Abstract

The effects of Gaussian-beam averaging with various apertures are derived for three-wave mixing in nonlinear two-level media. The model studies the phase transition from plane-wave to pure Gaussian saturation. Applications to phase conjugation, beat-frequency spectroscopy, and laser–optical-bistability instability phenomena are considered. The Gaussian beams are seen to reduce or wash out dynamic Stark splittings observed for plane-wave saturation. For strong saturation, the side-mode gain/absorption coefficient versus detuning yields slightly larger gain for the homogeneously broadened laser instability but substantially curtailed gain for the corresponding absorptive optical-bistability instability.

© 1984 Optical Society of America

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  1. G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian beams,” J. Appl. Phys. 39, 3597–3639 (1968).
    [CrossRef]
  2. A. G. Fox, Bell Labs. Tech. Memo MM70-1254-8, 1970 (unpublished).
  3. H. Maeda and K. Shimoda, “Theory of the inverted Lamb dip with a Gaussian beam,” J. Appl. Phys. 47, 1069–1071 (1976).
    [CrossRef]
  4. M. Sargent, “Effects of truncated Gaussian-beam variations in laser saturation spectroscopy,” J. Appl. Phys. 48, 243 (1977).
    [CrossRef]
  5. P. V. Avizonis, F. A. Hopf, W. D. Bomberger, S. F. Jacobs, A. Tomita, and K. H. Womack, “Optical phase conjugation in lithium formate crystal,” Appl. Phys. Lett. 31, 435–437 (1977).
    [CrossRef]
  6. A. Yariv, “Phase conjugate optics and real time holography,” IEEE J. Quantum Electron. QE-14, 650–660 (1978).
    [CrossRef]
  7. A. Yariv, D. Fekete, and D. M. Pepper, “Compensation for channel dispersion by nonlinear optical phase conjugation,” Opt. Lett. 4, 52–54 (1979).
    [CrossRef] [PubMed]
  8. B. Y. Zel’dovich, N. F. Pilipelskii, V. V. Ragul’skii, and V. V. Shkunov, “Wavefront reversal by nonlinear optics methods,” Sov. J. Quantum Electron. 8, 1021–1023 (1978).
    [CrossRef]
  9. M. Sargent and P. E. Toschek, “Unidirectional saturation spectroscopy, II. General lifetimes, interpretations and analogies,” Appl. Phys. 11, 107–120 (1976a).
    [CrossRef]
  10. M. Sargent, P. E. Toschek, and H. G. Danielmeyer, “Unidirectional saturation spectroscopy, I. Theory and short dipole lifetime limit,” Appl. Phys. 11, 55–62 (1976b).
    [CrossRef]
  11. S. T. Hendow and M. Sargent, “The role of population pulsations in single-mode laser instabilities,” Opt. Commun. 40, 385–390 (1982).
    [CrossRef]
  12. F. A. Hopf, A. Tomita, K. H. Womack, and J. L. Jewell, “Optical distortion in nonlinear phase conjugation by three-wave mixing,” J. Opt. Soc. Am. 69, 968–972 (1979).
    [CrossRef]
  13. R. L. Abrams and R. C. Lind, “Degenerate four-wave mixing in absorbing media,” Opt. Lett. 2, 94–96 (1978).
    [CrossRef] [PubMed]
  14. R. L. Abrams and R. C. Lind, “Degenerate four-wave mixing in absorbing media: errata,” Opt. Lett. 3, 205 (1978).
    [CrossRef] [PubMed]
  15. 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]
  16. A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Sys. Tech. J. 40, 453–488 (1961).
    [CrossRef]
  17. M. Sargent, “Spectroscopic techniques based on Lamb’s laser theory,” Phys. Rept. 43, 223 (1978).
    [CrossRef]
  18. T. Fu and M. Sargent, “Effects of signal detuning on phase conjugation,” Proc. Soc. Photo-Opt. Instrum. Eng. 190, 419 (1979).
  19. S. T. Hendow, S. Stuut, and M. Sargent, “Effects of transverse variations and propagation on beat-frequency spectroscopy,” J. Opt. Soc. Am. 72, 1734 (A) (1982).
  20. L. A. Lugiato and M. Milani, “Transverse effects and self pulsing in optical bistability,” Z. Phys. B50, 171–179 (1983).
    [CrossRef]

1983 (1)

L. A. Lugiato and M. Milani, “Transverse effects and self pulsing in optical bistability,” Z. Phys. B50, 171–179 (1983).
[CrossRef]

1982 (2)

S. T. Hendow and M. Sargent, “The role of population pulsations in single-mode laser instabilities,” Opt. Commun. 40, 385–390 (1982).
[CrossRef]

S. T. Hendow, S. Stuut, and M. Sargent, “Effects of transverse variations and propagation on beat-frequency spectroscopy,” J. Opt. Soc. Am. 72, 1734 (A) (1982).

