Abstract

Based on semi-classical theory, the noise performance of a multi-span fiber optical transmission system employing a cascaded phase-insensitive amplifier (PIA) and phase-sensitive amplifiers (PSAs) is investigated. Compared with the pure-PIA and pure-PSA based in-line amplification schemes, the copier + PSA scheme is found to improve the system NF by up to 6 and 3 dB, respectively, in an optimized long-haul fiber link. In addition, this cascaded configuration will significantly relax the requirement for accurate phase- and wavelength-locking which is rigorously needed in the pure-PSA configuration. This scheme is also modulation-format independent. As a proof of concept, the NF of a fiber parametric amplifier based copier + PSA cascade with inter-stage attenuation representing the fiber link is measured, which shows a 1.8-dB total NF improvement over the conventional EDFA cascade.

© 2010 OSA

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    [CrossRef]

2010 (1)

2008 (1)

2007 (2)

2006 (3)

C. J. McKinstrie, S. Radic, R. M. Jopson, and A. R. Chraplyvy, “Quantum noise limits on optical monitoring with parametric devices,” Opt. Commun. 259(1), 309–320 (2006).
[CrossRef]

C. J. McKinstrie, M. G. Raymer, S. Radic, and M. Vasilyev, “Quantum mechanics of phase-sensitive amplification in a fiber,” Opt. Commun. 257(1), 146–163 (2006).
[CrossRef]

P. L. Voss, K. G. Köprülü, and P. Kumar, “Raman-noise-induced quantum limits for χ(3) nondegenerate phase-sensitive amplification and quadrature squeezing,” J. Opt. Soc. Am. B 23(4), 598–610 (2006).
[CrossRef]

2005 (4)

2004 (3)

1999 (2)

W. Imajuku, A. Takada, and Y. Yamabayashi, “Low-noise amplification under the 3dB noise figure in high-gain phase-sensitive fibre amplifier,” Electron. Lett. 35(22), 1954–1955 (1999).
[CrossRef]

D. Levandovsky, M. Vasilyev, and P. Kumar, “Amplitude squeezing of light by means of a phase-sensitive fiber parametric amplifier,” Opt. Lett. 24(14), 984–986 (1999).
[CrossRef] [PubMed]

1998 (4)

M. Movassaghi, M. K. Jackson, V. M. Smith, and W. J. Hallam, “Noise figure of erbium-doped fiber amplifiers in saturated operation,” J. Lightwave Technol. 16(5), 812–816 (1998).
[CrossRef]

W. Imajuku and A. Takada, “Theoretical analysis of system limitation for AM-DD/NRZ optical transmission systems using in-line phase-sensitive amplifiers,” J. Lightwave Technol. 16(7), 1158–1170 (1998).
[CrossRef]

A. Takada and W. Imajuku, “In-line optical phase-sensitive amplifier employing pump laser injection locked to input signal light,” Electron. Lett. 34(3), 274–276 (1998).
[CrossRef]

W. Imajuku and A. Takada, “Error-free operation of in-line phase-sensitive amplifier,” Electron. Lett. 34(17), 1673–1674 (1998).
[CrossRef]

1997 (1)

J. A. Levenson, K. Bencheikh, D. J. Lovering, P. Vidakovic, and C. Simonneau, “Quantum noise in optical parametric amplification: a means to achieve noiseless optical functions,” Quantum Semiclass. Opt. 9(2), 221–237 (1997).
[CrossRef]

1993 (1)

1990 (1)

A. Mecozzi and P. Tombesi, “Parametric amplification and signal-to-noise ratio in optical transmission lines,” Opt. Commun. 75(3-4), 256–262 (1990).
[CrossRef]

1989 (1)

N. A. Olsson, “Lightwave systems with optical amplifiers,” J. Lightwave Technol. 7(7), 1071–1082 (1989).
[CrossRef]

1985 (1)

R. Loudon, “Theory of noise accumulation in linear optical-amplifier chains,” IEEE J. Quantum Electron. 21(7), 766–773 (1985).
[CrossRef]

1982 (1)

C. M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D Part. Fields 26(8), 1817–1839 (1982).
[CrossRef]

Abram, I.

Andrekson, P.

T. Torounidis and P. Andrekson, “Broadband single-pumped fiber-optic parametric amplifiers,” IEEE Photon. Technol. Lett. 19(9), 650–652 (2007).
[CrossRef]

Andrekson, P. A.

