Abstract

Beam distortion due to photorefraction limits the usability of lithium niobate (LiNbO3) crystals for frequency conversion applications. To prevent beam distortion in LiNbO3, 5mol.% magnesium-doped LiNbO3 (MgO:LN) is usually used. However, we show that strong beam distortion of green laser light can occur within seconds in MgO:LN, starting at light intensity levels in the 100mW/cm2 regime, if the crystal is heated by several degrees Celsius during or before illumination. Beam distortion does not occur in undoped congruent LiNbO3 (CLN) under the same conditions. We show that the pyroelectric effect together with an elevated photoconductivity compared to that of CLN causes this beam distortion and that this effect also influences frequency conversion experiments in the infrared even if no external heating is applied.

© 2011 Optical Society of America

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    [CrossRef]

2010 (2)

2009 (4)

M. Koesters, B. Sturman, D. Haertle, and K. Buse, “Kinetics of photorefractive recording for circular light beams,” Opt. Lett. 34, 1036–1038 (2009).
[CrossRef]

F. Johann and E. Soergel, “Quantitative measurement of the surface charge density,” Appl. Phys. Lett. 95, 232906 (2009).
[CrossRef]

J. Safioui, F. Devaux, and M. Chauvet, “Pyroliton: pyroelectric spatial soliton,” Opt. Express 17, 22209–22216 (2009).
[CrossRef] [PubMed]

M. Koesters, C. Becher, D. Haertle, B. Sturman, and K. Buse, “Charge transport properties of undoped congruent lithium niobate crystals,” Appl. Phys. B 97, 811–815 (2009).
[CrossRef]

2008 (1)

S. Gronenborn, B. Sturman, M. Falk, D. Haertle, and K. Buse, “Ultraslow shock waves of electron density in LiNbO3 crystals,” Phys. Rev. Lett. 101, 116601 (2008).
[CrossRef] [PubMed]

2007 (2)

D. S. Hum, R. K. Route, G. D. Miller, V. Kondilenko, A. Alexandrovski, J. Huang, K. Urbanek, R. L. Byer, and M. M. Fejer, “Optical properties and ferroelectric engineering of vapor-transport-equilibrated, near-stoichiometric lithium tantalate for frequency conversion,” J. Appl. Phys. 101, 093108(2007).
[CrossRef]

I. Breunig, M. Falk, B. Knabe, R. Sowade, K. Buse, P. Rabiei, and D. H. Jundt, “Second harmonic generation of 2.6 W green light with thermoelectrically oxidized undoped congruent lithium niobate crystals below 100 °C,” Appl. Phys. Lett. 91, 221110(2007).
[CrossRef]

2006 (1)

H. Furuya, A. Morikawa, K. Mizuuchi, and K. Yamamoto, “High-beam-quality continuous wave 3 W green-light generation in bulk periodically poled MgO:LiNbO3,” Jpn. J. Appl. Phys. 45, 6704–6707 (2006).
[CrossRef]

2005 (2)

D. Georgiev, V. P. Gapontsev, A. G. Dronov, M. Y. Vyatkin, A. B. Rulkov, S. V. Popov, and J. R. Taylor, “Watts-level frequency doubling of a narrow line linearly polarized Raman fiber laser to 589 nm,” Opt. Express 13, 6772–6776 (2005).
[CrossRef] [PubMed]

M. C. Wengler, U. Heinemeyer, E. Soergel, and K. Buse, “Ultraviolet light-assisted domain inversion in magnesium-doped lithium niobate crystals,” J. Appl. Phys. 98, 064104 (2005).
[CrossRef]

2004 (3)

E. Fazio, F. Renzi, R. Rinaldi, M. Bertolotti, M. Chauvet, W. Ramadan, A. Petris, and V. Vlad, “Screening-photovoltaic bright solitons in lithium niobate and associated single-mode waveguides,” Appl. Phys. Lett. 85, 2193–2195 (2004).
[CrossRef]

G. T. Niitsu, H. Nagata, and A. C. M. Rodrigues, “Electrical properties along the x and z axes of LiNbO3 wafers,” J. Appl. Phys. 95, 3116–3119 (2004).
[CrossRef]

U. Dörfler, T. Granzow, T. Woike, M. Wöhlecke, M. Imlau, and R. Pankrath, “Intensity and wavelength dependence of the photoconductivity in Cr-doped Sr0.61Ba0.39Nb2O6,” Eur. Phys. J. B 38, 19–24 (2004).
[CrossRef]

2003 (1)

2001 (1)

M. A. Ellabban, R. A. Rupp, and M. Fally, “Reconstruction of parasitic holograms to characterize photorefractive materials,” Appl. Phys. B 72, 635–640 (2001).
[CrossRef]

2000 (1)

Y. Furukawa, K. Kitamura, S. Takekawa, A. Miyamoto, M. Terao, and N. Suda, “Photorefraction in LiNbO3 as a function of [Li]/[Nb] and MgO concentrations,” Appl. Phys. Lett. 77, 2494–2496 (2000).
[CrossRef]

1998 (2)

N. Korneev, D. Mayorga, S. Stepanov, A. Gerwens, K. Buse, and E. Krätzig, “Enhancement of the photorefractive effect by homogeneous pyroelectric fields,” Appl. Phys. B 66, 393–396 (1998).
[CrossRef]

D. Kip, E. Krätzig, V. Shandarov, and P. Moretti, “Thermally induced self-focusing and optical beam interactions in planar strontium barium niobate waveguides,” Opt. Lett. 23, 343–345(1998).
[CrossRef]

1997 (3)

M. Mitchell and M. Segev, “Self-trapping of incoherent white light,” Nature 387, 880–883 (1997).
[CrossRef]

K. Buse, “Light-induced charge transport processes in photorefractive crystals II: materials,” Appl. Phys. B 64, 391–407 (1997).
[CrossRef]

G. D. Miller, R. G. Batchko, W. M. Tulloch, D. R. Weise, M. M. Fejer, and R. L. Byer, “42%-efficient single-pass cw second-harmonic generation in periodically poled lithium niobate,” Opt. Lett. 22, 1834–1836 (1997).
[CrossRef]

1996 (2)

1995 (1)

A. A. Zozulya and D. Z. Anderson, “Propagation of an optical beam in a photorefractive medium in the presence of a photogalvanic nonlinearity or an externally applied electric field,” Phys. Rev. A 51, 1520–1532 (1995).
[CrossRef] [PubMed]

1994 (2)

K. Buse, R. Pankrath, and E. Kraetzig, “Pyroelectrically induced photorefractive damage in Sr0.61Ba0.39Nb2O6:Ce,” Opt. Lett. 19, 260–262 (1994).
[CrossRef] [PubMed]