1979 (3)

1978 (5)

M. Sargent, “Spectroscopic techniques based on Lamb’s laser theory,” Phys. Rept. 43, 223 (1978).
[CrossRef]

R. L. Abrams and R. C. Lind, “Degenerate four-wave mixing in absorbing media,” Opt. Lett. 2, 94–96 (1978).
[CrossRef] [PubMed]

R. L. Abrams and R. C. Lind, “Degenerate four-wave mixing in absorbing media: errata,” Opt. Lett. 3, 205 (1978).
[CrossRef] [PubMed]

A. Yariv, “Phase conjugate optics and real time holography,” IEEE J. Quantum Electron. QE-14, 650–660 (1978).
[CrossRef]

B. Y. Zel’dovich, N. F. Pilipelskii, V. V. Ragul’skii, and V. V. Shkunov, “Wavefront reversal by nonlinear optics methods,” Sov. J. Quantum Electron. 8, 1021–1023 (1978).
[CrossRef]

1977 (3)

M. Sargent, “Effects of truncated Gaussian-beam variations in laser saturation spectroscopy,” J. Appl. Phys. 48, 243 (1977).
[CrossRef]

P. V. Avizonis, F. A. Hopf, W. D. Bomberger, S. F. Jacobs, A. Tomita, and K. H. Womack, “Optical phase conjugation in lithium formate crystal,” Appl. Phys. Lett. 31, 435–437 (1977).
[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]

1976 (3)

H. Maeda and K. Shimoda, “Theory of the inverted Lamb dip with a Gaussian beam,” J. Appl. Phys. 47, 1069–1071 (1976).
[CrossRef]

M. Sargent and P. E. Toschek, “Unidirectional saturation spectroscopy, II. General lifetimes, interpretations and analogies,” Appl. Phys. 11, 107–120 (1976a).
[CrossRef]

M. Sargent, P. E. Toschek, and H. G. Danielmeyer, “Unidirectional saturation spectroscopy, I. Theory and short dipole lifetime limit,” Appl. Phys. 11, 55–62 (1976b).
[CrossRef]

1968 (1)

G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian beams,” J. Appl. Phys. 39, 3597–3639 (1968).
[CrossRef]

1961 (1)

A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Sys. Tech. J. 40, 453–488 (1961).
[CrossRef]

Abrams, R. L.

Avizonis, P. V.

P. V. Avizonis, F. A. Hopf, W. D. Bomberger, S. F. Jacobs, A. Tomita, and K. H. Womack, “Optical phase conjugation in lithium formate crystal,” Appl. Phys. Lett. 31, 435–437 (1977).
[CrossRef]

Bomberger, W. D.

P. V. Avizonis, F. A. Hopf, W. D. Bomberger, S. F. Jacobs, A. Tomita, and K. H. Womack, “Optical phase conjugation in lithium formate crystal,” Appl. Phys. Lett. 31, 435–437 (1977).
[CrossRef]

Boyd, G. D.

G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian beams,” J. Appl. Phys. 39, 3597–3639 (1968).
[CrossRef]

Danielmeyer, H. G.

M. Sargent, P. E. Toschek, and H. G. Danielmeyer, “Unidirectional saturation spectroscopy, I. Theory and short dipole lifetime limit,” Appl. Phys. 11, 55–62 (1976b).
[CrossRef]

Fekete, D.

Fox, A. G.

A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Sys. Tech. J. 40, 453–488 (1961).
[CrossRef]

A. G. Fox, Bell Labs. Tech. Memo MM70-1254-8, 1970 (unpublished).

Fu, T.

T. Fu and M. Sargent, “Effects of signal detuning on phase conjugation,” Proc. Soc. Photo-Opt. Instrum. Eng. 190, 419 (1979).

Hendow, S. T.

S. T. Hendow, S. Stuut, and M. Sargent, “Effects of transverse variations and propagation on beat-frequency spectroscopy,” J. Opt. Soc. Am. 72, 1734 (A) (1982).

