Abstract

The Heisenberg equations of motion are used to calculate the phase noise of inverted-population-type traveling-wave laser amplifiers generally and semiconductor amplifiers specifically. Agreement with previously reported results in the limit of perfect inversion is demonstrated.

© 1992 Optical Society of America

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References

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  1. Y. Yamamoto, “Noise and error rate performance of semiconductor laser amplifiers in PCM–IM optical transmission systems,” IEEE J. Quantum. Electron. QE-16, 1073 (1980).
    [CrossRef]
  2. M. J. O’Mahony, “Semiconductor laser optical amplifiers for use in future fiber systems,” IEEE J. Lightwave Technol. 6, 531 (1988).
    [CrossRef]
  3. N. A. Olsson, “Lightwave systems with optical amplifiers,” IEEE J. Lightwave Technol. 7, 1071 (1989).
    [CrossRef]
  4. M. Gustavsson, A. Karlsson, L. Thylén, “Traveling wave semiconductor laser amplifier detectors,” IEEE J. Lightwave Technol. 8, 610 (1990).
    [CrossRef]
  5. M. Ikeda, O. Ohguchi, K. Yoshino, “Monolithic laser diode optical matrix switches,” in Proceedings of the Thirteenth European Conference on Optical Communications (Institute of Electrical and Electronics Engineers, New York, 1987), p. 227.
  6. J. Mellis, “Optical phase modulation using semiconductor laser amplifiers,” Trans. Inst. Electron. Inf. Commun. Eng. E73, 36 (1989); G. Grosskopf, R. Ludwig, R. Schnabel, H. G. Weber, “Semiconductor laser optical amplifier as phase modulator in a 140 Mb/s transmission experiment,” Trans. Inst. Electron. Inf. Commun. Eng. E73, 34 (1989).
  7. K. Schimoda, H. Takahashi, C. H. Townes, “Fluctuations in amplification of quanta with application to maser amplifiers,” J. Phys. Soc. Jpn. 12, 686 (1957).
    [CrossRef]
  8. L. Thylén, M. Gustavsson, T. K. Gustafson, I. Kim, A. Karlsson, “Calculation of photon and current fluctuations in traveling-wave semiconductor laser amplifiers,” IEEE J. Quantum Electron. 27, 1251 (1991).
    [CrossRef]
  9. R. Serber, C. H. Townes, “Limits on electromagnetic amplification due to complementarity,” in Advances in Quantum Electronics, J. R. Singer, ed. (Columbia U. Press, New York, 1961), p. 233.
  10. Y. Yamamoto, H. A. Haus, “Preparation, measurement and information capacity of optical quantum states,” Rev. Mod. Phys. 58, 1001 (1986).
    [CrossRef]
  11. P. A. M. Dirac, Proc. R. Soc. London Ser. A 114, 243 (1927).
    [CrossRef]
  12. L. Susskind, J. Glogower, Physics 1, 49 (1964).
  13. S. M. Barnett, D. T. Pegg, “Phase in quantum optics,” J. Phys. A 19, 3849 (1986).
    [CrossRef]
  14. D. T. Pegg, S. M. Barnett, “Phase properties of the quantized single-mode electromagnetic field,” Phys. Rev. A 39, 1665 (1989).
    [CrossRef] [PubMed]
  15. R. Loudon, The Quantum Theory of Light(Clarendon, Oxford, 1983).
  16. K. Hinton, “Optical carrier linewidth broadening in a traveling wave semiconductor laser amplifier,” IEEE J. Quantum Electron. 26, 1176 (1990).
    [CrossRef]
  17. Y. Yamamoto, S. Machida, “High-impedance suppression of pump fluctuation and amplitude squeezing in semiconductor lasers,” Phys. Rev. A 35, 5114 (1987).
    [CrossRef] [PubMed]
  18. After submission of this paper, a paper by K. Kikuchi, C.-E. Zah, T.-P. Lee [“Measurement and analysis of phase noise generated from semiconductor optical amplifiers,” IEEE J. Quantum Electron. 27, 416 (1991)] was drawn to our attention; in their paper phase noise in semiconductor laser amplifiers is also calculated, and, in addition to the noise sources described in the present paper, phase noise that is due to carrier fluctuations caused by beating between the signal and spontaneous emission is included. In our paper this noise source was not included, since it will not be appreciable for input powers below ≈0.1 μ W. The same comment applies to phase noise that is due to carrier fluctuations caused by total spontaneous recombination, as stated in the present paper. It is possible to include beat-noise-generated phase noise by using the analysis of Ref. 8.
    [CrossRef]

