Bragg Interactions in Periodic Media Citation Jaggard, Dwight L. (1977) Bragg Interactions in Periodic Media. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/6ctw-8m32. https://resolver.caltech.edu/CaltechTHESIS:11142025-203411358 Abstract The interaction of electromagnetic waves of wavelength λ with periodic structures of spatial period Λ. are studied. The emphasis of the work is on Bragg interactions where λ ≃ 2/\/N and the Bragg order N takes on the values 1, 2,... . An extended coupled waves (ECW) theory is developed for the case N ≥ 2 and the results of the theory are found to compare favorably with the exact results of Floquet theory. Numerous numerical results are displayed as Brillouin diagrams for the first few Bragg orders. Moreover, explicit expressions for coupling coefficients, bandgap shifts and bandgap widths are derived for singly periodic media. Particular note is taken of phase speeding effects. The effects of multiharmonic periodicities on the control of feedback strength are investigated. It is found that with proper phasing the feedback strength becomes zero and the bandgap disappears. Coupling parameters are calculated for typical multiharmonic periodicities for the first three Bragg orders. For odd Bragg orders, inverted bandgaps and phase slowing occur when the gain or loss of the media is modulated. Also average gain or loss affects the bandgap shape and the spatial or temporal growth or decay. Absolute instabilities are observed and expressions are derived for the instability frequencies, thresholds and growth rates. Under certain conditions, instabilities occur for structures with average loss. The results for the first and second Bragg orders are archetypical of all odd and even orders respectively. Applications of the ECW theory to higher-order DFB filters involve such phenomena as transient propagation, effects of periodicity profiles and the relative coupling due to boundaries and periodicities. The calculation of higher-order DFB laser parameters shows that the mode spectrum is asymmetrically shifted and the threshold gain is greatly dependent upon the periodicity profile. Approximate threshold parameters are calculated for high and low gain and for all Bragg orders. In addition, application of the ECW theory to holographic gratings and beam propagation is made. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Electrical Engineering and Applied Physics) Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Electrical Engineering Minor Option: Applied Physics Thesis Availability: Public (worldwide access) Research Advisor(s): Papas, Charles Herach Thesis Committee: Unknown, Unknown Defense Date: 29 July 1976 Record Number: CaltechTHESIS:11142025-203411358 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:11142025-203411358 DOI: 10.7907/6ctw-8m32 ORCID: Author ORCID Jaggard, Dwight L. 00-0002-0697-1612 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 17761 Collection: CaltechTHESIS Deposited By: Ben Maggio Deposited On: 19 Nov 2025 18:21 Last Modified: 19 Nov 2025 18:28 Thesis Files PDF - Final Version See Usage Policy. 50MB Repository Staff Only: item control page

BRAGG INTERACTIONS IN PERIODIC MEDIA Thesis by Dwight L. Jaggard In Partial Fulfillment of the Requirements For the Degree of Doctor of Philosophy California Institute of Technology <Pasadena, California 1977 (Submitted July 29, 1976) ii !!To 1/a n fie" e iii ACKNOWLEDGEMENTS Professor Charles Papas has provided fresh insights and constant encouragement during the course of this work at the California Institute of Technology ( Caltech). Discussions and work with Dr. Charles Elachi of the Jet Propulsion Laboratory generated new ideas. Professor S. R. Seshadri of the University of Wisconsin-Madison provided hospitality, helpful criticism and stimulating discussions during a recent visit. Dr. Gary Evans of R & D Associates, Professor Cavour Yeh of the University of California-Los Angeles and Professor John Pierce of Caltech have been available for consultation. I thank these people and many others for their help. A pleasant work envirornnent and computing time were provided by the Jet Propulsion Laboratory through the Physics Section. The expert typing was performed by Mrs . Virginia Conner. At various times I was supported by Caltech, the National Science Foundation and the Jet Propulsion Laboratory. iv ABSTRACT The interaction of electromagnetic waves of wavelength A with periodic structures of spatial period A. are studied. The emphasis of the work is on Bragg interactions where A""' 2/\./N and the Bragg order N takes on the values 1, 2,... . An extended coupled waves (ECW) theory is developed for the case N ;;;: 2 and the results of the theory are found to compare favorably with the exact results of Floquet theory. Numerous numerical results are displayed as Brillouin diagrams for the first few Bragg orders. Moreover, explicit expressions for coupling coefficients, bandgap shifts and bandgap widths are derived for singly periodic media. Particular note is taken of phase speeding effects. The effects of multiharmonic periodicities on the control of feedback strength are investigated. It is found that with proper phasing the feedback strength becomes zero and the bandgap disappears. Coupling parameters are calculated for typical multiharmonic periodicities for the first three Bragg orders. For odd Bragg orders, inverted bandgaps and phase slowing occur when the gain or loss of the media is modulated. Also average gain or loss affects the bandgap shape and the spatial or temporal growth or decay. Absolute instabilities are observed and expressions are derived for the instability frequencies, thresholds and growth rates. Under certain conditions, instabilities,occur for' structures with average loss. The results for the first and second Bragg orders are archetypical of all odd and even orders respectively. V Applications of t i e ECW theory to higher-order DFB filters involve such phenomena as transient propagation, effects of periodicity profiles and the relative coupling due to boundaries and periodicities. The calculation of higher-order DFB laser parameters shows that the mode spectrum is asymmetrically shifted and the threshold gain is greatly dependent upon the periodicity profile. Approximate threshold parameters .are calculated for high and low gain and for all Bragg orders. In addition, application of the ECW theory to holographic gratings and beam propagation is made. vi TABLE OF CONTENTS Page Chapter I: Introduction 1 Chapter II: Coupled Waves and Floquet Theory 5 A. Bragg Reflections 5 B. Coupled Waves Approach 8 1. TEM Waves in Passive Unbounded Media C. 8 2. Coupled Waves Dispersion Relation 12 Floquet Solution 13 1. TEM Waves in Passive Unbounded Media D. E. F. Chapter III: 13 2. Hill's Determinant 16 Relation of Coupled Waves Solution and Floquet Solution 19 Numerical Results 23 1. Limitations of Hill's Determinant 23 2. Brillouin Dia.grams for Lossless Passive Media 24 Modifications and Comments 31 1. Arbitrary Periodicities 31 2. Corrugated Surfaces 32 3. Com.ments 32 Higher -Order B r agg Intera ctions •. 33 A. 33 Extended Coupled Waves (ECW) Theory I. TEM Waves ih Passive Unbounded Media 2. ECW Dispersion Re.lations 33 43 vii Table of Contents (Cont'd) Page B. C. D. Chapter IV: Chapter V: Nunierical Examples 46 1. Second-Order Interaction 46 2. Third-Order Interaction 52 Mult:iharm.onic Perturbations in ECW Theory 56 th 1. N Order Bragg Interaction £1 and fN 57 with 2. Fourth-Order Bragg Interaction with f , f and £ 2 1 4 63 Comments on ECW Theory 68 Complex Periodic Media 70 A. Floquet Solution 71 B. ECW Theory for Complex Media 75 1. Analytic Expressions 75 2. First-Order Interaction 76 · 3. Second-Order Interaction 80 4. Third-Order Interaction , 84 C. Multiharmonic Periodicities 87 D. Comments on Com.pl.e x Periodic Media 88 Stability of Brag~ Interactions in A~tive 92 A. ECW Equations with Sources 93 B. Application of Stability Criteria to Bragg Resonances 99 Complex Coupling and Multiharmonic Periodicities 108 Comments on Stability U"l C. D. viii Table of Contents (Cont'd) Page Chapter VI: Applications of ECW Theory 113 A. DFB Filters 113 1. Effect of Longitudinal Boundaries 113 2. ECW Reflection and Transmission Coefficients 117 3. Born Approximation Reflection Coefficient 127 4. Transients in Periodic Slabs 129 5. Discussion of DFB Filters 133 Higher-Order DFB Lasers 136 1. ECW Result for Threshold Gain and Mode Spectra 136 2. High-Gain Approximation 137 3. Low-Gain Approximation 138 4. Discussion of Higher-Order DFB Lasers 142 Higher -Order Hologram Diffraction 142 1. ECW Equations for Transversely Periodic Media 144 2. ECW Reflection and Transmission Coefficients 146 3. Discussion of Holographic Gratings -147 Gaussian Bean1s in Periodic Media 148 B. C. D. Chapter VII: Conclusions 153 Appendix A: Asymptotic Form of Fields for Absolute Instabilities in Periodic Media 155 Appendix B: ECW Param.e ters for Square-Wave, Triangular Wave and Sawtooth Periodicities ]59 ix Table of Contents (Cont'd) Page Appendix C: Appendix D: Appendix E: References·: Reflection and Transmission Coefficients of DF'B Filter 165 Approximations for DFB Threshold and Spectrum 168 ECW Equations for Transversely Periodic Media 173 177 CHAPTER! INTRODUCTION This report investigates Bragg interactions in periodic media by using the example of electromagnetic waves propagating in spatially periodic dielectrics. The main purposes of this report are to develop physically meaningful approximate methods for higher-order Bragg ' interactions, to show the mathematic.al foundations of the coupled waves theory and to give physically meaningful explanations for several previously unexplained mathematical results. Exact and approximate theories are compared nurnerically and applications are made to both bounded and unbounded media, to lossless and lossy media and to both passive and active media. Examples are given which correspond both to wave-packets in space and time and to the steadystate response of plane waves. ' The history of wave propagation in periodic media started with Mathieu's equation and Hill 3 1 in 1868 and subsequent generalizations' by Floquet2 in the 1880' s. Although Mathieu's equation had its origin in . problems associated with elliptical boundaries, we will also show its connection to wave propagation in periodic media . 4 was first considered by Lord Rayleigh in 1887. This latter problem He considered the effect of periodic density variations upon the propagation of waves on a string. In the early 1900 1 s a different, more physical, approach was taken by Sir William Bragg. He derived the necessary spatial period for constructive reflection of X-rays by crystals. These ideas were formalized for quantum mechanical application,s by Bloch 1928. two books in the 1940's, one by McLachlan 5 44 i_n and ~e othe!' by Brillouin, 6 summarized previous work with Mathieu functions and 2 with waves in periodic 1nedia. The books also provide useful bibliographies. While most of the above work was concerned with exact solutions of differential equations, a second, independent approach was taken in the late 1940 1 s and early 1950 1 s. stressed the physical concept bations. This approach of wave coupling by periodic pertur- A mechanical device demonstrated this effect in 1949 by coupling torsional energy between two bicycle wheels which were periodically loaded with magnets. 7 In 1953 Pierce 9 used energy considerations to formulate what is now known as the coupled waves approach or the coupled mode theory. This approach has been popular because of its sirnplicity and intuitive appeal. Summaries of the coupled waves approach are given in texts by Pierce others. 14, 18 26 ' 10 and . . W1th1n the last twenty years the coupled waves approach has been successfully used in such diverse areas as holo- . . 15 22 68 69 81 82 83 84 85 . . gram diffraction, • • • ' • • • • waveguide couphng, 12 14 16 19 20 . 17 74 . • ' ' ' traveling-wave tubes, ' parametric devices, 13 14 18 56 21 • • • X-ray diffraction, distributed feedback (DFB) 1as er s, 14,23,24,25,30,35,36,58,70 an d o t·h.e r s. 10,14,22,26,27,28,29, 37 Extensive bibliographies on recent applications in optics and elec- . . 12 14 20 87 tromagnetic s are given 1n the references. ' • ' We note that the telegrapher• s equaMons which were developed before the cqupled waves theory are of the coupled waves form. These equatio~s are not approximations, however, since they exactly describe onedimensional transmission line problems. 3 The Floquet theory, which originated in the study of ordinary differential equations with periodic coefficients, has also been useful in the study of electromagnetic waves in periodic media. Although this theory is more cumbersome than the coupled waves approach, it provides an exact numerical solution. Extensions of the theory to include partial differential equations and finite length media have also been made. 44 45 • Applications of Floquet theory to electro-. magnetic waves have been made in the areas of traveling-wave • d'1c me d'1a, 11, 43 ' 56 1n • t egra t e d op t·1cs, ant ennas, 48 space- t·1me per10 32,58,63,64 corrugat e d structures, 8 • 60 and others. 11 ' 22 J 40 • 57 ' 61 ' 62, 75 Other exact methods that are used in plane-stratified material, such as the matrix method 78 88 or the method of invariant imbedding, , 89 will not be used here. The second chapter of this report contains the derivation of the first Bragg order coupled waves equations and the Floquet solution for electromagnetic waves in longitudinally periodic media. A si~ple explanation for phase speeding is given and the connection between the Floquet and coupled waves theory is explained. The dispersion relation is found as well as all pertinent coupling param.eters. Several Brillouin diagrams illustrate physical principles and compare the approximate and exact theories. The primary purpose of the third chapter is to extend the coupled waves concept to higher Bragg orders. The resulting ex- tended coupled waves (ECW) equations provide explicit dispersion and coupling information for every Bragg interaction. Numerical 4 examples again illustrate the results of both_ approximate and exact theories. A section is devoted to effects caused by perturbations of several frequencies and the resulting disappearance of bandgaps. Periodic media with loss or gain is covered in chapter four. Inverting and non-inverting bandgaps are found which depend upon Bragg order and coupling tYJ.>e. both considered. Index and gain/loss coupling are The effect of the periodicity upon average gain or loss near Bragg resonance is noted. The fifth chapter discusses the stability of active periodic media and gives explicit values for instability frequencies and thresholds at all Bragg orders. The stability criterion also speci- fies the correct root of the dispersion relation. Several applications of the ECW theory are given 1n successiv;e sections of chapter six. The reflection and transmission of transients are discussed arid demonstrated with detailed numerical examples. The extension of previous work to higher order DFB lasers is briefly covered. Diffraction efficiencies can be found when the ECW theory is applied to holographic gratings. Finally, the case of beam propagation in longitudinally and transversely periodic media is outlined. Conclusions of this report are given in chapter seven. 5 CHAPTER II COUPLED WAVES AND FLOQUET THEORY A. Bragg Reflections In order to gain physical insight into the problem of waves in periodic medi.a, we consider a plane wave incident upon a periodic structure as shown in Figure 2. I. It is apparent that the reflecting waves will constructively interfere if the reflections from. successive layers differ by an integral number of wave lengths, NA (N= 1, 2, 3, ... ). This result is usually stated as Bragg's Law, = 2 A. sin9 (N = 1, 2, ... ) A. = 2,r /k = wavelength of plane wave NA. where (ZA. I) A. = • 2,r /K = spatial period of structure N = Bragg order and where the velocity is assumed to be that of free space. N~ 2 are referred to as higher-order Bragg interactions. The cases For media with relative dielectric constant E:, we restate the result as = for normally incident waves. N/2 (N = 1, 2, . . . ) (2A. 2) Note that Bragg's Law does not account for the reflected wave amplitude, for the type of periodicity present or for the effect of slight variations of the wavenumber k from the value given by Bragg's Law. The latter effect is called phase 1nismatch and can be considered in a semi-quantitative way by the use of Fig. ·-2. 2. This figu,re shows an incident wave I which is reflected from a thre·e ,. layer -6- / / Fig. 2. I Bragg scattering of plane wave from periodic media. -7- a) I .... ..,__ R I b) I c) ► / Increasing Phase Mismatch I ~ ..t d) I .... --◄ R I = Incident Wave R = Reflected Wave Fig. 2. 2 The effect of phase mismatch upon the reflected wave. Zero phase mismatch a) indicates l , k E:·2 /K = N / 2. -8periodic medium to form a total reflected wave R. The wave R is a phasor sum of four sub-reflections each of which has phases relative to the first sub-reflection at the incident phase. are ignored in this simple model. Multiple reflections Parts a) -d) of Fig. 2. 2 show the relation of the strength of the reflected wave R to the phase mismatch. Fig. 2. 2a shows the constructive interference at the exact Bragg condition (i.e. zero phase mismatch) that produces large R. It is apparent that there is a considerable reflected wave R for slight phase mismatch (Fig. 2. 2b) whereas large phase mismatch will produce small R (Fig. 2. 2d). Therefore, from simple wave interference arguments, one can deduce Bragg's Law and the qualitative effects of phase mismatch. Other theories are needed, however, to account for wave amplitudes and the effect of the form of the periodicity. B. Coupled Waves Approach L TEM Waves in Passive ·Unbmmded Media Consider the case of a plane transverse electromagnetic (TEM) wave that propagates in a longitudinally periodic unbounded medium as shown in Fig. 2. 3. Assume a time variation of the form e - ·iwt_ Starting with Maxwell's equations 67 in a source-free, linear, iso;. tropic region, we find in the frequency don1a.in that w) = i w µ 0 H(z, N' x H (z, W) = -i WE: E: (z) E(z,W) 0 IJ • E (z, w) = 0 V x E (z, w) IJ • H (z, W) = 0 (2 . Bl) (2.B2) (2. B3) (-2. B4) -9- X TEM WAVE }-z~ y Fig. 2. 3 T EM. wave propagating in unbounded periodic media. -10- µo = = free-space permeability € = free-space permittivity = relative permittivity or relative dielectric w where 0 € (z) radian frequency constant and where E and H are the electric and magnetic field vectors. By combining (2. Bl), (2. B2) and (2. B3) we find the wave equation for no transverse variation 2 d E{z) + k 2 e(z) E(z) 2 dz E(z, w) = E(z)e where -iwt =0 (2. BS) = transverse component of E(z, w) = free-space wave number = w/c = 2rr/)._ e(z) = e[1 + ri f f. cos(pKz)], f =0, f = 1 0 1 p=0 p k ri s:: 1 is the perturbation. The periodic dielectric constant has been expanded in a Fourier cosine series. Assume that the electric field can be represented by just two waves, 10 12 15 19 ~ • ' • a forward wave ]'(z} and a backward wave B(z) which travel with positive and negative phase velocity ·along z. This assumption is intuitively appealing for Y) << 1 since these are the only two possible waves in the unperturbed case. Thus, we con- sider the transverse electric field E(z) = F(z) ei/3z + B(z) e -i/3z where /3 is the longitudinal wavenumber. interactions, /3/K"""' 1 /2. Then let For first-order Bragg (2. B6) -11- /3 .... /3 0 + b. j3 = K/2 + 6 /3 (2. B7) ~ e i6Az ~ -iM~z = F(z) '"-' • B(z) = B(z)e '"-' F(z) (2 . B8) Use equations (2. BS) - (2. B8) and the slowly varying approximation I I<< I / I Ftr Bu 2 /3 0 \ F' B' ) I/ I<< I/30 F' I ) \B" and 2 ( F B ) I 1 where . primes denote differentiation with respect to z, to find (2.B9) where the arguments in z have been dropped. The coupled waves ± if3 z 0 equations are found by equating the coefficients of e to · • ±ip/3 z 15 19 O zero. ' The terms proportional to e (p 2, 3, ... ) are = either ignored, termed non- synchronous or averaged over time and considered zero. 14 It will be shown later that these terms corre- spond to relatively unimportant coupling to other waves when /3 /K= 1 /2. The resulting coupled waves equations are, F'(z) - i 6 F(z) = i X B(z) (2.Bl0) = i X f(z) (2.Bll) -B'(z) - i 6 B(z) k2€-j3 2 where 0 6 = x = !l k2e 4 /30 = phase nrismatch (2.Bl2) = coupling coefficient (2.Bl3) -12• • f orm w1tn • -, th ose d er1ve • d e 1 sewh. ere. 10 ' 14 ' 1 5 ' Th ese equat ions agree 1n 18,20,23,30 The equations account for both the wave amplitudes and the phase mismatch as well as the interaction of the waves with the fundamental Fourier co1nponent of the dielectric periodicity. When o = 0, the waves F(z) and B(z) are coupled only through the perturbation Tj while the change in amplitude of one wave is proportional to the amplitude of the other wave. For .z ero perturbation, the wavenumber of F(z) and B(z) becomes equal to the phase mismatch. 2. ,Coupled .Waves Disp~1·si9n Relation t By differentiating the coupled waves equations, a wave equation [d 2 LTz2 is constructed. + ( 0 2 _ _2)] [F(z)J = O X B(z) .. (2. B14) Assuming a solution of the forin e±i.6.f3z, the dispersion relation is found to be (2. BIS) The approxim.ations for o and X are (2 . B16) (2 . Bl7) l when D./3 << K and where k 0 8 2 /K =½ and /3 0 = K/2. This produces the dispersion relation ~ K = 2 - (-R) The following properties are evident for real 8 _(2. Bl 8) and Tj. -131 1. Waves propagate without decay for ICike 2/K I> Iri/8 I which correspond to passband regions. 2. Stopbands or bandgaps occur for imaginary 6j3/K or for 1 ltike 2 /Kl<lri/SI. 3. Here waves decay. The coupling of waves is maximum at the bandgap 1 center (6k e2 /K = 0) where li/3 /K = i fl /8. The dispersion relation (2. Bl8) is plotted as a Brillouin diagram in Fig. 2. 