The Invariant Imbedding Solution for Electromagnetic Wave Propagation in Periodic, Almost Homogeneous, and Almost Periodic Media Citation Bedrosian, Gary (1977) The Invariant Imbedding Solution for Electromagnetic Wave Propagation in Periodic, Almost Homogeneous, and Almost Periodic Media. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/fqjc-b947. https://resolver.caltech.edu/CaltechTHESIS:11182025-162334808 Abstract The technique of invariant imbedding, as introduced for the problem of electromagnetic scattering by V. A. Ambarzumian in 1943, provides a very convenient method for the solution (albeit numerical in many cases) of plane wave scattering from a one-dimensional region of inhomogeneity. In the thirty-plus years which have intervened, the usefulness of this method has been extended in the case of electromagnetic properties of the region of inhomogeneity (dielectric constant, permeability, and conductivity). It is the purpose of this thesis to examine the invariant imbedding solution as it applies to periodic, almost periodic, and almost homogeneous media. The introduction of a complex number, Y , which is simply the reflection coefficient rotated by a fixed phase angle, is a new concept which allows the computation of the propagation constant for any periodic medium once the reflection and transmission properties for one cell are known, without any further complications such as matrix equations. The trajectory of the parameter, Y , also provides an interesting graphical representation of the properties of a periodic medium. The concepts derived for general periodic media are then applied to the important class of media whose reflection coefficients remain small, except perhaps at special frequencies. In particular, a small reflection approximation leads to the result that for any medium which is "almost homogeneous," there will be one special frequency, for each structure constant in the cosine expansion of the index of refraction, for which the reflection gets large. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Physics) Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Physics Thesis Availability: Public (worldwide access) Research Advisor(s): Papas, Charles Herach Thesis Committee: Unknown, Unknown Defense Date: 15 September 1976 Record Number: CaltechTHESIS:11182025-162334808 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:11182025-162334808 DOI: 10.7907/fqjc-b947 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 17766 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 19 Nov 2025 18:16 Last Modified: 19 Nov 2025 18:28 Thesis Files PDF - Final Version See Usage Policy. 29MB Repository Staff Only: item control page

THE INVARIANT IMBEDDING SOLUTION FOR ELECTROMAGNETIC WAVE PROPAGATION IN PERIODIC, ALMOST HOMOGENEOUS, AND ALMOST PERIODIC MEDIA Thesis by Gary Bedrosian In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1977 (Submitted 15 September 1976) -ii- TO: Professor Charles H. Papas, who gave this thesis direction; Gena Bedrosian, who put up with me while I wrote it; and . All of my Friends and Relativ_es, who have supported and encouraged me throughout my education. -iiiACKNOWLEDGMENTS I wish to express my appreciation to my thesis advisor, Professor Charles H. Papas, who suggested the topic and who has provided many invaluable insights and suggestions at every stage of my thesis research. I would also like to thank Professor Ralph W. Kavanagh, not only for the financial support of my graduate education as a teaching assistant under his direction, but also for the many hours of discussion in preparation for teaching the sophomore physics 1aboratory courses, which indirectly but substanti a 11 y furthered my graduate education. The computer time for this research was provided by a special student account available to Caltech students for unsupported research. Without that computer account, I would not have been able to test the equations numerically (especially the fast Fourier reconstruction of the index of refraction as represented in Chapter IV). The excellent job of typing this thesis was performed by Ruth Stratton. -ivABSTRACT The technique of invariant imbedding, as introduced for the problem of electromagnetic scattering by V. A. Ambarzumian in 1943, provides a very convenient method for the solution (albeit numerical in many cases) of plane wave scattering from a one-dimensional region of inhomogeneity. In the thirty-plus years which have intervened, the use- fulness of this method has been extended in the case of electromagnetic properties of the region of inhomogeneity (dielectric constant, permeability, and conductivity). It is the purpose of this thesis .to examine the invariant imbed- ding solution as it applies to periodic, almost periodic, and almost homogeneous media. The introduction of a complex number, Y , which is simply the reflection coefficient rotated by a fixed phase angle, is a new concept which allows the computation of the propagation constant for any periodic medium once the reflection and transmission properties for one cell are known, without any further complications such as matrix equations. The trajectory of the parameter, Y , also provides an in- teresting graphical representation of the properties of a periodic medium. The concepts derived for general periodic media are then applied to the important class of media whose reflection coefficients remain small, except perhaps at special frequencies. In particular, a small reflection approximation leads to the result that for any medium which is "almost homogeneous," there ~1/iVI be one special frequency, for each structure constant in the cosine expansion of the index of refraction, for which the reflection gets large. I -vTABLE OF CONTENTS List of Symbols Used vii Chapter I: Introduction 1 Chapter II: The Basic Equations and Relations 6 Definitions The Wave Equation The Invariant Imbedding Solution Some General Properties of the Transmission Coefficient An Alternative Invariant Imbedding Scheme Five Basic Relationships _ Relation to Quantum Mechanical Scattering in One Dimension Variation of All Electromagnetic Parameters Summary 6 II. l *II . 2 *II.3 11.4 tII.5 II.6 *II.7 tII.8 II.9 Chapter III: Periodic Media I I I. l The Periodic Recursion Formula III. 2 Fixed Points of the Recursion III. 3 Convergence to a Fixed Point; Method One III. 4 Convergence to a Fixed Point; Method Two I I I. 5 Calculation of the Propagation Const ant; STOP BAND III.6 Calculation of the Propagation Constant; PASS BAND III.7 Summary of Results for Lossless Media III.8 Lossy Periodic Media III.9 Multiply Periodic Media III. 10 Conclusions for Periodic Media Chapter IV: Almost Homogeneous Media IV.l The Almost Riccati Equation IV.2 Green's Function for the Almost Riccati Equation IV.3 Almost Inversion of the Reflection 7 11 19 22 28 30 33 39 42 42 47 50 51 59 61 63 66 68 77 80 80 82 84 -viIV.4 IV.5 IV.6 tIV.7 IV.8 Examples of Almost Inversion Modified Almost Inversion Purely Periodic, Almost Homogeneous Media Almost Periodic Functions STOP BANDS in Almost Periodic, Almost Homogeneous Media 87 95 99 106 108 Chapter V; Canel us ion 112 Appendix I: Inductive Proof of Reciprocity Relations 115 Appendix II: Careful Analysis of Higher Order STOP BANDS 121 References *Mostly Non-O_riginal Material tPartly Non-Original Material 124 -viiSymbols Used 1. Latin a Slab or cell length aT Transformed slab length A Fourier coefficient of the index of refraction B Fourier coefficient of the derivative of the index c Speed of light in free space k Wavenumber kavg kmax Average wavenumber in the slab kmm . Local minimum wavenumber in the slab kT Transformed wavenumber n Index of refraction navg Average index of refraction in the slab P Quasi-period of an orbit on the Y-plot r Local reflection coefficient or magnitude of R R Complex reflection coefficient RF Fixed reflection point t Local transmission coefficient or magnitude of T T Complex transmission coefficient V Quasi-velocity along a trajectory on the Y-plot X Rotated reflection coefficient XF Fixed value of X Y Standardized rotated reflection coefficient Yt Fixed value of Y Local maximum wavenumber in the slab -viiiz Slab length coordinate zT Transformed slab length coordinate Z Y - YF 2. Greek Cl Loss parameter s Propagation constant y Antisymmetry loss parameter 0 Phase shift (complex number) 6. Distance between interfaces € Dielectric constant n Y-plot trajectory shape parameter e Angle of incidence K Structure constant for periodic and almost periodic media µ Permeability ~ Parameter (small) (5 Conductivity ¢A,¢a, Loss angle parameters ¢R Phase of R ¢T Phase of T X Parameter 1/J Wave function, or phase angle w Radian frequency 3. Operators arg Complex phase operator exp Exponential function -ixRe Real part Im Imaginary part 0 Add one cell to Y ¢ Add one cell to X _, _ Chapter I Introduction The technique of invariant imbedding was first applied to electromagnetic scattering by Ambarzumian in 1943. 1 Since that time, there has been much work done both to extend the use of the technique in the solution of the wave-scattering problem (for instance, references 2 through 6) and to expound the theory behind the technique. 7 This thesis will be concerned with the application of Ambarzumian's technique to one-dimensional periodic and "almost periodic 11 materials. As the invariant imbedding approach of Ambarzumian applies to one-dimensional, linear wave-scattering problems, it may be stated with sufficient completeness very briefly as the following quasialgorithm: First, consider the simplest possible scattering problem, that is, the reflection from and transmission through a jump discontinuity in the properties of the medium conducting the wave, assuming uniformity on either side of the discontinuity. In optical terms, this would be transmission and reflection at an interface between media with differing indices of refraction. In quantum mechanics, a similar example would be the matching of wave solutions on two sides of a potential discontinuity. Second, assume that we know the solution for up to N such interfaces stacked together and examine what happens when we add one more interface at a distance 6 from the last. Finally, the solution of the simple case may be iteratively transformed by the derived relationship between Nth and (N+l)St solutions until we arrive at the final answer. In the case of a continuously-varying medium, -2- the limiting process which makes the t 1 s very small and the N correspondingly large yields first-order Riccati-type differential equations which, although hard to solve exactly, are very simple to approximate numerically to obtain the desired reflection and transmission properties of the inhomogeneous medium. The alternative methods for calculating the reflection and transmission are nearly always more difficult, both conceptually and mathematically. For instance, we could start with Maxwell's equations and reduce them to the second-order linear wave equation (when µ is constant), and solve for the proper ...,,ave solutions given k2 (z), assuming that k2 (z) is constant outside the slab and that the wave should look like the standard time-harmonic plane wave away from the region of inhomogeneity. Not only would this be a more difficult proposition numerically than invariant imbedding for almost all k2 (z), but the physical process involved at each point along z would tend to be obscured by the sophisticated techniques used to solve the problem. 11 Similarly, we could solve the quantum-mechanical one-dimensional square well II problem by matching solutions at the boundaries, but this method also tends to conceal the physical process until the final solution is examined. In contradistinction, the invariant imbedding formulation remains physically understandable at every point along the path to the solution and is capable of generating the solution in a straightforward manner for every applicable case. In this spirit, we will attempt to keep the mathematical formulation relatively simple -3- throughout. This much of the application of the invariant imbedding technique has previously been v1ell discussed by the authors of references 2 through 6. It would seem that, since we understand so well how to cal- culate the reflection and transmission coefficients for any given discrete distribution (and how to approximate accurately and easily in the continuous case), the problem is completely solved and there is no more to be done beyond, perhaps, finding some exact solutions previously unknown or solving interesting cases numerically. Indeed, the standard invariant imbedding formulation is capable of generating a numerical solution efficiently and accurately for any particular case of interest. However, the intent of this thesis is to extend the usefulness of the general invariant imbedding approach by deducing some general properties of the invariant imbedding solution and using these properties to derive new methods of exact and approximate solution of the problem based on the invariant imbedding solution. In particular, we will see how invariant imbedding leads to an interesting formulation of the problem of scattering from a periodic medium and how the Riccati equation may be reduced in many cases to a linear equation which greatly simplifies the analysis of any medium, in particular periodic and "almost periodic" media. Chapter II is concerned with reformulation of the invariant imbedding solution and the proof of general properties to be used later. It contains a substantial amount of non-original material. The contents of Section II.2, "The \:Jave Equation," are common knowledge to several -4- branches of physics and engineering. The mathematical ideas of Section II.3, The Invariant Imbedding Solution, 11 11 are available in many places, for instance Wave Pro pa gation in Turbulent Media by R. Adams and E. Denman (2). The notation used in this report is most similar to that found in Adams et al. (2). Sections II.5, native Invariant Imbedding Scheme, and II.8, 11 Electromagnetic Parameters, 11 11 11 An Alt~r- Variation of All contain as their basis background material (again, mainly using the notation of Adams (2)), but extra ideas have been added to the basic principles in the interest of later results. For instance, we have found it necessary to make a detailed comparison of the two schemes for invariant imbedding and to provide an explicit transformation which extends the results for normal incidence and constant permeability to the more general case. Finally, the similarity between electromagnetic and quantum mechanical scattering, as described in Section II.7, has been previously noted by Bellman et al. (3). Chapter III uses the general results of Chapter II to look at wave propagation in periodic media in a new way. cept cal led the 11 We introduce a con- Y-plot in the analysis of the propagation constant. 