Abstract

In this paper the quantum noise properties of phase-insensitive and phase-sensitive parametric processes are studied. Formulas for the field-quadrature and photon-number means and variances are derived, for processes that involve arbitrary numbers of modes. These quantities determine the signal-to-noise ratios associated with the direct and homodyne detection of optical signals. The consequences of the aforementioned formulas are described for frequency conversion, amplification, monitoring, and transmission through sequences of attenuators and amplifiers.

© 2005 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. J. Hansryd and P. A. Andrekson, �??Broad-band continuous-wave pumped fiber optical parametric amplifier with 49-dB gain and wavelength-conversion efficiency,�?? IEEE Photon. Technol. Lett. 13, 194�??191 (2001).
    [CrossRef]
  4. 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]
  5. K. Inoue, �??Tunable and selective wavelength conversion using fiber four-wave mixing with two pump lights,�?? IEEE Photon. Technol. Lett. 6, 1451�??1453 (1994).
    [CrossRef]
  6. K. Uesaka, K. K. Y. Wong, M. E. Marhic and L. G. Kazovsky, �??Wavelength exchange in a highly nonlinear dispersion-shifted fiber: theory and experiments,�?? IEEE J. Sel. Top. Quantum Electron. 8 560�??568 (2002).
    [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. Tech. Lett. 16, 551�??553 (2004).
    [CrossRef]
  8. J. Hansryd, P. A. Andrekson, M. Westlund, J. Li and P. O. Hedekvist, �??Fiber-based optical parametric amplifiers and their applications,�?? IEEE J. Sel. Top. Quantum Electron. 8, 506�??520 (2002).
    [CrossRef]
  9. S. Radic and C. J. McKinstrie, �??Two-pump fiber parametric amplifiers,�?? Opt. Fiber Technol. 9, 7�??23 (2003).
    [CrossRef]
  10. S. Radic and C. J. McKinstrie, �??Optical parametric amplification and signal processing in highly-nonlinear fibers,�?? IEICE Trans. Electron. E88C, 859�??869 (2005).
    [CrossRef]
  11. C. M. Caves, �??Quantum limits on noise in linear amplifiers,�?? Phys. Rev. D 26, 1817�??1839 (1982).
    [CrossRef]
  12. W. H. Louisell, Radiation and Noise in Quantum Electronics (McGraw-Hill, 1964).
  13. R. Loudon, The Quantum Theory of Light, 3rd Ed. (Oxford University Press, 2000).
  14. M. G. Raymer and M. Beck, �??Experimental quantum state tomography of optical fields and ultrafast statistical sampling,�?? in Lecture Notes in Physics, Vol. 649, edited by M. Paris and J. Rehacek (Springer-Verlag, 2004), pp. 235�??295.
  15. R. Loudon and P. L. Knight, �??Squeezed light,�?? J. Mod. Opt. 34, 709�??759 (1987).
    [CrossRef]
  16. C. J. McKinstrie, S. Radic and M. G. Raymer, �??Quantum noise properties of parametric amplifiers driven by two pump waves,�?? Opt. Express 12, 5037�??5066 (2004).
    [CrossRef] [PubMed]
  17. C. J. McKinstrie, S. Radic, R. M. Jopson and A. R. Chraplyvy, �??Quantum noise limits on optical monitoring with parametric devices,�?? submitted to Opt. Commun.
  18. C. J. McKinstrie and S. Radic, �??Phase-sensitive amplification in a fiber,�?? Opt. Express 12, 4973�??4979 (2004).
    [CrossRef] [PubMed]
  19. C. J. McKinstrie, M. G. Raymer, S. Radic and M. V. Vasilyev, �??Quantum mechanics of phase-sensitive amplification in a fiber,�?? submitted to Opt. Commun.
  20. R. Loudon, �??Theory of noise accumulation in linear optical-amplifier chains,�?? IEEE J. Quantum Electron. 21, 766�??773 (1985).
    [CrossRef]

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)

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

R. Loudon, �??Theory of noise accumulation in linear optical-amplifier chains,�?? IEEE J. Quantum Electron. 21, 766�??773 (1985).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (2)

K. Uesaka, K. K. Y. Wong, M. E. Marhic and L. G. Kazovsky, �??Wavelength exchange in a highly nonlinear dispersion-shifted fiber: theory and experiments,�?? IEEE J. Sel. Top. Quantum Electron. 8 560�??568 (2002).
[CrossRef]

