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Quantum noise properties of parametric amplifiers driven by two pump waves

C. J. McKinstrie, S. Radic, and M. G. Raymer

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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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Figures (10)
Fig. 1.
Fig. 1. Illustration of the constituent FWM processes.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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 − * − iγ 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 + = iγ 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 − * = − iγ ε ( P 1 P 2 ) 1 2 B 1 − * − iγ ε ( P 1 P 2 ) 1 2 B 1 +
− i ( β 2 − + γ P 2 ) B 2 − * − iγ P 2 B 2 + ,
d z B 2 + = iγ ε ( P 1 P 2 ) 1 2 B 1 − * + iγ ε ( P 1 P 2 ) 1 2 B 1 +
+ iγ 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 − * = − iδ C 1 − * − iκ C 1 + ,
d z C 1 + = iκ C 1 − * + iδ C 1 + ,
C ( z ) = M ( z ) C ( 0 ) ,
M ( z ) = [ cos ( kz ) − iδ sin ( kz ) k − iκ sin ( kz ) k iκ sin ( kz ) k cos ( kz ) + iδ 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 − = − iδ C 1 − + iκ C 2 − ,
d z C 2 − = iκ C 1 − + iδ C 2 − ,
M ( z ) = [ cos ( kz ) − iδ sin ( kz ) k iκ sin ( kz ) k iκ sin ( kz ) k cos ( kz ) + iδ 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 − * = − iδ C 1 − * − iκ C 2 + ,
d z C 2 + = iκ C 1 − * + iδ C 2 + ,
M ( z ) = [ cos ( kz ) − iδ sin ( kz ) k − iκ sin ( kz ) k iκ sin ( kz ) k cos ( kz ) + iδ 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 ) − iμ ( z ) μ ( z ) − iμ ( z ) iμ ( z ) μ ( z ) iμ ( z ) μ ( z ) μ ( z ) − iμ ( z ) μ ( z ) − iμ ( z ) iμ ( z ) μ ( z ) iμ ( 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 [ iβ ( ω ) z − iωt ] dω ( 2 π ) 1 2 ,
B ̂ 0 t z = ( 2 π h ̄ ω 0 n 0 c ) 1 2 ∫ a ̂ ( ω ) exp [ iβ ( ω ) z − iωt ] dω ( 2 π ) 1 2 ,
[ a ̂ ( ω ) , a ̂ † ( ω ′ ) ] = δ ( ω − ω ′ ) ,
∫ T ̂ 0 t z dz = h ̄ ω 0 ∫ a ̂ † ( ω ) a ̂ ( ω ) dω .
a ̂ ω z = a ̂ ( ω ) exp [ iβ ( ω ) z ]
a ̂ t z = ∫ a ̂ ( ω ) exp [ iβ ( ω ) z − iωt ] dω ( 2 π ) 1 2 .
∫ a ̂ † t z a ̂ t z dt = ∫ a ̂ † ω z a ̂ ω z dω
[ 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 ) dω
a ̂ α † = ∫ α ω z a ̂ † ω z dω ,
α ω z = α ( ω ) exp [ iβ ( 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 ) dω Δ 1 2 .
[ a k , a l † ] = δ kl ,
a ̂ j ( t , z ) = ( v L ) 1 2 ∑ k a ̂ k exp ⁡ [ iβ ( 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 − iα − iβ − iβ iα i δ 2 iβ iβ − iβ − iβ − i δ 3 − iγ iβ iβ iγ 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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