Abstract

We describe the passive Q-switching regime of a neodymium laser that contains a Cr4+:YAG saturable absorber inside the cavity. Two configurations of the laser cavity are modeled: the cavity containing a partial polarizer as an additional unit and the microchip laser cavity in which the functions of an active element and saturable absorber are combined in a single piece of a Nd3+:YAG/Cr4+:YAG crystal with reflecting facets. It is shown that both lasers are able to generate a giant pulse with a changing state of polarization. Both the kinetics of the state of polarization of the pulse and its impact on second-harmonic generation are treated numerically. We find that the up-conversion efficiency can be enhanced and that harmonic-pulse compression can be obtained by means of the proper orientations of the intracavity polarizing elements, for the first configuration, or by choosing the polarization of the longitudinal pump and angular orientation of the doubling crystal, for the second one. The ability of the doubling crystal to analyze the state of polarization of pulses with nonlinearly changing polarization is discussed.

© 2000 Optical Society of America

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  1. K. Spariosu, W. Chen, R. Stultz, M. Birnbaum, and A. V. Shestakov, “Dual Q switching and laser action at 1.05 and 1.44 μm in a Nd3+:YAG–Cr4+:YAG oscillator at 300 K,” Opt. Lett. 18, 814–816 (1993).
    [CrossRef] [PubMed]
  2. H. J. Eichler, A. Haase, M. R. Kokta, and R. Menzel, “Cr4+:YAG as passive Q-switch for a Nd:YALO oscillator with an average repetition rate of 2.7 kHz, TEM00 mode and 13 W output,” Appl. Phys. B 58, 409–411 (1994).
    [CrossRef]
  3. Y. Shimony, Z. Burshtein, A. Ben-Amar Baranga, Y. Kalisky, and M. Strauss, “Repetitively Q-switching of a CW Nd:YAG laser using Cr4+:YAG saturable absorbers,” IEEE J. Quantum Electron. 32, 305–310 (1996).
    [CrossRef]
  4. A. Sennaroglu, “Continuous-wave power transmission and thermal lensing of a saturable absorber subject to excited-state absorption,” Appl. Opt. 38, 3334–3337 (1999).
    [CrossRef]
  5. H. Eilers, K. R. Hoffman, M. Dennis, S. M. Jacobsen, and W. M. Yen, “Saturation of 1.064 μm absorption in Cr, Ca:Y3AL5O12 crystals,” Appl. Phys. Lett. 61, 2958–2960 (1992).
    [CrossRef]
  6. N. N. Il’ichev, A. V. Kir’yanov, P. P. Pashinin, and S. M. Shpuga, “Investigation of nonlinear-absorption anisotropy in YAG:Cr4+,” JETP 78, 768–776 (1994).
  7. N. N. Il’ichev, A. V. Kir’yanov, P. P. Pashinin, and S. M. Shpuga, “Changes in the profile and state of polarization of a short light pulse (λ = 1.06 μm) during propagation in a Cr4+:YAG crystal,” Quantum Electron. 24, 771–776 (1994).
    [CrossRef]
  8. A. Brignon, “Anisotropic properties of pulsed four-wave mixing in Cr4+:YAG saturable absorbers,” J. Opt. Soc. Am. B 13, 2154–2163 (1996).
    [CrossRef]
  9. S. Camacho-Lopez, R. P. M. Green, G. J. Crofts, and M. J. Damzen, “Intensity-induced birefringence in Cr4+:YAG,” J. Mod. Opt. 44, 209–219 (1997).
    [CrossRef]
  10. N. N. Il’ichev, A. V. Kir’yanov, and P. P. Pashinin, “Model of passive Q switching taking account of the anisotropy of nonlinear absorption in a crystal switch with phototropic centers,” Quantum Electron. 28, 147–151 (1998).
    [CrossRef]
  11. I. V. Klimov, I. A. Shcherbakov, and V. B. Tsvetkov, “Control of the Nd-laser output by Cr-doped Q-switches,” Laser Phys. 8, 232–237 (1998).
  12. A. V. Kir’yanov, V. Aboites, and N. N. Il’ichev, “A polarization-bistable neodymium laser with a Cr4+:YAG passive switch under the weak resonant signal control,” Opt. Comm. 169, 309–315 (1999).
    [CrossRef]
  13. S. Zhou, K. K. Lee, Y. C. Chen, and S. Li, “Monolithic self-Q-switched Cr, Nd:YAG laser,” Opt. Lett. 18, 511–512 (1993).
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  14. N. N. Il’ichev, A. V. Kir’yanov, E. S. Gulyamova, and P. P. Pashinin, “Polarization of a neodymium laser with a passive switch based on a Cr4+:YAG crystal,” Quantum Electron. 28, 17–20 (1998).
    [CrossRef]
  15. J. J. Zayhowski and C. Dill III, “Diode-pumped passively Q-switched picosecond microchip lasers,” Opt. Lett. 19, 1427–1429 (1994).
    [CrossRef] [PubMed]
  16. A. Agnesi, S. Dell’Acqua, C. Morello, G. Piccinno, G. C. Reali, and Z. Sun, “Diode-pumped neodymium lasers repetitively Q-switched by Cr4+:YAG solid-state saturable absorbers,” IEEE J. Sel. Top. Quantum Electron. 3, 45–52 (1997).
    [CrossRef]
  17. A. Agnesi, S. Dell’Acqua, E. Piccinini, G. Really, and G. Piccinno, “Efficient wavelength conversion with high-power passively Q-switched diode-pumped neodymium lasers,” IEEE J. Quantum Electron. 34, 1480–1484 (1998).
    [CrossRef]
  18. P. Peterson, A. Gavrielidis, M. P. Sharma, and T. Erneux, “Dynamics of passively Q-switched microchip lasers,” IEEE J. Quantum Electron. 35, 1247–1256 (1999).
    [CrossRef]
  19. M. Brunel, O. Emile, M. Vallet, F. Bretenaker, A. Le Floch, L. Fulbert, J. Marty, B. Ferrand, and E. Molva, “Experimental and theoretical study of monomode vectorial lasers passively Q-switched by Cr4+:yttrium aluminum garnet absorbers,” Phys. Rev. A 60, 4052–4058 (1999).
    [CrossRef]
  20. N. Zheludev, S. Saltiel, and P. Yankov, “Second-harmonic generators as a new class of light polarizers and analyzers,” Sov. J. Quantum Electron. 17, 948–952 (1987).
    [CrossRef]
  21. R. Dalgliesh, A. D. May, and G. Stephan, “Polarization states of a single-mode (microchip) Nd3+:YAG laser—Part I: theory,” IEEE J. Quantum Electron. 34, 1485–1492 (1998).
    [CrossRef]
  22. R. Dalgliesh, A. D. May, and G. Stephan, “Polarization states of a single-mode (microchip) Nd3+:YAG laser—PartII. comparison of theory and experiment,” IEEE J. Quantum Electron. 34, 1493–1502 (1998).
    [CrossRef]
  23. R. V. Goidin, V. S. Kichuk, V. V. Kravtsov, G. D. Laptev, E. G. Lariontsev, and V. V. Firsov, “Influence of the pump polarisation on the characteristics of radiation of a ring Nd:YAG chip laser,” Quantum Electron. 28, 358–360 (1998).
    [CrossRef]
  24. V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer-Verlag, Berlin, 1991).
  25. N. N. Il’ichev, A. V. Kir’yanov, E. S. Gulyamova, and P. P. Pashinin, “Polarization characteristics of a neodymium laser passively Q-switched with a Cr4+:YAG crystal,” Quantum Electron. 27, 298–301 (1997).
    [CrossRef]
  26. N. N. Il’ichev, E. S. Gulyamova, and P. P. Pashinin, “Passive Q switching of a neodymium laser by a Cr4+:YAG crystal switch,” Quantum Electron. 27, 972–977 (1997).
    [CrossRef]
  27. Y. Wang and R. Dragila, “Efficient conversion of picosecond laser pulses into second-harmonic frequency using group-velocity dispersion,” Phys. Rev. A 41, 5645–5649 (1990).
    [CrossRef] [PubMed]
  28. Y. Wang and B. Luther-Davies, “Frequency-doubling pulse compressor for picosecond high-power neodymium laser pulses,” Opt. Lett. 17, 1459–1461 (1992).
    [CrossRef] [PubMed]

