Abstract

In this work we present a numerical study of a three-level laser containing a polarizable saturable absorber inside the cavity. This model allows us to study the kinetics of solid-state lasers in a general form. The stability of Q-switching regime is analyzed by means of numerical solution of rate equations; and main results of our analysis let us suggest that under certain conditions, the laser may pass from unstable relaxation oscillations to a stable CW operation by changing the mutual orientation of the absorber and polarizer, or by choosing the pump level.

© 2001 Optical Society of America

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References

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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 (1993).
    [Crossref] [PubMed]
  2. H.J. Eichler, A. Haase, M.R. Kokta, and R. Menzel, “Cr4+:YAG as a passive Q-switch for a Nd:YAG oscillator with an average repetition rate of 2.7 KHz, TEM00 mode and 13 W output”, Appl. Phys. B58, 409 (1994).
    [Crossref]
  3. Y. Shimony, Z. Burstein, A. Ben-Amar Baranga, Y. Kalisky, and M. Strauss, “Repetively Q-switching of a CW Nd:YAG laser using Cr4+:YAG saturable absorbers”, IEEE J. Quant. Electron. QE-32, 305 (1996).
    [Crossref]
  4. A. Agnesi, S. Dell’Acqua, C. Morello, G. Piccino, 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 (1997).
    [Crossref]
  5. I.V. Klimov, I.A. Scherbakov, and V.B. Tsvetkov, “Control of the Nd:YAG laser output by Cr-doped Q-switches”, Laser Physics 8, 232 (1998).
  6. A.V. Kir’yanov, V. Aboites, and I.V. Mel’nikov, “Second-harmonic generation by Nd3+:YAG/Cr4+:YAG laser pulses with changing state of polarization”, J. Opt. Soc. Am. B 17, 1657 (2000).
    [Crossref]
  7. A.V. Kir’yanov, V. Aboites, and I.V. Mel’nikov, “Enhancing type-II second harmonic generation by the use of a laser beam with a rotating azimuth of polarization”, Appl. Phys. Lett. 78, 874 (2001).
    [Crossref]
  8. N.N. Ilichev, 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 (1998).
    [Crossref]
  9. M. Brunel, O. Emile, M. Vallet, F. Brtenaker, 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+:YAG absorbers”, Phys. Rev. A60, 4052 (1999).
  10. see, e.g., K.V. Yumashev, N.N. Posnov, I.A. Denisov, V.P. Mikhailov, and R. Moncorge, “Nonlinear spectroscopy and passive Q-switching of Cr4+:doped SrGd4(SiO4)3O and CaGd4(SiO4)3O crystals,” J. Opt. Soc. Am. B15, 1707 (1998), and references therein.

2001 (1)

A.V. Kir’yanov, V. Aboites, and I.V. Mel’nikov, “Enhancing type-II second harmonic generation by the use of a laser beam with a rotating azimuth of polarization”, Appl. Phys. Lett. 78, 874 (2001).
[Crossref]

2000 (1)

1999 (1)

M. Brunel, O. Emile, M. Vallet, F. Brtenaker, 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+:YAG absorbers”, Phys. Rev. A60, 4052 (1999).

1998 (3)

see, e.g., K.V. Yumashev, N.N. Posnov, I.A. Denisov, V.P. Mikhailov, and R. Moncorge, “Nonlinear spectroscopy and passive Q-switching of Cr4+:doped SrGd4(SiO4)3O and CaGd4(SiO4)3O crystals,” J. Opt. Soc. Am. B15, 1707 (1998), and references therein.

I.V. Klimov, I.A. Scherbakov, and V.B. Tsvetkov, “Control of the Nd:YAG laser output by Cr-doped Q-switches”, Laser Physics 8, 232 (1998).

N.N. Ilichev, 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 (1998).
[Crossref]

1997 (1)

A. Agnesi, S. Dell’Acqua, C. Morello, G. Piccino, 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 (1997).
[Crossref]

1996 (1)

Y. Shimony, Z. Burstein, A. Ben-Amar Baranga, Y. Kalisky, and M. Strauss, “Repetively Q-switching of a CW Nd:YAG laser using Cr4+:YAG saturable absorbers”, IEEE J. Quant. Electron. QE-32, 305 (1996).
[Crossref]

1994 (1)

H.J. Eichler, A. Haase, M.R. Kokta, and R. Menzel, “Cr4+:YAG as a passive Q-switch for a Nd:YAG oscillator with an average repetition rate of 2.7 KHz, TEM00 mode and 13 W output”, Appl. Phys. B58, 409 (1994).
[Crossref]

1993 (1)

Aboites, V.

