Abstract

In a parametric amplifier (PA) driven by two pump waves the signal sideband is coupled to three idler sidebands, all of which are frequency-converted (FC) images of the signal, and two of which are phase-conjugated (PC) images of the signal. If such a device is to be useful, the signal must be amplified, and the PC and FC idlers must be produced, with minimal noise. In this paper the quantum noise properties of two-sideband (TS) parametric devices are reviewed and the properties of many-sideband devices are determined. These results are applied to the study of two-pump PAs, which are based on the aforementioned four-sideband (FS) interaction. As a general guideline, the more sidebands that interact, the higher are the noise levels. However, if the pump frequencies are tuned to maximize the frequency bandwidth of the FS interaction, the signal and idler noise-figures are only slightly higher than the noise figures associated with the limiting TS interactions.

© 2004 Optical Society of America

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References

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  1. K. O. Hill, D. C. Johnson, B. S. Kawasaki and R. I. MacDonald, �??CW three-wave mixing in single-mode optical fibers,�?? J. Appl. Phys. 49, 5098�??5106 (1978).
    [CrossRef]
  2. R. H. Stolen and J. E. Bjorkholm, �??Parametric amplification and frequency conversion in optical fibers,�?? IEEE J. Quantum Electron. 18, 1062�??1072 (1982).
    [CrossRef]
  3. C. J. McKinstrie, S. Radic and A. R. Chraplyvy, �??Parametric amplifiers driven by two pump waves,�?? IEEE J. Sel. Top. Quantum Electron. 8, 538�??547 and 956 (2002), and references therein.
    [CrossRef]
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    [CrossRef] [PubMed]
  5. C. J. McKinstrie, H. Kogelnik, R. M. Jopson, S. Radic and A. V. Kanaev, �??Four-wave mixing in fibers with random birefringence,�?? Opt. Express 12, 2033�??2055 (2004) and refrences therein.
    [CrossRef] [PubMed]
  6. S. Radic, C. J. McKinstrie, R. M. Jopson, Q. Lin and G. P. Agrawal, �??Record performance of a parametric amplifier constructed with highly-nonlinear fiber,�?? Electron. Lett. 39, 838�??839 (2003).
    [CrossRef]
  7. T. Tanemura, C. S. Goh, K. Kikuchi and S. Y. Set, �??Highly efficient arbitrary wavelength conversion within entire C-band based on nondegenerate fiber four-wave mixing,�?? IEEE Photon. Technol. Lett. 16, 551�??553 (2004)
    [CrossRef]
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    [CrossRef]
  11. 11. J. P. Gordon,W. H. Louisell and L. .Walker, �??Quantum fluctuations and noise in parametric processes II,�?? Phys. Rev. 129, 481�??485 (1963).
    [CrossRef]
  12. W. H. Louisell, Radiation and Noise in Quantum Electronics (McGraw-Hill, New York, 1964).
  13. B. R. Mollow and R. J. Glauber, �??Quantum theory of parametric amplification I,�?? Phys. Rev. 160, 1076�??1096 (1967).
    [CrossRef]
  14. B. R. Mollow and R. J. Glauber, �??Quantum theory of parametric amplification II,�?? Phys. Rev. 160, 1097�??1108 (1967).
    [CrossRef]
  15. J. Mostowski and M. G. Raymer, �??Quantum statistics in nonlinear optics,�?? in Contemporary Nonlinear Optics, edited by G. P. Agrawal and R. W. Boyd (Academic Press, San Diego, 1992), and references therein.
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  24. S. Radic, C. J. McKinstrie, R. M. Jopson, A. H. Gnauck, J. C. Centanni and A. R. Chraplyvy, �??Multi-band bit-level switching in two-pump fiber parametric devices,�?? IEEE Photon. Technol. Lett. 16, 852�??854 (2004).
    [CrossRef]
  25. S. Radic, C. J. McKinstrie, A. R. Chraplyvy, G. Raybon, J. C. Centanni, C. G. Jorgensen, K. Brar and C. Headley, �??Continuous-wave parametric-gain synthesis using nondegenerate-pump four-wave mixing,�?? IEEE Photon. Technol. Lett. 14, 1406�??1408 (2002).
    [CrossRef]
  26. P. C. Becker, N. A. Olssen and J. R. Simpson, Erbium-Doped Fiber Amplifiers (Academic Press, San Diego, 1999).
  27. K. Rottwitt and A. J. Stentz, �??Raman amplification in lightwave communication systems,�?? in Optical Fiber Telecommunications IVA, edited by I. Kaminow and T. Li (Academic Press, San Diego, 2002), pp. 213�??257.
    [CrossRef]
  28. A. H. Nayfeh, Introduction to Perturbation Techniques (Wiley, New York, 1981).

