Abstract

The phase-conjugate laser (PCL) is analyzed as a network of energy flow with gain, saturation, and loss, Most PCL’s fit into one of three generic forms of energy coupling, defined by how the phase-conjugate mirror is pumped. The self-pumped PCL is a nonlinear feedback system with constraints not present in ordinary lasers. The internal and external powers are roots of a transcendental function that includes arbitrary gain and saturation. The PCL’s are compared for useful output power with an ordinary laser of the same gain/loss. Most PCL’s have maximum output as double-pass amplifiers (weak feedback and high extraction). The externally pumped PCL is inherently inefficient except at high gain. The self-pumped PCL’s are efficient at high and low gain and are multistable optical devices with discrete power levels. The multistability is generic and does not depend on any specific form of the reflectance function. The higher states have not been seen experimentally because they cannot develop from the lowest state but must be induced by hard excitation of the PCL.

© 1983 Optical Society of America

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References

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  1. P. A. Belanger, A. Hardy, and A. E. Siegman, “Resonant modes of optical cavities with phase-conjugate mirrors,” Appl. Opt. 19, 602–609 (1980); A. Hardy, P. A. Belanger, and A. E. Siegman, “Orthogonality properties of phase conjugate optical resonators,” Appl. Opt. 21, 1122–1124 (1982).
    [Crossref] [PubMed]
  2. I. M. Bel’dyugin, M. G. Galusbkin, and E. M. Zemskov, “Properties of resonators with wavefront-reversing mirrors,” Sov. J. Quantum Electron. 9, 20–23 (1979).
    [Crossref]
  3. M. G. Reznikov and A. I. Khizhnyak, “Properties of a resonator with a wavefront-reversing mirror,” Sov. J. Quantum Electron. 10, 764–765 (1980).
    [Crossref]
  4. I. M. Bel’dyugin and E. M. Zemskov, “Calculation of the field in a laser resonator with a wave-front-reversing mirror,” Sov. J. Quantum Electron. 10, 764–765 (1980).
    [Crossref]
  5. G. P. Agrawal and J. L. Boulnois, “Waveguide resonators with a phase-conjugate mirror,” Opt. Lett. 7, 159–161 (1982); J. L. Boulnois and G. P. Agrawal, “Mode discrimination and coupling losses in rectangular-waveguide resonators with conventional and phase-conjugate mirrors,” J. Opt. Soc. Am. 72, 853–860 (1982).
    [Crossref] [PubMed]
  6. J. F. Lam and W. P. Brown, “Optical resonators with phase-conjugate mirrors,” Opt. Lett. 5, 61–63 (1980).
    [Crossref] [PubMed]
  7. J. Feinberg and R. Hellwarth, “Phase-conjugating miror with continuous-wave gain,” Opt. Lett. 5, 519–521 (1980); errata 6, 257 (1981).
    [Crossref]
  8. J. Auyeung, D. Fekete, D. M. Pepper, and A. Yariv, “A theoretical and experimental investigation of the modes of optical resonators with phase-conjugate mirrors,” IEEE J. Quantum Electron. QE-15, 1180–1188 (1979).
    [Crossref]
  9. S. A. Lesnik, M. S. Soskin, and A. I. Khiznyak, “Laser with a stimulated-Brillouin-scattering complex-conjugate mirror,” Sov. Phys. Tech. Phys. 24, 1249–1250 (1979).
  10. A. Vanherzeele, J. L. Van Eck, and A. E. Siegman, “Mode-locked laser oscillation using self-pumped phase-conjugate reflection,” Opt. Lett. 6, 467–469 (1981).
    [Crossref] [PubMed]
  11. R. C. Lind and D. G. Steel, “Demonstration of the longitudinal modes and aberration-correction properties of a continuous-wave dye laser with a phase-conjugate mirror,” Opt. Lett. 6, 554–556 (1981).
    [Crossref] [PubMed]
  12. M. C. Gower and R. G. Caro, “KrF laser with a phase-conjugate Brillouin mirror,” Opt. Lett. 7, 162–164 (1982).
    [Crossref] [PubMed]
  13. B. Fischer, M. Cronin-Golomb, J. O. White, and A. Yariv, “Coherent oscillation by self-induced gratings in the photorefractive crystal BaTiO3,” Appl. Phys. Lett. 40, 450–452 (1982).
    [Crossref]
  14. R. K. Jain and G. J. Dunning, “Spatial and temporal properties of a continuous-wave phase-conjugate resonator based on the photorefractive crystal BaTiO3,” Opt. Lett. 7, 420–422 (1982).
    [Crossref] [PubMed]
  15. W. M. Grossman and D. M. Shemwell, “Practical constraints on phase conjugation by four-wave mixing,” presented at Lasers ’79 Conference.
  16. W. W. Rigrod, “Saturation effects in high-gain lasers,” J. Appl. Phys. 36, 2487–2490 (1965).
    [Crossref]
  17. A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975), Chap. 9.
  18. R. Hellwarth, “Generation of time-reversed wave fronts by nonlinear refraction,” J. Opt. Soc. Am. 67, 1–3 (1977).
    [Crossref]
  19. A. Yariv and D. M. Pepper, “Amplified reflection, phase conjugation, and oscillation in degenerate four-wave mixing,” Opt. Lett. 1, 16–18 (1977).
    [Crossref] [PubMed]
  20. J. H. Marburger and J. F. Lam, “Nonlinear theory of degenerate four-wave mixing,” Appl. Phys. Lett. 34, 389–391 (1979); “Effect of nonlinear index changes on degenerate four-wave mixing,” Appl. Phys. Lett. 35, 249–251 (1979).
    [Crossref]
  21. L. Casperson and A. Yariv, “The time behavior and spectra of relaxation oscillations in a high-gain laser,” IEEE J. Quantum Electron. QE-8, 69–73 (1972).
    [Crossref]

