Abstract

The four-wave mixing process in optical fibers is generally sensitive to dispersion uniformity along the fiber length. However, some specific phase matching conditions show increased robustness to longitudinal fluctuations in fiber dimensions, which affect the dispersion, even for signal and idler wavelengths far from the pump. In this paper, we present the method by which this point is found, how the fiber design characteristics impact on the stable point and demonstrate the stability through propagation simulations using the non-linear Schrödinger equation.

© 2013 Optical Society of America

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2013 (1)

2012 (3)

2010 (2)

2008 (4)

2007 (3)

2005 (3)

2002 (1)

J. Hansryd, P. Andrekson, M. Westlund, J. Li, and P. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron.8, 506–520 (2002).
[CrossRef]

2001 (2)

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Role of entanglement in two-photon imaging,” Phys. Rev. Lett.87, 123602 (2001).
[CrossRef] [PubMed]

J. E. Sharping, M. Fiorentino, A. Coker, P. Kumar, and R. S. Windeler, “Four-wave mixing in microstructure fiber.” Opt. Lett.26, 1048–1050 (2001).
[CrossRef]

2000 (2)

T. Jennewein, C. Simon, G. Weihs, H. Weinfurter, and A. Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett.84, 4729–4732 (2000).
[CrossRef] [PubMed]

F. Abdullaev, B. Umarov, M. Wahiddin, and D. Navotny, “Dispersion-managed solitons in a periodically and randomly inhomogeneous birefringent optical fiber,” J. Opt. Soc. Am. B17, 1117–1124 (2000).
[CrossRef]

1999 (1)

1998 (2)

1997 (2)

1996 (3)

N. Smith, F. Knox, N. Doran, K. Blow, and I. Bennion, “Enhanced power solitons in optical fibres with periodic dispersion management,” Electronics Letters32, 54–55 (1996).
[CrossRef]

D. Gillespie, “Exact numerical simulation of the Ornstein-Uhlenbeck process and its integral.” Phys. Rev. E54, 2084–2091 (1996).
[CrossRef]

S. Watanabe and M. Shirasaki, “Exact compensation for both chromatic dispersion and Kerr effect in a transmission fiber using optical phase conjugation,” J. Lightwave Technol.14, 243–248 (1996).
[CrossRef]

1994 (1)

1993 (1)

S. Watanabe, T. Naito, and T. Chikama, “Compensation of chromatic dispersion in a single-mode fiber by optical phase conjugation,” IEEE Photon. Technol. Lett.5, 92–95 (1993).
[CrossRef]

1992 (2)

C. Bennett, F. Bessette, and G. Brassard, “Experimental quantum cryptography,” Journal of Cryptology5, 3–28 (1992).
[CrossRef]

K. Inoue, “Four-wave mixing in an optical fiber in the zero-dispersion wavelength region,” J. Lightwave Technol.10, 1553–1561 (1992).
[CrossRef]

1991 (1)

1987 (1)

N. Shibata, R. Braun, and R. Waarts, “Phase-mismatch dependence of efficiency of wave generation through four-wave mixing in a single-mode optical fiber,” IEEE J. Quantum Electron.23, 1205–1210 (1987).
[CrossRef]

1985 (1)

M. Levenson, R. Shelby, A. Aspect, M. Reid, and D. Walls, “Generation and detection of squeezed states of light by nondegenerate four-wave mixing in an optical fiber,” Phys. Rev. A32, 1550–1562 (1985).
[CrossRef] [PubMed]

1982 (2)

G. J. Dunning and R. C. Lind, “Demonstration of image transmission through fibers by optical phase conjugation,” Opt. Lett.7, 558–560 (1982).
[CrossRef] [PubMed]

R. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron.18, 1062–1072 (1982).
[CrossRef]

Abdullaev, F.

Abouraddy, A. F.

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Role of entanglement in two-photon imaging,” Phys. Rev. Lett.87, 123602 (2001).
[CrossRef] [PubMed]

Agrawal, G. P.

G. P. Agrawal, Lightwave Technology: Telecommunication Systems (Wiley. com,2005).
[CrossRef]

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2006), 4th ed.