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

IEEE Photon. Tech. Lett. (1)

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. Tech. Lett. 16, 551�??553 (2004).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

J. Hansryd and P. A. Andrekson, �??Broad-band continuous-wave pumped fiber optical parametric amplifier with 49-dB gain and wavelength-conversion efficiency,�?? IEEE Photon. Technol. Lett. 13, 194�??191 (2001).
[CrossRef]

K. Inoue, �??Tunable and selective wavelength conversion using fiber four-wave mixing with two pump lights,�?? IEEE Photon. Technol. Lett. 6, 1451�??1453 (1994).
[CrossRef]

IEICE Trans. Electron. (1)

S. Radic and C. J. McKinstrie, �??Optical parametric amplification and signal processing in highly-nonlinear fibers,�?? IEICE Trans. Electron. E88C, 859�??869 (2005).
[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]

J. Mod. Opt. (1)

R. Loudon and P. L. Knight, �??Squeezed light,�?? J. Mod. Opt. 34, 709�??759 (1987).
[CrossRef]

Lecture Notes in Physics (1)

M. G. Raymer and M. Beck, �??Experimental quantum state tomography of optical fields and ultrafast statistical sampling,�?? in Lecture Notes in Physics, Vol. 649, edited by M. Paris and J. Rehacek (Springer-Verlag, 2004), pp. 235�??295.

Opt. Commun. (2)

C. J. McKinstrie, M. G. Raymer, S. Radic and M. V. Vasilyev, �??Quantum mechanics of phase-sensitive amplification in a fiber,�?? submitted to Opt. Commun.

C. J. McKinstrie, S. Radic, R. M. Jopson and A. R. Chraplyvy, �??Quantum noise limits on optical monitoring with parametric devices,�?? submitted to Opt. Commun.

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. D (1)

C. M. Caves, �??Quantum limits on noise in linear amplifiers,�?? Phys. Rev. D 26, 1817�??1839 (1982).
[CrossRef]

Other (2)

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

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

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Figures (11)

Fig. 1.
Fig. 1.

Illustration of the constituent two-mode processes in a four-mode parametric interaction driven by two pump waves

Fig. 2.
Fig. 2.

Illustration of the architecture of one stage in a communication link with phase-insensitive amplification. The attenuator ◁ is followed by a parametric amplifier ▷. The signal, idler and scattered modes are labeled 1, 2r and 2r+1, respectively.

Fig. 3.
Fig. 3.

Illustration of the constituent two-mode processes in a phase-sensitive parametric interaction driven by two pump waves.

Fig. 4.
Fig. 4.

Illustration of the architecture of one stage in a communication link with phase-sensitive amplification. The attenuator ◁ is followed by a parametric amplifier▷. The signal and scattered modes are labeled 1 and r+1, respectively.

Fig. 5.
Fig. 5.

Noise figures [Eqs. (51) and (52)] plotted as functions of the transmittance T. The solid line represents the signal, whereas the dashed curve represents the idler.

Fig. 6.
Fig. 6.

Noise figures [Eqs. (45), (46), (53) and (54)] plotted as functions of the relative phase x. The solid line represents the signal, whereas the dashed curve represents the idler.

Fig. 7.
Fig. 7.

Noise figures [Eqs. (61) and (62)] plotted as functions of the gain G. The solid curve represents the signal, whereas the dashed curve represents the idler.

Fig. 8.
Fig. 8.

Homodyne noise-figure [Eq. (86)] plotted as a function of the input phase ξ and the local-oscillator phase η. Dark shadings denote low noise figures, whereas light shadings denote high noise figures. The contour spacing is 4 dB.

Fig. 9.
Fig. 9.

Homodyne noise-figure [Eq. (86)] plotted as a function of (a) the input phase ξ, for the case in which the local-oscillator phase η=0, and (b) η, for the case in which ξ=0.

Fig. 10.
Fig. 10.

Direct noise-figure plotted as a function of the input phase ξ. The solid curve represents the exact noise figure [Eqs. (46) and (87)], whereas the dashed curve represents the approximate noise figure [Eq. (88)].

Fig. 11.
Fig. 11.

