10. T h e arrows indicate the leading and trailing edges of the depression i n the profile, which are used to obtain a measurement of the diameter of the holes at the surface.  Grating g22 g23 g24  P i t c h , A nm 571 ± 17 576 ± 16 573 ± 16  Hole radius, r n m 170 ± 1 0 187 ± 11 226 ± 14  r/A 0.300 ± 0.020 0.325 ± 0 . 0 2 1 0.394 ± 0 . 0 2 6  F i l l i n g fraction 0.278 ± 0 . 0 2 7 0.331 ± 0.026 0.489 ± 0.045  Table 3.5: Pitches and hole radii and corresponding air filling fractions estimated from A F M profiles from three gratings having nominally the same pitch but different electron beam exposure times.  126  Chapter 4 Optical Experiments  4.1  Introduction  T h e experimental setup described i n this chapter was used to collect specular reflectivity spectra from the gratings i n the geometry depicted i n Figure 4.1 below. For each grating, data were collected for a series of incidence angles 6, for each of (j> = 0 and <fi = 45°, and for each of s and p polarizations of the reflected light. T h i s was done by focusing white light from a tungsten quartz-halogen bulb delivered by an optical fiber down to a 200 fim diameter spot on the sample, which over-filled the 90 ^ m x 90 fj,m two-dimensional grating under study. T h e specularly reflected light was collected and focussed onto the input aperture of a Fourier Transform Spectrometer after passing through a polarizer to discriminate between s and p polarized light, and an aperture to eliminate light coming from outside the patterned area. Reflective optics were used throughout to avoid chromatic aberrations.  To aid i n alignment, a C C D  camera could be inserted into the beam path, at the intermediate focus, to image the surface of the sample. T h i s chapter describes the experimental apparatus, the procedure used for i n i t i a l alignment of its optical components, the angular resolution obtainable, and the procedure used for obtaining spectra on each grating. The optical set-up was designed and constructed by W . J . Mandeville. T h e optical experiments to collect the data used i n the present work were carried out by the author.  127  Figure 4.1: Schematic diagram of scattering geometry used i n specular reflectivity measurement. The dimensions of the sample and pattern depicted are the nominal values for grating 22 on sample A S U 721.  4.2  Experimental Apparatus  Figure  4.2 depicts the experimental optical apparatus used.  A l l of the components  were assembled on an optical table, except for the self-contained Fourier Transform Interferometer ( F T I R ) which was placed adjacent to the table. The light source used was a commercially available optical fiber illuminator consisting of a 100W quartz halogen light bulb w i t h the tungsten element emitting at 3200K. A pair of 1" diameter lenses of focal lengths 2.54 c m and 3.76 cm collimate and then focus the light onto the end of the optical fiber mounted on the fiber chuck and inserted into the fibre holder on the illuminator. T h e optical fibre used to deliver the light onto the turntable ring is a 2 meter long 100 /um-core multimode visible/near-IR fibre. The total power delivered by the fiber was measured to be approximately l m W using a Spectra Physics 404 power meter w i t h a spectral range 0.45 pm - 0.9 //m. Assuming a blackbody spectrum, this corresponds to approximately 4 m W of total power delivered by the fibre, and to approximately 2  128  Figure 4.2: Schematic diagram of experimental setup  129  m W i n the spectral range 0.77 pm - 1.67 fxm (6000 c m  - 1  - 13, 000 c m ) for which data - 1  were collected i n this work. The 9 — 29 apparatus is composed of two rotating stages. T h e output facet of the illuminating fibre, iris diaphragm, and the focusing ellipsoidal mirror are placed on a rotating annulus (with inner and outer diameter of 19 c m and 35 c m respectively) positioned to be concentric w i t h the rotating sample-holder which is described i n detail below. Three ellipsoidal mirrors were used to image from focal point to focal point. T h e sections of the ellipsoid were chosen for the mirrors such that the beam is turned by 90°. T h e first ellipsoidal mirror, E M I , images the output facet of the fibre onto the sample, creating a 2 X magnified spot. T h i s produces a 200 pra spot on the sample that slightly overfills the 90 / m i x 90 / m i gratings. T h e second ellipsoidal mirror, E M 2 , creates a 10X magnified image of the sample surface at the point C , 150 c m from the mirror. A square aperture is placed i n this image plane to ensure that only light reflected off the patterns is passed to the spectrometer. T h e aperture is so adjusted by viewing the image formed at this point using a C C D camera inserted into the beampath just downstream of the aperture. A polarizer cube is used to select a particular polarization of the collected light. Since the plane of incidence for the reflectivity measurement is parallel to the optical table, turning the polarizer such that it transmits light polarized w i t h the electric field perpendicular to the optical table corresponds to selecting the s-polarization. Conversely, having the polarizer axis parallel to the optical table selects the p-polarization. The t h i r d ellipsoidal mirror, E M 3 , then images the real image formed at point C onto the input aperture of the F T I R . Figure 4.3 shows a schematic diagram of the sample holder assembly. In addition to rotation concentric w i t h the outer ring, this assembly permits us to vary the angle made by the plane of incidence w i t h respect to the crystal axes of the photonic lattice,  130  without having to reposition the sample and re-align the setup. The translation and tip-tilt stages provide for convenient alignment.  4.2.1  Fourier Transform Spectrometer  A BOMEM  model DA8 Fourier Transform Interferometer was used i n this work . Fourier  transform spectroscopy allows for large bandwidth data acquisition w i t h a large range of spectral resolutions. T h e broad bandwidth feature is of particular value i n the present work where scans over a m i n i m u m range of 2000 c m  - 1  are essential for efficient data  collection. T h e system essentially consists of a Michelson interferometer augmented w i t h a thermally stabilized He-Ne laser, electro-optic transducers, and electromechanical actuators 131  to allow for automatic calibration and alignment of the optical components. Input and output optics are provided to introduce parallel radiation into the interferometer, and to subsequently couple parallel radiation out and focus it down onto a detector. Figure 4.4 shows a schematic diagram of the principal optical components. T h e parallel beam at the input of the interferometer is divided into two equalamplitude beams at the beamsplitter which, after travelling through different path lengths, are then superimposed and focussed onto a detector. A s the moveable mirror is scanned, the intensity falling on the detector is recorded as a function of the optical path delay between the two arms to form an interferogram. T h e power spectral density of the input light is obtained by (cosine) Fourier transformation of the interferogram. A n I n G a A s detector and a quartz beamsplitter combine to provide a spectral range of approximately 6000 c m of 7 c m  4.3  - 1  - 1  to 14,000 c m  - 1  Most spectra were obtained at a resolution  .  Alignment  A pointed piece of polished G a A s was mounted on the sample holder for alignment purposes. Using a level Helium-Neon laser beam, the fibre and ellipsoidal mirrors, E M I and E M 2 , were levelled and placed i n their approximate positions. In order to ensure that the sample was at the centre of rotation of the sample rotation stage, the following procedure was used. The fibre was pulled back from the focus of the elliptical mirror, E M I , i n order to illuminate the test sample w i t h a broad, defocussed spot such that the sample cast a shadow on a small screen placed behind it.  The  sample was then translated using the upper X Y Z translation stage on the sample holder assembly until the point of the shadow d i d not move as the sample was rotated. Having assured that the sample tip was aligned w i t h the rotation axis of the sample rotation stage, the centres of rotation of the sample rotation stage and turntable ring were aligned by adjusting the lower X Y translation stage until the shadow of the pointed tip  132  Scanning mirror  Figure 4.4: Schematic diagram of F T I R spectrometer  133  remained stationary on the screen fixed to the turntable when the turntable and sample holder were rotated through 360°. T h e ellipsoidal mirror, E M 2 , was then adjusted using the following procedure. T h e C C D camera was placed a focal distance away, and the mirror was adjusted until a feature (divot or scratch) on the sample was i n focus on the C C D camera. T h e ellipsoidal mirror, E M I , and fibre were then positioned such that a round, wellfocussed light spot was imaged by the C C D camera. Subsequently, the square aperture was set to the correct height using the He-Ne laser and placed on the optical table at the long focus of the ellipsoidal mirror, E M 3 . T h e square aperture was then illuminated and the ellipsoidal mirror adjusted such that the image of the square aperture was focussed down to a point at the entrance aperture of the F T I R . Another illuminated aperture serving as a point source was then placed at the focal point of the detector collection mirror i n the F T I R ( point D i n figure 4.4). T h e turning mirror, T M 5 was positioned and adjusted such that a sharp image, centred on the square aperture, was produced. Also, the center of the illuminated spot on the turning mirror T M 5 was verified to be aligned and level w i t h the square aperture using the Helium-Neon laser. T h e specularly reflected signal off a piece of G a A s was then verified to be falling near the centre of the of the spot produced by and co-aligned w i t h the light coming from the back-illuminated aperture inside the F T I R over the 1.5 m span between the aperture and T M 5 . Finally, the remaining turning mirrors were inserted, and the mirrors, T M 2 and T M 3 , were moved to adjust the path length such that the spot of white light was focussed at the position of the square aperture, as viewed using the C C D camera placed at that point.  