Andersen, T.

Andrekson, P.

J. Hansryd, P. Andrekson, M. Westlund, J. Li, and P. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron.8, 506–520 (2002).
[CrossRef]

Andrekson, P. A.

Aspect, A.

M. Levenson, R. Shelby, A. Aspect, M. Reid, and D. Walls, “Generation and detection of squeezed states of light by nondegenerate four-wave mixing in an optical fiber,” Phys. Rev. A32, 1550–1562 (1985).
[CrossRef] [PubMed]

Bang, O.

Bennett, C.

C. Bennett, F. Bessette, and G. Brassard, “Experimental quantum cryptography,” Journal of Cryptology5, 3–28 (1992).
[CrossRef]

Bennion, I.

N. Smith, F. Knox, N. Doran, K. Blow, and I. Bennion, “Enhanced power solitons in optical fibres with periodic dispersion management,” Electronics Letters32, 54–55 (1996).
[CrossRef]

Bessette, F.

C. Bennett, F. Bessette, and G. Brassard, “Experimental quantum cryptography,” Journal of Cryptology5, 3–28 (1992).
[CrossRef]

Bétourné, A.

Bibbona, E.

E. Bibbona, G. Panfilo, and P. Tavella, “The Ornstein-Uhlenbeck process as a model of a low pass filtered white noise,” Metrologia45, S117 (2008).
[CrossRef]

Birks, T. A.

Bjorkholm, J. E.

R. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron.18, 1062–1072 (1982).
[CrossRef]

Blow, K.

N. Smith, F. Knox, N. Doran, K. Blow, and I. Bennion, “Enhanced power solitons in optical fibres with periodic dispersion management,” Electronics Letters32, 54–55 (1996).
[CrossRef]

Bouwmans, G.

Boyd, R. W.

R. W. Boyd, Nonlinear Optics, vol. 5 of Electronics & Electrical (Academic Press, 2003).

Brassard, G.

C. Bennett, F. Bessette, and G. Brassard, “Experimental quantum cryptography,” Journal of Cryptology5, 3–28 (1992).
[CrossRef]

Braun, R.

N. Shibata, R. Braun, and R. Waarts, “Phase-mismatch dependence of efficiency of wave generation through four-wave mixing in a single-mode optical fiber,” IEEE J. Quantum Electron.23, 1205–1210 (1987).
[CrossRef]

Bussières, F.

Chen, J.

J. Chen, X. Li, and P. Kumar, “Two-photon-state generation via four-wave mixing in optical fibers,” Appl. Phys. Lett.72, 033801 (2005).

Chen, Y.

Chikama, T.

S. Watanabe, T. Naito, and T. Chikama, “Compensation of chromatic dispersion in a single-mode fiber by optical phase conjugation,” IEEE Photon. Technol. Lett.5, 92–95 (1993).
[CrossRef]

Coker, A.

Corbeil, J.-S.

Darmanyan, S.

Doran, N.

N. Smith, F. Knox, N. Doran, K. Blow, and I. Bennion, “Enhanced power solitons in optical fibres with periodic dispersion management,” Electronics Letters32, 54–55 (1996).
[CrossRef]

Douay, M.

Dunning, G. J.

Fiorentino, M.

François, P. L.

Friedland, L.

O. Yaakobi and L. Friedland, “Autoresonant four-wave mixing in optical fibers,” Phys. Rev. A82, 023820 (2010).
[CrossRef]

Frosz, M. H.

Furusawa, A.

A. Furusawa, “Unconditional quantum teleportation,” Science282, 706–709 (1998).
[CrossRef] [PubMed]

Gillespie, D.

D. Gillespie, “Exact numerical simulation of the Ornstein-Uhlenbeck process and its integral.” Phys. Rev. E54, 2084–2091 (1996).
[CrossRef]

Godbout, N.

Hansen, K.

Hansryd, J.

J. Hansryd, P. Andrekson, M. Westlund, J. Li, and P. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron.8, 506–520 (2002).
[CrossRef]

Hedekvist, P.

