Abstract

Renormalized solutions are obtained for an inverse-scattering problem that are equivalent to the second-order regular perturbation approximations for the exact (Gel’fand-Levitan-Marchenko) theory. We have developed an inversion 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 demonstrated for simulated-scattering data from Gaussian and parabolic profiles and homogeneous slabs.

© 1985 Optical Society of America

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References

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  1. I. Kay, “The inverse scattering problem,” Rep. No. EM-74 (Institute of Mathematical Sciences, New York University, N.Y., 1955).
  2. H. E. Moses, “Calculation of the scattering potential from reflection coefficients,” Phys. Rev. 102, 559–567 (1956).
    [CrossRef]
  3. I. Kay, H. E. Moses, Inverse Scattering Papers: 1955–1963 (Math Sci, Brookline, Mass., 1982).
  4. I. M. Gel’fand, B. M. Levitan, “On the determination of a differential equation by its spectral function,” Transl. Am. Math. Soc. Ser. 2 1, 253 (1955).
  5. V. A. Marchenko, “Concerning the theory of a differential operator of second order,”Dokl. Akad. Nauk SSSR 72, 457 (1950).
  6. M. Nayfeh, Perturbation Methods (Wiley, New York, 1973), p. 315.
  7. H. Bremmer, “The W.K.B. approximation as the first term of a geometrical-optical series,” Comm. Pure Appl. Math. 3, S169–S179 (1951).
  8. M. Nayfeh, Comm. Pure Appl. Math. 3, 315 (1951).
  9. A. K. Jordan, “Inverse scattering theory: exact and approximate solutions,” in Mathematical Methods and Applications of Scattering Theory, J. A. DeSanto, A. W. Saenz, W. W. Zachary, eds. (Springer-Verlag, New York, 1980), p. 318–326.
    [CrossRef]
  10. J. Hirsch, “An analytic solution to the synthesis problem for dielectric thin-film layers,” Opt. Acta 26, 1273–1279 (1979).
    [CrossRef]
  11. H. Kaiser, H. C. Kaiser, “Mathematical methods in the synthesis and identification of thin-film systems: errata,” Appl. Opt. 20, 1043–1049 (1981).
    [CrossRef] [PubMed]
  12. M. Nayfeh, Perturbation Methods (Wiley, New York, 1973), p. 367.
  13. R. B. Barrar, R. M. Redheffer, “On nonuniform dielectric media,” IEEE Trans. Antennas Propag. AP-7, 101–1071955).
  14. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), p. 132.

1981 (1)

1979 (1)

J. Hirsch, “An analytic solution to the synthesis problem for dielectric thin-film layers,” Opt. Acta 26, 1273–1279 (1979).
[CrossRef]

1956 (1)

H. E. Moses, “Calculation of the scattering potential from reflection coefficients,” Phys. Rev. 102, 559–567 (1956).
[CrossRef]

1955 (2)

I. M. Gel’fand, B. M. Levitan, “On the determination of a differential equation by its spectral function,” Transl. Am. Math. Soc. Ser. 2 1, 253 (1955).

R. B. Barrar, R. M. Redheffer, “On nonuniform dielectric media,” IEEE Trans. Antennas Propag. AP-7, 101–1071955).

1951 (2)

H. Bremmer, “The W.K.B. approximation as the first term of a geometrical-optical series,” Comm. Pure Appl. Math. 3, S169–S179 (1951).

M. Nayfeh, Comm. Pure Appl. Math. 3, 315 (1951).

1950 (1)

V. A. Marchenko, “Concerning the theory of a differential operator of second order,”Dokl. Akad. Nauk SSSR 72, 457 (1950).

Barrar, R. B.

R. B. Barrar, R. M. Redheffer, “On nonuniform dielectric media,” IEEE Trans. Antennas Propag. AP-7, 101–1071955).

Bremmer, H.

H. Bremmer, “The W.K.B. approximation as the first term of a geometrical-optical series,” Comm. Pure Appl. Math. 3, S169–S179 (1951).

Gel’fand, I. M.

I. M. Gel’fand, B. M. Levitan, “On the determination of a differential equation by its spectral function,” Transl. Am. Math. Soc. Ser. 2 1, 253 (1955).

Hirsch, J.

J. Hirsch, “An analytic solution to the synthesis problem for dielectric thin-film layers,” Opt. Acta 26, 1273–1279 (1979).
[CrossRef]

Jordan, A. K.

