Abstract

We consider the inverse-scattering problem of retrieving the structural parameters of a stratified medium consisting of dispersive materials, given knowledge of the complex reflection coefficient in a finite frequency range. It is shown that the inverse-scattering problem does not have a unique solution in general. When the dispersion is sufficiently small, such that the time-domain Fresnel reflections have durations less than the round-trip time in the layers, the solution is unique and can be found by layer peeling. Numerical examples with dispersive and lossy media are given, demonstrating the usefulness of the method for, e.g., terahertz technology.

© 2012 Optical Society of America

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  1. M. C. Kemp, P. F. Taday, B. E. Cole, J. A. Cluff, A. J. Fitzgerald, and W. R. Tribe, “Security applications of terahertz technology,” Proc. SPIE 5070, 44–52 (2003).
    [CrossRef]
  2. U. Møller, D. G. Cooke, K. Tanaka, and P. U. Jepsen, “Terahertz reflection spectroscopy of Debye relaxation in polar liquids,” J. Opt. Soc. Am. B 26, A113–A125 (2009).
    [CrossRef]
  3. Y. C. Shen, T. Lo, P. F. Taday, B. E. Cole, and W. R. Tribe, “Detection and identification of explosives using terahertz pulsed spectroscopic imaging,” Appl. Phys. Lett. 86, 241116 (2005).
    [CrossRef]
  4. A. D. van Rheenen and M. W. Haakestad, “Detection and identification of explosives hidden under barrier materials—what are the THz-technology challenges?” Proc. SPIE 8017, 801719 (2011).
    [CrossRef]
  5. I. M. Gel’fand and B. M. Levitan, “On the determination of a differential equation from its spectral function,” Transl. Am. Math. Soc. 1, 253–304 (1955).
  6. A. Boutet de Monvel and V. Marchenko, “New inverse spectral problem and its application,” in Inverse and Algebraic Quantum Scattering Theory (Lake Balaton, 1996), Vol. 488 of Lecture Notes in Physics (Springer, 1997), pp. 1–12.
  7. A. M. Bruckstein, B. C. Levy, and T. Kailath, “Differential methods in inverse scattering,” SIAM J. Appl. Math. 45, 312–335 (1985).
    [CrossRef]
  8. A. E. Yagle and B. C. Levy, “The Schur algorithm and its applications,” Acta Appl. Math. 3, 255–284 (1985).
    [CrossRef]
  9. G.-H. Song and S.-Y. Shin, “Design of corrugated waveguide filters by the Gel’fand–Levitan–Marchenko inverse-scattering method,” J. Opt. Soc. Am. A 2, 1905–1915 (1985).
    [CrossRef]
  10. P. V. Frangos and D. L. Jaggard, “A numerical-solution to the Zakharov–Shabat inverse scattering problem,” IEEE Trans. Antennas Propag. 39, 74–79 (1991).
    [CrossRef]
  11. R. Feced, M. N. Zervas, and M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. 35, 1105–1115 (1999).
    [CrossRef]
  12. J. Skaar, L. Wang, and T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” IEEE J. Quantum Electron. 37, 165–173 (2001).
    [CrossRef]
  13. A. Rosenthal and M. Horowitz, “Inverse scattering algorithm for reconstructing strongly reflecting fiber Bragg gratings,” IEEE J. Quantum Electron. 39, 1018–1026 (2003).
    [CrossRef]
  14. J. Skaar and O. H. Waagaard, “Design and characterization of finite length fiber gratings,” IEEE J. Quantum Electron. 39, 1238–1245 (2003).
    [CrossRef]
  15. O. H. Waagaard and J. Skaar, “Inverse scattering in multimode structures,” SIAM J. Appl. Math. 68, 311–333 (2007).
    [CrossRef]
  16. J. Skaar, L. Wang, and T. Erdogan, “Synthesis of thick optical thin-film filters with a layer-peeling inverse-scattering algorithm,” Appl. Opt. 40, 2183–2189 (2001).
    [CrossRef]
  17. H. M. Nussenzveig, Causality and Dispersion Relations(Academic, 1972), Chap. 1.
  18. P. U. Jepsen and B. M. Fischer, “Dynamic range in terahertz time-domain transmission and reflection spectroscopy,” Opt. Lett. 30, 29–31 (2005).
    [CrossRef]
  19. J. Skaar and R. Feced, “Reconstruction of gratings from noisy reflection data,” J. Opt. Soc. Am. A 19, 2229–2237 (2002).
    [CrossRef]
  20. L. Duvillaret, F. Garet, and J.-L. Coutaz, “Highly precise determination of optical constants and sample thickness in THz time-domain spectroscopy,” Appl. Opt. 38, 409–415 (1999).
    [CrossRef]

