Abstract

We measured the unclamped electro-optic coefficients r13 and r33 of near-stoichiometric and congruent lithium tantalate (LiTaO3) from λ=633nm down to the absorption edge at 275nm. Electro-optical coefficients up to r33=52±1pmV have been determined at λ=275nm. The reduced half-wave voltage at 275nm is vπ=300±10V for r33 and vπ=560±20V for the configuration employing rc=r33(none)3r13. We propose a two-oscillator polarization potential model that well describes our data in the UV. In the investigated wavelength range the values for stoichiometric and congruent LiTaO3 differ by less than 3%, which is below the accuracy of our measurements. In addition, the refractive indices in the UV were measured, and more precise Sellmeier parameters for the UV and visible were determined based on the previous and new refractive-index data.

© 2006 Optical Society of America

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  1. Y. Furukawa, K. Kitamura, K. Niwa, H. Hatano, P. Bernasconi, G. Montemezzani, and P. Günter, "Stoichiometric LiTaO3 for dynamic holography in near UV wavelength range," Jpn. J. Appl. Phys. Part 1 38, 1816-1819 (1999).
    [CrossRef]
  2. K. Kitamura, Y. Furukawa, Y. Ji, M. Zgonik, C. Medrano, G. Montemezzani, and P. Günter, "Photorefractive effect in LiNbO3 crystals enhanced by stoichiometry control," J. Appl. Phys. 82, 1006-1009 (1997).
    [CrossRef]
  3. P. Dittrich, B. Koziarska-Glinka, G. Montemezzani, P. Günter, S. Takekawa, K. Kitamura, and Y. Furukawa, "Deep-ultraviolet interband photorefraction in lithium tantalate," J. Opt. Soc. Am. B 21, 632-639 (2004).
    [CrossRef]
  4. M. Jazbinsek, M. Zgonik, S. Takekawa, M. Nakamura, K. Kitamura, and H. Hatano, "Reduced space-charge fields in near-stoichiometric LiTaO3 for blue, violet and near-ultraviolet light beams," Appl. Phys. B 75, 891-894 (2002).
    [CrossRef]
  5. A. Ashkin, G. Boyd, J. Dziedzic, R. Smith, A. Ballman, J. Levinstein, and K. Nassau, "Optically-induced refractive index inhomogeneities in LiNbO3 and LiTaO3," Appl. Phys. Lett. 9, 72-74 (1966).
    [CrossRef]
  6. K. Kitamura, Y. Furukawa, K. Niwa, V. Gopalan, and T. E. Mitchell, "Crystal growth and low coercive field 180 degrees domain switching characteristics of stoichiometric LiTaO3," Appl. Phys. Lett. 71, 3073-3075 (1998).
    [CrossRef]
  7. Y. W. Liu, K. Kitamura, S. Takekawa, M. Nakamura, Y. Furukawa, and H. Hatano, "Two-color photorefractive properties in near-stoichiometric lithium tantalate crystals," J. Appl. Phys. 95, 7637-7644 (2004).
    [CrossRef]
  8. K. Bastwöste, S. Schwalenberg, C. Bäumer, and E. Kratzig, "Temperature and composition dependence of birefringence of lithium-tantalate crystals determined by holographic scattering," Phys. Status Solidi A 199, R1-R3 (2003).
    [CrossRef]
  9. P. Dittrich, G. Montemezzani, and P. Günter, "Tunable optical filter for wavelength division multiplexing using dynamic interband photorefractive gratings," Opt. Commun. 214, 363-370 (2002).
    [CrossRef]
  10. H. Coufal, D. Pasaltis, and G. Sincerbox, Holographic Data Storage (Springer, 2000).
  11. P. Bernasconi, G. Montemezzani, M. Wintermantel, I. Biaggio, and P. Günter, "High-resolution, high-speed photorefractive incoherent-to-coherent optical converter," Opt. Lett. 24, 199-201 (1999).
    [CrossRef]
  12. P. Dittrich, G. Montemezzani, P. Bernasconi, and P. Günter, "Fast, reconfigurable light-induced waveguides," Opt. Lett. 24, 1508-1510 (1999).
    [CrossRef]
  13. J. Meyn and M. Fejer, "Tunable ultraviolet radiation by second-harmonic generation in periodically poled lithium tantalate," Opt. Lett. 22, 1214-1216 (1997).
    [CrossRef] [PubMed]
  14. P. Lenzo, E. Turner, E. Spencer, and A. Ballman, "Electrooptic coefficients and elastic-wave propagation in single-domain ferroelectric lithium tantalate," Appl. Phys. Lett. 8, 81-82 (1966).
    [CrossRef]
  15. J. Casson, K. Gahagan, D. Scrymgeour, R. Jain, J. Robinson, V. Gopalan, and R. Sander, "Electro-optic coefficients of lithium tantalate at near-infrared wavelengths," J. Opt. Soc. Am. B 21, 1948-1952 (2004).
    [CrossRef]
  16. K. S. Abedin and H. Ito, "Temperature-dependent dispersion relation of ferroelectric lithium tantalate," J. Appl. Phys. 80, 6561-6563 (1996).
    [CrossRef]
  17. M. Nakamura, S. Higuchi, S. Takekawa, K. Terabe, Y. Furukawa, and K. Kitamura, "Refractive indices in undoped and MgO-doped near-stoichiometric LiTaO3 crystals," Jpn. J. Appl. Phys., Part 1 41, L465-L467 (2002).
    [CrossRef]
  18. S. H. Wemple and M. DiDomenico, Jr., "Electrooptical and nonlinear optical properties of crystals," in Applied Solid State Science, R.Wolfe, ed. (Academic, 1972), Vol. 3, pp. 263-383.
  19. T. Fukuda, S. Matsumura, H. Hirano, and T. Ito, "Growth of LiTaO3 single-crystal for saw device applications," J. Cryst. Growth 46, 179-184 (1979).
    [CrossRef]
  20. Y. Furukawa, K. Kitamura, E. Suzuki, and K. Niwa, "Stoichiometric LiTaO3 single crystal growth by double crucible Czochralski method using automatic powder supply system," J. Cryst. Growth 197, 889-895 (1999).
    [CrossRef]
  21. C. Bäumer, C. David, A. Tunyagi, K. Betzler, H. Hesse, E. Kratzig, and M. Wohlecke, "Composition dependence of the ultraviolet absorption edge in lithium tantalate," J. Appl. Phys. 93, 3102-3104 (2003).
    [CrossRef]
  22. M. S. Shumate, "Interferometric measurements of large indices of refraction," Appl. Opt. 5, 327-331 (1966).
    [CrossRef] [PubMed]
  23. M. DiDomenico, Jr. and S. H. Wemple, "Oxygen-octahedra ferroelectrics. I. Theory of electro-optical and nonlinear optical effects," J. Appl. Phys. 40, 720-734 (1969).
    [CrossRef]
  24. M. Aillerie, N. Théofanous, and M. D. Fontana, "Measurement of the electrooptic coefficients: description and comparison of the experimental techniques," Appl. Phys. B 70, 317-334 (2000).
    [CrossRef]
  25. C. Bosshard, K. Sutter, R. Schlesser, and P. Günter, "Electro-optic effects in molecular crystals," J. Opt. Soc. Am. B 10, 867-885 (1993).
    [CrossRef]
  26. D. Haertle, G. Caimi, A. Haldi, G. Montemezzani, P. Günter, A. A. Grabar, I. M. Stoika, and Y. M. Vysochanskii, "Electro-optical properties of Sn2P2S6," Opt. Commun. 251, 333-343 (2003).
    [CrossRef]
  27. R. T. Smith and F. S. Welsh, "Temperature dependence of the elastic, piezoelectric, and dielectric constants of lithium tantalate and lithium niobate," J. Appl. Phys. 42, 2219-2230 (1971).
    [CrossRef]