J. Hansryd, P. Andrekson, M. Westlund, J. Li, and P. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron.8, 506–520 (2002).
[CrossRef]

Hilligsøe, K.

Hirano, M.

Hult, J.

Inoue, K.

K. Inoue, “Arrangement of fiber pieces for a wide wavelength conversion range by fiber four-wave mixing,” Opt. Lett.19, 1189–1191 (1994).
[CrossRef] [PubMed]

K. Inoue, “Four-wave mixing in an optical fiber in the zero-dispersion wavelength region,” J. Lightwave Technol.10, 1553–1561 (1992).
[CrossRef]

Jennewein, T.

T. Jennewein, C. Simon, G. Weihs, H. Weinfurter, and A. Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett.84, 4729–4732 (2000).
[CrossRef] [PubMed]

Karlsson, M.

Keiding, S.

Knight, J. C.

Knox, F.

N. Smith, F. Knox, N. Doran, K. Blow, and I. Bennion, “Enhanced power solitons in optical fibres with periodic dispersion management,” Electronics Letters32, 54–55 (1996).
[CrossRef]

Kudlinski, A.

Kumar, P.

J. Chen, X. Li, and P. Kumar, “Two-photon-state generation via four-wave mixing in optical fibers,” Appl. Phys. Lett.72, 033801 (2005).

J. E. Sharping, M. Fiorentino, A. Coker, P. Kumar, and R. S. Windeler, “Four-wave mixing in microstructure fiber.” Opt. Lett.26, 1048–1050 (2001).
[CrossRef]

Kuo, B. P.-P.

Lacroix, S.

Lægsgaard, J.

J. Lægsgaard, “Phase-matching conditions for single-pump parametric amplification in photonic crystal fibers,” J. Opt. A: Pure Appl. Opt.9, 1105–1112 (2007).
[CrossRef]

Laegsgaard, J.

Larsen, J.

Levenson, M.

M. Levenson, R. Shelby, A. Aspect, M. Reid, and D. Walls, “Generation and detection of squeezed states of light by nondegenerate four-wave mixing in an optical fiber,” Phys. Rev. A32, 1550–1562 (1985).
[CrossRef] [PubMed]

Li, J.

J. Hansryd, P. Andrekson, M. Westlund, J. Li, and P. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron.8, 506–520 (2002).
[CrossRef]

Li, X.

J. Chen, X. Li, and P. Kumar, “Two-photon-state generation via four-wave mixing in optical fibers,” Appl. Phys. Lett.72, 033801 (2005).

Lind, R. C.

Mamyshev, P. V.

Mamysheva, N. A.

Moselund, P. M.

Mussot, A.

Naito, T.

S. Watanabe, T. Naito, and T. Chikama, “Compensation of chromatic dispersion in a single-mode fiber by optical phase conjugation,” IEEE Photon. Technol. Lett.5, 92–95 (1993).
[CrossRef]

Navotny, D.

Nielsen, C.

Øksendal, B.

B. Øksendal, Stochastic Differential Equations, 6th ed. Universitext (Springer, Berlin, 2003).
[CrossRef]

Ould-Agha, Y.

Panfilo, G.

E. Bibbona, G. Panfilo, and P. Tavella, “The Ornstein-Uhlenbeck process as a model of a low pass filtered white noise,” Metrologia45, S117 (2008).
[CrossRef]

Pureur, V.

Quiquempois, Y.

Radic, S.

Reid, M.

M. Levenson, R. Shelby, A. Aspect, M. Reid, and D. Walls, “Generation and detection of squeezed states of light by nondegenerate four-wave mixing in an optical fiber,” Phys. Rev. A32, 1550–1562 (1985).
[CrossRef] [PubMed]

Russell, P. S.

Saleh, B. E. A.

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Role of entanglement in two-photon imaging,” Phys. Rev. Lett.87, 123602 (2001).
[CrossRef] [PubMed]

Sergienko, A. V.

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Role of entanglement in two-photon imaging,” Phys. Rev. Lett.87, 123602 (2001).
[CrossRef] [PubMed]

Sharping, J. E.

Shelby, R.