4 which clearly shows the regions of interest, namely the stopbands (ellipse) and passbands (part of hyperbola). Note that the coupled waves analysis gives much more information than Bragg's Law. However, the coupled waves approach is only valid around the first Bragg interaction and does not describe important wave interactions at higher Bragg orders. 41 42 , Further- more, we have assun1ed a solution that is based on only two waves, F(z) and B(z). This is strictly valid only as ·ri ..... 0 when 'F(z) and B(z) are the two eigenmodes of the media. C. Floquet Solution 1. TEM Waves in Passive Unbounded Media An exact solution to the wave equation (2. B5) may be constructed through the application of Floquet' s theorem. A form of this theorem is stated as follows: a linear differential equation with coefficients periodic in zwithaperiod.i\ has a solu.tionE(z) with the property E(z+Jt) = J/31\.E(z) where /3 is the fundamental wave number and fi. is the fundamental period; define ¢(z) such that E = ei/3z , ¢(z); ¢(z) is periodic in z since ¢(z+J\) = ¢(z). then A proof of the theorem -14- kJE - -_wv€ K cK i7Jl8 0.5 7Jl4 . /3 0.5 K Fig. 2. 4 Brillouin diagram near first Bragg interaction. Dotted line is imaginary part qf 13 /K. Dashed lines are for unperturbed media where 17 = O. -15. given . . severa I re f erences. 2 ' 5 ' 46 ' 47 1s 1n We now expand the function \6(z), mentioned above, in a . . 3 43-47 Fourier series. ' The resultant expansion for E(z) is E(z) ~ iH3+n'K)z =n:::-oo L.J a e n which is the constructed solution for the wave equation. (2.Cl) This solution is made up of . an infinite number of space harrnonics, a n , of order n, which propagate with longitudinal wave numbers (l3+nK) (see Fig. 2. 6). Substitute the solution (2. Cl) into the wave equation (2. B5) to find ~ f a } ei(l3+nK}z { [E:k2 - (f3+nK}2] a +k2E: '17 n 2 p=-oo p n-p =0 (2. C2) (n = 0, ±1, ±2, ... ) p = f -p . Algebraic manipulations transform + 6 where we have defined f this result to 00 D n a n p=-oo a f n-p p = 0 (2. C3) (2. C4) where In matrix form this can be rewritten as llnll· ~ = o (2. C5) where a -2 a a - = -1 ao al a2 (2. C6) -16- D_2 .f . "f llnll = 1 2 fl £2 £3 £4 D -1 fl £2 £3 f DO 1 £2 1 f . .. (2. C7) o' ' •f •f 3 4 f 2 fl DI fl £3 £2 f D2 1 • The non-triviality condition for the matrix equation requires det llnll = O (2. CS) which is the Floquet dispersion relation connecting f3 and k. For singly periodic media (i.e. fp = 0 for p ~ 2) an expression for the space harmonic ratios and for the dispersion relation can be found in terms of a rapidly convergent continued fraction. 2. 43 Hill' s D-e.t erminant 2 47 Hi11 • 3 • suggested an alternative form for the dispersion relation which is equivalent to (2. CS). The derivation for the case . . h t 'h,e under consideration h as b een given in a previous report 42 wit following result. 2 .!. = ~(O) sin ('IT k E: 2 /K) (2. C9) -17where the elements of the Hill's determinant L\(O) { by L\ pn = 1 2 - k t: p 2 K2 - k2 E: ri fln-p = det ll L\ lj are given p =n I p -1- n (2.C!O) 2 2 Figure 2. 5 shows the qualitative behavior of sin (TT j3/K) and the resulting bandgaps where j3 is cornplex. · 3-5 Several authors sugge s t an approximation for the infinite order determinant, J2 2 2 [2 k e '11 £ /K 2 2 2 2 [2 -4k e /K ] 1 L\(O) = 1 + TT cot(TT k € ~ /K) l6ke2/K (2. Cll) which is valid for ri f p << 1 (p = 1, 2, 3, ... ). Note the following points about the dispersion relation (2.C8) and its approximation (2. Cl 1). 1. This dispersion relation takes into account all Fourier components f p of the perturbation and is valid for all k and j3. 1 2. 2 Away from the Bragg interaction regions k € /K """'N /2 1 (N = 1~ 2, ... ), L\(0)""" 1 as ri-+ O. Hence k e 2 """'± j3. Thus, the periodic med'i um has little effect upon wave propagation in the passbands as ri -+ O. This was expected from Bragg's Law and the coupled waves approach. -18- 2 sin (-,,-/3/K) Bandgap Bandgap Fig. 2. 5 A sketch of the behavior of the dispersion relation which shows the bandgap location for Bragg 2 ' orders N 1, 2, 3 whenever sin (rrl3/K) > 1 or 2 sin (rrl3/K) < O. It is assumed that f , f and f 2 1 3 are significant in the perturbation. = -193. When sin 2 2 (1r/3/K) > 1 or sin (1r/3/K) < 0, /3 is complex. 1 From (2. C9) and Figure 2.5, this occurs for k € 2 /K """N /2 (N = 1, 2, 3, ... ). Hence, the dispersion relation agrees qualitatively with the position of the bandgaps predicted by both Bragg's Law and the coupled waves approach. Note that the Floquet solution provides all of the information of the coupled waves equations with the additional advantage of being an exact solution that is valid for all /3 and k. However, the Floquet solution lacks the intuitive appeal and siinplicity of the coupled In particular, ,.me has to use an approximation to waves approach. In addition, a truncation find the approximate bandgap placement. of an infinite determinant m .ust be perforrned. D. Relation of Coupled Waves Solution and F'loquet Solution The recent intere st2 5 • 51 ' 52 in the relation of the coupled waves solution and the Floquet solution is important for two reasons. First, it gives a more rigorous mathematical foundation for the coupled waves theory. Second, it can be used to expand the coupled • t erac t·.ions. waves approac h t o hi g h er-or d er B ragg 1n 53 'rh·1s w1·11 b e shown in Chapter III. Consider equation (2. C3) for the case of cosinusoidal per •turbations. = 0 Da +a +a n n n+l n-1 D n = ~ [1 - (j3+nK) fl k2e 2 (2.DI) J =± ± (n 0, 1, 2, . ~. ) (2.D2) -20- For f) << I I (n = 0, ±1, ±2, ... ) k € 2 /K ➔ ± ([3/K+n) (2. D3) The resulting Brillouin diagran~ is shown in Fig. 2. 6 to consist of an • f"1n1·t e num b er o f• space h armon1cs • 1n a ± w h"1ch propaga t e as e±i([3+nK)z . n From Bragg's Law, it is known that irnportant interactions I between F(z) and B(z) occur at k € 2 /K ""' !3/K ""'±½where the :f sign comes from considering waves of both positive and negative phase velocities. Therefore, consider only the a _ -, a:, a~ and a+; space 1 harmonics in the Floquet solution for the electric field E(z) = a _1- e -i([3-K)z -f- a + i[3z -I - -i[3z + i([3+K)z e -a e e 1- a o o +l (2.D4) Let f3 __, f3 0 + b.[3 = (±K/2 + b. [3) (2. D5) where the ± sign holds for space harmonics that intersect each other at [3 /K = ± ½. E(z) = (a - -1 e -il.if3z + a 0+ e ib.(3z) e i/3 0 z I., -- _____/ ~ F(z) ~ B(z) E(z) = (2.ri6) a) k-1€ K 73/K k./E 02 a, a+ 2 b) I N ,_. K I a+ a+ I -3 n =-I Fig. z. 6 n=-2 n=-3 /3/K a) Brillouin diagram for an unperturbed, homogeneous, infinite medium. b) Brillouin diagram for an infinite medium after introduction of a period perturbation. Part ·b) is derived from successive translation of a) along the 13 /K a:xis. -22This is exactly equivalent to the assumed form of the electric field in the coupled waves approach. The waves F(z) and B(z) are each the sum of two space harmonics which are in agreement with previously derived results. 51 However, the terms e ±in!3 z o (n=3, 4, 5, ... ) that were discarded in the coupled waves approach now have a clear meaning. These terms correspond to coupling to higher order space harqionics. 52 3 Explicitly, the terms e ±i. !3oz correspond to space harmonics a+; and a_; which (along with a+~ and a _t) couple 1 to the adjacent Bragg intersections at k E: 2 /K """+½, f3/K = ± 3/2. This truncated Floquet theory can also reproduce the coupled waves dispersion relation. at f3/K Consider the first Bragg intersection =½. Here only the a+0 and a - 1- space harmonics are irnportant. If we truncate the relation D· a. = 0 to include only these two space harmonics, we find the dispersion relation, (2.D8) <let where D _l = = = 13 /K 0 By evaluating the determinant we reproduce the results of the coupled waves theory, namely, -23(2. D9) Similar results are found at f3/K =-}if we use the a 0 and a+; space harmonics. We note the following. 1. The coupled waves expressions for F(z) and B(z) are each the sum of two space harmonics. 2. The coupled waves approach can be viewed as a truncation of the Floquet solution. This shows that previously discarded term.s correspond to coupling to higher-order space harmonics. 3. The coupled waves approach shows that wave coupling in periodic media can be viewed as coupling between the intersecting pairs of waves that make up F(z) and B(z). E. Numerical Results The dispersion relations are numerically compared by using the Hill's determinant and the coupled waves approach . 1. Limitations of Hill's Determinant The Hill's determinant dispersion relation sirr ( ,rf3 /K) (2. E 1) is limited by the number of significant places used by the computer. In the case of the Univac 1108 this is 9 places for co.i nplex calculations . The smallest number that can be used which is la:.rger than -248 uni ty is 1 + 1 0 - . Thus, if sin 2 (-rr/3 /K) ""' 1 + 10 -8 (2. E2) at the first bragg order, then sin (-rr/3 /K) cosh(-rr/3. /K) + i cos (1r/3 /K) sinh(Tr/3. / K) r 1 r 1 """ I + 10 - S where 13 = /3 r + (3.l (2. E3) 0 For 1r/3./K << 1 and 1rl3 /K =1r/2~ the above relation is approx1 r imated by (2. E4) through trigonometric expansions. ri/8 fro1n equation (2. D9). -2 l Thus, the Floquet numerical calculations should be limited to rJ > 10ri:?: 10 The maximum value of /3. /K is 4 at the first Bragg order. In this report, ' to account for this and any other cornputer errors. Similar arguments limit the numerical calculations at higher Bragg orders. 2. Brillouin Diagrams for .Lossless Passive Media Each Brillouin diagram is a plot of normalized frequency 1 1 [k e: 2 /K = w e: 2 /(cK)] versus normalized wavenumber [/3/KJ . In many of the cases the diagrams are expanded around the Bragg interaction region where Ak and A/3 replace k and /3. Figure 2. 7 illustrates the main features of Floquet the?ry for a co- =0 for p :#- 1) with ·17 =I.' The largest effect ., 1 of the periodicity is in the vicinity of the Bragg wavenumbers kE: 2 /K =N/2. sinusoidal perturbation (i.e. f p • =:. -25- k✓E / / / / / K / //.1. ................. \ / 0.05 .// 1.5 0.1 ...../0.15 .............................. / / / / / / / "'••··;·;:;,(.,•· .................\ / 0.05 ..9.J.........0.15 1.0 ······--··············· / / / / / / ....../, / / ............ ....... •••· . . / 0.5 0.05......9.J. . . . 0.15 ···•••••••••• r; = 1.0 0.5 1.0 1.5 2.0 /3 K Fig. 2. 7 Brillouin diagram. for first three Bragg orders when· 'r] = I. 0 using Floquet theo r y. {3 /K on separate scales . case . Dotted lines show imaginary parts of Dashed line represents the· _unperturbed A cosinusoidal perturbation is assumed,. -26where the dispersion relation deviates from the unperturbed case (dashed line) and 13 beco1nes complex. Note that the bandgap is shifted towards the larger wavenumber and that this shift increases w1"th B ragg or d er. Th"1s causes p h ase speec1·1ng 43 • an in• w h"1ch 1s crease in phase velocity (= w/13) due to an effective decrease in the dielectric constant. Physically, as the wave travels through the periodic medium, it speeds up and slows down with respect to the 1 unperturbed velocity c / € 2 . However, if we consider an average velocity (v) we find for singly periodic media (v) C = dz (2. ES) J€(lt'l') cos Kz) where /1. = 2rr/K = spatial period. For '1') < 1, expand the square root to find (v) = (v) = JO/1. (l - !l2 cosKz + -83 '1') 2 cos 2Kz + • • • )dz C /1. €2· C --r €2 [1+(3rr/16)'1') 2 4 + O(ri )] (2. E6) (2. E7) 2 Hence, the phase speeding is accounted for by effects O(ri ) for '1') < 1. Figure 2. 8 is an expansion of the first-order Bragg interaction region of Figure 2. 7. is superimposed. The coupled waves dispersion relation (2. Bl 9) Even for '1') = 1, the coupled waves theory closely predicts the correct coupling coefficient as indicated by the maximum value of 13 in the bandgap. However, the coupled waves theory does not predict the bandgap shift or phase speeding for first-order Bragg interactions . -27- 0 -0.1 -0.2 17 = 1.0 /3o -= 05 . K -0.3 I -c0.3 -0.2 -0.1 0 0.1 0.2 \, 0.3 .., ~/3 K Fig. 2. 8 Brillouin diagram of first Bragg interaction with 'r]::: 1. This compares Floquet theory (upper curve) with _coupled waves theory ( 1 owe r -·curve). Dotted and heavily dashed lines are imaginary parts- of ~13 /K. -28The coupled waves result (lower curve of Figure 2.8) can be used for perturbations other than 17 = 1 by multiplying each scale by 17. Figure 2. 9 shows similar results for 11 = 1. 0, O. 1, 0 . 01 from the Floquet theory. As 1l decreases, the coupled waves theory becomes a better approxim.ation to the Floquet theory. The curve for 1l = 0. 01 is also the coupled waves curve for all three cases since the Floquet theory and the c oupled waves theory are graphically- indistinguishable . Results of Figures 2. 8-2. 9 are summarized in Table 2. 1. Floquet Coupled Waves ri= 1 ri=0. 1 17=0.0l n=l 17=0. 1 fl=0. 01 Coupling x/K 0.1250 0.01250 o., 001250 0.1267 0.01250 0.001249 Bandgap Shift BGS o. 0 0.0 o.o 0.04 0.0005 <0.0001 0 .25 0 0.0250 0.00250 0.26 0.0255 0.00255 Bandgap Width w Table 2. 1 Summary of coupled waves and Floquet theory at the first Bragg order. The relative contributions of the different space harmonics are shown in Figure 2. 10. The upper and lower curves are the result of matrix truncation at 3x3 and 5x5 ele1nents respectively. Each truncated matrix is centered around the matrix element 6 00 in equation (2. ClO). The differences are not large and the 5x5 matrix produces dispersion characteristics that are ~ 1%diff~rent than those of the l 9xl 9 matrices used in Figures 2. 7-2.9. -29- ~k../E K 77 =1.0 77 = 0.1 0 .3 0.03 0.003 'Y} = 0.01- 0.2 0.02 0.002 0.1 0.01 0.001 0 .15 .015 .0015 -0.1 -0.01 -0.001 0.001249 -0.2 -0.02 -0.002 ~ f3o = 0.5 ·-0.3 -0.03 -0.003 K -0.1 -0.3 -0.2 -0.01 -0.03 -0.02 -0.003 -0.002 -0.001 0 0.1 0.01 0.001 ~ ~ 0.3 0.2 0.02 0.03 0.002 0.003 ~/3 Fig. 2. 9 Brillouin diagram at fir st Bragg order for rJ = l. 0 (top curve). ri = o. 1 (rrliddle curve). and 17 = O. 01 (bottom curve) using Floquet theory. Bottom curve also represents coupled waves solutions for all three cases. case. Note difference in scale £or each Imaginary b.!3 /K values are the elliptical curves with separate scale. K -30-- 0.3 0.2 0.1 0. 05 0.1 //0.15 •0. 1276 ~-;✓ 0 "- "' __,,... , __ ........ "" "'\, -0 .. 1 - ~.,,, 1 f3 -0.2 /K = 0. 5 CD 3 x 3 Matrix ® 5 x 5 Matrix / -0.3 0 / / / L -0.3 -0.2 -0.1 0.1 0.2 0.3 ... Fig. 2~ 10 The effect of limiting space harmonics in the Brillouin diagram. using Floquet theory at the first Bragg interaction. · -31 F. Modifications and Comrnents 1, Arbitrary Periodicities The coupled waves equations were derived for 00 E:(z) = e-( l+r) 6 f cos (pKz)) where only the f term played a role 1 p=O p in the fir st Bragg order calculations. For generality, consider the expan·sion of a completely arbitrary (although sn1ooth) €(z) as E: (z) = € [1 +ri where f 0 ~ [f cos (pKz )+ g_p sin(pKz) ]] (2. Fl) p=O p =0 =g 0 The analysis is similar to the previous calculations which lead to the coupled waves equations (2 . B 10 -13). Following the identical procedure we find 1 = iX+B(z) (2. F2) 1 = i X F(z) (2. F 3) F (z)-i6F(z) B (z) - i 6 B(z} where 2 2 6 -- e -13 0 2 f3 0 ±. = 17l±i172 4 k X (2.F4) k2e -13 0 - (2 . F5) The <lisper sion relation (2. D9) is modified by the substitutions 2 2 .!. fl -+ (111 + 112')2 2 X t -+ - X X (2. F6) (2, F7) -32All previous Brillouin diagrams can be used with the above substitutions for the periodicity given by (2. Fl). The fact that X- = (X + ) *, where the asterisk. denotes co1nplex conjugate, is a general result which holds for lossless sys terns. 2. 18 Corrugated Surfaces The previous results are st r ictly valid only for volume per turbations. · Similar results have been extended to surface perturba. 20 24 tions or corrugations. ' The extension :involves the assumption that the surface perturbation can be replaced by an equivalent volume perturbation. However, this assum.ption, known as the Rayleigh assumption, is valid only for Kd < O. 448 (where d = corrugation depth and K = periodicity wavenumber) as shown by Millar. 33 Physically this occurs because deep surface corrugations have proportionately less effect on the surface waves than do shallow corrugations. An exact Floquet analysis which solves the b01;mdary..- . value problem has been given . 3. 63 64 ' Comments This chapter has set the groundwork. for the next calculations. In addition the differences and similarities of the Floquet and coupled waves theory were discussed. In particular the coupled waves theory was seen to be an approximation of the exact theory, where certain space harmonics were retained and combined while others were discarded. A similar process will lead to descriptions at higher- order Bragg resonances ih the succeeding chapters. -33- CHAPTER III HIGHER-ORDER BRAGG INTERACTIONS This chapter is an extension of the previous chapter to higher order Bragg interactions (i.e. N ;;;: 2 in Bragg's Law). Present man- rnade dielectric periodicities other than superlattices are limited to the order of /i. ~ 1000 R.. Therefore, some applications in integrated optics require operation at higher -order Bragg interactions . Already higher order DFB lasers have been experimentally demonstrated~O, 7 71 0. Other optical applications include couplers arid filters . In section A, the coupled waves theory is extended to all Bragg orders for singly periodic media. Explicit expressions are given for all iinportant parameters relating to the bandgap and coupling. Numerical examples are give n in section B for the first three Bragg orders. Multiharm.onic periodicities and a fourth-order numerical example are given in section C. gaps is also shown. An example of disappearing band- In some of the illustrative numerical examples, we will use large values of the perturbation that may not be physically realizable. However, the objective is to dramatize the effects of the perturbation on the Brillouin diagram.. In addition, the extended .. ... coupled waves (ECW) examples are easily scaled for other values of 'Tl• A. Extended Coupled Waves (ECW) Theory 1. TEM Waves in Passive Unbounded Ivle dia By extending the assumptions made in Chapter II, we state the following assumptions for N th order Bragg interacti ons: · -34- 1. The most significant space harmonics are F (z) = a: eif3z 1 . - -i(B-NK)z - -if3z + i(r,+NK)z +a_Ne • andB (z)=a e +a_Ne . 1 0 2. To provide cross -coupling between F (z) and B (z) we 1 1 account also for the following pairs of space harm.onics - + + - + . (a_N+l' a+l ), (a_N+Z' a+ 2 ), ••• , (a_ 1 , aN-l) wluch are l slowly varying near k 8 2 /K = N /2. In this way we account for the intersecting space harm.onics between F (z) and 1 B (z) in the simplest possible manner. 1 3. Self-coupling which occurs between F (z), B (z) and their 1 1 adjacent space harm.onics must also be included. Assume fl << 1 although the theory may hold for fl -, 1 as 4. in the first order case. 5. All other space hannonics are ignored. The derivation is started by in.eluding the above m .entioned space harmonics of the Floquet theory in the expression for the electric field ""' i - -i(j3+nK)z NJl + i(f3+nK)z E(z) = Li a e + LJ a. n e n n=-(N+l) n::::0 Near the N th ( 3. Al) order Bragg interaction let {3 .... /3 0 + 6 {3 = (± NK/2 + L\ f.) ( 3. A.2) where the ± sign is used for space harmonics that intersect each other when {3 /K ~ 6. The electric field becomes -35- * N /2G • N~ • bP ] E(z) = 6 a- N e -i t-'z +a~ e 1 1_,z exp[i(l -2n/N)f3 z] n=-1 * N✓ 2 + n- r+ LJl LaN-n e n=- + o Tll ibif3z _, -ibif3z]exp - [i(l -2n/N)f3 zJ -,-- a_n e o (3. A3) (10 ) [a --N /2 e -H,f3z + a +N /2 e i6j3z] where N - 2 2 for N even N - 1 2 for Nodd - 1 for N = 1 j 1 for N even f O for Nodd This can be rewritten as (3.A4) where F 1-Zn/N(z) -i6j3z + i6j3z = a nNe ta n e + = aN -n e i6~z - + a -n e -i6~z S(z) Substitute the assumed form of the solution (3. A4) into the wave equation (2. BS) and use the slowly varying approximation as before. -36We drop the arguments in z for sin1plicity. { - (l -2n/N) 2 /3 2 F + 2i(l -2n/N)[3 F' , } J(l -Zn/N)/3oz o 1-2/n/N o l-2n/N + {- (l-2n/N)2/3; Bl-2n/N .• 2i(l-2n/N)l3oBl-2n/N}e-i(l-2n/N)/3oz 2 ·f- k E: 7J {F 2 l-2n/Ne i(l -2n/N)P O z + B (n l-2n/Ne -i(l -2n/N)(3 0 z} = -1, 0, 1, 2, •... ) (3. A5) which is analogous to (2. B9). Next,. for simplicity we limit the Fourier coefficients of the periodicity such that . f p = 0 for p -f. 1 (i.e. singly periodic media). Equate the coefficients of e ±i.(l -Zn/N)f3oz .• ii (3 •.A6) -371 We define interaction as meaning the region in the (ke 2 , f3) plane near the intersection of two space harmonics. We again note that higher - order interactions refer to interactions with frequency greater than 1 that of the fundamental Bragg frequency defined by k E: 2 /K = ½. Thus each wave, F(z) _2n/N and B(z) 1 _2n/N' defines an interaction. 1 The coupling diagram, Figure 3. 1, and the above set of coupled equations show that F (z) and B (z) are coupled through N intervening inter1 1 actions. Since each coupling is proportional to 17, the cross-coupling between F (z) and B (z) should be proportional ~o riN. 1 1 The adjacent interactions to F (z) and B (z) contribute term.s of order ri 1 1 2 to the phase mismatch and hence both the F(z)l±Z/N and B(z) 1 ±2 /N terms The F(z) +2 /N 1 need to be retained to obtain correct self-coupling. and B(z) + /N term.s will not contribute to the cross-coupling except 1 2 N+2 to order ri ~ This small contribution is ignored. To show the above statements mathematically, apply the slowly varying assumption and solve the inner N-1 and the outer two equations of (3.A6) to get all waves F(z) 1 _2 n/N' S(z), B(z) 1 _2n/N (n = -1, +1, 2, 3, .•• ) in terms of F (z) and B (z). 