@prefix vivo: .
@prefix edm: .
@prefix ns0: .
@prefix dcterms: .
@prefix skos: .
vivo:departmentOrSchool "Science, Faculty of"@en, "Physics and Astronomy, Department of"@en ;
edm:dataProvider "DSpace"@en ;
ns0:degreeCampus "UBCV"@en ;
dcterms:creator "Pacradouni, Vighen"@en ;
dcterms:issued "2010-01-16T17:45:37Z"@en, "2006"@en ;
vivo:relatedDegree "Doctor of Philosophy - PhD"@en ;
ns0:degreeGrantor "University of British Columbia"@en ;
dcterms:description "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. With 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 with energies and linewidths that disperse as the in-plane momentum (angle of incidence) of the exciting radiation is varied in the near-infrared. A numerical solution of Maxwell’s equations was developed to interpret these reflectivity spectra in terms of a \"photonic bandstructure\" for electromagnetic modes localized in 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."@en ;
edm:aggregatedCHO "https://circle.library.ubc.ca/rest/handle/2429/18270?expand=metadata"@en ;
skos:note "PHOTONIC BANDSTRUCTURE IN SEMICONDUCTOR SLAB WAVEGUIDES WITH STRONG TWO-DIMENSIONAL T E X T U R E by Vighen Pacradouni B . S c , Mcg i l l University, 1989 B .Eng . , M c g i l l University, 1991 M . S c , University of Br i t i sh Columbia, 1993 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F Doctor of Philosophy in T H E F A C U L T Y O F G R A D U A T E S T U D I E S (Physics) T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A A p r i l 2006 © Vighen Pacradouni, 2006 Abstract Fabrication techniques were developed and used to realize sub-100 nm-thick semiconduc-tor 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 wi th energies and linewidths that disperse as the in-plane mo-mentum (angle of incidence) of the exciting radiation is varied in the near-infrared. A numerical solution of Maxwell 's equations was developed to interpret these reflectivity spectra in terms of a \"photonic bandstructure\" for electromagnetic modes localized in 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 opti-cal 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. i i 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 Real Space Approach 20 2.2.1 Scattering from a periodically textured layer 20 2.2.2 Discretization and Integration of Maxwell 's Equations in Real Space 27 2.2.3 Eigenmodes on a Discrete Mesh 34 2.2.4 Significance of Using Pendry Modes 39 2.2.5 Computat ion 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 i i i 2.3.3 Bound Modes of Untextured Slab 52 2.3.4 Slab Waveguide with I D Grat ing 55 2.3.5 Waveguide with 2D weak 2nd order grating 62 2.3.6 2D Strong Texture 69 2.4 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 wi th Green's Function Based Model 87 2.6 Significance of Modification to Pendry's Method 92 2.7 Summary 96 3 Fabrication 98 3.1 Introduction and Overview 98 3.2 Processing Techniques and Equipment 102 3.2.1 Electron Beam Lithography 102 3.2.2 Etching 106 3.2.3 Oxidation I l l 3.2.4 Optical Microscopy 113 3.2.5 Atomic Force Microscope 114 3.3 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 Oxidat ion 120 3.4 Fabrication Results 121 iv 4 Optical Experiments 127 4.1 Introduction 127 4.2 Experimental Apparatus 128 4.2.1 Fourier Transform Spectrometer 131 4.3 Alignment 132 4.4 Angular 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 F i t t ing wi th Mode l 143 5.6 Extract ion of Mode Frequency and Linewidth from Spectra 145 5.6.1 Extract ion of Frequency and Linewidth from Experimental Spectra 145 5.6.2 Extract ion of Mode Parameters from Mode l Spectra 146 5.7 Bandstructure 149 5.8 Linewidths 151 5.8.1 Linewidth of Lowest Energy Band 151 5.8.2 Linewidths of 4th and 5th Bands 154 5.9 Effect of F i l l ing Fraction 155 6 Discussion 158 6.1 Introduction 158 6.2 Resonant Features in 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 ing Fraction 166 6.6 Gaps Away 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 v i List of Tables 3.1 Nominal and measured thicknesses of layers in sample A S U 721 116 3.2 Etch 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 126 5.1 Model , Nominal , and Measured sample structure parameters for grating 22 on sample A S U 721 145 v i i 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 2 1.2 Photonic bandstructure arising from 3D face-centred-cubic photonic lat-tice comprised of non-spherical \"atoms\" and dielectric contrast of — = 13.0 [1] 3 1.3 2D photonic crystal \"waveguide\" wi th 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 cylin-drical airholes separated by 120° into a block of material wi th 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 wi th non-spherical \"atoms\" 7 1.5 Schematic diagram of an asymmetric 3-layer slab waveguide wi th 2D pho-tonic crystal embedded in core 9 1.6 Schematic diagram of an asymmetric 3-layer slab waveguide 10 1.7 Schematic dispersion diagram of the modes of an asymmetric 3-layer slab waveguide. The 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). The dotted line represents the light line for the substrate, i.e., bottom semi-infinite layer 10 1.8 Schematic diagram of 2D photonic crystal wi th defect embedded in slab waveguide 14 v i i i 2.1 Schematic diagram of a two-dimensionaUy periodically textured slab struc-ture 21 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. The dielectric constant is sampled on the lattice indicated by the black dots. The fields are calculated on the sublattice indicated by the crossed squares. The planes shown span a unit cell of the periodicity in the in-plane direction. The light and dark grey shading of the planes indicates whether the E or H ' fields are computed there. The white hole in the center schematically depicts dielectric texture comprised of a circular airhole 33 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. The electric or magnetic fields are sampled only on those points wi th the crossed squares 43 2.4 Scheme of reciprocal lattice vectors corresponding to in-plane wavevector of launched modes 44 2.5 2D grating embedded in an arbitrary multilayer dielectric stack 49 2.6 Component scattering matrices for calculation of overall reflection (scat-tering) matrix from grating embedded in an arbitrary multilayer dielectric stack 50 2.7 Waveguide consisting of 120 nm-thick slab of material wi th dielectric con-stant e' = 12.25 52 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 ix 2.9 Plot 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 nm slab of material wi th 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 in the calculation of the spectra shown in Figure 2.10 below. The dots at the intersection of the vertical line wi th the dispersion curves indicate the appearance of a resonant feature in the spectra 54 2.10 Specular reflectivity spectrum for evanescent field incident on 120 nm slab of material wi th e' = 12.25, wi th 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 with 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 in the presence of the grating 56 2.12 Slab waveguide wi th \"weak\" I D texture: the structure consists of a 120 nm-thick slab of e' — 12.25 material, clad above and below by air, wi th a 10 nm deep I D square-toothed grating having a pitch A = 500 nm and an air filling fraction of 25% etched into the surface 56 2.13 Calculated specular reflectivity spectrum for evanescent light incident wi th in-plane wavevector /3 = (PQ/2)X = (TT/A)X. The solid curve is for s polarized light, the dashed curve is for p polarized light. The correspond-ing waveguide structure is depicted in Figure 2.12 57 x 2.14 (a) Calculated specular reflectivity spectrum for evanescent light inci-dent wi th in-plane wavevector, (3 = I.OIPGX, i.e., slightly detuned from the boundary between the 2nd and 3rd Bri l louin zones, (b) Calculated specular reflectivity spectrum for light incident wi th in-plane wavevec-tor, (3 = O.OIPGX, i.e., slightly detuned from the Bri l louin zone-centre, (c) Calculated 1st order diffraction coefficient, i.e., coefficient for scat-tering 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. The waveguide structure under study is depicted in Figure 2.12 2.15 (a) Calculated specular reflectivity spectrum for evanescent light incident wi th in-plane wavevector, (3 = PGX, at the edge of the 1st Br i l louin zone, (b) Calculated specular reflectivity spectrum for radiation incident wi th zero in-plane wavevector, i.e., at the Br i l louin zone-centre, (c) Calculated 1st order diffraction coefficient, i.e., coefficient for scattering from a field component at (3 = 0.01PGX to a field component at (3 = I.OIPQX. The solid curves are for 5 polarized light; the dashed curves are forp polarized light. The waveguide structure under study is depicted in Figure 2.12. . . 2.16 Calculated near-normal incidence specular reflectivity spectrum for light incident with in-plane wavevector, j3 = O.Ol/fcz, for s polarized light. The waveguide structure is identical to the one specified in Figure 2.12 except that the grating has an air filling fraction of 75% rather than 25% 2.17 Slab waveguide wi th \"strong\" I D texture: the structure consists of a 120 nm thick slab of e' = 12.25 material, clad above and below by air, wi th a I D square-toothed grating that completely penetrates the slab. The grating has a pitch, A = 500 nm, and an air filling fraction of 25% 2.18 Calculated near-normal incidence specular reflectivity spectrum for s po-larized light incident wi th in-plane wavevector, /3 = 0.01/?GX. The waveguide structure, depicted in Figure 2.17, consists of a 120 nm 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 nm of e' = 12.25 material wi th air above and below, having a square lattice of cylindrical airholes, 25 nm 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 64 2.20 Schematic in-plane bandstructure of resonant modes of waveguide with a square grating shown in 1st 2D Bri l louin zone 64 2.21 Symmetry directions for the square lattice 65 2.22 Calculated specular reflectivity spectrum for light incident on a 125 nm slab waveguide of e' = 12.25 material wi th air above and below, and a square lattice of cylindrical airholes, 25 nm 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 wi th in-plane wavevector fixed at P/PG = 0.01 in the T — X direction, which is defined in Figure 2.21 66 2.23 Expanded view of the spectrum depicted in Figure 2.22: (a) lowest fre-quency cluster (b) intermediate frequency cluster (c) high frequency clus-ter 67 2.24 Calculated specular reflectivity spectrum from same structure as in F ig -ure 2.23. The spectra were calculated for light incident with zero in-plane wavevector, P/PG = 0.00, corresponding to normal incidence 68 x i i 2.25 Calculated specular reflectivity spectrum from same structure as in F ig -ure 2.23. The spectra were calculated for light incident wi th in-plane wavevector fixed at P/PG = 0.01 in the [x + y] direction (i.e., along the r — M symmetry direction) 2.26 Calculated specular reflectivity spectrum for light incident on a 125 nm slab waveguide of e' = 12.25 material with air above and below, and a square lattice of cylindrical airholes, 25 nm deep embedded at its centre. The spectra were calculated for light incident wi th in-plane wavevector fixed at P/PG — 0.001 in 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 thin asymmetric slab waveguide: the structure con-sists 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 wi th 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 with hole radii given by r / A = 0.5 2.28 Calculated near-normal incidence specular reflectivity spectrum for light incident wi th in-plane wavevector, P = 0.03/?G, along the T — X direction for the structure depicted in Figure 2.27 wi th r / A = .2821. The 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 with in-plane wavevector, P = 0.03/?G, along the T — M direction for the structure depicted in Figure 2.27 with r / A = .2821. The solid line represents the s polarized spectrum, the dashed line depicts the p polarized spectrum x i i i 2.30 Calculated specular reflectivity spectra for light incident on the structure depicted in Figure 2.27 with r / A = .2821. (a) Normal incidence specular reflectivity, i.e.., light incident wi th (3 = 0. (b) Reflectivity for s polarized evanescent light incident wi th (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 in each of the x and y directions included in 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 (3-mc = 2ir/A. The structure used for the calculation consisted of a 120 nm slab of material wi th e' = 12.25 clad above and below by air (e' = 1.0). The different curves (lines) are for increasing numbers, (specifically: 12, 22, 32, 42, 62, and 122), from top to bottom on the graph, of mesh points in the z direction 2.32 Slab waveguide wi th \"strong\" I D texture: the structure consists of a slab of thickness d of e' = 12.25 material, clad above and below by air, wi th a I D square-toothed grating having a pitch, A = 500 nm, and an air filling fraction of 25% that completely penetrates the slab 2.33 Mode frequency (a) and width (b) vs. the \"number of elements\" (see text) used in F D R S calculation for the structure depicted in Figure 2.32 wi th d = 80 nm. The results are for the second-order lossy mode at the zone-centre. Plots wi th round markers and solid lines depict results wi th anti-aliasing spectral filtering of the e' — / / profile implemented; plots wi th square markers and dashed lines depict results without filtering of the e' — / / profile 2.34 (a) Mode frequency and (b) width, vs. square of the x — y mesh size normalized by the unit cell size (or grating pitch) used in the F D R S calculation for the structure depicted in Figure 2.32 wi th 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 wi th anti-aliasing spectral filtering of the e' — u! profile implemented. The dashed lines represent the best fit line obtained from a linear least squares fit. In (b) the rightmost three points were ignored in the fitting 2.35 Strongly 2D textured thin asymmetric slab waveguide: the structure con-sists of a 80 nm-thick slab of e' = 12.25 material which forms the guiding core, clad above by air and below by material wi th e' = 2.25. The core waveguide layer is penetrated by a 2D square array of holes on a pitch, A = 500 nm, and wi th hole radii given by r / A = 0.2821 2.36 Mode frequency (a) and width (b) vs. square of the x — y mesh size normalized by the unit cell size (or grating pitch) used in the F D R S calculation for the structure depicted in 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 with anti-aliasing spectral filtering of the e' — / / profile implemented. The 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 in the calculation xv 2.37 Near-normal incidence specular reflectivity spectra for light incident wi th in-plane wavevector 3 = 0.02/?G along the Y — X direction calculated wi th the F D R S model (lower plot) and the G F model (upper plot). The para-meters used for the F D R S model are as described in section 2.4.2 using 17 elements. The 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. The structure being modelled is the same as the one depicted in Figure 2.27. The resonance widths and shape are in excellent agreement. The mode frequencies agree to wi thin 1.5% 90 2.38 Location of the upper and lower edge of the 2nd order gap in 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 wi th 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%. The dashed lines are for the calculation wi th the G F model, the solid lines wi th the F D R S model which was verified to be converged to within 1% for the upper band and 0.2% for the lower band (using 17 \"elements\" in the calculations). The G F calculation included the three reciprocal lattice vectors at 0 and ±PG& 91 x v i 39 Calculated normal incidence specular reflectivity spectra for light inci-dent wi th zero in-plane wavevector. The calculation is performed (a) using Pendry's original approximations for k and (b) the approximations for k implemented in this work. The solid and dashed lines, which over-lap 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 wi th a square array of 300 nm diameter cylin-drical holes filled wi th a material having a dielectric constant of 2.25, all on a substrate with a dielectric constant of 2.25. The pitch of the grating is 500 nm .40 Calculated near-normal incidence specular reflectivity spectrum for light incident wi th in-plane wavevector, f3 — 0.01/?G, along the F — X direction. The 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. The light is incident on a film of material having a dielectric constant of 12.25 textured with a square array of 300 nm diam-eter cylindrical holes filled wi th a material having a dielectric constant of 2.25, al l on a substrate with a dielectric constant of 2.25. The pitch of the grating is 500 nm .1 Scanning electron microscope (SEM) image of freestanding a i r / G a A s / a i r slab waveguide patterned wi th 2D square lattice of airholes .2 High magnification (SEM) image of the same structure as in Figure 3.1 . .3 Schematic diagram of oxide-based high-index contrast waveguide with strong 2D texture xv i i 3.4 (a) Fie ld emission scanning electron microscope column, corresponding to the column of the Hitachi S-4100 S E M : (1) Electron source (2) Beam paths showing the effect of successive apertures in 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 Bz varies in the lens region; (c) Detai l showing geometry of beam aperture angle a at the sample 105 3.5 Schematic diagram of E C R plasma etcher 108 3.6 Oxidat ion 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 mi-croscope. The size of the grating is 90 /mi x 90 / /m. Some processing was performed on the image to enhance the contrast between the oxi-dized region, which extends out around the grating and is slightly darker compared to the unoxidized parts of the sample 122 3.10 4 iim x 4 pm A F M image of grating 22 125 3.11 Section through diameter of holes in image in Figure 3.10. The arrows indicate the leading and trail ing edges of the depression in the profile, which are used to obtain a measurement of the diameter of the holes at the surface 126 4.1 Schematic diagram of scattering geometry used in specular reflectivity measurement. The 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 xv i i i 5.1 Experimental (a) and calculated (b) specular reflectivity spectra on grat-ing 22 at incidence angle of 9 = 5° and 0 = 0° for p-polarization of reflected light 138 5.2 Experimental (a) and calculated (b) specular reflectivity spectra on grat-ing 22 at incidence angle of 9 = 5° and 0 = 0° for s-polarization of reflected light 139 5.3 Experimental (a) and calculated (b) specular reflectivity spectra on grat-ing 22 at incidence angle of 9 = 5° and 4> = 45° for p-polarization of reflected light 140 5.4 Experimental (a) and calculated (b) specular reflectivity spectra on grat-ing 22 at incidence angle of 9 = 5° and = 45° for s-polarization of reflected light 141 5.5 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 142 5.6 F i t to specular reflectivity spectrum obtained from grating 22 obtained wi th 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 wi th an A i r y function. The ini t ial fit to the Fabry-Perot fringes was performed over a wider range of energies (6000 c m - 1 to 11,500 c m - 1 ) not shown here. The fit using a Fano lineshape and A i r y function was performed using data points in the range 6000 c m - 1 - 9000 c m - 1 147 x ix 5.7 F i t to spectra generated by the G F model, (a) Specular reflectivity spec-t rum for reference (b) 1st order scattering to —QGX field component (c) 1st order scattering +3QX field component. In (b) and (c) the symbols correspond to the model values, and the smooth lines are the result of a fit wi th a Fano lineshape function. The spectra are for 10° angle of incidence along the T — X direction 148 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. The inset schematically depicts the bandstructure for the case of weak texture 150 5.9 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 in 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 in 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 in 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. 