T. Bartholomäus, K. Buse, C. Deuper, and E. Kratzig, “Pyroelectric coefficients of LiNbO3 crystals of different compositions,” Phys. Status Solidi A 142, K55–K57 (1994).
[CrossRef]

1993 (3)

K. Buse and K. H. Ringhofer, “Pyroelectric drive for light-induced charge transport in the photorefractive process,” Appl. Phys. A 57, 161–165 (1993).
[CrossRef]

K. Buse, “Thermal gratings and pyroelectrically produced charge redistribution in BaTiO3 and KNbO3,” J. Opt. Soc. Am. B 10, 1266–1275 (1993).
[CrossRef]

S. Sochava, K. Buse, and E. Krätzig, “Non-steady-state photocurrent technique for the characterization of photorefractive BaTiO3,” Opt. Commun. 98, 265–268 (1993).
[CrossRef]

1992 (1)

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation—tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[CrossRef]

1990 (1)

1985 (2)

R. S. Weis and T. K. Gaylord, “Lithium niobate: summary of physical properties and crystal structure,” Appl. Phys. A 37, 191–203 (1985).
[CrossRef]

W. Huafu, S. Guotong, and W. Zhongkang, “Photovoltaic effect in LiNbO3:Mg,” Phys. Status Solidi A 89, K211–K213 (1985).
[CrossRef]

1984 (1)

D. A. Bryan, R. Gerson, and H. E. Tomaschke, “Increased optical damage resistance in lithium niobate,” Appl. Phys. Lett. 44, 847–849 (1984).
[CrossRef]

1980 (1)

L. Kanaev, V. Malinovsky, and B. Sturman, “Investigation of photoinduced scattering in LiNbO3 crystals,” Opt. Commun. 34, 95–100 (1980).
[CrossRef]

1965 (1)

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
[CrossRef]

Alexandrovski, A.

D. S. Hum, R. K. Route, G. D. Miller, V. Kondilenko, A. Alexandrovski, J. Huang, K. Urbanek, R. L. Byer, and M. M. Fejer, “Optical properties and ferroelectric engineering of vapor-transport-equilibrated, near-stoichiometric lithium tantalate for frequency conversion,” J. Appl. Phys. 101, 093108(2007).
[CrossRef]

Anderson, D. Z.

A. A. Zozulya and D. Z. Anderson, “Propagation of an optical beam in a photorefractive medium in the presence of a photogalvanic nonlinearity or an externally applied electric field,” Phys. Rev. A 51, 1520–1532 (1995).
[CrossRef] [PubMed]

Bartholomäus, T.

T. Bartholomäus, K. Buse, C. Deuper, and E. Kratzig, “Pyroelectric coefficients of LiNbO3 crystals of different compositions,” Phys. Status Solidi A 142, K55–K57 (1994).
[CrossRef]

Batchko, R. G.

Becher, C.

M. Koesters, C. Becher, D. Haertle, B. Sturman, and K. Buse, “Charge transport properties of undoped congruent lithium niobate crystals,” Appl. Phys. B 97, 811–815 (2009).
[CrossRef]

Bertolotti, M.

E. Fazio, F. Renzi, R. Rinaldi, M. Bertolotti, M. Chauvet, W. Ramadan, A. Petris, and V. Vlad, “Screening-photovoltaic bright solitons in lithium niobate and associated single-mode waveguides,” Appl. Phys. Lett. 85, 2193–2195 (2004).
[CrossRef]

Brashaw, M. C.

Breunig, I.

I. Breunig, M. Falk, B. Knabe, R. Sowade, K. Buse, P. Rabiei, and D. H. Jundt, “Second harmonic generation of 2.6 W green light with thermoelectrically oxidized undoped congruent lithium niobate crystals below 100 °C,” Appl. Phys. Lett. 91, 221110(2007).
[CrossRef]

Bryan, D. A.

D. A. Bryan, R. Gerson, and H. E. Tomaschke, “Increased optical damage resistance in lithium niobate,” Appl. Phys. Lett. 44, 847–849 (1984).
[CrossRef]

Buse, K.

M. Koesters, B. Sturman, D. Haertle, and K. Buse, “Kinetics of photorefractive recording for circular light beams,” Opt. Lett. 34, 1036–1038 (2009).
[CrossRef]

M. Koesters, C. Becher, D. Haertle, B. Sturman, and K. Buse, “Charge transport properties of undoped congruent lithium niobate crystals,” Appl. Phys. B 97, 811–815 (2009).
[CrossRef]

S. Gronenborn, B. Sturman, M. Falk, D. Haertle, and K. Buse, “Ultraslow shock waves of electron density in LiNbO3 crystals,” Phys. Rev. Lett. 101, 116601 (2008).
[CrossRef] [PubMed]

I. Breunig, M. Falk, B. Knabe, R. Sowade, K. Buse, P. Rabiei, and D. H. Jundt, “Second harmonic generation of 2.6 W green light with thermoelectrically oxidized undoped congruent lithium niobate crystals below 100 °C,” Appl. Phys. Lett. 91, 221110(2007).
[CrossRef]

M. C. Wengler, U. Heinemeyer, E. Soergel, and K. Buse, “Ultraviolet light-assisted domain inversion in magnesium-doped lithium niobate crystals,” J. Appl. Phys. 98, 064104 (2005).
[CrossRef]

M. Luennemann, U. Hartwig, and K. Buse, “Improvements of sensitivity and refractive-index changes in photorefractive iron-doped lithium niobate crystals by application of extremely large external electric fields,” J. Opt. Soc. Am. B 20, 1643–1648 (2003).
[CrossRef]

N. Korneev, D. Mayorga, S. Stepanov, A. Gerwens, K. Buse, and E. Krätzig, “Enhancement of the photorefractive effect by homogeneous pyroelectric fields,” Appl. Phys. B 66, 393–396 (1998).
[CrossRef]

K. Buse, “Light-induced charge transport processes in photorefractive crystals II: materials,” Appl. Phys. B 64, 391–407 (1997).
[CrossRef]

T. Bartholomäus, K. Buse, C. Deuper, and E. Kratzig, “Pyroelectric coefficients of LiNbO3 crystals of different compositions,” Phys. Status Solidi A 142, K55–K57 (1994).
[CrossRef]

K. Buse, R. Pankrath, and E. Kraetzig, “Pyroelectrically induced photorefractive damage in Sr0.61Ba0.39Nb2O6:Ce,” Opt. Lett. 19, 260–262 (1994).
[CrossRef] [PubMed]

K. Buse and K. H. Ringhofer, “Pyroelectric drive for light-induced charge transport in the photorefractive process,” Appl. Phys. A 57, 161–165 (1993).
[CrossRef]

K. Buse, “Thermal gratings and pyroelectrically produced charge redistribution in BaTiO3 and KNbO3,” J. Opt. Soc. Am. B 10, 1266–1275 (1993).
[CrossRef]

S. Sochava, K. Buse, and E. Krätzig, “Non-steady-state photocurrent technique for the characterization of photorefractive BaTiO3,” Opt. Commun. 98, 265–268 (1993).
[CrossRef]

Byer, R. L.