1 1 The outer two equa- tions can be solved trivially for F(z) 1 +2 /N and B(z)l+Z/N" We begin by using matrix manipulations on the inner N-1 equations of (3. A6) Fl-Z/N = det IIA!l/det llcll Bl-Z/N = <let [1611/det llcll I (3. A 7) kV€ K XxXx-- ~y71" ~ N/ ~~~ 8 1+2/N 8 1 8 1-2/N 8 1-4/N for ECW theory~ 1 ~~~ Fl-4/N Fl-2/N FI F1+2/N . . - - - - - - - ~{31 K Fig~ 3~ 1 Coupling diagram at N cross-coupling between F ··XXXX 2 th and B 1 order Bragg interaction. Dominant is shown as well as self-coupling I w 0:, I -39- where Cl C C c? 0 C L., • \\c\\ = . 0 'A 1 C C (3 . A8) CZ C C C C 0 1 C c2 . \\A\\= 0 C AZ Gl C C Cz (3. A9) CZ C C C C 0 1. (3.AJ.O) \\G\\ = 0 C GZ C C n = k 2 e - (l-2n/N) Cz C C Cl 2 f3 2 0 2 = T) k e /2 G 2 = A 1 = - k 2 €T) F2 1 G 1 =A 2 = -k 2 eriB 12 Instead of solving (3. A 7) exactly, we approxirnate the deterrninants for T) << 1 when N ~ 2. <let \\ell Since C << C , we find n '2 f (n) """Tf' =rr ~:< 2 [k € - (1-Zn/N) 2 2 2 (30 ] (3.All) -40(N-1)/2 TT £2 (n) for N odd n=l N/2 where -1-1 * £2 (n) = TT f(n) for N even n=l 1 Approximat _e detllAII for N = 1 and <let IIGlj in a similar rnanner. (3.A.12) (3.Al3) The solution for F(z)l-Z/N becomes (3.Al4) Near the N th. Bragg interaction we approximate the free-space wave- number by The expression for F(z)l-Z/N can be approximated by F(z)l -2 /N = Ex'i:end the above formulation to include the N=l case by introducing the symbol ~N. (2.Al 5) -41-CN11NF (z) (-1) F(z) = - - - - "1' - + l-Z/N B(l-1/N) N+l (ri/2) N-1 B 1(z) 11* {4n(n-N)/N 2 } 2 (3.Al 6) N = 1 where ,N = { 01 N ~ 2 Similarly for B(z)l-Z/N we find the analogous result, -C B(z)l-2/N= T)NB (z) (-l)N+l (r1/2)N-l F (z) N 1 + - - - - - - - = - ,1~ 2 2 8(1-1/N) lT*(4n(n-N)/N } (3.Al 7) The expressions for F(z\+ 2 /N and B(z) 1 +2 /N are trivially fotmd from (3. A6) under the stated approximations as -CN(rJ/2)F l (z) F(z)l+2/N = 1-(1+2 /N/ CN17NF 1 (z) - -CN(1']/2)B 1 (z) _ CN 11N B 1 (z) l-(1+2/N) 2 - (3. Al 8) 8(1+1 /N) (3.Al9) 8(1+1/N) Expressions (3. A16-3. Al 9) are substituted into the following two equations from (3. A6) for F (z) and B (z) 1 1 2 2 , -k e: 'IJ ) , (k e:-f3o )Fl (z) + 2i f3o Fl (z) = 2 (F(z,)1-2/N+F(z 1+2/N' 2 (3. A20) Upon rearrangement the following extended coupled wave (ECW) equations are forrned. • . F 1' ( z) - i oN FI (z) = i X N B l ( z) (3.A22) -B '(z)-i oN B (z) = iXN F (z) 1 1 1 (3. A23) 6N where = = phase n1.isn1.atch I = N (3. A24) 2 /3 0 (3. A25) Tf [4n(n-N)/N 2 } 2 -:< = coupling coefficient = Bragg order = nK/2 The ECW equations agree in form with other work in diffraction • gra t·1ngs 22, 68, 69 w 11ere t·•h e per10 • d.1c1·t.y 1s . perpem.1cu 1· 1ar or h o I ograp h 1c to the propagation direction. However, this is the first time that analytic expressions have been derived for propagation in longitudinally varying media where the coupling, bandgap shift and bandgap width are given explicitly for all Bragg orders. Table 3. 1 presents num.erical factors found in the ECW equations for the first five Bragg orders. N rr [ ,:< 2 2 4n(n-N)/N •l g(N)= (4/N)(I-1/N) 2 2 1-C N(fl /2 / [N / [2(N -- l)] J • 1 1 0 2 I 1 2 2 1 - - · (ri/2) 3 3 ( 8 /9) 2 8/9 1 -16 (ri/2) 4 (3 I 4) 2 3/4 8 2 1 --(ri/2) 15 5 (16/25/ 16/25 1 -~(ri/2>2 48 1 9 2 Table 3. 1 Numerical values for factors 1n ECW equations. -432. ECW Dispersion Relations The dispersion relation is derived from the ECW equations as in the first order case. (3.A26) The following approximations may be made in the interaction region when !::. BN << K. (3. A2 7) (3.A28) where 1 The N th L:ikN = k - j30 /e-2 l:ij3N = /3 - /3 0 /30 = NK/2 order dispersion relation is explicitly !::. /3 N -K- (3. A29) Note that for real € and 'fl, the maximum imaginary part of L:i j3N/K occurs when cSN/K = O. This d e fines the norm.alized coupling in terms of the bandgap behavior since, -44- I I 613 I I 71 N Im{ KN}max = l·XKNI = 2N+Z • N • 2 }2 · Tr' [• 4n(n-N)/N (3 . A30) If oN/K = 0, then n1aximum coupling takes place at a wavenumber 1 that is displaced from the exact Bragg condition ke: a /K = N/2 for N :::: 2. This is referred to as bandgap shift (BGSN) and is defined by 6k e; ½ BGS = N N • K Io N C 'fl 2 N3 = N (3. A3 l) 3 2 (N 2 - 1) = 0 We note that this raises the Brillouin diagram and produces phase 2 speeding 0(77 ) as expected. The bandgap occurs whenever I 6N/K I < lxN/K I which causes 6/3N/K to become imaginary. The bandgap width (WN) is defined by TT * I (3. A32) N 2 2 • [ 4n(n-N) /N } Table 3. 2 summarizes some of the ECW results for the first five Bragg orders. N Coupling ';( /K N Bandgap shift BGSN 1 .!l. 0 8 2 4 2 n_ 12 4 -+ ..!L 3 3 243 2048 27 TI 256 4 - ..!L 9 5 2 ~ 15 4 5 n 1953125 18874368 Bandgap • width WN .!l. 2 n ; 125 !) 768 2 2 3 243 11 1024 ~ 9 1953125 !] 943 7184 Table 3. 2 Summary of the main features of the ECW theory for the first five Bragg orders, 5 -45We note the following points about the ECW theory: 1. The ECW derivation is an intuitively based theory that gives explicit values for xN/K, oN/K, BGSN and WN for any Bragg order. 2. The ECW theory predicts coupling coefficients that are proportional to (f)/2)N due to cross-coupling. This is expected because of the N interactions between F (z) and B (z) where each interaction couples 1 1 with strength Y)/2 to adjacent interactions. The sign of the coupling coefficient alternates with Bragg order. 3. The ECW theory predicts that 1naximum coupling occurs not at exact Bragg resonance but at a shifted frequency instead. N ri 2 This shift is proportional to for N :?: 2 and is due to the self-coupling of F (z) and B (z). 1 1 This bandgap shift accounts for the phase speeding effect that was first found in the Floquet results. 4. The bandgap width is proportional to (f)/2)N. Hence for large N, only a small range of frequencies will cause significant coupling between F (z) and B (z). 1 1 Also note that for large N, W N < 2 BGSN so that the longitudinal wavenurnber 13 may be real at exact Bragg resonance. Since bandgap width and coupling are proportional, it is impossible to attain large coupling and small band.gaps simultaneously. -46- . l 5. A simple scaling rule exists such that if rJ and tkN e: 2 /K are reduced by the same factor, then t l3N/K will be reduced by the identical factor. Thus ECW results for large perturbation can be directly applied to other perturbations. The extension to sinusoidal perturbations is made in the same manner as in the first order theory. periodicities of the type ri 1 cos(NKz) + ri 2 sin(NKz). The relative magnitude of the waves F(z)l-Zn/N' B(z)l-Zn/N (n S(z), = 2, 3, ... N* /2) can be found from equation (3.A6) in the same manner that F(z)l-Z/N and B(z)l-Z/N were found, The rei;mlts are in terms Qf F (z) and B (z) which are in turn related by 1 1 the boundary conditions . B. We will not need these results now. Numerical Examples 1. Second-Order Interaction The ECW approach uses the ten space harmonics shown in Fig . 3. 2 for N = 2. The explicit dispersion relation is = (3. Bl) and the phase mismatch and coupling are Xz where = 2 2 2 k e: (1-rJ /6) - 13 0 2 ~o = - k = K 2 E:!] 8 13 0 (3 . B2) 2 (3. B3) kVE K ~~,~~"}----an a') a_ 1 a1 a_') 'a-::i a_-~\ a_+l al a~ I ,j:s. -.J t -3 -2 Fig. 3. 2 -1 0 1 2 3 {3 / K Coupling diagram which shows the ten relevant space harmonics that are used in the ECW theory for secondorder interactions. -48Note the change in sign of the coupling coefficient from that of the first-order theory. The coupling is provided by S(z) and the bandgap shift is accounted for by B (z), S(z) and F (a). 2 2 Phase speeding is the result of the bandgap shift. Figure 3.3a,b,c shows a com.p arison of the Floquet and ECW theories at the second Bragg order for fl= 1. 0, O. 1 and O. 05 respectively. Although one would not e:,.,.'Pect the ECW theory to hold as fl ..... 1, Figure 3. 4 demonstrates the validity of the theory in this case. We note that for TJ = 1, the Floquet theory predicts a slightly larger bandgap than does the ECW theo1~y. of fl ~ O. 1, However, in the practical case the two theories become graphically indistinguishable. Table 3. 3 summarizes these results for second order Bragg interactions. Floquet ECW fl=0.05 fl=O. 1 ri= o. 0: fl=l.0 Tl = O. O Coupling 2 /K . 125 . 00125 Bandgap Shift BGS 2 . 0833 . 000833 .000208 . 10 . 00080 . 00022 _ Bandga.p Width w2 . 250 . 00250 .26 . 0025 . 00063 x fl= 1 . 0003125 . 133 . 00625 . 001:ZS . 000311 Table 3. 3 Summary of ECW and Floquet dispersion relations at the second Bragg order interaction. Figures 3. 4 and 3. 5 show the effect of truncating the ECW and Floquet theories. In Figure 3. 4, the outern1ost space harmonics (<a -,a;) and (a_;,a_1) of Figure 3.2) are not used for .fl= 0.1. 1 In this case the coupling between Fi and Bi i ::; given correctly since · the 0.5 0..005 0..0010 0. 004 0.3 0. 003 ~"' 0.2 /).kVE 0.1 K "'"' 0 "'"' a.cm 0.001 "' -:<" ~0.1 ~ECW -0.2 -0.3 / .15 - / '1 • l. 0 "--."' f:Jo!K • 1.0 "' 0 0 0.1 0.2 0.3 "'"' / / "'"' "\ / "' ,0. 0005 Q. 0015 ""- ECW / -a.cm 0.(0)6 0 tJ.P 6P K K "' o.cm "-. '1 • 0.05 /:Joi~· 1.0 O.<XXl4 "'"' O.<XXl2 0 ~efL~uET & 1/ " / '1 • 0.1 '"' / p 01K • 1.0 , -0.<m / ---0.00125 7 -0.001 -0.003 -0.3 -0.2 -0.1 o. (0)81 (c) (b) 0.4 • (a) -O.<XXl2 / -0.(m1 -0.0006 / -.... "' ~--j.,,.,,,✓ 0. (XX)l 0. <XXl2 / ✓-FLOlUET &"'ECW / -0.(XX)t 0 6P K Figo 3~ 3 Brillouin diagram at second Bragg order for a) 'I') = 1. 0, b) '17 = O. 1, and c) '11 = O. 05 showing Floquet and ECW results. Imaginary ~{3 /K values are the elliptical curves with separate scale. Note the close agreement of theories for '11 ~ O. 1. 3 ~ "'"' 0.(XXM " I .p.. '°I -50- 0.004 / 0.003 / 0.002 0.001 tl'I.Ji /0. 0005 o. ~ ~ - - - - - - , FLOQUET "-. & ECW -0.001 -0.002 ,,-0.1 -0.003 0. (X)l / ........•• ,"1 o. 00125 ........ ,,,,,, K / ~0/K • 1.0 ~ / -0.003 -0.002 -0.001 "'" "" 0 Ml. 0.001 0.002 0.003 K Fig. 3. 4 The results of neglecting the effects of F and B 2 2 in the ECW theory at the second Bragg interaction~ -51- 0.6 0.5 0.4 / 0.1 / _,,,, .. / . .. •·~ -~········ . •· ® ------- 0 -0.1 -0.2 -0.3 0.05 / -0.3 / / -0.2 / "'"' / (i) 3 x 3 MATRIX ® 5 x 5 MATRIX -0.1 0 ~/3 0.1 / ""' 0.2 : l 0.1363 " ~ 0.3 K Fig. 3. 5 The effect of limiting space harm.onics in the Brillouin diagran1 using Floquet theory at the second Bragg • interaction. -52cross-coupling is only dependent ori S. is not given correctly since S, F ri 2 2 and B However, the bandgap shift 2 to the self-coupling and bandgap shift. contribute terms of order Figure 3. 5 shows the effect of limiting the space harrnonic s in the Floquet theory for '17 = I. 0. The upper curve has a greatly enlarged bandgap width for a 3X3 matrix which includes the space harmonics at, a1 and a_~. adds the a: and a_~ space harmonics. The lower curve The resulting 5X5 matrix produces results that are graphically indistinguishable from the l 9X 19 matrix used for all other Floquet results. 2. Third-Order Interaction The results are. similar to the N=2 case. The explicit dispersion relation is (3. B4) = and the phase mismatch and coupling are 63 = X3 = where f30 = 2 3 2 k e ( 1 - 917 / 6 4) - f3 0 2 f3 0 2 8lk 8 1024 f3 r/ (3. B5) (3. B6) 0 3K/2 Figure 3. 6a, b displays the results of ECW and .F loquet calculations for 17 = 1. 0, 0. 5 at the third B1·agg interaction. Note that for the first time the half width (=WN/2) is less than the 1?andgap shift -53- 0.5 0.4 (a) 0.2 6kvi K 0.1 0 -0.I -0.2 -0.3 -0.3 -0.2 -0.1 0.1 0.2 0.3 0 6/3 K 0.07 0. 06 (b) "I• 0.5 o. 05 Po' K • I. 5 0. 04 6k..J'E 0.03 K ~· 0. 005 0. 010.,c , o.m ~-:•· ----~·· ·7" 0.0149 0.01 'v,, \_ECW 0 -0.01 / '\. / / '' -o. m-o. m-o. 01 o o. 01 o. 02 o. 03 6/3 K Fig. 5. 6 Brillouin diagram at third Bragg order for a) fl b) rt = 1. 0 and = o. 5 showing Floquet and ECW results. Imaginary l::.f3 /K values are the elliptical curves with separate scale. -54(= BSGN) which causes the entire bandgap to be above the exact Bragg wavenumbero This effect becomes more pronounced with increasing order and decreasing perturbation since WN a: (ri/2f whereas 2 BSGN a: (ri/2) . Although the ECW theory does not approximate the Floquet theory for N = 3 as well as it does for N = 2. (compare Figures 3, 3a and 3. 6a), the ECW approximation improves as Tl decreases. This is an expected result since in the ECW theory all waves except F 1 and B 1 are assumed to be slowly varying. However, as N increases, more of the waves contribute to the coupling between F 1 and B o Table 3o 4 summarizes the results of Figure 3. 6 . 1 Floquet ECW Tl = 1 r1 = O. 5 Tl = 1 Tl = 0. 5 Coupling 3 /K . 1186 . 0149 . 136 . 0155 Bandgap Shift BGS 3 . 105 .0264 • 16 . 030 Bandgap Width W3 . 211 . 0297 . 27 . 033 x Table 3. 4 Summary of ECW and Floquet dispersion relations at the third Bragg order interaction. Figure 3. 7 shows the effect of increasing the matrix order that is used in the Hill's determinant for the Floquet <lisper sion relation (2 . .El). It is apparent that space harmonics of order n > 2N+l must be used to insure accuracy of the Floquet result. shown for 7x7, 9X9 and l 7X 17 size matrices. Graphs are As before, l 9X 19 size matrices have been used in the Floquet theory for other figures _in this section. -55- 0.4 ....-.:~-- 0.3 / ........ ....... JD / ~;~«- ········-'::.":':, :-:-, , · , . 'X / 0.2 / I 0. 1451 \' ·....-o. 1636 0. 05 / 0. 1 : 0. b / 0. 1363 .-"/ .// / / / .. ,, ./ 0.1 ... -;;.""./ / :.,--............ ;,:·.;..._:.m ...... _,- ·-·--- ·~~- - - •- • ~ 0 / 1 / / • / -0.1 / / {J/K=l.5 / -0.2 / -0.3 / / 0 CD· 7 x 7 Matrix / ® 9 x 9 Matrix ® 17 x 17 Matrix / -0.3 TJ= 1 -0.2 -0.1 0 0.1 0.2 0.3 tJ./3 K Fig. 3~ 7 The effect of limiting space harmonics in the Brillouin diagram using Floquet theory at the third Bragg interaction. -56C. Multiharmonic Perturbations in ECW Theory Higher-order Bragg interactions are interactions in the region 1 of k e_2 /K ""'f3/K """N/2 where N::?: 2. However, there can be competing processes between different Fourier components, f p cos (pKz), of multiharmonic (i.e. p = 1, 2, 3, ... ) periodicities for N ': 2. In par- ticular we may find passbands where bandgaps existed :i.n the singly periodic case. This latter fact has been of interest in solid-state • theory where the Saxon-Hunter theorem 65 states that a forbidden level in an infinite lattice of pure type--A potentials and in a lattice of pure type-B potentials is also forbidden in any alloy containing bot·h type A and type B potentials. As applied to our problem, this theore1n im- plies that bandgaps formed by two dielectric periodicities f(Kz) and f (Kz) are just the bandgaps caused by f(Kz) and f(I<z) separately. number of counter-examples theorem. A 65 • 66 have been given . to the Saxon-Hunter The ECW approach will show e:>..rplicitly the effect of multi- harmonic periodicities upon the bandgap. The extension of the ECW theory is straightforward. However, the results for arbitrary multiharmonic periodicities becorne cumbersome. Several special cases will illustrate the general theory. From the previous sections we know that the waves between F (z) and B (z) are needed to couple energy from F (z) to B (z). 1 1 1 1 We also know that the waves F(z\+ /N and B(z) + 2 /N have to be included 2 1 to properly account for the bandgap shift. In addition, other waves that can couple significant energy between F (z) and B (z) or that 1 1 couple F (z) and B (z) to themselves have to be included. 1 1 -57Since the Fourier component [11 f p 2 cospK ] couples energy between intersections, p intersections apart with strength approxi mately proportional to 17 f , then coupling diagrams will again help p to show the waves that should be included in the ECW theory. Figure 3. 8 demonstrates son~e of the possible couplings for N = 2. We list the possible couplings and strengths shown in Figure 3. 8: >2 ; I. two first-order couplings Ct:: (ri.£ 2. o·ne second-order coupling cc 17 £ 2 3. one third- and one .first-order coupling cc(17 iif(rl £ ); 3 4. one third-, one second- and one first-order coupling 5. o·ne fourth- and one second-order coupling cc(ri f )(17 £ ); 2 4 6. one fourth- and two first-order couplings cc (17 f )(17 f )~ 4 1 1 The mathen~atical solution consists of writing a set of equations analogous to (3. A6) where the right hand terms are augmented by all of the possible couplings by each Fourier component of the periodicity, f . p N th Order Bragg Interaction with f i and fN Consider the periodicity 1nade up of Fourier components [17f1 cosKz + 17 fN cosNKz J at the N fu Bragg order. non-trivial case where O[.{ri\,)N] We assume the ~ O(rifN) so that contributions to the cross - coupling from the two Fourier components are of the same order. 2 The major bandgap shift is given by terms O[ (rif ) ]. 1 Figure 3. 9 is the coupling diagram ~hich shows the important space harmonics and couplings that, are used in this case. -4 -3 -2 -1 2 0 3 4 2-K Fig. 3. 8 Coupling diagram that shows several of the possible cross -couplings frorn F to B and self-couplings 1 1 The approximate magnitude of when N = 2" 1 the coupling strangths are shown. for F k,/E K ~~ -------,~fN ----- ,,."It,~~ XXXX··S ~~~ 81 81-2/N 81-4/N 8 1+2/N N/ ··XXXX~ 2 - - - _________,,,,\ Fig. 3. 9 Coupling diagram J: :C at the N when O[(T) £ )N] 1 ~~ • F1-4/N F1-2/N Fi • F1+2/N '\l\ th ~ O(T) fN). Bragg order .. {3/K I -60Since the coupling Tj fN only affects the equations for F (z) 1 and B (z) and the coupling Tj £ is used to couple F (z) to B (z) as 1 1 1 1 well as the coupling F (z) _, F(z)l-Z/N _, F (z) and F (z)->F(z\+z/N 1 1 1 -+ F (z), we can write by inspection from Figure 3. 9 and equation 1 (3.A6), (3.Cl) The only difference from the previous coupled equations (3. A6) is the change of notation Tj _, Tj.f 1 and the addition of the f) fN B 1 and Tj fN F 1 It is apparent from the N+3 coupled equations (3. Cl) that the terms. bandgap shift will be due to Tjf 1 as before but that the coupling co- efficient will be the algebraic sum of terms involving Tjf 1 and· TjfN. Solve the coupled equations as before to find (3. CZ) I - B (z)-i oN B (z) = iXN F (z) 1 1 1 6 N = (3. C3) 2 2 2 2 k e{l-CN(f)fl/2) [ ,N /2(N -I)J}- 13; 2/3 0 (3 .C4) -61k2e: [ T']fN + 2 j3 2 0 (3. C6) Table 3. 2 may be used to find the appropriate bandgap shift. The effective coupling coefficient is found by adding the values for x1 and XN• Hence the results in equations (3. C2-3. C6) could be written down by inspection. An interesting case arises when XN second Bragg order, = O. For example at the x2 = 0 implies (3. C7) which occurs when 17 f 2 = (17 f 1 )2 /2. Here the bandgap disappears but 2 the phase speeding effect is still given by the (T'] f ) / 12 term in (3. C7). 1 It is interesting to note that a periodic medium does not necessarily have bandgaps but it will still have the phase speeding property. In practical cases, if the effects of T']f 1 and 'rJfN cancel each other, other periodicities in the media may still cause bandgaps. Figure 3. 10a, b shows the ECW and Floquet results for ± Tl f 2 = (17 f 1 )2 /2 = 1/4 at the second Bragg order. the effects of 17 f bandgap. 1 and Tl f 2 In the first case cancel, and the ECW theory predicts no The Floquet theory shows a greatly diminished bandgap with a coupling coefficient x2 /K = O. 00568. The Floquet theory bandgap can be reduced by an order of magnitude, and perhaps more 2 • 2 by slightly adjusting the ratio f /17f away from f /17 f O. 5. For 1 2 2 1 = o.3or--.-----.----r--.-----.-------,---~ -62- (a) 0.25 0.05 • 0.20 0. 04 • "'- 0.15 0.10 6.kvi K 0. 002 _ ../0.006 --- / "" / / """'- 0.05 / / / "" 77 11 = 1/ {2 / · 7712 = 0. 25 {30 /K = 1.0 / / -0.15 A / / / / / -0.05 - "" " l<....-;:--'-;--::--:c-'-:-::--=-1::-::----1--_L__ -0. 15 -0. 10 • ECW / 0 -0.10 - / -0. 05 0 0. 05 "" " _ j __ 0. 10 __J_ ' 0. 15- 0. 4 r--.-----.----,---,-----.----,----.-~ 0.2 0.1 0 . -0.1 -0.2 77!2 = -0.25 {3 0 /K = LO -0. 3 -0. 3 -0. 2 -0. 1 0 6.(3 0. 1 0.2 0.3 -K = Fig. 3. 10 Brillouin diagra1n at second Bragg order for a) 'J1f 1 Jo.s, = = = O. 25 and b} ri£ ,Jo:s, 11£ 2 -0, 25 showing ECW and 1 2 Floquet results for multiharmonic periodicities. Imaginary 6[3/K values are the elliptical curves with separc1.te scale. r,£ Note the dependence of the bandgap upon the s_ign of ri£ 2 • -632 f /nf = O. 55, 2 1 x2 /K < O. 0002. The second case shows the large bandgap that occurs when the effect of 11£ 1 and 'rJ f 2 is additive. The vanishing of the bandgap is due to the relative phasing off 1 and fN and is therefore dependent upon the symmetry of the periodicity. Analogous effects have been noted in the electronic 5 86 stopbands of crystals 6 , and in stability diagrams of transverse magnetic (TM} wave propagation in periodic dielectrics. 2. Fourth-Order Bragg Interaction with f1, 57 f2 and f4 In the preceding case, the resulting coupling coefficient was a direct sum of the coupling coefficients of the two Fourier components. This is not always the case. order Bragg interaction where O(T] f ) 1 To illustrate, consider a fourth4 ~ O(T] f 2) 2 ~ O(T] f 4 }. That is, the Fourier com.ponents rif , nf and ri£ contribute values of the 2 1 4 same order of magnitude (i.e. O(T] f )) to the cross-coupling. 4 The details of possible couplings and the relevant space harmonics are shown in the coupling diagram of Figure 3. 11. Weaker couplings 4 such as those proportional to (T] f } ri f have been neglected. 2 1 From Figure 3. 11 and equation (3.A6) we write by inspection the following coupled equations. k{E K ~;TJf1,;---TJf2\ ~ ;--- .,.h ;"', ,.: r l 1 ~ n 0rr.,,t1\(.,,t1 \ 83/2-¾.-B, ~1/2-X- -~F112*-..·F1*_F3/2 I 0-fj::.. -5 -4/ -3/ -£ ?f O '( 2 "3 4 '5 ►/3! K Fig. 3. 11 Coupling diagram at fourth-order Bragg interaction for 2 ECW theory. Dominant couplings are shown when ('tjf 1 )4. (,')£ 2 ) and r,f are of the same order of magnitude. 4 -652 2 . I -k2E: Tl (k E: - 9/4 J3 0 )F¾ + 31/3 0 F¾ = (f F 1 ) 2 1 2 = -k2E:!] [fl(Ffl-F¾)+f2S+f4Bl] (3. C8) 1. 