Ann. Phys. (1)

W. Schottky, �??On spontaneous current fluctuations in different electrical conductors,�?? Ann. Phys. (Leipzig) 57, 541�??567 (1918).

Electron. Lett. (1)

S. Radic, C. J. McKinstrie, R. M. Jopson, Q. Lin and G. P. Agrawal, �??Record performance of a parametric amplifier constructed with highly-nonlinear fiber,�?? Electron. Lett. 39, 838�??839 (2003).
[CrossRef]

IEEE J. Quantum Electron. (2)

C. Freed and H. A. Haus, �??Photoelectron statistics produced by a laser operating below and above the threshold of oscillation,�?? IEEE J. Quantum Electron. 2, 190�??195 (1966).
[CrossRef]

R. H. 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)

C. J. McKinstrie, S. Radic and A. R. Chraplyvy, �??Parametric amplifiers driven by two pump waves,�?? IEEE J. Sel. Top. Quantum Electron. 8, 538�??547 and 956 (2002), and references therein.
[CrossRef]

IEEE Photon. Technol. Lett. (3)

T. Tanemura, C. S. Goh, K. Kikuchi and S. Y. Set, �??Highly efficient arbitrary wavelength conversion within entire C-band based on nondegenerate fiber four-wave mixing,�?? IEEE Photon. Technol. Lett. 16, 551�??553 (2004)
[CrossRef]

S. Radic, C. J. McKinstrie, R. M. Jopson, A. H. Gnauck, J. C. Centanni and A. R. Chraplyvy, �??Multi-band bit-level switching in two-pump fiber parametric devices,�?? IEEE Photon. Technol. Lett. 16, 852�??854 (2004).
[CrossRef]

S. Radic, C. J. McKinstrie, A. R. Chraplyvy, G. Raybon, J. C. Centanni, C. G. Jorgensen, K. Brar and C. Headley, �??Continuous-wave parametric-gain synthesis using nondegenerate-pump four-wave mixing,�?? IEEE Photon. Technol. Lett. 14, 1406�??1408 (2002).
[CrossRef]

J. Appl. Phys. (1)

K. O. Hill, D. C. Johnson, B. S. Kawasaki and R. I. MacDonald, �??CW three-wave mixing in single-mode optical fibers,�?? J. Appl. Phys. 49, 5098�??5106 (1978).
[CrossRef]

Opt. Express (2)

Opt. Fiber Technol. (1)

S. Radic and C. J. McKinstrie, �??Two-pump fiber parametric amplifiers,�?? Opt. Fiber Technol. 9, 7�??23 (2003).
[CrossRef]

Phys. Rev. (5)

W. H. Louisell, A. Yariv and A. E. Siegman, �??Quantum fluctuations and noise in parametric processes I,�?? Phys. Rev. 124, 1646�??1654 (1961)
[CrossRef]

11. J. P. Gordon,W. H. Louisell and L. .Walker, �??Quantum fluctuations and noise in parametric processes II,�?? Phys. Rev. 129, 481�??485 (1963).
[CrossRef]

R. J. Glauber, �??Coherent and incoherent states of the radiation field,�?? Phys. Rev. 131, 2766�??2788 (1963).
[CrossRef]

B. R. Mollow and R. J. Glauber, �??Quantum theory of parametric amplification I,�?? Phys. Rev. 160, 1076�??1096 (1967).
[CrossRef]

B. R. Mollow and R. J. Glauber, �??Quantum theory of parametric amplification II,�?? Phys. Rev. 160, 1097�??1108 (1967).
[CrossRef]

Phys. Rev. Lett. (1)

F. T. Arecchi, �??Measurement of the statistical distribution of Gaussian and laser sources,�?? Phys. Rev. Lett. 15, 912�??916 (1965).
[CrossRef]

Proc. IRE (2)

J. M. Manley and H. E. Rowe, �??Some general properties of nonlinear elements�??Part I. General energy relations,�?? Proc. IRE 44, 904�??913 (1956).
[CrossRef]

M. T. Weiss, �??Quantum derivation of energy relations analogous to those for nonlinear reactances,�?? Proc. IRE 45, 1012�??1013 (1957).

Other (8)

J. Mostowski and M. G. Raymer, �??Quantum statistics in nonlinear optics,�?? in Contemporary Nonlinear Optics, edited by G. P. Agrawal and R. W. Boyd (Academic Press, San Diego, 1992), and references therein.

R. Loudon, The Quantum Theory of Light, 3rd Ed. (Oxford University Press, Oxford, 2000).

L. D. Landau, E. M. Lifshitz and L. P. Pitaevskii, Electrodynamics of Continuous Media, 2nd Ed. (Butterworth-Heinemann, Oxford, 1984).