1991 (2)

L. Thylén, M. Gustavsson, T. K. Gustafson, I. Kim, A. Karlsson, “Calculation of photon and current fluctuations in traveling-wave semiconductor laser amplifiers,” IEEE J. Quantum Electron. 27, 1251 (1991).
[CrossRef]

After submission of this paper, a paper by K. Kikuchi, C.-E. Zah, T.-P. Lee [“Measurement and analysis of phase noise generated from semiconductor optical amplifiers,” IEEE J. Quantum Electron. 27, 416 (1991)] was drawn to our attention; in their paper phase noise in semiconductor laser amplifiers is also calculated, and, in addition to the noise sources described in the present paper, phase noise that is due to carrier fluctuations caused by beating between the signal and spontaneous emission is included. In our paper this noise source was not included, since it will not be appreciable for input powers below ≈0.1 μ W. The same comment applies to phase noise that is due to carrier fluctuations caused by total spontaneous recombination, as stated in the present paper. It is possible to include beat-noise-generated phase noise by using the analysis of Ref. 8.
[CrossRef]

1990 (2)

K. Hinton, “Optical carrier linewidth broadening in a traveling wave semiconductor laser amplifier,” IEEE J. Quantum Electron. 26, 1176 (1990).
[CrossRef]

M. Gustavsson, A. Karlsson, L. Thylén, “Traveling wave semiconductor laser amplifier detectors,” IEEE J. Lightwave Technol. 8, 610 (1990).
[CrossRef]

1989 (3)

J. Mellis, “Optical phase modulation using semiconductor laser amplifiers,” Trans. Inst. Electron. Inf. Commun. Eng. E73, 36 (1989); G. Grosskopf, R. Ludwig, R. Schnabel, H. G. Weber, “Semiconductor laser optical amplifier as phase modulator in a 140 Mb/s transmission experiment,” Trans. Inst. Electron. Inf. Commun. Eng. E73, 34 (1989).

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

D. T. Pegg, S. M. Barnett, “Phase properties of the quantized single-mode electromagnetic field,” Phys. Rev. A 39, 1665 (1989).
[CrossRef] [PubMed]

1988 (1)

M. J. O’Mahony, “Semiconductor laser optical amplifiers for use in future fiber systems,” IEEE J. Lightwave Technol. 6, 531 (1988).
[CrossRef]

1987 (1)

Y. Yamamoto, S. Machida, “High-impedance suppression of pump fluctuation and amplitude squeezing in semiconductor lasers,” Phys. Rev. A 35, 5114 (1987).
[CrossRef] [PubMed]

1986 (2)

S. M. Barnett, D. T. Pegg, “Phase in quantum optics,” J. Phys. A 19, 3849 (1986).
[CrossRef]

Y. Yamamoto, H. A. Haus, “Preparation, measurement and information capacity of optical quantum states,” Rev. Mod. Phys. 58, 1001 (1986).
[CrossRef]

1980 (1)

Y. Yamamoto, “Noise and error rate performance of semiconductor laser amplifiers in PCM–IM optical transmission systems,” IEEE J. Quantum. Electron. QE-16, 1073 (1980).
[CrossRef]

1964 (1)

L. Susskind, J. Glogower, Physics 1, 49 (1964).

1957 (1)

K. Schimoda, H. Takahashi, C. H. Townes, “Fluctuations in amplification of quanta with application to maser amplifiers,” J. Phys. Soc. Jpn. 12, 686 (1957).
[CrossRef]

1927 (1)

P. A. M. Dirac, Proc. R. Soc. London Ser. A 114, 243 (1927).
[CrossRef]

Barnett, S. M.

