ACKNOWLEDGMENTS

This work was supported in part by PronexrMCT, CPqD-Telebras, FAPESP, and CNPq.´

REFERENCES

1. H. C. Casey, Jr. and M. B. Panish, Heterostructure Lasers. Part A:Fundamental Principles, Academic Press, New York, 1978, p. 166.

2. D. I. Babic and S. W. Corzine, ‘‘Analytic Expressions for Reflec-tion Delay, Penetration Depth, and Absorptance of Quarter-WaveDielectric Mirrors,’’ IEEE J. Quantum Electron., Vol. 28, 1992, pp.514]524.

3. Adachi, ‘‘GaAs, AlAs and Al Ga As: Material Parameters forx 1yxUse in Research and Device Applications,’’ J. Appl. Phys., Vol. 58,1985, pp. R1]R29.

4. Scott, R. S. Geels, S. W. Corzine, and L. A. Coldren, ‘‘ModelingTemperature Effects and Spatial Hole Burning to Optimize Verti-cal-Cavity Surface-Emitting Laser Performance,’’ IEEE J. Quan-tum Electron., Vol. 29, 1993, pp. 1295]1308.

5. B. Tell, K. F. Brown-Goebeler, R. E. Leibenguth, F. M. Baez, andY. H. Lee, ‘‘Temperature Dependence of GaAs-AlGaAsVertical-Cavity Surface Emitting Lasers,’’ Appl. Phys. Lett., Vol.60, 1992, pp. 683]685.

Q 1998 John Wiley & Sons, Inc.CCC 0895-2477r98

INVERSION OF ARBITRARYELECTROMAGNETIC SCATTERINGDATA TO RECONSTRUCT THESCATTERING POTENTIAL OFA PLASMA MEDIUMYan Zhang,1 Jin Au Kong,1 and Arthur K. Jordan21 Department of Electric Engineering and Computer ScienceMassachusetts Institute of TechnologyCambridge, Massachusetts 021392 Remote Sensing DivisionNaval Research LaboratoryWashington, District of Columbia 20375

Recei ed 15 August 1997

ABSTRACT: A numerical method using matrix in¨ersion has been( )generalized to sol e the Gel’fand]Le¨itan]Marchenko GLM in¨erse

scattering problem for arbitrary forms of reflection coefficient data forplasma medium. An accurate representation of the reflection transientwas calculated using the Gauss]Legendre method with a small numberof sampling data points of the reflection coefficient. The reflectioncoefficient has also been deri ed in a closed form for a uniform plasmalayer, and the in¨ersion result accurately reconstructed the originalprofile. The algorithm can also be applied to reconstruct the 1-D profilefor inhomogeneous plasma medium from measured reflection coefficientdata. Q 1998 John Wiley & Sons, Inc. Microwave Opt Technol Lett 17:97]101, 1998.

Key words: in¨erse scattering; GLM integral equation; plasma layer

1. INTRODUCTION

The inverse scattering theory for one-dimensional scatteringŽ . w xpotential q x has been extensively studied 1, 3]5 . The

physical model is specified in Figure 1 in which the physicalŽ .properties, namely the permittivity « x, k , are to be ob-1

tained from the scattered data. An electromagnetic wave in

Figure 1 Configuration of electromagnetic inverse scattering prob-lem from an inhomogeneous medium

y-polarization is normally incident onto an inhomogeneousˆŽ .medium with the permittivity « x, k , so that the time-1

harmonic wave equation in the inhomogeneous medium is

­ 22Ž . Ž . Ž . Ž .E x , k q k x E x , k s 0 1y 1 y2­ x

where k is the wavenumber in free space, and the wavenum-ber in the inhomogeneous medium is

2 Ž . 2 Ž . Ž .k x s v « x , k m . 21 1 o

Since the plasma medium is assumed to be inhomoge-neous, the permittivity « is a function of position x as well1as the wavenumber in free space k. By defining the scatteringpotential as

