Abstract

The behavior of a single-mode class-C laser in the presence of an injected signal is investigated theoretically in detail. We have determined the ranges of input signal strength and the frequency detuning for which the oscillations of the cavity electric field and the atomic induced dipole moment are simultaneously stable and have only one oscillation component with the frequency equal to that of the input signal (injection-locked state). One can reproduce all the previous results corresponding to class-A and class-B lasers from our results by using their relevant conditions. Finally, the energy conservation law is demonstrated for both the below-threshold and the injection-locked states of class-C laser amplifiers.

© 2005 Optical Society of America

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  1. H. Risken, K. Nummedal, “Self-pulsing in lasers,” J. Appl. Phys. 39, 4662–4672 (1968).
    [CrossRef]
  2. R. Graham, H. Haken, “Quantum theory of light propagation in a fluctuating laser-active medium,” Z. Phys. 213, 420–426 (1968).
    [CrossRef]
  3. M. B. Spencer, W. E. Lamb, “Laser with a transmitting window,” Phys. Rev. A 5, 884–892 (1972).
    [CrossRef]
  4. D. K. Bandy, L. M. Narducci, L. A. Lugiato, “Coexisting attractors in a laser with an injected signal,” J. Opt. Soc. Am. B 2, 148–155 (1985).
    [CrossRef]
  5. Y. K. Park, G. Giuliani, R. L. Byer, “Stable single-axial-mode operation of an unstable-resonator Nd:YAG oscillator by injection locking,” Opt. Lett. 5, 96–98 (1980).
    [CrossRef] [PubMed]
  6. A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986).
  7. J. R. Tredicce, F. T. Arecchi, G. L. Lippi, G. P. Puccioni, “Instabilities in lasers with an injected signal,” J. Opt. Soc. Am. B 2, 173–183 (1985).
    [CrossRef]
  8. R. Loudon, M. Harris, T. J. Shepherd, J. M. Vaughan, “Laser-amplifier gain and noise,” Phys. Rev. A 48, 681–701 (1993).
    [CrossRef] [PubMed]
  9. C. Mayol, R. Toral, C. Mirasso, N. M. Natiello, “Class-A lasers with injected signal: bifurcation set and Lyapunov-potential function,” Phys. Rev. A 66, 013808–013819 (2002).
    [CrossRef]
  10. G. P. Agrawal, N. K. Dutta, Long-Wavelength Semiconductor Lasers (Van Nostrand Reinhold, New York, 1986).
  11. C. O. Weiss, R. Vilaseca, Dynamics of Lasers (VCH, Weinhemim, Country, 1991).
  12. C. C. Davis, Lasers and Electro-Optics (Cambridge U. Press, Cambridge, UK, 2000), p. 246.
  13. J. Jahanpanah, R. Loudon, “Theory of laser-amplifier linear gain,” Phys. Rev. A 56, 2255–2266 (1997).
    [CrossRef]
  14. F. Castelli, L. A. Lugiato, R. Pirovano, “Rabi resonance in frequency conversion by four-wave mixing in lasers and its connection with the multimode laser instability,” Phys. Rev. A 49, 4031–4037 (1994).
    [CrossRef] [PubMed]
  15. J. Jahanpanah, R. Loudon, “Theory of laser-amplifier injection-locking,” Phys. Rev. A 54, 5210–5226 (1996).
    [CrossRef] [PubMed]
  16. M. J. Collett, C. W. Gardiner, “Squeezing of intracavity and traveling-wave light fields produced in parametric amplification,” Phys. Rev. A 30, 1386–1391 (1984).
    [CrossRef]
  17. J. M. Liu, T. B. Simpson, “Four-wave mixing and optical modulation in a semiconductor laser,” IEEE J. Quantum Electron. 30, 957–965 (1994).
    [CrossRef]
  18. G. N. Pearson, M. Harris, C. A. Hill, J. M. Vaughan, A. M. Hornby, “Inter-transverse-mode injection locking and subthreshold gain measurements in a CO2 waveguide laser,” IEEE J. Quantum Electron. 31, 1064–1068 (1995).
    [CrossRef]

