Abstract

The effective nonlinear coefficients and the optimum phase-matching angles for three-wave mixing processes in biaxial crystals are calculated, and their analytical expressions are obtained. We carried out the calculations by taking into consideration the difference between the directions of the electric field vector E and the electric displacement vector D in biaxial crystals. Using the small-signal approximation, we also derived expressions for acceptance parameters and calculated their values for some typical three-wave interaction processes. In addition we obtained a theoretical curve for the energy conversion efficiency of three-wave interactions as a function of phase mismatch by numerically solving the coupled wave equations describing the processes. With this curve we are able to define and to compute the acceptance parameters for any type of three-wave mixing process. Finally, the numerical values of the phase-matching parameters are obtained for such biaxial crystals as KTiOPO4 and LiB3O5, and the results are graphically presented.

© 1992 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. F. C. Zumsteg, J. D. Bierlein, and T. E. Gier, “KxRb1−xTiOPO4: a new nonlinear material,” J. Appl. Phys 47, 4980 (1976).
    [Crossref]
  2. T. E. Gier, “Method for flux growth of KTiOPO4 and its analogues,” U.S. Patent4,231,838 (November1980).
  3. J. C. Jacco, G. M. Loiacono, M. Jaso, G. Mizell, and B. Greenberg, “Flux growth and properties of KTiOPO4,” J. Cryst. Growth 70, 484 (1984).
    [Crossref]
  4. A. A. Ballman, H. Brown, D. H. Olson, and C. E. Rice, “Growth of potassium titanyl phosphate (KTP) from molten tungstate melts,” J. Cryst. Growth 75, 390 (1986).
    [Crossref]
  5. H. Vanherzeele, J. D. Bierlein, and F C. Zumsteg, “Index of refraction measurement and parametric generation in hydrothermally grown TiOPO4,” Appl. Opt. 27, 3314 (1988).
    [Crossref] [PubMed]
  6. C. Chen, B. Wu, G. You, A. Jiang, and Y. Huang, “High efficiency and wide band second harmonic generation properties of the new crystal β-BaB2O4,” in Digest of International Quantum Electronics Conference (Optical Society of America, Washington, D.C., 1984), paper MCC5.
  7. C. Chen, B. Wu, A. Jiang, and G. You, “A new type ultraviolet SHG crystal β-BaB2O4,” Sci. Sin. Ser. B 28, 235 (1985).
  8. C. Chen, “Chinese lab grows new nonlinear optical borate crystals,” Laser Focus World 25 (11), 129 (1989).
  9. C. Chen, Y. Wu, A. Jiang, B. Wu, G. You, R. Li, and S. Lin, “New nonlinear optical crystal: LiB3O5,” J. Opt. Soc. Am. B 6, 616 (1989).
    [Crossref]
  10. S. K. Kurtz and T. T. Terry, “A powder technique for the evaluation of nonlinear optical materials,” J. Appl. Phys. 39, 3798 (1968).
    [Crossref]
  11. P. Gunter, P. M. Asbeck, and S. K. Kurtz, “Second harmonic generation with Ga1−xAlxAs lasers and KNbO3crystal,” Appl. Phys. Lett. 35, 461 (1979).
    [Crossref]
  12. M. H. Hobden, “Phase matched second harmonic generation in biaxial crystals,” J. Appl. Phys. 38, 4365 (1967).
    [Crossref]
  13. H. Ito, H. Naito, and H. Inaba, “Generalized study on angular dependence of induced second order nonlinear optical polarizations and phase matching in biaxial crystals,” J. Appl. Phys. 46, 3992 (1975).
    [Crossref]
  14. Y. S. Liu, L. Drafall, D. Dentz, and R. Belt, “Nonlinear optical phase matching properties of KTiOPO4,” Gen. Electr. Tech. Inf. Ser. Rep. 82CRD016 (General Electric Company, Schenectady, N.Y., 1982).
  15. J. Q. Yao and T. S. Fahlen, “Calculation of optimum phase match parameters for the biaxial crystal KTiOPO4,” J. Appl. Phys. 55, 65 (1984).
    [Crossref]
  16. T. Y. Fan, C. E. Huang, B. Q. Hu, R. C. Eckardt, Y. X. Fan, R. L. Byer, and R. S. Feigelson, “Second harmonic generation and accurate index of refraction measurements in flux grown KTiOPO4,” Appl. Opt. 26, 2390 (1987).
    [Crossref] [PubMed]
  17. J. D. Bierlein and H. Vanherzeele, “Potassium titanyl phosphate: properties and new applications,” J. Opt. Soc. Am. B 6, 622 (1989).
    [Crossref]
  18. J. Q. Yao and B. Xue, “Optimum parameters of high conversion efficiency intracavity frequency doubled laser with Gaussian-like beam,” in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1984), paper ThC6.
  19. J. Q. Yao and Y. Liu, “Intracavity frequency doubling of quasi-CW pumped YAG laser,” in Proceedings of Second Interdisciplinary Laser Science Conference (American Physical Society, New York, 1986), paper ThL63.
  20. J. Q. Yao and D. Sun, “Multi-frequency conversions using one biaxial crystal KTiOPO4,” in Proceedings of Third Interdisciplinary Laser Science Conference (American Physical Society, New York, 1987), paper WO9.
  21. J. Q. Yao, J. T. Lin, and W. Sheng, “Analysis of BBO and KTP crystals for optical parametric oscillation,” in Conference on Lasers and Electro-Optics, Vol. 7 of 1988 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1988), paper TuJ5.
  22. J. Q. Yao and K. C. Liu, “The optimum phase matching and experimental study of biaxial crystal KTP,” in Proceedings of Topical Meeting on Laser Materials and Spectroscopy (World Scientific, Singapore, 1988), paper WeO7.
  23. B. Wyncke and F. Brehat, “Calculation of the effective second-order nonlinear coefficients along the phase matching directions in acentric orthorhombic biaxial crystals,” J. Phys. B 22, 363 (1989).
    [Crossref]
  24. F. Brehat and B. Wyncke, “Calculation of double refraction walk-off angle along the phase matching directions in nonlinear biaxial crystals,” J. Phys. B 22, 1891 (1989).
    [Crossref]
  25. R. Morgan, “Phase matching considerations for generalized three wave mixing in nonlinear anisotropic crystals,” Appl. Opt. 29, 1259 (1990).
    [Crossref] [PubMed]
  26. M. Born and E. Wolf, Principles of Optics (Oxford U. Press, New York, 1970).
  27. C. L. Tang, W. Bosenberg, T. Ukaihi, R. Lane, and L. K. Cheng, “NLO materials displaying superior performance,” Laser Focus World (September1990), p. 87.
  28. J. T. Lin, J. L. Montgomery, and K. Kato, “Temperature-tuned non-critically phase-matched frequency conversion in LiB3O5 crystal,” Opt. Commun. 80, 159 (1990).
    [Crossref]
  29. S. Lin, Z. Sun, B. Wu, and C. Chen, “The nonlinear optical characteristics of a LiB3O5crystal,” J. Appl. Phys 67, 634 (1990).
    [Crossref]
  30. F. Hanson and D. Dick, “Blue parametric generation from temperature-tuned LiB3O5,” Opt. Lett. 16, 205 (1991).
    [Crossref] [PubMed]
  31. A. Borsutzky, R. Brunger, C. Huang, and R. Wallenstein, “Harmonic and sum-frequency generation of pulsed laser radiation in BBO, LBO and KD*P,” Appl. Phys. B 52, 55 (1991).
    [Crossref]