1999 (4)

A. Sennaroglu, “Continuous-wave power transmission and thermal lensing of a saturable absorber subject to excited-state absorption,” Appl. Opt. 38, 3334–3337 (1999).
[CrossRef]

A. V. Kir’yanov, V. Aboites, and N. N. Il’ichev, “A polarization-bistable neodymium laser with a Cr4+:YAG passive switch under the weak resonant signal control,” Opt. Comm. 169, 309–315 (1999).
[CrossRef]

P. Peterson, A. Gavrielidis, M. P. Sharma, and T. Erneux, “Dynamics of passively Q-switched microchip lasers,” IEEE J. Quantum Electron. 35, 1247–1256 (1999).
[CrossRef]

M. Brunel, O. Emile, M. Vallet, F. Bretenaker, A. Le Floch, L. Fulbert, J. Marty, B. Ferrand, and E. Molva, “Experimental and theoretical study of monomode vectorial lasers passively Q-switched by Cr4+:yttrium aluminum garnet absorbers,” Phys. Rev. A 60, 4052–4058 (1999).
[CrossRef]

1998 (7)

A. Agnesi, S. Dell’Acqua, E. Piccinini, G. Really, and G. Piccinno, “Efficient wavelength conversion with high-power passively Q-switched diode-pumped neodymium lasers,” IEEE J. Quantum Electron. 34, 1480–1484 (1998).
[CrossRef]

R. Dalgliesh, A. D. May, and G. Stephan, “Polarization states of a single-mode (microchip) Nd3+:YAG laser—Part I: theory,” IEEE J. Quantum Electron. 34, 1485–1492 (1998).
[CrossRef]

R. Dalgliesh, A. D. May, and G. Stephan, “Polarization states of a single-mode (microchip) Nd3+:YAG laser—PartII. comparison of theory and experiment,” IEEE J. Quantum Electron. 34, 1493–1502 (1998).
[CrossRef]

R. V. Goidin, V. S. Kichuk, V. V. Kravtsov, G. D. Laptev, E. G. Lariontsev, and V. V. Firsov, “Influence of the pump polarisation on the characteristics of radiation of a ring Nd:YAG chip laser,” Quantum Electron. 28, 358–360 (1998).
[CrossRef]

N. N. Il’ichev, A. V. Kir’yanov, and P. P. Pashinin, “Model of passive Q switching taking account of the anisotropy of nonlinear absorption in a crystal switch with phototropic centers,” Quantum Electron. 28, 147–151 (1998).
[CrossRef]

I. V. Klimov, I. A. Shcherbakov, and V. B. Tsvetkov, “Control of the Nd-laser output by Cr-doped Q-switches,” Laser Phys. 8, 232–237 (1998).

N. N. Il’ichev, A. V. Kir’yanov, E. S. Gulyamova, and P. P. Pashinin, “Polarization of a neodymium laser with a passive switch based on a Cr4+:YAG crystal,” Quantum Electron. 28, 17–20 (1998).
[CrossRef]

1997 (4)

S. Camacho-Lopez, R. P. M. Green, G. J. Crofts, and M. J. Damzen, “Intensity-induced birefringence in Cr4+:YAG,” J. Mod. Opt. 44, 209–219 (1997).
[CrossRef]

A. Agnesi, S. Dell’Acqua, C. Morello, G. Piccinno, G. C. Reali, and Z. Sun, “Diode-pumped neodymium lasers repetitively Q-switched by Cr4+:YAG solid-state saturable absorbers,” IEEE J. Sel. Top. Quantum Electron. 3, 45–52 (1997).
[CrossRef]

N. N. Il’ichev, A. V. Kir’yanov, E. S. Gulyamova, and P. P. Pashinin, “Polarization characteristics of a neodymium laser passively Q-switched with a Cr4+:YAG crystal,” Quantum Electron. 27, 298–301 (1997).
[CrossRef]

N. N. Il’ichev, E. S. Gulyamova, and P. P. Pashinin, “Passive Q switching of a neodymium laser by a Cr4+:YAG crystal switch,” Quantum Electron. 27, 972–977 (1997).
[CrossRef]

1996 (2)

Y. Shimony, Z. Burshtein, A. Ben-Amar Baranga, Y. Kalisky, and M. Strauss, “Repetitively Q-switching of a CW Nd:YAG laser using Cr4+:YAG saturable absorbers,” IEEE J. Quantum Electron. 32, 305–310 (1996).
[CrossRef]

A. Brignon, “Anisotropic properties of pulsed four-wave mixing in Cr4+:YAG saturable absorbers,” J. Opt. Soc. Am. B 13, 2154–2163 (1996).
[CrossRef]

1994 (4)

N. N. Il’ichev, A. V. Kir’yanov, P. P. Pashinin, and S. M. Shpuga, “Investigation of nonlinear-absorption anisotropy in YAG:Cr4+,” JETP 78, 768–776 (1994).