M. Levenson, R. Shelby, A. Aspect, M. Reid, and D. Walls, “Generation and detection of squeezed states of light by nondegenerate four-wave mixing in an optical fiber,” Phys. Rev. A32, 1550–1562 (1985).
[CrossRef] [PubMed]

Shibata, N.

N. Shibata, R. Braun, and R. Waarts, “Phase-mismatch dependence of efficiency of wave generation through four-wave mixing in a single-mode optical fiber,” IEEE J. Quantum Electron.23, 1205–1210 (1987).
[CrossRef]

Shirasaki, M.

S. Watanabe and M. Shirasaki, “Exact compensation for both chromatic dispersion and Kerr effect in a transmission fiber using optical phase conjugation,” J. Lightwave Technol.14, 243–248 (1996).
[CrossRef]

Simon, C.

T. Jennewein, C. Simon, G. Weihs, H. Weinfurter, and A. Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett.84, 4729–4732 (2000).
[CrossRef] [PubMed]

Slater, J. A.

Smith, N.

N. Smith, F. Knox, N. Doran, K. Blow, and I. Bennion, “Enhanced power solitons in optical fibres with periodic dispersion management,” Electronics Letters32, 54–55 (1996).
[CrossRef]

Stolen, R.

R. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron.18, 1062–1072 (1982).
[CrossRef]

Tavella, P.

E. Bibbona, G. Panfilo, and P. Tavella, “The Ornstein-Uhlenbeck process as a model of a low pass filtered white noise,” Metrologia45, S117 (2008).
[CrossRef]

Teich, M. C.

A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Role of entanglement in two-photon imaging,” Phys. Rev. Lett.87, 123602 (2001).
[CrossRef] [PubMed]

Thøgersen, J.

Thomsen, C. L.

Tittel, W.

Torounidis, T.

Umarov, B.

Vanvincq, O.

Virally, S.

Waarts, R.

N. Shibata, R. Braun, and R. Waarts, “Phase-mismatch dependence of efficiency of wave generation through four-wave mixing in a single-mode optical fiber,” IEEE J. Quantum Electron.23, 1205–1210 (1987).
[CrossRef]

Wadsworth, W. J.

Wahiddin, M.

Walls, D.

M. Levenson, R. Shelby, A. Aspect, M. Reid, and D. Walls, “Generation and detection of squeezed states of light by nondegenerate four-wave mixing in an optical fiber,” Phys. Rev. A32, 1550–1562 (1985).
[CrossRef] [PubMed]

Watanabe, S.

S. Watanabe and M. Shirasaki, “Exact compensation for both chromatic dispersion and Kerr effect in a transmission fiber using optical phase conjugation,” J. Lightwave Technol.14, 243–248 (1996).
[CrossRef]

S. Watanabe, T. Naito, and T. Chikama, “Compensation of chromatic dispersion in a single-mode fiber by optical phase conjugation,” IEEE Photon. Technol. Lett.5, 92–95 (1993).
[CrossRef]

Weihs, G.

T. Jennewein, C. Simon, G. Weihs, H. Weinfurter, and A. Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett.84, 4729–4732 (2000).
[CrossRef] [PubMed]

Weinfurter, H.

T. Jennewein, C. Simon, G. Weihs, H. Weinfurter, and A. Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett.84, 4729–4732 (2000).
[CrossRef] [PubMed]

Westlund, M.

J. Hansryd, P. Andrekson, M. Westlund, J. Li, and P. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron.8, 506–520 (2002).
[CrossRef]

Windeler, R. S.

Yaakobi, O.

O. Yaakobi and L. Friedland, “Autoresonant four-wave mixing in optical fibers,” Phys. Rev. A82, 023820 (2010).
[CrossRef]

Zeilinger, A.

T. Jennewein, C. Simon, G. Weihs, H. Weinfurter, and A. Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett.84, 4729–4732 (2000).
[CrossRef] [PubMed]

Appl. Phys. Lett. (1)

J. Chen, X. Li, and P. Kumar, “Two-photon-state generation via four-wave mixing in optical fibers,” Appl. Phys. Lett.72, 033801 (2005).

