Abstract

In this work, we introduce the periodic nonlinear Fourier transform (PNFT) method as an alternative and efficacious tool for compensation of the nonlinear transmission effects in optical fiber links. In the Part I, we introduce the algorithmic platform of the technique, describing in details the direct and inverse PNFT operations, also known as the inverse scattering transform for periodic (in time variable) nonlinear Schrödinger equation (NLSE). We pay a special attention to explaining the potential advantages of the PNFT-based processing over the previously studied nonlinear Fourier transform (NFT) based methods. Further, we elucidate the issue of the numerical PNFT computation: we compare the performance of four known numerical methods applicable for the calculation of nonlinear spectral data (the direct PNFT), in particular, taking the main spectrum (utilized further in Part II for the modulation and transmission) associated with some simple example waveforms as the quality indicator for each method. We show that the Ablowitz-Ladik discretization approach for the direct PNFT provides the best performance in terms of the accuracy and computational time consumption.

© 2016 Optical Society of America

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References

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2016 (1)

2015 (6)

P. J. Winzer, “Scaling optical fiber networks: challenges and solutions,” Opt. Photon. News 26, 28–35 (2015).
[Crossref]

S. T. Le, J. E. Prilepsky, and S. K. Turitsyn, “Nonlinear inverse synthesis technique for optical links with lumped amplification,” Opt. Express 23, 8317–8328 (2015).
[Crossref] [PubMed]

H. Bülow, “Experimental demonstration of optical signal detection using nonlinear fourier transform,” J. Lightwave Technol. 33, 1433–1439 (2015).
[Crossref]

Z. Dong, S. Hari, T. Gui, K. Zhong, M. I. Yousefi, C. Lu, P. A. Wai, F. R. Kschischang, and A. Lau, “Nonlinear frequency division multiplexed transmissions based on NFT,” IEEE Photon. Technol. Lett. 27, 1621–1623 (2015).
[Crossref]

S. Wahls and H. V. Poor, “Fast numerical nonlinear Fourier transforms,” IEEE Trans. Inf. Theory 61, 6957–6974 (2015).
[Crossref]

J. Frauendiener and C. Klein, “Computational approach to hyperelliptic Riemann surfaces,” Lett. Math. Phys. 105, 379–400 (2015).
[Crossref]

2014 (5)

J. E. Prilepsky, S. A. Derevyanko, K. J. Blow, I. Gabitov, and S. K. Turitsyn, “Nonlinear Inverse Synthesis and Eigenvalue Division Multiplexing in Optical Fiber Channels,” Phys. Rev. Lett. 113, 013901 (2014).
[Crossref] [PubMed]

S. T. Le, J. E. Prilepsky, and S. K. Turitsyn, “Nonlinear inverse synthesis for high spectral efficiency transmission in optical fibers,” Opt. Express 22, 26720–26741 (2014).
[Crossref] [PubMed]

M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, part I: mathematical tools,” IEEE Trans. Inf. Theory 60, 4312–4328 (2014).
[Crossref]

M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, part II: numerical methods,” IEEE Trans. Inf. Theory 60, 4329–4345 (2014).
[Crossref]

M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, part III: spectrum modulation,” IEEE Trans. Inf. Theory 60, 4346–4369 (2014).
[Crossref]

2013 (3)

2012 (1)

C. P. Olivier, B. M. Herbst, and M. A. Molchan, “A numerical study of the large limit of a Zakharov-Shabat eigenvalue problem with periodic potentials,” J. Phys. A: Math. Theor. 45, 255205 (2012).
[Crossref]

2009 (1)

2008 (2)

2006 (3)

S. L. Jansen, D. van den Borne, B. Spinnler, S. Calabro, H. Suche, P. M. Krummrich, W. Sohler, G. -D. Khoe, and H. de Waardt, “Optical phase conjugation for ultra long-haul phase-shift-keyed transmission,” J. Lightwave Technol. 24, 54–64 (2006).
[Crossref]

W. Shieh, W. Chen, and R.S. Tucker, “Polarisation mode dispersion mitigation in coherent optical orthogonal frequency division multiplexed systems,” Electron. Lett. 42, 996–997 (2006).
[Crossref]

