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Nonlinear response of two-dimensional waveguide-based photonic crystals and microstructured fibres Banaee, Mohamad G. 2002

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Nonlinear response of two-dimensional waveguide-based photonic crystals and microstructured fibres by Mohamad G. Banaee B.Sc, The University of Tehran-Iran, 1991 M.Sc, The University of Tehran-Iran, 1994 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES (Department of Physics and Astronomy) We accept this thesis as conforming to the required standard TH& UNIVERSITY OF BRITISH COLUMBIA October 11, 2002 © Mohamad G. Banaee, 2002 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia Vancouver, Canada DE-6 (2/88) ABSTRACT ii The nonlinear response of two-dimensional waveguide-based photonic crystals and photonic crystal fibres is in-vestigated in this thesis. First, the third order nonlin-ear response of planar waveguide-based photonic crystals is studied theoretically to estimate the influence of this nonlinear susceptibility on the specular reflectivity spec-trum of these structures. It is shown that coupling of the incident field with leaky photonic modes introduces features in the specular reflectivity spectrum that are de-pendent on the input intensity. By changing the input in-tensity, the reflectivity of this structure at a desired wave-length can, theoretically, be switched from zero to one. This nonlinear property might have potential application in all-optical switching devices. Both degenerate (single beam) and nondegenerate (dual beam, pump/probe) ge-ometries are considered. In the second part of the thesis, the output spectrum of a microstructured fibre, which has a two-dimensional photonic crystal cladding is investigated as a function of power when ~100 fs laser pulses are launched into the fibre at an 80 MHz repetition rate. For launched average powers of ~20 mW to ~100 mW, the output spectrum is dramatically shifted and broadened compared to the in-put laser spectrum. This occurs through a combination ABSTRACT iii of nonlinear optical effects that are not currently un-derstood. The detailed nature of the spectrum and the temporal properties of the light emitted from the pho-tonic crystal fibre depend on the excitation wavelength and the coupling geometry. Under some conditions the spectrum consists of a series of discrete red-shifted com-ponents that shift monotonically further to the red at higher input powers. In this case the temporal shape of the output beam is found to be strongly asymmetric, with a rapid leading edge, and a slower decay. Under other conditions, the spectrum is more like a continuum, and the behaviour in the time domain is symmetric. CONTENTS iv C O N T E N T S A B S T R A C T i i C O N T E N T S iv LIST O F T A B L E S vi LIST O F F I G U R E S vi i A C K N O W L E D G E M E N T S x i 1 I N T R O D U C T I O N 1 2 T H I R D - O R D E R N O N L I N E A R R E S P O N S E O F 2 D P L A N A R P H O T O N I C C R Y S T A L S 9 2.1 N O N L I N E A R P O L A R I Z A T I O N 9 2.2 G R E E N ' S F U N C T I O N F O R M A L I S M . . 11 2.3 D E G E N E R A T E K E R R E F F E C T 25 2.4 N O N D E G E N E R A T E K E R R E F F E C T . . 38 3 P H O T O N I C C R Y S T A L FIBRES 42 3.1 E X P E R I M E N T S E T U P 45 3.2 P H A S E M A T C H I N G 50 3.3 E X P E R I M E N T P R O C E D U R E 52 3.4 RESULTS 55 3.5 C O N C L U S I O N 67 CONTENTS v A F O U R I E R C O E F F I C I E N T S C A L C U L A T I O N 69 B L A B V I E W C O D E S 71 B I B L I O G R A P H Y 72 LIST OF TABLES vi LIST OF T A B L E S 2.1 List of common third-order nonlinear phe-nomena 10 3.1 Photonic crystal fibre specifications 46 3.2 Optical elements of the cross-correlation experiment 47 B . l LabView codes 71 LIST OF FIGURES vii 1.1 Schematic diagram of one-dimensional (ID), two-dimensional (2D), and three-dimensional (3D) photonic crystals 2 2.1 Two-dimensional waveguide-based photonic crystal with a square lattice of cylindrical holes 13 2.2 Linear specular reflectivity for S-polarized (solid line) and P-polarized (dashed line) incident plane waves 20 2.3 Dispersion diagram of the four lowest ly-ing modes along the X-direction of the square reciprocal lattice near zone-centre. . 21 2.4 Real space plot for Sl-mode at /3x/(3g = 0.01 22 2.5 Real space plot for S2-mode at (3x//3g = 0.01 23 2.6 Real space plot for Pl-mode at j3x/(3g = 0.01 23 2.7 Real space plot for S3-mode at (3x//3g — 0.01 24 2.8 The inverse Q of the same bands shown in Fig.(2.3) 25 2.9 Specular reflectivity of the. S3 mode for different incident intensities; The dashed line shows the linear reflectivity and the solid lines are for intensities of 765, 2342, and 3871 kW/cm2 for increasing blue shifts respectively 34 LIST OF FIGURES viii 2.10 Specular reflectivity for the SI mode for different incident intensities; The dashed line shows the linear reflectivity and the solid lines are for intensities of 30, 48,107, and 155 kW/cm2 for increasing blue shifts respectively 35 2.11 Bistable behaviour of the reflectivity for the S3 mode 36 2.12 Reflectivity in the vicinity of the S3 mode for nonzero and zero off-diagonal elements of the tensor 37 2.13 Nondegenerate response of the SI mode to different pump intensities, with the pump frequency fixed at 10744cm - 1. Dashed line shows the linear spectrum; solid line and dash-dot line are reflectivities pumped at 1720 and 305SkW/cm2 respectively 40 2.14 Nondegenerate response of the S3 mode to different pump intensities, with the pump frequency fixed at 10744cm - 1. The dashed line shows the the linear spectrum; the solid line and dash-dot line are reflectivi-ties pumped at 1720 and 3058kW/cm2 re-spectively. 41 3.1 A scanning electron micrograph, in cross section of the photonic crystal fibre used for the studies described in this chapter . 42 3.2 Dispersion of the photonic crystal fibre de-scribed in Table(3.1) provided by the fibre manufacturer 45 LIST OF FIGURES ix 3.3 Setup of the optical cross-correlation ex-periment 48 3.4 Phasematching geometry for sum genera-tion of the gate and fibre beams 51 3.5 Output of the Ti-sapp laser at 803 nm. . . 56 3.6 Output of the Ti-sapp laser at 829 nm. . . 57 3.7 Spectrum of the fibre with 20 mW input at 803 nm 57 3.8 Spectrum of the fibre with 100 mW input at 803 nm 58 3.9 Spectrum of the fibre with 100 mW input at 829 nm 58 3.10 Intensity versus wavelength at the peaks of the discrete spectra in Figs.(3.7) and (3.8). From left to right these spectra were obtained with nonlinear crystal phase match-ing angles and delays set to (21.43°, 19.8 ps), (22.93°, 26.4 ps), and (26.87°, 59.4 ps). 61 3.11 Intensity versus delay between the gate and fibre beams at the peak wavelengths of Fig. (3.10). The left spectrum corre-sponds to the left spectrum of Fig. (3.10), and so on 62 3.12 Three dimensional plot of intensity ver-sus delay and wavelength for the discrete spectra (composite of Figs. (3.10) and (3.11)). 62 LIST OF FIGURES x 3.13 Intensity versus wavelength at three dif-ferent wavelengths in the continuum spec-trum in Fig. (3.9). From left to right these spectra were obtained with nonlinear crys-tal phase matching angles and delays set to (24.08°, 29.7 ps), (25.58°, 39.6 ps), and (27.09°, 62.7 ps) 63 3.14 Intensity versus delay between the gate and fibre beams at the peak wavelengths of Fig. (3.13). The left spectrum corre-sponds to the left spectrum of Fig. (3.13). and so on 64 3.15 Three dimensional plot of intensity ver-sus delay and wavelength for the contin-uum spectrum (composite of Figs. (3.13) and (3.14)) 64 3.16 Delay versus peak up-converted wavelengths for discrete (squares) and continuum (cir-cles) spectra. The dashed line is the de-lay expected from Fig. (3.2) assuming all of the light propagates in the lowest order mode of the fibre 65 ACKNOWLEDGEMENTS xi F i r s t a n d foremost I w o u l d l ike t o t h a n k m y superv isor , P r o f . Jeff F . Y o u n g for his extensive assistance a n d en-couragement t h r o u g h o u t th is project . H i s advices were r e a l l y h e l p f u l for t h e o r e t i c a l a n d e x p e r i m e n t a l p r o b l e m s o l v i n g . I w o u l d also t h a n k a l l m e m b e r s of P h o t o n i c s N a n o s -t r u c t u r e lab for t h e i r assistance a n d guidance especia l ly A l l a n C o w a n a n d C h r i s t i n a K a i s e r . M o s t l y I w o u l d l ike t h a n k i n g m y dear wife for her i n -credib le s u p p o r t a n d u n d e r s t a n d i n g . CHAPTER i. INTRODUCTION 1 C H A P T E R 1 I N T R O D U C T I O N The term "Photonics" describes a technology in which transmission and processing of data occurs partially or entirely by means of photons. A motivation for this technology is that by replacing relatively slow electrons with photons as the carriers of information, the band-width and speed of communication and computation sys-tems can be dramatically increased. In telecommunica-tion systems, optical fibres with their huge optical band-width already carry most of the information, but switch-ing mostly relies on electronic or optoelectronic compo-nents. Even by using optoelectronic switches, optical sig-nals have to be converted to electronic signals and then the switched signals have to drive an optical transmitter. However, to realize the full benefits of optical technology it will be necessary to build all-optical networks. It is self-defeating to convert the signal from optical to elec-tronic forms every time it needs to be routed or switched. Photonic crystals (PCs) offer a promising solution for this problem. PCs are microstructured materials in which the dielectric constant is periodically modulated