Abstract

A numerical method is developed for solution of the Gel’fand–Levitan–Marchenko inverse scattering integral equations. The method is based on the fast inversion procedure of a Toeplitz–Hermitian matrix and special bordering technique. The method is highly competitive with the known discrete layer peeling method in speed and exceeds it noticeably in accuracy at high reflectance.

© 2007 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. 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).
  4. 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]
  5. 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]
  6. L. Poladian, "Iterative and noniterative design algorithms for Bragg gratings," Opt. Fiber Technol. 5, 215-222 (1999).
    [CrossRef]
  7. 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]
  8. F. Ahmad and M. Razzagh, "A numerical solution to the Gel'fand-Levitan-Marchenko equation," Appl. Math. Comput. 89, 31-39 (1998).
    [CrossRef]
  9. R. Feced, M. Zervas, and A. Muriel, "An efficient inverse scattering algorithm for the design of nonuniform Bragg gratings," IEEE J. Quantum Electron. 35, 1105-1115 (1999).
    [CrossRef]
  10. L. Poladian, "Simple grating synthesis algorithm," Opt. Lett. 25, 787-789 (2000).
    [CrossRef]
  11. L. Poladian, "Simple grating synthesis algorithm: errata," Opt. Lett. 251400 (2000).
    [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. J. Skaar and R. Feced, "Reconstruction of gratings from noisy reflection data," J. Opt. Soc. Am. A 19, 2229-2237 (2002).
    [CrossRef]
  14. 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]
  15. 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]
  16. A. Rosenthal and M. Horowitz, "Inverse scattering algorithm for reconstructing strongly reflecting fiber Bragg gratings," IEEE J. Quantum Electron. 39, 1018-1026 (2003).
    [CrossRef]
  17. O. V. Belai, L. L. Frumin, E. V. Podivilov, O. Y. Schwarz, and D. A. Shapiro, "Finite Bragg grating synthesis by numerical solution of Hermitian Gel'fand-Levitan-Marchenko equations," J. Opt. Soc. Am. B 23, 2040-2045 (2006).
    [CrossRef]
  18. N. Levinson, "The Winer RMS error criterion in filter design and prediction," J. Math. Phys. 25, 261-278 (1947).
    [CrossRef]
  19. W. F. Trench, "An algorithm for inversion of finite Toeplitz matrices," J. Soc. Ind. Appl. Math. 12, 512-522 (1964).
    [CrossRef]
  20. S. Zohar, "The solution of a Toeplitz set of linear equations," J. Assoc. Comput. Mach. 21, 272-276 (1974).
    [CrossRef]
  21. R. E. Blahut, Fast Algorithms for Digital Signal Processing (Addison-Wesley, 1985).
  22. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in Fortran (Cambridge U. Press, 1992).
  23. E. V. Podivilov, D. A. Shapiro, and D. A. Trubitsyn, "Exactly solvable profiles of quasi-rectangular Bragg filter with dispersion compensation," J. Opt. A, Pure Appl. Opt. 8, 788-795 (2006).
    [CrossRef]
  24. H. Bateman and A. Erdelyi, Higher Transcendental Functions (McGraw-Hill, 1953), Vol. 1.

2006 (2)

O. V. Belai, L. L. Frumin, E. V. Podivilov, O. Y. Schwarz, and D. A. Shapiro, "Finite Bragg grating synthesis by numerical solution of Hermitian Gel'fand-Levitan-Marchenko equations," J. Opt. Soc. Am. B 23, 2040-2045 (2006).
[CrossRef]

E. V. Podivilov, D. A. Shapiro, and D. A. Trubitsyn, "Exactly solvable profiles of quasi-rectangular Bragg filter with dispersion compensation," J. Opt. A, Pure Appl. Opt. 8, 788-795 (2006).
[CrossRef]

2003 (1)

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 (3)

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 (2)

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

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

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 (1)

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]

1985 (1)

1974 (1)

S. Zohar, "The solution of a Toeplitz set of linear equations," J. Assoc. Comput. Mach. 21, 272-276 (1974).
[CrossRef]

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).

1964 (1)

W. F. Trench, "An algorithm for inversion of finite Toeplitz matrices," J. Soc. Ind. Appl. Math. 12, 512-522 (1964).
[CrossRef]

1947 (1)

N. Levinson, "The Winer RMS error criterion in filter design and prediction," J. Math. Phys. 25, 261-278 (1947).
[CrossRef]

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. 1.

Belai, O. V.