2002

C. Mayol, R. Toral, C. Mirasso, N. M. Natiello, “Class-A lasers with injected signal: bifurcation set and Lyapunov-potential function,” Phys. Rev. A 66, 013808–013819 (2002).
[CrossRef]

1997

J. Jahanpanah, R. Loudon, “Theory of laser-amplifier linear gain,” Phys. Rev. A 56, 2255–2266 (1997).
[CrossRef]

1996

J. Jahanpanah, R. Loudon, “Theory of laser-amplifier injection-locking,” Phys. Rev. A 54, 5210–5226 (1996).
[CrossRef] [PubMed]

1995

G. N. Pearson, M. Harris, C. A. Hill, J. M. Vaughan, A. M. Hornby, “Inter-transverse-mode injection locking and subthreshold gain measurements in a CO2 waveguide laser,” IEEE J. Quantum Electron. 31, 1064–1068 (1995).
[CrossRef]

1994

J. M. Liu, T. B. Simpson, “Four-wave mixing and optical modulation in a semiconductor laser,” IEEE J. Quantum Electron. 30, 957–965 (1994).
[CrossRef]

F. Castelli, L. A. Lugiato, R. Pirovano, “Rabi resonance in frequency conversion by four-wave mixing in lasers and its connection with the multimode laser instability,” Phys. Rev. A 49, 4031–4037 (1994).
[CrossRef] [PubMed]

1993

R. Loudon, M. Harris, T. J. Shepherd, J. M. Vaughan, “Laser-amplifier gain and noise,” Phys. Rev. A 48, 681–701 (1993).
[CrossRef] [PubMed]

1985

1984

M. J. Collett, C. W. Gardiner, “Squeezing of intracavity and traveling-wave light fields produced in parametric amplification,” Phys. Rev. A 30, 1386–1391 (1984).
[CrossRef]

1980

1972

M. B. Spencer, W. E. Lamb, “Laser with a transmitting window,” Phys. Rev. A 5, 884–892 (1972).
[CrossRef]

1968

H. Risken, K. Nummedal, “Self-pulsing in lasers,” J. Appl. Phys. 39, 4662–4672 (1968).
[CrossRef]

R. Graham, H. Haken, “Quantum theory of light propagation in a fluctuating laser-active medium,” Z. Phys. 213, 420–426 (1968).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, N. K. Dutta, Long-Wavelength Semiconductor Lasers (Van Nostrand Reinhold, New York, 1986).

Arecchi, F. T.

Bandy, D. K.

Byer, R. L.

Castelli, F.

F. Castelli, L. A. Lugiato, R. Pirovano, “Rabi resonance in frequency conversion by four-wave mixing in lasers and its connection with the multimode laser instability,” Phys. Rev. A 49, 4031–4037 (1994).
[CrossRef] [PubMed]

Collett, M. J.

M. J. Collett, C. W. Gardiner, “Squeezing of intracavity and traveling-wave light fields produced in parametric amplification,” Phys. Rev. A 30, 1386–1391 (1984).
[CrossRef]

Davis, C. C.

C. C. Davis, Lasers and Electro-Optics (Cambridge U. Press, Cambridge, UK, 2000), p. 246.

Dutta, N. K.

G. P. Agrawal, N. K. Dutta, Long-Wavelength Semiconductor Lasers (Van Nostrand Reinhold, New York, 1986).

Gardiner, C. W.

M. J. Collett, C. W. Gardiner, “Squeezing of intracavity and traveling-wave light fields produced in parametric amplification,” Phys. Rev. A 30, 1386–1391 (1984).
[CrossRef]

Giuliani, G.

Graham, R.

R. Graham, H. Haken, “Quantum theory of light propagation in a fluctuating laser-active medium,” Z. Phys. 213, 420–426 (1968).
[CrossRef]

Haken, H.

R. Graham, H. Haken, “Quantum theory of light propagation in a fluctuating laser-active medium,” Z. Phys. 213, 420–426 (1968).
[CrossRef]

Harris, M.