on a length scale comparable to the desired optical wavelength of operation. They are usually viewed as the optical ana-logues of semiconductors: the periodic dielectric lattice modifies the propagation properties of the light in much CHAPTER i. INTRODUCTION 2 the same way as the microscopic atomic lattice of a semi-conductor crystal modifies the propagation properties of electrons. Multiple interference between waves scattered from each unit cell of the structure may actually result in a "photonic bandgap", or a range of frequencies within which no propagating electromagnetic modes exist. Fig. (1.1) shows a diagram of one, two, and three-dimensional photonic crystals. Figure 1.1: Schematic diagram of one-dimensional (ID), two-dimensional (2D), and three-dimensional (3D) photonic crystals. Although the bandgap property of photonic crystals facilitates the control and manipulation of light prop-agation, making optical switching elements from these structures requires the ability to dynamical tune their CHAPTER i. INTRODUCTION 3 properties. There are several ways to parametrically ad-just the photonic crystal bandstructure. Many of these methods involve incorporating foreign materials into the photonic crystals, materials that can have their refrac-tive index modified by varying some external parame-ter. Examples include liquid Crystals [1], magnetic ma-terial such as ferrite [2], and thermally-sensitive poly-meric material such as Poly(3-alkylthiophene)[3]. The alternative approach and the one that offers the ulti-mate in switching speed, involves using light itself to influence the propagation properties of photonic crystals made from materials that respond nonlinearly to light intensity[4, 5, 6, 7, 8, 9, 10, 11, 12]. Nonlinear optics of photonic crystals is therefore a rapidly growing area of research. The strong optical con-finement possible in these structures helps to reduce the optical power levels needed to produce usable nonlinear effects, as compared with bulk nonlinear optical materi-als. Another advantage of P C systems for nonlinear op-tics is that the photon dispersion relation and the density of electromagnetic states can be engineered over a wide range of relevant parameter space, and these factors are known to play important roles in nonlinear processes like harmonic generation and frequency mixing. Huge non-linear effects have already been reported [13, 14, 15, 16] in photonic-crystal fibres excited by unamplified pulsed laser systems. If research on the nonlinear-optical prop-erties of PCs and P C fibres can result in a quantitative understanding of these complex processes, then many possible all-optical devices might be realized.' Examples CHAPTER i. INTRODUCTION 4 include; limiters and switches, compact pulse compres-sors, and frequency converters [17, 18, 19, 20]. This thesis is concerned primarily with third-order nonlinear processes, one of which is the "optical Kerr effect" that effectively describes a material in which the refractive index is modified in proportion to the inten-sity of light in the material. That is n — n0 -\-n2I, where n0 refers to the linear refractive index of the material, ri2 is the nonlinear Kerr coefficient, and I is the light in-tensity. In recent years the influence of the optical Kerr effect on the band structure of bulk photonic crystals has been considered theoretically [21, 22, 23, 24] and experi-mentally observed in ID silicon crystals using free-carrier effects [25]. If free carriers are not involved, the Kerr ef-fect phenomena can be used at very high speeds limited only by the optical pulse duration. The main challenge in this area is to develop a quantitative and efficient non-linear numerical modelling tool, and use it to engineer structures that exhibit large nonlinear effects at moder-ate optical intensities that are compatible with optical communication systems (hundreds of milliwatts average power at GHz repetition rates). Severe difficulties in making three-dimensional pho-tonic crystals, and the existence of a mature technol-ogy for manufacturing electronic semiconductor circuits, make 2D planar photonic crystals an attractive solution for integrating various optical functions needed for pro-cessing information. Like bulk two-dimensional photonic crystals, planar photonic crystal semiconductor waveg-CHAPTER i. INTRODUCTION 5 uides, like in Fig.(1.1) (2D), can possess pseudo-photonic bandgaps that can be used to effectively localize pho-tonic modes [26, 27, 28, 29, 30, 31, 32, 33]. Electro-magnetic eigenmodes of two-dimensional planar photonic crystals (PPCs) can be classified as guided, leaky, or ra-diative. Both guided and leaky modes are substantially localized to the textured slab waveguide, while radia-tive modes are roughly uniformly distributed throughout all space. Guided modes are completely bound by total internal reflection to the waveguide core (they have an infinite lifetime) but leaky modes can couple to modes that propagate away from the waveguide, so they acquire a finite lifetime. Although the nonlinear properties of both guided and leaky eigenstates are of substantial in-terest, this thesis deals strictly with leaky modes. Leaky modes have been shown by this group [34] to strongly en-hance the conversion efficiency for second harmonic gen-eration from 2D photonic crystals. In that work, it was shown how the leaky modes of the 2DPPCs can really be thought of as microcavity resonances, similar to resonant photonic states in ID Fabry-Perot cavities. Chapter 2 of this thesis extends this second harmonic study to con-sider the way in which the third order optical Kerr effect can be used to optically modify leaky mode properties in 2D PPCs . Chapter 2 describes a Green's function formalism de-veloped and applied by the author to calculate the fields excited in the textured part of 2DPPCs by plane waves incident from the top half space. One of the advan-CHAPTER i. INTRODUCTION 6 tages of using a textured slab waveguide instead of a ID Fabry-Perot cavity to enhance these nonlinear ef-fects is the greater control the PPCs offer over the re-values (mode linewidth or full-width at half maximum divided by the centre frequency) and dispersion of the leaky modes. The fully self-consistent calculation shows intensity dependent changes of the reflectivity spectrum associated with the optical renormalization of leaky pho-tonic modes. The result of the calculation also shows that by changing the input intensity, the reflectivity of this structure at a desired wavelength can be switched from 0 to 1, which demonstrates the dynamic tunabil-ity of this structure with light intensity. Calculations also reveal that the magnitude of the optically induced frequency shift strongly depends on the bandwidth and spatial profiles of these leaky photonic eigenmodes. Both degenerate (a single strong excitation beam) and nonde-generate (one strong beam influencing the reflectivity of a weaker, independent probe beam) optical Kerr geome-tries are considered. Although preliminary experimental results related to these calculations have been obtained, they are not discussed in this thesis. The effects predicted in chapter 2 will ultimately be measured using pulsed laser sources to excite the sample, and both time-resolved and time-integrated spectroscopy will be used to characterize the nonlinear reflection. In order to develop the relevant experimental techniques on a simpler, but related system, the author set up a time-resolved optical excitation and probe system to charac-terize the nonlinear transmission of ~100 fs long laser CHAPTER i. INTRODUCTION 7 pulses through a commercially available photonic crys-tal fibre. Not only did this serve to help develop the apparatus needed to study the PPCs, it also revealed some novel results that might shed light on the nonlin-ear mechanisms at work in these fibres. Chapter 3 describes the apparatus developed by the author to measure the time-integrated spectrum and the temporal properties of the broadband light emitted from these fibres when excited by more than 20 mW of mode-locked laser pulses at a repetition rate of 80 MHz. The spectra are recorded using a Fourier transform spectrom-eter. The time-domain investigations involved an op-tical cross-correlation experiment in which a nonlinear B B O (Beta-Barium Borate) crystal is used to produce a sum frequency signal from a portion of the 100 fs T i -Sapp laser beam and the output beam from a photonic crystal fibre excited by the same 100 fs pulses. The ex-periment shows that the output spectrum of the fibre strongly depends on the input intensity and wavelength of the pump laser. At two different pump wavelengths, discrete shifts and supercontinuum generations are ob-served under different in-coupling conditions. The most interesting and novel result is that the temporal evolu-tion of the up-converted signal for these two situations is markedly different. When the output spectrum consists of a series of discrete, broadened peaks, their temporal profile is highly asymmetric with a sharp leading edge, followed by a relatively long decay. In the supercontin-uum case the temporal profile of the up-converted signal is symmetric. These observations may.help identify the CHAPTER i. INTRODUCTION 8 nature of the complex nonlinear mechanisms responsible for these broadenings. CHAPTER 2. THIRD-ORDER NONLINEAR RESPONSE OF 2D PLANAR PHOTONIC CRYSTALS 9 C H A P T E R 2 2.1 Nonlinear Polarization One way to treat theoretically nonlinear optical effects is to expand the polarization of a medium under the influence of an applied electric field as a power series: P(t) = P ( 0 )(0 + P(1){t) + P ( 2 ) ( t ) + P ( 3 ) ( t ) + ... (2.1) in which P^> (t) is proportional to the nth power of the applied electric field. In Eq.