Abstract

The inverse scattering problem for the one-dimensional Helmholtz wave equation is studied. The equation is reduced to a Fresnel set that describes multiple bulk reflection and is similar to the coupled-wave equations. The inverse scattering problem is equivalent to coupled Gel’fand–Levitan–Marchenko integral equations. In the discrete representation its matrix has Töplitz symmetry, and the fast inner bordering method can be applied for its inversion. Previously the method was developed for the design of fiber Bragg gratings. The testing example of a short Bragg reflector with deep modulation demonstrates the high efficiency of refractive-index reconstruction.

© 2008 Optical Society of America

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References

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  1. H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).
  2. R. Kashyap, Fiber Bragg Gratings (Academic, 1999).
  3. P. Sacks, J. Math. Phys. 45, 1699 (2004).
    [CrossRef]
  4. D. W. Huang and C. C. Yang, Appl. Opt. 38, 4494 (1999).
    [CrossRef]
  5. K. O. Hill and G. Meltz, J. Lightwave Technol. 15, 1263 (1997).
    [CrossRef]
  6. G. N. Balanis, J. Math. Phys. 23, 2562 (1982).
    [CrossRef]
  7. S. H. Gray, J. Math. Phys. 24, 1148 (1983).
    [CrossRef]
  8. A. M. Bruckstein and T. Kailath, SIAM Rev. 29, 359 (1987).
    [CrossRef]
  9. Y. Chen and V. Rokhlin, Inverse Probl. 8, 365 (1992).
    [CrossRef]
  10. Y. Zhang, J. A. Kong, and A. K. Jordan, Microwave Opt. Technol. Lett. 15, 277 (1998).
    [CrossRef]
  11. V. E. Zakharov and A. B. Shabat, Zh. Eksp. Teor. Fiz. 61, 118 (1971).
  12. O. V. Belai, L. L. Frumin, E. V. Podivilov, and D. A. Shapiro, J. Opt. Soc. Am. B 24, 1451 (2007).
    [CrossRef]
  13. H. Bremmer, Physica (Amsterdam) 15, 593 (1949).
    [CrossRef]
  14. R. Feced, M. N. Zervas, and M. A. Muriel, IEEE J. Quantum Electron. 35, 1105 (1999).
    [CrossRef]
  15. G. B. Xiao and K. Yashiro, IEEE Trans. Antennas Propag. 50, 807 (2002).
    [CrossRef]
  16. C. M. de Sterke, D. G. Salinas, and J. E. Sipe, Phys. Rev. E 54, 1969 (1996).
    [CrossRef]

2007 (1)

2004 (1)

P. Sacks, J. Math. Phys. 45, 1699 (2004).
[CrossRef]

2002 (1)

G. B. Xiao and K. Yashiro, IEEE Trans. Antennas Propag. 50, 807 (2002).
[CrossRef]

1999 (2)

D. W. Huang and C. C. Yang, Appl. Opt. 38, 4494 (1999).
[CrossRef]

R. Feced, M. N. Zervas, and M. A. Muriel, IEEE J. Quantum Electron. 35, 1105 (1999).
[CrossRef]

1998 (1)

Y. Zhang, J. A. Kong, and A. K. Jordan, Microwave Opt. Technol. Lett. 15, 277 (1998).
[CrossRef]

1997 (1)

K. O. Hill and G. Meltz, J. Lightwave Technol. 15, 1263 (1997).
[CrossRef]

1996 (1)

C. M. de Sterke, D. G. Salinas, and J. E. Sipe, Phys. Rev. E 54, 1969 (1996).
[CrossRef]

1992 (1)

Y. Chen and V. Rokhlin, Inverse Probl. 8, 365 (1992).
[CrossRef]

1987 (1)

A. M. Bruckstein and T. Kailath, SIAM Rev. 29, 359 (1987).
[CrossRef]

1983 (1)

S. H. Gray, J. Math. Phys. 24, 1148 (1983).
[CrossRef]

1982 (1)

G. N. Balanis, J. Math. Phys. 23, 2562 (1982).
[CrossRef]

1971 (1)

V. E. Zakharov and A. B. Shabat, Zh. Eksp. Teor. Fiz. 61, 118 (1971).

1969 (1)

H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).