G. N. Pearson, M. Harris, C. A. Hill, J. M. Vaughan, A. M. Hornby, “Inter-transverse-mode injection locking and subthreshold gain measurements in a CO2 waveguide laser,” IEEE J. Quantum Electron. 31, 1064–1068 (1995).
[CrossRef]

R. Loudon, M. Harris, T. J. Shepherd, J. M. Vaughan, “Laser-amplifier gain and noise,” Phys. Rev. A 48, 681–701 (1993).
[CrossRef] [PubMed]

Hill, C. A.

G. N. Pearson, M. Harris, C. A. Hill, J. M. Vaughan, A. M. Hornby, “Inter-transverse-mode injection locking and subthreshold gain measurements in a CO2 waveguide laser,” IEEE J. Quantum Electron. 31, 1064–1068 (1995).
[CrossRef]

Hornby, A. M.

G. N. Pearson, M. Harris, C. A. Hill, J. M. Vaughan, A. M. Hornby, “Inter-transverse-mode injection locking and subthreshold gain measurements in a CO2 waveguide laser,” IEEE J. Quantum Electron. 31, 1064–1068 (1995).
[CrossRef]

Jahanpanah, J.

J. Jahanpanah, R. Loudon, “Theory of laser-amplifier linear gain,” Phys. Rev. A 56, 2255–2266 (1997).
[CrossRef]

J. Jahanpanah, R. Loudon, “Theory of laser-amplifier injection-locking,” Phys. Rev. A 54, 5210–5226 (1996).
[CrossRef] [PubMed]

Lamb, W. E.

M. B. Spencer, W. E. Lamb, “Laser with a transmitting window,” Phys. Rev. A 5, 884–892 (1972).
[CrossRef]

Lippi, G. L.

Liu, J. M.

J. M. Liu, T. B. Simpson, “Four-wave mixing and optical modulation in a semiconductor laser,” IEEE J. Quantum Electron. 30, 957–965 (1994).
[CrossRef]

Loudon, R.

J. Jahanpanah, R. Loudon, “Theory of laser-amplifier linear gain,” Phys. Rev. A 56, 2255–2266 (1997).
[CrossRef]

J. Jahanpanah, R. Loudon, “Theory of laser-amplifier injection-locking,” Phys. Rev. A 54, 5210–5226 (1996).
[CrossRef] [PubMed]

R. Loudon, M. Harris, T. J. Shepherd, J. M. Vaughan, “Laser-amplifier gain and noise,” Phys. Rev. A 48, 681–701 (1993).
[CrossRef] [PubMed]

Lugiato, L. A.

F. Castelli, L. A. Lugiato, R. Pirovano, “Rabi resonance in frequency conversion by four-wave mixing in lasers and its connection with the multimode laser instability,” Phys. Rev. A 49, 4031–4037 (1994).
[CrossRef] [PubMed]

D. K. Bandy, L. M. Narducci, L. A. Lugiato, “Coexisting attractors in a laser with an injected signal,” J. Opt. Soc. Am. B 2, 148–155 (1985).
[CrossRef]

Mayol, C.

C. Mayol, R. Toral, C. Mirasso, N. M. Natiello, “Class-A lasers with injected signal: bifurcation set and Lyapunov-potential function,” Phys. Rev. A 66, 013808–013819 (2002).
[CrossRef]

Mirasso, C.

C. Mayol, R. Toral, C. Mirasso, N. M. Natiello, “Class-A lasers with injected signal: bifurcation set and Lyapunov-potential function,” Phys. Rev. A 66, 013808–013819 (2002).
[CrossRef]

Narducci, L. M.

Natiello, N. M.

C. Mayol, R. Toral, C. Mirasso, N. M. Natiello, “Class-A lasers with injected signal: bifurcation set and Lyapunov-potential function,” Phys. Rev. A 66, 013808–013819 (2002).
[CrossRef]

Nummedal, K.

H. Risken, K. Nummedal, “Self-pulsing in lasers,” J. Appl. Phys. 39, 4662–4672 (1968).
[CrossRef]

Park, Y. K.

Pearson, G. N.

G. N. Pearson, M. Harris, C. A. Hill, J. M. Vaughan, A. M. Hornby, “Inter-transverse-mode injection locking and subthreshold gain measurements in a CO2 waveguide laser,” IEEE J. Quantum Electron. 31, 1064–1068 (1995).
[CrossRef]

Pirovano, R.