Electronics Letters (1)

N. Smith, F. Knox, N. Doran, K. Blow, and I. Bennion, “Enhanced power solitons in optical fibres with periodic dispersion management,” Electronics Letters32, 54–55 (1996).
[CrossRef]

IEEE J. Quantum Electron. (2)

N. Shibata, R. Braun, and R. Waarts, “Phase-mismatch dependence of efficiency of wave generation through four-wave mixing in a single-mode optical fiber,” IEEE J. Quantum Electron.23, 1205–1210 (1987).
[CrossRef]

R. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron.18, 1062–1072 (1982).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

J. Hansryd, P. Andrekson, M. Westlund, J. Li, and P. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron.8, 506–520 (2002).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

S. Watanabe, T. Naito, and T. Chikama, “Compensation of chromatic dispersion in a single-mode fiber by optical phase conjugation,” IEEE Photon. Technol. Lett.5, 92–95 (1993).
[CrossRef]

J. Lightwave Technol. (4)

S. Watanabe and M. Shirasaki, “Exact compensation for both chromatic dispersion and Kerr effect in a transmission fiber using optical phase conjugation,” J. Lightwave Technol.14, 243–248 (1996).
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Illustration of the hybrid PCF fiber design under study (Fiber A). The gray area represents pure silica, the white areas represent air holes and the blue areas represent graded-index (parabolic) Ge-doped inclusions with a maximum index difference n0nSiO2 = 0.032. In this particular design, dAir/Λ = 0.6 and dGe/Λ = 0.7. The pitch is Λ0 = 1.03μm.

Fig. 2
Fig. 2

Linear phase matching curve obtained for fiber A (Fig. 1). The thick solid line is the phase matching at the nominal pitch while the two thinner solid lines illustrate the phase-matching for variations in the pitch Λ = Λ0 ± 1 % with nominal pitch Λ0 = 1.03 μm. The dashed line represents the stability condition Δk/∂Λ = 0. For this design, the stable point is at pump λ̃p = 602.6 nm. The outer set of phase-matched wavelengths are λ̃s =1,218.9 nm and λ̃i = 400.2 nm. The inner set of phase-matched wavelengths are λs = 906.7 nm and λi = 451.2 nm. The first ZDW for fiber A is located at 617.5 nm

Fig. 3
Fig. 3

Mapping of the stable point as a function in (Λ,λ̃p,λ̃s)-space for different dAir/Λ and Λ, all other parameters being the same as for fiber A. The projection of the 3D trajectories on each couple of parameters is also plotted on panels A, B and C to help illustrate the different trends.

Fig. 4
Fig. 4

Example of the phase matching curve with the stable point at different pump wavelengths (and corresponding pitch) for dAir/Λ = 0.6. The black circles denote the position of the stable set of wavelengths λ̃i,s for each pump wavelength λ̃p and the black “×” indicate the position of the ZDW. The dashed line represents the trivial solution λi,s = λp. We notice that, as the wavelength λ̃p gets shorter, the stable point moves closer to the ZDW as the pump then crosses into the abnormal dispersion regime. Missing points in the phase matching curves lie outside the simulation domain and thus have been discarded. It is to be noted that the pump is in the abnormal dispersion regime when it is located between the two ZDW for a given curve.

Fig. 5
Fig. 5

Illustration of the dependance of the expected gain 〈G〉 to spatial noise correlation length Lc. In these calculations, σΛ = 0.5%, z = 0.2m and Pp = 1kW. It is to be noted that all graphs share the same scale in order to better appreciate the change in gain as Lc is increased. One can also notice that the outer set of wavelengths, where the stability condition occurs, suffers very little gain degradation whereas the inner set vanishes.

Fig. 6
Fig. 6

(a) Values of the overlap integrals fmnpp and fmp as a function of wavelength for a fixed λp = 602.6 nm. The four-wave mixing term (blue) is symmetric (in frequency) and both curves cross at the pump wavelength with the value fmnpp = fmp = fpp = 1/Aeff,p. Black circles have been added on the curves at the set of wavelengths under study. As we can see, both factors vary by less than an order of magnitude over the entire domain. The abscissa represents λm, λn being the conjugate wavelength. (b) Gain at the stable point as a function of perturbation amplitude σΛ for Lc = 10 m and z = 0.2 m.

