Abstract

A model is developed for numerically calculating the nonlinear change of the refractive index in a Cr4+:YAG crystal under the powerful resonant (wavelength 1.06 μm) continuous-wave excitation. The anisotropy features of the Cr4+ centers’ population perturbation (bleaching) and the sample inhomogeneous heating (leading to the thermolensing effect) as the basic contributions in the refractive-index nonlinear change are addressed for the first time to our knowledge.

© 2003 Optical Society of America

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    [CrossRef]
  2. Y. Shimony, Z. Burshtein, A. Ben-Amar Baranga, Y. Kalisky, and M. Strauss, “Repetitive Q-switching of a cw Nd:YAG laser using Cr4+:YAG saturable absorbers,” IEEE J. Quantum Electron. 32, 305–310 (1996).
    [CrossRef]
  3. N. N. Il’ichev, A. V. Kir’yanov, and P. P. Pashinin, “Model of passive Q switching taking into account the anisotropy of nonlinear absorption in a crystal switch with phototropic centers,” Quantum Electron. 28, 147–151 (1998).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  37. A. J. Alcock, P. Scorah, and K. Hnatovsky, “Broadly tunable continuous-wave diode-pumped Cr4+:YAG laser,” Opt. Commun. 215, 153–157 (2003).
    [CrossRef]
  38. A. G. Okhrimchuk and A. V. Shestakov, “Performance of YAG:Cr4+ laser crystal,” Opt. Mater. 3, 1–13 (1994).
    [CrossRef]
  39. A. Brignon, P. Sillard, and J.-P. Huignard, “Vector phase conjugation in Cr4+:YAG by four-wave mixing with linearly-polarized pump beams,” Appl. Phys. B 63, 537–540 (1996).
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    [CrossRef]
  41. A. G. Okhrimchuk, Prokhorov General Physics Institute, Moscow, Russia (personal communication, 2002).

2003 (2)

2002 (2)

A. V. Kir’yanov, Yu. O. Barmenkov, M. del Rayo, and V. N. Filippov, “Ground-state absorption saturation and thermo-lensing effect as main sources of refractive index non-linear change in Cr4+:YAG at cw 1.06 mkm excitation,” Opt. Commun. 213, 151–162 (2002).
[CrossRef]

P. K. Mukhopadhyay, J. George, K. Ranganathan, S. K. Sharma, and T. P. S. Nathan, “An alternative approach to determine the fractional heat load in solid state laser materials: application to diode-pumped Nd:YVO4 laser,” Opt. Laser Technol. 34, 253–258 (2002).
[CrossRef]

2001 (2)

D. C. Brown and H. J. Hoffman, “Thermal, stress, and thermo-optic effects in high average power double-clad silica fiber lasers,” IEEE J. Quantum Electron. 37, 207–217 (2001).
[CrossRef]

S. Kuck, “Laser-related spectroscopy of ion-doped crystals for tunable solid-state lasers,” Appl. Phys. B 72, 515–562 (2001).
[CrossRef]

2000 (2)

A. G. Okhrimchuk and A. V. Shestakov, “Absorption saturation mechanism for YAG:Cr4+ crystals,” Phys. Rev. B 61, 988–995 (2000).
[CrossRef]

J. Song, C. Li, and K. Ueda, “Thermal influence of saturable absorber in passively Q-switched diode-pumped cw Nd:YAG/Cr4+:YAG laser,” Opt. Commun. 177, 307–316 (2000).
[CrossRef]

1999 (2)

1998 (4)

N. N. Il’ichev, A. V. Kir’yanov, and P. P. Pashinin, “Model of passive Q switching taking into account 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, “Polarisation of a neodymium laser with a passive switch based on a Cr4+:YAG crystal,” Quantum Electron. 28, 17–20 (1998).
[CrossRef]

M. Brunel, O. Emile, M. Vallet, F. Bretenaker, and A. LeFloch, “Experimental and theoretical study of monomode vectorial lasers passively Q-switched by Cr4+:yttrium aluminium garnet absorber,” Phys. Rev. A 60, 4052–4058 (1998).
[CrossRef]

N. N. Il’ichev, A. V. Kir’yanov, E. S. Gulyamova, and P. P. Pashinin, “Influence of the nonlinear anisotropy of absorption in a passive Cr4+:YAG switch on the energy and polarization characteristics of a neodymium laser,” Quantum Electron. 27, 298–301 (1998).
[CrossRef]

1997 (3)

1996 (5)

M. J. Damzen, S. Camacho-Lopez, and R. P. M. Green, “Wave-mixing and vector phase conjugation by polarization-dependent saturable absorption in Cr4+:YAG,” Phys. Rev. Lett. 76, 2894–2897 (1996).
[CrossRef] [PubMed]

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

L. C. Oliveira, T. Catunda, and S. C. Zilio, “Saturation effects in Z-scan measurements,” Jpn. J. Appl. Phys., 35, 2649–2652 (1996).
[CrossRef]

A. Brignon, P. Sillard, and J.-P. Huignard, “Vector phase conjugation in Cr4+:YAG by four-wave mixing with linearly-polarized pump beams,” Appl. Phys. B 63, 537–540 (1996).

