Abstract

An efficient method is presented here for reconstructing dispersionless refractive profiles from scattering data. This procedure makes use of known reflection data to determine the refractive index of a semi-infinite slab. The numerical method presented here avoids matrix inversion, successive differentiation, and the solution of auxiliary algebraic or differential equations. Instead, the method relies on leapfrogging through space and time to solve a recently developed integral equation. Results for several new profiles are displayed, and evidence of the robustness of the technique with respect to noise is exhibited.

© 1985 Optical Society of America

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References

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  1. I. M. Gel’fand, B. M. Levitan, “On the determination of a differential equation by its spectral function,” Am. Math. Soc. Transl. 1, 253–304 (1955).
  2. V. A. Marchenko, “Reconstruction of the potential energy from the phase of scattered waves,” Dokl. Akad. Nauk SSR 104, 695–698 (1955).
  3. I. Kay, “The inverse scattering problem,” Res. Rep. No. EM–74 (New York University, New York, N.Y., 1955).
  4. H. E. Moses, C. M. deRidder, “Properties of dielectrics from reflection coefficients,” Tech. Rep. 322 (MIT Lincoln Laboratory, Lexington, Mass., 1963).
  5. O. Hald, “Numerical solution of the Gel’fand–Levitan equation,” Linear Alg. Its Appl. 28, 99–111 (1979).
    [CrossRef]
  6. S. Coen, “Inverse scattering of a layered and dispersionless dielectric half-space, part I: reflection data from plane waves at normal incidence,” IEEE Trans. Antennas Propag. AP–29, 726–732 (1981).
    [CrossRef]
  7. K. R. Pechenick, J. M. Cohen, “Inverse scattering—exact solution of the Gel’fand–Levitan equation,” J. Math. Phys. 22, 1513–1516 (1981).
    [CrossRef]
  8. G. N. Balanis, “Inverse Scattering: determination of inhomogeneities in sound speed,” J. Math. Phys. 23, 2362–2568 (1982).
    [CrossRef]
  9. This is obtained from Eq. (2.8) of Ref. 8 with a change of variables and subsequent reordering.
  10. This is found by combining Eqs. (3.5) and (3.8) of Ref. 8.
  11. Here ΔK indicates the difference between the kernel function at any point and its value at an adjacent lattice point.
  12. H. N. Kritikos, D. L. Jaggard, D. B. Ge, “Numerical reconstruction of smooth dielectric profiles,” Proc. IEEE 70, 295–297 (1982).
    [CrossRef]
  13. J. A. Ware, K. Aki, “Continuous and discrete inverse scattering problems in a stratified elastic medium: I—plane waves at normal incidence,” J. Acoust. Soc. Am. 45, 911–921 (1969).
    [CrossRef]
  14. A. K. Jordan, “Applications of the Gel’fand–Levitan theory to profiles with bandpass characteristics,” Ph.D. dissertation (University of Pennsylvania, Philadelphia, Pa., 1972).
  15. P. B. Abraham, H. E. Moses, “Exact solutions of the one-dimensional acoustic wave equations for several new velocity profiles: transmission and reflection coefficients,” J. Acoust. Soc. Am. 71, 1391–1399 (1982).
    [CrossRef]
  16. D. L. Jaggard, P. Frangos, “The inverse scattering problem for layered dispersionless media with incomplete and imprecise data,” submitted to IEEE Trans. Antennas Propag.

1982 (3)

G. N. Balanis, “Inverse Scattering: determination of inhomogeneities in sound speed,” J. Math. Phys. 23, 2362–2568 (1982).
[CrossRef]

H. N. Kritikos, D. L. Jaggard, D. B. Ge, “Numerical reconstruction of smooth dielectric profiles,” Proc. IEEE 70, 295–297 (1982).
[CrossRef]

P. B. Abraham, H. E. Moses, “Exact solutions of the one-dimensional acoustic wave equations for several new velocity profiles: transmission and reflection coefficients,” J. Acoust. Soc. Am. 71, 1391–1399 (1982).
[CrossRef]

1981 (2)

S. Coen, “Inverse scattering of a layered and dispersionless dielectric half-space, part I: reflection data from plane waves at normal incidence,” IEEE Trans. Antennas Propag. AP–29, 726–732 (1981).
[CrossRef]

K. R. Pechenick, J. M. Cohen, “Inverse scattering—exact solution of the Gel’fand–Levitan equation,” J. Math. Phys. 22, 1513–1516 (1981).
[CrossRef]

1979 (1)

O. Hald, “Numerical solution of the Gel’fand–Levitan equation,” Linear Alg. Its Appl. 28, 99–111 (1979).
[CrossRef]

1969 (1)

J. A. Ware, K. Aki, “Continuous and discrete inverse scattering problems in a stratified elastic medium: I—plane waves at normal incidence,” J. Acoust. Soc. Am. 45, 911–921 (1969).
[CrossRef]

1955 (2)

I. M. Gel’fand, B. M. Levitan, “On the determination of a differential equation by its spectral function,” Am. Math. Soc. Transl. 1, 253–304 (1955).