1991 (2)

F. Hanson and D. Dick, “Blue parametric generation from temperature-tuned LiB3O5,” Opt. Lett. 16, 205 (1991).
[Crossref] [PubMed]

A. Borsutzky, R. Brunger, C. Huang, and R. Wallenstein, “Harmonic and sum-frequency generation of pulsed laser radiation in BBO, LBO and KD*P,” Appl. Phys. B 52, 55 (1991).
[Crossref]

1990 (4)

R. Morgan, “Phase matching considerations for generalized three wave mixing in nonlinear anisotropic crystals,” Appl. Opt. 29, 1259 (1990).
[Crossref] [PubMed]

C. L. Tang, W. Bosenberg, T. Ukaihi, R. Lane, and L. K. Cheng, “NLO materials displaying superior performance,” Laser Focus World (September1990), p. 87.

J. T. Lin, J. L. Montgomery, and K. Kato, “Temperature-tuned non-critically phase-matched frequency conversion in LiB3O5 crystal,” Opt. Commun. 80, 159 (1990).
[Crossref]

S. Lin, Z. Sun, B. Wu, and C. Chen, “The nonlinear optical characteristics of a LiB3O5crystal,” J. Appl. Phys 67, 634 (1990).
[Crossref]

1989 (5)

C. Chen, “Chinese lab grows new nonlinear optical borate crystals,” Laser Focus World 25 (11), 129 (1989).

C. Chen, Y. Wu, A. Jiang, B. Wu, G. You, R. Li, and S. Lin, “New nonlinear optical crystal: LiB3O5,” J. Opt. Soc. Am. B 6, 616 (1989).
[Crossref]

J. D. Bierlein and H. Vanherzeele, “Potassium titanyl phosphate: properties and new applications,” J. Opt. Soc. Am. B 6, 622 (1989).
[Crossref]

B. Wyncke and F. Brehat, “Calculation of the effective second-order nonlinear coefficients along the phase matching directions in acentric orthorhombic biaxial crystals,” J. Phys. B 22, 363 (1989).
[Crossref]

F. Brehat and B. Wyncke, “Calculation of double refraction walk-off angle along the phase matching directions in nonlinear biaxial crystals,” J. Phys. B 22, 1891 (1989).
[Crossref]

1988 (1)

1987 (1)

1986 (1)

A. A. Ballman, H. Brown, D. H. Olson, and C. E. Rice, “Growth of potassium titanyl phosphate (KTP) from molten tungstate melts,” J. Cryst. Growth 75, 390 (1986).
[Crossref]

1985 (1)

C. Chen, B. Wu, A. Jiang, and G. You, “A new type ultraviolet SHG crystal β-BaB2O4,” Sci. Sin. Ser. B 28, 235 (1985).

1984 (2)

J. C. Jacco, G. M. Loiacono, M. Jaso, G. Mizell, and B. Greenberg, “Flux growth and properties of KTiOPO4,” J. Cryst. Growth 70, 484 (1984).
[Crossref]

J. Q. Yao and T. S. Fahlen, “Calculation of optimum phase match parameters for the biaxial crystal KTiOPO4,” J. Appl. Phys. 55, 65 (1984).
[Crossref]

1979 (1)

P. Gunter, P. M. Asbeck, and S. K. Kurtz, “Second harmonic generation with Ga1−xAlxAs lasers and KNbO3crystal,” Appl. Phys. Lett. 35, 461 (1979).
[Crossref]

1976 (1)

F. C. Zumsteg, J. D. Bierlein, and T. E. Gier, “KxRb1−xTiOPO4: a new nonlinear material,” J. Appl. Phys 47, 4980 (1976).
[Crossref]

1975 (1)

H. Ito, H. Naito, and H. Inaba, “Generalized study on angular dependence of induced second order nonlinear optical polarizations and phase matching in biaxial crystals,” J. Appl. Phys. 46, 3992 (1975).
[Crossref]

1968 (1)

S. K. Kurtz and T. T. Terry, “A powder technique for the evaluation of nonlinear optical materials,” J. Appl. Phys. 39, 3798 (1968).
[Crossref]

1967 (1)

M. H. Hobden, “Phase matched second harmonic generation in biaxial crystals,” J. Appl. Phys. 38, 4365 (1967).
[Crossref]

Asbeck, P. M.

P. Gunter, P. M. Asbeck, and S. K. Kurtz, “Second harmonic generation with Ga1−xAlxAs lasers and KNbO3crystal,” Appl. Phys. Lett. 35, 461 (1979).
[Crossref]

Ballman, A. A.

A. A. Ballman, H. Brown, D. H. Olson, and C. E. Rice, “Growth of potassium titanyl phosphate (KTP) from molten tungstate melts,” J. Cryst. Growth 75, 390 (1986).
[Crossref]

Belt, R.

Y. S. Liu, L. Drafall, D. Dentz, and R. Belt, “Nonlinear optical phase matching properties of KTiOPO4,” Gen. Electr. Tech. Inf. Ser. Rep. 82CRD016 (General Electric Company, Schenectady, N.Y., 1982).

Bierlein, J. D.

Born, M.

M. Born and E. Wolf, Principles of Optics (Oxford U. Press, New York, 1970).

Borsutzky, A.

A. Borsutzky, R. Brunger, C. Huang, and R. Wallenstein, “Harmonic and sum-frequency generation of pulsed laser radiation in BBO, LBO and KD*P,” Appl. Phys. B 52, 55 (1991).
[Crossref]

Bosenberg, W.

C. L. Tang, W. Bosenberg, T. Ukaihi, R. Lane, and L. K. Cheng, “NLO materials displaying superior performance,” Laser Focus World (September1990), p. 87.

Brehat, F.