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 in by hand to serve as a guide to the eye 157 6.1 Schematic dispersion of resonant states attached to the core of a slab waveguide with square photonic lattice 160 xx 6.2 Dominant Fourier field components of eigenmodes in lowest four bands along the T — X (9 = 0) direction depicted in Figure 5.8: the solid black arrows represent the propagation vectors associated wi th 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 6.3 Dominant Fourier field components of eigenmodes in the four bands along the V — M direction depicted in Figure 5.8: the solid arrows represent the propagation vectors associated with 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 6.4 Calculated specular reflectivity spectra for s polarized (solid lines) and p polarized (dotted lines) for 9 = 2°, = 0 wi th (a) r / A = 0.29 and (b) r / A = 0.33. The other parameters in the model are as the fitted results for grating 22. The curves have been shifted vertically for clarity 167 6.5 Schematic depiction of the in 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 wi th (a) 25% (b) 50% (c) 75% air filling fraction. In (b) the lossy and lossless modes both modes have the same fraction of their total field energy (area under the curve) in the high-index region. The dielectric profile is indicated by the dashed line 169 x x i 6.6 Dominant Fourier field components of eigenmodes in the 4th and 5th lowest energy bands along T — X direction in Figure 5.8 in the vicinity of their anti-crossing: the solid arrows represent the propagation vectors associated wi th 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 6.7 Scattering amplitudes calculated using the F D R S Code for s polarized light incident at /3inc « 0.25/?G£ corresponding to 9 = 25° along the r — X direction, where the anti-crossing occurs. The various plots depict the scattering amplitudes to s polarized light wi th in-plane wavevector: (a) Pine (b) Pinc-Pcx (c) (3inc±(3Gy (d) i3inc + pGx (e) f3inc + pG[-x±y] (f) f3inc + PG[x ±y}. The structure is the same as that for grating 22 with the parameters as obtained from fitting wi th the G F code 173 A . l 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. The electric or magnetic fields are sampled only on the points wi th squares. A i and A 2 represent the primitive lattice vectors of the photonic crystal 187 A.2 Scheme of reciprocal lattice vectors corresponding to in-plane wavevector of launched modes for photonic crystal with oblique 2D lattice 188 x x i i Acknowledgements I would like to thank my contemporaries in the Photonic Nanostructures Group at U B C , postdoctoral researchers, Manoj Kanskar, Paul Paddon, and Jennifer Watson, graduate students, A lex Busch, Jody Mandeville, Francois Sfigakis, Murray McCutcheon, and A l l a n Cowan, as well as U B C Advanced Materials and Process Engineering Labora-tory research engineers, J i m MacKenzie 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 Tom Tiedje for growing the samples used throughout the course of the research described herein. I would like to gratefully acknowledge the Natural Sciences and Engineering Research Counci 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, Mark Halpern, Lome Whitehead, Jochen Meyer, David 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 McCutcheon 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. xx i i i Dedication To my loving parents and late grandmother. xxiv Chapter 1 Introduction Photonic crystals are structures in which the dielectric constant is given a periodic spatial variation, or \"texture\", as, for example, depicted schematically in Figure 1.1. For electromagnetic radiation wi th 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 ( ID) 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 John (indepen-dently) [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 in a periodic dielectric into the formalism of existing condensed matter band theory for electrons in a periodic atomic potential. Here, the periodic dielectric plays the role of a periodic potential that couples the freely propagating electromagnetic eigenmodes into photonic Bloch states that can be used to describe the E M excitations of the photonic crystal in analogy to the electronic Bloch states formed in \"normal\", i.e., atomic, crystals. Figure 1.2 depicts the photonic bandstructure in a face-centred-cubic photonic crystal [1]. A t the Br i l louin zone boundaries, the superposition of forward propagating and Bragg scattered radiation results in standing wave eigenstates with zero group velocity in the direction perpendicular to the zone boundaries. For a definition of Br i l louin zones, Bloch states, and their importance in solving problems involving potential functions wi th 1 e(r)= e( r + R) Figure 1.1: Schematic diagram of 3D photonic crystal concept. R is any one of the lattice vectors that define the crystal symmetry. 2 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]. The periodic dielectric generally alters the dispersion and eigenstates everywhere in the Br i l louin zone. The 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 within which there are no classical propagating electromagnetic modes with wavevectors along the given symmetry direction. Furthermore, if all of the gaps in 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. This means that no classical propagating radiation with a frequency within the bandgap is allowed inside the crystal, and consequently, that externally impinging radiation wi l l decay exponentially wi th distance into the crystal. The existence of a complete gap requires a crystal structure wi th a high degree of symmetry, as well as a large dielectric contrast in the texturing. The symmetry causes the energies of the modes not to vary greatly in 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 in al l of the different directions to overlap in some frequency band. A material with a full photonic bandgap is appealing as a platform for both 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 within the surrounding infinite photonic lattice for some range of frequencies, the light trapped at the defect cannot propagate away. The 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 wi th the electronic resonance. The 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 meV, 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 in such systems, where there are effectively no continuum electromagnetic modes to compete wi th the lasing mode below threshold. Whi le 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 # ® m • • • m r • • • • © c m w • • • • • m m m • mm o (ft \\ ^ • w •) o © ® m m m m m m m m o © o m o © o o o o o J o 'j o © O 1 \" ~) o © o o o © o o o o o © Figure 1.3: 2D photonic crystal \"waveguide\" with tight bend. a frequency within 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 in Figure 1.3 for the case of a 2D photonic crystal. Such lossless guiding around tight bends cannot be achieved with conventional index-guided waveguides, as the crit ical angle of total internal reflection is exceeded if the radius of curvature becomes too small. The ability of photonic crystal waveguides to guide light nimbly around corners on a photonic chip could be an essential tool in the large scale integration of photonic circuits. Even in the absence of a complete photonic bandgap, periodic dielectric texture con-fers the ability to drastically modify the dispersion of photons in a medium, beyond and separate from that attainable using composition control in untextured bulk material. This has already found application in dispersion compensation elements and compact delay components in optical fibre communication systems [10]. Furthermore, enhance-ment of gain in 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]. The properties of 2D and 3D photonic crystals have been studied theoretically for pure dielectrics [1, 2, 13, 14, 15, 16, 17, 18]. They have also been verified experimen-tally in 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 in Figure 1.2 actually represents a lattice formed by machining three sets of cylindrical airholes separated by 120° into a block of material wi th a dielec-tric constant e' = e/e 0 = 13.0 (where e0 is the permitt ivity of free space) at a 35° angle on a 2D periodic grid (see Figure 1.4). The resulting interconnecting network of voids forms a face-centred-cubic ( F C C ) lattice wi th non-spherical \"atoms\". Similar structures have found application in the microwave region as high-Q niters and broadband antennae platforms [19]. Semiconductors, in many ways, make ideal hosts for photonic crystals. The 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 appro-priate lattice geometries [1]. Furthermore, artificial semiconductor quantum wells, wires and dots can be formed in a number of ways to achieve tunable electronic properties (resonant frequencies, densities of states) throughout the near-infrared part of the spec-trum. In fact, it was recognized from the outset that incorporating photonic crystals into semiconductor hosts, rich in electronic resonances and ubiquitous in optical com-munication 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 nm required to operate in the near-infrared part of the spectrum has proven to be extremely challenging to realize in three dimensions. Several strategies have been suggested and attempted to date. Cheng and Scherer [20] have used an ion beam to mi l l three sets of cylinders, as in 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 wi th 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 with non-spherical \"atoms\". 7 depicted in 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 did sug-gest that these structures exhibit an optical gap near normal incidence consistent with model calculations. However, the limitations of ion mill ing preclude making the struc-tures 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]. The textured surface is then turned upside down and fused to a previously-laid 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. Al though this technique addresses the challenge of achieving high-contrast dielec-tric 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 wi th lattice constants from ~ 250 - 500 nm followed by infiltration of the resulting voids wi th a high-index material. However, this fabrication method produces crystals wi th lattice imperfections that result in poor stop-bands [22]. In summary, despite considerable effort, a good-quality bulk 3D photonic crystal useful in the optical or near-infrared spectrum is st i l l 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 proper-ties of photonic crystals simply by etching a 2D array of holes in a planar semiconductor waveguide. Such a planar waveguide-based photonic crystal is depicted in Figure 1.5. A n additional advantage to this approach is that it is compatible wi th the vast major-ity of existing opto-electronics components like semiconductor lasers, modulators, and couplers. The tradeoff is that it is impossible to perfectly localize light in 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 with 2D pho-tonic crystal embedded in core. This is because there are always solutions to Maxwell 's equations corresponding to radi-ation propagating in 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 wi th the intensity profile of one of its guided modes. The electromagnetic excita-tions 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 in 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 in bulk \\ / e / e o material with an index of refraction (=-y/e/e0) equal to that of their respective layers. The 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 in the absence of external sources. The variation of the fields in 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 in the g 0 (air) 2 Figure 1.6: Schematic diagram of an asymmetric 3-layer slab waveguide. C O Figure 1.7: Schematic dispersion diagram of the modes of an asymmetric 3-layer slab waveguide. The 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). The dotted line represents the light line for the substrate, i.e., bottom semi-infinite layer. 10 cladding layers decay exponentially away from the core such that no energy is carried away in 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 with fields that decay away from the core in 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 (TE) and transverse magnetic ( T M ) , based on whether they have their electric (TE) or magnetic ( T M ) field completely transverse to their direction of propagation, which lies in the plane of the slab. In the case considered here, there is only one T E polarized guided mode in the region of to — (3 space depicted in Figure 1.7. The details of planar waveguide theory can be found in reference [23]. Slab waveguides, themselves, can be classified into two types: \"soft\" guides and \"hard\" guides. \"Soft\" guides are comprised of a thicker (~ 1 fxm for near-infrared wavelengths) core, and cladding layers having a relatively small difference in refractive index from that of the core ( A n PS 0.1). \"Hard\" guides or membranes are comprised of a thin core (~ 0.1 ^ m for near-infrared wavelengths) wi th cladding layers having a larger refractive index difference ( A n ~ 2) from that of the core. \"Soft\" guides are found in the vast majority of opto-electronic applications, in part because they can be realized as semiconductor heterostructures which can be fabricated relatively easily using epitaxial growth techniques employing composition control in 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 thin guiding core to be made freestanding, and therefore, to have air for its cladding layers, or to be clad on one or both sides by a thick oxide layer (n ~ 1.5). From the point of view of embedding waveguides wi th 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 with dielectric con-stant 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 in \"soft\" guides, the holes are made as deep as possible but placed in a cladding layer. This is done at the expense of scattering strength. Neverthe-less, such \"soft\" guide based crystals are appealing due to their compatibility with the overwhelming majority of opto-electronic devices. A n early success in 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 in a G a A s / A l G a A s / I n G A s heterostructure. Photoluminescence excited in the vicinity of the crystal exhibited preferential propagation along certain of its symmetry directions. More experimental work in this area has been done by Labilloy et al. [25]. They have obtained evidence of a photonic bandgap and mapped dispersion by exciting broad-band photoluminescence in quantum dots embedded within a 2D 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 in the transmission spectrum to determine the extent of the photonic band gap along the symmetry direction of the crystal. They then use oscillations in the spectra outside the gap, arising from the varia-t ion 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 thin waveguide core. A n additional advantage is that they can sti l l 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 wi th 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 l imit of waveguide-based photonic crystals. The 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 in bulk dielectrics. This can be seen in the weakly textured case by applying a perturbative method [26, 27]. The dispersion of the bound modes is modified and gaps are opened up at the boundaries of the 2D Bri l louin 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 sti l l be introduced to create guides or cavities, but the photon confinement is achieved due to the photonic crystal in the in-plane directions, and due to the guiding properties of the slab structure in the transverse direction. Figure 1.8 schematically exemplifies a two-dimensionally textured waveguide wi th a point defect. However, in any geometry that lacks translational invariance in the transverse direc-t ion there wi 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 non-resonant states at those frequencies. This 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). From a practical 13 Figure 1.8: Schematic diagram of 2D photonic crystal wi th defect embedded in slab waveguide. point of view this may be framed in 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 in the vicinity of the pseudo-gap or the discrete defect states within the gap. Painter et al. [28] were able to demonstrate an optically pumped defect-mode laser in 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 in the centre, to form a linear, I D photonic crystal wi th defect for optical modes confined to the silicon channel. A localized photon was formed at the defect wi th an effective volume of < 0.1 pm3. The same group has also demonstrated low losses through tight bends using photonic crystals embedded in 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, in the sense that for any /? there are no modes wi th a neighbouring ui. This is also the case for the modes of a 3D photonic lattice as described in Figure 1.2. The addition of 2D texture to the slab waveguide causes some of the bound modes to mix wi th radiative modes in the continuum. Whi le some renormalized modes remain purely bound, the bandstructure for discrete modes now describes localized states which are generally only quasi-bound, or leaky. This leakiness can be described by an associated lifetime that has no analog in the case of pure 3D (or pure 2D) photonic crystals. It should be mentioned that such leaky modes also exist in the case of an untextured slab waveguide and are also associated wi th lifetimes. In this case, however, they occur in the continuum regime. Moreover, it should be emphasized that, in both cases, the lifetimes are not associated wi th an absorptive process. That is, the lifetimes do not describe a loss of electromagnetic energy, but rather the diminishment of its localization in the waveguide core. The points identified above provide the main motivation for this thesis, which thor-oughly characterizes the dispersion, polarization, and lifetime properties of discrete, slab modes of planar waveguides with strong 2D dielectric texture. From a theoretical point of view, the bound and quasi-bound modes in I D textured waveguides have been studied extensively with coupled mode theory ( C M T ) [32, 33, 34], which is a perturbative approach. Most of this work has been done in the context of Distributed Feedback Lasers ( D F B ' s ) , which incorporate weak I D texture in a slab waveguide geometry. This weak texture in D F B ' s results from particular conditions: either the index-contrast wi thin the grating is small, or it is placed away from the waveguide core, or both. Coupled Mode 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 2D texturing of a general planar waveguide, however, the true eigen-modes 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 vec-tor coupled-mode formalism has been presented by Paddon and Young [27]. Al though this formalism provides a transparent understanding of the non-trivial vector coupling which occurs in 2D waveguides and insight into the polarization properties that arise from symmetry, it remains l imited 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 prop-erties 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] in normal-incidence specular reflectivity spectra taken on a strongly 2D-textured waveguide [39]. This occurred when the energy of the incident light matched that of the photonic eigen-states at zone-centre. The experiments were performed on a \"hard\" waveguide-based photonic crystal [39], which was fabricated by etching a periodic 2D array of holes com-pletely through a 240 nm 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 wi th a finite-element model that was developed principally by the author. The 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 both difficult to fabri-cate and quite frail. As a result, a more robust structure and fabrication technique for \"hard\" waveguide-based photonic crystals was developed by the author in conjunction 16 with F . Sfigakis, a Master's student in the group. In addition, the use of a new and much-improved optical scattering setup, developed by W . J . Mandeville, a doctoral stu-dent in 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 (GF) based numerical model specific to the thin \"hard\" waveguide geometry, was developed by A . Cowan (another fellow doctoral student) and J .F . Young [40]. This much faster code allowed manual fit-t ing of the model-generated bandstructures (dispersions and lifetimes) to those obtained from experiment. This entire body of work, which provides for a clear interpretation of optical scattering data in terms of the dispersion, lifetime, and polarization properties of 2D photonic crystals embedded in thin, \"hard\" waveguides, is the subject of this thesis. The principal elements of this work include: (1) a numerical model for calculating specular reflectivity and scattering