11 The complex variable, Y , is equal to the complex reflection coefficient times a phase adjustment to standardize two special points in the Y-plane, called the 11 fixed points.i 1 The progression of the reflection coefficient as cells are added to the periodic structure is represented by a trajectory on the Y-plot. For STOP BANDS, the trajectories con- verge on one fixed point and diverge from the other; for PASS BANDS, the trajectories orbit a fixed point. In both cases, the propagation -5- constant may be computed knowing only the fixed points, vi a very simple formulas. These results are applied to both lossless and lossy periodic media. Chapter IV examines the class of problems for which t he wavenumber (k) does not vary much as a function of position in the sl ab, which 'v-/e call "almost homogeneous media." The very complicated (at least, difficult to solve exactly) differential equation for the reflection coefficient derived in Chapter II reduces to an almost embarrassingly simple linear approximation which possesses an equally simple Green's function. In these almost homogeneous cases, an inversion of the reflection coefficient as a function of frequency to reconstruct the index of refraction as a function of position is quite simple to perform (with the help of a digital computer). This method may even be used with modest success for cases which are not "almost homogeneous . " We finally use the approximate theory to examine the STOP BANDS of periodic and almost periodic media. The conclusion is that there exists only one non- vanishing STOP BAND corresponding to each structure constant in the cosine expansion of the index of refraction for either periodic or almost periodic media. Where applicable, all electromagnetic quantities are in MKS units. -6- Chapter II The Basic Equations and Relations II.l Definitions Our first task 111ill be to reformulate the invariant imbedding equations for electromagnetic scattering from a medium with constant permeability,µ , zero conductivity, 0 , and dielectric constant, s , which is a function only of the propagation direction (and perhaps frequency.) In this case, we may think of the problem as that of a section of space between z = -a/2 and z = +a/2 where the medium may be characterized by an index of refraction n(z;w) = (I I. l) c ✓ s(z;w)µ For now, we will drop the explicit dependence on w . gion of the "slab," n(z) is constant. Outside the re- For reasons which will be clear later, we will let nN ; z < -a/2 n(z) = (II.2) no ; z > +a/2 With this definition of n(z), we may also consider the local wavenumber, k(z), given by k ( z) = ± ( w/ c) n ( z) (II.3) where w/c wi 11 often be called the "free space wavenumber" because n =l in free space. -7II.2 The Wave Equation When the permeability is constant, Maxwell's equations reduce to a simple one-dimensional wave equation for harmonic waves (we assume an exp(-iwt) time dependence) propagating in a direction per- pendicular to both the electric and magnetic fields (which are, of course, also perpendicular to each other). The wave equation may be written in the simple form (II .4) where ~ is the (complex) length of -the electric field vector. The solutions where k2 (z) is constant are right- and left-hand traveling waves of the form ~ =~ 0 exp(±ikz) (II.5) where the (+,-) correspond to rig~t (+) and left(-) traveling waves (along the z-axis). We will assume for our scattering problem that a wave of unit electric field is incident from the left, traveling to the right. We will fix the phase of the incident wave so that it has zero phase angle at the front interface (z = -a/2). The scattering problem is to find the complex numbers R and T such that ~ = lexp[+ikN(z+i/2)] + z < -a/2 (II .6) T exp[+ik 0 (z-a/2)] z > +a/2 is a solution of the wave equation in the homogeneous region which matches correctly with a solution in the inhomogeneous region (see -8- Figure l). At this point we are not concerned with the exact nature of the solution in the inhomogeneous region. The phase convention adopted for the l ransmitted wave is chosen so that the complex number T carries all of the phase information at problem in which E(z) z = a/2. In the trivial is constant everyHhere (no reflec t ion), T = exp(ik 0 a). We will also consider the reverse problem of finding the reflection and transmission numbers (R' and T') when a unit wave is incident from the right (traveling to the left). In this case, we want our solution to be of the form z.:. -a/2 IT' exp[-ikN(z+a/2)] (II.7) exp[-ik 0 (z-a/2)] + R'exp[+ik 0 (z-a/2)]; z ~ +a/2 . We note in passing that in the trivial case of total homogeneity, T' = exp(ik 0 a)= T Although we will not find it convenient to obtain the complete solution by solving the wave equation, there are several important results which may be derived easily by integrating the wave equation. Suppose that we have two solutions of the wave equation, ~l and ~2 . Using a standard trick, (II.8) which reduces to = = 0 (II.9) R exp[-ikN(z+f)J exp[i kr/z + ½)] v-- z=-a/2 -6- Figure l z=O k(z) -.:..../ z=+a/2 ~ -6ko T exp[ik Q (z -a)] 2 I I \.0 -10Integration from z = -a/2 to z= +a/2 is trivial. The result is (II.10) We may use this result to examine two pairs of solutions. For the first pair, consider the forward and backward scattering solutions (equations II.6 and II.7). (l+R )(ik )(T) Substitution into (II.10) gives (T)(ik )(-l+R 1 (T )(ikN)(l-R) - (l+R)(ikN)(-::T 1 1 0 ) 0 1 = ) (II.11) which reduces to koT = kNT1 (II.12) or Equation (II.12) holds whether or not k(z) has an imaginary component. The second pair of solutions is a valid pair only when k2 (z) is real and k2 is positive in the homogeneous region. If this restric- tion holds, then the complex conjugate of a solution will also be a solution. The restriction k 2 > o in the homogeneous region is simply a convenience; we could as easily assume the opposite in either the right or left homogeneous regions and proceed from that assumption, but it is more usual to have k2 (z) positive in the asymptotic region. If we substitute the solution (II.6) and its complex conjugate in equation (I I. 10) , the res u1 t is = (II.13) -11- Reducing, we obtain (II.14) Had v-Je used instead the "reverse" solution of (II .7), we v-t0uld have obtained (R')*R' + (k N/k) o (T')*T' = 1 (II .15) The results given by equations (II.12), (II. 14), and (II. 15) are physically reasonable. The energy flux in a transverse electromag- netic wave, by Poynting's theorem, is proportional to the electric field strength times the magnetic field strength. At the same time, the magnetic field of our "reduced wave" is proportional to the wave number times the electric field (for fixed w and µ ). Therefore, the flux is proportional to the reduced wave amplitude squared times the wavenumber. Equation (II. 12) implies that the same percentage of flux will get through the slab in either direction, while equations (II.14) and (II. 15) imply a conservation of flux. The combination of all three equations implies R*R = (R')*R' (II.16) which implies that the same percentage of flux will also be reflected in either direction. II.3 The Invariant Imbedding Solution Now that we have some simple relationships (equations II.12, II.14, II.15, and II.16), we may proceed to re-derive the basic invariant imbedding solution for R, T, R', and T'. As a check of the -12invariant imbedding procedure, we expect to be able to prove these simple relationships from the invariant imbedding viewpoint. Although the general derivation of the invariant imbedding scheme has previously been discussed in many sources (references 2-6), we will find the derivation a useful prelude to later derivations for periodic, 11 almost periodic, 11 and "almost homogeneous 11 media. Consider the problem of an electromagnetic wave incident on a region where there are N abrupt discontinuities (interfaces) in n(z), but n(z) is constant between interfaces (Figure 2). The simp- lest non-trivial problem to solve is the problem of one interface. In this case, a= 0 and the reflection and transmission coefficients are real numbers given (respectively)by the standard Fresnel formulas: (II.17) (II.18) The derivation of these formulas is elementary, but beyond the scope of this report. 8 Assume now that we know the solutions, R1 and T1 , for a situation with I interfaces. no n(z) v1here z.l = The index of refraction is given by . ' n. ; nr ., 1 z > z = +a/2 1 Zi+ l < Z < Zi ' z < zI = -a/2 1 < i <I (II. 19) is the position of the ;th interface . Suppose that we wish to add another interface, thereby increasing the slab length, a , and R exp[-ikN(z-zN)] exp[+ikN(z-zN)] -a/2~zN nN N-1 nN-l 6 ZN-1 nN-2 6N-2 ZN-2 Figure 2 0 Z3 n2 62 22 n l +a/2=z t, 1 1 no I w __, I T exp[+ik (z-z )J 0 1 -14changing the index of refraction in the left hand asymptotic region (z < -a/2), The new index of refraction function is no . n.l ; n(z) = ' z > z1= +a/2 nI i+l < z < z.l 2 I+l < z < ZI nI+l ; z < ZI+l= -a/2 , l < i < I 2 where the new interface is added at z =zI+l z axis to keep the slab centered about convenience), (II.20) and we have shifted the z = 0 (merely a matter of The new reflection and t:ransmission coefficients wi 11 be calculated using the method of multiple reflections between interfaces I and I+l (see Figure 3), The local reflection and transmission coefficients at the (I+l)st interface are given by the real numbers r = (nI+l- nI)/ (nI+l+ nI) (II.21) t = 2nI+l/(nI+l+ nI) (II. 22) The local reverse coefficients are given by r' = (nI- nI+l)/(nI+ nI+l) (II.23) t' = 2n 1/(n 1+ nI+l) (II.24) These quantities are related by the Kirchhoff equations: r1 = -r (I I. 25) t = l + r (I I. 26) t' = l - r (I I. 27) t oR or oR ot 1 1 I 1 toR ot' 1 r 2 I+l ----<E ' \, '' ~ ·- ~ ; I' / / ,/ .............. ~ / / / / ., I' ,I ~ ZI ~ ~ I ,_, .... ~ - ~ .... -- - ...... ·- .............. _ . . _ ,~_......,....,__,_ ...---------- Figure 3 (N.B.: Vertical displace ment is for illustration only). '' '' ----~ 0 = exp(ik 16) 6 'Q,,_ to R1or 1 0R 1or 1 0T1 U7 I -' t oR o r I oT 1 1 I t 6T I -16The distance between interfaces, 6 , is given by (II.28) A wave propagating between the interfaces will gain the phase factor (II.29) We will now consider the multiple reflections which contribute to the new reflection coefficient, RI+l, and the new transmission coefficient, TI+l· The unit wave is incident from the left upon the (I+l)St interface. The first term of the infinite series for RI+l is simply the local reflection coefficient, r . The remainder of the wave, t , continues to the 1th interface gaining phase factor o . At the 1th interface, R1 is reflected and T1 gets through to the right asymptotic region. The reflected part propagates back to the (I+l)St interface, again gaining phase factor o . At the (I+l)St interface, t 1 gets through to give the second term of the series for RI+l (to R1ot 1) and r' is reflected back to become the second of the (infinite) multiple reflections. series for This process is illustrated in Figure 3. The RI+l and TI+l are given by RI+l = r + (l+r)oR 1o(l-r) + (l+r)oR 1o(-r)oR 1o(l-r) + ... = r+(l-r2)a2R I iiRi(-r)i I I i=O (II. 30) TI+l = (l-r)oT 1 + (l-r)oR 1o(-r)oT 1 + = (l-r)oT 1 I o2iRi(-r)i i=O (I I. 31 ) -17These expressions may be summed exactly with the use of the formula {X) a i = (1-a)-l I i=O (I I. 32) giving 2 2 RI+l = ( r + o RI) / ( l + ro RI) (I I. 33) 2 TI+l = tcST I/ (l + ro RI) The recursion relationships (II .33 ) solve exactly any problem with discrete changes in n(z). For the case of a continuous change in n(z), we will derive differential equations for The initial value conditions are qualtities R(z) and T(z). R(a/2) = 0 and T(a/2) = l . The R(-a/2) and T(-a/2) are the numbers which give the final answers for the reflection and transmission coefficients. Since the incremental interfaces are added in the negative z direction, we must be careful of our signs. large) when n(z) If v.;e let the t,,,'s get very small (and N very is a differentiable function, we may approximate the quantities in the discrete derivation: r(z) = (n(z-dz) - n(z))/(n(z-dz) +n(z)) (I I. 34) n(z-dz) = n(z) n'(z)dz (I I. 35) R(z-dz) = R(z) - R'(z)dz (I I. 36) In this spirit, we will expand all quantities in orders of (dz) and keep only the constant and linear terms: r(z) = -n'(z)dz/2n(z) (II .37) o2 = l + 2ik(z)dz (I I. 38) -18- ( l + ro 2R)-l = l + n'(z) R(z) dz 2n{z) (I I. 39) If we make a simple substitution of these quantities in the standard recursion fo rmula (II . 