S. T. Hendow and M. Sargent, “The role of population pulsations in single-mode laser instabilities,” Opt. Commun. 40, 385–390 (1982).
[CrossRef]

Hopf, F. A.

F. A. Hopf, A. Tomita, K. H. Womack, and J. L. Jewell, “Optical distortion in nonlinear phase conjugation by three-wave mixing,” J. Opt. Soc. Am. 69, 968–972 (1979).
[CrossRef]

P. V. Avizonis, F. A. Hopf, W. D. Bomberger, S. F. Jacobs, A. Tomita, and K. H. Womack, “Optical phase conjugation in lithium formate crystal,” Appl. Phys. Lett. 31, 435–437 (1977).
[CrossRef]

Jacobs, S. F.

P. V. Avizonis, F. A. Hopf, W. D. Bomberger, S. F. Jacobs, A. Tomita, and K. H. Womack, “Optical phase conjugation in lithium formate crystal,” Appl. Phys. Lett. 31, 435–437 (1977).
[CrossRef]

Jewell, J. L.

Kleinman, D. A.

G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian beams,” J. Appl. Phys. 39, 3597–3639 (1968).
[CrossRef]

Li, T.

A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Sys. Tech. J. 40, 453–488 (1961).
[CrossRef]

Lind, R. C.

Lugiato, L. A.

L. A. Lugiato and M. Milani, “Transverse effects and self pulsing in optical bistability,” Z. Phys. B50, 171–179 (1983).
[CrossRef]

Maeda, H.

H. Maeda and K. Shimoda, “Theory of the inverted Lamb dip with a Gaussian beam,” J. Appl. Phys. 47, 1069–1071 (1976).
[CrossRef]

Milani, M.

L. A. Lugiato and M. Milani, “Transverse effects and self pulsing in optical bistability,” Z. Phys. B50, 171–179 (1983).
[CrossRef]

Pepper, D. M.

Pilipelskii, N. F.

B. Y. Zel’dovich, N. F. Pilipelskii, V. V. Ragul’skii, and V. V. Shkunov, “Wavefront reversal by nonlinear optics methods,” Sov. J. Quantum Electron. 8, 1021–1023 (1978).
[CrossRef]

Ragul’skii, V. V.

B. Y. Zel’dovich, N. F. Pilipelskii, V. V. Ragul’skii, and V. V. Shkunov, “Wavefront reversal by nonlinear optics methods,” Sov. J. Quantum Electron. 8, 1021–1023 (1978).
[CrossRef]

Sargent, M.

S. T. Hendow, S. Stuut, and M. Sargent, “Effects of transverse variations and propagation on beat-frequency spectroscopy,” J. Opt. Soc. Am. 72, 1734 (A) (1982).

S. T. Hendow and M. Sargent, “The role of population pulsations in single-mode laser instabilities,” Opt. Commun. 40, 385–390 (1982).
[CrossRef]

T. Fu and M. Sargent, “Effects of signal detuning on phase conjugation,” Proc. Soc. Photo-Opt. Instrum. Eng. 190, 419 (1979).

M. Sargent, “Spectroscopic techniques based on Lamb’s laser theory,” Phys. Rept. 43, 223 (1978).
[CrossRef]

M. Sargent, “Effects of truncated Gaussian-beam variations in laser saturation spectroscopy,” J. Appl. Phys. 48, 243 (1977).
[CrossRef]

M. Sargent and P. E. Toschek, “Unidirectional saturation spectroscopy, II. General lifetimes, interpretations and analogies,” Appl. Phys. 11, 107–120 (1976a).
[CrossRef]

M. Sargent, P. E. Toschek, and H. G. Danielmeyer, “Unidirectional saturation spectroscopy, I. Theory and short dipole lifetime limit,” Appl. Phys. 11, 55–62 (1976b).
[CrossRef]

Shimoda, K.

H. Maeda and K. Shimoda, “Theory of the inverted Lamb dip with a Gaussian beam,” J. Appl. Phys. 47, 1069–1071 (1976).
[CrossRef]

Shkunov, V. V.

B. Y. Zel’dovich, N. F. Pilipelskii, V. V. Ragul’skii, and V. V. Shkunov, “Wavefront reversal by nonlinear optics methods,” Sov. J. Quantum Electron. 8, 1021–1023 (1978).
[CrossRef]

Stuut, S.