134  4.4  Angular Resolution  Because it is a focussed beam of light impinging on the grating, which is used to excite the modes of the structure, for a fixed frequency to, there is, i n actuality, an angular distribution of k vectors incident on the sample at once.  Consequently, this results,  again, i n an angular distribution of specularly reflected light about the nominal angle of reflection. T h e angular spread of k vectors incident on the sample can have a m a x i m a l value of ± 4 ° which is defined by the size of the mirror E M I . D u r i n g data collection this value was reduced by closing down the iris aperture, placed midway between the fibre and mirror, to a m a x i m u m of 3mm i n diameter. This allowed a AO of ± 3 ° . O f the cone of light reflecting off the sample, only that falling w i t h i n the detector acceptance angle is analyzed by the F T I R . T h i s acceptance angle is limited by the  //#  ( / / 4 ) of the off-axis parabolic collimating mirror i n the F T I R . This was calculated, by ray tracing, to correspond to an acceptance angle of ± 1 ° about the nominal angle of specular reflection off the sample. T h e result of the finite angular resolution is to smear out the sharp features expected in the collected spectra. A quantitative analysis of this effect is deferred to the next chapter.  4.5  Experimental Procedure  A series of specular reflectivity spectra was collected from each of the gratings at incident angles of 2.5°, 5°, 10°, 15°, 20°, 25°, 30°, 40°, and 50°, w i t h the polarizer rotated first i n the s, then i n the p position, for b o t h 0 = 0° and 4> = 45° (see Figure 4.1). T h e following procedure was used to position the sample for each incident angle. The C C D was inserted into the beam path and the grating translated until it appeared i n the field of view on the monitor. The input mirror, E M I , was rotated to position 28, and the  135  sample rotated until the signal on the detector, as monitored by a digital voltmeter, was maximized. T h e square aperture was then closed down until only light from the grating was imaged on the C C D . T h e C C D camera was then swung out of the way and a spectrum taken. W i t h the system properly aligned initially, it was possible to take spectra at all the incident angles for a particular polarization without having to adjust any of the optics, including the position of the square aperture. A series of specular reflectivity spectra were also taken off a piece of G a A s polished on one side. These were used to normalize out the input spectrum, the detector response, and absorption lines due to atmospheric molecules.  136  Chapter 5 Results  5.1  Introduction  T h e experimental setup described i n Chapter 4 was used to collect specular reflectivity spectra from the ten 2D gratings fabricated on sample A S U 721. The spectra from three of these gratings (grating numbers 22,23,24) exhibited features indicative of the photonic bandstructure expected near the second order gap. For grating 22, i n particular, a l l of these features remained clearly identifiable and w i t h i n the detector's spectral range over a large range of incident angles and for b o t h polarizations. Consequently, the bulk of this chapter presents the data obtained on grating 22. Spectra from gratings 23 and 24, which have the same pitch but larger hole sizes, are also presented i n order to show the effects of filling fraction on the ordering of the broad and narrow spectral features associated w i t h lossy modes.  5.2  Resonant Reflectivity  Figure 5.1 depicts the experimental and model specular reflectivity spectra obtained for p polarized light incident at 5° along the X (cp = 0°) direction of the square lattice. T h e spectra are plotted as a function of a "reduced frequency" or wavenumbers (in units of c m ) which we denote as - 1  Co =  LO/2TTC  (5.1)  where c is the speed of light i n free space. T h e experimental spectrum was obtained and normalized according to the procedure described i n the previous chapter. The resulting spectrum was then normalized again, this time, to its m a x i m u m value w i t h i n the range 137  Figure 5.1: Experimental (a) and calculated (b) specular reflectivity spectra on grating 22 at incidence angle of 9 = 5° and 0 = 0° for p-polarization of reflected light.  6000 c m  - 1  to 14,000 c m . - 1  The model spectrum was calculated using the Green's  function code referred to i n Chapter 2. The exact values of the parameters input to the program to describe the structure under study are discussed later i n this chapter. The broad features i n the spectra spaced at ~ 1000 c m  - 1  are Fabry-Perot fringes  arising from the optical cavity formed by the A l oxide layer. T h e y would be present even i n the absence of the grating. T h e sharp, bipolar feature is a Fano resonance [38], which indicates resonant coupling to a leaky slab mode attached to the textured waveguide structure. It is these Fano resonances that are used to probe the dispersion and lifetimes of the resonant modes of the textured waveguide. T h e precise Fano lineshape function is described i n section 5.6.1. A physical explanation of the reason behind the bipolar  138  1.04 (a)  0.5 H H  > I—I  H U  w  0.0 1.0  111111  1111 • i • • • • I • • • • i • • • 111 • 11111 • 111 • • • 11  0.5 H  0.0  ~|iiii|iiii[iiii|iiii|iiii|iiii|iiii|iiii[iiii|iiiniiii|iiii  6  7  8  9  FREQUENCY CO/2TCC  11, 12 [10 cm" ]  10  ,  Figure 5.2: Experimental (a) and calculated (b) specular reflectivity spectra on grating 22 at incidence angle of 9 = 5° and <>/ = 0° for s-polarization of reflected light.  nature of the Fano lineshape is again deferred to Chapter 6 (section 6.2).  5.3  Polarization Properties  Figure 5.2 shows the experimental and model spectra for reflected s polarized light. In this case, there are three sharp features superimposed on the Fabry-Perot fringes, giving a total of four features i n either polarization. Figures 5.3 and 5.4 show the experimental and model spectra for light incident at 9 = 5° along the r — M direction of the photonic crystal ((j> = 45°) for the p and  139  6  7  8  9  10  11  12  [loW] 1  CO/271C  Figure 5.3: Experimental (a) and calculated (b) specular reflectivity spectra on grating 22 at incidence angle of 9 = 5° and 4> = 45° for p-polarization of reflected light.  s polarizations respectively. The specular reflectivity spectra again exhibit four Fano resonances, but, i n this case, two features occur i n each polarization. In summary, then, the spectra exhibit well-defined polarization selection rules which are dependent on the orientation of the plane of incidence.  5.4  Dispersion  Figure 5.5 shows a series of s polarized specular reflectivity spectra, taken at various angles of incidence w i t h the in-plane propagation vector always oriented along the T — X direction (i.e., 4> — 0). The plots have been offset from one another for the sake of clarity,  140  (b)  Figure 5.4: Experimental (a) and calculated (b) specular reflectivity spectra on grating 22 at incidence angle of 9 = 5° and (f> = 45° for s-polarization of reflected light.  141  9=50° H  9=40°  i—i  > H U W  F R E Q U E N C Y co/27ic [lc/cm ] -1  Figure 5.5: Experimental specular reflectivity spectra for s polarization at various incident angles, 9, along the T — X direction (<j> = 0). T h e plots have been offset from one another for the sake of clarity; the vertical scale on the reflectivity traces is linear.  and each spectrum has been normalized to its m a x i m u m value i n the displayed spectral range. T h e basic dispersion of various bands leaking s polarized radiation is manifestly evident. There is a clear gap, or avoided crossing, near zone-centre (i.e., at small angles of incidence) where the bands flatten out. approximately 900 c m  - 1  T h e size of the gap near zone-centre is  which is of the order of 10% of the centre frequency. There is  also a fourth band occurring higher i n energy which anti-crosses w i t h the t h i r d band at about 25°. The linewidth of the modes varies from band to band and w i t h i n bands.  142  5.5  Fitting with Model  T h e G F model requires the following as input parameters: the layer thicknesses and materials of the slab waveguide structure, the grating pitch, hole radii and material filling the holes, the incidence angle or in-plane wavevector, and the frequency range for the calculation. One l i m i t a t i o n of this model is that there can only be one type of host material for the textured layer, whereas the core layer of the A S U 721 waveguide studied is nominally composed of a 45 n m thick G a A s layer as well as a 35 n m Alo.3Gao.7As layer. Consequently, a single material w i t h dielectric constant given by an average of those for G a A s and Alo.3Gao.7As, weighted according to thickness, was used i n the model for the core layer. T h a t is: e' (u) = a e' (CJ) s  GaAs  a  GaAs  + [1 -  a }e' GaAs  G a A s = ^GaAs/ (^GaAs + ^Al . Gao.7As) 0 3  (CJ)  Ah3G&orAs  (5.2) (5.3)  A l s o , the 2D array of holes that comprise the grating was etched part of the way into the cladding layer i n order to aid i n the oxidation. A s a result, an additional layer w i t h dielectric constant given by an average of aluminium oxide and air, weighted according to the air filling fraction of the holes, was placed between the core and uniform oxide layer i n the model. The nominal thickness for this layer was estimated to be 350 n m based on the etch time. The semiconductor dielectric constants, including dispersion, were taken from the literature [50]. T h e dielectric constant of the oxide was measured by our group, independently, to be 2.56 [51]. Altogether, the parameters that were allowed to vary i n the model were the pitch (A), the air filling fraction of the (assumed cylindrical) holes (Jf), the thicknesses of the core (t ), g  oxide  (£ xide), 0  and air-oxide layer( t ) , and the admixture of G a A s and a o  Alo.3Gao.7As i n the core layer (aGaAs)-  For each combination of these parameters a full set of reflectivity spectra were calculated for the series of incidence angles depicted i n figure 5.5, for b o t h the s and p 143  polarizations i n the T — X direction. These were compared manually w i t h those from the experiment. T h e model parameters yielding the best fit to a l l of the data are presented i n Table 5.1 along w i t h their nominal and/or measured values. T h e pitch of the photonic lattice was 550 nm, and the air filling fraction of the holes was 27%. These compare favourably (to w i t h i n 3.5%) to the nominal and measured values. T h e total thickness of the GaAsZAlo.3Gao.7As core layer from the best fit model results compares favourably w i t h b o t h the S E M measurement and the nominal value supplied by the sample grower. A s stated above, the G F model can not handle variation i n the grating layer in the transverse direction other than by averaging the dielectric constant of the layer. A 60% weight to the Alo.3Gao.7As content yielded the best result.  