2011

A. D. van Rheenen and M. W. Haakestad, “Detection and identification of explosives hidden under barrier materials—what are the THz-technology challenges?” Proc. SPIE 8017, 801719 (2011).
[CrossRef]

2009

2007

O. H. Waagaard and J. Skaar, “Inverse scattering in multimode structures,” SIAM J. Appl. Math. 68, 311–333 (2007).
[CrossRef]

2005

P. U. Jepsen and B. M. Fischer, “Dynamic range in terahertz time-domain transmission and reflection spectroscopy,” Opt. Lett. 30, 29–31 (2005).
[CrossRef]

Y. C. Shen, T. Lo, P. F. Taday, B. E. Cole, and W. R. Tribe, “Detection and identification of explosives using terahertz pulsed spectroscopic imaging,” Appl. Phys. Lett. 86, 241116 (2005).
[CrossRef]

2003

M. C. Kemp, P. F. Taday, B. E. Cole, J. A. Cluff, A. J. Fitzgerald, and W. R. Tribe, “Security applications of terahertz technology,” Proc. SPIE 5070, 44–52 (2003).
[CrossRef]

A. Rosenthal and M. Horowitz, “Inverse scattering algorithm for reconstructing strongly reflecting fiber Bragg gratings,” IEEE J. Quantum Electron. 39, 1018–1026 (2003).
[CrossRef]

J. Skaar and O. H. Waagaard, “Design and characterization of finite length fiber gratings,” IEEE J. Quantum Electron. 39, 1238–1245 (2003).
[CrossRef]

2002

2001

J. Skaar, L. Wang, and T. Erdogan, “Synthesis of thick optical thin-film filters with a layer-peeling inverse-scattering algorithm,” Appl. Opt. 40, 2183–2189 (2001).
[CrossRef]

J. Skaar, L. Wang, and T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” IEEE J. Quantum Electron. 37, 165–173 (2001).
[CrossRef]

1999

R. Feced, M. N. Zervas, and M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. 35, 1105–1115 (1999).
[CrossRef]

L. Duvillaret, F. Garet, and J.-L. Coutaz, “Highly precise determination of optical constants and sample thickness in THz time-domain spectroscopy,” Appl. Opt. 38, 409–415 (1999).
[CrossRef]

1991

P. V. Frangos and D. L. Jaggard, “A numerical-solution to the Zakharov–Shabat inverse scattering problem,” IEEE Trans. Antennas Propag. 39, 74–79 (1991).
[CrossRef]

1985

A. M. Bruckstein, B. C. Levy, and T. Kailath, “Differential methods in inverse scattering,” SIAM J. Appl. Math. 45, 312–335 (1985).
[CrossRef]

A. E. Yagle and B. C. Levy, “The Schur algorithm and its applications,” Acta Appl. Math. 3, 255–284 (1985).
[CrossRef]

G.-H. Song and S.-Y. Shin, “Design of corrugated waveguide filters by the Gel’fand–Levitan–Marchenko inverse-scattering method,” J. Opt. Soc. Am. A 2, 1905–1915 (1985).
[CrossRef]

1955

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

Boutet de Monvel, A.

A. Boutet de Monvel and V. Marchenko, “New inverse spectral problem and its application,” in Inverse and Algebraic Quantum Scattering Theory (Lake Balaton, 1996), Vol. 488 of Lecture Notes in Physics (Springer, 1997), pp. 1–12.

Bruckstein, A. M.

A. M. Bruckstein, B. C. Levy, and T. Kailath, “Differential methods in inverse scattering,” SIAM J. Appl. Math. 45, 312–335 (1985).
[CrossRef]

Cluff, J. A.