D. T. Pegg, S. M. Barnett, “Phase properties of the quantized single-mode electromagnetic field,” Phys. Rev. A 39, 1665 (1989).
[CrossRef] [PubMed]

S. M. Barnett, D. T. Pegg, “Phase in quantum optics,” J. Phys. A 19, 3849 (1986).
[CrossRef]

Dirac, P. A. M.

P. A. M. Dirac, Proc. R. Soc. London Ser. A 114, 243 (1927).
[CrossRef]

Glogower, J.

L. Susskind, J. Glogower, Physics 1, 49 (1964).

Gustafson, T. K.

L. Thylén, M. Gustavsson, T. K. Gustafson, I. Kim, A. Karlsson, “Calculation of photon and current fluctuations in traveling-wave semiconductor laser amplifiers,” IEEE J. Quantum Electron. 27, 1251 (1991).
[CrossRef]

Gustavsson, M.

L. Thylén, M. Gustavsson, T. K. Gustafson, I. Kim, A. Karlsson, “Calculation of photon and current fluctuations in traveling-wave semiconductor laser amplifiers,” IEEE J. Quantum Electron. 27, 1251 (1991).
[CrossRef]

M. Gustavsson, A. Karlsson, L. Thylén, “Traveling wave semiconductor laser amplifier detectors,” IEEE J. Lightwave Technol. 8, 610 (1990).
[CrossRef]

Haus, H. A.

Y. Yamamoto, H. A. Haus, “Preparation, measurement and information capacity of optical quantum states,” Rev. Mod. Phys. 58, 1001 (1986).
[CrossRef]

Hinton, K.

K. Hinton, “Optical carrier linewidth broadening in a traveling wave semiconductor laser amplifier,” IEEE J. Quantum Electron. 26, 1176 (1990).
[CrossRef]

Ikeda, M.

M. Ikeda, O. Ohguchi, K. Yoshino, “Monolithic laser diode optical matrix switches,” in Proceedings of the Thirteenth European Conference on Optical Communications (Institute of Electrical and Electronics Engineers, New York, 1987), p. 227.

Karlsson, A.

L. Thylén, M. Gustavsson, T. K. Gustafson, I. Kim, A. Karlsson, “Calculation of photon and current fluctuations in traveling-wave semiconductor laser amplifiers,” IEEE J. Quantum Electron. 27, 1251 (1991).
[CrossRef]

M. Gustavsson, A. Karlsson, L. Thylén, “Traveling wave semiconductor laser amplifier detectors,” IEEE J. Lightwave Technol. 8, 610 (1990).
[CrossRef]

Kikuchi, K.

After submission of this paper, a paper by K. Kikuchi, C.-E. Zah, T.-P. Lee [“Measurement and analysis of phase noise generated from semiconductor optical amplifiers,” IEEE J. Quantum Electron. 27, 416 (1991)] was drawn to our attention; in their paper phase noise in semiconductor laser amplifiers is also calculated, and, in addition to the noise sources described in the present paper, phase noise that is due to carrier fluctuations caused by beating between the signal and spontaneous emission is included. In our paper this noise source was not included, since it will not be appreciable for input powers below ≈0.1 μ W. The same comment applies to phase noise that is due to carrier fluctuations caused by total spontaneous recombination, as stated in the present paper. It is possible to include beat-noise-generated phase noise by using the analysis of Ref. 8.
[CrossRef]

Kim, I.

L. Thylén, M. Gustavsson, T. K. Gustafson, I. Kim, A. Karlsson, “Calculation of photon and current fluctuations in traveling-wave semiconductor laser amplifiers,” IEEE J. Quantum Electron. 27, 1251 (1991).
[CrossRef]

Lee, T.-P.

