Abstract

Experimental and theoretical results on recording of spatial gratings with interfering femtosecond pulses at 388nm and time-resolved Bragg-matched readout at 776nm are presented for LiNbO3 crystals. These include dependences of the diffraction intensity on the time delay between probe and pump pulses, on the pump intensity and angle, and on the crystal composition. The grating buildup involves instantaneous and quasi-permanent changes of the refractive index and the absorption coefficient. These changes are due to Kerr and two-photon absorption effects and the modulation of photoexcited carriers, respectively. A good agreement between theory and experiment is achieved. Our analysis provides an understanding of nonlinear optical phenomena in LiNbO3 crystals on the femtosecond time scale. The theory is applicable to a large range of optical materials with modestly wide, 3to5eV, bandgap.

© 2007 Optical Society of America

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    [CrossRef]

2005 (5)

O. Beyer, D. Maxein, K. Buse, B. Sturman, H. T. Hsieh, and D. Psaltis, "Femtosecond time-resolved absorption processes in lithium niobate crystals," Opt. Lett. 30, 1366-1368 (2005).
[CrossRef] [PubMed]

O. Beyer, D. Maxein, K. Buse, B. Sturman, H. T. Hsieh, and D. Psaltis, "Investigation of nonlinear absorption processes with femtosecond light pulses in lithium niobate crystals," Phys. Rev. E 71, 056603 (2005).
[CrossRef]

H. T. Hsieh, D. Psaltis, O. Beyer, D. Maxein, C. von Korff Schmising, K. Buse, and B. Sturman, "Femtosecond holography in lithium niobate crystals," Opt. Lett. 30, 2233-2235 (2005).
[CrossRef] [PubMed]

Z. Wang, X. Zhang, J. Xu, Q. Wu, H. Qiao, B. Tang, R. Rupp, Y. Kong, S. Chen, Z. Huang, B. Li, S. Liu, and L. Zhang, "Time-resolved femtosecond degenerate four-wave mixing in LiNbO3:Fe,Mg crystal," Chin. Phys. Lett. 22, 2831-2833 (2005).
[CrossRef]

H. T. Hsieh, D. Psaltis, O. Beyer, D. Maxein, K. Buse, and B. Sturman, "Enhanced temporal resolution in femtosecond dynamic-grating experiments," J. Appl. Phys. 97, 113107 (2005).
[CrossRef]

2004 (1)

L. Arizmendi, "Photonic applications of lithium niobate crystals," Phys. Status Solidi A 201, 253-283 (2004).
[CrossRef]

2002 (1)

2000 (1)

D. Berben, K. Buse, S. Wevering, P. Herth, M. Imlau, and Th. Woike, "Lifetime of small polarons in iron-doped lithium-niobate crystals," J. Appl. Phys. 87, 1034-1041 (2000).
[CrossRef]

1998 (1)

A. Othonos, "Probing ultrafast carrier and phonon dynamics in semiconductors," J. Appl. Phys. 83, 1789-1830 (1998).
[CrossRef]

1997 (1)

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

1996 (1)

R. DeSalvo, A. A. Said, D. J. Hagan, E. W. Van Stryland, and M. Sheik-Bahae, "Infrared to ultraviolet measurements of two-photon absorption and n2 in wide bandgap solids," IEEE J. Quantum Electron. 32, 1324-1333 (1996).
[CrossRef]

1993 (1)

1992 (1)

M. M. Fejer, G. L. 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]

1989 (1)

1979 (1)

C. T. Chen, D. M. Kim, and D. von der Linde, "Efficient hologram recording in LiNbO3:Fe using optical pulses," Appl. Phys. Lett. 34, 321-324 (1979).
[CrossRef]

1974 (1)

D. von der Linde, A. M. Glass, and K. F. Rodgers, "Multiphoton photorefractive processes for optical storage in LiNbO3," Appl. Phys. Lett. 25, 155-157 (1974).
[CrossRef]

1969 (1)

H. Kogelnik, "Coupled wave theory for thick hologram gratings," Bell Syst. Tech. J. 48, 2909-2947 (1969).

Arizmendi, L.

L. Arizmendi, "Photonic applications of lithium niobate crystals," Phys. Status Solidi A 201, 253-283 (2004).
[CrossRef]

Berben, D.

