Abstract

We present experimental and theoretical results of photorefractive one-dimensional self-focusing in an Fe-doped KNbO3 crystal. The self-focusing is investigated by Z-scan experiments with and without an applied electric field. We observe the strongest focusing effect for a ratio Iˆ/ID6 of the beam peak intensity to the so-called dark intensity (the intensity at which photoconductivity is equal to the conductivity in the dark). The good agreement between the presented theory and experiment in the Z-scan technique allows the precise determination of dark intensity, the photovoltaic field, and the electric field inside the crystal.

© 1998 Optical Society of America

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  1. A. Ashkin, G. D. Boyd, J. M. Dziedzic, R. G. Smith, A. A. Ballman, J. J. Levinstein, and K. Nassau, “Optically-induced refractive index inhomogeneities in LiNbO3 and LiTaO3,” Appl. Phys. Lett. 9, 72 (1966).
    [CrossRef]
  2. P. Günter and J.-P. Huignard, Photorefractive Materials and their Applications, Vol. 1 (Springer-Verlag, Berlin, 1988).
  3. M. Sheik-Bahae, A. A. Said, T. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760 (1990).
    [CrossRef]
  4. Q. W. Song, C.-P. Zhang, and P. J. Talbot, “Anisotropic light-induced scattering and ‘position dispersion’ in KNbO3:Fe crystal,” Opt. Commun. 98, 269 (1993).
    [CrossRef]
  5. Q. Song, C.-P. Zhang, and P. J. Talbot, “Self-defocusing, self-focusing and speckle in LiNbO3 and KNbO3:Fe crystals,” Appl. Opt. 32, 7266 (1993).
    [CrossRef] [PubMed]
  6. P. A. Marquez-Aguilar, J. J. Sanchez-Mondragon, S. Stepanov, and G. Bloch, “Z-scan experiments with cubic photorefractive crystal Bi12TiO20,” Opt. Commun. 118, 165 (1995).
    [CrossRef]
  7. N. Korneev, P. A. Marquez-Aguilar, J. J. Sanchez-Mondragon, S. Stepanov, M. Klein, and B. Wechsler, “Anisotropy of steady-state two-dimensional lenses in photorefractive crystals with drift nonlinearity,” J. Mod. Opt. 43, 311 (1995).
    [CrossRef]
  8. N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electro-optic crystals,” Ferroelectrics 22, 961 (1979).
    [CrossRef]
  9. 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, 1521 (1995).
    [CrossRef]
  10. A. Grunnet-Jepsen, I. Aubrecht, and L. Solymar, “Investigation of the internal field in photorefractive materials and measurement of the effective electro-optic coefficient,” J. Opt. Soc. Am. B 12, 921 (1995).
    [CrossRef]
  11. M. Zgonik, R. Schlesser, I. Biaggio, E. Voit, J. Tscherry, and P. Günter, “Materials constants of KNbO3 relevant for electro- and acousto-optics,” J. Appl. Phys. 74, 1287 (1993).
    [CrossRef]
  12. D. N. Christodoulides and M. I. Carvalho, “Bright, dark, and gray spatial soliton states in photorefractive media,” J. Opt. Soc. Am. B 12, 1628 (1995).
    [CrossRef]
  13. G. Montemezzani and P. Günter, “Profile of photorefractive one-dimensional bright spatial soliton,” Opt. Lett. 22, 451 (1997).
    [CrossRef] [PubMed]
  14. B. Zysset, I. Biaggio, and P. Günter, “Refractive indices of orthorhombic KNbO3. I. Dispersion and temperature dependence,” J. Opt. Soc. Am. B 9, 380 (1992).
    [CrossRef]
  15. M. Taya, M. C. Bashaw, M. M. Fejer, M. Segev, and G. C. Valley, “Observation of dark photovoltaic spatial solitons,” Phys. Rev. A 52, 3095 (1995).
    [CrossRef] [PubMed]
  16. D. Weaire, B. S. Wherrett, D. A. B. Miller, and S. D. Smith, “Effect of low-power nonlinear refraction on laser beam propagation in InSb,” Opt. Lett. 4, 331 (1974).
    [CrossRef]

1997 (1)

1995 (6)