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

1995 (1)

N. N. Il’ichev, A. V. Kir’yanov, P. P. Pashinin, V. A. Sandulenko, A. V. Sandulenko, and S. M. Shpuga, “Anisotropy of nonlinear absorption in a YAG:V3+ crystal,” Quantum Electron. 25, 1154–1157 (1995).
[CrossRef]

1994 (6)

L. C. Oliveira and S. C. Zilio, “Single-beam time-resolved Z-scan measurement of slow absorbers,” Appl. Phys. Lett. 65, 2121–2123 (1994).
[CrossRef]

A. G. Okhrimchuk and A. V. Shestakov, “Performance of YAG:Cr4+ laser crystal,” Opt. Mater. 3, 1–13 (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, TEMoo mode and 13 W output,” Appl. Phys. B 58, 409–411 (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, 768–777 (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 YAG:Cr4+ crystal,” Quantum Electron. 24, 771–776 (1994).
[CrossRef]

A. Sennaroglu, C. R. Pollock, and H. Nathel, “Continuous-wave self-mode-locked operation of a femtosecond Cr4+:YAG laser,” Opt. Lett. 19, 390–392 (1994).
[PubMed]

1993 (2)

K. Spariosu, W. Chen, R. Stultz, M. Birnbaum, and A. V. Shestakov, “Dual Q switching and laser action at 1.06 and 1.44 μm in a Nd3+:YAG–Cr4+:YAG oscillator at 300 K,” Opt. Lett. 18, 814–816 (1993).
[CrossRef] [PubMed]

N. N. Il’ichev, A. V. Kir’yanov, A. A. Malyutin, P. P. Pashinin, and S. M. Shpuga, “The anisotropy of nonlinear absorption induced by laser radiation in LiF:F2 crystal: the short pulse case,” Laser Phys. 3, 182–190 (1993).

1992 (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]

1991 (2)

N. N. Il’ichev, A. V. Kir’yanov, and A. A. Malyutin, “Self-induced change in the polarization of high-power resonant radiation in an LiF:F2 crystal,” Sov. J. Quantum Electron. 21, 844–848 (1991).
[CrossRef]

N. N. Il’ichev, A. V. Kir’yanov, and A. A. Malyutin, “Peculiarities of anisotropy and dichroizm induced by laser radiation in LiF:F2 crystal,” Laser Phys. 1, 311–323 (1991).

1971 (1)

1970 (1)

Alcock, A. J.

A. J. Alcock, P. Scorah, and K. Hnatovsky, “Broadly tunable continuous-wave diode-pumped Cr4+:YAG laser,” Opt. Commun. 215, 153–157 (2003).
[CrossRef]

Andrade, A. A.

Askar, A.

Atay, F. M.

Baesso, M. L.

Barmenkov, Yu. O.

A. V. Kir’yanov, Yu. O. Barmenkov, M. del Rayo, and V. N. Filippov, “Ground-state absorption saturation and thermo-lensing effect as main sources of refractive index non-linear change in Cr4+:YAG at cw 1.06 mkm excitation,” Opt. Commun. 213, 151–162 (2002).
[CrossRef]

Ben-Amar Baranga, A.

Y. Shimony, Z. Burshtein, A. Ben-Amar Baranga, Y. Kalisky, and M. Strauss, “Repetitive 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, and A. LeFloch, “Experimental and theoretical study of monomode vectorial lasers passively Q-switched by Cr4+:yttrium aluminium garnet absorber,” Phys. Rev. A 60, 4052–4058 (1998).
[CrossRef]

Brignon, A.

A. Brignon, P. Sillard, and J.-P. Huignard, “Vector phase conjugation in Cr4+:YAG by four-wave mixing with linearly-polarized pump beams,” Appl. Phys. B 63, 537–540 (1996).

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

Brown, D. C.

D. C. Brown and H. J. Hoffman, “Thermal, stress, and thermo-optic effects in high average power double-clad silica fiber lasers,” IEEE J. Quantum Electron. 37, 207–217 (2001).
[CrossRef]

Brunel, M.

M. Brunel, O. Emile, M. Vallet, F. Bretenaker, and A. LeFloch, “Experimental and theoretical study of monomode vectorial lasers passively Q-switched by Cr4+:yttrium aluminium garnet absorber,” Phys. Rev. A 60, 4052–4058 (1998).
[CrossRef]

Burshtein, Z.

Y. Shimony, Z. Burshtein, A. Ben-Amar Baranga, Y. Kalisky, and M. Strauss, “Repetitive 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]

M. J. Damzen, S. Camacho-Lopez, and R. P. M. Green, “Wave-mixing and vector phase conjugation by polarization-dependent saturable absorption in Cr4+:YAG,” Phys. Rev. Lett. 76, 2894–2897 (1996).
[CrossRef] [PubMed]

Cassanho, A.

Catunda, T.

Chen, W.