N. N. Il’ichev, A. V. Kir’yanov, P. P. Pashinin, and S. M. Shpuga, “Changes in the profile and state of polarization of a short light pulse (λ = 1.06 μm) during propagation in a Cr4+:YAG crystal,” Quantum Electron. 24, 771–776 (1994).
[CrossRef]

H. J. Eichler, A. Haase, M. R. Kokta, and R. Menzel, “Cr4+:YAG as passive Q-switch for a Nd:YALO oscillator with an average repetition rate of 2.7 kHz, TEM00 mode and 13 W output,” Appl. Phys. B 58, 409–411 (1994).
[CrossRef]

J. J. Zayhowski and C. Dill III, “Diode-pumped passively Q-switched picosecond microchip lasers,” Opt. Lett. 19, 1427–1429 (1994).
[CrossRef] [PubMed]

1993 (2)

1992 (2)

H. Eilers, K. R. Hoffman, M. Dennis, S. M. Jacobsen, and W. M. Yen, “Saturation of 1.064 μm absorption in Cr, Ca:Y3AL5O12 crystals,” Appl. Phys. Lett. 61, 2958–2960 (1992).
[CrossRef]

Y. Wang and B. Luther-Davies, “Frequency-doubling pulse compressor for picosecond high-power neodymium laser pulses,” Opt. Lett. 17, 1459–1461 (1992).
[CrossRef] [PubMed]

1990 (1)

Y. Wang and R. Dragila, “Efficient conversion of picosecond laser pulses into second-harmonic frequency using group-velocity dispersion,” Phys. Rev. A 41, 5645–5649 (1990).
[CrossRef] [PubMed]

1987 (1)

N. Zheludev, S. Saltiel, and P. Yankov, “Second-harmonic generators as a new class of light polarizers and analyzers,” Sov. J. Quantum Electron. 17, 948–952 (1987).
[CrossRef]

Aboites, V.

A. V. Kir’yanov, V. Aboites, and N. N. Il’ichev, “A polarization-bistable neodymium laser with a Cr4+:YAG passive switch under the weak resonant signal control,” Opt. Comm. 169, 309–315 (1999).
[CrossRef]

Agnesi, A.

A. Agnesi, S. Dell’Acqua, E. Piccinini, G. Really, and G. Piccinno, “Efficient wavelength conversion with high-power passively Q-switched diode-pumped neodymium lasers,” IEEE J. Quantum Electron. 34, 1480–1484 (1998).
[CrossRef]

A. Agnesi, S. Dell’Acqua, C. Morello, G. Piccinno, G. C. Reali, and Z. Sun, “Diode-pumped neodymium lasers repetitively Q-switched by Cr4+:YAG solid-state saturable absorbers,” IEEE J. Sel. Top. Quantum Electron. 3, 45–52 (1997).
[CrossRef]

Ben-Amar Baranga, A.

Y. Shimony, Z. Burshtein, A. Ben-Amar Baranga, Y. Kalisky, and M. Strauss, “Repetitively Q-switching of a CW Nd:YAG laser using Cr4+:YAG saturable absorbers,” IEEE J. Quantum Electron. 32, 305–310 (1996).
[CrossRef]

Birnbaum, M.

Bretenaker, F.

M. Brunel, O. Emile, M. Vallet, F. Bretenaker, A. Le Floch, L. Fulbert, J. Marty, B. Ferrand, and E. Molva, “Experimental and theoretical study of monomode vectorial lasers passively Q-switched by Cr4+:yttrium aluminum garnet absorbers,” Phys. Rev. A 60, 4052–4058 (1999).
[CrossRef]

Brignon, A.

Brunel, M.

M. Brunel, O. Emile, M. Vallet, F. Bretenaker, A. Le Floch, L. Fulbert, J. Marty, B. Ferrand, and E. Molva, “Experimental and theoretical study of monomode vectorial lasers passively Q-switched by Cr4+:yttrium aluminum garnet absorbers,” Phys. Rev. A 60, 4052–4058 (1999).
[CrossRef]

Burshtein, Z.

Y. Shimony, Z. Burshtein, A. Ben-Amar Baranga, Y. Kalisky, and M. Strauss, “Repetitively Q-switching of a CW Nd:YAG laser using Cr4+:YAG saturable absorbers,” IEEE J. Quantum Electron. 32, 305–310 (1996).
[CrossRef]

Camacho-Lopez, S.

S. Camacho-Lopez, R. P. M. Green, G. J. Crofts, and M. J. Damzen, “Intensity-induced birefringence in Cr4+:YAG,” J. Mod. Opt. 44, 209–219 (1997).
[CrossRef]

Chen, W.

Chen, Y. C.

Crofts, G. J.

S. Camacho-Lopez, R. P. M. Green, G. J. Crofts, and M. J. Damzen, “Intensity-induced birefringence in Cr4+:YAG,” J. Mod. Opt. 44, 209–219 (1997).
[CrossRef]

Dalgliesh, R.

R. Dalgliesh, A. D. May, and G. Stephan, “Polarization states of a single-mode (microchip) Nd3+:YAG laser—Part I: theory,” IEEE J. Quantum Electron. 34, 1485–1492 (1998).
[CrossRef]

R. Dalgliesh, A. D. May, and G. Stephan, “Polarization states of a single-mode (microchip) Nd3+:YAG laser—PartII. comparison of theory and experiment,” IEEE J. Quantum Electron. 34, 1493–1502 (1998).
[CrossRef]

Damzen, M. J.

S. Camacho-Lopez, R. P. M. Green, G. J. Crofts, and M. J. Damzen, “Intensity-induced birefringence in Cr4+:YAG,” J. Mod. Opt. 44, 209–219 (1997).
[CrossRef]

Dell’Acqua, S.