T. H. Stix, Waves in Plasmas (American Institute of Physics, New York, 1992).

W. H. Louisell, Radiation and Noise in Quantum Electronics (McGraw-Hill, New York, 1964).

P. C. Becker, N. A. Olssen and J. R. Simpson, Erbium-Doped Fiber Amplifiers (Academic Press, San Diego, 1999).

K. Rottwitt and A. J. Stentz, �??Raman amplification in lightwave communication systems,�?? in Optical Fiber Telecommunications IVA, edited by I. Kaminow and T. Li (Academic Press, San Diego, 2002), pp. 213�??257.
[CrossRef]

A. H. Nayfeh, Introduction to Perturbation Techniques (Wiley, New York, 1981).

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Figures (10)

Fig. 1.
Fig. 1.

Illustration of the constituent FWM processes.

Fig. 2.
Fig. 2.

Gains and noise figures of a two-mode parametric amplifier. The solid curves represent the signal, whereas the dashed curves represent the idler. Distance is normalized to the gain length [γε(P 1 P 2)1/2]-1.

Fig. 3.
Fig. 3.

Transmissions and noise figures of a two-mode frequency converter. The solid curves represent the signal, whereas the dashed curves represent the idler. Distance is normalized to the interaction length [γϕ(P 1 P 2)1/2]-1.

Fig. 4.
Fig. 4.

Gains and noise figures of a four-mode process with quadratic gain, for co-polarized pumps (ε = 2). The solid and long-dashed curves represent the 1- signal and 1+ idler, respectively, and the medium-dashed curves represent the 2- and 2+ idlers. Distance is normalized to the characteristic length (γP)-1.

Fig. 5.
Fig. 5.

Gains and noise figures of a four-mode process with quadratic gain, for cross-polarized pumps in a fiber with constant birefringence (ε = 2/3). The solid and long-dashed curves represent the 1- signal and 1+ idler, respectively, and the medium-dashed curves represent the 2- and 2+ idlers. Distance is normalized to the characteristic length (γP)-1.

Fig. 6.
Fig. 6.

Gains and noise figures of a four-mode process with quadratic gain, for cross-polarized pumps in a fiber with random birefringence (ε = 1). The solid curve represents the 1- signal, and the long-dashed curves represent the 1+, 2- and 2+ idlers. Distance is normalized to the characteristic length (γP)-1.

Fig. 7.
Fig. 7.

Gains and noise figures of a four-mode process with exponential gain, for cross-polarized pumps in a fiber with random birefringence (ε = 1). The solid curves represent the 1- signal, the long-dashed curves represent the 1+ and 2+ idlers, and the medium-dashed curves represent the 2- idler. Distance is normalized to the gain length (2γP)-1.

Fig. 8.
Fig. 8.

Gains and noise figures of a four-mode process driven by cross-polarized pumps in a fiber with random birefringence (ε= 1). The solid curves represent the 1- signal, and the long-, medium- and short-dashed curves represent the 1+, 2- and 2+ idlers, respectively. Frequencies are measured relative to the zero-dispersion frequency of the fiber. The vertical lines denote the pump frequencies.

Fig. 9.
Fig. 9.

Gains and noise figures of a four-mode process driven by cross-polarized pumps in a fiber with random birefringence (ε= 1). The long-dashed curves represent the 1+ signal, and the solid, medium- and short-dashed curves represent the 1-, 2- and 2+ idlers, respectively. Frequencies are measured relative to the zero-dispersion frequency of the fiber. The vertical lines denote the pump frequencies.

Fig. 10.
Fig. 10.

Transmissions and noise figures of a four-mode process driven by cross-polarized pumps in a fiber with random birefringence (ε = 1), plotted as functions of the pump frequency ω 1. The long-dashed curves represent the 1+ signal, and the solid, medium-and short-dashed curves represent the 1-, 2- and 2+ idlers, respectively. Frequencies are measured relative to the zero-dispersion frequency of the fiber.