33), and keep only terms of order dz , we obtain t(~)) dz+ R(z) + 2ik(z) R(z) dz R(z) - R (z)dz = 2 1 k I ( z ) R2 ( ) d + 2k{z) z z (II. 40) or, reducing, (II .41) Similarly, we get for T(z) T(z) - T1 (z) dz= T(z) + ik(z) T(z) dz -k'(z) k 1 (z) 2k(z) T(z) dz+ 2k(z) R(z) T(z) dz T'(z) = {}(1 -R(z)J ~(~)) - ik(z)} T(z) . (II.42) (II.43) Equation (II.43) is very convenient to integrate once R(z) din T = l r 1 _ R(z)J d( £n k) _ ik(z) dz ~ dz is known. (II.44) in(T(a/2)) - i n(T(-a/2)) = a/2 - i a/2 J k(z) dz + } J [l - R(z)J d( i ~/) dz -a/2 (II.45) -a/2 Our boundary condition on T is T(a/2) = l . a/2 kavg = 1/a J k(z) dz -a/2 If we define (I I. 46) -19then a/2 f T• = T(-a/2) = explik avg a - l2 -a/2 II.4 Some General Properties of the Transmission Coefficient Equation (II.47) reveals some important general properties of T. The first is that the major phase contribution when R is sma 11 comes 'from the exp(ik avg a) term. When we actually solve a problem) we should expect the phase of T to be near ( kavg a). The second is an interesting expression for the minimum magnitude that T could possibly have for a given k(z) (or n(z)). The magnitude of T is simply the exponential of the real part of the integral in equation (II.47) (when k is rea 1 ) : a/2 ITI = ex+} J Re[l - R(z)J d[~zk(z)l dz i (I I. 48) -a/2 Since the magnitude of the reflection is constrained to be less than or equal to one, the real part of (l -R) must lie between two and zero. Therefore, when k(z) is an increasing function, the integrand in (II.48) is positive, and when grand is negative. k(z) is a decreasing function) the inte- A conservative estimate of the minimum possible value of the magnitude of T results when v,ie estimate the maximum possible value of the integral. Without solving the problem in advance, we do not know what the function R(z) = -1 where k(z) R(z) looks li ke. is increasing and R(z) = +l However, if we let where k(z) is decreasing, then the integral obtains the maximum value it could -20- possibly have for any R(z) !Tl I or, integrating (over the regions where !Tl where the k'(z} > 0), (k . /k . /k max ) ·,, (k min . /k max ) (II •50) min l max l ) •(kmin 2 2 n ' n > k . mm and kmax are the values of k(z) and end (respectively) of portions of k(z) creasing function. where at the beginning k(z) is an in- In practice, this estimate is usually over- conservative, but it is completely reliable. that n(z) (II. 49) exp{-}· 2 d[ t n k(z)J} k'(z) >O > For instance, suppose is given by ; z < -a/2 n(z) = l +x(l + cos 2TT z/a) -a/2 ~ z ~ a/2 (II.51) ; z > a/2 l It is relatively easy to program a computer or programmable calculator to solve the Riccati equations (II.41 and II.43) for various values of k0 = (w/c) . We will examine some numerical solutions in detail later (Section III.9), but for the purposes of this section, we are interested in the minimum magnitude of T which we see when we solve numerically for as many ko as necessary. results for three cases, X = 1 ' X = . 05, X !Tl at minimum 1 .05 . 01 0.65 0.997 0.99987 The following table 1is ts and X = • 01 (km;/kma) 0.33 0. 909 0.98039 -21If the variation in k(z) is small, then (II.52) where ~ is a small positive number. When this is the case, then we also have a very useful estimate of the maximum maqnitude of R2 over the entire interval, not just at the end of the solution. wish to get estimates of the magnitudes of R(z) and Suppose we T(z) at some particular point, z , in the interval of integration of the Riccati equations. and If we recognize that by integrating the equations for R T from +a/2 to z , we have solved the problem of a distribu- tion of k(y) identical to the original from +a/2 to z and con- stant after that, then we have immediately from (II.14), 2 IR2 (z) I + (k 0 /k(z)) IT (z) I = 1 Since the estimates for the IT( z) I 11 new prob 1em wi 11 be 11 (I I. 54) -> l - 0 (t,;) (k 0 /k(z)) = (II.55) 1 - O( ~) due to the fact that we have the same constant k(y) for y (II .53) less than k(y) from +a/2 to z and a z , we easily obtain (II.56) This result is the basis of the useful approximation described in Chapter IV, in which we wish to be able to guarantee before we start that R2(z) will be 11 small 11 throughout the interval. -22II.5 An Alternative Invariant Imbedding Scheme The recursion relationships and differential equations (II.32, II.33, II.41, II.43) provide a convenient method for solving the electromagnetic scattering problem in one dimension. If we we re to program a computer to solve the problem, the most convenient way would be to use the recursion relations (II.32 and II.33) directly, le t ting the size of the intervals become as small as practicable. We could, there- fore, consider the task of solving these problems reduced to the almost trivial matter (in these days of computer abundance) of generating numerical solutions for interesting cases. However, we will obtain some very useful confirmations of the soundness of the invariant imbedding scheme (plus a new result which makes Chapter III possible) by considering an alternative invariant imbedding scheme in which we add slices from the right of the slab (rather than from the left as before). For convenience, we will call the "adding from the left method," method 11 A11 and the "adding from the right method," method 11 B. 11 In method 11 B11 we start at the Nth interface and work back to the 1st interface. We may use the simple Fresnel equations to generate RN, TN, Rt'.1' T'N . RN = (nN- nN-1)/(nN+ nN-1) (II .57) TN = 1 + RN (II .58) R'N = -RN (II .59) T(1 = l - RN (II.60) Assume now that we have the solution for interfaces N, N-1, • · · , I+l ,I. -23We wish to calculate R~_ 1 , T1_1 , R1_1 , and T1_1 . R1 _ and T _ may 1 1 1 be found using the (suitably modified) recursion relation s of method A (equations II.32 and II.33). This time, the multiple reflections are occurring at the ri ght- hand side of the slab (see Fi gure 4) As with method A, we will define local reflection and transmission coefficients and phase factor (II.61) t = l + r (I I. 62) (II. 63) Without further ado, the infinite series for R1_1 and T1_1 are + ... (I I. 64) (II.65) We may again sum these series with the help of equation (II.31) to give R1 + r o2 (T 1T1 - R1R1) 2 l - r oR (II.66) 1 (II .67) TI-1 = t o TI/(1 - r iRi) We will not have need of the differential equation for B. The differential equation for T(z) R(z) in method may be found in analogy with previous arguments : T1_1 = T(z+dz) = T(z) + (dT/dz)dz (II.68) r (II.69) = -(k'(z)/2k( z))dz T1oro R1oro T1 .-.. -- - - - - -- - - .... '\ I ~ ' ' ' ' ' '' '~ ~ - / .,,,. / / ~ -- - -- - ~ / / Figure 4 (N.B.: Vertical displacement is for illustration only.) - - - ... - - - -- - - - - - - - - - - - ... ~ ~ T1oroT 1~ RI 1 o= ex p(ik 1_1t) f., ;' / // - / ,,,, - ~r · t , TioroRi ot * r1ot -25t =l + r (II.70) T(z) + (dT/dz)dz = T(z) + ik(z) T(z)dz - (k'(z)/2k(z)) • R'(z) T(z)dz - (k'(z)/2k(z)) T(z)dz (II.71) (I I. 72) The boundary condition is T(-a/2) = l (as opposed to method A). We easily integrate (II.72) to obtain a/2 J [l +. R'(z)J d[in dzk{z)J dz} T = T(+a/2) = exp{ik avg a _l 2 • -a/2 Once R'(z) is known, T may be calculated by integration. Suppose we vJere to calculate TI by method tion (II.73) B . The trans forma- z' = -z gives the answer a/2 1 T = exp{ik a_ l J [l+R 11 (z 1 ) ] d[in k(z')] dz'} avg 2 dz' -a/2 R (z) is functionally the same as 11 K(-z) by the same argument. R(-z) from method A, and The integral expression for T1 a/2 1 T1 = exp{ik a+ [l + R(z)J d[in k(z)J dz} avg 2 dz -a/2 f (II.74) k(z) = becomes (II.75) If we compare this expression with (II.47), we easily calculate . (II.76) This result agrees with (II.12), which was derived directly from the reduced wave equation. -26- We should be careful to note the differences between the variables in the reduced \vave equation and the R and T coefficients in methods A and B. This difference is illustrated in Figure 5. If we were to solve the wave equation for ~(z), we would have information about the electric (and magnetic) field at any point in the slab. do not get the same information from We R(z) and T(z) in the invariant imbedding method, since it is the problem itself we are changing as a function of z . If we need specific information about the electric field at any point in the slab, one method to use would be to divide the slab at z , solve each half-slab separately with invariant imbedding, then finally use the method of multiple reflections at z to give both the final reflection and transmission coefficients and the value of the electric field in the multiple reflection region. not find it necessary to do this. We will However, it will be nice to have the mathematical formulation if we should need it later. Assume that we have used invariant imbedding to get the reflection and transmission coefficients. scripts A and 11 11 11 In particular, if we use sub- 8 to designate quantities found \vith methods A and B 11 8 respectively, we need T8(z), R (z), and RA(z). RA(z) may be found by solving (II.41), and R (z) may be found by making the transformation 8 21 = -z, solving (II.41), then making the transformation back again. Once R8(z) is known, T8 (z) is found from (II.72) or (II.73). Once these quantities are known, the series for the right- and left-going multiply-reflected waves are given (almost by inspection) by -27- Invariant Imb eddi ng - Method A I -a/2 +a/2 z z-axis Invariant Imbedding - Method B (> direction of integration l!.R'(z) -a/2 z +a/2 z-axis +a/2 z-axis Solution of the Wave Equation Two Independent Solutions -a/2 Figure 5 -28- ¢ = T8 (z)RA( z) + T8 (z)RA(z)R 8(z)RA(z) + T8(z)RA(z) R (z)RA(z)R (z)RA(z) + 8 8 (I I. 78) Once again, we invoke (II.31) to sum these series to get = T8 (z)/(l - RA(z)R 8(z)) (I I. 79) ¢ - = TB ( z) RA ( z) / ( l - RA ( z) RB ( z) ) (I I. 80) ¢+ The directions of the electric and magnetic field vectors may be found by elementary considerations (in the simple normal incidence case found here, the directions of the vectors remain constant). Their lengths (complex) may be found by II.6 E = ¢+ + ¢_ (I I.81) H = n/µ(¢+ - ¢_) (II.82) Five Basic Relationships We are now in a good position to derive the basic relationships among the parameters R, T, R 1 , and T 1 • We will show inductively in Appendix I that the following five relationships hold. As a matter of notatig.n, we let R = r exp(i cpR) (I I.83) T = t exp(i cpT) (II.84) R' = r exp(i ¢R ,) (I I.85) T' = t I exp ( i ¢T, ) (I I. 86) 1 ,-29The five relationships are l. r' = r (I I. 87) 2. t' = (k/kN )t (I I. 88) 3. a) 2 2 r + ( ko/k N) t = l (II.89) b) r'2 + (k /k )t 12 = l N o (II.90) 4. ¢T. = ¢r (I I. 91) 5. ¢R + ¢R1 = ±n + ¢T + ¢T, (I I. 92) The wave equati on has already given us rela t ions l through 4, and we -· have also shown 2 and 4 with the arguments of Section II.5 . 5 : may also be shown directly from Relation the wave equation. Al- though we prove relations l through 5 by induction for the disc rete case, there is no dependence on the number of slices or the size of the slices, so there is no problem passing to the limit which gives us the continuous case. In th.e light of the five basic relationships, let us reconsider the method B recursion for sion for R . Recall that the ba sic method B recur- R (II.66) is somewhat inconvenient because it requires simul- taneous calculation of T, T', and (II.32). R', unlike the method A recursion We may, however, si mpli.fy the method B recursion formula for R with the use of (II.93) R'I which are trivial results of the five relationships. (I I. 94) The method B -30- solution for R becomes (II.95) \vhich eliminates the R' and T' dependences (the "r in the equation 11 is the "local reflection coefficient," not the magnitude of R1). We still have an explicit dependence on T which we cannot eliminate, but we may at least eliminate the R' and T' dependences in the method B recursion for T: (I I. 96) (the "t" in the equation is the local transmission coefficient and the "r" is the local reflection coefficient). Although method Bis useful in proving the five relationships, it is not as convenient in practice as method A, so we will not pursue it further. II.7 Relation to Quantum Mechanical Scattering in One Dimension The non-relativistic one-dimensional wave equation in quantum mechanics is (I I. 97). where k 2 = 2m(E - V)/fi 2 (I I. 98) will put the equation in the same form as the retluced wave equation (II.4). when This similarity leads us to try the invariant imbedding scheme V is a function of z . -31- Suppose we have a situation where for z > 0 . This is a simple simplicity that E > V 0 = )2mE/n = 0 step potential barrier. 1jJ Assume for is as follows. 2 Let (II.99) a =J2m(E-V 0 )/n <JJ(z) z < 0 and V= V 11 Our solution for 0 k 11 V= 0 for 2 (II. l 00) exp(ik 0 z) + R exp(-ik 0 z) z < 0 (II.101) T exp ( i az) z > 0 R and T are connected by the continuity of 1jJ and 1jJ 1 at the boundary (z= 0): l + R = (II. l 02) T (II. l 03) The solution of these equations gives us the familiar result: R = (k 0 - a)/(k 0 + a) (I I. l 04) T = 2k /(k + a) (I I. 105) 0 Although 0 k(z) is calculated in hvo different ways for electromagnetic and quantum mechanical waves, the same formulas for the invariant imbedding scheme apply in either case once k(z) is known. All of the pre- viously derived theorems apply (with the restriction that k(z) must be real). As a simple test of this idea, we will solve the one-dimensional "square well II problem. We will let the potential barrier have height V0 and width extending from z = 0 to z = a . The solution will be of the form a , -32exp(ik 0 z) + R exp(-ik z) 0 ~ ; z < 0 (II. 106) = T exp[ik 0 (z-a)J z > a There are only two interfaces to consider, so the problem is very easy . Applying (II.32) and (II.33), R1 = (a - k0 )/( a + k ) 0 (II.107) T1 = 2a/(a + k0 ) (I I. l 08) k -a a-k o o exp (2'rna )' r+a + a+1< o a a o R= R = 2 k -a 2 1 - Cr+a) 0 T =T 2 = Cl, exp(2iaa)' 2k 0 ( ~k )(k + ) exp(i aa) a o oa k -a 2 1 - (~) o a (I I. l 09) (II.110) exp(2i aa) These expressions reduce to (k~-a 2 )[1-exp(2iaa)J R=----------- 4k 0 a exp(iaa) T = ----------- (k +a) 2 - (k -a) 2 exp{2i aa) 0 (I I. 111) (II.112) 0 They agree with the results obtained from the usual method of solution 14 , if the difference in phase convention for the outgoing wave is taken into account. If it should be the case that E < V0 , then a will be imaginary, but the solutions (II.111) and (II.112) will still be -33- correct. Moreover, relations l through 4 (II.87 - II.91) will hold no matter how large V0 II.8 Variation of All Electromagnetic Parameters In the general case, the wavenumber (complex) of a transverse electromagnetic wave is given by k = w/µs(l + i0/ws) (II.113) The positive square root is assumed, which means that exp(ikz) will "damp out as II z increases. vJe now assume that we have a slab of thickness 11 a 11 (betv1een z = -a/2 and z =+a/2) in which s, µ , and 0 are functions of z (and perhaps w), but constant outside the slab. A plane wave with time dependence exp(-iwt) is incident on the left face (z = -a/2) at incident angle eN . Part is reflected at exit angle eN , part is absorbed in the slab, and part exits from the right face (z = +a/2) at exit angle 00 . Let us see how the previous arguments are altered, without going through the complete derivation in detail again . For the moment, set 0=0 (k 2 (z) real). There \vill be two polarizations to consider; the "l" case in which the electric field is perpendicular to the plane of incidence, and the "II" case in which the electric field is parallel to the plane of incidence. As SN approaches zero, these two cases become degenerate . The local reflection coefficients are given by 8 -34ki(cos ei)'µ 1 - k1_1(cos e1_,)! i.t 1_1 (II. ll 4) rl = k/cos e )tµ + k _,(cos e _,:Vµ _ 1 1 1 1 11 (II.ll5) The Kirchhoff relations (II.25 - II.27) still apply . There is a sign convention in common usage 8 which reverses the sign of r ll , so that at zero incidence angle,the local reflection coefficients are equal in magnitude but have opposite signs . We will not use this convention. There remains the problem of determining the phase difference between adjacent interfaces. As a first guess, we might be tempted to say that the ray going between interfaces I and I-1 travels a distance 6 sec e1_1 (true), so therefore the phase factor gained across the interexp(ik 1_1 [; sec e1_1) (false). mediate region is this problem. Fi gure 6 illustrates We must remember to refer the phases to the z-axis (the line defined by x = 0, y = 0) and that the waves are no longer traveling along the z-axis, but at some angle to it. The phase of the wave at the z-axis on the front or bac k of the slab is the phase of the wavefront which intersects the surface of the slab at the z-axis. reflected rays 11 1 and 2 of Figure 6. 