from an infinite 2D photonic crystal embedded in 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 sym-metry of the problem; (2) a fabrication technique for making structurally robust and robustly guiding thin semiconductor slab waveguides based on a wet oxidation tech-nique; (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 in the Br i l louin zone; and (4) a comparison of these measurements with simulations based on a computationally effi-cient Green's function-based (GF) model. This G F model is validated using the slower, \"exact\" model presented here, for the case of th in \"hard\" waveguides studied in this work. Resulting from this body of work is the realization that, even in this extreme non-perturbative l imit of strongly textured thin waveguides, the photonic eigenstates in the vicinity of the second order T E gap can be understood qualitatively, and in some cases semi-quantitatively, as renormalized T E polarized slab modes. The organization of the remainder of this thesis is as follows. Chapter 2 presents 17 the finite difference model developed by the author. Results from the model are used to introduce certain concepts related to bandstructure arising in the slab waveguide geom-etry as well as the approach used in 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 exper-imental 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, us-ing simple symmetry and kinematical arguments . Chapter 7 presents conclusions and some direction for future work. 18 Chapter 2 Modelling 2.1 Introduction Two different techniques were used to model the linear optical scattering properties of thin porous slab waveguides in this work. The 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. However, this algorithm did not converge well in our geometry. The specular reflectivity spectra did not contain the correct number of resonant features expected from symmetry considerations, nor were the polarization selection rules obeyed. The problem was traced to the use of forward differences (rather than symmetric differences) in the integration scheme, which introduced an artificial chirality into the calculation. Consequently, a new integration scheme that converges well in our waveguide geom-etry was devised by the author in collaboration wi th P. Paddon. This new scheme was then implemented by the author. Additionally, the model was extended (by the author) to allow for efficient modelling of buried gratings. This new model is presented in detail in this chapter. The second model used in this work was developed by J .F Young and A . R Cowan. It is referred to as the Green's Function (GF) model. This model is more computationally efficient than the F D R S approach, wi th its l imitat ion being that it is only valid for 19 a textured layer that is th in 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 in Chapter 4. The 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 in 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 in section 2.5. The 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. Addit ionally, 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 Description of Finite Difference Real Space Approach 2.2.1 Scattering from a periodically textured layer We begin by considering the structure depicted in Figure 2.1. The dielectric function, e(r) in region II is periodic in the plane of the slab: where p is a vector in the x — y plane and Rj is a 2D real-space lattice vector also in the x — y plane. For simplicity, we assume that each material in the structure is linear, isotropic, and homogenous, so that it can be described in bulk form by a scalar dielectric e(p; z) = e(p + R*; z) 20 Figure 2.1: Schematic diagram of a two-dimensionally periodically textured slab struc-ture. 21 constant. Therefore, the dielectric function may be written as a Fourier series: e'(p; z) 4 ^ = 4 , » exp(tG,, r a • p) (2.2) where the G ; > m are the reciprocal lattice vectors of the dielectric modulation (\"photonic 27T 27T crystal\") , e.g., G / > m = l~^x + m~^~$> ^ o r a ^ S ( l u a r e lattice having a pitch A , where I and m are integers. The harmonic solutions to Maxwell 's equations in the homogeneous regions, I and III, for a given frequency can be expanded in 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 in regions I and III) where al l of the fields have a spatial-temporal variation of the form exp (ik • r) exp (—iut) subject to the dispersion relation: e'l/m^ = Ik| 2 = kl + hi + k> (2.3) and where e'i/in — 6 i/in/eo (2.4) represents the dielectric constant in region I or III, which does not vary spatially, and c is the speed of light in vacuum. There are, in general, two degenerate, orthogonal polarizations, labelled s and p according to whether the electric or magnetic field, respectively, lies completely in 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) = , n X *E0 exp(zk • r - itot) (2.5) | n x k | H W ( r ) = k ^ ( f i r k ) — exp(ik • r - iut) = — k x E ^ ( r ) (2.6) | n x k | phi pjuj and: H ( p ) ( r ) = , n X *H0 exp(ik • r - itot) (2.7) n x k 22 E ( p ) = k x ( n x k ) g o . r _ i u t ) = l k x H ( P ) ( r ) ( 2 8 ) | n x k | eto ecu where e and u are the permitt ivity and permeability of the medium, respectively, and E0 and HQ are normalization constants bearing units of electric and magnetic field respectively. For the special case of k parallel to n (i.e., normal incidence), the fields of the polarized plane waves lie entirely in the x — y plane, and hence the two independent polarizations may be defined wi th respect to any two orthogonal in-plane directions. For convenience, & and y are chosen, so our basis and the modes in this situation are defined as x or y polarized according to: E ( x ) (r) = E0 exp(ik • r - iut)x (2.9) H ^ f r ) = — E0ex.p(ik-T-iujt)y = — k x E ^ ( r ) (2.10) and H ( y ) ( r ) = H0exp(ik-r-icot)x (2.11) E ^ ( r ) = - ^ # 0 e x p ( z k • r - iut)y = - ^ k x H ( y ) ( r ) (2.12) The 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 + kzz, 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 in region I as well as a discrete set of plane waves travelling forward in region III. The in-plane wavevectors of the discrete set of scattered plane waves in both regions are given by (3 + G / > m . Physically, this arises from the fact that the periodic dielectric texture modulates the incident field, generating polarization with the same periodicity as the dielectric texture. This polarization wi l l , in 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 in terms of the following discrete set of transverse polarized plane waves on either side of the textured structure: (2.13) H } ^ ' J / / J i ) ( r ) = -[(/3 + G I > m ) + wffMz] x E<\"*>(r) (2.14) and (2.15) E g f = + G«.«) + ^ £ 7 / / / / ) 2 ] x H ^ ) ( r ) (2.16) where fi7/777) = ± [ z ' t / n ^ / c 2 - |/3 + G , , r a | 2 ] 1 / 2 (2-17) The quantity, wf^^111^, represents the ^-component of the wavevector for the par-ticular plane wave, and equation 2.17 was obtained from the dispersion relation, equa-t ion 2.3. When 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. When w^~'^Iiri is imaginary, then we have an evanescent 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 ight- (+) or left-going (—) and order them with respect to polarization and in-plane wave-vector in 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 in region III, which we denote by (ax)i for i = 1, 2, 3,.. . (Transmitted), where the subscript i, now labels both the polarization and reciprocal lattice vector. We 24 also wish to compute the amplitudes of the scattered left-going plane waves in region I which we denote by (OR )J (Reflected). The scattering problem can thus be written in matrix form as: a T = T + + a i n c (2.18) aR = T - + a i n c (2.19) where a j n c contains the amplitudes of the right-going (+) plane wave modes in region I (incident). Note that for the experiment under consideration this last column matrix is al l 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, in 2D, included in the calculation and the factor 2 arises from the two independent polarizations (s and p). The matrix element T^+, therefore, represents the scattering coefficient for plane wave i travelling toward the right (+) in region I to plane wave j leaving the textured layer in region III and travelling toward the right (+). The matrix element T^+, rep-resents the scattering coefficient from plane wave i travelling toward the right (+) in region I and reflected/diffracted back to left-going (—) plane wave j in region I. The problem, then, is to calculate the matrices T++ and T~+. This is accomplished in the present model by integrating Maxwell 's equations through the textured layer. However, in order to do so, one must perform the inverse scattering calculation using the following mathematical manipulation. a i n c = ( T + + ) - 1 a r (2.20) a * = T - + ( T + + ) - 1 a r (2.21) Now, by specifying a single plane wave of unit amplitude travelling toward the right in region III, integrating backwards, and projecting onto the forward and backward travelling plane wave basis in region I, we obtain a column of each of the matrices 25 (T\"1\"\"1\") -1 and T~+(T++)~1, respectively. The matrices may thus be constructed in their entirety by launching each of the basis plane waves i n region I. The matrices T + + and T~+ 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 in region I, (br)u and diffracted/reflected right-going modes in region III, (&R)J, we can define: b T = T ~ b m c (2.22) b R = T+~bmc (2.23) where b ; n c contains the amplitudes of the left-going (—) plane wave modes in region III (incident). The matr ix 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 in region I and travelling toward the left (—). The matrix element , represents the scattering coefficient from plane wave i travelling toward the left (—) in region III and reflected/diffracted back to right-going (+) plane wave j in 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 in 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 sym-metry in the x and y directions for al l z, Bloch's theorem may be exploited to ren-der 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; z) (2.24) 26 Therefore, we need only compute the fields on the finite unit cell of the photonic crystal. This makes it possible to discretize Maxwell 's equations on a finite number of mesh points wi th finite grid size so that they may be integrated forward or backward in 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 in the previous subsection, we need to be able to propagate the fields through the textured layer. That 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 in 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 in the absence of free charges and currents: V x E(r,t) dB(r,t)/dt (2.25) 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 0 [ l + Xe(r)]E(r, t) = e0e'(r)E(r, t) (2.29) B(r, t) = \"(r)H(r, t) = /x 0 [ l + Xm(r)]H(r, t) = /x0/x'(r)H(r, t) (2.30) 27 where Xe(r) and x m ( r ) are the electric and magnetic susceptibilities respectively, which are properties of the media, and eo and (io are the permitt ivity and permeability of free space, respectively. Equations 2.25 and 2.26 may be Fourier transformed with 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 in the previous section, the approach we take is to solve Maxwell 's equa-tions on a finite number of spatial mesh points. A s we shall, see this has the effect of approximating the spatial derivatives in the equations 2.25-2.26 wi th difference terms between neighbouring mesh points. Consequently, we now define a rectilinear mesh wi th spacings a, b, and g in the x, y, and z directions, respectively, and make the following approximations: [exp( 2 ; exp(-ikxa.. .. ^ 2 = «* (2.33) ftjy ~ [expi ,ikyb. 1 2 ; exp( = ^ ) ] / i 6 ^ Ky (2.34) [exp( ,ikzq. 2 > exp(-~ikzg., .. A 2 )\\hg = KZ (2.35) which are valid provided that: kxa « 1 (2.36) kyb « 1 (2.37) kzg « 1 (2.38) so that we define: K = KXX + Kyy + KZZ f« k. (2.39) Note that the greater the spatial frequency content in 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 wi th respect to the photonic crystal. Making the substitutions, k = K , in equations 2.31 and 2.32, carrying out the cross-products and expanding component-by-component yields the following set of six equa-tions: exp{iky^)Hz exp(-iky^)Hz exp(ikz^-)Hy exp(-ikz^-)Hy Dx « + + * * (2.40) ibu ibu igu igu exp(ikz^-)Hx exp(-ikz^-)Hx exp(ikx^)Hz exp(-ikx^)Hz Dv « * * * + ^ (2.41) igu igu iau iau exp(ikx^-)Hy exp(-ikx^)Hy exp(iky^-)Hx exp(-iky^)Hx Dz « * * -± + —A (2.42) ' - - • • ibu Bx « + : ^ (2.43) • • - - • igu ixp(-ikx?-)Ez Bv « ^ ^ ^ + * (2.44) „ „ , . , „ „ , . , ^ a a ; Bxp{-iky^)Ex Bz « ^ — ^ — -= + (2.45) iau iau ibu ibu 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. That is: exp{±ikx^)Ft & Fi(xT^,y,z) = F j ( r T a ) iau iau ibu exp(iky^)Ez exp(- -iky-)Ez exp(ikz^)Ey ibu ibu igu exp(ikz^)Ex exp(- -ikz^)Ex exp(ikx-)Ez igu igu iau exp(ikx^)Ey exp( a. ~ -ikx-)Ey exp(iky^)Ex iau iau ibu b exp(±iky-)Fl O FfayT^z) = *}(r =F C) ( 2 - 4 6 ) exp(±ikz^)Fl o F | ( x , y , 2 T | ) = * i (>T7) where i7} denotes a component of the E or H field, and a , £, and 7 are defined as 29 follows. a, g„ a = —x C = -v 7 = -z 2 2 2 The real space fields are then given by: Hz(r + C) - Hz(r - Q Hy(r + 7) - Hy(r - 7) ibu iguj Hx(r + 7) ~ Hx(v - 7) H z ( r + cx) - Hz{r - a ) igui iau> Hy(r + a) - ff„(r - a) ^ ( r + Q - ^ ( r - Q A, r Bx(T. By(V BJT iau) Ez(r + C) - ^ ( r - C ) + Ey(r + 7) - £„( r - 7) ibu igu Ex(r + 7) - E x ( r - 7) £ , ( r + a ) - £ z ( r - a ) *0W E y ( r + a ) - ^ ( r - a ) | Ex(T + Q - Ex(T - Q iau ibu) (2.47) (2.48) (2.49) (2.50) (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: Ex(v EJT Ez(r Hy(T HJT Hz(r + C) - Hz(r - Q Hv(r + 7) - Hv{r - 7) ibue(r) iguje(r) Hx(r + 7) - Hx(v - 7) Hz(r + a ) - ff,(r - a ) iguje(r) iaue(r) Hy(r + a ) - i J „ ( r - a ) #*(r + C) - #x(r - Q iaa;e(r) i6a;e(r) E z ( r + C) - Ez(r - _ 0 + Ey{v + 7) - ^ , ( r - 7) ibu>/j,(r) igufj.(r) Ex(r + 7) - Ex(r - 7) ^ ( r + a ) - £ z ( r - a ) igujfi(r) mu>/z(r) Ey(r + a) - E „ ( r - a) £ g ( r + fl - Ex(v - Q (2.54). (2.55) (2.56) (2.57) (2.58) (2.59) iawju(r) ' ibup(v) 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 (2.60) 30 c = l / V i o 7 ^ (2-61) e' = e/e 0 (2.62) H1 = fi/iMj (2.63) are made, where c is the speed of light in vacuum. The substitutions are made in order to make the equations for the E and H fields look similar. The result is the following set of four equations: Ex{v + 1) « £ ; x ( r - 7 ) + ^ u ' ( r ) F ; ( r ) + ^ { H'x(r + a + C)-H'x(r + a-C) + H'x(r-a + C)-H'x(r-a-C) e'(r + a)b e'(r — a)b H'{v)-H>{T-2ct) + ^ ( r + 2 a ) - ^ ( r ) 3 ^ e'(r — a) a e'(r + a) a ^ ( r + a + Q - g ; ( r - a + C) H'y{v + a - Q - H'y(r - a - <) e'(r + C)a e'(r - C)a H'g(r) - H'x(T - 2C) H'x(r + 2C) - J£(r) e ' ( r -C)& e'(r + C)6 } (2-65) c 2 £ x ( r + a + C) - Ex(r + a - C) £*(r - a + C) - #x(r - a - C) fx'(r + a)b fj,'(r — a)b Ey(v) - Ey(r - 2a) | ^ ( r + 2 a ) - ^ ( r ) /i'(r — a ) a u'(r + a ) a c 2 ^ ( r + 7 ) « F ; ( r - 7 ) - 6 ' ( r ) ^ ( r ) + — { | E ; ( r + a + C ) - ^ ( r - a + C) tfy(r + a - C) - #y(r - a - <) /x'(r + C)a A»'(r-C)a J%(r)-J%(r-2Q _ i%(r + 2Q - i%(r) /x'(r-C)6 ^'(r + C)& 1 1 ; 31 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 in 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 in 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 in the z direction in steps of g given the values of the E and H ' fields on one pair of planes. If these planes are aligned parallel wi th the slab of textured material depicted in Figure 2.1, that is, the coordinate system is aligned so that z points in 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 in Figure 2.2 for the case of a square photonic crystal. Specifically, to integrate the fields forward, one begins with 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 in which the E or H ' fields must be calculated is crucial and depends on the direction of integration for a given ini t ial ordering of the E and H ' planes. For example, in 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 wi th the ini t ial ly specified values E(r'), H'(r' + 7), and uses equations 2.66 and 2.67 to calculate H'(r' — 7). Then 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. The above expressions for the fields may be recast as a transfer matrix, M, that 32 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. The dielectric constant is sampled on the lattice indicated by the black dots. The fields are calculated on the sublattice indicated by the crossed squares. The planes shown span a unit cell of the periodicity in the in-plane direction. The light and dark grey shading of the planes indicates whether the E or H' fields are computed there. The white hole in 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. H'x(Ql,l>z) EX{QN,N;Z-J) EV(QN,N\\Z~~I) Hy(gNtN;z) where M is a AN2 x AN2 matrix, and are the A^ 2 2D (in-plane) lattice vectors of the discrete mesh on which the fields are evaluated. Note that, in principle, the matrix 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~ \\)C + (« - 1)(C + OL) + (j - 1)(< - a ) i,j = 1...N (2.69) 2.2.3 E i g e n m o d e s o n a D i s c r e t e M e s h In order to compute the matrices T++, T~+, T+~ a basis set of AN2-vectors is chosen in 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 in region III are integrated separately back through the textured layer using equations 2.64 - 2.67, and the resulting 4A^ 2-vector is projected onto each of the basis vectors in region I. Similarly, to construct the matrices T , T+~ the basis vectors that represent left-going plane waves in region I are integrated forward through the textured layer using equations = M Ey{QiXz + l) Ex(eNtN; z + 7) Ey(QNtN; z + 7) H'x(QN,m Z + 2 7 ) H'y(QN,N;z + 2 7 ) (2.68) 34 2.64 - 2.67 and the resulting 4iV 2 -vector is projected onto each of the basis vectors in region III. Whi le one might be tempted to use the basis vectors defined in equations 2.5 to 2.8, which are polarized plane waves, these are not, in fact, eigenvectors of the discretized version of Maxwell 's equations applied to a uniform medium with e' = ^j/ni-Formally, in order to construct a basis set of the eigenvectors of the discretized Maxwell 's equations in regions I and III one must use the set of eigenvectors of the matrix M in equation 2.68, defined by the equations 2.64 - 2.67, where the spatially constant value of e': or e'IH is used in equations 2.64 - 2.67. <—> Unfortunately, the matrix M is not Hermitean. This means that its right eigenvec-tors, defined by: M v ( i ) = A ( i ) v w (2.70) and its left eigenvectors, defined by: u m M = A w u ( i ) i f (2.71) where the superscript H denotes the Hermitean conjugate, are not the same, and neither set is orthogonal. That is: umu(j) ^ § i j (2.73) This nominally presents a problem when performing the projection operations. However, it is t r iv ia l to show that and, therefore, the projection of an arbitrary 47V 2-vector, a, e.g., a wavefield integrated through the textured region, onto the basis of right eigenvectors is given by AN2 a = J2vjVij) (2.75) 35 where rjj = u{j)H3L (2.76) Thus, each right eigenvector, for which we seek an expansion coefficient, has a unique left eigenvector that can be used in equation 2.76 to find that expansion coefficient. Note that right and left eigenvectors have no correlation wi th right and left going plane waves. B o t h right and left going plane waves are right eigenvectors of the transfer matrix 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 in terms of the basis of plane waves. Pendry [18] has shown that, to a good approximation, the right eigenvectors of M in a uniform medium are given by the 2D in-plane Discrete Fourier Transform ( D F T ) of s and p polarized plane waves described in Fourier space as in equations 2.13 to 2.16, where K, the approximation made for k in deriving the transfer matrix equations, is substituted for k. That 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,,m) = n x « f £ £ (2.77) H ' ( . , ± , / / / / i ) ( G i m ) = Ki/m x g ( . , ± , / / / / i ) ( G j m ) ( 2 . 