2) • BI (k 2"'" - °4 /3 B 1 - 1/3 1 0 2 0 2 2 2 I (k E:- 9/4 A )B3-3iA B3 r--o z r--o z where the arguments with respect to z have been dropped. Solve the coupled equations as before to find (3. C9) (3. ClO) (3. Cll) 16 [ B½ = 9 F 1 l.!l:. _!L 2 2l 3 J 16 f 1£2 - 8 f 1£2 - 8 f 1 (3. Cl2) (3. Cl3) -66These values are substituted into the equations for F (z) and Bl (z) 1 in (3. CS) to find the coupled equations for F (z) and B (z). 1 1 The and x . are 4 4 k2e [ 1 - ~ f2] - /32 · 15 1 o resulting values for 6 = (3. Cl4) (3. C15) where j3 0 = 2K and where smaller order terms are neglected. As expected, the above expressions reduce to the first-order case if f 1 = f 2 = 0, to the second-order case if £1 = f 4 = 0, and the fourth-order case if f term f 2 = f 4 = O. However, the presence of the mixed 2 f means that the above results cannot be obtained from 1 2 simple superposition of the terms from each component. As in the previous case, a proper choice of f , f , and f would allow us to 1 2 4 eliminate the bandgap. The different terms in the expression of x can be easily ex4 plained as shown in Fig. 3.11. There are four possible ways to achieve a fourth-order coupling with the three Fourier co1nponents f , f , and £ . 1 2 4 I. These are: four couplings through f ; 1 2. , a .ny combination of one coupling through f couplings through f 3. 1 2 and two ; one coupling through f • 4 The1if 2 term in equation (3. Cl4) has a large numerical co- efficient because this coupling can occur in three different ways. -67- 0. 12 0. 10 0.08 I 0.06 0. 0311 0.04 0.02 0 / / .-0.02 / / 77f1 = 0.5 / -0.04 17 f2 = 0. 2 77f4 = 0. 1 / / -0.06 / / f3olK = 2.0 / -0. 06 -0. 04 -0. 02 0 0. 02 ~/j 0. 04 0. 06 0. 08 K Fig. 3. 12 Brillouin diagram at fourth Bragg order when ,i£ 1 = O. 5, ri£ 2 = o. 2, and ri£4 = o. 1 showing ECW and Floquet results. Imaginary !':,.f, /K values are the elliptical curves with separate scale. -68Fig. 3. 12 compares the ECW and the Floquet solution for the values 17f 1 = 0. 5, rif 2 = 0. 2 and rif 4 = 0. 05. Note the close approximation of the ECW theory to the Floquet solution at the fourth Bragg order with several har1nonic periodicities even when the perturbation is relatively large. D. Comments on ECW Theory The predictive abilities of the ECW tb.eory have been sum- marized at the end of Section A. The nurnerical examples dem.onstrate the accuracy of the ECW theory at higher Bragg orders and for rnultiharmonic periodicities. The ECW theory is particularly attrac- tive for small perturbations where the Floquet theory is cumbersome and time consuming. where often 17 ~ 10 -3 This is relevant to present optical applications . The extension to higher Bragg orders opens up the possibility of using the many first-order results of the literature at higher Bragg orders by the use of the proper oN and ~• Several extensions of this nature will be made in a later chapter. Two interesting results have been explained in this chapter. Firs1; the phase speeding effect was observed to be a result of selfcoupling and occurs regardless of the value of the higher frequency Fourier components f . p Second, the disappearing bandgaps were explained through simple interference effects of different periodicity frequencies. This opens up the possibility of controlli1;1g feedback through the relative phases of two harmonics since feedback or coupling strength is directly proportional to bandgap width. -69It is anticipated that a similar analysis could be applied to bounded media~ space-time periodic media and active media. The latter case will be covered in detail in the next two chapters. Parts of this chapter have been summarized elsewhere 53 the results have been confirmed through calculations that use the I method of multiple scales , . 95 and -70CHAPTER IV COMPLEX PERIODIC MEDIA Although the study of wave propagation in periodic structures has been a popular subject since late last century, most of the analytical studies have dealt with simple {passive and lossless) periodic media. . 23 30 58 The recent development of DFB lasers ' • has extended the applicability of periodic structures to the case of com.plex {e: = e: r + ie:.,1 T] = T] r + iT].)1 periodic media. of DFB lasers Kogelnik and- Shank 23 In their analysis briefly discuss the Brillouin diagram near the first-order Bragg interaction when the gain coefficient is modulated. 25 58 Using Floquet theory, Wang ' presents a rnore detailed analysis of the first-order Brillouin diagram for active periodic media. Higher-order 71 and multiply-resonant ?O DFB lasers have been mentioned very recently in the literature. In this chapter we investigate in detail both the ECW and the Floquet theory for complex media at the first few Bragg orders. Analogous results will hold for higher even and odd orders. In section A we use the exact Floquet solution t:o plot the Brillouin diagram. for several values of the perturbation and average d:i.electric constant. In section B we use the approximate but sim.p le ECW solution to study in detail the changes in the Brillouin diagram for the first three Bragg orders in singly periodic media. A short discussion of multiharmonic periodicities is given in section C, and section D summarizes the results of the chapter. As in the previous chapter, some of the parameters in the numerical examples will be larger than those that might be _ -71experimentally obtained. However, this exaggeration is again used to dramatize the periodicity effects. The ECW examples are easily scaled for other values of gain/loss and perturbation. A. • Floguet Solution We consider the case of a TEM wave propagating along the z axis in an unbounded medium with a periodically modulated dielectric constant: E:{z) = E: r + ie. + E: ~('ll +iri.) f cospKz 1 rp r 1p p ( 4. Al) where E: r 'llr and f 0 = Re[e} E:. 1 = Re[ri} '1'11· = Im[ ri} = = o, f 1 =1, p=l,2,3, . . . . and I'll Im[e} fl:;;;1,lri-fl:;;;1 rp p lp p for Index coupling will refer to the case where ri. = 0 1 and gain/loss coupling will refer to the case where ri r = O. Note that gain (loss) media implies E:. < 0 {E:. > 0) for e -iwt excitation. 1 1 The Hill's determinant equation (4. A2) of chapter Il still holds with minor changes in the matrix elements 6 : pn 1 p=n 2 - k e 2K2 p - k2 E: ,,,1 f ' In-pl 2 E: E: (4. A3) r p,:Jn These elements make up the Hill's determinant 6(0) ·= <let II t.ll. -72Several examples of the Floquet solution for singly periodic media are given in the Brillouin diagram Figure 4. I for real frequencies. The example for simple periodic media (real E: and r,) was found in chapter III and is given in curve 1 of Figure 4. la. are two main characteristics. First, note the presence of bandgaps in frequency where (3 becomes complex. inverting bandgaps. There These are known as non.- Second, note the phase speeding that is a result of upward bandgap shift. The Brillouin diagram changes drastically for gain or loss coupling where the perturbation is imaginary. In curve 2 of Figure 4. la we plotted the case for real frequency where E: = E: r and ri = irJ l.. This corresponds to media with successive amplifying and lossy layers while the average gain (or loss) is zero. Note the inverting bandgaps (i . e. bandgaps in longitudinal wavenumber) at odd Bragg orders . - The pattern of non-inverting and inverting bandgaps alternate with Bragg order. However, toward lower frequency. bandgaps of both types are shifted This latter effect can be understood from the average velocity which was given by (4. A4) (v) = for Ir, I < 1. It is !3-pparent that phase speeding (positive bandgap shift) occurs for ri = ri r and phase slowing (negative bandgap shift) = irJ..l This also follows naturally from the definition occurs for ri of BGSN that was given in the previous chapter (3 . A3 If. Let us now consider the case of complex media where only the real part of the dielectric constant is sinusoidally perturbed -73- --' a) )2 1.5 j k¾) Rel~r 1.0 )2 I ,,, / / / 1 0.5 / ~2 /,,, / I = 1.0 €, / € =0 I r 11 I / ------ _,,, 71= i 1.0 2 { € ./ E I r = 0 0~----;:-'-;:-------:-'~----,-1-----c:,-1,-,---!,--=-1,-,---l..---1--_J 0 0. 5 1.0 1.5 2.0 0 ±0.05 ±0. 10 ±0. 15 Im{¾} Re{~} b) 1. 5 0.5 0"'--------::-'-:-------:-''-::-------'--------L----"--L----1.-_.1...-~ 0. 5 1.0 ~{f} 1.5 2.0 . 0 ±0.05±0.10±0._15 ~{¾} Fig. 4~ 1 Brillouin diagram for first three Bragg orders using Floquet theory and real frequency using: a) index coupling (c_u rve 1) and V,ain/ loss coupling (curve 2) when e.1 fr: r = O; b) similar results for finite. ei /er • The light dashed lines are the unperturbed values ('1"1=0) . -74- = e: r + ie:.,1 17 = T\ r ). but average gain or loss is present (i.e. e: The corresponding Brillouin diagram for real frequency is shown in curve I of Figure 4. lb. For e:. -:J. 0, we observe that Re[f3/K} is no l longer a constant across the Bragg region and the effective spatial . 25 76 gain or loss increases appreciably near the Bragg frequencies. ' The unperturbed value ('1"]=0) is shown by the light dashed lines. When the gain or loss is sinusoidally perturbed, the spatial gain or loss diminishes near the odd-order Bragg resonances and is enhanced near the even-order resonances. This case is illustrated in curve 2 of Fig. 4. lb where there is an average imaginary dielectric constant and imaginary perturbation. Again the light dashed lines represent the unperturbed values: We see that the spatial gain can be either enhanced or diminished near Bragg resonances. This behavior depends upon Bragg order and perturbation or coupling type. l The sign of Im[f3/K} and Im[k e:;/I-<} (not shown) have not been specified because the correct root of the dispersion relation is dependent upon the stability of the wave. For passive m .edia, e:. > 0 l and the correct signs are chosen to indicate spatially and temporally decaying waves since no sources are present. does not support instabilities. Hence, the media However, instabilities may arise in active periodic media where €. < 0 for some (or all) z. l In this case an analysis of wavepackets in the periodic medium must be made. 73 This case is taken up in the following chapter. For the present we will only examine monochromatic waves and will not specify the correct sign of the dispersion relation roots. -75B. ECW Theory for Complex Media 1. Analytic Expressions . Since we again assum.e a solution to the wave equation of the form N*/z E(z) =~ n=-1 [F(z) 1- 2 n + B(z)l -Zn/N /N exp[i(l-2n/N)!3 z] o exp[-i(l -2n/N)!3 0 z] + (~) S(z)] where the symbols have the same meaning as in chapter III. (4.Bl) This implies that the gain/loss and the perturbation of the 1nedia are small. Hence for a singly periodic effective dielectric constant E: (z) = E: r +iE:.1 + E: r (f] r + ir, 1. ) cosKz (4.B2) we assume lri r I. lri. I, le.le r I<< 1 1 1 However, as in the case of simple n,edia, we might expect the ECW dispersion relations to be a good approximation to the Floquet results even as Iri r I. lri.1 I . Ie.1 1e r l -- 1 • Following the identical analysis that led to the ECW expressions (3. A20-3. A25) we find immediately that for the N th Bragg order 2 1'1 l}-132o lz(N -d· kze (l+it;./e ){1-cNC(ri Hri.)/2J2 [ r 1 r r 2 130 1 (4. B3) -761 TT where 13 = NK/2. 0 affected by 8.. l * [ 4n(n-N) /N 2 } 2 Note that the coupling coefficient XN is not The dispersion relation as before is = 2. (4. B4) (4. B5) First-Order Interactions At the fir st order Bragg interaction we have the approximate expressions I 61 K ~ Xl K ~ 6k 8 2 1 r i 8. + K l (4. B6) 48 r 17r + i 17.1 8 (4.B7) which give I (~f = K ; I L\ k 8 2 i r K 8. )2 .6k 1 8;) _l ( 4; +i ( K 28 r r 17 2 - 17.2 i 17 r 17.1 64 32 r 1 (4. B8) where tik = 1 I k - 13 /8 2 o r Note that at the first Bragg order, 8. only affects the phase mismatch 1 x 6 , and 17i only affects the coupling . For index coupling, the 1 1 maxim.um value of Im[ti13 /K} occurs exactly at Bragg resonance 1 e; l L'ik •• 1 -77/K = O. The value is { I 6/\ Im 1 Kf = I max c~ e/ + ri/ J½ I r 64 Hence, near Bragg resonance, the effect of spatial gain or loss and the perturbation add as the sum of their squares £or index coupling. This enhancement of spatial gain or loss has been observed in the Floquet solution of Figure 4. 1 b and is shown in Figure 4. 2a £or the ECW solution for several values of e.l le r when 'l'1'I ::2 O. 1. Far fro1n the Bragg resonance Im.[6[3 /K} = ± o /K, the unperturbed value. 1 1 1 The temporal gain or loss, Im[L'ik e; /K}. remains constant across 1 the bandgap with value e. I 4e . i r For gain or loss coupling, 6f3 /K is real if E\ = O. 1 x12 changes sign and therefore This produces an inverted bandgap as shown in the previous Floquet results. The ECW results are shown in Figure 4. 2b £or several values of e. I e l r when T] = i O. 1. Two classes of behavior appear that depend upon the sign of A= [le.le I -lri./21]. l r 1 2 2 .!. For A< 0, a bandgap of width 2[fl. /64 - e. /(4e )]2 appears in l l r Re[ti13 /K}, and Im[lif3 /K} = 0 at Bragg resonance. 1 1 e; The two roots 1 of Iin[L'ik 1 /K} are of opposite sign. For A> 0, there is no band- gap and the gain or loss coupling diminishes the average spatial gain .or loss near Bragg resonance. This is due to the £act that the coefficients of 11~ and e~ have opposite sign in (4. B8). 1 l. In this case, l the two roots of Im[tik 1 e; /K} are of the same sign. Thus, the effect of finite E:. can either enhance ('ll = 'l1 ) or l r reduce (11 = i 'lli) the effective spatial gain or loss near the first Bragg -78- Fig. 4. 2 Brillouin diagrarn at fir st Bragg order using: a) index coupling; b) gain/loss coupling; and c), d) both couplings in the ECW theory. ±0 .01 :t0. 01 1 2 ~~ , 0.005 ~~ ±0.005 1 2 11 E '"- 0.03 il· // '" '' "' -0.01 f/ -0 .02 1- -0.03 r / t:r:j I-'· / ~/ / '' /. ' '' /J '- '' / ' -0. 02 ~ -C. 01 • ' ', '' 0.01 '' '' ' o. 01 o. 02 ' '' -0.03. / ' '' ' ,80/K = 0.5 ' , '} ""- i 0.1 'l r;/ r., = 0 "-, 2 ..:/,r =±0. 025 "\. 3 r./ cr =±0.05 '-, </ <,. = ±0.l '\. -0.0 2 ";·- --i= ± 0.2 ', _J_ -0. 03 -0. 02 - 0. 01 6 1mj () I K ~• o. 02 -0.01 Re j6/3) . / bl ',, "' .B/K = 0.5 '\. '?= 0. 1 'l </r.r= 0 '-, 2 r. ./ r. = ±0.01 'r r '\,., -' </<r=±0,03 '\. 4 <./f. =±0.05 '\. / / 1 ltl"" / / ,/ 1// // - 0. 03 OQ '' ' /( / 0.03 // o. 01 ~ 5 // '' 0.02 4 E al '' 3 0 o. 01 o. 02 o. 03 0 ±0. 01 1/1 6 Re ±0. 02 ±0. 03 I lm!t.tl -.J ~ N I ±0.01 ~I~ ,cC . 005 E ;;;-1 :- - ±.0.005 ~~ E 2 o. 03 ' , lfl 0.02 cl '' - "" <l 0.03 '' ' '' ~0.01 "' Lfl '' <l i"" '' ~ '' "' ' ' ::. -0. 03 -0. 02 di ' ''' '' ' ', 'Ji= 0 1)2 )3 </<, = 0 ' , /.,_= 0.1 '\. / " "\. '\. 0. 01 0 . 02 16/3) '' 130/K = 0.5 2 'Ji= 0.0 1 3 " · = 0. 05 4 'Ji= ' 0.1 5 .,i = o. 2 - 0. 01 Re 0. 01 ', ', = ,80/K 0.5 c/ r.r =- 0 'l = 0. 1 '-.. -0.01 0.02 '- '- " I l .,r = 0 2 .,r = 0. 01 3 ., r = 0. 02 4 ., = o. 05 5 .,r_ - 0.1 0 6.,r=0.20 ,'\,., '\.'\. ''-, '\. ' '-.. ',. 0.03 0 ±0.005 0. 01 ±0.01 ±0.01 5 im!n!f Re !t./31 K l 0.02 ' '\. '\,., o. o3 o 1/' -+:. o.oos ± 0.01 Im j 6:/ :::.o. 01s -80resonance. For both cases of coupling, increased €. /e: l r tends to mask the effect of the coupling or perturbation. Figures 4. 2c, d show the effect of having complex coupling (17 = 17 r t i 17.) in a media with no average gain or loss. l As the ratio 17. /17 is increased, the bandgap region changes from the shape l r typical of index coupling to the shape typical of gain or loss coupling 1 for Re[t.f3 /.K} and Re[t.k e;/K}. 1 1 The values of Im[t.f3 /K} and 1 1 Im [ 6k E:; /K} t .e nd to peak in the Bragg interaction region. 1 3. Second-Order Interaction The second-order Bragg interaction and all even-order interactions produce very different <lisper sion relations from those of the first-order for gain or loss coupling. Floquet results of Figure 4. 1. This was seen in the The change is due to the t·erm (i 17. /N l in the dispersion relation which takes on different signs depending upon the oddness or evenness of N. In the second-order Bragg region, the approximate relations are 1 62 K -::::=. Xz 6k e: 2 2 r (17 r ti 17.l ) K 12 K ~ - 2 (17 r +if).>2] i E:. ~ l t--1 1 6 28 r (4. B9) (17 ti 17./ r l (4.BI0) 8 1 where 6k 2 =k-f3 0 /e z r (3 0 = K. Note that the perturbation affects not only the coupling x , 2 but also the phase mismatch o . 2 In fact, 'I') now acts to. modify the gain or loss from e. I (2 e; ) to the effective value e. (l-ri2 /6)/(2e: ). 1 r 1 r The Brillouin diagram ·£or the case of index coupling is shown in Figure 4. 3a for various values of e:i/ E:r with 17 = O. 1. The result -81is similar to the first-order case with the exception of the bandgap shift. Again, at the center of the bandgap, the effective gain or loss and the perturbation add as the sum of their squares. The spatial gain or loss is greatly enhanced near Bragg resonance when I 8. le l I << I 111 r 2 whereas the temporal gain or loss remains constant. In the case of gain or loss coupling, a difference occurs since x2 is proportional to '11 4 in (4. B 10) and the band gap shift changes sign which causes the Brillouin diagram to be a mirror image of the index coupling case about the axis o /K = O. 2 That is, at the second Bragg order, the difference between index and gain or loss coupling is a matter only of positive or negative bandgap shift. This is demon- strated in Figure 4. 3b for '11 = i O. 1 and several values of 8. / 8 • l r 2 This dependence of the phase mismatch upon ri accounts for the phase speeding and slowing effects shown by previous Floquet results. The temporal gain or loss is constant as in the index coupling case. 1 Figures 4.3c,dare similar n-tlrror images about Re[6kz8J/K} for 8. /8 1 = 0 and various ratios fl.1 /fl r . r =0 The symmetry is expected since fl always enters as an even power in the dispersion relation. 1 From (4.B9-10) it is evident that Im[613 /K} 2 only when = 0 at Lk 2 8;"/K = 0 I fl.l I = Ifl r I. This corresponds to curve 5 and is similar in shape to the gain or loss coupling Brillouin diagram at the first 1 Bragg order. fl r ~ fl.. l For Im[6k e 2 /K} =.± · ,n -n. /12 Im[Lf3 /K} = 0 for 'Ir ·11 • 2 2 r Thus for co1nplex coupling, the perturbation will mask the spatial gain or loss for some frequency near Bragg resonance, while a large spatial gain or loss will occur for nearby frequencies. In -82- Fig. 4. 3 Brillouin diagrarn at second Bragg order using: a) index coupling; b) gain/loss coupling; and c}, d) both couplings in the ECW theory. ±0.003 ~ 0 . 003 ~I~ __ :.0.002 :::0 . 002 ~e'.'. .§ -:.0.001 0.003 '' <I ;"' 0.002 3 • ' J~ al lfi I 2 _t.0.001 bl / / '' / / '' :fj" / / '' 0.001 / / ' / '' '' 0.002 f / / '' / ' 0.001 / '' / / ' / ' / '' ,:, '' / / ' / / / / / -0. 00lf- / / 8 0 /K = LO / / -0.001 ' / ' r= 0 ' / <r = ±0. 001 'i ' 'r = ±0.003 4 </ <r = ±0.005 5 , /, ,=±0.025, -0. 002 -0. 003 h:j ... ,._,. ... I-' • ,. --- -0.001 0 l 0.001 0. 002 '0.003 0 -0.002f- I "l = i 0.1 1 t /1:r = 0 ' / ' r :- ±0. 001 </ <r =' ±0.003 4 f /<r:: ±0. 005 ' , 5 , ;1 ,, = ±0,025 / / / / - 0. 003 r / / -0.004 ±0.001 ±0, 002 -0.001 G. 001 O'Q • Re 1m/ Af f Ref/f ~ .:.0.002 ~I::: ..:0.001 0. 002 O. 003 ll,tl lmj~j ::o.003 . - - - . - - , - --,-- --,-- - , - - - , - - -,--- - , - -,--- _C.003 •(,-) / ~ - , . -- , w ~E E ;t0 . 002 t0 .001 ~ di cl G.003 ~~ \so o.ooc '' o. 002 '' <l . w "' '' ' ', L_t j" o. 001 <l I ' w "' 0.001 '' /4 '' '' '' '' ' '' - 0.001 60/K=l.0 '7 1 .,i -0.002 = r, , ,, 0.01 '\. =0. 01 =0. 02 'I r " -0.003 '1 ; = 0.05" 0.1 , 'lj - 0.003 - _,. ... I ,. " ~" -0. 001 0 /3l 6 Rej 0 . 0 01 = O. 002 ' ' /30/K =1.0 </<, = 0 7i o. ! l '1 , =0 </ <r = 0 '1 , = 0.1 1 ") i ,:; 0 - 0. 001 '0. 003 0 l (X) O. 02 4 1 , = 0.05 5 '7 r = 0.1 I , ', 21 -0.0 04 ±0. 0005 .!:O. 0010 1m j6: f / , , I ,' I I 31 If ",..,,., " ... ,,. ... -0.001 0 Ref'! ! 0. 001 0.002 o.003 0 ± 0.0005 .:.. n "'"" Im 16:1 -84all cases of both index and gain or loss coupling, the temporal gain or loss is peaked in the vicinity of Bragg resonance. 4. Third-Order Interaction From the structure of the dispersion relation it is apparent that odd Bragg order interactions will exhibit similar Brillouin diagrams for index and gain or loss coupling. To exemplify this, consider the third Bragg order approximate parameters, 1 L.\k fl -i 3 r K - 256 ~ 243 2048 (ri +iri.) r l L\k3 = k - f3o /e r2 f3 0 = 3 K/2 03 K ~ X3 K 27 ( 1 E\ )( . )2 +€ f]r +i f]i r i 38. l - :rer 3 (4. Bl l) (4.B12) 1 where Figure 4. 4a, b de1nonstrates the dispersion characteristics for index and gain or loss coupling respectively. Except for the bandgap shift, the results are nearly identical to the first-order case. That is, the perturbation and gain or loss add at the center of the bandgap to enhance the spatial gain or loss for index coupling. In the case of gain or loss coupling the Brillouin diagram exhibits two distinct behaviors which depend upon strength of the gain or loss relative to the perturbation. For small average loss or gain, the effective spatial loss or gain is zero at the bandgap center. The case of both index and gain or loss coupling is shown in Figure 4.4c, d. In all third-order cases, the temporal gain or loss is peaked in the Bragg region just as for the first-order case. -85 - Fig. 4. 4 Brillouin diagra1n at third Bragg order using: a) index coupling, b) gain/loss coupling and c /, d) both couplings in the ECW theory. r , 0.0001 "'I" Jl E I~ 1 : " / 0 .000!3 t l 0 .00012 - :;; "> >< 0.00011 <Ji ~ l ,t 0.000 10 ro/!1£: _ , _ .:.0.00005 - 0.0008 \\ \ ] - O.(.:l09 ~1 r-:--- <i - fJ0/ K =1. 5 "1 =0.1 ~ -0 .00J.l "' =0 2 <jl< r =;:0.00005 1 </<r /r C.00009 f-- 0~ - 0.CivlO J 0.00010 I _ .).0000 5 -0.0012 3 </<r = ..::.0. 0001 1-rj 4 </ < r = ±0 . 0002 f O.OOOO ~ f-- -0. 000..i - 0.000£ - 0.000 1 0 -o.uci; ~ 0 . 0001 0.000:.! 0 . 0003 0 1 R, j t.k F I • 3 <1 1 <, = =0.0001 4 <, / <, ., .:0.0002 -0 . 0003 - 0 . 0002 -0.0001 ±0.0002 '"1~,1 .... . OQ .:::0.000 .1 / Re 0 0.0001 0.0002 0.00 03 l/lkf'-1 .::.0.0000 5 .:.0 .00 ~1 lt.,S l I Kl I 00 0-- >+'>- • >+'>- :0 , 0()003 t "'I" :0 .0000 , / I "-. 2 ,0.00 001 ..______ "°1~ ~ 1 l I 1~ ,5! +0 . 0 0005 C 0~ 0 0 f••'l ! '; -0.0 008 o.ovv : -0.0009 0 . 00011 3 <l - =0 . 00010 o.ooo c ii! / I -0.0003 -0.0002 - 0 . 