11 11 11 Consider the These two rays are originally in phase at the first contact with the slab at point P. Ray l is immediately reflected and travels a distance (II.ll6) dl -- in medium I to the wavefront which intersects point Q , the exit point of ray 2. We will assume that point P lies on the z-axis, so we will have to refer phases to that point. Ray 2 travels a distance eI- ·1 -· - Figure 6 -36- d2 = 26 sec e1_1 (II.117) in medium I-1 to the wavefront originally mentioned. by ray l, referred back to point P, is zero. The phase gained The phase gained by ray 2, referred back to point P again, is (II.118) By virtue of Sne 11 s law, this reduces to 1 p = 2kI-l 6(sec eI-l - sin eI-l tan e1_1 )=2k 1_1 6cose 1_1 . (II.119) Alternatively, we could think of ray (2) as having started at such a point higher up on the slab so as to exit at point P . In any case, the phase factor gained is given by (II.12O) We may ask whether the phase factor gained by the transmitted wave is simply o , as equation (II. 120) would imply. Similar arguments will verify this conjecture. The invariant imbedding analysis remains remarkably unchanged. l1/here before we had the expression 11 1< 111 in differences (or derivatives), we replace it by 11 k1cos eI/µ 1 in the perpendicular polarization ("11') 11 case or by "krfµ 1cos 0111 in the parallel polarization ( 11 11 Hhere we had the expression replace it by 11 kI6I 11 (or k(z)dz 11 11 ) 11 ) case. in phase factors, we k 61cos 0111 in either case. This replacement leads to 1 two interesting transformations for the two polarizations. 11 -37For l let a/2 f aT = µ(z)/µ 0 dz -a/2 z ZT = -aT/2 + µ(y)/µo dy -a/2 (II. 121) f kT = k(z)[cos e(z)]µ/µ(z) The Riccati equations for (II.122) (0(z) from Snell's Law) (II.123) R and T are given in the general case by ~z = (1- R2(z)) d[k(z) cos e(z)/µ(z)J - 2ik(z) cos e(z) aL 2k(z) cos e(z)/µ(zY dz (II.124) ~z = (1 - R(z)) d[k(z) cos e(z)/µ(z)J - ik(z) cos e(z) aL 2k(z) cos e(z)/µ(z) dz Since dz dzT -l µo Fr= (rz) =µ{TT (I I. 125) the transformed Riccati equations are (II.126) (II.127) This somewhat surprising result implies that any problem with perpendicularly polarized waves may be transformed to a problem for which the incidence is normal and µ(z) is constant. -38For II let µ(z) cos 2 e(z)/µ 0 dz (II.128) z f zT = -aT/2 + µ(y) [cos2 e(y)]/µ0 dy -a/2 (II.129) (II.130) The Riccati equations are dR - (l - R2(z)) d[k(z)/cos e(z) 1.l (z)J - 2ik(z)cos e(z) dz 2k(z)/cos e(z) µ(z) dz dT d[k(z)/cos e(z) µ(z)J . ( ) ( ) dz= (l - R(z)) 2k(z)/cos e(z) µ(z) dz - ,k z cos e z (II. 131) • ( II.132 ) It is a trivial matter to verify that the transformation defined by equations (II.128) through (II.130) will reduce (II.131) and (II.132) to the standard form of (II. 126) and (II. 127). However, the parallel fonnulation encounters serious trouble when the factor, cos e , goes to zero or becomes imaginary due to Snell's law ("total internal reflection"). The transformation to zT becomes non-invertible and we are left with the full Riccati equations (II.131) and (II.132). The case of "total internal reflection" (cos2 e :;,_ 0) is interesting because it is an example of how relation 5 (II.92) breaks down. We may always make a piecewise transformation similar to (II.128) th r ough (II.130). If we do that, then k~ will be real even if cos 2e is negative. The wave equation guarantees that relations 1 through 4 -39- (II.87 through II.91) will continue to hold (in a slightly altered form which we will see later) as long as the homogeneous regions have cos 2 e > 0 . However, relationship 5 relies on the unit magnitude of the phase factor, 6 , which is not the case when Regardless of any res t rictions on kT is not real. kT , the altered versio n of (II.76) will always be true. We may introduce 0 IO in the previous formulations with little change in the results, except we must be prepared for any quantity to become complex as a result, thereby invalidating all relationships except the connection between transmission coefficients as derived from the integrated Riccati e~uations of methods A and B: l (II.133) II (I I. 134) I I. 9 Summary The condition that n0sin e0 < n(z) applies to many cases of interest for which the plane wave originates in a region where n(z) is approximately unity , propagates through a re9ion where n(z) is greater than or equal to unity, and exits into a region where n(z) is again close to unity. Unless otherwise specified, we will assume that this is the case, because we may then always transform to the normal incidence, constant permeability case. If the conductivity is zero, we have the following relationships for perpendicular and parallel polarization: -40- 1 r l. r' 2. t' = (µNk cos e /µ kNcos eN) t (II. 136) 3. r 2 + (µNk 0 cos e / µ k1•1cos eN) t2 = l (I I. 137) 4. ¢T = ¢T' (I I. 138) 5. ¢R + ¢R' = ±TT+ ¢T + ¢T 1 (II.139) l. r' = r (I I. 140) 2. t' = (µNk cos eN /µ kNcos 00) t 3. r2+ (µNk cos eN/µ kNcos 00) 4. ¢T = ¢T' (II.143) 5. ¢R + ¢R' = ±TT + </>T + ¢T I (I I. 144) = (I I. 135) 0 0 0 0 0 I • II : 0 0 (II.141) 0 0 t2 = l (I I. 142) It is appropriate to ask the question whether there could be other relations connecting R, R', T, and T' when k(z) (II.76) or its alternatives for oblique incidence. is complex, besides A simple thought experiment will show that there can be no other general relations when the slab is arbitrarily conductive. Suppose we have a slab which is re- flective at the ends but highly absorptive in the middle. tive middle effectively "decouples" The attenua- R from R' so that we could change the parameters on one side 1t✓ ithout changing the reflection from the other side. In contrast, if there is no absorption, it is impossible to change the parameters anywhere without affecting all quantities. Of -41- course, if the slab is only slightly absorptive at frequencies of interest, then all the relations will be approximately true; the amount by which they are not true is a measure of the lossiness. We will use this idea in our discussion of periodic media in the next chapter. In the following chapters, we will generally restrict our attention to cases vlith zero conductivity and no regions of total internal 11 reflection." Since it is always possible, with these restrictions, to make the transformation to the constant permeability, normal incidence case, we will assume that we have done so. As a final comment on the results of this chapter, we remark that the invariant imbedding analysis will remain correct even if we make all quantities complex, so that we should have no difficulty using invariant imbedding in more sophisticated models of interaction with matter, such as the complex permeability of a ferromagnetic material, 9 as used for example by Sommerfeld in Optics. -42Chapter I I I Periodic Media III.l The Periodic Recursion Formula ~4e will consider the class of problems for which the index of refraction is a periodic function 1-'dthin the slab (defined between z = -aM/2 and z = +aM/2) , with M peri ads. has length a ( see Figure 7). Each "ce 11 " of the slab We might worry about whether the index of refraction function fits continuously at the junctions of the cells with each other and at the b10 juncti ans with the "background, or 11 homogeneous index of refraction outside the slab. The junctions with the background will be taken into account when the problem of just one cell is solved, whether or not the cell fits continuously with the background. The junctions between the cells will be handled by the method of this section. A necessary preliminary step will be to show that the index of refraction of a thin zone intermediate to two semi-infinite zones will not affect the reflection and transmission in the limit as the size of the intermediate zone goes to zero. Let the index of refraction be given by n(z) = na n. ; z < -t:,/2 nb ; z > +t:,/2 l -6/2 < z ,:s_ +6/2 The standard invariant imbedding results are (III.l) -43- R2 = R = )/(n.+n ) exp(2ik.6) (nb-ni)/(nb+ni) + (n.-n l a l a . l (n. -n ) (nb-ni) l + exp(2iki6) (nb+ni) (n~+na) l a 2n.l 2nb ) exp(ik. 6) (nb+ni) (n l.+n a l T2 = T = (nb-ni) (n l. -n a ) l + ) exp(2iki6) (nb+ni) (n.+n l a As (III.2) (III.3) 6 ➔ 0, these solutions become R= (nb-n.)(n.+n ) + (n.-n )(nb+n.) l la l a l = 2nbni + 2nl; (III.4) 4nbni 2nb· n.·1 + 2n an.l (III.5) T = ---~-~ = As we readily see, Rand Tare independent of n. l intermediate zone goes to zero. as the size of the This result enables us to link cells together without worrying what is the intermediate index of refraction in the zero distance bet\-'/een cells. \>Je will find it convenient to as- sume that the index of refraction is the same between the cells as to the right of the first cell. When we solve the problem for L cells, we will assume that the index of refraction to the left is the same as the index to the right. If this is not so, one final application of the recursion relations (II.31) and (II.33) will give the correct final answer. In other words, we will assume that the background is identical on both sides, with the understanding that one simple final calculation will correct the result if this assumption is false. In case we wish n(z) -44 - I < one ce l { I of length a ~----------------------------~-z PERIOD! C SLAB n(z) REFLECTION AND TRANSM ISSION FOR ONE CELL '------------------------------~-z Figure 7 -45to have a non-zero distance between cells (constant to maintain periodicity), we should incorporate the extra distance into the defined cell. We could, of course, simply solve the problem with our standard recursion formulas (II.32) and (II.33) and/or the Ricatti equations (II.41) and (II.43), but that would be inefficient and would not lend itself to a general solution. Suppose, instead, that we have used invariant imbedding (or any other method) to calculate R1 , T1 , R1, and T1 for one cell. late The invariant imbedding method may be used to calcu- RM and TM for M cells. Suppose that we know the solutions, RL and TL, for L cells, and we wish to add one more cell (see Figure 8). The method of multiple reflections in the zero-distance region between the new cell and the L original cells is straightforward (and analogous to the previous examples of multiple reflection). The recursion formulas obtained are Rl + RL(T, Tl- Rl Rl) 1 - Rl RL (III.6) (III.7) As a matter of notation, let r = !R1 I (III.8) cpR = arg(R 1 ) (III.9) t = ITl I (III.10) ¢T = arg(T 1) (III.11) R1 ~~ ~~ l ., Added Tl RL l ~ Figure 8 ~ ~I Region of Mu ltiple Reflections I I One Ce1l l• r (Zero Distance) t L Cells 1• ! E t TL I ~ -47Using the reciprocity relationships (II.87- II.92), we may rewrite the periodic recursion formulas: r + Rlexp(2icpT- icpR) 1 + rRLexp(2i cpT- i¢R) (III.12) (III .13) III.2 Fixed Points of the Recursion In general, we should not expect the sequence verge. IRLI to con- If the sequence is to converge, it will converge to a fixed point of the transformation, RF , where RF = RF obeys the equation r + RF exp(2icpT- icpR) exp(i¢R) [ - - - - - - ~ - ] . l + rRFexp(2i¢T- i¢R) (III.14) This is simply a quadratic equation whose solutions are [exp(2icpT)-l] ± {[exp(2i¢T)-1J2+ 4r2exp(2icpT)} 112 RF = - - - - - ~ - - - - - - - - ~ ~ - . (III. 15) 2r exp(2i¢T- i¢R) We will leave it for later discussion to show that the sequence converges to RF only if IRF I = 1 . RL Let us see what happens if IRFI = 1 : (III .16) Substitution of (III.16) into (III.14) yields -48- (III.17) Let (III.18) With a small amount of manipulation, assuming ¢R is real, F (III.19) So far, ¢R has not been determined. It will be equal to whatever it F must equal to make (III.19) true. range of angles which ~ Figure 9 illustrates the maximum may obtain given r. It is obvious from the diagram and elementary geometry that (III.20) l~I:;.. arcsin(r) If we convert ¢r to the region ±TI , then we have a very simple relationship which must hold if and only if jRFI = l : * (III.21) l<t>rl :5.. arcsin(r) It is more convenient to rewrite (III.21) so that it is not necessary to convert ¢r back to the region ±TI : (I I I. 22) The alternative, which holds if and only if IRFI I 1*, is (III.23) Since the condition that the sequence {IRL1} converges to unity means *See also next section. -49 - . .... .. . . I rna gi r. a ry Axis . . . .. a . .. a l : ~e ( ±a) i s the maximum range of the phase of (l + re' ) , vJ here is a rb it ra ry. a = a resin ( r). Figure 9 Rea l Axi s .. e -50- that no flux will get through in the limit as we add many cells, we will call condition o:n.22) the 11 STOP BAND condition. 11 The obvious corollary is to call condition(rn.23) the PASS BAND condition. 