Equations (202)

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i z A 1 = β 1 ( i t ) A 1 + γ ( A 1 2 + ε A 2 2 ) A 1 ,
i z A 2 = β 2 ( i t ) A 2 + γ ( ε A 1 2 + A 2 2 ) A 2 ,
A 1 ( z ) = P 1 1 2 exp [ i ϕ 1 ( z ) ] ,
A 2 ( z ) = P 1 1 2 exp [ i ϕ 2 ( z ) ] ,
A 1 z t = [ P 1 1 2 + B 1 + ( z ) exp ( iωt ) + B 1 ( z ) exp ( iωt ) ] exp [ i ϕ 1 ( z ) ] ,
A 2 z t = [ P 2 1 2 + B 2 + ( z ) exp ( iωt ) + B 2 ( z ) exp ( iωt ) ] exp [ i ϕ 2 ( z ) ] ,
d z B 1 * = i ( β 1 + γ P 1 ) B 1 * P 1 B 1 +
iγε ( P 1 P 2 ) 1 2 B 2 * iγε ( P 1 P 2 ) 1 2 B 2 + ,
d z B 1 + = P 1 B 1 * + i ( β 1 + + γ P 1 ) B 1 +
+ iγε ( P 1 P 2 ) 1 2 B 2 * + iγε ( P 1 P 2 ) 1 2 B 2 + ,
d z B 2 * = ε ( P 1 P 2 ) 1 2 B 1 * ε ( P 1 P 2 ) 1 2 B 1 +
i ( β 2 + γ P 2 ) B 2 * P 2 B 2 + ,
d z B 2 + = ε ( P 1 P 2 ) 1 2 B 1 * + ε ( P 1 P 2 ) 1 2 B 1 +
+ P 2 B 2 * + i ( β 2 + + γ P 2 ) B 2 + ,
d z ( P 1 P 1 + P 2 P 2 + ) = 0 .
B 1 * ( z ) = C 1 * ( z ) exp [ i ( β 1 + β 1 ) z 2 ] ,
B 1 + ( z ) = C 1 + ( z ) exp [ i ( β 1 + β 1 ) z 2 ]
d z C 1 * = C 1 * C 1 + ,
d z C 1 + = C 1 * + C 1 + ,
C ( z ) = M ( z ) C ( 0 ) ,
M ( z ) = [ cos ( kz ) sin ( kz ) k sin ( kz ) k sin ( kz ) k cos ( kz ) + sin ( kz ) k ]
d z ( P 1 P 1 + ) = 0
B 1 * ( z ) = C 1 * ( z ) exp { i [ β 2 + β 1 + γ ( P 2 + P 1 ) ] z 2 } ,
B 2 * ( z ) = C 2 * ( z ) exp { i [ β 2 + β 1 + γ ( P 2 + P 1 ) ] z 2 }
d z C 1 = C 1 + C 2 ,
d z C 2 = C 1 + C 2 ,
M ( z ) = [ cos ( kz ) sin ( kz ) k sin ( kz ) k sin ( kz ) k cos ( kz ) + sin ( kz ) k ]
d z ( P 1 + P 2 ) = 0
B 1 * ( z ) = C 1 * ( z ) exp { i [ β 2 + β 1 + γ ( P 2 P 1 ) ] z 2 } ,
B 2 + ( z ) = C 2 + ( z ) exp { i [ β 2 + β 1 + γ ( P 2 P 1 ) ] z 2 }
d z C 1 * = C 1 * C 2 + ,
d z C 2 + = C 1 * + C 2 + ,
M ( z ) = [ cos ( kz ) sin ( kz ) k sin ( kz ) k sin ( kz ) k cos ( kz ) + sin ( kz ) k ]
d z ( P 1 P 2 + ) = 0
d z B = LB ,
B j ( z ) B j ( 0 ) exp [ λ j ( 0 ) z + l jj ( 1 ) z ] + k j B k ( 0 ) l jk ( 1 ) exp [ λ k ( 0 ) z ] exp [ λ j ( 0 ) z ] λ k ( 0 ) λ j ( 0 ) .
B ( z ) = M ( z ) B ( 0 ) ,
μ jk ( z ) { exp [ λ j ( 0 ) z + l jj ( 1 ) z ] if k = j , l jk ( 1 ) exp [ λ k ( 0 ) z ] exp [ λ j ( 0 ) z ] λ k ( 0 ) λ j ( 0 ) if k j .