A.V. Kir’yanov, V. Aboites, and I.V. Mel’nikov, “Enhancing type-II second harmonic generation by the use of a laser beam with a rotating azimuth of polarization”, Appl. Phys. Lett. 78, 874 (2001).
[Crossref]

A.V. Kir’yanov, V. Aboites, and I.V. Mel’nikov, “Second-harmonic generation by Nd3+:YAG/Cr4+:YAG laser pulses with changing state of polarization”, J. Opt. Soc. Am. B 17, 1657 (2000).
[Crossref]

Agnesi, A.

A. Agnesi, S. Dell’Acqua, C. Morello, G. Piccino, 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 (1997).
[Crossref]

Ben-Amar Baranga, A.

Y. Shimony, Z. Burstein, A. Ben-Amar Baranga, Y. Kalisky, and M. Strauss, “Repetively Q-switching of a CW Nd:YAG laser using Cr4+:YAG saturable absorbers”, IEEE J. Quant. Electron. QE-32, 305 (1996).
[Crossref]

Birnbaum, M.

Brtenaker, F.

M. Brunel, O. Emile, M. Vallet, F. Brtenaker, 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+:YAG absorbers”, Phys. Rev. A60, 4052 (1999).

Brunel, M.

M. Brunel, O. Emile, M. Vallet, F. Brtenaker, 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+:YAG absorbers”, Phys. Rev. A60, 4052 (1999).

Burstein, Z.

Y. Shimony, Z. Burstein, A. Ben-Amar Baranga, Y. Kalisky, and M. Strauss, “Repetively Q-switching of a CW Nd:YAG laser using Cr4+:YAG saturable absorbers”, IEEE J. Quant. Electron. QE-32, 305 (1996).
[Crossref]

Chen, W.

Dell’Acqua, S.

A. Agnesi, S. Dell’Acqua, C. Morello, G. Piccino, 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 (1997).
[Crossref]

Denisov, I.A.

see, e.g., K.V. Yumashev, N.N. Posnov, I.A. Denisov, V.P. Mikhailov, and R. Moncorge, “Nonlinear spectroscopy and passive Q-switching of Cr4+:doped SrGd4(SiO4)3O and CaGd4(SiO4)3O crystals,” J. Opt. Soc. Am. B15, 1707 (1998), and references therein.

Eichler, H.J.

H.J. Eichler, A. Haase, M.R. Kokta, and R. Menzel, “Cr4+:YAG as a passive Q-switch for a Nd:YAG oscillator with an average repetition rate of 2.7 KHz, TEM00 mode and 13 W output”, Appl. Phys. B58, 409 (1994).
[Crossref]

Emile, O.

M. Brunel, O. Emile, M. Vallet, F. Brtenaker, 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+:YAG absorbers”, Phys. Rev. A60, 4052 (1999).

Ferrand, B.

M. Brunel, O. Emile, M. Vallet, F. Brtenaker, 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+:YAG absorbers”, Phys. Rev. A60, 4052 (1999).

Fulbert, L.

M. Brunel, O. Emile, M. Vallet, F. Brtenaker, 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+:YAG absorbers”, Phys. Rev. A60, 4052 (1999).

Gulyamova, E.S.

N.N. Ilichev, 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 (1998).
[Crossref]

Haase, A.

H.J. Eichler, A. Haase, M.R. Kokta, and R. Menzel, “Cr4+:YAG as a passive Q-switch for a Nd:YAG oscillator with an average repetition rate of 2.7 KHz, TEM00 mode and 13 W output”, Appl. Phys. B58, 409 (1994).
[Crossref]

Ilichev, N.N.

N.N. Ilichev, 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 (1998).
[Crossref]

Kalisky, Y.

Y. Shimony, Z. Burstein, A. Ben-Amar Baranga, Y. Kalisky, and M. Strauss, “Repetively Q-switching of a CW Nd:YAG laser using Cr4+:YAG saturable absorbers”, IEEE J. Quant. Electron. QE-32, 305 (1996).
[Crossref]

Kir’yanov, A.V.