11 11 We have not given a rigorous proof yet, but conditions (III.22) and (III.23) are enough to make a PASS BAND/STOP BAND analysis of any (lossless) periodic medium. One need only find r and 1 as a function of frequency and then use (rir.22) and(III.23) to determine whether any particular frequency is in ¢ a PASS BAND or a STOP BAND . We will do this later for several media. III.3 Convergence to a Fixed Point; Method One .. Suppose that RL is "very close 11 to RF . ~Je may write (III.24) where lt;I << l . ~ terms of Using (III.12) to calculate RL+l and expanding in , we obtain If .,.,e write (III. 26) then It; I I (III.27) The 0( ~2 ) term is very small when lt;I is small, but it is not zero. define We -51- 1 - rRFexp (-i </JR) =--------- (I I I. 28) 1 + rRFexp(2i¢T- i¢R) If we examine P simultaneously with RF , we obtain the following re- sults: A. IRF I = l In this case the STOP BAND criterion (III.22) holds. two solutions for RF . For one of them, !Pl > l . IP! < l ; this will be the one to which {RL} There are For the other, converges . .. Here the PASS BAND criterion (III.23) holds. For both solutions of RF , IP! = l . v!e cannot prove it with this simple analysis, but the 0(~ 2 ) term will prevent convergence, and instead leads to orbits around RF . The solution in which we are interested lies inside the unity circle (jRFI = l). The other solution is not interesting because it lies outside the unity circle and is therefore not possible for usual media. III.4 Convergence to a Fixed Point; Method Two ~~e may think of equation (III. 12) as a transformation which maps points in the complex plane to points in the complex plane. formation is parameterized by the numbers r , </JR , and ¢T This transPart of the transformation is a simple rotation which we may factor out by noting r + [Rlexp(-i¢R)J exp(2i¢T) l + r[Rlexp(-i¢R)J exp(2i¢T) (III.29) -52(I I I. 30) and write r + \ exp(2i ¢T) =------- (II I. 31 ) We are only interested in those points which lie inside the unity circle and only those transformations for which O .s_ r < l . The inverse trans- formation is r-X =XL= exp(-2i¢ )[ L+l J T 1-r\+l It is very easy to prove _that both ¢ (I I I. 32) and ¢ -1 will leave points inside the unity circle. Let Z be any complex number with magni- tude less than one, and let ijJ = arg(Z) . Then r2 + 1z12 < l + r21zl2 (III .33) r2 + 2r IZIcos ijJ + IZj 2 < l + 2r IZIcos 1jJ + r2 IZI2 (I I I. 34) (III. 35) t~e conclude that for any ¢T and r < l , the transformation r + X exp(2i¢T) (I I I. 36) l + rX exp(2i¢T) is one-to-one and onto inside the unity circle, (This topic is also mentioned by Bellman et al. (3).) The fixed points of ¢ reveal where the transformation may converge after many applications. If we call the fixed point of ¢,XF, then XF obeys -53- (III.37) This is a quadratic equation, whose solutions are . . 2 sin ¢ sin ¢r 1/2 XF exp ( i ¢r) = i [ r TJ ~ [1 2" J r (III. 38) This is a slightly inconvenient result because it depends directly on ¢r , so we will make one last chan ge of variable. Let Evidently, 0 is one-to-one and onto in the same sense as we wish to calculate RL given ¢ . If YL , it is easy: (III .41) The fixed points of 0 Tl = are calculated using sin ¢r r ' (III.42) giving YF = in ± [l _ n2Jl/2 (III.43) The fixed points lie either on the imaginary axis (Jnl > l: PASS BAND) or on the unity circle (lnl < 1: STOP BAND). Let us consider now the transformation (which we tentatively assume is a member of the class of transformations {e} ) defined by -54(I I I. 45) We may calculate YL+ 2 two ways and compare the answers: l + ~ Ylexp(i¢T) (I I I. 46) = ..-------- ~ exp(-i¢T) + YL 1 r2exp(2i¢T) + exp(4i¢T) + r exp(i¢T) +rexp(3i¢T) YL .(III. 47) 2 . l + r exp(2i¢T) ---------- + y r exp(i¢.T) + r exp(3i¢T) L Examination of (III.46) and (III.47) reveals the following connections between r, ¢T' r', and ¢f r' = r l + exp(2i¢T) (I I I.48) l + r2exp(2i¢T) sin <1>+ sin ¢r = r r n (I I I. 49) We would like to use equations (III.48) and (III.49) to find r and ¢T given r' = and ¢r so that we may decompose any transformation into two "smaller" transformations. Of course, we may always invert (III.48) and (III.49) numerically to obtain the desired results. It is not important that we actually obtain a closed-form inversion of (III.48) and (III.49), only that we recognize that we may find the solutions if necessary. When mate inversion is very simple: r' and ¢r are small, then the approxi- -55r = -l r I 2 (I I I. 50) (III.51) The point of this argument is that v,e may always decompose any 0,1, '+'T• r into 2M smaller transformations; in the limit, we may consider 0,1, to be infinitely-many infinitesimal transformations which preserve '+'T' r the fixed points (or, equ·ivalently, n) . An infinitesimal transformation w"ill have the parameters E;:¢T, where E;: is a small real number. If r t;;r and itself is small, then t;; v,rill be (1/N), where N is the number of "smaller" transformations \vhich are equivalent to the transformation 11hich is parameterized by and r The infinitesimal transformation may be written cpT. YL +[; = n = r + \exp(i[;¢T) [ l + t;;r \ exp ( i [;cpT) J exp(i[;cpT) (I I I. 52) Since sin(t;:¢T)/t;;r ~ ¢r/r (III.53) we easily expand (III.52) in powers of t;; , keeping up to linear terms, to obtain (I I I. 54) The form of (III.54) very closely resembles the Riccati equation for R(z) (II.41), which is not surprising. dependence on t;; and r It is interesting that explicit occurs in the expression for (YL+E;: - YL) only -56- as the multiplier, (;r), which implies that the path of evolution of YL+; is independent of ; and r (for fixed evolution is linearly dependent on (;r). coordinate with n ), but the "speed 11 of If we associate a quasi-time ; , so that adding a cell is equivalent to advancing quasi-time by some increment, then we may define a velocity function (III. 55) The value of r may vary, but the shape of the velocity field depends only on n. When to one cell added. r is small, one "unit" of quasi-time corresponds When r is close to unity, the correspondence is not so easy, but may be found using (III.48); in any event, the relation between quasi-time and adding one cell is simply a multiplicative constant. Straightforward substitution of (III.43) into (III.55) yields (I I I. 56) We now have the necessary background for a detailed analysis of the paths of YL (and therefore RL) for the PASS BAND and STOP BAND cases. A. In I < l (PASS BAND) For simplicity, we will assume that n is positive. The fixed point v1hich is inside the unity circle is given by (III.43) as (III.57) Let (III.58) The velocity field is -57(I I I. 59) Z has a small magnitude (Y is close to VF), the lin ea r term of When (III.59) dominates and the orbit path circulates about VF (Z = 0). The quadratic term causes the orbit to depart somewhat from a purely circular form. cross. Since 0 is continuous and invertible, no t\.'/O orbits may We can see from (III.59) that an orbit cannot converge to the For these fixed point; neither can it get outside the unity circle . reasons, the orbit may neither shrink nor expand each time it completes one cycle around YF. The orbits simply repeat themselves indefinitely. Figure 10 illustrates two orbits whic~were calculated using r + YL exp(i¢T) = [-----] (I I I. 60) exp(i ¢T) 1 + r YL exp(i¢T) for r = 0.1 and n = 1.4 . Two starting values, Y0 , were used: 0.0 and -(0.B)i.The first starting value corresponds to the physical process of adding cells, starting with no cells (no reflection). The ends and tips of the arrows in Figure 10 are located at the points YL B. In! < l (STOP BAND) Again for simplicity, we assume that n is positive. b10 fixed points of concern, both on the unity circle. There are Let (III.61) YF2 = in - / l - n For YFl let: 2 (III.62) -58- ~ ~ ~ / ~ ~ ,)C' / I "' "\ YF t I t \ / " ' \ ... I unity ci rel e Orbits in the Y-plane for r = 0. 1 ;.,'l'T -- go lass less medium Figure 1o >PASSBAND Re(Y) -59- (III.63) The velocity field is given by (I I I. 64) Clearly, the orbits near YFl will be exponentially converging toward = y - y Fl. For YF2 let: z YF2 (III.66) The velocity field is given by 2 V(Y)=r(+2.l-n J 2 Z-Z) (III.67) The orbits near YF 2 will be exponentially diyerging from YF 2 . It is easy to see that the paths of the orbits for the STOP BAND case originate at YF 2 and converge toward YFl" Figure 11 illustrates the paths as calculated with (III.60) for r = 0.1 and n = 0.7. One path beginning at Y0 = 0 was calculated (the physical case); the rest were chosen to begin near YF 2 to Figure 10. III.5 The meaning of the arrows is identical Calculation of the Propagation Constant; STOP BAND The propagation constant, S , is a complex number which is defined by the ratio (suitably averaged) of transmission coefficients for L and L+l cells: (III.67) -60Im(Y) .,,,. , ~ r .,,,. .,.,.... ...- ~ ~ -,),,,..~"3,,,-. ~ ... _ - -~~ t ~ ~ ~ ~ ~~ YF - \ fl \ l 1 / l \ i l I I \ ' \ / ~ .,,,,,?!"" i / :r Re(Y) ~ unity circle Orbits in the Y-pl an_e for r = 0.1 = 40 T lossless medium Figure 11 \ /' I <P \ l / " \ IJ \ l > STOP BAND -61- exp(iSa) t = (I I I. 70) In view of the fact that the transmission is expected to gain the phase ¢T for each cell, and ¢T is close to nn (the Bragg condition) for the STOP BANDS, we may write our final solution for B in the STOP BAND as where the absolute value operator and the (nn/a) term account for the case of cos ¢T < 0 . The positive imaginary component of S represents an ~xponential average decrease in transmission for the STOP BAND, which is the reason it is ca 11 ed the "STOP BAND" vJhen lnl > 1, we are in the PASS BAND and the situation is not as easy, because the reflection coefficient never converges to RF We must therefore de- velop a technique to do the proper averaging. III.6 Calculation of the Propagation Constant; PASS BAND From the definition of S , we must compute the following average: L s = limit i/al I £n(TM/TM+l) = limit i/al £n(T 1/\+l). (III.72) L ➔ co As M= 1 L ➔ co L approaches infinity, RL will make many orbits around fixed point, as discussed in Section III.4. It is sufficient to calcu- late the average of £n(TM/TM+l) as one orbit is made. ation RF , the Since the rel- between the "quasi-time" to get once around the orbit and the number of cells to get once around the orbit is simply a multiplicative constant, we need only average £n(TM/TM+l) over quasi-time once around -62the orbit. We must first calculate the quasi-period of one orbit. Th e quasi-period is simply the contour integral of v- 1 (Y) around the orbit: P= f v- 1(Y) dY v- 1(Y) has only one simple pole at YF. (III.73) By Cauchy s integral theorem, 1 using equation (III.59), we easily have p = _2_TTr,i===== = 2i r/n 2- l TI (III.74) r } n2 - l This equation is supported by Figure lO. When r is small (O. l is small enough), one unit of quasi-time corresponds to one cell added. From the figure, we see that approximately 32 cells are needed for one .orbit. Equation (III.74) predicts 32.5 units of quasi-time. If we wish to compute the average of any function, G, of RL around one orbit, the equation to use is (III. 75) If G(R) is a nonsingular function everywhere within the unity circle (or even just along and inside the orbit), then we have the simple and convenient result (III.76) For the case at hand, G(R) (III.77) -63- The restriction on the magnitudes of r and R (less than one) kee ps G(R) "well-behaved" in the region of integration: no singul arities or branch cuts. The propagation constant is given by (I I I. 78) Using the formula for the fixed point inside the unity circle, equation (III. 14), we obtain (I I I. 79) Since we are following orbits which neither expand nor contract, it follows that B cannot have an imaginary component . Since this is the case, it makes sense to find the cosine of Ba: cos(Ba) = Re[exp(iBa)J = cos ¢T/t (I I I. 80) Our final answer . for the propagation constant in the PASS BAND case is B = a where by 11 -1 arccos[cos ¢Tit] arccos we mean that 11 as close as possible to it follows that Ba= ¢T ¢T (I I I. 81 ) (Ba) is to be adjusted by 2mr to be Clearly, for the trivial case t = l , The adjustment of (Ba) to the proper quad- rant allows us to make a Brillouin diagram in the traditional sense. III.7 Summary of Results for Lossless Media If we have a periodic, lossless medium, then an analysis of one period (by any method) will yield all the n·ecessary information to -64- calculate the PASS and STOP BANDS for propagation in the medium, including the propagation constant. In particular, we need only calculate the fraction of flux reflected from one period (r 2 ) and the phase of transmission through one period (¢T). If we define t = / 1-r2 we have the following formulas: A. PASS BAND: Jsin ¢Tl/r > l or Jcos ¢rift < 1 s = a-1 arccos[cos ¢Tit] When r (III.81) is small, then the reflection from the periodic medium repeats every (I I I. 82) cells. If M, the actual number of cells in the whole slab, is much less than P, then (MBa) is not necessarily a good estimate of the phase gained in transmission through B. STOP BAND: Isin ¢rl /r < l S Whether r = or M cells. Icos ¢rl /t > l 2 2 nTI/a + i/a ,Q,n[lcos ¢rl/t+/cos ¢T/t - 1 ]. (III.83) is small or is close to one, exp(iMBa) is a good estimate to use to calculate the transmission coefficient after M cells. Figure 12 is a Brillouin 10 11 ' diagram of the first several PASS and STOP BANDS for the periodic medium with cell index of refraction given by n(z) (see also Figure 15). = 2 + cos(2Tiz/a) (II I. 84) The form of this diagram is very similar to one I -65wa PASS BAND 4 C •• STOP BAND 3 PASS BAND 3 B= k avg 2,JJ c = -- -STOPGAND 2 .. TI •• II-!-" ..... :__ - PASS BAND 2 ~ .. L - - - - - /• ••• ··- ... imaginary ·•... component STOP BAND l n/2 .. .. PASS BAND l 0 1T 2n 31T Brillouin Diagram : n(z) = 2 + cos(2nz/a) Figure 12 41T Sa -66- done by D. Jaggard and G. Evans of the California Institute on a similar periodic medium using a different approach. 11 III . 