(2.1), terms with n > 2 are responsible for nonlinear polarization. The third order nonlinear polarization, transformed to the frequency do-main is: P ( 3 ) M = X ( 3 ) ( - ^ l a ; ! , ^ , ^ ) ^ ^ ) ^ ) ^ ) (2.2) with bja = u\ + 0J2 + W3 and where X* ^ (—coa] wi> ^ 2 , ^ 3 ) represents the third-order susceptibility of the medium, which is in general dispersive. Table (2.1) summarizes the common third-order nonlinear phenomena [35], and associates names with the different physical responses. The distinguishing feature of each distinct process is the CHAPTER 2. THIRD-ORDER NONLINEAR RESPONSE OF 2D PLANAR PHOTONIC CRYSTALS 10 Table 2 . 1 : List of common third-order nonlinear phenomena. degeneracy and signs of the frequency arguments of the general third-order susceptibility. Process D.C. Kerr effect (quadratic electro-optic effects) X ( 3 ) ( - ^ : 0 ,0 , w) D .C . Induced second harmonic generation Third harmonic generation X ( 3 ) ( - 3 o ; : U,U,UJ) Four wave mixing X ( 3 ) ( - ^ 4 ; Ui,U2,OJ3) Third-order sum and difference frequency mixing X ( 3 ) ( -^3i±wi,a ;2 , a ;3) Optical Kerr effect (optically-induced birefringnece) Cross-phase modulation, Stimulated Raman scattering Stimulated Brillouin scattering Optical Kerr effect (intensity-dependent refractive index) Degenerate four wave mixing Self and cross-phase modulation Self-focusing X ( 3 ) ( - u ; : UJ, -u,u>) Two photon absorption, ionisation, and emission x ( 3 ) ( - ^ i ; - w 2 , w 2 , ^ i ) In the following section a Green's function formalism is explained, which can be used to calculate the linear and nonlinear response of two-dimensional wave-guide CHAPTER 2. THIRD-ORDER NONLINEAR RESPONSE OF 2D PLANAR PHOTONIC CRYSTALS 11 based photonic crystals. The nonlinearity considered in sections (2.3) and (2.4) is the optical Kerr effect in which the refraction index of the medium depends on the in-tensity of light (I), n = n0 + ml. For the degenerate Kerr effect, 712 in the above equa-tion is related to the third-order susceptibility of the medium[35] as: SRex^i—u> : UJ, —OJ, UJ) 7 1 2 = A 2 • 2.2 Green's function formalism The model developed by the author1 to treat the third order nonlinear response of planar waveguide based pho-tonic crystals is an extension of the linear optical model [33, 37, 38]developed previously to describe the disper-sion of photonic eigenstates in this geometry. This sec-tion therefore starts with a brief review of the general Green's function formalism used to efficiently solve the Maxwell's equations in this geometry. The following sec-tion extends the discussion to describe how the third-order nonlinearity was incorporated. The numerical model calculates the reflectivity spectrum of plane waves inci-dent from the top half space for different incident field 'The work described in this chapter was recently pub-lished in Ref[36] CHAPTER 2. THIRD-ORDER NONLINEAR RESPONSE OF 2D PLANAR PHOTONIC CRYSTALS 12 strengths. The situation in which the 2DPC is excited only by one strong pump beam, and the other situation where it is excited by a strong pump and a weak sig-nal beam at a different wavelength, are considered sepa-rately. Fig. (2.1) shows a schematic diagram of the specific 2DPC which was used in the nonlinear simulations de-scribed next. It consists of an 80 nm thick GaAs core layer that is textured by a square lattice of air holes (with 164nm radius), which penetrate the core layer completely on a pitch of 500 nm. Beneath this 2DPC was an 1800 nm thick alumina layer and a GaAs substrate. The fab-rication process used to make samples like this consists of the following steps: Molecular beam epitaxy (MBE) is used to grow single-crystal layers of Alo.9sGao.02As and GaAs on the GaAs substrate. Etching the waveguide through a mask of P M M A (polymethyl-methacrylate) makes the holes that define the photonic crystal. The holes are formed in the mask using electron beam lithography system and then the etching is done with an electron cyclotron res-onance (ECR) plasma etcher using chlorine. After etch-ing, the structure is exposed to a moist environment at 425°C which converts the Alo.98Gao.02As layer into an aluminium oxide (alumina) layer with a refraction in-dex of n~1.6 [39, 40] (the linear refraction index of GaAs is around ~3.4 for the range of wavelengths considered here). This oxidization results in stronger confinement of the light in the GaAs layer due to the higher (compared CHAPTER 2. THIRD-ORDER NONLINEAR RESPONSE OF 2D PLANAR PHOTONIC CRYSTALS 13 to the Alo.98Gao.02As layer) refractive index contrast be-tween the GaAs waveguide layer and the layer beneath. Figure 2.1: Two-dimensional waveguide-based photonic crystal with a square lattice of cylindrical holes. We are seeking an integral solution of Maxwell's equa-tions in the textured area of the two-dimensional pho-tonic crystal etched with a square lattice of air holes. When this photonic crystal is excited by an incident field with electric field Einc(f), the total field inside the grat-ing will satisfy an inhomogeneous equation derived from the general Maxwell's equations [41], u? U) 2 V x V x ^ ( r ) - - y ^ ( f ) = 47T^X{r)E{r) (2.3) CHAPTER 2. THIRD-ORDER NONLINEAR RESPONSE OF 2D PLANAR PHOTONIC CRYSTALS 14 where we assume a harmonic, e~%U)t time dependence for all fields. x(f) is a combination of the susceptibility of the untextured background and that which describes the scattering centres etched into the dielectric media: x ( 0 = X*(r) + Ax(r) (2.4) XB(T) represents the linear susceptibility of the back-ground and Ax(r) represents the linear and nonlinear susceptibility of the textured core layer. Therefore Eq. (2.3) is expressed as: V x V x E(r) - ^ ( 1 + 4TTX_ ( f )£(f ) ) = 4 T F ^ A X(r)E(f) (2.5) The Green's function solution of the above equation is[42]: E(r) = Ehom(r) + f dr' GB (f,P) A x{r')E{?) (2.6) where Ehom(f) is a solution of the homogeneous wave equation. These homogeneous solutions are the fields generated in the background, untextured multi-layer, when driven by an incident plane wave from the top half space. GB {r,r') is the solution of Eq.(2.5) for a point source in the textured region . If the medium is a slab waveg-uide structure and is textured only in the core layer with a periodic pattern, then (Appendix A) Eq.(2.6) Fourier transforms in the plane to yield: CHAPTER 2. THIRD-ORDER NONLINEAR RESPONSE OF 2D PLANAR PHOTONIC CRYSTALS 15 Ec{u> : Pine + A»; z) = E^iu : A n c + Pn; z)+ S = ( W : + A>> 2 , : ftnc + An z ' ) (2.7) —* where f3n is the reciprocal lattice vector of each Fourier component of the (in this case) square lattice, pn = pg(ix + jy),P9 = T (2.8) where A is the pitch of real space lattice. In all of these equations {3inc is the in-plane wave vector of the incident field. A detailed derivation of the Green's function that propagates the perturbing polarization into the same tex-tured slab area can be found elsewhere [33, 37, 38], and the result is: gc {UJ : P,z,z) = — j ^ x {[${z - z'y"^z-^ + 6(z' - z)e-^c(P)(z-z') + r -iuc{0){z+z'-L) , iuc(p)(z+z'+L), L Sup ' Sdown ' r s u r r S i m A ^ m - ' + 2 L ) + e - i ^ ' « ^ - ' - 2 i ) ) ] i - ] s ( / 3 ) s ( / 3 ) + [0(z - Z ' ) e ^ m z - Z , ) + *»rp*™e ]p c + ( /5 )p c + ( /3 )+ Up CHAPTER 2. THIRD-ORDER NONLINEAR RESPONSE OF 2D PLANAR PHOTONIC CRYSTALS 16 -iwc(P)(z-z'-2L) [6{z' - z^Mz-z1) + W i w ^ ]pc_(/3)pc_(/5)+ ^Pdown^ - - C ( / 3 ) ( , + , ' + i ) p c + ( / 3 ) p c _ ( / 3 ) ] - — - z')zz (2.9) The various reflection coefficients in this Green's func-tion, r s , r s , , r0 , and r D , are the net reflection ^ »ixp7 "down' yup' Pdown coefficients of the s and p polarized fields inside the tex-tured layer as they try to propagate upwards and down-wards respectively. They include all multiple reflections from the untextured background. The other parameters appearing in the Green's function include D, = l - r a rSd e^2L o o Up o down ( 2 - 1 0 ) where L is thickness of the slab. The unit vectors s((3), —* —* —* pc+(f3), pc-(f3), and z(/3) are basis vectors specific to each Fourier component of the electric field. They are defined as: s(/3) = k(/3) x z(/3) (2.11) CHAPTER 2. THIRD-ORDER NONLINEAR RESPONSE OF 2D PLANAR PHOTONIC CRYSTALS 17 and and finally uc(/3) = \jCSlec — (32, UJ = u/c, and k is the unit vector in the direction of the incident in-plane wave vector. — * In Eq.(2.7) AP is the total polarization inside the tex-tured layer resulting from the incident field. In the linear situation this polarization is written as: AP(u : Anc + AO =Ax ( 1 ) ( - w ; w : / I - : A n c + Afc) (2.13) — * where summation over the repeated Afc is implied. If the thickness of the grating is very small relative to the wavelength of the incident field, the electric field, po-larization, and susceptibility can be considered constant in the z direction, and after doing the integration, the solution for the field components in the grating layer is: E(UJ : A n c + Pn) = Ehom(uj : A n c + &)+ G (w : A n c + pn) Ax ( 1 ) ( - w ; w : /3n - At)^ (w : Anc + Ac) (2.14) CHAPTER 2. THIRD-ORDER NONLINEAR RESPONSE OF 2D PLANAR PHOTONIC CRYSTALS 18 Dropping the frequency and wave vector arguments, this can be expressed more simply as: Fourier components, the above equation transforms to a coupled linear system of equations, that has a self-consistent solution. Usually we do use this calculation to model the spec-ular reflectivity spectrum of two-dimensional photonic crystals as a function of the incident frequency and in-plane wavevector (equivalent to the incident angle). This is because such spectra are relatively easy to obtain ex-perimentally, and they, together with the simulations, provide a means of determining the dispersion of leaky modes of the photonic crystal. To get the specular re-flectivity from Eq.