T h i s corresponds  to thicknesses of 33 n m and 50 n m for the G a A s and Alo.3Gao.7As layers respectively, which are somewhat off from their respective nominal values of 45 n m and 35 nm. N o more can be said since we are unable to measure the layer structure w i t h i n the core. T h e total thickness of the oxide layer used i n the model is i n good agreement w i t h the measured value. Best agreement was obtained when it was assumed that the 2D array of holes penetrated 350 n m into the oxide layer.  T h i s was modelled as a layer w i t h  a reduced dielectric constant according to the air fill fraction of the holes which were assumed to have vertical sidewalls. T h i s penetration into the oxide is not inconsistent w i t h the depths registered by the probe during A F M microscopy, which were i n excess of 200 n m , when it is considered that the non-vertical probe prevents the probe tip from reaching the b o t t o m of the holes. In summary, the parameters describing the physical waveguide geometry and composition that were input to the model to provide the best agreement w i t h the measured spectra were w i t h i n 3.5% of the measured values where measurement was possible. T h i s same set of parameters was then used to generate spectra at various angles of incidence oriented along the T — M direction ((f) = 45). The excellent agreement w i t h the corresponding experimental data suggested that no further iteration of the parameters  144  Parameter  M o d e l value  N o m i n a l value  Measured value  grating pitch air filling fraction G a A s thickness Alo.3Gao.7As thickness total core thickness oxide layer total thickness air-oxide layer thickness  550 n m 0.27 33 n m 50 n m 83 n m 1900 n m 350 n m  575 n m none 45 n m 35 n m 80 n m 1600 n m none  (571 ± 17) n m (0.278 ± 0.027) n m none none (90 ± 10) n m (1800 ± 8 0 ) n m > 200 n m  Table 5.1: Model, Nominal, and Measured sample structure parameters for grating 22 on sample A S U 721 was necessary. The bandstructure generated using these best-fit model results and that obtained from the experimental spectra are both plotted i n Figure 5.8, which is presented later i n this chapter i n section 5.7.  5.6  Extraction of Mode Frequency and Linewidth from Spectra  5.6.1  Extraction of Frequency and Linewidth from Experimental Spectra  The specular reflectivity spectra i n the vicinity of the relevant resonant features were fit w i t h a function comprised of the sum of an A i r y function [42], which models the Fabry-Perot fringes, and a Fano resonance, which models the feature associated w i t h the B l o c h state of the photonic crystal. The Airy-like function is given by: A,~S L 2 r [ l - c o s ( d a ) + V)] A(u)=b + sx-— . ,_ ' 1 + r - 2r cos(d£ + 2  v  A  y  4  n  0  , (5.4) r  A S  2  For a simple 3-layer structure the parameters r and d would represent the reflectivity of the interfaces and the optical path through the cavity respectively. A n offset, b, scale factor, s, and phase factor, ip, were added to account for the fact that there are additional layers above the oxide cladding layer that is primarily responsible for the Fabry-Perot fringes. T h i s function was used to fit the spectra for the values of the parameters i n 145  a region away from any Fano resonances.  It was combined w i t h the Fano lineshape  function given by:  (q + 2(Co-Co )/r)  2  F(u)  =  Fc  Q  1 + (2(a) - £ )/r)  (5.5)  2  0  to yield: R{u)  = A(u>) +  (5.6)  F(Q)  which was fit to the entire spectrum, allowing only the Fano lineshape parameters, q, T, £>o, FQ to vary. Finally, a fit was performed using R(d)) but allowing all of the parameters to vary. T h i s procedure usually resulted i n a good fit to the experimental spectrum. One such fit to an experimental spectrum obtained from grating 22 is depicted i n F i g ure 5.6. O n l y the lower frequency Fano-like resonance was fit over the range (6000 c m - 9000 c m ) . F r o m this fitting procedure the mode was found to lie at 7450 c m - 1  to have a linewidth of 105 c m  5.6.2  - 1  - 1  - 1  and  .  Extraction of Mode Parameters from Model Spectra  A slightly different approach was used i n order to extract the widths and positions of the resonances from the G F model. In addition to specular reflectivity spectra, the model is also capable of generating scattering spectra for the four field components w i t h in-plane propagation constants / 3  i n c  ± (3QX and / 3  i n c  ±  day.  T h e quasi-bound modes of the system again manifest themselves as Fano resonances i n the spectra, but this approach has the advantage that the modes appear almost background-free.  T h i s is the case because the non-resonant background is first order i n  the dielectric texture for these components, while it is of order unity i n the specular component. Figure 5.7 shows sample fits to first order scattering spectra w i t h the Fano lineshape function F(u),  together w i t h the corresponding specular spectrum.  Notice  that the first order spectra contain only a weak background, which simplifies and reduces the number of parameters required to obtain good fits. Furthermore, as discussed 146  6000  7000 8000 E N E R G Y [cm ]  9000  1  Figure 5.6: F i t to specular reflectivity spectrum obtained from grating 22 obtained w i t h light incident at 9 = 50° along the V — X direction. T h e solid line represents the experimental spectrum while the dashed line represents the "best fit" spectrum generated by a Fano lineshape function summed w i t h an A i r y function. T h e initial fit to the Fabry-Perot fringes was performed over a wider range of energies (6000 c m to 11,500 c m ) not shown here. The fit using a Fano lineshape and A i r y function was performed using data points i n the range 6000 c m - 9000 c m . - 1  - 1  - 1  - 1  above, away from zone-centre, each of the bands are rapidly dominated by a particular Fourier component. Consequently, by fitting to the dominant component for the band i n question, the neighbouring resonances associated w i t h the other bands remain small. A n d therefore, the estimates for the errors on the mode frequency and linewidth arising from the first-order fitting procedure are much smaller than the errors associated w i t h fitting the experimental specular reflectivity spectra. It should be mentioned here that while these first-order diffracted spectra provide " cleaner" signals for extracting mode frequencies and linewidths from model calculations, such first-order diffracted spectra are considerably more difficult to obtain experimentally t h a n specular reflectivity spectra. T h e first-order diffracted light must travel down the t h i n untextured waveguide core to the edge of the sample, and since the waveguide core is on the surface of the sample there is considerable loss due to scattering from surface  147  10.0  FREQUENCY  CO/2JIC  r 10.5 1  [10 cm"] 3  < U  on  (b)  W  H on  11.0  < U  (c)  O H on  11.0 FREQUENCY  CO/2TIC  [10 cm''] 3  Figure 5.7: F i t to spectra generated by the G F model, (a) Specular reflectivity spectrum for reference (b) 1st order scattering to —QG field component (c) 1st order scattering +PG field component. In (b) and (c) the symbols correspond to the model values, and the smooth lines are the result of a fit w i t h a Fano lineshape function. The spectra are for 10° angle of incidence along the V — X direction. X  X  148  debris. Also, since there is no lateral confinement, the diffracted beam spreads i n the in-plane directions. T h i s all necessitates creating a cleaved edge whence the diffracted light is to be extracted very near the textured region. M a k i n g such precisely positioned cleaves without destroying the grating proved to be extremely challenging, especially since aluminum oxide flakes and crumbles upon cleaving.  5.7  Bandstructure  T h e bandstructure for these modes was obtained by extracting the energies, u>i, at which the resonant features occur and plotting these as a function of the magnitude of the i n plane propagation vector, given by P = — sin(0). Figure 5.8 shows the dispersion of the c L  resonant modes of the structure along the T — X and T — M symmetry directions of the square B r i l l o u i n zone. For comparison, the inset schematically depicts the bandstructure for a similar structure had the array of holes been etched only a small fraction of the way through the guiding core. In that case, the corresponding dispersion curves would effectively represent the bare T E polarized slab modes, zone-folded to the first B r i l l o u i n zone by the in-plane Fourier components of the square grating. T h e polarization of the features i n the reflectivity measurement and i n the calculations, remains the same w i t h i n each band along a given symmetry direction of the crystal, and so the curves i n Figure 5.8 have been labelled as either s or p polarized. A l o n g the F — X direction, there are a total of 5 curves, w i t h four labelled s and one labelled p.  