8 Lossy Periodic Media If the periodic medium is lossy, then the derivations of the preceding sections are no longer correct past equation (III.?). quantity (T 1T1 - R1R1) is no longer of unit magnitude. The vie will find it convenient to define the following quantities (I I I. 85) .. ¢Y = 2¢a - ¢R - ¢R' ± TI (I I I. 86) (III.87) (I I I. 88) The recursion relation for YL becomes (I I I. 89) As the losses in the medium go to zero, the "loss parameters" approach the following limits: =1 (I I I. 90) = T ct> (III.91) =1 (III.92) ¢y = 0 (III.93) a ¢ y a -67reducing equation (III. 89) to the lossless form (III.60). The fixed points of the n~w recursion obey the equation (I I I. 94) The solutions may be computed trivially. As compared with the lossless fixed points, there are tv/0 general comments to make . For the PASS BAND fixed points, both will move off the imaginary axis, but the fixed points will remain inside and outside the unity circle, respectively. For the STOP BAND fixed points, the convergent fixed point will move inside the unity circle, while the divergent fixed point vlill move out.. side the unity circle. (Th ese results would not be true if the magni- tude of a were not less than or equal to one.) Once we have found VF (or RF), it is a simple matter to compute the propagation constant by substitution of RF in exp(i Sa) (III.95) When the medium is very lossy, convergence to RF is very rapid. When the medium is only slightly lossy, then the STOP BAND case still converges to RF, but the PASS BAND case displays a decaying orbit behavio~ There is no problem using RF even in this event, because either the orbit will not be decaying very rapidly, in which case equation (III.76) is approximately true, or the orbit is decaying rapidly, in which case RL converges to RF quickly. Of course, the distinction between PASS and STOP BANDS is blurred somewhat in a lossy medium, since enough cell s will eliminate the bulk of the transmit t ed wave in either case; the greater the "lossiness, 11 the more the distinction between PASS and -68- STOP BANDS is lost. When the losses are slight, then the PASS BAND path on the Y-plane retains its general orbit feature. A good approxi- mate criterion to use to separate the PASS and STOP BANDS is to determine whether !sin( ¢a )l/r is greater than one (PASS) or less than one (STOP); however, this should only be used to get an idea of the PASS and STOP BANDS and not to provide the fine distinction that was possible in the lossless case. Figures 13 and 14 illustrate the Y-orbits for two cases of slightly lossy, symmetric (y = l) cells, which are identical in the limit as the losses go to zero with the cases illustrated in Figures 10 and 11. III.9 Multiply Periodic Media If an inhomogeneous slab has a periodic structure with cell length a , so that n(z + Na) = n(z) , (III.96) and an average index of refraction in the cell, navg' then we expect to have STOP BANDS whenever the phase gained through one cell is an integral multiple of TT (but not zero): (III.97) This formula helps us to understand the positions of the STOP BANDS in Figure 12. As (w/c) gets larger, the reflection coefficient gets smaller for cases with continuous variation in n(z), so we expect that the width of the STOP BANDS will get smaller as (w/c) increases. feature is also apparent in Figure 12. When r This is small and almost constant as a function of (w/c), the \vidth of the STOP BAND is given Im ( Y) -69- f t \ unity circle Orbits in the Y-plane for r = 0.1 <Pa = 80 a= !::'. 0.99 y = 1 <P y = 0. 01 ° Figure 13 <Pr > lossy PASS BAND t Im(Y} ~ .... ..... \ ...... ~ \ r ~ , ~ \ ~ -· \ \ ' ~:>-. -70- " \\ " ~ ,-r / ~ - ·OIi"- ...-;,,--~ - - ~ ~- - J :J:..- -- ·-"'-- ·- , '~ · - - - - 1- .. Re ( Y) ~ ~ Orbits i n the Y-pl ane for r = 0. l ¢Cl = 4 0 "' ¢r (J, = 0. 99 ¢y = 0 . 01 ° ' y ::: l Fi gure 14 > lossy STOP BAND -71- approximately by 6kSTOP-N = 2 arcsin(r)/anavg = 2r/anavg, (III.98) because the phase for small reflection cases is very nearly (I I I. 99) (Recall the arguments of Section II.4). The index of refraction in any periodic medium with symmetric cells of length a may be written in the Fourier cosine series form: 00 (I I I. l 00) where (III .101) (We will generally restrict our attention to indices of refraction which ah1ays remain greater than or equal to one, but this restriction is not necessary.) Equation (III.97) may be written in terms of the lattice structure numbers, K£ , as We expect all STOP BANDS to exist for not there is a component in n(z) £ = l to infinity \vhether or with corresponding K£, but we also expect that the widths (strengths) of the STOP BANDS will be strongly influenced by the presence of a non-zero component in responding K£ n(z) with cor- -72- To illustrate this idea, consider the two cell indices of refraction given by = 2 + cos (Kl z) = 2 + l (III.103) l 2 COS(K 1z) + 2 COS(K 3z) (I I I. l 04) These two functions have identical averages (n avg = 2), minimum values (n min . = l), and maximum values (n max = 3). (x= l). They are shown in Figure 15 Consider also the index of refraction function given by = 2 - -2l cos(K l z) - -2l cos(K 3z) We have not illustrated nc(z) (III.105) because it is so similar to n8 (z). In fact, if we had one slab composed of many cells of type B, and another composed of many cells of type C, they would differ only at the very ends, since nc(z) is n8 (z) shifted by a/2 We expect that the PASS and STOP BANDS should be identical for the two slabs \'lith cells of types B and C. Figures 16, 17, and 18 represent the computed magnitude of reflection from one cell i,,iith index of refraction given by equations (III.103) , (III.104), and (III.105) respectively, where we have assumed a background index of unity. Note in particular that IR(w/c)I is not iden- tical for n8 (z) and nc(z). We expect this because, among other reasons, nc(z) has abrupt discontinuities which prevent I R(w/c) I from becoming smaller as (w/c) becomes large. (In fact, the maxima of IR(w/c)I will approach 0.8 for case C as (w/c) becomes large.) not plotted We have ¢T(w/c), but of course it was also calculated so that -· 73n(z) 1+2x ( l+x) + x cos (2 ·1:z/a) l+x -a/ n (z) ! 0 a/2 z l+2X (l+X) + ~ x[cos(2rTz/a) + cos(6TTz/a)J l+X --!------------r-'"'-----~--~ z 0 -a/2 \ a/2 Fiqure 15 .\ -74- I Ri l l .8 .6 n I .4 .2 'IT 2TI 3TI 4 11 5n 6TT 7Tr wa C Magnitude of Re flection versus Free-Spa ce Wavenu mber Times Cell Si ze Index of Refraction in the cell: 2tcos(2nz/a); lzl < a/2 Ba ckground Index: l. Figu re 16 -75- IR[ .8 .6 .4 wa Lhr 1T 2TT 3TT 5TT 6n 7TT 0 C Magnitude of Reflection versus Free-Space Wavenumber Times Cell Size. Index of Refraction in the Cell: 2+}cos(2nz/a)+}cos(6rrz/a); Jzl<a/2. Background Index: l. Figure 17 -76- J Ri .8 .6 I .4 .2 0 TT 2TT 3rr 4TT 51T 7rr 6TT wa C Magnitude of Reflection versus Free-Space Wavenumber Times Cell Size. Index of Refraction in the Cell: 2 -} cos(2nz/a) - } cos(6nz/a); lzl < a/2 . Background Index: Figure 18 l . -77- S(w/c) could be computed. In all cases, ¢T(w/c) is close to the ex- pected value: ( I I I. l 06) but the smal 1 differences betv1een the expected and actual values determine the exact nature of the STOP BANDS. Figure 19 reveals the STOP BANDS (shaded areas) for the three cell types. It is interesting to note that, as predicted, the STOP BANDS for cell types Band Care identical [in fact, so is S(w/c)], even though both are different for the two cases. I R(w/c) I and ¢T(w/c) This result gives us greater confi.. dence in our theory of periodic media, especially the arguments of Section III. 1, since the calculations for the cell type C case assumed a unit index of refraction between eel ls when the 11 actual II value is three. Figure 19 also gives instancial evidence for our claim that some STOP BANDS will be enhanced and others diminished by the addition of higher-order components in n(z). The exact amount of enhancement is a matter for calculation in each patticular case. III. 10 Conclusions for Periodic Media We have seen that electromagnetic wave propagation in any periodic medium may be calculated very conveniently in terms of a propagation constant (S) once the reflection and transmission properties of one cell are determined. We have done our calculations assuming that the inci- dence is normal and the permeability is constant. However, the trans- formations of Section II.8 extend all results to the more general case in which all electromagnetic parameters are variable. Although we have H 3n 5n/2 STOP GMWS Fi gure 19 TI TI/2 ~ ?rr/ 2 3n/2 211 4n wa C C wa I -...J 0:, I -79dropped the explicit dependence on w, it is also possible to assume that the electromagnetic parameters are functions of w as well as z ; in this case, the formulas for the propagation constant and the criteria for PASS and STOP BANDS will be the same, but the form of the Brillouin diagram will be altered somewhat because longer a constant, but depends on navg is no w. Chapter IV will deal with media which are "almost homogeneous" and, perhaps, "almost periodic." We will see that, if we are so fortunate as to have a medium which is both "almost homogeneous" and purely periodic, then the calculation of the propagation constant becomes particularly easy. Based on the material of Section 111.9, we pose the interesting question of what would happen to the PASS and STOP BANDS if the index of refraction has cosine components at Kvalues which are not rational multiples; in other words, the index of refraction is an "almost periodic" function. The res·ults of this chap- ter are, unfortunately, not suited to answering this question, since we assumed from the outset that we were dealing with a purely periodic index of refraction. However, we will find the results of the next chapter very convenient for handling the important subset of "almost periodic" cases \'lhich are also almost homogeneous. 11 11 -80- Chapter IV Almost Homogeneous Media IV. l The Almost Riccati Equation The main difficulty in integrating the Riccati differential equa- tion for R(z) (equation II.41) is the R2 term. could simply ignore this term entirely. to consider those cases for which the It would be nice if we The purpose of this chapter is 2 R term may~ priori be neglected. We \<Jill consider first exactly what we mean by an "almost homogeneous medium. 11 t~e mean by 11 almost homogeneous" that the index of re- fraction is very nearly constant, or (IV.l) where l;(z) is small compared with unity. s For simplicity, we vlill let = max[il;(z)IJ be the measure of II al most homogeneity. 11 (IV.2) We wi 11 deal later with the situation for which there are many oscillations in n(z) will assume that there are only 11 a fev1 11 oscillations in length of the slab. For no1--1, we n(z) over the In this case, the results of Chapter II (equ ation I I. 50) give ITI -> l - Ll; IT21 -> I where 1 - 2LE,: R2 I < 2Ls (IV.3) (IV.4) (IV.5) L is the small number of oscillations, and we have dropped a 11 terms beyond the linear terms in ,; i~ith an "almost homogeneous medium," we need an "almost Riccati ~<le may find this equation by equation" to use to find the reflection. substitution of appropriate approximations in the standard Riccati equation for reflection (11.41): dR _ l [ dz - 2 l - O(,;)] n avg dn/ dz . ( [l + O(,;)] - 2inavg w/c) [l + O(,;)] R(z). The two terms in equation (IV.6) do two different things. term generates local reflections as acts to change R(z) n(z) varies. (IV.6) The first The second term at right angles . to itself in the complex plane; in other words, it is a rotation. Since the two terms do two different things, there is no problem taking them to two different orders in Since R is small (of order ,; . .; 112 ), we might want to eliminate the second term in (IV.6) completely. However, vJith no knowledge of the size of (dn/dz), this is inadvisable, not to mention the loss of the "rotation" effect which would result. Therefore, whatever the size of (dn/dz), we vlill take each term in equation (IV.6) to lo1t1est order in ,; . This yields the 11 almost Riccati equation 11 : dR dz = dn/dz 2 navg - 2in avg (w/c) R(z) (IV. 7) One of the wonderful things about.the "almost Riccati equation'' is that it is a linear equation, in the sense that if solution for n(z) n(z) = n avg + n 1 (z) and R (z) 2 R (z) 1 is a is a solution for = navg + n 2 (z) , then (IV.8) -82- is a sol uti on for (IV.9) as long as a and b are not large numbers (compared with unity). This may be verified by direct substitution in (IV.7). IV.2 Green's Function for the Almost Riccati Equation Since the almost Riccati equation is linear, it makes good sense to attempt to find a Green's function to reduce it to a simple integration over a source function. 11 11 This is actually fairly easy, so we will not needlessly obfuscate the physical process involved in finding the Green's function with unnecessary mathematical ballast. Consider the partitioning of the function n(z) - navg into many pieces _which are constant _except over a very small region and which are differentiabie where n(z) is differentiable. If we have M pieces, then n(z) (IV.10) = M dn/~z = I dnl/dz (IV.11) L=l The differential equation for RL(z)*begins with the initial value RL(+a/2) = O , at the right hand boundary of the slab. is zero until we reach Since (dnl/dz) zl (the z-coordinate at the region where (dnl/dz) is non-zero), RL(zl+) = 0 . In the very small region in which (dnl/dz) is non-zero, we get a contribution to R from the first term bf the almost Riccati equation (IV.7): *~(z) is a generalization of R 1 and R2 in equation (IV.8). -83- (IV.12) The "phase shifting" term has negligible effect over the small region (zL-'zL+) . As we continue to solve the almost Riccati equation in the negative (dz) direction, we only have the phase shifting term as a contributor, since (dnl/dz) is zero beyond zl-. tor multiplier to This adds a phase fac- RL(zl-) to give the final answer for one piece: RL(-a/2) RL(zl-) 2in avg (w/c) dz] (IV.13) Putting equations (IV.12) and (IV.13) together, we obtain zl+ x f - f (dnl/navgdz) dz (IV.14) ZLTo get the reflection for the original problem, we need only sum over all L : M R= = exp[ian w/c)J I exp[2inavgzl(w/c)J _avg L=l zl+ . x f(dnl/navgdz) .dz. zl- f - (IV.15) If we make the number of pieces, M , very large, then we may make the regions (zL-'zL+) very small, small enough so that (dn/dz) is constant over each region. Since only one (dnl/dz) is non-zero over each region, -84- (dnl/dz) is nearly constant over the small region (or is a delta function if n(z) has a jump discontinuity in the region), so zl+ I (IV.16) ZL- The 6ZL is just the factor we need to convert the sum to an integral. Over each interval, (dn/dz) is equal to (dnl/dz). The final integrated form is a/2 J exp[2inavg(w/c)z][-} (dn/navgdz)J dz. R = exp[ianavg(w/c)J -a/2 (IV.17) If we consider the function (IV.18) to be the source function 11 11 giving rise to reflection, then the Green's function is G(w/c;z) = -exp[ian avg (w/c)J exp[2in avg (w/c)zl- (IV.19) giving us the formal solution to the problem a/2 R(w/c) = G(w/c;z) S(z) dz -a/2 f (IV. 20) IV.3 Almost Inversion of the Reflection The second wonderful thing about the almost Riccati equation is that its Green's function is just a Fourier transform with slightly altered coordinates and a slightly different constant. Rec a 11 that the standard Fourier transform pairs are related by the equations -85co f( v) = (2n):.. 