Abstract

We derive analytical expressions for the quantum-noise spectra of a singly resonant active frequency-doubling laser with single-frequency operation by using a linearized input–output method. For operation of a practical solid-state laser, we analyze separately the contributions of the various noise sources to the final spectrum. The change in resonance in the spectrum from an overdamped- (at higher μ) to an underdamped- (at smaller μ) driven second-order oscillator where μ is the nonlinear coupling coefficient, was experimentally observed with a diode-pumped Nd:YVO4+KTP single-frequency-doubling laser. The experimental result is in good agreement with the theoretical expectation.

© 2000 Optical Society of America

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  1. K. I. Martin, W. A. Clarkson, and D. C. Hanna, “3 W of single-frequency output at 532 nm by intracavity frequency doubling of a diode-bar-pumped Nd:YAG ring laser,” Opt. Lett. 21, 875–877 (1996).
    [CrossRef] [PubMed]
  2. P. J. Hardman, W. A. Clarkson, and D. C. Hanna, “High-power diode-bar-pumped intracavity-frequency-doubling Nd:YLF ring laser,” Opt. Commun. 156, 49–52 (1998).
    [CrossRef]
  3. R. Paschotta, M. Collett, P. Kurz, K. Fiedler, H. A. Bachor, and J. Mlynek, “Bright squeezed light from a singly resonant frequency doubler,” Phys. Rev. Lett. 72, 3807–3810 (1994).
    [CrossRef] [PubMed]
  4. M. J. Collett and R. B. Leven, “Two-photon-loss model of intracavity second-harmonic generation,” Phys. Rev. A 43, 5068–5072 (1990).
    [CrossRef]
  5. C. W. Gardiner and M. J. Collett, “Input and output in damped quantum sustems: quantum stochastic differential equations and master equation,” Phys. Rev. A 31, 3761–3774 (1985).
    [CrossRef] [PubMed]
  6. Y. Yamamoto and N. Imoto, “Internal and external field fluctuations of a laser oscillator. I. Quantum mechanical Langevin treatment,” IEEE J. Quantum Electron. QE-22, 2032–2042 (1986).
    [CrossRef]
  7. A. G. White, T. C. Ralph, and H.-A. Bachor, “Active versus passive squeezing by second-harmonic generation,” J. Opt. Soc. Am. B 13, 1337–1346 (1996).
    [CrossRef]
  8. J. Maeda, T. Numata, and S. Kogoshi, “Amplitude squeezing from singly resonant frequency-doubling laser,” IEEE J. Quantum Electron. 33, 1057–1067 (1997).
    [CrossRef]
  9. Y. Yamamoto, S. Machida, and O. Nilsson, “High-impedance suppression of pump fluctuation and amplitude squeezing in semiconductor lasers,” Phys. Rev. A 35, 5114–5151 (1987).
    [CrossRef] [PubMed]
  10. T. C. Ralph, C. C. Harb, and H.-A. Bachor, “Intensity noise of injection locked lasers: quantum theory using a linearised input–output method,” Phys. Rev. A 54, 4359–4369 (1996).
    [CrossRef] [PubMed]
  11. C. C. Harb, T. C. Ralph, E. H. Huntington, I. Freitag, D. E. McClelland, and H.-A. Bachor, “Intensity-noise properties of injection-locked lasers,” Phys. Rev. A 54, 4370–4381 (1996).
    [CrossRef] [PubMed]

1998

P. J. Hardman, W. A. Clarkson, and D. C. Hanna, “High-power diode-bar-pumped intracavity-frequency-doubling Nd:YLF ring laser,” Opt. Commun. 156, 49–52 (1998).
[CrossRef]

1997

J. Maeda, T. Numata, and S. Kogoshi, “Amplitude squeezing from singly resonant frequency-doubling laser,” IEEE J. Quantum Electron. 33, 1057–1067 (1997).
[CrossRef]

1996

K. I. Martin, W. A. Clarkson, and D. C. Hanna, “3 W of single-frequency output at 532 nm by intracavity frequency doubling of a diode-bar-pumped Nd:YAG ring laser,” Opt. Lett. 21, 875–877 (1996).
[CrossRef] [PubMed]