After submission of this paper, a paper by K. Kikuchi, C.-E. Zah, T.-P. Lee [“Measurement and analysis of phase noise generated from semiconductor optical amplifiers,” IEEE J. Quantum Electron. 27, 416 (1991)] was drawn to our attention; in their paper phase noise in semiconductor laser amplifiers is also calculated, and, in addition to the noise sources described in the present paper, phase noise that is due to carrier fluctuations caused by beating between the signal and spontaneous emission is included. In our paper this noise source was not included, since it will not be appreciable for input powers below ≈0.1 μ W. The same comment applies to phase noise that is due to carrier fluctuations caused by total spontaneous recombination, as stated in the present paper. It is possible to include beat-noise-generated phase noise by using the analysis of Ref. 8.
[CrossRef]

Loudon, R.

R. Loudon, The Quantum Theory of Light(Clarendon, Oxford, 1983).

Machida, S.

Y. Yamamoto, S. Machida, “High-impedance suppression of pump fluctuation and amplitude squeezing in semiconductor lasers,” Phys. Rev. A 35, 5114 (1987).
[CrossRef] [PubMed]

Mellis, J.

J. Mellis, “Optical phase modulation using semiconductor laser amplifiers,” Trans. Inst. Electron. Inf. Commun. Eng. E73, 36 (1989); G. Grosskopf, R. Ludwig, R. Schnabel, H. G. Weber, “Semiconductor laser optical amplifier as phase modulator in a 140 Mb/s transmission experiment,” Trans. Inst. Electron. Inf. Commun. Eng. E73, 34 (1989).

O’Mahony, M. J.

M. J. O’Mahony, “Semiconductor laser optical amplifiers for use in future fiber systems,” IEEE J. Lightwave Technol. 6, 531 (1988).
[CrossRef]

Ohguchi, O.

M. Ikeda, O. Ohguchi, K. Yoshino, “Monolithic laser diode optical matrix switches,” in Proceedings of the Thirteenth European Conference on Optical Communications (Institute of Electrical and Electronics Engineers, New York, 1987), p. 227.

Olsson, N. A.

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

Pegg, D. T.

D. T. Pegg, S. M. Barnett, “Phase properties of the quantized single-mode electromagnetic field,” Phys. Rev. A 39, 1665 (1989).
[CrossRef] [PubMed]

S. M. Barnett, D. T. Pegg, “Phase in quantum optics,” J. Phys. A 19, 3849 (1986).
[CrossRef]

Schimoda, K.

K. Schimoda, H. Takahashi, C. H. Townes, “Fluctuations in amplification of quanta with application to maser amplifiers,” J. Phys. Soc. Jpn. 12, 686 (1957).
[CrossRef]

Serber, R.

R. Serber, C. H. Townes, “Limits on electromagnetic amplification due to complementarity,” in Advances in Quantum Electronics, J. R. Singer, ed. (Columbia U. Press, New York, 1961), p. 233.

Susskind, L.

L. Susskind, J. Glogower, Physics 1, 49 (1964).

Takahashi, H.

K. Schimoda, H. Takahashi, C. H. Townes, “Fluctuations in amplification of quanta with application to maser amplifiers,” J. Phys. Soc. Jpn. 12, 686 (1957).
[CrossRef]

Thylén, L.

L. Thylén, M. Gustavsson, T. K. Gustafson, I. Kim, A. Karlsson, “Calculation of photon and current fluctuations in traveling-wave semiconductor laser amplifiers,” IEEE J. Quantum Electron. 27, 1251 (1991).
[CrossRef]

M. Gustavsson, A. Karlsson, L. Thylén, “Traveling wave semiconductor laser amplifier detectors,” IEEE J. Lightwave Technol. 8, 610 (1990).
[CrossRef]

Townes, C. H.

K. Schimoda, H. Takahashi, C. H. Townes, “Fluctuations in amplification of quanta with application to maser amplifiers,” J. Phys. Soc. Jpn. 12, 686 (1957).
[CrossRef]

R. Serber, C. H. Townes, “Limits on electromagnetic amplification due to complementarity,” in Advances in Quantum Electronics, J. R. Singer, ed. (Columbia U. Press, New York, 1961), p. 233.