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]

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]

M. J. Damzen, S. Camacho-Lopez, and R. P. M. Green, “Wave-mixing and vector phase conjugation by polarization-dependent saturable absorption in Cr4+:YAG,” Phys. Rev. Lett. 76, 2894–2897 (1996).
[CrossRef] [PubMed]

del Rayo, M.

A. V. Kir’yanov, Yu. O. Barmenkov, M. del Rayo, and V. N. Filippov, “Ground-state absorption saturation and thermo-lensing effect as main sources of refractive index non-linear change in Cr4+:YAG at cw 1.06 mkm excitation,” Opt. Commun. 213, 151–162 (2002).
[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]

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, TEMoo 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, and A. LeFloch, “Experimental and theoretical study of monomode vectorial lasers passively Q-switched by Cr4+:yttrium aluminium garnet absorber,” Phys. Rev. A 60, 4052–4058 (1998).
[CrossRef]

Filippov, V. N.

A. V. Kir’yanov, Yu. O. Barmenkov, M. del Rayo, and V. N. Filippov, “Ground-state absorption saturation and thermo-lensing effect as main sources of refractive index non-linear change in Cr4+:YAG at cw 1.06 mkm excitation,” Opt. Commun. 213, 151–162 (2002).
[CrossRef]

George, J.

P. K. Mukhopadhyay, J. George, K. Ranganathan, S. K. Sharma, and T. P. S. Nathan, “An alternative approach to determine the fractional heat load in solid state laser materials: application to diode-pumped Nd:YVO4 laser,” Opt. Laser Technol. 34, 253–258 (2002).
[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]

M. J. Damzen, S. Camacho-Lopez, and R. P. M. Green, “Wave-mixing and vector phase conjugation by polarization-dependent saturable absorption in Cr4+:YAG,” Phys. Rev. Lett. 76, 2894–2897 (1996).
[CrossRef] [PubMed]

Gulyamova, E. S.

N. N. Il’ichev, A. V. Kir’yanov, E. S. Gulyamova, and P. P. Pashinin, “Polarisation 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, “Influence of the nonlinear anisotropy of absorption in a passive Cr4+:YAG switch on the energy and polarization characteristics of a neodymium laser,” Quantum Electron. 27, 298–301 (1998).
[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, TEMoo mode and 13 W output,” Appl. Phys. B 58, 409–411 (1994).
[CrossRef]

Hnatovsky, K.

A. J. Alcock, P. Scorah, and K. Hnatovsky, “Broadly tunable continuous-wave diode-pumped Cr4+:YAG laser,” Opt. Commun. 215, 153–157 (2003).
[CrossRef]

Hoffman, H. J.

D. C. Brown and H. J. Hoffman, “Thermal, stress, and thermo-optic effects in high average power double-clad silica fiber lasers,” IEEE J. Quantum Electron. 37, 207–217 (2001).
[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]

Huignard, J.-P.

A. Brignon, P. Sillard, and J.-P. Huignard, “Vector phase conjugation in Cr4+:YAG by four-wave mixing with linearly-polarized pump beams,” Appl. Phys. B 63, 537–540 (1996).

Il’ichev, N. N.

N. N. Il’ichev, A. V. Kir’yanov, and P. P. Pashinin, “Model of passive Q switching taking into account 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, “Influence of the nonlinear anisotropy of absorption in a passive Cr4+:YAG switch on the energy and polarization characteristics of a neodymium laser,” Quantum Electron. 27, 298–301 (1998).
[CrossRef]

N. N. Il’ichev, A. V. Kir’yanov, E. S. Gulyamova, and P. P. Pashinin, “Polarisation 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, P. P. Pashinin, V. A. Sandulenko, A. V. Sandulenko, and S. M. Shpuga, “Anisotropy of nonlinear absorption in a YAG:V3+ crystal,” Quantum Electron. 25, 1154–1157 (1995).
[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, 768–777 (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 YAG:Cr4+ crystal,” Quantum Electron. 24, 771–776 (1994).
[CrossRef]

N. N. Il’ichev, A. V. Kir’yanov, A. A. Malyutin, P. P. Pashinin, and S. M. Shpuga, “The anisotropy of nonlinear absorption induced by laser radiation in LiF:F2 crystal: the short pulse case,” Laser Phys. 3, 182–190 (1993).

N. N. Il’ichev, A. V. Kir’yanov, and A. A. Malyutin, “Self-induced change in the polarization of high-power resonant radiation in an LiF:F2 crystal,” Sov. J. Quantum Electron. 21, 844–848 (1991).
[CrossRef]

N. N. Il’ichev, A. V. Kir’yanov, and A. A. Malyutin, “Peculiarities of anisotropy and dichroizm induced by laser radiation in LiF:F2 crystal,” Laser Phys. 1, 311–323 (1991).

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]

Jenssen, H. P.

Kalisky, Y.

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

Kir’yanov, A. V.