2004

2003

K. Bastwöste, S. Schwalenberg, C. Bäumer, and E. Kratzig, "Temperature and composition dependence of birefringence of lithium-tantalate crystals determined by holographic scattering," Phys. Status Solidi A 199, R1-R3 (2003).
[CrossRef]

C. Bäumer, C. David, A. Tunyagi, K. Betzler, H. Hesse, E. Kratzig, and M. Wohlecke, "Composition dependence of the ultraviolet absorption edge in lithium tantalate," J. Appl. Phys. 93, 3102-3104 (2003).
[CrossRef]

D. Haertle, G. Caimi, A. Haldi, G. Montemezzani, P. Günter, A. A. Grabar, I. M. Stoika, and Y. M. Vysochanskii, "Electro-optical properties of Sn2P2S6," Opt. Commun. 251, 333-343 (2003).
[CrossRef]

2002

M. Nakamura, S. Higuchi, S. Takekawa, K. Terabe, Y. Furukawa, and K. Kitamura, "Refractive indices in undoped and MgO-doped near-stoichiometric LiTaO3 crystals," Jpn. J. Appl. Phys., Part 1 41, L465-L467 (2002).
[CrossRef]

P. Dittrich, G. Montemezzani, and P. Günter, "Tunable optical filter for wavelength division multiplexing using dynamic interband photorefractive gratings," Opt. Commun. 214, 363-370 (2002).
[CrossRef]

M. Jazbinsek, M. Zgonik, S. Takekawa, M. Nakamura, K. Kitamura, and H. Hatano, "Reduced space-charge fields in near-stoichiometric LiTaO3 for blue, violet and near-ultraviolet light beams," Appl. Phys. B 75, 891-894 (2002).
[CrossRef]

2000

M. Aillerie, N. Théofanous, and M. D. Fontana, "Measurement of the electrooptic coefficients: description and comparison of the experimental techniques," Appl. Phys. B 70, 317-334 (2000).
[CrossRef]

1999

Y. Furukawa, K. Kitamura, E. Suzuki, and K. Niwa, "Stoichiometric LiTaO3 single crystal growth by double crucible Czochralski method using automatic powder supply system," J. Cryst. Growth 197, 889-895 (1999).
[CrossRef]

Y. Furukawa, K. Kitamura, K. Niwa, H. Hatano, P. Bernasconi, G. Montemezzani, and P. Günter, "Stoichiometric LiTaO3 for dynamic holography in near UV wavelength range," Jpn. J. Appl. Phys. Part 1 38, 1816-1819 (1999).
[CrossRef]

P. Bernasconi, G. Montemezzani, M. Wintermantel, I. Biaggio, and P. Günter, "High-resolution, high-speed photorefractive incoherent-to-coherent optical converter," Opt. Lett. 24, 199-201 (1999).
[CrossRef]

P. Dittrich, G. Montemezzani, P. Bernasconi, and P. Günter, "Fast, reconfigurable light-induced waveguides," Opt. Lett. 24, 1508-1510 (1999).
[CrossRef]

1998

K. Kitamura, Y. Furukawa, K. Niwa, V. Gopalan, and T. E. Mitchell, "Crystal growth and low coercive field 180 degrees domain switching characteristics of stoichiometric LiTaO3," Appl. Phys. Lett. 71, 3073-3075 (1998).
[CrossRef]

1997

K. Kitamura, Y. Furukawa, Y. Ji, M. Zgonik, C. Medrano, G. Montemezzani, and P. Günter, "Photorefractive effect in LiNbO3 crystals enhanced by stoichiometry control," J. Appl. Phys. 82, 1006-1009 (1997).
[CrossRef]

J. Meyn and M. Fejer, "Tunable ultraviolet radiation by second-harmonic generation in periodically poled lithium tantalate," Opt. Lett. 22, 1214-1216 (1997).
[CrossRef] [PubMed]

1996

K. S. Abedin and H. Ito, "Temperature-dependent dispersion relation of ferroelectric lithium tantalate," J. Appl. Phys. 80, 6561-6563 (1996).
[CrossRef]

1993

1979

T. Fukuda, S. Matsumura, H. Hirano, and T. Ito, "Growth of LiTaO3 single-crystal for saw device applications," J. Cryst. Growth 46, 179-184 (1979).
[CrossRef]

1971

R. T. Smith and F. S. Welsh, "Temperature dependence of the elastic, piezoelectric, and dielectric constants of lithium tantalate and lithium niobate," J. Appl. Phys. 42, 2219-2230 (1971).
[CrossRef]

1969

M. DiDomenico, Jr. and S. H. Wemple, "Oxygen-octahedra ferroelectrics. I. Theory of electro-optical and nonlinear optical effects," J. Appl. Phys. 40, 720-734 (1969).
[CrossRef]

1966

M. S. Shumate, "Interferometric measurements of large indices of refraction," Appl. Opt. 5, 327-331 (1966).
[CrossRef] [PubMed]

A. Ashkin, G. Boyd, J. Dziedzic, R. Smith, A. Ballman, J. Levinstein, and K. Nassau, "Optically-induced refractive index inhomogeneities in LiNbO3 and LiTaO3," Appl. Phys. Lett. 9, 72-74 (1966).
[CrossRef]

P. Lenzo, E. Turner, E. Spencer, and A. Ballman, "Electrooptic coefficients and elastic-wave propagation in single-domain ferroelectric lithium tantalate," Appl. Phys. Lett. 8, 81-82 (1966).
[CrossRef]

Abedin, K. S.

