Abstract

A nonlinear method based on the inversion of the Riccati equation is presented here for the one-dimensional nondispersive inverse-scattering problem. This method avoids the significant errors in both amplitude and phase that plague most linearized (e.g., Born or its varients) inversion schemes. Instead, a nonlinear approximation to the Riccati equation is used for the accurate determination of the refractive-index amplitude from reflection data. This information is subsequently used to stretch the coordinates so as to remove the phase-accumulation error. The resulting refractive-index reconstructions are therefore accurate both in amplitude and in longitudinal placement as evidenced by the excellent comparison with exact theory. The method is applicable to both continuous and discontinuous refractive profiles and is supported by experimental measurements.

© 1985 Optical Society of America

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  1. J. W. Strutt, “On the electromagnetic theory of light,” Philos. Mag. 12, 81–101 (1881).
  2. M. Born, “Quantenmechanik der Stossvorgänge,” Z. Phys. XXXVII, 803–827 (1926).
  3. B. Ambarzumian, “On the problem of the diffuse reflection of light,” J. Phys. (Moscow) VIII, 65–75 (1944).(Here the Riccati equation is derived using an invarient embedding method. The result is expressed as an integral equation for the reflection coefficient rather than the usual differential equation.)
  4. J. R. Pierce, “A note on the transmission line equation in terms of impedance,” Bell Syst. Tech. J. 22, 263–265 (1943).(In this short note, the Riccati differential equation is derived using circuit concepts and physical insight. It is noted that the same equation describes the electron-optics equation for paraxial trajectories.)
    [CrossRef]
  5. L. R. Walker, N. Wax, “Non-uniform transmission lines and reflection coefficients,” J. Appl. Phys. 17, 1043–1045 (1946).
    [CrossRef]
  6. S. A. Schelkunoff, “Remarks concerning wave propagation in stratified media,” in The Theory of Electromagnetic Waves, M. Kline, ed. (Interscience, New York, 1951), pp. 181–192.
  7. For a modern derivation of the Riccati equation, see, e.g., F. T. Ulaby, R. K. Moore, A. K. Fung, Fundamentals and Radiometry, Vol. 1 of Microwave Remote Sensing (Addison-Wesley, Reading, Mass., 1981), pp. 82–84.
  8. L. M. Brekhovskikh, Waves in Layered Media (Academic, New York, 1960).
  9. C. H. Greenewalt, W. Brandt, D. D. Friel, “Iridescent colors of hummingbird feathers,” J. Opt. Soc. Am. 50, 1005–1013 (1960).
    [CrossRef]
  10. I. A. Kozlov, “The limit of applicability of the first approximation for determining the reflection coefficient in the theory of nonuniform lines,” Radio Eng. Electron. Phys. 14, 132–133 (1969).
  11. I. S. Gaydabura, “A method for linearization of the equation of an inhomogeneous line,” Radio Eng. Electron. Phys. 16, 1625–1627 (1971).
  12. M. Lahlou, J. K. Cohen, N. Bleistein, “Highly accurate inversion methods for three-dimensional stratified media,” Soc. Ind. Appl. Math. 43, 726—758 (1983).
    [CrossRef]
  13. The techniques include invariant embedding, use of the impedance concept and a direct manipulation of Maxwell’s equations or the Helmholtz equation. See Refs. 5–8 for examples.
  14. The dispersive case for a cold plasma is treated by D. B. Ge, D. L. Jaggard, H. N. Kritikos, “Perturbational and high frequency profile inversion,” IEEE Trans. Antennas Propag. AP-31, 804–803 (1983),while the nondispersive case is discussed in Ref. 8.
    [CrossRef]
  15. Note that r(p) must satisfy energy conservation and causality. However, r̂(p) is a mathematical transformation that does not necessarily satisfy the restriction |r̂(p)|<1.
  16. J. W. Strutt, “On the reflection of light from a regularly stratified medium,” Proc. R. Soc. London Ser. A 93, 565–577 (1917).
    [CrossRef]
  17. S. M. Rytov, “Diffraction of light by ultrasonic wave,” Izv. Akad. Nauk SSSR Ser. Fiz. 2, 223–259 (1937).
  18. See, e.g., I. Kay, “The inverse scattering problem,” NYU Res. Rep. No. EM-74 (New York University, New York, 1955).
  19. See, e.g., D. L. Jaggard, K. E. Olson, “Numerical reconstruction of dispersionless refractive profiles,” J. Opt. Soc. Am. A 2, 1931–1936 (1985).
    [CrossRef]
  20. D. L. Jaggard, P. V. Frangos, “Profile reconstruction for dispersionless layered dielectrics with imprecise and band-limited data,” IEEE Trans. Antennas Propag. (to be published).
  21. H. D. Ladouceur, A. K. Jordan, “Renormalization of an inverse-scattering theory for inhomogeneous dielectrics,” J. Opt. Soc. Am. A 2, 1916–1921 (1985).
    [CrossRef]