F. Castelli, L. A. Lugiato, R. Pirovano, “Rabi resonance in frequency conversion by four-wave mixing in lasers and its connection with the multimode laser instability,” Phys. Rev. A 49, 4031–4037 (1994).
[CrossRef] [PubMed]

Puccioni, G. P.

Risken, H.

H. Risken, K. Nummedal, “Self-pulsing in lasers,” J. Appl. Phys. 39, 4662–4672 (1968).
[CrossRef]

Shepherd, T. J.

R. Loudon, M. Harris, T. J. Shepherd, J. M. Vaughan, “Laser-amplifier gain and noise,” Phys. Rev. A 48, 681–701 (1993).
[CrossRef] [PubMed]

Siegman, A. E.

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

Simpson, T. B.

J. M. Liu, T. B. Simpson, “Four-wave mixing and optical modulation in a semiconductor laser,” IEEE J. Quantum Electron. 30, 957–965 (1994).
[CrossRef]

Spencer, M. B.

M. B. Spencer, W. E. Lamb, “Laser with a transmitting window,” Phys. Rev. A 5, 884–892 (1972).
[CrossRef]

Toral, R.

C. Mayol, R. Toral, C. Mirasso, N. M. Natiello, “Class-A lasers with injected signal: bifurcation set and Lyapunov-potential function,” Phys. Rev. A 66, 013808–013819 (2002).
[CrossRef]

Tredicce, J. R.

Vaughan, J. M.

G. N. Pearson, M. Harris, C. A. Hill, J. M. Vaughan, A. M. Hornby, “Inter-transverse-mode injection locking and subthreshold gain measurements in a CO2 waveguide laser,” IEEE J. Quantum Electron. 31, 1064–1068 (1995).
[CrossRef]

R. Loudon, M. Harris, T. J. Shepherd, J. M. Vaughan, “Laser-amplifier gain and noise,” Phys. Rev. A 48, 681–701 (1993).
[CrossRef] [PubMed]

Vilaseca, R.

C. O. Weiss, R. Vilaseca, Dynamics of Lasers (VCH, Weinhemim, Country, 1991).

Weiss, C. O.

C. O. Weiss, R. Vilaseca, Dynamics of Lasers (VCH, Weinhemim, Country, 1991).

IEEE J. Quantum Electron.

J. M. Liu, T. B. Simpson, “Four-wave mixing and optical modulation in a semiconductor laser,” IEEE J. Quantum Electron. 30, 957–965 (1994).
[CrossRef]

G. N. Pearson, M. Harris, C. A. Hill, J. M. Vaughan, A. M. Hornby, “Inter-transverse-mode injection locking and subthreshold gain measurements in a CO2 waveguide laser,” IEEE J. Quantum Electron. 31, 1064–1068 (1995).
[CrossRef]

J. Appl. Phys.

H. Risken, K. Nummedal, “Self-pulsing in lasers,” J. Appl. Phys. 39, 4662–4672 (1968).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Lett.

Phys. Rev. A

M. B. Spencer, W. E. Lamb, “Laser with a transmitting window,” Phys. Rev. A 5, 884–892 (1972).
[CrossRef]

R. Loudon, M. Harris, T. J. Shepherd, J. M. Vaughan, “Laser-amplifier gain and noise,” Phys. Rev. A 48, 681–701 (1993).
[CrossRef] [PubMed]

C. Mayol, R. Toral, C. Mirasso, N. M. Natiello, “Class-A lasers with injected signal: bifurcation set and Lyapunov-potential function,” Phys. Rev. A 66, 013808–013819 (2002).
[CrossRef]

J. Jahanpanah, R. Loudon, “Theory of laser-amplifier linear gain,” Phys. Rev. A 56, 2255–2266 (1997).
[CrossRef]

F. Castelli, L. A. Lugiato, R. Pirovano, “Rabi resonance in frequency conversion by four-wave mixing in lasers and its connection with the multimode laser instability,” Phys. Rev. A 49, 4031–4037 (1994).
[CrossRef] [PubMed]