Direct noise-figure plotted as a function of the relative phase x. The solid curve represents the exact noise figure [Eqs. (46) and (93)], whereas the dashed curve represents the approximate noise figure [Eq. (94)].

Equations (163)

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a j ( z ) = μ ̄ ( z ) a j ( 0 ) + ν ̄ ( z ) a k ( 0 ) ,
a k ( z ) = ν ̄ * ( z ) a j ( 0 ) + μ ̄ * ( z ) a k ( 0 ) .
d jk = ( μ ̄ 2 ν ̄ 2 ) ( a j a j a k a k ) + 2 ( μ ̄ ν ̄ * a j a k + μ ̄ * ν ̄ a j a k ) .
d j = a j a l e i ϕ + a j a l e i ϕ ,
d j ( ϕ ) = 2 α l q j ( ϕ ) ,
δ d j 2 ( ϕ ) = 4 α l 2 δ q j 2 ( ϕ ) + 1 l n j .
a 1 + ( z ) = ν ( z ) a 1 ( 0 ) + μ ( z ) a 1 + ( 0 ) .
a 1 + ( z ) = μ ( z ) a 1 + ( 0 ) + ν ( z ) a 2 ( 0 ) ,
a 1 + ( z ) = μ ̄ ( z ) a 1 + ( 0 ) + ν ̄ ( z ) a 2 + ( 0 ) .
a 2 ( z ) = v 21 ( z ) a 1 ( 0 ) + μ 22 ( z ) a 2 ( 0 ) + ν 23 ( z ) a 3 ( 0 ) + μ 24 ( z ) a 4 ( 0 ) ,
a 1 ( z r ) = μ ̄ ( z r z r 1 ) a 1 ( z r 1 ) + ν ̄ ( z r z r 1 ) a 2 r + 1 ( z r 1 ) ,
a 1 ( z r ) = μ ( z r z r ) a 1 ( z r ) + ν ( z r z r ) a 2 r ( z r ) ,
a 1 ( z r ) = μ 11 ( z r , z r 1 ) a 1 ( z r 1 ) + ν 12 r ( z r , z r ) a 2 r ( z r )
+ μ 12 r + 1 ( z r , z r 1 ) a 2 r + 1 ( z r 1 ) ,
a 1 ( z s ) = μ 11 ( z s ) a 1 ( 0 ) + r = 1 s [ ν 12 r ( z s ) a 2 r ( 0 ) + μ 12 r + 1 ( z s ) a 2 r + 1 ( 0 ) ] .
a 1 + ( z ) = μ ( z ) a 1 + ( 0 ) + ν ( z ) a 1 + ( 0 ) ,
a 2 ( z ) = μ 21 ( z ) a 1 ( 0 ) + ν 21 ( z ) a 1 ( 0 ) + μ 22 ( z ) a 2 ( 0 )
+ ν 22 ( z ) a 2 ( 0 ) + μ 23 ( z ) a 3 ( 0 ) + ν 23 ( 0 ) a 3 ( 0 ) ,
a 1 ( z r ) = μ ̄ ( z r z r 1 ) a 1 ( z r 1 ) + ν ̄ ( z r z r 1 ) a r + 1 ( z r 1 ) ,
a 1 ( z r ) = μ ( z r z r ) a 1 ( z r ) + ν ( z r z r ) a 1 ( z r ) ,
a 1 ( z r ) = μ 11 ( z r , z r 1 ) a 1 ( z r 1 ) + ν 11 ( z r , z r 1 ) a 1 ( z r 1 )
+ μ 1 r + 1 ( z r , z r 1 ) a r + 1 ( z r 1 ) + ν 1 r + 1 ( z r , z r 1 ) a r + 1 ( z r 1 ) ,
a 1 ( z s ) = μ 11 ( z s ) a 1 ( 0 ) + ν 11 ( z s ) a 1 ( 0 )
+ r = 1 s [ μ 1 r + 1 ( z s ) a r + 1 ( 0 ) + ν 1 r + 1 ( z s ) a r + 1 ( 0 ) ] ,