M. C. Kemp, P. F. Taday, B. E. Cole, J. A. Cluff, A. J. Fitzgerald, and W. R. Tribe, “Security applications of terahertz technology,” Proc. SPIE 5070, 44–52 (2003).
[CrossRef]

Cole, B. E.

Y. C. Shen, T. Lo, P. F. Taday, B. E. Cole, and W. R. Tribe, “Detection and identification of explosives using terahertz pulsed spectroscopic imaging,” Appl. Phys. Lett. 86, 241116 (2005).
[CrossRef]

M. C. Kemp, P. F. Taday, B. E. Cole, J. A. Cluff, A. J. Fitzgerald, and W. R. Tribe, “Security applications of terahertz technology,” Proc. SPIE 5070, 44–52 (2003).
[CrossRef]

Cooke, D. G.

Coutaz, J.-L.

Duvillaret, L.

Erdogan, T.

J. Skaar, L. Wang, and T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” IEEE J. Quantum Electron. 37, 165–173 (2001).
[CrossRef]

J. Skaar, L. Wang, and T. Erdogan, “Synthesis of thick optical thin-film filters with a layer-peeling inverse-scattering algorithm,” Appl. Opt. 40, 2183–2189 (2001).
[CrossRef]

Feced, R.

J. Skaar and R. Feced, “Reconstruction of gratings from noisy reflection data,” J. Opt. Soc. Am. A 19, 2229–2237 (2002).
[CrossRef]

R. Feced, M. N. Zervas, and M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. 35, 1105–1115 (1999).
[CrossRef]

Fischer, B. M.

Fitzgerald, A. J.

M. C. Kemp, P. F. Taday, B. E. Cole, J. A. Cluff, A. J. Fitzgerald, and W. R. Tribe, “Security applications of terahertz technology,” Proc. SPIE 5070, 44–52 (2003).
[CrossRef]

Frangos, P. V.

P. V. Frangos and D. L. Jaggard, “A numerical-solution to the Zakharov–Shabat inverse scattering problem,” IEEE Trans. Antennas Propag. 39, 74–79 (1991).
[CrossRef]

Garet, F.

Gel’fand, I. M.

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

Haakestad, M. W.

A. D. van Rheenen and M. W. Haakestad, “Detection and identification of explosives hidden under barrier materials—what are the THz-technology challenges?” Proc. SPIE 8017, 801719 (2011).
[CrossRef]

Horowitz, M.

A. Rosenthal and M. Horowitz, “Inverse scattering algorithm for reconstructing strongly reflecting fiber Bragg gratings,” IEEE J. Quantum Electron. 39, 1018–1026 (2003).
[CrossRef]

Jaggard, D. L.

P. V. Frangos and D. L. Jaggard, “A numerical-solution to the Zakharov–Shabat inverse scattering problem,” IEEE Trans. Antennas Propag. 39, 74–79 (1991).
[CrossRef]

Jepsen, P. U.

Kailath, T.

A. M. Bruckstein, B. C. Levy, and T. Kailath, “Differential methods in inverse scattering,” SIAM J. Appl. Math. 45, 312–335 (1985).
[CrossRef]

Kemp, M. C.

M. C. Kemp, P. F. Taday, B. E. Cole, J. A. Cluff, A. J. Fitzgerald, and W. R. Tribe, “Security applications of terahertz technology,” Proc. SPIE 5070, 44–52 (2003).
[CrossRef]

Levitan, B. M.

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

Levy, B. C.

A. E. Yagle and B. C. Levy, “The Schur algorithm and its applications,” Acta Appl. Math. 3, 255–284 (1985).
[CrossRef]

A. M. Bruckstein, B. C. Levy, and T. Kailath, “Differential methods in inverse scattering,” SIAM J. Appl. Math. 45, 312–335 (1985).
[CrossRef]

Lo, T.

Y. C. Shen, T. Lo, P. F. Taday, B. E. Cole, and W. R. Tribe, “Detection and identification of explosives using terahertz pulsed spectroscopic imaging,” Appl. Phys. Lett. 86, 241116 (2005).
[CrossRef]

Marchenko, V.

