Abstract

We have studied the dynamical properties of the optical parametric oscillator when it is inserted into the pump-laser cavity. The temporal evolutions of the population inversion and of the pump and signal intracavity powers are modeled by a simple system of coupled equations. Numerical simulations show that the optical parametric oscillator can produce one or several pulses, depending on the signal cavity finesse and the pumping level of the laser medium. The main characteristics of the pulses are deduced from an analytical study of the equations. The energy transfer efficiency from the laser medium to the signal pulses is predicted to be up to 80%.

© 1996 Optical Society of America

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References

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  1. J. Falk, J. M. Yarborough, and E. O. Ammann, "Internal optical parametric oscillation," IEEE J. Quantum Electron. QE-7, 359–369 (1971).
    [CrossRef]
  2. M. K. Oshman and S. E. Harris, "Theory of optical parametric oscillation internal to the laser cavity," IEEE J. Quantum Electron. QE-4, 491–502 (1968).
    [CrossRef]
  3. L. R. Marshall, A. D. Hays, J. Kasinski, and R. Burnham, "Highly efficient optical parametric oscillators," in Eye-safe Lasers: Components, Systems, and Applications, A. M. Johnson, ed., Proc. SPIE 1419,141–152 (1991).
  4. L. R. Marshall, A. Kaz, and R. Burnham, "Nonlinear cavity dumping," in Conference on Lasers and Electro-Optics, Vol. 11 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 436–437.
  5. A. Kaz and L. R. Marshall, "Continuouswave diode-pumped lasers and parametric oscillator," in Conference on Lasers and Electro-Optics, Vol. 11 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 244–245.
  6. T. Debuisschert, J. Raffy, J.-P. Pocholle, and M. Papuchon, "Dynamics of the pulsed intracavity OPO," in Advanced Solid-State Lasers, B. H. T. Chai and S. A. Payne, eds., Vol. 24 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1995), pp. 158–162.
  7. W. E. Lamb, Jr., "Theory of an optical maser," Phys. Rev. A 134, 1429–1450 (1964).
    [CrossRef]
  8. M. Sargent III, M. O. Scully, and W. E. Lamb, Jr., Laser Physics, 5th ed. (Addision-Wesley, Reading, Mass., 1987).
  9. J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, "Interactions between light waves in a nonlinear dielectric," Phys. Rev. 127, 1918–1939 (1962).
    [CrossRef]
  10. R. L. Byer, in Treatise in Quantum Electronics, H. Rabin and C. L. Tang, eds. (Academic, N.Y., 1975), p. 587.
  11. A. E. Siegmann, Lasers (University Science, Mill Valley, Calif., 1986).
  12. C. Cohen Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons et Atomes. Introduction à l'Electrodynamique Quantique (InterEditions, Editions du Centre National de la Recherche Scientifique, Paris, 1987).
  13. C. Fabre, E. Giacobino, A. Heidmann, and S. Reynaud, "Noise characteristics of a non-degenerate optical parametric oscillator. Application to quantum noise reduction," J. Phys. France 50, 1209–1225 (1989).
    [CrossRef]
  14. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).
  15. J. Raffy, T. Debuisschert, J.-P. Pocholle, and M. Papuchon, "Tunable IR laser source with optical parameteric oscillators in series," Appl. Opt. 33, 985–987 (1994).
    [CrossRef] [PubMed]

1995

T. Debuisschert, J. Raffy, J.-P. Pocholle, and M. Papuchon, "Dynamics of the pulsed intracavity OPO," in Advanced Solid-State Lasers, B. H. T. Chai and S. A. Payne, eds., Vol. 24 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1995), pp. 158–162.

1994

1993

L. R. Marshall, A. Kaz, and R. Burnham, "Nonlinear cavity dumping," in Conference on Lasers and Electro-Optics, Vol. 11 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 436–437.

A. Kaz and L. R. Marshall, "Continuouswave diode-pumped lasers and parametric oscillator," in Conference on Lasers and Electro-Optics, Vol. 11 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 244–245.

1991

L. R. Marshall, A. D. Hays, J. Kasinski, and R. Burnham, "Highly efficient optical parametric oscillators," in Eye-safe Lasers: Components, Systems, and Applications, A. M. Johnson, ed., Proc. SPIE 1419,141–152 (1991).