J. Jahanpanah, R. Loudon, “Theory of laser-amplifier injection-locking,” Phys. Rev. A 54, 5210–5226 (1996).
[CrossRef] [PubMed]

M. J. Collett, C. W. Gardiner, “Squeezing of intracavity and traveling-wave light fields produced in parametric amplification,” Phys. Rev. A 30, 1386–1391 (1984).
[CrossRef]

Z. Phys.

R. Graham, H. Haken, “Quantum theory of light propagation in a fluctuating laser-active medium,” Z. Phys. 213, 420–426 (1968).
[CrossRef]

Other

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

G. P. Agrawal, N. K. Dutta, Long-Wavelength Semiconductor Lasers (Van Nostrand Reinhold, New York, 1986).

C. O. Weiss, R. Vilaseca, Dynamics of Lasers (VCH, Weinhemim, Country, 1991).

C. C. Davis, Lasers and Electro-Optics (Cambridge U. Press, Cambridge, UK, 2000), p. 246.

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Figures (8)

Fig. 1
Fig. 1

Representation of the laser cavity showing the notation for field amplitudes inside and outside the cavity, the rates of the pumping energy and spontaneous emission, and the mirror loss rates.

Fig. 2
Fig. 2

Normalized cavity mean photon number as a function of the normalized input flux at the CAF detunings δ CA , indicated on the curves, for a below-threshold amplifier ( C = 0.5 ) , zero normalized signal detunings ω = 0 , the parameter m = γ C / γ = 1 , and a symmetrical cavity ( γ 1 = γ 2 = γ C ) .

Fig. 3
Fig. 3

Normalized population inversion for the same parameters as in Fig. 2.

Fig. 4
Fig. 4

Variation of transmission intensity gain with CAF detuning in the laser below threshold with the pumping rates C indicated on the curves for ω = 0 , | β S | 2 = 0.5 γ C n S , m = 1 , and a symmetrical cavity.

Fig. 5
Fig. 5

Energy conservation in a below-threshold class-C laser amplifier ( C = 0.5 ) with a symmetrical cavity and parameter m = 1 , showing the variations of the output fluxes and change in spontaneous emission rate with (a) CAF detuning δ CA at fixed normalized signal detuning ( ω = 0 ) and with (b) normalized signal detuning ω at fixed CAF detuning ( δ CA = 0.5 ) . The rates are normalized by the value of the input signal flux, chosen to be | β S | 2 = γ C n S , so output fluxes are expressed as gains.

Fig. 6
Fig. 6

Variation of the normalized intracavity mean photon number with normalized input signal flux in the injection-locked state of a class-C laser amplifier with C = 2 , δ CA = 0 , γ = γ = γ C   ( m = n = 1 ) and for the values of detuning ω indicated. The dashed curves are the boundaries of the stability region shown by the shaded area.

Fig. 7
Fig. 7

Variation of the normalized population inversion with normalized input signal flux for the same values of the laser parameters shown in Fig. 6.

Fig. 8
Fig. 8

Variation of transmission and reflection energy gains through the cavity mirrors together with the spontaneous-emission radiation rate in all directions with CAF detuning δ CA for the fixed signal detuning ( ω = 0 ) , | β S | 2 = γ C n S , m = 1 and at the normalized pumping rate C = 2 .

Equations (54)