D. S. Hum, R. K. Route, G. D. Miller, V. Kondilenko, A. Alexandrovski, J. Huang, K. Urbanek, R. L. Byer, and M. M. Fejer, “Optical properties and ferroelectric engineering of vapor-transport-equilibrated, near-stoichiometric lithium tantalate for frequency conversion,” J. Appl. Phys. 101, 093108(2007).
[CrossRef]

G. D. Miller, R. G. Batchko, W. M. Tulloch, D. R. Weise, M. M. Fejer, and R. L. Byer, “42%-efficient single-pass cw second-harmonic generation in periodically poled lithium niobate,” Opt. Lett. 22, 1834–1836 (1997).
[CrossRef]

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation—tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[CrossRef]

Chauvet, M.

J. Safioui, F. Devaux, and M. Chauvet, “Pyroliton: pyroelectric spatial soliton,” Opt. Express 17, 22209–22216 (2009).
[CrossRef] [PubMed]

E. Fazio, F. Renzi, R. Rinaldi, M. Bertolotti, M. Chauvet, W. Ramadan, A. Petris, and V. Vlad, “Screening-photovoltaic bright solitons in lithium niobate and associated single-mode waveguides,” Appl. Phys. Lett. 85, 2193–2195 (2004).
[CrossRef]

Clark, W. W.

Cole, B.

Deuper, C.

T. Bartholomäus, K. Buse, C. Deuper, and E. Kratzig, “Pyroelectric coefficients of LiNbO3 crystals of different compositions,” Phys. Status Solidi A 142, K55–K57 (1994).
[CrossRef]

Devaux, F.

Dörfler, U.

U. Dörfler, T. Granzow, T. Woike, M. Wöhlecke, M. Imlau, and R. Pankrath, “Intensity and wavelength dependence of the photoconductivity in Cr-doped Sr0.61Ba0.39Nb2O6,” Eur. Phys. J. B 38, 19–24 (2004).
[CrossRef]

Dronov, A. G.

Ellabban, M. A.

M. A. Ellabban, R. A. Rupp, and M. Fally, “Reconstruction of parasitic holograms to characterize photorefractive materials,” Appl. Phys. B 72, 635–640 (2001).
[CrossRef]

Falk, M.

S. Gronenborn, B. Sturman, M. Falk, D. Haertle, and K. Buse, “Ultraslow shock waves of electron density in LiNbO3 crystals,” Phys. Rev. Lett. 101, 116601 (2008).
[CrossRef] [PubMed]

I. Breunig, M. Falk, B. Knabe, R. Sowade, K. Buse, P. Rabiei, and D. H. Jundt, “Second harmonic generation of 2.6 W green light with thermoelectrically oxidized undoped congruent lithium niobate crystals below 100 °C,” Appl. Phys. Lett. 91, 221110(2007).
[CrossRef]

Fally, M.

M. A. Ellabban, R. A. Rupp, and M. Fally, “Reconstruction of parasitic holograms to characterize photorefractive materials,” Appl. Phys. B 72, 635–640 (2001).
[CrossRef]

Fazio, E.

E. Fazio, F. Renzi, R. Rinaldi, M. Bertolotti, M. Chauvet, W. Ramadan, A. Petris, and V. Vlad, “Screening-photovoltaic bright solitons in lithium niobate and associated single-mode waveguides,” Appl. Phys. Lett. 85, 2193–2195 (2004).
[CrossRef]

Fejer, M. M.

J. S. Pelc, C. Langrock, Q. Zhang, and M. M. Fejer, “Influence of domain disorder on parametric noise in quasi-phase-matched quantum frequency converters,” Opt. Lett. 35, 2804–2806 (2010).
[CrossRef] [PubMed]

D. S. Hum, R. K. Route, G. D. Miller, V. Kondilenko, A. Alexandrovski, J. Huang, K. Urbanek, R. L. Byer, and M. M. Fejer, “Optical properties and ferroelectric engineering of vapor-transport-equilibrated, near-stoichiometric lithium tantalate for frequency conversion,” J. Appl. Phys. 101, 093108(2007).
[CrossRef]

G. D. Miller, R. G. Batchko, W. M. Tulloch, D. R. Weise, M. M. Fejer, and R. L. Byer, “42%-efficient single-pass cw second-harmonic generation in periodically poled lithium niobate,” Opt. Lett. 22, 1834–1836 (1997).
[CrossRef]

M. Taya, M. C. Brashaw, and M. M. Fejer, “Photorefractive effects in periodically poled ferroelectrics,” Opt. Lett. 21, 857–859 (1996).
[CrossRef] [PubMed]

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation—tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[CrossRef]

Furukawa, Y.

Y. Furukawa, K. Kitamura, S. Takekawa, A. Miyamoto, M. Terao, and N. Suda, “Photorefraction in LiNbO3 as a function of [Li]/[Nb] and MgO concentrations,” Appl. Phys. Lett. 77, 2494–2496 (2000).
[CrossRef]

Furuya, H.

H. Furuya, A. Morikawa, K. Mizuuchi, and K. Yamamoto, “High-beam-quality continuous wave 3 W green-light generation in bulk periodically poled MgO:LiNbO3,” Jpn. J. Appl. Phys. 45, 6704–6707 (2006).
[CrossRef]

Gapontsev, V. P.

Gaylord, T. K.

R. S. Weis and T. K. Gaylord, “Lithium niobate: summary of physical properties and crystal structure,” Appl. Phys. A 37, 191–203 (1985).
[CrossRef]

Georgiev, D.

Gerson, R.

D. A. Bryan, R. Gerson, and H. E. Tomaschke, “Increased optical damage resistance in lithium niobate,” Appl. Phys. Lett. 44, 847–849 (1984).
[CrossRef]

Gerwens, A.

N. Korneev, D. Mayorga, S. Stepanov, A. Gerwens, K. Buse, and E. Krätzig, “Enhancement of the photorefractive effect by homogeneous pyroelectric fields,” Appl. Phys. B 66, 393–396 (1998).
[CrossRef]

Goldberg, L.

