Inverse Scatter

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1916 J. Opt. Soc. Am. A/Vol. 2, No. 11/November 1985

Renormalization of an inverse-scattering theory for

inhomogeneous dielectrics

H. D. Ladouceur

Chemistry Division, Naval Research Laboratory, Washington, D.C. 20375

A. K. Jordan

Space Science Division, Naval Research Laboratory, Washington, D.C. 20375

Received May 6, 1985; accepted July 25, 1985

Renormalized solutions are obtained for an inverse-scattering problem that are equivalent to the second-orderregular perturbation approximations for the exact (Gel'fand-Levitan-Marchenko) theory. We have developed aninversion method for reconstruction the permittivity profiles of inhomogeneous dielectric slabs from reflection-coefficient data. Solutions with increased radii of convergence are obtained. Numerical examples are demonstrat-ed for simulated-scattering data from Gaussian and parabolic profiles and homogeneous slabs.

INTRODUCTION

Inverse-scattering theory seeks to reconstruct the unknownphysical properties of an object from information containedin the scattering data. In this paper we consider the recon-struction of the dielectric profile of an inhomogeneous slabfrom the electromagnetic reflection coefficient. Approxi-mate solutions to this problem are often obtained by consid-ering a limited range of frequencies or wavelengths. Forexample, the wave equation for the electric field can beconverted into an integral equation, which is then solved byiteration. In the first iteration the Born approximation isthe solution obtained by assuming small phase shifts in thescattered field. The radius of convergence or this approx-imation is limited to small values of the wave number, sothat inversion methods based on this approximation willhave a limited range of applicability.

In this paper we investigate a method to increase theradius of convergence of approximate solutions of inverse-scattering problems by using renormalization. We base ourinversion method on the exact theory that has been devel-oped by Kay,' Moses, 2 and Kay and Moses 3 from the mathe-matical investigations of Gel'fand and Levitan4 and Mar-chenko.5 The search for approximate solutions to be usedfor the electric field within inhomogeneous dielectric layersled us to consider the method of multiple scales. 6

By considering the continuity of the electric field and itsderivative at the boundary of the slab and by using therenormalized electric field, a rapidly convergent solution isobtained that is equivalent to a second-order perturbationtreatment of the exact inverse-scattering theory. In thelowest order, the renormalized electric field is equivalent tothe Wentzel-Kramers-Brillouin (WKB) approximation. 7This theory is demonstrated by numerical examples usingdata obtained from a numerical simulation of scatteringfrom Gaussian and parabolic dielectric profiles and homoge-neous slabs.

INVERSION THEORY

We consider the idealized physical model shown in Fig. 1. Atime-harmonic plane-polarized electromagnetic wave ofwave number k = 27r/X

(1)

is normally incident from the left upon an inhomogeneousdielectric slab of thickness L. The permittivitiy relative tofree space Er s a function of the geometric distance x; we firstdemonstrate our method with er defined by the truncatedGaussian function

-bx)ebX2 l /211.0, Ixi L/2 (2)

where b is chosen to ensure continuity of er at Ixi = /L/2.The wave amplitude E (x, k) satisfies the scalar Helmholtzequation

d2E(x, k) + k 2Er(r) E(x, k) = 0.dx 2

The reflected wave amplitude is

E8 c(x,k) = r(k) e-ix, x -

(3)

(4)

where r(k) is the complex reflection coefficient.The electromagnetic path length s in the slab is

s = (x, k) = | k / r(X) dx'J-/

(5)

which is the Liouville transformation between geometric xspace and path length s space.8 The inverse problem thatwe consider is based on the Helmholtz Eq. (3) with an arbi-trary relative permittivity er(X). If Er(X)s

amonotonic

func-tion of x and independent of k, differential Eq. (3) can betransformed into a Schrodinger-type equation in s spacewhose potential function q(s) can be determined uniquely if