1989

C. Fabre, E. Giacobino, A. Heidmann, and S. Reynaud, "Noise characteristics of a non-degenerate optical parametric oscillator. Application to quantum noise reduction," J. Phys. France 50, 1209–1225 (1989).
[CrossRef]

1975

R. L. Byer, in Treatise in Quantum Electronics, H. Rabin and C. L. Tang, eds. (Academic, N.Y., 1975), p. 587.

1971

J. Falk, J. M. Yarborough, and E. O. Ammann, "Internal optical parametric oscillation," IEEE J. Quantum Electron. QE-7, 359–369 (1971).
[CrossRef]

1968

M. K. Oshman and S. E. Harris, "Theory of optical parametric oscillation internal to the laser cavity," IEEE J. Quantum Electron. QE-4, 491–502 (1968).
[CrossRef]

1964

W. E. Lamb, Jr., "Theory of an optical maser," Phys. Rev. A 134, 1429–1450 (1964).
[CrossRef]

1962

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, "Interactions between light waves in a nonlinear dielectric," Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Ammann, E. O.

J. Falk, J. M. Yarborough, and E. O. Ammann, "Internal optical parametric oscillation," IEEE J. Quantum Electron. QE-7, 359–369 (1971).
[CrossRef]

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, "Interactions between light waves in a nonlinear dielectric," Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Bloembergen, N.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, "Interactions between light waves in a nonlinear dielectric," Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Burnham, R.

L. R. Marshall, A. Kaz, and R. Burnham, "Nonlinear cavity dumping," in Conference on Lasers and Electro-Optics, Vol. 11 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 436–437.

L. R. Marshall, A. D. Hays, J. Kasinski, and R. Burnham, "Highly efficient optical parametric oscillators," in Eye-safe Lasers: Components, Systems, and Applications, A. M. Johnson, ed., Proc. SPIE 1419,141–152 (1991).

Byer, R. L.

R. L. Byer, in Treatise in Quantum Electronics, H. Rabin and C. L. Tang, eds. (Academic, N.Y., 1975), p. 587.

Debuisschert, T.

T. Debuisschert, J. Raffy, J.-P. Pocholle, and M. Papuchon, "Dynamics of the pulsed intracavity OPO," in Advanced Solid-State Lasers, B. H. T. Chai and S. A. Payne, eds., Vol. 24 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1995), pp. 158–162.

J. Raffy, T. Debuisschert, J.-P. Pocholle, and M. Papuchon, "Tunable IR laser source with optical parameteric oscillators in series," Appl. Opt. 33, 985–987 (1994).
[CrossRef] [PubMed]

Ducuing, J.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, "Interactions between light waves in a nonlinear dielectric," Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Dupont-Roc, J.

C. Cohen Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons et Atomes. Introduction à l'Electrodynamique Quantique (InterEditions, Editions du Centre National de la Recherche Scientifique, Paris, 1987).

Fabre, C.

C. Fabre, E. Giacobino, A. Heidmann, and S. Reynaud, "Noise characteristics of a non-degenerate optical parametric oscillator. Application to quantum noise reduction," J. Phys. France 50, 1209–1225 (1989).
[CrossRef]

Falk, J.

J. Falk, J. M. Yarborough, and E. O. Ammann, "Internal optical parametric oscillation," IEEE J. Quantum Electron. QE-7, 359–369 (1971).
[CrossRef]

Giacobino, E.

C. Fabre, E. Giacobino, A. Heidmann, and S. Reynaud, "Noise characteristics of a non-degenerate optical parametric oscillator. Application to quantum noise reduction," J. Phys. France 50, 1209–1225 (1989).
[CrossRef]

Grynberg, G.

C. Cohen Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons et Atomes. Introduction à l'Electrodynamique Quantique (InterEditions, Editions du Centre National de la Recherche Scientifique, Paris, 1987).

Harris, S. E.

M. K. Oshman and S. E. Harris, "Theory of optical parametric oscillation internal to the laser cavity," IEEE J. Quantum Electron. QE-4, 491–502 (1968).
[CrossRef]

Hays, A. D.