7 8 ) H ' ^ / ^ ( Q , m ) = n x (2.79) E ( p . ± ^ ( G l , m ) = K\\^1 x H ' < * ± . ' / ' \" > ( G l > m ) (2.80) where the s or p superscript denotes the polarization of the plane wave, the I/III superscripts denote the region in 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 KL M . ± we begin wi th kx = {Gltm + /3)-x; ky = {Glim + 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 kz corresponding to the right-and left-going waves, using e ^ / i 7 as appropriate for the particular region. Finally, for each such value of kz, we use equation 2.35 to obtain the corresponding two values of the z-components of K J ^ . ± . However, the dispersion relation that must be used is not precisely that for actual plane waves in a homogeneous medium given by equation 2.3. Rather, it is modified such that K is again substituted for k, as in the definition of the polarized basis set of plane waves itself (equations 2.13 - 2.16). This new dispersion relation is given by: 2 2 2 wVoeoCj /m = K H K = - ^ [ l - coskxa] + —[1 - coskyb] + — [1 - coskzg] (2.82) a o g Explici t ly , the two ( ± ) solutions for kz in each region (I/III) and for each reciprocal lattice vector G ; > m are given by: ki,m,±j/ni = i c o s - K l - ^ V o ^ / m 2 2 + 9-[i - C os (a [G , , m + p] • x)] + 9-[l - cos(&[G,,m + p] • y)]} (2.83) Notice that we have dropped the normalization constants present in equations 2.13 -2.16 since they are superfluous for vectors to be used as a basis set in subsequent cal-culations in 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 by equation 2.60, which is relevant to the transfer matrix 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 ftxS/P,±,I/III){Gi E . (s/p,±,I/III) V (2.84) (G;,m) ~(s/p,±,I/III) , ~{s/p,±,I/III) H ' ( G/ , m ) The real-space right eigenvectors are then obtained by projecting these onto the mesh points: -i R EX{QIX,Z0) Ey(Qi,i,zo) H'x(QiX'zo) Ex{gjtn;z0) H'x{Qj,n'i ZO) EX{QN,N'I ZO) Ey(QN,Ni zo) H'X(QN,N'I ZO) H'y{QN,N'i Zo) R,I/III J J (G,, m ,«/p,±) E; (s/p,±,I/III) E . (8/p,±,I/M). — (s/p,±,I/III), ~(s/p,±,I/III) , e(i[/3+G,,m]- e i i l) g(s/p,±,/////)j G / , m ) E . H (s/p,±,i/in)(r x y l « ! , m j s/p,±,I/III) (s/p,±,I/III) , e(i[/3+G / i m]-^. n) (s/p,±,I/III), K'x ( Q , m ) (8/p,±,I/III) -i R H (G*,m) (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: g(s/p,±,/////) ( -l L E (s/p,±,I/III) V 1 G/,m) (s/p,±,I/III) H ' x (Q,m) ~(s/p,±,I/III), 0 - 1 1 0 0 0 0 0 0 0 0 0 0 c tl/lll 0 I/III ^(p/s,±,I/III)(r X E. (p/s,±,I/III)l G l,m) —(p/s,±,I/III) it (2.86) (notice the swapping of s and p subscripts). The real space left eigenvectors may then be constructed in the same manner as in equation 2.85. 2.2.4 S ign i f i cance o f 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 u n ; E and B = yUoH, assume harmonically time-varying fields, and drop the time varying exponential, e~~ZU}t, from both sides of equations 2.25 and 2.26, they become : V x E ( r ) = zcj / i 0 H(r) (2.87) V x H ( r ) = -iooenniE(r) (2.88) Without making any approximations, we may again proceed as in section 2.2.2, carry out the curls, and eliminate the ^-components of the fields to produce an equation, for example, for the Ey field: (2.89) For an s polarized plane wave it is a simple matter to show that the difference term in the curly braces in equation 2.89 is identically zero. This is because: H^xs\\x,y,z) = H^eikxXeikyyeik* H^s\\x,y,z) = H^eik*xeikyyeik> 39 (2.90) (2.91) so that {^diHy{r) ~ W2Hx{T)} = - ^ ^ ^ ^ - kvHis)\\ (2-92) but since from equations 2.5 - 2.8 we have H X S ) = kzkx (2.93) = kzky (2.94) it follows that {——HV(T) -—Hx(r)} = 0. oyox oy1 When Maxwell 's equations are discretized through the procedure described in sec-t ion 2.2.2, the analogous equation to equation 2.89 is obtained by rearranging equa-tion 2.65 to yield the difference equation corresponding to the ^-derivative: ^ ( r + 7 ) - ^ ( r - 7 ) = - ^ ' ( r ^ r ) + 9j { (2.95) HY(T + a + C)-HV(T-a + C) _ H'(r + a - Q - H'y{v - cc - fl e'(r + C)a e'(r - C)a H'x(v) - H'x(v - 2C) H'x(r + 2Q-H'x(r)^ e ' ( r - < ) 6 + e>(r + C)b 1 1 ' For a plane wave in a uniform medium the translated fields, e.g., H'y(r + a — £) in equation 2.95 may be rewritten in terms of a phase factor multiplying an untranslated field. Assuming a square array of mesh points, i.e., a = b, we obtain: £ , ( r + 7 ) - ^ ( r - 7 ) = - ^ ' ( r ) i £ ( r ) + -f-{ (2.97) [eik*aeIK^H'y(r) - e~ik*aeik^ H'y(r)] _ [eik*<*e-ikyCH^ _ e -i**« e -i*i,C#/( r )] + [H'x(v)-e-^H'x{r)} + mr)-e2ikyCK(r)}} (2-98) The term in curly braces can be further simplified to yield 4sm(kyC)[H'x(r) s in (^C) - H'y{r) sm{kxa)} (2.99) 40 and making use of the fact that H'x(r) = K Z K x ; H'y(r) = KZKV (2.100) the term in curly braces becomes: 4 sin(A;j/C)^{Kx sm(ky() — K,ysm(kxa} (2.101) which, using equations 2.33 - 2.34, reduces to , • > s f 2 s i n ( L a ) > N 2sm(kv() . ., ,, ^ , n 4sm(kyC)KZ{ sm(ky() s i n ^ a ) } = 0 (2.102) Ct CL Therefore, the terms in curly braces in equation 2.95, which represent the discrete analog to the double partial derivative terms in curly braces in equation 2.89, cancel to yield identically zero as was shown for the exact continuous case. Thus these \"Pendry\" modes wi l l propagate as eigenmodes of the discretized integra-tion scheme and wi l l not mix into one another as they propagate. It is important to note that had we used \"true\" plane waves, that is, H'x(r) = kzkx;H'y(r) = kzky (2.103) then we would have obtained Asm(kyC)kz{kxsm(ky() - A ^ s i n ^ a : ) } fa 0 (2.104) which is only approximately equal to zero to the extent that kxa, kyb 1. This 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. The same also holds true for al l similar terms in curly braces in 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 + + ) _ 1 , T _ + ( T + + ) _ 1 , and hence T++, T~+, T , T+~ is described in detail for the case of a square photonic lattice. The method can be generalized to accommodate other types of photonic lattices, and an outline for how this may be done is presented in Appendix A . Of course, only a finite number of plane waves are included in the calculation, wi th 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 in two-dimensions, the size of the T matrices that result is 2A'' 2 x 2A'' 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. The scheme of real-space lattice points and reciprocal lattice vectors used for the square lattice is depicted in Figures 2.3 and 2.4, respectively. Given / 3 i n c , the value of kz is calculated from the dispersion relation, for each recip-rocal lattice vector, Gitm, using equation 2.82 wi th the value of the dielectric constant on the right hand side of the structure, e'in. A s previously mentioned, there wi l l be two solutions for kz, each of which wi l l be either purely real or purely imaginary for the 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 kz. For the case of real kz corresponding to propagating waves, its sign corresponds to that of the group velocity; for imaginary kz corresponding to evanescent fields, its sign determines whether the fields are growing or decaying in 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 in equation 2.84. These \"plane waves\" are then projected onto the N x N real-space mesh points on the unit cell depicted in 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. The electric or magnetic fields are sampled only on those points with the crossed squares. 43 y Figure 2.4: Scheme of reciprocal lattice vectors corresponding to in-plane wavevector of launched modes 44 -1 / / / H'x(Qi,i) H v ( Q l t l ) . Hy(Sj,i) _ EX(QN,N) EV{QN,N) H'X(QN,N) HV{QN,N) S/P, + ) III s/P,+) III (Gm,n,s/p,+) J {Gm,n) fi(s/p, + ,IH) {s/p, + JII)(r s ~(s/p,+,III) H ' x (G m j 7 l ) E. H MP,+,III) (Gm ] 7j) R e[i(/3+Gm,n)-ei,i] E x (s/p,+,I/III)l Gr ^(s/p,+,I/III)(n x ~(s/p,+,I/III) x \\^xm,n) ( s /p ) +, / / / / 7 ) / J/ H ' H ' ( G m ) n ) it fi(s/p,+,I/III) E-IX E (s/p,+,I/III) y (Gmtn) {Gm,n) ~(s/p,+,I/III) , H'{x/P' J ( G m , n ) ~(s/p,+,I/III) , it [i(/3+Gm,„) (2.105) where the superscript J i 7 indicates that it is a plane wave basis mode in region III. The subscripts Gm s , where the subscripts on the 4-vectors continue to identify the particular launched mode. Erp(Gr,s) EVVOP(Gr,s) H'xPr°P(Gr,s) HyPr°P(Gr,S) III = e x p [ - z ( G r , s + (3) • £ •,] Erp{Qhd Ef°P^u) in (2.107) 46 The matr ix elements are then computed by projecting the resulting Fourier trans-formed 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). This is done by taking the inner product with each of the corresponding left eigenvectors for that basis. Specifically, the elements of the matrix ( T + + ) _ 1 are obtained by projection onto the right-going modes on the left hand side of the structure (region I): { T + + ) (GT,s,u),(Gm,n,q) V L x ( G r , s ) H ' H ' ( G r , s ) J EPrOP(Gr,s) H'rp(Gr,s) H'yPT°P(Gr,s) III (2.108) T++ may then be obtained by matrix inversion. The elements of the product matrix T~+{T++)~1 are obtained by projection onto the left-going modes in region I. [ r - + ( r + + ) - 1 ] ( G r , . f U ) , ( G T I V ' 7 ) ( G r , s ) By\"' ^(G r j S) X (Gr,s) H ' y (G r > s ) -l L \\ H -l / / / E r p ( G r , s ) EPT0P(Gr,s) r,s) H?IOP(Gr (2.109) Hence T~+ may be obtained by matrix multiplication of the above result wi th 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 eigen-vectors on the right hand side(region III): 47 ( G r , s , u ) , ( G m , „ , g ) - , / / / ) - 777) Gr, s) {u H ' X - 777) ( G r , s ) V •—(u H ' -,III) ( G r ] S ) ^ p r o p ( G , , s ) ^ r o p (G r , s ) ^ ; p r o p ( G , , s ) # I p r o p ( G r , s ) [T + ( T ) 1 ] ( G r , „ u ) , ( G r o > n , g ) — V H ' H K + , 7 7 7 ) / (u,+,III) El« , t , i J JUp \\ x v G r , s ) (u,+,777), ( G r , s ) J / ^ p r o p ( G , , s ) ^ P r 0 P ( G r , S ) iY>°P(G r, s) ^ p r o p ( G r , s ) (2.110) (2.111] The matrix T , is obtained by matrix 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 in the middle of a dielectric stack as depicted in Figure 2.5. The 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 in the calculation described in 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, in a multiple scattering cal-culation to yield intermediate matrices C++, C~+, C+~, C , which describe scattering off the grating and dielectric stack to the right of the grating. These latter matrices are then combined wi th L++, L~+, L+~, L , which describe the scattering from the entire 48 49 T R ++ R Figure 2.6: Component scattering matrices for calculation of overall reflection (scatter-ing) matrix from grating embedded in an arbitrary multilayer dielectric stack. dielectric stack to the left of the grating, in a multiple scattering calculation to yield F ~ + , the matrix that describes reflection and diffraction from the entire buried grating structure. The expression for C~+ is given by: C~+ = T~+ + T—[I - P;+1R-+P+_1T+1]-1P-+1R-+P++1T++ (2.112) and the expression for F ~ + is given by: F~* = L~+ + L [I - (P-_1C-+P;_1L+-)]-1P-_1C-+P;_1L++ (2.113) where, the P^±1 are diagonal matrices accounting for the phase acquired or exponen-t ial growth or decay of the plane wave corresponding to each polarization and Fourier component in the calculation as it \"propagates\" right (+) or left (—) in the layers to the immediate left (g — 1) or right (g + 1) of the grating layer. 50 The 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 in a multiple scattering calculation using Fresnel [42] reflection and transmission coefficients at each interface and plane wave propagation through each layer. This 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 wi th layer thickness. Moreover, the multiple scattering calculation yields a result which is exact for all of the basis functions included in the calculation. 2.3 Results 2.3.1 Introduction In this section we illustrate how reflectivity calculations can be used to reveal the na-ture of the photonic eigenmodes of two-dimensionally strongly-textured high-index con-trast semiconductor waveguides. In order to introduce the various concepts involved, we proceed first through an investigation of the untextured, weakly and strongly one-dimensionally 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. The structure is depicted in Figure 2.7. The s and p spectra are somewhat different, as expected, with both exhibiting fringes arising from the Fabry-Perot cavity formed within the high-index material due to reflections from the interfaces. The maxima in 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 in 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 with dielectric constant e'2 = 12.25 constitutes a slab waveguide which supports bound electromagnetic modes. These modes have fields in the cladding layers above and below the slab that decay exponentially away from the slab and thus carry away no energy. Bound modes have a dispersion curve falling between the light lines for the refractive index of the core and cladding layers, as depicted in Figure 1.7. These modes are, in a sense, bare modes that wi l l be mixed by a 2D texture to form the renormalized eigenstates of the planar 2D 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 with that particular wavevector. The specular reflectivity spectrum wi l l then exhibit a pole when the energy and in-plane wavevector match the dispersion of one of these bound modes. This corresponds to 52 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 wi th 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. This spectrum corresponds to probing ui — 3 space along the line depicted in Figure 2.9. A feature (pole) appears in 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 in Figure 2.9. Figure 2.9: Plot 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 nm slab of material with 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 u — 3 space that is explored in the calculation of the spectra shown in Figure 2.10 below. The dots at the intersection of the vertical line with the dispersion curves indicate the appearance of a resonant feature in the spectra. 0 0 10 40 50 54 'I 1 1 1 1 1 1 ^ 1 7 8 9 10 11 12 13 F R E Q U E N C Y co/2rcc [ 103 cm\" l] Figure 2.10: Specular reflectivity spectrum for evanescent field incident on 120 nm slab of material wi th e' = 12.25, wi th 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. 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 in Figure 2.12. Gaps open up at the Br i l louin zone boundaries as the various orders (PG, 2PG) of the dielectric modulation couple together the bound modes which existed in degenerate pairs at ± / ? G / 2 and ±PG fc»r the untextured structure. It should be noted that one such dispersion plot exists for each of the T E and T M polarized modes, wi th the T M dispersion plot shifted higher in frequency. Finally, we note that the periodicity of the dielectric allows us to describe the mode dispersion in a zone-folded scheme in the first Br i l louin zone. When the specular reflectivity calculations are performed with incident (evanescent) light having in-plane wavevector given by (PG/2)X, two features are seen to occur in each polarization as shown in Figure 2.13. These occur again as poles in the reflectivity spectra indicating that the excitations remain completely bound to the guide. The eigenmodes in the vicinity of the second order gap at in-plane wavevector PGX, 55 Figure 2.11: Schematic dispersion plot for single-mode slab waveguide with 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 in the presence of the grating. e / = 1 . 0 Figure 2.12: Slab waveguide with \"weak\" I D texture: the structure consists of a 120 nm-thick slab of e' = 12.25 material, clad above and below by air, wi th a 10 nm deep I D square-toothed grating having a pitch A = 500 nm and an air filling fraction of 25% etched into the surface. 5G — i — i I I i 5000 6000 7000 8000 9000 FREQUENCY co/2rc [ cm\"1] Figure 2.13: Calculated specular reflectivity spectrum for evanescent light incident wi th 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. The corresponding waveguide structure is depicted in Figure 2.12. although primarily composed of superpositions of field components at ±/?G£> are never-theless 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 inci-dent wi th in-plane wavevector, /3 = 1.0ipGx, i.e., slightly detuned from the boundary between the 2nd and 3rd Br i l louin zones, for both the s and p polarizations. The Bloch states again manifest themselves as resonant features in the spectra, however, they are no longer true poles. They have non-zero linewidths owing to their leaky nature. The zeroth order radiative component of the Bloch states in the vicinity of the 2nd order gap allows us to excite them with 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 both the s and p polarizations. Once again, two features per polarization occur in the specular reflectivity spectra when the energy and in-plane wavevector match the dispersion of the modes. Here, the features occur as bipolar resonances in the specular reflectivity spectra, which are bounded by unity as required by energy (flux) conservation for propagating incident and reflected 57 4000 Figure 2.14: (a) Calculated specular reflectivity spectrum for evanescent light incident wi th in-plane wavevector, (3 = 1.0ipGx, i.e., slightly detuned from the boundary between the 2nd and 3rd Bri l louin zones, (b) Calculated specular reflectivity spectrum for light incident wi th in-plane wavevector, (3 = 0.0ipGx, i.e., slightly detuned from the Br i l louin zone-centre, (c) Calculated 1st order diffraction coefficient, i.e., coefficient for scattering from a field component at (3 = 0.0ipGx to a field component at (3 = 1.0ipGx. The solid curves are for s polarized light; the dashed curves are for p polarized light. The waveguide structure under study is depicted in Figure 2.12. 58 fields. This is like hit t ing a diffraction grating near-normal incidence but wi th a wavelength such that there are no propagating diffracted orders. The diffraction is only into these evanescent \"bound\" modes. Figure 2.14 (c) depicts the calculated 1st order scattering coefficient. This is the coefficient for scattering from a field component at (3 = Ancident^ to a field component at (3 — Ancident^ + PGX- In this case this corresponds to scattering from (3 = O.OIAJX to (3 = I . O I A J X . This is the amount of the incoming radiation at near-normal incidence, wi th in-plane wavevector (3 = 0 .0 lA?x , that is scattered by the grating to light having E , and H fields which are evanescent in the direction normal to the slab (£) and which vary sinusoidally in the x direction wi th 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. This scattering coefficient also displays resonant features when the energy and in-plane wavevector match the dispersion of the Bloch states. It is important to note that, in al l of the spectra shown in 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. This occurs because symmetry requires an exact cancellation of the radiative component of that particular state at the precise centre of the Bri l louin zone. Figure 2.15 shows the results of the reflectivity calculations carried out exactly at the zone-centre ((3 — 0). There are sti l l two distinct Bloch states at zone-centre, as evidenced by the two features in the evanescent specular reflectivity spectrum. However, only one of them has a non-zero, zeroth order, radiative component which allows us to couple to it wi th radiation from above the grating. Nevertheless, we can sti l l couple to the other Bloch state wi th an evanescent field, phase matched to the field component of the Bloch state at ~ QGx. 59 Figure 2.15: (a) Calculated specular reflectivity spectrum for evanescent light incident wi th in-plane wavevector, j3 — (3QX, at the edge of the 1st Br i l louin zone, (b) Calcu-lated specular reflectivity spectrum for radiation incident wi th zero in-plane wavevector, i.e., at the Bri l louin 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. The waveguide structure under study is depicted in Figure 2.12. 