B. Deconinck and J. K. Nathan, “Computing spectra of linear operators using the Floquet–Fourier–Hill method,” J. Comput. Phys. 219, 296–321 (2006).
[Crossref]

2004 (1)

B. Deconinck, H. Heil, A. Bobenko, M. Van Hoeij, and M. Schmies, “Computing Riemann theta functions,” Mathematics of Computation 73, 1417–1442 (2004).
[Crossref]

1997 (1)

J. A. C. Weideman and B. M. Herbst, “Finite difference methods for an AKNS eigenproblem,” Math. Comput. Simulat. 43, 77–88 (1997).
[Crossref]

1993 (1)

A. Hasegawa and T. Nyu, “Eigenvalue communication,” J. Lightwave Technol. 11, 395–399 (1993).
[Crossref]

1992 (1)

G. Boffetta and A. Osborne, “Computation of the direct scattering transform for the nonlinear Schrödinger equation,” J. Comput. Phys. 102, 252–264 (1992).
[Crossref]

1991 (1)

A. Hasegawa and Y. Kodama, “Guiding-center soliton,” Phys. Rev. Lett. 66, 161 (1991).
[Crossref] [PubMed]

1988 (1)

E. R. Tracy and H. H. Chen, “Nonlinear self-modulation: an exactly solvable model,” Phys. Rev. A 37, 815–839 (1988).
[Crossref]

1981 (1)

Y. Ma and M. J. Ablowitz, “The periodic cubic Schrödinger equation,” Stud. Appl. Math. 65, 113–158 (1981).
[Crossref]

1976 (1)

O. R. Its and V. P. Kotlyarov., “Explicit formulas for the solutions of a nonlinear Schrödinger equation,” Doklady Akad. Nauk Ukrainian SSR, ser. A, 10, 965–968 (1976).

1972 (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,” Soviet Physics-JETP 34, 62–69 (1972).

Ablowitz, M. J.

Y. Ma and M. J. Ablowitz, “The periodic cubic Schrödinger equation,” Stud. Appl. Math. 65, 113–158 (1981).
[Crossref]

Agrawal, G. P.

G. P. Agrawal, Fibre-Optic Communication Systems, 4th ed. (Wiley, 2010).
[Crossref]

G. P. Agrawal, Nonlinear Fiber Optics, 5th ed. (Academic, 2013).

Ania-Castanon, J. D.

S. T. Le, J. E. Prilepsky, M. Kamalian, P. Rosa, M. Tan, J. D. Ania-Castanon, P. Harper, and S. K. Turitsyn, “Modified nonlinear inverse synthesis for optical links with distributed Raman amplification,” in European Conference on Optical Communication (ECOC), Valencia, Spain, paper Tu.1.1.3 (2015).

Aref, V.

V. Aref, H. Bülow, K. Schuh, and W. Idler, “Experimental demonstration of nonlinear frequency division multiplexed transmission,” in European Conference on Optical Communication (ECOC), Valencia, Spain (2015).

H. Bülow, V. Aref, and W. Idler, “Transmission of waveforms determined by 7 eigenvalues with PSK-modulated spectral amplitudes,” preprint arXiv:1605.08069 (2016).

Blow, K. J.

J. E. Prilepsky, S. A. Derevyanko, K. J. Blow, I. Gabitov, and S. K. Turitsyn, “Nonlinear Inverse Synthesis and Eigenvalue Division Multiplexing in Optical Fiber Channels,” Phys. Rev. Lett. 113, 013901 (2014).
[Crossref] [PubMed]

Bobenko, A.

B. Deconinck, H. Heil, A. Bobenko, M. Van Hoeij, and M. Schmies, “Computing Riemann theta functions,” Mathematics of Computation 73, 1417–1442 (2004).
[Crossref]

Boffetta, G.

G. Boffetta and A. Osborne, “Computation of the direct scattering transform for the nonlinear Schrödinger equation,” J. Comput. Phys. 102, 252–264 (1992).
[Crossref]

Bülow, H.

H. Bülow, “Experimental demonstration of optical signal detection using nonlinear fourier transform,” J. Lightwave Technol. 33, 1433–1439 (2015).
[Crossref]

V. Aref, H. Bülow, K. Schuh, and W. Idler, “Experimental demonstration of nonlinear frequency division multiplexed transmission,” in European Conference on Optical Communication (ECOC), Valencia, Spain (2015).