S. T. Hendow, S. Stuut, and M. Sargent, “Effects of transverse variations and propagation on beat-frequency spectroscopy,” J. Opt. Soc. Am. 72, 1734 (A) (1982).

Tomita, A.

F. A. Hopf, A. Tomita, K. H. Womack, and J. L. Jewell, “Optical distortion in nonlinear phase conjugation by three-wave mixing,” J. Opt. Soc. Am. 69, 968–972 (1979).
[CrossRef]

P. V. Avizonis, F. A. Hopf, W. D. Bomberger, S. F. Jacobs, A. Tomita, and K. H. Womack, “Optical phase conjugation in lithium formate crystal,” Appl. Phys. Lett. 31, 435–437 (1977).
[CrossRef]

Toschek, P. E.

M. Sargent, P. E. Toschek, and H. G. Danielmeyer, “Unidirectional saturation spectroscopy, I. Theory and short dipole lifetime limit,” Appl. Phys. 11, 55–62 (1976b).
[CrossRef]

M. Sargent and P. E. Toschek, “Unidirectional saturation spectroscopy, II. General lifetimes, interpretations and analogies,” Appl. Phys. 11, 107–120 (1976a).
[CrossRef]

Womack, K. H.

F. A. Hopf, A. Tomita, K. H. Womack, and J. L. Jewell, “Optical distortion in nonlinear phase conjugation by three-wave mixing,” J. Opt. Soc. Am. 69, 968–972 (1979).
[CrossRef]

P. V. Avizonis, F. A. Hopf, W. D. Bomberger, S. F. Jacobs, A. Tomita, and K. H. Womack, “Optical phase conjugation in lithium formate crystal,” Appl. Phys. Lett. 31, 435–437 (1977).
[CrossRef]

Yariv, A.

Zel’dovich, B. Y.

B. Y. Zel’dovich, N. F. Pilipelskii, V. V. Ragul’skii, and V. V. Shkunov, “Wavefront reversal by nonlinear optics methods,” Sov. J. Quantum Electron. 8, 1021–1023 (1978).
[CrossRef]

Appl. Phys. (2)

M. Sargent and P. E. Toschek, “Unidirectional saturation spectroscopy, II. General lifetimes, interpretations and analogies,” Appl. Phys. 11, 107–120 (1976a).
[CrossRef]

M. Sargent, P. E. Toschek, and H. G. Danielmeyer, “Unidirectional saturation spectroscopy, I. Theory and short dipole lifetime limit,” Appl. Phys. 11, 55–62 (1976b).
[CrossRef]

Appl. Phys. Lett. (1)

P. V. Avizonis, F. A. Hopf, W. D. Bomberger, S. F. Jacobs, A. Tomita, and K. H. Womack, “Optical phase conjugation in lithium formate crystal,” Appl. Phys. Lett. 31, 435–437 (1977).
[CrossRef]

Bell Sys. Tech. J. (1)

A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Sys. Tech. J. 40, 453–488 (1961).
[CrossRef]

IEEE J. Quantum Electron. (1)

A. Yariv, “Phase conjugate optics and real time holography,” IEEE J. Quantum Electron. QE-14, 650–660 (1978).
[CrossRef]

J. Appl. Phys. (3)

G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian beams,” J. Appl. Phys. 39, 3597–3639 (1968).
[CrossRef]

H. Maeda and K. Shimoda, “Theory of the inverted Lamb dip with a Gaussian beam,” J. Appl. Phys. 47, 1069–1071 (1976).
[CrossRef]

M. Sargent, “Effects of truncated Gaussian-beam variations in laser saturation spectroscopy,” J. Appl. Phys. 48, 243 (1977).
[CrossRef]

J. Opt. Soc. Am. (2)

F. A. Hopf, A. Tomita, K. H. Womack, and J. L. Jewell, “Optical distortion in nonlinear phase conjugation by three-wave mixing,” J. Opt. Soc. Am. 69, 968–972 (1979).
[CrossRef]

S. T. Hendow, S. Stuut, and M. Sargent, “Effects of transverse variations and propagation on beat-frequency spectroscopy,” J. Opt. Soc. Am. 72, 1734 (A) (1982).