(2.14), which provides only the fields inside the textured layer, we need another Green's func-tion that propagates the polarization associated with the fields in the grating layer to fields propagating in the up-per half space of photonic crystal. This propagator is E = Ehom+ME (2.15) with M=G^X^ • By considering a finite numbers of given by [33, 37, 38]: 9UHS (w : (3,z,z') = 2-KiCJ1 eiu0(p)(z-zt)eiwc(p)(L/2-z') X s CHAPTER 2. THIRD-ORDER NONLINEAR RESPONSE OF 2D PLANAR PHOTONIC CRYSTALS 19 + ^ ( P c + ( / 3 ) p c + ( « + P c + ( « P c - ( / 5 ) w e ^ ^ ^ 2 ^ ' ) ] Up (2.16) and finally the specular field of interest is: ErjHs{u '• Pine + Pn]z) = ^UH'S(u : Pine + Pn\ *) + -L/2 dZ' 9 U H S ^U '' ^inC + 7^)-^PiUJ ' Anc + Pn\ A (2.17) The specular reflectivity spectrum of the structure de-scribed in Fig.(2.1) in the near infrared is shown in Fig.(2.2). For this calculation the in-plane wave vector of the inci-dent field was oriented along the X-direction of recipro-—* _ ^ cal lattice Brillouin zone with fiinc = Q.Qipgx. There are three s-polarized and one p-polarized modes with unit reflection in this range of frequencies. These modes rep-resent resonance coupling of the incident field to leaky modes of this two-dimensional photonic crystal. In gen-eral these modes have Fano-like line shapes, but for cer-tain filling factors (7rr 2 /A 2 ) and in-plane wave vectors the modes can have Lorentzian line shapes like the high-est energy mode, S3, in Fig. (2.2). Fig. (2.3) shows the dispersion diagram of these leaky modes in the X-direction of the first Brillouin zone close to the zone centre. These are the four lowest energy bands at zone centre, and the corresponding eigenstates CHAPTER 2. THIRD-ORDER NONLINEAR RESPONSE OF 2D PLANAR PHOTONIC CRYSTALS 20 >-0.7 > 0.6 i— O 0.5 uu 0.3 0.2 0.1 0 0.95 1.05 G)/27tc (cm 1) f — — ^ \ \ • \ ; • \\ u / • \\ Figure 2.2: Linear specular reflectivity for S-polarized (solid line) polarized (dashed line) incident plane waves. 1.1 x 10 4 and P-are principally composed of different linear combinations of fields with wavevectors close to the four smallest re-ciprocal lattice vectors that describe the square lattice [37]. Figs. (2.4) through (2.7) show real space diagrams of the electric field magnitude within one unit cell associ-ated with the four modes represented in the specular re-flectivity plot of Fig.(2.2). The real space electric fields were obtained by summing all 81 Fourier field compo-nents, which were used in the linear simulation. The high-energy S3 mode is largely concentrated in the air CHAPTER 2. THIRD-ORDER NONLINEAR RESPONSE OF 2D PLANAR PHOTONIC CRYSTALS 21 holes, the low energy mode SI is largely localized in the dielectric background, and the two intermediate modes S2, and P I are more uniformly distributed within the unit cell. The white circle shows the perimeter of the air hole in each unit cell. This is a general feature of the field configuration in dielectric media. To minimize the electromagnetic energy, the lowest frequency mode,Sl, concentrates its displacement field in regions of high di-electric constant. Although the higher-order modes tend to concentrate their displacement fields in regions of high dielectric constant, they should also remain orthogonal to the lower-order mode and each other, therefore their CHAPTER 2. THIRD-ORDER NONLINEAR RESPONSE OF 2D PLANAR PHOTONIC CRYSTALS 22 Figure 2.4: Real space plot for Sl-mode at 0x/Pg = 0-01 field distributions cannot be understood as simply as those of the lowest-order mode[44]. In the following section we describe how the inclusion of a third order nonlinear response can modify these re-flectivity spectra. The basic effect responsible for these altered reflectivity spectra is the effective modification of the refraction index of the core material at high inten-sities (the Kerr effect). With this in mind, the effects should be greatest when strong fields are created inside the textured slab area. From the real space plots one would therefore expect that the SI mode should exhibit a stronger nonlinear sensitivity than the other modes that are not so concentrated in the material. However, the net nonlinear response depends not just on the fraction CHAPTER 2. THIRD-ORDER NONLINEAR RESPONSE OF 2D PLANAR PHOTONIC CRYSTALS 23 0 100 200 300 400 500 Figure 2.5: Real space plot for S2-mode at 0X/Pg = 0.01 100 200 300 400 500 Figure 2.6: Real space plot for Pl-mode at /?x//33 = 0.01 CHAPTER 2. THIRD-ORDER NONLINEAR RESPONSE OF 2D PLANAR PHOTONIC CRYSTALS 24 Figure 2.7: Real space plot for S3-mode at Px//3g = 0.01 of the field inside the material part of the textured core, but also on the total field strength associated with the leaky mode. The internal field for each mode also depends on its linewidth. Modes with smaller linewidth (or high Q-value) support higher internal fields. Fig.(2.8) shows the variation of inverse Q(Q — 6UJ/UJ0, or full-width at half maximum divided by the centre frequency) of the four bands shown in Fig. (2.2) above. This figure shows that the SI and S3 modes have very high Q-values, especially near zone centre. They are therefore expected to yield a much larger nonlinear responses, than the relatively broad S2 and P I modes. The infinite Q values sug-gested for modes SI and S3 at zone centre in Fig.(2.8) are CHAPTER 2. THIRD-ORDER NONLINEAR RESPONSE OF 2D PLANAR PHOTONIC CRYSTALS 25 an artifact of the calculation that assumes infinite plane wave excitation. Finite-size crystals excited by finite-size beams will have large, but non-infinite lifetimes at zone centre. 0.03 0.02 O 0.01 h S2-mode ll iBfc' ES'i P-mode S3-mode >i«> , > l l i l S1-mode 0.05 0.1 0.15 Figure 2.8: The inverse Q of the same bands shown in Fig.(2.3). 2.3 Degenerate Kerr effect Starting with the degenerate optical Kerr effect in which a single strong pump field with frequency UJ, and in-plane wave vector /% n c, illuminates the two-dimensional pho-tonic crystal, the total polarization in the thin grating region can be expressed as: CHAPTER 2. THIRD-ORDER NONLINEAR RESPONSE OF 2D PLANAR PHOTONIC CRYSTALS 26 AP(u : 4 c + AO =Ax ( 1 ) (-w; w : 0n - Pk)E{u : 4 c + 4 + <-> X ( 3 ) (-w; -w, a; : - /5m + A - A) x 5 ( W : 4 c + Pm)E*(w • - A n a ~ Pl)E{uJ : 4 c + A c ) (2.18) where again summation is implied over repeated indices in product terms. By dropping the frequency and wave vector arguments, this can be written as: AP, = AXME, + XiXEaE*sEJ, (2.19) The third-order polarization can be written as: APJ3> = 0$£? 7 , (2.20) where explicitly the 6^ tensor is: : & - Ac) = x2*7(-w; w> -w>w: A - 4 + A - A)x £a(o; : 4 c + 4 ) ^ > : - 4 c - A), (2.21) and therefore the total polarization is: AP, = (A X # + 0 $ ) £ 7 = x ^ / ^ 7 , (2-22) CHAPTER 2. THIRD-ORDER NONLINEAR RESPONSE OF 2D PLANAR PHOTONIC CRYSTALS 27 where xe^* 1S the total effective susceptibility in the tex-tured area, which explicitly depends on the intensity of the incident light. Here we assume that the matrix elements of the x ^ tensor are determined by the symmetry of the bulk ma-terial that makes up the photonic crystal (GaAs in this case). For 43m (GaAs) and m3m (Si, Ge) group sym-metry, the only nonzero elements of the x ^ tensor can be categorized in four groups [43]: Y ( 3 ) = (3) = (3) A-XXXX A.yvyv A.ZZZZ1 (3) xxyy _ y ( 3 ) A.XXZZ - Y( 3 ) A.yyxx - Y( 3 ) A.yyzz ~ Y ( 3 ) A.ZZXX ~ Y ( 3 ) A.zzyy (3) xyxy - y( 3 ) A.XZXZ - y ( 3 ) A.yxyx - y ( 3 ) A.yzyz - Y( 3 ) ^zyzy - Y( 3 ) A.ZXZX1 (3) xyyx _ Y ( 3 ) A.XZZX - y ( 3 ) A,yxxy - v( 3 ) A.yzzy ~ Y ( 3 ) A.zyyz - y ( 3 ) A.zxxz' X, X; X. (2.23) where the subscripts refer to the principal atomic crystal axes. In this case the 6^ matrix components can be written as: 0(3) v ( 3 ) TP fi* , (3) fi* i v ( 3 ) fi J?* wxx Axxxx-^x-^x 1 A-xyyx y y 1 A-xzzx z z 0(3) _ (3) fi fi* i Y ( 3 ) fi fi* xy A.xxyy & y 1 A-xyxy y x 0(3) _ (3) fi fi* , (3) E ; c;* — Xxxzz^x^z ' A a ^ a ^ - ^ - ^ x CHAPTER 2. THIRD-ORDER NONLINEAR RESPONSE OF 2D PLANAR PHOTONIC CRYSTALS 28 0(3) _ v ( 3 ) / ? / ? * ( _ Y ( 3 ) f? JP* "yx A.yyxx^y^x 1 A-yxyx^x^y # ( 3 ) (3) JP p* j _ (3) v ( 3 ) i ? / ? * 2/y A-yxxy^x^x 1 A-yyyyV y 1 At/.z,jiyx-/z-'-yz 0(3) _ (3) p ft* \ v(3) 1 7 2/2 — Xyyzz^V^Z I A-yzyZ^Z^y 0(3) _ (3) v ( 3 ) ZT> £ > * ^^a; ~~ Xzzxx^z^x ^ A.zxzxJ-Jx-L^z 0(3) _ (3) P P * i v ( 3 ) P ^zy — A-zzyy^z^y ^ A-zyzy^y^z ®zz ~ xixxz^xEx + Xzyyz^yEy + xi]zz^zEz (2.24) The Green's function tensor in this formalism, Eq.(2.9), is expressed in a coordinate system with (k, s, z) basis vectors associated with each Fourier component of the field, but the 6^ tensor is naturally expressed in the fixed (x, y, z) coordinate system. Therefore we need a transformation from one coordinate system to another. Suppose we calculate the third-order polarization in the x < y coordinate system, z (2.25) CHAPTER 2. THIRD-ORDER NONLINEAR RESPONSE OF 2D PLANAR PHOTONIC CRYSTALS 29 The transformation of this vector to the new coordinate k s proceeds as follows; z system < AP'(Vn) =T0n)AP^0n), --T 0n) 0n-0k)E0k), (2.26) or A P '(Vn) =T {fin) 0n ~ Ac) T 0k) T 0k)E0k), --T 0n) 6?(3) 0n ~0k) f 1 0k)E'0k), (2.27) but this vector in the new coordinate system can be cal-culated from: AP ' l (/3n) =6 ' ((3n-(3k)E>({3k), (2.28) therefore the transformation of the 6^ tensor can be calculated from the following equation: e ' 0 n - Ac) = T0n) 6 0n - Ac)T -\Pk) (2-29) CHAPTER 2. THIRD-ORDER NONLINEAR RESPONSE OF 2D PLANAR PHOTONIC CRYSTALS 30 in which: (okk0n-pk) eka0n-fa) e k z 0 n - p k ) \ QskiPn - Pk) &ss{Pn - Pk) 6sz{Pn ~ Pk) IMA*-AO ozs0n-pk) 6 x s 0 n - $ h ) ) T(Pn) fk0n)± k0n).y k(A).Z \ s(&).x s(^ ).y s(/3„).z V z(/?