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α out = γ 1 1 / 2 α ,
β out = γ 2 1 / 2 α - β in .
γ C = γ 1 + γ 2 2 .
G T = α out β in 2 = γ 1 | α S | 2 | β S | 2 ,
G R = β out β in 2 = γ 2 | α S | 2 | β S | 2 - γ 2 1 / 2 α S * β S + α S β S * | β S | 2 + 1 ,
α ˙ + ( γ C + i ω C ) α = gd + γ 2 1 / 2 β in ,
D ˙ + γ D = γ D P - g ( α * d + α d * ) ,
d ˙ + ( γ + i ω A ) d = g α D ,
D = D 0 = γ γ C g 2   ( 1 + δ CA 2 ) ,
δ CA = ω C - ω A γ C + γ
| α | 2 = | α L | 2 = D P D 0 - 1 γ γ 2 g 2   ( 1 + δ CA 2 ) = ( C - 1 ) n s ,
n s = γ γ 2 g 2   ( 1 + δ CA 2 ) .
ω L = γ ω C + γ C ω A γ + γ C = ω C - γ C δ CA = ω A + γ δ CA .
β in = β S exp ( - i ω S t ) = β S exp [ - i ( ω L + ω ) t ] ,
α = α S exp ( - i ω S t ) = α S exp [ - i ( ω L + ω ) t ] ,
d = d S exp ( - i ω S t ) = d S exp [ - i ( ω L + ω ) t ] ,
[ γ C + i ( γ C δ CA - ω ) ] α S = gd S + γ 2 1 / 2 β S ,
γ D = γ D P - g ( α S * d S + α S d S * ) ,
[ γ - i ( γ δ CA + ω ) ] d S = gD α S .
d S = γ 2 1 / 2 gD β S [ γ C + i ( γ C δ CA - ω ) ] [ γ - i ( γ δ CA + ω ) ] - g 2 D .
α S = [ γ - i ( γ δ CA + ω ) ] γ 2 1 / 2 β S [ γ C + i ( γ C δ CA - ω ) ] [ γ - i ( γ δ CA + ω ) ] - g 2 D .
( 1 + δ CA 2 ) 2 D 3 - ( 1 + δ CA 2 ) [ ( C + 2 ) ( 1 + δ CA 2 ) + 2 ( m - 1 ) δ CA ω - 2 m ω 2 ] D 2 + { [ 1 + δ CA 2 + ( m - 1 ) δ CA ω - m ω 2 ] 2 + 2 C ( 1 + δ CA 2 ) [ 1 + δ CA 2 + ( m - 1 ) δ CA ω - m ω 2 ] + ( m + 1 ) 2 ω 2 + ( 1 + δ CA 2 ) ( γ 2 | β S | 2 / γ C 2 n S ) } D - C { [ 1 + δ CA 2 + ( m - 1 ) δ CA ω - m ω 2 ] 2 + ( m + 1 ) 2 ω 2 } = 0 ,
α S = | α S | exp ( i ϕ S ) , β S = | β S | exp ( i ϕ in ) .
| α S | 2 n S = 1 + [ δ CA + m ω ] 2 [ ( 1 + δ CA 2 ) ( 1 - D ) + ( m - 1 ) δ CA ω - m ω 2 ] 2 + ( m + 1 ) 2 ω 2 γ 2 | β S | 2 γ C 2 n S .
D 3 - ( C + 2 ) D 2 + ( 1 + 2 C + ω 2 + γ 2 | β S | 2 / γ C 2 n S ) D
- C ( 1 + ω 2 ) = 0 ,
| α S | 2 n S = 1 ( 1 - D ) 2 + ω 2 γ 2 | β S | 2 γ C 2 n S ,
| α out | 2 + | β out | 2 + γ D = γ D P + | β in | 2 ,
G T + G R + γ ( D - D P ) | β S | 2 = 1 ,
α S = α S + i α S , d S = d S + id S ,
β S = β S + i β S .
α S α S + δ α S = α S + δ α S 0 exp ( λ t ) ,
α S α S + δ α S = α S + δ α S 0 exp ( λ t ) ,
d S d S + δ d S = d S + δ d S 0 exp ( λ t ) ,