Ž . 2 2 Ž . Ž .q x , k s k y k x , 31

Ž .Eq. 3 can be written as a Schrodinger-type wave equation:¨

­ 22Ž . Ž . Ž . Ž . Ž .E x , k y q x , k E x , k s yk E x , k . 4y y y2­ x

We consider a special case in which the inhomogeneousregion is considered to be a plasma medium with the disper-sion relation

k2pŽ . Ž .« x , k s « 1 y 51 o 2ž /k

where k is the plasma wavenumber which is defined asp

e2mo2 Ž . Ž .k s N x 6p m

Ž .in which N x is the electron density, and e and m are theŽ . Ž . Ž .electron charge and mass, respectively. From Eqs. 2 , 3 , 5 ,

Ž .and 6 ,

Ž . 2N x e moŽ . Ž .q x , k s 7m

which indicates that the scattering potential for a plasmamedium depends on position only. The scalar electric field

Ž . Ž .E x, k in the wave equation 4 can be transformed toy

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 17, No. 2, February 5 1998 97

Ž .C x, t in the time domain with the transformation of

Ž . Ž . yick t Ž .C x , t s c E x , k e dk 8H y

where c is the speed of light in free space. Thus, the waveequation in the time domain is

­ 2 1 ­ 2

Ž . Ž . Ž . Ž . Ž .C x , t y C x , t y q x C x , t s 0. 92 2c­ x ­ t

Ž .The field C x, t in the inhomogeneous region II can beŽ . Ž .expressed in terms of C x, t in free space region I :o

xŽ . Ž . Ž . Ž . Ž .C x , t s C x , t q K x , y C y , t dy 10Ho o

yx

Ž .where the total field C x, t in region I is the sum of theoŽ . Žincident impulse d x y ct and the reflection transient R x q

.ct :

Ž . Ž . Ž . Ž .c x , t s d x y ct q R x q ct . 11o

Ž . Ž .With Eqs. 9 ] 11 , considering the spectral properties of theboundary value problem, it can be shown that the kernel

Ž . w xfunction K x, ct satisfies the following integral equation 2 ,Ž .which is called the Gel’fand]Levitan]Marchenko GLM in-

tegral equation:

xŽ . Ž . Ž . Ž .K x , ct q R x q ct q K x , y R y q ct dy s 0,H

yx

Ž .for x ) 0, t ) 0 12

with the boundary conditions

Ž .R x q ct s 0, for x q ct F 0Ž .K x , y s 0, for y ) x or y F yx

and

Ž .d q xŽ . Ž .K x , x s . 13

dx 2

Ž .Once the kernel function has been solved from Eq. 12Ž .with a given reflection transient R x q ct , the scatteringŽ .potential can be calculated with Eq. 13 . However, only for

special cases, such as the reflection transient being derivedfrom the reflection coefficient in the form of a rational

Ž .function, can the analytical solution to Eq. 13 be obtained.In our previous work, a numerical method based on matrix

inversion was developed to solve the GLM inverse problemw xfor arbitrary forms of the reflection coefficient 1 . In this

paper, considering a homogeneous plasma layer without los-ing generality, we will derive the reflection coefficient inclosed form, and then calculate the reflection transient usingthe Fourier transform and the Gauss]Legendre integralmethod. Using the reflection transient for the plasma slab, wewill calculate the scattering potential and compare it with theoriginal profile.

2. REFLECTION COEFFICIENT OF AHOMOGENEOUS PLASMA LAYER

In this section, we consider a plasma layer with uniformelectron density, so that the scattering potential in the layeris constant. As we will show later, the reflection coefficient of

the homogeneous profile is not a rational function; therefore,w xthe numerical approach 1 is used to reconstruct the profile.

The numerical method can be applied to the general inverseproblem for 1-D plasma medium. The configuration of the1-D plasma layer is shown in Figure 1, in which the depth of

Ž .the slab is l and the electron density N x is constant. TheŽ .outside of the plasma layer regions I and III is the free

ik xŽ .space. For the normal incident wave E x s ye , the waveˆequation is

­ 22Ž . Ž . Ž . Ž .y q q x E x , k s k E x , k 142ž /­ x

Ž . Ž .where q x s 0 for regions I and III, and q x is a constantin region II.