a j ( z ) = k [ μ j k ( z ) a k ( 0 ) + ν j k ( z ) a k ( 0 ) ] ,
l [ μ j l ( z ) ν k l ( z ) μ k l ( z ) ν j l ( z ) ] = 0 ,
l [ μ j l ( z ) μ k l * ( z ) ν j l ( z ) ν k l * ( z ) ] = δ j k .
l [ μ j l ( z ) 2 ν j l ( z ) 2 ] = 1 .
a j ( z ) = α j ( z ) + v j ( z ) ,
α j ( z ) = μ j i ( z ) α i ( 0 ) + v j i ( z ) α i * ( 0 )
v j ( z ) = k [ μ j k ( z ) a k ( 0 ) + v j k ( z ) a k ( 0 ) ]
a j 2 = α j 2 + 2 α j v j + v j 2 ,
a j a j = α j 2 + α j v j + α j * v j + v j v j ,
a j a j = α j 2 + α j v j + α j * v j + v j v j ,
( a j a j ) 2 = α j 4 + [ α j 2 ( v j ) 2 + α j 2 ( v j v j + v j v j ) + ( α j * ) 2 v j 2 ]
+ ( v j v j ) 2 + 2 α j 2 ( α j v j + α j * v j ) + 2 α j 2 v j v j
+ ( α j v j + α j * v j ) v j v j + v j v j ( α j v j + α j * v j ) .
v j 0 = k v j k 1 k ,
v j 0 = k μ j k * 1 k ,
v j 2 0 = k [ μ j k ν j k 0 + 2 1 2 ν j k 2 2 k + l k ν j k ν j l 1 k 1 l ] ,
v j v j 0 = k [ μ j k 2 0 + 2 1 2 μ j k * ν j k 2 k + l k μ j k * ν j l 1 k 1 l ] ,
v j v j 0 = k [ 2 1 2 μ j k * ν j k 2 k + ν j k 2 0 + l k μ j l * ν j k 1 k 1 l ] ,
( v j ) 2 0 = k [ 2 1 2 ( μ j k * ) 2 2 k + μ j k * ν j k * 0 + l k μ j k * μ j l * 1 k 1 l ] ,
q j = ( α j e i ϕ + α j * e i ϕ ) 2 ,
δ q j 2 = [ ( k μ j k ν j k ) e i 2 ϕ + k ( μ j k 2 + ν j k 2 ) + ( k μ j k ν j k ) * e i 2 ϕ ] 4 ,
δ q j 2 = k λ j k 2 4 .
n j = α j 2 + k ν j k 2 ,
δ n j 2 = α j 2 k ( μ j k ν j k ) * + α j 2 k ( μ j k 2 + ν j k 2 ) + ( α j 2 ) * k μ j k ν j k
+ 2 k μ j k ν j k 2 + k l > k μ j k * ν j l + μ j l * ν j k 2 ,
δ n j 2 = α j 2 k λ j k ' 2 + 2 k μ j k ν j k 2 + k l > k U j k * ν j l + μ j l * ν j k 2 .
α j ( z ) = i [ μ j i ( z ) α i ( 0 ) + ν j i ( z ) α i * ( 0 ) ] ,
S i = 4 α i 2 ,
S i = α i 2 .
S j = 4 α j 2 cos 2 ( ϕ j ϕ ) k λ j k 2 ,
S j = 4 α j 2 k λ j k 2 .
S j = ( α j 2 + k ν j k 2 ) 2 α j 2 k λ j k ' 2 + 2 k μ j k ν j k 2 + k l > k μ j k * ν j l + μ j l * ν j k 2 ,
S j α j 2 k λ j k ' 2 .
F 1 ( z ) = 1 T ,
F 2 ( z ) = 1 ( 1 T ) ,
α 1 ( z ) 2 = T α 1 2 + ( 1 T ) α 2 2 + 2 [ T ( 1 T ) ] 1 2 α 1 α 2 cos ξ ,