M. Taya, M. C. Bashaw, M. M. Fejer, M. Segev, and G. C. Valley, “Observation of dark photovoltaic spatial solitons,” Phys. Rev. A 52, 3095 (1995).
[CrossRef] [PubMed]

D. N. Christodoulides and M. I. Carvalho, “Bright, dark, and gray spatial soliton states in photorefractive media,” J. Opt. Soc. Am. B 12, 1628 (1995).
[CrossRef]

P. A. Marquez-Aguilar, J. J. Sanchez-Mondragon, S. Stepanov, and G. Bloch, “Z-scan experiments with cubic photorefractive crystal Bi12TiO20,” Opt. Commun. 118, 165 (1995).
[CrossRef]

N. Korneev, P. A. Marquez-Aguilar, J. J. Sanchez-Mondragon, S. Stepanov, M. Klein, and B. Wechsler, “Anisotropy of steady-state two-dimensional lenses in photorefractive crystals with drift nonlinearity,” J. Mod. Opt. 43, 311 (1995).
[CrossRef]

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, 1521 (1995).
[CrossRef]

A. Grunnet-Jepsen, I. Aubrecht, and L. Solymar, “Investigation of the internal field in photorefractive materials and measurement of the effective electro-optic coefficient,” J. Opt. Soc. Am. B 12, 921 (1995).
[CrossRef]

1993 (3)

M. Zgonik, R. Schlesser, I. Biaggio, E. Voit, J. Tscherry, and P. Günter, “Materials constants of KNbO3 relevant for electro- and acousto-optics,” J. Appl. Phys. 74, 1287 (1993).
[CrossRef]

Q. W. Song, C.-P. Zhang, and P. J. Talbot, “Anisotropic light-induced scattering and ‘position dispersion’ in KNbO3:Fe crystal,” Opt. Commun. 98, 269 (1993).
[CrossRef]

Q. Song, C.-P. Zhang, and P. J. Talbot, “Self-defocusing, self-focusing and speckle in LiNbO3 and KNbO3:Fe crystals,” Appl. Opt. 32, 7266 (1993).
[CrossRef] [PubMed]

1992 (1)

1990 (1)

M. Sheik-Bahae, A. A. Said, T. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760 (1990).
[CrossRef]

1979 (1)

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electro-optic crystals,” Ferroelectrics 22, 961 (1979).
[CrossRef]

1974 (1)

1966 (1)

A. Ashkin, G. D. Boyd, J. M. Dziedzic, R. G. Smith, A. A. Ballman, J. J. Levinstein, and K. Nassau, “Optically-induced refractive index inhomogeneities in LiNbO3 and LiTaO3,” Appl. Phys. Lett. 9, 72 (1966).
[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, 1521 (1995).
[CrossRef]

Ashkin, A.

A. Ashkin, G. D. Boyd, J. M. Dziedzic, R. G. Smith, A. A. Ballman, J. J. Levinstein, and K. Nassau, “Optically-induced refractive index inhomogeneities in LiNbO3 and LiTaO3,” Appl. Phys. Lett. 9, 72 (1966).
[CrossRef]

Aubrecht, I.

Ballman, A. A.

A. Ashkin, G. D. Boyd, J. M. Dziedzic, R. G. Smith, A. A. Ballman, J. J. Levinstein, and K. Nassau, “Optically-induced refractive index inhomogeneities in LiNbO3 and LiTaO3,” Appl. Phys. Lett. 9, 72 (1966).
[CrossRef]

Bashaw, M. C.

M. Taya, M. C. Bashaw, M. M. Fejer, M. Segev, and G. C. Valley, “Observation of dark photovoltaic spatial solitons,” Phys. Rev. A 52, 3095 (1995).
[CrossRef] [PubMed]

Biaggio, I.

M. Zgonik, R. Schlesser, I. Biaggio, E. Voit, J. Tscherry, and P. Günter, “Materials constants of KNbO3 relevant for electro- and acousto-optics,” J. Appl. Phys. 74, 1287 (1993).
[CrossRef]

B. Zysset, I. Biaggio, and P. Günter, “Refractive indices of orthorhombic KNbO3. I. Dispersion and temperature dependence,” J. Opt. Soc. Am. B 9, 380 (1992).
[CrossRef]

Bloch, G.

