Abstract

We examine the problem of fiber Bragg grating reconstruction from its reflection coefficient. A direct numerical method of solving the Gel’fand–Levitan–Marchenko integral equations for the problem is developed. The method is based on a bordering procedure, Cholesky decomposition, and piecewise-linear approximation. It is tested using high-reflectance homogeneous and hyperbolic secant profiles. The proposed method is shown to concede the popular discrete layer peeling technique in efficiency but surpasses it in accuracy and stability at high reflectance.

© 2006 Optical Society of America

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References

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  1. G. P. Agrawal, Fiber-Optic Communication Systems (Wiley, 2002).
    [CrossRef]
  2. R. Kashyap, Fiber Bragg Gratings (Academic, 1999).
  3. M. Sumetsky and B. J. Eggleton, "Fiber Bragg gratings for dispersion compensation in optical communication systems," J. Opt. Fiber. Commun. Rep. 2, 256-278 (2005).
    [CrossRef]
  4. V. E. Zakharov and A. B. Shabat, "Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media," Zh. Eksp. Teor. Fiz. 61, 118-134 (1971).
  5. L. Poladian, "Simple grating synthesis algorithm," Opt. Lett. 25, 787-789 (2000).
    [CrossRef]
  6. L. Poladian, "Simple grating synthesis algorithm," Opt. Lett. 25, 1400 (2000), errata.
    [CrossRef]
  7. 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]
  8. P. Frangos and D. Jaggard, "A numerical solution to the Zakharov-Shabat inverse scattering problem," IEEE Trans. Antennas Propag. 39, 74-79 (1991).
    [CrossRef]
  9. C. Papachristos and P. Frangos, "Design of corrugated optical waveguide filters through a direct numerical solution of the coupled Gel'fand-Levitan-Marchenko integral equations," J. Opt. Soc. Am. A 19, 1005-1012 (2002).
    [CrossRef]
  10. C. Papachristos and P. Frangos, "Synthesis of single- and multi-mode planar optical waveguides by a direct numerical solution of the Gel'fand-Levitan-Marchenko integral equations," Opt. Commun. 203, 27-37 (2002).
    [CrossRef]
  11. G. B. Xiao and K. Yashiro, "An efficient algorithm for solving Zakharov-Shabat inverse scattering problem," IEEE Trans. Antennas Propag. 50, 807-811 (2002).
    [CrossRef]
  12. P. V. Frangos and D. L. Jaggard, "The reconstruction of stratified dielectric profiles using successive approximations," IEEE Trans. Antennas Propag. 35, 1267-1272 (1987).
    [CrossRef]
  13. J. Skaar and R. Feced, "Reconstruction of gratings from noisy reflection data," J. Opt. Soc. Am. A 19, 2229-2237 (2002).
    [CrossRef]
  14. J. Capmany and J. Marti, "Design of fibre grating dispersion compensators using a novel iterative solution of the Gel'fand-Levitan-Marchenko coupled equations," Electron. Lett. 32, 918-919 (1996).
    [CrossRef]
  15. E. Peral, J. Capmany, and J. Marti, "Iterative solution to the Gel'fand-Levitan-Marchenko coupled equations and application to synthesis of fiber gratings," IEEE J. Quantum Electron. 32, 2078-2084 (1996).
    [CrossRef]
  16. L. Poladian, "Iterative and noniterative design algorithms for Bragg gratings," Opt. Laser Technol. 5, 215-222 (1999).
  17. P. V. Frangos and D. L. Jaggard, "Inverse scattering: solution of coupled Gelfand-Levitan-Marchenko integral equations using successive kernel approximations," IEEE Trans. Antennas Propag. 43, 547-552 (1995).
    [CrossRef]
  18. 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]
  19. F. Ahmad and M. Razzagh, "A numerical solution to the Gel'fand-Levitan-Marchenko equation," Appl. Math. Comput. 89, 31-39 (1998).
    [CrossRef]
  20. A. Rosenthal and M. Horowitz, "Inverse scattering algorithm for reconstructing strongly reflecting fiber Bragg gratings," IEEE J. Quantum Electron. 39, 1018-1026 (2003).
    [CrossRef]
  21. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in Fortran (Cambridge U. Press, 1992).
  22. H. Kogelnik, "Coupled wave theory for thick hologram gratings," Bell Syst. Tech. J. 48, 2909-2947 (1969).
  23. H. Bateman and A. Erdelyi, Higher Transcendental Functions (McGraw-Hill, 1953), Vol. 2.
  24. D. A. Shapiro, "Family of exact solutions for reflection spectrum of Bragg grating," Opt. Commun. 215, 295-301 (2003).
    [CrossRef]
  25. J. Skaar and O. H. Waagaard, "Design and characterization of finite-length fiber grating," IEEE J. Quantum Electron. 39, 1238-1245 (2003).
    [CrossRef]