A. Agnesi, S. Dell’Acqua, E. Piccinini, G. Really, and G. Piccinno, “Efficient wavelength conversion with high-power passively Q-switched diode-pumped neodymium lasers,” IEEE J. Quantum Electron. 34, 1480–1484 (1998).
[CrossRef]

A. Agnesi, S. Dell’Acqua, C. Morello, G. Piccinno, G. C. Reali, and Z. Sun, “Diode-pumped neodymium lasers repetitively Q-switched by Cr4+:YAG solid-state saturable absorbers,” IEEE J. Sel. Top. Quantum Electron. 3, 45–52 (1997).
[CrossRef]

Dennis, M.

H. Eilers, K. R. Hoffman, M. Dennis, S. M. Jacobsen, and W. M. Yen, “Saturation of 1.064 μm absorption in Cr, Ca:Y3AL5O12 crystals,” Appl. Phys. Lett. 61, 2958–2960 (1992).
[CrossRef]

Dill III, C.

Dragila, R.

Y. Wang and R. Dragila, “Efficient conversion of picosecond laser pulses into second-harmonic frequency using group-velocity dispersion,” Phys. Rev. A 41, 5645–5649 (1990).
[CrossRef] [PubMed]

Eichler, H. J.

H. J. Eichler, A. Haase, M. R. Kokta, and R. Menzel, “Cr4+:YAG as passive Q-switch for a Nd:YALO oscillator with an average repetition rate of 2.7 kHz, TEM00 mode and 13 W output,” Appl. Phys. B 58, 409–411 (1994).
[CrossRef]

Eilers, H.

H. Eilers, K. R. Hoffman, M. Dennis, S. M. Jacobsen, and W. M. Yen, “Saturation of 1.064 μm absorption in Cr, Ca:Y3AL5O12 crystals,” Appl. Phys. Lett. 61, 2958–2960 (1992).
[CrossRef]

Emile, O.

M. Brunel, O. Emile, M. Vallet, F. Bretenaker, A. Le Floch, L. Fulbert, J. Marty, B. Ferrand, and E. Molva, “Experimental and theoretical study of monomode vectorial lasers passively Q-switched by Cr4+:yttrium aluminum garnet absorbers,” Phys. Rev. A 60, 4052–4058 (1999).
[CrossRef]

Erneux, T.

P. Peterson, A. Gavrielidis, M. P. Sharma, and T. Erneux, “Dynamics of passively Q-switched microchip lasers,” IEEE J. Quantum Electron. 35, 1247–1256 (1999).
[CrossRef]

Ferrand, B.

M. Brunel, O. Emile, M. Vallet, F. Bretenaker, A. Le Floch, L. Fulbert, J. Marty, B. Ferrand, and E. Molva, “Experimental and theoretical study of monomode vectorial lasers passively Q-switched by Cr4+:yttrium aluminum garnet absorbers,” Phys. Rev. A 60, 4052–4058 (1999).
[CrossRef]

Firsov, V. V.

R. V. Goidin, V. S. Kichuk, V. V. Kravtsov, G. D. Laptev, E. G. Lariontsev, and V. V. Firsov, “Influence of the pump polarisation on the characteristics of radiation of a ring Nd:YAG chip laser,” Quantum Electron. 28, 358–360 (1998).
[CrossRef]

Fulbert, L.

M. Brunel, O. Emile, M. Vallet, F. Bretenaker, A. Le Floch, L. Fulbert, J. Marty, B. Ferrand, and E. Molva, “Experimental and theoretical study of monomode vectorial lasers passively Q-switched by Cr4+:yttrium aluminum garnet absorbers,” Phys. Rev. A 60, 4052–4058 (1999).
[CrossRef]

Gavrielidis, A.

P. Peterson, A. Gavrielidis, M. P. Sharma, and T. Erneux, “Dynamics of passively Q-switched microchip lasers,” IEEE J. Quantum Electron. 35, 1247–1256 (1999).
[CrossRef]

Goidin, R. V.

R. V. Goidin, V. S. Kichuk, V. V. Kravtsov, G. D. Laptev, E. G. Lariontsev, and V. V. Firsov, “Influence of the pump polarisation on the characteristics of radiation of a ring Nd:YAG chip laser,” Quantum Electron. 28, 358–360 (1998).
[CrossRef]

Green, R. P. M.

S. Camacho-Lopez, R. P. M. Green, G. J. Crofts, and M. J. Damzen, “Intensity-induced birefringence in Cr4+:YAG,” J. Mod. Opt. 44, 209–219 (1997).
[CrossRef]

Gulyamova, E. S.

N. N. Il’ichev, A. V. Kir’yanov, E. S. Gulyamova, and P. P. Pashinin, “Polarization of a neodymium laser with a passive switch based on a Cr4+:YAG crystal,” Quantum Electron. 28, 17–20 (1998).
[CrossRef]

N. N. Il’ichev, E. S. Gulyamova, and P. P. Pashinin, “Passive Q switching of a neodymium laser by a Cr4+:YAG crystal switch,” Quantum Electron. 27, 972–977 (1997).
[CrossRef]

N. N. Il’ichev, A. V. Kir’yanov, E. S. Gulyamova, and P. P. Pashinin, “Polarization characteristics of a neodymium laser passively Q-switched with a Cr4+:YAG crystal,” Quantum Electron. 27, 298–301 (1997).
[CrossRef]

Haase, A.

H. J. Eichler, A. Haase, M. R. Kokta, and R. Menzel, “Cr4+:YAG as passive Q-switch for a Nd:YALO oscillator with an average repetition rate of 2.7 kHz, TEM00 mode and 13 W output,” Appl. Phys. B 58, 409–411 (1994).
[CrossRef]

Hoffman, K. R.

H. Eilers, K. R. Hoffman, M. Dennis, S. M. Jacobsen, and W. M. Yen, “Saturation of 1.064 μm absorption in Cr, Ca:Y3AL5O12 crystals,” Appl. Phys. Lett. 61, 2958–2960 (1992).
[CrossRef]

Il’ichev, N. N.