1985

1983

M. Lahlou, J. K. Cohen, N. Bleistein, “Highly accurate inversion methods for three-dimensional stratified media,” Soc. Ind. Appl. Math. 43, 726—758 (1983).
[CrossRef]

The dispersive case for a cold plasma is treated by D. B. Ge, D. L. Jaggard, H. N. Kritikos, “Perturbational and high frequency profile inversion,” IEEE Trans. Antennas Propag. AP-31, 804–803 (1983),while the nondispersive case is discussed in Ref. 8.
[CrossRef]

1971

I. S. Gaydabura, “A method for linearization of the equation of an inhomogeneous line,” Radio Eng. Electron. Phys. 16, 1625–1627 (1971).

1969

I. A. Kozlov, “The limit of applicability of the first approximation for determining the reflection coefficient in the theory of nonuniform lines,” Radio Eng. Electron. Phys. 14, 132–133 (1969).

1960

1946

L. R. Walker, N. Wax, “Non-uniform transmission lines and reflection coefficients,” J. Appl. Phys. 17, 1043–1045 (1946).
[CrossRef]

1944

B. Ambarzumian, “On the problem of the diffuse reflection of light,” J. Phys. (Moscow) VIII, 65–75 (1944).(Here the Riccati equation is derived using an invarient embedding method. The result is expressed as an integral equation for the reflection coefficient rather than the usual differential equation.)

1943

J. R. Pierce, “A note on the transmission line equation in terms of impedance,” Bell Syst. Tech. J. 22, 263–265 (1943).(In this short note, the Riccati differential equation is derived using circuit concepts and physical insight. It is noted that the same equation describes the electron-optics equation for paraxial trajectories.)
[CrossRef]

1937

S. M. Rytov, “Diffraction of light by ultrasonic wave,” Izv. Akad. Nauk SSSR Ser. Fiz. 2, 223–259 (1937).

1926

M. Born, “Quantenmechanik der Stossvorgänge,” Z. Phys. XXXVII, 803–827 (1926).

1917

J. W. Strutt, “On the reflection of light from a regularly stratified medium,” Proc. R. Soc. London Ser. A 93, 565–577 (1917).
[CrossRef]

1881

J. W. Strutt, “On the electromagnetic theory of light,” Philos. Mag. 12, 81–101 (1881).

Ambarzumian, B.

B. Ambarzumian, “On the problem of the diffuse reflection of light,” J. Phys. (Moscow) VIII, 65–75 (1944).(Here the Riccati equation is derived using an invarient embedding method. The result is expressed as an integral equation for the reflection coefficient rather than the usual differential equation.)

Bleistein, N.

M. Lahlou, J. K. Cohen, N. Bleistein, “Highly accurate inversion methods for three-dimensional stratified media,” Soc. Ind. Appl. Math. 43, 726—758 (1983).
[CrossRef]

Born, M.

M. Born, “Quantenmechanik der Stossvorgänge,” Z. Phys. XXXVII, 803–827 (1926).

Brandt, W.

Brekhovskikh, L. M.

L. M. Brekhovskikh, Waves in Layered Media (Academic, New York, 1960).

Cohen, J. K.

M. Lahlou, J. K. Cohen, N. Bleistein, “Highly accurate inversion methods for three-dimensional stratified media,” Soc. Ind. Appl. Math. 43, 726—758 (1983).
[CrossRef]

Frangos, P. V.