Bencheikh, K.

J. A. Levenson, K. Bencheikh, D. J. Lovering, P. Vidakovic, and C. Simonneau, “Quantum noise in optical parametric amplification: a means to achieve noiseless optical functions,” Quantum Semiclass. Opt. 9(2), 221–237 (1997).
[CrossRef]

Bogris, A.

Z. Tong, A. Bogris, M. Karlsson, and P. A. Andrekson, “Full characterization of the signal and idler noise figure spectra in single-pumped fiber optical parametric amplifiers,” Opt. Express 18(3), 2884–2893 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-3-2884 .
[CrossRef] [PubMed]

Z. Tong, A. Bogris, C. Lundström, C. J. McKinstrie, M. Vasilyev, M. Karlsson, and P. A. Andrekson, “Modeling and measurement of noise figure in a cascaded non-degenerate phase-sensitive parametric amplifier,” to appear in Opt. Express .

Caves, C. M.

C. M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D Part. Fields 26(8), 1817–1839 (1982).
[CrossRef]

Chau, F. S.

Chraplyvy, A. R.

C. J. McKinstrie, S. Radic, R. M. Jopson, and A. R. Chraplyvy, “Quantum noise limits on optical monitoring with parametric devices,” Opt. Commun. 259(1), 309–320 (2006).
[CrossRef]

Devgan, P.

R. Tang, P. Devgan, V. S. Grigoryan, and P. Kumar, “Inline frequency-non-degenerate phase-sensitive fibre parametric amplifier for fibre-optic communication,” Electron. Lett. 41(19), 1072–1074 (2005).
[CrossRef]

Devgan, P. S.

Grangier, P.

Grigoryan, V. S.

Hallam, W. J.

Hedekvist, P.-O.

Imajuku, W.

W. Imajuku, A. Takada, and Y. Yamabayashi, “Low-noise amplification under the 3dB noise figure in high-gain phase-sensitive fibre amplifier,” Electron. Lett. 35(22), 1954–1955 (1999).
[CrossRef]

A. Takada and W. Imajuku, “In-line optical phase-sensitive amplifier employing pump laser injection locked to input signal light,” Electron. Lett. 34(3), 274–276 (1998).
[CrossRef]

W. Imajuku and A. Takada, “Error-free operation of in-line phase-sensitive amplifier,” Electron. Lett. 34(17), 1673–1674 (1998).
[CrossRef]

W. Imajuku and A. Takada, “Theoretical analysis of system limitation for AM-DD/NRZ optical transmission systems using in-line phase-sensitive amplifiers,” J. Lightwave Technol. 16(7), 1158–1170 (1998).
[CrossRef]

Jackson, M. K.

Jopson, R.

Jopson, R. M.

C. J. McKinstrie, S. Radic, R. M. Jopson, and A. R. Chraplyvy, “Quantum noise limits on optical monitoring with parametric devices,” Opt. Commun. 259(1), 309–320 (2006).
[CrossRef]

Kanaev, A.

Karlsson, M.

Kogelnik, H.

Köprülü, K. G.

Kumar, P.

Kylemark, P.

Lasri, J.

Levandovsky, D.

Levenson, J. A.

J. A. Levenson, K. Bencheikh, D. J. Lovering, P. Vidakovic, and C. Simonneau, “Quantum noise in optical parametric amplification: a means to achieve noiseless optical functions,” Quantum Semiclass. Opt. 9(2), 221–237 (1997).
[CrossRef]

J. A. Levenson, I. Abram, Th. Rivera, and P. Grangier, “Reduction of quantum noise in optical parametric amplification,” J. Opt. Soc. Am. B 10(11), 2233–2238 (1993).
[CrossRef]

Loudon, R.

R. Loudon, “Theory of noise accumulation in linear optical-amplifier chains,” IEEE J. Quantum Electron. 21(7), 766–773 (1985).
[CrossRef]

Lovering, D. J.

J. A. Levenson, K. Bencheikh, D. J. Lovering, P. Vidakovic, and C. Simonneau, “Quantum noise in optical parametric amplification: a means to achieve noiseless optical functions,” Quantum Semiclass. Opt. 9(2), 221–237 (1997).
[CrossRef]

Lundström, C.

Z. Tong, A. Bogris, C. Lundström, C. J. McKinstrie, M. Vasilyev, M. Karlsson, and P. A. Andrekson, “Modeling and measurement of noise figure in a cascaded non-degenerate phase-sensitive parametric amplifier,” to appear in Opt. Express .