The solutions to the wave equation in different regionsare

Ž . ik x Ž . yi k x Ž .E x s e q r k e , in region I 151

Ž . ik1 x yi k1 x Ž .E x s Ae q Be , in region II 162

Ž . ik x Ž .E x s Ce , in region III 173

2 2where k s k y k . By matching the boundary conditions'1 p

at x s 0 and x s l so that the tangential electric and mag-netic fields are continuous, we get

Ž .1 q r k s A q BŽ . Ž .k y kr k s k A y k B 181 1

and

Aeik1 l q Beyik1 l s Ceikl

ik1 l y i k1 l ikl Ž .k Ae y k Be s kCe . 191 1

Ž .Solving for r k by eliminating A, B, and C, we get thereflection coefficient

1 y g 2

Ž . Ž .r k s 202Ž . Ž .1 q g q i2g cot k l1

where

2kp Ž .g s k rk s 1 y . 21(1 2k

Ž . Ž .Using the symmetry property r yk s r* k , the reflectiontransient can be calculated by

1 `yi ktŽ . Ž .R t s r k e dkH2p y`

1 `� w Ž .x Ž . w Ž .x Ž .4s Re r k cos kt q Im r k sin kt dkH

p 0

Ž .22

where t s ct.

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 17, No. 2, February 5 199898

3. NUMERICAL RESULTS OF THEPROFILE RECONSTRUCTION

Ž .In the GLM integral equation 12 , by changing the variablet s ct and discretizing y and t into N q 1 grid points with

Ž .equal spacing h s 2 xrN, we get y s yx q 2 xrN j for theŽ . Žjth point and t s yx q 2 xrN i for the ith point i, j s

. Ž .0, 1, 2, . . . , N . The variable x is fixed to x s x rM m formaxeach solution of the GLM integral equation at step m, whereM is the total number of grid points, x is the maximummaxvalue of x, and m s 0, 1, 2, . . . , M is the step number. There-

Ž .fore, the corresponding difference equation of 12 is

N

Ž .K q R q c hK R s 0 23Ým , i m , i j m , j j , ijs0

where

0.5, for j s 0 or N Ž .c s 24j ½ 1, otherwise

and

x x 2 xmax max maxŽ . Ž .K s K x , t s K m , y m q mi 25m , i ž /M M MN

2 xmaxŽ . Ž .R s R x q t s R mi 26m , i ž /MN

x x 2 xmax max maxŽ . Ž .K s K x , y s K m , y m q mj 27m , j ž /M M MN

2 x i q jmaxŽ . Ž .R s R y q t s R m y 1 . 28j, i ž /ž /M N

Ž .Let K s d K ; then Eq. 23 becomesm, i i j m , j

N

w x Ž .d q c hR K s yR . 29Ý i j j j, i m , j m , ijs0

Thus, we get

y1 Ž .K s A ? R 30m , i m i

w x w xwith A s d q c hR and R s y R . Once the ker-i j j j, i m m , inel K has been solved for each m, the potential is foundm, ito be

K y K 2 Mm , N my1, N Ž . Ž .q s 2 s K y K . 31m m , N my1, ND x xmax

Ž .To calculate the reflection transient R x from the reflec-Ž . Ž .tion coefficient r k , the Fourier transform 22 is performed.

However, it is not suitable to use the FFT algorithm in whichŽ . Ž .both r k and R x are uniformly discretized because a more

Ž .accurate R x is expected for solving the GLM equation. TheŽ .Gauss]Legendre method is used to evaluate the integral 22 ;

thus,

Nk1Ž . � w Ž .x Ž . w Ž .x Ž .4R t f w Re r k cos k t q Im r k sin k tÝ j j j j jp js1

Ž .32

where N is the number of sampling points for k between 0kand the cutoff wavenumber k , and w is the weightingc jcoefficient for the corresponding location of k .j

Ž . Ž .Figure 2 Reflection coefficient r k for normal plane incident wave on a uniform plasma layer with scattering potential q x s 1.0Ž y2 . Ž .m and width l s 1.0 m

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 17, No. 2, February 5 1998 99

Ž . Ž . Ž y2 .Figure 3 Reflection transient R t for normal incident impulse on a uniform plasma layer with scattering potential q x s 1.0 mŽ . Ž .and width l s 1.0 m . The peak value of R t was normalized to unity

Ž . Ž y2 . Ž .Figure 4 Reconstructed profile for a uniform plasma layer with scattering potential q x s 1.0 m , width l s 1.0 m , andŽ . Ž .different cutoff frequencies k for reflection coefficient r k . The number of sampling points for r k is N s 64c k

