8
Synthesis And Analysis Of Large-Scale Integrated Photonic DevicesAnd Circuits

Lakshman S. Tamil

Arthur K. Jordan

University of Texas at Dallas

Optical devices and circuits can be synthesized from specified transmission characteristics using the methods of inverse scattering. Both analytical and numerical inverse scattering techniques that have been developed to synthesize optical devices and circuits are discussed. Large-scale guided-wave structures such as optical logic gates and optical interconnects can be synthesized using the techniques discussed here. Finite difference-based frequency-domain analysis technique has been used to verify the results obtained by these inverse scattering techniques.

INTRODUCTION

The conventional method of designing optical guided-wave devices or structures is to assume a refractive index profile and solve the governing differential equation to find the various propagating modes and their propagation characteristics. If the propagation characteristics do not meet the expected behavior, the refractive index is changed and the propagation characteristics are again evaluated; this is repeated until the expected propagation behavior of the modes is obtained. This being an iterative procedure, it is time consuming. Also, to obtain certain arbitrary transmission characteristics, one may not be able to guess the correct initial refractive index profile.

The procedure discussed in this paper, as opposed to the direct method, starts with the required propagation characteristics of the guided-wave device and obtains the refractive index profile as the end result. We achieved this by transforming the wave equation for both the TE and TM modes in the planar waveguide to a Schrodinger-type equation and then applying the inverse scattering theory as formulated by Gelfand, Levitan, and Marchenko (Gelfand and Levitan, 1955; Marchenko, 1950). The inverse scattering problem encountered here has a direct analogy to the inverse scattering problem of the quantum mechanics. The refractive index profile of the planar waveguide is contained in the potential of the Schrodinger-type equation and the propagating modes are the bound states of the quantum mechanics (Marcuse, 1972). In general the guided-wave devices are based on channel waveguides; however, we have considered here planar waveguides for mathematical simplicity. The theory presented here can be extended to channel waveguide structures, though it is nontrivial.



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Large-Scale Structures in Acoustics and Electromagnetics: Proceedings of a Symposium 8 Synthesis And Analysis Of Large-Scale Integrated Photonic DevicesAnd Circuits Lakshman S. Tamil Arthur K. Jordan University of Texas at Dallas Optical devices and circuits can be synthesized from specified transmission characteristics using the methods of inverse scattering. Both analytical and numerical inverse scattering techniques that have been developed to synthesize optical devices and circuits are discussed. Large-scale guided-wave structures such as optical logic gates and optical interconnects can be synthesized using the techniques discussed here. Finite difference-based frequency-domain analysis technique has been used to verify the results obtained by these inverse scattering techniques. INTRODUCTION The conventional method of designing optical guided-wave devices or structures is to assume a refractive index profile and solve the governing differential equation to find the various propagating modes and their propagation characteristics. If the propagation characteristics do not meet the expected behavior, the refractive index is changed and the propagation characteristics are again evaluated; this is repeated until the expected propagation behavior of the modes is obtained. This being an iterative procedure, it is time consuming. Also, to obtain certain arbitrary transmission characteristics, one may not be able to guess the correct initial refractive index profile. The procedure discussed in this paper, as opposed to the direct method, starts with the required propagation characteristics of the guided-wave device and obtains the refractive index profile as the end result. We achieved this by transforming the wave equation for both the TE and TM modes in the planar waveguide to a Schrodinger-type equation and then applying the inverse scattering theory as formulated by Gelfand, Levitan, and Marchenko (Gelfand and Levitan, 1955; Marchenko, 1950). The inverse scattering problem encountered here has a direct analogy to the inverse scattering problem of the quantum mechanics. The refractive index profile of the planar waveguide is contained in the potential of the Schrodinger-type equation and the propagating modes are the bound states of the quantum mechanics (Marcuse, 1972). In general the guided-wave devices are based on channel waveguides; however, we have considered here planar waveguides for mathematical simplicity. The theory presented here can be extended to channel waveguide structures, though it is nontrivial.