1982 (4)

1981 (2)

1980 (5)

1979 (4)

I. M. Bel’dyugin, M. G. Galusbkin, and E. M. Zemskov, “Properties of resonators with wavefront-reversing mirrors,” Sov. J. Quantum Electron. 9, 20–23 (1979).
[Crossref]

J. Auyeung, D. Fekete, D. M. Pepper, and A. Yariv, “A theoretical and experimental investigation of the modes of optical resonators with phase-conjugate mirrors,” IEEE J. Quantum Electron. QE-15, 1180–1188 (1979).
[Crossref]

S. A. Lesnik, M. S. Soskin, and A. I. Khiznyak, “Laser with a stimulated-Brillouin-scattering complex-conjugate mirror,” Sov. Phys. Tech. Phys. 24, 1249–1250 (1979).

J. H. Marburger and J. F. Lam, “Nonlinear theory of degenerate four-wave mixing,” Appl. Phys. Lett. 34, 389–391 (1979); “Effect of nonlinear index changes on degenerate four-wave mixing,” Appl. Phys. Lett. 35, 249–251 (1979).
[Crossref]

1977 (2)

1972 (1)

L. Casperson and A. Yariv, “The time behavior and spectra of relaxation oscillations in a high-gain laser,” IEEE J. Quantum Electron. QE-8, 69–73 (1972).
[Crossref]

1965 (1)

W. W. Rigrod, “Saturation effects in high-gain lasers,” J. Appl. Phys. 36, 2487–2490 (1965).
[Crossref]

Agrawal, G. P.

Auyeung, J.

J. Auyeung, D. Fekete, D. M. Pepper, and A. Yariv, “A theoretical and experimental investigation of the modes of optical resonators with phase-conjugate mirrors,” IEEE J. Quantum Electron. QE-15, 1180–1188 (1979).
[Crossref]

Bel’dyugin, I. M.

I. M. Bel’dyugin and E. M. Zemskov, “Calculation of the field in a laser resonator with a wave-front-reversing mirror,” Sov. J. Quantum Electron. 10, 764–765 (1980).
[Crossref]

I. M. Bel’dyugin, M. G. Galusbkin, and E. M. Zemskov, “Properties of resonators with wavefront-reversing mirrors,” Sov. J. Quantum Electron. 9, 20–23 (1979).
[Crossref]

Belanger, P. A.

Boulnois, J. L.

Brown, W. P.

Caro, R. G.

Casperson, L.

L. Casperson and A. Yariv, “The time behavior and spectra of relaxation oscillations in a high-gain laser,” IEEE J. Quantum Electron. QE-8, 69–73 (1972).
[Crossref]

Cronin-Golomb, M.