L. R. Marshall, A. D. Hays, J. Kasinski, and R. Burnham, "Highly efficient optical parametric oscillators," in Eye-safe Lasers: Components, Systems, and Applications, A. M. Johnson, ed., Proc. SPIE 1419,141–152 (1991).

Heidmann, A.

C. Fabre, E. Giacobino, A. Heidmann, and S. Reynaud, "Noise characteristics of a non-degenerate optical parametric oscillator. Application to quantum noise reduction," J. Phys. France 50, 1209–1225 (1989).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).

Kasinski, J.

L. R. Marshall, A. D. Hays, J. Kasinski, and R. Burnham, "Highly efficient optical parametric oscillators," in Eye-safe Lasers: Components, Systems, and Applications, A. M. Johnson, ed., Proc. SPIE 1419,141–152 (1991).

Kaz, A.

A. Kaz and L. R. Marshall, "Continuouswave diode-pumped lasers and parametric oscillator," in Conference on Lasers and Electro-Optics, Vol. 11 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 244–245.

L. R. Marshall, A. Kaz, and R. Burnham, "Nonlinear cavity dumping," in Conference on Lasers and Electro-Optics, Vol. 11 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 436–437.

Lamb, W. E.

W. E. Lamb, Jr., "Theory of an optical maser," Phys. Rev. A 134, 1429–1450 (1964).
[CrossRef]

M. Sargent III, M. O. Scully, and W. E. Lamb, Jr., Laser Physics, 5th ed. (Addision-Wesley, Reading, Mass., 1987).

Marshall, L. R.

L. R. Marshall, A. Kaz, and R. Burnham, "Nonlinear cavity dumping," in Conference on Lasers and Electro-Optics, Vol. 11 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 436–437.

A. Kaz and L. R. Marshall, "Continuouswave diode-pumped lasers and parametric oscillator," in Conference on Lasers and Electro-Optics, Vol. 11 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 244–245.

L. R. Marshall, A. D. Hays, J. Kasinski, and R. Burnham, "Highly efficient optical parametric oscillators," in Eye-safe Lasers: Components, Systems, and Applications, A. M. Johnson, ed., Proc. SPIE 1419,141–152 (1991).

Oshman, M. K.

M. K. Oshman and S. E. Harris, "Theory of optical parametric oscillation internal to the laser cavity," IEEE J. Quantum Electron. QE-4, 491–502 (1968).
[CrossRef]

Papuchon, M.

T. Debuisschert, J. Raffy, J.-P. Pocholle, and M. Papuchon, "Dynamics of the pulsed intracavity OPO," in Advanced Solid-State Lasers, B. H. T. Chai and S. A. Payne, eds., Vol. 24 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1995), pp. 158–162.

J. Raffy, T. Debuisschert, J.-P. Pocholle, and M. Papuchon, "Tunable IR laser source with optical parameteric oscillators in series," Appl. Opt. 33, 985–987 (1994).
[CrossRef] [PubMed]

Pershan, P. S.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, "Interactions between light waves in a nonlinear dielectric," Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Pocholle, J.-P.

T. Debuisschert, J. Raffy, J.-P. Pocholle, and M. Papuchon, "Dynamics of the pulsed intracavity OPO," in Advanced Solid-State Lasers, B. H. T. Chai and S. A. Payne, eds., Vol. 24 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1995), pp. 158–162.

J. Raffy, T. Debuisschert, J.-P. Pocholle, and M. Papuchon, "Tunable IR laser source with optical parameteric oscillators in series," Appl. Opt. 33, 985–987 (1994).
[CrossRef] [PubMed]

Raffy, J.

T. Debuisschert, J. Raffy, J.-P. Pocholle, and M. Papuchon, "Dynamics of the pulsed intracavity OPO," in Advanced Solid-State Lasers, B. H. T. Chai and S. A. Payne, eds., Vol. 24 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1995), pp. 158–162.

J. Raffy, T. Debuisschert, J.-P. Pocholle, and M. Papuchon, "Tunable IR laser source with optical parameteric oscillators in series," Appl. Opt. 33, 985–987 (1994).
[CrossRef] [PubMed]

Reynaud, S.