M ( z ) [ 1 iγPz iγPz iγεPz iγεPz iγPz 1 + iγPz iγεPz iγεPz iγεPz iγεPz 1 iγPz iγPz iγεPz iγεPz iγPz 1 + iγPz ] .
M ( z ) [ μ ( z ) ( z ) μ ( z ) ( z ) ( z ) μ ( z ) ( z ) μ ( z ) μ ( z ) ( z ) μ ( z ) ( z ) ( z ) μ ( z ) ( z ) μ ( z ) ] ,
[ a ̂ j , a ̂ k ] = δ jk ,
D ω k = ω 2 n 2 ( ω ) c 2 k 2 ,
E + t z = E 0 t z exp [ i ϕ 0 t z ] ,
B + t z = B 0 t z exp [ i ϕ 0 t z ]
t T 0 + z S 0 =0,
T 0 = ( D ω ) 0 E 0 2 4 π ω 0 ,
S 0 = ( D k ) 0 E 0 2 4 π ω 0
E ̂ 0 t z = ( 2 π h ̄ ω 0 n 0 c ) 1 2 a ̂ ( ω ) exp [ ( ω ) z iωt ] ( 2 π ) 1 2 ,
B ̂ 0 t z = ( 2 π h ̄ ω 0 n 0 c ) 1 2 a ̂ ( ω ) exp [ ( ω ) z iωt ] ( 2 π ) 1 2 ,
[ a ̂ ( ω ) , a ̂ ( ω ) ] = δ ( ω ω ) ,
T ̂ 0 t z dz = h ̄ ω 0 a ̂ ( ω ) a ̂ ( ω ) .
a ̂ ω z = a ̂ ( ω ) exp [ ( ω ) z ]
a ̂ t z = a ̂ ( ω ) exp [ ( ω ) z iωt ] ( 2 π ) 1 2 .
a ̂ t z a ̂ t z dt = a ̂ ω z a ̂ ω z
[ a ̂ ( ω ) , a ̂ ( ω , z ) ] = δ ( ω ω ) ,
[ a ̂ t z , a ̂ ( t , z ) ] = δ ( t t ) .
i z a ̂ = β ( i t ) a ̂ ,
S ̂ 0 t z = h ̄ ω 0 a ̂ t z a ̂ t z
m ̂ T t z = t T 2 t + T 2 a ̂ ( t , z ) a ̂ ( t , z ) dt .
{ α } = exp ( a ̂ α a ̂ α ) { 0 } ,
a α = α ( ω ) a ̂ ( ω )
a ̂ α = α ω z a ̂ ω z ,
α ω z = α ( ω ) exp [ ( w ) z ]
a ̂ ω z { α } = α ω z { α } ,
a ̂ t z { α } = α t z { α } ,
m ̄ T t z = t T 2 t + T 2 α ( t , z ) 2 dt .
δ m T 2 t z = m ̄ T t z .
a ̂ k = ω k Δ 2 ω k + Δ 2 a ̂ ( ω ) Δ 1 2 .
[ a k , a l ] = δ kl ,
a ̂ j ( t , z ) = ( v L ) 1 2 k a ̂ k exp [ ( ω k ) z i ω k t ] .
m ̂ T ( t , z ) = ( Tv / L ) k , l sinc ( ω lk T / 2 ) a ̂ k a ̂ l exp ( i β lk z i ω lk t ) ,
α j = exp ( α j a ̂ j α j * a ̂ j ) 0 .
a ̂ j α j = α j α j ,
m ̄ T = ( Tv / L ) m ̄ j ,
δ m T 2 = m ¯ T .
B = MA ,
M ( z ) = [ μ * ( z ) v * ( z ) v ( z ) μ ( z ) ]
μ 2 v 2 = 1 .
n 1 = μ 2 a 1 a 1 + v 2 a 2 a 2 + μ * v a 1 a 2 + μ v * a 2 a 1 ,
n 2 = v 2 a 1 a 1 + μ 2 a 2 a 2 + μ v * a 1 a 2 + μ * v a 2 a 1 .
n 1 = μ 2 m 1 + v 2 ( m 2 + 1 ) ,
n 2 = v 2 ( m 1 + 1 ) + μ 2 m 2 .
n 1 2 = ( μ 2 a 1 a 1 + v 2 a 2 a 2 ) 2 + ( μ * v a 1 a 2 + μ v * a 2 a 1 ) 2
+ ( μ 2 a 1 a 1 + v 2 a 2 a 2 ) ( μ * v a 1 a 2 + μ v * a 2 a 1 )
+ ( μ * v a 1 a 2 + μ v * a 2 a 1 ) ( μ 2 a 1 a 1 + v 2 a 2 a 2 ) .
δ n 1 2 = μ 2 v 2 ( 2 m 1 m 2 + m 1 + m 2 + 1 ) .
ψ = m 1 = 0 [ p ( m 1 ) ] 1 2 exp ( i m 1 ϕ ) m 1 , 0 ,
n 1 = μ 2 m ̄ 1 + v 2 ,