α 2 ( z ) 2 = ( 1 T ) α 1 2 + T α 2 2 2 [ T ( 1 T ) ] 1 2 α 1 α 2 cos ξ ,
S 1 ( z ) = 4 G n 1 ( 2 G 1 ) ,
S 2 ( z ) = 4 ( G 1 ) n 1 ( 2 G 1 ) ,
F 1 ( z ) = 1 + ( G 1 ) G ,
F 2 ( z ) = 1 + G ( G 1 ) .
S 1 ( z ) = [ G n 1 + G 1 ] 2 [ G ( 2 G 1 ) n 1 + G ( G 1 ) ] ,
S 2 ( z ) = [ ( G 1 ) n 1 + G 1 ] 2 [ ( G 1 ) ( 2 G 1 ) n 1 + G ( G 1 ) ] .
F 1 ( z ) 1 + ( G 1 ) G ,
F 2 ( z ) = 1 + G ( G 1 ) .
S j ( z ) = 4 κ j i 2 n i k κ j k 2 ,
F j ( z ) = k κ j k 2 κ j i 2 ,
S j ( z ) = [ κ j i 2 n i + k κ j k 2 σ j k ] 2 [ κ j i 2 k κ j k 2 n i + k l > k κ j k κ j l 2 σ k l ] .
F j ( z ) k κ j k 2 κ j i 2 .
S 1 ( z ) = 4 G T n 1 ( 2 G 1 ) ,
F 1 ( z ) = ( 2 G 1 ) G T ,
S 1 ( z ) = [ G T n 1 + G 1 ] 2 [ G ( 2 G 1 ) T n 1 + G ( G 1 ) ] .
F 1 ( z ) ( 2 G 1 ) G T .
S 1 ( z ) = 4 n 1 [ 1 + 2 s ( G 1 ) ] ,
F 1 ( z ) = 1 + 2 s ( G 1 ) .
S 1 ( z ) = [ n 1 + s ( G 1 ) ] 2 { [ 1 + 2 s ( G 1 ) ] n 1 + s ( G 1 ) [ 1 + s ( G 1 ) ] } .
F 1 = 1 + 2 s ( G 1 ) .
S 1 ( z ) = 4 T f G T i n 1 [ 1 + 2 T f ( G 1 ) ] ,
F 1 ( z ) = [ 1 + 2 T f ( G 1 ) ] ( T f G T i ) ,
S 1 ( z ) = [ T f G T i n 1 + T f ( G 1 ) ] 2 { ( T f G T i ) [ 1 + 2 T f ( G 1 ) ] n 1
+ T f ( G 1 ) [ 1 + T f ( G 1 ) ] } .
F 1 ( z ) [ 1 + 2 T f ( G 1 ) ] ( T f G T i ) .
α 1 ( z ) 2 = α 1 ( 0 ) 2 { 2 G 1 + 2 [ G ( G 1 ) ] 1 2 cos ξ } ,
ϕ 1 ( z ) = ϕ 1 ( 0 ) + ϕ μ + tan 1 [ ( G 1 ) 1 2 sin ξ G 1 2 + ( G 1 ) 1 2 cos ξ ] ,
n 1 ( z ) = H ( ξ ) n 1 + G 1 ,
δ q 1 2 ( z ) = H ( η ) 4 ,
δ n 1 2 ( z ) = H ( ξ ) H ( ζ ) n 1 + 2 G ( G 1 ) ,
ζ = ξ tan 1 [ ( G 1 ) 1 2 sin ξ G 1 2 + ( G 1 ) 1 2 cos ξ ] .
S 1 ( z ) = 4 n 1 H ( ξ ) cos 2 [ ( ζ η ) 2 ] H ( η ) ,
F 1 ( z ) = H ( η ) { H ( ξ ) cos 2 [ ( ζ η ) 2 ] } ,
S 1 ( z ) = [ H ( ξ ) n 1 + G 1 ] 2 [ H ( ξ ) H ( ζ ) n 1 + 2 G ( G 1 ) ] .
F 1 ( z ) H ( ζ ) H ( ξ ) .
α 1 ( z ) 2 = G α 1 2 + ( G 1 ) α 2 2 + 2 [ G ( G 1 ) ] 1 2 α 1 α 2 cos ξ ,
α 2 ( z ) 2 = ( G 1 ) α 1 2 + G α 2 2 + 2 [ G ( G 1 ) ] 1 2 α 1 α 2 cos ξ ,
S j ( z ) = 4 α j ( z ) 2 ( 2 G 1 ) ,
F j ( 2 ) = ( 2 G 1 ) α j ( 0 ) α j ( z ) 2 ,
S j ( z ) = [ [ α j ( z ) ] 2 + G 1 ] 2 [ ( 2 G 1 ) α j ( z ) 2 + G ( G 1 ) ] .