A. Boutet de Monvel and V. Marchenko, “New inverse spectral problem and its application,” in Inverse and Algebraic Quantum Scattering Theory (Lake Balaton, 1996), Vol. 488 of Lecture Notes in Physics (Springer, 1997), pp. 1–12.

Møller, U.

Muriel, M. A.

R. Feced, M. N. Zervas, and M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. 35, 1105–1115 (1999).
[CrossRef]

Nussenzveig, H. M.

H. M. Nussenzveig, Causality and Dispersion Relations(Academic, 1972), Chap. 1.

Rosenthal, A.

A. Rosenthal and M. Horowitz, “Inverse scattering algorithm for reconstructing strongly reflecting fiber Bragg gratings,” IEEE J. Quantum Electron. 39, 1018–1026 (2003).
[CrossRef]

Shen, Y. C.

Y. C. Shen, T. Lo, P. F. Taday, B. E. Cole, and W. R. Tribe, “Detection and identification of explosives using terahertz pulsed spectroscopic imaging,” Appl. Phys. Lett. 86, 241116 (2005).
[CrossRef]

Shin, S.-Y.

Skaar, J.

O. H. Waagaard and J. Skaar, “Inverse scattering in multimode structures,” SIAM J. Appl. Math. 68, 311–333 (2007).
[CrossRef]

J. Skaar and O. H. Waagaard, “Design and characterization of finite length fiber gratings,” IEEE J. Quantum Electron. 39, 1238–1245 (2003).
[CrossRef]

J. Skaar and R. Feced, “Reconstruction of gratings from noisy reflection data,” J. Opt. Soc. Am. A 19, 2229–2237 (2002).
[CrossRef]

J. Skaar, L. Wang, and T. Erdogan, “Synthesis of thick optical thin-film filters with a layer-peeling inverse-scattering algorithm,” Appl. Opt. 40, 2183–2189 (2001).
[CrossRef]

J. Skaar, L. Wang, and T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” IEEE J. Quantum Electron. 37, 165–173 (2001).
[CrossRef]

Song, G.-H.

Taday, P. F.

Y. C. Shen, T. Lo, P. F. Taday, B. E. Cole, and W. R. Tribe, “Detection and identification of explosives using terahertz pulsed spectroscopic imaging,” Appl. Phys. Lett. 86, 241116 (2005).
[CrossRef]

M. C. Kemp, P. F. Taday, B. E. Cole, J. A. Cluff, A. J. Fitzgerald, and W. R. Tribe, “Security applications of terahertz technology,” Proc. SPIE 5070, 44–52 (2003).
[CrossRef]

Tanaka, K.

Tribe, W. R.

Y. C. Shen, T. Lo, P. F. Taday, B. E. Cole, and W. R. Tribe, “Detection and identification of explosives using terahertz pulsed spectroscopic imaging,” Appl. Phys. Lett. 86, 241116 (2005).
[CrossRef]

M. C. Kemp, P. F. Taday, B. E. Cole, J. A. Cluff, A. J. Fitzgerald, and W. R. Tribe, “Security applications of terahertz technology,” Proc. SPIE 5070, 44–52 (2003).
[CrossRef]

van Rheenen, A. D.

A. D. van Rheenen and M. W. Haakestad, “Detection and identification of explosives hidden under barrier materials—what are the THz-technology challenges?” Proc. SPIE 8017, 801719 (2011).
[CrossRef]

Waagaard, O. H.

O. H. Waagaard and J. Skaar, “Inverse scattering in multimode structures,” SIAM J. Appl. Math. 68, 311–333 (2007).
[CrossRef]

J. Skaar and O. H. Waagaard, “Design and characterization of finite length fiber gratings,” IEEE J. Quantum Electron. 39, 1238–1245 (2003).
[CrossRef]

Wang, L.

J. Skaar, L. Wang, and T. Erdogan, “Synthesis of thick optical thin-film filters with a layer-peeling inverse-scattering algorithm,” Appl. Opt. 40, 2183–2189 (2001).
[CrossRef]

J. Skaar, L. Wang, and T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” IEEE J. Quantum Electron. 37, 165–173 (2001).
[CrossRef]

Yagle, A. E.