000 1 Re 0 "11 =0 .0 2 'I i= 0.005 3 "lj = 0 .0 1 I 0.0001 0.0002 0.0003 0 l~\'t!I ( Il ± 0,0001 ± 0.0002 .±0 . 0003 Im !Af! -0. 0 01 0 - >< ~ 0. 000 10 0.0009 [ ~1· -0.COll -0.00U - 0 . 0013 - 0.0003 -0.0002 - 0 . 0001 Re 0 0.0001 0.000 2 0 . 0003 0 _!:0.00005 !0 .0001 .:0 .00015 )A•F I 1rn /t>/3/ -87C. Multiharmonic Periodicities Recent experimental work in higher-order DFB lasers and in multiply-periodic DFB lasers 70 41 71 • make the results of multi- harmonic ECW theorytimely. As in the passive-lossless case, ECW theory cannot give explicit dispersion relations for arbitrary Bragg order N, since the number of significant Fourier components (f ) p have to be specified. However, the extension is straightforward, although laborious, for any spe cific case. Consider the case where f 1 and fN are the significant Fourier components of the periodicity and (4.Cl) where Following the derivation of the previous chapter we find the slightly modified ECW equations: (4.C2) where (4. C 3) I TT * ( 4n(n-N)/N 2}2 ( 4. C4) -88The resulting dispersion relation is (6N)2 . _ \ K (4. C5) By varying the phase between the Fourier components, XN can be varied while 6N remains constant. + This allows control of the coupling - .!. and bandgap width [= 2 (XN XN ) 2 J without change in the bandgap l shift. For N = 2, the explicit dispersion relation is (4. C6) Figure 4. 5 shows two examples of multiharrnonic Brillouin diagram.s at the second Bragg order. In Figure 4. 5a, the index coupling case 2 is shown when (71 f ) /2 = 17 2 f = O. 1 and E\ = O. 1 1 2 and Im[llf3 /K} both vary as sin(8/2). 2 max The bandga.p width Figure 4. 5b represents the case of index coupling ·with spatial wavenumber K and gain/loss coupling with wavenumber 2K. The result shows not only the Re [ 6[3 /K} typical of gain/loss coupling, but also the effective 2 gain/loss typical of index coupling. In both cases shown, 1 Im[li~e; /K} = 0 for Re[6f3/KJ. similar if (e/er/ << x2+ x2- /K 2 • D. The results for lossy media are Com.ments on Complex Periodic Media The extension of chapter III results t c complex media has been mathematically straightforward for both the Floquet and the ECW results. However, interesting new features appeared in this 0.08 0.04 1 / / / / / / / / / / / / / / / / /30 /K "' 1. 0 / 0 / / / f l:lk~1 Re l--K- / / 0. 12 E./E I r =0 -0.04 1 8"' 0 -0.08 2 8 = -rr /4 3 0 = -rr/2 4 8 = 3-rr/4 -o. 12 5 8 = -rr -o. 12 -0.08 0 -0.04 0.04 o. 12 0 0.08 0.02 Re{¥} 0.06 '' '' 0.04 / '' / '' 0.02 Re In'-;,; l '' ' / / / / / / "' / -0. 02 1 •• 3 / f3 /K = 1. 0 0 '' E./E I '' r =0 771 fl = 'JD.2 ' ' ' , 1)2 f2 = i 0. 1 2 8 = -rr/4, 31r/4 ' ' / / 8 = 0,-rr / / / / / '" '' 3 8 = -rr/2 / / / -0.04 / / 'X ··O. 06 / / 0 -0.06 / / / / / -0.04 0.06 Im{¥} b) '' 0.04 -0.02 0 0.02 0.04 0.06 ' 0 ±.0.01 Re{¥} ±0.02 ±0.03 Im{~} Fig. 4 . 5 Brillouin diagram at second Bragg order using multiharmonic periodicities in the ECW theory: a) index coupling with relative phase 9 between two periodicities; b) index and gain/loss coupling. 1 Im (6ke 2 /K} :: O for Re ll'.\f3/K}. r Note -90chapter: 1. Even and odd Bragg orders produce non-inverting and inverting bandgaps, respectively., for gain/loss coupling. Index coupling always produces the usual non-inverting bandgaps. 2. Phase speeding and positive bandgap shifts are produced by index coupling while gain/loss coupling produces phase slowing and negative bandgap shifts. 3. The average effective spatial gain or loss (i.e. Im [llf3N/K}) is significantly enhanced or diminished near Bragg resonance whenever le./e l<lrilN. 1 r En- hancement or reduction depends upon coupling type and Bragg order. I 4. Temporal gain/loss (i.e. Im[lll~ e//K}) is either constant or is peaked at the bandgap center. 5. Multiharmonic periodicities offer great flexibility in changing the bandgap shape through phase and coupling variationso If the Fourier components f p are dynamically generated, this opens the po:ssibility of quickly controlling the feedback strength in DFB lasers or filters. Possible applications are in the areas of microwave and optical filters, switches and modulators. In this chapter, Floquet and ECW results were not directly compared because little or no graphical differences appear whenever .:.91- for the first few Bragg orders. The accuracy of the ECW results is sin1ilar to that of the passive-lossless case. The only notable differ- ence occurred in the first order gain/loss coupling case where the ECW results predict Im[613 /K} = 0 at exact Bragg resonance. 1 The Floquet result gives a small but finite value here for Im[6!3 . /K}. l Portions of this chapter have been summarized elsewhere. 96 -92CHAPTER V STABILITY OF BRAGG INTERACTIONS IN ACTIVE PERIODIC MEDIA The problem of wave stability was briefly mentioned in the previous chapter with regard to choosing the correct root of the dispersion relation for the frequency and longitudinal wavenumber. It is evident that the preceding monochromatic plane wave analysis is inadequate to predict the correct spatial a n d temporal behavior of waves in periodic media in all but the simplest (passive and loss less) cases. Our purpose is to forrn.ulate the stability problem in such a way that the preceding ECW equations can unam.biguously define wave characteristics. The problem of wave stability was greatly clarified through the original work of Sturrock 72 which provided a method for distin- guishing between growing waves (unstable media) and decaying waves (stable media) through the dispersion relation. Sturrock further divided unstable media into convectively unstable and absolutely unstable media. The former referred to waves which grew and then decayed at a given point in space like waves in traveling-wave amplifiers. The latter referred to waves which grow everywhere in space as those waves found in backward wave oscillators. extended by Briggs 73 This work was and was formulated as a mathematical pre - scription which unambiguously classified waves according to their stability. A large number of electron-stream interactions demon- strated the usefulness of Briggs I method in plasma physics. The -93stability criterion was again extended to include time -varying media by Cassedy 56 who examined the stability of parametric interactions due to space -time periodicities. In this chapter, the stability of time -independent active periodic media is studied at the first- and higher-order Bragg resonances. The stability classification depends only upon the dispersion relation or the Brillouin diagram and hence the ECW theory of the previous two chapters will be used. suggested by Briggs. We will follow the procedure This requires the formulation of the field response as a function of a source, localized in space and time. In section A the ECW equations are developed with sources present and the stability criterion is stated~ The application of the criterion to the fir st two Bragg resonances is carried out in detail in section B. These resonances are the archetypes for all odd- and even-order Bragg interactions. Explicit values for insta- bility threshold, frequency and growth rate are given. Section C contains a brief explanation of the effects of complex coupling (i.e. 17 = 'l1 + i '!7.) and multiharmonic periodicities upon instability param- r eters. A. l A short discussion of the results is given in section D. ECW Equations with Source~ We carry out the usual manipulations with the Maxwell equa - tions that include a current density source J(z). homogeneous wave equation periodic media is 67 The resulting in- for a TEM wave in longitudinally -94(5. Al) where f E(z, W) 1 = f {E(z, t)} eiwt dt . J(z,t) J (z. w) J l -oo e(z) = e- +ie. + E: 6(T] +iT].) f cospKz r 1 r p r 1p p Assume that E(z) is made up of the N+3 waves F(z)l+Zn/N' F 1(z), ••• , B 1 (z), B(z)l+Zn/N that are used in the ECW formulation. We then find the following N+3 ECW equations for singly periodic media at the N [k 2 th E: - Bragg order. 2 f3 o ]B 1 = 2 -k E:TJ B . JiO +2/N)f3 z 0 - 2 - 1-iwµo 1+2/N e (5. A2) where the arguments with respect to z and t have been dropped and the primes denote differentiation with respect to z. densities J ± The current (z)l.- n/N are the portions of J(z) which are in phase 2 synchronism with the rest of the equation. We accomplish this by truncation in wavenumber space such that J ± ~] _Jf3-(l-2n/N)f3 )=J(f3T(l-2n/N)f3 )rect[[l3:i:(l-2n/N)f3 ]/6(3 + o T o o 1- 2 n 1N' (n = -1, 0, 1, 2, ... ) (5. A3) -95where J (13) 00 J J(z)e = ·R -li---Z dz -oo rect(l3) = ,., L'i 13 113 I < ½ G << I 13 I > .!.2 130 As we shall see, the exact truncation details are unimportant since the current will only contribute to the field at wavenumber 13 = 13 0 • Since we are particularly i_nter ested in the waves F (z) and B (z), 1 1 assume that the current is slowly varying in wavenumber over the N+3 significant waves. Thus, we consider only the current con.tribu- + hons J 1 (z) and J -(z) and ignore current contributions O('Y']). 1 Using the above approximation and the other usual ECW approximations we find the coupled equations, These equations can be solved by taking Fourier transforms with respect to z and solving for F (L'il3, w) and B (6f.>, w). 1 1 The inverse transforms produce the time and space variation, w[ ( 613 +6 )Jr ( 6!3 +13 , W) -xJ - ( 613 -13 , W)} j ( lll3z -wt) cffil3 d W 0 1 D ( L'il3 ~ w) 0 ( 2 11/ (5. A5) -96- w[ (.6(3 Hi )J 1- ( .6(3 -(3d w) -x J: ( .6(3 +f3o• w)} i (.6(3 z -wt) _d.6(3dw D(.6(3, W) (2 ,r) 2 (5. A6) where (5. A 7) D(.6(3, w) = is the dispersion relation and the radian frequency w is related to the 1 wavenumber by w = c(ko +t.k) le r·z and (3-(3 o = t.(3 as before. All sub- scripts N have been dropped for simplicity. •Assume that the current is turned on at t = 0, hence J(z, t) = 0 for t < O. It is apparent that the integrand of (5. A5-6 ), excluding the currents, is the Green's function for a periodic medium in the ECW approximation. It is of similar form to an exact expression which accounts for all space harmonics and current components. 75 As pointed out by both Sturrock and Briggs, it is necessary to investigate wave packets in space and time due to localized sources. Therefore, the previous ECW equations (5. A4) with a source turned on at t = 0 are a useful start. The remaining problem is to find how F (z, t) and B (z, t) behave asym.ptotically by choosing the contours 1 1 of integration correctly and unambiguously. Note that the poles of the integrand, or the roots of D(t.(3, w), determine the field response. For causality, it is necessary to carry out the frequency integration above all singularities. in Figure 5. la. The integration contour is shown Note that no roots appear in the region enclosed by the c ontour for t < O. This contour is the rotated Bromwich contour that is used for Laplace transforrn.s. The contour in the wavenumber plane is ambiguous since the roots of D{t.(3, w) may cross the real t.(3 -97axis for different values of w(= w +i w. )~ This difficulty has been r l 73 remedied by Briggs by use of the following physical criterion: for a source with sufficiently large temporal growth rate (i.e. large w. l = s > 0),- the waves away from the source rnust spatially decay. Therefore, if the contour is chosen in the w plane to satisfy causality for w. =s, the roots of D(6f3, w) will detern1ine the response for F(z, t) l and B(z, t) and must represent spatially decaying waves~ shows this condition. Figure 5. lb Since it is convenient to integrate along the axis w. = 0, we depress the contour around singularities down to the l real waxis (Figure 5.lc). As w. _,. 0 for fixed w., l the roots of l D(6f3, w} will: 1) stay in the same half plane of 6f3; 2) cross the real 6f3 axis; or 3) merge from opposite sides of the real 6f3 axis in pairs for some w. = a > 0. l spectively to: As shown by Briggs, the above cases lead re - 1) decaying waves (stable media); 2) amplifying waves (convectively unstable media); or 3) tiine growing waves (absolutely unstable media). In the first case, the contour in the 6f3 plane along real 6f3 axis is not deformed (i.e. decaying roots) while in the second case the contour is deformed (i.e. arn.plifying roots) by root crossings (Figure 5. ld). In the third case, as w. _, a for some W = w (Figure l r 0 5. le), the merging root behavior in the 6f3 plane (Figure 5. lf) prevents the distortion of the contour since the integration must be carried out between the merging roots. The details of the integration are carried out in Appendix A for absolute instabilities with the proper contours shown in Figure A. 1. that the field varies as It can be shown (equation (A. 7)) -98- a) IW·I b) lWj s X X ----l----1----j.-----1-Wr X X -+------+-------1--Wr ~ ~ - - Roots of D(~/3,w)=O c) IW·I d) z < 0 Decaying Roots e) IWj f) Merging Roots x Fig. 5. 1 Integration contours in a) w plane for causality, b) wplane for t > 0 and time growing source, c) w plane with deformation to axis, d) Lif3 plane which corresponds to c) with changing roots, r e) w plane for absolute instability at w. = <1", f) li.!3 plane correw 1 sponding to e ). -99I F(z, t) o::. i [ 6l3 Z - W0 t] at e e t2 (5. A8) Note that the frequency and wavenumber of the instability are just I I that of the merging roots (i.e. 613 = 613 , W = W . t·1me as e CH . th e waves grow 1n = w0 + i a). Therefore, The absolute instability paramet ," rs are deterinined from the mapping of the dispersion equation roots in the 613 plane for variable w. and different constant values of W • 1 r B. Application of Stability Criteria to Bragg Resonances The Brillouin diagrams .take on different characteristics for even and odd Bragg orders in the case of index (T) = T) ) and gain r (T) = i T).)1 coupling. Thus, we will examine in detail the fir st and second Bragg resonances for singly periodic media since they are the archetypes of all even and odd Bragg order n interactions. The first-order Brillouin diagrarns are repeated in Figure 5. 2a, b for the case of index and gain coupling respectively with various values of average gain or loss. Figures 5. 3 and 5. 4 show a mapping of the roots of D(6l3, w) in the.6!3 plane as w. varies from l large positive values to zero for several values of w . r Normalized l frequency 6k E: 2 /K is used instead of w. r Figure 5 . 3a shows the case of no average gain or loss (E:. = 0) and index coupling which corre1 sponds to curve 1 of Figure 5. 2a. The merging root behavior is l l noticed for Re [ 6k E: 2 /K} = T) / 8 when Ir.n [ 6k E: 2 /K} = O. r r r Thus, the instability has no temporal growth and the medium is .actually stable. l This is also noted from the fact that Im [ 6 k E: 2 /K} = 0 for all Re fo13/K} r and hence no instability is possible. Therefore, the physical notion :.:.0.01 :.0. 01 l 2 ;;;1::: - - .:0 . 005 3 4 ;;;I;:; , l 5 - - : 0.005 ll 0.03 ', al / '' - ~ - 0.01 ' '' "' lf!" ''' -0.01 <J ! '' ii! ' '' "3/21' ' ', // 3 </' <r' =±0.03 4 <.!< = =V.05 / / ,/,. ' ' ',,, ' ' - 0. 02 0.01 -0. 01 Reill!\ Fig~ 5. 2 o. 02 '' 0.01 0,02 ' , ' -0.03 ' - 0. 03 1m jll/l '' ', I I-' ' '' .B/K = 0,5 '1 = i 0.1 -0.02 "\,,. / ' ,, bl -0. 01 .B/K = o.s, ~ '"' 0. 1 "\,,. ( ./ ( = 0 ....... 2 ( ./ ' ( ' = ::t0,01 - 0.02 -0. 03 0.03 / ~ } 0.02 -0. 03 / / ''' - 0. 02 0 0 ' "\,,. .............. </ ', l 2 3 4 5 =O </ ,;,,=±0.025 </ <r = ±0. 05 </ <r=±0.l </<r=±0.2 - 0. 01 0. 01 '' '' 0. 02 '' '' 0. 03 0 ± 0. 01 =0. 02 Re !'3/31 1mjll/\ Brillouin diagr am at the first Bragg order for the ECW theory showing a ) index and b} gain/loss coupling. ±0. 03 -101of the stability of passive periodic media is confirmed. Since there are no root crossings, the proper choice of the sign of Im [ 6!3 /K} is that which indicates spatially decaying waves. Figure 5. 3b shows the case of index coupling with average loss(€? O)which correspbnds to curve 3 of Figure 5. 2a. There are no root crossings or mergings 1 The proper signs for Im [ 6k E: 2 /K} r and hence the media is stable. and Im[6l3/K} indicate temporally and spatially decaying outgoing waves. The case of index coupling with average gain {E:. < 0) which l corresponds to curve 3 of Figure 5. 2a is shown in Figure 5. 3c. In 1 all cases, similar curves exist for Re [6k E: 2 /K} < 0 which are mirror r images about the imaginary 6!3/K axis. Note that the root crossing 1 occurs for 6k E: 2 /K = i E:. /4 E: ± 17 /8. r 1 r r Thus, absolute instabilities occur for index coupling with (the threshold condition) positive average 1 gain at the bandgap edges corresponding to Re [ 6k E: 2 /K} = ± 17 /8. r r 1 The normalized temporal growth rate is Im [ 6k E: 2 /K] = - E:. / 4c and r 1 r the longitudinal wavenumber r:3 = l3 0 is real,. (Bandgap edges will refer to the lossless case for this chapter.) Figure 5. 4 shows similar mappings of the dispersion equation roots for gain coupling. In Figure 5. 4a, b, c the merging root be1 havior occurs at the value L'ik e 2 /K = -i e. / 4 E: r 1 r + i I ' 1 I/ 8. '!"'I. 1 Thus, for gain modulation, absolute instability occurs at the center of the bandgap for average gain {E:. < 0) and average loss (e. > 0) whenever the l threshold €. / € l r l < l 11-l l /2 is satisfied. The result for zero average gain is not surprising since the inverted bandgaps are similar to those of parametric instabilities and backward wave oscillators. However, note that gain-coupled media may also have instabilities -102- a) b) 0.01 0.01 I I I ------1-----1 I -0.01 -0.01 I I 3 -0.01 0 0. 01 -0.01 0.01 0 Rejt./3/Kl RejC./3/K} c) d) 0.02 0.01 1/ = 0. l <:. If 0.01 ~ C!t s 0 I- .§ 1 -0.01 0 \I\ = -0.03 0 Re jt.,8/KI 0.01 3 ,~4 - ------- I I -0.01 2 - - -i-- I I I 0 0.01 0.02 Re jt.k '-~ /I<\ Fig. 5. 3 Locus of the roots of the dispersion relation in the ti~ /K plane at the first Bragg order for index coupling with average gain :::: o.o, b) e./e == 0.03 and c) e./e == -0.03. Locus of r 1 r 1 r 6.k E: 2 /K for the above is given in d). Similar loci in the D./3/K plane r l are produced for Re l 6ke 2 /K} < O. a) e./e 11 -103- a) 4 b) 4 2 0.01 0.01 0 -0.01 -0.01 3 4 2 -0. 01 -0.01 0.01 0 0.01 0 Re jt.,/3/K} Re !1\/3/ Kf c) d) 4 0.02 I- 'T) ~ i 0. 1 I- 1 0 I- - - I -- I 3 2 ~ - •- 4 ·- - -- I I -0.01 0.01 0 I 0 I I 0.01 0.02 Re !M.JE; / Kl Re !ll/3/ Kf Fig. 5. 4 Locus of the r oots of the dispersion relation in the ti~ /K plane at the first Bragg orde r for gain coupling with ave r age gain a ) e,. /e 1 ir = o. 0, b) ~-1 / e.r = o. 05 and c) e.1 /er= -0. 05 . Locus of ilk E: z /K for the above is giv en in d}: r - -104when the average gain is negative as long as the gain periodically takes on p·o sitive values (i.e. the average loss is less than some fraction of the variation). As evidenced by the structure of the dispersion relations of chapters Ill and IV and the repeated Brillouin diagram of Figure 5. 5, index and gain coupling at the second Bragg resonance display similar characteristics, to the first-order index coupling diagrams (i.e. no inverted bandgaps appear). Thus, the stability of both index and gain coupling at the second Bragg resonance should be similar to that of the first order index coupling case since the stability criteria is dependent only upon the dispersion relation. The only difference should be in the sign of the bandgap shift,and hence the relative instability frequency; which is positive or negative for index or gain coupling respectively. Indeed this is true, The mapping of the complex roots (not shown) of D(6j3, w) as w.-+ 0 resembles 1 Figure 5.3. The absolute instability occurs at the bandgap edges for index 1 or gain coupling with temporal growth rate given by Im(6k e: 2 /K} at the r bandgap. The threshold condition requires positive gain (e:. <O) for abso1 lute instabilities at the second Bragg resonance for both coupling types. The results of odd and even Bragg resonances show different stability characteristics as exemplified by the specific cases of N= 1, 2 discussed above. The mathematical conditions for absolute instability parameters can be found from the above considerations and the ECW dispersion relation. For any Bragg order N, oscillation takes place at (5. Bl) with temporal growth rate Im[oN/K} = + Im{xN/K} at or above the threshold ( 5. B2) ±0 . 003 :-:0.003 :.0 . 002 ~I~ - -r.0.001 -- =0.002 E ..'; 0 .001 ~I~ 1 2 3 4 5 o~ bl al 0.0031 I ~ , ~ : - 1~ <J I / ' ', / / / 0.002 / / / ~ "' 11: / / 0.001 ~ a: / 0.001 0.002 ', '' '' '' '' / / .....I '' 0 IJ1 / / -0.001 6 / K = LO 0 "1 =. I 0.1 -0. 001 ~- </<r= 0 ::~::::~:~~! 3 4 t/( r = ±0.005 s </( , = ±0.02:s, -0. 003 - 0. 002 -0. 001 0 RejAj') 0. 001 o. 002 o.' 003 0 I I ;· ±0. 001 ±0. 002 im l6:l -0. 002 l (/ <r;,. 0 2 t"idr :..: ±0.001 3 , /t"r =±0. 003 , / <r = ±0.005 '-, ' / ,, =± 0.025 -0.003 ~- -0. 003 -o. 002 -0. 001 0 o. 001 o. 002 o. 003 Re jA!) Figo 5. 5 Brillouin diagram of the second Bragg order for the ECW the ory, showing a) index and b ) gain/loss coupling. I ±0. 001 ±0. 002 1ml6:l -1061 = 0 Im[61)1//K} (5 . B3) The results for the first few Bragg resonances are summarized in Table 5. 1. It is noted that with incre~sing Bragg order, the threshold approaches zero for all couplings since fl is usually small. Also the instability frequencies tend to cluster above (index c oupling) or below (gain coupling) the exact Bragg resonance for higher Bragg orders. This is due to the bandgap shift. The fir st two Bragg resonances have been shown to be arche- typical of all Bragg resonances. In particular we found that for index coupling, absolute instabilities occur for positive average gain at both bandgap e dges with temporal g rowth rates given by w. = 0 at the 1 bandgap. For gain coupling, the behavior is identical to the index coupling c ase for N even, with the exception of the instability £re quency which is shifted due to the bandgap shift. For gain coupling with odd N, absolute instabilities occur only at the center of the bandgap with temporal growth rates equal to the value of w . at the bandgap. 1 The correct choice for the sign of Im[6l3/K} depends upon the position of the roots .of the dispersion relation as w. _, 0 or at the 1 merging root point w. = 0. 