A.V. Kir’yanov, V. Aboites, and I.V. Mel’nikov, “Enhancing type-II second harmonic generation by the use of a laser beam with a rotating azimuth of polarization”, Appl. Phys. Lett. 78, 874 (2001).
[Crossref]

A.V. Kir’yanov, V. Aboites, and I.V. Mel’nikov, “Second-harmonic generation by Nd3+:YAG/Cr4+:YAG laser pulses with changing state of polarization”, J. Opt. Soc. Am. B 17, 1657 (2000).
[Crossref]

N.N. Ilichev, 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 (1998).
[Crossref]

Klimov, I.V.

I.V. Klimov, I.A. Scherbakov, and V.B. Tsvetkov, “Control of the Nd:YAG laser output by Cr-doped Q-switches”, Laser Physics 8, 232 (1998).

Kokta, M.R.

H.J. Eichler, A. Haase, M.R. Kokta, and R. Menzel, “Cr4+:YAG as a passive Q-switch for a Nd:YAG oscillator with an average repetition rate of 2.7 KHz, TEM00 mode and 13 W output”, Appl. Phys. B58, 409 (1994).
[Crossref]

Le Floch, A.

M. Brunel, O. Emile, M. Vallet, F. Brtenaker, 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+:YAG absorbers”, Phys. Rev. A60, 4052 (1999).

Marty, J.

M. Brunel, O. Emile, M. Vallet, F. Brtenaker, 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+:YAG absorbers”, Phys. Rev. A60, 4052 (1999).

Mel’nikov, I.V.

A.V. Kir’yanov, V. Aboites, and I.V. Mel’nikov, “Enhancing type-II second harmonic generation by the use of a laser beam with a rotating azimuth of polarization”, Appl. Phys. Lett. 78, 874 (2001).
[Crossref]

A.V. Kir’yanov, V. Aboites, and I.V. Mel’nikov, “Second-harmonic generation by Nd3+:YAG/Cr4+:YAG laser pulses with changing state of polarization”, J. Opt. Soc. Am. B 17, 1657 (2000).
[Crossref]

Menzel, R.

H.J. Eichler, A. Haase, M.R. Kokta, and R. Menzel, “Cr4+:YAG as a passive Q-switch for a Nd:YAG oscillator with an average repetition rate of 2.7 KHz, TEM00 mode and 13 W output”, Appl. Phys. B58, 409 (1994).
[Crossref]

Mikhailov, V.P.

see, e.g., K.V. Yumashev, N.N. Posnov, I.A. Denisov, V.P. Mikhailov, and R. Moncorge, “Nonlinear spectroscopy and passive Q-switching of Cr4+:doped SrGd4(SiO4)3O and CaGd4(SiO4)3O crystals,” J. Opt. Soc. Am. B15, 1707 (1998), and references therein.

Molva, E.

M. Brunel, O. Emile, M. Vallet, F. Brtenaker, 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+:YAG absorbers”, Phys. Rev. A60, 4052 (1999).

Moncorge, R.

see, e.g., K.V. Yumashev, N.N. Posnov, I.A. Denisov, V.P. Mikhailov, and R. Moncorge, “Nonlinear spectroscopy and passive Q-switching of Cr4+:doped SrGd4(SiO4)3O and CaGd4(SiO4)3O crystals,” J. Opt. Soc. Am. B15, 1707 (1998), and references therein.

Morello, C.

A. Agnesi, S. Dell’Acqua, C. Morello, G. Piccino, 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 (1997).
[Crossref]

Pashinin, P.P.

N.N. Ilichev, 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 (1998).
[Crossref]

Piccino, G.

A. Agnesi, S. Dell’Acqua, C. Morello, G. Piccino, 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 (1997).
[Crossref]

Posnov, N.N.

see, e.g., K.V. Yumashev, N.N. Posnov, I.A. Denisov, V.P. Mikhailov, and R. Moncorge, “Nonlinear spectroscopy and passive Q-switching of Cr4+:doped SrGd4(SiO4)3O and CaGd4(SiO4)3O crystals,” J. Opt. Soc. Am. B15, 1707 (1998), and references therein.