B. Wyncke and F. Brehat, “Calculation of the effective second-order nonlinear coefficients along the phase matching directions in acentric orthorhombic biaxial crystals,” J. Phys. B 22, 363 (1989).
[Crossref]

F. Brehat and B. Wyncke, “Calculation of double refraction walk-off angle along the phase matching directions in nonlinear biaxial crystals,” J. Phys. B 22, 1891 (1989).
[Crossref]

Brown, H.

A. A. Ballman, H. Brown, D. H. Olson, and C. E. Rice, “Growth of potassium titanyl phosphate (KTP) from molten tungstate melts,” J. Cryst. Growth 75, 390 (1986).
[Crossref]

Brunger, R.

A. Borsutzky, R. Brunger, C. Huang, and R. Wallenstein, “Harmonic and sum-frequency generation of pulsed laser radiation in BBO, LBO and KD*P,” Appl. Phys. B 52, 55 (1991).
[Crossref]

Byer, R. L.

Chen, C.

S. Lin, Z. Sun, B. Wu, and C. Chen, “The nonlinear optical characteristics of a LiB3O5crystal,” J. Appl. Phys 67, 634 (1990).
[Crossref]

C. Chen, “Chinese lab grows new nonlinear optical borate crystals,” Laser Focus World 25 (11), 129 (1989).

C. Chen, Y. Wu, A. Jiang, B. Wu, G. You, R. Li, and S. Lin, “New nonlinear optical crystal: LiB3O5,” J. Opt. Soc. Am. B 6, 616 (1989).
[Crossref]

C. Chen, B. Wu, A. Jiang, and G. You, “A new type ultraviolet SHG crystal β-BaB2O4,” Sci. Sin. Ser. B 28, 235 (1985).

C. Chen, B. Wu, G. You, A. Jiang, and Y. Huang, “High efficiency and wide band second harmonic generation properties of the new crystal β-BaB2O4,” in Digest of International Quantum Electronics Conference (Optical Society of America, Washington, D.C., 1984), paper MCC5.

Cheng, L. K.

C. L. Tang, W. Bosenberg, T. Ukaihi, R. Lane, and L. K. Cheng, “NLO materials displaying superior performance,” Laser Focus World (September1990), p. 87.

Dentz, D.

Y. S. Liu, L. Drafall, D. Dentz, and R. Belt, “Nonlinear optical phase matching properties of KTiOPO4,” Gen. Electr. Tech. Inf. Ser. Rep. 82CRD016 (General Electric Company, Schenectady, N.Y., 1982).

Dick, D.

Drafall, L.

Y. S. Liu, L. Drafall, D. Dentz, and R. Belt, “Nonlinear optical phase matching properties of KTiOPO4,” Gen. Electr. Tech. Inf. Ser. Rep. 82CRD016 (General Electric Company, Schenectady, N.Y., 1982).

Eckardt, R. C.

Fahlen, T. S.

J. Q. Yao and T. S. Fahlen, “Calculation of optimum phase match parameters for the biaxial crystal KTiOPO4,” J. Appl. Phys. 55, 65 (1984).
[Crossref]

Fan, T. Y.

Fan, Y. X.

Feigelson, R. S.

Gier, T. E.

F. C. Zumsteg, J. D. Bierlein, and T. E. Gier, “KxRb1−xTiOPO4: a new nonlinear material,” J. Appl. Phys 47, 4980 (1976).
[Crossref]

T. E. Gier, “Method for flux growth of KTiOPO4 and its analogues,” U.S. Patent4,231,838 (November1980).

Greenberg, B.

J. C. Jacco, G. M. Loiacono, M. Jaso, G. Mizell, and B. Greenberg, “Flux growth and properties of KTiOPO4,” J. Cryst. Growth 70, 484 (1984).
[Crossref]

Gunter, P.

P. Gunter, P. M. Asbeck, and S. K. Kurtz, “Second harmonic generation with Ga1−xAlxAs lasers and KNbO3crystal,” Appl. Phys. Lett. 35, 461 (1979).
[Crossref]

Hanson, F.

Hobden, M. H.

M. H. Hobden, “Phase matched second harmonic generation in biaxial crystals,” J. Appl. Phys. 38, 4365 (1967).
[Crossref]

Hu, B. Q.

Huang, C.

A. Borsutzky, R. Brunger, C. Huang, and R. Wallenstein, “Harmonic and sum-frequency generation of pulsed laser radiation in BBO, LBO and KD*P,” Appl. Phys. B 52, 55 (1991).
[Crossref]

Huang, C. E.

Huang, Y.

C. Chen, B. Wu, G. You, A. Jiang, and Y. Huang, “High efficiency and wide band second harmonic generation properties of the new crystal β-BaB2O4,” in Digest of International Quantum Electronics Conference (Optical Society of America, Washington, D.C., 1984), paper MCC5.

Inaba, H.

H. Ito, H. Naito, and H. Inaba, “Generalized study on angular dependence of induced second order nonlinear optical polarizations and phase matching in biaxial crystals,” J. Appl. Phys. 46, 3992 (1975).
[Crossref]

Ito, H.

H. Ito, H. Naito, and H. Inaba, “Generalized study on angular dependence of induced second order nonlinear optical polarizations and phase matching in biaxial crystals,” J. Appl. Phys. 46, 3992 (1975).
[Crossref]

Jacco, J. C.

J. C. Jacco, G. M. Loiacono, M. Jaso, G. Mizell, and B. Greenberg, “Flux growth and properties of KTiOPO4,” J. Cryst. Growth 70, 484 (1984).
[Crossref]

Jaso, M.

J. C. Jacco, G. M. Loiacono, M. Jaso, G. Mizell, and B. Greenberg, “Flux growth and properties of KTiOPO4,” J. Cryst. Growth 70, 484 (1984).
[Crossref]

Jiang, A.

C. Chen, Y. Wu, A. Jiang, B. Wu, G. You, R. Li, and S. Lin, “New nonlinear optical crystal: LiB3O5,” J. Opt. Soc. Am. B 6, 616 (1989).
[Crossref]

C. Chen, B. Wu, A. Jiang, and G. You, “A new type ultraviolet SHG crystal β-BaB2O4,” Sci. Sin. Ser. B 28, 235 (1985).

C. Chen, B. Wu, G. You, A. Jiang, and Y. Huang, “High efficiency and wide band second harmonic generation properties of the new crystal β-BaB2O4,” in Digest of International Quantum Electronics Conference (Optical Society of America, Washington, D.C., 1984), paper MCC5.

Kato, K.

J. T. Lin, J. L. Montgomery, and K. Kato, “Temperature-tuned non-critically phase-matched frequency conversion in LiB3O5 crystal,” Opt. Commun. 80, 159 (1990).
[Crossref]

Kurtz, S. K.