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Large-Scale Structures in Acoustics and Electromagnetics: Proceedings of a Symposium The wave equations for the inhomogeneous planar optical waveguides can be derived from the Maxwell's equations. If we take z as the propagation direction and let to represent the frequency of laser radiation, we have the following wave equations for one-dimensional inhomogeneous planar waveguides (Tamir, 1990) for TE modes and for TM modes. The planar waveguide we are considering here has a refractive index that varies continuously in the x direction. For the planar optical waveguide shown in Figure 8.1, our problem is to find the refractive index profile function in the core for a set of specified propagation constants. We assume that this planar waveguide has a refractive index profile guiding N modes. The propagation constants are , in which is the value of n(x) as and n1 = sup n(x). Designing an optical waveguide is analogous to the inverse problem encountered in quantum mechanics. We are trying to get the potential function from the given bound states and scattering data. The wave equation for the TE modes can be easily transformed to an equivalent Schrodinger equation by letting and We can see in our case the potential function V(x) is continuous and . The TE mode cases have been solved by Yukon and Bendow (1980) and Jordan and Lakshmanasamy (1989). We now need to transfer the wave equation for the TM modes to Schrodinger-type equation to apply the inverse scattering method. In (8.2), the first derivative of Ex can be eliminated if we let The wave equation then becomes

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Large-Scale Structures in Acoustics and Electromagnetics: Proceedings of a Symposium Figure 8.1 The physical structure of an inhomogeneous symmetrical planar optical waveguide showing the reflection and transmission of electromagnetic wave. We are now able to obtain the equivalent Schrodinger equation by setting the potential function as and letting

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Large-Scale Structures in Acoustics and Electromagnetics: Proceedings of a Symposium The TM mode case has been solved by Tamil and Lin (1993). INVERSE SCATTERING THEORY The inverse scattering theory of Kay and Moses (Kay, 1955) provides us with a way to obtain the potential from the reflection coefficient that characterizes the propagation properties of the planar waveguide. As the potential we defined vanishes at infinity, we can apply the Gelfand-Levitan-Marchenko (G-L-M) equation to solve our problem. Let us consider a time-dependent formulation of the scattering. We take the Fourier transform of (8.7) (the transform pairs are and ) to obtain in which t is the time variable with the velocity of light . The incident plane wave is represented by the unit impulse which will produce the reflected transient wave function where are the discrete eigenvalues of Schrodinger-type (8.7), r( k) is the complex reflection coefficient, and An are arbitrary constants normalizing the wave equation such that The reflected transient is produced only after the incident unit impulse has interacted with the inhomogeneous core of the optical waveguide and therefore A linear transform independent of k can now relate the wave amplitude Ψ(x,t) in the core region with the wave amplitude Ψ0(x,t) in the exterior region

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Large-Scale Structures in Acoustics and Electromagnetics: Proceedings of a Symposium Here the exterior field is From physical consideration, since is a rightward moving transient Thus the kernel K(x,t) = 0, for t > x and K(x,t) = 0 for . We substitute (8.16) into (8.15) and using (8.14) and (8.17) yield the integral equation By substituting (8.15) into (8.10) the kernel K(x,t) satisfies a differential equation of the same form as (8.10) provided the following conditions are imposed and We now can see how the solution of the integral (8.18) for the function K(x,t) can lead to the synthesis of optical waveguides. DESIGN EXAMPLE 1:ZERO REFLECTION COEFFICIENT The reflection coefficient characterizes the propagation properties of the guided-wave optical devices. The zero reflection coefficient characterizes a system with propagating modes only, whereas a non-zero reflection coefficient characterizes a system with both guided and nonguided modes. Let us first consider the special case of zero reflection coefficient (Kay and Moses, 1956). We substitute (8.12) for r(k) = 0 in Gelfand-Levitan-Marchenko (8.18)

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Large-Scale Structures in Acoustics and Electromagnetics: Proceedings of a Symposium It is clear from the above equation that the solution for K(x,t) should have the form (Kay and Moses, 1956) Substituting (8.22) into (8.21) produces a system of equations for fn(x): where n = 1, 2, . . . N. This system can be conveniently written as where [f] and [B] are column vectors with fn, and Bn = An exp(KnX) respectively, and [A] is a square matrix with elements in which is a Kronecker delta. The solution for f is f = -A-1B and then from (8.22) K(x,x) = ETf where E is the column vector with element En = exp(KnX) and T denotes transpose. Now, and so when written with subscript notation and the summation convention. The K(x,x) given by (8.22) can be recognized in the form and therefore the potential V(x) according to (8.20) is