In the vicinity of the zone-centre, there are three s polarized curves and  one p polarized curve. T w o of these exhibit large dispersion, w i t h one having positive dispersion (frequency increasing w i t h (3) and the other having negative dispersion. T h e remaining s polarized curve and the p polarized curve b o t h have little dispersion. A w a y from zone-centre, i n the V — X direction, the fourth and fifth curves, which are b o t h s polarized, anti-cross near P/PG ~ 0.25. T h e frequency gap at the point of closest approach is comparable to the gap which occurs at zone-centre. 149  Figure 5.8: Mode frequency versus magnitude of in-plane wavevector for experiment (solid lines and markers) and model (dotted lines and hollow markers) for s (circles) and p (squares) polarized radiation. T h e inset schematically depicts the bandstructure for the case of weak texture.  150  A l o n g the Y — M direction, there are two moderately dispersive pairs of curves w i t h each pair containing one s and one p polarized band. One pair have a positive dispersion while the other pair have a negative dispersion. There is excellent agreement between the model and experiment except for the fifth curve i n the Y — X direction. This is partly due to the fact that it lies at or above the electronic bandgap of G a A s (11,450 c m ) , which forms half of the waveguide core. In - 1  this region, the model used for the dielectric constant of G a A s neglects the imaginary part and is not strictly valid even for the real part. T h e discrepancy is also due to the omission from the model of higher order Fourier components of the field, which become significant at higher energies.  5.8  Linewidths  5.8.1  Linewidth of Lowest Energy Band  Figure 5.9 shows the variation of the linewidth of the lowest energy band as a function of the in-plane propagation vector for b o t h experiment and theory. Except very near the zone-centre, where the calculated value approaches zero, the measured linewidth agrees w i t h the model to w i t h i n a factor of two. A w a y from zone-centre, the experimental and calculated linewidths b o t h exhibit a non-monotonic variation. It should be noted that while the calculation assumes pure plane waves incident w i t h well-defined k vectors, and an infinite, perfectly periodic photonic crystal that is free from imperfections, none of these assumptions is strictly true i n the experiment. The finite size of the photonic crystal, the non-zero detector acceptance angle A9, and non-uniformities i n the photonic crystal w i l l all contribute to a smearing out of the spectra, i n a convoluted manner, that limits the resolution of the experiment. Qualitatively, these factors account for the fact that the experimental linewidth does not seem to be approaching zero at the zone-centre as it does i n the calculation and as dictated by symmetry considerations. A n approximate quantitative treatment of their effects on 151  Figure 5.9: Experimental (rilled circles and solid lines) and model (filled squares and dotted lines) linewidths of the lowest frequency band appearing along the T - X direction i n Figure 5.8.  152  the measured linewidths is given below. In Chapter 4 it was stated that there is an angular spread of ± 3 ° of k vectors exciting the modes of the structure and that the detector acceptance is ± 1 ° . Since (3 = <Dsin(0), this corresponds to A / ? = Q cos (9) A9.  Consequently, one would expect the reflectivity  spectrum to contain a continuous spread of Fano resonances corresponding to modes w i t h i n a frequency band: Au>  spr  = a) cos (6>)A6>^j.  (5.7)  Furthermore, because the photonic crystal is of finite size, the Fourier Transform of the dielectric modulation is not just a series of delta functions translated i n Fourier space by the reciprocal lattice vectors of the crystal. For a finite crystal, this function must be convolved i n each of the x and y directions w i t h the sine function, —-p—— Du) where D is the size of the crystal. For the value, D = 90 fim, corresponding to the size of the photonic crystals i n this work, the main lobe of the sine function has a full w i d t h at half m a x i m u m of approximately A/3Q — 60 c m  - 1  . The effect on the eigenstates of the  crystal is that they are no longer made up of discrete field components w i t h in-plane propagation vectors given by ±np x G  ± mPcy for integer m and n. Rather, to a good  approximation, they contain significant field components i n a range spanning Apo i n each of the x and y directions, centred at ±nPox  ± mPoy-  T h e components i n the  band near zero in-plane wavevector w i l l be phase-matched to radiating components i n the cladding air layer. W h e n white light is used to probe along P = d)sin#, reflectivity w i l l be enhanced when the in-plane wavevector is matched to the near-zone-centre field components of the eigenstates of the finite-sized crystal. T h i s occurs i n the frequency range: A£  f i n  = AP ^  (5.8)  G  T h e experimental linewidth can be approximated by adding i n quadrature A c D  spr  and  Ad)fi to the inherent w i d t h of the mode. Near the zone-centre, this adds approximately n  153  50 c m  - 1  to the inherent linewidth, bringing it into much better agreement w i t h the  experiment . 1  A w a y from zone-centre, the dispersion increases then becomes roughly constant. However, this is largely offset by the u and cos# factors i n equation 5.7, which b o t h decline, so that the net effect is to add about 80 — 90 c m  _ 1  to the inherent linewidth  leaving it i n reasonable agreement w i t h the experiment. Finally, it should be re-iterated that non-uniformities i n b o t h the periodicity, as well as i n the sizes and shapes of the airholes that comprise the photonic crystal are also expected to have the effect of smearing out the spectra. However, the lack of any available quantitative characterization of these non-uniformities precludes any further comment i n this regard.  5.8.2  Linewidths of 4th and 5th Bands  Figure 5.10 shows the experimental and model linewidths as a function of in-plane wavevector along the T — X direction for the fourth and fifth lowest bands. T h e feature associated w i t h the fifth band remains resolved for a l l of the incident angles, and the associated linewidth varies smoothly i n b o t h the experiment and the model. The model predicts that, for the fourth band, the linewidth of the associated feature vanishes at the anti-crossing and varies considerably on either side of it. T h e experiment is qualitatively consistent w i t h this: there is a larger variation i n the linewidth i n the fourth band than i n the fifth band, and the resonance associated w i t h the fourth band vanishes at 6 = 25°. Once again, due to averaging effects, we expect the linewidth to vanish from the spectrum when it becomes too small ( < 80 c m  - 1  - 90 c m  - 1  , see footnote above).  This treatment would be rigorous if the resonances in reflection were strictly Lorentzian in character. The fact that they are generally Fano-like complicates the situation. In particular the averaging of Fano lineshapes tends to diminish the visibility of the features as much as it increases their apparent widths. This is consistent with the experimental results that consistently have the resonant features "disappear" when their line widths decrease below the instrumentally limited resolution calculated as above. More realistic (and complicated) averaging of the theoretical spectra bear this out but this is beyond the scope of this thesis. 1  154  200 H B  (a)  100 H  (b) 4^®—  0.4  0.5 X  Figure 5.10: Experimental (dots and solid lines) and model (open circles and dotted lines) linewidths of 5th lowest (a), and 4th lowest (b) bands i n Figure 5.8 vs. in-plane wavevector along the T - X direction. The crossed circle represents a zero linewidth that was inferred from the absence of any feature i n the spectrum.  A crossed circle has, therefore, been used for the experimental data point i n the fourth band at 0/6G ~ 0.25, which corresponds to 9 = 25°, i n order to indicate an unknown but small linewidth. Thus, b o t h model and experiment are consistent i n demonstrating that one band involved i n an anti-crossing away from zone-centre shows no dramatic variations i n linewidth, while the other band exhibits a widely varying linewidth that vanishes near the point i n the B r i l l o u i n zone where m a x i m u m coupling of the bands occurs.  5.9  Effect of Filling Fraction  Figure  5.11 depicts s polarized experimental spectra for different incident angles ob-  tained using the grating analyzed above (no. 22) and two other gratings (nos. 23, 24)  155  having the same pitch but larger hole sizes. T h e spectra have been offset vertically from one another for the sake of clarity. A l l the samples yield a series of Fano resonances that exhibit qualitatively similar dispersion, but there are obvious quantitative differences due to the differing fill fractions. T h e resonance associated w i t h the lowest energy band has been highlighted i n each set of spectra: it shifts up i n energy w i t h increasing air filling fraction, as expected. In fact, a l l four bands shift to higher energy w i t h more air i n the crystal.  Subtle differences i n the precise dispersion of the bands i n these  other samples were not pursued due to the difficulty i n extracting accurate fits to nearly overlapping Fano features i n many of the spectra.  