P. Gunter, P. M. Asbeck, and S. K. Kurtz, “Second harmonic generation with Ga1−xAlxAs lasers and KNbO3crystal,” Appl. Phys. Lett. 35, 461 (1979).
[Crossref]

S. K. Kurtz and T. T. Terry, “A powder technique for the evaluation of nonlinear optical materials,” J. Appl. Phys. 39, 3798 (1968).
[Crossref]

Lane, R.

C. L. Tang, W. Bosenberg, T. Ukaihi, R. Lane, and L. K. Cheng, “NLO materials displaying superior performance,” Laser Focus World (September1990), p. 87.

Li, R.

Lin, J. T.

J. T. Lin, J. L. Montgomery, and K. Kato, “Temperature-tuned non-critically phase-matched frequency conversion in LiB3O5 crystal,” Opt. Commun. 80, 159 (1990).
[Crossref]

J. Q. Yao, J. T. Lin, and W. Sheng, “Analysis of BBO and KTP crystals for optical parametric oscillation,” in Conference on Lasers and Electro-Optics, Vol. 7 of 1988 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1988), paper TuJ5.

Lin, S.

S. Lin, Z. Sun, B. Wu, and C. Chen, “The nonlinear optical characteristics of a LiB3O5crystal,” J. Appl. Phys 67, 634 (1990).
[Crossref]

C. Chen, Y. Wu, A. Jiang, B. Wu, G. You, R. Li, and S. Lin, “New nonlinear optical crystal: LiB3O5,” J. Opt. Soc. Am. B 6, 616 (1989).
[Crossref]

Liu, K. C.

J. Q. Yao and K. C. Liu, “The optimum phase matching and experimental study of biaxial crystal KTP,” in Proceedings of Topical Meeting on Laser Materials and Spectroscopy (World Scientific, Singapore, 1988), paper WeO7.

Liu, Y.

J. Q. Yao and Y. Liu, “Intracavity frequency doubling of quasi-CW pumped YAG laser,” in Proceedings of Second Interdisciplinary Laser Science Conference (American Physical Society, New York, 1986), paper ThL63.

Liu, Y. S.

Y. S. Liu, L. Drafall, D. Dentz, and R. Belt, “Nonlinear optical phase matching properties of KTiOPO4,” Gen. Electr. Tech. Inf. Ser. Rep. 82CRD016 (General Electric Company, Schenectady, N.Y., 1982).

Loiacono, G. M.

J. C. Jacco, G. M. Loiacono, M. Jaso, G. Mizell, and B. Greenberg, “Flux growth and properties of KTiOPO4,” J. Cryst. Growth 70, 484 (1984).
[Crossref]

Mizell, G.

J. C. Jacco, G. M. Loiacono, M. Jaso, G. Mizell, and B. Greenberg, “Flux growth and properties of KTiOPO4,” J. Cryst. Growth 70, 484 (1984).
[Crossref]

Montgomery, J. L.

J. T. Lin, J. L. Montgomery, and K. Kato, “Temperature-tuned non-critically phase-matched frequency conversion in LiB3O5 crystal,” Opt. Commun. 80, 159 (1990).
[Crossref]

Morgan, R.

Naito, H.

H. Ito, H. Naito, and H. Inaba, “Generalized study on angular dependence of induced second order nonlinear optical polarizations and phase matching in biaxial crystals,” J. Appl. Phys. 46, 3992 (1975).
[Crossref]

Olson, D. H.

A. A. Ballman, H. Brown, D. H. Olson, and C. E. Rice, “Growth of potassium titanyl phosphate (KTP) from molten tungstate melts,” J. Cryst. Growth 75, 390 (1986).
[Crossref]

Rice, C. E.

A. A. Ballman, H. Brown, D. H. Olson, and C. E. Rice, “Growth of potassium titanyl phosphate (KTP) from molten tungstate melts,” J. Cryst. Growth 75, 390 (1986).
[Crossref]

Sheng, W.

J. Q. Yao, J. T. Lin, and W. Sheng, “Analysis of BBO and KTP crystals for optical parametric oscillation,” in Conference on Lasers and Electro-Optics, Vol. 7 of 1988 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1988), paper TuJ5.

Sun, D.

J. Q. Yao and D. Sun, “Multi-frequency conversions using one biaxial crystal KTiOPO4,” in Proceedings of Third Interdisciplinary Laser Science Conference (American Physical Society, New York, 1987), paper WO9.

Sun, Z.

S. Lin, Z. Sun, B. Wu, and C. Chen, “The nonlinear optical characteristics of a LiB3O5crystal,” J. Appl. Phys 67, 634 (1990).
[Crossref]

Tang, C. L.

C. L. Tang, W. Bosenberg, T. Ukaihi, R. Lane, and L. K. Cheng, “NLO materials displaying superior performance,” Laser Focus World (September1990), p. 87.

Terry, T. T.

S. K. Kurtz and T. T. Terry, “A powder technique for the evaluation of nonlinear optical materials,” J. Appl. Phys. 39, 3798 (1968).
[Crossref]

Ukaihi, T.

C. L. Tang, W. Bosenberg, T. Ukaihi, R. Lane, and L. K. Cheng, “NLO materials displaying superior performance,” Laser Focus World (September1990), p. 87.

Vanherzeele, H.

Wallenstein, R.

A. Borsutzky, R. Brunger, C. Huang, and R. Wallenstein, “Harmonic and sum-frequency generation of pulsed laser radiation in BBO, LBO and KD*P,” Appl. Phys. B 52, 55 (1991).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Oxford U. Press, New York, 1970).

Wu, B.

S. Lin, Z. Sun, B. Wu, and C. Chen, “The nonlinear optical characteristics of a LiB3O5crystal,” J. Appl. Phys 67, 634 (1990).
[Crossref]

C. Chen, Y. Wu, A. Jiang, B. Wu, G. You, R. Li, and S. Lin, “New nonlinear optical crystal: LiB3O5,” J. Opt. Soc. Am. B 6, 616 (1989).
[Crossref]

C. Chen, B. Wu, A. Jiang, and G. You, “A new type ultraviolet SHG crystal β-BaB2O4,” Sci. Sin. Ser. B 28, 235 (1985).

C. Chen, B. Wu, G. You, A. Jiang, and Y. Huang, “High efficiency and wide band second harmonic generation properties of the new crystal β-BaB2O4,” in Digest of International Quantum Electronics Conference (Optical Society of America, Washington, D.C., 1984), paper MCC5.

Wu, Y.

Wyncke, B.