D. L. Jaggard, P. V. Frangos, “Profile reconstruction for dispersionless layered dielectrics with imprecise and band-limited data,” IEEE Trans. Antennas Propag. (to be published).

Friel, D. D.

Fung, A. K.

For a modern derivation of the Riccati equation, see, e.g., F. T. Ulaby, R. K. Moore, A. K. Fung, Fundamentals and Radiometry, Vol. 1 of Microwave Remote Sensing (Addison-Wesley, Reading, Mass., 1981), pp. 82–84.

Gaydabura, I. S.

I. S. Gaydabura, “A method for linearization of the equation of an inhomogeneous line,” Radio Eng. Electron. Phys. 16, 1625–1627 (1971).

Ge, D. B.

The dispersive case for a cold plasma is treated by D. B. Ge, D. L. Jaggard, H. N. Kritikos, “Perturbational and high frequency profile inversion,” IEEE Trans. Antennas Propag. AP-31, 804–803 (1983),while the nondispersive case is discussed in Ref. 8.
[CrossRef]

Greenewalt, C. H.

Jaggard, D. L.

See, e.g., D. L. Jaggard, K. E. Olson, “Numerical reconstruction of dispersionless refractive profiles,” J. Opt. Soc. Am. A 2, 1931–1936 (1985).
[CrossRef]

The dispersive case for a cold plasma is treated by D. B. Ge, D. L. Jaggard, H. N. Kritikos, “Perturbational and high frequency profile inversion,” IEEE Trans. Antennas Propag. AP-31, 804–803 (1983),while the nondispersive case is discussed in Ref. 8.
[CrossRef]

D. L. Jaggard, P. V. Frangos, “Profile reconstruction for dispersionless layered dielectrics with imprecise and band-limited data,” IEEE Trans. Antennas Propag. (to be published).

Jordan, A. K.

Kay, I.

See, e.g., I. Kay, “The inverse scattering problem,” NYU Res. Rep. No. EM-74 (New York University, New York, 1955).

Kozlov, I. A.

I. A. Kozlov, “The limit of applicability of the first approximation for determining the reflection coefficient in the theory of nonuniform lines,” Radio Eng. Electron. Phys. 14, 132–133 (1969).

Kritikos, H. N.

The dispersive case for a cold plasma is treated by D. B. Ge, D. L. Jaggard, H. N. Kritikos, “Perturbational and high frequency profile inversion,” IEEE Trans. Antennas Propag. AP-31, 804–803 (1983),while the nondispersive case is discussed in Ref. 8.
[CrossRef]

Ladouceur, H. D.

Lahlou, M.

M. Lahlou, J. K. Cohen, N. Bleistein, “Highly accurate inversion methods for three-dimensional stratified media,” Soc. Ind. Appl. Math. 43, 726—758 (1983).
[CrossRef]

Moore, R. K.

For a modern derivation of the Riccati equation, see, e.g., F. T. Ulaby, R. K. Moore, A. K. Fung, Fundamentals and Radiometry, Vol. 1 of Microwave Remote Sensing (Addison-Wesley, Reading, Mass., 1981), pp. 82–84.

Olson, K. E.

Pierce, J. R.

J. R. Pierce, “A note on the transmission line equation in terms of impedance,” Bell Syst. Tech. J. 22, 263–265 (1943).(In this short note, the Riccati differential equation is derived using circuit concepts and physical insight. It is noted that the same equation describes the electron-optics equation for paraxial trajectories.)
[CrossRef]

Rytov, S. M.

S. M. Rytov, “Diffraction of light by ultrasonic wave,” Izv. Akad. Nauk SSSR Ser. Fiz. 2, 223–259 (1937).

Schelkunoff, S. A.

S. A. Schelkunoff, “Remarks concerning wave propagation in stratified media,” in The Theory of Electromagnetic Waves, M. Kline, ed. (Interscience, New York, 1951), pp. 181–192.

Strutt, J. W.

J. W. Strutt, “On the reflection of light from a regularly stratified medium,” Proc. R. Soc. London Ser. A 93, 565–577 (1917).
[CrossRef]

J. W. Strutt, “On the electromagnetic theory of light,” Philos. Mag. 12, 81–101 (1881).

Ulaby, F. T.