D. Berben, K. Buse, S. Wevering, P. Herth, M. Imlau, and Th. Woike, "Lifetime of small polarons in iron-doped lithium-niobate crystals," J. Appl. Phys. 87, 1034-1041 (2000).
[CrossRef]

Beyer, O.

O. Beyer, D. Maxein, K. Buse, B. Sturman, H. T. Hsieh, and D. Psaltis, "Investigation of nonlinear absorption processes with femtosecond light pulses in lithium niobate crystals," Phys. Rev. E 71, 056603 (2005).
[CrossRef]

H. T. Hsieh, D. Psaltis, O. Beyer, D. Maxein, K. Buse, and B. Sturman, "Enhanced temporal resolution in femtosecond dynamic-grating experiments," J. Appl. Phys. 97, 113107 (2005).
[CrossRef]

H. T. Hsieh, D. Psaltis, O. Beyer, D. Maxein, C. von Korff Schmising, K. Buse, and B. Sturman, "Femtosecond holography in lithium niobate crystals," Opt. Lett. 30, 2233-2235 (2005).
[CrossRef] [PubMed]

O. Beyer, D. Maxein, K. Buse, B. Sturman, H. T. Hsieh, and D. Psaltis, "Femtosecond time-resolved absorption processes in lithium niobate crystals," Opt. Lett. 30, 1366-1368 (2005).
[CrossRef] [PubMed]

Boggers, T. F.

Bohnert, K. M.

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, 1992).

Buse, K.

H. T. Hsieh, D. Psaltis, O. Beyer, D. Maxein, K. Buse, and B. Sturman, "Enhanced temporal resolution in femtosecond dynamic-grating experiments," J. Appl. Phys. 97, 113107 (2005).
[CrossRef]

O. Beyer, D. Maxein, K. Buse, B. Sturman, H. T. Hsieh, and D. Psaltis, "Investigation of nonlinear absorption processes with femtosecond light pulses in lithium niobate crystals," Phys. Rev. E 71, 056603 (2005).
[CrossRef]

H. T. Hsieh, D. Psaltis, O. Beyer, D. Maxein, C. von Korff Schmising, K. Buse, and B. Sturman, "Femtosecond holography in lithium niobate crystals," Opt. Lett. 30, 2233-2235 (2005).
[CrossRef] [PubMed]

O. Beyer, D. Maxein, K. Buse, B. Sturman, H. T. Hsieh, and D. Psaltis, "Femtosecond time-resolved absorption processes in lithium niobate crystals," Opt. Lett. 30, 1366-1368 (2005).
[CrossRef] [PubMed]

D. Berben, K. Buse, S. Wevering, P. Herth, M. Imlau, and Th. Woike, "Lifetime of small polarons in iron-doped lithium-niobate crystals," J. Appl. Phys. 87, 1034-1041 (2000).
[CrossRef]

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

Byer, R. L.

M. M. Fejer, G. L. 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]

Chen, C. T.

C. T. Chen, D. M. Kim, and D. von der Linde, "Efficient hologram recording in LiNbO3:Fe using optical pulses," Appl. Phys. Lett. 34, 321-324 (1979).
[CrossRef]

Chen, S.

Z. Wang, X. Zhang, J. Xu, Q. Wu, H. Qiao, B. Tang, R. Rupp, Y. Kong, S. Chen, Z. Huang, B. Li, S. Liu, and L. Zhang, "Time-resolved femtosecond degenerate four-wave mixing in LiNbO3:Fe,Mg crystal," Chin. Phys. Lett. 22, 2831-2833 (2005).
[CrossRef]

DeSalvo, R.

R. DeSalvo, A. A. Said, D. J. Hagan, E. W. Van Stryland, and M. Sheik-Bahae, "Infrared to ultraviolet measurements of two-photon absorption and n2 in wide bandgap solids," IEEE J. Quantum Electron. 32, 1324-1333 (1996).
[CrossRef]

Dragonmir, A.

Fejer, M. M.

M. M. Fejer, G. L. 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]

Glass, A. M.

D. von der Linde, A. M. Glass, and K. F. Rodgers, "Multiphoton photorefractive processes for optical storage in LiNbO3," Appl. Phys. Lett. 25, 155-157 (1974).
[CrossRef]

Grunnet-Jepsen, A.