0740-3232/85/111916-06$02.00 1985 Optical Society of America

H. D. Ladouceur and A. K. Jordan

ikx -L/2,i,,(x, k) = e , i = F1, x


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1918 J. Opt. Soc. Am. A/ Vol. 2, No. 11/November 1985

slab can be expressed in terms of the electromagneticjlength s as

d 2 EWx dE(s)d2 + g(x) ds + E(s) = 0,

where

(x)1 r(\-3/2 der(x)gX) 2k tEr(x)], dx

If the function g(x) is small in magnitude,

max jg(x)j = y) < 1, I xj < L/2,

then we can apply a regular perturbation expansion in rers of y to Eq. (20). However, such an expansion wouldbe uniformly valid because of the presence of secular tein the higher-order approximations. Moreover, the expfunctional form of g(x) is unknown in the inverse proble

In performing the transformation of the Helmholtz Eqinto the Schr6dinger Eq. (8), it was implicity assumed th

can be determined from Eq. (5). The method of multscales is used to provide an effective technique for sumrperturbation sequences when the explicit functional for:Er(x) s not known. Using this method, we seek an apprmate solution for the electric field within the slab in the f(

E[s(x, k), x] = EO(s, ) + yE,(s, x) + y2 E 2 (s, x) + .. ., (

where s and x are regarded as the fast and slow varialrespectively. The independent variable s in Eq. (20) isplaced by the set (so, x), where we formally set s so.using the chain rule, the s derivative is transformed acccing to

d = d,, .yds a, Ox

Thus the ordinary differential equation for E(s, k) becompartial differential equation (PDE) for E(s, x, k). This Pcan now be written through the first order in the perturtion parameter y as

o: a Eo(s, x) + Eo(s, x) = 0,

ly : 2E ) + E,(s, X) = g(x) aE 0 (s, x)

2 022- EO(s,). (

The general solution of Eq. (23) is given by

EO[s(x, k)x] = Ao(x) eis + Bo(x)e-is. (

Since we are interested only in rightward-traveling warwe set Bo(x) = 0. The zero-order solution is coupled ,wthe first-order equation for the electric-field amplituThis coupling leads to secular growth in E1(s, x), whviolates energy conservation. In order to eliminate tgrowth, the functional form of Ao(x) must be determirfrom

( [E(x)L3/2 der /( aOA dEo =-,,X]A (x)+2 [e(x)]11 - 0,(t2 r dx r Ox O

path which has the solution

Ao(x) = [c,(x)]-11,

(20) so that the renormalized form for Eo(s, x) becomes

Eo(s, ) = Eis[Er(x)]-1/4, s = s(x, k).

(28)

(29)

The functional form (28) was used to transform the original(21) Helmholtz Eq. (3) into a Schr6dinger equation. The multi-ple-scale technique demonstrates that this functional formarises from denying secular growth in the first-order correc-tion for the electric-field amplitude. This process of sum-

(22) ming perturbation expansions to make them more uniformly)ow- valid is called renormalization.1 21not Equation (20) can be converted into an integral equationerms by using an appropriate Green function. Consideration ofilicit the asymptotic form of the wave solution gives an exactm. expression for the reflection coefficient[. 3) ~L/2 der diatx r(k) = - J [Er(x)]3 2 d rdE[s(x, k), x]lds. (30)

t~iple 2-L/2dxsning Substituting the renormalized solution (29) into Eq. (30)m of and applying .the Fourier transform gives the approximateroxi- solution for Er(s) in s space,

Er([) = exp [4 J (z)dz] (31)where the R(z) term in the exponent represents the Fourier

Jes, transform of the scattering data, as is discussed in the nextre- section.By A complete inversion method should also be able to obtain

)rd- Eras a function of the geometric distance x, as discussed byKay.' The Born approximation relates Er n x space to theFourier transform of r(k). It is possible with our inversion

:24) method to transform from s space to x space if the slabthickness L is known. The WKB approximation corre-

esa sponds to the first term of the perturbation solution in kDE space.ba- Since wewant to relate a discrete set of data points for Er(s)

to the corresponding points within the slab in x space, weconsider the inhomogeneous region to be composed of thindielectric slabs of thickness A, so that xj = -L/2 + (1 - 1) A, 1= 1, . .. , N. The number of data points N can be deter-mined since we also know the beginning and the end pointsof the slab in both x and s spaces; namely, xi = -L/2 corre-sponds to s = 0, and XN = L/2 corresponds to s = k -L/2i\I/(x') dx', as shown in Fig. 3 below. In this example, N can

25) be chosen conveniently to divide the region in s space whereEr iffers from the permittivity of free space; in this example,N is approximately 50. In order to reconstruct the dielectricprofile in x space, we expand the profile in a Maclaurin series

vf;) about the center of the slab

Er(x) = Er(0) + Er'(0) x + Er"(0) x 2/2 + .... (32)

The data points for Er(s) obtained from Eq. (31) can betransformed to the equivalent data points for Er(x) by fittingto a least-squares polynomial whose coefficients can be re-lated to Eq. (32).