K. S. Abedin and H. Ito, "Temperature-dependent dispersion relation of ferroelectric lithium tantalate," J. Appl. Phys. 80, 6561-6563 (1996).
[CrossRef]

Aillerie, M.

M. Aillerie, N. Théofanous, and M. D. Fontana, "Measurement of the electrooptic coefficients: description and comparison of the experimental techniques," Appl. Phys. B 70, 317-334 (2000).
[CrossRef]

Ashkin, A.

A. Ashkin, G. Boyd, J. Dziedzic, R. Smith, A. Ballman, J. Levinstein, and K. Nassau, "Optically-induced refractive index inhomogeneities in LiNbO3 and LiTaO3," Appl. Phys. Lett. 9, 72-74 (1966).
[CrossRef]

Ballman, A.

A. Ashkin, G. Boyd, J. Dziedzic, R. Smith, A. Ballman, J. Levinstein, and K. Nassau, "Optically-induced refractive index inhomogeneities in LiNbO3 and LiTaO3," Appl. Phys. Lett. 9, 72-74 (1966).
[CrossRef]

P. Lenzo, E. Turner, E. Spencer, and A. Ballman, "Electrooptic coefficients and elastic-wave propagation in single-domain ferroelectric lithium tantalate," Appl. Phys. Lett. 8, 81-82 (1966).
[CrossRef]

Bastwöste, K.

K. Bastwöste, S. Schwalenberg, C. Bäumer, and E. Kratzig, "Temperature and composition dependence of birefringence of lithium-tantalate crystals determined by holographic scattering," Phys. Status Solidi A 199, R1-R3 (2003).
[CrossRef]

Bäumer, C.

K. Bastwöste, S. Schwalenberg, C. Bäumer, and E. Kratzig, "Temperature and composition dependence of birefringence of lithium-tantalate crystals determined by holographic scattering," Phys. Status Solidi A 199, R1-R3 (2003).
[CrossRef]

C. Bäumer, C. David, A. Tunyagi, K. Betzler, H. Hesse, E. Kratzig, and M. Wohlecke, "Composition dependence of the ultraviolet absorption edge in lithium tantalate," J. Appl. Phys. 93, 3102-3104 (2003).
[CrossRef]

Bernasconi, P.

Betzler, K.

C. Bäumer, C. David, A. Tunyagi, K. Betzler, H. Hesse, E. Kratzig, and M. Wohlecke, "Composition dependence of the ultraviolet absorption edge in lithium tantalate," J. Appl. Phys. 93, 3102-3104 (2003).
[CrossRef]

Biaggio, I.

Bosshard, C.

Boyd, G.

A. Ashkin, G. Boyd, J. Dziedzic, R. Smith, A. Ballman, J. Levinstein, and K. Nassau, "Optically-induced refractive index inhomogeneities in LiNbO3 and LiTaO3," Appl. Phys. Lett. 9, 72-74 (1966).
[CrossRef]

Caimi, G.

D. Haertle, G. Caimi, A. Haldi, G. Montemezzani, P. Günter, A. A. Grabar, I. M. Stoika, and Y. M. Vysochanskii, "Electro-optical properties of Sn2P2S6," Opt. Commun. 251, 333-343 (2003).
[CrossRef]

Casson, J.

Coufal, H.

H. Coufal, D. Pasaltis, and G. Sincerbox, Holographic Data Storage (Springer, 2000).

David, C.

C. Bäumer, C. David, A. Tunyagi, K. Betzler, H. Hesse, E. Kratzig, and M. Wohlecke, "Composition dependence of the ultraviolet absorption edge in lithium tantalate," J. Appl. Phys. 93, 3102-3104 (2003).
[CrossRef]

DiDomenico, M.

M. DiDomenico, Jr. and S. H. Wemple, "Oxygen-octahedra ferroelectrics. I. Theory of electro-optical and nonlinear optical effects," J. Appl. Phys. 40, 720-734 (1969).
[CrossRef]

S. H. Wemple and M. DiDomenico, Jr., "Electrooptical and nonlinear optical properties of crystals," in Applied Solid State Science, R.Wolfe, ed. (Academic, 1972), Vol. 3, pp. 263-383.

Dittrich, P.

Dziedzic, J.

A. Ashkin, G. Boyd, J. Dziedzic, R. Smith, A. Ballman, J. Levinstein, and K. Nassau, "Optically-induced refractive index inhomogeneities in LiNbO3 and LiTaO3," Appl. Phys. Lett. 9, 72-74 (1966).
[CrossRef]

Fejer, M.

Fontana, M. D.

M. Aillerie, N. Théofanous, and M. D. Fontana, "Measurement of the electrooptic coefficients: description and comparison of the experimental techniques," Appl. Phys. B 70, 317-334 (2000).
[CrossRef]

Fukuda, T.

T. Fukuda, S. Matsumura, H. Hirano, and T. Ito, "Growth of LiTaO3 single-crystal for saw device applications," J. Cryst. Growth 46, 179-184 (1979).
[CrossRef]

Furukawa, Y.

Y. W. Liu, K. Kitamura, S. Takekawa, M. Nakamura, Y. Furukawa, and H. Hatano, "Two-color photorefractive properties in near-stoichiometric lithium tantalate crystals," J. Appl. Phys. 95, 7637-7644 (2004).
[CrossRef]

P. Dittrich, B. Koziarska-Glinka, G. Montemezzani, P. Günter, S. Takekawa, K. Kitamura, and Y. Furukawa, "Deep-ultraviolet interband photorefraction in lithium tantalate," J. Opt. Soc. Am. B 21, 632-639 (2004).
[CrossRef]

M. Nakamura, S. Higuchi, S. Takekawa, K. Terabe, Y. Furukawa, and K. Kitamura, "Refractive indices in undoped and MgO-doped near-stoichiometric LiTaO3 crystals," Jpn. J. Appl. Phys., Part 1 41, L465-L467 (2002).
[CrossRef]