L. Solymar, D. J. Webb, and A. Grunnet-Jepsen, The Physics and Applications of Photorefractive Materials (Clarendon, 1996).

Hagan, D. J.

R. DeSalvo, A. A. Said, D. J. Hagan, E. W. Van Stryland, and M. Sheik-Bahae, "Infrared to ultraviolet measurements of two-photon absorption and n2 in wide bandgap solids," IEEE J. Quantum Electron. 32, 1324-1333 (1996).
[CrossRef]

Herth, P.

D. Berben, K. Buse, S. Wevering, P. Herth, M. Imlau, and Th. Woike, "Lifetime of small polarons in iron-doped lithium-niobate crystals," J. Appl. Phys. 87, 1034-1041 (2000).
[CrossRef]

Hsieh, H. T.

O. Beyer, D. Maxein, K. Buse, B. Sturman, H. T. Hsieh, and D. Psaltis, "Investigation of nonlinear absorption processes with femtosecond light pulses in lithium niobate crystals," Phys. Rev. E 71, 056603 (2005).
[CrossRef]

H. T. Hsieh, D. Psaltis, O. Beyer, D. Maxein, K. Buse, and B. Sturman, "Enhanced temporal resolution in femtosecond dynamic-grating experiments," J. Appl. Phys. 97, 113107 (2005).
[CrossRef]

O. Beyer, D. Maxein, K. Buse, B. Sturman, H. T. Hsieh, and D. Psaltis, "Femtosecond time-resolved absorption processes in lithium niobate crystals," Opt. Lett. 30, 1366-1368 (2005).
[CrossRef] [PubMed]

H. T. Hsieh, D. Psaltis, O. Beyer, D. Maxein, C. von Korff Schmising, K. Buse, and B. Sturman, "Femtosecond holography in lithium niobate crystals," Opt. Lett. 30, 2233-2235 (2005).
[CrossRef] [PubMed]

Huang, Z.

Z. Wang, X. Zhang, J. Xu, Q. Wu, H. Qiao, B. Tang, R. Rupp, Y. Kong, S. Chen, Z. Huang, B. Li, S. Liu, and L. Zhang, "Time-resolved femtosecond degenerate four-wave mixing in LiNbO3:Fe,Mg crystal," Chin. Phys. Lett. 22, 2831-2833 (2005).
[CrossRef]

Imlau, M.

D. Berben, K. Buse, S. Wevering, P. Herth, M. Imlau, and Th. Woike, "Lifetime of small polarons in iron-doped lithium-niobate crystals," J. Appl. Phys. 87, 1034-1041 (2000).
[CrossRef]

Jain, R. K.

R. K. Jain and M. B. Klein, "Degenerate four-wave mixing in semiconductors," in Optical Phase Conjugation, R. A. Fisher, ed. (Academic, 1983) Chap. 10.

Jermann, F.

Jundt, D. H.

M. M. Fejer, G. L. 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]

Kim, D. M.

C. T. Chen, D. M. Kim, and D. von der Linde, "Efficient hologram recording in LiNbO3:Fe using optical pulses," Appl. Phys. Lett. 34, 321-324 (1979).
[CrossRef]

Klein, M. B.

R. K. Jain and M. B. Klein, "Degenerate four-wave mixing in semiconductors," in Optical Phase Conjugation, R. A. Fisher, ed. (Academic, 1983) Chap. 10.

Kogelnik, H.

H. Kogelnik, "Coupled wave theory for thick hologram gratings," Bell Syst. Tech. J. 48, 2909-2947 (1969).

Kong, Y.

Z. Wang, X. Zhang, J. Xu, Q. Wu, H. Qiao, B. Tang, R. Rupp, Y. Kong, S. Chen, Z. Huang, B. Li, S. Liu, and L. Zhang, "Time-resolved femtosecond degenerate four-wave mixing in LiNbO3:Fe,Mg crystal," Chin. Phys. Lett. 22, 2831-2833 (2005).
[CrossRef]

Li, B.