McKinstrie, C. J.

C. J. McKinstrie, S. Radic, R. M. Jopson, and A. R. Chraplyvy, “Quantum noise limits on optical monitoring with parametric devices,” Opt. Commun. 259(1), 309–320 (2006).
[CrossRef]

C. J. McKinstrie, M. G. Raymer, S. Radic, and M. Vasilyev, “Quantum mechanics of phase-sensitive amplification in a fiber,” Opt. Commun. 257(1), 146–163 (2006).
[CrossRef]

C. J. McKinstrie, M. Yu, M. G. Raymer, and S. Radic, “Quantum noise properties of parametric processes,” Opt. Express 13(13), 4986–5012 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?uri=OE-13-13-4986 .
[CrossRef] [PubMed]

C. J. McKinstrie, H. Kogelnik, R. Jopson, S. Radic, and A. Kanaev, “Four-wave mixing in fibers with random birefringence,” Opt. Express 12(10), 2033–2055 (2004), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-10-2033 .
[CrossRef] [PubMed]

Z. Tong, A. Bogris, C. Lundström, C. J. McKinstrie, M. Vasilyev, M. Karlsson, and P. A. Andrekson, “Modeling and measurement of noise figure in a cascaded non-degenerate phase-sensitive parametric amplifier,” to appear in Opt. Express .

Mecozzi, A.

A. Mecozzi and P. Tombesi, “Parametric amplification and signal-to-noise ratio in optical transmission lines,” Opt. Commun. 75(3-4), 256–262 (1990).
[CrossRef]

Movassaghi, M.

Olsson, N. A.

N. A. Olsson, “Lightwave systems with optical amplifiers,” J. Lightwave Technol. 7(7), 1071–1082 (1989).
[CrossRef]

Radic, S.

C. J. McKinstrie, S. Radic, R. M. Jopson, and A. R. Chraplyvy, “Quantum noise limits on optical monitoring with parametric devices,” Opt. Commun. 259(1), 309–320 (2006).
[CrossRef]

C. J. McKinstrie, M. G. Raymer, S. Radic, and M. Vasilyev, “Quantum mechanics of phase-sensitive amplification in a fiber,” Opt. Commun. 257(1), 146–163 (2006).
[CrossRef]

C. J. McKinstrie, M. Yu, M. G. Raymer, and S. Radic, “Quantum noise properties of parametric processes,” Opt. Express 13(13), 4986–5012 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?uri=OE-13-13-4986 .
[CrossRef] [PubMed]

C. J. McKinstrie, H. Kogelnik, R. Jopson, S. Radic, and A. Kanaev, “Four-wave mixing in fibers with random birefringence,” Opt. Express 12(10), 2033–2055 (2004), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-10-2033 .
[CrossRef] [PubMed]

Raymer, M. G.

C. J. McKinstrie, M. G. Raymer, S. Radic, and M. Vasilyev, “Quantum mechanics of phase-sensitive amplification in a fiber,” Opt. Commun. 257(1), 146–163 (2006).
[CrossRef]

C. J. McKinstrie, M. Yu, M. G. Raymer, and S. Radic, “Quantum noise properties of parametric processes,” Opt. Express 13(13), 4986–5012 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?uri=OE-13-13-4986 .
[CrossRef] [PubMed]

Rivera, Th.

Simonneau, C.

J. A. Levenson, K. Bencheikh, D. J. Lovering, P. Vidakovic, and C. Simonneau, “Quantum noise in optical parametric amplification: a means to achieve noiseless optical functions,” Quantum Semiclass. Opt. 9(2), 221–237 (1997).
[CrossRef]

Smith, V. M.

Sunnerud, H.

Takada, A.

W. Imajuku, A. Takada, and Y. Yamabayashi, “Low-noise amplification under the 3dB noise figure in high-gain phase-sensitive fibre amplifier,” Electron. Lett. 35(22), 1954–1955 (1999).
[CrossRef]

A. Takada and W. Imajuku, “In-line optical phase-sensitive amplifier employing pump laser injection locked to input signal light,” Electron. Lett. 34(3), 274–276 (1998).
[CrossRef]

W. Imajuku and A. Takada, “Error-free operation of in-line phase-sensitive amplifier,” Electron. Lett. 34(17), 1673–1674 (1998).
[CrossRef]

W. Imajuku and A. Takada, “Theoretical analysis of system limitation for AM-DD/NRZ optical transmission systems using in-line phase-sensitive amplifiers,” J. Lightwave Technol. 16(7), 1158–1170 (1998).
[CrossRef]

Tang, R.