P. A. Marquez-Aguilar, J. J. Sanchez-Mondragon, S. Stepanov, and G. Bloch, “Z-scan experiments with cubic photorefractive crystal Bi12TiO20,” Opt. Commun. 118, 165 (1995).
[CrossRef]

Boyd, G. D.

A. Ashkin, G. D. Boyd, J. M. Dziedzic, R. G. Smith, A. A. Ballman, J. J. Levinstein, and K. Nassau, “Optically-induced refractive index inhomogeneities in LiNbO3 and LiTaO3,” Appl. Phys. Lett. 9, 72 (1966).
[CrossRef]

Carvalho, M. I.

Christodoulides, D. N.

Dziedzic, J. M.

A. Ashkin, G. D. Boyd, J. M. Dziedzic, R. G. Smith, A. A. Ballman, J. J. Levinstein, and K. Nassau, “Optically-induced refractive index inhomogeneities in LiNbO3 and LiTaO3,” Appl. Phys. Lett. 9, 72 (1966).
[CrossRef]

Fejer, M. M.

M. Taya, M. C. Bashaw, M. M. Fejer, M. Segev, and G. C. Valley, “Observation of dark photovoltaic spatial solitons,” Phys. Rev. A 52, 3095 (1995).
[CrossRef] [PubMed]

Grunnet-Jepsen, A.

Günter, P.

Hagan, D. J.

M. Sheik-Bahae, A. A. Said, T. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760 (1990).
[CrossRef]

Klein, M.

N. Korneev, P. A. Marquez-Aguilar, J. J. Sanchez-Mondragon, S. Stepanov, M. Klein, and B. Wechsler, “Anisotropy of steady-state two-dimensional lenses in photorefractive crystals with drift nonlinearity,” J. Mod. Opt. 43, 311 (1995).
[CrossRef]

Korneev, N.

N. Korneev, P. A. Marquez-Aguilar, J. J. Sanchez-Mondragon, S. Stepanov, M. Klein, and B. Wechsler, “Anisotropy of steady-state two-dimensional lenses in photorefractive crystals with drift nonlinearity,” J. Mod. Opt. 43, 311 (1995).
[CrossRef]

Kukhtarev, N. V.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electro-optic crystals,” Ferroelectrics 22, 961 (1979).
[CrossRef]

Levinstein, J. J.

A. Ashkin, G. D. Boyd, J. M. Dziedzic, R. G. Smith, A. A. Ballman, J. J. Levinstein, and K. Nassau, “Optically-induced refractive index inhomogeneities in LiNbO3 and LiTaO3,” Appl. Phys. Lett. 9, 72 (1966).
[CrossRef]

Markov, V. B.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electro-optic crystals,” Ferroelectrics 22, 961 (1979).
[CrossRef]

Marquez-Aguilar, P. A.

N. Korneev, P. A. Marquez-Aguilar, J. J. Sanchez-Mondragon, S. Stepanov, M. Klein, and B. Wechsler, “Anisotropy of steady-state two-dimensional lenses in photorefractive crystals with drift nonlinearity,” J. Mod. Opt. 43, 311 (1995).
[CrossRef]

P. A. Marquez-Aguilar, J. J. Sanchez-Mondragon, S. Stepanov, and G. Bloch, “Z-scan experiments with cubic photorefractive crystal Bi12TiO20,” Opt. Commun. 118, 165 (1995).
[CrossRef]

Miller, D. A. B.

Montemezzani, G.

Nassau, K.

A. Ashkin, G. D. Boyd, J. M. Dziedzic, R. G. Smith, A. A. Ballman, J. J. Levinstein, and K. Nassau, “Optically-induced refractive index inhomogeneities in LiNbO3 and LiTaO3,” Appl. Phys. Lett. 9, 72 (1966).
[CrossRef]

Odulov, S. G.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electro-optic crystals,” Ferroelectrics 22, 961 (1979).
[CrossRef]

Said, A. A.

M. Sheik-Bahae, A. A. Said, T. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760 (1990).
[CrossRef]

Sanchez-Mondragon, J. J.

