Abstract

We have calculated transfer functions for frequency and intensity fluctuations in an injection-locked solid-state laser. At modulation frequencies well below the locking frequency we find significant frequency-noise reduction, and at modulation frequencies above the locking frequency we find that the frequency noise is that of the free-running slave laser. Our intensity-noise theory predicts substantial damping of relaxation oscillations in the slave laser. To verify these results we have measured the frequency and intensity noise of a 5-W, injection-locked Nd:YAG laser.

© 1995 Optical Society of America

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References

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  1. W. Koechner, Solid State Laser Engineering, 3rd ed. (Springer-Verlag, New York, 1992), p. 79.
    [CrossRef]
  2. M. Zhu and J. L. Hall, "Stabilization of optical phase/frequency of a laser system: application to a commercial dye laser with an external stabilizer," J. Opt. Soc. Am. B 10, 802 (1993).
    [CrossRef]
  3. T. J. Kane and R. L. Byer, "Monolithic, unidirectional single-mode Nd:YAG ring laser," Opt. Lett. 10, 65 (1985).
    [CrossRef] [PubMed]
  4. R. Adler, "A study of locking phenomena in oscillators," Proc. IRE 34, 351 (1946).
    [CrossRef]
  5. C. D. Nabors, A. D. Farinas, T. Day, S. T. Yang, E. K. Gustafson, and R. L. Byer, "Injection locking of a 13-W cw Nd:YAG ring laser," Opt. Lett. 14, 1189 (1989).
    [CrossRef] [PubMed]
  6. S. T. Yang, C. C. Pohalski, E. K. Gustafson, R. L. Byer, R. S. Feigelson, R. J. Raymakers, and R. K. Route, "6.5-W, 532-nm radiation by cw resonant external-cavity second-harmonic generation of an 18-W Nd:YAG laser in LiNb3O5," Opt. Lett. 16, 1493 (1991).
    [CrossRef] [PubMed]
  7. A. D. Farinas, E. K. Gustafson, and R. L. Byer, "Design and characterization of a 5.5-W, cw, injection-locked, fiber-coupled laser-diode pumped Nd:YAG miniature-slab laser," Opt. Lett. 19, 114 (1994).
    [CrossRef]
  8. A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Voght, R. Weiss, S. Whitcomb, and M. E. Zucker, "LIGO: the laser interferometer gravitational-wave observatory," Science 256, 325 (1992).
    [CrossRef] [PubMed]
  9. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), p. 923.
  10. Ref. 9, p. 1129.
  11. M. E. Hines, J. R. Collinet, and J. G. Ondria, "FM noise suppression of an injection phase-locked oscillator," IEEE Trans. Microwave Theory Tech. MTT-16, 738 (1968).
    [CrossRef]
  12. T. Day, E. K. Gustafson, and R. L. Byer, "Sub-hertz relative frequency stabilization of two-diode laser-pumped Nd:YAG lasers locked to a Fabry—Perot interferometer," IEEE J. Quantum Electron. 28, 1106 (1992).
    [CrossRef]
  13. Ref. 9, p. 954.
  14. R. W. P. Drever, J. L. Hall, F. V. Kawalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, "Laser phase and frequency stabilization using an optical resonator," Appl. Phys. B 31, 97 (1983).
    [CrossRef]
  15. T. Day, A. D. Farinas, and R. L. Byer, "Demonstration of a low bandwidth 1.06 μm optical phase-locked loop for coherent homodyne communication," IEEE Photon. Technol. Lett. 2, 294 (1990).
    [CrossRef]
  16. J. Smith, Modern Communications Circuits (McGraw-Hill, New York, 1986), p. 295.
  17. T. J. Kane, "Intensity noise in diode-pumped single-frequency Nd:YAG lasers and its control by electronic feedback," IEEE Photon. Technol. Lett. 2, 244 (1990).
    [CrossRef]
  18. A. L. Schawlow and C. H. Townes, "Infrared and optical masers," Phys. Rev. 112, 1940 (1958).
    [CrossRef]

1994 (1)

1993 (1)

1992 (2)

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Voght, R. Weiss, S. Whitcomb, and M. E. Zucker, "LIGO: the laser interferometer gravitational-wave observatory," Science 256, 325 (1992).
[CrossRef] [PubMed]

T. Day, E. K. Gustafson, and R. L. Byer, "Sub-hertz relative frequency stabilization of two-diode laser-pumped Nd:YAG lasers locked to a Fabry—Perot interferometer," IEEE J. Quantum Electron. 28, 1106 (1992).
[CrossRef]