A. K. Jordan, “Inverse scattering theory: exact and approximate solutions,” in Mathematical Methods and Applications of Scattering Theory, J. A. DeSanto, A. W. Saenz, W. W. Zachary, eds. (Springer-Verlag, New York, 1980), p. 318–326.
[CrossRef]

Kaiser, H.

Kaiser, H. C.

Kay, I.

I. Kay, H. E. Moses, Inverse Scattering Papers: 1955–1963 (Math Sci, Brookline, Mass., 1982).

I. Kay, “The inverse scattering problem,” Rep. No. EM-74 (Institute of Mathematical Sciences, New York University, N.Y., 1955).

Levitan, B. M.

I. M. Gel’fand, B. M. Levitan, “On the determination of a differential equation by its spectral function,” Transl. Am. Math. Soc. Ser. 2 1, 253 (1955).

Marchenko, V. A.

V. A. Marchenko, “Concerning the theory of a differential operator of second order,”Dokl. Akad. Nauk SSSR 72, 457 (1950).

Moses, H. E.

H. E. Moses, “Calculation of the scattering potential from reflection coefficients,” Phys. Rev. 102, 559–567 (1956).
[CrossRef]

I. Kay, H. E. Moses, Inverse Scattering Papers: 1955–1963 (Math Sci, Brookline, Mass., 1982).

Nayfeh, M.

M. Nayfeh, Comm. Pure Appl. Math. 3, 315 (1951).

M. Nayfeh, Perturbation Methods (Wiley, New York, 1973), p. 315.

M. Nayfeh, Perturbation Methods (Wiley, New York, 1973), p. 367.

Redheffer, R. M.

R. B. Barrar, R. M. Redheffer, “On nonuniform dielectric media,” IEEE Trans. Antennas Propag. AP-7, 101–1071955).

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), p. 132.

Appl. Opt. (1)

Comm. Pure Appl. Math. (2)

H. Bremmer, “The W.K.B. approximation as the first term of a geometrical-optical series,” Comm. Pure Appl. Math. 3, S169–S179 (1951).

M. Nayfeh, Comm. Pure Appl. Math. 3, 315 (1951).

Dokl. Akad. Nauk SSSR (1)

V. A. Marchenko, “Concerning the theory of a differential operator of second order,”Dokl. Akad. Nauk SSSR 72, 457 (1950).

IEEE Trans. Antennas Propag. (1)

R. B. Barrar, R. M. Redheffer, “On nonuniform dielectric media,” IEEE Trans. Antennas Propag. AP-7, 101–1071955).

Opt. Acta (1)

J. Hirsch, “An analytic solution to the synthesis problem for dielectric thin-film layers,” Opt. Acta 26, 1273–1279 (1979).
[CrossRef]

Phys. Rev. (1)

H. E. Moses, “Calculation of the scattering potential from reflection coefficients,” Phys. Rev. 102, 559–567 (1956).
[CrossRef]

Transl. Am. Math. Soc. Ser. 2 (1)

I. M. Gel’fand, B. M. Levitan, “On the determination of a differential equation by its spectral function,” Transl. Am. Math. Soc. Ser. 2 1, 253 (1955).

Other (6)

I. Kay, “The inverse scattering problem,” Rep. No. EM-74 (Institute of Mathematical Sciences, New York University, N.Y., 1955).

I. Kay, H. E. Moses, Inverse Scattering Papers: 1955–1963 (Math Sci, Brookline, Mass., 1982).

M. Nayfeh, Perturbation Methods (Wiley, New York, 1973), p. 315.

A. K. Jordan, “Inverse scattering theory: exact and approximate solutions,” in Mathematical Methods and Applications of Scattering Theory, J. A. DeSanto, A. W. Saenz, W. W. Zachary, eds. (Springer-Verlag, New York, 1980), p. 318–326.
[CrossRef]

M. Nayfeh, Perturbation Methods (Wiley, New York, 1973), p. 367.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), p. 132.

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Figures (3)

Fig. 1
Fig. 1

Physical model for electromagnetic reflection from an inhomogeneous dielectric slab. The slab width is L, the permittivity r(x) is a Gaussian function r(x) = A exp(−bx2), and the reflection coefficient is r(k).

Fig. 2
Fig. 2

Simulated reflection data to be inverted to obtain the profile of permittivity r(x). Reflection coefficient r(k) data given for 256 discrete values of k. In this example, A = 1.80, b = 0.59, and L = 2.0. The Born approximation is valid for k 0.6. Insert depicts r(x) drawn to scale.