T. C. Ralph, C. C. Harb, and H.-A. Bachor, “Intensity noise of injection locked lasers: quantum theory using a linearised input–output method,” Phys. Rev. A 54, 4359–4369 (1996).
[CrossRef] [PubMed]

C. C. Harb, T. C. Ralph, E. H. Huntington, I. Freitag, D. E. McClelland, and H.-A. Bachor, “Intensity-noise properties of injection-locked lasers,” Phys. Rev. A 54, 4370–4381 (1996).
[CrossRef] [PubMed]

A. G. White, T. C. Ralph, and H.-A. Bachor, “Active versus passive squeezing by second-harmonic generation,” J. Opt. Soc. Am. B 13, 1337–1346 (1996).
[CrossRef]

1994

R. Paschotta, M. Collett, P. Kurz, K. Fiedler, H. A. Bachor, and J. Mlynek, “Bright squeezed light from a singly resonant frequency doubler,” Phys. Rev. Lett. 72, 3807–3810 (1994).
[CrossRef] [PubMed]

1990

M. J. Collett and R. B. Leven, “Two-photon-loss model of intracavity second-harmonic generation,” Phys. Rev. A 43, 5068–5072 (1990).
[CrossRef]

1987

Y. Yamamoto, S. Machida, and O. Nilsson, “High-impedance suppression of pump fluctuation and amplitude squeezing in semiconductor lasers,” Phys. Rev. A 35, 5114–5151 (1987).
[CrossRef] [PubMed]

1986

Y. Yamamoto and N. Imoto, “Internal and external field fluctuations of a laser oscillator. I. Quantum mechanical Langevin treatment,” IEEE J. Quantum Electron. QE-22, 2032–2042 (1986).
[CrossRef]

1985

C. W. Gardiner and M. J. Collett, “Input and output in damped quantum sustems: quantum stochastic differential equations and master equation,” Phys. Rev. A 31, 3761–3774 (1985).
[CrossRef] [PubMed]

Bachor, H. A.

R. Paschotta, M. Collett, P. Kurz, K. Fiedler, H. A. Bachor, and J. Mlynek, “Bright squeezed light from a singly resonant frequency doubler,” Phys. Rev. Lett. 72, 3807–3810 (1994).
[CrossRef] [PubMed]

Bachor, H.-A.

A. G. White, T. C. Ralph, and H.-A. Bachor, “Active versus passive squeezing by second-harmonic generation,” J. Opt. Soc. Am. B 13, 1337–1346 (1996).
[CrossRef]

T. C. Ralph, C. C. Harb, and H.-A. Bachor, “Intensity noise of injection locked lasers: quantum theory using a linearised input–output method,” Phys. Rev. A 54, 4359–4369 (1996).
[CrossRef] [PubMed]

C. C. Harb, T. C. Ralph, E. H. Huntington, I. Freitag, D. E. McClelland, and H.-A. Bachor, “Intensity-noise properties of injection-locked lasers,” Phys. Rev. A 54, 4370–4381 (1996).
[CrossRef] [PubMed]

Clarkson, W. A.

P. J. Hardman, W. A. Clarkson, and D. C. Hanna, “High-power diode-bar-pumped intracavity-frequency-doubling Nd:YLF ring laser,” Opt. Commun. 156, 49–52 (1998).
[CrossRef]

K. I. Martin, W. A. Clarkson, and D. C. Hanna, “3 W of single-frequency output at 532 nm by intracavity frequency doubling of a diode-bar-pumped Nd:YAG ring laser,” Opt. Lett. 21, 875–877 (1996).
[CrossRef] [PubMed]

Collett, M.

R. Paschotta, M. Collett, P. Kurz, K. Fiedler, H. A. Bachor, and J. Mlynek, “Bright squeezed light from a singly resonant frequency doubler,” Phys. Rev. Lett. 72, 3807–3810 (1994).
[CrossRef] [PubMed]

Collett, M. J.

M. J. Collett and R. B. Leven, “Two-photon-loss model of intracavity second-harmonic generation,” Phys. Rev. A 43, 5068–5072 (1990).
[CrossRef]

C. W. Gardiner and M. J. Collett, “Input and output in damped quantum sustems: quantum stochastic differential equations and master equation,” Phys. Rev. A 31, 3761–3774 (1985).
[CrossRef] [PubMed]

Fiedler, K.