1991 (1)

1990 (2)

T. Day, A. D. Farinas, and R. L. Byer, "Demonstration of a low bandwidth 1.06 μm optical phase-locked loop for coherent homodyne communication," IEEE Photon. Technol. Lett. 2, 294 (1990).
[CrossRef]

T. J. Kane, "Intensity noise in diode-pumped single-frequency Nd:YAG lasers and its control by electronic feedback," IEEE Photon. Technol. Lett. 2, 244 (1990).
[CrossRef]

1989 (1)

1985 (1)

1983 (1)

R. W. P. Drever, J. L. Hall, F. V. Kawalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, "Laser phase and frequency stabilization using an optical resonator," Appl. Phys. B 31, 97 (1983).
[CrossRef]

1968 (1)

M. E. Hines, J. R. Collinet, and J. G. Ondria, "FM noise suppression of an injection phase-locked oscillator," IEEE Trans. Microwave Theory Tech. MTT-16, 738 (1968).
[CrossRef]

1958 (1)

A. L. Schawlow and C. H. Townes, "Infrared and optical masers," Phys. Rev. 112, 1940 (1958).
[CrossRef]

1946 (1)

R. Adler, "A study of locking phenomena in oscillators," Proc. IRE 34, 351 (1946).
[CrossRef]

Abramovici, A.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Voght, R. Weiss, S. Whitcomb, and M. E. Zucker, "LIGO: the laser interferometer gravitational-wave observatory," Science 256, 325 (1992).
[CrossRef] [PubMed]

Adler, R.

R. Adler, "A study of locking phenomena in oscillators," Proc. IRE 34, 351 (1946).
[CrossRef]

Althouse, W. E.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Voght, R. Weiss, S. Whitcomb, and M. E. Zucker, "LIGO: the laser interferometer gravitational-wave observatory," Science 256, 325 (1992).
[CrossRef] [PubMed]

Byer, R. L.

Collinet, J. R.

M. E. Hines, J. R. Collinet, and J. G. Ondria, "FM noise suppression of an injection phase-locked oscillator," IEEE Trans. Microwave Theory Tech. MTT-16, 738 (1968).
[CrossRef]

Day, T.

T. Day, E. K. Gustafson, and R. L. Byer, "Sub-hertz relative frequency stabilization of two-diode laser-pumped Nd:YAG lasers locked to a Fabry—Perot interferometer," IEEE J. Quantum Electron. 28, 1106 (1992).
[CrossRef]

T. Day, A. D. Farinas, and R. L. Byer, "Demonstration of a low bandwidth 1.06 μm optical phase-locked loop for coherent homodyne communication," IEEE Photon. Technol. Lett. 2, 294 (1990).
[CrossRef]

C. D. Nabors, A. D. Farinas, T. Day, S. T. Yang, E. K. Gustafson, and R. L. Byer, "Injection locking of a 13-W cw Nd:YAG ring laser," Opt. Lett. 14, 1189 (1989).
[CrossRef] [PubMed]

Drever, R. W. P.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Voght, R. Weiss, S. Whitcomb, and M. E. Zucker, "LIGO: the laser interferometer gravitational-wave observatory," Science 256, 325 (1992).
[CrossRef] [PubMed]

R. W. P. Drever, J. L. Hall, F. V. Kawalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, "Laser phase and frequency stabilization using an optical resonator," Appl. Phys. B 31, 97 (1983).
[CrossRef]

Farinas, A. D.

Feigelson, R. S.

Ford, G. M.

R. W. P. Drever, J. L. Hall, F. V. Kawalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, "Laser phase and frequency stabilization using an optical resonator," Appl. Phys. B 31, 97 (1983).
[CrossRef]

Gürsel, Y.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Voght, R. Weiss, S. Whitcomb, and M. E. Zucker, "LIGO: the laser interferometer gravitational-wave observatory," Science 256, 325 (1992).
[CrossRef] [PubMed]

Gustafson, E. K.

Hall, J. L.

M. Zhu and J. L. Hall, "Stabilization of optical phase/frequency of a laser system: application to a commercial dye laser with an external stabilizer," J. Opt. Soc. Am. B 10, 802 (1993).
[CrossRef]

R. W. P. Drever, J. L. Hall, F. V. Kawalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, "Laser phase and frequency stabilization using an optical resonator," Appl. Phys. B 31, 97 (1983).
[CrossRef]

Hines, M. E.