Opt. Commun. (1)

S. T. Hendow and M. Sargent, “The role of population pulsations in single-mode laser instabilities,” Opt. Commun. 40, 385–390 (1982).
[CrossRef]

Opt. Lett. (4)

Phys. Rept. (1)

M. Sargent, “Spectroscopic techniques based on Lamb’s laser theory,” Phys. Rept. 43, 223 (1978).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

T. Fu and M. Sargent, “Effects of signal detuning on phase conjugation,” Proc. Soc. Photo-Opt. Instrum. Eng. 190, 419 (1979).

Sov. J. Quantum Electron. (1)

B. Y. Zel’dovich, N. F. Pilipelskii, V. V. Ragul’skii, and V. V. Shkunov, “Wavefront reversal by nonlinear optics methods,” Sov. J. Quantum Electron. 8, 1021–1023 (1978).
[CrossRef]

Z. Phys. (1)

L. A. Lugiato and M. Milani, “Transverse effects and self pulsing in optical bistability,” Z. Phys. B50, 171–179 (1983).
[CrossRef]

Other (1)

A. G. Fox, Bell Labs. Tech. Memo MM70-1254-8, 1970 (unpublished).

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Figures (7)

Fig. 1
Fig. 1

Beam geometry. Gaussian beam propagates along the z axis perpendicular to the ρ plane. Beam has waist w0; medium has length L; aperture has radius a. The Gaussian profile shown is for the field amplitude rather than the intensity.

Fig. 2
Fig. 2

Interaction geometry. The three scalar, classical, electromagnetic fields have amplitudes A1, A2, and A3 and the frequencies ν1, ν2, and ν3, respectively. A1 and A3, the signal and conjugate waves, respectively, are weak relative to the pump wave A2. A3 arises within the medium from the interactions among A1, A2, and the medium. Angles θ and θ′ are small.

Fig. 3
Fig. 3

Reflectance versus signal detuning. The ordinate is intensity reflectance for wave 3 (coupling of wave 1 into wave 3; see Fig. 2), R = |r|2, where r is the amplitude reflection coefficient. The abscissa, ΔT2, is signal detuning from line center. The top curve is the fully truncated (plane-wave) case of r0 = 0.001, the bottom curve is the untruncated (full-Gaussian-wave) case of r0 = 2, and the intermediate curves are the sequential truncations r0 = 0.707, 1.0, 1.41.

Fig. 4
Fig. 4

Reflectance versus pump intensity. The abscissa is pump intensity I2, and the leftmost curve is the fully truncated (plane-wave) case; otherwise the same as Fig. 3.

Fig. 5
Fig. 5

Normalized absorption coefficient (thin medium). The ordinate, αeffL/α0L, is a normalized absorption coefficient for wave 1 (energy being coupled away from wave 1 to wave 3; see Fig. 2), where α0 is the unsaturated absorption coefficient, αeff is the effective absorption coefficient, and L is the interaction length. The abscissa, ΔT2, is signal detuning from line center. The bottom curve is the fully truncated (plane-wave) case of r0 = 0.01, the top curve is the un-truncated (full-Gaussian-wave) case of r0 = 16, and the intermediate curves are the sequential truncations of r0 = 0.707, 1.0, 2.0. This is the thin-medium case (α0L = 0.01). The pump intensity I2 = 2.

Fig. 6
Fig. 6

Normalized absorption coefficient (thick medium). The thick-medium case (α0L = 8); otherwise identical to Fig. 5.

Fig. 7
Fig. 7

Side-mode gain (absorption) in a laser (absorber) cavity for truncation radii r0 = 0 (plane wave, greatest negative area), 0.707, 1, 2, 16 (essentially full Gaussian, has least negative area).

Equations (47)

Equations on this page are rendered with MathJax. Learn more.