R. Paschotta, M. Collett, P. Kurz, K. Fiedler, H. A. Bachor, and J. Mlynek, “Bright squeezed light from a singly resonant frequency doubler,” Phys. Rev. Lett. 72, 3807–3810 (1994).
[CrossRef] [PubMed]

Freitag, I.

C. C. Harb, T. C. Ralph, E. H. Huntington, I. Freitag, D. E. McClelland, and H.-A. Bachor, “Intensity-noise properties of injection-locked lasers,” Phys. Rev. A 54, 4370–4381 (1996).
[CrossRef] [PubMed]

Gardiner, C. W.

C. W. Gardiner and M. J. Collett, “Input and output in damped quantum sustems: quantum stochastic differential equations and master equation,” Phys. Rev. A 31, 3761–3774 (1985).
[CrossRef] [PubMed]

Hanna, D. C.

P. J. Hardman, W. A. Clarkson, and D. C. Hanna, “High-power diode-bar-pumped intracavity-frequency-doubling Nd:YLF ring laser,” Opt. Commun. 156, 49–52 (1998).
[CrossRef]

K. I. Martin, W. A. Clarkson, and D. C. Hanna, “3 W of single-frequency output at 532 nm by intracavity frequency doubling of a diode-bar-pumped Nd:YAG ring laser,” Opt. Lett. 21, 875–877 (1996).
[CrossRef] [PubMed]

Harb, C. C.

C. C. Harb, T. C. Ralph, E. H. Huntington, I. Freitag, D. E. McClelland, and H.-A. Bachor, “Intensity-noise properties of injection-locked lasers,” Phys. Rev. A 54, 4370–4381 (1996).
[CrossRef] [PubMed]

T. C. Ralph, C. C. Harb, and H.-A. Bachor, “Intensity noise of injection locked lasers: quantum theory using a linearised input–output method,” Phys. Rev. A 54, 4359–4369 (1996).
[CrossRef] [PubMed]

Hardman, P. J.

P. J. Hardman, W. A. Clarkson, and D. C. Hanna, “High-power diode-bar-pumped intracavity-frequency-doubling Nd:YLF ring laser,” Opt. Commun. 156, 49–52 (1998).
[CrossRef]

Huntington, E. H.

C. C. Harb, T. C. Ralph, E. H. Huntington, I. Freitag, D. E. McClelland, and H.-A. Bachor, “Intensity-noise properties of injection-locked lasers,” Phys. Rev. A 54, 4370–4381 (1996).
[CrossRef] [PubMed]

Imoto, N.

Y. Yamamoto and N. Imoto, “Internal and external field fluctuations of a laser oscillator. I. Quantum mechanical Langevin treatment,” IEEE J. Quantum Electron. QE-22, 2032–2042 (1986).
[CrossRef]

Kogoshi, S.

J. Maeda, T. Numata, and S. Kogoshi, “Amplitude squeezing from singly resonant frequency-doubling laser,” IEEE J. Quantum Electron. 33, 1057–1067 (1997).
[CrossRef]

Kurz, P.

R. Paschotta, M. Collett, P. Kurz, K. Fiedler, H. A. Bachor, and J. Mlynek, “Bright squeezed light from a singly resonant frequency doubler,” Phys. Rev. Lett. 72, 3807–3810 (1994).
[CrossRef] [PubMed]

Leven, R. B.

M. J. Collett and R. B. Leven, “Two-photon-loss model of intracavity second-harmonic generation,” Phys. Rev. A 43, 5068–5072 (1990).
[CrossRef]

Machida, S.

Y. Yamamoto, S. Machida, and O. Nilsson, “High-impedance suppression of pump fluctuation and amplitude squeezing in semiconductor lasers,” Phys. Rev. A 35, 5114–5151 (1987).
[CrossRef] [PubMed]

Maeda, J.

J. Maeda, T. Numata, and S. Kogoshi, “Amplitude squeezing from singly resonant frequency-doubling laser,” IEEE J. Quantum Electron. 33, 1057–1067 (1997).
[CrossRef]

Martin, K. I.