2005 (1)

M. Sumetsky and B. J. Eggleton, "Fiber Bragg gratings for dispersion compensation in optical communication systems," J. Opt. Fiber. Commun. Rep. 2, 256-278 (2005).
[CrossRef]

2003 (3)

D. A. Shapiro, "Family of exact solutions for reflection spectrum of Bragg grating," Opt. Commun. 215, 295-301 (2003).
[CrossRef]

J. Skaar and O. H. Waagaard, "Design and characterization of finite-length fiber grating," IEEE J. Quantum Electron. 39, 1238-1245 (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]

2002 (4)

C. Papachristos and P. Frangos, "Synthesis of single- and multi-mode planar optical waveguides by a direct numerical solution of the Gel'fand-Levitan-Marchenko integral equations," Opt. Commun. 203, 27-37 (2002).
[CrossRef]

G. B. Xiao and K. Yashiro, "An efficient algorithm for solving Zakharov-Shabat inverse scattering problem," IEEE Trans. Antennas Propag. 50, 807-811 (2002).
[CrossRef]

C. Papachristos and P. Frangos, "Design of corrugated optical waveguide filters through a direct numerical solution of the coupled Gel'fand-Levitan-Marchenko integral equations," J. Opt. Soc. Am. A 19, 1005-1012 (2002).
[CrossRef]

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

2001 (1)

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]

2000 (2)

1999 (1)

L. Poladian, "Iterative and noniterative design algorithms for Bragg gratings," Opt. Laser Technol. 5, 215-222 (1999).

1998 (1)

F. Ahmad and M. Razzagh, "A numerical solution to the Gel'fand-Levitan-Marchenko equation," Appl. Math. Comput. 89, 31-39 (1998).
[CrossRef]

1996 (2)

J. Capmany and J. Marti, "Design of fibre grating dispersion compensators using a novel iterative solution of the Gel'fand-Levitan-Marchenko coupled equations," Electron. Lett. 32, 918-919 (1996).
[CrossRef]

E. Peral, J. Capmany, and J. Marti, "Iterative solution to the Gel'fand-Levitan-Marchenko coupled equations and application to synthesis of fiber gratings," IEEE J. Quantum Electron. 32, 2078-2084 (1996).
[CrossRef]

1995 (1)

P. V. Frangos and D. L. Jaggard, "Inverse scattering: solution of coupled Gelfand-Levitan-Marchenko integral equations using successive kernel approximations," IEEE Trans. Antennas Propag. 43, 547-552 (1995).
[CrossRef]

1991 (1)

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

1987 (1)

P. V. Frangos and D. L. Jaggard, "The reconstruction of stratified dielectric profiles using successive approximations," IEEE Trans. Antennas Propag. 35, 1267-1272 (1987).
[CrossRef]

1985 (1)

1971 (1)

V. E. Zakharov and A. B. Shabat, "Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media," Zh. Eksp. Teor. Fiz. 61, 118-134 (1971).