F. Brehat and B. Wyncke, “Calculation of double refraction walk-off angle along the phase matching directions in nonlinear biaxial crystals,” J. Phys. B 22, 1891 (1989).
[Crossref]

B. Wyncke and F. Brehat, “Calculation of the effective second-order nonlinear coefficients along the phase matching directions in acentric orthorhombic biaxial crystals,” J. Phys. B 22, 363 (1989).
[Crossref]

Xue, B.

J. Q. Yao and B. Xue, “Optimum parameters of high conversion efficiency intracavity frequency doubled laser with Gaussian-like beam,” in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1984), paper ThC6.

Yao, J. Q.

J. Q. Yao and T. S. Fahlen, “Calculation of optimum phase match parameters for the biaxial crystal KTiOPO4,” J. Appl. Phys. 55, 65 (1984).
[Crossref]

J. Q. Yao and Y. Liu, “Intracavity frequency doubling of quasi-CW pumped YAG laser,” in Proceedings of Second Interdisciplinary Laser Science Conference (American Physical Society, New York, 1986), paper ThL63.

J. Q. Yao and B. Xue, “Optimum parameters of high conversion efficiency intracavity frequency doubled laser with Gaussian-like beam,” in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1984), paper ThC6.

J. Q. Yao, J. T. Lin, and W. Sheng, “Analysis of BBO and KTP crystals for optical parametric oscillation,” in Conference on Lasers and Electro-Optics, Vol. 7 of 1988 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1988), paper TuJ5.

J. Q. Yao and D. Sun, “Multi-frequency conversions using one biaxial crystal KTiOPO4,” in Proceedings of Third Interdisciplinary Laser Science Conference (American Physical Society, New York, 1987), paper WO9.

J. Q. Yao and K. C. Liu, “The optimum phase matching and experimental study of biaxial crystal KTP,” in Proceedings of Topical Meeting on Laser Materials and Spectroscopy (World Scientific, Singapore, 1988), paper WeO7.

You, G.

C. Chen, Y. Wu, A. Jiang, B. Wu, G. You, R. Li, and S. Lin, “New nonlinear optical crystal: LiB3O5,” J. Opt. Soc. Am. B 6, 616 (1989).
[Crossref]

C. Chen, B. Wu, A. Jiang, and G. You, “A new type ultraviolet SHG crystal β-BaB2O4,” Sci. Sin. Ser. B 28, 235 (1985).

C. Chen, B. Wu, G. You, A. Jiang, and Y. Huang, “High efficiency and wide band second harmonic generation properties of the new crystal β-BaB2O4,” in Digest of International Quantum Electronics Conference (Optical Society of America, Washington, D.C., 1984), paper MCC5.

Zumsteg, F C.

Zumsteg, F. C.

F. C. Zumsteg, J. D. Bierlein, and T. E. Gier, “KxRb1−xTiOPO4: a new nonlinear material,” J. Appl. Phys 47, 4980 (1976).
[Crossref]

Appl. Opt. (3)

Appl. Phys. B (1)

A. Borsutzky, R. Brunger, C. Huang, and R. Wallenstein, “Harmonic and sum-frequency generation of pulsed laser radiation in BBO, LBO and KD*P,” Appl. Phys. B 52, 55 (1991).
[Crossref]

Appl. Phys. Lett. (1)

P. Gunter, P. M. Asbeck, and S. K. Kurtz, “Second harmonic generation with Ga1−xAlxAs lasers and KNbO3crystal,” Appl. Phys. Lett. 35, 461 (1979).
[Crossref]

J. Appl. Phys (2)

F. C. Zumsteg, J. D. Bierlein, and T. E. Gier, “KxRb1−xTiOPO4: a new nonlinear material,” J. Appl. Phys 47, 4980 (1976).
[Crossref]

S. Lin, Z. Sun, B. Wu, and C. Chen, “The nonlinear optical characteristics of a LiB3O5crystal,” J. Appl. Phys 67, 634 (1990).
[Crossref]

J. Appl. Phys. (4)

J. Q. Yao and T. S. Fahlen, “Calculation of optimum phase match parameters for the biaxial crystal KTiOPO4,” J. Appl. Phys. 55, 65 (1984).
[Crossref]

S. K. Kurtz and T. T. Terry, “A powder technique for the evaluation of nonlinear optical materials,” J. Appl. Phys. 39, 3798 (1968).
[Crossref]

M. H. Hobden, “Phase matched second harmonic generation in biaxial crystals,” J. Appl. Phys. 38, 4365 (1967).
[Crossref]

H. Ito, H. Naito, and H. Inaba, “Generalized study on angular dependence of induced second order nonlinear optical polarizations and phase matching in biaxial crystals,” J. Appl. Phys. 46, 3992 (1975).
[Crossref]

J. Cryst. Growth (2)

J. C. Jacco, G. M. Loiacono, M. Jaso, G. Mizell, and B. Greenberg, “Flux growth and properties of KTiOPO4,” J. Cryst. Growth 70, 484 (1984).
[Crossref]

A. A. Ballman, H. Brown, D. H. Olson, and C. E. Rice, “Growth of potassium titanyl phosphate (KTP) from molten tungstate melts,” J. Cryst. Growth 75, 390 (1986).
[Crossref]

J. Opt. Soc. Am. B (2)

J. Phys. B (2)

B. Wyncke and F. Brehat, “Calculation of the effective second-order nonlinear coefficients along the phase matching directions in acentric orthorhombic biaxial crystals,” J. Phys. B 22, 363 (1989).
[Crossref]

F. Brehat and B. Wyncke, “Calculation of double refraction walk-off angle along the phase matching directions in nonlinear biaxial crystals,” J. Phys. B 22, 1891 (1989).
[Crossref]

Laser Focus World (2)

C. L. Tang, W. Bosenberg, T. Ukaihi, R. Lane, and L. K. Cheng, “NLO materials displaying superior performance,” Laser Focus World (September1990), p. 87.

C. Chen, “Chinese lab grows new nonlinear optical borate crystals,” Laser Focus World 25 (11), 129 (1989).

Opt. Commun. (1)

J. T. Lin, J. L. Montgomery, and K. Kato, “Temperature-tuned non-critically phase-matched frequency conversion in LiB3O5 crystal,” Opt. Commun. 80, 159 (1990).
[Crossref]

Opt. Lett. (1)

Sci. Sin. Ser. B (1)

C. Chen, B. Wu, A. Jiang, and G. You, “A new type ultraviolet SHG crystal β-BaB2O4,” Sci. Sin. Ser. B 28, 235 (1985).

Other (9)

C. Chen, B. Wu, G. You, A. Jiang, and Y. Huang, “High efficiency and wide band second harmonic generation properties of the new crystal β-BaB2O4,” in Digest of International Quantum Electronics Conference (Optical Society of America, Washington, D.C., 1984), paper MCC5.