M. E. Hines, J. R. Collinet, and J. G. Ondria, "FM noise suppression of an injection phase-locked oscillator," IEEE Trans. Microwave Theory Tech. MTT-16, 738 (1968).
[CrossRef]

Hough, J.

R. W. P. Drever, J. L. Hall, F. V. Kawalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, "Laser phase and frequency stabilization using an optical resonator," Appl. Phys. B 31, 97 (1983).
[CrossRef]

Kane, T. J.

T. J. Kane, "Intensity noise in diode-pumped single-frequency Nd:YAG lasers and its control by electronic feedback," IEEE Photon. Technol. Lett. 2, 244 (1990).
[CrossRef]

T. J. Kane and R. L. Byer, "Monolithic, unidirectional single-mode Nd:YAG ring laser," Opt. Lett. 10, 65 (1985).
[CrossRef] [PubMed]

Kawalski, F. V.

R. W. P. Drever, J. L. Hall, F. V. Kawalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, "Laser phase and frequency stabilization using an optical resonator," Appl. Phys. B 31, 97 (1983).
[CrossRef]

Kawamura, S.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Voght, R. Weiss, S. Whitcomb, and M. E. Zucker, "LIGO: the laser interferometer gravitational-wave observatory," Science 256, 325 (1992).
[CrossRef] [PubMed]

Koechner, W.

W. Koechner, Solid State Laser Engineering, 3rd ed. (Springer-Verlag, New York, 1992), p. 79.
[CrossRef]

Munley, A. J.

R. W. P. Drever, J. L. Hall, F. V. Kawalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, "Laser phase and frequency stabilization using an optical resonator," Appl. Phys. B 31, 97 (1983).
[CrossRef]

Nabors, C. D.

Ondria, J. G.

M. E. Hines, J. R. Collinet, and J. G. Ondria, "FM noise suppression of an injection phase-locked oscillator," IEEE Trans. Microwave Theory Tech. MTT-16, 738 (1968).
[CrossRef]

Pohalski, C. C.

Raab, F.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Voght, R. Weiss, S. Whitcomb, and M. E. Zucker, "LIGO: the laser interferometer gravitational-wave observatory," Science 256, 325 (1992).
[CrossRef] [PubMed]

Raymakers, R. J.

Route, R. K.

Schawlow, A. L.

A. L. Schawlow and C. H. Townes, "Infrared and optical masers," Phys. Rev. 112, 1940 (1958).
[CrossRef]

Shoemaker, D.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Voght, R. Weiss, S. Whitcomb, and M. E. Zucker, "LIGO: the laser interferometer gravitational-wave observatory," Science 256, 325 (1992).
[CrossRef] [PubMed]

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), p. 923.

Sievers, L.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Voght, R. Weiss, S. Whitcomb, and M. E. Zucker, "LIGO: the laser interferometer gravitational-wave observatory," Science 256, 325 (1992).
[CrossRef] [PubMed]

Smith, J.

J. Smith, Modern Communications Circuits (McGraw-Hill, New York, 1986), p. 295.

Spero, R. E.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Voght, R. Weiss, S. Whitcomb, and M. E. Zucker, "LIGO: the laser interferometer gravitational-wave observatory," Science 256, 325 (1992).
[CrossRef] [PubMed]

Thorne, K. S.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Voght, R. Weiss, S. Whitcomb, and M. E. Zucker, "LIGO: the laser interferometer gravitational-wave observatory," Science 256, 325 (1992).
[CrossRef] [PubMed]

Townes, C. H.

A. L. Schawlow and C. H. Townes, "Infrared and optical masers," Phys. Rev. 112, 1940 (1958).
[CrossRef]

Voght, R. E.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Voght, R. Weiss, S. Whitcomb, and M. E. Zucker, "LIGO: the laser interferometer gravitational-wave observatory," Science 256, 325 (1992).
[CrossRef] [PubMed]

Ward, H.

R. W. P. Drever, J. L. Hall, F. V. Kawalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, "Laser phase and frequency stabilization using an optical resonator," Appl. Phys. B 31, 97 (1983).
[CrossRef]

Weiss, R.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Voght, R. Weiss, S. Whitcomb, and M. E. Zucker, "LIGO: the laser interferometer gravitational-wave observatory," Science 256, 325 (1992).
[CrossRef] [PubMed]

Whitcomb, S.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Voght, R. Weiss, S. Whitcomb, and M. E. Zucker, "LIGO: the laser interferometer gravitational-wave observatory," Science 256, 325 (1992).
[CrossRef] [PubMed]

Yang, S. T.