Reali, G.C.

A. Agnesi, S. Dell’Acqua, C. Morello, G. Piccino, 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 (1997).
[Crossref]

Scherbakov, I.A.

I.V. Klimov, I.A. Scherbakov, and V.B. Tsvetkov, “Control of the Nd:YAG laser output by Cr-doped Q-switches”, Laser Physics 8, 232 (1998).

Shestakov, A.V.

Shimony, Y.

Y. Shimony, Z. Burstein, A. Ben-Amar Baranga, Y. Kalisky, and M. Strauss, “Repetively Q-switching of a CW Nd:YAG laser using Cr4+:YAG saturable absorbers”, IEEE J. Quant. Electron. QE-32, 305 (1996).
[Crossref]

Spariosu, K.

Strauss, M.

Y. Shimony, Z. Burstein, A. Ben-Amar Baranga, Y. Kalisky, and M. Strauss, “Repetively Q-switching of a CW Nd:YAG laser using Cr4+:YAG saturable absorbers”, IEEE J. Quant. Electron. QE-32, 305 (1996).
[Crossref]

Stultz, R.

Sun, Z.

A. Agnesi, S. Dell’Acqua, C. Morello, G. Piccino, 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 (1997).
[Crossref]

Tsvetkov, V.B.

I.V. Klimov, I.A. Scherbakov, and V.B. Tsvetkov, “Control of the Nd:YAG laser output by Cr-doped Q-switches”, Laser Physics 8, 232 (1998).

Vallet, M.

M. Brunel, O. Emile, M. Vallet, F. Brtenaker, 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+:YAG absorbers”, Phys. Rev. A60, 4052 (1999).

Yumashev, K.V.

see, e.g., K.V. Yumashev, N.N. Posnov, I.A. Denisov, V.P. Mikhailov, and R. Moncorge, “Nonlinear spectroscopy and passive Q-switching of Cr4+:doped SrGd4(SiO4)3O and CaGd4(SiO4)3O crystals,” J. Opt. Soc. Am. B15, 1707 (1998), and references therein.

Appl. Phys. (1)

H.J. Eichler, A. Haase, M.R. Kokta, and R. Menzel, “Cr4+:YAG as a passive Q-switch for a Nd:YAG oscillator with an average repetition rate of 2.7 KHz, TEM00 mode and 13 W output”, Appl. Phys. B58, 409 (1994).
[Crossref]

Appl. Phys. Lett. (1)

A.V. Kir’yanov, V. Aboites, and I.V. Mel’nikov, “Enhancing type-II second harmonic generation by the use of a laser beam with a rotating azimuth of polarization”, Appl. Phys. Lett. 78, 874 (2001).
[Crossref]

IEEE J. Quant. Electron. (1)

Y. Shimony, Z. Burstein, A. Ben-Amar Baranga, Y. Kalisky, and M. Strauss, “Repetively Q-switching of a CW Nd:YAG laser using Cr4+:YAG saturable absorbers”, IEEE J. Quant. Electron. QE-32, 305 (1996).
[Crossref]

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

A. Agnesi, S. Dell’Acqua, C. Morello, G. Piccino, 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 (1997).
[Crossref]

J. Opt. Soc. Am. (1)

see, e.g., K.V. Yumashev, N.N. Posnov, I.A. Denisov, V.P. Mikhailov, and R. Moncorge, “Nonlinear spectroscopy and passive Q-switching of Cr4+:doped SrGd4(SiO4)3O and CaGd4(SiO4)3O crystals,” J. Opt. Soc. Am. B15, 1707 (1998), and references therein.

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

Laser Physics (1)

I.V. Klimov, I.A. Scherbakov, and V.B. Tsvetkov, “Control of the Nd:YAG laser output by Cr-doped Q-switches”, Laser Physics 8, 232 (1998).

Opt. Lett. (1)

Phys. Rev. (1)

M. Brunel, O. Emile, M. Vallet, F. Brtenaker, 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+:YAG absorbers”, Phys. Rev. A60, 4052 (1999).

Quantum Electron. (1)

N.N. Ilichev, 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 (1998).
[Crossref]

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Figures (6)

Fig. 1.
Fig. 1.