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Large-Scale Structures in Acoustics and Electromagnetics: Proceedings of a Symposium Given N modes with desired propagation constants, we can obtain a potential function as given by (8.29). Here we have N degrees of freedom due to N arbitrary constants . For TE modes the refractive index profiles is simply given by in which k0 is the free space wave number. Whereas for TM modes, obtaining the refractive index profile is more complicated because it is a solution to a nonlinear differential equation [(8.8)]. The nonlinear differential equation can only be solved numerically. First we transform (8.8) into a convenient form by setting . We then obtain This is a constant coefficient equation that yields the refractive index profile provided the potential V(x) is given. DESIGN EXAMPLE 2: NON-ZERO REFLECTION COEFFICIENT In the previous section, we took advantage of assuming that the reflection coefficient is zero, which simplified the problem a lot. Now we are going to solve the problem with non-zero reflection coefficient. We take the rational function approximation for our scattering data. We represent our reflection coefficient using a three-pole rational function of transverse wave number k (Jordan and Lakshmanasamy, 1989). The three poles are as follows: one pole on the upper imaginary axis of the complex k plane, which represents discrete spectrum of function R(x + t) [(8.12)] characterizing the propagating mode; two symmetric poles lie in the lower half of the k plane, which represent the continuous spectrum of R(x + t) characterizing the unguided modes; the three-pole reflection coefficient can be written as where r0 can be determined by the normalization condition r(0)= -1. This condition ensures total reflection at k = 0. k1, k2 have the following forms: and . The third pole on the positive imaginary axis is k3 =ta.

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Large-Scale Structures in Acoustics and Electromagnetics: Proceedings of a Symposium Figure 8.2 Permitted regions of the complex k plane for the pole positions in a three-pole reflection coefficient. The pole positions are confined to certain "allowed regions" that are determined by the law of conservation of energy, which can be represented by for all real k; see Figure 8.2 and refer to Jordan and Lakshmanasamy (1989) for details. It has been shown that the reconstructed potential function V(x) has following form: in which a and b are column vectors, and are given by where

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Large-Scale Structures in Acoustics and Electromagnetics: Proceedings of a Symposium Matrix A(x) is given by where So, it is possible to construct the potential from the three poles of reflection coefficient using the above equations. DESIGN EXAMPLE 3:NONRATIONAL REFLECTION COEFFICIENT The refractive index profiles reconstructed for the cases discussed above go to zero asymptotically and they approximately model the actual refractive index profiles used in practice. The refractive index profiles used in practice are truncated and the truncations form the core-cladding boundary. For a doubly truncated refractive index profile modeling a planar optical waveguide, the reflection coefficient is not a rational function of the complex wavenumber, but a more complicated form (Mills and Tamil, 1991, 1992; Lamb, 1980). Reconstructing refractive index profiles for nonrational reflection coefficients is not possible in analytical closed forms and so numerical techniques must be used. Discretization of the G-L-M Equation To solve the G-L-M (8.18) by numerical methods, the space-time diagram is discretized into square grids rotated by 45° with respect to abscissa, as shown in Figure 8.3. The interval Δt = 2Δx, and x = mΔx, m = 0,1,... N, where N is the total number of grid points along the x direction, and t = nΔt - (m / 2)Δt, n = 0,1,..., m. The G-L-M integral equation can then be discretized as

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Large-Scale Structures in Acoustics and Electromagnetics: Proceedings of a Symposium Figure 8.3 Discretized grid diagram in a space-time plane for numerical reconstruction. where y = lΔt-(m/2)Δt. The subscript m in Km(n) represents the grid position along the x direction and the argument n represents the grid position along the t direction. C(1) is the coefficient for numerical integration; if the trapezoidal role is used,

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Large-Scale Structures in Acoustics and Electromagnetics: Proceedings of a Symposium Iteration Scheme With Relaxation The Gelfand-Levitan-Marchenko equation is an integral equation of the second kind and can be solved numerically in an iterative manner. We rewrite (8.40) as where the superscript i in represents the i-th iteration result. It is worth pointing out that the iterative process involves only the grid points on the m-th column in Figure 8.3. In (8.12) the poles on the positive imaginary axis are in the discrete spectrum and correspond to the guided modes. The exponential term in (8.12) then grows rapidly as (x + t) increases and in order to improve the convergence, the relaxation technique is used (Press et al., 1989), so that (8.42) is revised as where is the relaxation factor. If lies between 0 and 1.0, it is called the under-relaxation method; if lies between 1.0 and 2.0, it is called the over-relaxation method. In our computations, provides the desired results. Initial Values For K(x, t) The convergence of analytical solutions to the G-L-M equation has been proved (Szu et al., 1976). However, the convergence of its discretized form cannot be ascertained, because of the additional errors due to truncation and discretization. Good initial values for K(x,t) are important for the numerical iterative scheme, in particular when a bound state corresponding to the propagating mode exists. The Born approximation has been used by other authors to provide initial trial values for K(x, t) for cases where there are no bound states. However, for cases discussed here, where there are bound states, the Born approximation when used to provide the initial values for K(x, t) fails to reconstruct the potential correctly. Although for shorter lengths of the potential the reconstruction is in agreement with the actual value, the method fails for larger lengths. The leapfrogging algorithm (Jordan and Ladoucer, 1987) provides an effective initial value for K(x,t). To obtain the leapfrogging algorithm, we substitute (8.20) into (8.10) where Ψ(x,t) has been replaced by K(x,t) yielding

