Abstract

A simple scheme of a profile inversion from the reflection data is obtained for a half-space weakly lossy medium. We have developed two approximations for reconstructing the permittivity and the conductivity profiles, both of which have closed forms.

© 1993 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. Rayleigh (J. W. Strutt), “On the reflection of light from a regularly stratified medium,” Proc. R. Soc. London A 93, 565–577 (1917).
    [CrossRef]
  2. M. Born, “Quantenmechanik der Stossvorgange,” Z. Phys. 37, 803–827 (1926).
  3. 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.
  4. E. F. Bolinder, “Fourier transforms and tapered transmission lines,” Proc. IRE 44, 557–562 (1956).
  5. D. B. Ge, D. L. Jaggard, H. N. Kritikos, “Perturbation and high frequency profile Inversion,”IEEE Trans. Antennas Propagat. AP-31, 804–808 (1983).
    [CrossRef]
  6. D. L. Jaggard, Y. Kim, “Accurate one-dimensional inverse scattering using a nonlinear renormalization technique,” J. Opt. Soc. Am. A 2, 1922–1930 (1985).
    [CrossRef]
  7. 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]
  8. A. K. Jordan, H. D. Ladouceur, “Renormalization of an inverse scattering theory for discontinuous profiles,” Phys. Rev. A 36, 4245–4253 (1987).
    [CrossRef] [PubMed]
  9. D. B. Ge, L. J. Chen, “A direct profile inversion for weakly conducting layered medium,”IEEE Trans. Antennas Propagat. AP-39, 907–909 (1991).
    [CrossRef]
  10. A. G. Tijhuis, Electromagnetic Inverse Profiling: Theory and Numerical Implementation (VNU Science Press, Utrecht, The Netherlands, 1987).

1991 (1)

D. B. Ge, L. J. Chen, “A direct profile inversion for weakly conducting layered medium,”IEEE Trans. Antennas Propagat. AP-39, 907–909 (1991).
[CrossRef]

1987 (1)

A. K. Jordan, H. D. Ladouceur, “Renormalization of an inverse scattering theory for discontinuous profiles,” Phys. Rev. A 36, 4245–4253 (1987).
[CrossRef] [PubMed]

1985 (2)

1983 (1)

D. B. Ge, D. L. Jaggard, H. N. Kritikos, “Perturbation and high frequency profile Inversion,”IEEE Trans. Antennas Propagat. AP-31, 804–808 (1983).
[CrossRef]

1956 (1)

E. F. Bolinder, “Fourier transforms and tapered transmission lines,” Proc. IRE 44, 557–562 (1956).

1926 (1)

M. Born, “Quantenmechanik der Stossvorgange,” Z. Phys. 37, 803–827 (1926).

1917 (1)

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

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

Bolinder, E. F.

E. F. Bolinder, “Fourier transforms and tapered transmission lines,” Proc. IRE 44, 557–562 (1956).

Born, M.

M. Born, “Quantenmechanik der Stossvorgange,” Z. Phys. 37, 803–827 (1926).

Chen, L. J.

D. B. Ge, L. J. Chen, “A direct profile inversion for weakly conducting layered medium,”IEEE Trans. Antennas Propagat. AP-39, 907–909 (1991).
[CrossRef]

Ge, D. B.

D. B. Ge, L. J. Chen, “A direct profile inversion for weakly conducting layered medium,”IEEE Trans. Antennas Propagat. AP-39, 907–909 (1991).
[CrossRef]

D. B. Ge, D. L. Jaggard, H. N. Kritikos, “Perturbation and high frequency profile Inversion,”IEEE Trans. Antennas Propagat. AP-31, 804–808 (1983).
[CrossRef]

Jaggard, D. L.

D. L. Jaggard, Y. Kim, “Accurate one-dimensional inverse scattering using a nonlinear renormalization technique,” J. Opt. Soc. Am. A 2, 1922–1930 (1985).
[CrossRef]

D. B. Ge, D. L. Jaggard, H. N. Kritikos, “Perturbation and high frequency profile Inversion,”IEEE Trans. Antennas Propagat. AP-31, 804–808 (1983).
[CrossRef]

Jordan, A. K.

A. K. Jordan, H. D. Ladouceur, “Renormalization of an inverse scattering theory for discontinuous profiles,” Phys. Rev. A 36, 4245–4253 (1987).
[CrossRef] [PubMed]

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]

Kim, Y.

Kritikos, H. N.

D. B. Ge, D. L. Jaggard, H. N. Kritikos, “Perturbation and high frequency profile Inversion,”IEEE Trans. Antennas Propagat. AP-31, 804–808 (1983).
[CrossRef]

Ladouceur, H. D.

A. K. Jordan, H. D. Ladouceur, “Renormalization of an inverse scattering theory for discontinuous profiles,” Phys. Rev. A 36, 4245–4253 (1987).
[CrossRef] [PubMed]

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]

Rayleigh,

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

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.

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

Tijhuis, A. G.

A. G. Tijhuis, Electromagnetic Inverse Profiling: Theory and Numerical Implementation (VNU Science Press, Utrecht, The Netherlands, 1987).