F j ( z ) ( 2 G 1 ) α j ( 0 ) α j ( z ) 2 .
α 1 ( z ) 2 = T α 1 ( 0 ) 2 { 2 G 1 + 2 [ G ( G 1 ) ] 1 2 cos ξ } ,
ϕ 1 ( z ) = ϕ 1 ( 0 ) + ϕ μ + ϕ μ ̄ + tan 1 [ ( G 1 ) 1 2 sin ξ G 1 2 + ( G 1 ) 1 2 cos ξ ] ,
n 1 ( z ) = H ( ξ ) T n 1 + G 1 ,
δ q 1 2 ( z ) = H ( η ) 4 ,
δ n 1 2 ( z ) = H ( ξ ) H ( ζ ) T n 1 + 2 G ( G 1 ) ,
S 1 ( z ) = 4 T n 1 H ( ξ ) cos 2 [ ( ζ η ) 2 ] H ( η ) ,
F 1 ( z ) = H ( η ) { H ( ξ ) T cos 2 [ ( ζ η ) 2 ] } ,
S 1 ( z ) = [ H ( ξ ) T n 1 + G 1 ] 2 [ H ( ξ ) H ( ζ ) T n 1 + 2 G ( G 1 ) ] .
F 1 ( z ) H ( ζ ) [ H ( ξ ) T ] .
S 1 ( z ) = 4 α 1 ( 0 ) 2 [ 1 + s ( L 1 ) ] ,
F 1 ( z ) = 1 + s ( L 1 ) .
S 1 ( z ) { α 1 ( 0 ) 2 + [ s ( L 1 ) 1 ] 4 } 2 [ 1 + s ( L 1 ) ] α 1 ( 0 ) 2 + { [ 1 + s ( L 1 ) ] 2 2 } 8 .
F 1 ( z ) 1 + s ( L 1 ) .
a j ( z ) = u j ( z ) + v j ( z ) ,
u j ( z ) = μ j 1 ( z ) a 1 ( 0 ) + ν j 1 ( z ) a 1 ( 0 )
v j ( z ) = k > 1 [ μ j k ( z ) a k ( 0 ) + ν j k ( z ) a k ( 0 ) ]
a j 2 = u j 2 + 2 u j v j + v j 2 ,
a j a j = u j u j + u j v j + u j v j + v j v j ,
a j a j = u j u j + u j v j + u j v j + v j v j ,
( a j a j ) 2 = ( u j u j ) 2 + [ u j 2 ( v j ) 2 + u j u j v j v j + u j u j v j v j + ( u j ) 2 v j 2 ]
+ ( v j v j ) 2 + u j u j ( u j v j + u j v j ) + ( u j v j + u j v j ) u j u j
+ 2 u j u j v j v j + ( u j v j + u j v j ) v j v j + v j v j ( u j v j + u j v j ) .
q j = u j e i ϕ + u j e i ϕ 2 ,
δ q j 2 = δ q 2 ( u j ) + δ q 2 ( v j ) ,
δ q 2 ( v j ) = k > 1 λ j k 2 4 ,
n j = u j u j + v j v j ,
δ n j 2 = δ n 2 ( u j ) + u j 2 ( v j ) 2 + u j u j v j v j
+ u j u j v j v j + ( u j ) 2 v j 2 + δ n 2 ( v j ) ,
v j 2 = k > 1 μ j k ν j k ,
v j v j = k > 1 μ j k 2 ,
v j v j = k > 1 ν j k 2 ,
( v j ) 2 = k > 1 μ j k * ν j k *
δ n 2 ( v j ) = 2 k > 1 μ j k ν j k 2 + k > 1 l > k μ j k * ν j l + μ j l * ν j k 2 .
q ( u j ) = ( α j e i ϕ + α j * e i ϕ ) 2 ,
δ q 2 ( u j ) = λ j 1 2 4 ,
n ( u j ) = α j 2 + ν j 1 2 ,
δ n 2 ( u j ) = α j 2 λ j 1 ' 2 + 2 μ j 1 ν j 1 2 ,
δ n j 2 = α j 2 λ j 1 ' 2 + 2 μ j 1 ν j 1 2