T. E. Gier, “Method for flux growth of KTiOPO4 and its analogues,” U.S. Patent4,231,838 (November1980).

J. Q. Yao and B. Xue, “Optimum parameters of high conversion efficiency intracavity frequency doubled laser with Gaussian-like beam,” in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1984), paper ThC6.

J. Q. Yao and Y. Liu, “Intracavity frequency doubling of quasi-CW pumped YAG laser,” in Proceedings of Second Interdisciplinary Laser Science Conference (American Physical Society, New York, 1986), paper ThL63.

J. Q. Yao and D. Sun, “Multi-frequency conversions using one biaxial crystal KTiOPO4,” in Proceedings of Third Interdisciplinary Laser Science Conference (American Physical Society, New York, 1987), paper WO9.

J. Q. Yao, J. T. Lin, and W. Sheng, “Analysis of BBO and KTP crystals for optical parametric oscillation,” in Conference on Lasers and Electro-Optics, Vol. 7 of 1988 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1988), paper TuJ5.

J. Q. Yao and K. C. Liu, “The optimum phase matching and experimental study of biaxial crystal KTP,” in Proceedings of Topical Meeting on Laser Materials and Spectroscopy (World Scientific, Singapore, 1988), paper WeO7.

Y. S. Liu, L. Drafall, D. Dentz, and R. Belt, “Nonlinear optical phase matching properties of KTiOPO4,” Gen. Electr. Tech. Inf. Ser. Rep. 82CRD016 (General Electric Company, Schenectady, N.Y., 1982).

M. Born and E. Wolf, Principles of Optics (Oxford U. Press, New York, 1970).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (14)

Fig. 1
Fig. 1

Phase-matching curve of KTP for SHG.

Fig. 2
Fig. 2

Effective nonlinear coefficient deff of KTP versus wavelength for SHG.

Fig. 3
Fig. 3

Acceptance parameters of KTP for SHG versus wavelength: (A) (Δθ)l, (B) (Δφ)l, (C) (Δλ)l.

Fig. 4
Fig. 4

Walk-off angles of KTP versus wavelength for SHG.

Fig. 5
Fig. 5

Efficiency η of KTP versus (A) θ, (B) φ, and (C) λ for 1.064–0.532-μm SHG.

Fig. 6
Fig. 6

(A) Phase-matching curves and (B) deff of a 0.532-μm pumped OPO with KTP.

Fig. 7
Fig. 7

Acceptance parameters of a 0.532-μm pumped OPO with KTP: (A) (Δθ)l, (B) (Δφ)l, (C) (Δλ)l.

Fig. 8
Fig. 8

Walk-off angle of a 0.532-μm pumped OPO with KTP.

Fig. 9
Fig. 9

(A) Phase-matching curves and (B) deff of LBO versus wavelength for SHG.

Fig. 10
Fig. 10

Acceptance parameters of LBO for SHG versus wavelength: (A) (Δθ)l, (B) (Δφ)l, (C) (Δλ)l.

Fig. 11
Fig. 11

Walk-off angle of LBO for SHG versus wavelength.

Fig. 12
Fig. 12

(A) Phase-matching curves and (B) deff of a 0.355-μm pumped OPO with LBO.

Fig. 13
Fig. 13

Acceptance angles (A) (Δθ)l and (B) (Δφ)l of a 0.355-μm pumped OPO with LBO.

Fig. 14
Fig. 14

Walk-off angle of a 0.355-μm pumped OPO with LBO.

Equations (75)