McClelland, D. E.

C. C. Harb, T. C. Ralph, E. H. Huntington, I. Freitag, D. E. McClelland, and H.-A. Bachor, “Intensity-noise properties of injection-locked lasers,” Phys. Rev. A 54, 4370–4381 (1996).
[CrossRef] [PubMed]

Mlynek, J.

R. Paschotta, M. Collett, P. Kurz, K. Fiedler, H. A. Bachor, and J. Mlynek, “Bright squeezed light from a singly resonant frequency doubler,” Phys. Rev. Lett. 72, 3807–3810 (1994).
[CrossRef] [PubMed]

Nilsson, O.

Y. Yamamoto, S. Machida, and O. Nilsson, “High-impedance suppression of pump fluctuation and amplitude squeezing in semiconductor lasers,” Phys. Rev. A 35, 5114–5151 (1987).
[CrossRef] [PubMed]

Numata, T.

J. Maeda, T. Numata, and S. Kogoshi, “Amplitude squeezing from singly resonant frequency-doubling laser,” IEEE J. Quantum Electron. 33, 1057–1067 (1997).
[CrossRef]

Paschotta, R.

R. Paschotta, M. Collett, P. Kurz, K. Fiedler, H. A. Bachor, and J. Mlynek, “Bright squeezed light from a singly resonant frequency doubler,” Phys. Rev. Lett. 72, 3807–3810 (1994).
[CrossRef] [PubMed]

Ralph, T. C.

A. G. White, T. C. Ralph, and H.-A. Bachor, “Active versus passive squeezing by second-harmonic generation,” J. Opt. Soc. Am. B 13, 1337–1346 (1996).
[CrossRef]

T. C. Ralph, C. C. Harb, and H.-A. Bachor, “Intensity noise of injection locked lasers: quantum theory using a linearised input–output method,” Phys. Rev. A 54, 4359–4369 (1996).
[CrossRef] [PubMed]

C. C. Harb, T. C. Ralph, E. H. Huntington, I. Freitag, D. E. McClelland, and H.-A. Bachor, “Intensity-noise properties of injection-locked lasers,” Phys. Rev. A 54, 4370–4381 (1996).
[CrossRef] [PubMed]

White, A. G.

Yamamoto, Y.

Y. Yamamoto, S. Machida, and O. Nilsson, “High-impedance suppression of pump fluctuation and amplitude squeezing in semiconductor lasers,” Phys. Rev. A 35, 5114–5151 (1987).
[CrossRef] [PubMed]

Y. Yamamoto and N. Imoto, “Internal and external field fluctuations of a laser oscillator. I. Quantum mechanical Langevin treatment,” IEEE J. Quantum Electron. QE-22, 2032–2042 (1986).
[CrossRef]

IEEE J. Quantum Electron.

J. Maeda, T. Numata, and S. Kogoshi, “Amplitude squeezing from singly resonant frequency-doubling laser,” IEEE J. Quantum Electron. 33, 1057–1067 (1997).
[CrossRef]

Y. Yamamoto and N. Imoto, “Internal and external field fluctuations of a laser oscillator. I. Quantum mechanical Langevin treatment,” IEEE J. Quantum Electron. QE-22, 2032–2042 (1986).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Commun.

P. J. Hardman, W. A. Clarkson, and D. C. Hanna, “High-power diode-bar-pumped intracavity-frequency-doubling Nd:YLF ring laser,” Opt. Commun. 156, 49–52 (1998).
[CrossRef]

Opt. Lett.

