Abstract

What we believe to be a new mathematical procedure has been proposed for analyzing the rotational Maker-fringe data taking account of the effects of multiple reflections and interferences of the fundamental and second-harmonic beams in anisotropic plane-parallel plates. In the present formalisms partial overlaps of the fundamental and second-harmonic beams are fully included. A comparison between numerical simulations based on the proposed formalisms and experimental data obtained for a high-quality (0001) plane-parallel plate of 6H-SiC revealed the validity of our theory. We have determined the magnitudes of the nonlinear optical coefficients of 6H-SiC as d31=d32=6.7±0.8pmV, d15=d24=6.4±1.1pmV, and d33=9.7±8.0pmV.

© 2008 Optical Society of America

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  1. P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M. Savage, “Effects of dispersion and focusing on the production of optical harmonics,” Phys. Rev. Lett. 8, 21-22 (1962).
    [CrossRef]
  2. N. Bloembergen and P. S. Pershan, “Light waves at the boundary of nonlinear media,” Phys. Rev. 128, 606-622 (1962).
    [CrossRef]
  3. J. Jerphagnon and S. K. Kurtz, “Maker fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals,” J. Appl. Phys. 41, 1667-1681 (1970).
    [CrossRef]
  4. W. N. Herman and L. M. Hayden, “Maker fringes revisited: second-harmonic generation from birefringent or absorbing materials,” J. Opt. Soc. Am. B 12, 416-427 (1995).
    [CrossRef]
  5. R. Morita, Y. Kaneda, A. Sugihashi, T. Kondo, N. Ogasawara, S. Umegaki, and R. Ito, “Multiple-reflection effects in optical second-harmonic generation,” Jpn. J. Appl. Phys., Part 2 27, L1134-L1136 (1988).
    [CrossRef]
  6. M. Ohashi, T. Kondo, K. Kumata, S. Fukatsu, S. S. Kano, Y. Shiraki, and R. Ito, “Determination of quadratic nonlinear optical coefficient of AlxGa1−xAs system by the method of reflected second harmonics,” J. Appl. Phys. 74, 596-601 (1993).
    [CrossRef]
  7. N. Hashizume, M. Ohashi, T. Kondo, and R. Ito, “Optical harmonic generation in multilayered structures: a comprehensive analysis,” J. Opt. Soc. Am. B 12, 1894-1904 (1995).
    [CrossRef]
  8. I. Shoji, T. Kondo, A. Kitamoto, M. Shirane, and R. Ito, “Absolute scale of second-order nonlinear-optical coefficients,” J. Opt. Soc. Am. B 14, 2268-2294 (1997).
    [CrossRef]
  9. I. Shoji, T. Kondo, and R. Ito, “Second-order nonlinear susceptibilities of various dielectric and semiconductor materials,” Opt. Quantum Electron. 34, 797-833 (2002).
    [CrossRef]
  10. N. A. Sanford, A. V. Davydov, D. V. Tsvetkov, A. V. Dmitriev, S. Keller, U. K. Mishra, S. P. DenBaars, S. S. Park, J. Y. Han, and R. J. Molnar, “Measurement of second order susceptibilities of GaN and AlGaN,” J. Appl. Phys. 97, 053512 (2005).
    [CrossRef]
  11. H. Hellwig and L. Bohatý, “Multiple reflections and Fabry-Perot interference corrections in Maker fringe experiments,” Opt. Commun. 161, 51-56 (1999).
    [CrossRef]
  12. N. A. Sanford and J. A. Aust, “Nonlinear optical characterization of LiNbO3. I. Theoretical analysis of Maker fringe patterns for x-cut wafers,” J. Opt. Soc. Am. B 15, 2885-2909 (1998).
    [CrossRef]
  13. J. A. Aust, “Maker-fringe analysis and electric-field poling of lithium niobate,” Ph.D. dissertation (University of Colorado, 1999).
  14. V. Gopalan, N. A. Sanford, J. A. Aust, K. Kitamura, and Y. Furukawa, in Handbook of Advanced Electronic and Photonic Materials and Devices: Ferroelectrics and DielectricsH.S.Nalwa, ed. (Academic, 2001), Vol. 4, p. 57.
    [CrossRef]
  15. J. E. Sipe, “New Green-function formalism for surface optics,” J. Opt. Soc. Am. B 4, 481-489 (1987).
    [CrossRef]
  16. R. J. Gehr, G. L. Fischer, R. W. Boyd, and J. E. Sipe, “Nonlinear optical response of layered composite materials,” Phys. Rev. A 53, 2792-2798 (1996).
    [CrossRef]
  17. P. S. Bechthold and S. Haussühl, “Nonlinear optical properties of orthorhombic barium formate and magnesium barium fluoride,” Appl. Phys. 14, 403-410 (1977).
    [CrossRef]
  18. S. X. Dou, M. H. Jiang, Z. S. Shao, and X. T. Tao, “Maker fringes in biaxial crystals and the nonlinear optical coefficients of thiosemicarbazide cadmium chloride monohydrate,” Appl. Phys. Lett. 54, 1101-1103 (1989).
    [CrossRef]
  19. P. T. B. Shaffer, “Refractive index, dispersion, and birefringence of silicon carbide polytypes,” Appl. Opt. 10, 1034-1036 (1971).
    [CrossRef]
  20. A. N. Pikhtin and A. D. Yaskov, “Refractive index and birefringence of semiconductors with the wurtzite structure,” Sov. Phys. Semicond. 15, 8-10 (1981).
  21. D. A. Roberts, “Simplified characterization of uniaxial and biaxial nonlinear optical crystals: a plea for standardization of nomenclature and conventions,” IEEE J. Quantum Electron. 28, 2057-2074 (1992).
    [CrossRef]
  22. S. Singh, J. R. Potopowicz, L. G. Van Uitert, and S. H. Wemple, “Nonlinear optical properties of hexagonal silicon carbide,” Appl. Phys. Lett. 19, 53-56 (1971).
    [CrossRef]
  23. P. M. Lundquist, W. P. Lin, G. K. Wong, M. Razeghi, and J. B. Ketterson, “Second harmonic generation in hexagonal silicon carbide,” Appl. Phys. Lett. 66, 1883-1885 (1995).
    [CrossRef]
  24. S. Niedermeier, H. Schillinger, R. Sauerbrey, B. Adolph, and F. Bechstedt, “Second-harmonic generation in silicon carbide polytypes,” Appl. Phys. Lett. 75, 618-620 (1999).
    [CrossRef]