156  i i i i | i n i | i •i i | i i i  1  6  7  8  9  | i n i | i i i i | n i r] i i i i -  i  10  FREQUENCY  11  CO/2TTC  12  13  14  [io cm- ] 3  1  Figure 5.11: Experimental specular reflectivity spectra for s polarized light (solid lines) and p polarized lines (dotted lines) (a) grating 22 (b) grating 23 (c) grating 24 which have the same pitch but different air filling fractions of (a) 0.28 (b) 0.33 (c) 0.49. The spectra have been offset vertically from one another for the sake of clarity. The vertical scale on the reflectivity traces is linear. The dark, thick lines have been drawn i n by hand to serve as a guide to the eye.  157  Chapter 6 Discussion  6.1  Introduction  T h i s chapter provides an interpretation of the experimental data presented i n the previous chapter based on symmetry considerations as well as knowledge gained from the numerical models. The results can be interpreted entirely i n terms of renormalized (coupled) bound slab modes of some effective slab waveguide. T h e effective waveguide can be thought of as one where the dielectric constant i n the textured guiding core is averaged according to the air filling fraction of the holes. In the spectral range of interest, the effective waveguide supports only a single, T E polarized mode, the frequency and in-plane wavevector of which follows some dispersion relation: ^'guide = fiPguide)  where  P ide gu  (6-1)  is the magnitude of the in-plane wavevector,  (3  g u i d e  ,  of the guided mode,  and Co is given by equation 5.1. T h e dispersion relation is isotropic w i t h i n the plane of the guide. Light incident from above the grating at some frequency Co at incident angle, 9, and azimuthal angle, 4>, will have an in-plane momentum given by: Pine  =  &  sin(0)  cos(0)x + Co sin(0) cos(<fi)y  (6.2)  T h e presence of the square grating causes this light to be scattered by the reciprocal lattice vectors of the photonic crystal, Gj i = t  and / are integers. T h a t is, /3  inc  —> /3  inc  JPG  X  +  WGVI  where  PQ  =  2TT/A,  and j  + G^i. T h e significant reciprocal lattice vectors  for our structure i n the spectral range of interest are:  ±PG , X  ±PGV,  ±PG  X  ±  PGV-  A  resonant feature appears i n the specular reflectivity spectrum whenever the frequency 158  of incident radiation is such that one (or more) of the scattered components fall on the dispersion curve for the renormalized  guided mode at that same frequency.  M a n y of the interesting properties revealed i n this study can be qualitatively explained by ignoring the renormalization of these slab mode energies and focusing just on the restrictions imposed by symmetry. Therefore, many of the following features also apply to weakly 2D textured slab waveguides, where the renormalization is minimal.  6.2  Resonant Features in Reflectivity  T h e reflectivity spectrum is comprised of b o t h resonant and non-resonant components. T h e non-resonant, slowly varying, component can be thought of as reflection off the "average waveguide structure".  T h a t is, by treating the textured layer as a uniform  layer w i t h a dielectric constant taken to be the average of the dielectric constants of the semiconductor and air, weighted by their respective filling fractions. Because the layer structure is such that it contains a cavity formed by the oxide layer sandwiched between two semiconductors (namely, the substrate and the "average waveguide layer"), this slowly varying component exhibits Fabry-Perot fringes. T h e resonant, bipolar and asymmetric feature i n the reflectivity is a Fano [38] resonance.  The asymmetric and  bipolar lineshape arises from interference between contributions to the reflected field from the directly scattered component described above and from a component which is doubly-scattered from the incoming radiation from above the grating to a bound mode supported by the guide and back to radiation scattered out the top of the grating. T h e phase of the doubly-scattered contribution varies by TT radians as the frequency is tuned through the phase-matching condition for coupling to the bound mode, whereas the phase of the directly scattered contribution varies very little. Thus the two contributions sum constructively on one side of the phase-matching condition and destructively on the other side, thereby giving rise to a bipolar and asymmetric lineshape.  159  Figure 6.1: Schematic dispersion of resonant states attached to the core of a slab waveguide w i t h square photonic lattice.  6.3  Polarization Properties and Basic Dispersion  F r o m the fits to the Fano resonances obtained for a wide range of incident angles and polarizations, the photonic bandstructure along the two high-symmetry directions in the second B r i l l o u i n zone was obtained as i n Figure 5.8.  Concentrating on the four  prominent bands emanating from the zone-centre i n Figure 5.8, their dispersion and polarization properties can be schematically summarized as i n Figure 6.1. Starting along the T — X direction, three bands have s polarization and one has p polarization. T h e modes associated w i t h a l l four of these bands are p r i m a r i l y composed of a superposition of TE polarized field components w i t h in-plane spatial variation given by {PincX ± PG ) X  a  n  d {Binci ± PGV)-  These field components are depicted i n Figure 6.2.  A w a y from zone-centre, the upwardly dispersing band along r — X is primarily composed of a T E slab mode at (Pi  + PG) , which can only radiate s polarized radiation. X  nc  Similarly, the downwardly dispersing, lowest energy band is dominated by the T E polarized (Pi —PG)  X  nc  slab mode which also can only radiate in the s polarization. T h e two flat  bands are composed principally of either symmetric or antisymmetric superpositions of  160  Figure 6.2: Dominant Fourier field components of eigenmodes i n lowest four bands along the r — X {6 = 0) direction depicted i n Figure 5.8: the solid black arrows represent the propagation vectors associated w i t h the in-plane variation of the field components; dashed lines represents the electric fields and their x and y components; the grey arrows represent the incident in-plane wavevector, i.e., detuning from zone-centre  161  the T E slab modes at (Pi x ± P y) • T h e symmetric combination w i l l generate a zerothnc  G  order polarization w i t h no x component, since the principal x components of the two respective T E slab modes cancel exactly, again leaving only an s polarized residual field that can couple to the emitted radiation. O n the other hand, the antisymmetric superposition will have the y components cancel, yielding a field entirely along the x direction that can only couple to p polarized radiation. T h i s fully explains b o t h the dispersion and polarization of the four bands along the T — X direction using only symmetry and the fact that the photonic eigenstates represented by these bands are composed primarily of T E slab modes. W i t h the aid of Figure 6.3, a similar analysis explains why i n the F — M direction there are two pairs of moderately dispersive bands, w i t h one pair having positive and the other negative dispersion w i t h respect to detuning away from zone-centre, and w i t h each pair being comprised of one s and one p polarized band. In this case, the four relevant field components, (@^[x + y]±PGx),  (^^[x  -\-y)±  Pcy),  occur i n pairs lying symmetrically about the in-plane direction of propagation of the exciting field.  Here, the symmetric (anti-symmetric) superpositions of the modes at  (@jf[x + y] + PGX) and (^f[x  + y] + Pay) give resultant fields polarized perpendicular  (parallel) to the direction of propagation of the radiating field, {^f[x  + y]), and thus  can couple only to s (p) polarized radiation. It is also evident from the diagram that as the detuning from zone-centre, Pi , nc  increases, the in-plane wavevector of these modes  increases so that they and the pair of eigentstates of the photonic crystal they principally comprise, exhibit positive dispersion. T h e same applies to the modes at (^-[x  + y]  — PGX)  and (^-[x+y]  — PGIJ)-  However,  in this case, the in-plane wavevector of these modes decreases when the detuning from zone-centre increases so that these modes comprise the pair of eigenstates exhibiting negative dispersion.  162  Figure 6.3: Dominant Fourier field components of eigenmodes i n the four bands along the r — M direction depicted i n Figure 5.8: the solid arrows represent the propagation vectors associated w i t h the in-plane variation of the field components; the grey arrows represent the incident in-plane wavevector, i.e., detuning from zone-centre; the dashed lines represents the electric fields and their x and y components.  163  More generally, the polarization of leaked radiation will only be either s or p polarized for excitation along directions that possess mirror symmetry. T h i s can be seen from the following argument. A l o n g such directions, parity requires that field components of the eigenmode lying symmetrically about the plane of symmetry occur i n either symmetric or anti-symmetric pairs of equal amplitude. Such pairs necessarily superimpose to give resultant electric fields which are either perpendicular or parallel to the symmetry direction and hence can only drive polarization i n the zeroth order which is either purely s or p polarized. T h e remaining field components, which lie along the line of mirror symmetry, can only drive purely s or p polarization at zeroth order according to whether they are T E or T M , respectively. Thus, it is an important property of these structures that, for light incident along the crystal symmetry directions (</> = 0° and 4> — 45° i n the present case), there is no s-p or p-s scattering i n the specular direction. T h a t is to say that the polarization of the radiative component of the eigenmodes is a "good quantum number". Since the preceding arguments d i d not refer at all to the magnitude of the in-plane wavevector, it follows that the polarization of the radiative component of the eigenmodes remains the same w i t h i n a curve along a given symmetry direction i n the bandstructure plot of Figure 5.8. Notice, however, that a given band cannot be given a unique polarization label because, away from directions of high symmetry, leaky eigenmodes will, i n general, radiate an elliptically polarized field. Indeed, the fourth highest energy band shown i n Figure 5.8 radiates s polarized light along the T - X direction and p polarized light along the T - M direction. Finally, although beyond the scope of the experimental results presented i n this work, it is important to note that none of the above arguments depends upon the radiative nature of the exciting and reflected fields. T h e same arguments apply equally well to the truly bound eigenmodes below the light line, for example, i n the vicinity of the first order gap of a similar structure w i t h 2D square lattice. Thus, we can generalize the  164  preceding statement to say that along directions of the photonic crystal which possess mirror symmetry, the polarization of the field component lying i n the first B r i l l o u i n zone is a "good quantum number" for the B l o c h states i n that band.  6.4  Gaps and Symmetry Influences on Linewidths  T h e s polarized spectra around the second order gaps near zone-centre contain two narrow features and one broad feature which has approximately the same w i d t h and energy as the lone feature occurring i n the p polarization.  