E ( z , ρ , t ) = 1 2 n = 1 3 E n ( z , ρ ) exp ( - i ν n t ) + c . c . , = 1 2 n = 1 3 A n ( z ) U n ( z , ρ , t ) + c . c . ,
U n ( z , ρ , t ) = exp ( - ρ 2 / w 0 2 ) exp [ - i ( ν n t - K n z ) ] .
P ( z , ρ , t ) = 1 2 n = 1 3 P n ( z , ρ ) U n ( z , t ) + c . c . ,
U n ( z , t ) = exp [ - i ( ν n t - K n z ) ] .
× × E = - 1 c 2 2 E t 2 - 1 0 c 2 2 P t 2 .
2 E = 1 c 2 2 E t 2 + 1 0 c 2 2 P t 2 .
2 E z 2 - 1 c 2 2 E t 2 = 1 0 c 2 2 P t 2 .
0 a d ρ exp ( - ρ 2 / w 0 2 ) 0 2 π ρ d ϕ 1 τ 0 τ d t exp ( i ν 1 t ) × ( n = 1 3 { - i 2 K n d A n ( z ) d z exp ( - ρ 2 / w 0 2 ) exp [ i ( K n z - ν n t ) ] } = n = 1 3 { 1 0 c 2 P n ( z , ρ ) exp [ i ( K n z - ν n t ) ] } ) .
B exp [ 1 2 i ( Δ ν ) τ ] sinc [ 1 2 ( Δ ν ) τ ] ,
d A 1 ( z ) d z = i K 1 2 0 P 1 ( z ) ,
P 1 ( z ) = 0 a ρ d ρ exp ( - ρ 2 / w 0 2 ) P 1 ( z , ρ ) .
d A 3 ( z ) d z = i K 3 2 0 P 3 ( z ) .
P ( z , ρ , t ) = p ρ a b ( z , ρ , t ) + c . c . ,
0 a d ρ 0 2 π ρ d ϕ 1 τ 0 τ d t [ 1 2 n = 1 3 P n ( z , ρ ) U n ( z , t ) ] × U 1 * ( z , ρ , t ) = 0 a d ρ 0 2 π ρ d ϕ 1 τ 0 τ d t p ρ a b ( z , ρ , t ) U 1 * ( z , ρ , t ) .
0 a d ρ ρ 1 τ 0 τ d t { 1 2 n = 1 3 P n ( z , ρ ) exp [ - i ( ν n t - K n z ) ] } × exp ( - ρ 2 / w 0 2 ) exp [ i ( ν 1 t - K 1 z ) ] = 0 a d ρ ρ 1 τ 0 τ d t p ρ a b ( z , ρ , t ) U 1 * ( z , ρ , t ) .
P 1 ( z ) = 2 p N 1 τ 0 a d ρ ρ 0 τ d t ρ a b ( z , ρ , t ) U 1 * ( z , ρ , t ) ,
N 1 = 0 a d ρ ρ exp ( - 2 ρ 2 / w 0 2 )
V a b ( z , ρ , t ) - 1 2 p n = 1 3 E n ( z , ρ ) exp ( - i ν n t ) .
ρ ˙ a b = - ( i ω + γ ) ρ a b + i V a b ( ρ a a - ρ b b ) ,
ρ ˙ a a = λ a - γ a ρ a a - ( i V a b ρ b a + c . c . ) ,
ρ ˙ b b = λ b - γ b ρ b b + ( i V a b ρ b a + c . c . ) .
ρ a b ( z , ρ , t ) = N exp ( - i ν 1 t ) m p m + 1 ( z , ρ ) exp ( - i m Δ t ) ,
ρ α α = N k n α k ( z , ρ ) exp ( i k Δ t ) ,             α = a or b ,
ρ a a - ρ b b = N k d k ( z , ρ ) exp ( i k Δ t ) ,
P 1 ( z ) = 2 p N N 1 τ 0 a d ρ ρ 0 τ d t × [ exp ( - i ν 1 t ) m p m + 1 ( z , ρ ) exp ( - i m Δ t ) ] × U 1 * ( z , ρ , t ) = 2 p N N 1 τ 0 a d ρ ρ 0 τ d t × [ exp ( - i ν 1 t ) m p m + 1 ( z , ρ ) exp ( - i m Δ t ) ] × exp ( - ρ 2 / w 0 2 ) exp [ i ( ν 1 t - K 1 z ) ] .