Y. Furukawa, K. Kitamura, K. Niwa, H. Hatano, P. Bernasconi, G. Montemezzani, and P. Günter, "Stoichiometric LiTaO3 for dynamic holography in near UV wavelength range," Jpn. J. Appl. Phys. Part 1 38, 1816-1819 (1999).
[CrossRef]

Y. Furukawa, K. Kitamura, E. Suzuki, and K. Niwa, "Stoichiometric LiTaO3 single crystal growth by double crucible Czochralski method using automatic powder supply system," J. Cryst. Growth 197, 889-895 (1999).
[CrossRef]

K. Kitamura, Y. Furukawa, K. Niwa, V. Gopalan, and T. E. Mitchell, "Crystal growth and low coercive field 180 degrees domain switching characteristics of stoichiometric LiTaO3," Appl. Phys. Lett. 71, 3073-3075 (1998).
[CrossRef]

K. Kitamura, Y. Furukawa, Y. Ji, M. Zgonik, C. Medrano, G. Montemezzani, and P. Günter, "Photorefractive effect in LiNbO3 crystals enhanced by stoichiometry control," J. Appl. Phys. 82, 1006-1009 (1997).
[CrossRef]

Gahagan, K.

Gopalan, V.

J. Casson, K. Gahagan, D. Scrymgeour, R. Jain, J. Robinson, V. Gopalan, and R. Sander, "Electro-optic coefficients of lithium tantalate at near-infrared wavelengths," J. Opt. Soc. Am. B 21, 1948-1952 (2004).
[CrossRef]

K. Kitamura, Y. Furukawa, K. Niwa, V. Gopalan, and T. E. Mitchell, "Crystal growth and low coercive field 180 degrees domain switching characteristics of stoichiometric LiTaO3," Appl. Phys. Lett. 71, 3073-3075 (1998).
[CrossRef]

Grabar, A. A.

D. Haertle, G. Caimi, A. Haldi, G. Montemezzani, P. Günter, A. A. Grabar, I. M. Stoika, and Y. M. Vysochanskii, "Electro-optical properties of Sn2P2S6," Opt. Commun. 251, 333-343 (2003).
[CrossRef]

Günter, P.

P. Dittrich, B. Koziarska-Glinka, G. Montemezzani, P. Günter, S. Takekawa, K. Kitamura, and Y. Furukawa, "Deep-ultraviolet interband photorefraction in lithium tantalate," J. Opt. Soc. Am. B 21, 632-639 (2004).
[CrossRef]

D. Haertle, G. Caimi, A. Haldi, G. Montemezzani, P. Günter, A. A. Grabar, I. M. Stoika, and Y. M. Vysochanskii, "Electro-optical properties of Sn2P2S6," Opt. Commun. 251, 333-343 (2003).
[CrossRef]

P. Dittrich, G. Montemezzani, and P. Günter, "Tunable optical filter for wavelength division multiplexing using dynamic interband photorefractive gratings," Opt. Commun. 214, 363-370 (2002).
[CrossRef]

P. Dittrich, G. Montemezzani, P. Bernasconi, and P. Günter, "Fast, reconfigurable light-induced waveguides," Opt. Lett. 24, 1508-1510 (1999).
[CrossRef]

P. Bernasconi, G. Montemezzani, M. Wintermantel, I. Biaggio, and P. Günter, "High-resolution, high-speed photorefractive incoherent-to-coherent optical converter," Opt. Lett. 24, 199-201 (1999).
[CrossRef]

Y. Furukawa, K. Kitamura, K. Niwa, H. Hatano, P. Bernasconi, G. Montemezzani, and P. Günter, "Stoichiometric LiTaO3 for dynamic holography in near UV wavelength range," Jpn. J. Appl. Phys. Part 1 38, 1816-1819 (1999).
[CrossRef]

K. Kitamura, Y. Furukawa, Y. Ji, M. Zgonik, C. Medrano, G. Montemezzani, and P. Günter, "Photorefractive effect in LiNbO3 crystals enhanced by stoichiometry control," J. Appl. Phys. 82, 1006-1009 (1997).
[CrossRef]

C. Bosshard, K. Sutter, R. Schlesser, and P. Günter, "Electro-optic effects in molecular crystals," J. Opt. Soc. Am. B 10, 867-885 (1993).
[CrossRef]

Haertle, D.

D. Haertle, G. Caimi, A. Haldi, G. Montemezzani, P. Günter, A. A. Grabar, I. M. Stoika, and Y. M. Vysochanskii, "Electro-optical properties of Sn2P2S6," Opt. Commun. 251, 333-343 (2003).
[CrossRef]

Haldi, A.

D. Haertle, G. Caimi, A. Haldi, G. Montemezzani, P. Günter, A. A. Grabar, I. M. Stoika, and Y. M. Vysochanskii, "Electro-optical properties of Sn2P2S6," Opt. Commun. 251, 333-343 (2003).
[CrossRef]

Hatano, H.

Y. W. Liu, K. Kitamura, S. Takekawa, M. Nakamura, Y. Furukawa, and H. Hatano, "Two-color photorefractive properties in near-stoichiometric lithium tantalate crystals," J. Appl. Phys. 95, 7637-7644 (2004).
[CrossRef]

M. Jazbinsek, M. Zgonik, S. Takekawa, M. Nakamura, K. Kitamura, and H. Hatano, "Reduced space-charge fields in near-stoichiometric LiTaO3 for blue, violet and near-ultraviolet light beams," Appl. Phys. B 75, 891-894 (2002).
[CrossRef]

Y. Furukawa, K. Kitamura, K. Niwa, H. Hatano, P. Bernasconi, G. Montemezzani, and P. Günter, "Stoichiometric LiTaO3 for dynamic holography in near UV wavelength range," Jpn. J. Appl. Phys. Part 1 38, 1816-1819 (1999).
[CrossRef]

Hesse, H.

C. Bäumer, C. David, A. Tunyagi, K. Betzler, H. Hesse, E. Kratzig, and M. Wohlecke, "Composition dependence of the ultraviolet absorption edge in lithium tantalate," J. Appl. Phys. 93, 3102-3104 (2003).
[CrossRef]

Higuchi, S.

M. Nakamura, S. Higuchi, S. Takekawa, K. Terabe, Y. Furukawa, and K. Kitamura, "Refractive indices in undoped and MgO-doped near-stoichiometric LiTaO3 crystals," Jpn. J. Appl. Phys., Part 1 41, L465-L467 (2002).
[CrossRef]

Hirano, H.