A. V. Kir’yanov, V. Aboites, and N. N. Il’ichev, “A polarization-bistable neodymium laser with a Cr4+:YAG passive switch under the weak resonant signal control,” Opt. Comm. 169, 309–315 (1999).
[CrossRef]

N. N. Il’ichev, A. V. Kir’yanov, and P. P. Pashinin, “Model of passive Q switching taking account of the anisotropy of nonlinear absorption in a crystal switch with phototropic centers,” Quantum Electron. 28, 147–151 (1998).
[CrossRef]

N. N. Il’ichev, A. V. Kir’yanov, E. S. Gulyamova, and P. P. Pashinin, “Polarization of a neodymium laser with a passive switch based on a Cr4+:YAG crystal,” Quantum Electron. 28, 17–20 (1998).
[CrossRef]

N. N. Il’ichev, E. S. Gulyamova, and P. P. Pashinin, “Passive Q switching of a neodymium laser by a Cr4+:YAG crystal switch,” Quantum Electron. 27, 972–977 (1997).
[CrossRef]

N. N. Il’ichev, A. V. Kir’yanov, E. S. Gulyamova, and P. P. Pashinin, “Polarization characteristics of a neodymium laser passively Q-switched with a Cr4+:YAG crystal,” Quantum Electron. 27, 298–301 (1997).
[CrossRef]

N. N. Il’ichev, A. V. Kir’yanov, P. P. Pashinin, and S. M. Shpuga, “Investigation of nonlinear-absorption anisotropy in YAG:Cr4+,” JETP 78, 768–776 (1994).

N. N. Il’ichev, A. V. Kir’yanov, P. P. Pashinin, and S. M. Shpuga, “Changes in the profile and state of polarization of a short light pulse (λ = 1.06 μm) during propagation in a Cr4+:YAG crystal,” Quantum Electron. 24, 771–776 (1994).
[CrossRef]

Jacobsen, S. M.

H. Eilers, K. R. Hoffman, M. Dennis, S. M. Jacobsen, and W. M. Yen, “Saturation of 1.064 μm absorption in Cr, Ca:Y3AL5O12 crystals,” Appl. Phys. Lett. 61, 2958–2960 (1992).
[CrossRef]

Kalisky, Y.

Y. Shimony, Z. Burshtein, A. Ben-Amar Baranga, Y. Kalisky, and M. Strauss, “Repetitively Q-switching of a CW Nd:YAG laser using Cr4+:YAG saturable absorbers,” IEEE J. Quantum Electron. 32, 305–310 (1996).
[CrossRef]

Kichuk, V. S.

R. V. Goidin, V. S. Kichuk, V. V. Kravtsov, G. D. Laptev, E. G. Lariontsev, and V. V. Firsov, “Influence of the pump polarisation on the characteristics of radiation of a ring Nd:YAG chip laser,” Quantum Electron. 28, 358–360 (1998).
[CrossRef]

Kir’yanov, A. V.

A. V. Kir’yanov, V. Aboites, and N. N. Il’ichev, “A polarization-bistable neodymium laser with a Cr4+:YAG passive switch under the weak resonant signal control,” Opt. Comm. 169, 309–315 (1999).
[CrossRef]

N. N. Il’ichev, A. V. Kir’yanov, and P. P. Pashinin, “Model of passive Q switching taking account of the anisotropy of nonlinear absorption in a crystal switch with phototropic centers,” Quantum Electron. 28, 147–151 (1998).
[CrossRef]

N. N. Il’ichev, A. V. Kir’yanov, E. S. Gulyamova, and P. P. Pashinin, “Polarization of a neodymium laser with a passive switch based on a Cr4+:YAG crystal,” Quantum Electron. 28, 17–20 (1998).
[CrossRef]

N. N. Il’ichev, A. V. Kir’yanov, E. S. Gulyamova, and P. P. Pashinin, “Polarization characteristics of a neodymium laser passively Q-switched with a Cr4+:YAG crystal,” Quantum Electron. 27, 298–301 (1997).
[CrossRef]

N. N. Il’ichev, A. V. Kir’yanov, P. P. Pashinin, and S. M. Shpuga, “Changes in the profile and state of polarization of a short light pulse (λ = 1.06 μm) during propagation in a Cr4+:YAG crystal,” Quantum Electron. 24, 771–776 (1994).
[CrossRef]

N. N. Il’ichev, A. V. Kir’yanov, P. P. Pashinin, and S. M. Shpuga, “Investigation of nonlinear-absorption anisotropy in YAG:Cr4+,” JETP 78, 768–776 (1994).

Klimov, I. V.

I. V. Klimov, I. A. Shcherbakov, and V. B. Tsvetkov, “Control of the Nd-laser output by Cr-doped Q-switches,” Laser Phys. 8, 232–237 (1998).

Kokta, M. R.

H. J. Eichler, A. Haase, M. R. Kokta, and R. Menzel, “Cr4+:YAG as passive Q-switch for a Nd:YALO oscillator with an average repetition rate of 2.7 kHz, TEM00 mode and 13 W output,” Appl. Phys. B 58, 409–411 (1994).
[CrossRef]

Kravtsov, V. V.

R. V. Goidin, V. S. Kichuk, V. V. Kravtsov, G. D. Laptev, E. G. Lariontsev, and V. V. Firsov, “Influence of the pump polarisation on the characteristics of radiation of a ring Nd:YAG chip laser,” Quantum Electron. 28, 358–360 (1998).
[CrossRef]

Laptev, G. D.

R. V. Goidin, V. S. Kichuk, V. V. Kravtsov, G. D. Laptev, E. G. Lariontsev, and V. V. Firsov, “Influence of the pump polarisation on the characteristics of radiation of a ring Nd:YAG chip laser,” Quantum Electron. 28, 358–360 (1998).
[CrossRef]

Lariontsev, E. G.

R. V. Goidin, V. S. Kichuk, V. V. Kravtsov, G. D. Laptev, E. G. Lariontsev, and V. V. Firsov, “Influence of the pump polarisation on the characteristics of radiation of a ring Nd:YAG chip laser,” Quantum Electron. 28, 358–360 (1998).
[CrossRef]

Le Floch, A.

M. Brunel, O. Emile, M. Vallet, F. Bretenaker, A. Le Floch, L. Fulbert, J. Marty, B. Ferrand, and E. Molva, “Experimental and theoretical study of monomode vectorial lasers passively Q-switched by Cr4+:yttrium aluminum garnet absorbers,” Phys. Rev. A 60, 4052–4058 (1999).
[CrossRef]

Lee, K. K.

Li, S.

Luther-Davies, B.

Marty, J.

M. Brunel, O. Emile, M. Vallet, F. Bretenaker, A. Le Floch, L. Fulbert, J. Marty, B. Ferrand, and E. Molva, “Experimental and theoretical study of monomode vectorial lasers passively Q-switched by Cr4+:yttrium aluminum garnet absorbers,” Phys. Rev. A 60, 4052–4058 (1999).
[CrossRef]

May, A. D.