Tombesi, P.

A. Mecozzi and P. Tombesi, “Parametric amplification and signal-to-noise ratio in optical transmission lines,” Opt. Commun. 75(3-4), 256–262 (1990).
[CrossRef]

Tong, Z.

Z. Tong, A. Bogris, M. Karlsson, and P. A. Andrekson, “Full characterization of the signal and idler noise figure spectra in single-pumped fiber optical parametric amplifiers,” Opt. Express 18(3), 2884–2893 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-3-2884 .
[CrossRef] [PubMed]

Z. Tong, A. Bogris, C. Lundström, C. J. McKinstrie, M. Vasilyev, M. Karlsson, and P. A. Andrekson, “Modeling and measurement of noise figure in a cascaded non-degenerate phase-sensitive parametric amplifier,” to appear in Opt. Express .

Torounidis, T.

T. Torounidis and P. Andrekson, “Broadband single-pumped fiber-optic parametric amplifiers,” IEEE Photon. Technol. Lett. 19(9), 650–652 (2007).
[CrossRef]

Vasilyev, M.

Vasilyev, M. V.

Vidakovic, P.

J. A. Levenson, K. Bencheikh, D. J. Lovering, P. Vidakovic, and C. Simonneau, “Quantum noise in optical parametric amplification: a means to achieve noiseless optical functions,” Quantum Semiclass. Opt. 9(2), 221–237 (1997).
[CrossRef]

Voss, P. L.

P. L. Voss, K. G. Köprülü, and P. Kumar, “Raman-noise-induced quantum limits for χ(3) nondegenerate phase-sensitive amplification and quadrature squeezing,” J. Opt. Soc. Am. B 23(4), 598–610 (2006).
[CrossRef]

P. L. Voss and P. Kumar, “Raman-effect induced noise limits on χ(3) parametric amplifiers and wavelength converters,” J. Opt. B Quantum Semiclassical Opt. 6(8), S762–S770 (2004).
[CrossRef]

Yamabayashi, Y.

W. Imajuku, A. Takada, and Y. Yamabayashi, “Low-noise amplification under the 3dB noise figure in high-gain phase-sensitive fibre amplifier,” Electron. Lett. 35(22), 1954–1955 (1999).
[CrossRef]

Yu, M.

Zhou, G.

Electron. Lett. (4)

R. Tang, P. Devgan, V. S. Grigoryan, and P. Kumar, “Inline frequency-non-degenerate phase-sensitive fibre parametric amplifier for fibre-optic communication,” Electron. Lett. 41(19), 1072–1074 (2005).
[CrossRef]

W. Imajuku, A. Takada, and Y. Yamabayashi, “Low-noise amplification under the 3dB noise figure in high-gain phase-sensitive fibre amplifier,” Electron. Lett. 35(22), 1954–1955 (1999).
[CrossRef]

W. Imajuku and A. Takada, “Error-free operation of in-line phase-sensitive amplifier,” Electron. Lett. 34(17), 1673–1674 (1998).
[CrossRef]

A. Takada and W. Imajuku, “In-line optical phase-sensitive amplifier employing pump laser injection locked to input signal light,” Electron. Lett. 34(3), 274–276 (1998).
[CrossRef]

IEEE J. Quantum Electron. (1)

R. Loudon, “Theory of noise accumulation in linear optical-amplifier chains,” IEEE J. Quantum Electron. 21(7), 766–773 (1985).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

T. Torounidis and P. Andrekson, “Broadband single-pumped fiber-optic parametric amplifiers,” IEEE Photon. Technol. Lett. 19(9), 650–652 (2007).
[CrossRef]

J. Lightwave Technol. (4)

J. Opt. B Quantum Semiclassical Opt. (1)

P. L. Voss and P. Kumar, “Raman-effect induced noise limits on χ(3) parametric amplifiers and wavelength converters,” J. Opt. B Quantum Semiclassical Opt. 6(8), S762–S770 (2004).
[CrossRef]

J. Opt. Soc. Am. B (2)

Opt. Commun. (3)

A. Mecozzi and P. Tombesi, “Parametric amplification and signal-to-noise ratio in optical transmission lines,” Opt. Commun. 75(3-4), 256–262 (1990).
[CrossRef]