1 If the roots have crossed the real 6!3 /K axis the outgoing waves are amplifying and if they do not cross, the outgoing waves are decaying. Thus, for the case of absolute instabil- ities, there will be both amplifying and decaying waves at frequencies adjacent to the instability (see Figure 5 . 3 ~4). However, this behavior is over shadowed by the absolute instability and is not important for infinite media . The proper sign of Im [ 6 /K} is chosen by the fact that Thre shold Coupling Order = 17r N =l = i 'll-1 N=l E:. = _1 E: r N= 2 E:. -E: 1 r F r equency E: . 'll 'll 'll 'll = 'llr = i f).1 'll = 'llr Tl = i 171. Tl = 'llr 'll = i'17.1 Tabl e 5 . I N=2 N=3 E: . 1 -e- = ill 1· E: . 1 E: r - 4e r E:. __1_ + 4E: r 0 2 2 2 'llr 'llr -±-12 8 0 2 2 _.2. ± _ 1_ 12 8 0 r T). 0 2 3 2 43ri r ...L r 256 __._ 2 048 3 81 I ,\ 1 / 512 2 27'rj. 1 256 2 7ri = l~ I 8 E:i ( 'll; -- 1 - 2 E: 6 r ) r/ ) E: 1_ . ( 1 + _1_ __ 2 E: 6 r 3 E:. - 4E: : ( l - I 9'll 2 ) ..... 6: -.J 0 I --...1 '2. E:. 1 E: r N= 4 E:. 1 E: r N=4 E: . -E: 1 r N ±- 1j. = E:. 1 'llr 8 -e 1 = 0 r Growth Rate = [ 9T]/ ] 1 +64 - 2 0 0 2 'llr 15 3e.1 ( 1 + _1_ 9'l"j~) + 2431ri.1 _ __ 4 E: r 64 204 8 4 Tlr ± -9- 2 4 2'rj . '17. _ _ 1_ ±-115 9 E:. 2'rj - E:: ( 1 , 8. - -E: 1 r ( 2 ) 1: 2 ) 217 . I + I~ Threshold, instability fr equency and tempo r al growth rate for index and gain coupling at the fi rst fou r Bragg orders . All frequency - 1 and gr owth rat e v alues are in u n its of 6. k E: 2 / K~ r I3 -108the dominant time behavior is controlled by the highest root in the W plane. This is equal to the value W. = er that occurs when the l roots of 6j3/K merge. In all cases where the gain is not large enough to support absolute instabilities, the medium is stable and the outgoing waves decay spatially due to coupling and losses. Thus, because there are no axis eras sings of the roots of D (.613, W ) in the 6j3 /K plane, the sign of Im[6j3/K} > 0 is specified for outgoing waves. The sign of Im[o/K} is chosen to correspond to the least lossy wave (i.e. Im [ 6 /K} > 0) since this root produces the dominant field contribution. Thus, for stable media the periodicity will enhance the spatial decay and not affect the temporal decay for index and even-order gain coupling. For odd-order gain coupling the spatial and tem.poral decay are both diminished. C. Complex Coupling and Multiharmonic Periodicities In practical cases, both index and gain periodicity will occur together (Tl= fl r +if] . ). This case, referred to as complex coupling, l will usually cause absolute instabilities to occur at two frequencies due to finite f] . r The explicit conditions are involved for arbitrary Bragg order but reduce to the following form at first order: absolute 1 instabilities will occur in pairs at the bandgap edges (Re [tik E: 2 /K} = ±f] /8) 1 r r 1 with a growth rate Im[6k E: 2/K} == -e./4e +1ri-l/8; the threshold is 1 r 1 r 1 e ./ E: = \ri-1 /2 and thus cannot be reduced below the gain coupling value. l r l . At even Bragg resonances, where the threshold was pre viously zero, a reduction in threshold can take place for the case of complex couplings. As an example consider the second Bragg -109resonance. Figure 5. 6 repeats the Brillouin diagram for complex coupling and E:. /e 1 coupling (i.e. fl r = O. For equal amounts of index and gain = fl.)1 the curves give inverted bandgaps. r Thus, one expects absolute instabilities at exact Bragg resonance. It can be shown that the instability threshold is reduced to e./e 1 r for fl = 5 lfl. fl 1 r r I /6 (average loss). Two absolute instabilities occur f:. fl. and e./e 1 1 r = 0 at the frequencies where Im[ti13 /K} is • 2 zero and where it is maximum. Thus, as the ratio -n. /-n approaches '11 'Ir • zero or infinity, the instability frequencies merge to the center of the bandgap and as fl/flr -+ 1 the instability frequencies tend to merge in pairs toward exact Bragg resonance. e; /K} The growth rates 1 are found from the value of Im[6k 2 at the bandgap center. The application of the stability criteria to multiharmonic periodicities is similar to the previous analysis (chapters III, IV) with the proper phase mismatch oN and coupling ')(N substituted into the dispersion relation. Although there is little or no effect at the fir st Bragg order for changes from sinusoidal periodicities to other typical periodicities, the effects at higher Bragg orders (N;;::: 2) may be significant due to Fourier components fN. It can be shown 2 (Appendix B) that the coupling is increased from values O(fl ) for sinusoidal periodicities to values O(T]) for sawtooth periodicities at the second Bragg resonance. At the third Bragg order, the coupling can be increased 3 • fro1n values O(T] ) to values O('ll) for square-, triangular - or sawtooth-wave periodicities. This implies that instability :.0.003 , - - , - - - , - - - , - - . . . , . . . - - - , - - - - , - - - , - - , - - - , - - - , - - - , _ 0.003 .;.0 .00 2 - - ..:.0 . 002 ':'.'I:: ':'.'I':: _ 0,001 c.0 .001 di cl o. 002 0.003 ~ ~ O 0 0:' <J i~ I • ~~I" ~ "" (§_ 0. 001 0.001 • ', '' /' '' '' I '' '' I-"' I-"' 0 ' I - 0.001 -0. 001 ~- 1 • r '=' 0 3 -0.002 ~; - ::. 0.01 'r O. 02 = =0. 02 • r =- 0.0 5 :::~:~5'·, - 0.003 -0.002 -0.001 0 6 Re / 0.001 /31 0. 002 0.' 003 0 ', ', • r = 0.1 :-: 0.0005 .:. 0.0 010 - 0. 003 -0. 002 -0. 001 0,001 6 1m / 6 t l Re/ !1 o. 002 o. 003 0 I I ,I ,' I I I I 3/ :--. 0.0005 .:. 0,001 0 Im jt>f l Fig. 5. 6 Brillouin diagram at second Bragg order for ECW theory when both index and gain/loss coupling are present. -111- thresholds can be changed at higher Bragg orders by the use of nonsinusoidal periodicities, Typically, 111 I ~ 0(10- 2 -10- 5 ) in integrated optics applications, D. Comments on Stability In this chapter we have found the stability criteria for time- independent periodic media, This media may support either decay- ing (stable) · or temporally growing (absolutely unstable) waves in basically dispersionless dielectrics, The ECW equations provided explicit values of threshold, instability frequency and temporal growth rates for absolute instabilities, Average positive gain was required for oscillation at the bandgap edges for index coupling and even-order gain coupling, For odd·-order gain coupling, the average gain could be negative if the gain periodically took on positive values and the oscillation took place at the bandgap center, In some cases both types of coupling or multiharmonic periodicities could reduce thresholds. There has been some question as to the effect of boundaries upon DFB oscillation, It is apparent from the preceding sections that instability takes place when certain modes of the active periodic media achieve threshold, No boundaries are needed since feedback action is produced by the periodicity, It appears that for frequencies other than the instability frequencies, waves may either decay or be convectiv~ly amplified if the structure is short enough such that the absolute instabilities do not occur, In this case, the active periodic structure might be -112used as a filter-amplifier. However, each individual system must be considered along with the boundary conditions to determine stab 1·i 1·t y, 35, 73, 74 The results of this chapter have been summarized. 97 -113CHAPTER VI APPLICA T IONS OF ECW THEORY The results of previous chapters have depended only upon the dispersion diagram of unbounded periodic media. several cases of bounded media will be discussed. In this chapter The use of the ECW equations allows behavior at all Bragg orders to be approximated easily. The purpose of this chapter is not an exhaustive coverage of periodic structures but ra~er an indication of a wide range of problems which may be solved by the use of the previous theory. In section A, longitudinally bounded passive media (i.e. DFB filters) will be covered along with examples of transients in a periodic slab. In section B the characteristics of higher-order DFB lasers will be given. Section C will cover the case of holographic grating diffraction and the last section will characterize the propagation of Gaussian beams in periodic media. A. DFB Filters 1. Effect of Longitudinal Boundaries We consider waves in a periodic slab of length J,, The ECW equations account for Bragg coupling or scattering fro1n the periodicities but not for the coupling or scattering due to the boundaries. The equations can be easily modified. The relative dielectric con- stant is now E:(z) = E: r + iE:. + rect(z/J,) E: 1 i.J(ri +if) . ) f cospKz rp r 1p p -114where rect(z/1,) I z I < J/2 = {: lzl > L/2 For simplicity, consider singly periodic media with no average loss or gain (i.e. f p = 0 for p ,J. 1, e:.1 = 0). The additional Fourier com- ponents available for coupling or scattering arise from rect(z/ 1,) and have not been accounted for previously. These components are pro- portional to ·the Fourier transform, 00 'I'] J-oo rect(z/ ~.) cos Kz e i~z dz (6 . Al) _ ~ { sin(/3+-K)i,/2 + sin(Q-K)li/2 } - 2 (/3+K)~/2 (f3-K)li/2 Thus, the additional available perturbation per unit length is proportional to the right-hand side of (6. Al) divided by fue length ii. As in the case of multi.harmonic periodicities, only the strongest effect (i.e. first order in 'I")) of the boundary perturbation of (6. Al) upon F and B 1 are considered. 1 For higher .. order Bragg interactions, the coupling diagram of Figure 6.1 is helpful. Not only is there the usual cross-coupling through terms 0(17N) and self-coupling through terms O(r,h, but there is cross- and self-coupling proportional to T)4! and ri@, respectively, where sin((:3+ 1(/2) 1, (~+K/2 )£. (6. A2) C N { sin((3+IS:)£/2 + . (tJ+K)l,/2 are the boundary effects. ~.in(/3 - K).t/2} (/3-K)t/2 2 This is due to th~ terms [-k e:r'l7( ®F 1+ if? B )/2.] 1 2 and [-k e:rn ( (8) B +if? F )/2] which are added respectively to the F 1 1 1 and B 1 equations of the ECW equations (3. A6). Consequently, fue k~ K ~ ,--... ,-7]~~ ~x >QQ< 81+2/N 8 1 81-2N Fl-2/N F1 t -N/2 N/2 Fig. 6. 1 Coupling diagram with ~ supe r imposed for a given Kt. Cross ~oupling occurs due to both the periodicity perturbation and the boundary perturbation. The self-coupling due to the boundary effect (8) is small and is not shown. F1+2/N ►/3/K I ,__. ,__. Ul I -116modified ECW equations become identical to the former ECW equations with the changes oN _, 6N and XN -- xN where ~oN K ~XN K oN = K - !i!L 8 @ (6. A3) ' ' XN + Nn_ w = K 8 In each case the first term of the above equation expresses the effect of the periodicity while the second term expresses the effect of the boundary. For higher-order interactions, the latter effect predomi- nates in coupling when Kt<< (17/2)l-N while the opposite is true for Kt>> (ri/2) 1 -N. This is expected since for thick slabs the extra Fourier components introduced by the truncated periodic media are tightly clustered around the components introduced by the periodicity of the infinite rnedia and no new effects are observed. However, for thin slabs, the truncation produces Fourier components at many multiples of the Bragg wavenumber which are capable of directly coupling F nates. 1 to B . 1 In this case the boundary couplin_g (ex: 17 t) predomi- Note that the above inequalities denot e the region where the results of the infinite media can be directly applied to the longitudinally bounded case. Also note that the bou ndary effects can be reduced if the perturbation is gradually truncated at the slab ends. A similar effect has been noted in quantum mechanical scattering. 79 The boundary effect s ddom affects the phase mismatch since it is negligible when Kl. >> 17 - I. -1172. ECW Reflection and Transmission Coefficients In this subsection we consider the general case of transversely and longitudinally bounded structures where each coupled wave represents a different mode. different for each mode. Hence, the longitudinal wavenumbers n1ay be 31 For simplicity the numerical examples will correspond to the special case of coupling between waves of identical modes in thick slabs although the analytic results are given for the general case using the ECW approximations. Figure 6. 2 schematically shows the configuration. Consider i /3 z a positive phase velocity wave F (z) U (x)e P which couples to a 1 p ip z negative phase velocity wave B (z) U (x) e q . The subscripts p, q . q 1 refer to the mode number of the tran sverse distribution U(x). phase match condition for significant coupling at the N th The Bragg order is (3 p + 13 = NK q ( 6. A 4) where K = 2TI" / /\ and /\ is the fundamental spatial period. The boundary conditions are found from Figure 6. 3 by use of the ECW equations which correspond to continuity of the electric fi~ld at z:±1,/2. >'< e-if31,/2 + RN ei{3t/2 = i•12 F(z=-t/2) e-i(l-2n/N)t/2 n=-1 1-211/N +B (z=-t/Z) ei(l-2n/N)t/2 l-2n/N I + ( O) S(z= -t/2) • (6.AS) -118- i _J INCIDENT PULSE •. ___.-,___ ·; : ;~ ':: , ' :' . ~ i?ME ~TED PULSE ij;;;;f ,...___,....._. TRANSMITTED PULSE PERIODIC STRUCTURE -J./2 ~ - - ~ 1/2 Fig. 6. 2 Configuration of pulses and periodic slab. -119~:< T Ne i(3 i,/2 = ~2 F(z=t/2)ei(l-2n/N)£/2 n=-1 1~2n/N + B(z= £,/Z) e-i(l-2n/N)£/2 l-2n/N + ( where N-2 -2- ~:, N /2 = (~) = N-1 1 0 ) S(z=.l/2) for N even -r for Nodd 1 for N = 1 1 for for .{ 0 (6. A6) N even Nodd and RN and TN are the reflection and transm.ission coefficients. Similar equations result from m .atching the derivatives (or the magnetic field) at z = ± .l/2. The only additional equations needed are the ECW equations modified for transversely bounded media. (6. A 7) where the coupling and phase misrnatch are assumed to be known. Instead of solving the above equations exactly, a perturbation schem.e is introduced which significantly simplifies the calculation and eliminates the boundary coupling which has been accounted for previously. Introduce the ordering parameter ). _ which will eventually be set to unity. 77 Consider a power series solution of the form - :F'iO)+AF:1)+),__2Ff)+ ... } = Bi°)+ AB?)+ AZ Biz)+._.. -120- € ( z) F1-2/N e i( l-2/N)t30 z F I eitioz • • • eiPz ► ... ◄ - 81-2/N e-i {1-2/ N) J%z 81 e-iJ3oz • • • z =-.i/2 z=+.i/2 Fig. 6. 3 ECW waves in periodic slab for coupling between identical modes {i.e. /3 p =/3 q =/3). -121 - for the waves and let the smallness be specified by A.fl. waves are solved in terms of F and B 1 1 If all the (see equation (3 . A6) ), then the electric field continuity equation bec01nes, e -if3 i/2 + ~ eif3 i/2 where near the N th f3 = f3 o +L':i/3 N f3 0 = NK/2 Bragg resonance. By equating terms O(A.) and phase matching, the approximate boundary conditions become 19 (6. A12) The reflection and transmission coefficients are solved using the boundary conditions and the ECW equations . out in Appendix C. R T The details are carried We find -_ N - (6.Al3) = (6 . Al4} N -122.- where = ( [ ~Xpq ~ Xqp - 6 +?; . l. q p )] 2 z If the two coupled modes are identical, then '5 = 6 = 6 p q Xpq =Xqp =X• and The subscript N has been dropped on 6 and X for convenience. From the above equations for ~ and TN we observe the following: 1. maximum reflection R 2. I N 1 X )2 ;, '• pq qp = i tanh(x max (6.Al5) minimum. transmission 1 TNI min . = 1/cosh(xpq Xqp )2 t and (6. Al6) for 6+6 p q (6.Al7} where the corresponding phase of ~ is (Zn- I hr /2; 4. 2 l RN!. + 2 lTNI ::: 1 for passive and lossless media. These results are demonstrated in. Figure 6. 4 for the case X P., = 2 where the boundary effects are neglected. The ff1agnitudes of the reflection and transrnission coefficients are shown as a function of oi-(i. e. frequency deviation from bandgap center)o Note that the half-width of the main reflection maximum is equal to t4e ba.ndgap width. Hence, there is significant reflection outside of the bandgap due to the deviation of Re(A/3/K} as well as Im.(l'll3/K} away from their TRANSMISSION COEFFICIENT AMPLITUDE REFLECTION COEFFICIENT AMPLITUDE 1.0 c) xi = 2 Xi = 2 0.8 0.6 w 0 ::, ,_ ii'. ...... ::; N w ~ 0.4 ~ I n II 0.2 0 I I II 0 . ~ ~ \~~fVV~{MV\LJW 20 O 20 40 60 NORMALIZED FREQUENCY 81 Fig. 6. 4 Magnitude of reflection and transmission coefficients as a function of phase mismatch for fixed coupling. Boundary effects are neglected. -124unperturbed values. Figure 6. 5 shows the equimagnitude and equi- phase curves of the reflection coefficient plotted as a function of xt and oi where again the boundary effects are neglected. Note that Figure 6. 5a corresponds to a horizontal line (x.£. = 2) across Figure 6. 4. If the coupled modes are different (i.e. p f. q), tl1e curves in Figure 6. 4 and 6. 5 are valid if o is replaced by (o 1 (X pq Xqp )2 . + o )/2 and X by p q . Similar results for media with average gain have been . discussed for the f:i,rst-order index coup~ing case when the modes are .1"d ent·1ca1. 35 It is of interest to note the change in the reflection coefficient with Bragg order. For small coupling XNi, << 1 , (6.Al8) Thus. the variation of RN with N is RN I i{ a: (ri/2)N Ki+17 sin[(N±l)K f,/2. max J} (6.Al9) where the first term expresses the dependence upon the periodicity perturbation and the second terrn expresses the effect of the boundary perturbation. For other typical periodicities, tl1.e results of Appendix B will provide appropriate coupling or phase mismatch. In particular, note that for thick slabs, the coupling is constant at odd Bragg orders for square-wave periodicities. This implies that the maxi~um reflection coefficient is also constant at odd Bragg orders. These results are in agreernent with well-known ex.act results that use matrix calculations. 78 Figure 6. 6 exemplifies fuese concepts for three typical 0 0 I I I I 0 I I I I o. 6 I ">I ;,< 4 I/ I I II; WI/ 5 6r-rr/2 I I 5L I I I I I I I - 'I' ""I X 'I ' I '! I I '1 I 2 I i I I I I I 0,.:> 1 I 0.1 II I I ~ O. llE[=---_j_ _L__ _ _~ ; : - - - - - : ; - - - - - - 0, 1 0.5 I 2 I 2 3 8.Q rr I I I I I I I I I I I I I I I I I 1 I 3 8.Q 7r Fig . 6 . 5 Equimagnitude and equiphase curves of the reflection coefficient as a function of phase mismatch and coupling. Boundary effects are neglected. I I I I i t I I 5rr/ 2 I I I I •I ,.I I/ I / I : I I I I 0. 5 I ' 31 'I I I I I / I '/ 2rr I I I I I / I / 4//5rr/ / 4/I/ -1r/ 4-7r 4~ I , 3 -377:/ 8 TI I I / rr/2 I v 3v/2 lo ..... N ln I -126- 0JV' I ¾~772 /-- 'I I Q) ±"-'773 - -0 ::> a_ E 'Lfl_J7 <( C Q) u '+- '+Q.) l I JL II 0 0 C 0 - , ~ u (1) 4-- Q) et: Fig. 6. 6 Sketches of reflection coefficient a 1nplitude for x.t ~ O. 5 for a) sinusoidal, b) square-wave and c) sawtooth-vva.ve periodicities. In all cases the boundal'y effects are neglected. Note th , small reflection at N Bragg order when fN = O. The bandgap shift is too small to observe on this scale. ► -127periodicities with index coupling. Note that periodicities with large Fourier components fN will have large coupling and reflection co~fficients at the N th Bragg order. Thus, periodicities with odd Fourier components (e.g. square- or triangular-wave) will have large reflection coefficients only at odd Bragg orders. The use of dynamically generated (e.g. acoustic waves or electro-optic effect) multiharmonic periodicities opens up the possibility of controlling the feedback strength in DFB filters. turn, varies the passband. This, in From the results of chapter IV it is ap- parent that the passband could vary as sin(0 /2) where 9 is the relative phase between Fourier components of the periodicity at the second Bragg resonance (see Figure 4. 5). If the boundary effects are sig- nificant or predominant, then the previous results will be modified. That is, for thin slabs, the reflection will be significantly increased l. at higher Bragg orders when [risin[(N±l)K + t:.k£; ].£/2} is no longer negligible. 3. Born Approximation Reflection Coefficient I In order to summarize the properties of reflections from both thick and thin slabs, it is useful to find the Bo1·n approximation to the reflection coefficient. Consider the wave equation for some transverse component of the electric field E(z). 2 d 2 [ dz 2 + k e r J E(z) = -AT'lk 2 e cosKz E(z)= -AS(z) E(z) r where the ordering parameter A has again been used to show the smallness of 17 . Assuming a solution E(z) = E(O)(z) + ~ E(l\,:) + :>i. 2 E(Z)(z) + ... (6. A20) -128- we find 00 = J S(z')g(z,z')E(n)(z')dz' (6 . A2 l) - 00 where e g(z, z 1 ) = il3z . e i13 I z-z' I l 213 1 and where 13 = k ~; for _this computation. A straightforward calculation shows, (l) E i'nRi, (z) _. = -'½f-=- e 1 13.z [sin(@+K/2)Z+sin(@-K/2)i.,J (l3+K/2)i., (13 - K/2)£. (6. A22) Thus E(l) represents a wave traveling to the left which is produced by the wave E(O) traveling to. the right. The approximation which uses the terms E(O) and E( 1 ) is known as the Born approximati.on. The Born reflection coefficient is then the coefficient of e -il3z in E( 1 ), or = ~ [sin(@+K/2)1, + sin(@ - K/2)£.J 4 (l3+K/Z)£. (13-K/2 ).l 1 2 i'nke: 'I r .t = (6. A2.3) 4 1 I ri k e:; i., I/ 4 << 1 which is valid for or for I ri/ 41 << 1 at the first Bragg resonance at higher Bragg resonances. At the first Bragg order, for 1-x.t I << 1 and lri I << 1, the Born approximation and the n10dified ECW approximation become equal. Explic itly we find 1 R(l) I "'"' max i17ke:/t 4 "'"'R 1 I n1ax (6.A24) -129at the bandgap center for Kl PIO. For higher Bragg orders, the Born approximation produces a series in powers of 'I'] which accounts only for the depth of the perturbation (i.e. the end effects) and not the length over which it acts. A series in riKt is needed to account for This latter series 'occurs through the significant periodic effects. modified ECW equations. Thus, the Born approximation is useful at the first Bragg order or at higher orders where the slab satisfies the relation ~.e, << fll-N. For thick slabs, the ECW theory provides the proper phase mismatch, coupling and reflection. 