H. Bülow, V. Aref, and W. Idler, “Transmission of waveforms determined by 7 eigenvalues with PSK-modulated spectral amplitudes,” preprint arXiv:1605.08069 (2016).

Calabro, S.

Carena, A.

Chan, T. H.

Q. Zhang, T. H. Chan, and A. Grant, “Spatially periodic signals for fiber channels,” in IEEE International Symposium on Information Theory (ISIT), Honolulu, HI, USA, pp. 2804–2808 (2014).

Chandrasekhar, S.

A. R. C. X. Liu, P. J. Winzer, R. W. Tkach, and S. Chandrasekhar, “Phase-conjugated twin waves for communication beyond the Kerr nonlinearity limit,” Nat. Photonics 7, 560–568 (2013).
[Crossref]

X. Chen, X. Liu, S. Chandrasekhar, B. Zhu, and R. W. Tkach, “Experimental demonstration of fiber nonlinearity mitigation using digital phase conjugation,” in Technical Digest of Optical Fiber Communication Conference and Exposition and the National Fiber Optic Engineers Conference (OFC/NFOEC), paper OTh3C.1 (2012).

Chen, H. H.

E. R. Tracy and H. H. Chen, “Nonlinear self-modulation: an exactly solvable model,” Phys. Rev. A 37, 815–839 (1988).
[Crossref]

Chen, W.

W. Shieh, W. Chen, and R.S. Tucker, “Polarisation mode dispersion mitigation in coherent optical orthogonal frequency division multiplexed systems,” Electron. Lett. 42, 996–997 (2006).
[Crossref]

Chen, X.

X. Chen, X. Liu, S. Chandrasekhar, B. Zhu, and R. W. Tkach, “Experimental demonstration of fiber nonlinearity mitigation using digital phase conjugation,” in Technical Digest of Optical Fiber Communication Conference and Exposition and the National Fiber Optic Engineers Conference (OFC/NFOEC), paper OTh3C.1 (2012).

Curri, V.

Cvijetic, M.

M. Cvijetic and I. B. Djordjevic, Advanced Optical Communication Systems and Networks (Artech House, 2013).

de Waardt, H.

Deconinck, B.

B. Deconinck and J. K. Nathan, “Computing spectra of linear operators using the Floquet–Fourier–Hill method,” J. Comput. Phys. 219, 296–321 (2006).
[Crossref]

B. Deconinck, H. Heil, A. Bobenko, M. Van Hoeij, and M. Schmies, “Computing Riemann theta functions,” Mathematics of Computation 73, 1417–1442 (2004).
[Crossref]

B. Deconinck, M. S. Patterson, and C. Swierczewski, “Computing the Riemann constant vector,” preprint, available online at https://depts.washington.edu/bdecon/papers/pdfs/rcv.pdf (2015).

C. Swierczewski and B. Deconinck, “Computing Riemann theta functions in Sage with applications,” Math. Comput. Simulat., Available online at http://dx.doi.org/10.1016/j.matcom.2013.04.018 (2013).

Derevyanko, S. A.

J. E. Prilepsky, S. A. Derevyanko, K. J. Blow, I. Gabitov, and S. K. Turitsyn, “Nonlinear Inverse Synthesis and Eigenvalue Division Multiplexing in Optical Fiber Channels,” Phys. Rev. Lett. 113, 013901 (2014).
[Crossref] [PubMed]

J. E. Prilepsky, S. A. Derevyanko, and S. K. Turitsyn, “Nonlinear spectral management: linearization of the lossless fiber channel,” Opt. Express 21, 24344–24367 (2013).
[Crossref] [PubMed]

Djordjevic, I. B.

M. Cvijetic and I. B. Djordjevic, Advanced Optical Communication Systems and Networks (Artech House, 2013).

Dong, Z.

Z. Dong, S. Hari, T. Gui, K. Zhong, M. I. Yousefi, C. Lu, P. A. Wai, F. R. Kschischang, and A. Lau, “Nonlinear frequency division multiplexed transmissions based on NFT,” IEEE Photon. Technol. Lett. 27, 1621–1623 (2015).
[Crossref]

Doran, N. J.