Blahut, R. E.

R. E. Blahut, Fast Algorithms for Digital Signal Processing (Addison-Wesley, 1985).

Capmany, 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]

Erdelyi, A.

H. Bateman and A. Erdelyi, Higher Transcendental Functions (McGraw-Hill, 1953), Vol. 1.

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.

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. Zervas, and A. Muriel, "An efficient inverse scattering algorithm for the design of nonuniform Bragg gratings," IEEE J. Quantum Electron. 35, 1105-1115 (1999).
[CrossRef]

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.

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]

Frumin, L. L.

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

Kashyap, R.

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

Levinson, N.

N. Levinson, "The Winer RMS error criterion in filter design and prediction," J. Math. Phys. 25, 261-278 (1947).
[CrossRef]

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]

Muriel, A.

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

Papachristos, C.

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]

Podivilov, E. V.

O. V. Belai, L. L. Frumin, E. V. Podivilov, O. Y. Schwarz, and D. A. Shapiro, "Finite Bragg grating synthesis by numerical solution of Hermitian Gel'fand-Levitan-Marchenko equations," J. Opt. Soc. Am. B 23, 2040-2045 (2006).
[CrossRef]

E. V. Podivilov, D. A. Shapiro, and D. A. Trubitsyn, "Exactly solvable profiles of quasi-rectangular Bragg filter with dispersion compensation," J. Opt. A, Pure Appl. Opt. 8, 788-795 (2006).
[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]

Schwarz, O. Y.

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.

O. V. Belai, L. L. Frumin, E. V. Podivilov, O. Y. Schwarz, and D. A. Shapiro, "Finite Bragg grating synthesis by numerical solution of Hermitian Gel'fand-Levitan-Marchenko equations," J. Opt. Soc. Am. B 23, 2040-2045 (2006).
[CrossRef]

E. V. Podivilov, D. A. Shapiro, and D. A. Trubitsyn, "Exactly solvable profiles of quasi-rectangular Bragg filter with dispersion compensation," J. Opt. A, Pure Appl. Opt. 8, 788-795 (2006).
[CrossRef]

Shin, S. Y.

Skaar, J.

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.

Teukolsky, S. A.

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

Trench, W. F.

W. F. Trench, "An algorithm for inversion of finite Toeplitz matrices," J. Soc. Ind. Appl. Math. 12, 512-522 (1964).
[CrossRef]

Trubitsyn, D. A.

E. V. Podivilov, D. A. Shapiro, and D. A. Trubitsyn, "Exactly solvable profiles of quasi-rectangular Bragg filter with dispersion compensation," J. Opt. A, Pure Appl. Opt. 8, 788-795 (2006).
[CrossRef]

Vetterling, W. T.

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

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).

Zervas, M.

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

Zohar, S.

S. Zohar, "The solution of a Toeplitz set of linear equations," J. Assoc. Comput. Mach. 21, 272-276 (1974).
[CrossRef]

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]

IEEE J. Quantum Electron. (4)

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

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]

IEEE Trans. Antennas Propag. (2)

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

J. Assoc. Comput. Mach. (1)

S. Zohar, "The solution of a Toeplitz set of linear equations," J. Assoc. Comput. Mach. 21, 272-276 (1974).
[CrossRef]

J. Math. Phys. (1)

N. Levinson, "The Winer RMS error criterion in filter design and prediction," J. Math. Phys. 25, 261-278 (1947).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

E. V. Podivilov, D. A. Shapiro, and D. A. Trubitsyn, "Exactly solvable profiles of quasi-rectangular Bragg filter with dispersion compensation," J. Opt. A, Pure Appl. Opt. 8, 788-795 (2006).
[CrossRef]

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

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

J. Soc. Ind. Appl. Math. (1)

W. F. Trench, "An algorithm for inversion of finite Toeplitz matrices," J. Soc. Ind. Appl. Math. 12, 512-522 (1964).
[CrossRef]

Opt. Fiber Technol. (1)

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

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 (5)

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

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

R. E. Blahut, Fast Algorithms for Digital Signal Processing (Addison-Wesley, 1985).

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. 1.

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Figures (5)

Fig. 1
Fig. 1

(a) Reflection spectrum of GHS grating and (b) group delay characteristics for testing examples, k 0 L = 5 × 10 4 , F = 3 , Q = 1 (dashed curve), 2 (dotted curve), 3 (solid curve).