C. J. McKinstrie, S. Radic, R. M. Jopson, and A. R. Chraplyvy, “Quantum noise limits on optical monitoring with parametric devices,” Opt. Commun. 259(1), 309–320 (2006).
[CrossRef]

C. J. McKinstrie, M. G. Raymer, S. Radic, and M. Vasilyev, “Quantum mechanics of phase-sensitive amplification in a fiber,” Opt. Commun. 257(1), 146–163 (2006).
[CrossRef]

Opt. Express (8)

G. Zhou and F. S. Chau, “Nondispersive optical phase shifter array using microelectromechanical systems based gratings,” Opt. Express 15(17), 10958–10963 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-17-10958 .
[CrossRef] [PubMed]

R. Tang, P. S. Devgan, V. S. Grigoryan, P. Kumar, and M. Vasilyev, “In-line phase-sensitive amplification of multi-channel CW signals based on frequency nondegenerate four-wave-mixing in fiber,” Opt. Express 16(12), 9046–9053 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-12-9046 .
[CrossRef] [PubMed]

Z. Tong, A. Bogris, M. Karlsson, and P. A. Andrekson, “Full characterization of the signal and idler noise figure spectra in single-pumped fiber optical parametric amplifiers,” Opt. Express 18(3), 2884–2893 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-3-2884 .
[CrossRef] [PubMed]

C. J. McKinstrie, H. Kogelnik, R. Jopson, S. Radic, and A. Kanaev, “Four-wave mixing in fibers with random birefringence,” Opt. Express 12(10), 2033–2055 (2004), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-10-2033 .
[CrossRef] [PubMed]

C. J. McKinstrie, M. Yu, M. G. Raymer, and S. Radic, “Quantum noise properties of parametric processes,” Opt. Express 13(13), 4986–5012 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?uri=OE-13-13-4986 .
[CrossRef] [PubMed]

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[CrossRef] [PubMed]

Z. Tong, A. Bogris, C. Lundström, C. J. McKinstrie, M. Vasilyev, M. Karlsson, and P. A. Andrekson, “Modeling and measurement of noise figure in a cascaded non-degenerate phase-sensitive parametric amplifier,” to appear in Opt. Express .

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Other (7)

M. V. Vasilyev, “Phase-sensitive amplification in optical fibers,” in Frontiers in Optics, Technical Digest (Optical Society of America, 2005), paper FThB1.

Z. Tong, A. Bogris, C. Lundström, C. J. McKinstrie, M. Vasilyev, M. Karlsson, and P. A. Andrekson, “Noise figure measurements in phase-insensitive and phase-sensitive fiber parametric amplifier cascade,” in Optical Fiber Communications Conference, paper OWT4 (2010).

M. E. Marhic, Fiber optical parametric amplifiers, oscillators and related devices (Cambridge University, 2007), Chap. 14.

E. Desuvire, Erbium-doped fiber amplifiers (John Wiley and Sons, 1994), Chap. 2.

C. Lundström, J. Kakande, P. A. Andrekson, Z. Tong, M. Karlsson, P. Petropoulos, F. Parmigiani, and D. J. Richardson, “Experimental comparison of gain and saturation characteristics of a parametric amplifier in phase-sensitive and phase-insensitive mode,” in European Conference on Optical Communications, paper Mo. 1.1.1 (2009).

R. Weerasuriya, S. Sygletos, S. K. Ibrahim, R. Phelan, J. O’Carroll, B. Kelly, J. O’Gorman, and A. D. Ellis, “Generation of frequency symmetric signals from a BPSK input for phase sensitive amplification,” in Optical Fiber Communications Conference, paper OWT6 (2010).

G. Baxter, S. Frisken, D. Abakoumov, H. Zhou, I. Clarke, A. Bartos, and S. Poole, “Highly programmable wavelength selective switch based on liquid crystal on silicon switching elements,” in Optical Fiber Communication Conference, paper OTuF2 (2009).

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Figures (5)

Fig. 1
Fig. 1

Schematic of a non-degenerate PSA with loss and phase-shifter.

Fig. 2
Fig. 2

(a) Type A and (b) Type B copier + PSA amplified multi-span transmission link. DPC/PC: Dispersion and polarization compensation / Phase shifter; OIL: Optical injection-locked laser; PLL: Phase locking loop.