A s shown by the model  results i n 2.3.5, it is expected that, at normal incidence, the two broad features become degenerate and retain a non-zero w i d t h whereas the narrow features become infinitely narrow and vanish. T h i s behaviour may also be arrived at by the following symmetry argument. Right near zone-centre, the modes i n the four bands being considered here cannot be dominated by just one or two slab modes.  Rather, they have to be made up of  appropriate combinations of a l l four T E slab modes at ±6G  X  and ±/?G2/ that preserve  the required symmetry. T w o of these modes necessarily generate polarization w i t h no radiative component due to the perfect cancellation of the T E slab modes' contributions to the zeroth order polarization at 3 = 0. T h e other two modes have a net non-zero zeroth order polarization, and they are necessarily degenerate at the zone-centre since there can be no preferred polarization for such leaked radiation at 3 = 0. Thus, nondegenerate modes at zone-centre must be infinitely long-lived, and  finite-lifetime  modes  must be degenerate at zone-centre. The size of the gaps at zone-centre (for extremely weak 2D square texture, all four of the zone-centre modes would have almost identical energies) are a measure of how much the zone-centre eigenmodes differ i n their respective amounts of localization i n the dielectric versus the air. The size of the gap and the energetic ordering of degenerate and non-degenerate modes at zone-centre are b o t h influenced considerably by the air  165  filling fraction of the photonic lattice.  6.5  Effect of F i l l i n g F r a c t i o n  A s mentioned i n the previous chapter (section 5.9), the lowest energy feature shifts up in Figure 5.11 as the air-filling fraction of the 2D grating is increased. T h i s is because the frequencies of the bound slab modes of the "effective untextured" waveguide w i l l tend to increase as more low-index material is introduced into the core, guiding layer, thereby reducing its average dielectric constant.  These bound modes also become less  tightly bound to the (textured) guiding core, and therefore, interact less w i t h the grating. Consequently the w i d t h of the gap, which is a measure of the scattering induced by the grating, w i l l also be reduced for gratings w i t h larger filling fraction. Hence, the effect of having a larger air-filling fraction on the gap is to increase its centre-frequency and decrease its size. B o t h of these effects result i n an increase of the frequency of the lowest band, which is consistent w i t h the experimental results. Figure 6.4 depicts model specular reflectivity spectra for light incident along the r — X direction at 0 — 10° for two different fill fractions. A s the hole radius is increased, the lossy mode is clearly seen to shift from the highest to the middle s polarized feature. Increasing the hole radius further would, i n principle, shift it to the lowest feature. However, this is not possible i n our structure; since the holes completely penetrate the guiding core, increasing the normalized hole radius ( r / A ) much beyond 0.35 reduces the average index of the guiding core so much that it ceases to support any bound modes. We can conclude that the hole size for our structure is such that the lossy mode is the one that is either second highest or t h i r d highest i n frequency of the s polarized modes. T h e frequency ordering of the lossy and non-lossy modes at zone-centre comes about as a result of the real-space field intensity distribution of the eigenmode over the unit cell. For simplicity we revert to consideration of the specular reflectivity spectra from  166  Figure 6.4: Calculated specular reflectivity spectra for s polarized (solid lines) and p polarized (dotted lines) for 9 = 2°, <j> = 0 w i t h (a) r / A = 0.29 and (b) r / A = 0.33. The other parameters i n the model are as the fitted results for grating 22. The curves have been shifted vertically for clarity.  167  the waveguide structures w i t h I D texture presented i n section 2.3.4. Right at zone-centre, for a grating having reflection symmetry, such as the square grating under consideration, parity requires that the eigenmodes be composed of a symmetric or anti-symmetric superposition of field components having in-plane variation given by e  .  ±l/3aX  A t the second order T E gap, the E field is i n the y direction and  dominated by components at ± / ? G  :  E ( x , z) = [E e y if}GX  y0  ± Ey^-^y^z).  (6.3)  However, each of the ±PG components can scatter w i t h the same scattering amplitude, to a zeroth order Fourier component which is phase matched to radiate away into the cladding regions. In the case of the eigenmode comprised of the symmetric superposition, the fields scattered back to zeroth order w i l l add, thereby enhancing the radiated field and giving rise to a lossy eigenmode. Conversely, for the eigenmode comprised of the anti-symmetric superposition, the fields scattered back to the zeroth order component w i l l cancel exactly, so that no radiating field is produced and the mode remains lossless. Slightly, away from zone-centre the cancellation ceases to be exact so that the "lossless" mode appears as a very narrow feature w i t h small loss. T h e frequency ordering arises from the fact that the in-plane intensity profile is different for the two eigenmodes. Figure 6.5 schematically depicts the electric field intensity profile arising from symmetric and antisymmetric superpositions of field components having in-plane variation given by e ® ±l  GX  over a unit cell of a I D grating. T h i s is done  for three different gratings having air filling fractions of 25%, 50%, and 75%. For the case of 25% fill fraction, the lossy, symmetric superposition (left), has a greater proportion of its field energy (area under the curve) concentrated i n the low dielectric portion than does the lossless eigenmode comprised of the anti-symmetric superposition (right). Consequently, the lossy mode appears as the high-frequency (or high-energy) mode. Conversely, at a filling fraction of 75%, the situation is reversed and hence the lossy mode appears at low frequency. T h e transition occurs at a filling fraction of 50%. Here, 168  -A/2  A/2-A/2  A/2  Figure 6.5: Schematic depiction of the i n plane electric field intensity distribution over the unit cell for a I D grating of the symmetric, lossy (left) and anti-symmetric, non-lossy (right) band edge states for a grating w i t h (a) 25% (b) 50% (c) 75% air filling fraction. In (b) the lossy and lossless modes b o t h modes have the same fraction of their total field energy (area under the curve) i n the high-index region. T h e dielectric profile is indicated by the dashed line.  the lossy and lossless modes b o t h modes have the same fraction of their total field energy i n the high-index region, and are, therefore, degenerate.  6.6  Gaps A w a y from Zone-Centre  Figure 5.10 shows the experimental and model linewidths as a function of in-plane wavevector along the T — X direction for the fourth and fifth lowest bands. B o t h the experiment and model exhibit a linewidth going to zero well away from the zone-centre  169  and boundary. This corresponds to the anti-crossing of these bands which occurs near Pine/PG ~ 0.25 i n Figure 5.8. The energy of closest approach is comparable to the gap which occurs at zone-centre.  T h i s implies that the corresponding modes are strongly  mixed. A s described above, away from the zone-centre, the 4th band is dominated by the Pinc + PG T E - l i k e Fourier component corresponding to a bound mode traveling i n the x  X  +x direction. F r o m kinematic considerations, one would expect the next higher set of modes at zone-centre to be symmetric and anti-symmetric superpositions of T E polarized field components at Pi  ± (PG ± PGV)-  x  X  nc  This is because these reciprocal lattice vectors  have the next largest magnitude and the dispersion of the untextured slab modes, which is isotropic i n the in-plane directions, is monotonically increasing, and also because the lowest order T M modes occur at much higher frequency, owing to the thinness of the waveguide core. For these modes, symmetry considerations for detuning i n the Y — X direction are similar to those for detuning i n the T — M direction for the modes at the "second order"  and  (±PG  X  ±PGV)  gap. Consequently, these modes will appear as two  pairs of bands having positive and negative dispersion, w i t h each pair having one band leaking s polarized radiation and one band leaking p polarized radiation. T h e 5th band must be one of the modes w i t h negative dispersion, which, i n turn, implies that it must be composed of in-phase or out of phase superpositions of Pi x nc  — PQ(X ±y).  Since it is  leaking only s polarized radiation, the superposition must be such that it cancels out the E  y  field, which implies that it must be the symmetric superposition (see Figure 6.6.) Therefore, i n the vicinity of the anti-crossing away from zone-centre, the modes i n  the 4th and 5th bands must be comprised of a significant admixture of T E polarized field components at Pi x nc  — PG(X ± y) and Pi x nc  + P G - Right at the anti-crossing, i.e., X  the point of closest approach of the 4th and 5th bands, which occurs experimentally at PIPG ~ 0.25, these three components scatter back to pi  x nc  so as to produce a zero net field.  