„).x z0n).y Z0n).Z ) 0 0n - Pk) = (8xx0n-Pk) Oxy0n-Pk) B x z 0 n - 0k) \ e y x 0 n - 0 k ) Qyy0n-Pk) Oyz0n - 0k) { e z x 0 n - fa) 9zy0n - Ac) 6zz0n - Ac) ) The transformations of the two sets of basis vectors for each Fourier component are: W - 0 n \ \Pn\ y I A I A z(/5n) = Z (2.30) 2. THIRD-ORDER NONLINEAR RESPONSE OF 2D PLANAR PHOTONIC CRYSTALS 31 The explicit form of each component of the 6^ tensor in the new coordinate system is thus: Okk0n - 0k) = T|TMTlK.(& - 0k) + I At I I A i l I A i l —> — * — * i.Pk)y f{Pn)x a ,fi Q ^  \Pn)yQ (fi ^ 1 |Afc| \Pn\ \Pn\ —* — * — * M A * - A) = - At) + %f M A . - 0k)}-|Afc| \Pn\ I A i l {Pk)xAPn)x * /fi fi \ , (AJyn ,fi B \] T ? T L ^ 7 ^ i / l P n - Pk) + -rzj-VyyKPn - Pk)\ lAfel |Ai| |Ai| — * — * M A » - A) = - f ^ M A i - 0k) + - A) |Ai| |Pn| — * — * —+ M A . - At) = ^ r [ ^ e - ( A \ - ft) - ~ | y M A > -|Afc| |Ai| |Ai| — * — * —> (Pk)y[(Pn)yQ ,fi fi s {Pn)xQ (fi fi ^ \-TWTtixyKPn ~ Pk) ~ ^T^yyKPn - Pk)\ \Pk\ | A . I IA - A) = % [ % M A > - A) - % M A > - A)] I Ac I |Ai| I A* | (Pk)x{{Pn)y Q (fi fis [Pn)xQ ,fi fi^ lAfcl I A i l \Pn\ CHAPTER 2. THIRD-ORDER NONLINEAR RESPONSE OF 2D PLANAR PHOTONIC CRYSTALS 32 Qsz(Pn - Pk) = —^-f-&xz{Pn ~ Pk) T^—QyziPn ~ Pk) \Pn\ \Pn\ 0zk(Pn - Pk) = ^ O z X ( P n ~ Pk) + ^OzyiPn ~ Pk) \Pk\ \Pk OzsiPn - Pk) = ^ifOzx{Pn - pk) - ^OzyiPn " Pk) \Pk\ \Pk Ozz(Pn-Pk)=Ozz(Pn-Pk) (2.31) Finally, the self-consistent solution of the field that incorporates this third-order nonlinearity is calculated using a similar equation to that which was used in the linear situation (Eq. (2.15)): Ea = Ehaom + G a / 1 ( X # + e^)E, (2.32) or E = [ l - G $ 1 ) + tf\1$hom (2.33) where 9 tensor in Eq.(2.33) depends on the field solu-tion ,E, through Eq.(2.21). Therefore we can only solve CHAPTER 2. THIRD-ORDER NONLINEAR RESPONSE OF 2D PLANAR PHOTONIC CRYSTALS 33 Eq.(2.33) iteratively. To start the iteration loop we cal-++(3) culate the Q tensor by using Enom as the field solu-tion. Then new values of the field are found by solving Eq.(2.33). This solution then is used to find new values ++(3) of $ tensor. We repeat this process until we reach a self-consistent solution. For having a fast convergence, the average of the last two field values are used to guess the solution in each iteration step. Numerical values of the diagonal elements of the GaAs third order susceptibility are taken to be —9.74x 10 - n esw [45]. This value was the result of a susceptibility mea-surement at 1064nm or 9398cm" 1 wavenumber (which is very close to the range of resonance modes considered in this simulation). Unfortunately we are unaware of any report of the measured values for the off-diagonal ele-ments of the third-order GaAs susceptibility. We there-fore assume that the off-diagonal elements are equal to the diagonal elements for most of the calculations. Fig. (2.9) shows the reflectivity spectrum for the S3 mode at different incident intensities. By increasing the input intensity, the peak of the mode shifts to higher energies (blue shifts). This is consistent with the nega-tive value of third-order susceptibility in this range which means that the light reduces the effective refractive in-dex of the material. The S3 mode has a Q « 3000, and it responds fairly strongly to the field even though most of the S3 mode is localized in the air hole. As the graph shows, the reflectivity at 10744.5cm - 1 increases CHAPTER 2. THIRD-ORDER NONLINEAR RESPONSE OF 2D PLANAR PHOTONIC CRYSTALS 34 from « 0.25 at low powers to around 1.0 at input power « 38Q0kW/cm2. Figure 2.9: Specular reflectivity of the S3 mode for different incident intensities; The dashed line shows the linear reflectivity and the solid lines are for intensities of 765, 2342, and 3871 kW/cm'2 for increasing blue shifts respectively. Fig.(2.10) shows the degenerate response of the SI mode at different incident intensities. This mode has a relatively narrow, Fano-like line shape and a Q-value of ~ 12000, significantly higher than the S3 mode. It also has its field localized primarily in the GaAs part of the textured layer. A l l of these factors result in the SI spectrum shifting the same amount, measured in units of linewidths, as the S3 spectrum, at much lower abso-CHAPTER 2. THIRD-ORDER NONLINEAR RESPONSE OF 2D PLANAR PHOTONIC CRYSTALS 35 lute field strengths. It only requires an incident inten-sity of 155kW/cm2 to change the reflectivity from 0 to 0.8 at 9679cm - 1 . In the case of the S3 mode, incident intensities of ~ 3871kW/cm2 are required to change the reflectivity from 1 to 0.4. Figure 2.10: Specular reflectivity for the SI mode for different incident inten-sities; The dashed line shows the linear reflectivity and the solid lines are for intensities of 30, 48,107, and 155 kW/cm2 for increas-ing blue shifts respectively. The P I and S2 modes also blue shift at high incident powers, but a very high input power is needed to in-duce a given percentage (with respect to the linewidth) shift because they have a relatively small Q value ( « 50). CHAPTER 2. THIRD-ORDER NONLINEAR RESPONSE OF 2D PLANAR PHOTONIC CRYSTALS 36 •j | 1 i . . — - r - , . . , - I T I . . l . . l . . , l . i - . ^ . T W . S i , . . , . . 1 1 i i r ) 2< < 1 1 1 i 1 1 > 3000 3500 4000 4500 5000 5500 6000 6500 7000 INCIDENT P O W E R (kW/cm 2 ) Figure 2.11: Bistable behaviour of the reflectivity for the S3 mode. Fig. (2.9) shows that the line shape of the S3 mode be-comes multi-valued for incident intensities higher than ~ 4Q00kW/cm2. This suggests the possibility of observ-ing bistable behaviour at a fixed frequency as a func-tion of the optical power. Fig. (2.11) illustrates such bistable behaviour for the S3 mode at 10745.5cm - 1. A clear hysterisis loop is evident in the range of 4700 to 6QQ0kW/ cm2. The bistablity behaviour simply shows the dynamic response of the reflectivity to the applied incident field. In other words, the reflectivity can be switched from one value to another (all-optical switch-ing) simply by changing the applied incident power. CHAPTER 2. THIRD-ORDER NONLINEAR RESPONSE OF 2D PLANAR PHOTONIC CRYSTALS 37 As mentioned above, the values of the off-diagonal el-ements of the third-order susceptibility of GaAs are not known. The simulations in Figs. (2.9-2.11) were obtained by setting the off-diagonal elements equal to the diago-nal elements. To indicate the significance of this assump-tion, Fig.(2.12) shows the simulated reflectivity for the S3 mode obtained by setting the diagonal elements equal to zero (dash-dot line) and when they are equal to the diagonal elements (solid line). 0 i 1 1 1 1 1.0732 1.0736 1.074 1.0744 1.0748 co/27ic(cm"1) x 1 ° 4 Figure 2.12: Reflectivity in the vicinity of the S3 mode for nonzero and zero off-diagonal elements of the x^ 3' tensor. Clearly there is a considerable difference between the two approximations. It is obviously important in this case to get values for the off diagonal elements, and per-CHAPTER 2. THIRD-ORDER NONLINEAR RESPONSE OF 2D PLANAR PHOTONIC CRYSTALS 38 haps quantitative comparisons with these types of reflec-tivity experiments might be a way of measuring them. 2.4 Nondegenerate Kerr effect We now consider the nondegenerate optical Kerr effect in which a strong pump beam with frequency UJP and in-plane wavevector Ppump is used to modify the effective bandstructure as probed by a weak signal beam that can in general have a frequency cus and in-plane wavevector —* Pprobe- For simplicity in the following discussion, it was assumed that the two beams have the same in-plane wave vector (O.OiPgx). The nondegenerate simulation proceeds as follows. The fields generated by the pump beam alone are first solved for in the textured layer using the algorithm described in the preceding section. The effective susceptibility in the presence of the strong pump beam, including the Kerr effect is then used to calculate the reflectivity spectrum of the weak probe beam using: Ea{ws, Pprobe + Pn) •= E^°m(us, Pprobe + Pn) + Gafl(us, Pprobe + Pn)x [Ax$ Us • Pn - @k)E>y{Us • Pprobe + Pk) + fo\ —* —* —* —* X^i-Us] Wp, -Wp, U8 : P n - Pm + P i - Pk) X Ea{^p '• Ppump + Prn)E§(uJp : —Ppump — Pi)x CHAPTER 2. THIRD-ORDER NONLINEAR RESPONSE OF 2D PLANAR PHOTONIC CRYSTALS 39 — * —* E7(tUs : fiprobe + P*)]j (2.34) The effective susceptibility, including the third order optically-induced changes, felt by the signal beam is: (2.35) Fig. (2.13) shows the specular reflectivity of the signal beam in the vicinity of the SI mode when the medium is excited by a pump beam resonant with the S3 mode at 10744cm - 1 . Fig.(2.14) also shows the result when pump-ing at the same frequency, but when the signal beam is tuned in the vicinity of the S3 mode. Comparison of these two figures shows that the S3 mode shifts considerably more, in absolute terms, than the SI mode when the pump is resonant with the S3 mode. This is probably because of the different distribu-tions of the optical fields within the unit cell for photonic eigenstates associated with the SI and S3 bands. Wi th reference to the real space plots of the field distributions in Figs.