For a modern derivation of the Riccati equation, see, e.g., F. T. Ulaby, R. K. Moore, A. K. Fung, Fundamentals and Radiometry, Vol. 1 of Microwave Remote Sensing (Addison-Wesley, Reading, Mass., 1981), pp. 82–84.

Walker, L. R.

L. R. Walker, N. Wax, “Non-uniform transmission lines and reflection coefficients,” J. Appl. Phys. 17, 1043–1045 (1946).
[CrossRef]

Wax, N.

L. R. Walker, N. Wax, “Non-uniform transmission lines and reflection coefficients,” J. Appl. Phys. 17, 1043–1045 (1946).
[CrossRef]

Bell Syst. Tech. J.

J. R. Pierce, “A note on the transmission line equation in terms of impedance,” Bell Syst. Tech. J. 22, 263–265 (1943).(In this short note, the Riccati differential equation is derived using circuit concepts and physical insight. It is noted that the same equation describes the electron-optics equation for paraxial trajectories.)
[CrossRef]

IEEE Trans. Antennas Propag.

The dispersive case for a cold plasma is treated by D. B. Ge, D. L. Jaggard, H. N. Kritikos, “Perturbational and high frequency profile inversion,” IEEE Trans. Antennas Propag. AP-31, 804–803 (1983),while the nondispersive case is discussed in Ref. 8.
[CrossRef]

Izv. Akad. Nauk SSSR Ser. Fiz.

S. M. Rytov, “Diffraction of light by ultrasonic wave,” Izv. Akad. Nauk SSSR Ser. Fiz. 2, 223–259 (1937).

J. Appl. Phys.

L. R. Walker, N. Wax, “Non-uniform transmission lines and reflection coefficients,” J. Appl. Phys. 17, 1043–1045 (1946).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Phys. (Moscow)

B. Ambarzumian, “On the problem of the diffuse reflection of light,” J. Phys. (Moscow) VIII, 65–75 (1944).(Here the Riccati equation is derived using an invarient embedding method. The result is expressed as an integral equation for the reflection coefficient rather than the usual differential equation.)

Philos. Mag.

J. W. Strutt, “On the electromagnetic theory of light,” Philos. Mag. 12, 81–101 (1881).

Proc. R. Soc. London Ser. A

J. W. Strutt, “On the reflection of light from a regularly stratified medium,” Proc. R. Soc. London Ser. A 93, 565–577 (1917).
[CrossRef]

Radio Eng. Electron. Phys.

I. A. Kozlov, “The limit of applicability of the first approximation for determining the reflection coefficient in the theory of nonuniform lines,” Radio Eng. Electron. Phys. 14, 132–133 (1969).

I. S. Gaydabura, “A method for linearization of the equation of an inhomogeneous line,” Radio Eng. Electron. Phys. 16, 1625–1627 (1971).

Soc. Ind. Appl. Math.

M. Lahlou, J. K. Cohen, N. Bleistein, “Highly accurate inversion methods for three-dimensional stratified media,” Soc. Ind. Appl. Math. 43, 726—758 (1983).
[CrossRef]

Z. Phys.

M. Born, “Quantenmechanik der Stossvorgänge,” Z. Phys. XXXVII, 803–827 (1926).

Other

Note that r(p) must satisfy energy conservation and causality. However, r̂(p) is a mathematical transformation that does not necessarily satisfy the restriction |r̂(p)|<1.

See, e.g., I. Kay, “The inverse scattering problem,” NYU Res. Rep. No. EM-74 (New York University, New York, 1955).

D. L. Jaggard, P. V. Frangos, “Profile reconstruction for dispersionless layered dielectrics with imprecise and band-limited data,” IEEE Trans. Antennas Propag. (to be published).

The techniques include invariant embedding, use of the impedance concept and a direct manipulation of Maxwell’s equations or the Helmholtz equation. See Refs. 5–8 for examples.

S. A. Schelkunoff, “Remarks concerning wave propagation in stratified media,” in The Theory of Electromagnetic Waves, M. Kline, ed. (Interscience, New York, 1951), pp. 181–192.

For a modern derivation of the Riccati equation, see, e.g., F. T. Ulaby, R. K. Moore, A. K. Fung, Fundamentals and Radiometry, Vol. 1 of Microwave Remote Sensing (Addison-Wesley, Reading, Mass., 1981), pp. 82–84.