T. Fukuda, S. Matsumura, H. Hirano, and T. Ito, "Growth of LiTaO3 single-crystal for saw device applications," J. Cryst. Growth 46, 179-184 (1979).
[CrossRef]

Ito, H.

K. S. Abedin and H. Ito, "Temperature-dependent dispersion relation of ferroelectric lithium tantalate," J. Appl. Phys. 80, 6561-6563 (1996).
[CrossRef]

Ito, T.

T. Fukuda, S. Matsumura, H. Hirano, and T. Ito, "Growth of LiTaO3 single-crystal for saw device applications," J. Cryst. Growth 46, 179-184 (1979).
[CrossRef]

Jain, R.

Jazbinsek, M.

M. Jazbinsek, M. Zgonik, S. Takekawa, M. Nakamura, K. Kitamura, and H. Hatano, "Reduced space-charge fields in near-stoichiometric LiTaO3 for blue, violet and near-ultraviolet light beams," Appl. Phys. B 75, 891-894 (2002).
[CrossRef]

Ji, Y.

K. Kitamura, Y. Furukawa, Y. Ji, M. Zgonik, C. Medrano, G. Montemezzani, and P. Günter, "Photorefractive effect in LiNbO3 crystals enhanced by stoichiometry control," J. Appl. Phys. 82, 1006-1009 (1997).
[CrossRef]

Kitamura, K.

P. Dittrich, B. Koziarska-Glinka, G. Montemezzani, P. Günter, S. Takekawa, K. Kitamura, and Y. Furukawa, "Deep-ultraviolet interband photorefraction in lithium tantalate," J. Opt. Soc. Am. B 21, 632-639 (2004).
[CrossRef]

Y. W. Liu, K. Kitamura, S. Takekawa, M. Nakamura, Y. Furukawa, and H. Hatano, "Two-color photorefractive properties in near-stoichiometric lithium tantalate crystals," J. Appl. Phys. 95, 7637-7644 (2004).
[CrossRef]

M. Nakamura, S. Higuchi, S. Takekawa, K. Terabe, Y. Furukawa, and K. Kitamura, "Refractive indices in undoped and MgO-doped near-stoichiometric LiTaO3 crystals," Jpn. J. Appl. Phys., Part 1 41, L465-L467 (2002).
[CrossRef]

M. Jazbinsek, M. Zgonik, S. Takekawa, M. Nakamura, K. Kitamura, and H. Hatano, "Reduced space-charge fields in near-stoichiometric LiTaO3 for blue, violet and near-ultraviolet light beams," Appl. Phys. B 75, 891-894 (2002).
[CrossRef]

Y. Furukawa, K. Kitamura, K. Niwa, H. Hatano, P. Bernasconi, G. Montemezzani, and P. Günter, "Stoichiometric LiTaO3 for dynamic holography in near UV wavelength range," Jpn. J. Appl. Phys. Part 1 38, 1816-1819 (1999).
[CrossRef]

Y. Furukawa, K. Kitamura, E. Suzuki, and K. Niwa, "Stoichiometric LiTaO3 single crystal growth by double crucible Czochralski method using automatic powder supply system," J. Cryst. Growth 197, 889-895 (1999).
[CrossRef]

K. Kitamura, Y. Furukawa, K. Niwa, V. Gopalan, and T. E. Mitchell, "Crystal growth and low coercive field 180 degrees domain switching characteristics of stoichiometric LiTaO3," Appl. Phys. Lett. 71, 3073-3075 (1998).
[CrossRef]

K. Kitamura, Y. Furukawa, Y. Ji, M. Zgonik, C. Medrano, G. Montemezzani, and P. Günter, "Photorefractive effect in LiNbO3 crystals enhanced by stoichiometry control," J. Appl. Phys. 82, 1006-1009 (1997).
[CrossRef]

Koziarska-Glinka, B.

Kratzig, E.

C. Bäumer, C. David, A. Tunyagi, K. Betzler, H. Hesse, E. Kratzig, and M. Wohlecke, "Composition dependence of the ultraviolet absorption edge in lithium tantalate," J. Appl. Phys. 93, 3102-3104 (2003).
[CrossRef]

K. Bastwöste, S. Schwalenberg, C. Bäumer, and E. Kratzig, "Temperature and composition dependence of birefringence of lithium-tantalate crystals determined by holographic scattering," Phys. Status Solidi A 199, R1-R3 (2003).
[CrossRef]

Lenzo, P.

P. Lenzo, E. Turner, E. Spencer, and A. Ballman, "Electrooptic coefficients and elastic-wave propagation in single-domain ferroelectric lithium tantalate," Appl. Phys. Lett. 8, 81-82 (1966).
[CrossRef]

Levinstein, J.

A. Ashkin, G. Boyd, J. Dziedzic, R. Smith, A. Ballman, J. Levinstein, and K. Nassau, "Optically-induced refractive index inhomogeneities in LiNbO3 and LiTaO3," Appl. Phys. Lett. 9, 72-74 (1966).
[CrossRef]

Liu, Y. W.

Y. W. Liu, K. Kitamura, S. Takekawa, M. Nakamura, Y. Furukawa, and H. Hatano, "Two-color photorefractive properties in near-stoichiometric lithium tantalate crystals," J. Appl. Phys. 95, 7637-7644 (2004).
[CrossRef]

Matsumura, S.

T. Fukuda, S. Matsumura, H. Hirano, and T. Ito, "Growth of LiTaO3 single-crystal for saw device applications," J. Cryst. Growth 46, 179-184 (1979).
[CrossRef]

Medrano, C.

K. Kitamura, Y. Furukawa, Y. Ji, M. Zgonik, C. Medrano, G. Montemezzani, and P. Günter, "Photorefractive effect in LiNbO3 crystals enhanced by stoichiometry control," J. Appl. Phys. 82, 1006-1009 (1997).
[CrossRef]

Meyn, J.

Mitchell, T. E.

K. Kitamura, Y. Furukawa, K. Niwa, V. Gopalan, and T. E. Mitchell, "Crystal growth and low coercive field 180 degrees domain switching characteristics of stoichiometric LiTaO3," Appl. Phys. Lett. 71, 3073-3075 (1998).
[CrossRef]

Montemezzani, G.