A. V. Kir’yanov, Yu. O. Barmenkov, M. del Rayo, and V. N. Filippov, “Ground-state absorption saturation and thermo-lensing effect as main sources of refractive index non-linear change in Cr4+:YAG at cw 1.06 mkm excitation,” Opt. Commun. 213, 151–162 (2002).
[CrossRef]

N. N. Il’ichev, A. V. Kir’yanov, and P. P. Pashinin, “Model of passive Q switching taking into account 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, “Polarisation 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, “Influence of the nonlinear anisotropy of absorption in a passive Cr4+:YAG switch on the energy and polarization characteristics of a neodymium laser,” Quantum Electron. 27, 298–301 (1998).
[CrossRef]

N. N. Il’ichev, A. V. Kir’yanov, P. P. Pashinin, V. A. Sandulenko, A. V. Sandulenko, and S. M. Shpuga, “Anisotropy of nonlinear absorption in a YAG:V3+ crystal,” Quantum Electron. 25, 1154–1157 (1995).
[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 YAG:Cr4+ 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, 768–777 (1994).

N. N. Il’ichev, A. V. Kir’yanov, A. A. Malyutin, P. P. Pashinin, and S. M. Shpuga, “The anisotropy of nonlinear absorption induced by laser radiation in LiF:F2 crystal: the short pulse case,” Laser Phys. 3, 182–190 (1993).

N. N. Il’ichev, A. V. Kir’yanov, and A. A. Malyutin, “Peculiarities of anisotropy and dichroizm induced by laser radiation in LiF:F2 crystal,” Laser Phys. 1, 311–323 (1991).

N. N. Il’ichev, A. V. Kir’yanov, and A. A. Malyutin, “Self-induced change in the polarization of high-power resonant radiation in an LiF:F2 crystal,” Sov. J. Quantum Electron. 21, 844–848 (1991).
[CrossRef]

Koechner, W.

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, TEMoo mode and 13 W output,” Appl. Phys. B 58, 409–411 (1994).
[CrossRef]

Kuck, S.

S. Kuck, “Laser-related spectroscopy of ion-doped crystals for tunable solid-state lasers,” Appl. Phys. B 72, 515–562 (2001).
[CrossRef]

LeFloch, A.

M. Brunel, O. Emile, M. Vallet, F. Bretenaker, and A. LeFloch, “Experimental and theoretical study of monomode vectorial lasers passively Q-switched by Cr4+:yttrium aluminium garnet absorber,” Phys. Rev. A 60, 4052–4058 (1998).
[CrossRef]

Li, C.

J. Song, C. Li, and K. Ueda, “Thermal influence of saturable absorber in passively Q-switched diode-pumped cw Nd:YAG/Cr4+:YAG laser,” Opt. Commun. 177, 307–316 (2000).
[CrossRef]

Malyutin, A. A.

N. N. Il’ichev, A. V. Kir’yanov, A. A. Malyutin, P. P. Pashinin, and S. M. Shpuga, “The anisotropy of nonlinear absorption induced by laser radiation in LiF:F2 crystal: the short pulse case,” Laser Phys. 3, 182–190 (1993).

N. N. Il’ichev, A. V. Kir’yanov, and A. A. Malyutin, “Self-induced change in the polarization of high-power resonant radiation in an LiF:F2 crystal,” Sov. J. Quantum Electron. 21, 844–848 (1991).
[CrossRef]

N. N. Il’ichev, A. V. Kir’yanov, and A. A. Malyutin, “Peculiarities of anisotropy and dichroizm induced by laser radiation in LiF:F2 crystal,” Laser Phys. 1, 311–323 (1991).

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, TEMoo mode and 13 W output,” Appl. Phys. B 58, 409–411 (1994).
[CrossRef]

Mukhopadhyay, P. K.

P. K. Mukhopadhyay, J. George, K. Ranganathan, S. K. Sharma, and T. P. S. Nathan, “An alternative approach to determine the fractional heat load in solid state laser materials: application to diode-pumped Nd:YVO4 laser,” Opt. Laser Technol. 34, 253–258 (2002).
[CrossRef]

Nathan, T. P. S.

P. K. Mukhopadhyay, J. George, K. Ranganathan, S. K. Sharma, and T. P. S. Nathan, “An alternative approach to determine the fractional heat load in solid state laser materials: application to diode-pumped Nd:YVO4 laser,” Opt. Laser Technol. 34, 253–258 (2002).
[CrossRef]

Nathel, H.

Naumov, S.

Okhrimchuk, A. G.

A. G. Okhrimchuk and A. V. Shestakov, “Absorption saturation mechanism for YAG:Cr4+ crystals,” Phys. Rev. B 61, 988–995 (2000).
[CrossRef]

A. G. Okhrimchuk and A. V. Shestakov, “Performance of YAG:Cr4+ laser crystal,” Opt. Mater. 3, 1–13 (1994).
[CrossRef]

Oliveira, L. C.

L. C. Oliveira, T. Catunda, and S. C. Zilio, “Saturation effects in Z-scan measurements,” Jpn. J. Appl. Phys., 35, 2649–2652 (1996).
[CrossRef]

L. C. Oliveira and S. C. Zilio, “Single-beam time-resolved Z-scan measurement of slow absorbers,” Appl. Phys. Lett. 65, 2121–2123 (1994).
[CrossRef]

Pashinin, P. P.