1969 (1)

H. Kogelnik, "Coupled wave theory for thick hologram gratings," Bell Syst. Tech. J. 48, 2909-2947 (1969).

Agrawal, G. P.

G. P. Agrawal, Fiber-Optic Communication Systems (Wiley, 2002).
[CrossRef]

Ahmad, F.

F. Ahmad and M. Razzagh, "A numerical solution to the Gel'fand-Levitan-Marchenko equation," Appl. Math. Comput. 89, 31-39 (1998).
[CrossRef]

Bateman, H.

H. Bateman and A. Erdelyi, Higher Transcendental Functions (McGraw-Hill, 1953), Vol. 2.

Capmany, J.

J. Capmany and J. Marti, "Design of fibre grating dispersion compensators using a novel iterative solution of the Gel'fand-Levitan-Marchenko coupled equations," Electron. Lett. 32, 918-919 (1996).
[CrossRef]

E. Peral, J. Capmany, and J. Marti, "Iterative solution to the Gel'fand-Levitan-Marchenko coupled equations and application to synthesis of fiber gratings," IEEE J. Quantum Electron. 32, 2078-2084 (1996).
[CrossRef]

Eggleton, B. J.

M. Sumetsky and B. J. Eggleton, "Fiber Bragg gratings for dispersion compensation in optical communication systems," J. Opt. Fiber. Commun. Rep. 2, 256-278 (2005).
[CrossRef]

Erdelyi, A.

H. Bateman and A. Erdelyi, Higher Transcendental Functions (McGraw-Hill, 1953), Vol. 2.

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]

Feced, R.

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in Fortran (Cambridge U. Press, 1992).

Frangos, P.

C. Papachristos and P. Frangos, "Synthesis of single- and multi-mode planar optical waveguides by a direct numerical solution of the Gel'fand-Levitan-Marchenko integral equations," Opt. Commun. 203, 27-37 (2002).
[CrossRef]

C. Papachristos and P. Frangos, "Design of corrugated optical waveguide filters through a direct numerical solution of the coupled Gel'fand-Levitan-Marchenko integral equations," J. Opt. Soc. Am. A 19, 1005-1012 (2002).
[CrossRef]

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

Frangos, P. V.

P. V. Frangos and D. L. Jaggard, "Inverse scattering: solution of coupled Gelfand-Levitan-Marchenko integral equations using successive kernel approximations," IEEE Trans. Antennas Propag. 43, 547-552 (1995).
[CrossRef]

P. V. Frangos and D. L. Jaggard, "The reconstruction of stratified dielectric profiles using successive approximations," IEEE Trans. Antennas Propag. 35, 1267-1272 (1987).
[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.

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

Jaggard, D. L.

P. V. Frangos and D. L. Jaggard, "Inverse scattering: solution of coupled Gelfand-Levitan-Marchenko integral equations using successive kernel approximations," IEEE Trans. Antennas Propag. 43, 547-552 (1995).
[CrossRef]

P. V. Frangos and D. L. Jaggard, "The reconstruction of stratified dielectric profiles using successive approximations," IEEE Trans. Antennas Propag. 35, 1267-1272 (1987).
[CrossRef]

Kashyap, R.

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

Kogelnik, H.

H. Kogelnik, "Coupled wave theory for thick hologram gratings," Bell Syst. Tech. J. 48, 2909-2947 (1969).

Marti, J.

E. Peral, J. Capmany, and J. Marti, "Iterative solution to the Gel'fand-Levitan-Marchenko coupled equations and application to synthesis of fiber gratings," IEEE J. Quantum Electron. 32, 2078-2084 (1996).
[CrossRef]

J. Capmany and J. Marti, "Design of fibre grating dispersion compensators using a novel iterative solution of the Gel'fand-Levitan-Marchenko coupled equations," Electron. Lett. 32, 918-919 (1996).
[CrossRef]

Papachristos, C.