(2.4) and (2.7), these distributions are consider-ably different for the two modes. The modified index profile induced through excitation of the S3 mode will thus overlap more with the S3 mode than with the SI mode. It is interesting to note that although the abso-lute shift is much less for the SI mode, the relative shift with respect to the linewidth is almost the same for the CHAPTER 2. THIRD-ORDER NONLINEAR RESPONSE OF 2D PLANAR PHOTONIC CRYSTALS 40 Figure 2.13: Nondegenerate response of the S i mode to different pump inten-sities, with the pump frequency fixed at 10744cm - 3. Dashed line shows the linear spectrum; solid line and dash-dot line are reflec-tivities pumped at 1720 and 305&kW/cm'2 respectively. two bands. The result of the degenerate and nondegenerate simu-lations presented in this chapter, and published in Ref[36] suggests the potential for using 2DPCs as optical switch-ing and optical limiting components. Simply by changing the power of an incident field, the reflectivity of the struc-ture can be parametrically controlled without relying on electronic controls. The major problem in using this par-ticular phenomenon in applications is the need for a rel-CHAPTER 2. THIRD-ORDER NONLINEAR RESPONSE OF 2D PLANAR PHOTONIC CRYSTALS 41 Figure 2.14: Nondegenerate response of the S3 mode to different pump inten-sities, with the pump frequency fixed at 10744cm"1. The dashed line shows the the linear spectrum; the solid line and dash-dot line are reflectivities pumped at 1720 and 3058kW/cm'2 respectively. atively high incident power. This can be traced to the relatively small off-resonant, intrinsic third order suscep-tibility of GaAs assumed in these simulations. The re-quired power levels could be reduced by either introduc-ing free-carriers (excited by the incident field) or by tun-ing the pump and/or probe beams near a direct bandgap in the host semiconductor heterostructure[46]. Both of these approaches would increase the effective third-order susceptibility over that assumed here, but at the expense of causing some absorption of the light. This may or may not be tolerable depending on the specific application. CHAPTER 3. PHOTONIC CRYSTAL FIBRES 42 CHAPTER 3 PHOTONIC CRYSTAL FIBRES The concept of photonic crystals was incorporated in op-tical fibre technology first by Russell et al., and subse-quently by a large number of research groups around the world [47, 48, 49, 50, 51]. Fig. (3.1) shows a scanning electron micrograph of the cross section of a silica-based "photonic crystal fibre1" (PCF) used in the work de-scribed below. 2um Figure 3.1: A scanning electron micrograph, in cross section of the photonic crystal fibre used for the studies described in this chapter In conventional optical fibres, a relatively high index ^ade by Crystal Fibre Company in Denmark CHAPTER 3. PHOTONIC CRYSTAL FIBRES 43 core is surrounded by a lower index cladding glass in order to support bound optical modes that propagate only in the core, due to total internal reflection from the core-cladding interface. By surrounding the core of the fibre by a periodic array of air holes, as in Fig. (3.1), it is also possible to support bound modes that prop-agate only in the core, but the group velocity disper-sion (GVD), -^(1/vg), of PCFs can differ markably from conventional fibres. The G V D parameter indicates how fast a short pulse will spread as it propagates through the fibre (this is a big problem in fibre optic communi-cation systems, and is the main reason why long-haul fibre systems operate at the zero dispersion wavelength of typical silica single mode fibres at 1.55 microns). In conventional fibres, the G V D is "normal" (negative) for wavelengths shorter than 1.3(j,m and "anomalous" (posi-tive) for longer wavelengths. By engineering the air hole size and pitch in photonic crystal fibres, one can create fibres with anomalous group velocity dispersion in the visible and near infrared part of the spectrum. The G V D spectrum of a fibre is important not only in optical communication systems, but also in the context of the nonlinear propagation of light through fibres. Much work has been done over the past 20 years concerning the influence of nonlinear response on the propagation of light through single mode fibres (SMFs). In fact, a viable solution to the negative effects of pulse broadening actu-ally makes use of the third order nonlinear response of silica. When a fibre is operated in the anomalous disper-sion regime, it is possible to excite optical solitons that CHAPTER 3. PHOTONIC CRYSTAL FIBRES 44 propagate without distortion by cancelling the effect of group velocity dispersion through self-phase modulation [13, 14, 52]. P C F s offer a particularly attractive medium in which to study guided wave nonlinear optics for two reasons. First, the high index contrast between the ~1 fim di-ameter silica core and the surrounding photonic crystal lattice is relatively large compared to conventional SMF, thus reducing the effective mode diameter from ~10 fim to 1.8/im. The smaller mode volume means that the elec-tric field strength in the silica is proportionately higher for a given guided power level. Second, by being able to tune the dispersion, it is possible to make fibres in which the anomalous dispersion region and the near-zero dis-persion point are in the near infrared part of the spec-trum, near 800 nm. This means that relatively conve-nient short-pulse laser systems can be used to study the propagation of light through such fibres. Already, several groups have reported that the output spectrum from PCFs like that shown in Fig. (3.1) can as-sume a broad, continuum shape when pumped with laser pulses ranging in duration from 40 fs to 0.8 ns, at aver-age power levels of a few mW [15, 16]. The shape of the nonlinearly shifted output spectrum varies from fibre to fibre, and is strongly dependent on the pump wavelength and power. Although it is clear that several nonlinear phenomena are likely operating in concert to produce such extraordinary spectra, an exact explanation for the effect has yet to be determined. In the present work, CHAPTER 3. PHOTONIC CRYSTAL FIBRES 45 two distinctly different forms of broad output spectra were observed, and the temporal nature of the output fields was measured and correlated with the spectra. 3.1 Experiment setup Fig. (3.2) shows the fibre's linear dispersion properties. Table (3.1) summarizes the important properties of the photonic crystal fibre shown in Fig. (3.1). Figure 3.2: Dispersion of the photonic crystal fibre described in Table(3.1) pro-The objective of the experiments described below was to characterize the spectral and temporal properties of the light emanating from the PCF when ~100 fs pulses 1.45 0.4 0.6 O.ti I 1.2 1.4 I ft 1.8 Wuvdcnglh (Jim) vided by the fibre manufacturer CHAPTER 3. PHOTONICCRYSTALFIBRES 46 Material Pure silica Cladding diameter 80 ± 5fim Coating diameter 195 ± 5[i,rn Coating material Acrylate Core diameter 1.0±0.2^m Piteh(distance between air holes) 1.6±0.2/Ltm Ratio between air hole diameter and pitch > 0.85 Numerical aperture > 0.5@633nm Zero dispersion wavelength 600 ± 100nmandll50 ± 150nra Attenuation < 0.5dB/m@635nm < ldB/m@94Anm « ldJB/m@1250nm < 10dJ5/m@1390nm Table 3.1: Photonic crystal fibre specifications. ranging in wavelength from 760nm to 850nra are launched into it using a 40X microscope objective. Fig. (3.3) shows the setup of the experiments and ta-ble(3.2) describes all of the optical elements. The T i -sapphire laser is a Tsunami model 3960-S1S that emits 100/s pulses at a mode-locking repetition rate of 82MHz. The centre wavelength of the laser can be varied from 750 to 830nm with a maximum average power of ~ 800mW at 800nra. CHAPTER 3. PHOTONIC CRYSTAL FIBRES 47 Table 3.2: Optical elements of the cross-correlation experiment By using a beam splitter (BS), this laser pulse is di-vided into two parts. The part that is directed towards the retro-reflector RF\ on the translation stage is referred to as the gate beam. It is focused by L3 onto the surface of the nonlinear crystal (NLC) used to up-convert the output in order to measure its temporal behaviour. Optical Specifications elements M i - M10 Dielectric mirrors (IR) 25.4 mm Diameter, 1/5 wave, 0.5 — 18 fim M n , M i 2 Dielectric mirrors (UV) 25.4 mm Diameter, 1/5 wave, 250 — 600 nm H W P Half-wave plate P L Polarizer £ 1 Objective lens 40X, Newport F-L40B Lens with respectively 10 and 15 cm focal length U V lens Focal length 7.5 cm, A R coated, 250-430 nm N L C Beta-Barium Borate (BBO) crystal 5 x 5 x 0.5 mm, cut at 38° BG40 filter P M Parabolic mirror 90° off axis, 28.75 mm focal length BS 50/50 Beam spliter RFi, RF2 Newport Gold-coated retro-reflector D L Delay line translation stage CHAPTER 3. PHOTONIC CRYSTAL FIBRES 48 Another part of the laser beam passes through a half-wave plate (HWP) and a linear polarizer (PL) and is coupled by L\ onto the cleaved input facet of the pho-tonic crystal fibre. Ideally, the laser should be optically isolated from the fibre by using a Faraday rotator, which CHAPTER 3. PHOTONICCRySTALFIBR,ES 49 prevents reflections from the fibre facet going back to the laser. However, an isolator was not available at the time of this experiment, and back reflections were not found to be a problem. A parabolic mirror (PM) collimates the output of the fibre and lens L2 focuses this beam (referred to as the fibre beam) onto the nonlinear crystal at the same point as the gate beam is focused. A C C D camera positioned to display the surface of the nonlin-ear crystal helps to overlap the two beams (fibre and the gate) at the same point. The nonlinear crystal is B B O 2 , and it rests on a motorized rotation stage that has an accuracy of 0.01°. If the gate and fibre signals overlap in time and space inside the B B O crystal, a pulse at the sum frequency can be generated. By varying the relative arrival time of the gate and fibre signals on the B B O crystal, it is possible to map out the temporal profile of the fibre signal, which is generally much longer than the gate pulses. Lens L 4 collimates the up-converted signal and mirrors Mn,Mu, and lens L 5 images it on the entrance of a monochro-mater (Digikrom DK242 from C V I Laser Corporation). Filter F\ helps to block any IR beam passing through the monochromator. A photomultiplier tube (Model 9784A made by Thorn E M I Electron Tubes Ltd.) at the exit of the monochromator is used to monitor the sum gen-eration signal in a photon counting mode (Hamamatsu model C3866). 