B. Fischer, M. Cronin-Golomb, J. O. White, and A. Yariv, “Coherent oscillation by self-induced gratings in the photorefractive crystal BaTiO3,” Appl. Phys. Lett. 40, 450–452 (1982).
[Crossref]

Dunning, G. J.

Feinberg, J.

Fekete, D.

J. Auyeung, D. Fekete, D. M. Pepper, and A. Yariv, “A theoretical and experimental investigation of the modes of optical resonators with phase-conjugate mirrors,” IEEE J. Quantum Electron. QE-15, 1180–1188 (1979).
[Crossref]

Fischer, B.

B. Fischer, M. Cronin-Golomb, J. O. White, and A. Yariv, “Coherent oscillation by self-induced gratings in the photorefractive crystal BaTiO3,” Appl. Phys. Lett. 40, 450–452 (1982).
[Crossref]

Galusbkin, M. G.

I. M. Bel’dyugin, M. G. Galusbkin, and E. M. Zemskov, “Properties of resonators with wavefront-reversing mirrors,” Sov. J. Quantum Electron. 9, 20–23 (1979).
[Crossref]

Gower, M. C.

Grossman, W. M.

W. M. Grossman and D. M. Shemwell, “Practical constraints on phase conjugation by four-wave mixing,” presented at Lasers ’79 Conference.

Hardy, A.

Hellwarth, R.

Jain, R. K.

Khizhnyak, A. I.

M. G. Reznikov and A. I. Khizhnyak, “Properties of a resonator with a wavefront-reversing mirror,” Sov. J. Quantum Electron. 10, 764–765 (1980).
[Crossref]

Khiznyak, A. I.

S. A. Lesnik, M. S. Soskin, and A. I. Khiznyak, “Laser with a stimulated-Brillouin-scattering complex-conjugate mirror,” Sov. Phys. Tech. Phys. 24, 1249–1250 (1979).

Lam, J. F.

J. F. Lam and W. P. Brown, “Optical resonators with phase-conjugate mirrors,” Opt. Lett. 5, 61–63 (1980).
[Crossref] [PubMed]

J. H. Marburger and J. F. Lam, “Nonlinear theory of degenerate four-wave mixing,” Appl. Phys. Lett. 34, 389–391 (1979); “Effect of nonlinear index changes on degenerate four-wave mixing,” Appl. Phys. Lett. 35, 249–251 (1979).
[Crossref]

Lesnik, S. A.

S. A. Lesnik, M. S. Soskin, and A. I. Khiznyak, “Laser with a stimulated-Brillouin-scattering complex-conjugate mirror,” Sov. Phys. Tech. Phys. 24, 1249–1250 (1979).

Lind, R. C.

Marburger, J. H.

J. H. Marburger and J. F. Lam, “Nonlinear theory of degenerate four-wave mixing,” Appl. Phys. Lett. 34, 389–391 (1979); “Effect of nonlinear index changes on degenerate four-wave mixing,” Appl. Phys. Lett. 35, 249–251 (1979).
[Crossref]

Pepper, D. M.

J. Auyeung, D. Fekete, D. M. Pepper, and A. Yariv, “A theoretical and experimental investigation of the modes of optical resonators with phase-conjugate mirrors,” IEEE J. Quantum Electron. QE-15, 1180–1188 (1979).
[Crossref]

A. Yariv and D. M. Pepper, “Amplified reflection, phase conjugation, and oscillation in degenerate four-wave mixing,” Opt. Lett. 1, 16–18 (1977).
[Crossref] [PubMed]

Reznikov, M. G.

M. G. Reznikov and A. I. Khizhnyak, “Properties of a resonator with a wavefront-reversing mirror,” Sov. J. Quantum Electron. 10, 764–765 (1980).
[Crossref]

Rigrod, W. W.

W. W. Rigrod, “Saturation effects in high-gain lasers,” J. Appl. Phys. 36, 2487–2490 (1965).
[Crossref]

Shemwell, D. M.

W. M. Grossman and D. M. Shemwell, “Practical constraints on phase conjugation by four-wave mixing,” presented at Lasers ’79 Conference.

Siegman, A. E.

Soskin, M. S.