C. Papachristos and P. Frangos, "Synthesis of single- and multi-mode planar optical waveguides by a direct numerical solution of the Gel'fand-Levitan-Marchenko integral equations," Opt. Commun. 203, 27-37 (2002).
[CrossRef]

C. Papachristos and P. Frangos, "Design of corrugated optical waveguide filters through a direct numerical solution of the coupled Gel'fand-Levitan-Marchenko integral equations," J. Opt. Soc. Am. A 19, 1005-1012 (2002).
[CrossRef]

Peral, E.

E. Peral, J. Capmany, and J. Marti, "Iterative solution to the Gel'fand-Levitan-Marchenko coupled equations and application to synthesis of fiber gratings," IEEE J. Quantum Electron. 32, 2078-2084 (1996).
[CrossRef]

Poladian, L.

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in Fortran (Cambridge U. Press, 1992).

Razzagh, M.

F. Ahmad and M. Razzagh, "A numerical solution to the Gel'fand-Levitan-Marchenko equation," Appl. Math. Comput. 89, 31-39 (1998).
[CrossRef]

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]

Shabat, A. B.

V. E. Zakharov and A. B. Shabat, "Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media," Zh. Eksp. Teor. Fiz. 61, 118-134 (1971).

Shapiro, D. A.

D. A. Shapiro, "Family of exact solutions for reflection spectrum of Bragg grating," Opt. Commun. 215, 295-301 (2003).
[CrossRef]

Shin, S. Y.

Skaar, J.

J. Skaar and O. H. Waagaard, "Design and characterization of finite-length fiber grating," 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, "On the synthesis of fiber Bragg gratings by layer peeling," IEEE J. Quantum Electron. 37, 165-173 (2001).
[CrossRef]

Song, G. H.

Sumetsky, M.

M. Sumetsky and B. J. Eggleton, "Fiber Bragg gratings for dispersion compensation in optical communication systems," J. Opt. Fiber. Commun. Rep. 2, 256-278 (2005).
[CrossRef]

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in Fortran (Cambridge U. Press, 1992).

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in Fortran (Cambridge U. Press, 1992).

Waagaard, O. H.

J. Skaar and O. H. Waagaard, "Design and characterization of finite-length fiber grating," IEEE J. Quantum Electron. 39, 1238-1245 (2003).
[CrossRef]

Wang, L.

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]

Xiao, G. B.

G. B. Xiao and K. Yashiro, "An efficient algorithm for solving Zakharov-Shabat inverse scattering problem," IEEE Trans. Antennas Propag. 50, 807-811 (2002).
[CrossRef]

Yashiro, K.

G. B. Xiao and K. Yashiro, "An efficient algorithm for solving Zakharov-Shabat inverse scattering problem," IEEE Trans. Antennas Propag. 50, 807-811 (2002).
[CrossRef]

Zakharov, V. E.

V. E. Zakharov and A. B. Shabat, "Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media," Zh. Eksp. Teor. Fiz. 61, 118-134 (1971).

Appl. Math. Comput. (1)

F. Ahmad and M. Razzagh, "A numerical solution to the Gel'fand-Levitan-Marchenko equation," Appl. Math. Comput. 89, 31-39 (1998).
[CrossRef]

Bell Syst. Tech. J. (1)

H. Kogelnik, "Coupled wave theory for thick hologram gratings," Bell Syst. Tech. J. 48, 2909-2947 (1969).

Electron. Lett. (1)

J. Capmany and J. Marti, "Design of fibre grating dispersion compensators using a novel iterative solution of the Gel'fand-Levitan-Marchenko coupled equations," Electron. Lett. 32, 918-919 (1996).
[CrossRef]

IEEE J. Quantum Electron. (4)