112 f F(z) exp(ivz) dz (IV.21) - co co F(z) = (2nf l/2 I f( \) ) exp(-ivz) dv (IV.22) - co If we make th e identification of S(z) with F( z), with the additional (obvious) extension that S(z) = 0 outside th e slab, then co (IV.23) - co and (IV.24) or f(2navgw/c) = -(2 n) -l/2 exp(-ianavgw/c) R( w/c) (IV.25) It is then a si mple matter to get back the original source function by the inverse Fourier transform: co S(z) = -l/2n J exp(-ianavgw/c) R(w/c) exp(-2inavgzw/c)(2navg)d( w/c). - co (IV.26) Suppos e that the index of refraction is represented by a Fourier series as co n(z) = J A(K) exp(-iKz) dK (IV.27) -co where co A(K) = l/2n J n(z) exp (iKz) dz - co (IV.2 8) -86- The derivative of n(z) is given by 00 I dn/dz = B(K) exp(-iKz) dK (IV.29) -oo where B(K) -iK A(K) = (IV. 30) B(K) may also be obtained from the Fourier inversion of dn/dz 00 B(K) = l/2TT f (dn/dz) exp(iKz) dz (IV.31) - co Comparing this with the integral for ·R(w/c), we have (IV.32) or -i exp(-iKa/2) R(w/c) 2TT(w/c) (IV.33) where K = 2n avg w/c (IV. 34) The inversion is complete with the final integral 00 n(z) = n0 + I -i • exe(-iKa/2) ZTT(w/c) R(w/c ) exp ("lKZ ) dK. (IV.35) - oo The additive constant, n0 , must be determined by some extra information, such as knowledge of the index of refraction in the left homogeneous region. vJe have assumed that navg is somehow knmvn before the inversion. This will probably not be the case, but no matter. In any event, navg -87- will not be very different from the background index of refraction because the medium is almost homogeneous. The average index was chosen as the representative index of refraction for substitution in the almost Riccati equation (over, say, the left background index) because it might tend to minimize the errors in the approximation, but mostly because of previous results relating the phase of the transmission coefficient to the average index of refraction (equation II.47). To find a better guess for navg than the background index, it is a simple matter to iterate equation (IV.35) with navg as calculated from the previous iteration. Since we are approximating by dropping O( ~) terms anyway, there is no point in doing the iteration more than once, if even once. The other term to worry about in the inversion is exp(-iKa/2), since we do not necessarily know what is the slab length before the inversion. Here we are in luck, because we recognize this term to be a simple translation operator, which shifts n(z) to the left by a/2 . Since the origin of the z-axis is arbitrary anyway, this is no problem. If we wish, we may ignore this factor in the inversion, which means that the slab will be located between zero and a after inversion, rather than between -a/2 and +a/2. IV.4 Examples of Almost Inversion A good question to ask at this point is hO\'I well does the inversion work? This question is best answered by way of example. der three index of refraction functions. We will consi- They are essentially the same as the three functions of Section III.9, except that the coefficients of the cosine terms are reduced by two orders ·of magnitude and the constant -88- In terms of Figure 15, the x parameter term is adjusted accordingly. is l/100. The three indices of refraction are n1 (z) = 1.01 + 0.01 cos(21T z/a) (IV.36) n2 (z) = 1.01 + 0. 005[cos(2 1Tz/a) + cos(61Tz/a)J (IV.37) 0.005[cos(2 1Tz/a) + cos(61Tz/a)J (IV~38) The full invariant imbedding Riccati equation (as programmed on a digital computer) was used to calculate R( w/c) for the relevant argu- ments and a fast Fourier transform was calculated to reconstruct n(z). The magnitude of R( w/c) is illustrated in Figures 20, 22, and 24. The phase is not plotted, but is close to (wanavg/c) ± TI/2 (recall equation II.92 for symmetric n(z)). Figures 21, 23, and 25 illustrate the fast Fourier reconstruction of n(z) for the three cases (respectively). The 11 high frequency wiggles 11 in n(z) as reconstructed (which are especi- ally noticeable in Figure 25) are due totally to the arbitrary cut-off of the frequencies used in the fast Fourier transform. Except for the non-essential '\viggles," the reconstruction of n(z) is faithful for these almost homogeneous cases. The next question to ask is how poorly does the reconstruction work for cases which are not almost homogeneous by any stret ch of the imagination? Good examples of this would be the original functions of Section III.9. Since ~ve already wrote the computer program to handle the almost homogeneous reconstruction, there is no reason not to try it out on the other cases as well. The results are illustrated in -89- I Ri 15 10 (0 I 5 0 X ---- 0 Tf 2TT 3rr 4TT 5,r 6,r 7TI (w/c)a Ma gni tude of Reflection versus Free-Space Wavenumber (times Cell Size) for n(z) = 1.01 + 0.01 cos(2,rz/a). Figure 20 - 90 - n L02 LOl 0 n(z) a/2 a from Reflection Inversion (Reflection on Figure 20) Figure 21 z -91 - ! RI 25 (' I\ 20 15 ~10 I 0 X .__,, 5 0 0 Tf 2TT 3Tf 4 Tf 57T 67T 7TT (w/c )a Magnitude of Reflection versus Free- Space Wavenumber (times Cell Size) .for n(z) = 1.01 + 0.005[cos (2nz/a) + cos(6nz/ a)]. Figure 22 -92- n 1.0 l.O 0 n(z) a/2 a from Reflection Inversion (Reflection on Figure 22) Figure 23 z -93- IR I I' 4 3 2 N I 0 0 0 Tr 2Tr 3Tr 4Tr 61r 7rr (w/c)a Magnitude of Reflection versus Free-Space Wavenumber (times Cell Size) for n(z) = 1.01 -0.005[cos(21rz/a) + cos(61rz /a)]. Figure 24 .- 94- n l.02 l. 01 I 0 n(z) a/2 a from Reflection Inversion (Reflection on Figure 24). Figure 25 z -95- Figures 26 through 28 (respectively). As expected, they are not faith- ful reproductions of the original functions, but they do still shoi,-1 the general features. The inversion represented in Figure 28 is particu- larly bad because of the discontinuities in the original n(z) function which keep the reflection coefficient high even as frequency is increased. An interesting feature of all three reconstructions is that the graph "jerks 11 near z = 2a, which marks the end of the slab, although the variation as reconstructed continues beyond that point. The reason that the slab as reconstructed seems to end at z = 2a (rather than at z = a) is that the inversion assumed that navg = l when it was really twice as much. The factor of two, if inserted in the inversion, shrinks the slab size back to its original length, 11 a 11 • The general similarity between the indices of refraction as reconstructed and as originally defined leads us to attempt to find a somewhat better method of inversion for the large-variation cases, but which remains the same as the old inversion for the almost homogeneous cases, which were inverted satisfactorily. IV. 5 Modified Al most Inversion In the derivation of the almost inversion, the source function was defined to be S(z) =} (dn/dz)/navg (IV.18) This 11 source function 11 would be more accurately represented as S(z) =} d[ t n(n)]/dz (IV.39) for the large variation cases (but reduces to the same definition for -96 - n 2 l. 5 0 n( z ) a/2 a 3a/2 2a from Reflection I nversion (Reflection on Figure 16). Figure 26 z - 97- r. 2 l. 5 l 0 a/2 a 3a/2 2a n(z) from Reflection Inversion (Reflection on Figure 17). Figure 27 z ··98- n 2 l. 5 -~----,..-,-----r-----,-----,-----.---------:>--0 a/2 a 3a/2 2a n(z) from Reflection Inversi on (Reflection on Figure 18) Figure 28 z -99the small variation cases). will find If we use the second definition, then we tn[n(z)J when we reconstruct. Of course, the over-all addi- tive constant will be different (so that in(n) is matched in the left homogeneous region). Taking the exponential, the modified inversion becomes 00 n(z) = no exp[ f -i 2TT(w/c) exp(-iKa/ 2 ) R(w/c) exp(iKz) dK] (IV .40) where n0 is a multiplicative constant which provides normalization based on n(z) region where at a known point (presumably in the left homogeneous R would be measured) . . The results of the modified inversion in the three cases are shown in Figures 29 through 31. The two improvements which may be seen over the original inversion are the increase in height (coming closer to the original nmax ) and the somewhat subtler effect of shaping the peaks to conform more closely (but still not exactly) with the original functions. The factor of two in the slab length is still a problem, but this time it can more easily be solved because it is immediately apparent from the figures that the average index of refraction in the slabs is approximately two. The third case (with the discontinuities) is more or less hopeless with either method, but there is still some useful information about n(z) in the reconstruction. IV.6 Purely Periodic, Almost Homogeneous Media If we are ever faced with the problem of a purely periodic, almost homogeneous medium, we are really in luck. For the purposes of this analysis, let us assume that the index of refraction in one cell is -100- n 3 2 0 a/2 a 3a/2 2a n(z) from Modified Reflection Inversion (Reflection on Figure 16) Figure 29 z -101- 3 2 0 n(z) a/2 a 3a/2 2a from Modified Reflection Inversion (Reflection on Figure 17) Figure 30 z -102- n 3 2 0 a/2 a 3a/2 2a n(z) from Modified Reflection Inversion (Reflection on Figure 18) Figure 31 z -103- given by n(z) = navg + X cos(2nz/a) v1here a is now the ce 11 size. 11 The source function is 11 S(z) = - (IV.41) TIX sin(2nz/a) navga (IV. 42) Because of parity considerations, the Green's function integral reduces to a/2 2inX R(w/c) = - - exp(ianavgw/c) navga f0 sin(2nz/a) sin(2zn avgw/c)dz(IV. .43) This definite integral is trivial. Evaluation of it gives inX sin(n-an w/c) R(w/c) = 2n exp(ianavgw/c)[ TT- an a~7c avg avg sin(-n+an w/c) avg J TT + an w/ c • avg (IV .44) Since x is small, R remains small for all values of (w/c). The PASS BANDS in this case will be very large and, since the reflection is very small, the propagation constant (real part) is given very closely by S =n avg (w/c) everywhere. (IV .45) The STOP BANDS are expected to be located near the regions where sin(¢T) = sin(Sa) = 0. The first STOP BAND brackets the propa- gation constant S = n/a , giving the free-space wavenumber at the first STOP BAND, (w/c) = n/anavg· The magnitude of the reflection co- efficient there is a maximum (therefore almost constant) and has the simple form -104r = JRI = nX/2n avg . (IV.46) Based on this, the maximum imaginary component of B in the first STOP BAND is Im( S) = max a- l in [ 1 J1 - r2 +J-l 1- r 2 - l J (IV.47) or, expanding based on the 11 sma 11 ness 11 of r , Im(B) ma x = a- 1r = TTX/2n avg a (IV. 48) The width of the first STOP BAND (from equation III.98) is (IV.49) A similarly straightforward analysis is not possible for STOP BANDS two and beyond, because the magnitude of the reflection coefficient is not nearly constant in the vicinity of (w/c)N = kSTOP-N = NTT/anavg (N ~ 2) (IV.50) in fact, the reflection goes to zero, so we must be more careful in our investigation than for the first STOP BAND. calculate is The relevant parameter to n , given by (IV.51) where we have added the absolute value operator for convenience. Suppose that the incident wave has a wavenumber near the STOP BAND expected wavenumber: (w/c) = Nn/anavg + s/anavg (N ~ 2) (IV.52) -105- s is a small number. The magnitude of reflection is sin[(N+l )n + s] r = 2n nX avg - (N+l)n + This expression is easily expanded in terms of order in s (IV.53) s to give (to first s ): (IV.54) At the same time, the sine of the transmission phase is approximately !sin ¢Tl = lsin(Nn + s)I = Isl (IV.55) giving the value of n near the STOP BAND expected wavenumber: (IV. 56) Since X is small, n remains large (greater than one), which in turn means that the STOP BAND criterion is never satisfied for any expected STOP BANDS beyond the first. The sin(¢T) = 0 condition is repeatedly met, but the magnitude of reflection goes to zero quickly enough for STOP BANDS two and up that they are reduced to mere ghosts, making themselves known only through very small perturbations in the (real) propagation constant near the STOP wavenumbers. Of course, as the reflec- tion becomes non-infinitesimal, the STOP BANDS for all N will reappear. However, they will be very small compared with the first STOP BAND and very difficult to detect in an experiment; still mere ghosts of STOP BANDS. (This matter is treated further in Appendix II.) -106- This result should not be surprising, because the almost inversion should reproduce an infinite, purely sinusoidal inde x of refraction by way of a delta function for R(w/c). On the other hand, it might not have been the case, since there are many cycles (instead of the few assumed in the almost Riccati equation derivation) and we have no~ priori guarantee that the reflection will stay small, which we need to do a proper almost inversion. The minimum value of n is found at the (expected) STOP BANDS. The result represented by equation (IV.56) and the theory of Y-orbits presented in Chapter III guarante~ that the reflection will not become large for any values of (w/c) except near the first STOP BAND for each sinusoidal term in n(z). of refraction, with cell length In general, a purely periodic index a , may be written n(z) = n + I Ar cos(2Nnz/a + 6N) avg N=l 1 . (IV.57) As long as the AN s remain small, n(z) is almost homogeneous in the 1 cell, and we are out of the STOP BANDS for all terms, then the almost Riccati equation serves to calculate the reflection and transmission for one cell, from which the PASS BAND formula (III.Bl) may be used to calculate the (real) propagation constant. The propagation constant in one of the STOP BANDS may be calculated similarly with the STOP BAND formula (III.83). IV.7 Almost Periodic Functions Our main use of the properties of almost periodic functions will be confined to the idea that we wish to have a medium which is almost -107- periodic (but not quite) and examine what, if anything, can be said about the propagation constant. The first concern is with what an almost periodic function will 11 look like 11 • Our prototype almost periodic function will be (IV.58) where Ka and KB are not rational multiples of each other. We can see that this function cannot be purely periodic, because if we think that we have found a period, P , then 2 = f(O) = f(P) = cos(K P) + cos(K P) a 8 (IV .59) which implies that K p = a 2MTT and KBP = 2NTT ' (IV.60) or (IV .61) which contradicts the hypothesis about Ka and KB . Of course, the 11 almost periodic 11 nature of f(x) arises because we may always let (IV.62) where (M/N) is a rational approximation to (Ka/KB) taken to any accuracy we desire, and ~ is a correspondingly small number. If this is the case, then any period near Pa = 2MTT/K a and may be considered an 11 almost period. 