Phys. Rev. A

M. J. Collett and R. B. Leven, “Two-photon-loss model of intracavity second-harmonic generation,” Phys. Rev. A 43, 5068–5072 (1990).
[CrossRef]

C. W. Gardiner and M. J. Collett, “Input and output in damped quantum sustems: quantum stochastic differential equations and master equation,” Phys. Rev. A 31, 3761–3774 (1985).
[CrossRef] [PubMed]

Y. Yamamoto, S. Machida, and O. Nilsson, “High-impedance suppression of pump fluctuation and amplitude squeezing in semiconductor lasers,” Phys. Rev. A 35, 5114–5151 (1987).
[CrossRef] [PubMed]

T. C. Ralph, C. C. Harb, and H.-A. Bachor, “Intensity noise of injection locked lasers: quantum theory using a linearised input–output method,” Phys. Rev. A 54, 4359–4369 (1996).
[CrossRef] [PubMed]

C. C. Harb, T. C. Ralph, E. H. Huntington, I. Freitag, D. E. McClelland, and H.-A. Bachor, “Intensity-noise properties of injection-locked lasers,” Phys. Rev. A 54, 4370–4381 (1996).
[CrossRef] [PubMed]

Phys. Rev. Lett.

R. Paschotta, M. Collett, P. Kurz, K. Fiedler, H. A. Bachor, and J. Mlynek, “Bright squeezed light from a singly resonant frequency doubler,” Phys. Rev. Lett. 72, 3807–3810 (1994).
[CrossRef] [PubMed]

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Figures (10)

Fig. 1
Fig. 1

Single-frequency-doubling ring laser. DM, dichroic mirror (reflectivity, R1 for the fundamental, ∼0 for the harmonic); NLC, nonlinear crystal.

Fig. 2
Fig. 2

Diode-pumped single-frequency-doubling laser described by the quantum model: γf, γt, and γs, spontaneous-emission rates; Γ, pump rate; G stimulated-emission coefficient; Vout, output harmonic noise; Vcin, second-harmonic input noise; Vpump, noise entering the laser from its pump source; Vab, quantum (or vacuum) noise, as not all of the pump field is absorbed; Vspont32, Vspont21, noise from spontaneous emission; Vdipole, noise from dipole fluctuations; Vlosses, noise from intracavity losses; NLC, nonlinear crystal.

Fig. 3
Fig. 3

Spectra of the output harmonic field in the limit well above threshold. Vpump=1, μα02=4.27×106 s-1.

Fig. 4
Fig. 4

Frequency dependence of the various noise sources. (a) Resonance overdamped (ωr<γr), with μ˜=2×1013 s-1; (b) resonance underdamped (ωr>γr) with μ˜=2×1011 s-1; other parameters, those in Table 1 with Vpump=1. a, Noise of the harmonic with all contributions added; b, contribution from VCin; c, contribution from Vpump; d, contribution from Vspont32; e, shows the contribution from Vdipole; f, contribution from Vlosses.

Fig. 5
Fig. 5

Harmonic spectra for different pump noise from that of Fig. 4(a). Resonance overdamped (ωr<γr), with μ˜=2×1013 s-1; (b) resonance underdamped (ωr>γr), with μ˜=2×1011 s-1. a, curve a pump noise Vpump=1; b, Vpump=20 dB; c, Vpump=30 dB; d Vpump=50 dB.

Fig. 6
Fig. 6

Intracavity fundamental photon number α02 per atom of the lasing mode versus μ˜.

Fig. 7
Fig. 7

a, Frequency ωr of the oscillations and b, damping rate γr versus μ˜.

Fig. 8
Fig. 8

Harmonic spectra for three values of μ˜ with Vpump=50 dB: a, μ˜=1013; b, μ˜=1012; c, μ˜=1011.

Fig. 9
Fig. 9

Experimental setup of a Nd:YVO4 ring laser for intracavity frequency doubling. M1–M4, mirrors; TGG; terbium gallium garnet crystal.

Fig. 10
Fig. 10

Harmonic noise spectra with various nonlinear Coupling strengths: a, Intensity noise spectrum of theory and experiment for the 8-mW harmonic output, with the parameter μ˜ set to 2×1013 s-1; b, for the 0.85-mW harmonic output; parameter μ˜ is set to 1×1012 s-1; c, for the 70-µW harmonic output; parameter μ˜ set to 2×1011 s-1.