A. E. Yagle and B. C. Levy, “The Schur algorithm and its applications,” Acta Appl. Math. 3, 255–284 (1985).
[CrossRef]

Zervas, M. N.

R. Feced, M. N. Zervas, and M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. 35, 1105–1115 (1999).
[CrossRef]

Acta Appl. Math.

A. E. Yagle and B. C. Levy, “The Schur algorithm and its applications,” Acta Appl. Math. 3, 255–284 (1985).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

Y. C. Shen, T. Lo, P. F. Taday, B. E. Cole, and W. R. Tribe, “Detection and identification of explosives using terahertz pulsed spectroscopic imaging,” Appl. Phys. Lett. 86, 241116 (2005).
[CrossRef]

IEEE J. Quantum Electron.

R. Feced, M. N. Zervas, and M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. 35, 1105–1115 (1999).
[CrossRef]

J. Skaar, L. Wang, and T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” IEEE J. Quantum Electron. 37, 165–173 (2001).
[CrossRef]

A. Rosenthal and M. Horowitz, “Inverse scattering algorithm for reconstructing strongly reflecting fiber Bragg gratings,” IEEE J. Quantum Electron. 39, 1018–1026 (2003).
[CrossRef]

J. Skaar and O. H. Waagaard, “Design and characterization of finite length fiber gratings,” IEEE J. Quantum Electron. 39, 1238–1245 (2003).
[CrossRef]

IEEE Trans. Antennas Propag.

P. V. Frangos and D. L. Jaggard, “A numerical-solution to the Zakharov–Shabat inverse scattering problem,” IEEE Trans. Antennas Propag. 39, 74–79 (1991).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Lett.

Proc. SPIE

M. C. Kemp, P. F. Taday, B. E. Cole, J. A. Cluff, A. J. Fitzgerald, and W. R. Tribe, “Security applications of terahertz technology,” Proc. SPIE 5070, 44–52 (2003).
[CrossRef]

A. D. van Rheenen and M. W. Haakestad, “Detection and identification of explosives hidden under barrier materials—what are the THz-technology challenges?” Proc. SPIE 8017, 801719 (2011).
[CrossRef]

SIAM J. Appl. Math.

A. M. Bruckstein, B. C. Levy, and T. Kailath, “Differential methods in inverse scattering,” SIAM J. Appl. Math. 45, 312–335 (1985).
[CrossRef]

O. H. Waagaard and J. Skaar, “Inverse scattering in multimode structures,” SIAM J. Appl. Math. 68, 311–333 (2007).
[CrossRef]

Transl. Am. Math. Soc.

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

Other

A. Boutet de Monvel and V. Marchenko, “New inverse spectral problem and its application,” in Inverse and Algebraic Quantum Scattering Theory (Lake Balaton, 1996), Vol. 488 of Lecture Notes in Physics (Springer, 1997), pp. 1–12.

H. M. Nussenzveig, Causality and Dispersion Relations(Academic, 1972), Chap. 1.

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Figures (15)

Fig. 1.
Fig. 1.

Planar structure consisting of N + 1 layers. The thicknesses and refractive indices of the layers are d j and n j ( ω ) , respectively.

Fig. 2.
Fig. 2.

Power reflection coefficient and squared magnitude of the window function for the numerical example.

Fig. 3.
Fig. 3.

Incident and reflected pulse at z = 0 . The incident pulse is given by Eq. (21), which is the time-domain representation of the window function. The reflected pulse is given by the convolution of the incident pulse and h 1 ( t ) . The amplitude of the incident pulse has been scaled a factor 1 / 10 in the figure.

Fig. 4.
Fig. 4.

Exact and retrieved refractive index of the first layer: (a) real part and (b) imaginary part of refractive index. The maximum error in the retrieved refractive index is 6 · 10 4 .

Fig. 5.
Fig. 5.

Incident and retrieved reflected pulse at z = d 1 . The amplitude of the incident pulse has been scaled a factor 1 / 100 in the figure.

Fig. 6.
Fig. 6.

Exact and retrieved refractive index of the second layer: (a) real part and (b) imaginary part of refractive index.

Fig. 7.
Fig. 7.

Exact and retrieved refractive index of the first layer when d min = 0.25 λ s : (a) real part and (b) imaginary part of refractive index.