60 Figure 2.16 shows the near-normal incidence specular reflectivity spectrum for the same structure but wi th the grating having an-air filling fraction of 75% rather than 25%. Notice that the broader mode is now the lower-frequency mode. The frequency (energy) ordering of the lossy, and non-lossy modes are dependent upon the duty cycle (air filling fraction) of the grating. The frequency ordering is basically due to the differing distributions of the real space field intensities of the eigenmodes in the high and low-index regions of the unit cell. This effect is discussed in greater detail in 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 8250 FREQUENCYCO/2TIC [cm\"'] Figure 2.16: Calculated near-normal incidence specular reflectivity spectrum for light incident wi th in-plane wavevector, (3 = O.Olpcx, for s polarized light. The waveguide structure is identical to the one specified in Figure 2.12 except that the grating has an air filling fraction of 75% rather than 25%. It should be mentioned that, in comparison to the photonic crystals discussed later, these structures are considered \"weakly textured\" because although the dielectric con-trast between the teeth and trenches in the gratings is large, the grating itself occupies 61 a small fraction (~ 20%) of the total core thickness, resulting in a relatively small inter-action of the mode with the grating. The width of the gap is approximately 1.5 percent of its centre frequency, which, nevertheless, is st i l l larger than that corresponding to gratings used in 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 in 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 with \"strong\" I D texture: the structure consists of a 120 nm thick slab of e' = 12.25 material, clad above and below by air, wi th a I D square-toothed grating that completely penetrates the slab. The 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 nm deep, is buried in the core of our 125 nm waveguide of e' = 12.25 material. The struc-ture, which is depicted schematically in Figure 2.19, also illustrates the ability of the model to perform calculations incorporating buried gratings. G2 i . o H | 0.8 H 0.6H § 0.2 11 F R E Q U E N C Y GV2TC [103 cm\"1] Figure 2.18: Calculated near-normal incidence specular reflectivity spectrum for s po-larized light incident wi th in-plane wavevector, (3 = O.Ol/fex. The waveguide structure, depicted in Figure 2.17, consists of a 120 nm 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, in the vicinity of the 2D Bri l louin zone-centre, there are now four field components resembling bound modes of the waveguide wi th 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, al l four of these couple to a zeroth order component, which radiates out normally from the structure, as in the I D case, via the ±BQX and ±6GV components of the texture. Figure 2.20 schematically depicts the expected 2D bandstructure in the vicinity of the zone-centre as consisting of four band sheets shown in the reduced 2D Bri l louin 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. The radiative, zeroth order component of the Bloch state once again allows us to probe these modes wi th radiation incident from above the grating in a specular reflectivity calculation. Figure 2.22 depicts the specular reflectivity spectrum for light incident from above this structure with in-plane wavevector along the T — X symmetry direction. The symmetry directions for the square lattice are depicted in Figure 2.21. Three clusters 63 Figure 2.19: Slab waveguide consisting of 125 nm of e' = 12.25 material with air above and below, having a square lattice of cylindrical airholes, 25 nm 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. W 2 * Figure 2.20: Schematic in-plane bandstructure of resonant modes of waveguide with a square grating shown in 1st 2D Bri l louin zone. 64 of four bipolar features appear corresponding to Fano resonances [38] resulting from the four leaky modes described above. The precise Fano lineshape function is described in Chapter 5 (section 5.6.1), and a physical explanation of the reason behind the bipolar nature of the Fano lineshape is given in Chapter 6 (section 6.2). The 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 in the vicinity of each cluster. We note that in the lowest frequency cluster (where the modes are comprised primarily of T E field components) three features occur in the s polarization, while one occurs in the p polar-ization. Conversely, in the highest frequency cluster (where the modes are comprised primarily of T M field components) three features occur in the p polarization while one occurs in the s polarization. We now examine the properties of the modes occurring in the lowest frequency cluster in greater detail. Figure 2.24 shows reflectivity spectra for light incident wi th zero 65 8 9 10 11 FREQUENCY (O/2TTC [103 cm\"1] Figure 2.22: Calculated specular reflectivity spectrum for light incident on a 125 nm slab waveguide of e' = 12.25 material wi th air above and below, and a square lattice of cylindrical airholes, 25 nm 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 wi th in-plane wavevector fixed at P/PG = 0.01 in the T — X direction, which is defined in Figure 2.21. 66 i.• H 0 .6 -U FLE 0 .4 -0 .2 -0 .0 -9600 1.0 H £ 0.8 -j i—i > H 0 .6- | S 0.2' 0.0' (a) 7700 7750 7800 7850 7900 7950 8000 F R E Q U E N C Y CO/2TIC [ cm\"1] > r 9640 9680 9720 F R E Q U E N C Y CO/2TIC [ cm\" ] 11.50 11.55 —I— 11.60 — 1 — 11.65 F R E Q U E N C Y co/27ic [103 cm\"1] (b) (c) 11.70 Figure 2.23: Expanded view of the spectrum depicted in 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 in each of the s and p (or rather, x and y) polarized spectra, become degenerate. 1.0-H i—i > P U W w 0.8 0.6 H 0.4 H 0.2-T T 7700 7750 7800 7850 7900 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 in F ig -ure 2.23. The spectra were calculated for light incident with zero in-plane wavevector, P/PG = 0.00, corresponding to normal incidence. Figure 2.25 depicts reflectivity spectra for light incident wi th in-plane wavevector oriented along the direction + ( M symmetry direction). In this case, two resonances appear in each polarization (s and p). Figure 2.26 depicts three specular reflectivity spectra, in the vicinity of the T E gap, for off-normal incident light along the X direction, for three different diameters of the airholes. The 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 in Chapter 6. 68 i.oH 0.2- j 0.0 J 1 1 1 1 1 7800 7820 7840 7860 7880 7900 FREQUENCYco/27tc [cm\"1] Figure 2.25: Calculated specular reflectivity spectrum from same structure as in F ig -ure 2.23. The spectra were calculated for light incident with in-plane wavevector fixed at P/PG — 0.01 in the [x + y] direction (i.e., along the T — M symmetry direction). 2.3.6 2D Strong Texture We now investigate structures with a \"strong\" 2D grating. A s in 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 in the present work. The waveguide 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 of material with e' = 2.25. The semi-infinite half-plane below is also 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 wi th 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, with 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 nm slab waveguide of e' = 12.25 material wi th air above and below, and a square lattice of cylindrical airholes, 25 nm deep embedded at its centre. The spectra were calculated for light incident with in-plane wavevector fixed at 3/(3G — 0.001 in 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 70 Figure 2.27: Strongly 2D textured thin asymmetric slab waveguide: the structure con-sists 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 wi th 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 with 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 in Figure 2.27 along the two symmetry directions. The 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. This 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 nor-mal incidence. As in the case of weak texture, a lone feature appears, as the two narrow features vanish and the broad features overlap exactly. Finally, as in 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 wi th in-plane wavevector, j3 = 0.03(3G, along the F—X direction for the structure depicted in Figure 2.27 wi th r / A = .2821. The solid line represents the s polarized spectrum, the dashed line depicts the p polarized spectrum. - i 1 1 r * 8.5 9.0 9.5 10.0 F R E Q U E N C Y W/2TCC [10 3 cm\"1] Figure 2.29: Calculated near-normal incidence specular reflectivity spectrum for light incident wi th in-plane wavevector, Q = 0.03/fc, along the T - M direction for the structure depicted in Figure 2.27 wi th r / A = .2821. The solid line represents the s polarized spectrum, the dashed line depicts the p polarized spectrum. 72 components in al l four modes at zone-centre. This spectrum is shown in Figure 2.30 (b). The two lossless modes manifest themselves as poles while the two lossy modes appear as degenerate resonances wi th finite linewidth. The nature of the various photonic eigenstates corresponding to each of the resonant features in the specular reflectivity spectra is described in Chapter 6. 73 - i 1 1 r~ 8.5 9.0 9.5 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 FREQUENCYco/27ic [10 3 cm\"1] Figure 2.30: Calculated specular reflectivity spectra for light incident on the structure depicted in Figure 2.27 wi th r / A = .2821. (a) Normal incidence specular reflectivity, i.e.., light incident wi th f3 = 0. (b) Reflectivity for s polarized evanescent light incident wi th P = PGx. 74 2.4 Convergence We recall that in 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. The 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, in principle, several distinct issues with respect to convergence having to do with the choice of the number of mesh points, and the number of basis wave functions to include in the calculation. Firstly, for any given basis state, the accuracy of the propagation algorithm is l imited 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 thin sinusoidal gratings, higher order scattering requires, in principle, that an infinite number of plane waves be included. Of course, for any given structure, there wi l l only be significant scattering to a finite subset of these. The 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 wi l l depend on the structure. The thi rd issue has to do with the relationship between the number of mesh points used for the field and the number of basis functions used in the calculation. We recall that the basis functions are essentially plane waves wi th 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 in each direction (x and y) be at least equal to the number of Fourier components (or basis functions) in that direction. This 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 in the calculation, then the fields at the end of the integration may contain significant higher spatial frequency content than that included in 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 matrix 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 wi th the same number of mesh points as basis functions. However, this would mean that the basis functions wi th the highest in-plane propagation vectors are not integrated very N 2TY A accurately because kx/yax/y = (~2~~^~)( jy) = 7 1 ^ Since these higher order field components are relatively small, the significance of this error should be small. However, in the waveguide structures of interest here, these higher order components sometimes become erroneously enhanced and thereby introduce false modes in the reflectivity spec-tra. In some cases, this leads to spurious features appearing in the spectral range in 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 gradu-ally move up and out of the range of interest. This 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 in this regime on the structures studied is only in the range of a couple of percent. No such spurious features arise in the spectra presented in this section and so the calculations were carried out with the same number of mesh points as basis functions. A fourth issue concerns the inherent pitfalls in attempting to represent the contin-uous permitt ivity 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 in terms of its permitt ivity and perme-ability (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. The 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 wi th 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 in 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 in the in-plane directions, the size of the in-plane unit cell for the calculation, may, in 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 in 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 in-plane propagation constant for which it is desired to find the bound mode frequency in the calculation. This further allows only one mesh-point per unit cell to be used in the integration of the fields. A s a result the F D R S code becomes much less computationally expensive than in the textured case. However, purely for convenience, in 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 in the present work. Only 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. The in-plane wave vector of the incident evanescent light was set to (3-mc = 2ir/A. The untextured waveguide structure used in the calculation consisted of a 120 nm slab of material with 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 in the calculation in each of the x and y directions. The different curves (lines) from top to bottom on the graph are for increasing numbers (specifically: 12, 22, 32, 42, 62, and 122) of mesh points in the z direction. A s the number of z steps increases the curves approach a straight line, wi th 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 in each of the x, and y directions. Since there is no scattering to higher order in-plane Fourier components in the case of the untextured waveguide, it is sufficient to include in the calculation only the one in-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, in 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 in the calculation to yield quantitatively accurate results. A s in the untextured case, the number of mesh points in the in-plane (x — y), as well as in the normal (z) direction may 78 1.0 1.1 1.2 1.3 1.4 LOG(NO. OF MESH 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 in each of the x and y directions included in 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-mc = 27r/A. The structure used for the calculation consisted of a 120 nm slab of material wi th e' = 12.25 clad above and below by air (e' = 1.0). The different curves (lines) are for increasing numbers, (specifically: 12, 22, 32, 42, 62, and 122), from top to bottom on the graph, of mesh points in the z direction. 79 also be varied. Rather than exploring the entire parameter space, which is computa-tionally expensive, particularly for two-dimensionally textured structures, the approach adopted in the work described below is to vary these parameters in a related fashion. This 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 in figure 2.4. Thus, in 2D, there wi l l 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 in 2D. 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). The additional two pairs of planes account for the fact that one pair of planes is required in each of the untextured regions, so that the number of planes in the textured region remains N. Therefore, in 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 in equations 2.33-2.35 are given by: where d is the thickness of the textured region. The 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\", with 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 in Figure 2.32 with slab and grating thickness, d = 80 nm. The plots wi th the square markers and dashed lines in Figure 2.33a and Figure 2.33b represent the frequency and width respectively of the lossy mode at the zone-centre (P = 0) as a function of the number of elements used in the F D R S calculation. These plots indicate that the convergence behaviour of this model wi th respect to both the a = b=V2 A N (2.114) d (2.115) 80 mode frequency and linewidth possesses a strong oscillatory component. e2'=12.25 ej =1.0 A=500 nm G ! =1.0 Figure 2.32: Slab waveguide with \"strong\" I D texture: the structure consists of a slab of thickness d of e' = 12.25 material, clad above and below by air, wi th 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 in this work is a Kaiser window [43] with /?Kaiser = 2.3 and - ^ K a i s e r s e t equal to the number of elements used in the F D R S calculation (in order to optimally satisfy the Nyquist criterion). The plots with the round markers and solid lines in Figure 2.33a and Figure 2.33b represent the corresponding results of the F D R S calculations when they are carried out wi th this spectral pre-filtering of the e' — / / profile implemented. Monotonic convergence wi th 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 calcu-lation it is useful to re-plot the above convergence results wi th spectral pre-filtering as 81 a o OH a X H Q 11.2-11.1 -11.0 10.9-10.8 10.7-10.6 300-280-260-240-220-200-180-* ' , , , \" ' l \" M \" \" ,\". L ^ r T ; '.; \\. i: i * • i i i I i i i i I i i i i I i i i i I i i i i I i i i i i i i i i ) i i i i I i i i i I i i i i I | i i i i | I I i i | i i i i | i i i i | i i i i | i i i i | i i I I | (a) (b) 10 20 30 40 50 NO. OF ELEMENTS Figure 2.33: Mode frequency (a) and width (b) vs. the \"number of elements\" (see text) used in F D R S calculation for the structure depicted in Figure 2.32 wi th d = 80 nm. The results are for the second-order lossy mode at the zone-centre. Plots with round markers and solid lines depict results with anti-aliasing spectral filtering of the e' — // profile implemented; plots wi th 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). This plot is shown in Figure 2.34. The dashed lines represent the best fit line obtained from a least-squares fit. The 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 par-ticular 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 with respect to the square of the mesh size, but when the mesh size squared decreases below a certain point (~ 10 x 10~ 3 on the graph, corresponding to 15 elements), both 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 wi th 15 elements was sufficient to produce results that are converged to within 1% for the mode frequencies, and to wi thin 1.7% for the width. Similar convergence behaviour is exhibited in the case of two-dimensional strong texture. Figure 2.36 depicts the convergence of the mode frequency and width for the second-order lossy mode at the zone-centre for the structure depicted in Figure 2.35. This structure is very similar to those used in the experimental work described later in this thesis. Here it is found that a computation wi th 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 within 1% for the mode frequency and to wi thin 7% for the width. Finally, it is perhaps not surprising that the F D R S calculations converge roughly linearly wi th the square of the mesh size since this is the leading error term in the approximations in equations 2.33 to 2.35. 83 e o u w 11.1 -E » ^ ^ 11.0-j X N -10.9-i N 10.8-j N s v \\ N 10.7 \\ N X V I I I I I '[• 1 I I I I 1 I T T | I I I I | I I I T | (a) 296-292-X H • • • * / m • ' i i i i I i i i 1 r | T T i i | i i i i | i 1 (b) 10 20 30 40 X-Y MESH SIZE SQR ( a / A f xlO\"3 Figure 2.34: (a) Mode frequency and (b) width, vs. square of the x — y mesh size normalized by the unit cell size (or grating pitch) used in the F D R S calculation for the structure depicted in Figure 2.32 wi th 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 wi th an-ti-aliasing spectral filtering of the e' — / / profile implemented. The dashed lines represent the best fit line obtained from a linear least squares fit. In (b) the rightmost three points were ignored in the fitting. 84 A=500 nm e3'=2.25 Figure 2.35: Strongly 2D textured thin asymmetric slab waveguide: the structure con-sists of a 80 nm-thick slab of e' = 12.25 material which forms the guiding core, clad above by air and below by material with e' = 2.25. The core waveguide layer is penetrated by a 2D square array of holes on a pitch, A = 500 nm, and wi th hole radii given by r / A = 0.2821. 85 • g 10.20 d o 10.15^ KlO.lOd ® 10.05 i 110.00-3 9! 9.95 A 230 -j 220-1 S 210^ g 200^ 190^ n—i—i—i—r -i—i—i—i—i—i—r (a) T T (b) 5 10 15 20 X-YMESH SIZE SQR ( a / A f xlO\"3 Figure 2.36: Mode frequency (a) and width (b) vs. square of the x — y mesh size normalized by the unit cell size (or grating pitch) used in the F D R S calculation for the structure depicted in 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 with anti-aliasing spectral filtering of the e' — fj,' profile implemented. The 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 in the calculation. 