I. Phillips, M. Tan, M. F. Stephens, M. McCarthy, E. Giacoumidis, S. Sygletos, P. Rosa, S. Fabbri, S. T. Le, T. Kanesan, S. K. Turitsyn, N. J. Doran, P. Harper, and A. D. Ellis, “Exceeding the nonlinear-shannon limit using Raman laser based amplification and optical phase conjugation,” in Optical Fiber Communication Conference and Exhibition (OFC), Los Angeles, USA, paper M3C.1 (2014).
[Crossref]

Elliott, C. F.

C. F. Elliott, Handbook of Digital Signal Processing: Engineering Applications (Academic, 2013).

Ellis, A. D.

I. Phillips, M. Tan, M. F. Stephens, M. McCarthy, E. Giacoumidis, S. Sygletos, P. Rosa, S. Fabbri, S. T. Le, T. Kanesan, S. K. Turitsyn, N. J. Doran, P. Harper, and A. D. Ellis, “Exceeding the nonlinear-shannon limit using Raman laser based amplification and optical phase conjugation,” in Optical Fiber Communication Conference and Exhibition (OFC), Los Angeles, USA, paper M3C.1 (2014).
[Crossref]

Essiambre, R. J.

R. J. Essiambre, R. W. Tkach, and R. Ryf, “Fiber nonlinearity and capacity: single mode and multimode fibers,” in Optical Fiber Telecommunications, Vol. VIB: Systems and Networks, 6th ed., Ch. 1, edts. I. Kaminow, T. Li, and A.E. Willner, eds., pp. 1–43 (Academic, 2013).
[Crossref]

Fabbri, S.

I. Phillips, M. Tan, M. F. Stephens, M. McCarthy, E. Giacoumidis, S. Sygletos, P. Rosa, S. Fabbri, S. T. Le, T. Kanesan, S. K. Turitsyn, N. J. Doran, P. Harper, and A. D. Ellis, “Exceeding the nonlinear-shannon limit using Raman laser based amplification and optical phase conjugation,” in Optical Fiber Communication Conference and Exhibition (OFC), Los Angeles, USA, paper M3C.1 (2014).
[Crossref]

Forghieri, F.

Frauendiener, J.

J. Frauendiener and C. Klein, “Computational approach to hyperelliptic Riemann surfaces,” Lett. Math. Phys. 105, 379–400 (2015).
[Crossref]

Gabitov, I.

J. E. Prilepsky, S. A. Derevyanko, K. J. Blow, I. Gabitov, and S. K. Turitsyn, “Nonlinear Inverse Synthesis and Eigenvalue Division Multiplexing in Optical Fiber Channels,” Phys. Rev. Lett. 113, 013901 (2014).
[Crossref] [PubMed]

Geisler, A.

A. Geisler and C. G. Schaeffer, “Implementation of eigenvalue multiplex transmission with a real Fiber Link using the discrete nonlinear Fourier spectrum,” in Proceedings of Photonic Networks (2016), Leipzig, Germany, pp. 1–6.

Giacoumidis, E.

I. Phillips, M. Tan, M. F. Stephens, M. McCarthy, E. Giacoumidis, S. Sygletos, P. Rosa, S. Fabbri, S. T. Le, T. Kanesan, S. K. Turitsyn, N. J. Doran, P. Harper, and A. D. Ellis, “Exceeding the nonlinear-shannon limit using Raman laser based amplification and optical phase conjugation,” in Optical Fiber Communication Conference and Exhibition (OFC), Los Angeles, USA, paper M3C.1 (2014).
[Crossref]

Grant, A.