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

sin 2 θ cos 2 φ n i 2 n x i 2 + sin 2 θ sin 2 φ n i 2 n y i 2 + cos 2 θ n i 2 n z i 2 = 0 ,
ω 1 n 1 + ω 2 n 2 = ω 3 n 3 ,
ω 1 n 1 + ω 2 n 2 = ω 3 n 3 ,
P ( ω 3 ) = â i d ijk â j â k E ( ω 1 ) E ( ω 2 ) = d eff E ( ω 1 ) E ( ω 2 ) ,
d eff = a i d ijk a j a k .
D j e 1 ( ω i ) = b j e 1 D e 1 ( ω i ) , D j e 2 ( ω i ) = b j e 2 D e 2 ( ω i ) ,
b ̂ e 1 = ( cos θ cos φ cos δ i sin φ sin δ i cos θ sin φ cos δ i + cos φ sin δ i sin θ cos δ i ) = ( b 1 e 1 b 2 e 1 b 3 e 1 ) ,
b ̂ e 2 = ( cos θ cos φ cos δ i sin φ cos δ i cos θ sin φ sin δ i + cos φ cos δ i sin θ sin δ i ) = ( b 1 e 2 b 2 e 2 b 3 e 2 ) ,
cot 2 δ i = cot 2 Ω i sin 2 θ cos 2 θ cos 2 φ + sin 2 φ cos θ sin 2 φ , tan Ω i = n 3 ( ω i ) n 1 ( ω i ) [ n 2 2 ( ω i ) n 1 2 ( ω i ) n 3 2 ( ω i ) n 2 2 ( ω i ) ] 1 / 2 ,
( E 1 E 2 E 3 ) = ( 1 / n 1 2 0 0 0 1 / n 2 2 0 0 0 1 / n 3 2 ) ( D 1 D 2 D 3 ) .
E e 1 ( ω i ) [ ( b 1 e 1 ) 2 n 1 4 ( ω i ) + ( b 2 e 1 ) 2 n 2 4 ( ω i ) + ( b 3 e 1 ) 2 n 3 4 ( ω i ) ] 1 / 2 D e 1 ( ω i ) = P ( ω i ) D e 1 ( ω i ) ,
E e 2 ( ω i ) = [ ( b 1 e 2 ) 2 n 1 4 ( ω i ) + ( b 2 e 2 ) 2 n 2 4 ( ω i ) + ( b 3 e 2 ) 2 n 3 4 ( ω i ) ] 1 / 2 D e 2 ( ω i ) = Q ( ω i ) D e 2 ( ω i ) .
â e 1 = 1 E e 1 ( ω i ) ( E 1 e 1 ( ω i ) E 2 e 1 ( ω i ) E 3 e 1 ( ω i ) ) = 1 P ( ω i ) ( n 1 2 ( ω i ) b 1 e 1 n 2 2 ( ω i ) b 2 e 1 n 3 2 ( ω i ) b 3 e 1 ) = ( a 1 e 1 a 2 e 1 a 3 e 1 ) ,
â e 2 = 1 Q ( ω i ) ( n 1 2 ( ω i ) b 1 e 2 n 2 2 ( ω i ) b 2 e 2 n 3 2 ( ω i ) b 3 e 2 ) = ( a 1 e 2 a 2 e 2 a 3 e 2 ) ,
â e 1 â e 1 = ( ( a 1 e 1 ) 2 ( a 2 e 1 ) 2 ( a 3 e 1 ) 2 2 a 2 e 1 a 3 e 1 2 a 1 e 1 a 3 e 1 2 a 1 e 1 a 2 e 1 ) = ( A 11 A 12 A 13 A 14 A 15 A 16 ) ,
â e 1 â e 2 = ( a 1 e 1 a 1 e 2 a 2 e 1 a 2 e 2 a 3 e 1 a 3 e 2 a 2 e 1 a 3 e 2 + a 3 e 1 a 2 e 2 a 1 e 1 a 3 e 2 + a 3 e 1 a 1 e 2 a 1 e 1 a 2 e 2 + a 2 e 1 a 1 e 2 ) = ( A 21 A 22 A 23 A 24 A 25 A 26 ) .
d eff ( I ) = ( a 1 e 2 a 2 e 2 a 3 e 2 ) d ijk ( A 11 A 12 A 13 A 14 A 15 A 16 ) ,
d eff ( II ) = ( a 1 e 2 a 2 e 2 a 3 e 2 ) d ijk ( A 21 A 22 A 23 A 24 A 25 A 26 ) .
α ω i = arccos ( [ b 1 e i ( ω i ) n 1 ( ω i ) ] 2 + [ b 2 e i ( ω i ) n 2 ( ω i ) ] 2 + [ b 3 e i ( ω i ) n 3 ( ω i ) ] 2 { [ b 1 e i n 1 2 ( ω i ) ] 2 + [ b 2 e i n 2 2 ( ω i ) ] 2 + [ b 3 e i n 3 2 ( ω i ) ] 2 } 1 / 2 ) ,
Δ K = k ( ω 3 ) k ( ω 2 ) k ( ω 1 ) = 2 π λ 3 n ( ω 3 ) 2 π λ 2 n ( ω 2 ) 2 π λ 1 n ( ω 1 ) .
Δ K = Δ K θ = θ m φ = φ m + Δ K θ θ = θ m φ = φ m Δ θ + 1 2 2 Δ K θ 2 θ = θ m φ = φ m ( Δ θ ) 2 + ,
Δ K = Δ K θ = θ m φ = φ m + Δ K φ θ = θ m φ = φ m Δ φ + 1 2 2 Δ K φ 2 θ = θ m φ = φ m ( Δ φ ) 2 + .
Δ K θ = 2 π λ 3 n ( ω 3 ) θ 2 π λ 2 n ( ω 2 ) θ 2 π λ 1 n ( ω 1 ) θ ,
2 Δ K θ 2 = 2 π λ 3 2 n ( ω 3 ) θ 2 2 π λ 2 2 n ( ω 2 ) θ 2 2 π λ 1 2 n ( ω 1 ) θ 2 .
n ( ω i ) = [ 2 B i ± ( B i 2 4 C i ) 1 / 2 ] 1 / 2 ,
n ( ω i ) θ = 2 2 { B i ± [ B i ± ( B i 2 4 C i ) 1 / 2 ] 1 / 2 } 3 / 2 × [ B i θ ± B i B i θ 2 C i θ ( B i 2 4 C i ) 1 / 2 ] ,
2 n ( ω i ) θ 2 = θ [ n ( ω i ) θ ] .
B i = ( sin 2 θ ) ( cos 2 φ ) [ n y 2 ( ω i ) + n z 2 ( ω i ) ] ( sin 2 θ ) ( sin 2 φ ) [ n x 2 ( ω i ) + n z 2 ( ω i ) ] ( cos 2 θ ) × [ n x 2 ( ω i ) + n y 2 ( ω i ) ] ,
C i = ( sin 2 θ ) ( cos 2 φ ) n y 2 ( ω i ) n z 2 ( ω i ) + ( sin 2 θ ) ( sin 2 φ ) × n x 2 ( ω i ) n z 2 ( ω i ) + ( cos 2 θ ) n x 2 ( ω i ) n y 2 ( ω i ) .
B i θ = [ n x 2 ( ω i ) + n y 2 ( ω i ) n z 2 ( ω i ) n x 2 ( ω i ) sin 2 φ n y 2 ( ω i ) cos 2 φ ] sin 2 θ ,
C i θ = [ n y 2 ( ω i ) n z 2 ( ω i ) cos 2 φ + n x 2 ( ω i ) n z 2 ( ω i ) sin 2 φ n x 2 ( ω i ) n y 2 ( ω i ) ] sin 2 θ .
( Δ K / 2 ) l = ± π
Δ K θ Δ θ + 1 2 2 Δ K θ 2 ( Δ θ ) 2 = 2 π l
Δ K θ Δ θ + 1 2 2 Δ K θ 2 ( Δ θ ) 2 = 2 π l .
B i φ = [ n y 2 ( ω i ) n x 2 ( ω i ) ] sin 2 θ sin 2 φ ,