L. M. Brekhovskikh, Waves in Layered Media (Academic, New York, 1960).

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Figures (6)

Fig. 1
Fig. 1

Refractive-profile reconstruction of a homogeneous slab with refractive index of 2. a, Exact (solid line) and perturbational (dashed line) reconstruction. b, Exact (solid line) and coordinate-stretched version of perturbational (dashed line) reconstruction. c, Exact (solid line) and nonlinear without coordinate-stretch (dashed line) reconstructions. d, Exact (solid line) and nonlinear renormalization with coordinate-stretch (dashed line) reconstructions. Note that the exact and the nonlinear renormalization reconstructions are graphically indistinguishable.

Fig. 2
Fig. 2

Refractive-profile reconstruction of an asymmetric slab. The solid, dashed, and dotted lines represent the exact, nonlinear, and perturbational reconstructions of Eqs. (35), (37), and (38), respectively.

Fig. 3
Fig. 3

Velocity-profile reconstruction of a homogeneous slab. The solid line represents the exact velocity profile. The reconstruction of Lahlou et al.12 (dotted line) and the reconstruction using our nonlinear renormalization method (dashed line) are compared.

Fig. 4
Fig. 4

Refractive-profile reconstruction of the one-pole case. The solid, dashed, and dotted lines represent the exact, nonlinear renormalized, and perturbational reconstructions of Eqs. (46), (48), and (47), respectively.

Fig. 5
Fig. 5

Refractive-profile reconstruction of the three-pole–one zero case. (a) Exact (solid line) and perturbational (dashed line) reconstruction. (b) Exact (solid line) and coordinate-stretched version of perturbational (dashed line) reconstructions. (c) Exact (solid line) and nonlinear without coordinate-stretch (dashed line) reconstruction. (d) Exact (solid line) and nonlinear renormalization with coordinate-stretch (dashed line) reconstructions.

Fig. 6
Fig. 6

Refractive-profile reconstruction of the thirteen-pole–seven-zero case. The solid, dashed, and dotted lines represent the exact, nonlinear renormalization, and perturbational reconstructions, respectively.

Equations (52)