P 1 ( z ) = 2 p N N 1 0 a d ρ ρ p 1 ( z , ρ ) exp ( - ρ 2 / w 0 2 ) exp ( - i K 1 z ) .
P 1 ( z ) = i N p 2 N 1 0 a d ρ ρ D 1 exp ( - ρ 2 / w 0 2 ) exp ( - i K 1 z ) 1 + I 2 ρ L 2 { E 1 - ( γ / 2 ) F ( Δ ) [ I 2 ρ ( D 2 * + D 1 ) E 1 + C E 2 2 ( D 3 * + D 2 ) E 3 * ] 1 + ( γ / 2 ) F ( Δ ) I 2 ρ ( D 3 * + D 1 ) } ,
I 2 ρ = | p A 2 | 2 T 1 T 2 exp ( - 2 ρ 2 / w 0 2 ) = I 2 exp ( - 2 ρ 2 / w 0 2 ) , E n = A n ( z ) exp ( i K n z ) exp ( - ρ 2 / w 0 2 ) , D n = 1 γ + i ( ω - ν n ) , L n = γ 2 γ 2 + ( ω - ν n ) 2 , F ( Δ ) = 1 2 T 1 ( 1 γ a + i Δ + 1 γ b + i Δ ) , T 1 = 1 2 ( 1 γ a + 1 γ b ) , T 2 = 1 γ , C = ( p ) 2 T 1 T 2 .
P 1 ( z ) = i N p 2 N 1 D 1 0 a d ρ ρ exp ( - ρ 2 / w 0 2 ) exp ( - i K 1 z ) 1 + I 2 L 2 exp ( - 2 ρ 2 / w 0 2 ) × A 1 exp ( i K 1 z ) exp ( - ρ 2 / w 0 2 ) - exp ( - 3 ρ 2 / w 0 2 ) exp ( i K 1 z ) × f 1 I 2 A 1 + f 3 * C A 2 2 A 3 * exp [ i ( 2 K 2 - K 1 - K 3 ) z ] 1 + f 2 I 2 exp ( - 2 ρ 2 / w 0 2 ) .
P 1 ( z ) = B 0 0 a d ρ ρ 1 1 + B 4 exp ( - 2 ρ 2 / w 0 2 ) × [ A 1 exp ( - 2 ρ 2 / w 0 2 ) - B 13 exp ( - 4 ρ 2 / w 0 2 ) 1 + B 2 exp ( - 2 ρ 2 / w 0 2 ) ] .
d u = - 4 w 0 2 ρ exp ( - 2 ρ 2 / w 0 2 ) d ρ , d ρ ρ exp ( - 2 ρ 2 / w 0 2 ) = - w 0 2 4 d u .
B 0 0 a d ρ ρ exp ( - 2 ρ 2 / w 0 2 ) A 1 1 + B 4 exp ( - 2 ρ 2 / w 0 2 ) = B 0 A 1 1 u a ( w 0 2 4 d u ) A 1 1 + B 4 u = B 0 A 1 w 0 2 4 B 4 ln 1 + B 4 1 + B 4 u a .
- B 0 0 a d ρ ρ × B 13 exp ( - 4 ρ 2 / w 0 2 ) [ 1 + B 4 exp ( - 2 ρ 2 / w 0 2 ) ] [ 1 + B 2 exp ( - 2 ρ 2 / w 0 2 ) ] = - B 0 B 13 1 u a ( - w 0 2 4 d u ) u ( 1 + B 4 u ) ( 1 + B 2 u ) = + B 0 B 13 w 0 2 4 ( B 4 - B 2 ) [ 1 B 4 ln ( 1 + B 4 ) 1 + B 4 u a - 1 B 2 ln ( 1 + B 2 ) 1 + B 2 u a ] .
P 1 ( z ) = B 0 A 1 w 0 2 4 B 4 ln [ 1 + B 4 ( 1 + B 4 u a ) ] + B 0 ( B 1 + B 3 ) w 0 2 4 ( B 4 - B 2 ) { 1 B 4 ln [ 1 + B 4 ( 1 + B 4 u a ) ] - 1 B 2 ln [ 1 + B 2 ( 1 + B 2 u a ) ] } .
P 1 ( z ) = B 0 w 0 2 4 { 1 B 4 [ A 1 + B 1 ( B 4 - B 2 ) ] ln [ 1 + B 4 ( 1 + B 4 u a ) ] - B 1 B 2 ( B 4 - B 2 ) ln [ 1 + B 2 ( 1 + B 2 u a ) ] } + B 0 w 0 2 B 3 4 ( B 4 - B 2 ) × { 1 B 4 ln [ 1 + B 4 ( 1 + B 4 u a ) ] - 1 B 2 ln [ 1 + B 2 ( 1 + B 2 u a ) ] } .