S. A. Lesnik, M. S. Soskin, and A. I. Khiznyak, “Laser with a stimulated-Brillouin-scattering complex-conjugate mirror,” Sov. Phys. Tech. Phys. 24, 1249–1250 (1979).

Steel, D. G.

Van Eck, J. L.

Vanherzeele, A.

White, J. O.

B. Fischer, M. Cronin-Golomb, J. O. White, and A. Yariv, “Coherent oscillation by self-induced gratings in the photorefractive crystal BaTiO3,” Appl. Phys. Lett. 40, 450–452 (1982).
[Crossref]

Yariv, A.

B. Fischer, M. Cronin-Golomb, J. O. White, and A. Yariv, “Coherent oscillation by self-induced gratings in the photorefractive crystal BaTiO3,” Appl. Phys. Lett. 40, 450–452 (1982).
[Crossref]

J. Auyeung, D. Fekete, D. M. Pepper, and A. Yariv, “A theoretical and experimental investigation of the modes of optical resonators with phase-conjugate mirrors,” IEEE J. Quantum Electron. QE-15, 1180–1188 (1979).
[Crossref]

A. Yariv and D. M. Pepper, “Amplified reflection, phase conjugation, and oscillation in degenerate four-wave mixing,” Opt. Lett. 1, 16–18 (1977).
[Crossref] [PubMed]

L. Casperson and A. Yariv, “The time behavior and spectra of relaxation oscillations in a high-gain laser,” IEEE J. Quantum Electron. QE-8, 69–73 (1972).
[Crossref]

A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975), Chap. 9.

Zemskov, E. M.

I. M. Bel’dyugin and E. M. Zemskov, “Calculation of the field in a laser resonator with a wave-front-reversing mirror,” Sov. J. Quantum Electron. 10, 764–765 (1980).
[Crossref]

I. M. Bel’dyugin, M. G. Galusbkin, and E. M. Zemskov, “Properties of resonators with wavefront-reversing mirrors,” Sov. J. Quantum Electron. 9, 20–23 (1979).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (2)

B. Fischer, M. Cronin-Golomb, J. O. White, and A. Yariv, “Coherent oscillation by self-induced gratings in the photorefractive crystal BaTiO3,” Appl. Phys. Lett. 40, 450–452 (1982).
[Crossref]

J. H. Marburger and J. F. Lam, “Nonlinear theory of degenerate four-wave mixing,” Appl. Phys. Lett. 34, 389–391 (1979); “Effect of nonlinear index changes on degenerate four-wave mixing,” Appl. Phys. Lett. 35, 249–251 (1979).
[Crossref]

IEEE J. Quantum Electron. (2)

L. Casperson and A. Yariv, “The time behavior and spectra of relaxation oscillations in a high-gain laser,” IEEE J. Quantum Electron. QE-8, 69–73 (1972).
[Crossref]

J. Auyeung, D. Fekete, D. M. Pepper, and A. Yariv, “A theoretical and experimental investigation of the modes of optical resonators with phase-conjugate mirrors,” IEEE J. Quantum Electron. QE-15, 1180–1188 (1979).
[Crossref]

J. Appl. Phys. (1)

W. W. Rigrod, “Saturation effects in high-gain lasers,” J. Appl. Phys. 36, 2487–2490 (1965).
[Crossref]

J. Opt. Soc. Am. (1)

Opt. Lett. (8)

A. Yariv and D. M. Pepper, “Amplified reflection, phase conjugation, and oscillation in degenerate four-wave mixing,” Opt. Lett. 1, 16–18 (1977).
[Crossref] [PubMed]

R. K. Jain and G. J. Dunning, “Spatial and temporal properties of a continuous-wave phase-conjugate resonator based on the photorefractive crystal BaTiO3,” Opt. Lett. 7, 420–422 (1982).
[Crossref] [PubMed]

A. Vanherzeele, J. L. Van Eck, and A. E. Siegman, “Mode-locked laser oscillation using self-pumped phase-conjugate reflection,” Opt. Lett. 6, 467–469 (1981).
[Crossref] [PubMed]

R. C. Lind and D. G. Steel, “Demonstration of the longitudinal modes and aberration-correction properties of a continuous-wave dye laser with a phase-conjugate mirror,” Opt. Lett. 6, 554–556 (1981).
[Crossref] [PubMed]