1949 (1)

H. Bremmer, Physica (Amsterdam) 15, 593 (1949).
[CrossRef]

Balanis, G. N.

G. N. Balanis, J. Math. Phys. 23, 2562 (1982).
[CrossRef]

Belai, O. V.

Bremmer, H.

H. Bremmer, Physica (Amsterdam) 15, 593 (1949).
[CrossRef]

Bruckstein, A. M.

A. M. Bruckstein and T. Kailath, SIAM Rev. 29, 359 (1987).
[CrossRef]

Chen, Y.

Y. Chen and V. Rokhlin, Inverse Probl. 8, 365 (1992).
[CrossRef]

de Sterke, C. M.

C. M. de Sterke, D. G. Salinas, and J. E. Sipe, Phys. Rev. E 54, 1969 (1996).
[CrossRef]

Feced, R.

R. Feced, M. N. Zervas, and M. A. Muriel, IEEE J. Quantum Electron. 35, 1105 (1999).
[CrossRef]

Frumin, L. L.

Gray, S. H.

S. H. Gray, J. Math. Phys. 24, 1148 (1983).
[CrossRef]

Hill, K. O.

K. O. Hill and G. Meltz, J. Lightwave Technol. 15, 1263 (1997).
[CrossRef]

Huang, D. W.

Jordan, A. K.

Y. Zhang, J. A. Kong, and A. K. Jordan, Microwave Opt. Technol. Lett. 15, 277 (1998).
[CrossRef]

Kailath, T.

A. M. Bruckstein and T. Kailath, SIAM Rev. 29, 359 (1987).
[CrossRef]

Kashyap, R.

R. Kashyap, Fiber Bragg Gratings (Academic, 1999).

Kogelnik, H.

H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).

Kong, J. A.

Y. Zhang, J. A. Kong, and A. K. Jordan, Microwave Opt. Technol. Lett. 15, 277 (1998).
[CrossRef]

Meltz, G.

K. O. Hill and G. Meltz, J. Lightwave Technol. 15, 1263 (1997).
[CrossRef]

Muriel, M. A.

R. Feced, M. N. Zervas, and M. A. Muriel, IEEE J. Quantum Electron. 35, 1105 (1999).
[CrossRef]

Podivilov, E. V.

Rokhlin, V.

Y. Chen and V. Rokhlin, Inverse Probl. 8, 365 (1992).
[CrossRef]

Sacks, P.

P. Sacks, J. Math. Phys. 45, 1699 (2004).
[CrossRef]

Salinas, D. G.

C. M. de Sterke, D. G. Salinas, and J. E. Sipe, Phys. Rev. E 54, 1969 (1996).
[CrossRef]

Shabat, A. B.

V. E. Zakharov and A. B. Shabat, Zh. Eksp. Teor. Fiz. 61, 118 (1971).

Shapiro, D. A.

Sipe, J. E.

C. M. de Sterke, D. G. Salinas, and J. E. Sipe, Phys. Rev. E 54, 1969 (1996).
[CrossRef]

Xiao, G. B.

G. B. Xiao and K. Yashiro, IEEE Trans. Antennas Propag. 50, 807 (2002).
[CrossRef]

Yang, C. C.

Yashiro, K.

G. B. Xiao and K. Yashiro, IEEE Trans. Antennas Propag. 50, 807 (2002).
[CrossRef]

Zakharov, V. E.

V. E. Zakharov and A. B. Shabat, Zh. Eksp. Teor. Fiz. 61, 118 (1971).

Zervas, M. N.

R. Feced, M. N. Zervas, and M. A. Muriel, IEEE J. Quantum Electron. 35, 1105 (1999).
[CrossRef]

Zhang, Y.

Y. Zhang, J. A. Kong, and A. K. Jordan, Microwave Opt. Technol. Lett. 15, 277 (1998).
[CrossRef]

Appl. Opt. (1)

Bell Syst. Tech. J. (1)

H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).