P. A. Marquez-Aguilar, J. J. Sanchez-Mondragon, S. Stepanov, and G. Bloch, “Z-scan experiments with cubic photorefractive crystal Bi12TiO20,” Opt. Commun. 118, 165 (1995).
[CrossRef]

N. Korneev, P. A. Marquez-Aguilar, J. J. Sanchez-Mondragon, S. Stepanov, M. Klein, and B. Wechsler, “Anisotropy of steady-state two-dimensional lenses in photorefractive crystals with drift nonlinearity,” J. Mod. Opt. 43, 311 (1995).
[CrossRef]

Schlesser, R.

M. Zgonik, R. Schlesser, I. Biaggio, E. Voit, J. Tscherry, and P. Günter, “Materials constants of KNbO3 relevant for electro- and acousto-optics,” J. Appl. Phys. 74, 1287 (1993).
[CrossRef]

Segev, M.

M. Taya, M. C. Bashaw, M. M. Fejer, M. Segev, and G. C. Valley, “Observation of dark photovoltaic spatial solitons,” Phys. Rev. A 52, 3095 (1995).
[CrossRef] [PubMed]

Sheik-Bahae, M.

M. Sheik-Bahae, A. A. Said, T. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760 (1990).
[CrossRef]

Smith, R. G.

A. Ashkin, G. D. Boyd, J. M. Dziedzic, R. G. Smith, A. A. Ballman, J. J. Levinstein, and K. Nassau, “Optically-induced refractive index inhomogeneities in LiNbO3 and LiTaO3,” Appl. Phys. Lett. 9, 72 (1966).
[CrossRef]

Smith, S. D.

Solymar, L.

Song, Q.

Song, Q. W.

Q. W. Song, C.-P. Zhang, and P. J. Talbot, “Anisotropic light-induced scattering and ‘position dispersion’ in KNbO3:Fe crystal,” Opt. Commun. 98, 269 (1993).
[CrossRef]

Soskin, M. S.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electro-optic crystals,” Ferroelectrics 22, 961 (1979).
[CrossRef]

Stepanov, S.

N. Korneev, P. A. Marquez-Aguilar, J. J. Sanchez-Mondragon, S. Stepanov, M. Klein, and B. Wechsler, “Anisotropy of steady-state two-dimensional lenses in photorefractive crystals with drift nonlinearity,” J. Mod. Opt. 43, 311 (1995).
[CrossRef]

P. A. Marquez-Aguilar, J. J. Sanchez-Mondragon, S. Stepanov, and G. Bloch, “Z-scan experiments with cubic photorefractive crystal Bi12TiO20,” Opt. Commun. 118, 165 (1995).
[CrossRef]

Talbot, P. J.

Q. W. Song, C.-P. Zhang, and P. J. Talbot, “Anisotropic light-induced scattering and ‘position dispersion’ in KNbO3:Fe crystal,” Opt. Commun. 98, 269 (1993).
[CrossRef]

Q. Song, C.-P. Zhang, and P. J. Talbot, “Self-defocusing, self-focusing and speckle in LiNbO3 and KNbO3:Fe crystals,” Appl. Opt. 32, 7266 (1993).
[CrossRef] [PubMed]

Taya, M.

M. Taya, M. C. Bashaw, M. M. Fejer, M. Segev, and G. C. Valley, “Observation of dark photovoltaic spatial solitons,” Phys. Rev. A 52, 3095 (1995).
[CrossRef] [PubMed]

Tscherry, J.

M. Zgonik, R. Schlesser, I. Biaggio, E. Voit, J. Tscherry, and P. Günter, “Materials constants of KNbO3 relevant for electro- and acousto-optics,” J. Appl. Phys. 74, 1287 (1993).
[CrossRef]

Valley, G. C.

M. Taya, M. C. Bashaw, M. M. Fejer, M. Segev, and G. C. Valley, “Observation of dark photovoltaic spatial solitons,” Phys. Rev. A 52, 3095 (1995).
[CrossRef] [PubMed]

Van Stryland, E. W.

M. Sheik-Bahae, A. A. Said, T. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760 (1990).
[CrossRef]

Vinetskii, V. L.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electro-optic crystals,” Ferroelectrics 22, 961 (1979).
[CrossRef]

Voit, E.

M. Zgonik, R. Schlesser, I. Biaggio, E. Voit, J. Tscherry, and P. Günter, “Materials constants of KNbO3 relevant for electro- and acousto-optics,” J. Appl. Phys. 74, 1287 (1993).
[CrossRef]

Weaire, D.