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Z. Dong, S. Hari, T. Gui, K. Zhong, M. I. Yousefi, C. Lu, P. A. Wai, F. R. Kschischang, and A. Lau, “Nonlinear frequency division multiplexed transmissions based on NFT,” IEEE Photon. Technol. Lett. 27, 1621–1623 (2015).
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Figures (6)

Fig. 1
Fig. 1 ISI mitigation in linear regime by using the linear filtering.
Fig. 2
Fig. 2 ISI mitigation in nonlinear regime with three neighboring pulses using BP.
Fig. 3
Fig. 3 Processing window in the case of a burst-mode signal and periodic signal with cyclic extension.
Fig. 4
Fig. 4 a) The error, and b) the normalized (to the number of samples) runtime, for finding the main spectrum of the plane wave, Eq. (18), with q0 = 5, μ = 3 for Ablowitz-Ladik, layer peeling (on this method’s runtime refer to the text), Crank-Nicolson and spectral methods vs. the number of temporal samples.
Fig. 5
Fig. 5 a) The error, and b) the normalized (to the number of samples) runtime of finding the main spectrum of a rectangular pulse train [see Fig. 6(b)] with amplitude A = 3, extent T = 2 and period T0 = 2π for the Ablowitz-Ladik, layer peeling (see the text for runtime), Crank-Nicolson and spectral methods vs. the number of samples.
Fig. 6
Fig. 6 main spectrum of a) a plane wave with q0 = 5, μ = 3, and b) a rectangular pulse train with T = 2, A = 3 and T0 = 2π

Equations (37)