Z. Wang, X. Zhang, J. Xu, Q. Wu, H. Qiao, B. Tang, R. Rupp, Y. Kong, S. Chen, Z. Huang, B. Li, S. Liu, and L. Zhang, "Time-resolved femtosecond degenerate four-wave mixing in LiNbO3:Fe,Mg crystal," Chin. Phys. Lett. 22, 2831-2833 (2005).
[CrossRef]

Liu, S.

Z. Wang, X. Zhang, J. Xu, Q. Wu, H. Qiao, B. Tang, R. Rupp, Y. Kong, S. Chen, Z. Huang, B. Li, S. Liu, and L. Zhang, "Time-resolved femtosecond degenerate four-wave mixing in LiNbO3:Fe,Mg crystal," Chin. Phys. Lett. 22, 2831-2833 (2005).
[CrossRef]

Magel, G. L.

M. M. Fejer, G. L. 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]

Maxein, D.

H. T. Hsieh, D. Psaltis, O. Beyer, D. Maxein, C. von Korff Schmising, K. Buse, and B. Sturman, "Femtosecond holography in lithium niobate crystals," Opt. Lett. 30, 2233-2235 (2005).
[CrossRef] [PubMed]

O. Beyer, D. Maxein, K. Buse, B. Sturman, H. T. Hsieh, and D. Psaltis, "Femtosecond time-resolved absorption processes in lithium niobate crystals," Opt. Lett. 30, 1366-1368 (2005).
[CrossRef] [PubMed]

O. Beyer, D. Maxein, K. Buse, B. Sturman, H. T. Hsieh, and D. Psaltis, "Investigation of nonlinear absorption processes with femtosecond light pulses in lithium niobate crystals," Phys. Rev. E 71, 056603 (2005).
[CrossRef]

H. T. Hsieh, D. Psaltis, O. Beyer, D. Maxein, K. Buse, and B. Sturman, "Enhanced temporal resolution in femtosecond dynamic-grating experiments," J. Appl. Phys. 97, 113107 (2005).
[CrossRef]

McInerney, J. G.

Mullen, R. A.

Nikogosyan, D. N.

Othonos, A.

A. Othonos, "Probing ultrafast carrier and phonon dynamics in semiconductors," J. Appl. Phys. 83, 1789-1830 (1998).
[CrossRef]

Otten, J.

Psaltis, D.

H. T. Hsieh, D. Psaltis, O. Beyer, D. Maxein, K. Buse, and B. Sturman, "Enhanced temporal resolution in femtosecond dynamic-grating experiments," J. Appl. Phys. 97, 113107 (2005).
[CrossRef]

O. Beyer, D. Maxein, K. Buse, B. Sturman, H. T. Hsieh, and D. Psaltis, "Investigation of nonlinear absorption processes with femtosecond light pulses in lithium niobate crystals," Phys. Rev. E 71, 056603 (2005).
[CrossRef]

H. T. Hsieh, D. Psaltis, O. Beyer, D. Maxein, C. von Korff Schmising, K. Buse, and B. Sturman, "Femtosecond holography in lithium niobate crystals," Opt. Lett. 30, 2233-2235 (2005).
[CrossRef] [PubMed]

O. Beyer, D. Maxein, K. Buse, B. Sturman, H. T. Hsieh, and D. Psaltis, "Femtosecond time-resolved absorption processes in lithium niobate crystals," Opt. Lett. 30, 1366-1368 (2005).
[CrossRef] [PubMed]

Qiao, H.

Z. Wang, X. Zhang, J. Xu, Q. Wu, H. Qiao, B. Tang, R. Rupp, Y. Kong, S. Chen, Z. Huang, B. Li, S. Liu, and L. Zhang, "Time-resolved femtosecond degenerate four-wave mixing in LiNbO3:Fe,Mg crystal," Chin. Phys. Lett. 22, 2831-2833 (2005).
[CrossRef]

Rodgers, K. F.

D. von der Linde, A. M. Glass, and K. F. Rodgers, "Multiphoton photorefractive processes for optical storage in LiNbO3," Appl. Phys. Lett. 25, 155-157 (1974).
[CrossRef]

Rupp, R.