P. Dittrich, B. Koziarska-Glinka, G. Montemezzani, P. Günter, S. Takekawa, K. Kitamura, and Y. Furukawa, "Deep-ultraviolet interband photorefraction in lithium tantalate," J. Opt. Soc. Am. B 21, 632-639 (2004).
[CrossRef]

D. Haertle, G. Caimi, A. Haldi, G. Montemezzani, P. Günter, A. A. Grabar, I. M. Stoika, and Y. M. Vysochanskii, "Electro-optical properties of Sn2P2S6," Opt. Commun. 251, 333-343 (2003).
[CrossRef]

P. Dittrich, G. Montemezzani, and P. Günter, "Tunable optical filter for wavelength division multiplexing using dynamic interband photorefractive gratings," Opt. Commun. 214, 363-370 (2002).
[CrossRef]

P. Dittrich, G. Montemezzani, P. Bernasconi, and P. Günter, "Fast, reconfigurable light-induced waveguides," Opt. Lett. 24, 1508-1510 (1999).
[CrossRef]

P. Bernasconi, G. Montemezzani, M. Wintermantel, I. Biaggio, and P. Günter, "High-resolution, high-speed photorefractive incoherent-to-coherent optical converter," Opt. Lett. 24, 199-201 (1999).
[CrossRef]

Y. Furukawa, K. Kitamura, K. Niwa, H. Hatano, P. Bernasconi, G. Montemezzani, and P. Günter, "Stoichiometric LiTaO3 for dynamic holography in near UV wavelength range," Jpn. J. Appl. Phys. Part 1 38, 1816-1819 (1999).
[CrossRef]

K. Kitamura, Y. Furukawa, Y. Ji, M. Zgonik, C. Medrano, G. Montemezzani, and P. Günter, "Photorefractive effect in LiNbO3 crystals enhanced by stoichiometry control," J. Appl. Phys. 82, 1006-1009 (1997).
[CrossRef]

Nakamura, M.

Y. W. Liu, K. Kitamura, S. Takekawa, M. Nakamura, Y. Furukawa, and H. Hatano, "Two-color photorefractive properties in near-stoichiometric lithium tantalate crystals," J. Appl. Phys. 95, 7637-7644 (2004).
[CrossRef]

M. Nakamura, S. Higuchi, S. Takekawa, K. Terabe, Y. Furukawa, and K. Kitamura, "Refractive indices in undoped and MgO-doped near-stoichiometric LiTaO3 crystals," Jpn. J. Appl. Phys., Part 1 41, L465-L467 (2002).
[CrossRef]

M. Jazbinsek, M. Zgonik, S. Takekawa, M. Nakamura, K. Kitamura, and H. Hatano, "Reduced space-charge fields in near-stoichiometric LiTaO3 for blue, violet and near-ultraviolet light beams," Appl. Phys. B 75, 891-894 (2002).
[CrossRef]

Nassau, K.

A. Ashkin, G. Boyd, J. Dziedzic, R. Smith, A. Ballman, J. Levinstein, and K. Nassau, "Optically-induced refractive index inhomogeneities in LiNbO3 and LiTaO3," Appl. Phys. Lett. 9, 72-74 (1966).
[CrossRef]

Niwa, K.

Y. Furukawa, K. Kitamura, K. Niwa, H. Hatano, P. Bernasconi, G. Montemezzani, and P. Günter, "Stoichiometric LiTaO3 for dynamic holography in near UV wavelength range," Jpn. J. Appl. Phys. Part 1 38, 1816-1819 (1999).
[CrossRef]

Y. Furukawa, K. Kitamura, E. Suzuki, and K. Niwa, "Stoichiometric LiTaO3 single crystal growth by double crucible Czochralski method using automatic powder supply system," J. Cryst. Growth 197, 889-895 (1999).
[CrossRef]

K. Kitamura, Y. Furukawa, K. Niwa, V. Gopalan, and T. E. Mitchell, "Crystal growth and low coercive field 180 degrees domain switching characteristics of stoichiometric LiTaO3," Appl. Phys. Lett. 71, 3073-3075 (1998).
[CrossRef]

Pasaltis, D.

H. Coufal, D. Pasaltis, and G. Sincerbox, Holographic Data Storage (Springer, 2000).

Robinson, J.

Sander, R.

Schlesser, R.

Schwalenberg, S.

K. Bastwöste, S. Schwalenberg, C. Bäumer, and E. Kratzig, "Temperature and composition dependence of birefringence of lithium-tantalate crystals determined by holographic scattering," Phys. Status Solidi A 199, R1-R3 (2003).
[CrossRef]

Scrymgeour, D.

Shumate, M. S.

Sincerbox, G.

H. Coufal, D. Pasaltis, and G. Sincerbox, Holographic Data Storage (Springer, 2000).

Smith, R.

A. Ashkin, G. Boyd, J. Dziedzic, R. Smith, A. Ballman, J. Levinstein, and K. Nassau, "Optically-induced refractive index inhomogeneities in LiNbO3 and LiTaO3," Appl. Phys. Lett. 9, 72-74 (1966).
[CrossRef]

Smith, R. T.

R. T. Smith and F. S. Welsh, "Temperature dependence of the elastic, piezoelectric, and dielectric constants of lithium tantalate and lithium niobate," J. Appl. Phys. 42, 2219-2230 (1971).
[CrossRef]

Spencer, E.

P. Lenzo, E. Turner, E. Spencer, and A. Ballman, "Electrooptic coefficients and elastic-wave propagation in single-domain ferroelectric lithium tantalate," Appl. Phys. Lett. 8, 81-82 (1966).
[CrossRef]

Stoika, I. M.

D. Haertle, G. Caimi, A. Haldi, G. Montemezzani, P. Günter, A. A. Grabar, I. M. Stoika, and Y. M. Vysochanskii, "Electro-optical properties of Sn2P2S6," Opt. Commun. 251, 333-343 (2003).
[CrossRef]

Sutter, K.

Suzuki, E.

Y. Furukawa, K. Kitamura, E. Suzuki, and K. Niwa, "Stoichiometric LiTaO3 single crystal growth by double crucible Czochralski method using automatic powder supply system," J. Cryst. Growth 197, 889-895 (1999).
[CrossRef]

Takekawa, S.