C. Fabre, E. Giacobino, A. Heidmann, and S. Reynaud, "Noise characteristics of a non-degenerate optical parametric oscillator. Application to quantum noise reduction," J. Phys. France 50, 1209–1225 (1989).
[CrossRef]

Sargent, M.

M. Sargent III, M. O. Scully, and W. E. Lamb, Jr., Laser Physics, 5th ed. (Addision-Wesley, Reading, Mass., 1987).

Scully, M. O.

M. Sargent III, M. O. Scully, and W. E. Lamb, Jr., Laser Physics, 5th ed. (Addision-Wesley, Reading, Mass., 1987).

Siegmann, A. E.

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

Tannoudji, C. Cohen

C. Cohen Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons et Atomes. Introduction à l'Electrodynamique Quantique (InterEditions, Editions du Centre National de la Recherche Scientifique, Paris, 1987).

Yarborough, J. M.

J. Falk, J. M. Yarborough, and E. O. Ammann, "Internal optical parametric oscillation," IEEE J. Quantum Electron. QE-7, 359–369 (1971).
[CrossRef]

Appl. Opt.

IEEE J. Quantum Electron.

J. Falk, J. M. Yarborough, and E. O. Ammann, "Internal optical parametric oscillation," IEEE J. Quantum Electron. QE-7, 359–369 (1971).
[CrossRef]

M. K. Oshman and S. E. Harris, "Theory of optical parametric oscillation internal to the laser cavity," IEEE J. Quantum Electron. QE-4, 491–502 (1968).
[CrossRef]

J. Phys. France

C. Fabre, E. Giacobino, A. Heidmann, and S. Reynaud, "Noise characteristics of a non-degenerate optical parametric oscillator. Application to quantum noise reduction," J. Phys. France 50, 1209–1225 (1989).
[CrossRef]

Phys. Rev.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, "Interactions between light waves in a nonlinear dielectric," Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Phys. Rev. A

W. E. Lamb, Jr., "Theory of an optical maser," Phys. Rev. A 134, 1429–1450 (1964).
[CrossRef]

Other

M. Sargent III, M. O. Scully, and W. E. Lamb, Jr., Laser Physics, 5th ed. (Addision-Wesley, Reading, Mass., 1987).

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).

R. L. Byer, in Treatise in Quantum Electronics, H. Rabin and C. L. Tang, eds. (Academic, N.Y., 1975), p. 587.

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

C. Cohen Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons et Atomes. Introduction à l'Electrodynamique Quantique (InterEditions, Editions du Centre National de la Recherche Scientifique, Paris, 1987).

L. R. Marshall, A. D. Hays, J. Kasinski, and R. Burnham, "Highly efficient optical parametric oscillators," in Eye-safe Lasers: Components, Systems, and Applications, A. M. Johnson, ed., Proc. SPIE 1419,141–152 (1991).

L. R. Marshall, A. Kaz, and R. Burnham, "Nonlinear cavity dumping," in Conference on Lasers and Electro-Optics, Vol. 11 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 436–437.

A. Kaz and L. R. Marshall, "Continuouswave diode-pumped lasers and parametric oscillator," in Conference on Lasers and Electro-Optics, Vol. 11 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 244–245.

T. Debuisschert, J. Raffy, J.-P. Pocholle, and M. Papuchon, "Dynamics of the pulsed intracavity OPO," in Advanced Solid-State Lasers, B. H. T. Chai and S. A. Payne, eds., Vol. 24 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1995), pp. 158–162.

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Figures (24)

Fig. 1
Fig. 1

Synoptic scheme of the IOPO. The OPO cavity (mirrors M1, M2) is inserted into the laser cavity (mirrors M1, M3).

Fig. 2
Fig. 2

Population inversion evolution versus time when Rs is equal to 0.7 and σ is equal to 10.

Fig. 3
Fig. 3

Intracavity pump-power evolution versus time when Rs is equal to 0.7 and σ is equal to 10. One obtains curve (a) without starting the OPO; curve (b), with the start of the OPO.

Fig. 4
Fig. 4

Intracavity signal-power evolution versus time when Rs is equal to 0.7 and σ is equal to 10. One pulse is produced.

Fig. 5
Fig. 5

Energy transfer efficiency versus time when Rs is equal to 0.7 and σ is equal to 10.