R. Dalgliesh, A. D. May, and G. Stephan, “Polarization states of a single-mode (microchip) Nd3+:YAG laser—PartII. comparison of theory and experiment,” IEEE J. Quantum Electron. 34, 1493–1502 (1998).
[CrossRef]

R. Dalgliesh, A. D. May, and G. Stephan, “Polarization states of a single-mode (microchip) Nd3+:YAG laser—Part I: theory,” IEEE J. Quantum Electron. 34, 1485–1492 (1998).
[CrossRef]

Menzel, R.

H. J. Eichler, A. Haase, M. R. Kokta, and R. Menzel, “Cr4+:YAG as passive Q-switch for a Nd:YALO oscillator with an average repetition rate of 2.7 kHz, TEM00 mode and 13 W output,” Appl. Phys. B 58, 409–411 (1994).
[CrossRef]

Molva, E.

M. Brunel, O. Emile, M. Vallet, F. Bretenaker, A. Le Floch, L. Fulbert, J. Marty, B. Ferrand, and E. Molva, “Experimental and theoretical study of monomode vectorial lasers passively Q-switched by Cr4+:yttrium aluminum garnet absorbers,” Phys. Rev. A 60, 4052–4058 (1999).
[CrossRef]

Morello, C.

A. Agnesi, S. Dell’Acqua, C. Morello, G. Piccinno, G. C. Reali, and Z. Sun, “Diode-pumped neodymium lasers repetitively Q-switched by Cr4+:YAG solid-state saturable absorbers,” IEEE J. Sel. Top. Quantum Electron. 3, 45–52 (1997).
[CrossRef]

Pashinin, P. P.

N. N. Il’ichev, A. V. Kir’yanov, and P. P. Pashinin, “Model of passive Q switching taking account of the anisotropy of nonlinear absorption in a crystal switch with phototropic centers,” Quantum Electron. 28, 147–151 (1998).
[CrossRef]

N. N. Il’ichev, A. V. Kir’yanov, E. S. Gulyamova, and P. P. Pashinin, “Polarization of a neodymium laser with a passive switch based on a Cr4+:YAG crystal,” Quantum Electron. 28, 17–20 (1998).
[CrossRef]

N. N. Il’ichev, E. S. Gulyamova, and P. P. Pashinin, “Passive Q switching of a neodymium laser by a Cr4+:YAG crystal switch,” Quantum Electron. 27, 972–977 (1997).
[CrossRef]

N. N. Il’ichev, A. V. Kir’yanov, E. S. Gulyamova, and P. P. Pashinin, “Polarization characteristics of a neodymium laser passively Q-switched with a Cr4+:YAG crystal,” Quantum Electron. 27, 298–301 (1997).
[CrossRef]

N. N. Il’ichev, A. V. Kir’yanov, P. P. Pashinin, and S. M. Shpuga, “Investigation of nonlinear-absorption anisotropy in YAG:Cr4+,” JETP 78, 768–776 (1994).

N. N. Il’ichev, A. V. Kir’yanov, P. P. Pashinin, and S. M. Shpuga, “Changes in the profile and state of polarization of a short light pulse (λ = 1.06 μm) during propagation in a Cr4+:YAG crystal,” Quantum Electron. 24, 771–776 (1994).
[CrossRef]

Peterson, P.

P. Peterson, A. Gavrielidis, M. P. Sharma, and T. Erneux, “Dynamics of passively Q-switched microchip lasers,” IEEE J. Quantum Electron. 35, 1247–1256 (1999).
[CrossRef]

Piccinini, E.

A. Agnesi, S. Dell’Acqua, E. Piccinini, G. Really, and G. Piccinno, “Efficient wavelength conversion with high-power passively Q-switched diode-pumped neodymium lasers,” IEEE J. Quantum Electron. 34, 1480–1484 (1998).
[CrossRef]

Piccinno, G.

A. Agnesi, S. Dell’Acqua, E. Piccinini, G. Really, and G. Piccinno, “Efficient wavelength conversion with high-power passively Q-switched diode-pumped neodymium lasers,” IEEE J. Quantum Electron. 34, 1480–1484 (1998).
[CrossRef]

A. Agnesi, S. Dell’Acqua, C. Morello, G. Piccinno, G. C. Reali, and Z. Sun, “Diode-pumped neodymium lasers repetitively Q-switched by Cr4+:YAG solid-state saturable absorbers,” IEEE J. Sel. Top. Quantum Electron. 3, 45–52 (1997).
[CrossRef]

Reali, G. C.

A. Agnesi, S. Dell’Acqua, C. Morello, G. Piccinno, G. C. Reali, and Z. Sun, “Diode-pumped neodymium lasers repetitively Q-switched by Cr4+:YAG solid-state saturable absorbers,” IEEE J. Sel. Top. Quantum Electron. 3, 45–52 (1997).
[CrossRef]

Really, G.

A. Agnesi, S. Dell’Acqua, E. Piccinini, G. Really, and G. Piccinno, “Efficient wavelength conversion with high-power passively Q-switched diode-pumped neodymium lasers,” IEEE J. Quantum Electron. 34, 1480–1484 (1998).
[CrossRef]

Saltiel, S.

N. Zheludev, S. Saltiel, and P. Yankov, “Second-harmonic generators as a new class of light polarizers and analyzers,” Sov. J. Quantum Electron. 17, 948–952 (1987).
[CrossRef]

Sennaroglu, A.

Sharma, M. P.

P. Peterson, A. Gavrielidis, M. P. Sharma, and T. Erneux, “Dynamics of passively Q-switched microchip lasers,” IEEE J. Quantum Electron. 35, 1247–1256 (1999).
[CrossRef]

Shcherbakov, I. A.

I. V. Klimov, I. A. Shcherbakov, and V. B. Tsvetkov, “Control of the Nd-laser output by Cr-doped Q-switches,” Laser Phys. 8, 232–237 (1998).

Shestakov, A. V.

Shimony, Y.

Y. Shimony, Z. Burshtein, A. Ben-Amar Baranga, Y. Kalisky, and M. Strauss, “Repetitively Q-switching of a CW Nd:YAG laser using Cr4+:YAG saturable absorbers,” IEEE J. Quantum Electron. 32, 305–310 (1996).
[CrossRef]

Shpuga, S. M.