Fig. 7
Fig. 7

Illustration of the dependance of the expected gain 〈G〉 to pump power Pp. In these calculations, σΛ = 0.5%, z = 0.2m and Lc = 5 × 10−3 m.

Fig. 8
Fig. 8

Evolution of the mean phase-matching condition (Δsp + Δip)/2 = 0 for different pump powers. The thicker solid line represents the linear phase-matching condition and “×” marks the position of the ZDW. The vertical line illustrates the position of the stable pump λ̃p.

Fig. 9
Fig. 9

Plots of the expected gain 〈G〉 as a function of pump wavelength λp and signal/idler wavelength λs,i for fiber A. In these calculations, σΛ = 0.5%, z = 0.2 m, Pp = 1kW and Lc = 5 mm. The FWHM (in log-space) of the stable point gain in terms of λp is about 6 nm.

Fig. 10
Fig. 10

Numerical simulation data. (a) Example of noise profiles used for pitch perturbations in the MGNLSE simulations (the same step-size resolution has been used to generate both profiles). The standard deviation used for the OUP is 0.5% of the nominal pitch and the mean is zero. Initial conditions are random and normally distributed with the same standard deviation as the process and zero mean. (b) MGNLSE simulations for Lc = 1 mm (10 simulations) and Lc = 10 cm (30 simulations). The results shown are the average of the gain for simulations with different random scale profiles with the same correlation length Lc with z = 200 mm, a pulse duration of 200 ps and a peak power of Pp =1 kW.

Equations (21)

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A p z = in 2 ω p c f p p | A p | 2 A p
A s z = in 2 ω s c [ 2 f s p | A p | 2 A s + f s i p p A p 2 A i * exp ( i Δ k z ) ]
A i z = in 2 ω i c [ 2 f i p | A p | 2 A i + f i s p p A p 2 A s * exp ( i Δ k z ) ]
Δ k = β s + β i 2 β p
A s z = in 2 ω s c [ 2 f s p P p A s + f s i p p P p A i * exp ( i θ z ) ]
A i * z = in 2 ω i c [ 2 f i p * P p A i * + f i s p p * P p A s exp ( i θ z ) ]
i B s z = κ s p 2 B s γ s i p p P p B i *
i B i * z = κ i p * 2 B i * + γ i s p p * P p B s
κ s p = 2 ( γ 2 γ s p ) P p Δ k mean Δ k var ( z )
κ i p = 2 ( γ 2 γ i p ) P p Δ k mean Δ k var ( z )
B z = [ G + H W ( z ) ] B
B = [ B s B i * ] , G = [ i Δ s p i γ s i p p P p i γ i s p p * P p i Δ i p * ] , H = [ i 0 0 i ] .
L c , X = σ X 2 0 R X ( ζ ) d ζ
σ W 2 = i = 1 N ( 1 2 Δ k X i | X i = X i , 0 ) 2 σ X i 2
σ W 2 = ( 1 2 Δ k Λ | Λ = Λ 0 ) 2 σ Λ 2
R ( ζ ) = R ( 0 ) exp ( | ζ | L c ) .
G = [ 2 Im ( Δ s p ) 0 0 ω s h P p 0 2 Im ( Δ i p ) 0 ω i h P p 0 0 Im ( Δ s p + Δ i p ) Re ( Δ s p + Δ i p ) 2 ω i h P p 2 ω s h P p Re ( Δ s p + Δ i p ) Im ( Δ s p + Δ i p ) ]
P ( z ) = exp ( G z + H B ( z ) ) P ( 0 ) ,
P ( z ) = exp ( G z + H 2 2 f ( z ) ) P ( 0 ) .
f ( z ) = 2 L c 2 σ W 2 [ exp ( z L c ) 1 + z L c ] .
G = 1 2 ( 1 + { cosh [ g ( z ) ] + f ( z ) g ( z ) sinh [ g ( z ) ] } exp [ f ( z ) ] )

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