Z. Wang, X. Zhang, J. Xu, Q. Wu, H. Qiao, B. Tang, R. Rupp, Y. Kong, S. Chen, Z. Huang, B. Li, S. Liu, and L. Zhang, "Time-resolved femtosecond degenerate four-wave mixing in LiNbO3:Fe,Mg crystal," Chin. Phys. Lett. 22, 2831-2833 (2005).
[CrossRef]

Said, A. A.

R. DeSalvo, A. A. Said, D. J. Hagan, E. W. Van Stryland, and M. Sheik-Bahae, "Infrared to ultraviolet measurements of two-photon absorption and n2 in wide bandgap solids," IEEE J. Quantum Electron. 32, 1324-1333 (1996).
[CrossRef]

Sheik-Bahae, M.

R. DeSalvo, A. A. Said, D. J. Hagan, E. W. Van Stryland, and M. Sheik-Bahae, "Infrared to ultraviolet measurements of two-photon absorption and n2 in wide bandgap solids," IEEE J. Quantum Electron. 32, 1324-1333 (1996).
[CrossRef]

Smirl, A. L.

Solymar, L.

L. Solymar, D. J. Webb, and A. Grunnet-Jepsen, The Physics and Applications of Photorefractive Materials (Clarendon, 1996).

Sturman, B.

O. Beyer, D. Maxein, K. Buse, B. Sturman, H. T. Hsieh, and D. Psaltis, "Investigation of nonlinear absorption processes with femtosecond light pulses in lithium niobate crystals," Phys. Rev. E 71, 056603 (2005).
[CrossRef]

H. T. Hsieh, D. Psaltis, O. Beyer, D. Maxein, K. Buse, and B. Sturman, "Enhanced temporal resolution in femtosecond dynamic-grating experiments," J. Appl. Phys. 97, 113107 (2005).
[CrossRef]

O. Beyer, D. Maxein, K. Buse, B. Sturman, H. T. Hsieh, and D. Psaltis, "Femtosecond time-resolved absorption processes in lithium niobate crystals," Opt. Lett. 30, 1366-1368 (2005).
[CrossRef] [PubMed]

H. T. Hsieh, D. Psaltis, O. Beyer, D. Maxein, C. von Korff Schmising, K. Buse, and B. Sturman, "Femtosecond holography in lithium niobate crystals," Opt. Lett. 30, 2233-2235 (2005).
[CrossRef] [PubMed]

Tang, B.

Z. Wang, X. Zhang, J. Xu, Q. Wu, H. Qiao, B. Tang, R. Rupp, Y. Kong, S. Chen, Z. Huang, B. Li, S. Liu, and L. Zhang, "Time-resolved femtosecond degenerate four-wave mixing in LiNbO3:Fe,Mg crystal," Chin. Phys. Lett. 22, 2831-2833 (2005).
[CrossRef]

Valley, G.

Van Stryland, E. W.

R. DeSalvo, A. A. Said, D. J. Hagan, E. W. Van Stryland, and M. Sheik-Bahae, "Infrared to ultraviolet measurements of two-photon absorption and n2 in wide bandgap solids," IEEE J. Quantum Electron. 32, 1324-1333 (1996).
[CrossRef]

von der Linde, D.

C. T. Chen, D. M. Kim, and D. von der Linde, "Efficient hologram recording in LiNbO3:Fe using optical pulses," Appl. Phys. Lett. 34, 321-324 (1979).
[CrossRef]

D. von der Linde, A. M. Glass, and K. F. Rodgers, "Multiphoton photorefractive processes for optical storage in LiNbO3," Appl. Phys. Lett. 25, 155-157 (1974).
[CrossRef]

von Korff Schmising, C.

Wang, Z.

Z. Wang, X. Zhang, J. Xu, Q. Wu, H. Qiao, B. Tang, R. Rupp, Y. Kong, S. Chen, Z. Huang, B. Li, S. Liu, and L. Zhang, "Time-resolved femtosecond degenerate four-wave mixing in LiNbO3:Fe,Mg crystal," Chin. Phys. Lett. 22, 2831-2833 (2005).
[CrossRef]

Webb, D. J.

L. Solymar, D. J. Webb, and A. Grunnet-Jepsen, The Physics and Applications of Photorefractive Materials (Clarendon, 1996).

Wevering, S.