Gómez Sarabia, C. M.

Gordon, J. P.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
[CrossRef]

Granzow, T.

U. Dörfler, T. Granzow, T. Woike, M. Wöhlecke, M. Imlau, and R. Pankrath, “Intensity and wavelength dependence of the photoconductivity in Cr-doped Sr0.61Ba0.39Nb2O6,” Eur. Phys. J. B 38, 19–24 (2004).
[CrossRef]

Gronenborn, S.

S. Gronenborn, B. Sturman, M. Falk, D. Haertle, and K. Buse, “Ultraslow shock waves of electron density in LiNbO3 crystals,” Phys. Rev. Lett. 101, 116601 (2008).
[CrossRef] [PubMed]

Guotong, S.

W. Huafu, S. Guotong, and W. Zhongkang, “Photovoltaic effect in LiNbO3:Mg,” Phys. Status Solidi A 89, K211–K213 (1985).
[CrossRef]

Haertle, D.

M. Koesters, B. Sturman, D. Haertle, and K. Buse, “Kinetics of photorefractive recording for circular light beams,” Opt. Lett. 34, 1036–1038 (2009).
[CrossRef]

M. Koesters, C. Becher, D. Haertle, B. Sturman, and K. Buse, “Charge transport properties of undoped congruent lithium niobate crystals,” Appl. Phys. B 97, 811–815 (2009).
[CrossRef]

S. Gronenborn, B. Sturman, M. Falk, D. Haertle, and K. Buse, “Ultraslow shock waves of electron density in LiNbO3 crystals,” Phys. Rev. Lett. 101, 116601 (2008).
[CrossRef] [PubMed]

Hartwig, U.

Heinemeyer, U.

M. C. Wengler, U. Heinemeyer, E. Soergel, and K. Buse, “Ultraviolet light-assisted domain inversion in magnesium-doped lithium niobate crystals,” J. Appl. Phys. 98, 064104 (2005).
[CrossRef]

Huafu, W.

W. Huafu, S. Guotong, and W. Zhongkang, “Photovoltaic effect in LiNbO3:Mg,” Phys. Status Solidi A 89, K211–K213 (1985).
[CrossRef]

Huang, J.

D. S. Hum, R. K. Route, G. D. Miller, V. Kondilenko, A. Alexandrovski, J. Huang, K. Urbanek, R. L. Byer, and M. M. Fejer, “Optical properties and ferroelectric engineering of vapor-transport-equilibrated, near-stoichiometric lithium tantalate for frequency conversion,” J. Appl. Phys. 101, 093108(2007).
[CrossRef]

Hum, D. S.

D. S. Hum, R. K. Route, G. D. Miller, V. Kondilenko, A. Alexandrovski, J. Huang, K. Urbanek, R. L. Byer, and M. M. Fejer, “Optical properties and ferroelectric engineering of vapor-transport-equilibrated, near-stoichiometric lithium tantalate for frequency conversion,” J. Appl. Phys. 101, 093108(2007).
[CrossRef]

Imlau, M.

U. Dörfler, T. Granzow, T. Woike, M. Wöhlecke, M. Imlau, and R. Pankrath, “Intensity and wavelength dependence of the photoconductivity in Cr-doped Sr0.61Ba0.39Nb2O6,” Eur. Phys. J. B 38, 19–24 (2004).
[CrossRef]

Johann, F.

F. Johann and E. Soergel, “Quantitative measurement of the surface charge density,” Appl. Phys. Lett. 95, 232906 (2009).
[CrossRef]

Jundt, D. H.

I. Breunig, M. Falk, B. Knabe, R. Sowade, K. Buse, P. Rabiei, and D. H. Jundt, “Second harmonic generation of 2.6 W green light with thermoelectrically oxidized undoped congruent lithium niobate crystals below 100 °C,” Appl. Phys. Lett. 91, 221110(2007).
[CrossRef]

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation—tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[CrossRef]

Kanaev, L.

L. Kanaev, V. Malinovsky, and B. Sturman, “Investigation of photoinduced scattering in LiNbO3 crystals,” Opt. Commun. 34, 95–100 (1980).
[CrossRef]

King, V.

Kip, D.

Kitamura, K.

Y. Furukawa, K. Kitamura, S. Takekawa, A. Miyamoto, M. Terao, and N. Suda, “Photorefraction in LiNbO3 as a function of [Li]/[Nb] and MgO concentrations,” Appl. Phys. Lett. 77, 2494–2496 (2000).
[CrossRef]

Knabe, B.

I. Breunig, M. Falk, B. Knabe, R. Sowade, K. Buse, P. Rabiei, and D. H. Jundt, “Second harmonic generation of 2.6 W green light with thermoelectrically oxidized undoped congruent lithium niobate crystals below 100 °C,” Appl. Phys. Lett. 91, 221110(2007).
[CrossRef]

Koechner, W.

W. Koechner, Solid-State Laser Engineering (Springer, 1999).

Koesters, M.

M. Koesters, C. Becher, D. Haertle, B. Sturman, and K. Buse, “Charge transport properties of undoped congruent lithium niobate crystals,” Appl. Phys. B 97, 811–815 (2009).
[CrossRef]

M. Koesters, B. Sturman, D. Haertle, and K. Buse, “Kinetics of photorefractive recording for circular light beams,” Opt. Lett. 34, 1036–1038 (2009).
[CrossRef]

Kondilenko, V.

D. S. Hum, R. K. Route, G. D. Miller, V. Kondilenko, A. Alexandrovski, J. Huang, K. Urbanek, R. L. Byer, and M. M. Fejer, “Optical properties and ferroelectric engineering of vapor-transport-equilibrated, near-stoichiometric lithium tantalate for frequency conversion,” J. Appl. Phys. 101, 093108(2007).
[CrossRef]

Korneev, N.

N. Korneev, D. Mayorga, S. Stepanov, A. Gerwens, K. Buse, and E. Krätzig, “Enhancement of the photorefractive effect by homogeneous pyroelectric fields,” Appl. Phys. B 66, 393–396 (1998).
[CrossRef]

Kraetzig, E.

Kratzig, E.

T. Bartholomäus, K. Buse, C. Deuper, and E. Kratzig, “Pyroelectric coefficients of LiNbO3 crystals of different compositions,” Phys. Status Solidi A 142, K55–K57 (1994).
[CrossRef]

Krätzig, E.