4. Transients in Periodic Slabs Consider an incident pulse f(t) with frequency spectrum • 00 F(w) = J f (t)e iwt <lt (6.A25) - 00 where the reflected and transmitted pulses are, respectively, 1 r(t) = Zn 00 J F(w) RN(w)e- ·wt d w 1 (6. A26) -oo l t(t) = Zir 00 - iwt J-oo F(w) T N(w )e dw • This case was shown in Figure 6. 2. (6. AZ 7) Numerical inversion has been used by a number of authors to obtain tin1.e response to radiation and scattering problems. The Cooley-Tukey fast Fourier transform. (FFT)BO has been used in the numerical examples with 211 = 2048 samples to calculate r(t) and t(t). The cases of reflection and transmission of rectangular and Gaussian pulses of several center frequencies and widths have been carried out. Normalized pulse lengths 'r were chosen to be O. 25 -130and 2. 0 time units where each time unit corresponds to the transit time of the pulse across the slab length 1-. As in previous sections, the numerical results are valid at the first Bragg order and all higher Bragg orders where the bour1dary effects can be ignored. Figure 6. 7 displays the envelopes of the reflected and trans1nitted pulses which result from. an incident pulse of length 'T = O. 25 . Several values of coupling are used and the carrier frequency is at the bandgap center (6N = O) _. In all the illustrations, the time value t = 0 corresponds to the instant when the center of the pulse is at the first boundary. Fo_r weak coupling, the reflected pulse is spread ove r 2 . 25 time units (Figure 6. 7 a, c) because the energy is r eflected from. successive striations and the echo from the last boundary has a round trip tiine of 2 units. Since the 'successive reflections are relatively weak, t h e reflected pulse is quasi-rectangular while the transmitted pulse is similar to the time delayed incident pulse . As the coupling is increased, multiple interference leads to transients over a longer time period (Figure 6. 7 i, j). The reflected pulse reaches a rnaxiinum at O. 25 units when the entire incident pulse has just entered the slab. The subsequent fall-off in the reflected pulse am.p litude is due to the fact that a large portion of the signed has already been reflected due to the first few striations. For large coupling, the transmitted pulse consists mainly of two peaks, O. 25 units apart, which correspond to differentiation (i.e. high-pass filter) of the incident pulse. The same coupling str ength sequence is given in ·Figure 6. 8 for a rectangular pulse of length 'T = 2. For weak coupling (Figure 6. _8a), the refle cted pulse is similar to the autocorrelation of the -131- 0 REFLECTED OUTPUT .030 a) .025 Xi. = 0,2 .020 TRANSMITTED OUTPUT l •2 ·-b) 1 IN PUT 1.0 "":fil:ot- r =0.25 :::> ..... ,015 0,6 ~ ,010 O,,f. ,005 0.2 - i2 <( 0, 12 c) 0 1,2 d) 1,0 Xi.= 0,5 0.4 0.02 0.2 0.6 0 1..-...L.--l~..L--J 0,24 e) 0.20 - 0 '---'-L=,___.__j 1.2 f) 1.0 Xi = 1,0 0 0.16 0.8 - I- 0,12 - 0.6 ~ :t 0,08 <( 0 2 • 0 0 L........L.--l....,.,....._--' 0.30 g) 0,25 1.2 h) 0 0.20 UJ t:: ~ :t <I: 0.6 0.10 0.4 0.05 0.2 i) 0 ,.__._"-,l-'<""""-J 1.2 j) x.Q = 5 1.0 1.0 0.8 0.8 - 0.6 0,6 0.4 0.4 0.2 0 ,., -· 0 L-.L.U<:11!.J.~ -2 xi= 2 0,8 0 L_.J____J_~LI>..a- o:::> ..........'--".__..___j 1.0 ...:::> i2 o. 15 1.2 x..i. = 1.0 0.4 ·- o.o4 ~ Xi. = 0 . 5 0,8 - 0.08 ::> ..... 0,06 ~ :t 0.04 <( ::> Xi.= 0.2 o~-~~-~~ 0 L--..L----<>"'--''---J 0.10 0 0.8 0 +2 Xi. = 5 o,_......___,""""'""""'""' +4 +6 -2 0 +2 +4 +6 NORMALIZED TIME Fig. 6. 7 Reflected and transmitted pulses for different values of coupling coefficient. Incident rectangular pulse has length O. 25 tirne units and a carrier frequency at ~ in vertical scales. I max . Note the difference -132-- 0 TRANSMITTED OUTPUT REFLECTED OUTPUT 0,2-4 a) 1.2 b) 0.20 1.0 - 0.16 0.8 ::::, 1 INPUT .....r.lL--r "- 2 0 0,6 - !:: iC :£ < 0.08 0.4 0.0-4 0.2 xJ. = 0.2 0~------ 0 1.2 d) 1.0 UJ o.a i2 0.6 - ...g ~ x1 = 0.5 0.4 0.2 -· 0 ........~_....._,,_..,___, e) 1.2 1.0 1,2 1,0 0,8 0,8 0.6 0,6 0,-4 0.4 0.2 0.2 - X1 = 1 0 1.2 9) 1,0 ~ __,_......,.___, h) X1 = 2 1.2 1.0 x1 = 2 0.8 ~ 0.6 0.6 ~ 0.-4 0.4 0.2 0.2 - I- 0 L...LJ...-.l.:L-L..k>.-.1 1.2 0 '--·"'---'-'-"'-'...... 1.2 d 1.0 = 5 1.0 - w 0 0,8 0.8 - !:: _, 0.6 - 0.6 0,-4 - 0.4 - ::::, "~ < 0.2 j) XJ. = 5 0,2 - o.__._~~ -2 X1 = 1 0 .___ 0.8 :::, F) 0 .J 0 .____...... _~~-"'""~ -2 0 +2 +-4 +6 +2 +4 +6 NORMALIZED TIME Fig. 6. 8 Reflected and transmitted pulses for different values of coupling coefficient. Incident rectangular pulse has length 2. 0 time units and a carrier frequency at RN in vertical scales. Irna.x. Note the difference -133incident pulse. This indicates that the reflection coefficient is similar to the incident pulse spectrum. For strong coupling, the transmitted. pulse is again similar to the absolute value of the derivative of the incident pulse with characteristic i n terfering echo s . Note that the reflected pulse may have an amplitude larger than unity (Figure 6. 8g, i) due to constructive addition of successive reflections, • Figure 6. 9 displays the reflected and transmitted pulses for a Gaussian i~cident pulse of width 'f = 2 (width is taken at the 1 / e values). Since a Gaussian pulse contains smaller higher frequency spectral components, the reflected and transn1itted pulses are grossly similar to the incident pulse. For narrowed Gaussian pulses (not shown here), the results are somewhat similar to those obtained for a rectangular pulse of the same vvidth. Several examples of pulses with carrier frequencies at the first several zeros where. pulse. 29 of the reflection coefficient have been given else- In general, the transrnitted pulse is similar to the incident However, the reflected pulse is broken up into a long pulse train which can be considerably longer than the incident pulse \vidth. 5. Discussion of DFB Filters In the preceding subsections, the ECW theory was .applied to longitudinally bounded media. The EC W equations were 1nodified to directly account for the coupling between F (z) and B (z) due to the 1 1 boundary. This changed the coupling so that it consisted of two parts with the following physical rneaning. One part of the coupling, due only to the perturbation is proportional to [riNKi,} and hence increases -134TRANSMITTED OUTPUT REFLECTED OUTPUT 1.0 l.O a) b) o.s IN PUT I ~r=2 0.8 0 xJ = 1 ... 0.6 0.6 - 0.-4 - 0.4 0.2 0.2 0 0 0 ::, !:: Is:! ~ <( 1.0 ~~-=h __J 0.40 c) d) 0.8 - 0.32 ... 0.6 ... 0.2-4 x1 = 2 0 ::, ~ ~ <( 0.-4 0 . 16 0.2 - 0.08 0L-_ --4 _ _.J.,.c__ _ J...._ _ _~_::,..--=-"---' -2 +'4, +6 0L__ __1._c___ -4 __J__ __._~.ll--~<="--~-:-+6 'J -2 +4 NORMALIZED TIME Fig. 6. 9 Reflected and transmitted pulses for incident Gaussian pulse of width 2 time units and carrier frequency at RN difference in vertical scales. I . Note 1nax -135with slab thickness since it is due to repeated reflections in the dielectric. The second part of the coupling is due to the above n1entione d l boundary effects and is proportional to [ri sin[(N±l}K+fike: 2 ] i,/2}. r This accounts only for the interference effects due to the periodic ity tnmcation and may become zero f o r certain lengths at particular frequencies. The application of the b01mdary conditions was then introduced in such a way that end effects were not accounted for again. The validity of this approach was tested by using the exact results of Bedrosian. 89 90 • The cornparison was rna.de on periodic media of length K~ = 26,r with f] = 0. 05. The ECW and exact invariant ilnbedding technique agreed to within 1'% in the two cases where the .had a large effect (R 2 I I 2 = 2 . 55Xl0- ) and where it 2 rnax 0 = 5. 88x 10- 2 } at the second Bragg order? boundary effect was insignificant (R max Comparisons to the Born approximation showed that at higher Bragg orders, the Born approxin1ation ignored the riNK.R, terrn of the coupling. However, at tl1e f:i.r st Bragg order the Born and 'EC W ap- proximations are in very close agreement. For cases where fl --+ 1, the full Floquet theory must be used. This is straightforward in principle but laborious to carry out. The disadvantage is that lengthy numerical calculations have to be carried out and the intuitive appeal of the ECW theory is lost. We have shown that periodic slab::; may be useful as edge differentiations or multiple pulse generators. The transient re- sults may be useful in the fields of microwave and optical filters and for sounders of subsurface layers. -136- B. Higher-Order DFB Lasers Ever since Kogelnik and Shank's demonstration tion 23 54 and descrip- of DFB lasers in the early .1970 1 s there has been great interest in this field. Since then much work has been done at the first Bragg or d er. 14,24,25,30,35,36,58,70 Inl 9 ?zB·Jo r khl o m andSh ank41 • demonstrated the first higher-order second and third Bragg resonances. DFB laser with output at the More recently there has been further experim.ental work at the second Bragg :resonance multiple frequency DFB lasers. 71 and with 70 In this section the approxin~ate 1node spectrurn and threshold gain are given as a function of Bragg order. The results, which use the ECW equations, are analogous to the coupled wave results of Kogelnik and Shank. sketched here. 1. 23 Hence, the derivation will be only briefly The details are given in Appendix D. ECW Result for Threshold Gains and Mode Spectra Oscillation will take place when output occurs for zero input. This can be alternately stated as the condition where the reflection or transmission coefficient becomes infinite. We use the ECW re- flection coefficient discussed in previous sections and set the denom.inator equal to zero or (6. Bl) This can be put into the form -13 7- = -1 (6. B2) The complex solution to the above transcend e ntal equation produces the threshold gain and the mode spectrum when solved for amplitude and phase. 2. High-Gain Approximation The high-gain approximation assurr1es that the periodicity has little effect on the prop.a gation, or, for 0 _N K = 6kNJe; K oN K 6 tiN I( 2 2 N [ 'N 11 N +-~ E: 2 4 1 8(N2 -1} 32(N -1} r C N 173 N3 i E:. - -i KgN - J (6. B 3) (6.B4) the condition is X N << (6.BS) This is approximated for singly periodic media by the inequality (6.B6) Fron~ Appendix D, the threshold gain approximation is given by (6.B7) and the longitudinal mode spectrum is given by -138where the first term is usually· sm.all and can be neglected. These expressions · neglect boundary effects which can be accounted for by a straightforward modification as in the pr e vi ous section. Equation (6. B8) is first solved for t.N, the norrnalized fr e quency deviation from. the bandgap center and then (6.B7) is solved for gN' the nonn alized average gain. It is apparent that for singly periodic media, the average gain will increase drastically with B r agg order. However., bounda r y effects or multiharrnonic periodicities m.ay reduce tlb1e threshold gain by increasing the coupling. The mode spectrum. is asyn1metric and is sketched i n Figure 6. 10 for higher Bragg ord e rs. For index coupling (Figure 6. I Oa), the gain symmet r ically pushes the modes outward from the usual two mirror cavity case (shown by dashed lines). However , the bandgap shift produces an asymmetry and shifts the entire spectrmn toward higher wavenum.bers. As in the first order case, no oscillation takes place at exact Bragg resonance . Figure 6.10b, c shows the analogous information for gain coupling at ocld and even Bragg orders, respectively. For odd Bragg orde r s, oscillation is possible near Bragg resonance, but the ba.ndgap shift preve nts oscillation exactly 1 at 6 ke; ,£, = O. As expected from the Brillouin diagran.1.s, even - order ga:i.n/los s coupling is similar to even-order index coupling e x cept for the sign of the bandgap shift. 3. Low-Gain Approxir.nation Low gain implies the inequality for singly periodic media le./e I << I ri121N· 1 r (6. B 9) -139- a) - . .,.. I I I I I t I -I I I • I I -- I I I I I I I b) z - <( C!) 0 _J 0 :c 0 V, I.LI 0:::: :::c I- c) - I -- I I. I I I I I I I -31rl 2 --rr l I ' - ..,,, 2 0 ~k I ' 1T/ 2 -- I .,,. ' I I 37T/ 2 vi-~Jl. __,... Fig. 6. 10 Sketch of mode spectrum in higher-orde1· (N ;;:: 2) DFB lase;s for: a) index coupling (N = 2, 3. 4, ..• ); b) gain coupling -~~ (N; 3, 5, 7, ... ); and c) gain coupling (N = 2, 4, 6, •.. ) in the highgain approximation. -140Again the details of the approximation are given in Appendix D. Boundary effects are neglected since low gain implies a large coupling-length product. Hence the boundary effects are small. Consider first the case of index coupling. We find that the lowest-order longitudinal mode frequency is given by !::, 2 (6. BIO) N which becomes (6.Bll) for long structures. ba.ndgap edges. Therefore, the oscillation takes place near the This agrees with the stability analysis of the pre- vious chapter as expected. Identical reasoning holds for the ca.se of gain/loss coupling at even Bragg ord e rs. The threshold gain condition for index coupling and for evenorder gain/loss coupling is given by 3 3 (6. Bl2) This is approximated by E:. l E: l (6. Bl3) r for singly periodic media. Hence the threshold gain varies inversely 2 with (n/2) N for singly periodic media and inversely with the length cubed. Thus the threshold gain condition predicts that the average gain approaches zero as the length becomes infinite. in agreement with the stability criteria results. Again this is -141For gain/loss coupling at odd Bragg orders the lowestorder longitudinal oscillation takes place at the bandgap center 0 = (6. Bl4) with the threshold condition 9n the perturbation (6. Bl5) for zero average gain. Thus for DFB oscillation to start with 8. = 0, 1 the necessary critical length .R,c varies as J3 J., (6. Bl6) C \3/ith Bragg order. The mode spectrum (6. B14,) agrees with the stability prediction and (6. Bl5) implies that negative average gain can produce oscillation of .R, > J., • This is verified by the a p proximate C threshold gain condition on the average gain which is ±6 = (6.Bl7) and becornes (6 . BJ.8) as the length increases. For singly periodic m .edia the threshold N gain varies as (f]/2) . This also agrees with the stability criteria predictions since this condition is identical to equation (5 . B3). Os c:illation aga in takes place at the band.gap center given by °" 0 (6. B 19) -142Note that only the lowest-order (m = 0) longitudinal mode characteristics are given in the low-gain approximation. 4. Discussion of Higher-Order D:F'B Lasers In all cases, the previous results are extensions of the wellknown work of Kogelnik and Shank 23 to higher Bragg orders. The ECW theory gives the correct XN and ON to use in the theories. Numerical results have been given at the first Bragg order for both transversely unbounded 23 35 24 30 • and bounded • m.ed.ia and so can be used at higher orders for guided modes. The ECW theory shows that the m.ode spectrum. is asyrnmetrically shifted from the exact Bragg resonance for high gain and has characteristic differences which depend upon coupling type and Bragg order. Note that in the low-gain case, the threshold gain and fre- quency of the lowest-order longitudinal n:1.ode is also predicted by the stability criteria without regard for the boundary conditions. In this case, enough coupling or feedback is available to produce oscillation without boundary coupling. In general, threshold gains increase drastically with Bragg order unless multi.harmonic periodicities are used or un~ess boundary perturbations play a significant role. The low-gain threshold varies a.s (T]/2)-ZN or as (r1/2f for the case of index and even-order gain coupling or odd-order gain coupling, respectively, for singly periodic 1nedia. C~ Higher-Order Hologram Diffraction There has been interest in transversely varying periodic media {Figure 6-. 11 }. This configuration represents electrom.agnetic -143- X t A t ::::::======::=======: e(x) = E(l+77cos Kx) z Fig. 6. 11 Configuration of TE wave obliquely incident upon a holographic grating. The original wave and the Bragg reflected wave em.erge as TN and ¾ respectively. -144wave diffraction by acoustic waves or by holographic gratings. former problem was solved by Chu and Tamir Floquet and the coupled waves approach. dealt with exhaustively by Kogelnik 15 22 The who used both the The latter problem. was who developed a coupled waves approach that was valid near the first Bragg resonance. More recently variations of these proble1ns have been investigated theo• 1 an_d experirnenta • 11 y b y severa l aut h ors. 68 • 69 ' 81-84 rehcal..y In this section the ECW equatio_n s are discussed for transversely periodic media. The derivation is siinilar to that of chapter Ill and the details are given in Appendix E. The approximate boundary conditions are applied and the results are given for the strengths of the undiffracted and Bragg diffracted waves in holographic gratings. As in previous chapters, the ECW results will hold for all Bragg orders. 1. ECW Equations for Transversely Periodic Media Consider solutions to the wave equation for the transverse electric (TE) field [ : :2 - + a:: + 1/ e(x)JE(x, z) ~O (6. Cl) where E: (x) - e: r + ie..1 + c rp6 {1l r +i17.) f cos pKz 1p p is the periodic relative dielectric constant . Following the example 5 • h 1s • sue o f K oge 1111.kl we cons1·d er a wave o f £orm. ei[J3x+kz z] •w 1nc cessively scattered from the periodic dielectric to a wave of forn-i e i[(/3 -NK) x+kz z] . The first wave is designated as F ( z ) and t h e Bragg 1 -145scattered wave as B {z). 1 chapter III we let Using the same ECW assumptions as in l N* /2 . F{ ) 1[(1-2n)/N)f3 0 xt6f3x+kz zJ E{ x, z ) -_ ,;;;, L.J z 1 -Zn/N e o n=-1 ) + B{ zl-2n/Ne -(l -2n/N)f3 0 xt6f3x+kz z] 0 + { ~) S{z) ei[L'if3x+lczoz] be the assumed TE field. (6. C2) ! Substituting the above expression into the wave equation (6 . Cl) for singly periodic lossless media, we find the following ECW equations {see Appendix E). (6. C3) where 2 k XN = 1 zN+l kz {sin29 )N-1 * 2 2 11 ( 4n(n-N) /N } 0 k = ko + t.kN f3 = f3 o + L'if3N z = kz f3 0 = NK/2 k2e = kr. 2 +f3 2 zo 0 0 o + L'ikz 1 sin 8 0 {-l}N+l kze TIN 0 k z = f3o/ko 82 0 N (6. C5) -146\ssuming a solution of the form e i.6k z z gives the dispersion relation (6. C6) C'he similarities and differences between the longitudinally periodic LP) and the transversely periodic (TP) case are given below . 1. The sign differences in the TP case cause the dispersion relation to be real for real periodicities and passive lossless dielectrics (i.e. '!"). l = 0 = €.l ). Hence no bandgaps appear in the longitudinal wavenumber as in the LP case. 2. Maximum phase matching between F (z) and B (z) occurs 1 1 slightly away from the exact Bragg condition for higher order Bragg interactions in both the TP and LP cases. The first Bragg order results are found from (6. C4-5) as 61 K """ [tik 1 Je _ .6l3 1 sin8 0 ] K K ! _ l_ _ .6l3 cos8 K2 (6. C7) 0 (6. C8) 8 cos 8 0 sin 8 0 Small changes in frequency or angle are accounted for by the first and second terms of the phase mismatch. away from normal incidence. The third term is negligible Results for other Bragg orders and for co1nplex periodicities can be found easily from previous results. A similar derivation for TM waves can also be made. 2. ECW Reflection and Transmission Coefficients The slab configuration of Fig. 6. 11 shows the undiffracted wave proportional to TN and the Bragg diffracted wave proportional -147to RN. The incident wave is of unit a1nplitude. The appropriate boundary conditions for the TP slab are approximately 15 = 1 F(l/2) B(-t/2) = 0 B(l/2) F(-1,/2) } (6. C9) These equations indicate that the Bragg diffracted wave increases in value from zero to RN while the undiffracted wave decreases from unity to TN• Note that any reflected ,wave in the region z < - l/2 is ignoredo The ECW equations (6 9 C3) are solved with the boundary conditions imposed by (6. C9). The method of solution is analogous to that of the LP case (see Appendix C). The results are i ·xn sin 6 kz .e N__ 6k ZN RN = TN tik cos 6.kzN .e + i oN sin t.kz .e ZN N = 6k (6. ClO) (6. Cll) ZN for coupling between identical m.odes (i.e. o p = o q , Xpq = Xqp =x). Note that for exact phase matching, the reflection and transmi.s sion coefficients vary as sines and cosines. Hence, the energy is alter- nately shifted between the undiffracted wave F (z) and the Bragg 1 diffracted wave B (z), along the z coordinate. 1 Away from exact phase match the energy transfer is incomplete. 3. Discussion of Holographic Gratings We note that the development of the ECW equations for the TP case is identical to that of the LP case. The ECW equations are 19 22 69 of the same form as that of several other authors ~ • • The -148advantage of the present derivation is the simplicity of the explicit expressions for phase mismatch and coupling. In particular the deviation frorn the exact Bragg condition that provides maximum coupling or highest diffraction efficiency is easily found. In the preceding derivation we found coupling between the waves F (z) and B (z). 1 1 This means that the diffracted and undiffracted waves, the amplitudes of which are given by RN and TN' are symmetric with respect to the z axis. A similar theory could be devel- oped for coupling between any other set of two waves. For example, one could find a set of coupled equations for coupling between F (z) 1 and B(z)l+Zq/N where q = 0, 1, 2, ... represents orders. different spectral In this case t he undiffracted wave propagates at the angle 0 °"' arc sin (f3 /k 0 Z ) whereas the diffracted wave propagates at the 0 angle 0 ""'arcsin[f3 (1+2q/N)/kz ] with respect to the z axis. 0 0 The 2 diffraction efficiency DE is usually defined by the relation DE = IT NI • The theory is approximate since only a few space harm.o nics were used, only the approximate boundary conditions were applied and because the (assumed small) reflected wave for z < --J.,/2 was ignored. D. Gaussian Beams in Periodic Media With the advent of the las er, it is useful to consider Gaussian beam scattering from periodic structures. 