N. N. Il’ichev, A. V. Kir’yanov, and P. P. Pashinin, “Model of passive Q switching taking into account 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, “Influence of the nonlinear anisotropy of absorption in a passive Cr4+:YAG switch on the energy and polarization characteristics of a neodymium laser,” Quantum Electron. 27, 298–301 (1998).
[CrossRef]

N. N. Il’ichev, A. V. Kir’yanov, E. S. Gulyamova, and P. P. Pashinin, “Polarisation 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, P. P. Pashinin, V. A. Sandulenko, A. V. Sandulenko, and S. M. Shpuga, “Anisotropy of nonlinear absorption in a YAG:V3+ crystal,” Quantum Electron. 25, 1154–1157 (1995).
[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, 768–777 (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 YAG:Cr4+ crystal,” Quantum Electron. 24, 771–776 (1994).
[CrossRef]

N. N. Il’ichev, A. V. Kir’yanov, A. A. Malyutin, P. P. Pashinin, and S. M. Shpuga, “The anisotropy of nonlinear absorption induced by laser radiation in LiF:F2 crystal: the short pulse case,” Laser Phys. 3, 182–190 (1993).

Pilla, V.

Pollock, C. R.

Ranganathan, K.

P. K. Mukhopadhyay, J. George, K. Ranganathan, S. K. Sharma, and T. P. S. Nathan, “An alternative approach to determine the fractional heat load in solid state laser materials: application to diode-pumped Nd:YVO4 laser,” Opt. Laser Technol. 34, 253–258 (2002).
[CrossRef]

Rice, D. K.

Sandulenko, A. V.

N. N. Il’ichev, A. V. Kir’yanov, P. P. Pashinin, V. A. Sandulenko, A. V. Sandulenko, and S. M. Shpuga, “Anisotropy of nonlinear absorption in a YAG:V3+ crystal,” Quantum Electron. 25, 1154–1157 (1995).
[CrossRef]

Sandulenko, V. A.

N. N. Il’ichev, A. V. Kir’yanov, P. P. Pashinin, V. A. Sandulenko, A. V. Sandulenko, and S. M. Shpuga, “Anisotropy of nonlinear absorption in a YAG:V3+ crystal,” Quantum Electron. 25, 1154–1157 (1995).
[CrossRef]

Scorah, P.

A. J. Alcock, P. Scorah, and K. Hnatovsky, “Broadly tunable continuous-wave diode-pumped Cr4+:YAG laser,” Opt. Commun. 215, 153–157 (2003).
[CrossRef]

Sennaroglu, A.

Sharma, S. K.

P. K. Mukhopadhyay, J. George, K. Ranganathan, S. K. Sharma, and T. P. S. Nathan, “An alternative approach to determine the fractional heat load in solid state laser materials: application to diode-pumped Nd:YVO4 laser,” Opt. Laser Technol. 34, 253–258 (2002).
[CrossRef]

Shestakov, A. V.

Shimony, Y.

Y. Shimony, Z. Burshtein, A. Ben-Amar Baranga, Y. Kalisky, and M. Strauss, “Repetitive 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, V. A. Sandulenko, A. V. Sandulenko, and S. M. Shpuga, “Anisotropy of nonlinear absorption in a YAG:V3+ crystal,” Quantum Electron. 25, 1154–1157 (1995).
[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 YAG:Cr4+ 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, 768–777 (1994).

N. N. Il’ichev, A. V. Kir’yanov, A. A. Malyutin, P. P. Pashinin, and S. M. Shpuga, “The anisotropy of nonlinear absorption induced by laser radiation in LiF:F2 crystal: the short pulse case,” Laser Phys. 3, 182–190 (1993).

Sillard, P.

A. Brignon, P. Sillard, and J.-P. Huignard, “Vector phase conjugation in Cr4+:YAG by four-wave mixing with linearly-polarized pump beams,” Appl. Phys. B 63, 537–540 (1996).

Song, J.

J. Song, C. Li, and K. Ueda, “Thermal influence of saturable absorber in passively Q-switched diode-pumped cw Nd:YAG/Cr4+:YAG laser,” Opt. Commun. 177, 307–316 (2000).
[CrossRef]

Sorokin, E.

Sorokina, I. T.

Spariosu, K.

Strauss, M.

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

Stultz, R.

Tenorio, E.

Ueda, K.

J. Song, C. Li, and K. Ueda, “Thermal influence of saturable absorber in passively Q-switched diode-pumped cw Nd:YAG/Cr4+:YAG laser,” Opt. Commun. 177, 307–316 (2000).
[CrossRef]

Vallet, M.

M. Brunel, O. Emile, M. Vallet, F. Bretenaker, and A. LeFloch, “Experimental and theoretical study of monomode vectorial lasers passively Q-switched by Cr4+:yttrium aluminium garnet absorber,” Phys. Rev. A 60, 4052–4058 (1998).
[CrossRef]

Wintner, E.

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]

Zilio, S. C.