M. C. Gower and R. G. Caro, “KrF laser with a phase-conjugate Brillouin mirror,” Opt. Lett. 7, 162–164 (1982).
[Crossref] [PubMed]

G. P. Agrawal and J. L. Boulnois, “Waveguide resonators with a phase-conjugate mirror,” Opt. Lett. 7, 159–161 (1982); J. L. Boulnois and G. P. Agrawal, “Mode discrimination and coupling losses in rectangular-waveguide resonators with conventional and phase-conjugate mirrors,” J. Opt. Soc. Am. 72, 853–860 (1982).
[Crossref] [PubMed]

J. F. Lam and W. P. Brown, “Optical resonators with phase-conjugate mirrors,” Opt. Lett. 5, 61–63 (1980).
[Crossref] [PubMed]

J. Feinberg and R. Hellwarth, “Phase-conjugating miror with continuous-wave gain,” Opt. Lett. 5, 519–521 (1980); errata 6, 257 (1981).
[Crossref]

Sov. J. Quantum Electron. (3)

I. M. Bel’dyugin, M. G. Galusbkin, and E. M. Zemskov, “Properties of resonators with wavefront-reversing mirrors,” Sov. J. Quantum Electron. 9, 20–23 (1979).
[Crossref]

M. G. Reznikov and A. I. Khizhnyak, “Properties of a resonator with a wavefront-reversing mirror,” Sov. J. Quantum Electron. 10, 764–765 (1980).
[Crossref]

I. M. Bel’dyugin and E. M. Zemskov, “Calculation of the field in a laser resonator with a wave-front-reversing mirror,” Sov. J. Quantum Electron. 10, 764–765 (1980).
[Crossref]

Sov. Phys. Tech. Phys. (1)

S. A. Lesnik, M. S. Soskin, and A. I. Khiznyak, “Laser with a stimulated-Brillouin-scattering complex-conjugate mirror,” Sov. Phys. Tech. Phys. 24, 1249–1250 (1979).

Other (2)

W. M. Grossman and D. M. Shemwell, “Practical constraints on phase conjugation by four-wave mixing,” presented at Lasers ’79 Conference.

A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975), Chap. 9.

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Figures (8)

Fig. 1
Fig. 1

Three generic forms of the PCL. The forms are distinguished by the origin of the PCM pump power. After passing through the PCM, pump power may be recirculated to a gain medium. Optimal power-extraction point for applications is always the end of the resonator opposite the PCM.

Fig. 2
Fig. 2

Self-pumped PCL geometry used in Ref. 10. The depleted pump goes back through the YAG rod and is regenerated. Optimal output extraction point for applications is A; in the experiment of Ref. 10 power was extracted at B and measured.

Fig. 3
Fig. 3

Schematic diagram of a PCL showing normalized power levels in both directions; symbols defined as in the text. Self-pumped devices have nonlinear feedback through the pump path from ordinary mirror to PCM.

Fig. 4
Fig. 4

Graphical solution for the lumped-parameter laser model. At a steady-state solution, saturated gain cancels cavity losses, or G A = 1. In (a) an ordinary laser has A constant, with one possible solution. In (b) a self-pumped PCL has A(Pi) smoothly increasing; with low gain and high losses, only one solution is possible. In (c), with higher gain and lower losses, multiple solutions are possible. If the loss function A(Pi) is not pathological, then the existence of multiple power states requires an inflection point in G(Pi), which occurs at sufficiently high gain.

Fig. 5
Fig. 5

Calculated performance of a Type 1 externally pumped PCL as a function of unsaturated gain in decibels (4.343 Go). The intrinsic loss a = 0.02. The pump power is proportional to the conventional laser power, Px = 0.6 P ¯ o. Four values are used for the PCM coupling content κ. Output transmittance to is adjusted (for each Go, a, Px, κ) to give maximum power. The graphs show (a) maximum output power and pump power, (b) the corresponding optimal output reflectance, (c) break-even beam quality, (d) pump-to-signal ratio, and (e) PCM reflectance.

Fig. 6
Fig. 6

Calculated performance of an ordinary laser (with an ideal backreflector) as a function of unsaturated gain in decibels (4.343 Go). Two values are used for intrinsic loss. Output transmittance is adjusted (for each Go, a) to give maximum power. The graphs show (a) maximum output power and (b) the corresponding optimal output reflectance.