N. Korneev, D. Mayorga, S. Stepanov, A. Gerwens, K. Buse, and E. Krätzig, “Enhancement of the photorefractive effect by homogeneous pyroelectric fields,” Appl. Phys. B 66, 393–396 (1998).
[CrossRef]

D. Kip, E. Krätzig, V. Shandarov, and P. Moretti, “Thermally induced self-focusing and optical beam interactions in planar strontium barium niobate waveguides,” Opt. Lett. 23, 343–345(1998).
[CrossRef]

S. Sochava, K. Buse, and E. Krätzig, “Non-steady-state photocurrent technique for the characterization of photorefractive BaTiO3,” Opt. Commun. 98, 265–268 (1993).
[CrossRef]

Langrock, C.

Leach, L.

Leite, R. C. C.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
[CrossRef]

Luennemann, M.

Magel, G. A.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation—tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[CrossRef]

Malinovsky, V.

L. Kanaev, V. Malinovsky, and B. Sturman, “Investigation of photoinduced scattering in LiNbO3 crystals,” Opt. Commun. 34, 95–100 (1980).
[CrossRef]

Márquez Aguilar, P. A.

Mayorga, D.

N. Korneev, D. Mayorga, S. Stepanov, A. Gerwens, K. Buse, and E. Krätzig, “Enhancement of the photorefractive effect by homogeneous pyroelectric fields,” Appl. Phys. B 66, 393–396 (1998).
[CrossRef]

Miller, G. D.

D. S. Hum, R. K. Route, G. D. Miller, V. Kondilenko, A. Alexandrovski, J. Huang, K. Urbanek, R. L. Byer, and M. M. Fejer, “Optical properties and ferroelectric engineering of vapor-transport-equilibrated, near-stoichiometric lithium tantalate for frequency conversion,” J. Appl. Phys. 101, 093108(2007).
[CrossRef]

G. D. Miller, R. G. Batchko, W. M. Tulloch, D. R. Weise, M. M. Fejer, and R. L. Byer, “42%-efficient single-pass cw second-harmonic generation in periodically poled lithium niobate,” Opt. Lett. 22, 1834–1836 (1997).
[CrossRef]

Miller, M. J.

Mitchell, M.

M. Mitchell and M. Segev, “Self-trapping of incoherent white light,” Nature 387, 880–883 (1997).
[CrossRef]

Miyamoto, A.

Y. Furukawa, K. Kitamura, S. Takekawa, A. Miyamoto, M. Terao, and N. Suda, “Photorefraction in LiNbO3 as a function of [Li]/[Nb] and MgO concentrations,” Appl. Phys. Lett. 77, 2494–2496 (2000).
[CrossRef]

Mizuuchi, K.

H. Furuya, A. Morikawa, K. Mizuuchi, and K. Yamamoto, “High-beam-quality continuous wave 3 W green-light generation in bulk periodically poled MgO:LiNbO3,” Jpn. J. Appl. Phys. 45, 6704–6707 (2006).
[CrossRef]

Monson, B.

Moore, R. S.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
[CrossRef]

Moretti, P.

Morikawa, A.

H. Furuya, A. Morikawa, K. Mizuuchi, and K. Yamamoto, “High-beam-quality continuous wave 3 W green-light generation in bulk periodically poled MgO:LiNbO3,” Jpn. J. Appl. Phys. 45, 6704–6707 (2006).
[CrossRef]

Nagata, H.

G. T. Niitsu, H. Nagata, and A. C. M. Rodrigues, “Electrical properties along the x and z axes of LiNbO3 wafers,” J. Appl. Phys. 95, 3116–3119 (2004).
[CrossRef]

Neurgaonkar, R. R.

Niitsu, G. T.

G. T. Niitsu, H. Nagata, and A. C. M. Rodrigues, “Electrical properties along the x and z axes of LiNbO3 wafers,” J. Appl. Phys. 95, 3116–3119 (2004).
[CrossRef]

Pankrath, R.

U. Dörfler, T. Granzow, T. Woike, M. Wöhlecke, M. Imlau, and R. Pankrath, “Intensity and wavelength dependence of the photoconductivity in Cr-doped Sr0.61Ba0.39Nb2O6,” Eur. Phys. J. B 38, 19–24 (2004).
[CrossRef]

K. Buse, R. Pankrath, and E. Kraetzig, “Pyroelectrically induced photorefractive damage in Sr0.61Ba0.39Nb2O6:Ce,” Opt. Lett. 19, 260–262 (1994).
[CrossRef] [PubMed]

Pelc, J. S.

Petris, A.

E. Fazio, F. Renzi, R. Rinaldi, M. Bertolotti, M. Chauvet, W. Ramadan, A. Petris, and V. Vlad, “Screening-photovoltaic bright solitons in lithium niobate and associated single-mode waveguides,” Appl. Phys. Lett. 85, 2193–2195 (2004).
[CrossRef]

Popov, S. V.

Porto, S. P. S.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
[CrossRef]

Rabiei, P.

I. Breunig, M. Falk, B. Knabe, R. Sowade, K. Buse, P. Rabiei, and D. H. Jundt, “Second harmonic generation of 2.6 W green light with thermoelectrically oxidized undoped congruent lithium niobate crystals below 100 °C,” Appl. Phys. Lett. 91, 221110(2007).
[CrossRef]

Ramadan, W.

E. Fazio, F. Renzi, R. Rinaldi, M. Bertolotti, M. Chauvet, W. Ramadan, A. Petris, and V. Vlad, “Screening-photovoltaic bright solitons in lithium niobate and associated single-mode waveguides,” Appl. Phys. Lett. 85, 2193–2195 (2004).
[CrossRef]

Renzi, F.

E. Fazio, F. Renzi, R. Rinaldi, M. Bertolotti, M. Chauvet, W. Ramadan, A. Petris, and V. Vlad, “Screening-photovoltaic bright solitons in lithium niobate and associated single-mode waveguides,” Appl. Phys. Lett. 85, 2193–2195 (2004).
[CrossRef]

Rinaldi, R.

E. Fazio, F. Renzi, R. Rinaldi, M. Bertolotti, M. Chauvet, W. Ramadan, A. Petris, and V. Vlad, “Screening-photovoltaic bright solitons in lithium niobate and associated single-mode waveguides,” Appl. Phys. Lett. 85, 2193–2195 (2004).
[CrossRef]

Ringhofer, K. H.

K. Buse and K. H. Ringhofer, “Pyroelectric drive for light-induced charge transport in the photorefractive process,” Appl. Phys. A 57, 161–165 (1993).
[CrossRef]

Rodrigues, A. C. M.