Fig. 6
Fig. 6

Detail of the temporal profile of the intracavity signal pulse when Rs is equal to 0.7 and σ is equal to 10.

Fig. 7
Fig. 7

Population inversion evolution versus time when Rs is equal to 0.95 and σ is equal to 10.

Fig. 8
Fig. 8

Intracavity pump-power evolution versus time when Rs is equal to 0.95 and σ is equal to 10. One obtains curve (a) without starting the OPO; curve (b), with the start of the OPO.

Fig. 9
Fig. 9

Intracavity signal-power evolution versus time when Rs is equal to 0.95 and σ is equal to 10. Two pulses are produced.

Fig. 10
Fig. 10

Energy transfer efficiency versus time when Rs is equal to 0.95 and σ is equal to 10.

Fig. 11
Fig. 11

Detail of the temporal profile of the first intracavity signal pulse when Rs is equal to 0.95 and σ is equal to 10.

Fig. 12
Fig. 12

Energy transfer efficiency from the laser medium to the signal pulse versus Rs. The curves with white squares correspond to σ = 10. The curves with black squares correspond to σ = 20. For a given value of Rs, the existence of several branches corresponds to several successive signal pulses (dashed curves). The upper branches (solid curves) indicate the total energy transfer efficiency.

Fig. 13
Fig. 13

Illustration of the buildup of the pump pulse. A linear depletion of the population inversion induces a Gaussian profile of the pump pulse.

Fig. 14
Fig. 14

Illustration of the buildup of the signal pulse. A linear depletion of the pump pulse induces a Gaussian profile of the signal pulse.

Fig. 15
Fig. 15

Comparison of the rise time at half-maximum of the signal pulse calculated analytically (solid curve) and numerically (squares). The white squares are calculated for σ = 10; the black squares, for σ = 20. The abscissa gives the intracavity pump power at the starting time of the signal pulse.

Fig. 16
Fig. 16

Comparison of the decrease time at half-maximum of the signal pulse calculated analytically (solid curve) and numerically (squares). The white squares are calculated for σ = 10; the black squares, for σ = 20. The abscissa gives the intracavity pump power at the starting time of the signal pulse.

Fig. 17
Fig. 17

Comparison of the FWHM of the signal pulse calculated analytically (solid curve) and numerically (squares). The white squares are calculated for σ = 10; the black squares, for σ = 20. The abscissa gives the intracavity pump power at the starting time of the signal pulse.

Fig. 18
Fig. 18

Comparison of the peak power of the signal pulse calculated analytically (solid curve) and numerically (squares). The white squares are calculated for σ = 10; the black squares, for σ = 20. The abscissa gives the intracavity pump power at the starting time of the signal pulse.

Fig. 19
Fig. 19

Starting time of the signal pulse as a function of the signal reflection coefficient. The solid curves are calculated numerically, and the dashed curves are calculated analytically. The white squares denote a pumping level of 10. The black squares denote a pumping level of 20.

Fig. 20
Fig. 20

Intracavity pump-power level at the starting time of the signal pulse as a function of the signal reflection coefficient. The solid curves are calculated numerically, and the dashed curves are calculated analytically. The white squares denote a pumping level of 10. The black squares denote a pumping level of 20.

Fig. 21
Fig. 21

Intracavity signal peak power as a function of the signal reflection coefficient for a pumping level of 10. The solid curve is calculated numerically, and the dashed curve is calculated analytically.

Fig. 22
Fig. 22

FWHM of the signal pulse as a function of the signal reflection coefficient for a pumping level of 10. The solid curve is calculated numerically, and the dashed curve is calculated analytically.

Fig. 23
Fig. 23

Intracavity signal peak power as a function of the signal reflection coefficient for a pumping level of 20. The solid curve is calculated numerically, and the dashed curve is calculated analytically.

Fig. 24
Fig. 24

FWHM of the signal pulse as a function of the signal reflection coefficient for a pumping level of 20. The solid curve is calculated numerically, and the dashed curve is calculated analytically.