Fig. 3
Fig. 3

Reconstructed profile of permittivity r(s) drawn to the same scale as that of the insert of Fig. 2. This is a plot of 512 data points obtained by applying the renormalized inversion theory to the data points of Fig. 2.

Tables (2)

Tables Icon

Table 1 Comparison of Reconstructed Profiles with Input Gaussian Profiles

Tables Icon

Table 2 Comparison of Inversion Results for Parameters A, b for Gaussian and Parabolic Profiles and Homogeneous Slabsa

Equations (37)

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E in ( x , k ) = e ikx , i 1 , x L / 2 ,
r ( x ) = { A e b x 2 , | x | L / 2 1.0 , | x | L / 2 ,
d 2 E ( x , k ) d x 2 + k 2 r ( r ) E ( x , k ) = 0 .
E s c ( x , k ) = r ( k ) e ikx , x L / 2 ,
s s ( x , k ) = L / 2 x k r ( x ) d x
r ( x , k ) = 1 q ( x ) / k 2
ψ ( s , k ) = { r [ x ( s ) ] } 1 / 4 E ( s , k ) ,
d 2 ψ ( s , k ) d s 2 + [ k 2 q ( s ) ] ψ ( s , k ) = 0 ,
2 ψ s 2 Ψ ( s , t ) 2 t 2 Ψ ( s , t ) q ( s ) Ψ ( s , t ) = 0 ,
Ψ i n ( s , t ) = δ ( s t ) ,
R ( s + t ) = 1 2 π r ( k ) exp [ i k ( s + t ) d k .
R ( z ) = 0 , z L / 2 ,
Ψ ( s , t ) = Ψ 0 ( s , t ) + K ( s , z ) Ψ 0 ( z , t ) d z
Ψ 0 ( s , t ) = δ ( s t ) + R ( s + t ) .
Ψ ( s , t ) = 0 , s > t .
K ( s , t ) + R ( s + t ) + t s K ( s , z ) R ( z + t ) d z = 0 ,
K ( s , s ) = 0 and 2 d d s K ( s , s ) = q ( s ) .
K ( s , t ) = n = 1 δ n K n ( s , t ) , R ( s ) = n = 1 δ n R n ( s )
q ( s ) = 2 d d s R ( 2 s ) + 4 [ R ( 2 s ) ] 2 ,
d 2 E ( x ) d s 2 + g ( x ) d E ( s ) d s + E ( s ) = 0 ,
g ( x ) 1 2 k [ r ( x ) ] 3 / 2 d r ( x ) d x .
max | g ( x ) | γ < 1 , | x | L / 2 ,
E [ s ( x , k ) , x ] = E 0 ( s , x ) + γ E 1 ( s , x ) + γ 2 E 2 ( s , x ) + ,
d d s = s + γ x .
γ 0 : s 2 E 0 ( s , x ) + E 0 ( s , x ) = 0 , γ 1 : 2 E 1 ( s , x ) s 2 + E 1 ( s , x ) = g ( x ) E 0 ( s , x ) s 2 r ( x ) 2 s x E 0 ( s , x ) .
E 0 [ s ( x , k ) x ] = A 0 ( x ) e is + B 0 ( x ) e is .
{ 1 2 [ r ( x ) ] 3 / 2 d r d x A 0 ( x ) + 2 [ r ( x ) ] 1 / 2 A 0 x } E 0 s = 0 ,
A 0 ( x ) = [ r ( x ) ] 1 / 4 ,
E 0 ( s , x ) = is [ r ( x ) ] 1 / 4 , s = s ( x , k ) .
r ( k ) = i 2 L / 2 L / 2 [ r ( x ) ] 3 / 2 d r d x d d s { E [ s ( x , k ) , x ] } d s .
r ( s ) = exp [ 4 2 s R ( z ) d z ] ,
r ( x ) = r ( 0 ) + r ( 0 ) x + r ( 0 ) x 2 / 2 +
d r d x ( x , k ) = i k 2 { r ( x ) [ 1 + r ] 2 + [ 1 r ] 2 ] , | x | L 2 ,
r ( k p ) = a 0 2 + m = 1 M 2 a m cos ( ϕ m ) sin ( ϕ m ) i b m sin ( ϕ m ) + 1 2 a M cos ( π k p ) ,
p = m [ i = 1 N x i 4 / N ( x ¯ 2 ) 2 ] { i = 1 N [ ln i ( x i ) ] 2 / N ( ln ¯ ) 2 } .
r ( x ) A ( 1 b x 2 ) .
( k L ) [ r ( x ) 1 ] 1 .

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