GENERATION OF SCATTERING DATAThe experimental reflection data are simulated by numeri-cal integration of the exact Riccati equation for the complex



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Vol. 2, No. 11/November 1985/J. Opt. Soc. Am. A 1919

3.0

2.5 REFLECT

2.0-

~1.5

1.0

0.5

0.00 '0 6 10

Fig. 2. Simulated reflection data to be inverted to obtain the prodiscrete values of k. In this example, A =1.80, b = 0.59, andL = 2.0.scale.

reflection coefficient r(k). The direct-scattering problem ismodeled by a single slab whose permittivity profile is aGaussian function centered at the midpoint of the inhomo-geneous region, as is shown in Fig. 1. The magnitude of therelative permittivity Er s matched to the relative permittivi-ty of free space at the boundaries of the slab Ixl = L/2. Thereflection coefficient at the left face of the slab r(k) = r(k, -L/2), which is used as a data point for inversion, is obtainedfrom the solution to the Riccati equation' 3

dr -k 1 [ L(x, k) = iEr(x) [1 + r]2 + [1-r] 2 ], jxl ' (33)

dx 2 2

with the boundary condition r(k, + L/2) = 0. The reflectioncoefficient r(k) is calculated at discrete points xl hrough theslab for each value of wave number k.

We use a fourth-order Runge-Kutta method with 500 in-crements in x to compute the complex reflection coefficientsr(k) at discrete wave numbers, k, = (1 - p) 6, p = 1, ... M(in our case M = 256 points were used for k = 0 to 25, so that

the increment a = 0.098). The modulus of r(k) is shown inFig. 2 for a typical set of parameters.

For use in the inversion method, this set of reflection datais then represented by a complex Fourier series

aoM-2r(kp) = 2+ E am cos(Gm) sin() -ibm sin((m)

m=1

+ aM cos(rkp),2

15 20 25Tave Numberfile of permittivity Er(x). Reflection coefficient r(k) data given for 256The Born approximation is valid fork < 0.6. Insert depicts er(x) drawn to

with the series at the approximate wave numbers; the imagi-nary part of the reflection data is represented by a sineseries, so the r*(k) = r(-k), k real. This continuous repre-sentation of the complex reflection coefficient enables inter-polation and accurate numerical determination of the mini-ma and the maxima of r(k). It is interesting to note that theFourier transform ofEq. (34) gives an analytic expression forthe function R(s), which is the kernel needed for the solutionof the integral Eq. (16).

DISCUSSION OF RESULTS

We have demonstrated an inversion method for scatteringdata from a smooth dielectric profile. The scattering data ofFig. 2 were used in our inversion method to reconstruct thepermittivity profile of Fig. 3. Table 1 compares the recon-structed results with several known Gaussian profiles; theparameters A, b were obtained by a regression analysis onGaussian profiles in order to test the accuracy of this inver-sion method for increasing gradients in the dielectric inho-mogeneity. This inhomogeneity is measured by the param-eter -y, which can be shown from Eq. (21) to be equal to b/2.As expected, good agreement was obtained for smaller val-ues of b, and poorer agreement as b was increased

The correlation coefficient p tests how closely a linearregression model for a Gaussian function agrees with ourreconstructed profiles

(34)

where 0bm mkp7r/p, M = 256, and P is the half-period of thereflection coefficient. We note that our inversion algorithmassumes that r(k) is periodic; as is shown in Fig. 2, P = 25

approximately. The real part of the reflection coefficient isrepresented by a truncated cosine series whose coefficientsare determined by requiring that the data points collocate