V. A. Marchenko, “Reconstruction of the potential energy from the phase of scattered waves,” Dokl. Akad. Nauk SSR 104, 695–698 (1955).

Abraham, P. B.

P. B. Abraham, H. E. Moses, “Exact solutions of the one-dimensional acoustic wave equations for several new velocity profiles: transmission and reflection coefficients,” J. Acoust. Soc. Am. 71, 1391–1399 (1982).
[CrossRef]

Aki, K.

J. A. Ware, K. Aki, “Continuous and discrete inverse scattering problems in a stratified elastic medium: I—plane waves at normal incidence,” J. Acoust. Soc. Am. 45, 911–921 (1969).
[CrossRef]

Balanis, G. N.

G. N. Balanis, “Inverse Scattering: determination of inhomogeneities in sound speed,” J. Math. Phys. 23, 2362–2568 (1982).
[CrossRef]

Coen, S.

S. Coen, “Inverse scattering of a layered and dispersionless dielectric half-space, part I: reflection data from plane waves at normal incidence,” IEEE Trans. Antennas Propag. AP–29, 726–732 (1981).
[CrossRef]

Cohen, J. M.

K. R. Pechenick, J. M. Cohen, “Inverse scattering—exact solution of the Gel’fand–Levitan equation,” J. Math. Phys. 22, 1513–1516 (1981).
[CrossRef]

deRidder, C. M.

H. E. Moses, C. M. deRidder, “Properties of dielectrics from reflection coefficients,” Tech. Rep. 322 (MIT Lincoln Laboratory, Lexington, Mass., 1963).

Frangos, P.

D. L. Jaggard, P. Frangos, “The inverse scattering problem for layered dispersionless media with incomplete and imprecise data,” submitted to IEEE Trans. Antennas Propag.

Ge, D. B.

H. N. Kritikos, D. L. Jaggard, D. B. Ge, “Numerical reconstruction of smooth dielectric profiles,” Proc. IEEE 70, 295–297 (1982).
[CrossRef]

Gel’fand, I. M.

I. M. Gel’fand, B. M. Levitan, “On the determination of a differential equation by its spectral function,” Am. Math. Soc. Transl. 1, 253–304 (1955).

Hald, O.

O. Hald, “Numerical solution of the Gel’fand–Levitan equation,” Linear Alg. Its Appl. 28, 99–111 (1979).
[CrossRef]

Jaggard, D. L.

H. N. Kritikos, D. L. Jaggard, D. B. Ge, “Numerical reconstruction of smooth dielectric profiles,” Proc. IEEE 70, 295–297 (1982).
[CrossRef]

D. L. Jaggard, P. Frangos, “The inverse scattering problem for layered dispersionless media with incomplete and imprecise data,” submitted to IEEE Trans. Antennas Propag.

Jordan, A. K.

A. K. Jordan, “Applications of the Gel’fand–Levitan theory to profiles with bandpass characteristics,” Ph.D. dissertation (University of Pennsylvania, Philadelphia, Pa., 1972).

Kay, I.

I. Kay, “The inverse scattering problem,” Res. Rep. No. EM–74 (New York University, New York, N.Y., 1955).

Kritikos, H. N.

H. N. Kritikos, D. L. Jaggard, D. B. Ge, “Numerical reconstruction of smooth dielectric profiles,” Proc. IEEE 70, 295–297 (1982).
[CrossRef]

Levitan, B. M.

I. M. Gel’fand, B. M. Levitan, “On the determination of a differential equation by its spectral function,” Am. Math. Soc. Transl. 1, 253–304 (1955).

Marchenko, V. A.

V. A. Marchenko, “Reconstruction of the potential energy from the phase of scattered waves,” Dokl. Akad. Nauk SSR 104, 695–698 (1955).

Moses, H. E.

P. B. Abraham, H. E. Moses, “Exact solutions of the one-dimensional acoustic wave equations for several new velocity profiles: transmission and reflection coefficients,” J. Acoust. Soc. Am. 71, 1391–1399 (1982).
[CrossRef]

H. E. Moses, C. M. deRidder, “Properties of dielectrics from reflection coefficients,” Tech. Rep. 322 (MIT Lincoln Laboratory, Lexington, Mass., 1963).

Pechenick, K. R.