n 2 = v 2 ( m ̄ 1 + 1 ) .
δ n 1 2 = μ 4 m ̄ 1 + μ 2 v 2 ( m ̄ 1 + 1 ) ,
δ n 2 2 = v 4 m ̄ 1 + μ 2 v 2 ( m ̄ 1 + 1 ) .
F 1 = m ̄ 1 [ G 2 m ̄ 1 + G ( G 1 ) ( m ̄ 1 + 1 ) ] [ G m ̄ 1 + ( G 1 ) ] 2 ,
F 2 = m ̄ 1 [ ( G 1 ) 2 m ̄ 1 + G ( G 1 ) ( m ̄ 1 + 1 ) ] ( G 1 ) 2 + ( m ̄ 1 + 1 ) 2 .
M ( z ) = [ μ ( z ) v ( z ) v * ( z ) μ * ( z ) ] .
μ 2 + v 2 = 1 .
n 1 = μ 2 a 1 a 1 + v 2 a 2 a 2 + μ * v a 1 a 2 + μ v * a 2 a 1 ,
n 2 = v 2 a 1 a 1 + μ 2 a 2 a 2 μ * v a 1 a 2 μ v * a 2 a 1 .
n 1 = μ 2 m 1 + v 2 m 2 ,
n 2 = v 2 m 1 + μ 2 m 2 .
n 1 2 = ( μ 2 a 1 a 1 + v 2 a 2 a 2 ) 2 + ( μ * v a 1 a 2 + μ v * a 2 a 1 ) 2
+ ( μ 2 a 1 a 1 + v 2 a 2 a 2 ) ( μ * v a 1 a 2 + μ v * a 2 a 1 )
+ ( μ * v a 1 a 2 + μ v * a 2 a 1 ) ( μ 2 a 1 a 1 + v 2 a 2 a 2 ) .
δ n 1 2 = μ 2 v 2 ( 2 m 1 m 2 + m 1 + m 2 ) .
n 1 = μ 2 m ̄ 1 ,
n 2 = v 2 m ̅ 1 .
δ n 1 2 = μ 4 m ̄ 1 + μ 2 v 2 m ̅ 1 ,
δ n 2 2 = v 4 m ̅ 1 + μ 2 v 2 m ̅ 1 .
F 1 = 1 T ,
F 2 = 1 ( 1 T ) .
B = MA ,
μ 11 2 μ 12 2 + μ 13 2 = 1 .
μ 21 2 + μ 22 2 μ 23 2 = 1 .
n 1 = μ 11 2 a 1 a 1 + μ 12 2 a 2 a 2 + μ 13 2 a 3 a 3 + μ 11 μ 12 * a 1 a 2 + μ 12 μ 11 * a 2 a 1
+ μ 12 μ 13 * a 2 a 3 + μ 13 μ 12 * a 3 a 2 + μ 13 μ 11 * a 3 a 1 + μ 11 μ 13 * a 1 a 3 ,
n 2 = μ 21 2 a 1 a 1 + μ 22 2 a 2 a 2 + μ 23 2 a 3 a 3 + μ 21 * μ 22 a 1 a 2 + μ 22 * μ 21 a 2 a 1
+ μ 22 * μ 23 a 2 a 3 + μ 23 * μ 22 a 3 a 2 + μ 23 * μ 21 a 3 a 1 + μ 21 * μ 13 a 1 a 3 .
n 1 = μ 11 2 m 1 + μ 12 2 ( m 2 + 1 ) + μ 13 2 m 3 ,
n 2 = μ 21 2 ( m 1 + 1 ) + μ 22 2 m 2 + μ 23 2 ( m 3 + 1 ) .
δ n 1 2 = μ 11 μ 12 2 ( 2 m 1 m 2 + m 1 + m 2 + 1 )
+ μ 12 μ 13 2 ( 2 m 2 m 3 + m 2 + m 3 + 1 )
+ μ 13 μ 11 2 ( 2 m 3 m 1 + m 3 + m 1 ) ,
δ n 2 2 = μ 21 μ 22 2 ( 2 m 1 m 2 + m 1 + m 2 + 1 )
+ μ 22 μ 23 2 ( 2 m 2 m 3 + m 2 + m 3 + 1 )
+ μ 23 μ 21 2 ( 2 m 3 m 1 + m 3 + m 1 ) .
n 1 = μ 11 2 m ̄ 1 + μ 12 2 ,
n 2 = μ 21 2 ( m ̄ 1 + 1 ) + μ 23 2 .
δ n 1 2 = μ 11 4 m ¯ 1 + μ 11 μ 12 2 ( m ¯ 1 + 1 ) + μ 12 μ 13 2 + μ 13 μ 11 2 m ¯ 1 ,
δ n 2 2 = μ 21 4 m ¯ 1 + μ 21 μ 22 2 ( m ¯ 1 + 1 ) + μ 22 μ 23 2 + μ 23 μ 21 2 m ¯ 1 .
F 1 = m ¯ 1 [ μ 11 4 m ¯ 1 + μ 11 μ 12 2 ( m ¯ 1 + 1 ) + μ 12 μ 13 2 + μ 13 μ 11 2 m ¯ 1 ] [ μ 11 2 m ¯ 1 + μ 12 2 ] 2 ,
F 2 = m ¯ 1 [ μ 21 4 m ¯ 1 + μ 21 μ 22 2 ( m ¯ 1 + 1 ) + μ 22 μ 23 2 + μ 23 μ 21 2 m ¯ 1 ] [ μ 21 2 ( m ¯ 1 + 1 ) + μ 23 2 ] 2 .