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Large-Scale Structures in Acoustics and Electromagnetics: Proceedings of a Symposium and introduce new variables u and v, defined as and (see Figure 8.3). With this coordinate transformation, the partial differential equation (8.44) can be rewritten as so that its discretization gives the following equation (Jordan and Ladoucer, 1987), which relates the grid point Km(n) with the other five grid points, as shown in Figure 8.3. Note that Km(n) on the LHS of (8.48) is at the ''current'' reconstruction column m, while the remaining five grid points on the RHS are all located within its left region, which are either on the boundary whose values are provided by K(x,-x)=-R (0) or grid points that have already been reconstructed by the step-by-step marching algorithm marching in the x direction. (8.48) does not provide values for Km(m), m = 1,2, ....N and a different procedure should be adopted to find those values. Solving (8.42) for Km(m) yields which provides initial trial values for Km(m). Furthermore, we obtain To summarize, (8.48), (8.49), and (8.50) can provide the initial trial values for K(x,t) necessary for the iterative numerical solution of the G-L-M equation.

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Large-Scale Structures in Acoustics and Electromagnetics: Proceedings of a Symposium Reconstruction of the Potential v(x) The potential in its discretized form can be expressed using (8.20) as This expression can be used to reconstruct the potential when the values of Km(m), m = 0,1,... N are already evaluated. This reconstructs the potential at every point in x except at x = 0, corresponding to the grid point m = 0. To evaluate the potential at the origin we substitute (8.18) into (8.20), yielding Because R(t) = 0 for , we obtain at the origin which is an exact formula for recovering the potential at the origin. It is interesting to note that the perturbation expansion theory derives the approximate solutions (Kritikos et al., 1982) and which are called the Born and the modified Born approximation, respectively. At the origin, the Born expression provides an approximate reconstruction, even though there exists a discontinuity at the boundary. The numerical inverse scattering theory can now be summarized in the following steps: Compute the potential at the origin, v(0) using (8.53); Set the initial trial values for Km(n), n=0,1,...m on the current column m using (8.48), (8.49) and (8.50); Iteratively calculate Km(n), n= 1,2,...m for each value of n on the current column m using (8.42) with an appropriate choice of relaxation factor ; Evaluate potential v(m-1) using (8.51); Move the current column from m to m + 1, and repeat the steps (b) to (d).

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Large-Scale Structures in Acoustics and Electromagnetics: Proceedings of a Symposium DISCUSSION We have developed a method based on inverse scattering theory that can be used to design planar optical waveguides that transmit a prescribed number of TE or TM modes with prescribed propagation constants. To demonstrate some practical examples for the zero reflection case, let us compute the refractive index profiles for two cases: the single mode case and the N mode case. For the single mode case, (8.23) becomes Then, the potential has the form where A1 is an arbitrary constant, and note that K1 can be obtained from For a desired propagation constant β1, we can get a set of refractive index profiles corresponding to different arbitrary choice of A1; see Figure 8.4. We use the following data relating to waveguide: , wavelength λ = 0.8 μm and . We obtained the refractive index profiles by solving (8.31) using the potential V(x) obtained from (8.57). Runge-Kutta's fourth-order approximation is applied in solving the differential equation (8.31) (Levy and Baggott, 1976). We can see from Figure 8.4 that the maximum value of refractive index lies on the positive side of x = 0 when A1 < 2K1, on the negative side of x = 0 when A1 > 2K1, and at x = 0 when A1 = 2K1. Substituting A1 = 2K1 into (8.57) yields This potential is everywhere negative and goes to 0 as x goes to infinity. Also the potential is symmetric about its minimum point. We can truncate the potential at the point where the potential is 1 percent of its maximum value to find the width of the core d. The refractive index profile corresponding to this potential is shown by the continuous line in Figure 8.4. Similarly, for the N mode case, we need to construct the potential first using (8.25) and then solve the nonlinear differential (8.31) for the refractive index profiles. For a set of prescribed propagation constants, every arbitrary choice of normalization constants will produce a different potential and a corresponding refractive index profile. In order to construct a symmetric refractive index profile with single peak, we found that the normalization constants must satisfy the following equation (Deift and Trubowitz, 1979):