K. R. Pechenick, J. M. Cohen, “Inverse scattering—exact solution of the Gel’fand–Levitan equation,” J. Math. Phys. 22, 1513–1516 (1981).
[CrossRef]

Ware, J. A.

J. A. Ware, K. Aki, “Continuous and discrete inverse scattering problems in a stratified elastic medium: I—plane waves at normal incidence,” J. Acoust. Soc. Am. 45, 911–921 (1969).
[CrossRef]

Am. Math. Soc. Transl. (1)

I. M. Gel’fand, B. M. Levitan, “On the determination of a differential equation by its spectral function,” Am. Math. Soc. Transl. 1, 253–304 (1955).

Dokl. Akad. Nauk SSR (1)

V. A. Marchenko, “Reconstruction of the potential energy from the phase of scattered waves,” Dokl. Akad. Nauk SSR 104, 695–698 (1955).

IEEE Trans. Antennas Propag. (1)

S. Coen, “Inverse scattering of a layered and dispersionless dielectric half-space, part I: reflection data from plane waves at normal incidence,” IEEE Trans. Antennas Propag. AP–29, 726–732 (1981).
[CrossRef]

J. Acoust. Soc. Am. (2)

J. A. Ware, K. Aki, “Continuous and discrete inverse scattering problems in a stratified elastic medium: I—plane waves at normal incidence,” J. Acoust. Soc. Am. 45, 911–921 (1969).
[CrossRef]

P. B. Abraham, H. E. Moses, “Exact solutions of the one-dimensional acoustic wave equations for several new velocity profiles: transmission and reflection coefficients,” J. Acoust. Soc. Am. 71, 1391–1399 (1982).
[CrossRef]

J. Math. Phys. (2)

K. R. Pechenick, J. M. Cohen, “Inverse scattering—exact solution of the Gel’fand–Levitan equation,” J. Math. Phys. 22, 1513–1516 (1981).
[CrossRef]

G. N. Balanis, “Inverse Scattering: determination of inhomogeneities in sound speed,” J. Math. Phys. 23, 2362–2568 (1982).
[CrossRef]

Linear Alg. Its Appl. (1)

O. Hald, “Numerical solution of the Gel’fand–Levitan equation,” Linear Alg. Its Appl. 28, 99–111 (1979).
[CrossRef]

Proc. IEEE (1)

H. N. Kritikos, D. L. Jaggard, D. B. Ge, “Numerical reconstruction of smooth dielectric profiles,” Proc. IEEE 70, 295–297 (1982).
[CrossRef]

Other (7)

A. K. Jordan, “Applications of the Gel’fand–Levitan theory to profiles with bandpass characteristics,” Ph.D. dissertation (University of Pennsylvania, Philadelphia, Pa., 1972).

D. L. Jaggard, P. Frangos, “The inverse scattering problem for layered dispersionless media with incomplete and imprecise data,” submitted to IEEE Trans. Antennas Propag.

This is obtained from Eq. (2.8) of Ref. 8 with a change of variables and subsequent reordering.

This is found by combining Eqs. (3.5) and (3.8) of Ref. 8.

Here ΔK indicates the difference between the kernel function at any point and its value at an adjacent lattice point.

I. Kay, “The inverse scattering problem,” Res. Rep. No. EM–74 (New York University, New York, N.Y., 1955).

H. E. Moses, C. M. deRidder, “Properties of dielectrics from reflection coefficients,” Tech. Rep. 322 (MIT Lincoln Laboratory, Lexington, Mass., 1963).

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Figures (7)

Fig. 1
Fig. 1

Geometry of the problem considered here showing the incident and reflected waves eikz and r(k)eikz, respectively. The (known) reflection coefficient is r(k), and the (unknown) refractive index is n(z). Here k is the wave number of the incident wave, and 2 is the coordinate.

Fig. 2
Fig. 2

Diagram indicating change of variables x = ξ + η, and y = ξη). The heavy line indicates the path of integration for the Balanis integral Eq. (4) and its transformed version, Eq. (9). The shaded region is the future light cone bounded by ξ = 0 and η = 0.

Fig. 3
Fig. 3

Diagrams of the η–ξ plane used to understand the calculation of Km,n by the leapfrogging method. The upper plot shows the diagonals ξ = pdη (p = 1, 2, 3,…) along which values of Km,n are related by Eq. (16). By Eq. (19), Km,n is zero along the η axis. Here p = m + n − 2. The lower plot shows the four points (oversized dots) at which values of Km,n are related by Eq. (18).