E. Peral, J. Capmany, and J. Marti, "Iterative solution to the Gel'fand-Levitan-Marchenko coupled equations and application to synthesis of fiber gratings," IEEE J. Quantum Electron. 32, 2078-2084 (1996).
[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 grating," IEEE J. Quantum Electron. 39, 1238-1245 (2003).
[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]

IEEE Trans. Antennas Propag. (4)

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

P. V. Frangos and D. L. Jaggard, "Inverse scattering: solution of coupled Gelfand-Levitan-Marchenko integral equations using successive kernel approximations," IEEE Trans. Antennas Propag. 43, 547-552 (1995).
[CrossRef]

G. B. Xiao and K. Yashiro, "An efficient algorithm for solving Zakharov-Shabat inverse scattering problem," IEEE Trans. Antennas Propag. 50, 807-811 (2002).
[CrossRef]

P. V. Frangos and D. L. Jaggard, "The reconstruction of stratified dielectric profiles using successive approximations," IEEE Trans. Antennas Propag. 35, 1267-1272 (1987).
[CrossRef]

J. Opt. Fiber. Commun. Rep. (1)

M. Sumetsky and B. J. Eggleton, "Fiber Bragg gratings for dispersion compensation in optical communication systems," J. Opt. Fiber. Commun. Rep. 2, 256-278 (2005).
[CrossRef]

J. Opt. Soc. Am. A (3)

Opt. Commun. (2)

D. A. Shapiro, "Family of exact solutions for reflection spectrum of Bragg grating," Opt. Commun. 215, 295-301 (2003).
[CrossRef]

C. Papachristos and P. Frangos, "Synthesis of single- and multi-mode planar optical waveguides by a direct numerical solution of the Gel'fand-Levitan-Marchenko integral equations," Opt. Commun. 203, 27-37 (2002).
[CrossRef]

Opt. Laser Technol. (1)

L. Poladian, "Iterative and noniterative design algorithms for Bragg gratings," Opt. Laser Technol. 5, 215-222 (1999).

Opt. Lett. (2)

Zh. Eksp. Teor. Fiz. (1)

V. E. Zakharov and A. B. Shabat, "Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media," Zh. Eksp. Teor. Fiz. 61, 118-134 (1971).

Other (4)

G. P. Agrawal, Fiber-Optic Communication Systems (Wiley, 2002).
[CrossRef]

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

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in Fortran (Cambridge U. Press, 1992).

H. Bateman and A. Erdelyi, Higher Transcendental Functions (McGraw-Hill, 1953), Vol. 2.

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Figures (4)

Fig. 1
Fig. 1

Fourier transform of the reflection coefficient of a homogeneous FBG at coupling parameter q 0 = 2 (solid curve), 4 (dashed curve), 6 (dotted curve).

Fig. 2
Fig. 2

Solution A 2 ( x , z ) , given by Eq. (16), of the example GLM equation A 2 ( x , z ) at x = L 2 , x z x from the bottom upwards at q 0 = 2 , 4, 6.

Fig. 3
Fig. 3

Numerical solution of the GLM equations. (a) Reconstructed coupling coefficient of finite homogeneous grating q ( x ) = q 0 at N = 1024 , q 0 = 2 , 4, 6 (from the bottom upwards). (b) The same at q 0 = 6 and N = 64 , 128, 256, 512, 1024 (from the bottom upwards).

Fig. 4
Fig. 4

Comparison of the reconstruction of a hyperbolic secant profile by different methods: GLM direct solution (+), DLP (dashed curves), and exact profile (solid curves) at L = 1 32 : (a) N L = 40 , (b) 60. Insets at the bottom show the differences between numerical and analytical profiles: new method (solid curves) and the DLP (dashed curves).