2Beta-Barium Borate CHAPTER 3. PHOTONIC CRYSTAL FIBRES 50 A computer running Labview software controls the ro-tation stage, monochromator, and photon counting sys-tem. The Labview codes used in this experiment are summarized in Appendix B. 3.2 Phase Matching Efficient up-conversion can only occur if the B B O crys-tal is oriented at an angle that allows phase matching of the nonlinear conversion process [35, 53]. The B B O crystal is a uniaxial crystal characterized by ordinary (nG) and extraordinary (n e) refractive indexes, both of which exhibit considerable dispersion across the visible and near infrared part of the spectrum. This dispersion is described by Sellmeier's equation[54]: o 0.01878 = 2 - 7 3 5 9 + A 2 _ - 0.01354A' o 0.01224 ~ , , n e = 2 ' 3 7 5 3 + A 2 - 0.01667 " ° " 0 1 5 1 6 A t 3" 1) where A is in units of fxm. B B O is transparent between 0.19 — 3.5/im. Fig.(3.4) shows the geometry of the in-teraction of gate and fibre beams within the nonlinear crystal. The phase matching condition depends on the polar-ization of the interacting beams. The "Type I" configu-ration used in this experiment is such that the gate and fibre beams are polarized along the ordinary axis, and CHAPTER 3. PHOTONIC CRYSTAL FIBRES 51 Normal direction' Entrance Exit surface surface Figure 3.4: Phasematching geometry for sum generation of the gate and fibre beams. the sum frequency signal is generated in the extraordi-nary polarization. The phase matching condition for sum frequency generation in this situation is then given by: s m <*"> = (iK.) . ( 3 ' 2 ) where 9m is the phase matching angle defined in Fig.(3.4) and n2s and n2es are the squared refractive indices of the ordinary and extraordinary beams at the sum gen-eration wavelength. Also ns is the refractive index of the upconverted beam which depends on the angle and CHAPTER 3. PHOTONIC CRYSTAL FIBRES 52 wavelengths of the interacted beams. There is a unique phase matching angle for each com-bination of gate and fibre wavelengths and you can up-convert different fibre wavelengths by tuning the B B O crystal. There is a Maple code described in Refs.[55, 56] which calculates the phase matching angle, 6m, for the sum generation situation. By entering the gate and fi-bre wavelengths and also the angle between these two beams, the Maple code gives you the desired rotation angle of the nonlinear crystal. In this experiment the angle between the gate and fibre beams was set to 14°. 3.3 Experiment procedure Excellent alignment of all optics is required to overlap the gate and fibre pulses temporally and physically inside the B B O crystal. Under optimum conditions the typical up-converted count rate from the P M T is ~ 50, 000 units for average fibre output powers of 2.0 mW. The following briefly outlines the alignment procedure used to detect these weak signals: 1-For upconversion of each portion at output spectrum of the fibre, changing the delay between gate and fibre beam is necessary. This is because of group velocity dis-persion of the fibre. The gate beam must be incident on the translatable retro-reflector RFi so that changing the delay does not alter the spatial overlap of the two beams on the B B O crystal surface. This requires the gate beam and the retro-reflector's axis both to be parallel to the CHAPTER 3. PHOTONIC CRYSTAL FIBRES 53 axis of translation stage. By installing an iris before mir-ror M 7 , we make sure that the gate beam passes through the aperture for all positions of the translation stage. The beam splitter and mirror M 2 are used to align the translation axis of the moving stage. 2-As explained above, the B B O crystal has to be ro-tated over a range of angles (~10 degress) in order to measure the temporal evolution of the full spectrum of the fibre emission. The two beams must obviously not be allowed to walk off each other under this rotation. Therefore, the two beams must strike the crystal pre-cisely on the axis of the rotation state it is mounted on. To fulfill this requirement a thin piece of Teflon tape was attached to the surface of the B B O crystal. At first by moving the crystal on its translation stage and looking at the location of the gate beam on the monitor during the rotation, the axis of rotation can be found easily. Then by aligning mirror M10, the fibre beam can be focused at the same point. The next step is to adjust the optical paths of two beams. First we set the input power to the fibre at a low level, ~10 mW, in order that the output spectrum of the fibre exhibits minimal broadening. Under this condition the gate and fibre beams are at the same wavelength (we typically align the system at 800nra). Measuring the separate paths with a ruler is not sufficiently accurate to ensure that the two pulses will overlap in the B B O crys-tal. By inserting a fast Silicon PIN (P-type, Intrinsic, N-type) detector (Newport, model 818-BB-21) after the CHAPTER 3. PHOTONICCRYSTALFIBRES 54 filterFi to intercept both beams, and by adjusting the delay line while observing the 50 ohm terminated signal on an oscilloscope (Tektronix TDS 350) it is possible to get the relative delay to within ~100 ps, which means the uncertainty in the zero delay between the beams is ~±1 .5cm. To assist in fine-tuning the delay stage, a chopper was installed after the mirror M§ to modulate the fibre out-put at 270Hz. This facilitated using a lock-in ampli-fier to monitor the up-converted signal as measured on a sensitive silicon detector (UDT-455 UV) placed directly after the filter F\. Previous researchers using the same basic setup[55, 56], employed a mini-shaker with a sec-ond retro-reflector to periodically sweep the delay over a 20 ps interval while searching for the overlap condition. In the present work, the chopper/lock-in technique was found to be easier to work with. Experience showed that by moving the delay line a few mm from the position in which the gate and fibre beams had nominally equal lengths as measured by the fast P IN detector, the upconverted signal can be detected on the lock-in amplifier. After the zero delay position is deter-mined in this way, the chopper is turned off and mirrors Mj and M\Q are tweaked, and the B B O crystal is rotated in 0.01° steps to maximize the D C upconverted signal at the optimal pulse overlap. When the power into the fibre is increased, and the output spectrum shifts, the crystal has to be rotated and CHAPTER 3. PHOTONIC CRYSTAL FIBRES 55 a new zero delay must be procedure outlined above, optics after the filter F\ is Refs.[55, 56]. determined using the same The alignment of collection similar to that described in 3.4 Results Figs. (3.5) and (3.6) show the output spectra from the Ti-sapp laser when it was tuned to 803nra and 829nm with ~13.5nra and ~8.5nra bandwidths respectively. Before doing the cross-correlation experiment the time-integrated output spectrum of the photonic crystal fibre was measured using a Fourier transform spectrometer ( B O M E M DA8). When the fibre was pumped at 803nm, the output spectrum consists of a series of well-defined peaks, each much broader than the input spectrum, and all shifted to the red. The higher the pump power, the larger the maximum shift, and the larger number of satel-lite peaks. Fig. (3.7) shows the output spectrum of the fibre when it was pumped at 803nra with 2QmW input power. In this case the output power of the fibre was ~1.0mW. The second peak in this figure was centred at 885nm with a ~ 25nm bandwidth. Also Fig.(3.8) shows the spectrum with lOOmW input power and ~2.2mW output power. There are two sharp peaks in this figure around 942 and 1128nra with respectively 31 and 38nra bandwidths. CHAPTER 3. PHOTONIC CRYSTAL FIBRES 56 9000 • 8000 • 7000 • 6000 • 3* 5000 >. " S £Z £ 4000 • 3000 • 2000 • 1000 • 0 r. 600 800 1000 1200 1400 1600 Wavelength (nm) Figure 3.5: Output of the Ti-sapp laser at 803 nm. In contrast, when the fibre pumped at 829nm, the out-put, is more uniformly shifted to the red, but again, the maximum shift increases with increasing launch power. Fig. (3.9) shows the output spectrum of the fibre when it was pumped at 829nra and the input and output powers were 100 and 2.0mW respectively. At the highest pow-ers used in the experiment, lOOmW, the output spec-trum extended as far as 1300nm, and this maximum shift was approximately the same for the two different wavelengths. CHAPTER 3. PHOTONIC CRYSTAL FIBRES 57 600 800 1000 1200 1400 1600 Wavelength (nm) Figure 3.6: Output of the Ti-sapp laser at 829 nm. 600 800 1000 1200 Wavelength(nm) 1400 1600 Figure 3.7: Spectrum of the fibre with 20 mW input at 803 nm. CHAPTER 3. PHOTONIC CRYSTAL FIBRES 58 12000| r Wavelength (nm) Figure 3.8: Spectrum of the fibre with 100 mW input at 803 nm. Figure 3.9: Spectrum of the fibre with 100 mW input at 829 nm. CHAPTER 3. PHOTONIC CRYSTAL FIBRES 59 Silica is a "Kerr medium", which means that the local refractive index depends on the local intensity of light, just as in the 2DPCs discussed in Chapter 2. While in Chapter 2 we only considered harmonic (constant am-plitude) excitation, here we are explicitly dealing with a pulsed excitation. In a Kerr medium a spatial and tem-poral variation of the intensity will result in a time and space dependent refraction index. When a short pulse passes through a length L of this medium it experiences an intensity-dependent phase shift described by [35, 53]: (/>(t) = -n2I{t)u0L/c (3.3) where cu0 is the frequency of the optical carrier. The pulse experiences a corresponding spectral broadening equal to: Su{t) = -W) (3.4) Assuming the pulse has a hyperbolic secant profile (as is the case for the Ti-Sapp laser Tsunami user's manual) with width r 0 ; I(t) = I0sech2{t/r0) (3.5) then the maximum frequency shift due to the "Kerr ef-fect" is: n2LJoIoL f . OUJmax = (3.6) CHAPTER 3. PHOTONICCRYSTALFIBRES 60 In the present case r o ~100/s , and in silica n2 — 3 x 10 - 2 0 m 2 jW. At an average power of 2mW inside the ~l[im core of the fibre, the maximum wavelength shift attributable to the conventional Kerr effect is SXmax — 79nm. This shows that Kerr-related (third order) self-phase modulation cannot explain the ~500nra shift ob-served in both the discrete and continuum broadening regimes. To reveal the -temporal behaviour of the output of the photonic crystal fibre, the cross-correlation measurement is necessary (a common experiment to characterize short, non-transform limited pulses). The setup of this exper-iment was shown in Fig. (3.3). It can be used both to obtain spectra from the monochromator at a fixed gate-signal delay, and to obtain time-delay data at a fixed setting of the monochromator. Fig. (3.10) shows the spec-trum of the upconverted signal at the satellite peaks lo-cated near 885 nm (see Fig.