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k ( z ) = { k 0 , z < 0 k 0 n ( z ) , z 0 ,
[ d 2 d z 2 + k 2 ( z ) ] ψ ( z ) = 0
ψ ( z ) = e i k 0 z + r ( k 0 ) e i k 0 z .
d r ( z , k 0 ) d z = 1 2 [ 1 r 2 ( z , k 0 ) ] d d z { ln [ k ( z ) ] } Amplitude Term 2 i k ( z ) r ( z , k 0 ) . Phase Term
r ( d , k 0 ) = r 0 ( d > 0 )
r ( k 0 ) = r ( z , k 0 ) z = 0 .
d r ( z , k 0 ) d z = 1 2 d d z ñ ( z ) 2 i k 0 r ( z , k 0 ) ,
r ( k 0 ) = 1 2 d 0 d d z { ñ ( z ) } exp { 2 i k 0 z } d z ,
ñ ( z ) = 1 2 1 { 1 p r ( p ) } ,
| r ( k ) | 1
| k 0 d [ n ( z ) 1 ] | 1.
1 [ 1 r 2 ( z , k 0 ) ] d r ( z , k 0 ) d z = 1 2 d d z { ln [ k ( z ) ] } 2 i k ( z ) r ( z , k 0 ) [ 1 r 2 ( z , k 0 ) ] ,
d r ̂ ( z , k 0 ) d z = 1 2 d d z { ln [ n ( z ) ) } Amplitude Term 2 i k 0 n ( z ) r ̂ ( z , k 0 ) , Phase Term
r ̂ ( z , k 0 ) tanh 1 [ r ( z , k 0 ) ] r ( z , k 0 ) [ 1 r 2 ( z , k 0 ) ] 1
r ̂ ( d , k 0 ) = r ̂ 0 ( d > 0 ) .
r ̂ ( k 0 ) = r ̂ ( z , k 0 ) z = 0 = tanh 1 [ r ( k 0 ) ]
r ̂ ( k 0 ) = 1 2 0 d d d z { ln [ k ( z ) ] } exp [ 2 i k 0 0 z n ( z ) d z ] d z .
x 0 z n ( z ) d z .
r ̂ ( k 0 ) = 1 2 0 d d x { ln [ k ( x ) ] } exp [ 2 i k 0 x ] d x ,
k ( z ) = k ( z ) x = 0 z n ( z ) d z .
r ̂ ( p ) = 1 2 0 d d x [ ln k ( x ) ] e p x d x .
n ( x ) = exp { 2 0 x 1 [ r ( p ) ] d x } = exp { 2 1 [ 1 p r ( p ) ] } ,
n ( z ) = { exp { 2 0 x 1 [ r ̂ ( p ) ] d x } x = 0 z n ( z ) d z exp { 2 1 [ r ̂ ( p ) p ] } x = 0 z n ( z ) d z ,
n ¯ ( z ) = ñ ( x ) x = 0 z ñ ( x ) d x
n ( z ) = { 1 , z < 0 2 , 0 z 1 1 , z > 1
r ( k 0 ) = 1 3 ( 1 e i 4 k 0 ) ( 1 1 9 e i 4 k 0 )
r ( p ) = 1 3 ( 1 e 2 p ) ( 1 1 9 e 2 p ) ,
r ( p ) = 1 3 + 8 27 e 2 p + 8 243 e 4 p +
1 { tanh 1 [ r ( p ) ] } = δ ( x ) tanh 1 ( 1 3 ) + 1 3 δ ( x 2 ) +
n ( x ) = { 1 , x < 0 e ( ln 2 ) = 2 , 0 x 2 e ( ln 2 2 / 3 ) = 1.027 , x > 2
x = 0 z 2 d z = 2 z ( 0 x 2 ) .
n ( z ) = { 1 , z < 0 2 , 0 z 1 1.027 z > 1 ,
ñ ( z ) = { 1 , z < 0 5 / 3 , 0 z 2 29 / 27 z > 2 ,
n ¯ ( z ) = { 1 , z < 0 5 / 3 , 0 z 1.20 29 / 27 , z > 1.20
n ( z ) = { 1 , z < 0 100 , 0 z 1 50 , z > 1 .
r ( p ) = 99 101 + 1 3 e 100 p 1 33 101 e 100 p .
n ( z ) = { 1 , z < 0 100 , 0 z 1 51.34 , z 1 ,
n ( z ) = { 1 , z < 0 2.96 , 0 z 100 2.93 , z 100 .
c ( z ) / c 0 = n ( z ) 1 ,
n ( z ) = { 1 , z < 0 1 / 3 , 0 z 1000 1 , z 1000 ,
r ( p ) = 1 2 [ 1 exp ( 1000 3 p ) ] [ 1 1 4 exp ( 1000 3 p ) ] .
n ( z ) = { 1 , z < 0 1 3 , 0 z 1000 0.91 , z 1000
ñ ( z ) = { 1 , 0 , 3 4 , z < 0 0 z 1000 / 3 1000 / 3 z 2000 / 3 15 16 , 63 64 , 2000 / 3 z 1000 1000 z 4000 3 ,
c ( z ) / c 0 = n ( z ) 1 .
r ( p ) = 1 p + 1 ,
n ( z ) = [ 1 + 3 z ] 2 / 3 .
ñ ( z ) = 1 + 2 e z
n ( z ) = exp [ 0 x ( x ) 1 ( 1 e 2 x ) d x ] x = 0 z n ( z ) d z ,
r ( p ) = 4 p p 3 + 2 ( 1 + 2 ) p 2 + 4 ( 1 + 2 ) p + 8 ,
ñ ( z ) = 1 2 [ ( 1 + 2 ) e 2 z cos ( 2 z ) 2 e 2 z sin ( 2 z ) ( 1 + 2 ) e 2 z ]
n ( z ) exp { 0 x ( x ) 1 [ e A x e B x + 2 e C x cos ( D x ) 2 e E x cos ( F x ) ] d x } ,
x = 0 z n ( z ) d z , A = 0.756 , B = 3.904 , C = 2.036 , D = 2.536 , E = 0.462 , F = 1.355 .

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