IEEE J. Quantum Electron. (1)

R. Feced, M. N. Zervas, and M. A. Muriel, IEEE J. Quantum Electron. 35, 1105 (1999).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

G. B. Xiao and K. Yashiro, IEEE Trans. Antennas Propag. 50, 807 (2002).
[CrossRef]

Inverse Probl. (1)

Y. Chen and V. Rokhlin, Inverse Probl. 8, 365 (1992).
[CrossRef]

J. Lightwave Technol. (1)

K. O. Hill and G. Meltz, J. Lightwave Technol. 15, 1263 (1997).
[CrossRef]

J. Math. Phys. (3)

G. N. Balanis, J. Math. Phys. 23, 2562 (1982).
[CrossRef]

S. H. Gray, J. Math. Phys. 24, 1148 (1983).
[CrossRef]

P. Sacks, J. Math. Phys. 45, 1699 (2004).
[CrossRef]

J. Opt. Soc. Am. B (1)

Microwave Opt. Technol. Lett. (1)

Y. Zhang, J. A. Kong, and A. K. Jordan, Microwave Opt. Technol. Lett. 15, 277 (1998).
[CrossRef]

Phys. Rev. E (1)

C. M. de Sterke, D. G. Salinas, and J. E. Sipe, Phys. Rev. E 54, 1969 (1996).
[CrossRef]

Physica (Amsterdam) (1)

H. Bremmer, Physica (Amsterdam) 15, 593 (1949).
[CrossRef]

SIAM Rev. (1)

A. M. Bruckstein and T. Kailath, SIAM Rev. 29, 359 (1987).
[CrossRef]

Zh. Eksp. Teor. Fiz. (1)

V. E. Zakharov and A. B. Shabat, Zh. Eksp. Teor. Fiz. 61, 118 (1971).

Other (1)

R. Kashyap, Fiber Bragg Gratings (Academic, 1999).

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Figures (3)

Fig. 1
Fig. 1

Diagram of the scattering problem. Right and left arrows denote forward and backward waves, respectively.

Fig. 2
Fig. 2

Absolute value of the reflection coefficient r as a function of spatial frequency k 0 of incident wave at α = 1 (dashed curve) and α = 0 (solid curve).

Fig. 3
Fig. 3

Refractive index n as a function of coordinate x: solid curve, theoretical dependence; diamonds, numerical reconstruction with N = 2 13 ; upper curves, α = 1 ; lower curves, α = 0 .

Equations (14)

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E + k 2 ( x ) E = 0 , k ( x ) = ω n ( x ) c = k 0 n ( x ) ,
E ( x ) = 1 k ( A e i S + B e i S ) , E ( x ) = i k ( A e i S B e i S ) ,
A = n 2 n B e 2 i S , B = n 2 n A e 2 i S ,
ξ = 0 x n ( x ) d x .
d A d ξ = q ( ξ ) e 2 i k 0 ξ B , d B d ξ = q ( ξ ) e 2 i k 0 ξ A ,
q ( ξ ) = d ln n d ξ
Λ = 0 L n ( x ) d x
n ( ξ ) = n ( 0 ) exp [ 2 0 ξ q ( ξ ) d ξ ] , x = 0 ξ d ξ n ( ξ ) .
q m = q ( ξ m ) , ξ m = m Λ N , m = 1 , , N .
n ( x ) = n ¯ + δ e ( x L w ) 2 ( α + sin κ x ) , x L 2 ,
n ( x ) = n ¯ + n ̃ ( x ) , n ¯ = 0 L n ( x ) d x L ,
n ̃ ( x ) = n ( x ) n ¯ = n 0 ( x ) + n 1 ( x ) sin ( κ x + φ ( x ) ) ,
A = q * e 2 i Ω x B , B = q e 2 i Ω x A , q = κ n 1 ( x ) 4 n ¯ e i θ ( x ) ,
θ ( x ) = φ ( x ) 2 k 0 0 x n 0 ( x ) d x ,

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