Tables (1)

Tables Icon

Table 1 Parameters of a Nd:YVO4 Single-Frequency-Doubling Ring Laser

Equations (67)

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Hˆa=ωaaˆaˆ.
Hˆlaser1=ig(aˆσˆ23-aˆσˆ23+),
Hˆb=ωbbˆbˆ.
Hˆpump2=igp(bˆσˆ14-bˆσˆ14+),
Hˆc=ωccˆcˆ.
Hˆshg3=i u2(aˆ2cˆ-aˆ2cˆ),
aˆ=G˜2(σˆ3-σˆ2)aˆ-κlaˆ-μaˆaˆ2+2μaˆδCˆ+2κlδAˆl-G˜δΛˆp,
σˆ˙1=-4Gpκbσˆ1(κb+2Gpσˆ1)2(BˆBˆ+BˆBˆ)+4Gpκb(κb+2Gpσˆ1)2(δΛˆqδΛˆq+δΛˆqδΛˆq)+8Gpκb (κb-2Gpσˆ1)(κb+2Gpσˆ1)2(δΛˆqBˆ+δΛˆqBˆ)+γsσˆ2-γsδΛˆ,
σˆ˙2=G˜2(σˆ3-σˆ2)(aˆaˆ+σˆaˆ)-G˜(δΛˆpaˆ+δΛˆppaˆ)+γtσˆ3-γsσˆ2+γsδΛˆ-γtδΛˆt,
σˆ˙3=G˜2(σˆ3-σˆ2)(aˆaˆ+σˆaˆ)-G˜(δΛˆpaˆ+δΛˆpaˆ)+4Gpκbσˆ1(κb+2Gpσˆ1)2(BˆBˆ+BˆBˆ)-4Gpκb(κb+2Gpσˆ1)2(δΛˆqδΛˆq+δΛˆqδΛˆq)-8Gpκb (κb-2Gpσˆ1)(κb+2Gpσˆ1)2(δΛˆqBˆ+δΛˆqBˆ)-γtσˆ3+γtδAˆt,
G˜=2g2γP,
Gp=gp22γQ
δΛˆ=δCˆsσˆ12+σˆ12+δCˆs=σˆ2δXˆCx,
δΛˆ=δCˆsσˆ23+σˆ23+δCˆt=σˆ3δXˆCt,
δΛˆp=(δCˆP-δCˆP)σ23,
δΛˆq=(δCˆQ-δCˆQ)σ14,
δΛˆQ=δΛˆq+δΛˆq=σˆ1+σˆ4δXˆq,
δΛˆP=δΛˆp+δΛˆp=σˆ3+σˆ2δXˆp.
δΛˆ(t)δΛˆ(t)=σˆ2δ(t-t),
δΛˆt(t)δΛˆt(t)=σˆ3δ(t-t),
δΛˆQ(t)δΛˆQ(t)=σˆ1+σˆ4δ(t-t),
δΛˆP(t)δΛˆP(t)=σˆ3+σˆ2δ(t-t).
α˙=G2(J3-J2)α-κlα-μ˜α*α2,
J˙1=-ΓJ1+γsJ2,
J˙2=G(J3-J2)αα*+γtJ3-γsJ2,
J˙3=-G(J3-J2)αα*-γtJ3+ΓJ1,
Γ=8Gpκb(κb+2GpJ1)2BˆBˆ
μ˜=Nμ.
G=NG˜=sρc,
aˆ(t)=Nα0+δαˆ(t),σˆi(t)=NJi0+δσˆi(t),
Bˆ(t)=NB0+δBˆ(t),
δXˆ˙=G(δσˆ3-δσˆ2)α0-μ˜α0*α0δXˆa-μ˜α02δXˆa+2κlδXˆAl+2μ˜α0*δXˆc-G(J3+J2)δXˆp,
δσˆ˙1=-Γ1-ηδσˆ1+γδσˆ2-ΓJ10ηδXˆB
+γsJ2δXˆCx+[Γ(1-η)(J4+J1)]1/2δXˆq,
δσˆ˙2=G(δσˆ2)α02+G(J30-J20)α0δXˆa+γtδσˆ3-γδσˆ2+γsJ2δXˆCx-γtJ3δXˆCt-G(J3+J2)α0δXˆp,
δσˆ˙3=-G(δσˆ3-δσˆ2)α02-G(J30-J20)α0δXˆa
-γtδσˆ3+Γ1-ηδσˆ1+ΓJ1ηδXˆB
+γsJ2δXˆCx-[Γ(1-η)(J4+J1)]1/2δXˆq