w i t h appropriate amplitudes  Consequently, the leaky component vanishes, as  170  Figure 6.6: Dominant Fourier field components of eigenmodes i n the 4th and 5th lowest energy bands along T — X direction in Figure 5.8 i n the vicinity of their anti-crossing: the solid arrows represent the propagation vectors associated w i t h the in-plane variation of the field components; the grey arrows represent the incident in-plane wavevector, i.e., detuning from zone-centre; dashed lines represents the electric fields and their x and y components.  171  evidenced by the disappearance of the Fano resonance i n the specular reflectivity spectra. A t the anti-crossing the mode i n the 4th band is completely b o u n d to the slab and has infinite lifetime. The above description was verified by looking at scattering coefficients produced by the F D R S model. T h e dominant components of the mode correspond to the dominant scattering amplitudes from Pi x nc  to P x+(nx+my)PGinc  Figure 6.7 depicts the spectra of  the various scattering amplitudes calculated w i t h in-plane wavevector value of Pi /PG nc  ~  0.25, corresponding to 9 = 25°, at which the anti-crossing occurs as inferred from the fact that the resonant feature i n the 4th band disappears from the reflectivity spectrum i n Figure 5.5. T h e results clearly show the largest amplitudes for the components at Pinc% — PG{ ^V) X  and PinJt + PGX, w i t h those for the former two being equal as expected  from symmetry considerations.  6.7  E x t r i n s i c Influences o n L i n e w i d t h s  Referring now to Figure 5.9, the calculated w i d t h is clearly seen to be approaching zero at zone-centre; therefore, this mode becomes completely bound there.  T h e oscillating  behavior of the linewidth is due to the presence of the substrate and would not exist i n the case of a three layer a i r / A l G a A s / A l oxide guide. T h i s can be understood i n terms of a simple model wherein the grating is taken to be a t h i n source emitting radiation leaked by the mode downward toward the substrate w i t h the perpendicular component of the propagation vector given by:  k\ = (V^^f - p\ c  (6.4)  T h i s component is reflected at the oxide/substrate boundary w i t h a TT phase change. Consequently, when the product  k±_d ^ is such that it returns i n phase w i t h the field i n Q  e  the grating, the radiation is enhanced, the mode becomes more leaky and the linewidth is increased. Conversely, when the reflected field returns out of phase w i t h the field i n  172  o  o  •  o  o  •  o  (f) o o o  (e)  Q ID H  •  o  o  o  o  •  o  o  o  i—i  OH  (d)  s  o  o  o  o  •  o  o  o  o  a i—i  (c)  w  H H < U  o  •  o  o  o  o  •  o  (b) •  o  o  o  o  o  o  o  o  o  (a)  o  o  o  •  o  o  o  o  111111111111111111 • 11111111111111111111111111111111  8  9  10  3  11  -1  12  co/27Cc [10 c m ] Figure 6.7: Scattering amplitudes calculated using the F D R S Code for s polarized light incident at f3 « 0.25p x corresponding to 0 = 25° along the T — X direction, where the anti-crossing occurs. T h e various plots depict the scattering amplitudes to s polarized light w i t h in-plane wavevector: (a) f3 (b) f3 - p x (c) (3 ± p y (d) (3 + p x (e) (3 + P [—x ± y] (f) (3 + P [x ± y]. T h e structure is the same as that for grating 22 w i t h the parameters as obtained from fitting w i t h the G F code. inc  G  inc  inc  G  inc  inc  G  173  G  inc  G  inc  G  the grating, radiation is suppressed, the mode becomes less leaky and the linewidth is decreased. T h i s effect has previously been predicted by Benisty et al. [52]. It should also be noted that since the only other significant Fourier components of the B l o c h state i n this band are evanescent i n the cladding layer and hence are not appreciably influenced by the oxide layer thickness when it exceeds 500 nm, the thickness of the cladding layer could be used to tune the mode lifetime without significantly altering its energy. For these bands, then, the first order Fourier component of the dielectric modulation couples the vertical oxide cavity to the 2D in-plane "cavity" formed by the distributed feedback.  W h e n a high-index-contrast photonic lattice is used, the first-order Fourier  component of the "scattering potential" is larger than i n the weakly textured case . T h e larger diffraction rate which results manifests itself not only i n broader linewidths but also an enhanced coupling between the vertical and distributed in-plane cavities.  174  Chapter 7  Summary and Conclusions  7.1  Summary  A numerical model was developed for calculating specular reflectivity and scattering from an infinite 2D photonic crystal embedded i n an arbitrary dielectric slab waveguide structure.  A discrete set of plane wave-like basis functions is integrated through the  textured structure using Maxwell's equations on a discrete real-space mesh.  Resonant  features i n the reflectivity and scattering spectra are used to obtain information regarding the dispersion, polarization, and lifetimes of the bound and resonant electromagnetic modes of the textured waveguide.  W i t h run-times on the order of two hours on a  current P C (AMD Duron 1600 1.6 G H z , 1 G B R A M ) the model is found to converge to w i t h i n 1% for the mode frequencies and to w i t h i n 7% for the linewidths. A fabrication procedure was developed for making structurally robust and robustly guiding t h i n semiconductor slab waveguides w i t h 2D square photonic lattices of airholes that completely penetrate the core layer. The procedure involved using electron beam lithography and E C R plasma etching to define the texture i n an 80 n m thick G a A s / A l o . 3 G a . 7 A s core on the surface of a ~ 1.8/z m layer of Alo.9sGao.02As grown on a 0  G a A s substrate using Molecular B e a m Epitaxy. T h e texture consisted of a square lattice of cylindrical airholes. These varied i n pitch from 500 n m to 575 n m and i n air filling fraction from 0.28 to 0.49. T h e sample was then placed i n a warm, moist environment to oxidize the Alo.9sGao.02As layer i n the region beneath the 2D lattice through the holes. Polarized, angle-resolved specular reflectivity spectra were measured at a series of incident angles for energies between 6000 c m  175  - 1  and 14,000 c m  - 1  .  T h e experimental  spectra bear a strong resemblance to those calculated using the numerical models. B o t h the experimental and model spectra were fit w i t h a combination of an A i r y function and Fano-lineshape function i n order to extract the energy and lifetimes of the modes excited at each angle of incidence. These were used to plot the dispersion of mode frequencies and lifetimes. The bandstructure so obtained exhibits large (1000 c m  - 1  at 9500 c m ) - 1  gaps and drastic modification of the dispersion over a large portion of the B r i l l o u i n zone. B o t h of these effects are much more pronounced than those that are typical of weakly textured slab waveguides. T h e spectra exhibited well-defined polarization properties along the symmetry directions of the crystal i n that the polarization of leaked radiation was either purely s or purely p . Near zone-centre, i n the vicinity of the second order gap, and for detuning along the T — X direction of the crystal, three modes appeared i n the s polarization and one i n the p polarization. One of each of the s and p polarized modes exhibited little dispersion over a large portion of the 1st B r i l l o u i n zone. For detuning along the T — M direction, two moderately dispersive pairs of features appeared, w i t h one of each pair appearing i n the s and p polarization. There was also generally good agreement between the model and experiment for the linewidths of the resonant features associated w i t h the modes, except where the linewidths were predicted to vanish (or become narrower than ~ 80 c m ) ; i n this - 1  case the lower limit on the experimental resolution made a quantitative comparison impossible. Nevertheless, linewidths (lifetimes) were observed that varied considerably w i t h i n a given band and between bands i n a manner that agreed qualitatively w i t h the model. In particular, for one of the modes, there was an oscillation of the linewidth as a function of the in-plane wavevector which was clearly evident i n b o t h the model and experimental spectra. Finally, an anti-crossing was observed away from any high symmetry points of the  176  crystal, simultaneously w i t h the vanishing of the linewidth of one of the features associated w i t h the anti-crossing bands.  7.2  Conclusions  Features i n specular reflectivity spectra from 2D textured semiconductor slab waveguidebased photonic crystals can be used to study the intrinsic properties of resonant electromagnetic modes localized i n the vicinity of the textured core. B y comparing model calculations w i t h the experimental results, it has been clearly shown that i n order to fully characterize these electromagnetic excitations their dispersion, lifetimes, and polarization must be determined. Analysis of the results indicates that for the "membrane-like" structures studied i n this work (thin, high-index core, low-index cladding and texture, A n ~ 1.8) the electromagnetic excitations can be largely interpreted as renormalized (mixed) T E and T M polarized slab modes characteristic of a fictitious effective slab waveguide where the core has a uniform refractive index given by the volume-averaged refractive index of the textured core of the actual structure. T h e degree of mixing introduced by the 2D texture, as inferred from the w i d t h of the anti-crossings, is ~ 10%. T h e polarization properties can be explained entirely i n terms of symmetry arguments regarding the coupling of the zeroth order radiative component w i t h the bound-slabmode-like T E field components at ±/3GX and  ±PGV-  The gross dispersion properties can  be explained qualitatively from kinematical considerations. T h e oscillation of the linewidth w i t h in-plane wavevector was explained by the coupling of the Bloch states of the photonic crystal embedded i n the waveguide to the Fabry-Perot modes of the cavity formed by the oxide layer via the zeroth order Fourier component of the Block state that radiates down into the oxide and undergoes multiple reflections there. More generally, the detailed shape of the bandstructure and the variation of the  177  resonant mode lifetimes is rich, and affords considerable flexibility for future device designers. A l l bands exhibit zero group-velocity at zone-centre, and small group velocity out to at least 10% of the zone boundary. It has been shown that the lifetimes of degenerate bands at zone-centre are, i n general, finite. These degenerate bands at zone-centre offer small group velocity and strong coupling to external radiation modes. T h i s makes them interesting for detector, emitter, and coupling applications.  