Tables (4)

Tables Icon

Table 1 Normalized Values of the Starting Time of the Signal Pulse, Intracavity Pump Power, Rise Time and Decrease Time at Half-Maximum, FWHM, and Intracavity Peak Power of the Signal Pulse as a Function of the Signal Reflection Coefficienta

Tables Icon

Table 2 Same as Table 1, but Calculated Numerically for a Pumping Level of 20

Tables Icon

Table 3 Same as Table 1, but Calculated Analytically for a Pumping Level of 10

Tables Icon

Table 4 Same as Table 1, but Calculated Analytically for a Pumping Level of 20

Equations (115)

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ω p = ω s + ω id .
E j ± ( z , t ) = A j ( t ) exp ( ± i k j z ) exp [ i ( ω j t + ϕ j ± ) ] × exp [ ( r 2 / w j 2 ) ] , j = p , s
E id ± ( z , t ) = A id ± ( z , t ) exp ( ± i k id z ) exp [ i ( ω id t + ϕ id ± ) ] × exp [ ( r 2 / w id 2 ) ] .
d A p d t + 1 τ p A p = ω p 2 0 n p 2 Im ( P p + ) ω p 2 0 n p 2 Im ( P p ) ω p 2 0 n p 2 Im ( P laser ) ,
d A s d t + 1 τ s A s = ω s 2 0 n s 2 Im ( P s + ) ω s 2 0 n s 2 Im ( P s ) ,
n id c A id ± t ± A id ± z + α id A id + = ω id 2 0 n id c Im ( P id ± ) .
P p NL ( z , t ) = 0 χ NL E s ( z , t ) E id ( z , t ) ,
P s NL ( z , t ) = 0 χ NL E p ( z , t ) E id * ( z , t ) ,
P id NL ( z , t ) = 0 χ NL E p ( z , t ) E s * ( z , t ) .
1 w id 2 = 1 w ¯ id 2 = 1 w s 2 + 1 w p 2 .
Im ( P id ± ) = 0 χ NL A p A s sin ( Δ ϕ ± ) .
Δ ϕ ± = ϕ p ± ϕ i ± ϕ s ± .
k p = k s + k i .
A id + ( z , t ) = B + ( t ) α id [ 1 exp ( α id z ) ] ,
A id ( z , t ) = B ( t ) α id { 1 exp [ α id ( z 1 ) ] } ,
B ± ( t ) = μ 0 ω id c 2 n id Im ( P id ± ) .
Im ( P p ± ) = 0 χ NL 2 [ 2 ( 1 + w p 2 w ¯ p 2 ) ] ( l OPO 2 L laser ) A s 2 A p × ω id 2 c n id l OPO 2 ( 1 α id l OPO 3 ) sin 2 ( Δ ϕ ± ) ,
1 w ¯ p 2 = 1 w s 2 + 1 w id 2 .
Im ( P s ± ) = 0 χ NL 2 [ 2 ( 1 + w s 2 w ¯ s 2 ) ] ( l OPO 2 L OPO ) A p 2 A s × ω id 2 c n id l OPO 2 ( 1 α id l OPO 3 ) sin 2 ( Δ ϕ ± ) ,
1 w ¯ s 2 = 1 w p 2 + 1 w id 2 .
P laser = i 2 ћ γ l laser L laser ρ ¯ a a A p .
d ρ ¯ a a d t = λ a γ a [ ρ ¯ a a ( 1 + A p 2 2 ћ 2 γ γ a ) ] .
F j = 1 2 n j c 0 A p 2 π w j 2 2 1 ћ ω k , j = ( p , s ) .
d ρ ¯ a a d t = λ a γ a [ ρ ¯ a a ( 1 + F p F sat ) ] .
F sat = ћ γ n p c 0 2 2 ω p τ a π w p 2 2 = 1 2 σ e τ a π w p 2 2 .
ρ ¯ a a th = 0 c n p ћ γ ( 1 r p ) ω p 2 l laser .