L. C. Oliveira, T. Catunda, and S. C. Zilio, “Saturation effects in Z-scan measurements,” Jpn. J. Appl. Phys., 35, 2649–2652 (1996).
[CrossRef]

L. C. Oliveira and S. C. Zilio, “Single-beam time-resolved Z-scan measurement of slow absorbers,” Appl. Phys. Lett. 65, 2121–2123 (1994).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. B (3)

S. Kuck, “Laser-related spectroscopy of ion-doped crystals for tunable solid-state lasers,” Appl. Phys. B 72, 515–562 (2001).
[CrossRef]

A. Brignon, P. Sillard, and J.-P. Huignard, “Vector phase conjugation in Cr4+:YAG by four-wave mixing with linearly-polarized pump beams,” Appl. Phys. B 63, 537–540 (1996).

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, TEMoo mode and 13 W output,” Appl. Phys. B 58, 409–411 (1994).
[CrossRef]

Appl. Phys. Lett. (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]

L. C. Oliveira and S. C. Zilio, “Single-beam time-resolved Z-scan measurement of slow absorbers,” Appl. Phys. Lett. 65, 2121–2123 (1994).
[CrossRef]

IEEE J. Quantum Electron. (2)

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

D. C. Brown and H. J. Hoffman, “Thermal, stress, and thermo-optic effects in high average power double-clad silica fiber lasers,” IEEE J. Quantum Electron. 37, 207–217 (2001).
[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. (1)

J. Opt. Soc. Am. B (3)

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, 768–777 (1994).

Jpn. J. Appl. Phys. (1)

L. C. Oliveira, T. Catunda, and S. C. Zilio, “Saturation effects in Z-scan measurements,” Jpn. J. Appl. Phys., 35, 2649–2652 (1996).
[CrossRef]

Laser Phys. (2)

N. N. Il’ichev, A. V. Kir’yanov, and A. A. Malyutin, “Peculiarities of anisotropy and dichroizm induced by laser radiation in LiF:F2 crystal,” Laser Phys. 1, 311–323 (1991).

N. N. Il’ichev, A. V. Kir’yanov, A. A. Malyutin, P. P. Pashinin, and S. M. Shpuga, “The anisotropy of nonlinear absorption induced by laser radiation in LiF:F2 crystal: the short pulse case,” Laser Phys. 3, 182–190 (1993).

Opt. Commun. (3)

A. J. Alcock, P. Scorah, and K. Hnatovsky, “Broadly tunable continuous-wave diode-pumped Cr4+:YAG laser,” Opt. Commun. 215, 153–157 (2003).
[CrossRef]

J. Song, C. Li, and K. Ueda, “Thermal influence of saturable absorber in passively Q-switched diode-pumped cw Nd:YAG/Cr4+:YAG laser,” Opt. Commun. 177, 307–316 (2000).
[CrossRef]

A. V. Kir’yanov, Yu. O. Barmenkov, M. del Rayo, and V. N. Filippov, “Ground-state absorption saturation and thermo-lensing effect as main sources of refractive index non-linear change in Cr4+:YAG at cw 1.06 mkm excitation,” Opt. Commun. 213, 151–162 (2002).
[CrossRef]

Opt. Laser Technol. (1)

P. K. Mukhopadhyay, J. George, K. Ranganathan, S. K. Sharma, and T. P. S. Nathan, “An alternative approach to determine the fractional heat load in solid state laser materials: application to diode-pumped Nd:YVO4 laser,” Opt. Laser Technol. 34, 253–258 (2002).
[CrossRef]

Opt. Lett. (4)

Opt. Mater. (1)

A. G. Okhrimchuk and A. V. Shestakov, “Performance of YAG:Cr4+ laser crystal,” Opt. Mater. 3, 1–13 (1994).
[CrossRef]

Phys. Rev. A (1)

M. Brunel, O. Emile, M. Vallet, F. Bretenaker, and A. LeFloch, “Experimental and theoretical study of monomode vectorial lasers passively Q-switched by Cr4+:yttrium aluminium garnet absorber,” Phys. Rev. A 60, 4052–4058 (1998).
[CrossRef]

Phys. Rev. B (1)

A. G. Okhrimchuk and A. V. Shestakov, “Absorption saturation mechanism for YAG:Cr4+ crystals,” Phys. Rev. B 61, 988–995 (2000).
[CrossRef]

Phys. Rev. Lett. (1)

M. J. Damzen, S. Camacho-Lopez, and R. P. M. Green, “Wave-mixing and vector phase conjugation by polarization-dependent saturable absorption in Cr4+:YAG,” Phys. Rev. Lett. 76, 2894–2897 (1996).
[CrossRef] [PubMed]

Quantum Electron. (5)

N. N. Il’ichev, A. V. Kir’yanov, E. S. Gulyamova, and P. P. Pashinin, “Polarisation 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, P. P. Pashinin, V. A. Sandulenko, A. V. Sandulenko, and S. M. Shpuga, “Anisotropy of nonlinear absorption in a YAG:V3+ crystal,” Quantum Electron. 25, 1154–1157 (1995).
[CrossRef]