Zhu, M.

Zucker, M. E.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Voght, R. Weiss, S. Whitcomb, and M. E. Zucker, "LIGO: the laser interferometer gravitational-wave observatory," Science 256, 325 (1992).
[CrossRef] [PubMed]

Appl. Phys. B (1)

R. W. P. Drever, J. L. Hall, F. V. Kawalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, "Laser phase and frequency stabilization using an optical resonator," Appl. Phys. B 31, 97 (1983).
[CrossRef]

IEEE J. Quantum Electron. (1)

T. Day, E. K. Gustafson, and R. L. Byer, "Sub-hertz relative frequency stabilization of two-diode laser-pumped Nd:YAG lasers locked to a Fabry—Perot interferometer," IEEE J. Quantum Electron. 28, 1106 (1992).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

T. J. Kane, "Intensity noise in diode-pumped single-frequency Nd:YAG lasers and its control by electronic feedback," IEEE Photon. Technol. Lett. 2, 244 (1990).
[CrossRef]

T. Day, A. D. Farinas, and R. L. Byer, "Demonstration of a low bandwidth 1.06 μm optical phase-locked loop for coherent homodyne communication," IEEE Photon. Technol. Lett. 2, 294 (1990).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

M. E. Hines, J. R. Collinet, and J. G. Ondria, "FM noise suppression of an injection phase-locked oscillator," IEEE Trans. Microwave Theory Tech. MTT-16, 738 (1968).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Lett. (4)

Phys. Rev. (1)

A. L. Schawlow and C. H. Townes, "Infrared and optical masers," Phys. Rev. 112, 1940 (1958).
[CrossRef]

Proc. IRE (1)

R. Adler, "A study of locking phenomena in oscillators," Proc. IRE 34, 351 (1946).
[CrossRef]

Science (1)

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Voght, R. Weiss, S. Whitcomb, and M. E. Zucker, "LIGO: the laser interferometer gravitational-wave observatory," Science 256, 325 (1992).
[CrossRef] [PubMed]

Other (5)

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), p. 923.

Ref. 9, p. 1129.

Ref. 9, p. 954.

W. Koechner, Solid State Laser Engineering, 3rd ed. (Springer-Verlag, New York, 1992), p. 79.
[CrossRef]

J. Smith, Modern Communications Circuits (McGraw-Hill, New York, 1986), p. 295.

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Figures (10)

Fig. 1
Fig. 1

Magnitude of the phase-modulation transfer functions. Response of the injection-locked laser to phase variation in the master laser, Hm(ωmod), and phase variation in the free-running slave laser, Hs(ωmod), plotted versus normalized frequency.

Fig. 2
Fig. 2

Magnitude of the pump modulation transfer function Gp(ωmod) plotted for three master-slave power ratios, assuming a Nd:YAG laser pumped three times above threshold, with total cavity losses of 17% and 750-MHz axial mode spacing. The presence of the injected signal damps the relaxation-oscillation peak.

Fig. 3
Fig. 3

Damping ratio ζ of the relaxation oscillations plotted versus the injection-locking power ratio. The system is over-damped for all practical injection-locking power ratios.

Fig. 4
Fig. 4

Magnitude of the master-laser modulation transfer function Gm(ωmod) plotted for two master–slave power ratios. The effect of intensity noise in the master laser is greatest near the slave laser’s relaxation-oscillation frequency but is kept small by the damping effect of the injected signal.

Fig. 5
Fig. 5

Schematic of the injection-locked laser. The 300-mW nonplanar ring oscillator (NPRO) is used to injection lock the 5-W laser-diode-pumped ring laser. The optical isolator prevents perturbation of the master laser by the slave laser. A piezoelectric transducer (PZT) is used to vary the free-running frequency of the slave laser.

Fig. 6
Fig. 6

Schematic of the laser diagnostics. The laser diagnostic output is used as the transmitter in a phase-locked loop. The control signal applied to the local oscillator, the 40-mW nonplanar ring oscillator (NPRO), is a measure of the frequency noise of the transmitter.

Fig. 7
Fig. 7

Measured spectral density of frequency noise. The frequency noise of the master laser is reproduced in the injection-locked output below 1 kHz. The frequency noise of the local oscillator, which sets a resolution limit on the experiment, dominates above 1 kHz.