Fig. 3
Fig. 3

Calculated (a) Type A link NF and (b) Type B link NF vs. the span number for the cascaded copier + PSA, pure-PIA and pure-PSA transmission systems at different link gains. Black circles shows the approximate results based on Eqs. (14) and (20), respectively.

Fig. 4
Fig. 4

Experimental setup. NFA: Noise figure analyzer; ESA: Electrical spectrum analyzer; OSA: Optical spectrum analyzer; PC: Polarization controller; VOA: Variable optical attenuator. Inset (a) shows the principal scheme and inset (b) shows the output optical spectrum.

Fig. 5
Fig. 5

Measured PSA gain and NF spectra of different cascaded amplification cases.

Equations (38)

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[ A s A i * ] = [ μ ν ν * μ * ] [ A s 0 + n s A i 0 * + n i * ] = G [ A s 0 + n s A i 0 * + n i * ] ,
[ A s A i * ] = T ^ [ A s 0 + n s A i 0 * + n i * ] ,
T ^ A = [ T s 0 0 T i ] A + [ 1 T s n s ' 1 T i n i * ' ] ,
[ A s A i * ] = [ e j ϕ s 0 0 e j ϕ i ] [ A s 0 + n s A i 0 * + n i * ] = D [ A s 0 + n s A i 0 * + n i * ] ,
N F = S N R i n S N R o u t = < I i n > 2 / < Δ I i n 2 > < I o u t > 2 / < Δ I o u t 2 > ,
[ A s , 1 A i , 1 * ] = L ^ A , 1 [ A s 0 + n s n i * ] = [ T s , 1 μ 1 e j ϕ s , 1 T s , 1 ν 1 e j ϕ s , 1 T i , 1 ν 1 * e j ϕ i , 1 T i , 1 μ 1 * e j ϕ i , 1 ] [ A s 0 + n s n i * ] + [ 1 T s , 1 n s , 1 ' e j ϕ s , 1 1 T i , 1 n i , 1 * ' e j ϕ i , 1 ] ,
< I o u t , 1 > = R 0 G T | A s 0 | 2 B o = R 0 | A s 0 | 2 B o , < Δ I o u t , 1 2 > = 4 R 0 2 | A s 0 | 2 [ 2 T ( G 1 ) + 1 ) ] h v Δ f / 2.
[ A s , 2 A i , 2 * ] = L ^ A , 2 L ^ A , 1 [ A s 0 + n s n i * ] .
G s , 2 ( 2 n π ) = G i , 2 ( 2 n π ) = ( | μ 2 | + | ν 2 | ) 2 4 | μ 2 | 2 ,
< I o u t , 2 > = R 0 ( G T ) 2 | A s 0 | 2 B o = R 0 | A s 0 | 2 B o , < Δ I o u t , 2 2 > = 4 R 0 2 | A s 0 | 2 [ 3 + ( 2 | μ 2 | 2 1 ) T 2 | μ 2 | 2 T 2 T ] h v Δ f / 2.
[ A s , m A i , m * ] = L ^ A , m L ^ A , 2 L ^ A , 1 [ A s 0 + n s n i * ] .
φ μ , l 1 + φ μ , l + φ ν , l 1 φ ν , l + ϕ s , l 1 + ϕ i , l 1 = 2 n π ,   n = 0 ,   1 ,   2 ,   ... ,
[ A s , l A i , l * ] = | L ^ A , l | | L ^ A , l 1 | | L ^ A , 2 | | L ^ A , 1 | [ A s 0 + n s n i * ] ,
| L ^ A , l | [ A s + n s A i * + n i * ] = T [ | μ 2 | | ν 2 | | ν 2 | | μ 2 | ] | L ^ A , l 1 | | L ^ A , 2 | | L ^ A , 1 | [ A s 0 + n s A i * + n i * ] + 1 T [ n s , l ' n i , l * ' ] ,
< I o u t , m > = R 0 ( G T ) m | A s 0 | 2 B o = R 0 | A s 0 | 2 B o , < Δ I o u t , m 2 > 4 R 0 2 | A s 0 | 2 [ 3 T + i = 2 m ( G i 1 + T i 1 2 ) T i 1 ( 1 T ) ] h v Δ f / 2.