Y. W. Liu, K. Kitamura, S. Takekawa, M. Nakamura, Y. Furukawa, and H. Hatano, "Two-color photorefractive properties in near-stoichiometric lithium tantalate crystals," J. Appl. Phys. 95, 7637-7644 (2004).
[CrossRef]

P. Dittrich, B. Koziarska-Glinka, G. Montemezzani, P. Günter, S. Takekawa, K. Kitamura, and Y. Furukawa, "Deep-ultraviolet interband photorefraction in lithium tantalate," J. Opt. Soc. Am. B 21, 632-639 (2004).
[CrossRef]

M. Nakamura, S. Higuchi, S. Takekawa, K. Terabe, Y. Furukawa, and K. Kitamura, "Refractive indices in undoped and MgO-doped near-stoichiometric LiTaO3 crystals," Jpn. J. Appl. Phys., Part 1 41, L465-L467 (2002).
[CrossRef]

M. Jazbinsek, M. Zgonik, S. Takekawa, M. Nakamura, K. Kitamura, and H. Hatano, "Reduced space-charge fields in near-stoichiometric LiTaO3 for blue, violet and near-ultraviolet light beams," Appl. Phys. B 75, 891-894 (2002).
[CrossRef]

Terabe, K.

M. Nakamura, S. Higuchi, S. Takekawa, K. Terabe, Y. Furukawa, and K. Kitamura, "Refractive indices in undoped and MgO-doped near-stoichiometric LiTaO3 crystals," Jpn. J. Appl. Phys., Part 1 41, L465-L467 (2002).
[CrossRef]

Théofanous, N.

M. Aillerie, N. Théofanous, and M. D. Fontana, "Measurement of the electrooptic coefficients: description and comparison of the experimental techniques," Appl. Phys. B 70, 317-334 (2000).
[CrossRef]

Tunyagi, A.

C. Bäumer, C. David, A. Tunyagi, K. Betzler, H. Hesse, E. Kratzig, and M. Wohlecke, "Composition dependence of the ultraviolet absorption edge in lithium tantalate," J. Appl. Phys. 93, 3102-3104 (2003).
[CrossRef]

Turner, E.

P. Lenzo, E. Turner, E. Spencer, and A. Ballman, "Electrooptic coefficients and elastic-wave propagation in single-domain ferroelectric lithium tantalate," Appl. Phys. Lett. 8, 81-82 (1966).
[CrossRef]

Vysochanskii, Y. M.

D. Haertle, G. Caimi, A. Haldi, G. Montemezzani, P. Günter, A. A. Grabar, I. M. Stoika, and Y. M. Vysochanskii, "Electro-optical properties of Sn2P2S6," Opt. Commun. 251, 333-343 (2003).
[CrossRef]

Welsh, F. S.

R. T. Smith and F. S. Welsh, "Temperature dependence of the elastic, piezoelectric, and dielectric constants of lithium tantalate and lithium niobate," J. Appl. Phys. 42, 2219-2230 (1971).
[CrossRef]

Wemple, S. H.

M. DiDomenico, Jr. and S. H. Wemple, "Oxygen-octahedra ferroelectrics. I. Theory of electro-optical and nonlinear optical effects," J. Appl. Phys. 40, 720-734 (1969).
[CrossRef]

S. H. Wemple and M. DiDomenico, Jr., "Electrooptical and nonlinear optical properties of crystals," in Applied Solid State Science, R.Wolfe, ed. (Academic, 1972), Vol. 3, pp. 263-383.

Wintermantel, M.

Wohlecke, M.

C. Bäumer, C. David, A. Tunyagi, K. Betzler, H. Hesse, E. Kratzig, and M. Wohlecke, "Composition dependence of the ultraviolet absorption edge in lithium tantalate," J. Appl. Phys. 93, 3102-3104 (2003).
[CrossRef]

Zgonik, M.

M. Jazbinsek, M. Zgonik, S. Takekawa, M. Nakamura, K. Kitamura, and H. Hatano, "Reduced space-charge fields in near-stoichiometric LiTaO3 for blue, violet and near-ultraviolet light beams," Appl. Phys. B 75, 891-894 (2002).
[CrossRef]

K. Kitamura, Y. Furukawa, Y. Ji, M. Zgonik, C. Medrano, G. Montemezzani, and P. Günter, "Photorefractive effect in LiNbO3 crystals enhanced by stoichiometry control," J. Appl. Phys. 82, 1006-1009 (1997).
[CrossRef]

Appl. Opt.

Appl. Phys. B

M. Aillerie, N. Théofanous, and M. D. Fontana, "Measurement of the electrooptic coefficients: description and comparison of the experimental techniques," Appl. Phys. B 70, 317-334 (2000).
[CrossRef]

M. Jazbinsek, M. Zgonik, S. Takekawa, M. Nakamura, K. Kitamura, and H. Hatano, "Reduced space-charge fields in near-stoichiometric LiTaO3 for blue, violet and near-ultraviolet light beams," Appl. Phys. B 75, 891-894 (2002).
[CrossRef]

Appl. Phys. Lett.

A. Ashkin, G. Boyd, J. Dziedzic, R. Smith, A. Ballman, J. Levinstein, and K. Nassau, "Optically-induced refractive index inhomogeneities in LiNbO3 and LiTaO3," Appl. Phys. Lett. 9, 72-74 (1966).
[CrossRef]

K. Kitamura, Y. Furukawa, K. Niwa, V. Gopalan, and T. E. Mitchell, "Crystal growth and low coercive field 180 degrees domain switching characteristics of stoichiometric LiTaO3," Appl. Phys. Lett. 71, 3073-3075 (1998).
[CrossRef]

P. Lenzo, E. Turner, E. Spencer, and A. Ballman, "Electrooptic coefficients and elastic-wave propagation in single-domain ferroelectric lithium tantalate," Appl. Phys. Lett. 8, 81-82 (1966).
[CrossRef]

J. Appl. Phys.

K. S. Abedin and H. Ito, "Temperature-dependent dispersion relation of ferroelectric lithium tantalate," J. Appl. Phys. 80, 6561-6563 (1996).
[CrossRef]

C. Bäumer, C. David, A. Tunyagi, K. Betzler, H. Hesse, E. Kratzig, and M. Wohlecke, "Composition dependence of the ultraviolet absorption edge in lithium tantalate," J. Appl. Phys. 93, 3102-3104 (2003).
[CrossRef]

M. DiDomenico, Jr. and S. H. Wemple, "Oxygen-octahedra ferroelectrics. I. Theory of electro-optical and nonlinear optical effects," J. Appl. Phys. 40, 720-734 (1969).
[CrossRef]

Y. W. Liu, K. Kitamura, S. Takekawa, M. Nakamura, Y. Furukawa, and H. Hatano, "Two-color photorefractive properties in near-stoichiometric lithium tantalate crystals," J. Appl. Phys. 95, 7637-7644 (2004).
[CrossRef]