P 1 ( z ) = ( i N p 2 N 1 D 1 ) w 0 2 A 1 4 { 1 I 2 L 2 [ 1 + f 1 , I 2 ( I 2 L 2 - f 2 I 2 ) ] × ln [ 1 + I 2 L 2 ( 1 + I 2 L 2 u a ) ] - f 1 I 2 f 2 I 2 ( I 2 L 2 - f 2 I 2 ) ln [ 1 + f 2 I 2 ( 1 + f 2 I 2 u a ) ] } × ( i N p 2 N 1 D 1 ) w 0 2 f 3 * C A 2 2 A 3 * sinc θ 2 4 ( I 2 L 2 - f 2 I 2 ) × { 1 I 2 L 2 ln [ 1 + I 2 L 2 ( 1 + I 2 L 2 u a ) ] - 1 f 2 I 2 ln [ 1 + f 2 I 2 ( 1 + f 2 I 2 u a ) ] } .
P 1 ( z ) = i N p 2 D 1 ( 1 - u a ) I 2 { 1 L 2 [ L 2 - f 2 + f 1 ( L 2 - f 2 ) ] ln [ 1 + I 2 L 2 ( 1 + I 2 L 2 u a ) ] - 1 f 2 [ f 1 ( L 2 - f 2 ) ] ln [ 1 + f 2 I 2 ( 1 + f 2 I 2 u a ) ] } A 1 + i N p 2 D 1 f 3 * ( 1 - u a ) I 2 2 ( L 2 - f 2 ) sinc θ 2 × { 1 L 2 ln [ 1 + I 2 L 2 ( 1 + I 2 L 2 u a ) ] - 1 f 2 ln [ 1 + f 2 I 2 ( 1 + f 2 I 2 u a ) ] } A 3 * .
d A 3 d z = - α 3 A 3 - i κ 3 * A 1 * ,
d A 1 * d z = - α 1 * A 1 * + i κ 1 A 3 .
- α 1 * = i K 1 2 0 ( i N 2 D 1 h ( 1 - u a ) I 2 { 1 L 2 L 2 - f 2 + f 1 L 2 - f 2 ln [ 1 + I 2 L 2 ( 1 + I 2 L 2 u a ) ] - 1 f 2 f 1 L 2 - f 2 ln [ 1 + f 2 I 2 ( 1 + f 2 I 2 u a ) ] } ) * .
α 1 * = α 0 γ D 1 * ( 1 - u a ) I 2 { 1 L 2 L 2 - f 2 * + f 1 * L 2 - f 2 * ln [ 1 + I 2 L 2 ( 1 + I 2 L 2 u a ) ] - 1 f 2 * f 1 * L 2 - f 2 * ln [ 1 + f 2 * I 2 ( 1 + f 2 * I 2 u a ) ] } ,
i κ 1 = - i K 1 2 0 ( i N p 2 D 1 f 3 * sinc θ 2 h ( 1 - u a ) I 2 ( L 2 - f 2 ) × { 1 L 2 ln [ 1 + I 2 L 2 ( 1 + I 2 L 2 u a ) ] - 1 f 2 ln [ 1 + f 2 I 2 ( 1 + f 2 I 2 u a ) ] } ) * κ 1 = i α 0 γ D 1 * f 3 ( 1 - u a ) I 2 ( L 2 - f 2 * ) sinc θ 2 × { 1 L 2 ln [ 1 + I 2 L 2 ( 1 + I 2 L 2 u a ) ] - 1 f 2 * ln [ 1 + f 2 * I 2 ( 1 + f 2 * I 2 u a ) ] } .
α 3 = α 0 γ D 3 ( 1 - u a ) I 2 ( 1 L 2 L 2 - f 2 * + f 3 L 2 - f 2 * ln [ 1 + I 2 L 2 ( 1 + I 2 L 2 u a ) ] - 1 f 2 * { f 3 L 2 - f 2 * ln [ 1 + f 2 * I 2 ( 1 + f 2 * I 2 u a ) ] } ) ,
κ 3 * = - i α 0 γ D 3 A 2 2 f 1 * ( 1 - u a ) I 2 2 ( L 2 - f 2 * ) sinc θ 2 { 1 L 2 ( 1 + I 2 L 2 1 + I 2 L 2 u a ) - 1 f 2 * ln [ 1 + f 2 * I 2 ( 1 + f 2 I 2 * u a ) ] } .
r = A 3 ( L ) A 1 * ( 0 ) = - i κ 3 * L exp ( - a L ) sinh w L w L ,
a = α 3 + α 1 * 2 , α = α 3 - α 1 * 2 , w = ( α 2 - κ 1 κ 3 * ) 1 / 2 ,
t 1 = A 1 * ( L ) A 1 * ( 0 ) = exp ( - a L ) ( cosh w L + α L sinh w L w L ) .

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