N F A , m 5 2 + m 2 .
[ A s , 1 A i , 1 * ] = L ^ B , 1 [ A s 0 + n s n i * ] = [ T s , 1 μ 1 T s , 1 ν 1 T i , 1 ν 1 * T i , 1 μ 1 * ] [ A s 0 + n s n i * ] + [ 1 T s , 1 μ 1 n s , 1 ' + 1 T i , 1 ν 1 n i , 1 * ' 1 T s , 1 ν 1 * n s , 1 ' + 1 T i , 1 μ 1 * n i , 1 * ' ] ,
< I o u t , 1 > = R 0 G T | A s 0 | 2 B o = R 0 | A s 0 | 2 B o , < Δ I o u t , 1 2 > = 4 R 0 2 G T | A s 0 | 2 [ T ( 2 G 1 ) + ( 1 T ) ( 2 G 1 ) ] h v Δ f / 2.
[ A s , 2 A i , 2 * ] = L ^ B , 2 L ^ B , 1 [ A s 0 + n s n i * ] .
< I o u t , 2 > = R 0 ( G T ) 2 | A s 0 | 2 B o = R 0 | A s 0 | 2 B o , < Δ I o u t , 2 2 > = 4 R 0 2 G T | A s 0 | 2 [ 2 G + 2 | μ 2 | 2 2 | μ 2 | 2 T 1 ] h v Δ f / 2.
< I o u t , m > = R 0 ( G T ) m | A s 0 | 2 B o = R 0 | A s 0 | 2 B o , < Δ I o u t , m 2 > 4 R 0 2 | A s 0 | 2 [ 2 G + i = 2 m ( G i 1 + T i 1 2 ) T i 2 ( 1 T ) ] h v Δ f / 2.
N F B , m 3 G 2 + m G 2 .
N F A , m = 1 + 2 m ( 1 1 / G ) ,   N F B , m = 1 + 2 m G ( 1 1 / G ) ,
N F A , m = 1 + m ( 1 1 / G ) ,   N F B , m = 1 + m G ( 1 1 / G ) ,
N F = 1 G + P i n ( S o u t S i n ) 2 h v I o u t 2 ,
K ^ [ A s 0 + n s A i 0 * + n i * ] = [ 1 0 0 0 ] [ A s 0 + n s A i 0 * + n i * ] + [ 0 n i * ' ] ,
[ A s , m A i , m * ] = L ^ A , P I A m L ^ A , P I A 2 L ^ A , P I A 1 [ A s 0 + n s n i * ] , [ A s , m A i , m * ] = L ^ B , P I A m L ^ B , P I A 2 L ^ B , P I A 1 [ A s 0 + n s n i * ] .
< Δ I o u t , m 2 > = 4 R 0 2 | A s 0 | 2 [ 1 + i = 1 m 2 ( 1 T ) ] h v Δ f / 2 ,
< Δ I o u t , m 2 > = 4 R 0 2 | A s 0 | 2 [ 1 + i = 1 m 2 G ( 1 T ) ] h v Δ f / 2 ,
[ A s , m A s , m * ] = L ^ A , P S A m L ^ A , P S A 2 L ^ A , P S A 1 [ A s 0 + n s A s 0 * + n s * ] ,
[ A s , m A s , m * ] = L ^ B , P S A m L ^ B , P S A 2 L ^ B , P S A 1 [ A s 0 + n s A s 0 * + n s * ] ,
< Δ I o u t , m 2 > = 4 R 0 2 | A s 0 | 2 [ 1 + i = 1 m ( 1 T ) ] h v Δ f / 2 ,
< Δ I o u t , m 2 > = 4 R 0 2 | A s 0 | 2 [ 1 + i = 1 m G ( 1 T ) ] h v Δ f / 2 ,
[ A s , m A i , m * ] = L ^ A , P S A m L ^ A , P S A 2 L ^ A , P S A 1 [ A s 0 + n s A i 0 * + n i * ] ,
[ A s , m A i , m * ] = L ^ B , P S A m L ^ B , P S A 2 L ^ B , P S A 1 [ A s 0 + n s A i 0 * + n i * ] ,
< Δ I o u t , m 2 > = 4 R 0 2 | A s 0 | 2 [ 1 T + ( G m + T m 2 ) T m + i = 1 m 1 ( G i + T i 2 ) T i ( 1 T ) ] h v Δ f / 2 ,
< Δ I o u t , m 2 > = 4 R 0 2 | A s 0 | 2 [ ( G m + T m 2 ) T m + i = 1 m ( G i + T i 2 ) T i 1 ( 1 T ) ] h v Δ f / 2 ,
N F A s , m m + 2 ,   N F B s , m m G .

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