86 2.5 Comparison with Green's Function Based Model Due to run 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 in this thesis was used for validation of the more computationally efficient Green's Function (GF) 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. Maxwell 's equations are then arranged so that the polarization arising from A x 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. The polarization is then re-expressed in 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 in 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 in 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 in any actual calculation wi l l be finite and may, therefore, affect the accuracy of the results; and 2) that the accuracy of the results wi 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 thin slab, by comparing it wi th results from the F D R S model. Figure 2.37 compares the near-normal (P = 0.02/3^ along the F — X direction) specu-lar 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). The structure used in the comparison is the same as that depicted in Figure 2.35 and is intended to be a somewhat simplified version of that used in the experimental work described in this thesis. It consists of an 80 nm slab of material with e' = 12.25 completely penetrated by a 2D square array of cylindrical holes on a 500 nm pitch and 25% air filling fraction, all sitting on a semi-infinite substrate wi th e' = 2.25. The number of mesh points and plane waves used for the F D R S model are as described in section 2.4.2 using 17 elements. The G F calculation included the nine reciprocal lattice vectors at 0, ±PG%, ±PGV, ±PG% ± PGV-The 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 in excellent agreement, both with respect to the locations and widths of the resonances. The 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 pro-file), epitaxial growth tolerances (layer thickness and alloy composition), uncertainties in 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 po-larized features were extracted from the results of both calculations and found to agree to wi thin 6.6%. Similar calculations on the same structure at precisely normal incidence (P = 0) using both models yielded spectra, exhibiting, as expected, a single broad de-generate feature in each of the s and p polarized spectra. Here, the agreement between the two models was within 1.0% with respect to the absolute frequency, and to within 88 6.5% with respect to the width. To address the limitations imposed by the thin-grating approximation made in the G F model we consider a symmetric planar waveguide consisting of a slab of varying thickness of e' = 12.25-material, wi th air (e' = 1.0) above and below, completely pene-trated by a I D grating having an air-filling fraction of 25%. The structures are similar to that depicted in Figure 2.17. Figure 2.38 plots the location of the upper and lower edges of the 2nd order gap as calculated wi th the G F model (dotted curve), and F D R S model (solid curve) for varying thickness of the grating (=slab). The F D R S model was verified to be converged to within 1% for the upper band and 0.2% for the lower band (using 17 \"elements\" in the calculations). The G F calculation included the three reciprocal —* lattice vectors at 0 and ± / ? G X . It is clear that for thin gratings the two methods agree well. When the waveguide-grating is 80 nm thick the two methods agree to within 1.4% for the upper edge, and 0.27% for the lower edge. However, when the waveguide is 250 nm thick (a thickness greater then those that are usually of interest in the context of studying photonic crystal membranes, which is the object of the present work) the two methods agree to within 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, in 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 in this example. Furthermore, different layer structures may result in 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 1 1 1 1 ' 1 1 1 8000 9000 10000 11000 12000 FREQUENCY co/27ic [cm\" ] Figure 2.37: Near-normal incidence specular reflectivity spectra for light incident wi th in-plane wavevector P = 0.02/?G along the T — X direction calculated with the F D R S model (lower plot) and the G F model (upper plot). The parameters used for the F D R S model are as described in section 2.4.2 using 17 elements. The G F calculation included —* the nine reciprocal lattice vectors at 0, ± /?G^> ±PGV, ±PG% ± PGV- The solid lines represent the s polarized spectra, and the dashed lines represent the p polarized spectra. The structure being modelled is the same as the one depicted in Figure 2.27. The resonance widths and shape are in excellent agreement. The mode frequencies agree to within 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 in 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 with 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%. The dashed lines are for the calculation with the G F model, the solid lines wi th the F D R S model which was verified to be converged to within 1% for the upper band and 0.2% for the lower band (using 17 \"elements\" in the calculations). The G F calculation included the three reciprocal lattice vectors at 0 and ±PGX-91 2.6 Significance of Modification to Pendry's Method A s 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 in the approximations Pendry makes for the wavevector. Instead of equations 2 . 3 3 to 2 . 3 5 , Pendry defines two approximate wavevectors: kx fa [exp(ikxa) — l]/ia = KX ky fa [exp(ikyb) - l]/ib = « J ( 2 . 1 1 6 ) kz fa [exp(ikzg) - l]/ig = « f and kx fa — [exp(—ikxa) — l]/ia = KX ky fa -[exp(-ikyb) -l]/ib = K* ( 2 . 1 1 7 ) kz fa ~[exp(-ikzg) - l]/ig = Substitution of ( 2 . 1 1 6 ) into ( 2 . 3 1 ) and ( 2 . 1 1 7 ) into ( 2 . 3 2 ) , Fourier transformation to real space, and elimination of the z components in a manner similar to the derivation presented in section 2 . 2 yields, for example, for the Ex component of the fields: Ex(r + g) = Ex(r) + ^ ^ ( r ) ^ ( r ) + ^ { , H'x(v + a - b) - H'x(r + a) H'x(r - b) - H'x(r) e ' (r + a)6 e'(r)b Hy(T - s) - Hv(r) Hy(r) - Hy(r + * e'(r)a e'(r + a)a Comparing this with equation 2 . 6 4 derived in section 2 . 2 : } ( 2 . 1 1 8 ) £ * ( r + 7) = ^ ( r - 7 ) + ^ ' ( r K ( r ) + ^ { c z a Hx(r + a + C)-Hx(r + a - C ) | H't(r - a + Q - H'x(T - a - Q e'(r + a)b ' e'(v — a)b H'y{v)-H'y(v-2a) H>(r + 2a) - Hy(v) e ' (r — a) a + e '(r + a) a } ( 2 . 1 1 9 ) 9 2 we see that both contain the four terms in braces corresponding to discrete, numerical derivatives arising from V x H in Maxwell 's Equations. However, Pendry's approxima-tions for k lead to forward or backward difference terms for the derivatives rather than the centred difference terms that arise in the approximations for k used in the present work. This introduces an artificial chirality for the scattering to Fourier components in the ±x and ±y directions in the plane, and this asymmetry prevents the model from pro-ducing specular reflectivity spectra wi th 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 in this work. The specular reflectivity spectra are for light incident from air on a 80 nm film of material having a refractive index of 3.5, textured wi th a square array of 300 nm diameter cylindrical holes filled wi th 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 wi th 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 both the s and p reflectivity. However, it produces three distinct features in each of these spectra rather than one degenerate feature. More generally, for the case of ra-diation incident wi th a wavevector P = 0.01/?G along the T — X symmetry direction, Pendry's original scheme yields three features in each polarization rather than the 3s+lp combination required by the symmetry of the 2D square lattice. 93 10.4 10.6 10.8 11.0 11.2 11.4 11.6 11.8 12.0 12.2 Frequency .84 to a damp but otherwise inert atmosphere at high temperature. The oxygen in the water reacts to form an oxide wi th 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 wi th the aluminium composition of the alloy. The exact nature of the oxide formed is an outstanding issue in the literature, and possibly dependent on oxidation temperature and other conditions. Potential candidates include: ( A l x G a i _ x ) 2 in a similar form to 7-AI2O3, crystalline phases such as a - A l 2 0 3 , /3-A\\203, ( 5 -A l 2 0 3 , amorphous alumina A 1 0 y , hydrates ( A l 2 0 3 ) - n H 2 0 , and hydroxides A l m ( O H ) n . 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 in 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 with copper or brass) up to 500°C. A quartz oven tube is used in order to support temperatures up to 525°C. The valves and other Pyrex glassware used away from the oven are capable of withstanding temperatures up to 200°C. Nitrogen ( N 2 ) , which does not react with G a A s or A l G a A s at a temperature 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. The 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). Two parallel controllers were used in order to increase the maximum controlled gas throughput possible. The boiler is heated by the combination of hemispherical heating mantle and heating tape. The 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. The oven temperature is automatically regulated by a Lindberg Proportional-Integral-Derivative (PID) programmable controller based on a temperature measurement fed back to it from a thermocouple in contact wi th the quartz tube in the centre of the oven. The exhaust gases containing toxic arsine are bubbled through a flask of water, dubbed the bubbler (BB) , to partially dissolve and remove arsine, before being vented up the fumehood exhaust. Two 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. The holes in 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 and under the patterns and the rest of the sample. This 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 Atomic 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 in \"Tapping Mode\". Atomic force microscopy consists of scanning a pointed probe mounted on a cantilever across the sample surface. The tip position is measured with the use of a laser beam which is reflected off the backside of the cantilever, near the t ip, and onto an array of four photodetectors as depicted in Figure 3.7. The vertical position of the tip is inferred from the difference signal between the top and bottom pairs of photodetectors. In \"Tapping Mode\" operation, oscillation is induced near the cantilever's resonant frequency using a piezoelectric crystal. When the oscillating tip comes in 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 in 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. The 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 \"Tap-ping Mode\" A F M was selected over S E M in 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 both the radius of curvature of the very end of the tip and its aspect ratio l imit 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 with the topography of the sample since some portions of the sample may be inaccessible to the tip due to its non-zero width and non-vertical profile. 3.3 Fabrication Procedure 3.3.1 SEM measurement of Layer Structure The layer structure for the as-grown sample is depicted in 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 Al0.7Gao.3^4s layers. These measured thicknesses, along with their nominal values, are given in Table 3.1. 3.3.2 Cleaving A small, approximately 5mm x 5mm, piece of sample was cleaved off the quarter wafer by scribing the wafer at the desired cleave locations, placing it over a thin 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 No. Mater ia l Nominal Thickness [nm] Measured Thickness [nm] 1 G a A s 45 90 ± 10 2 Alo .7Gao.3As 35 3 Alo.98Gao.02As 1600 1800 ± 8 0 4 G a A s substrate substrate Table 3.1: Nominal and measured thicknesses of layers in sample A S U 721 116 with 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 with tweezers while deionized (DI) water from a gun was sprayed on the surface . It was subsequently sprayed liberally wi th acetone from a squirt bottle to remove the water, and then sprayed wi th methanol to remove the acetone. Finally, the sample surface was blown wi th dry nitrogen gas. After cleaning, the sample was placed on the chuck of the photo-resist spinner. The resist used was Polymethylmethacrylate ( P M M A ) 950K molecular weight. It is shipped by the supplier, Olin, in 4% solution in 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 rpm 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 in a fumehood. Exposure The sample was then fixed to a ~ 7 m m stainless steel stub using a carbon based con-ductive glue with the resist-coated side facing up. In order to provide greater electrical contact and mechanical support, two thin 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 2B pencil gently against them. These provided small features useful in 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 main chamber through the air-lock and locked into place on the translation/rotation/t i l t-stage. The tip was then flashed and left for 2.5 hours to allow the adsorption on the tip to reach its stable configuration. The electron beam was then turned on. The electron beam and apertures were aligned and the astigmatism adjustment performed as per the procedures specified in the S E M manual. The 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. The 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. The sample was then translated such that each of its edges was, in turn, brought into the field of view. B y thus verifying that all of the edges remained in 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 wi th 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 writ ing 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 in this paragraph was repeated for each of the patterns defined in 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 wi th dry nitrogen gas to remove the graphite particles. It was then immersed in developer solution (from O C G Y Inc.) consisting principally of 118 methyl isobutyl ketone ( M I B K ) diluted in isopropyl alcohol ( IPA), M I B K : I P A 1:3, for 90 seconds. The development process was stopped by immersing the sample in an I P A rinse for 60 seconds. This was followed by immersion of the sample in DI water for 30 seconds followed by drying wi th nitrogen gas. The sample was then immersed in a high-contrast developer solution consisting of ethoxy ethanol (EE) diluted in methanol, EE:methanol 3:7, for 15 seconds to remove any thin film of P M M A remaining at the bottom of the developed holes. This final development process was stopped by immersing the sample in methanol for 30 seconds. This was followed, once again, by rinsing of the sample in DI water for 30 seconds followed by drying wi th 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 The sample holder that shipped with the etcher is designed to hold 3\"-diameter wafers. Consequently, a 3\" silicon wafer was placed in 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. The load-lock cover was closed and the load-lock pumped down. The 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 in Table 3.2. After the chamber was pumped down to a base pressure of ~ 1 x 10~ 7 Torr, the process gases were allowed to flow for 30 seconds before the microwave source was ac-tivated 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 with nitrogen gas for 30 seconds in 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 mTorr A r flow rate 20.1 seem CI2 flow rate 1.6 seem BCI3 flow rate 2.2 seem Microwave power 100 Watts R F Bias 100 Volts Chuck temperature 5°C Backside He pressure 5 Torr Pre-etch purge time 45 seconds Etch time 160 seconds Table 3.2: Etch conditions The sample was then removed from the air lock, rinsed, and left in a beaker of acetone for two hours in order to remove the remaining P M M A mask. This was immediately followed by rinsing in methanol and DI water for 30 seconds, each followed by drying wi th 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. The system was then purged with 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 sti l l on, the oven was turned on and the P I D controller was programmed to ramp the temperature up to 400°C in 30 minutes. Meanwhile the boiler was filled wi th fresh DI 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 in two hours. The boiler, heating tape, and oven were turned off, and the sample and oven were allowed to cool overnight wi th the controlled 100 seem nitrogen purge on. The sample was then removed from the oxidation oven. 3.4 Fabrication Results Lithography, etching, and oxidation were performed, as described previously in this chapter, on sample A S U 721 so as to produce twenty-four patterns arranged on a 0.5 m m grid in 6 rows of 4 gratings each. The doses and pitches programmed into the lithography software are given in Table 3.3. Note that two copies of each pitch-dose combination were produced for the sake of redundancy. The 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. This region was oxidized outward from the grating, and we can see that it extends out approximately 10 / im 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 /mi x 4 (im image of grating 22 is shown in Figure 3.10. The 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 mea-surement using a helium-neon laser, and from observation in the S E M . The image shows good uniformity in both the hole size and pitch over its field of view. This 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 micro-scope. The 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 Grat ing Dose [fC] Pi tch nm] g l 35 550 g2 50 550 g3 65 550 g 4 25 500 g5 35 500 g6 45 500 g7 30 525 g8 45 525 g9 60 525 gio 45 575 g l l 60 575 g l 2 75 575 g l 3 30 550 g l 4 50 550 g l 5 65 550 g l 6 25 500 g l 7 35 500 g l 8 45 500 g l 9 30 525 g20 45 525 g21 60 525 g22 45 575 g23 60 575 g24 75 575 Table 3.3: Pitches and doses specified for e-beam lithography 123 Grat ing Extent of oxidation out from edge [/im] Condit ion g l - bad g2 - bad g3 - bad g 4 - bad g5 12 top left corner bad g6 12 2 bad patches g7 6 good g8 12 good g9 14 good gio 8 good g l l - good g l 2 15 good g l 3 3 spot on left edge g l 4 9 good g l 5 13 good g l6 12 good g l 7 9 good g l8 12 good g l 9 9 good g20 10 good g21 15 spot on left edge g22 10 good g23 11 good g24 10 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 in 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 diame-ters 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 in Figure 3.11. Table 3.5 gives the measured hole diameters and pitches of the gratings as well as their air filling fractions, calculated with the assumption of cylindrical holes. 125 o m CM 1 ' t 1.00 2 .00 3 .00 4 .00 JJM Figure 3.11: Section through diameter of holes in image in Figure 3.10. The arrows indicate the leading and trail ing edges of the depression in the profile, which are used to obtain a measurement of the diameter of the holes at the surface. Grat ing Pi tch, A nm Hole radius, r nm r / A F i l l ing fraction g22 571 ± 17 170 ± 1 0 0.300 ± 0.020 0.278 ± 0 . 0 2 7 g23 576 ± 16 187 ± 11 0.325 ± 0 . 0 2 1 0.331 ± 0.026 g24 573 ± 16 226 ± 14 0.394 ± 0 . 0 2 6 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 The experimental setup described in this chapter was used to collect specular reflectivity spectra from the gratings in the geometry depicted in Figure 4.1 below. For each grating, data were collected for a series of incidence angles 6, for each of (j> = 0 and = 45° (see Figure 4.1). The 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 unti l it appeared in the field of view on the monitor. The input mirror, E M I , was rotated to position 28, and the 135 sample rotated unti l the signal on the detector, as monitored by a digital voltmeter, was maximized. The square aperture was then closed down unti l only light from the grating was imaged on the C C D . The 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 The experimental setup described in 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, in particular, al l of these features remained clearly identifiable and within the detector's spectral range over a large range of incident angles and for both 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 in order to show the effects of filling fraction on the ordering of the broad and narrow spectral features associated with 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. The spectra are plotted as a function of a \"reduced frequency\" or wavenumbers (in units of c m - 1 ) which we denote as Co = LO/2TTC (5.1) where c is the speed of light in free space. The experimental spectrum was obtained and normalized according to the procedure described in the previous chapter. The resulting spectrum was then normalized again, this time, to its maximum value within 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 in Chapter 2. The exact values of the parameters input to the program to describe the structure under study are discussed later in this chapter. The broad features in the spectra spaced at ~ 1000 c m - 1 are Fabry-Perot fringes arising from the optical cavity formed by the A l oxide layer. They would be present even in the absence of the grating. The 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. The precise Fano lineshape function is described in section 5.6.1. A physical explanation of the reason behind the bipolar 138 H > I—I H U w 1.04 0.5 H 0.0 1.0 0.5 H 111111 1111 • i • • • • I • • • • i • • • 111 • 11111 • 111 • • • 11 (a) 0.0 ~ | i i i i | i i i i [ i i i i | i i i i | i i i i | i i i i | i i i i | i i i i [ i i i i | i i i n i i i i | i i i i 6 7 8 9 10 , 11, 12 FREQUENCY CO/2TCC [10 cm\" ] 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 in 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 CO/271C [loW1] 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, in this case, two features occur in 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 D i s p e r s i o n Figure 5.5 shows a series of s polarized specular reflectivity spectra, taken at various angles of incidence wi th 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 H i — i > H U W 9=50° 9=40° 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 inci-dent angles, 9, along the T — X direction ( = 0). The 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 maximum value in the displayed spectral range. The 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. The size of the gap near zone-centre is approximately 900 c m - 1 which is of the order of 10% of the centre frequency. There is also a fourth band occurring higher in energy which anti-crosses wi th the third band at about 25°. The linewidth of the modes varies from band to band and within bands. 