41N - (x2)2]

(35)p _[In Ei(x,)]2/N (Ine)2}

where m is the slope of the regression model (m = b forGaussian profile), Er(xi) are the reconstructed values of the



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1920 J. Opt. Soc. Am. A/ Vol. 2, No. 11/November 1985

2.0

1.5 H

0

i)

0

1.0 k

0.5 H

0.0 _-20 -10 S position

Fig. 3. Reconstructed profile of permittivity Er(S) drawn to the same scale as that of the insert of Fig. 2. This is a plot of 512 data points ob-tained by applying the renormalized inversion theory to the data points of Fig. 2.

permittivity profile at the distance xi in the slab, and N is thenumber of increments. The average value of In e is In E, andthe average value of xi is x. In terms of a linear regressionanalysis p = 1 indicates that the reconstructed profile agreesperfectly with the model profile.

Table 2 tests the accuracy of our method when applied tothree different profiles: Gaussian, parabola, and homoge-neous slab. Simulated numerical scattering data were gen-

erated by solving for each of these profiles, as characterizedby equivalent A and b parameters. The inversion methodwas then applied to the three sets of data, and profiles of E,were obtained in s space and in x space. By assuming thatthe unknown profile for Er(x) can be developed as a series ofthe form (32), the coefficients of the least-squares polynomi-al can be related to the parameters A, b of an assumedGaussian, parabolic, or homogeneous slab model. A regres-sion analysis with the three model profiles was used to com-pare the profile parameters A, b; the correlation coefficientsp are also shown in Table 2. We note that the b parametersfor the parabola and the homogeneous slab do not ensurecontinuity of Er at Ixl = L/2. For example, the approxima-tion for the Gaussian profile is written

ase,(x) - A(1 - bx2 ). (36)

The elements along the diagonal showresults that are due to"good guesses," i.e., the least-squares analysis compared thereconstructed profile with the same model profile that wasused in the direct calculation of the reflection data. The off-diagonal elements shows results comparing "wrong" regres-sion models with different direct models. Note that thismatrix is not symmetric.

From these results we see that, to second order, it is notclear whether the Gaussian or the parabola is the better fit;the homogeneous slab is clearly a poor fit. The qualitativefeatures of the slab can be reconstructed by the

inversionalgorithm, but the magnitude of Er is too large by 20-30%.

This is expected since the homogeneous slab presents asharp discontinuity that violates condition (22) for conver-gence of the perturbation series. We conjecture that ourinversion method can be extended to discontinuities andsteep gradients by boundary-layer theory.

The results of Table 1 show that the renormalized inver-

Table 1. Comparison of Reconstructed Profiles withInput Gaussian Profiles

Input Parameters Reconstructed ParametersA b A b P

1.50 0.41 1.59 0.41 0.991.80 0.59 1.94 0.60 0.974.00 1.39 4.70 1.25 0.956.00 1.79 7.19 1.55 0.958.00 2.08 9.54 1.59 0.94

10.00 2.30 11.54 1.87 0.92

Table 2. Comparison of Inversion Results forParameters A, b for Gaussian and Parabolic Profiles

and Homogeneous SlabsaRegression Model

Direct Model Gaussian Parabola Slab

Gaussian A = 1.94 A = 1.92 A= 1.62b = 0.60 b = 0.47 b -0.004P = 0.97 p = 0.96 p =0.009

Parabola A = 1.64 A = 1.61 A = 2.16b = 0.71 b = 0.53 b -0.07p = 0.96 p = 0.97 p =0.174

Slab A = 3.40 A = 3.35 A 2.99b = 0.43 b = 0.31 b -1.81p = 0.43 p = 0.43 p 0.519

a Input Gaussian-profile parameters, A = 1.80, b = 0.59,L = 20.

SLAB

FREE SPACEFREE SPACE

RECONSTRUCTED DIELECTRIC PROFILE


0 in 20


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Vol. 2, No. 11/November 1985/J. Opt. Soc. Am. A 1921

sion theory has a larger radius of convergence than the Bornapproximation. Our inversion method also includes phaseinformation that is neglected in the Born approximation14

(kL)[fr(x) - 1]

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