2005 (1)

N. A. Sanford, A. V. Davydov, D. V. Tsvetkov, A. V. Dmitriev, S. Keller, U. K. Mishra, S. P. DenBaars, S. S. Park, J. Y. Han, and R. J. Molnar, “Measurement of second order susceptibilities of GaN and AlGaN,” J. Appl. Phys. 97, 053512 (2005).
[CrossRef]

2002 (1)

I. Shoji, T. Kondo, and R. Ito, “Second-order nonlinear susceptibilities of various dielectric and semiconductor materials,” Opt. Quantum Electron. 34, 797-833 (2002).
[CrossRef]

2001 (1)

V. Gopalan, N. A. Sanford, J. A. Aust, K. Kitamura, and Y. Furukawa, in Handbook of Advanced Electronic and Photonic Materials and Devices: Ferroelectrics and DielectricsH.S.Nalwa, ed. (Academic, 2001), Vol. 4, p. 57.
[CrossRef]

1999 (3)

S. Niedermeier, H. Schillinger, R. Sauerbrey, B. Adolph, and F. Bechstedt, “Second-harmonic generation in silicon carbide polytypes,” Appl. Phys. Lett. 75, 618-620 (1999).
[CrossRef]

J. A. Aust, “Maker-fringe analysis and electric-field poling of lithium niobate,” Ph.D. dissertation (University of Colorado, 1999).

H. Hellwig and L. Bohatý, “Multiple reflections and Fabry-Perot interference corrections in Maker fringe experiments,” Opt. Commun. 161, 51-56 (1999).
[CrossRef]

1998 (1)

1997 (1)

1996 (1)

R. J. Gehr, G. L. Fischer, R. W. Boyd, and J. E. Sipe, “Nonlinear optical response of layered composite materials,” Phys. Rev. A 53, 2792-2798 (1996).
[CrossRef]

1995 (3)

1993 (1)

M. Ohashi, T. Kondo, K. Kumata, S. Fukatsu, S. S. Kano, Y. Shiraki, and R. Ito, “Determination of quadratic nonlinear optical coefficient of AlxGa1−xAs system by the method of reflected second harmonics,” J. Appl. Phys. 74, 596-601 (1993).
[CrossRef]

1992 (1)

D. A. Roberts, “Simplified characterization of uniaxial and biaxial nonlinear optical crystals: a plea for standardization of nomenclature and conventions,” IEEE J. Quantum Electron. 28, 2057-2074 (1992).
[CrossRef]

1989 (1)

S. X. Dou, M. H. Jiang, Z. S. Shao, and X. T. Tao, “Maker fringes in biaxial crystals and the nonlinear optical coefficients of thiosemicarbazide cadmium chloride monohydrate,” Appl. Phys. Lett. 54, 1101-1103 (1989).
[CrossRef]

1988 (1)

R. Morita, Y. Kaneda, A. Sugihashi, T. Kondo, N. Ogasawara, S. Umegaki, and R. Ito, “Multiple-reflection effects in optical second-harmonic generation,” Jpn. J. Appl. Phys., Part 2 27, L1134-L1136 (1988).
[CrossRef]

1987 (1)

1981 (1)

A. N. Pikhtin and A. D. Yaskov, “Refractive index and birefringence of semiconductors with the wurtzite structure,” Sov. Phys. Semicond. 15, 8-10 (1981).