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 17, No. 2, February 5 1998100

Ž . Ž y2 . Ž .Figure 5 Reconstructed profile for a uniform plasma layer with scattering potential q x s 1.0 m and width l s 1.0 m . For theŽ . Ž y1 .reflection coefficient r k , the cutoff wavenumber is k s 100.0 m and the number of sampling points is N s 256c k

In the numerical simulation, we consider a uniform plasmaŽ . Ž y2 .layer with scattering potential q t s 1.0 m and width

Ž . Ž .l s 1.0 m . The reflection coefficient r k was calculatedŽ .from Eq. 20 , and the curves are shown in Figure 2. In

Ž .Figure 3, the reflection transient R t is calculated from Eq.Ž .32 for which the number of points is N s 256 and thek

Ž y1.cutoff wavenumber is k s 100.0 m . The reflection tran-cŽ .sient R t is the echo for an ideal normal incident impulse.

Ž .With the reflection transient R t , the GLM integral equa-Ž .tion is solved for the kernel function K x, t , and thus the

Ž . Ž .scattering potential q x is calculated with Eq. 31 . TheŽ .accuracy of q x depends on the cutoff frequency k as wellc

Ž .as the number of sampling points N for r k . The larger thekvalues of k and N , the more details can be obtained for thec kscattering potential. The numerical results of scattering po-tential reconstruction with different cutoff frequencies areshown in Figure 4. The number of sampling points waschosen to be N s 64 for all of the cases in Figure 4. Thek

Ž y1.curve with k s 40 m agrees with the slab profile bettercthan others since a wider band is included.

To get a more accurate result to reconstruct the scatteringŽ . Ž y1.potential q x , we increase k and N to be k s 100 mc k c

and N s 256. A good agreement is shown in Figure 5 inkŽ .comparison with the reconstructed q x and the expected

profile. The computing time on a DEC alpha machine wasabout 10 min CPU time for this case.

4. CONCLUSIONS

Ž .The Gel’fand]Levitan]Marchenko GLM inverse theory hasbeen applied to reconstruct the scattering potential of 1-Dplasma medium from the reflection coefficient. The numeri-cal algorithm based on matrix inversion to solve the GLMintegral equation has been developed, and is able to calculatethe scattering potential from the general form of the reflec-tion coefficient. As an example, the reflection coefficient fora homogeneous plasma layer is no longer a rational function;

thus, the closed solution for scattering potential is not feasi-ble. The reflection transient of the plasma layer has beencalculated from the reflection coefficient using the Fouriertransform with the Gauss]Legendre integral method, and thescattering potential has been reconstructed by applying theGLM inverse theory. With the choice of the number ofsampling points and cutoff frequency for the reflection coef-ficient in the example, a good agreement has been achievedfor the reconstruction of the scattering potential in compari-son with the assumed profile.

ACKNOWLEDGMENT

This work was supported by ONR Contracts N00014-92-J-4098and N00014-97-I-0172.

REFERENCES

1. Y. Zhang, J. A. Kong, and A. K. Jordan, ‘‘Numerical Solution ofGel’fand-Levitan-Marchenko Integral Equation for Electromag-netic Inverse Scattering Theory Using Matrix Inversion,’’ Mi-crowa¨e Opt. Technol. Lett., Vol. 15, Aug. 5, 1997.

2. I. M. Gel’fand and B. M. Levitan, ‘‘On the Determination of aDifferential Equation from Its Spectral Function,’’ Trans. Amer.Math. Soc., Ser. 2, Vol. 1, 1955, pp. 253]304.

3. I. Kay, ‘‘The Inverse Scattering Problem When the ReflectionCoefficient is a Rational Function,’’ Commun. Pure Appl. Math.,Vol. XIII, 1960, pp. 371]393.

4. A. K. Jordan and H. N. Kritikos, ‘‘An Application of One-Dimen-sional Inverse-Scattering Theory for Inhomogeneous Regions,’’IEEE Trans. Antennas Propagat., Vol. AP-21, Nov. 1973, pp.909]911.

5. A. K. Jordan and S. Lakshmanasamy, ‘‘Inverse Scattering TheoryApplied to the Design of Single-Mode Planar Optical Wave-guides,’’ J. Opt. Soc. Amer. A, Vol. 6, Aug. 1989.

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