F 1 1 + ( μ 12 2 + μ 13 2 ) μ 11 2 ,
F 2 1 + ( μ 22 2 + μ 23 2 ) μ 21 2 .
k μ jk 2 s jk = 1 ,
n j = k μ jk 2 ( m k + σ jk ) ,
δ n j 2 = k , l > k μ jk μ jl 2 ( 2 m k m l + m k + m l + σ kl ) ,
n j = μ ji 2 m ̄ i + k μ jk 2 σ jk
δ n j 2 = μ ji 4 m ¯ i + k i μ ji μ jk 2 m ¯ i + k , l > k μ jk μ jl 2 σ kl ,
F j = m ¯ i ( μ ji 4 m ¯ i + k i μ ji μ jk 2 m ¯ i + k , l > k μ jk μ jl 2 σ kl ) ( μ ji 2 m ¯ i + k μ jk 2 σ jk ) 2 .
F j 1 + k i μ jk 2 μ ji 2 .
F 1 ± = 2 + 2 ε 2 ,
F 2 ± = 2 + 2 ε 2 .
F 1 ± = F 2 ± = 4 .
d z B = LB ,
d z 0 B ( 0 ) L ( 0 ) B ( 0 ) = 0 ,
d z 0 B ( 1 ) L ( 0 ) B ( 1 ) = d z 1 B ( 0 ) + L ( 1 ) B ( 0 ) ,
B j ( 0 ) ( z 0 ) = C j ( 0 ) exp [ λ j ( 0 ) z 0 ] ,
d z 0 B j ( 1 ) λ j ( 0 ) B j ( 1 ) = d z 1 C j ( 0 ) exp [ λ j ( 0 ) z 0 ] + k l jk ( 1 ) C k ( 0 ) exp [ λ k ( 0 ) z 0 ] .
d z 1 C j ( 0 ) = l jj ( 1 ) C j ( 0 ) ,
C j ( 0 ) ( z 1 ) = C j ( 0 ) ( 0 ) exp [ l jj ( 1 ) z 1 ] .
B j ( 1 ) ( z 0 ) = k j l jk ( 1 ) C k ( 0 ) [ λ k ( 0 ) z 0 ] exp [ λ j ( 0 ) z 0 ] λ k ( 0 ) λ j ( 0 ) .
B ( z ) = M ( z ) B ( 0 ) ,
M ( z ) 1 + Lz .
d z G j = i β e H j ,
d z H j = i ( β e + 2 γP ) G j + i 2 γεP G k ,
[ d zz 2 + β e ( β e + 2 γ P ) ] G 1 + 2 β e γε P G 2 = 0 ,
2 β e γε P G 1 + [ d zz 2 + β e ( β e + 2 γ P ) ] G 2 = 0 .
k ± 2 = β e [ β e + 2 γ ( 1 ± ε ) P ] .
G 1 = 1 2 cos ( k + z ) + i σ β e 2 k + sin ( k + z ) + 1 2 cos ( k z ) i σ β e 2 k sin ( k z ) ,
H 1 = σ 2 cos ( k + z ) + i k + 2 β e sin ( k + z ) + σ 2 cos ( k z ) i k 2 β e sin ( k z ) ,
G 2 = 1 2 cos ( k + z ) + i σ β e 2 k + sin ( k + z ) 1 2 cos ( k z ) i σ β e 2 k sin ( k z ) ,
H 2 = σ 2 cos ( k + z ) + i k + 2 β e sin ( k + z ) σ 2 cos ( k z ) i k 2 β e sin ( k z ) .
C 1 * = 1 σ 4 cos ( k + z ) + i 4 ( σ β e k + k + β e ) sin ( k + z )
+ 1 σ 4 cos ( k z ) + i 4 ( σ β e k k β e ) sin ( k z ) ,
C 1 + = 1 + σ 4 cos ( k + z ) + i 4 ( σ β e k + + k + β e ) sin ( k + z )
+ 1 + σ 4 cos ( k z ) + i 4 ( σ β e k + k β e ) sin ( k z ) ,
C 2 * = 1 σ 4 cos ( k + z ) + i 4 ( σ β e k + k + β e ) sin ( k + z )
1 σ 4 cos ( k z ) i 4 ( σ β e k k β e ) sin ( k z ) ,
C 2 + = 1 + σ 4 cos ( k + z ) + i 4 ( σ β e k + + k + β e ) sin ( k + z )