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Large-Scale Structures in Acoustics and Electromagnetics: Proceedings of a Symposium Figure 8.4 The reconstructed refractive index profiles for a single prescribed TM mode with β1 = 17.2 and various A1 = 2K1 = 3.7386, 0.4, and 0.7 corresponding to the solid, dashed, and dotted curves, respectively. where for the reflectionless case. Here N is the number of guided modes in the planar waveguide. For the case N = 5, using sets of arbitrary normalization constants we have computed the refractive index profiles and these are shown in Figure 8.5. The symmetric profile obtained using the condition (8.60) is shown by a continuous line in the figure. To demonstrate the reconstruction of the potential from a three-pole reflection coefficient (a case of non-zero rational reflection coefficients) we have chosen here two examples. In example 1, the poles are determined by the following parameters: a = 1.0, c1 = 0.8, and c2 = 0.499; example 2 has different unguided modes characterized by c1 = 0.05, c2 = 0.1, and the same propagating mode characterized by a = 1.0. Figure 8.6 shows the plots of potential functions for examples 1 and 2. In example 2, we see that the potential is everywhere negative.

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Large-Scale Structures in Acoustics and Electromagnetics: Proceedings of a Symposium Figure 8.5 Reconstructed refractive index profiles for five prescribed TM modes with correspondence to (dashed curve) and for An satisfying (8.60) (solid curve).

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Large-Scale Structures in Acoustics and Electromagnetics: Proceedings of a Symposium Figure 8.6 Potentials of a waveguide characterized by a three-pole rational reflection coefficient. The solid curve corresponds to a = 1.0, c1 = 0.8, c2 = 0.499; the dashed curve corresponds to a = 1.0, c1 = 0.05, c2 = 0.1. Figure 8.7 shows the refractive index profiles for TM mode in both the above discussed examples obtained by substituting the potentials into the nonlinear differential equation (8.31) and solving for . We notice that a depressed cladding is obtained in example 1, and we also see that the profiles we found here resemble the profiles we normally find in practical optical waveguides (Okoshi, 1976). Introducing a truncated potential to model the planar waveguide (Mills and Tamil, 1992), it can be shown that both propagating and non-propagating modes appear when the reflectionless potential v(x) = -2sech2 (x) is truncated at a point on the left x = x1. Based on the Jost solutions corresponding to the untruncated potential v0 = -2sech2(x), the reflection coefficient from the left for the truncated potential can be derived (Mills and Tamil, 1992) as

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Large-Scale Structures in Acoustics and Electromagnetics: Proceedings of a Symposium Figure 8.7 Reconstructed refractive index profiles corresponding to the potentials shown in Figure 8.6. which has two poles in the complex k-plane located at and Since , both poles are located on the imaginary axis, so that k2 = ik corresponds to the guided mode. The exponential factor exp(i2kx1) in (8.62) represents a shift x1 on the x axis relative to the corresponding untruncated potential. Equation (8.62) can then be rewritten as in which the phase shift factor has been excluded. The characteristic function is

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Large-Scale Structures in Acoustics and Electromagnetics: Proceedings of a Symposium Using (8.53), the potential at the truncation location is This is a case of nonrational reflection coefficient. Figure 8.8 gives the potential, assuming x1 = -1.0, where the asterisks show the potential obtained by numerical reconstruction, and the exact potential is plotted in solid line for comparison. Again good agreement is achieved. Figure 8.8 Potential -2sech2 (x) truncated at the left, x1 = -1.0. Solid curve, exact potential; circles, numerical reconstruction.