Wechsler, B.

N. Korneev, P. A. Marquez-Aguilar, J. J. Sanchez-Mondragon, S. Stepanov, M. Klein, and B. Wechsler, “Anisotropy of steady-state two-dimensional lenses in photorefractive crystals with drift nonlinearity,” J. Mod. Opt. 43, 311 (1995).
[CrossRef]

Wei, T.

M. Sheik-Bahae, A. A. Said, T. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760 (1990).
[CrossRef]

Wherrett, B. S.

Zgonik, M.

M. Zgonik, R. Schlesser, I. Biaggio, E. Voit, J. Tscherry, and P. Günter, “Materials constants of KNbO3 relevant for electro- and acousto-optics,” J. Appl. Phys. 74, 1287 (1993).
[CrossRef]

Zhang, C.-P.

Q. W. Song, C.-P. Zhang, and P. J. Talbot, “Anisotropic light-induced scattering and ‘position dispersion’ in KNbO3:Fe crystal,” Opt. Commun. 98, 269 (1993).
[CrossRef]

Q. Song, C.-P. Zhang, and P. J. Talbot, “Self-defocusing, self-focusing and speckle in LiNbO3 and KNbO3:Fe crystals,” Appl. Opt. 32, 7266 (1993).
[CrossRef] [PubMed]

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, 1521 (1995).
[CrossRef]

Zysset, B.

Appl. Opt. (1)

Appl. Phys. Lett. (1)

A. Ashkin, G. D. Boyd, J. M. Dziedzic, R. G. Smith, A. A. Ballman, J. J. Levinstein, and K. Nassau, “Optically-induced refractive index inhomogeneities in LiNbO3 and LiTaO3,” Appl. Phys. Lett. 9, 72 (1966).
[CrossRef]

Ferroelectrics (1)

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electro-optic crystals,” Ferroelectrics 22, 961 (1979).
[CrossRef]

IEEE J. Quantum Electron. (1)

M. Sheik-Bahae, A. A. Said, T. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760 (1990).
[CrossRef]

J. Appl. Phys. (1)

M. Zgonik, R. Schlesser, I. Biaggio, E. Voit, J. Tscherry, and P. Günter, “Materials constants of KNbO3 relevant for electro- and acousto-optics,” J. Appl. Phys. 74, 1287 (1993).
[CrossRef]

J. Mod. Opt. (1)

N. Korneev, P. A. Marquez-Aguilar, J. J. Sanchez-Mondragon, S. Stepanov, M. Klein, and B. Wechsler, “Anisotropy of steady-state two-dimensional lenses in photorefractive crystals with drift nonlinearity,” J. Mod. Opt. 43, 311 (1995).
[CrossRef]

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

Opt. Commun. (2)

Q. W. Song, C.-P. Zhang, and P. J. Talbot, “Anisotropic light-induced scattering and ‘position dispersion’ in KNbO3:Fe crystal,” Opt. Commun. 98, 269 (1993).
[CrossRef]

P. A. Marquez-Aguilar, J. J. Sanchez-Mondragon, S. Stepanov, and G. Bloch, “Z-scan experiments with cubic photorefractive crystal Bi12TiO20,” Opt. Commun. 118, 165 (1995).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. A (2)

M. Taya, M. C. Bashaw, M. M. Fejer, M. Segev, and G. C. Valley, “Observation of dark photovoltaic spatial solitons,” Phys. Rev. A 52, 3095 (1995).
[CrossRef] [PubMed]

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, 1521 (1995).
[CrossRef]

Other (1)

P. Günter and J.-P. Huignard, Photorefractive Materials and their Applications, Vol. 1 (Springer-Verlag, Berlin, 1988).

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Figures (8)

Fig. 1
Fig. 1

Numerical calculation of the Z-scan transmission T as a function of the normalized displacement of the sample relative to the beam waist z/z0 for different values of the beam-intensity-to-dark-intensity ratio Iˆ/ID and for a maximum phase shift ΔΦ0=1.