Yamamoto, Y.

Y. Yamamoto, S. Machida, “High-impedance suppression of pump fluctuation and amplitude squeezing in semiconductor lasers,” Phys. Rev. A 35, 5114 (1987).
[CrossRef] [PubMed]

Y. Yamamoto, H. A. Haus, “Preparation, measurement and information capacity of optical quantum states,” Rev. Mod. Phys. 58, 1001 (1986).
[CrossRef]

Y. Yamamoto, “Noise and error rate performance of semiconductor laser amplifiers in PCM–IM optical transmission systems,” IEEE J. Quantum. Electron. QE-16, 1073 (1980).
[CrossRef]

Yoshino, K.

M. Ikeda, O. Ohguchi, K. Yoshino, “Monolithic laser diode optical matrix switches,” in Proceedings of the Thirteenth European Conference on Optical Communications (Institute of Electrical and Electronics Engineers, New York, 1987), p. 227.

Zah, C.-E.

After submission of this paper, a paper by K. Kikuchi, C.-E. Zah, T.-P. Lee [“Measurement and analysis of phase noise generated from semiconductor optical amplifiers,” IEEE J. Quantum Electron. 27, 416 (1991)] was drawn to our attention; in their paper phase noise in semiconductor laser amplifiers is also calculated, and, in addition to the noise sources described in the present paper, phase noise that is due to carrier fluctuations caused by beating between the signal and spontaneous emission is included. In our paper this noise source was not included, since it will not be appreciable for input powers below ≈0.1 μ W. The same comment applies to phase noise that is due to carrier fluctuations caused by total spontaneous recombination, as stated in the present paper. It is possible to include beat-noise-generated phase noise by using the analysis of Ref. 8.
[CrossRef]

IEEE J. Lightwave Technol. (3)

M. J. O’Mahony, “Semiconductor laser optical amplifiers for use in future fiber systems,” IEEE J. Lightwave Technol. 6, 531 (1988).
[CrossRef]

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

M. Gustavsson, A. Karlsson, L. Thylén, “Traveling wave semiconductor laser amplifier detectors,” IEEE J. Lightwave Technol. 8, 610 (1990).
[CrossRef]

IEEE J. Quantum Electron. (3)

L. Thylén, M. Gustavsson, T. K. Gustafson, I. Kim, A. Karlsson, “Calculation of photon and current fluctuations in traveling-wave semiconductor laser amplifiers,” IEEE J. Quantum Electron. 27, 1251 (1991).
[CrossRef]

K. Hinton, “Optical carrier linewidth broadening in a traveling wave semiconductor laser amplifier,” IEEE J. Quantum Electron. 26, 1176 (1990).
[CrossRef]

After submission of this paper, a paper by K. Kikuchi, C.-E. Zah, T.-P. Lee [“Measurement and analysis of phase noise generated from semiconductor optical amplifiers,” IEEE J. Quantum Electron. 27, 416 (1991)] was drawn to our attention; in their paper phase noise in semiconductor laser amplifiers is also calculated, and, in addition to the noise sources described in the present paper, phase noise that is due to carrier fluctuations caused by beating between the signal and spontaneous emission is included. In our paper this noise source was not included, since it will not be appreciable for input powers below ≈0.1 μ W. The same comment applies to phase noise that is due to carrier fluctuations caused by total spontaneous recombination, as stated in the present paper. It is possible to include beat-noise-generated phase noise by using the analysis of Ref. 8.
[CrossRef]

IEEE J. Quantum. Electron. (1)

Y. Yamamoto, “Noise and error rate performance of semiconductor laser amplifiers in PCM–IM optical transmission systems,” IEEE J. Quantum. Electron. QE-16, 1073 (1980).
[CrossRef]