1 + σ 4 cos ( k z ) i 4 ( σ β e k + k + β e ) sin ( k z ) .
d z G j = i β je H j ,
d z H j = i ( β je + 2 γ P j ) G j + i 2 γε ( P j P k ) 1 2 G k ,
[ d zz 2 + β 1 e ( β 1 e + 2 γ P 1 ) ] G 1 + 2 β 1 e γε ( P 1 P 2 ) 1 2 G 2 = 0 ,
2 β 2 e γε ( P 1 P 2 ) 1 2 G 1 + [ d zz 2 + β 2 e ( β 2 e + 2 γ P 2 ) ] G 2 = 0 .
2 k ± 2 = β 1 e ( β 1 e + 2 γ P 1 ) + β 2 e ( β 2 e + 2 γ P 2 )
+ { [ β 1 e ( β 1 e + 2 γ P 1 ) β 2 e ( β 2 e + 2 γ P 2 ) ] 2
+ 4 [ 4 β 1 e β 2 e ( γε ) 2 P 1 P 2 ] } 1 2 .
G 1 = cos ( k + z ) 1 α 1 + α 1 + i σ β 1 e sin ( k + z ) k + ( 1 α 1 + α 1 )
+ cos ( k z ) 1 α 1 α 1 + + i σ β 1 e sin ( k z ) k ( 1 α 1 α 1 + ) ,
H 1 = σ cos ( k + z ) 1 α 1 + α 1 + i k + sin ( k + z ) β 1 e ( 1 α 1 + α 1 )
+ σ cos ( k z ) 1 α 1 α 1 + + i k sin ( k z ) β 1 e ( 1 α 1 α 1 + ) ,
G 2 = α 1 + cos ( k + z ) 1 α 1 + α 1 + i σ α 1 + β 1 e sin ( k + z ) k + ( 1 α 1 + α 1 )
+ α 1 cos ( k z ) 1 α 1 α 1 + + i σ α 1 β 1 e sin ( k z ) k ( 1 α 1 α 1 + ) ,
H 2 = σ α 1 + β 1 e cos ( k + z ) β 2 e ( 1 α 1 + α 1 ) + i α 1 + k + sin ( k + z ) β 2 e ( 1 α 1 + α 1 )
+ σ α 1 β 1 e cos ( k z ) β 2 e ( 1 α 1 α 1 + ) + i α 1 k sin ( k z ) β 2 e ( 1 α 1 α 1 + ) ,
α 1 ± = [ k ± 2 β 1 e ( β 1 e + 2 γ P 1 ) ] 2 β 1 e γε ( P 1 P 2 ) 1 2 .
C 1 * = ( 1 σ ) cos ( k + z ) 2 ( 1 α 1 + α 1 ) + ( σ β 1 e k + k + β 1 e ) i sin ( k + z ) 2 ( 1 α 1 + α 1 ) + ( + ) ,
C 1 + = ( 1 + σ ) cos ( k + z ) 2 ( 1 α 1 + α 1 ) + ( σ β 1 e k + + k + β 1 e ) i sin ( k + z ) 2 ( 1 α 1 + α 1 ) + ( + ) ,
C 2 * = α 1 + ( 1 σ β 1 e β 2 e ) cos ( k + z ) 2 ( 1 α 1 + α 1 ) + ( σ β 1 e k + k + β 2 e ) i α 1 + sin ( k + z ) 2 ( 1 α 1 + α 1 ) ,
C 2 + = α 1 + ( 1 + σ β 1 e β 2 e ) cos ( k + z ) 2 ( 1 α 1 + α 1 ) + ( σ β 1 e k + + k + β 2 e ) i α 1 + sin ( k + z ) 2 ( 1 α 1 + α 1 ) .
B = MA ,
k μ jk 2 s jk = 1 ,
j μ jk 2 s jk = 1 ,
d z B = LB ,
b j ( z ) = n e j ( n ) f k ( n ) * exp [ λ ( n ) z ] b k ( 0 ) ,
μ jk = n e j ( n ) f k ( n ) * exp [ λ ( n ) z ] .
L = [ i δ 1 i δ 2 i δ 3 i δ 4 ] ,
l kj = l jk s jk .
l ki ( n + 1 ) = j l kj l ji ( n )
= j l jk s jk l ij ( n ) s ij
= j l ij ( n ) l jk s ij s jk .
l ki ( n + 1 ) = l ik ( n + 1 ) s ik ,
μ kj = μ jk s jk .

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