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Large-Scale Structures in Acoustics and Electromagnetics: Proceedings of a Symposium The results obtained by inverse scattering theory can be verified by a finite difference-based analysis scheme. Using this method we find the propagation constants of guided modes of an optical waveguide with arbitrary refractive index profile. Owing to its simplicity and flexibility, this method is proved to be very effective. For demonstration purposes we consider here a symmetric planar waveguide. We have compared in Table 8.1 the propagation constants of various modes that we used in reconstructing the refractive index profile of the waveguide against the propagation constants obtained by analysis for the normalized frequency at which the propagation constants are prescribed. We see that the last two columns of the table agree very well. This shows that the inverse technique outlined here can be used to synthesize waveguides with prescribed modes. Table 8.1 Prescribed TM Mode Spectra Used in Reconstructing Refractive Index of Planar Waveguide and Spectra Obtained by Analysis Using Finite Difference Scheme Number of Modes Mode Number γ Prescribed Mode Spectra βγ / k0 βγ / k0 Obtained by Authors' Analysis N = 1 0 2.18997 2.18995 N = 2 0 2.20556 2.20553   1 2.18417 2.18398 N = 3 0 2.20926 2.20916   1 2.19140 2.19100   2 2.18061 2.18036 N = 5 0 2.21288 2.21266   1 2.20003 2.19968   2 2.18998 2.18968   3 2.18278 2.18254   4 2.17845 2.17797 N = 7 0 2.21466 2.21452   1 2.20473 2.20449   2 2.19630 2.19606   3 2.18927 2.18915   4 2.18397 2.18379   5 2.18010 2.17997   6 2.17778 2.17753 The method demonstrated here can be extended to the synthesis of optical devices (Mills and Tamil, 1993, 1994) with specified transmission characteristics. REFERENCES Deift, P., and E. Trubowitz, 1979, "Inverse scattering on the line,"Commun. Pure Appl. Math.32, 121-251.

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Large-Scale Structures in Acoustics and Electromagnetics: Proceedings of a Symposium Gelfand, I.M., and B.M. Levitan, 1955, "On the determination of a differential equation by its spectral function,"Trans. Am. Math.Soc. Ser.21, 253-304. Jordan, A.K., and H. Ladoucer, 1987, "A renormalized inverse scattering theory for discontinuous profiles,"Phy. Rev.A36, 4245-4253. Jordan A.K., and S. Lakshmanasamy, 1989, "Inverse scattering theory applied to the design of single-mode planar optical waveguides,"J. Opt. Soc. Am.A 6, 1206-1212. Kay, I., 1955, The Inverse Scattering Problem, Rep. EM-74, New York, N.Y.: New York University. Kay, I., and H. Moses, 1956, "Reflectionless transmission through dielectrics and scattering potentials,"J. Appl. Phys.27, 1503-1508. Kritikos, H.N., D.L. Jaggard, and D.B. Gc, 1982, "Numeric reconstruction of smooth dielectric profiles,"Proc. IEEE70, 295-297. Lamb, Jr., G.L., 1980, Elements of Soliton Theory, New York: Wiley and Sons. Levy, H., and E.A. Baggott, 1976, Numerical Solution of DifferentialEquations, New York: Springer-Verlag. Marchenko, V.A., 1950, "Concerning the theory of a differential operator of second order,"Dokl. Akad. Nauk SSSRT2, 457-463. Marcuse, D., 1972, Light Transmission Optics, Princeton, N.J.: Van Nostrand Reinhold. Mills, D.W., and L.S. Tamil, 1991, "A new approach to the design of graded-index guided-wave devices,"IEEE Microwave Guided WaveLett.1, 87-88. Mills, D.W., and L.S. Tamil, 1992, "Analysis of planar waveguides using scattering data,"J. Opt. Soc. Am.A 9, 1769-1778. Mills, D.W., and L.S. Tamil, 1993, "Synthesis of guided-wave optical interconnects,"IEEE J. Quantum Electron.29, 2825-2834. Mills, D.W., and L.S. Tamil, 1994, "Coupling in multilayer optical waveguides: An approach based on scattering data,"J. Lightwave Technol .9, 1560-1568. Okoshi, T., 1976, Optical Fibers, New York: Academic Press. Press, H.H., B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling, 1989, Numerical Recipes, New York: Cambridge University Press. Szu, H.H., C.E. Carroll, C.C. Yang, and S. Ahn, 1976, "A new functional equation in the plasma inverse scattering problem and its analytical properties,"J. Math. Phys.17, 1236-1247. Tamil, L., and Y. Lin, 1993, "Synthesis and analysis of optical planar waveguides with prescribed TM modes,"J. Opt. Soc. Am.A 9, 1953-1962. Tamir, T., 1990, Guided-Wave Optoelectronics, New York: Springer-Verlag, Chap. 2. Yukon, S.P., and B. Bendow, 1980, "Design of waveguides with prescribed propagation constants,"J. Opt. Soc. Am.70, 172-179.