J. Phys. A (1)

S. M. Barnett, D. T. Pegg, “Phase in quantum optics,” J. Phys. A 19, 3849 (1986).
[CrossRef]

J. Phys. Soc. Jpn. (1)

K. Schimoda, H. Takahashi, C. H. Townes, “Fluctuations in amplification of quanta with application to maser amplifiers,” J. Phys. Soc. Jpn. 12, 686 (1957).
[CrossRef]

Phys. Rev. A (2)

D. T. Pegg, S. M. Barnett, “Phase properties of the quantized single-mode electromagnetic field,” Phys. Rev. A 39, 1665 (1989).
[CrossRef] [PubMed]

Y. Yamamoto, S. Machida, “High-impedance suppression of pump fluctuation and amplitude squeezing in semiconductor lasers,” Phys. Rev. A 35, 5114 (1987).
[CrossRef] [PubMed]

Physics (1)

L. Susskind, J. Glogower, Physics 1, 49 (1964).

Proc. R. Soc. London Ser. A (1)

P. A. M. Dirac, Proc. R. Soc. London Ser. A 114, 243 (1927).
[CrossRef]

Rev. Mod. Phys. (1)

Y. Yamamoto, H. A. Haus, “Preparation, measurement and information capacity of optical quantum states,” Rev. Mod. Phys. 58, 1001 (1986).
[CrossRef]

Trans. Inst. Electron. Inf. Commun. Eng. (1)

J. Mellis, “Optical phase modulation using semiconductor laser amplifiers,” Trans. Inst. Electron. Inf. Commun. Eng. E73, 36 (1989); G. Grosskopf, R. Ludwig, R. Schnabel, H. G. Weber, “Semiconductor laser optical amplifier as phase modulator in a 140 Mb/s transmission experiment,” Trans. Inst. Electron. Inf. Commun. Eng. E73, 34 (1989).

Other (3)

R. Serber, C. H. Townes, “Limits on electromagnetic amplification due to complementarity,” in Advances in Quantum Electronics, J. R. Singer, ed. (Columbia U. Press, New York, 1961), p. 233.

M. Ikeda, O. Ohguchi, K. Yoshino, “Monolithic laser diode optical matrix switches,” in Proceedings of the Thirteenth European Conference on Optical Communications (Institute of Electrical and Electronics Engineers, New York, 1987), p. 227.

R. Loudon, The Quantum Theory of Light(Clarendon, Oxford, 1983).

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Figures (3)

Fig. 1
Fig. 1

Phasor diagram with n ^ ¯ > 1, showing angle Δ Φ ^.

Fig. 2
Fig. 2

Phase noise according to Eq. (21), calculated with the following parameters: amplifier length, 300 μm; confinement factor, 0.5; gain coefficient, 1.3 × 10−12 m3/s; electron density at transparency, 1.6 × 1024 m−3; internal losses, 20 cm−1; λ, 1.55 μm; detection bandwidth, 300 MHz. The four curves represent input powers of (a) −50 dm, (b) −45 dBm, (c) −40 dBm, and (d) −34 dBm. The dotted curve (d) represents the highest input power for which the phase noise, as given by the first term in Eq. (21), dominates the phase-noise contribution that is due to carrier fluctuations caused by total spontaneous recombination.

Fig. 3
Fig. 3

Ratio of the total variance according to Eq. (21) and the first term on the right-hand side of Eq. (21), in decibels, versus gain, for the same parameters as in Fig. 2.

Equations (30)