τ p = 2 n p L laser c 1 ( 1 r p ) .
Δ ϕ + = Δ ϕ = π / 2 .
τ p d A p d t = ( ρ a a ¯ ρ a a th ¯ 1 ) A p τ p ω p 0 n p 2 Im ( P p ) .
F p th = 2 n p n s n id c 3 0 ( 1 r s ) ( 1 + α id l OPO 3 ) h ω p ω s ω id χ NL 2 l OPO 2 π ( w s 2 + w p 2 ) 2 .
τ s = 2 n s L OPO c = 1 ( 1 r s ) .
τ s d A s d t = ( F p F p th 1 ) A s .
N = ρ ¯ a a / ρ ¯ a a th ,
P p = F p / F p th ,
P s = F s / F p th .
d P s d t = P s ( P p 1 ) ,
β p d P p d t = P p ( N 1 ) F P s P p ,
β a d N d t = σ N ( 1 + x P p ) .
σ = λ a τ a ρ ¯ a a th .
F = 1 r s 1 r p .
x = F p th F sat .
β p = τ p / τ s .
β a = 2 τ a / τ s .
E j = ( ћ ω j 2 0 L j ) 1 / 2 { b j exp [ i ( k j r ω j t ) ] b j + × exp [ i ( k j r ω j t ) ] , j = ( s , id ) .
[ b k , b j + ] = δ k j .
H = i ћ χ B p ( b id + b s + b id b s ) .
N s = b s + b s .
d N s d t = 1 i ћ [ N s , H ] 2 τ s N s .
d N s d t = χ B p ( b s + b id + + b s b id ) .
d b id + d t = 1 i ћ [ b id + , H ] 1 τ id b id + .
b id + = τ id χ B p b s .
τ s 2 d N s d t = τ id τ s χ 2 B p 2 ( N s + 1 2 ) N s .
H q = ћ ω s ( N s + 1 / 2 ) .
H = 1 / 2 0 n s 2 | E s | 2
V = 2 L OPO ( π w s 2 / 2 ) .
H cl = ћ ω s ( n s / c ) 2 L OPO F s .
Δ P s = 1 2 c n s 1 2 L OPO 1 F p th .
Δ P p = 1 2 c n p 1 2 L laser 1 F p th .
d P s d t = P s ( P p 1 ) + P p × Δ P s ,
β p dP p d t = P p ( N 1 ) F P s P p + N × Δ P p ,
β a d N d t = σ N ( 1 + x P p ) .
E s = T s τ s 2 F p th d t P s ( t ) .
N ( Δ T ) = π w p 2 2 l laser ρ ¯ a a th N ( Δ T ) .
Eff ( t 3 ) = x F β a N ( Δ T ) t 3 d t P s ( t ) .
N ( t 1 ) = N ( 0 ) exp [ ( t 1 / β a ) ] + σ { 1 exp [ ( t 1 / β a ) ] } .
N ( t 2 ) = [ 1 N ( Δ T ) Δ t p on ] t 2 + 1 .
P p ( t 2 ) = P p max exp [ ( t 2 2 / Δ τ p 2 ) ] .
Δ τ p = [ 2 β p Δ t p on N ( Δ T ) 1 ] 1 / 2 .
P p max = β a [ N ( Δ T ) 1 ] x × Δ t p on .
Δ t p on = β p C p 2 ,
C p = { 1 + π 8 [ N ( Δ T ) 1 ] } 1 / 2 + { π 8 [ N ( Δ T ) 1 ] } 1 / 2 .
Δ τ p = β p C p { [ 2 N ( Δ T ) 1 ] 1 / 2 } ,
P p max = β a [ N ( Δ T ) 1 ] x β p C p 2 .
F p max = c 2 n p L laser π w p 2 2 l laser 1 C p 2 [ ρ ¯ a a ( Δ T ) ρ ¯ a a th ] .
P p ( t 3 ) = [ 1 P p ( T 2 ) Δ t s on ] t 3 + 1 , Δ t s on t 3 0 .
P s ( t 3 ) = P s max exp [ ( t 3 2 / Δ τ s 2 ) ] ,
Δ τ s = { 2 Δ t s on [ P p ( T 2 ) 1 ] } 1 / 2 .
P s max = β p F [ P p ( T 2 ) 1 ] Δ t s on .