Fig. 7
Fig. 7

Calculated performance of a Type 3 self-pumped PCL as a function of unsaturated gain in decibels (4.343 Go). The intrinsic loss is a = 0.02; the internal beam splitter, which diverts pump power to the PCM, has reflectance = 0.55; and the output transmittance is to = 0.95. Eight values are used for the PCM coupling constant κ, covering the bifurcation region where the PCL develops multiple states. The graphs show (a) output power, (b) break-even beam quality, (c) pump-to signal ratio, and (d) PCM reflectance. Solid lines are stable states; dashed lines are unstable states.

Fig. 8
Fig. 8

Transition region where the self-pumped PCL develops multiple-power states. Output power is calculated as a function of the PCM coupling constant for a Type 3 PCL. The intrinsic loss a = 0.02, the diverting reflectance = 0.55, and output transmittance to = 0.95. Six values are used for the unsaturated gain in decibels (4.343 Go). Solid lines are stable power states; dashed lines are unstable states.

Tables (1)

Tables Icon

Table 1 Rigrod Model Parameters for the Phase-Conjugate Laser

Equations (39)

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γ ( z ) = γ o / [ 1 + P + ( Z ) + P - ( Z ) ] .
γ ( z ) = 1 P + d P + d z = - 1 P - d P - d z ,
P + P - = constant .
D + E + F o = 1 - a ,
P p = D P 4 + P x
F c = ( 1 - a ) r c ,
P 1 = F o P 4 ,             P 3 = F c P 2 .
P 4 = G o + 1 2 ln ( F o F c ) [ 1 + ( F o / F c ) 1 / 2 ] [ 1 - ( F o F c ) 1 / 2 ] ,
G o = γ o L .
P e = G o + 1 2 ln ( F o F c ) .
P i = ( P + P - ) 1 / 2 = ( F o ) 1 / 2 P 4 = ( F o ) 1 / 2 [ 1 + ( F o / F c ) 1 / 2 ] [ 1 - ( F o F c ) 1 / 2 ] P e ,
P o = E P 4 = E [ 1 + ( F o / F c ) 1 / 2 ] [ 1 - ( F o F c ) 1 / 2 ] P e .
1 exp ( 2 G o ) F o < F c < 1 F o .
( 1 - a ) P 2 + P p ( r c + t c ) ( 1 - a ) P 2 ,
1 + ρ r c + t c ,
ρ P p ( 1 - a ) P 2 = D ( 1 - a ) ( F c F o ) 1 / 2 + P x ( 1 - a ) P 2 .
P o P e + P x / ( 1 + t c / r c ) .
P x δ P x , D δ D , F o F o + ( 1 - δ ) D , κ κ / δ
G = exp [ G o / ( 1 + P i ) σ ] ,
A = [ F o F c ( P i ) ] 1 / 2 ,
G A = 1 .
P o = P ¯ o β e 2 + P x
β e [ P ¯ o / ( P o - P x ) ] 1 / 2 .
r c = tan 2 ( κ P p ) .
t c = 1 + r c .
U ˙ = N B U - U / θ , N ˙ = r - N B U - N / τ .
θ = n L c - 1 ½ ln ( F o F c ) ,
U o = r / r o - 1 B τ , N o = 1 / B θ ,
r o = 1 / B θ τ
U ˙ = U o + U 1 ( t ) , N ˙ = N o + N 1 ( t ) ,
U ˙ 1 = N 1 B U o + ( U o θ / θ 2 ) U 1 + O ( U 1 2 ) , N ˙ 1 = N 1 B U 1 - N o B U 1 - N 1 / τ + O ( N 1 2 ) ,
θ d θ d U | U o .
U 1 ( t ) = exp ( - α t + i ω t ) U 1 ( 0 ) , N 1 ( t ) = exp ( - α t + i ω t ) N 1 ( 0 ) ,
α = 1 2 ( r / r 0 τ - μ ) ,
ω 2 = - α 2 + r / r o - 1 τ θ ( 1 - θ μ ) - μ / τ ,
μ U o θ θ 2 = - U o c 2 n L d ln F c d U | U o .
r / r o max ( τ μ , 1 + θ μ 1 - θ μ ) .
θ μ 1.
μ = c n L ( 1 r c + r c arctan r c ) .