142 5.5 F i t t i n g w i t h M o d e l The 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 imitat ion 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 nm thick G a A s layer as well as a 35 nm Alo .3Gao.7As layer. Consequently, a single material wi th dielectric constant given by an average of those for G a A s and Alo .3Gao.7As, weighted according to thickness, was used in the model for the core layer. That is: Also, the 2D array of holes that comprise the grating was etched part of the way into the cladding layer in order to aid in the oxidation. A s a result, an additional layer with 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 in the model. The nominal thickness for this layer was estimated to be 350 nm based on the etch time. The semiconductor dielectric constants, including dispersion, were taken from the literature [50]. The dielectric constant of the oxide was measured by our group, independently, to be 2.56 [51]. Altogether, the parameters that were allowed to vary in the model were the pitch (A), the air filling fraction of the (assumed cylindrical) holes (Jf), the thicknesses of the core (tg), oxide ( £ 0 x i d e ) , and air-oxide layer( t a o ) , and the admixture of G a A s and Alo .3Gao.7As in the core layer (aGaAs)-For each combination of these parameters a full set of reflectivity spectra were cal-culated for the series of incidence angles depicted in figure 5.5, for both the s and p e's(u) = aGaAse'GaAs(CJ) + [1 - aGaAs}e'Ah3G&orAs(CJ) (5.2) a GaAs = ^GaAs/ (^GaAs + ^Al0.3Gao.7As) (5.3) 143 polarizations in the T — X direction. These were compared manually wi th those from the experiment. The model parameters yielding the best fit to al l of the data are pre-sented in Table 5.1 along wi th their nominal and/or measured values. The pitch of the photonic lattice was 550 nm, and the air filling fraction of the holes was 27%. These compare favourably (to within 3.5%) to the nominal and measured values. The total thickness of the GaAsZAlo .3Gao.7As core layer from the best fit model results compares favourably with both the S E M measurement and the nominal value supplied by the sam-ple grower. A s stated above, the G F model can not handle variation in 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. This corresponds to thicknesses of 33 nm and 50 nm for the G a A s and Alo .3Gao.7As layers respectively, which are somewhat off from their respective nominal values of 45 nm and 35 nm. No more can be said since we are unable to measure the layer structure within the core. The total thickness of the oxide layer used in the model is in good agreement with the measured value. Best agreement was obtained when it was assumed that the 2D array of holes penetrated 350 nm into the oxide layer. This was modelled as a layer wi th a reduced dielectric constant according to the air fill fraction of the holes which were assumed to have vertical sidewalls. This penetration into the oxide is not inconsistent wi th the depths registered by the probe during A F M microscopy, which were in excess of 200 nm, when it is considered that the non-vertical probe prevents the probe tip from reaching the bottom 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 with the measured spectra were within 3.5% of the measured values where measurement was possible. This 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 wi th the corresponding experimental data suggested that no further iteration of the parameters 144 Parameter Mode l value Nominal value Measured value grating pitch 550 nm 575 nm (571 ± 17) nm air filling fraction 0.27 none (0.278 ± 0.027) nm G a A s thickness 33 nm 45 nm none Alo.3Gao.7As thickness 50 nm 35 nm none total core thickness 83 nm 80 nm (90 ± 10) nm oxide layer total thickness 1900 nm 1600 nm (1800 ± 8 0 ) nm air-oxide layer thickness 350 nm none > 200 nm 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 in Figure 5.8, which is presented later in this chapter in 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 in the vicinity of the relevant resonant features were fit wi th 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 wi th the Bloch state of the photonic crystal. The Airy- l ike function is given by: A,~S L 2 r 2 [ l - c o s ( d a ) + V)] , r A S A(u)=b + s x - -A—n 0 v . ,_ (5.4) y ' 1 + r 4 - 2 r 2 cos(d£ + 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. This function was used to fit the spectra for the values of the parameters in 145 a region away from any Fano resonances. It was combined with the Fano lineshape function given by: F(u) = Fc (q + 2(Co-CoQ)/r)2 1 + (2(a) - £0)/r)2 (5.5) to yield: R{u) = A(u>) + F(Q) (5.6) 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. This procedure usually resulted in a good fit to the experimental spectrum. One such fit to an experimental spectrum obtained from grating 22 is depicted in F ig -ure 5.6. Only the lower frequency Fano-like resonance was fit over the range (6000 c m - 1 - 9000 c m - 1 ) . From this fitting procedure the mode was found to lie at 7450 c m - 1 and to have a linewidth of 105 c m - 1 . 5.6.2 Extraction of Mode Parameters from Model Spectra A slightly different approach was used in 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 wi th in-plane propagation constants / 3 i n c ± (3QX and / 3 i n c ± day. The quasi-bound modes of the system again manifest themselves as Fano resonances in the spectra, but this approach has the advantage that the modes appear almost background-free. This is the case because the non-resonant background is first order in the dielectric texture for these components, while it is of order unity in the specular component. Figure 5.7 shows sample fits to first order scattering spectra wi th the Fano lineshape function F(u), together with the corresponding specular spectrum. Notice that the first order spectra contain only a weak background, which simplifies and re-duces the number of parameters required to obtain good fits. Furthermore, as discussed 146 6000 7000 8000 9000 E N E R G Y [cm 1 ] Figure 5.6: F i t to specular reflectivity spectrum obtained from grating 22 obtained wi th light incident at 9 = 50° along the V — 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 with an A i r y function. The ini t ia l fit to the Fabry-Perot fringes was performed over a wider range of energies (6000 c m - 1 to 11,500 c m - 1 ) not shown here. The fit using a Fano lineshape and A i r y function was performed using data points in the range 6000 c m - 1 - 9000 c m - 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 in question, the neighbouring resonances associated with 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 wi th 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 than specular reflectivity spectra. The first-order diffracted light must travel down the th in 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 < U on W H on < U O H on 10.0 FREQUENCY CO/2JIC [103 cm\"1] r 10.5 (b) 11.0 (c) 11.0 FREQUENCY CO/2TIC [103 cm''] 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 —QGX field component (c) 1st order scattering +PGX field component. In (b) and (c) the symbols correspond to the model values, and the smooth lines are the result of a fit wi th a Fano lineshape function. The spectra are for 10° angle of incidence along the V — X direction. 148 debris. Also, since there is no lateral confinement, the diffracted beam spreads in the in-plane directions. This all necessitates creating a cleaved edge whence the diffracted light is to be extracted very near the textured region. Making 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 The 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 in-plane propagation vector, given by PL = — sin(0). Figure 5.8 shows the dispersion of the c resonant modes of the structure along the T — X and T — M symmetry directions of the square Br i l louin 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 Br i l louin zone by the in-plane Fourier components of the square grating. The polarization of the features in the reflectivity measurement and in the calcu-lations, remains the same within each band along a given symmetry direction of the crystal, and so the curves in Figure 5.8 have been labelled as either s or p polarized. Along the F — X direction, there are a total of 5 curves, wi th 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. Two of these exhibit large dispersion, wi th one having positive dispersion (frequency increasing with (3) and the other having negative dispersion. The remaining s polarized curve and the p polarized curve both have little dispersion. Away from zone-centre, in the V — X direction, the fourth and fifth curves, which are both s polarized, anti-cross near P/PG ~ 0.25. The 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. The inset schematically depicts the bandstructure for the case of weak texture. 150 Along the Y — M direction, there are two moderately dispersive pairs of curves wi th 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 in 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 - 1 ) , which forms half of the waveguide core. In 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. The 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 both experiment and theory. Except very near the zone-centre, where the calculated value approaches zero, the measured linewidth agrees wi th the model to within a factor of two. Away from zone-centre, the experimental and calculated linewidths both exhibit a non-monotonic variation. It should be noted that while the calculation assumes pure plane waves incident with well-defined k vectors, and an infinite, perfectly periodic photonic crystal that is free from imperfections, none of these assumptions is strictly true in the experiment. The finite size of the photonic crystal, the non-zero detector acceptance angle A9, and non-uniformities in the photonic crystal wi l l all contribute to a smearing out of the spectra, in a convoluted manner, that limits the resolution of the experiment. Quali ta-tively, these factors account for the fact that the experimental linewidth does not seem to be approaching zero at the zone-centre as it does in 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 in 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 = s p r = 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 in Fourier space by the reciprocal lattice vectors of the crystal. For a finite crystal, this function must be convolved in each of the x and y directions wi th 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 in this work, the main lobe of the sine function has a full width at half maximum 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 wi th in-plane propagation vectors given by ±npGx ± mPcy for integer m and n. Rather, to a good approximation, they contain significant field components in a range spanning Apo in each of the x and y directions, centred at ±nPox ± mPoy- The components in the band near zero in-plane wavevector wi l l be phase-matched to radiating components in the cladding air layer. When white light is used to probe along P = d)sin#, reflectivity wi 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. This occurs in the frequency range: A £ f i n = APG^ (5.8) The experimental linewidth can be approximated by adding in quadrature AcD s p r and Ad)fi n to the inherent width of the mode. Near the zone-centre, this adds approximately 153 50 c m - 1 to the inherent linewidth, bringing it into much better agreement wi th the experiment 1 . Away from zone-centre, the dispersion increases then becomes roughly constant. However, this is largely offset by the u and cos# factors in equation 5.7, which both decline, so that the net effect is to add about 80 — 90 cm _ 1 to the inherent linewidth leaving it in reasonable agreement wi th the experiment. Finally, it should be re-iterated that non-uniformities in both the periodicity, as well as in 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 in 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. The feature associated wi th the fifth band remains resolved for al l of the incident angles, and the associated linewidth varies smoothly in both 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. The experiment is qualitatively consistent wi th this: there is a larger variation in the linewidth in the fourth band than in the fifth band, and the resonance associated with 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). 1This 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. 154 200 H B 100 H 4 ^ ® — (a) (b) 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 in 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 in the spectrum. A crossed circle has, therefore, been used for the experimental data point in the fourth band at 0/6G ~ 0.25, which corresponds to 9 = 25°, in order to indicate an unknown but small linewidth. Thus, both model and experiment are consistent in demonstrating that one band involved in an anti-crossing away from zone-centre shows no dramatic variations in linewidth, while the other band exhibits a widely varying linewidth that vanishes near the point in the Bri l louin zone where maximum 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. The 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. The resonance associated wi th the lowest energy band has been highlighted in each set of spectra: it shifts up in energy wi th increasing air filling fraction, as expected. In fact, al l four bands shift to higher energy wi th more air in the crystal. Subtle differences in the precise dispersion of the bands in these other samples were not pursued due to the difficulty in extracting accurate fits to nearly overlapping Fano features in many of the spectra. 156 i i i i | i n i | i • i i | i i i1 i | i n i | i i i i | n i r-] i i i i 6 7 8 9 10 11 12 13 14 FREQUENCY CO/2TTC [io3cm-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 in by hand to serve as a guide to the eye. 157 Chapter 6 Discussion 6.1 Introduction This chapter provides an interpretation of the experimental data presented in the pre-vious chapter based on symmetry considerations as well as knowledge gained from the numerical models. The results can be interpreted entirely in terms of renormalized (cou-pled) bound slab modes of some effective slab waveguide. The effective waveguide can be thought of as one where the dielectric constant in the textured guiding core is aver-aged 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) (6-1) where Pguide 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. The dispersion relation is isotropic within the plane of the guide. Light incident from above the grating at some frequency Co at incident angle, 9, and azimuthal angle, 4>, wi l l have an in-plane momentum given by: Pine = & sin(0) cos(0)x + Co sin(0) cos( /3inc + G^i. The significant reciprocal lattice vectors for our structure in the spectral range of interest are: ±PGX, ±PGV, ±PGX ± PGV- A resonant feature appears in 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. Many of the interesting properties revealed in this study can be qualitatively ex-plained 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 The reflectivity spectrum is comprised of both resonant and non-resonant components. The non-resonant, slowly varying, component can be thought of as reflection off the \"average waveguide structure\". That is, by treating the textured layer as a uniform layer with 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. The resonant, bipolar and asymmetric feature in 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. The 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 wi th square photonic lattice. 6.3 Polarization Properties and Basic Dispersion From 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 Br i l louin zone was obtained as in Figure 5.8. Concentrating on the four prominent bands emanating from the zone-centre in Figure 5.8, their dispersion and polarization properties can be schematically summarized as in Figure 6.1. Starting along the T — X direction, three bands have s polarization and one has p polarization. The modes associated wi th al l four of these bands are primarily composed of a superposition of TE polarized field components wi th in-plane spatial variation given by {PincX ± PGX) a n d {Binci ± PGV)- These field components are depicted in Figure 6.2. Away from zone-centre, the upwardly dispersing band along r — X is primarily com-posed of a T E slab mode at (Pinc + PG)X, which can only radiate s polarized radiation. Similarly, the downwardly dispersing, lowest energy band is dominated by the T E polar-ized (Pinc—PG)X slab mode which also can only radiate in the s polarization. The two flat bands are composed principally of either symmetric or antisymmetric superpositions of 160 Figure 6.2: Dominant Fourier field components of eigenmodes in lowest four bands along the r — X {6 = 0) direction depicted in Figure 5.8: the solid black arrows represent the propagation vectors associated with 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 (Pincx ± PGy) • The symmetric combination wi l l generate a zeroth-order polarization wi th 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 super-position wi l l have the y components cancel, yielding a field entirely along the x direction that can only couple to p polarized radiation. This fully explains both 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 in the F — M direction there are two pairs of moderately dispersive bands, wi th one pair having positive and the other negative dispersion wi th respect to detuning away from zone-centre, and wi th 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 in 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, Pinc, 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. The 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 in the four bands along the r — M direction depicted in Figure 5.8: the solid arrows represent the propagation vectors associated with 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 wi l l only be either s or p polar-ized for excitation along directions that possess mirror symmetry. This can be seen from the following argument. Along such directions, parity requires that field components of the eigenmode lying symmetrically about the plane of symmetry occur in either sym-metric or anti-symmetric pairs of equal amplitude. Such pairs necessarily superimpose to give resultant electric fields which are either perpendicular or parallel to the symme-try direction and hence can only drive polarization in the zeroth order which is either purely s or p polarized. The 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° in the present case), there is no s-p or p-s scattering in the specular direction. That is to say that the polarization of the radiative component of the eigenmodes is a \"good quantum number\". Since the preceding arguments did 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 within a curve along a given symmetry direction in the bandstructure plot of Figure 5.8. Notice, however, that a given band cannot be given a unique polar-ization label because, away from directions of high symmetry, leaky eigenmodes wi l l , in general, radiate an elliptically polarized field. Indeed, the fourth highest energy band shown in 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 in this work, it is important to note that none of the above arguments depends upon the radiative nature of the exciting and reflected fields. The same arguments apply equally well to the truly bound eigenmodes below the light line, for example, in the vicinity of the first order gap of a similar structure wi th 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 in the first Br i l louin zone is a \"good quantum number\" for the Bloch states in that band. 6.4 Gaps and Symmetry Influences on Linewidths The s polarized spectra around the second order gaps near zone-centre contain two narrow features and one broad feature which has approximately the same width and energy as the lone feature occurring in the p polarization. A s shown by the model results in 2.3.5, it is expected that, at normal incidence, the two broad features become degenerate and retain a non-zero width whereas the narrow features become infinitely narrow and vanish. This behaviour may also be arrived at by the following symmetry argument. Right near zone-centre, the modes in 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 al l four T E slab modes at ±6GX and ±/?G2/ that preserve the required symmetry. Two of these modes necessarily generate polarization wi th no radiative component due to the perfect cancellation of the T E slab modes' contributions to the zeroth order polarization at 3 = 0. The 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, non-degenerate 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 in their respective amounts of localization in the dielectric versus the air. The size of the gap and the energetic ordering of degenerate and non-degenerate modes at zone-centre are both influenced considerably by the air 165 filling fraction of the photonic lattice. 