1977 (1)

P. S. Bechthold and S. Haussühl, “Nonlinear optical properties of orthorhombic barium formate and magnesium barium fluoride,” Appl. Phys. 14, 403-410 (1977).
[CrossRef]

1971 (2)

S. Singh, J. R. Potopowicz, L. G. Van Uitert, and S. H. Wemple, “Nonlinear optical properties of hexagonal silicon carbide,” Appl. Phys. Lett. 19, 53-56 (1971).
[CrossRef]

P. T. B. Shaffer, “Refractive index, dispersion, and birefringence of silicon carbide polytypes,” Appl. Opt. 10, 1034-1036 (1971).
[CrossRef]

1970 (1)

J. Jerphagnon and S. K. Kurtz, “Maker fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals,” J. Appl. Phys. 41, 1667-1681 (1970).
[CrossRef]

1962 (2)

P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M. Savage, “Effects of dispersion and focusing on the production of optical harmonics,” Phys. Rev. Lett. 8, 21-22 (1962).
[CrossRef]

N. Bloembergen and P. S. Pershan, “Light waves at the boundary of nonlinear media,” Phys. Rev. 128, 606-622 (1962).
[CrossRef]

Adolph, B.

S. Niedermeier, H. Schillinger, R. Sauerbrey, B. Adolph, and F. Bechstedt, “Second-harmonic generation in silicon carbide polytypes,” Appl. Phys. Lett. 75, 618-620 (1999).
[CrossRef]

Aust, J. A.

V. Gopalan, N. A. Sanford, J. A. Aust, K. Kitamura, and Y. Furukawa, in Handbook of Advanced Electronic and Photonic Materials and Devices: Ferroelectrics and DielectricsH.S.Nalwa, ed. (Academic, 2001), Vol. 4, p. 57.
[CrossRef]

J. A. Aust, “Maker-fringe analysis and electric-field poling of lithium niobate,” Ph.D. dissertation (University of Colorado, 1999).

N. A. Sanford and J. A. Aust, “Nonlinear optical characterization of LiNbO3. I. Theoretical analysis of Maker fringe patterns for x-cut wafers,” J. Opt. Soc. Am. B 15, 2885-2909 (1998).
[CrossRef]

Bechstedt, F.

S. Niedermeier, H. Schillinger, R. Sauerbrey, B. Adolph, and F. Bechstedt, “Second-harmonic generation in silicon carbide polytypes,” Appl. Phys. Lett. 75, 618-620 (1999).
[CrossRef]

Bechthold, P. S.

P. S. Bechthold and S. Haussühl, “Nonlinear optical properties of orthorhombic barium formate and magnesium barium fluoride,” Appl. Phys. 14, 403-410 (1977).
[CrossRef]

Bloembergen, N.

N. Bloembergen and P. S. Pershan, “Light waves at the boundary of nonlinear media,” Phys. Rev. 128, 606-622 (1962).
[CrossRef]

Bohatý, L.

H. Hellwig and L. Bohatý, “Multiple reflections and Fabry-Perot interference corrections in Maker fringe experiments,” Opt. Commun. 161, 51-56 (1999).
[CrossRef]

Boyd, R. W.

R. J. Gehr, G. L. Fischer, R. W. Boyd, and J. E. Sipe, “Nonlinear optical response of layered composite materials,” Phys. Rev. A 53, 2792-2798 (1996).
[CrossRef]

Davydov, A. V.

N. A. Sanford, A. V. Davydov, D. V. Tsvetkov, A. V. Dmitriev, S. Keller, U. K. Mishra, S. P. DenBaars, S. S. Park, J. Y. Han, and R. J. Molnar, “Measurement of second order susceptibilities of GaN and AlGaN,” J. Appl. Phys. 97, 053512 (2005).
[CrossRef]

DenBaars, S. P.

N. A. Sanford, A. V. Davydov, D. V. Tsvetkov, A. V. Dmitriev, S. Keller, U. K. Mishra, S. P. DenBaars, S. S. Park, J. Y. Han, and R. J. Molnar, “Measurement of second order susceptibilities of GaN and AlGaN,” J. Appl. Phys. 97, 053512 (2005).
[CrossRef]

Dmitriev, A. V.