(3.7)), 939 nm, and 1128nra (see Fig.(3.8)) respectively. The spectra in Fig.(3.10) were obtained with the delay (between the gate and fi-bre beams) fixed at Ops, the monochromator bandwidth (resolution) set to 1 nm, and the nonlinear crystal phase matching angle set to optimize the upconversion at the quoted peak wavelengths in the time-integrated spectra. The time of counting for each wavelength for these spec-tra was 3 seconds. CHAPTER 3. PHOTONIC CRYSTAL FIBRES 61 Wavelength (nm) Figure 3.10: Intensity versus wavelength at the peaks of the discrete spectra in Figs.(3.7) and (3.8). From left to right these spectra were obtained with nonlinear crystal phase matching angles and delays set to (21.43°, 19.8 ps), (22.93°, 26.4 ps), and (26.87°, 59.4 ps). Fig. (3.11) shows the temporal delay curves obtained by fixing the monochromator (with a 1 nm bandpass) at the peak wavelengths of Fig. (3.10), and varying the delay between the gate and fibre beams. Each point in this time-delay data was obtained by averaging for 10 seconds. Fig. (3.12) superimposes the time-delay and spectral data on a single plot. The corresponding set of plots obtained when the gate beam was at 829nra and the B B O phase matching angle and the delay were optimized for upconversion at 990 nm, 1060 nm, and 1140nm (of the spectrum shown in Fig.(3.9)) are shown in Figs.(3.13-3.15). CHAPTER 3. PHOTONIC CRYSTAL FIBRES 62 Delay (ps) Figure 3.11: Intensity versus delay between the gate and fibre beams at the peak wavelengths of Fig. (3.10). The left spectrum corresponds to the left spectrum of Fig.(3.10), and so on. Figure 3.12: Three dimensional plot of intensity versus delay and wavelength for the discrete spectra (composite of Figs.(3.10) and (3.11)). CHAPTER 3. PHOTONIC CRYSTAL FIBRES 63 440 450 460 470 480 490 500 Wavelength (nm) Figure 3.13: Intensity versus wavelength at three different wavelengths in the continuum spectrum in Fig.(3.9). From left to right these spectra were obtained with nonlinear crystal phase matching angles and delays set to (24.08°, 29.7 ps), (25.58°, 39.6 ps), and (27.09°, 62.7 ps)-The delay values needed to optimize the signal strength, with the B B O crystal set to upconvert different wave-lengths, can be plotted as a function of wavelength to estimate the group velocity delay characteristics of the fibre. This information is shown in Fig. (3.16), which shows the delay values as a function of the peak in the detected wavelengths for each of the B B O angle settings. Also shown in Fig. (3.16) is the delay that one would expect assuming all of the light propagates in the low-est order bound mode of the fibre (dashed line). This latter curve was obtained by graphically measuring the slope of the dispersion curve of the lowest bound mode in Fig.(3.2). CHAPTER 3. PHOTONIC CRYSTAL FIBRES 64 50 60 70 Delay (ps) 100 Figure 3.14: Intensity versus delay between the gate and fibre beams at the peak wavelengths of Fig.(3.13). The left spectrum corresponds to the left spectrum of Fig.(3.13), and so on. 100 50 Delay (ps) 500 -50 440 480 4 6 C Wavelength (nm) Figure 3.15: Three dimensional plot of intensity versus delay and wavelength for the continuum spectrum (composite of Figs.(3.13) and (3.14)). I CHAPTER 3. PHOTONIC CRYSTAL FIBRES 65 25 20 • 1 1 1 1 1 o o • • , ' ' ** -ob- • • • < < > ' 800 850 900 950 1000 1050 1100 1150 Wavelength (nm) Figure 3.16: Delay versus peak up-converted wavelengths for discrete (squares) and continuum (circles) spectra. The dashed line is the delay ex-pected from Fig.(3.2) assuming all of the light propagates in the lowest order mode of the fibre The close overlap of these two curves shows that all of the light is propagating in the lowest order fibre mode, at least in the range of 800 — 1300nm. Although the exact mechanism responsible for these spectra is yet unclear, the correlation of the asymmetric behaviour of time delay results for the discrete spectra, Fig. (3.11), versus the symmetric time-delay behaviour for the continuous spectrum, Fig. (3.14), provides new and potentially useful information that might help to determine the nature of the underlying mechanism. A simple calculation shows that the asymmetric time-delay data might be evidence of a very rapidly varying phase, with respect to frequency shift, in the case of the dis-CHAPTER 3. PHOTONICCRYSTALFIBRES 66 Crete nonlinear spectra. The symmetric time-delay data in the continuum case is consistent with a slowly varying phase, again with respect to frequency, in the continuum spectrum. To reveal the exact mechanism behind these results, additional experiments that measure the time-delay spec-tra at all frequencies would help. CHAPTER 3. PHOTONIC CRYSTAL FIBRES 67 3.5 Conclusion A Green's function formalism was used to describe the third-order nonlinear response of the planar photonic crystal waveguides, which contain Kerr materials. A n iterative numerical solution for degenerate Kerr effect showed the dependence of reflectivity spectrum in the vicinity of leaky photonic crystal modes with the incident intensity of a single pump beam. It was shown that the Q value and line shape of these modes (which can be en-gineered by changing the thickness of the oxide cladding, the air filling-fraction and the in-plane wavevector of the incident field) have a great influence on reflectivity spec-trum of this structure at higher powers. The nondegen-earate Kerr effect was also considered to study the reflec-tivity spectrum of a relatively weak signal beam when a strong pump beam changed the effective susceptibility of the medium. In this case the overlapping between the field distributions of the pump and signal beams in the texture area had a great influence. Observed bistablity behaviour and the shifting of reflectivity at high powers may be useful in all-optical applications such as power-limiting and optical switching. The temporal and spectral response of a photonic crys-tal fibre, pumped by a lOOfs Ti-Sapp laser, was also ob-served by using time-integrated spectroscopy and an op-tical cross-correlation measurement. The output spec-trum of the fibre measured by a Fourier transform spec-trometer showed discrete (pump wavelength at 803 nm) and continuous spectral broadening (pump wavelength CHAPTER 3. PHOTONIC CRYSTAL FIBRES 68 at 829 nm) that depend on pump wavelength and pump intensity. A simple calculation proved that the self-phase modulation cannot be the only nonlinear mechanism which caused ~ 500nm spectral broadening when the fibre pumped with 100 mW input power. The time-resolved spec-troscopy of the up-conversion of different portions of the output spectrum of the fibre showed asymmetric tempo-ral behaviour for the discrete spectrum and symmetric behaviour for the continuous spectrum. More experi-ments would be required to gain a full microscopic un-derstanding of the observed strong nonlinear effects. APPENDIX A. FOURIER COEFFICIENTS CALCULATION 69 APPENDIX A C O E F F I C I E N T S The susceptibility of a two-dimensional photonic crystal with a square lattice in real space is: X(r) = Xb + {Xg - Xb)0{R - \ f\) where Xb and Xg a r e the susceptibilities of the back-ground and the grating respectively and R is the radius of the holes. The Fourier coefficient of susceptibility can be calculated from: X(G) = xtSao + J dfe^%R - \f\) where G are the reciprocal lattice vectors and Vceu is the volume of the unit cell, which in this case will be area of the unit cell. The above integration can be written as: J dre^-f6{R-\r\) = J°° rO(R - \r\)dr d(j)eiGsCos<t> POO , , „ , /"27T / q re(R-\r\)drJQ but the angular part of the integration can be written in terms of a Bessel function: J0(x) = ^ dfe**0™* APPENDIX A. FOURIER COEFFICIENTS CALCULATION 70 therefore J dfeidy9{R - \r\) = 2rr j\j0(Gr)dr By using the following recurrence relation for Bessel func-tions: -~[xJ\{x)} = XJQ(X) dx finally the susceptibility in Fourier space can be written as: X(G) = » f c o + 2 n R ( * a , Xb)MGR) < J Vcell APPENDIX B. LABVIEW CODES 71 APPENDIX B L A B V I E W C O D E S A l l necessary Lab View codes, which were used in the cross-correlation esperiment, are summerized in the fol-lowing table. rotate, vi Rotate nonlinear crystal with 0.01° accuracy Read.vi and write.vi Read and write specific parameters for monochromator. They are used for changing slit size and position of grating. 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