C i φ = [ n x 2 ( ω i ) n y 2 ( ω i ) ] n z 2 ( ω i ) sin 2 θ sin 2 φ
Δ K φ Δ φ + 1 2 2 Δ K φ 2 ( Δ φ ) 2 = 2 π l ,
Δ K φ Δ φ + 1 2 2 Δ K φ 2 ( Δ φ ) 2 = 2 π l .
1 λ 1 + 1 λ 2 = 1 λ 3 , λ 1 = λ 2 .
Δ λ 1 = Δ λ 2 , Δ λ 3 = Δ λ 1 / 2 .
Δ λ 1 = ( λ 3 + Δ λ 3 ) ( λ 2 + Δ λ 2 ) ( λ 3 + Δ λ 3 ) ( λ 2 + Δ λ 2 ) λ 1
Δ λ 2 = ( λ 3 + Δ λ 3 ) ( λ 1 + Δ λ 1 ) ( λ 3 + Δ λ 3 ) ( λ 1 + Δ λ 1 ) λ 2 .
Δ λ 1 = ( λ 20 + k Δ λ 3 ) ( λ 30 + Δ λ 3 ) λ 20 λ 30 + ( k 1 ) Δ λ 3 λ 10 = f ( Δ λ 3 ) .
Δ K = Δ K λ 3 = λ 30 + d Δ K d λ 3 λ 3 = λ 30 Δ λ 3 + d 2 ( Δ K ) d λ 3 2 λ 3 = λ 30 ( Δ λ 3 ) 2 + .
d Δ K d λ 3 = 2 π [ λ 1 d n ( ω 1 ) d λ 3 n ( ω 1 ) d λ 1 d λ 3 λ 1 2 λ 2 d n ( ω 2 ) d λ 3 n ( ω 2 ) d λ 2 d λ 3 λ 2 2 + λ 3 d n ( ω 3 ) d λ 3 n ( ω 3 ) d λ 3 d λ 3 λ 3 2 ] ,
d λ 1 d λ 3 = d Δ λ 1 d Δ λ 3 = ( λ 20 λ 30 ) ( k λ 30 + λ 20 ) + 2 ( λ 20 λ 30 ) k Δ λ 3 + k ( k 1 ) Δ λ 3 2 [ λ 20 λ 30 + ( k 1 ) Δ λ 3 ] 2 ,
d λ 2 d λ 3 = d Δ λ 2 d Δ λ 3 = k .
d n ( ω i ) d λ 3 = 2 2 { B i ± [ B i ± ( B i 2 4 C i ) 1 / 2 ] 1 / 2 } 3 / 2 × [ d B i d λ 3 ± B i d B i d λ 3 2 d C i d λ 3 ( B i 2 4 C i ) 1 / 2 ] .
d B i d λ 3 = 2 ( sin 2 θ ) ( cos 2 φ ) [ n y 3 ( ω i ) d n y ( ω i ) d λ 3 + n z 3 ( ω i ) d n z ( ω i ) d λ 3 ] + 2 ( sin 2 θ ) ( sin 2 φ ) × [ n x 3 ( ω i ) d n x ( ω i ) d λ 3 + n z 3 ( ω i ) d n z ( ω i ) d λ 3 ] + 2 ( cos 2 θ ) × [ n x 3 ( ω i ) d n x ( ω i ) d λ 3 + n y 3 ( ω i ) d n y ( ω i ) d λ 3 ] ,
d C i d λ 3 = 2 ( sin 2 θ ) ( cos 2 φ ) [ n y 3 ( ω i ) n z 2 ( ω i ) d n y ( ω i ) d λ 3 + n y 2 ( ω i ) n z 3 ( ω i ) d n z ( ω i ) d λ 3 ] 2 ( sin 2 θ ) ( sin 2 φ ) × [ n x 3 ( ω i ) n z 2 ( ω i ) d n x ( ω i ) d λ 3 + n x 2 ( ω i ) n z 3 ( ω i ) d n z ( ω i ) d λ 3 ] 2 ( cos 2 θ ) [ n x 3 ( ω i ) n y 2 ( ω i ) d n x ( ω i ) d λ 3 + n x 2 ( ω i ) n y 3 ( ω i ) d n y ( ω i ) d λ 3 ] .
n j ( ω i ) = ( M j + N j λ i 2 P j Q j λ i 2 ) 1 / 2 ,
d n j ( ω i ) d λ 3 = 1 2 ( M j + N j λ i 2 P j Q j λ i 2 ) 1 / 2 × [ 2 λ i d λ i d λ 3 N j ( λ i P j ) 2 2 Q j λ i d λ i d λ 3 ] .
d Δ K d λ 3 Δ λ 3 + 1 2 d 2 Δ K d λ 3 2 ( Δ λ 3 ) 3 = 2 π l ,
d Δ K d λ 3 Δ λ 3 + 1 2 d 2 Δ K d λ 3 2 ( Δ λ 3 ) 2 = 2 π l .
Δ K = Δ K T T = T 0 Δ T + 1 2 2 Δ K T 2 T = T 0 ( Δ T ) 2 + .
Δ K = 2 π [ 1 λ 3 n ( ω 3 ) T 1 λ 2 n ( ω 2 ) T 1 λ 1 n ( ω 1 ) T ] Δ T .
( Δ T ) l = [ 1 λ 3 n ( ω 3 ) T 1 λ 2 n ( ω 2 ) T 1 λ 1 n ( ω 1 ) T ] 1 .
η sinc 2 [ ( Δ K ) l / 2 ] ,
× × E + 1 c 2 2 E t 2 = μ 0 2 P t 2 .
E j = A j ( z ) exp [ i ( k j z ω j t ) ] ,
d A 1 d z = i ω 1 2 μ 0 d eff k 1 cos 2 α 1 A 3 A 2 * exp [ i ( k 3 k 2 k 1 ) z ] ,
d A 2 d z = i ω 2 2 μ 0 d eff k 2 cos 2 α 2 A 3 A 1 * exp [ i ( k 3 k 2 k 1 ) z ] ,
d A 3 d z = i ω 3 2 μ 0 d eff k 3 cos 2 α 3 A 1 A 2 exp [ i ( k 3 k 2 k 1 ) z ] ,
A j ( z ) = ρ j ( z ) exp ( i γ j )
d [ ρ 1 ( z ) ] d z = μ 0 ω 1 2 d eff k 1 cos 2 α 1 ρ 3 ( z ) ρ 2 ( z ) sin [ β ( z ) ] ,
d [ ρ 2 ( z ) ] d z = μ 0 ω 2 2 d eff k 2 cos 2 α 2 ρ 3 ( z ) ρ 1 ( z ) sin [ β ( z ) ] ,
d [ ρ 3 ( z ) ] d z = μ 0 ω 3 2 d eff k 3 cos 2 α 3 ρ 1 ( z ) ρ 2 ( z ) sin [ β ( z ) ] ,
d [ β ( z ) ] d z = Δ k + μ 0 d eff ( ω 3 2 k 3 cos 2 α 3 ρ 1 ρ 2 ρ 3 ω 2 2 k 2 cos 2 α 2 ρ 1 ρ 3 ρ 2 ω 1 2 k 1 cos 2 α 1 ρ 2 ρ 3 ρ 1 ) cos [ β ( z ) ] ,
β ( z ) = γ 3 γ 1 γ 2 .
d ijk = ( 0 0 0 0 d 15 0 0 0 0 d 24 0 0 d 31 d 32 d 33 0 0 0 ) .
d eff ( I ) = C 21 d 15 A 15 + C 22 d 24 A 14 + C 23 ( d 31 A 11 + d 32 A 12 + d 33 A 13 ) ,
d eff ( II ) = C 21 d 15 A 25 + C 22 d 24 A 24 + C 23 ( d 31 A 21 + d 32 A 22 + d 33 A 23 ) .
d ijk = ( 0 0 0 0 0 d 15 d 31 d 33 d 32 0 0 0 0 0 0 d 24 0 0 ) .
d eff ( I ) = C 21 d 15 A 16 + C 22 ( d 31 A 11 + d 32 A 13 + d 33 A 12 ) + C 23 d 24 A 14 ,
d eff ( II ) = C 21 d 15 A 26 + C 22 ( d 31 A 21 + d 32 A 23 + d 33 A 22 ) + C 23 d 24 A 24 .

Metrics