Isolated bands at zone-  centre are necessarily infinitely-long-lived right at the zone-centre, and have relatively long lifetimes out to ~ 10% of the way to the zone boundary. T h i s combination of small group velocity and long lifetimes is of particular interest i n active laser devices that incorporate 2D photonic crystals. T h e anti-crossing observed away from the zone-centre, is i n a certain respect, "accidental" i n that it depends on the details of the bare slab mode dispersion and, unlike the gaps (anticrossings) occurring at high symmetry points of the photonic lattice, could not be predicted purely from symmetry considerations of the 2D lattice alone. In summary, the complete understanding of the complex (energy and lifetime) dispersion and polarization properties of this interesting new class of materials, obtained by developing and using a numerical solution of Maxwell's equations to interpret broadband specular reflectivity spectra, has shown that this bandstructure for photons can be qualitatively explained by applying symmetry and kinematical considerations to the renormalization of the slab waveguide modes by the high index contrast 2D scattering potential.  7.3  Future Directions  Work i n this area is proceeding along several lines. One is the inclusion of electronic resonances through incorporation of quantum wells, wires, and dots i n appropriately designed slab waveguide-based photonic crystals. Some theoretical work has also been  178  done [53] on simultaneously texturing the magnetic permeability of materials. In addition to allowing for greater scattering, this also allows an additional degree of freedom i n engineering photonic bandstructure, as well as tunability via application of an external magnetic field. Yet another direction involves the search for purely photonic effects through the design and realization of more elaborate structures, such as the integration of several layers of slab waveguide crystals and the inclusion of periodic defects. W o r k is already underway i n our group using the latter strategy. A modified version of b o t h the F D R S code and the G F code, capable of modelling infinite photonic lattices w i t h periodic defects embedded i n slab waveguides, has been developed. These structures hold promise for engineering very weakly dispersive bands predicted to play important roles i n enhancement of non-linear optical cross-sections [11].  179  Bibliography  J . D . Joannopoulos, Robert D . Meade, and Joshua N . W i n n . 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Labilloy, C . Weisbuch, C . J . M . Smith, T . F . Krauss, D . Cassagne, A . B r a u d , and C . Jouanin. R a d i a t i o n losses of waveguide-based two-dimensional photonic crystals: positive role, of the substrate. Appl. Phys. Lett., 76:532, 2000. M . M . Sigalas, C . M . Soukoulis, R . Biswas, and K . M . H o . Effect of the magnetic permeability on photonic band gaps. Phys. Rev. B, 56:959, 1997.  183  Appendix A Generalization of FDRS Method to an Arbitrary 2 D Lattice  T h e most general 2D Bravais lattice of no particular symmetry is the oblique net, which may be specified by a pair of non-collinear primitive lattice vectors, A i , A of arbitrary 2  length and orientation. A s before, we define a mesh for taking difference terms, however, this time we use an oblique, i.e., non-rectilinear, mesh given by:  a = ax b = bb — bcosipx  (A.l)  + bsmipy  (A.2)  g = gz  (A.3)  where: b = cosipx + sin tpy  (A.4)  Here again, we presently make no assumption regarding the orientation of the 2 D photonic crystal w i t h respect to our coordinate axes. We now generalize equations 2.33 2.35 i n accordance w i t h our oblique mesh to: k-a  —-  |a|  ,ik a.  r  ,—ik o,.,..  x  x  / A r-\  A  = k RJ exp(——) - e x p ( — - — ) \ / i a = K 2 2  (A.5)  = k ~ [ e x p ( ^ ) - exp(—^)]/z& = K  b  (A.6)  ± K  (A.7)  x  X  b  = k « [ e x p ( ^ ) - exp(^-)]/zg z  Z  so that we may write: K  = K X + Kbb + K Z X  Z  184  (A.8)  We note from equation A . 2 that: — cotipx  y=^  (A.9)  so that we may write k •b k = k •y = —  k • x c o t V ' = k — k cotip.  y  b  (A.10)  x  We may now proceed as before w i t h equations 2.31 - 2.32, carrying out the crossproducts, making the approximations for k, and expanding component-by-component to get a set of six equations similar to equations 2.40 - 2.45. We take as an example the equation for the x component of the B field. (kxE)  = k E -k E  x  y  z  z  = uB  y  (A. 11)  x  Using equation A . 10 this becomes: kbE — k E z  x  z  cot tp — k E z  = uB .  y  (A. 12)  x  Now, making the approximations for k using equations A . 5 - A . 7 , we obtain: exp(ik ~)E —A ibu b  B  x  «  exp(-ikX)E —A ibu  z  exp(ik 9-)E z  igu  exp(ik ^)E A iau  z  x  z  exp(-ik ^)E A o t ip iau x  c o  t ib H  z  C  exp(-ik 9-)E " ~ igu z  y  - +  y  (A.13)  We now note that: exp{±ik -) b  * \  h  \  _i_-i  = exp(±zk • - ) = exp[±2k r  (bcosipx  + bsinipy)  J = exp[zbk  (b x x  + b y) J y  2  (A.14) and therefore, similarly to the second of expressions 2.46 exp(±ik ^)F b  l  ^F (xT^,yTj,z) l  = F (r =p C) t  (A.15)  where, F\ denotes a component of the E , D , B or H field, <=> denotes Fourier transformation, and £ is now defined as: C = \ 185  (A.16)  We now take the inverse Fourier transform of equation A . 13, making use of equation A . 15 and the first and t h i r d of equations 2.46, to yield:  _g,(r + Q -  M  g.(r + «) - S,(r - a) ^(r  E,(r - 0  +  i&w  iau  +  7  ) - g,(r - ) 7  igui (A.17)  Isolating the E  y  q,(r  +  7  terms i n the preceding equation, we obtain:  )-^(,- ) ^ 7  < l ( r ) g i ( r ) +  g . ( r + 0 - fr(r- < )  zgu;  + c o t  ^ ( r + «) - S.(r - a)  z6a;  iau (A.18)  In order to obtain an equation resembling equation 2.65 we need to eliminate the z-fields. Consequently, using the same procedure as for B  above, we first obtain the expression  x  for the D  component given by:  z  ^ H (r  + a ) - Hyjv - a )  y  H (r  + Q - H (r  x  x  iau  - Q  |  ^  H (r x  ibu  + a) - H (r  -  x  a)  iau (A.19)  so that: E() z  r  2  «  g  y ( + oQ ~ # y ( iaue(r) r  r  ~ «)  #*(  r  + Q - # » ( r - C) ibue(r)  | c Q t  ^H (r x  + a ) - H {r iawe(r) x  -  a)  (A.20) In this manner, expressions similar to those given i n equations 2.64 - 2.67 may be obtained that again allow for the E and H fields defined on a discrete mesh on a pair of planes to be propagated forward or backward i n the z direction. A s before, these equations require evaluation of the fields at r, r ± ( a ± £ ) , r ± 2£, r ± 2 a , as well as evaluation of e' and p! at r, r ± £, and r ± a.  However, now ot and £ are no longer  defined to be orthogonal to one another. Figure A . l depicts the mesh points w i t h i n the unit cell of the photonic crystal on which evaluation of the fields (squares) and e' and / / (black dots) are required. Notice that i n this scheme (as before), we take a and b along the body diagonals of the p r i m itive cell of (the underlying Bravais lattice of) the photonic crystal we wish to model. 186  Explicitly: a=(A  + A )/A^  (A.21)  b = (Ai-A )/iV  (A.22)  1  2  2  where N  2  ( N along each primitive lattice vector) is the number of mesh points we desire  for the fields. W e then orient our rectilinear coordinate axes such that x lies along a.  ty  Figure A . l : Scheme of real-space points on which the fields and dielectric constant are sampled on the unit cell. T h e dielectric constant is sampled at points represented by the black dots. T h e electric or magnetic fields are sampled only on the points w i t h squares. A i and A represent the primitive lattice vectors of the photonic crystal. 2  The reciprocal lattice vectors for the general 2D oblique lattice w i t h primitive lattice vectors, A  1  ;  A , are given by: 2  G  L  r ~ r - ( ^ " **y) A  = -J—A lx^-2y  —  A  (A.23)  2x^-\y  A  2-7T G  S  =  - A — ^•\x^2y  A  — 4 - 4 - ^ — s\2xA\  - ^y) A  (- ) A  24  y  The reciprocal lattice is similar to the direct lattice but rotated by 90 degrees. It is depicted i n figure A . 2 . A s before, the polarized plane wave like basis modes may be 187  Figure A . 2 : Scheme of reciprocal lattice vectors corresponding to in-plane wavevector of launched modes for photonic crystal w i t h oblique 2D lattice.  188  formed from a finite set of the reciprocal lattice vectors using equations 2.77 - 2.80, where the expression for K i n equation A . 8 may be used, and where h = {G  lim  + P)-b.  (A.25)  The dispersion relation for this mesh is obtained from:  Ld /j, eoe' K K ' 2  H  Q  I/NI  = ^\l-cosk a] a 2  + ^[l-cosk b] b  x  + ~[l-cosk g] 9  b  2  z  + ^- cosip sin ab  2  sin  ^ 2  (A.26) where: |Ai| -|A | 2  V> =  ,  |A  ' 1  1 n '  I K  '  + A ||A -A | 2  1  ,  2  2  C 0 S  ,  v  A.27  2  '  T h i s dispersion relation may be solved, as before, for k to yield two solutions which z  are then used to generate K ' S corresponding to incoming and outgoing plane waves. Z  189  

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