D. Berben, K. Buse, S. Wevering, P. Herth, M. Imlau, and Th. Woike, "Lifetime of small polarons in iron-doped lithium-niobate crystals," J. Appl. Phys. 87, 1034-1041 (2000).
[CrossRef]

Woike, Th.

D. Berben, K. Buse, S. Wevering, P. Herth, M. Imlau, and Th. Woike, "Lifetime of small polarons in iron-doped lithium-niobate crystals," J. Appl. Phys. 87, 1034-1041 (2000).
[CrossRef]

Wu, Q.

Z. Wang, X. Zhang, J. Xu, Q. Wu, H. Qiao, B. Tang, R. Rupp, Y. Kong, S. Chen, Z. Huang, B. Li, S. Liu, and L. Zhang, "Time-resolved femtosecond degenerate four-wave mixing in LiNbO3:Fe,Mg crystal," Chin. Phys. Lett. 22, 2831-2833 (2005).
[CrossRef]

Xu, J.

Z. Wang, X. Zhang, J. Xu, Q. Wu, H. Qiao, B. Tang, R. Rupp, Y. Kong, S. Chen, Z. Huang, B. Li, S. Liu, and L. Zhang, "Time-resolved femtosecond degenerate four-wave mixing in LiNbO3:Fe,Mg crystal," Chin. Phys. Lett. 22, 2831-2833 (2005).
[CrossRef]

Zhang, L.

Z. Wang, X. Zhang, J. Xu, Q. Wu, H. Qiao, B. Tang, R. Rupp, Y. Kong, S. Chen, Z. Huang, B. Li, S. Liu, and L. Zhang, "Time-resolved femtosecond degenerate four-wave mixing in LiNbO3:Fe,Mg crystal," Chin. Phys. Lett. 22, 2831-2833 (2005).
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Schematic diagram of a grating recording–diffraction pulse experiment: λ p and λ r are the wavelengths of pump and readout (probe) pulses, M are dichroic mirrors, L is a long-focus ( 500 mm ) lens placed at 400 mm in front of the sample, D trans and D diff are photodetectors, DS is a probe delay stage, and BC is the Berek compensator. The grating vector K is parallel to the x axis and perpendicular to the light fringes depicted inside the crystal. The polarization of the pump pulses and the polar c axis are parallel to the y axis of the laboratory coordinate system.

Fig. 2
Fig. 2

Dependences η ( Δ t ) and η ( Δ t ) on the picosecond scale for the thin sample, I p 0 270 GW cm 2 , and θ p out = 4 ° . The solid curve is a theoretical prediction for the instantaneous response; see Sections 4, 5.

Fig. 3
Fig. 3

Dependences η ( Δ t ) and η ( Δ t ) for the thick ( d = 1 mm ) sample; the other experimental parameters are given in the caption of Fig. 2. The left (before the break at Δ t 2 ps ) and right parts of the figure refer to the linear picosecond and logarithmic picosecond-to-nanosecond time scales, respectively. The solid curve is a theoretical prediction for thin samples; see Sections 4, 5.

Fig. 4
Fig. 4

Dependence η ( Δ t ) for three differently doped samples of approximately the same thickness ( d 1 mm ) , I p 0 260 GW cm 2 , and orthogonal polarizations of the pump and probe pulses.

Fig. 5
Fig. 5

Broadening factor, i.e., the ratio of the 1 e width of the peak of η ( Δ t ) to the 1 e width of the pump pulse versus the pump half-angle for the thin sample; squares are experimental data, and the solid curve is the theoretical prediction; see Section 4.

Fig. 6
Fig. 6

Intensity dependences of the peak and plateau values of η for the thin sample, θ p out = 4 ° , and identically polarized pump and probe beams. The curves represent power laws with the exponents κ.

Fig. 7
Fig. 7

(a) Wave-vector diagram for recording and readout. (b) Propagation and overlap of pump and readout pulses of a square-wave shape. The symbols used are introduced in the text.

Fig. 8
Fig. 8

Dependence f G ( θ p ) for D r D p = 6 ; curves 1–4 are plotted for D p v W = 5 , 10, 20, and 40.

Fig. 9
Fig. 9

Dependence F ( 2 Δ t W , 2 d δ v W ) on Δ t W for several values of 2 d δ v W ; curves 1 to 5 are plotted for 2 d δ v W = 0.1 , 2, 4, 6, and 8, respectively.