Equations (38)

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

ψ 1 i ω ψ 1 = q * ψ 2 , ψ 2 + i ω ψ 2 = q ψ 1 .
0 = A 1 ( x , y ) + x R ( z + y ) A 2 * ( x , z ) d z , y < x ,
0 = A 2 ( x , z ) + R ( x + z ) + x R ( z + y ) A 1 * ( x , y ) d y ,
z < x .
q ( x ) = 2 lim z x 0 A 2 ( x , z ) .
0 = A 1 ( x , x s ) + s 2 x R ( τ s ) A 2 * ( x , τ x ) d τ , s > 0 ,
0 = A 2 ( x , τ x ) + R ( τ ) + 0 τ R ( τ s ) A 1 * ( x , x s ) d s ,
τ < 2 x .
A 2 ( x , τ x ) 0 2 x Φ ( x τ , σ ) A 2 ( x , σ x ) d σ = R ( τ ) .
Φ ( x τ , σ ) = 0 min ( τ , σ ) R ( τ s ) R * ( σ s ) d s , 0 < τ , σ < 2 x .
u n ( m ) = A 2 ( x m , τ n x m ) , Φ n k ( m ) = Φ ( x m τ n , σ k ) ,
x m = m h 2 , m = 1 , , N ,
τ n = ( n 1 2 ) h , σ k = ( k 1 2 ) h ,
n , k = 1 , , m .
G ( m ) u ( m ) = b ( m ) .
Φ n k ( m ) = ( Φ k n ( m ) ) * = h l = 1 m R ̃ n l R ̃ k l * ,
R ̃ 0 = R 0 2 , R ̃ k = R k , k 0 .
Φ n k ( m ) = Φ n k ( m 1 ) , n , k = 1 , , m 1 ,
Φ n m ( m ) = h l = 1 n R ̃ n l R ̃ m l * , n = 1 , , m 1 .
L n k ( m ) = L n k ( m 1 ) , n , k = 1 , , m 1 ,
L m k ( m ) = G n k ( m ) n = 1 k 1 L m n ( m ) ( L k n ( m ) ) * L k k ( m ) , L k m ( m ) = 0 ,
k = 1 , , m 1 ,
L m m ( m ) = G m m ( m ) k = 1 m 1 L m k ( m ) 2 , k = 1 , , m 1 .
u m ( m ) = k = 1 m [ ( L ( m ) ) 1 ] m k b k ( m ) L m m ( m ) , u m 1 ( m ) = u m 1 ( m 1 ) u m ( m ) L m 1 , m ( m ) * L m 1 , m 1 ( m ) ,
q ( m h 2 ) = { 3 u m ( m ) u m 1 ( m ) , m > 1 4 u 1 ( 1 ) + 2 R ( 0 ) , m = 1 } .
A 2 ( x , x 0 ) = A 2 ( x , x h 2 ) + A 2 ( x , x h 2 ) A 2 ( x , x 3 h 2 ) 2 ,
r ( ω ) = q 0 sinh ν L ν cosh ν L i ω sinh ν L ,
A 2 ( x , z ) = q 0 ν sinh ( ν x ) e i ω z d ω 2 π .
A 2 ( x , z ) = q 0 2 I 0 ( q 0 x 2 z 2 ) Θ ( x 2 z 2 ) ,
Θ ( s ) = { 0 , at s < 0 1 , at s > 0 } .
q ( x ) = N L cosh ( x L ) ,
r ( ω ) = i π sinh ( π N ) Γ ( 1 2 + i ω L ) Γ ( 1 2 i ω L ) Γ ( 1 2 i ( ω L N ) ) Γ ( 1 2 i ( ω L + N ) ) .
Ψ ( x , ω ) = e i ω x at x > L
Ψ ( x , ω ) = a ( ω ) e i ω x + b ( ω ) e i ω x at x < 0 .
r ( ω ) = b ( ω ) a ( ω ) , t ( ω ) = 1 a ( ω ) .
a ( z ) , b ( z ) 0 only at 0 < z < 2 L .
A 1 ( x , z ) = f * ( x z ) at x > L .
R ( z ) + 0 2 L R ( z s ) f ( s ) d s = 0 , z > 2 L .

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