Abstract

We propose a new method for the recovery of the refractive-index profile of a single-mode or multimode optical guided structure. We solve the inverse problem using the Wigner distribution and reduce it to the solution of a linear system of equations.

© 1998 Optical Society of America

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References

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  1. S. I. Hosain, J. P. Meunier, E. Bourillot, F. De Fornel, J. P. Goudonnet, “Review of the basic methods for characterizing integrated-optic waveguides,” Fiber Integr. Opt. 14, 89–107 (1995).
    [CrossRef]
  2. D. Dragoman, M. Dragoman, “Integrated optic-devices characterization with the Wigner transform,” IEEE J. Sel. Top. Quantum Electron. 2, 181–186 (1996).
    [CrossRef]
  3. H. O. Bartelt, K. H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
    [CrossRef]
  4. H. Weber, “Wave optical analysis of the phase space analyzer,” J. Mod. Opt. 39, 543–559 (1992).
    [CrossRef]
  5. M. Conner, Y. Li, “Optical generation of the Wigner distribution of 2-D real signals,” Appl. Opt. 24, 3825–3829 (1985).
    [CrossRef] [PubMed]
  6. A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983), Chap. 12.
  7. D. Dragoman, “Wigner distribution function in nonlinear optics,” Appl. Opt. 35, 4142–4146 (1996).
    [CrossRef] [PubMed]

1996

D. Dragoman, M. Dragoman, “Integrated optic-devices characterization with the Wigner transform,” IEEE J. Sel. Top. Quantum Electron. 2, 181–186 (1996).
[CrossRef]

D. Dragoman, “Wigner distribution function in nonlinear optics,” Appl. Opt. 35, 4142–4146 (1996).
[CrossRef] [PubMed]

1995

S. I. Hosain, J. P. Meunier, E. Bourillot, F. De Fornel, J. P. Goudonnet, “Review of the basic methods for characterizing integrated-optic waveguides,” Fiber Integr. Opt. 14, 89–107 (1995).
[CrossRef]

1992

H. Weber, “Wave optical analysis of the phase space analyzer,” J. Mod. Opt. 39, 543–559 (1992).
[CrossRef]

1985

1980

H. O. Bartelt, K. H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

Bartelt, H. O.

H. O. Bartelt, K. H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

Bourillot, E.

S. I. Hosain, J. P. Meunier, E. Bourillot, F. De Fornel, J. P. Goudonnet, “Review of the basic methods for characterizing integrated-optic waveguides,” Fiber Integr. Opt. 14, 89–107 (1995).
[CrossRef]

Brenner, K. H.

H. O. Bartelt, K. H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

Conner, M.

De Fornel, F.

S. I. Hosain, J. P. Meunier, E. Bourillot, F. De Fornel, J. P. Goudonnet, “Review of the basic methods for characterizing integrated-optic waveguides,” Fiber Integr. Opt. 14, 89–107 (1995).
[CrossRef]

Dragoman, D.

D. Dragoman, “Wigner distribution function in nonlinear optics,” Appl. Opt. 35, 4142–4146 (1996).
[CrossRef] [PubMed]

D. Dragoman, M. Dragoman, “Integrated optic-devices characterization with the Wigner transform,” IEEE J. Sel. Top. Quantum Electron. 2, 181–186 (1996).
[CrossRef]

Dragoman, M.

D. Dragoman, M. Dragoman, “Integrated optic-devices characterization with the Wigner transform,” IEEE J. Sel. Top. Quantum Electron. 2, 181–186 (1996).
[CrossRef]

Goudonnet, J. P.

S. I. Hosain, J. P. Meunier, E. Bourillot, F. De Fornel, J. P. Goudonnet, “Review of the basic methods for characterizing integrated-optic waveguides,” Fiber Integr. Opt. 14, 89–107 (1995).
[CrossRef]

Hosain, S. I.

S. I. Hosain, J. P. Meunier, E. Bourillot, F. De Fornel, J. P. Goudonnet, “Review of the basic methods for characterizing integrated-optic waveguides,” Fiber Integr. Opt. 14, 89–107 (1995).
[CrossRef]

Li, Y.

Lohmann, A. W.

H. O. Bartelt, K. H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

Love, J. D.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983), Chap. 12.

Meunier, J. P.

S. I. Hosain, J. P. Meunier, E. Bourillot, F. De Fornel, J. P. Goudonnet, “Review of the basic methods for characterizing integrated-optic waveguides,” Fiber Integr. Opt. 14, 89–107 (1995).
[CrossRef]

Snyder, A. W.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983), Chap. 12.