N. N. Il’ichev, A. V. Kir’yanov, P. P. Pashinin, and S. M. Shpuga, “Changes in the profile and state of polarization of a short light pulse (λ = 1.06 μm) during propagation in a Cr4+:YAG crystal,” Quantum Electron. 24, 771–776 (1994).
[CrossRef]

N. N. Il’ichev, A. V. Kir’yanov, P. P. Pashinin, and S. M. Shpuga, “Investigation of nonlinear-absorption anisotropy in YAG:Cr4+,” JETP 78, 768–776 (1994).

Spariosu, K.

Stephan, G.

R. Dalgliesh, A. D. May, and G. Stephan, “Polarization states of a single-mode (microchip) Nd3+:YAG laser—Part I: theory,” IEEE J. Quantum Electron. 34, 1485–1492 (1998).
[CrossRef]

R. Dalgliesh, A. D. May, and G. Stephan, “Polarization states of a single-mode (microchip) Nd3+:YAG laser—PartII. comparison of theory and experiment,” IEEE J. Quantum Electron. 34, 1493–1502 (1998).
[CrossRef]

Strauss, M.

Y. Shimony, Z. Burshtein, A. Ben-Amar Baranga, Y. Kalisky, and M. Strauss, “Repetitively Q-switching of a CW Nd:YAG laser using Cr4+:YAG saturable absorbers,” IEEE J. Quantum Electron. 32, 305–310 (1996).
[CrossRef]

Stultz, R.

Sun, Z.

A. Agnesi, S. Dell’Acqua, C. Morello, G. Piccinno, G. C. Reali, and Z. Sun, “Diode-pumped neodymium lasers repetitively Q-switched by Cr4+:YAG solid-state saturable absorbers,” IEEE J. Sel. Top. Quantum Electron. 3, 45–52 (1997).
[CrossRef]

Tsvetkov, V. B.

I. V. Klimov, I. A. Shcherbakov, and V. B. Tsvetkov, “Control of the Nd-laser output by Cr-doped Q-switches,” Laser Phys. 8, 232–237 (1998).

Vallet, M.

M. Brunel, O. Emile, M. Vallet, F. Bretenaker, A. Le Floch, L. Fulbert, J. Marty, B. Ferrand, and E. Molva, “Experimental and theoretical study of monomode vectorial lasers passively Q-switched by Cr4+:yttrium aluminum garnet absorbers,” Phys. Rev. A 60, 4052–4058 (1999).
[CrossRef]

Wang, Y.

Y. Wang and B. Luther-Davies, “Frequency-doubling pulse compressor for picosecond high-power neodymium laser pulses,” Opt. Lett. 17, 1459–1461 (1992).
[CrossRef] [PubMed]

Y. Wang and R. Dragila, “Efficient conversion of picosecond laser pulses into second-harmonic frequency using group-velocity dispersion,” Phys. Rev. A 41, 5645–5649 (1990).
[CrossRef] [PubMed]

Yankov, P.

N. Zheludev, S. Saltiel, and P. Yankov, “Second-harmonic generators as a new class of light polarizers and analyzers,” Sov. J. Quantum Electron. 17, 948–952 (1987).
[CrossRef]

Yen, W. M.

H. Eilers, K. R. Hoffman, M. Dennis, S. M. Jacobsen, and W. M. Yen, “Saturation of 1.064 μm absorption in Cr, Ca:Y3AL5O12 crystals,” Appl. Phys. Lett. 61, 2958–2960 (1992).
[CrossRef]

Zayhowski, J. J.

Zheludev, N.

N. Zheludev, S. Saltiel, and P. Yankov, “Second-harmonic generators as a new class of light polarizers and analyzers,” Sov. J. Quantum Electron. 17, 948–952 (1987).
[CrossRef]

Zhou, S.

Appl. Opt. (1)

Appl. Phys. B (1)

H. J. Eichler, A. Haase, M. R. Kokta, and R. Menzel, “Cr4+:YAG as passive Q-switch for a Nd:YALO oscillator with an average repetition rate of 2.7 kHz, TEM00 mode and 13 W output,” Appl. Phys. B 58, 409–411 (1994).
[CrossRef]

Appl. Phys. Lett. (1)

H. Eilers, K. R. Hoffman, M. Dennis, S. M. Jacobsen, and W. M. Yen, “Saturation of 1.064 μm absorption in Cr, Ca:Y3AL5O12 crystals,” Appl. Phys. Lett. 61, 2958–2960 (1992).
[CrossRef]

IEEE J. Quantum Electron. (5)

Y. Shimony, Z. Burshtein, A. Ben-Amar Baranga, Y. Kalisky, and M. Strauss, “Repetitively Q-switching of a CW Nd:YAG laser using Cr4+:YAG saturable absorbers,” IEEE J. Quantum Electron. 32, 305–310 (1996).
[CrossRef]

A. Agnesi, S. Dell’Acqua, E. Piccinini, G. Really, and G. Piccinno, “Efficient wavelength conversion with high-power passively Q-switched diode-pumped neodymium lasers,” IEEE J. Quantum Electron. 34, 1480–1484 (1998).
[CrossRef]

P. Peterson, A. Gavrielidis, M. P. Sharma, and T. Erneux, “Dynamics of passively Q-switched microchip lasers,” IEEE J. Quantum Electron. 35, 1247–1256 (1999).
[CrossRef]

R. Dalgliesh, A. D. May, and G. Stephan, “Polarization states of a single-mode (microchip) Nd3+:YAG laser—Part I: theory,” IEEE J. Quantum Electron. 34, 1485–1492 (1998).
[CrossRef]

R. Dalgliesh, A. D. May, and G. Stephan, “Polarization states of a single-mode (microchip) Nd3+:YAG laser—PartII. comparison of theory and experiment,” IEEE J. Quantum Electron. 34, 1493–1502 (1998).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

A. Agnesi, S. Dell’Acqua, C. Morello, G. Piccinno, G. C. Reali, and Z. Sun, “Diode-pumped neodymium lasers repetitively Q-switched by Cr4+:YAG solid-state saturable absorbers,” IEEE J. Sel. Top. Quantum Electron. 3, 45–52 (1997).
[CrossRef]

J. Mod. Opt. (1)

S. Camacho-Lopez, R. P. M. Green, G. J. Crofts, and M. J. Damzen, “Intensity-induced birefringence in Cr4+:YAG,” J. Mod. Opt. 44, 209–219 (1997).
[CrossRef]

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

JETP (1)

N. N. Il’ichev, A. V. Kir’yanov, P. P. Pashinin, and S. M. Shpuga, “Investigation of nonlinear-absorption anisotropy in YAG:Cr4+,” JETP 78, 768–776 (1994).