+γtJ3δXˆCt+G(J3+J2)α0δXˆp,
δXˆa=δαˆ+δαˆ,δXˆAl=δAˆl+δAˆl,δXˆc=δCˆ+δCˆ .
1-η=κb-2GpJ10κb+2GpJ10,
η=8GpJ10κb(2GpJ10+κb)2.
Cˆout=μaˆ2-δCˆm,
δXˆCout=2μ˜α0δXˆa-δXˆc.
δXCout={[2μ˜α02-iω-F1(ω)]δXc+2μ˜α0ΓJ1F2(ω)(1-η)δXq+2μ˜α0ΓJ1F2(ω)ηδXB+2μ˜α0γsJ2F3(ω)δXCx+2μ˜α0γtJ3F4(ω)δXCt+2μ˜α0G(J3+J2)[1-F4(ω)]δXp+2μ˜α02klδXAl}/[iω+F1(ω)+2μ˜α02]},
F1(ω)
=G2a02(J3-J2)(2iω+γs+2Γ˜)(iω+Γ˜)(iω+γs+2Ga02+γt)+γs(Ga02+γt),
F2(ω)
=Ga0(iω+γs-γt)(iω+Γ˜)(iω+γs+2Ga02+γt)+γs(Ga02+γt),
F3(ω)
=Ga0(iω+2Γ˜+γt)(iω+Γ˜)(iω+γs+2Ga02+γt)+γs(Ga02+γt),
F4(ω)
=Ga0(iω+2Γ˜+γs)(iω+Γ˜)(iω+γs+2Ga02+γt)+γs(Ga02+γt),
Vout(ω)=|δXout|2
={[2μ˜α02-iω-F1(ω)]2Vcin+4μ˜α02ΓJ1[F2(ω)]2(1-η)Vab+4μ˜α02ΓJ1[F2(ω)]2ηVpump+4μ˜α02γsJ2[F3(ω)]2Vspont21+4μ˜α02γtJ3[F4(ω)]2Vspont32+4μ˜α02G(J3+J2)[1-F4(ω)]2Vdipole+8μ˜α02klVlosses}/[iω+F1(ω)+2μ˜α02]2,
Vout(ω)
=ω2ω2+(4μ˜α02)2
+2(2μ˜α02)2(ω2+γs2)[4μ˜α02(γs+2Γ˜)-2ω2]2+ω2(8μ˜α02+γs+2Γ˜)2
×[(1-η)+ηVpump]
+2(2μ˜α02)2[ω2+(2Γ˜)2][4μ˜α02(γs+2Γ˜)-2ω2]2+ω2(8μ˜α02+γs+2Γ˜)2.
Vout(ω)=ω2ω2+(4μ˜α02)2+2(2μ˜α02)2(4μ˜α02)2+ω2×[(1-η)+ηVp].
Vout(ω)=({[2μ˜a02(Ga02+Γ˜+γt)+ω2-2Ga02(μ˜a02+κl)]2+ω2(2μ˜a02-Ga02-Γ˜-γt)2}+{4μ˜α02ΓJ1G2a02[1+η(Vpump-1)]}+(4μ˜α02γtJ3G2a02)+{4μ˜α02G(J3+J2)[(γt+Γ˜)2+ω2]}+{8μ˜α02kl[(Ga02+Γ˜+γt)2+ω2]})/[(ωr2-ω2)2+ω2(γL)2];
ωr=[2μ˜a02(Ga02+Γ˜+γt)+2Ga02(μ˜a02+κl)]1/2,
γL=2μ˜a02+Ga02+Γ˜+γt,
Vout(ω)=({[2km(Ga02+Γ˜+γt)+ω2-2km(μ˜a02+κl)]2+ω2(2km-Ga02-Γ˜-γt)2}+{2kmΓJ1G2a02[1-η(Vpump-1)]}+(2kmγtG2a02J3)+{2kmG(J3+J2)([γt+Γ˜)2+ω2]}+{4kmkl[(Ga02+Γ˜+γt)2+ω2]})/[(ωr2-ω2)2+ω2(γL)2],
ωr=[2Ga02(km+κl)]1/2
γl=Ga02+Γ˜+γt.

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