Weber, H.

H. Weber, “Wave optical analysis of the phase space analyzer,” J. Mod. Opt. 39, 543–559 (1992).
[CrossRef]

Appl. Opt.

Fiber Integr. Opt.

S. I. Hosain, J. P. Meunier, E. Bourillot, F. De Fornel, J. P. Goudonnet, “Review of the basic methods for characterizing integrated-optic waveguides,” Fiber Integr. Opt. 14, 89–107 (1995).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

D. Dragoman, M. Dragoman, “Integrated optic-devices characterization with the Wigner transform,” IEEE J. Sel. Top. Quantum Electron. 2, 181–186 (1996).
[CrossRef]

J. Mod. Opt.

H. Weber, “Wave optical analysis of the phase space analyzer,” J. Mod. Opt. 39, 543–559 (1992).
[CrossRef]

Opt. Commun.

H. O. Bartelt, K. H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

Other

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983), Chap. 12.

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Equations (23)

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W r , p =   ϕ r + r ϕ * r - r exp i 2 pr d r ,
r 2 ϕ r + k 2 n 2 r - β 2 ϕ r = 0 ,
  r 2 ϕ r + r + k 2 n 2 r + r ϕ r + r - β 2 ϕ r + r ϕ * r - r - ϕ r + r r 2 ϕ * r - r + k 2 n 2 r - r ϕ * r - r - β 2 ϕ * r - r exp i 2 pr d r = 0 .
n 2 r + r - n 2 r - r = j = 0 2 2 j + 1 ! r 2 j + 1 r 2 j + 1 n 2 r ,
2 p r W r , p + k 2 j = 0 - 1 j 2 2 j 2 j + 1 ! × r 2 j + 1 n 2 · p 2 j + 1 W r , p = 0 .
  n 2 r p 1 r · p W d r d p = 0
2     p 2 W r , p d r d p + k 2     n 2 r p × p W + r r · p W d r d p = 0
4     r 1 p 2 W r , p d r d p + k 2     n 2 r p 1 p × 2 r p W + r 2 r · p W d r d p = 0
2 N     r N - 1 1 N + 1 p 2 W r , p d r d p + k 2     n 2 r p 1 N + 1 N r N - 1 p W + r N r · p W d r d p = 0 .
  n 2 r K j r d r = C j ,   j = 0 , 1 , , N ,
K j r =   p 1 j + 1 j r j - 1 p W + r j r · p W d p , C j = - 2 j / k 2     r j - 1 1 j + 1 p 2 W r , p d r d p .
K j r = n = 0 N   k jn Ψ n r ,
K j r = w r n = 0 N   k jn Ψ n r .
n 2 r = m = 0 N   n m Ψ m r ,
KN = C ,
r W = W r + d r , p - W r , p / d r , p W = W r , p + d p - W r , p / d p .
ϕ j x = H j 2 a x exp - ax 2 ,
W x , p = π 2 a 1 / 2 exp - 2 ax 2 - p 2 2 a .
C 1 = - 2 a π k 2 π 2 a 1 / 2 , C 3 = - 3 π 2 k 2 π 2 a 1 / 2 , C 5 = - 15 π 8 ak 2 π 2 a 1 / 2 , K 1 x = a π 4 x 2 - 1 / a exp - 2 ax 2 , K 3 x = a π 4 x 4 - 3 x 2 / a exp - 2 ax 2 , K 5 x = a π 4 x 6 - 5 x 4 / a exp - 2 ax 2 .
n 2 = - a π k 2 2 a π 1 / 4 ,
n 2 x = n 0 - a π k 2 2 a π 1 / 4 Ψ 2 x = n 0 - a k 2 4 ax 2 - 1 .
W x , p = 8 a π 2 a 1 / 2 x 2 + p 2 4 a 2 - 1 4 a × exp - 2 ax 2 - p 2 2 a .
C 1 = - 12 a π k 2 π 2 a 1 / 2 , C 3 = - 15 π k 2 π 2 a 1 / 2 , C 5 = - 135 π 8 ak 2 π 2 a 1 / 2 , K 1 x = 8 a π 4 ax 4 - 3 x 2 exp - 2 ax 2 , K 3 x = 8 a π 4 ax 6 - 5 x 4 exp - 2 ax 2 , K 5 x = 8 a π 4 ax 8 - 7 x 6 exp - 2 ax 2 .

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