N. N. Il’ichev, A. V. Kir’yanov, and P. P. Pashinin, “Model of passive Q switching taking into account 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, 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 YAG:Cr4+ crystal,” Quantum Electron. 24, 771–776 (1994).
[CrossRef]

N. N. Il’ichev, A. V. Kir’yanov, E. S. Gulyamova, and P. P. Pashinin, “Influence of the nonlinear anisotropy of absorption in a passive Cr4+:YAG switch on the energy and polarization characteristics of a neodymium laser,” Quantum Electron. 27, 298–301 (1998).
[CrossRef]

Sov. J. Quantum Electron. (1)

N. N. Il’ichev, A. V. Kir’yanov, and A. A. Malyutin, “Self-induced change in the polarization of high-power resonant radiation in an LiF:F2 crystal,” Sov. J. Quantum Electron. 21, 844–848 (1991).
[CrossRef]

Other (5)

V. B. Tsvetkov, I. V. Klimov, and I. A. Shcherbakov, “Influence of saturation anisotropy on output polarization of Nd:YAG laser with Cr 4+-doped Q-switch,” in Advanced Solid-State Lasers, M. E. Fermann and L. R. Marshall, eds., Vol. 68 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2002), pp. 445–449.

N. N. Il’ichev, A. V. Kir’yanov, A. A. Malyutin, P. P. Pashinin, and S. M. Shpuga, “Anisotropy of nonlinear absorption in LiF: F2 and YAG:Cr 4+ crystals,” in Book of Abstracts (Kuban State University, Krasnodar, Russia, 1993), pp. 38–39.

H. Liu, J. Dawes, P. Dekker, and J. Piper, “Measurement of polarization-dependent loss mechanisms in Cr 4+ :YAG,” in Advanced-Solid-State Photonics, OSA Topics in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2003), pp. 195–197.

I. T. Sorokina, S. Naumov, E. Sorokin, and A. G. Okhrimchuk, “The mechanisms of slow bleaching in YAG:Cr 4+ under cw pumping,” in Laser Optics 2000: Solid State Lasers, V. I. Ustugov, ed., Proc. SPIE 4350, 99–105 (2001).

A. G. Okhrimchuk, Prokhorov General Physics Institute, Moscow, Russia (personal communication, 2002).

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Figures (11)

Fig. 1
Fig. 1

(a) Schematic of orientations of Cr4+PC centers in Cr4+:YAG lattice (given by its main crystallographic axes). (b), (c) Orientation (given by angle θ) of incidence beam polarization azimuth E with respect to the lattice in cases of (100)-cut (b) and (111)-cut (c) crystals, respectively.

Fig. 2
Fig. 2

(a) Schematic of propagation of laser beam in Cr4+:YAG crystal: Z, direction of propagation; L, sample length. (b) Schematic dividing of Cr4+:YAG for calculating TL effect. Pumped region (I) is limited by beam radius (ω), and unpumped region (II) is between the core and crystal border; b, crystal radius.

Fig. 3
Fig. 3

Examples of modeling [(100)-cut Cr4+:YAG, sample 1] of (a) transient transmission coefficient and refractive-index change due to (b) PP and (c) TL effects versus radial coordinate r. Curves 1 (filled symbols) and 2 (empty symbols) correspond to incidence azimuth orientation θ=0, 45°, respectively. P=1 W.

Fig. 4
Fig. 4

Examples of modeling (sample 1) of (a), (b), transient transmission coefficient and refractive-index change due to (c), (d) PP and (e), (f) TL effects versus longitudinal coordinate Z. (a), (c), (e), (100)-Cut Cr4+:YAG, and (b), (d), (f), (111)-cut Cr4+:YAG. Labels 1, 2, and 3 answer to incidence power P=0.2, 1.0, and 5.0 W, respectively. Sets of curves are calculated for θ=0°, 7.5°, 15°, 22.5°, 30°, 37.5°, and 45°.  

Fig. 5
Fig. 5

Angular dependences of Cr4+:YAG (sample 1) transient transmittance. Graphs (a), (b), and (c) correspond to incidence power P=0.2, 1.0, and 5.0 W, respectively. Curves 1 and 2 represent (100)-cut and (111)-cut crystals.

Fig. 6
Fig. 6

Angular dependences of refractive-index nonlinear change in Cr4+:YAG (sample 1) due to PP effect. Graphs (a), (b), and (c) correspond to incidence power P=0.2, 1.0, and 5.0 W, respectively. Curves 1 and 2 represent (100)-cut and (111)-cut crystals.

Fig. 7
Fig. 7

Angular dependences of refractive-index nonlinear change in Cr4+:YAG (sample 1) due to TL effect. Graphs (a), (b), and (c) correspond to incidence power P=0.2, 1.0, and 5.0 W, respectively. Curves 1 and 2 represent (100)-cut and (111)-cut crystals.

Fig. 8
Fig. 8

Angular dependences of overall (PP plus TL) nonlinear change in Cr4+:YAG (sample 1) due to TL effect. Graphs (a), (b), and (c) correspond to incidence power P=0.2, 1.0, and 5.0 W, respectively. Curves 1 and 2 represent (100)-cut and (111)-cut crystals.