Fig. 8
Fig. 8

Calculated spectral density of frequency noise. Phase-modulation transfer functions are used to model the spectral density of frequency noise of the injection-locked slave laser, given the measured master-laser and free-running slave-laser frequency noise. Low frequency noise in the free-running slave laser is suppressed.

Fig. 9
Fig. 9

Measured relative intensity noise, RIN, in the master, the free-running slave, and the injection-locked slave lasers. The free-running slave laser’s intensity-noise peak (near 100 kHz) is eliminated in the injection-locked slave laser.

Fig. 10
Fig. 10

Calculated relative intensity noise, RIN, in the three lasers. Intensity-noise transfer functions are used to model the intensity noise in the injection-locked slave laser, given the measured intensity noise in the master and the free-running slave lasers.

Equations (23)

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d ( t ) d t + [ γ c / 2 + j ( ω ω c ) ] ( t ) = j ω 2 P ( t ) + ( 2 γ e V c ) 1 / 2 e ( t ) ,
d P ( t ) d t + [ Δ ω a / 2 + j ( ω ω a ) ] P ( t ) = j κ 2 ω V c Δ N ( t ) ( t ) ,
d Δ N ( t ) d t + γ 2 Δ N ( t ) + R p ( t ) = j V c 4 [ ( t ) P * ( t ) * ( t ) P ( t ) ] ,
P ( t ) j κ ω Δ ω a V c 1 1 + 2 j ( ω ω a ) / Δ ω a Δ N ( t ) ( t ) .
d E ( t ) d t + γ c γ 0 2 E ( t ) = γ e E m ( t ) cos [ ϕ ( t ) ϕ m ( t ) ] ,
d ϕ ( t ) d t + ω m ω s ( t ) = γ e E m ( t ) E ( t ) sin [ ϕ ( t ) ϕ m ( t ) ] ,
d ϕ ( t ) d t + ω m ω s ( t ) = ω lock sin [ ϕ ( t ) ϕ m ( t ) ] ,
ω lock γ e E m E s T oc Δ f ax P m / P s ,
ω lock 2 π = 1 MHz ( T 10 % ) ( Δ f ax 600 MHz ) ( 100 P m P s ) 1 / 2 .
Δ ϕ ϕ ϕ m = sin 1 ( ω s ω m ω lock ) .
H m ( ω mod ) ϕ ˆ ϕ m = 1 1 + j ω mod ω lock cos Δ ϕ .
H s ( ω mod ) ϕ ˆ ϕ s = j ω mod ω lock cos Δ ϕ 1 + j ω mod ω lock cos Δ ϕ .
S f , i l ( ω mod ) = [ | H m ( ω mod ) S f , m ( ω mod ) | 2 + | H s ( ω mod ) S f , s ( ω mod ) | 2 ] 1 / 2 .
d n ( t ) d t = K N ( t ) n ( t ) γ c n ( t ) + ( V c 2 ω ) ( 2 γ e V c ) 1 / 2 × [ e ( t ) * ( t ) + e * ( t ) ( t ) ] .
d n ( t ) d t = K N ( t ) n ( t ) γ c n ( t ) + 2 Δ f ax cos Δ ϕ [ n ( t ) n e ( t ) ] 1 / 2 ,
d N ( t ) d t = R p ( t ) γ 2 N ( t ) K N ( t ) n ( t ) .
N 0 = N th 2 Δ f ax K cos Δ ϕ n e / n ss
n 0 = n ss + 2 r γ 2 γ c Δ f ax K cos Δ ϕ n e / n ss
G p ( ω mod ) n ˆ 1 R p 1 = 1 γ c ω sp 2 ω sp 2 ω mod 2 + 2 j ω mod γ sp ,
ω sp 2 = ( r 1 ) γ 2 γ c + 2 ( r + 1 ) Δ f ax γ 2 cos Δ ϕ n e / n ss ( r 1 ) γ 2 γ c ,
γ sp = r γ 2 2 + Δ f ax ( r γ 2 γ c + 1 ) cos Δ ϕ n e n ss r γ 2 2 + γ e ( r γ 2 γ c + 1 ) cos Δ ϕ P m P s
G m ( ω mod ) Δ P / P Δ P m / P m = r γ 2 γ e cos Δ ϕ P m P s × ( 1 + j ω mod r γ 2 ) ω sp 2 ω mod 2 + 2 j ω mod γ sp ,
S RIN , i l ( ω mod ) = [ | G p ( ω mod ) S N , p ( ω mod ) | 2 + | G m ( ω mod ) S RIN , m ( ω mod ) | 2 ] 1 / 2

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