G. T. Niitsu, H. Nagata, and A. C. M. Rodrigues, “Electrical properties along the x and z axes of LiNbO3 wafers,” J. Appl. Phys. 95, 3116–3119 (2004).
[CrossRef]

Route, R. K.

D. S. Hum, R. K. Route, G. D. Miller, V. Kondilenko, A. Alexandrovski, J. Huang, K. Urbanek, R. L. Byer, and M. M. Fejer, “Optical properties and ferroelectric engineering of vapor-transport-equilibrated, near-stoichiometric lithium tantalate for frequency conversion,” J. Appl. Phys. 101, 093108(2007).
[CrossRef]

Rulkov, A. B.

Rupp, R. A.

M. A. Ellabban, R. A. Rupp, and M. Fally, “Reconstruction of parasitic holograms to characterize photorefractive materials,” Appl. Phys. B 72, 635–640 (2001).
[CrossRef]

Safioui, J.

Salamo, G. J.

Sánchez Mondragón, J. J.

Segev, M.

M. Mitchell and M. Segev, “Self-trapping of incoherent white light,” Nature 387, 880–883 (1997).
[CrossRef]

Shandarov, V.

Sharp, E. J.

Sochava, S.

S. Sochava, K. Buse, and E. Krätzig, “Non-steady-state photocurrent technique for the characterization of photorefractive BaTiO3,” Opt. Commun. 98, 265–268 (1993).
[CrossRef]

Soergel, E.

F. Johann and E. Soergel, “Quantitative measurement of the surface charge density,” Appl. Phys. Lett. 95, 232906 (2009).
[CrossRef]

M. C. Wengler, U. Heinemeyer, E. Soergel, and K. Buse, “Ultraviolet light-assisted domain inversion in magnesium-doped lithium niobate crystals,” J. Appl. Phys. 98, 064104 (2005).
[CrossRef]

Sowade, R.

I. Breunig, M. Falk, B. Knabe, R. Sowade, K. Buse, P. Rabiei, and D. H. Jundt, “Second harmonic generation of 2.6 W green light with thermoelectrically oxidized undoped congruent lithium niobate crystals below 100 °C,” Appl. Phys. Lett. 91, 221110(2007).
[CrossRef]

Stepanov, S.

N. Korneev, D. Mayorga, S. Stepanov, A. Gerwens, K. Buse, and E. Krätzig, “Enhancement of the photorefractive effect by homogeneous pyroelectric fields,” Appl. Phys. B 66, 393–396 (1998).
[CrossRef]

C. M. Gómez Sarabia, P. A. Márquez Aguilar, J. J. Sánchez Mondragón, S. Stepanov, and V. Vysloukh, “Dynamics of photoinduced lens formation in a photorefractive Bi12TiO20 crystal under an external dc electric field,” J. Opt. Soc. Am. B 13, 2767–2774 (1996).
[CrossRef]

Sturman, B.

M. Koesters, B. Sturman, D. Haertle, and K. Buse, “Kinetics of photorefractive recording for circular light beams,” Opt. Lett. 34, 1036–1038 (2009).
[CrossRef]

M. Koesters, C. Becher, D. Haertle, B. Sturman, and K. Buse, “Charge transport properties of undoped congruent lithium niobate crystals,” Appl. Phys. B 97, 811–815 (2009).
[CrossRef]

S. Gronenborn, B. Sturman, M. Falk, D. Haertle, and K. Buse, “Ultraslow shock waves of electron density in LiNbO3 crystals,” Phys. Rev. Lett. 101, 116601 (2008).
[CrossRef] [PubMed]

L. Kanaev, V. Malinovsky, and B. Sturman, “Investigation of photoinduced scattering in LiNbO3 crystals,” Opt. Commun. 34, 95–100 (1980).
[CrossRef]

Suda, N.

Y. Furukawa, K. Kitamura, S. Takekawa, A. Miyamoto, M. Terao, and N. Suda, “Photorefraction in LiNbO3 as a function of [Li]/[Nb] and MgO concentrations,” Appl. Phys. Lett. 77, 2494–2496 (2000).
[CrossRef]

Takekawa, S.

Y. Furukawa, K. Kitamura, S. Takekawa, A. Miyamoto, M. Terao, and N. Suda, “Photorefraction in LiNbO3 as a function of [Li]/[Nb] and MgO concentrations,” Appl. Phys. Lett. 77, 2494–2496 (2000).
[CrossRef]

Taya, M.

Taylor, J. R.

Terao, M.

Y. Furukawa, K. Kitamura, S. Takekawa, A. Miyamoto, M. Terao, and N. Suda, “Photorefraction in LiNbO3 as a function of [Li]/[Nb] and MgO concentrations,” Appl. Phys. Lett. 77, 2494–2496 (2000).
[CrossRef]

Tomaschke, H. E.

D. A. Bryan, R. Gerson, and H. E. Tomaschke, “Increased optical damage resistance in lithium niobate,” Appl. Phys. Lett. 44, 847–849 (1984).
[CrossRef]

Tulloch, W. M.

Urbanek, K.

D. S. Hum, R. K. Route, G. D. Miller, V. Kondilenko, A. Alexandrovski, J. Huang, K. Urbanek, R. L. Byer, and M. M. Fejer, “Optical properties and ferroelectric engineering of vapor-transport-equilibrated, near-stoichiometric lithium tantalate for frequency conversion,” J. Appl. Phys. 101, 093108(2007).
[CrossRef]

Vlad, V.

E. Fazio, F. Renzi, R. Rinaldi, M. Bertolotti, M. Chauvet, W. Ramadan, A. Petris, and V. Vlad, “Screening-photovoltaic bright solitons in lithium niobate and associated single-mode waveguides,” Appl. Phys. Lett. 85, 2193–2195 (2004).
[CrossRef]

Vyatkin, M. Y.

Vysloukh, V.

Weis, R. S.

R. S. Weis and T. K. Gaylord, “Lithium niobate: summary of physical properties and crystal structure,” Appl. Phys. A 37, 191–203 (1985).
[CrossRef]

Weise, D. R.

Wengler, M. C.

M. C. Wengler, U. Heinemeyer, E. Soergel, and K. Buse, “Ultraviolet light-assisted domain inversion in magnesium-doped lithium niobate crystals,” J. Appl. Phys. 98, 064104 (2005).
[CrossRef]

Whinnery, J. R.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
[CrossRef]

Wöhlecke, M.

U. Dörfler, T. Granzow, T. Woike, M. Wöhlecke, M. Imlau, and R. Pankrath, “Intensity and wavelength dependence of the photoconductivity in Cr-doped Sr0.61Ba0.39Nb2O6,” Eur. Phys. J. B 38, 19–24 (2004).
[CrossRef]

Woike, T.