6.5 Effec t o f F i l l i n g F r a c t i o n A s mentioned in 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. This is because the frequencies of the bound slab modes of the \"effective untextured\" waveguide wi 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 with the grating. Consequently the width of the gap, which is a measure of the scattering induced by the grating, w i l l also be reduced for gratings with 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. Bo th of these effects result in an increase of the frequency of the lowest band, which is consistent wi th 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, in principle, shift it to the lowest feature. However, this is not possible in 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 third highest in frequency of the s polarized modes. The 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°, = 0 wi th (a) r / A = 0.29 and (b) r / A = 0.33. The other parameters in the model are as the fitted results for grating 22. The curves have been shifted vertically for clarity. 167 the waveguide structures wi th I D texture presented in 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 sym-metric 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 in the y direction and dominated by components at ± / ? G : E ( x , z) = [Ey0eif}GXy ± Ey^-^y^z). (6.3) However, each of the ±PG components can scatter wi th 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 wi 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 wi 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 with small loss. The frequency ordering arises from the fact that the in-plane intensity profile is differ-ent 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. This 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 propor-t ion of its field energy (area under the curve) concentrated in 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. The 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 in 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 wi th (a) 25% (b) 50% (c) 75% air filling fraction. In (b) the lossy and lossless modes both modes have the same fraction of their total field energy (area under the curve) in the high-index region. The dielectric profile is indicated by the dashed line. the lossy and lossless modes both modes have the same fraction of their total field energy in the high-index region, and are, therefore, degenerate. 6.6 G a p s A w a y f r o m Z o n e - C e n t r e 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. Bo th 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 in Figure 5.8. The energy of closest approach is comparable to the gap which occurs at zone-centre. This implies that the corresponding modes are strongly mixed. As described above, away from the zone-centre, the 4th band is dominated by the Pincx + PGX TE- l ike Fourier component corresponding to a bound mode traveling in the +x direction. From 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 Pincx ± (PGX ± PGV)- This is because these reciprocal lattice vectors have the next largest magnitude and the dispersion of the untextured slab modes, which is isotropic in 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 in the Y — X direction are similar to those for detuning in the T — M direction for the modes at the \"second order\" (±PGX and ±PGV) gap. Consequently, these modes wi l l appear as two pairs of bands having positive and negative dispersion, with each pair having one band leaking s polarized radiation and one band leaking p polarized radiation. The 5th band must be one of the modes wi th negative dispersion, which, in turn, implies that it must be composed of in-phase or out of phase superpositions of Pincx — PQ(X ±y). Since it is leaking only s polarized radiation, the superposition must be such that it cancels out the Ey field, which implies that it must be the symmetric superposition (see Figure 6.6.) Therefore, in the vicinity of the anti-crossing away from zone-centre, the modes in the 4th and 5th bands must be comprised of a significant admixture of T E polarized field components at Pincx — PG(X ± y) and Pincx + P G X - Right at the anti-crossing, i.e., the point of closest approach of the 4th and 5th bands, which occurs experimentally at PIPG ~ 0.25, these three components scatter back to pincx wi th appropriate amplitudes so as to produce a zero net field. Consequently, the leaky component vanishes, as 170 Figure 6.6: Dominant Fourier field components of eigenmodes in the 4th and 5th lowest energy bands along T — X direction in Figure 5.8 in the vicinity of their anti-crossing: the solid arrows represent the propagation vectors associated with 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 in the specular reflectivity spectra. A t the anti-crossing the mode in the 4th band is completely bound 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. The dominant components of the mode correspond to the dominant scattering amplitudes from Pincx to Pincx+(nx+my)PG- Figure 6.7 depicts the spectra of the various scattering amplitudes calculated with in-plane wavevector value of Pinc/PG ~ 0.25, corresponding to 9 = 25°, at which the anti-crossing occurs as inferred from the fact that the resonant feature in the 4th band disappears from the reflectivity spectrum in Figure 5.5. The results clearly show the largest amplitudes for the components at Pinc% — PG{X^V) and PinJt + PGX, wi th those for the former two being equal as expected from symmetry considerations. 6.7 E x t r i n s i c Inf luences o n L i n e w i d t h s Referring now to Figure 5.9, the calculated width is clearly seen to be approaching zero at zone-centre; therefore, this mode becomes completely bound there. The oscillating behavior of the linewidth is due to the presence of the substrate and would not exist in the case of a three layer a i r / A l G a A s / A l oxide guide. This can be understood in terms of a simple model wherein the grating is taken to be a th in source emitting radiation leaked by the mode downward toward the substrate wi th the perpendicular component of the propagation vector given by: k\\ = (V^^f - p\\ (6.4) c This component is reflected at the oxide/substrate boundary wi th a TT phase change. Consequently, when the product k±_dQ^e is such that it returns in phase wi th the field in 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 with the field in 172 Q ID H i—i OH s a i—i w H H < U o o • (f) o o o o o • • o o (e) o o o • o o o o o (d) o o • o o o o • o (c) o o o o • o o o o (b) • o o o o o o o o (a) o • o o o o 111111111111111111 • 11111111111111111111111111111111 8 9 10 11 12 3 -1 co/27Cc [10 cm ] Figure 6.7: Scattering amplitudes calculated using the F D R S Code for s polarized light incident at f3inc « 0.25pGx corresponding to 0 = 25° along the T — X direction, where the anti-crossing occurs. The various plots depict the scattering amplitudes to s polarized light wi th in-plane wavevector: (a) f3inc (b) f3inc - pGx (c) (3inc ± pGy (d) (3inc + pGx (e) (3inc + PG[—x ± y] (f) (3inc + PG[x ± y]. The structure is the same as that for grating 22 wi th the parameters as obtained from fitting wi th the G F code. 173 the grating, radiation is suppressed, the mode becomes less leaky and the linewidth is decreased. This 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 Bloch state in this band are evanescent in 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. When a high-index-contrast photonic lattice is used, the first-order Fourier component of the \"scattering potential\" is larger than in the weakly textured case . The larger diffraction rate which results manifests itself not only in broader linewidths but also an enhanced coupling between the vertical and distributed in-plane cavities. 174 C h a p t e r 7 S u m m a r y a n d C o n c l u s i o n s 7.1 S u m m a r y A numerical model was developed for calculating specular reflectivity and scattering from an infinite 2D photonic crystal embedded in 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 in 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 wi th in 1% for the mode frequencies and to within 7% for the linewidths. A fabrication procedure was developed for making structurally robust and robustly guiding thin semiconductor slab waveguides wi th 2D square photonic lattices of air-holes that completely penetrate the core layer. The procedure involved using electron beam lithography and E C R plasma etching to define the texture in an 80 nm thick GaAs/Alo .3Ga 0 .7As core on the surface of a ~ 1.8/z m layer of Alo.9sGao.02As grown on a G a A s substrate using Molecular Beam Epitaxy. The texture consisted of a square lattice of cylindrical airholes. These varied in pitch from 500 nm to 575 nm and in air filling fraction from 0.28 to 0.49. The sample was then placed in a warm, moist environment to oxidize the Alo.9sGao.02As layer in 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 - 1 and 14,000 c m - 1 . The experimental 175 spectra bear a strong resemblance to those calculated using the numerical models. Bo th the experimental and model spectra were fit wi th a combination of an A i r y function and Fano-lineshape function in 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 Br i l louin zone. B o t h of these effects are much more pronounced than those that are typical of weakly textured slab waveguides. The spectra exhibited well-defined polarization properties along the symmetry direc-tions of the crystal in that the polarization of leaked radiation was either purely s or purely p . Near zone-centre, in the vicinity of the second order gap, and for detuning along the T — X direction of the crystal, three modes appeared in the s polarization and one in the p polarization. One of each of the s and p polarized modes exhibited little dispersion over a large portion of the 1st Br i l louin zone. For detuning along the T — M direction, two moderately dispersive pairs of features appeared, with one of each pair appearing in the s and p polarization. There was also generally good agreement between the model and experiment for the linewidths of the resonant features associated wi th the modes, except where the linewidths were predicted to vanish (or become narrower than ~ 80 c m - 1 ) ; in this case the lower l imit on the experimental resolution made a quantitative comparison impossible. Nevertheless, linewidths (lifetimes) were observed that varied considerably wi thin a given band and between bands in a manner that agreed qualitatively with 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 in both the model and experimental spectra. Finally, an anti-crossing was observed away from any high symmetry points of the 176 crystal, simultaneously with the vanishing of the linewidth of one of the features associ-ated wi th the anti-crossing bands. 7.2 C o n c l u s i o n s Features in specular reflectivity spectra from 2D textured semiconductor slab waveguide-based photonic crystals can be used to study the intrinsic properties of resonant elec-tromagnetic modes localized in the vicinity of the textured core. B y comparing model calculations wi th the experimental results, it has been clearly shown that in order to fully characterize these electromagnetic excitations their disper-sion, lifetimes, and polarization must be determined. Analysis of the results indicates that for the \"membrane-like\" structures studied in 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. The degree of mixing introduced by the 2D texture, as inferred from the width of the anti-crossings, is ~ 10%. The polarization properties can be explained entirely in terms of symmetry arguments regarding the coupling of the zeroth order radiative component with the bound-slab-mode-like T E field components at ±/3GX and ±PGV- The gross dispersion properties can be explained qualitatively from kinematical considerations. The oscillation of the linewidth with in-plane wavevector was explained by the cou-pling of the Bloch states of the photonic crystal embedded in 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, in general, finite. These degenerate bands at zone-centre offer small group velocity and strong coupling to external radiation modes. This 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. This combination of small group velocity and long lifetimes is of particular interest in active laser devices that incorporate 2D photonic crystals. The anti-crossing observed away from the zone-centre, is in a certain respect, \"acci-dental\" in 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) dis-persion and polarization properties of this interesting new class of materials, obtained by developing and using a numerical solution of Maxwell 's equations to interpret broad-band 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 F u t u r e D i r e c t i o n s Work in this area is proceeding along several lines. One is the inclusion of electronic resonances through incorporation of quantum wells, wires, and dots in 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 addi-t ion to allowing for greater scattering, this also allows an additional degree of freedom in 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 de-sign and realization of more elaborate structures, such as the integration of several layers of slab waveguide crystals and the inclusion of periodic defects. Work is already under-way in our group using the latter strategy. A modified version of both the F D R S code and the G F code, capable of modelling infinite photonic lattices wi th periodic defects embedded in slab waveguides, has been developed. These structures hold promise for en-gineering very weakly dispersive bands predicted to play important roles in enhancement of non-linear optical cross-sections [11]. 179 Bibliography J . D . Joannopoulos, Robert D . Meade, and Joshua N . W i n n . Photonic Crystals: Molding the Flow of Light. 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B, 56:959, 1997. 183 Appendix A Generalization of FDRS Method to an Arbitrary 2 D Lattice The 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 2 of arbitrary 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 ( A . l ) b = bb — bcosipx + 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 2D photonic crystal with respect to our coordinate axes. We now generalize equations 2.33 -2.35 in accordance wi th our oblique mesh to: k-a r ,ikxa. ,—ikxo,.,.. A / A r-\\ —- = kx RJ exp(——) - exp(—-— ) \\ / ia = KX (A.5) |a| 2 2 = kb ~ [ e x p ( ^ ) - e x p ( — ^ ) ] / z & = Kb (A.6) = kz « [ e x p ( ^ ) - exp(^-)]/zg ± KZ (A.7) so that we may write: K = KXX + Kbb + KZZ (A.8) 184 We note from equation A . 2 that: so that we may write y=^ — cotipx (A.9) k • b ky = k • y = — k • xco tV ' = kb — kxcotip. (A.10) We may now proceed as before wi th equations 2.31 - 2.32, carrying out the cross-products, 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)x = kyEz-kzEy = uBx (A. 11) Using equation A . 10 this becomes: kbEz — kxEz cot tp — kzEy = uBx. (A. 12) Now, making the approximations for k using equations A . 5 - A . 7 , we obtain: exp(ikb~)Ez exp(-ikX)Ez exp(ikx^)Ez exp(-ikx^)Ez Bx « —A —A A c o t ib H A C ot ip ibu ibu iau iau exp(ikz9-)Ey exp(-ikz9-)Ey - + \" ~ (A.13) igu igu We now note that: * \\ h \\ r _i_-i (bcosipx + bsinipy) (bxx + byy) exp{±ikb-) = e x p ( ± z k • - ) = e x p [ ± 2 k J = exp[zbk 2 J (A.14) and therefore, similarly to the second of expressions 2.46 exp(±ikb^)Fl ^Fl(xT^,yTj,z) = Ft(r =p C) (A.15) where, F\\ denotes a component of the E , D , B or H field, <=> denotes Fourier transfor-mation, and £ is now defined as: C = \\ (A.16) 185 We now take the inverse Fourier transform of equation A . 13, making use of equation A . 15 and the first and third of equations 2.46, to yield: M _g,(r + Q - E,(r - 0 g.(r + «) - S,(r - a) + ^(r + 7) - g,(r - 7 ) i&w iau igui (A.17) Isolating the Ey terms in the preceding equation, we obtain: q,(r + 7 ) - ^ ( , - 7 ) ^ < l ( r ) g i ( r ) + g . ( r + 0 - fr(r- < ) + c o t ^ ( r + «) - S.(r - a) zgu; 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 Bx above, we first obtain the expression for the Dz component given by: ^ Hy(r + a ) - Hyjv - a ) Hx(r + Q - Hx(r - Q | ^ Hx(r + a) - Hx(r - a) iau ibu iau (A.19) so that: Ez(r) « g y ( r + oQ ~ # y ( r ~ « ) #*( r + Q - # » ( r - C) | c Q t ^ H x ( r + a ) - Hx{r - a ) 2 iaue(r) ibue(r) iawe(r) (A.20) In this manner, expressions similar to those given in 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 in 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 within the unit cell of the photonic crystal on which evaluation of the fields (squares) and e' and / / (black dots) are required. Notice that in this scheme (as before), we take a and b along the body diagonals of the prim-itive cell of (the underlying Bravais lattice of) the photonic crystal we wish to model. 186 Explic i t ly : a = ( A 1 + A 2 ) / A ^ (A.21) b = ( A i - A 2 ) / i V (A.22) where N2 (N along each primitive lattice vector) is the number of mesh points we desire for the fields. We 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. The dielectric constant is sampled at points represented by the black dots. The electric or magnetic fields are sampled only on the points wi th squares. A i and A 2 represent the primitive lattice vectors of the photonic crystal. The reciprocal lattice vectors for the general 2D oblique lattice with primitive lattice vectors, A 1 ; A 2 , are given by: G L = -J—A r ~ r - ( ^ \" A**y) (A.23) Alx^-2y — A2x^-\\y 2-7T G S = - A — A — 4 - 4 - ^ - A^y) (A-24) ^•\\x^2y — s\\2xA\\y The reciprocal lattice is similar to the direct lattice but rotated by 90 degrees. It is depicted in figure A.2. As 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 with 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 in equation A .8 may be used, and where h = {Glim + P)-b. (A.25) The dispersion relation for this mesh is obtained from: Ld2/j,Qeoe'I/NIKHK = ^\\l-coskxa] + ^ [l-coskbb] + ~[l-coskzg] + ^- cosip sin sin ^ ' a2 b2 9 ab 2 2 (A.26) where: | A i | 2 - | A 2 | 2 I K , C 0 S V > = , ' 1 n ' ' , A.27 | A 1 + A 2 | | A 1 - A 2 | v ' This dispersion relation may be solved, as before, for kz to yield two solutions which are then used to generate K Z ' S corresponding to incoming and outgoing plane waves. 189 "@en ;
edm:hasType "Thesis/Dissertation"@en ;
vivo:dateIssued "2006-05"@en ;
edm:isShownAt "10.14288/1.0085505"@en ;
dcterms:language "eng"@en ;
ns0:degreeDiscipline "Physics"@en ;
edm:provider "Vancouver : University of British Columbia Library"@en ;
dcterms:publisher "University of British Columbia"@en ;
dcterms:rights "For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use."@en ;
ns0:scholarLevel "Graduate"@en ;
dcterms:title "Photonic bandstructure in semiconductor slab waveguides with strong two-dimensional texture"@en ;
dcterms:type "Text"@en ;
ns0:identifierURI "http://hdl.handle.net/2429/18270"@en .