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i q z β 2 2 q t t + γ q | q | 2 = 0 ,
y ( t ) = m x m h 0 ( t ) * q 0 ( t m T ) = m x m q ( t m T ) = m y m ( t m T ) .
H ( ω ) = H 0 1 ( ω ) = exp ( i ω 2 β 2 L 2 ) , h ( t ) = i 2 π β 2 L exp ( i t 2 2 β 2 L ) ,
y ^ ( t ) = W ( t ) × m y m ( t m T ) = y n ( t ) + W ( t ) × y n + 1 ( t n T T ) + W ( t ) × y n 1 ( t n T + T ) = y n ( t ) + [ W ( t T ) W ( t T + T d T T w ) ] y n + 1 ( t n T T ) + [ W ( t + T ) W ( t + T T d T T w ) ] y n 1 ( t n T + T ) = y n ( t ) + W ( t T ) × y n + 1 ( t n T T ) W ( t T + T d T T w ) × y n + 1 ( t n T T ) y n + 1 r ( t ) + W ( t + T ) × y n 1 ( t n T + T ) W ( t + T T d T T w ) × y n 1 ( t n T + T ) y n 1 r ( t ) = y n ( t ) + y n + 1 ( t n T T ) + y n 1 ( t n T + T ) y n 1 r ( t ) y n + 1 r ( t ) ,
x ^ ( t ) = h ( t ) * y ( t ) = x n q 0 ( t n T ) + x n + 1 q 0 ( t n T T ) × x n + 1 q 0 ( t n T + T ) + x n 1 r ( t ) + x n + 1 r ( t ) ,
i q z + q t t + 2 q | q | 2 = 0 ,
t T s t , z Z s z , q γ Z s 2 q .
[ i t i q ( t , z ) i q * ( t , z ) i t ] [ ϕ 1 ϕ 2 ] = λ [ ϕ 1 ϕ 2 ] .
Ψ 1 ( t , λ ) ( 0 1 ) e i λ t , t + , and Ψ 2 ( t , λ ) ( 1 0 ) e i λ t , t .
Ψ i ( t , λ ) = ( Ψ i 1 ( t , λ ) Ψ i 2 ( t , λ ) ) , Ψ ˜ i ( t , λ ) = ( Ψ i 2 * ( t , λ * ) Ψ i 1 * ( t , λ * ) ) , i = 1 , 2 .
Ψ 2 ( t , λ ) = a ( λ ) Ψ ˜ 1 ( t , λ ) + b ( λ ) Ψ 1 ( t , λ ) , Ψ ˜ 2 ( t , λ ) = a * ( λ ) Ψ 1 ( t , λ ) + b * ( λ ) Ψ ˜ 1 ( t , λ ) .
r ( λ ) = b ( λ ) a ( λ ) .
r ( λ , z ) = r ( λ , z 0 ) e 4 i λ ( z z 0 ) , λ ; γ n ( λ n , z ) = γ n ( λ n , z 0 ) e 4 i λ n ( z z 0 ) , λ n + ,
M 1 * ( τ , τ ) + τ d y R ( τ + y ) M 2 ( τ , y ) = 0 , M 2 * ( τ , τ ) + R ( τ + τ ) + τ d y R ( τ + y ) M 1 ( τ , y ) = 0 .
R s ( τ ) = i n γ n e i λ n τ , R r d ( τ ) = 1 2 π d λ r ( λ ) e i λ τ ,
Φ ( t , t 0 ; λ ) = [ ϕ 1 ( t , t 0 ; λ ) ϕ ˜ 1 ( t , t 0 ; λ ) ϕ 2 ( t , t 0 ; λ ) ϕ ˜ 2 ( t , t 0 ; λ ) ] .
𝕄 = { λ | Tr M ( t 0 ; λ ) = ± 2 } = { λ j } j = 1 2 g ,
λ j ± = μ 2 ± i | q 0 | 2 j 2 4 .
𝔸 = { μ ( t , z ) | M 11 ( t ; μ ( t , z ) ) = 0 } = { μ j ( t , z ) } j = 1 g 1 .
t μ j ( t , z ) = 2 i σ j Π n = 1 2 g μ j λ n Π n = 1 , n j g 1 μ j μ n z μ j ( t , z ) = 2 ( n = 1 , n j g 1 μ n 1 2 n = 1 2 g λ n ) t μ j ( t , z ) ,
q ( t , z ) = q ( 0 , 0 ) Θ ( W | τ ) Θ ( W + | τ ) e i k 0 z i ω 0 t ,
Θ ( W | τ ) = m g 1 exp ( 2 π i m T W + π i m T τ m ) .
Ψ t = [ i λ q ( t , z ) q * ( t , z ) i λ ] Ψ = P ( q ) Ψ , Ψ | t = t 0 = [ 1 0 0 1 ] .
U ( q n ) = exp ( Δ t [ i λ q n q n * i λ ] ) = ( cos k Δ t i λ k sin k Δ t q n k sin k Δ t q n * k sin k Δ t cos k Δ t + i λ k sin k Δ t ) ,
M ( t 0 ; λ ) = Ψ ( t N ) = n = 0 N 1 U ( q n ) Ψ ( t 0 ) = n = 0 N 1 U ( q n ) .
Ψ ( t n + Δ t ) Ψ ( t n ) Δ t = P ( q n + 1 ) Ψ ( q n + 1 ) + P ( q n ) Ψ ( t n ) 2 , Ψ ( 0 ) = I ,
Ψ ( t n + 1 ) = ( I Δ t 2 P ( q n + 1 ) ) 1 ( I + Δ t 2 P ( q n ) ) Ψ ( t n ) .
M ( t 0 ; λ ) = n = 1 M ( I Δ t 2 P ( q n + 1 ) ) 1 ( I + Δ t 2 P ( q n ) ) .
Ψ ( t n + 1 ) = α n 1 ( ω Δ t q n Δ t q n * ω 1 ) Ψ ( t n ) .
M ( t 0 ; λ ) = n = 1 N α n 1 ( ω Δ t q n Δ t q n * ω 1 ) .
Φ 1 ( t ) = e i μ t Φ ^ 1 ( t ) , Φ 2 ( t ) = e i μ t Φ ^ 2 ( t ) ,
[ i ( t μ ) i q ( t , z ) i q * ( t , z ) i ( t μ ) ] [ Φ ^ 1 Φ ^ 2 ] = λ [ Φ ^ 1 Φ ^ 2 ] .
Φ ^ 1 = n = N N a n 1 e i n k t , Φ ^ 2 = n = N N a n 2 e i n k t , q ( t , z ) = n = N N q n e i n k t ,
[ D i Q i Q D ] [ A 1 A 2 ] = λ [ A 1 A 2 ] ,
D = k diag ( N , N + 1 , , N 1 , N ) + μ I 2 N + 1 , A 1 = ( a N 1 , a N + 1 1 , , a N 1 1 , a N 1 ) T , A 2 = ( a N 2 , a N + 1 2 , , a N 1 2 , a N 2 ) T ,
M rec = [ e i λ ( T 0 T ) ( cosh k T + i λ T k sinh k T ) A T k sinh T k A * T k sinh k T e i λ ( T 0 T ) ( cosh k T i λ T k sinh k T ) ] ,
Tr M req = cos [ λ ( T 0 T ) ] cosh ( k T ) + λ k sin [ λ ( T 0 T ) ] sinh ( k T ) .

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