85 94 Several authors • 91 - have considered Gaussian beam scattering at dielectric interfaces or in periodic media. The method of beam reflection at interfa c es was developed by Brekhovskikh. 91 We will use this formulation and -149a similar one used by Tamir and Bertoni. 92 In this section, the formulation for Gaussian beam propagation in LP and TP slabs will be outlined. The previously derived ECW reflection and transmission coefficients will be used. examples will be given in a future report. Numerical However, the general characteristics may be determined from the previous transient analysis for certain cases. Consider the aperture Gaussian beam E ap (x, -h) of Figure 6.12 that is formed at the plane z = -h. E ap (x', 0) = e -(x' /w) 2 (6. Dl) The propagating wave is given approximately by E ap (x, z) I e = -(xcos8/w/ . ik,./e[xsin'9+(z+h)cos9] I e 1Tz w · . z=-h . (6. D2) z=-h The incident wave E. (x, z) is given by_ the Fourier transforrn of the inc spectral am.p litude <P (k ) X E. (x, z) = .1_ Joo ~ (k )e i [kx x + kz (z+h)] dk inc x x 2 'IT -oo (6. D3) -h<z<O where k = z The spectral amplitude is given in terms of the aperture field by the relation <P (k X ) = 00 J-oo E ap (x, -h)e -ik- X ·'X dx (6. D4) -150- X z C Fig. 6. 12 Gaussian beam incident upon LP or TP slab near Bragg resonance. -151- ii? (k ) X e-[w(kx-kJ°e sin8 0 )/2 cos8 0 = cos8 J2 (6 . D5) 0 Two assumptions are made ~ First, assume that the beam is well-collirnated, or k F w >> l (6.D6) where w represents the beam width. This limits significant spectral 1 amplitude to values about k X = k e 2 sin8 • Hence, k O Z is usually real and the contribution to surface or lateral waves can be ignored. This is also necessary for the ECW approximations. Second, the incident and scattered waves are near Bragg resonance, or LP media TP media (6 . D7) The transmitted and reflected fields ET (x, z) and ER (x, z) are found from the integral of the product of the spectral amplitude (at z = 0) and the appropriate coefficient; = 2,rI J T(kX )ii? (kX )e +i[ly+kz{z+h)] dkX. 00 ET(x, z) -oo 1 J R(k )ii? {k )li[kxx-kz{z+h )] dk = 2,r X X X (6. DB) . 00 ER(x, z) (6 ~ D9) -oo where the ( lupper) signs hold for ( LTPP) media. ower The expressions for R(k ) and T(k ) can be found from equax X tions (6.Al3, 14) and (6 . CIO, 11) for LP and TP media, respectively. The values of phase mismatch and coupling have been derived for arbitrary angle of incidence in TP media and at normal incidence for -152LP media. The extension of the ~p results to arbitrary angles of incidence is a straightforward derivation and will not be given here. The explicit results near the first Bragg resonance are, 2 tk 1 •x sin8 -,., - - o 'coso 0 K2 J X1 1 K °"' sc2se 0 (LP) media TP for c?s8 ) ( sm9 0 for LP and TP media (6. DIO) (6.Dll) 0 where sin9 tk 0 X = { k XO /~, EJ l for LP media /k e2 _ 0 for TP media /3 = k = { 0 k X 2 X [k e - k X 0 /30 = 0 /30 0 /3 0 2 J½ for LP media for TP media K/2 Note that for LP media the expression~ for the transmitted and reflected fields are similar to the transient analysis if the time coordinate is replaced by the space coordinate. In particular, Figures 6. 7-9 show the spatial dispersion of pulses in space for rectangular and Gaussian input beams at exact Bragg frequencies and non-normal incidence if the normalized time is replaced by normalized distance. In this section the derivation for spat_ially bounded beams was given a form that is easy to compute under the ECW assumptions. It was shown that temporal and spatial dispersion are similar and that the results of the transient analysis could be extended to include beam propagation. -153CHAPTER VII CONCLUSIONS This report establishes an approximate method of calculating the properties of the Brillouin diagram at all Bragg orders for waves in periodic media. The method is introduced in the second chapter where the connection between the Floquet and the coupled waves theory is shown and demonstrated with several numerical examples at the first Bragg resonance. In the third chapter, the idea of cross- and self-coupling helps to extend the coupled waves theory to all Bragg resonances by the use of coupling diagrams. These results explicitly show the dependence of the bandgap width, bandgap shift and coupling upon Bragg order. Floquet dispersion relations. The results closely match the exact Fur thermore, the ECW theory accounts for multiharmonic periodicities and demonstrates the idea of disappearing bandgaps under certain conditions. The fourth and fifth chapters deal with active or lossy dielectrics. Inverted bandgaps occur only for certain types of coupling and at certain Bragg orders. The stability of active periodic media has characteristics that depended on the nature (i.e. inverting· or noninverting) of these bandgaps. Absolute stabilities are found to occur under certain conditions and only at certain frequencies in active periodic media. These threshold conditions and mode spectra agree with results found from an entirely different analysis, in the appropriate limit, at the first Bragg order. The advantage of the stability analysis is that only the dispersion relation is needed to fully describe the stability characteristics. -154The last chapter includes a few applications of the preceding theory to finite length media. The topics of reflection, transmission, transients, DFB lasers, holographic gratings and beams in periodic media are briefly discussed and illustrated. The ECW theory demonstrates some of the power and versatility of the coupled waves formalism. It is anticipated that space- time periodic media and variable-frequency or almost-periodic media may be treated in a similar manner. The case of transversely bounded media can be treated using a previously developed approach for the guided modes. 32 The result is that each power of 17 will be mul- tiplied by an overlap integral. Hence the coupling will be somewhat decreased. Exact theories such as the Floquet theory, matrix theory or the method of invariant imbedding will be useful for specific cases where exact results are needed. However, the intuitively appealing ECW theory gives results explicitly without lengthy computations and is surprisingly accurate in the cases treated here. -155APPENDIX A ASYMPTOTIC FORM: OF FIELDS FOR ABSOLUTE INSTABILITIES IN PERIODIC MEDIA Consider the following equation for F(z, t) which is proportional to the electric field 00 F(z, t) =J 00 J -oo -oo S(.6.@, w)ei(.6.f3z- wt) D(t.(3, W) (A. 1) where S(.6.(3, w) is the normalized source and D(.6.j3, u,l) is the dispersion relation. We are interested in the asymptotic value of F(z, t) for large times when an absolute instability occurs. This happens when the roots of D(.6.(3, W) merge in the .6.!3 plane from opposite half-planes separated by the line Im[.6.~} = O. We assume that the source is analytic in .6.{3 and that it is turned on at the time t = O. We follow the work of previous authors 73 74 , and expand the dispersion relation about the instability at w = w' and 6f3 = 413 ' in a Taylor series. Then the integration is first carried out in the .6.(3 plane and finally in the W plane. The Taylor expansion of D(.6.(3, w)/S(43, w) is, D(.6.~ 1 w) S(.6.!3,W) where :::,,, D f ( 43 - llf3 1 / D2 I = 1 D2 2 = 2 2 a (D/S) a (llf3 ) 2 o(D/S! aw + n i (w - w 1 > I llf3=.6.f3 , I w=w' We rearrange the dispersion relation to find (A. 2) -J.56- F(z, w) = 1 1 4rriD D (w-w') 2 1 2 (A. 3) d(6(3) where 00 F(z, t) = -iwt dw F(z, w) e J -oo 2'IT and the contour C' is shown in Figure A. la. The first pole of (A. 3) is located in the upper half plane and contributes for z > 0 while the second pole is located in the lower half plane and contributes for z < O. For z > 0, equation (A. 3) becomes 1 F(z, w) ei[6(3'+iDz(w-w') 2 /D 1 ]z = 2i D (A. 4) 1 D ( w- w')z 1 2 To carry out the integration in w, we <;1.epress the contour around the singularities down to the real w axis. This contour C 11 is shown in Figure A. lb where two poles at w = ± w source contributions. s represent the harmonic This second integral is 1 F(z, t) = e e-[Dz(w-w') 2 /D 1 ]ze-iwt iil{3 1 Z en (w-w 1 ) 2 dw 2'IT (A. 5) 1 The branch cut of ( w-w1 ) 2 provides the major contribution. Hence, we approximate the integral (A. 5) as t-: oo by i6(3'z e F(z, t) = Zi D D I z {oJ O e 0 -[Dz(w-w')½/D 1 ] +[Dz(w-w 1 )½/D]z • e .!. 1 (w-w 1 )2 o (w- w')z -iwt dw X e 21r (A. 6) J -157- a) iW·I b) --...~~.....----➔--~~..........~...........-__..wr Wo Fig. A. 1 a) Contour C' in 6f3-plane showing poles when 6f3' = O. b) Contour Ctt in w-plane showing 2 poles due to source at ±w S and branch cut at instability frequency w . 0 -158- F(z,t) = w' = w where e illf3 I z -i(.c.b t · e 0 + ia Let (w. - a) = -q l Ja e - q t cosh[D1(-iq).!. /D1]zdq (A.S) 2 F(z, t) = o (-i)2 q2 Letting the upper limit tend to infinity we get F(z, t) _, as (A. 9) Thus, the electric field is time-growing with the wavenumber and frequency of the absolute instability. For two instabilities at different frequencies, there will b e two branch cuts in Figure A. lb and the F(z, t) will have two similar contributions, each of the form of (A. 9) . -159APPENDIX B ECW PARAMETERS FOR SQUARE- WAVE, TRIANGULAR-WAVE AND SAWTOOTH PERIODICITIES For applications to instabilities and DFB filters and oscillators it is convenient to have the phase mismatch 6N and the coupling XN at the first three Bragg orders for square-wave, triangular-wave and sawtooth periodicities. It has been shown that the major contribution to N th order coupling is from the Fouriei- component fN of the periodicity. Hence for the three periodicities under consideration, the results at first and third orders are a straightforward application of the results of chapter III since each wave contains odd harmonics. In addition the sawtooth wave contains both even and odd harmonics so previous results can be used at all Bragg orders for this periodicity. The computation that needs to be carried out is for parameters at evenorder resonances with odd-order Fourier components. We will carry out these computations for N = 2 in the lossless case (e=e ). r The three periodicities have the following Fourier decompo- p odd Square-wave p even Triangular-wave f Sawtooth f p odd p p with the normalization f (B. 1) (B. 2) p even = 1 (B. 3) = 1. An appropriate sine or cosine Fourier -160series is used for periodicities that are odd or even in z, re spec - tively. = At the first Bragg resonance, ...;,e find that X /K ri/8 and 1 1 2 o1 /K = 6k € /K by the use of (3.A24, 3.A25) for all three periodicities . This is identical to the sinusoidal case. At the second Bragg resonance, we find x /K -ri 2 /8 and 2 2 6ke½ /K-ri /12 for sinusoidal periodicities. The other perio- = o2 /K = dicities present need to be calculated. We consider the seventeen waves B , B , ... , S, ... , F , F for periodicities with fp 8 8 7 7 p even and N = 2. = 0 for Thus, we need only the eleven waves B , B , ... , 8 6 S, ... F , F since no significant coupling can take place through 8 6 Bq , F q with q odd. Using the ECW approximations, we need to solve the following equations. = -1f7 F 1 = -¥ (f5Fl+f7Bl) = -15 F 4 3 F -t (f3F l+fSBl) = -1 (f 1 F 1+f3 B 1) 2 I 26k 7c 2i F Fl + = -1[f5 (B6 + F 4)+f 7(F8+ B6) 1 ke"z +f3(F4+B2) + fl(Fz+S) J s I 2i B + 1 --r k e2 = -t [f 1( B 1 + F 1)] = -°t [f5(B6 +F 4)+f7(Bg+F 6) + f (B +F ) + f (B +S)] 2 3 4 1 2 = -1 (fl B 1 + f 3 F 1) (continued on next pag e ) -161- = - °t(f3Bl + f5Fl) = -¥(f5Bl +f7F2) --.!lf B 2 7 1 (B. 4) where self- and cross-coupling O(ri2) has been included. The equations (B. 4) are solved to give the usual coupled equations for F 1 and B 1 with (B. 5) • In (3. B6), if the terms drop off as f p 0. J (B. 6) 1 /p or faster, only the first two ex: terms need to be kept in the seri es for accuracies of order 1% in phase mismatch. In (3. B5) the series can be summed explicitly by using Schlafli' s Polynomials wave. 98 Sn of order 11 for the square- and triangular- Two terms will again give accuracies of order I%. For square- waves we find l Xz K 2 02 = "13" [-1+2 s2 ] , K = £1 ke: z 2 Yz (.911) (B. 7) = -ri- - 1z (. 990) (B. 8) ri while for triangular waves l Xz K = 2 62 2L[-1+2S] , 8 3 K 2 where rr - 8 s2 = ---u;- S3 32-3,r 64 = ::,., o. 1168 .. '1ke: z 0 2 ::,., 0.3736 ... 2 -162Thus, we have accounted for the major self- and crosscoupling terms O(r/) which can be approximated by only considering the seven waves B , B , B , S 2 1 4 F , F , F for periodicities with 2 1 4 odd Fourier components f . p The case of the sawtooth-wave 1nust be treated separately since it contains both even and odd Fourie-r components and must be expanded in a sine series. coupling from rif 1 To consider the correction to the self- and to the cross-coupling from rif 2 we consider only the first three Fourier components f , f , f and the waves 1 2 3 B , B , B , S, F , F , F for simplicity. 2 1 1 2 3 3 The equations for N =2 in the ECW approximation are then, -15 F -BF = - -frf3 Fl 4 - - -n-£2 Fl 3 f ]"" = - ..11.. 2i I I I 2 !!lk ~ F I + 2i FI k ea (B. 9) The solution for the coupling and phase mismatch are -163Xz (B. 10) 1 [£ 2 + O(ri fp)] K = oz fz 2 3f z 3 !L[ 2 = ~12 fl -16 - To - ... J 1 K bike 2 2 ± where it is understood that X = ± i X• (B. 11) The approximations to 1% in- volves keeping only the first term in (B . 10) for ri << 1) and the first two ter1ns in the series in (B . 11 ). Xz K = 1 t.k e 2 02 .!l. = K 8 Thus, for the ·sawtooth-wave, K 2 - ..!L 12 (0~953) (B. 12) The approximation that uses only the first term in both (B. 10) and (B. 11) can be gotten directly from (3. A24, 3. A25) with~ 5% accuracy. At the third Bragg order a.ll three of the periodicities considered here contain the Fourier c: omponent f . 3 cross-coupling will be due tori f due tori f . 1 3 Hence, the major and the major self-coupling will be The phase mismatch and coupling from (3.A24, 3.A25) are 63 K X3 K 1 = tik e 2 K 2 27ri2fl 256 3ri f3 = 8 (B. 13) (B. 14) The error is again on the order of 1% except for the phase mismatch of the sawtooth-wave where the error~ 5%,. The results of this appendix and comparisons to the sine wave are found in Tables B . 1 and B. 2. -164- N =I ~ rul.J .!1 .!1 8 8 'V'v 14 .!1 .!1 8 N =2 2 :.!L. 8 :!L (0. 77) ..:.!L (0. 93) 8 8 .!1 N =3 243ri3 2048 ]1 8 24 ..!1 Table B. I 2 8 2 .TI. 8 8 Normalized coupling coefficient xN/K for sine-, square-, triangular- _and sawtooth-wave periodicities. For the sawtooth-wave the" coupling is x± = ± i X• Vv fl_f7_i N =1 N =2 6kE: 1- _.-TI.:_ K 12 ..!c. N =3 i1kE: 1- !C. -r - 12 (0. 9J) 2 ~ 6kE: 2 .. ~ 6kE: ·z- K 256 K 2 _~ 256 6kE: t -X- - 2 717 2 256. ..!,. 6kE: 2 K _ 2 J:L_ 12 Table B. 2 Normalized phase mismatch oN/K for sine-,square- , triangular- and sawtooth-wave periodicities. ~165APPENDIX C REFLECTION AND TRANSMISSION COEFFICIENTS OF DFB FILTER Consider the following coupled equations = ( C. l) = which describe waves in a DFB filter. The primes denote differen- tiation with respect to the coordinate z. Differentiating (C. 1) and eliminating B (z) and F (z) respectively, produces 1 1 Ir ( ~) r) I - i ( op - oq) ( + ( oq op - Xpq Xqp ) ( ~ ) = 0 (C. 2) where the subscripts and arguments have been dropped for simplicity and the primes denote differentiation with respect to the coordinate z. The solutions to ( C. 2) are o -o - i( pz g )z Fe (C. 3) (C. 4) where D = + [ ~q Xqp - ( op +o 9 ] 1.2 2 ) (C. 5) Apply the boundary conditions F(ii/2)= l F(ii/2)= T } (C. 6) B(-ii/2) = R B(f,/2) = 0 -166to find the following equations ( c. 7) (C. 8) ( c. 9) (C. 10) Use (C . 7-8) to solve for f _ [ 1 2 and use (C . 9-10) to solve for b . 2 e(i 6 .l/2 + D.l/2) _ T e( - i 6 .l/2 ± D.l/2) ] + e [ where "o 1 D.l Re(i 6 .l/2 e Dt - e + D.l/2) - e -Dt ( c. 11) -D.l ] (C.12) = (o p -o q )/2 Two other equations for f 1 and b into the coupled equations2 ( C. J. ). 1 2 can be found by substituting ( C. 3, 4) This produces the coupled equations (C.13) (C.14) Equations (C.11-14) provide the necessary relations to solve for T and R, the transmission and reflection coefficients. manipulations, the results are, After afgebraic -167- R = (c. 15) o +o D coth Di. - i( P g) 2 T i(o - o )t/2 p q De = (c. 16) (o +o ) D cosh Di. - i E 2 9 sinh D.Q, Note that the equations hold for all Bragg orders when the proper o and X are used. Also the equations hold for coupling between waves of different transverse modes, or in general for any coupled system where the phase mismatch and coupling might be different for the waves F(z) and B(z). -168APPENDIX D APPROXIMATIONS FOR DFB THRESHOLD AND SPECTRUM 1. High-Gain Approximation Oscillation will occur in a DFB laser when the reflection or transmission coefficient becomes infinite. DN coth DN i, where = This implies the condition i oN DN = 2 ]2 [xN - 0 N2 oN = 6N - i gN (D. 1) Equation (D. 1) can be written in exponential form (D. 2) which is a complex equation. Under the high-gain approximation gN >> XN and expanding the expression for DN we find i oN-- (D. 3) Either root of (D. 3) when substituted into (D. 2) yields (D. 4) Taking the amplitude of (D. 4) we find the threshold gain condition (D . 5) -169where the value 6N is found from. equating the phase of (D . 4), or, (m=O, ±1, ±2, ... ) (D. 6) This latter result produces rnultiple values for the spectrum or oscillation frequencies which are called longitudinal 1nodes. Since tan - l (6N/ gN) << 1, the mode spectrum can be further approximated by 6k,/e_, r --K-- 'N 17 = 2 N 3 + - - -2 - - (D. 7) 32(N - 1) 0 where ( rr /2 ) =( 0 index coupling (all N), gain coupling (N even) • rr /2 gain coupling (N odd) for index or gain coupling . and singly periodic media . The high-gain approximation yield~ the entire spectrum and a transcendental equation for the threshold gain of each longitudinal mode . 2. Low-Gain Approximation It is more convenient to start with an alternative expression for index coupling or even-order gain coupling. F 1 = f e D )(, + f n 1 2 e Consider the solutions - Di, _ b Di,+ b e 2 B1 - l e (D . 8) -Di, (D. 9) to the coupled equations I - Bl - i oN Bl = i XN Fl where DN 2 ½ = [xN2 - oNJ } (D.10) -170Since there are no outside sources for DFB lasers, we may use the boundary conditions B(-l/2) = 0 = F(£,J2) (D. 11) This implies the relation = = -D i, -e N· (D. 12) When (D. 12) is substituted into the coupled equations (D. 10) along with expressions (D. 8, 9), the following equation can be derived. {D. 13) Either of the exact equations (D. 1) or (D. 13) may be used for threshold calculations. Consider the low-gain approximation where gN << XN for index coupling and even-order gain coupling. Expand the hyperbolic func- (D. 14) where From the real part of the above equation, (D. 15) For /, large, oscillation takes place just outside the bandgap edges given by (D. 16) -171From the imaginary part of (D. 14) the gain condition becomes (D. 17) The low-gain approximation for odd-order gain coupling is most easily derived from equation (D. 1). First consider the neces- sary condition on the perturbation for no average gain (i.e. g N = 0). An expansion of the hyperbolic function in (D. 1) produces (D. 18) Equating the real and imaginary parts produces the results (D.19) (D. 20) Thus oscillation takes place at the bandgap center. The necessary length i, for a given perturbation, is the critical length C (D. 2 I) For gain perturbations or lengths less than those given by (D. 20), no oscillation takes place. To find the threshold condition on the average gain, equation (D. 22) Keep all terms in nJ to get -172- lxNI + gN 2 6 = ± (D. 23) lxNl.e and L'.1N = 0 (D. 24) He1,1ce oscillation takes place at the bandgap center with the approximate gain condition (D. 25) when .e is large. Note that the low-gain approximation gives explicit values for the spectrum and thresholds of the average gain and perturbation for the lowest order (m=O) longitudinal mode only. All previous results match those of Kogelnik and Shank first Bragg order. orders. 23 at the This appendix extends their results to all Bragg -173- APPENDIX E ECW EQUATIONS FOR TRANSVERSELY PERIODIC MEDIA Consider the wave equation dz + dz 2 [ dz 2 dx: + 2 ] • k e:(x) E(x, z)_ = 0 (E. I) where E: (x) = · e: ( I +r] cosKz) for transverse singly periodic media . . Assume a TE field of the form E(x, z) (E. 2) where N-2 -z- for N even N-1 -z- for Nodd -1 for N = 1 for N even for Nodd This accounts for positive group velocity space harmonics involved in cross- and self-coupling of F (z) and B (z). 1 1 A similar expression exists for negative group velocity space harmonics. The following set of N+3 coupled equations exist when (E. 2) is substituted into (E. 1) for higher-order Bragg interactions. -174- (E . 3) 2 2 • 2 ( k e:-kz - (-(3 +t:.(3) }B o where o I = kz 0 + C:lk (3 = (30 + 6(3 k = k + 6k k 2 e: = kzo + (3 2 0 k z 0 0 2 z = NK/2 + 6(3 The primes denote differentiation with respect to the coordinate z and the arguments with respect to z have been dropped for simplicity. The above equations are solved using the ECW assumptions and approximations of chapter III. First, solve for F(z)l-Z/N and B(z\_z/N· The result is 2 similar to (3.Al6-17) with the change 17 _, 17/sin where e·O l sin6, 0 = (3 0 /k O·e: 2 • Hence, N-1 .'.!]2 · ) Bl(z) -i 11 N F (z) 9 ·N 1 + 2sm 0 8(1-1/N)sinZeo TT,,< (4JJ(n-N)/N2} 2 (-l)N+l( = F(z) 1-2/N (E. 4) -175N-1 (-l)N+l ( !2 ) + 2s1n 8 0 ~ 2 2 TT [ 4n(n-N /N -CNriNB (z) 1 = 2 8( 1-1 /N)sin 8 CN = (E. 5) J. 0 where Fl(z) N =1 {~ N;;::z I (N-1)/2 ? TT~< r(n) n=l for Nodd N/2 TT n=l 1 Second, trivially solve f(n) for N even for N =1 fdf F(z)l+Z/N and B(z)l+Z/N" The result is again identical to the previous results of (3.Al8-19) with the change fl _, ri/ sin 2 e0 . F(z)l+2/N = CNT")NF (z) 1 2 8(1 +I /N)sin 8 (E. 6) 0 B(z)1+2/N = CN fl N B 1 (z) 2 8(1+1/N)sin e (E. 7) 0 S_ubstituting (E. 4-7) into the equations for F (z) and B (z) in (E. 3) 1 1 produces the ECW coupled equations for TP media. 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