1112 • 13 (IV .63) -108For a definite example, consider the function f(x) = cos(TTx) + cos(v'2TTx) (IV.64) This function is represented graphically in Figure 32 for the range of argument x= 0 to x= 10. One 11 We will let K 8 reasonab l e rational approximation of 11 Associated with this approximation are the P = 14TT/12 = 7/2 Cl and K /K a 11 = TT and 8 = K (J, = ✓2TT . ll is 7/ 5 = M/N. almost periods 11 (IV .65) PB = l OTT/ TT = 10 Examination of Figure 32 reveals that a number near 10 is indeed a good es ti mate of the first 11 a l most period. 11 A better rational estimate of /2 is (M/N) = 75/53 = 1.415 This approximation gives an "almost period 11 of P = 106. Figure 33 8 illustrates our almost periodic function in the region from x= 100 to x=llO. x = 106. It is readily seen that In fact, f(x) gets very close to 2 near f(x) reaches a maximum around x = 106.04, obtain- ing the value f(l06.04) = l .986. similarity to f(x-106.04), as may be seen by comparing f(x) in the region between x = 2 and between x = 108 and However, f(x) quickly loses its x = 3 (Figure 32) with f(x) in the region x = 109 (Figure 33). IV.8 STOP BANDS in Almost Periodic, Almost Homogeneous Media All of the work has already been done to find the STOP BANDS in an almost periodic, almost homogeneous medium. Although the results are somewhat disappointing, since we would like to see something -109- ... .. .. . . . .. . . . . . .. • • 0 ... .. . .. ....... It 1:1 • • ~ ... . . . . . . . . . . . . . . .. .•••······::;:i .,.,.....•••••• • H • • • • • • • • 0 • • • • • • • • • • • • • • o ~....... q .......... ... ... ... ' .... '° - . . .... ... .. . ... . .. ... . . . .........................••.,.,..,.. + {r ••• " ••-.•~a N (Y) 1/l 0 Lt) u •r- LL 1/l 0 u ······................•••••• N 0 N I -110- •..................... ' . . .. . . . . . • • • V) 0 u + -.- ......... X F V) 0 u N 0 r-l " ...... . r-l --a .. . .. r-l N 0 N LL -11 l 11 speci al II happen v.;i th al most periodic structures, they are comforting in the sense that it would be a big surprise if it really mattered (in a physical sense) v,hether the structure of a medium is periodic or only 11 almost periodic. 11 Let the index of refraction be given in the general case by 00 (IV.66) where the Ka are 1\.1ell-separated 11 constants (we will define this more precisely later), not necessarily rational multiples, and 00 I X « l (IV .67) a=l a There will be one and only one non-vanishing STOP BAND associated with each a , centered at (IV. 68) with width ~(w/c) a = 21 Xa Ka /n 2avg (IV .69) and maximum imaginary component of the propagation constant Im(Q) = xa Ka/4n avg . µ max-a (IV.70) These results apply as long as condition (IV.67) holds and the STOP BANDS do not overlap; in other words MINQ[I KµQ- Ka IJ > xa Ka /n avg µ (for all a ) (IV.71) We need not worry about whether the K.'s are or are not rational multiples of each other; the formulas work just as well in any event. -112- Chapter V Conclusion It has been the purpose of this thesis to make a general investigation of the properties of the invariant imbedding solution for electromagnetic wave propagation in general, periodic, almost homogeneous, and almost periodic media. While the analysis was (almost) exclusively made for electromagnetic waves, a section of the second chapter (Section II.7) extends all results to quantum mechanical waves, or in a more general sense to any wave which may be characterized by the standard wave equation (equation II.4). Chapter II contained a reformulation of the invariant imbedding solution and the derivations of some very important general properties of the solution. These general properties, which related the reflec- tion and transmission coefficients in two directions through the region of inhomogeneity, and related the minimum transmitted flux with the range of variation of the local wavenumber, provided the basis for Chapters III and IV. Chapter III was a new way of looking at waves in a periodic medium. Assuming that we have already solved the problem of reflec- tion and transmission for a single cell, an invariant imbedding recursion using the cell as the basic unit of recursion gave the propagation constant according to very simple formulas which did not involve any matrix operations. The computation of the PASS BANDS .and STOP BANDS was seen to be completely t~ivial once the transmission and reflection coefficients for one cell are known as a function of frequency. The -113- general treatment was extended to include absorptive media as well, al though the sharp distinction between PASS and STOP BANDS was lost. Chapter IV looked at those situations for \vhich the reflection coefficient is small. In this case, a very simple exponential Green's function is available to transform the differential equation for the reflection coefficient into a definite integral. The reflection was also transformed back to get the original index of refraction function. It was possible simply to use this same method when the reflection was large, and we have seen that the resulting reconstructed index of refraction has the same general appearance as the original, although the fact that the reflection was not small made the shape progressively worse as the reflection failed to get smaller with increasing frequency. Finally, we used the small reflection theory to show that an almost periodic medium will not exhibit interference between structure constants to produce STOP BANDS at frequencies beyond the fundamental for each structure constant, in the limit as the variation in wavenumber with distance is small. We have not addressed ourselves to the problem of a large-variation type of almost periodic medium, since neither the methods of Chapter III nor Chapter IV are applicable to that case. It is very possible that there could be an interference effect between irrationally related structure constants to produce STOP BANDS at frequencies which are sums of integral multiples of the fundamental frequencies, when the variation in wavenumber with distance becomes large. This \vould be a very difficult thing to calculate, even numerically, since no finite slice of an almost periodic structure may -114- be considered as representative of the whole, in contradistinction to a purely periodic medium. -115Appendix I Inductive Proof of Reciprocity Relations We v1ill prove inductively, using (II.32L (II.33L (II.66), and (II.67), the five reciprocity relationships: l. r' = r (AI. l) 2. t' = ( k/kN )t (AI.2) 3 a) r'2 + (k/kN)t 2 = l (AI.3) b) r'2 + ( kN/k o) t ' 2 = l (AI.4) (AI . 5) (AI.6) Inductive Proof: I. - When there is only one interface (N=l), ~he transmission and reflection coefficients are R = (k 1-k 0 )/(k 1+k 0 ) (AI. 7) T = 2k,J(k +k ) 1 0 (AI.8) R' = (k 0 -k )/(k +k ) 1 1 0 (AI.9} T' = 2k (k +k ) 1 0 0 (AI.10) All of the relations follow trivially. II. Assume that we have shown (AI .1-6) for I interfaces. to verify that they will also hold for I+l interfaces. l~e wish It will be con- venient to use Method A (II.32 and II.33) to calculate Rl+l and TI+l, -116- and Method 8 (II.66 and II.67) to calculate R~+l and Ti+i· We should note a change of notation (of subscripts) for method B due to the fact that we will be adding interfaces in the usual direction (that is, l to N) when calculating the reverse coefficients (R' and T'). Let rl = (kI+l-kI)/(kI+l+ kl) (AI. 11) tr = 1 + rr (AI.12) r'I = -rr (AI. 13) t• = 1 + r' = 1 - r I I I (AI.14) cS = exp[ik (AI.15) 1(z 1-zI+l)J The invariant imbedding recursions give RI+l = TI+l = RI+l = r + cS 2 R1 1 (AI.16) 2 l + r o RI 1 t 1 oTI (AI. 17) 2 1 + r 2 o R1 Ri + rii(TFr- RIRI) 2 l - r I1 0 RI (AI. 18) t II oT'I (AI.19) Straightforward calculation will verify the relationships. 1: 2 2 r = IRI+ 1 1 = ir 1+ o R1 i/[l +r 1o R1 1 (AI.20) 2 IRi + rio (T T1- R R1)[ = 2 11 + r 1o R1 1 (AI. 21) r' = 1 1 -117- We need only compare numerators as denominators are equal. From the previous step in the recursion, we have 1 = (k/k 1) T1 (AI.22) T RiRI = -exp(2i¢T )1Rr1 2 (AI .23) I (AI. 24) TiTI - RIRI = exp(2i¢T) r Substitution of (Ar.24) in (Ar.21) and division by (Ar.20) gives r'/r = 2 IR 1 - rro exp(2i¢T 1)1 (AI.25) 2 Irr+ o -Rrl A little manipulation and the use of (Ar.6) gives 2 l-a2exp(2i¢rr) I • Irr+ o- 1Rr lexp(-i¢R ) I 1 r' /r = Since lol = l and lx*I = !xi (AI.26) for any complex number x , l • I ( rt a2 IR1 I exp ( i ¢R ) )* I r '/r = - - - 2 - - - ~ - - - I- =l (AI. 27) l(r 1+ o IR 1 1 exp(i¢R ))I I (This would not work if k1 had a complex nature . ) 2 and 4: (AI.28) (This works even if k1 is complex, but if it is, 2 and 4 must be com- bined . ) . -118- 3a: (k I+l + k )2 I ] R* R} I I 3b: Follows immediately from 1, 2, and 3a. 5: Let (AI. 30) i_/J Then = arg( o) (AI. 31) -119- (AI. 32) <PT' =~+<PT' + 8 0 I+l I (AI.33) From the first step of the iteration, we have (AI.34) With this, equation (AI.18) reduces to R' - r o2exp ( 2i <Pr ) -· I 1 1 RI+l = 2 l + rlo RI (AI.35) 2 eN = arg( r + o RI) (AI.36) Let Then (AI. 37) If we examine the numerator in (AI.35), we can manipulate it with the help of l, 4, and (AI.34) to read: -o 2exp(2i¢T )[IR 1 1o- 2exp(-i¢R) + r 1 J. I I (AI.38) Then (AI. 39) We finally obtain the desired result: -120- ¢RI I+l = ±n + ¢T !+1 + ¢T' 1+1 (This proof depends heavily on the fact that This completes the inductive proof. (AI. 40) lo! = l , or k1 real.) -121Appendix II Careful Analysis of Higher Order Stop Bands We have seen in Section IV.6 that the magnitude of reflection for wavenumber (w/c) = NTT/an avg + ~/an avg where ~ (N ~ 2) (IV.52) is a small number, is approximately {IV.54) In that section we assumed that the phase of transmission was ¢T = NTT + ~ (AII. 1) which is the basic approximation from equation (II.47). However, since the reflection vanishes where we expect the STOP BANDS, we should compute a correction based on equation (II.47) for n(z) ¢T = NTT + ~ + real: a/2 dz ½ f Im[R(z)] (dn/dz) navg (AII.2) -a/2 The function R(w/c,z) is a simple matter to calculate with the almost Riccati equation. Recall that R(w/c,a/2) = 0 and R(w/c). ~(w/c,-a/z) = The integrated form of R{w/c,z) is a/2 R{w/c,z) = TTX exp[-2iznavgw/c] sin(2TTz'/a)exp[2iz'navgw/c]dz'. navga z (AII.3) f For convenience, we let -122a= navg a(w/c) = NTT + ~ (AII.4) The imaginary part of R(w/c,z) is easily integrated to become Im[R(w/c,z)] = TTX [cos( 2az/a}(sin(TT-a) _ sin(TT+a ) 4navg TT-a TT+a sin[2(TT-a) !J sin[2(TT+a) 2 ------+ a TT+a TT-a _ sin( 2az/a)(-cos(TT-a) _ cos(TT+a) TT-a TT+a + cos[2(TT-a ) ~] cos[2(TT+a) ~] •• a + + a )].(AII.5) TT-a TT a One more integral will give us the desired final answer. The (dn/dz) term is a sine, so some of the terms in the i~tegral will drop out by parity considerations, others will cancel. The result is 2 2- - - 2 + (cos(TT-a) + cos(TT+a)) ¢T = NTT + ~ + -TT~ 2 - [TI+a TI-a TI-a TI+a 4n avg x2 x (sin(TT-a) _ sin(TT+a))] . TI-a Expanding in terms of 2 x , \-Je obtain ~ TI+a (AII.6) and retaining only low-order terms in ~ and (AI I. 7) Equation (AII.7) implies that the STOP BANDS are somewhat shifted from NTI/an_avg . * They will be extremely small, so r will be approximately * If 1>1e let ~ =- (NTIX2 )/[(N 2-l)navg] 2 , we \-Jill be right in the middle of the STOP BAND. -123constant, and equal to (AI I. 8) The maximum imaginary component of S is NTTx 3 (AII.9) Im(S)max = -2----'--'--2_3__ (N -1) anavg and the width of the STOP BAND is approximately ( AI I. 10) Compared with the first STOP BAND, since are "ghosts II indeed. x is a small number, these -124References l. Ambarzumian, V. A., 11 Diffuse Reflection of Light by a Foggy Medium," Comptes Rendus ( Dokl ady) de L I Acad_emy des Sciences de l I URSS 38, 229-232 (1943). 2. Adams, R. N. and E. D. Denman, Wave Propagation and Turbulent Media (New York: American Elsevier, 1966). 3. Bellman, R. and R. Kalaba, 11 Invariant Imbedding and Wave Propagation in Stochastic Media," in Electromagnetic lfave Propagation, Desirant and Michaels (eds.) (New York: Academic Press, 1960), pp. 243-253. 4. Papas, C. H., "Plane Inhomogeneous Dielectric Slab, 11 Caltech Antenna Laboratory Note, March 1954. 5. Kritikos, H. N., K.S.H. Lee, and C. H. Papas, "Electromagnetic Reflectivity of Non-Uniform Jet Streams, Caltech Antenna Laboratory Report, August 1966. 6. Carlisle, G. vJ., "Reflection from and Transmission through a Plane Inhomogeneous Medium, 11 Douglas Paper No. 4148, October 1966. 7. Bellman, R. and G. i~ing, An Introduction to Invariant Imbedding (New York: l~iley, 1975). 8. Klein, M. V., Optics (New York: ~Jiley, 1970). 9. Sommerfeld, A., Optics (New York: Academic Press, 1964). 10. Brillouin, L., Wave Propagation in Periodic Structures (New York: Dover, 1953). l l. Jaggard, D. L. and G. A. Evans, 11 Coupl ed ~faves and Floquet Approach to Periodic Structures, 11 Caltech Antenna Laboratory Report, August 1975. -125- 12. Bohr, H., Almost Periodic Functions (New York: Chelsea, 1951). 13. Besicovitch, A. S., Almost Periodic Functions (Cambridge: University Press, 1932). 14. Schiff, L.I., Quantum Mechanics (New York: McGraw-Hill, 1968).