Fig. 10
Fig. 10

Height F max and normalized 1 e width Δ t 1 e W of the peak of F ( Δ t ) as functions of the dimensionless parameter 2 d δ v W .

Equations (31)

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E p = exp ( i ω p t ) [ A 1 exp ( i k 1 r ) + A 2 exp ( i k 2 r ) ] + c.c. ,
E r = exp ( i ω r t ) [ A 3 exp ( i k 3 r ) + A 4 exp ( i k 4 r ) ] + c.c. ,
2 E p , r ϵ p , r c 2 2 E p , r t 2 = 0 ,
ϵ = n 2 + 2 n ( Δ n + i λ 4 π Δ α ) ,
Δ n = n K exp ( i K r ) + c.c. , Δ α = α K exp ( i K r ) + c.c. ,
( t + v 3 ) A 3 = 2 π c λ r n r ( i n K r λ r 4 π α K r ) A 4 ,
( t + v 4 ) A 4 = 2 π c λ r n r ( i n K r λ r 4 π α K r ) A 3 ,
( t + v i ) A i = 0 .
z ̂ i = x sin θ i + z cos θ i ,
x ̂ i = x cos θ i z sin θ i ,
t ̂ i = t ( x sin θ i + z cos θ i ) v i ,
A i = A i 0 exp ( S i ) , S i = 2 ( x ̂ i 2 + y 2 ) D i 2 + 2 ( t ̂ i t i 0 ) 2 W i 2 ,
Δ n = n 2 I p , Δ α = β I p ,
Δ n r t = f r β p I p 2 ω p , Δ α r t = σ r β p I p 2 ω p ,
A 4 z ̂ = 2 π λ r ( i n K r λ r 4 π α K r ) A 3 .
A 4 ( z = d ) = 2 π λ r z ̂ 0 z ̂ d ( i n K r λ r 4 π α K r ) A 3 d z ̂ .
A 4 ( z = d ) 2 d x ̂ d t ̂ d y ,
A 3 2 d x ̂ 3 d t ̂ 3 d y = π 3 2 W D r 2 A 3 0 2 8 .
S 1 + S 2 + S 3 2 x ̂ 2 D 2 ( 1 + 2 q 2 ) + 2 y 2 D 2 + 4 ( t ̂ W 2 q x ̂ D + z ̂ δ v W ) 2 + 2 ( t ̂ W Δ t W 4 q x ̂ D ) 2 ,
1 D 2 = 1 D r 2 + 2 D p 2 , q = θ p D v W .
η = η 0 f G exp [ ( 2 Δ t W ) 2 ] ,
η 0 = ( 2 π d λ r ) 2 I 1 0 I 2 0 [ n 2 r 2 + ( β r λ r 4 π ) 2 ] ,
f G = D p 2 D p 2 + 2 D r 2 1 3 + 14 q 2 ,
W = W 3 + 14 q 2 2 + 4 q 2 .
F ( a , b ) = 3 π 4 b 2 exp [ ( u a ) 2 ] [ erf ( u b ) erf ( u ) ] 2 d u .
( I p 2 ) K = 2 A 1 0 A 2 0 * [ I 1 0 exp ( 3 S 1 S 2 ) + I 2 0 exp ( S 1 3 S 2 ) ] .
n K r f r = α K r σ r = π β p W I p 0 A 1 0 A 2 0 * 2 ω p exp [ ( 2 x ̂ 2 ) ( 4 D p 2 + 3 θ p 2 v 2 W 2 ) 8 y 2 D p 2 ] .
η plat = η 0 f G ,
η 0 = π 2 ( 2 π d λ r ) 2 W 2 I 1 0 I 2 0 ( β p I p 0 ω p ) 2 [ f r 2 + ( σ r λ r 4 π ) 2 ] ,
f G = ( 1 + 4 D r 2 D p 2 ) 1 [ 1 + 3 θ p 2 D p 2 D r 2 v 2 W 2 ( D p 2 + 4 D r 2 ) ] 1 2 .
n 2 r 2 + ( β r λ r 4 π ) 2 1.5 × 10 10 cm 4 GW 2 .

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