Fig. 4
Fig. 4

Diagrams of the η–ξ plane used in the calculation of Km,n. The upper plot shows the order in which the η–ξ plane is filled starting at the origin and using the leapfrogging method through successive use of relations (16) and (18). The lower plot shows the points used in the iteration scheme where n = 2, 3,… in Eq. (16).

Fig. 5
Fig. 5

Profile reconstruction using the leapfrogging method and additional iterations for the reflection coefficient r(k) = i0.5/(k + i0.5). The exact refractive index (solid line) is n(z) = (1 + 3z)−2/3 for z ≥ 0. The dashed line represents the leapfrogging method with n = 1 in Eq. (16), whereas the iterated method with n = 2, 3 in Eq. (16) is graphically indistinguishable from the exact result.

Fig. 6
Fig. 6

Butterworth reflection coefficients rN(k) for N = 1, 2, 3, 4 as indicated (upper plot) and defined by Eq. (23). Corresponding profile reconstructions of the refractive index n(z) with N = 1, 2, 3, 4 as indicated (lower plot). Here one iteration is used with n = 2 in Eq. (16).

Fig. 7
Fig. 7

Profile reconstruction using the leapfrogging method and iteration for noisy impulse response. The exact refractive index (solid line) is given by n(z) = (1 + 3z)−2/3 as in Fig. 5. The dashed lines show the reconstruction with S/N = 1.0 (upper plot) and S/N = 0.25 (lower plot).

Equations (30)

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[ d 2 d z 2 + k 2 n 2 ( z ) ] ψ ( k , z ) = 0
ψ ( k , z ) = e ikz + r ( k ) e ikz ( z < 0 ) ,
R ( y ) = 1 2 π Γ r ( k ) e iky d k .
K ( x , y ) y x [ 1 + K ( x , y ) ] R ( y + y ) d y = 0 ,
2 K ( x , y ) x 2 2 K ( x , y ) y 2 2 [ 1 + K ( x , x ) ] [ K ( x , y ) x d K ( x , x ) d x ] = 0 ,
K ( x , x ) = 0 ,
n [ z ( x ) ] = [ 1 + K ( x , x ) ] 2 ( x > 0 ) ,
z ( x ) = 0 x [ 1 + K ( x , x ) ] 2 d x .
K ( ξ η ) 2 0 ξ [ 1 + K ( ξ + η , ξ ξ ) ] R ( 2 ξ ) d ξ = 0 ,
2 K ( ξ , η ) ξ η 1 [ 1 + K ( ξ , 0 ) ] × [ K ( ξ , η ) ξ + K ( ξ , η ) η ] d K ( ξ , 0 ) d ξ | ξ = ξ + η = 0 ,
K ( 0 , η + ξ ) = 0 ,
n [ z ( x ) ] = [ 1 + K ( ξ , 0 ) ] 2 | ξ = x
z ( x ) = 0 x [ 1 + K ( ξ , 0 ) ] 2 d ξ .
R ( 2 ξ ) R [ ( 2 m 2 ) d ] R 2 m 1
K ( ξ , η ) K [ ( m 1 ) d , ( n 1 ) d ] K m , n .
K m , n 2 d ν = 2 m 1 [ 1 + K υ + n 1 , m ν + 1 ] R 2 υ 1 + d ( 1 + K m + n 1 , 1 ) R 2 m 1 = 0 ( m , n 1 )
K m + 1 , n + 1 = K m , n + 1 + K m + 1 , n K m , n + 1 [ 1 + K m + n 1 , 1 ] × ( K m + 1 , n + K m , n + 1 2 K m , n ) ( K m + n , 1 K m + n 1 , 1 ) , ( m , n 1 )
K m + 1 , n + 1 = K m , n + 1 + K m + 1 , n K m , n ( m , n 1 )
K 1 , n + m 1 = 0 ,
n q ( m ) = [ 1 + K m , 1 ] 2 ,
q ( m ) = i = 2 m d ( 1 + K i , 1 ) 2 ,
d = { d i m d / 2 i = m .
r N ( k ) = ( 1 ) N j = 1 N k j j = 1 N ( k k j ) ,
r ( k ) = i 0.5 k + i 0.5 ,
R ( y ) = { ½ e 1 / 2 y y > 0 0 y 0 .
K ( x , y ) = { [ ( x 2 y 2 ) / 4 + ( x + y ) / 2 ] x > | y | 0 x < | y | .
n [ z ( x ) ] = [ x 2 + 2 x + 1 ] 1 ( x > 0 ) ,
z ( x ) = x + x 2 + x 3 / 3 ,
n ( z ) = [ 1 + 3 z ] 2 / 3 ( z > 0 ) .
k j = exp [ i π ( 2 N 2 j 1 ) / 2 n ] ( j = 0 , 1 , 2 , , N 1 ) .

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