IEEE Trans. Antennas Propagat. (2)

D. B. Ge, D. L. Jaggard, H. N. Kritikos, “Perturbation and high frequency profile Inversion,”IEEE Trans. Antennas Propagat. AP-31, 804–808 (1983).
[CrossRef]

D. B. Ge, L. J. Chen, “A direct profile inversion for weakly conducting layered medium,”IEEE Trans. Antennas Propagat. AP-39, 907–909 (1991).
[CrossRef]

J. Opt. Soc. Am. A (2)

Phys. Rev. A (1)

A. K. Jordan, H. D. Ladouceur, “Renormalization of an inverse scattering theory for discontinuous profiles,” Phys. Rev. A 36, 4245–4253 (1987).
[CrossRef] [PubMed]

Proc. IRE (1)

E. F. Bolinder, “Fourier transforms and tapered transmission lines,” Proc. IRE 44, 557–562 (1956).

Proc. R. Soc. London A (1)

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

Z. Phys. (1)

M. Born, “Quantenmechanik der Stossvorgange,” Z. Phys. 37, 803–827 (1926).

Other (2)

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.

A. G. Tijhuis, Electromagnetic Inverse Profiling: Theory and Numerical Implementation (VNU Science Press, Utrecht, The Netherlands, 1987).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (1)

Fig. 1
Fig. 1

Comparisons of reconstructed profiles (dashed curves) with exact profiles (solid curves).

Equations (21)

Equations on this page are rendered with MathJax. Learn more.

d 2 φ i d z 2 + k 2 cos 2 θ φ i = 0 ,
d 2 φ d z 2 + k 2 [ ɛ ( z ) - sin 2 θ ] φ - j k η 0 σ ( z ) φ = 0 ,
d 2 φ s d z 2 + k 2 cos 2 θ φ s = - k 2 [ ɛ ( z ) - 1 ] φ + j k η 0 σ ( z ) φ
G ( z , z ) = j 2 k cos θ exp ( - j k z - z cos θ ) .
φ s ( z , k ) = - j k 2 cos θ 0 + [ ɛ ( z ) - 1 ] φ ( z , k ) × exp ( - j k z - z cos θ ) d z - η 0 2 cos θ × 0 + σ ( z ) φ ( z , k ) exp ( - j k z - z cos θ ) d z .
φ i ( z , k ) = exp ( - j k z cos θ )
ϕ ( z , k ) = exp ( - j k z cos θ ) + r ( k , θ ) exp ( j k z cos θ ) ,
r ( k , θ ) = - j k cos θ 2 cos 2 θ 0 + [ ɛ ( z ) - 1 ] φ ( z , k ) × exp ( - j k z cos θ ) d z - η 0 2 cos θ × 0 + σ ( z ) φ ( z , k ) exp ( - j k z cos θ ) d z .
k ˜ = k cos θ ,
r ( k ˜ , θ ) = - j k ˜ 2 cos 2 θ 0 + [ ɛ ( z ) - 1 ] φ ( z , k ˜ ) exp ( - j k ˜ z ) d z - η 0 2 cos θ 0 + σ ( z ) φ ( z , k ˜ ) exp ( - j k ˜ z ) d z .
R ( t ˜ , θ ) = - 1 2 cos 2 θ 0 + [ ɛ ( z ) - 1 ] Φ ( z , t ˜ - z ) t ˜ d z - η 0 2 cos θ 0 + σ ( z ) Φ ( z , t ˜ - z ) d z ,
Φ ( z , t ˜ ) Φ i ( z , t ˜ ) = δ ( t ˜ - z ) .
Φ ( z , t ˜ - z ) δ ( t ˜ - 2 z ) = 1 2 δ ( z - t ˜ 2 ) , t ˜ Φ ( z , t ˜ - z ) - 1 4 δ ( z - t ˜ 2 ) , - + f ( t ˜ ) δ ( t ˜ ) d t ˜ = f ( 0 ) ,             - + f ( t ˜ ) δ ( t ˜ ) d t ˜ = - f ( 0 ) .
R ( t ˜ , θ ) - 1 4 cos 2 θ d d t ˜ [ ɛ ( t ˜ 2 ) - 1 ] - η 0 4 cos θ σ ( t ˜ 2 ) ,
R ( 2 z , θ ) - 1 8 cos 2 θ d d z [ ɛ ( z ) - 1 ] - η 0 4 cos θ σ ( z ) ,
r ( k , θ ) = r ( k ˜ cos θ , θ ) cos θ R ( t cos θ , θ ) ,
R ( 2 z cos θ , θ ) - 1 8 cos 3 θ d d z [ ɛ ( z ) - 1 ] - η 0 4 cos 2 θ σ ( z ) .
σ ( z ) = 4 η 0 ( cos θ 1 - cos θ 2 ) [ cos 3 θ R ( 2 z cos θ , θ ) ] θ 1 θ 2 ,
ɛ ( x ) = 1 + 4 cos θ 1 cos θ 2 cos θ 2 - cos θ 1 [ cos θ 0 2 z cos θ R ( t , θ ) d t ] | θ 1 θ 2 ,
ɛ ( x ) = { ɛ 1 + ( ɛ 2 - ɛ 1 ) x d 0 x d ɛ 2 x > d ,
σ ( x ) = { σ 1 + ( σ 2 - σ 1 ) x d 0 x d σ 2 x > d .

Metrics