U. Dörfler, T. Granzow, T. Woike, M. Wöhlecke, M. Imlau, and R. Pankrath, “Intensity and wavelength dependence of the photoconductivity in Cr-doped Sr0.61Ba0.39Nb2O6,” Eur. Phys. J. B 38, 19–24 (2004).
[CrossRef]

Wong, K. K.

K. K. Wong, Properties of Lithium Niobate (INSPEC, 2002).

Wood, G. L.

Yamamoto, K.

H. Furuya, A. Morikawa, K. Mizuuchi, and K. Yamamoto, “High-beam-quality continuous wave 3 W green-light generation in bulk periodically poled MgO:LiNbO3,” Jpn. J. Appl. Phys. 45, 6704–6707 (2006).
[CrossRef]

Zhang, Q.

Zhongkang, W.

W. Huafu, S. Guotong, and W. Zhongkang, “Photovoltaic effect in LiNbO3:Mg,” Phys. Status Solidi A 89, K211–K213 (1985).
[CrossRef]

Zozulya, A. A.

A. A. Zozulya and D. Z. Anderson, “Propagation of an optical beam in a photorefractive medium in the presence of a photogalvanic nonlinearity or an externally applied electric field,” Phys. Rev. A 51, 1520–1532 (1995).
[CrossRef] [PubMed]

Appl. Opt. (1)

Appl. Phys. A (2)

K. Buse and K. H. Ringhofer, “Pyroelectric drive for light-induced charge transport in the photorefractive process,” Appl. Phys. A 57, 161–165 (1993).
[CrossRef]

R. S. Weis and T. K. Gaylord, “Lithium niobate: summary of physical properties and crystal structure,” Appl. Phys. A 37, 191–203 (1985).
[CrossRef]

Appl. Phys. B (4)

M. A. Ellabban, R. A. Rupp, and M. Fally, “Reconstruction of parasitic holograms to characterize photorefractive materials,” Appl. Phys. B 72, 635–640 (2001).
[CrossRef]

N. Korneev, D. Mayorga, S. Stepanov, A. Gerwens, K. Buse, and E. Krätzig, “Enhancement of the photorefractive effect by homogeneous pyroelectric fields,” Appl. Phys. B 66, 393–396 (1998).
[CrossRef]

K. Buse, “Light-induced charge transport processes in photorefractive crystals II: materials,” Appl. Phys. B 64, 391–407 (1997).
[CrossRef]

M. Koesters, C. Becher, D. Haertle, B. Sturman, and K. Buse, “Charge transport properties of undoped congruent lithium niobate crystals,” Appl. Phys. B 97, 811–815 (2009).
[CrossRef]

Appl. Phys. Lett. (5)

E. Fazio, F. Renzi, R. Rinaldi, M. Bertolotti, M. Chauvet, W. Ramadan, A. Petris, and V. Vlad, “Screening-photovoltaic bright solitons in lithium niobate and associated single-mode waveguides,” Appl. Phys. Lett. 85, 2193–2195 (2004).
[CrossRef]

F. Johann and E. Soergel, “Quantitative measurement of the surface charge density,” Appl. Phys. Lett. 95, 232906 (2009).
[CrossRef]

I. Breunig, M. Falk, B. Knabe, R. Sowade, K. Buse, P. Rabiei, and D. H. Jundt, “Second harmonic generation of 2.6 W green light with thermoelectrically oxidized undoped congruent lithium niobate crystals below 100 °C,” Appl. Phys. Lett. 91, 221110(2007).
[CrossRef]

D. A. Bryan, R. Gerson, and H. E. Tomaschke, “Increased optical damage resistance in lithium niobate,” Appl. Phys. Lett. 44, 847–849 (1984).
[CrossRef]

Y. Furukawa, K. Kitamura, S. Takekawa, A. Miyamoto, M. Terao, and N. Suda, “Photorefraction in LiNbO3 as a function of [Li]/[Nb] and MgO concentrations,” Appl. Phys. Lett. 77, 2494–2496 (2000).
[CrossRef]

Eur. Phys. J. B (1)

U. Dörfler, T. Granzow, T. Woike, M. Wöhlecke, M. Imlau, and R. Pankrath, “Intensity and wavelength dependence of the photoconductivity in Cr-doped Sr0.61Ba0.39Nb2O6,” Eur. Phys. J. B 38, 19–24 (2004).
[CrossRef]

IEEE J. Quantum Electron. (1)

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Figures (13)

Fig. 1
Fig. 1

Pyroelectrically induced refractive index change Δ n ill (normalized to Δ n hom ) versus ξ = z / w . (a) refractive index change plotted for t = 5 τ di and η = 0.1 l with l = 0 , 0.5 , 1 , 1.5 , , 4 (from smallest to largest refractive index change). The curves for l = 2 , , 4 overlap each other. (b) refractive index change plotted for t = 0 , 1 , , 10 × τ di , and η = 0.001 (from inside to outside). In both graphs I ¯ is shown as red dashed line.

Fig. 2
Fig. 2

Numerically simulated normalized refractive index change Δ n ill / Δ n hom . The ratio of ϵ 11 / ϵ 33 is 2.9, as is the case for MgO:LN.

Fig. 3
Fig. 3

Numerically simulated refractive index change Δ n ill ( y = 0 , z ) in MgO:LN for varying illumination times t = 1 τ di , 2 τ di , 4 τ di , 8 τ di , 16 τ di , 32 τ di , 64 τ di (smallest to largest Δ n ill ).

Fig. 4
Fig. 4

Cut E z ( y = 0 , z , t ) through the 2D electric field profile along the z axis for the beam self-heated case for different times (multiples of τ di ) and slablike crystal shape. For t = 0 the curve is parabolic around y = z = 0 ; for longer times a flat-top profile starts to develop. The electric field is normalized to E pyro from Eq. (27).

Fig. 5
Fig. 5

Setup for the observation of beam distortion. The laser is a 532 nm frequency-doubled continuous-wave Nd:YAG laser.

Fig. 6
Fig. 6

Shape of a laser beam after passing an MgO: LiNb O 3 crystal (sample YamMgOL N 1 ) that is heated from 40 ° C to 50 ° C . (The c axis is parallel to the vertical dimension in the photographs).

Fig. 7
Fig. 7

OPD map of CTIMgOL N 4 for Δ T = 2 ° C and P = 150 mW : (a) unprocessed, (b) after 10 s , and (c) after 90 s of illumination. Data are normalized to 2 L .