d S d S + δ d S = d S + δ d S 0 exp ( λ t ) ,
D D + δ D = D + δ D 0 ,
λ 5 + b 4 λ 4 + b 3 λ 3 + b 2 λ 2 + b 1 λ + b 0 = 0 ,
λ 1 + λ 2 + λ 3 + λ 4 + λ 5 = - b 4 ,
λ 1 ( λ 2 + λ 3 + λ 4 + λ 5 ) + λ 2 ( λ 3 + λ 4 + λ 5 )
+ λ 3 ( λ 4 + λ 5 ) + λ 4 λ 5 = b 3 ,
λ 1 ( λ 2 λ 3 + λ 2 λ 4 + λ 2 λ 5 + λ 3 λ 4 + λ 3 λ 5 + λ 4 λ 5 )
+ λ 2 ( λ 3 λ 4 + λ 3 λ 5 + λ 4 λ 5 ) + λ 3 λ 4 λ 5 = - b 2 ,
λ 1 ( λ 2 λ 3 λ 4 + λ 2 λ 3 λ 5 + λ 3 λ 4 λ 5 + λ 2 λ 4 λ 5 )
+ λ 2 λ 3 λ 4 λ 5 = b 1 ,
λ 1 λ 2 λ 3 λ 4 λ 5 = - b 0 .
b 0 > 0
b 3 b 4 3 - b 2 b 4 2 + b 1 b 4 - b 0 > 0
( b 1 b 2 - b 3 b 0 ) ( b 3 b 4 - b 2 ) - ( b 1 b 4 - b 0 ) 2 > 0 .
G T + G R + γ ( D - D 0 ) | β S | 2 = 1 + 2 γ C | α L | 2 | β S | 2 = 1 + 2 ( C - 1 ) ( | β S | 2 / γ C n S ) .
b 4 = 2 γ + γ + 2 γ C ,
b 3 = ( γ + γ C ) 2 + 2 ( γ γ + γ γ C + γ γ C ) + ( γ δ CA + ω ) 2 + ( γ C δ CA - ω ) 2 - 2 γ γ C ( 1 + δ CA 2 ) D + γ γ ( 1 + δ CA 2 )   | α S | 2 n S ,
b 2 = γ ( γ + γ C ) 2 + 2 γ γ C ( γ + γ + γ C ) + γ γ γ C ( 1 + δ CA 2 ) C + ( γ + 2 γ C ) × ( γ δ CA + ω ) 2 + ( γ + 2 γ ) ( γ C δ CA - ω ) 2 - 2 γ γ C ( 1 + δ CA 2 ) γ + 3 2   γ + γ C D + γ γ ( γ + 2 γ C ) ( 1 + δ CA 2 )   | α S | 2 n S ,
b 1 = γ γ C ( 2 γ γ + 2 γ γ C + γ γ C ) + γ γ γ C ( γ + γ C ) × ( 1 + δ CA 2 ) ( C - D ) + γ C ( γ C + 2 γ ) ( γ δ CA + ω ) 2 + [ γ ( 2 γ + γ ) + ( γ δ CA + ω ) 2 ] ( γ C δ CA - ω ) 2 - 2 γ γ C ( 1 + δ CA 2 ) [ γ ( γ + γ C ) + γ γ C + ( γ C δ CA - ω ) ( γ δ CA + ω ) - 0.5 γ γ C ( 1 + δ CA 2 ) D ] D + γ γ [ γ C ( 2 γ + γ C ) + ( γ C δ CA - ω ) 2 ] ( 1 + δ CA 2 )   | α S | 2 n S + γ 2 γ γ C ( γ δ CA + ω ) [ ( γ - γ C ) δ CA + 2 ω ] γ 2 + ( γ δ CA + ω ) 2 - 1 × ( 1 + δ CA 2 ) 2 D | α S | 2 n S ,
b 0 = γ [ γ 2 + ( γ δ CA + ω ) 2 ] [ γ C 2 + ( γ C δ CA - ω ) 2 ] + γ 2 γ ( 1 + δ CA 2 ) [ γ C 2 + ( γ C δ CA - ω ) 2 ]   | α S | 2 n S + γ γ γ C ( 1 + δ CA 2 ) [ γ γ C ( 1 + δ CA 2 ) ( 1 - D ) + ( γ C δ CA - ω ) ( γ δ CA + ω ) ] C - 3 γ γ γ C ( 1 + δ CA 2 ) [ γ γ C + ( γ C δ CA - ω ) × ( γ δ CA + ω ) ] D + 2 γ 2 γ γ C 2 ( 1 + δ CA 2 ) 2 D 2 + γ 2 γ γ C ( 1 + δ CA 2 ) 2 × ( γ + γ C ) ( γ δ CA + ω ) ω γ 2 + ( γ δ CA + ω ) 2 - γ C D | α S | 2 n S .

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