Fig. 2
Fig. 2

Numerical results for maximum difference in the Z-scan transmission ΔTmax/ΔΦ0 normalized to the maximum phase shift as a function of the beam-intensity-to-dark-intensity ratio Iˆ/ID. The curves are plotted for two different phase shifts, ΔΦ0=2 and ΔΦ0=0.1. As seen in the figure, ΔTmax is maximal for Iˆ/ID6 and scales with ΔΦ0 in almost the entire range of intensities as long as ΔΦ0<2.

Fig. 3
Fig. 3

Distance Δz/z0 between the maximum and the minimum of the Z-scan transmission T as a function of the beam-intensity-to-dark-intensity ratio Iˆ/ID. Results of numerical calculation are plotted for three different phase shifts, ΔΦ0=0.1, 1, and 2. As seen in the figure, the distance Δz/z0 is maximal for Iˆ/ID4.

Fig. 4
Fig. 4

Scheme of the photorefractive Z-scan experiment. The KNbO3 crystal is moved through the focus of the one-dimensional Gaussian beam with intensity I, produced by a cylindrical lens. The center of the beam is collected on a photodiode by the spherical lens after traversing the aperture. The crystal can be illuminated by an additional homogeneous light beam I0 with the beam splitter.

Fig. 5
Fig. 5

Z-scan transmission T measurement for a beam intensity Iˆ=10 W/cm2 with no external field (Eext=0) and without additional homogeneous illumination (I0=0). The measurements are denoted by points, and the solid curve is the theoretical fit with the photovoltaic field parameter Eph=1.3 kV/cm and the dark intensity ID=2 W/cm2.

Fig. 6
Fig. 6

Maximum difference ΔTmax of the Z-scan transmission measured for different beam intensities Iˆ, with no external field (Eext=0), and no additional illumination (I0=0). The measurements are denoted by points, and the solid curve is the theoretical fit with the photovoltaic field parameter Eph=1.4kV/cm and the dark intensity ID=2.5 W/cm2. The dashed curve is the theoretical fit with the same parameters as above; however, the saturation of the nonlinearity is neglected.

Fig. 7
Fig. 7

Maximum difference ΔTmax of the Z-scan transmission for different external fields Eext. A beam intensity Iˆ=10 W/cm2 and no additional illumination (I0=0) were used for these measurements, which are denoted by points. The solid curve is the theoretical fit with the photovoltaic field Eph=1.4 kV/cm and the screening factor η=0.4.

Fig. 8
Fig. 8

Z-scan transmission T for a beam intensity Iˆ=10 W/cm2 with an applied external field Eext=5 kV/cm. The measurements were done with a homogeneous illumination I0=25 mW/cm2 (crosses) and without it (triangles). The solid curve is a theoretical fit for Iˆ/ID=3.4, Eph=1.4 kV/cm, Eext=5 kV/cm, and η=0.64, and the dashed curve is plotted for Iˆ/ID=4, Eph=1.4 kV/cm, Eext=5 kV/cm, and η=0.4. The values for the ratios Iˆ/ID are calculated from previous experiments that used Eq. (4), and the screening reduction factors η are the result of a fit.

Equations (19)

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

ESC+ ln(I+ID)ESC
=-(E0-Eph) x ln(I+ID)+kBTe 2II+ID,
E0=ηVext/d,
Eph=γμ NALph,
ID=βs(λ)+s(λ0)s(λ) I0,
ESC=-(E0-Eph) II+ID+kBTe I/xI+ID.
ΔΦNL=ΔΦ0 I/ID1+I/ID,
ΔΦ0=-πlλ n3reff(E0-Eph),
ΔΦNL=ΔΦ0IID,
l/nz0,
ΔΦmaxkw02/l,
ψ(x, z)=ψ0w0w(z) exp-x2w2(z)-ikx22R(z)+δ(z),
δ(z)=-12 arctanww0,
w(z)=w01+(z/z0)2,
R(z)=z[1+(z0/z)2].
ψout(x)=ψ(x, zout)exp[iΔΦNL(x)],
ψ˜out(xa)=1iλR0 dxψout(x)exp(-ikxax/R0).
T=P(Δn)P(Δn=0)=aperturedxa|ψ˜out(xa)|2/P(0).
T=1+ΔΦmax2 1(1+x2)(9+x2)-ΔΦmax sign(x)×2(1+x2)(9+x2)-2x2-6(1+x2)(9+x2),

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