Fig. 8.
Fig. 8.

Exact and retrieved refractive index of the first layer in the presence of noise: (a) real part and (b) imaginary part of refractive index.

Fig. 9.
Fig. 9.

Incident and retrieved reflected pulse at z = d 1 in the presence of noise. The incident pulse (including the noise) has been scaled a factor 1 / 100 in the figure.

Fig. 10.
Fig. 10.

Exact and retrieved refractive index of the second layer in the presence of noise: (a) real part and (b) imaginary part of refractive index.

Fig. 11.
Fig. 11.

Power reflection coefficient and squared magnitude of window function for the structure with three layers.

Fig. 12.
Fig. 12.

Incident and reflected pulse at z = 0 for the structure with three layers. The amplitude of the incident pulse has been scaled a factor 1 / 5 in the figure.

Fig. 13.
Fig. 13.

Exact and retrieved refractive index of the first layer for the structure with three layers: (a) real part and (b) imaginary part of refractive index.

Fig. 14.
Fig. 14.

Exact and retrieved refractive index of the second layer for the structure with three layers: (a) real part and (b) imaginary part of refractive index.

Fig. 15.
Fig. 15.

Exact and retrieved refractive index of the third layer: (a) real part and (b) imaginary part of refractive index.

Equations (27)

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T j ρ = 1 1 ρ j [ 1 ρ j ρ j 1 ] ,
ρ j ( ω ) = n j 1 ( ω ) n j ( ω ) n j 1 ( ω ) + n j ( ω ) ,
T j d = [ exp [ i ω n j ( ω ) d j / c ] 0 0 exp [ i ω n j ( ω ) d j / c ] ] ,
M = T N d T N ρ T N 1 d T 2 ρ T 1 d T 1 ρ T 0 d .
r ( ω ) = M 21 M 22 ,
t ( ω ) = det M M 22 .
r ( ω ) = 2 d 0 / c h ( t ) exp ( i ω t ) d t ,
[ u 0 ( ω ) v 0 ( ω ) ] = [ 1 r ( ω ) ] .
[ u 1 ( ω ) v 1 ( ω ) ] = T 0 d [ u 0 ( ω ) v 0 ( ω ) ] .
r 1 ( ω ) = v 1 ( ω ) u 1 ( ω ) .
r 1 ( ω ) = ρ 1 ( ω ) + 2 d 1 / c h 1 ( t ) exp ( i ω t ) d t ,
h 1 ( t ) = 1 2 π r 1 ( ω ) exp ( i ω t ) d ω .
ρ 1 ( ω ) = 0 2 d 1 / c h 1 ( t ) exp ( i ω t ) d t .
r 2 ( ω ) = v 2 ( ω ) u 2 ( ω ) , [ u 2 ( ω ) v 2 ( ω ) ] = T 1 d T 1 ρ [ u 1 ( ω ) v 1 ( ω ) ] .
w ( t ) = 1 2 π ω 1 ω 2 W ( ω ) exp ( i ω t ) d ω .
ω 2 ω 1 π c d min ,
ρ 1 ( ω ) W ( ω ) = t w 2 d 1 / c + t w h 1 ( t ) * w ( t ) exp ( i ω t ) d t .
exp [ 2 Im ( n ) ω d / c ] Δ n 2 n ϱ ,
d c 2 ω Im ( n ) ln ( Δ n 2 n ϱ ) .
n ( ω ) = n c 1 + χ ( ω ) n c 2 ,
χ ( ω ) = F ω 0 2 ω 0 2 ω 2 i G ω .
w ( t ) = cos ( ω c t ) exp [ ( t / τ ) 2 ] .
2 Δ c = 2 π 2 ω max
ρ 1 1 ( ω ) = 0 h 1 1 ( t ) exp ( i ω t ) d t
ρ 1 1 ( ω ) = j = 0 h 1 1 [ j ] exp ( i ω j 2 Δ / c )
ρ ¯ 1 1 = h 1 1 [ 0 ] .
ρ 1 2 ( ω ) = exp ( i ω 2 Δ / c ) ρ 1 1 ( ω ) ρ ¯ 1 1 1 ρ ¯ 1 1 ρ 1 1 ( ω ) .

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