K. Kitamura, Y. Furukawa, Y. Ji, M. Zgonik, C. Medrano, G. Montemezzani, and P. Günter, "Photorefractive effect in LiNbO3 crystals enhanced by stoichiometry control," J. Appl. Phys. 82, 1006-1009 (1997).
[CrossRef]

R. T. Smith and F. S. Welsh, "Temperature dependence of the elastic, piezoelectric, and dielectric constants of lithium tantalate and lithium niobate," J. Appl. Phys. 42, 2219-2230 (1971).
[CrossRef]

J. Cryst. Growth

T. Fukuda, S. Matsumura, H. Hirano, and T. Ito, "Growth of LiTaO3 single-crystal for saw device applications," J. Cryst. Growth 46, 179-184 (1979).
[CrossRef]

Y. Furukawa, K. Kitamura, E. Suzuki, and K. Niwa, "Stoichiometric LiTaO3 single crystal growth by double crucible Czochralski method using automatic powder supply system," J. Cryst. Growth 197, 889-895 (1999).
[CrossRef]

J. Opt. Soc. Am. B

Jpn. J. Appl. Phys. Part 1

Y. Furukawa, K. Kitamura, K. Niwa, H. Hatano, P. Bernasconi, G. Montemezzani, and P. Günter, "Stoichiometric LiTaO3 for dynamic holography in near UV wavelength range," Jpn. J. Appl. Phys. Part 1 38, 1816-1819 (1999).
[CrossRef]

Jpn. J. Appl. Phys., Part 1

M. Nakamura, S. Higuchi, S. Takekawa, K. Terabe, Y. Furukawa, and K. Kitamura, "Refractive indices in undoped and MgO-doped near-stoichiometric LiTaO3 crystals," Jpn. J. Appl. Phys., Part 1 41, L465-L467 (2002).
[CrossRef]

Opt. Commun.

P. Dittrich, G. Montemezzani, and P. Günter, "Tunable optical filter for wavelength division multiplexing using dynamic interband photorefractive gratings," Opt. Commun. 214, 363-370 (2002).
[CrossRef]

D. Haertle, G. Caimi, A. Haldi, G. Montemezzani, P. Günter, A. A. Grabar, I. M. Stoika, and Y. M. Vysochanskii, "Electro-optical properties of Sn2P2S6," Opt. Commun. 251, 333-343 (2003).
[CrossRef]

Opt. Lett.

Phys. Status Solidi A

K. Bastwöste, S. Schwalenberg, C. Bäumer, and E. Kratzig, "Temperature and composition dependence of birefringence of lithium-tantalate crystals determined by holographic scattering," Phys. Status Solidi A 199, R1-R3 (2003).
[CrossRef]

Other

H. Coufal, D. Pasaltis, and G. Sincerbox, Holographic Data Storage (Springer, 2000).

S. H. Wemple and M. DiDomenico, Jr., "Electrooptical and nonlinear optical properties of crystals," in Applied Solid State Science, R.Wolfe, ed. (Academic, 1972), Vol. 3, pp. 263-383.

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Figures (4)

Fig. 1
Fig. 1

Interferometric setup used to measure the refractive indices and electro-optical coefficients (eo-coeff.). For the determination of refractive indices, the crystal was mounted onto a rotation stage. The unclamped electro-optic coefficients were measured by applying a low-frequency sine voltage.

Fig. 2
Fig. 2

Refractive indices of (a) CLT and (b) SLT. The dashed curves and the data above 400 nm are taken from literature.[16, 17] The solid curves correspond to the new coefficients (Table 2). There is a small correction at lower wavelengths due to the new data in the UV.

Fig. 3
Fig. 3

Measured dispersion of the electro-optic coefficients r 33 and r 13 of CLT and SLT. The solid curves correspond to Eq. (12), while the dashed curves correspond to Eq. (11) that describes the dispersion at longer wavelengths. The parameters for the theoretical curves are listed in Table 3.

Fig. 4
Fig. 4

Absorption α c in CLT and SLT for the extraordinary polarization and reduced half-wave voltage v π for the higher electro-optic coefficient r 33 and for the configuration employing r c . Since v π is similar for both investigated crystals, only the one of SLT is plotted. To compare the absorption, a vertical dotted line is plotted at λ = 275 nm .

Tables (3)

Tables Icon

Table 1 Crystals Investigated in This Paper

Tables Icon

Table 2 Sellmeier Parameters Used to Describe the Dispersion of n o and n e in CLT and SLT Crystals a

Tables Icon

Table 3 Parameters Used to Describe the Dispersion of the Electro-Optic Coefficient a

Equations (15)

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

n = α 2 + 2 ( 1 cos θ ) ( 1 α ) 2 ( 1 cos θ α ) ,
n 2 ( λ ) = A + B λ 2 C D λ 2 .
n 2 ( ω ) 1 = i f i ω i 2 ω 2 .
n 2 1 = S γ λ γ 2 1 ( λ γ λ ) 2 + S ϵ λ ϵ 2 1 ( λ ϵ λ ) 2 = E d γ E γ E γ 2 E 2 + E d ϵ E ϵ E ϵ 2 E 2 .
n 2 ( λ ) 1 = A + S ϵ λ ϵ 2 1 ( λ ϵ λ ) 2 D λ 2 .
n 2 ( E ) 1 = E d γ E γ + E d ϵ E ϵ E ϵ 2 E 2 E IR 2 E 2 .
φ = 2 π L λ n 3 r eff E + 4 π L λ ( n 1 ) d eff E ,
r eff = δ I δ V d L λ π n 3 Δ I + 2 n 1 n 3 d eff .
r = ϵ 0 ( ϵ 1 ) d d P ( 1 n 2 ) ,
n 2 1 = S 0 λ 0 1 ( λ 0 λ ) 2 = E d E 0 E 0 2 E 2 ,
r eff ( λ ) = ϵ 0 ( ϵ 1 ) [ 1 1 n ( λ ) 2 ] 2 d E 0 d P E d [ ( 1 K ) + ( 1 + K ) ( λ 0 λ ) 2 ] ,
r eff ( λ ) = ϵ 0 ( ϵ 1 ) [ 1 1 + A n ( λ ) 2 ] 2 d E ϵ d P E d ϵ [ ( 1 K 1 K 2 ) + ( 1 + K 1 + 2 K 2 ) ( λ ϵ λ ) 2 K 2 ( λ ϵ λ ) 4 ] .
K 1 = d ln E d ϵ d ln E ϵ
K 2 = d ln ( E d γ E γ ) d ln E ϵ R
V π = λ n 3 r d L = v π d L ,

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