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i d a ^ d t = [ a ^ , H ^ ] ,
Δ Φ ^ = a ^ 2 / a 1 ,
a ^ = a ^ 1 + i a ^ 2 ,
a ^ = a ^ 1 - i a ^ 2 ,
H ^ = ω 0 π π + ω a ^ a ^ + i g ( π a ^ - a ^ π ) ,
π ^ = 1 2 ,             π ^ = 2 1 ,
i d a ^ 2 d t = - i ω a ^ 1 - i g ( π ^ - π ^ ) / 2
d a ^ 2 d t = - ω a ^ 1 - ( i g / 2 ) ( π ^ - π ^ ) ,
d a ^ 1 d t = ω a ^ 2 - ( g / 2 ) ( π ^ + π ^ ) ;
d π ^ d t = - i ω 0 π ^ + g a ^ ( 1 - 2 π ^ π ^ ) .
i d a ^ 2 2 d t = - i ω ( a ^ 1 a ^ 2 + a ^ 2 a ^ 1 ) + g ( π ^ a ^ 2 - a ^ 2 π ^ ) .
π ^ ( t + Δ t ) = π ^ ( t ) - { g [ 2 π ^ ( t ) π ^ ( t ) - 1 ] a ^ ( t ) × exp [ i ( ω - ω 0 ) Δ t ] - 1 i ( ω - ω 0 ) } ,
a ^ ( t + Δ t ) = a ^ ( t ) - { g π ^ ( t ) exp [ i ( ω - ω 0 ) Δ t ] - 1 i ( ω - ω 0 ) } .
a ^ 2 ( t + Δ t ) = a ^ 2 ( t ) - ( i g / 2 ) Δ t [ π ^ ( t ) - π ( t ) ] ,
d a ^ 2 2 ¯ d t = - ω ( a ^ 2 a ^ 1 ¯ + a ^ 1 a ^ 2 ¯ ) - i g ( π ^ a ^ 2 ¯ - a ^ 2 π ^ ¯ ) .
d a ^ 2 2 ¯ d t = - ω ( a ^ 2 a ^ 1 ¯ + a ^ 1 a ^ 2 ¯ ) - i g { i g π ^ π ( t ) ¯ - g Q ( t ) [ i 2 - 2 i a ^ 2 2 ¯ ( t ) ] } Δ t .
Var ( a ^ 2 ) a ^ 2 2 ¯ - a ^ 2 2 ¯ = a ^ 2 2 ¯ .
d Var ( a ^ 2 ) d t = - ω ( a ^ 2 a ^ 1 ¯ + a ^ 1 a ^ 2 ¯ ) + 2 g 2 Q ( t ) Δ t Var ( a ^ 2 ) + g 2 Δ t [ π ^ π ¯ ( t ) Q t 2 ] .
d Var ( a ^ 2 ) d t = - ω ( a ^ 2 a ^ 1 ¯ + a ^ 1 a ^ 2 ¯ ) + g f Var ( a ^ 2 ) + g f 4 Q ,
Var ( a ^ 2 ) = Var [ a ^ 2 ( 0 ) ] exp ( g f t ) + exp ( g f t ) - 1 4 Q .
Var ( Δ Φ ^ ) = Var [ a ^ 2 ( 0 ) ] a 1 ( 0 ) 2 + G - 1 G ( 2 n sp - 1 ) 4 a 1 2 ( 0 ) .
a 1 2 ( 0 ) = n in = n t T ,
Var ( Δ Φ ^ ) = [ 1 4 n t + G - 1 G ( 2 n sp - 1 ) 4 n t ] 2 B .
Var ( Δ Φ ^ ) A = ( π Δ ν ) / B ,
Var ( Δ Φ ^ ) B = Var ( Δ Φ ^ ) A + [ ( K - 1 ) / ( 4 n t ) ] 2 B .
Var ( Δ Φ ^ ) C = π Δ ν B + ( K - 1 ) 2 B 4 n t + [ G - 1 G ( 2 n sp - 1 ) 4 n t ] 2 B K .
Var ( δ N ) rec = 2 τ sp 2 ( I / e ) B ,
δ n ref = γ ( δ N ) / V a ,
Var ( Δ Φ ) = ( k 0 L ) 2 γ 2 2 I B τ sp 2 / ( e V a 2 ) ,
L = 300 μ m ,             k 0 = 4.05 μ m - 1 , V a = 0.2 × 300 μ m 3 ,             γ = - 2.8 × 10 - 26 m 3 , B = 300 MHz ,             τ sp = 1 ns ,             I = 40 mA ,

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