Fig. 8
Fig. 8

Nonuniform refractive index change Δ n ill ( y = 0 , z , t ) in CTIMgOL N 4 for different illumination times t = 5 , 10 , 20 , 40 , 80 , 160 s . Experimental parameters were Δ T = 3 ° C and P = 150 mW .

Fig. 9
Fig. 9

Normalized maximum refractive index change Δ n ill ( y = 0 , z = 0 , t ) versus illumination time t (squares) in sample CTI MgO LN 4 . The solid line represents the result of the numerical simulation for τ di = 14 s .

Fig. 10
Fig. 10

Δ n ill ( y = 0 , z , t = 20 s ) of sample CTI MgO LN 4 for different laser powers P = 600 , 300 , 150 , 75 , 40 , 20 mW (largest to smallest refractive index change). Experimental parameters were Δ T = 3 ° C and the illumination time t = 20 s .

Fig. 11
Fig. 11

Schematic of the apparatus for measurement of photo conductivity and bulk-photovoltaic current.

Fig. 12
Fig. 12

Specific photoconductivity versus intensity for CLN (squares) and MgO:LN (circles).

Fig. 13
Fig. 13

Bulk-photovoltaic coefficient β versus intensity for CLN (squares) and MgO:LN (circles).

Tables (1)

Tables Icon

Table 1 Crystals Used in the Experiments and Their Short Names

Equations (41)

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j = j phv + j diff + j drift = β I z ^ + ( k b T / e ) μ e ν + σ E .
σ = κ I + σ d ,
j = σ ϕ + ( k b T / e ) μ e ν + β I z ^ .
Δ n o , e = n o , e 3 r 13 , 33 E z 2 ,
E pyro = 1 ϵ 33 ϵ 0 p 3 Δ T z ^ ,
E ( ξ ) = E pv I ¯ ( ξ ) I ¯ ( ξ ) + η z ^ + E w d I ¯ ( ξ ) / d ξ I ¯ ( ξ ) + η z ^ ,
j = σ E .
ρ t + · j = 0 .
· [ ϵ ϵ 0 E t + σ E ] = 0
ϵ ϵ 0 E t + σ E = j d ,
E ( t ¯ , ξ ) = U H { η I ¯ ( ξ ) + η + I ¯ ( ξ ) I ¯ ( ξ ) + η exp [ t ¯ ( I ¯ + η ) ] } z ^ ,
Δ n = 1 2 n e 3 r 33 U H { η I ¯ ( ξ ) + η + I ¯ ( ξ ) I ¯ ( ξ ) + η exp [ t ¯ ( I ¯ + η ) ] } .
Δ n = Δ n hom + Δ n ill .
Δ n hom = 1 2 n e 3 r 33 U H
Δ n ill = 1 2 n e 3 r 33 U H ( I ¯ ( ξ ) I ¯ ( ξ ) + η { exp [ t ¯ ( I ¯ + η ) ] 1 } ) .
ρ t = { κ ( E 0 z ^ E pv ) · I · [ ( κ I + σ d ) ϕ int ] } .
ρ ϵ 0 = ϵ 11 x 2 ϕ int + ϵ 11 y 2 ϕ int + ϵ 33 z 2 ϕ int .
τ di = ϵ 0 ( ϵ 33 + ϵ 33 ϵ 11 ) κ I + σ d .
( 1 k th t 2 ) Δ T = q ( r ) λ th ,
q ( r ) = α I ( r ) = α I 0 I ¯ ( r ) ,
2 ( Δ T ) = q ( r ) / λ th .
T 0 = 2 P α / ( π λ th ) .
¯ 2 ( Δ T ¯ ) ( r ¯ ) = I ¯ ( r ¯ ) .
Δ T ( r ) = α P 4 π λ th [ E i ( 2 R 2 w 2 ) + E i ( 2 r 2 w 2 ) 2 ln ( r R ) ] .
Δ T ( 0 ) α P 4 π λ th [ γ Euler + ln ( 2 R 2 w 2 ) ] ,
E z , pyro p 3 T ϵ 0 ( ϵ 33 + ϵ 33 ϵ 11 ) for     t τ di ,
E pyro p 3 α P [ γ Euler + ln ( 2 R 2 w 2 ) ] 4 π λ th ϵ 0 ( ϵ 33 + ϵ 33 ϵ 11 ) .
ρ pyro ( r ) ( r , t = 0 ) = · ( Δ P s ) = · ( p 3 Δ T ( r ) z ^ ) ,
f π w 2 λ th α P L ( d n / d T )
p ( y ) / p 3 = a 0 + m = 1 a m cos ( m K g y + ν m ) ,
2 ϕ = z ^ · [ p ( y ) Δ T ( r ) ] ϵ ϵ 0 ,
ϕ ( r ) = Φ 0 ( x , z ) + m = 1 Φ m ( x , z ) cos ( m K g y + ν m ) .
2 ϕ = p ( y ) ϵ ϵ 0 z ^ · [ Δ T ( r ) ] .
( t 2 m 2 K m 2 ) Φ m ( x , z ) = p 3 a m ϵ ϵ 0 z ^ · t [ Δ T ( r ) ] ,
( d 2 d ξ 2 m 2 K g 2 w 2 ) Φ m ( ξ ) = p 3 T 0 a m ϵ ϵ 0 w d Δ T ¯ ( ξ ) d ξ .
Φ m ( ξ ) = p 3 T 0 a m ϵ ϵ 0 1 m 2 K g 2 w 2 d Δ T ¯ ( ξ ) d ξ .
E z , m ( z ) = 1 w Φ m ( y , ξ ) ξ = p 3 α I 0 w 2 a m ϵ ϵ 0 λ th 1 m 2 K g 2 w 2 I ¯ ( ξ ) .
Δ n e , PPMgOLN = p 3 T 0 ϵ ϵ 0 n e 3 r 33 eff 2 [ a 0 + m = 1 m = a m 2 cos ( m K g y + ν m ) ] [ E z , 0 + m = 1 m = E z , m cos ( m K g y + ν m ) ] .
Δ n e , PPMgOLN p 3 T 0 ϵ ϵ 0 n e 3 r 33 eff 2 [ a 0 2 Δ T ¯ ( ξ ) + a 1 2 2 K g 2 w 2 cos 2 ( K g y ) I ¯ ( ξ ) ] .
Δ n e , PPMgOLN Δ n e , u 8 π 2 1 ( K g w ) 2 I ¯ ( ξ ) Δ T ¯ ( ξ ) .
a 0 2 = ( 2 D 1 ) 2 > 8 π 2 1 ( K g w ) 2 ,

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