+ ( α j 2 + μ j 1 ν j 1 ) k > 1 ( μ j k ν j k ) * + ( α j 2 + μ j 1 2 ) k > 1 ν j k 2
+ ( α j 2 + ν j 1 2 ) k > 1 μ j k 2 + ( α j 2 + μ j 1 ν j 1 ) * k > 1 μ j k ν j k
+ 2 k > 1 μ j k ν j k 2 + k > 1 l > k μ j k * ν j l + μ j l * ν j k 2 .
α j 2 λ j k ' 2 = α j 2 μ j k * ν j k * + α j 2 μ j k 2 + α j 2 ν j k 2 + ( α j * ) 2 μ j k ν j k ,
μ j 1 * ν j l + μ j l * ν j 1 2 = μ j 1 ν j 1 μ j l * ν j l * + μ j 1 2 ν j l 2 + μ j l 2 ν j 1 2 + μ j 1 * ν j 1 * μ j l ν j l ,
u j ( z ) = j [ μ j l ( z ) a i ( 0 ) + ν j i ( z ) a i ( 0 ) ] ,
v j ( z ) = k i [ μ j k ( z ) a k ( 0 ) + μ j k ( z ) a k ( 0 ) ] .
a 1 ( z 1 " ) = μ μ ̄ a 1 ( z 0 ) + ν a 2 ( z ' 1 ) + μ ν ̄ a 3 ( z 0 ) .
a 1 ( z 2 ) = ( μ μ ̄ ) 2 a 1 ( z 0 ) + ( μ μ ̄ ) ν a 2 ( z 1 ) + ( μ μ ̄ ) μ ν ̄ a 3 ( z 0 ) + ν a 4 ( z 2 ) + μ ν ̄ a 5 ( z 1 ) .
a 1 ( z s " ) = ( μ μ ̄ ) s a 1 ( z 0 ) + r = 1 s ( μ μ ̄ ) s r [ ν a 2 r ( z ' r ) + μ ν ̄ a 2 r + 1 ( z " r 1 ) ] .
a 1 ( z 1 " ) = μ ̄ μ a 1 ( z 0 " ) + μ ̄ * ν a 1 ( z 0 " ) + ν ̄ μ a 2 ( z 0 " ) + ν ̄ * ν a 2 ( z 0 " ) .
a 1 ( z 2 ) = μ ̄ 2 ( μ 2 + ν 2 ) a 1 ( z 0 ) + μ ̄ 2 ( 2 μ ν ) a 1 ( z 0 ) + μ ̄ ν ̄ ( μ 2 + ν 2 ) a 2 ( z 0 )
+ μ ̄ ν ̄ ( 2 μ ν ) a 2 ( z 0 ) + ν ̄ μ a 3 ( z 1 " ) + ν ̄ ν a 3 ( z 1 " ) .
a 1 ( z 2 " ) = μ ̄ s [ p s ( μ , ν ) a 1 ( z 0 " ) + q s ( μ , ν ) a 1 ( z 0 " ) ] + r = 1 s μ ̄ s r ν ̄
× [ p s + 1 r ( μ , ν ) a r + 1 ( z r 1 ) + q s + 1 r ( μ , ν ) a r + 1 ( z r 1 ) ] ,
r = 0 s ν 1 r + 1 2 = [ s ( L 1 ) ( 1 T 2 s ) ( 1 + T ) ] 4 .
r = 0 s μ 1 r + 1 ν 1 r + 1 2 = { ( 1 T 2 s ) 2 + ( L 1 ) 2 [ s 2 T 2 ( 1 T 2 s ) ( 1 T 2 )
+ T 4 ( 1 T 4 s ) ( 1 T 4 ) ] } 16 .
r = 1 s μ 11 ν 1 r + 1 + μ 1 r + 1 ν 11 2 = ( L 1 ) [ s 2 T s + 1 ( 1 T s ) ( 1 T )
+ T 2 s + 2 ( 1 T 2 s ) ( 1 T 2 ) ] 4 .
r = q + 1 s μ 1 q + 1 ν 1 r + 1 + μ 1 r + 1 ν 1 q + 1 2 = ( L 1 ) 2 [ ( s q ) 2 T s + 2 q ( 1 T s q ) ( 1 T )
+ T 2 s + 4 2 q ( 1 T 2 s 2 q ) ( 1 T 2 ) ] 4 ,
2 k μ 1 k ν 1 k 2 + k l > k μ 1 k ν 1 l + μ 1 l ν 1 k 2 { [ 1 + s ( L 1 ) ] 2 2 } 8 ,

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