Fig. 2
Fig. 2

Root-mean-square error σ of the first-order (triangles) and second-order (boxes) reconstruction as a function of 1 N in logarithmic coordinates for N = 128 4096 , Q = 1 , L = 5 × 10 4 k 0 , F = 1 . The straight lines show the least-squares linear fitting.

Fig. 3
Fig. 3

Envelope α as a function of coordinate reconstructed by second-order TIB (solid curve) and by DLP (crosses): from the top down Q = 3 , 2 , 1 .

Fig. 4
Fig. 4

Comparison of the second-order TIB method with GHS profile [Eq. (19)]: the deviation of numerical calculations from the analytical formula as a function of coordinate. The number near each curve denotes the value of grating strength Q .

Fig. 5
Fig. 5

Deviation of the spatial frequency of the grating [Eq. (21)] from κ calculated by TIB method (solid curve) and DLP (crosses) at Q = 3 .

Equations (41)

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

δ n ( x ) n = 2 α ( x ) cos [ κ x + θ ( x ) ] ,
ψ 1 i ω ψ 1 = q * ψ 2 , ψ 2 + i ω ψ 2 = q ψ 1 ,
A 1 ( x , t ) + x R ( t + y ) A 2 * ( x , y ) d y = 0 ,
A 2 ( x , t ) + x R ( t + y ) A 1 * ( x , y ) d y = R ( x + t ) , x > t .
R ( t ) = 1 2 π r ( ω ) e i ω t d ω
q ( x ) = 2 lim t x 0 A 2 ( x , t ) .
u ( x , s ) + s 2 x R * ( τ s ) v ( x , τ ) d τ = 0 ,
v ( x , τ ) + 0 τ R ( τ s ) u ( x , s ) d s = R ( τ ) .
q ( x ) = 2 v ( x , 2 x 0 ) .
s k = h ( k 1 2 ) , k = 1 , , m ,
τ n = h ( n 1 2 ) , n = 1 , , m ,
x m = m h 2 , m = 1 , , N .
u k ( m ) + h n = k m R n k * v n ( m ) = 0 ,
v n ( m ) + h k = 1 n R n k u k ( m ) = R n ,
n , k = 1 , , m , m = 1 , , N .
q ( m ) = 2 v m ( m ) .
G ( m ) w ( m ) = b ( m ) ,
w ( m ) = ( u ( m ) v ( m ) ) .
G ( m ) = ( E h R h R E ) .
R = ( R 0 0 0 0 R 1 R 0 0 0 R 2 R 1 R 0 0 0 R m 1 R m 2 R m 3 R 0 ) .
f ( m ) = ( f 1 ( m ) f 2 m ( m ) ) ,
G ( m ) f ( m ) = ( 1 0 0 ) .
f ̃ ( m ) = ( f 2 m * ( m ) f 1 * ( m ) ) .
G ( m ) f ̃ ( m ) = ( 0 0 1 ) .
f ( m ) = ( y ( m ) z ( m ) ) .
y ( m + 1 ) = c m ( y ( m ) 0 ) + d m ( 0 z ̃ ( m ) ) ,
z ( m + 1 ) = c m ( z ( m ) 0 ) + d m ( 0 y ̃ ( m ) ) .
y 1 ( 1 ) = 1 1 h 2 R 0 2 , z 1 ( 1 ) = h R 0 1 h 2 R 0 2 .
c m = 1 1 β ( m ) 2 , d m = β ( m ) 1 β ( m ) 2 ,
β ( m ) = h ( R m y 1 ( m ) + R m 1 y 2 ( m ) + + R 1 y m ( m ) ) .
q ( x ) = Q L ( sech x L ) 1 2 i F .
α ( x ) = δ n max 2 n sech x L ,
θ ( x ) = 2 F ln ( cosh x L ) π 2 ,
κ ( x ) = κ + d θ d x = κ + 2 F L tanh x L ,
r ( ω ) = 2 2 i F Q Γ ( d ) Γ ( d * ) Γ ( f ) Γ ( g ) Γ ( f + ) Γ ( g + ) ,
d = 1 2 + i [ ω L F ] ,
f ± = 1 2 i [ ω L ± F 2 + Q 2 ] ,
g ± = 1 i [ F ± F 2 + Q 2 ] .
r ( ω ) 2 = cosh 2 π Q 2 + F 2 cosh 2 π F cosh 2 π Q 2 + F 2 + cosh 2 π ω L .
n total ( l N m + l + 1 m N 2 ) log 2 N ,
n total N 2 m log 2 N + m N .

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