A) Nd4+:YAG laser containing saturable absorber. 1 & 2- cavity mirrors, 3- Nd3+:YAG crystal, 4- Cr4+:YAG crystal with nonlinear anisotropy described by θ, 5- glass plate as a partial polarizer (at angular position b, to set linear anisotropy of the cavity). B) Mutual angular orientations of Cr4+:YAG crystal axes (θ,θ+π/2), and ellipse azimuth of the elliptically polarized laser output Eout (ϕ). Such orientations are related with the longitudinal axis of cavity (Z).

Fig. 2.
Fig. 2.

A) Three energy level scheme for laser transitions: N1,2,3 - ground, upper and pump population level densities of laser. B) Two energy level scheme for saturable absorber: n1,2 - ground-state population of ions oriented along [100] and [010] axis of Cr4+:YAG crystal, correspondingly.

Fig. 3.
Fig. 3.

Boundaries between stable and unstable regions of the laser. The unstable region is surrounded by stable regions. For very small values of the relative absorber concentration the laser is seen to be always stable, but for larger values it is readily seen a region of instability. For the density of the same value as the density of lasing three-level particles, atoms or ions the device does not raise.

Fig. 4.
Fig. 4.

Diagram of a different kind of stability regions. We fixed the value gs/ga=0.1 and let the pump power vary along the horizontal axis. Again, there is a unstable region surrounded by stable regions. But this diagram shows that under certain conditions, the laser can pass from pulsing to stable CW oscillations simply by raising the pump power and by mutual orientation of the absorber and the partial polarizer.

Fig. 5.
Fig. 5.

Typical pulsing behaviour of laser output. It is readily seen the self-induced pulsations of the output power. In particular, it follows that the generated quasi-period of pulses is equal to the lifetime of the photon in the cavity, and the pulse width is defined by the ratio of the length of the lasing element, to that of the absorber. The phase difference is equal to 1.5×10-3.

Fig. 6.
Fig. 6.

Period doubling of output train. The difference between this and Fig. 5 on their parameter sets, is the self-induced anisotropy of the saturable absorption. In this case, the phase difference is equal to 10-3. It is noticing that, if being taken at a longer time scale, the both trains look like a decaying sequence of pulses.

Equations (14)

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

N 0 = N 1 + N 2 + N 3 .
d F d t = γ τ s p N 2 + g a F ( N 2 N 1 ) g s F ( n 1 cos 2 ( θ ϕ ) + n 2 sin 2 ( θ ϕ ) )
F ( 1 n 1 r + α x cos 2 ϕ + α y sin 2 ϕ ) ,
d N 3 d t = 1 τ p P P t h ( N 1 N 3 ) N 3 τ 32
d N 2 d t = N 3 τ 32 N 2 P sp g a F ( N 2 N 1 ) ,
d n 1 , 2 d t = g s F { cos 2 ( θ ϕ ) sin 2 ( θ ϕ ) } n 1 , 2 + n 0 n 1 , 2 τ s .
φ ( t ) = 1 2 arctan { sin 2 θ cos 2 θ α y α x 2 g s ( n 1 n 2 ) } ,
G = τ 32 τ p P P t h .
F s = 1 g a τ 32 1 + 2 G 2 + 3 G ( G 1 + G + τ 32 τ s p ) + N 0 τ 32 ( G 1 + G 1 + 2 G 2 + 3 G τ 32 τ s p ) ×
( g s g a n 0 1 + ½ g s F s sin 2 2 ( θ ϕ ) 1 + g s F s + ½ g s 2 F s 2 sin 2 2 ( θ ϕ ) + 1 2 g a 1 n 1 r + α x cos 2 ϕ + α y sin 2 ϕ 2 g a ) 1 ,
N 2 = ( 1 + G 2 + 3 G ) N 0 +
1 + 2 G 2 + 3 G ( g s g a n 0 1 + ½ g s F s sin 2 2 ( θ ϕ ) 1 + g s F s + ½ g s 2 F s 2 sin 2 2 ( θ ϕ ) + 1 2 g s 1 n 1 r + α x cos 2 ϕ + α y sin 2 ϕ 2 g s ) ,
N 3 = G 1 + 2 G ( N 0 N 2 ) ,
n 1 , 2 = n 0 [ 1 + g s F s { cos 2 ( θ ϕ ) sin 2 ( θ ϕ ) } ] 1

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