Fig. 9
Fig. 9

Dependences of (a) transient transmittance and (b)–(d) nonlinear change of refractive index in Cr4+:YAG versus saturation parameter A. Graphs (b), (c), and (d) correspond to PP-induced, TL-induced, and overall refractive-index change. Curves (1, 1) and (2, 2) correspond to incidence azimuth of θ=0° and 45° and the case of (100)-cut Cr4+:YAG. Curves (3, 3) and (4, 4) answer to incidence azimuth of θ=30° and 0° and the case of (111)-cut Cr4+:YAG. Solid and dashed curves are for samples 1 and 2, respectively.

Fig. 10
Fig. 10

Dependences of nonlinear change of refractive index due to PP (upper curves) and TL (lower curves) in Cr4+:YAG versus contrast K. Graphs (a), (b), and (c) correspond to P=0.2, 1.0, and 5.0 W, respectively. Labeling of curves is the same as in Fig. 9. The vertical line shows the case of sample 1 (K=7.3).

Fig. 11
Fig. 11

Dependences of (a), (d) transient transmittance and nonlinear change of refractive index due to (b), (e) PP and (c), (f) TL effects in (100)-cut Cr4+:YAG versus beam waist (Z=0) value. Graphs (a), (b), and (c), and (d), (e), and (f) correspond to samples 2 and 3, respectively. Solid and dashed curves are, respectively, for P=1.0 and 5.0 W. Labeling of curves, -θ=45° (1), (3) and θ=0° (2), (4).  

Tables (1)

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Table 1 Parameters Used for Numerical Calculations

Equations (39)

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k=α+β=α01+A+β.
Is=hνσaτf,
α(I, θ)=σaN0i=1M(aie0)2/[1+A(aie0)2]=α0F(I, θ),
F(I, θ)=i=1M(aie0)2/[1+A(aie0)2]
Iz=-[α0F(I, θ)+β]I,
ψz=-δα02 F(I, θ).
I0(P, Z, r)=2Pπω2(Z)exp-2r2ω2(Z),
ψ0(Z, r)=κr22R(Z),
AG(P, Z, r)=2Pπω2(Z)Isexp-2r2ω2(Z).
F[100](P, Z, r, θ)=cos2(θ)1+A cos2(θ)+sin2(θ)1+A sin2(θ),
F[111](P, Z, r, θ)
=23cos2(θ)1+A 23cos2(θ)+23sin2(30°-θ)1+A 23sin2(30°-θ)+23cos2(60°-θ)1+A 23cos2(60°-θ),
T(P, Z, θ)=0Iout(P, Z, θ, r)rdr0I0(P, Z, r)rdr,
ΔnPP(P, Z, θ)
=0ΔψPP(P, Z, θ, r)I0(P, Z, r)rdr0I0(P, Z, r)rdrλp2πL,
ΔψPP(P, Z, θ, r)=ψout(P, Z, θ, r)-ψ0(Z, r).
1rrr TI(r)r=-Q0K,regionI,
1rrr TII(r)r=0,regionII,
dTII(r=b)dr=hK [TC-T(r=b)],
TI(P, Z, θ, r)=T0(P, Z, θ)-Q0(P, Z, θ, r)r24K,0rω(Z),
TII(P, Z, θ, r)=T0(P, Z, θ)-Q0(P, Z, θ, r)ω2(Z)4K-Q0(P, Z, θ, r)ω2(Z)2Klnrω(Z),ω(Z)rb.
T0(P, Z, θ)=TC+Q0(P, Z, θ, 0)ω2(Z)4K×1+2 lnbω(Z)+2Kbh.
I*(P, Z, θ, z, r)=2PT*(P, Z, θ, z, r)πω2(Z)exp-2r2ω2(Z).
Q*(P, Z, θ, z, r)=k*(P, Z, θ, z, r)I*(P, Z, θ, z, r).
Q0(P, Z, θ, r)=1L0Lk*(P, Z, θ, z, r)×I*(P, Z, θ, z, r)dz.
Δn(I)TL=dndT (T0-Tr),
dsdT=dndT+(n0-1)(1+υ)αT+n03αTCr,t.
ΔnTL=Δn(I)TL+Δn(II)TL+Δn(III)TL.
ξ=1-ε λPλL,
PPabs=ξP,
ΔnTL(P, Z, θ)=0dndT+(n0-1)(1+υ)αT+n03αCr,t(T0-Tr)I0(P, Z, r)rdr0I0(P, Z, r)rdr,
Fth(P, Z, θ)=4κπω2(Z)ξ[1-T(P, Z, θ)]P dsdT  ,
ΔnTotal=ΔnPP+ΔnTL,
α0L=-lnT0inT0fin,
βL=-ln T0fin.
K=ln(T0in)/ln(T0fin),
ΔnPP=Δn(Linear)PP+Δn(Nonlinear)PP-nres+Δnres(I),
ΔnTL=Δn(Linear)TL+Δn(Nonlinear)TLdSdT (T-TC)+dSdT [T0-T2(I)].
T2=Q0(P)r24K.

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