Δ t s on = ( { 1 + π 8 [ P p ( T 2 ) 1 ] } 1 / 2 + { π 8 [ P p ( T 2 ) 1 ] } 1 / 2 ) 2 .
exp ( t up 2 / Δ τ s 2 ) = 1 / 2 .
t up = { 2 log 2 [ P p ( T 2 ) 1 ] } 1 / 2 × ( { 1 + π 8 [ P p ( T 2 ) 1 ] } 1 / 2 + { π 8 [ P p ( T 2 ) 1 ] } 1 / 2 ) .
F s max = n p L laser n s L OPO 1 Δ t s on [ F p ( T 2 ) F p th ] .
F s max = c 2 n s L OPO 1 Δ t s on 1 C p 2 π w p 2 2 l laser × [ ρ ¯ a a ( Δ T ) ρ ¯ a a th ] .
β p ( d P p / d t ) = P p P s max F .
P p = exp ( α t 3 ) for t 3 0 ,
α = P s max F β p .
P s ( t 3 ) = P s max exp { ( 1 / α ) [ 1 exp ( α t 3 ) ] t 3 } for t 3 0 .
exp { ( 1 / α ) [ 1 exp ( α t down ) ] t down } = 1 / 2 .
α = [ P p ( T 2 ) 1 ] ( { 1 + π 8 [ P p ( T 2 ) 1 ] } 1 / 2 + { π 8 [ P p ( T 2 ) 1 ] } 1 / 2 ) 2 .
P s ( t 0 ) = Δ P s .
P s ( t 2 ) = Δ P s exp { π 2 P s max Δ τ p [ erf ( t 2 Δ τ p ) erf ( t 0 Δ τ p ) ] ( t 2 t 0 ) } .
[ 1 N ( Δ T ) Δ t p on ] T 2 = F P s ( T 2 ) .
F s out = T s F s .
E s out = T s τ s 2 F p th Δ t s on + d t 3 P s ( t 3 ) .
E s out = T s τ s 2 F p th β p F P p max .
E s out = 1 C p 2 π w p 2 2 l laser [ ρ ¯ a a ( Δ T ) ρ ¯ a a th ] .
N ( T ) = π w p 2 2 l laser ρ ¯ a a ( Δ T ) .
Eff s = 1 C p 2 [ N ( Δ T ) 1 N ( Δ T ) ] .
P n ( t ) = 2 exp [ i ( ν n t + ϕ n ) ] 1 N 0 2 L laser d z U n * ( z ) ρ a b ( z , t ) .
ρ a b = i 2 ħ γ ρ a a E n ( t ) U n ( z ) exp [ i ( ν n t + ϕ n ) ] .
E n ( t ) U n ( z ) = A p ( t ) exp ( i k p z ) .
P n ( t ) = i 2 N ħ γ A p ρ a a ¯ 2 l laser .
N = 0 2 L laser d z | exp ( i k p z ) | 2 = 2 L laser .
P laser ( t ) = i 2 ħ γ A p ρ a a ¯ l laser L laser .
d ρ a a d t = λ a γ a ρ a a R ρ a a .
R = 1 2 γ [ E n ( t ) ħ ] 2 | U n ( z ) | 2 .
E n ( t ) U n ( z ) = A p ( t ) [ exp ( i k p z ) + exp ( i k p z ) ] .
d ρ ¯ a a d t = λ a γ a [ ρ ¯ a a ( 1 + A p 2 ) 2 ħ 2 γ γ a ] .
x β p d P p d t + β a d N d t = σ N x P p .
β a [ 1 N ( Δ t p on ) ] + β p x [ P p max P p ( Δ t p on ) ] = x Δ t p on 0 d t 2 P p ( t 2 ) + σ × Δ t p on Δ t p on 0 d t 2 N ( t 2 ) .
Δ t p on = β p C p 2 ,
C p = { 1 + π 8 [ N ( Δ T ) 1 ] } 1 / 2 + { π 8 [ N ( Δ T ) 1 ] } 1 / 2 .
β p d P p d t + F d P s d t = F P s + N × P p P p .
β p [ 1 P p ( T 2 ) ] + F [ P s max P s ( Δ t s on ) ] = F Δ t s on 0 d t 3 P s ( t 3 ) .
Δ t s on = ( { 1 + π 8 [ P p ( T 2 ) 1 ] } 1 / 2 + { π 8 [ P p ( T 2 ) 1 ] } 1 / 2 ) 2 .

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