Laser Phys. (1)

I. V. Klimov, I. A. Shcherbakov, and V. B. Tsvetkov, “Control of the Nd-laser output by Cr-doped Q-switches,” Laser Phys. 8, 232–237 (1998).

Opt. Comm. (1)

A. V. Kir’yanov, V. Aboites, and N. N. Il’ichev, “A polarization-bistable neodymium laser with a Cr4+:YAG passive switch under the weak resonant signal control,” Opt. Comm. 169, 309–315 (1999).
[CrossRef]

Opt. Lett. (4)

Phys. Rev. A (2)

Y. Wang and R. Dragila, “Efficient conversion of picosecond laser pulses into second-harmonic frequency using group-velocity dispersion,” Phys. Rev. A 41, 5645–5649 (1990).
[CrossRef] [PubMed]

M. Brunel, O. Emile, M. Vallet, F. Bretenaker, A. Le Floch, L. Fulbert, J. Marty, B. Ferrand, and E. Molva, “Experimental and theoretical study of monomode vectorial lasers passively Q-switched by Cr4+:yttrium aluminum garnet absorbers,” Phys. Rev. A 60, 4052–4058 (1999).
[CrossRef]

Quantum Electron. (6)

R. V. Goidin, V. S. Kichuk, V. V. Kravtsov, G. D. Laptev, E. G. Lariontsev, and V. V. Firsov, “Influence of the pump polarisation on the characteristics of radiation of a ring Nd:YAG chip laser,” Quantum Electron. 28, 358–360 (1998).
[CrossRef]

N. N. Il’ichev, A. V. Kir’yanov, E. S. Gulyamova, and P. P. Pashinin, “Polarization characteristics of a neodymium laser passively Q-switched with a Cr4+:YAG crystal,” Quantum Electron. 27, 298–301 (1997).
[CrossRef]

N. N. Il’ichev, E. S. Gulyamova, and P. P. Pashinin, “Passive Q switching of a neodymium laser by a Cr4+:YAG crystal switch,” Quantum Electron. 27, 972–977 (1997).
[CrossRef]

N. N. Il’ichev, A. V. Kir’yanov, P. P. Pashinin, and S. M. Shpuga, “Changes in the profile and state of polarization of a short light pulse (λ = 1.06 μm) during propagation in a Cr4+:YAG crystal,” Quantum Electron. 24, 771–776 (1994).
[CrossRef]

N. N. Il’ichev, A. V. Kir’yanov, and P. P. Pashinin, “Model of passive Q switching taking account of the anisotropy of nonlinear absorption in a crystal switch with phototropic centers,” Quantum Electron. 28, 147–151 (1998).
[CrossRef]

N. N. Il’ichev, A. V. Kir’yanov, E. S. Gulyamova, and P. P. Pashinin, “Polarization of a neodymium laser with a passive switch based on a Cr4+:YAG crystal,” Quantum Electron. 28, 17–20 (1998).
[CrossRef]

Sov. J. Quantum Electron. (1)

N. Zheludev, S. Saltiel, and P. Yankov, “Second-harmonic generators as a new class of light polarizers and analyzers,” Sov. J. Quantum Electron. 17, 948–952 (1987).
[CrossRef]

Other (1)

V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer-Verlag, Berlin, 1991).

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Figures (7)

Fig. 1
Fig. 1

(a) Schematic of laser A; (b) schematic of laser B.

Fig. 2
Fig. 2

Scenarios for laser A (see Table 1) intracavity intensity I (curves 2) and polarization azimuth ϕ (curves 1). (a) θ=40°, β=10; (b) 50; (c) 30°; and (d) θ=45°, β=30°.

Fig. 3
Fig. 3

Numerical data for (a) type I synchronism and (b) type II synchronism SHG converters. The data correspond to Fig. 2(c).

Fig. 4
Fig. 4

(a), (b) Dependencies of SHG peak intensity (in units of pump amplitude); (c), (d) SHG pulse duration on SHG crystal angular position ψ. Curves 1 and 2 correspond to the laser dynamics in Fig. 2(d) and Fig. 2(c), respectively. (a), (c) Type I synchronism; (b), (d) Type II synchronism.

Fig. 5
Fig. 5

One of the scenarios of the GP envelope (curve 1) and polarization azimuth (curve 2) for laser B. Numerical data for θ = 40°, kY-kX=0.055.

Fig. 6
Fig. 6

Experimental oscillation traces of (a) a fundamental (1.064 µm) GP from a microchip laser; (b) a SH (0.532 µm) pulse after a KTP doubling crystal. Horizontal scale: 200 ps/div. Data from Ref. 15.

Fig. 7
Fig. 7

Theoretical SH pulses generated by GP propagating along a KTP crystal cut for Type II synchronism. (a) SH pulse corresponding to scenario shown in Fig. 5; (b) the same for input GP without polarization rotation. Labels 1–4 correspond to snapshots at increasing normalized distances ξ measured with respect to doubling crystal.

Tables (2)

Tables Icon

Table 1 Input Parameters for Laser A

Tables Icon

Table 2 Input Parameters for Laser B

Equations (13)

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

dFadt=FatR2σaNala-2σsls[ns(1) cos2(θ-φ)+ns(2) sin2(θ-φ)]-ln1r-αx cos2 φ-αy sin2 φ,
dNadt=-γσaNaFac,
dns(1)dt=-σsns(1)FacK cos2(θ-φ)+ns0-ns(1)τs,
dns(2)dt=-σsns(2)FacK sin2(θ-φ)+ns0-ns(2)τs,
φ(t)=12arctansin(2θ)cos(2θ)-αY-αX2lsσs[ns(1)(t)-ns(2)(t)],
φ(t)=12 arctansin(2θ)cos(2θ)-kY-kX2lsσs[ns(1)(t)-ns(2)(t)].
dApdξ=-ApAs sin η,
dAsdξ=Ap2 sin η,
dηdξ=δ-2As-Ap2Ascos η,
dAp(1)dξ=-Ap(2)As sin η,
dAp(2)dξ=-Ap(1)As sin η,
dAsdξ=Ap(1)Ap(2) sin η,
dηdξ=δ-Ap(2)AsAp(1)+Ap(1)AsAp(2)-Ap(1)Ap(2)Ascos η,

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