N. A. Sanford, A. V. Davydov, D. V. Tsvetkov, A. V. Dmitriev, S. Keller, U. K. Mishra, S. P. DenBaars, S. S. Park, J. Y. Han, and R. J. Molnar, “Measurement of second order susceptibilities of GaN and AlGaN,” J. Appl. Phys. 97, 053512 (2005).
[CrossRef]

Dou, S. X.

S. X. Dou, M. H. Jiang, Z. S. Shao, and X. T. Tao, “Maker fringes in biaxial crystals and the nonlinear optical coefficients of thiosemicarbazide cadmium chloride monohydrate,” Appl. Phys. Lett. 54, 1101-1103 (1989).
[CrossRef]

Fischer, G. L.

R. J. Gehr, G. L. Fischer, R. W. Boyd, and J. E. Sipe, “Nonlinear optical response of layered composite materials,” Phys. Rev. A 53, 2792-2798 (1996).
[CrossRef]

Fukatsu, S.

M. Ohashi, T. Kondo, K. Kumata, S. Fukatsu, S. S. Kano, Y. Shiraki, and R. Ito, “Determination of quadratic nonlinear optical coefficient of AlxGa1−xAs system by the method of reflected second harmonics,” J. Appl. Phys. 74, 596-601 (1993).
[CrossRef]

Furukawa, Y.

V. Gopalan, N. A. Sanford, J. A. Aust, K. Kitamura, and Y. Furukawa, in Handbook of Advanced Electronic and Photonic Materials and Devices: Ferroelectrics and DielectricsH.S.Nalwa, ed. (Academic, 2001), Vol. 4, p. 57.
[CrossRef]

Gehr, R. J.

R. J. Gehr, G. L. Fischer, R. W. Boyd, and J. E. Sipe, “Nonlinear optical response of layered composite materials,” Phys. Rev. A 53, 2792-2798 (1996).
[CrossRef]

Gopalan, V.

V. Gopalan, N. A. Sanford, J. A. Aust, K. Kitamura, and Y. Furukawa, in Handbook of Advanced Electronic and Photonic Materials and Devices: Ferroelectrics and DielectricsH.S.Nalwa, ed. (Academic, 2001), Vol. 4, p. 57.
[CrossRef]

Han, J. Y.

N. A. Sanford, A. V. Davydov, D. V. Tsvetkov, A. V. Dmitriev, S. Keller, U. K. Mishra, S. P. DenBaars, S. S. Park, J. Y. Han, and R. J. Molnar, “Measurement of second order susceptibilities of GaN and AlGaN,” J. Appl. Phys. 97, 053512 (2005).
[CrossRef]

Hashizume, N.

Haussühl, S.

P. S. Bechthold and S. Haussühl, “Nonlinear optical properties of orthorhombic barium formate and magnesium barium fluoride,” Appl. Phys. 14, 403-410 (1977).
[CrossRef]

Hayden, L. M.

Hellwig, H.

H. Hellwig and L. Bohatý, “Multiple reflections and Fabry-Perot interference corrections in Maker fringe experiments,” Opt. Commun. 161, 51-56 (1999).
[CrossRef]

Herman, W. N.

Ito, R.

I. Shoji, T. Kondo, and R. Ito, “Second-order nonlinear susceptibilities of various dielectric and semiconductor materials,” Opt. Quantum Electron. 34, 797-833 (2002).
[CrossRef]

I. Shoji, T. Kondo, A. Kitamoto, M. Shirane, and R. Ito, “Absolute scale of second-order nonlinear-optical coefficients,” J. Opt. Soc. Am. B 14, 2268-2294 (1997).
[CrossRef]

N. Hashizume, M. Ohashi, T. Kondo, and R. Ito, “Optical harmonic generation in multilayered structures: a comprehensive analysis,” J. Opt. Soc. Am. B 12, 1894-1904 (1995).
[CrossRef]

M. Ohashi, T. Kondo, K. Kumata, S. Fukatsu, S. S. Kano, Y. Shiraki, and R. Ito, “Determination of quadratic nonlinear optical coefficient of AlxGa1−xAs system by the method of reflected second harmonics,” J. Appl. Phys. 74, 596-601 (1993).
[CrossRef]

R. Morita, Y. Kaneda, A. Sugihashi, T. Kondo, N. Ogasawara, S. Umegaki, and R. Ito, “Multiple-reflection effects in optical second-harmonic generation,” Jpn. J. Appl. Phys., Part 2 27, L1134-L1136 (1988).
[CrossRef]

Jerphagnon, J.

J. Jerphagnon and S. K. Kurtz, “Maker fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals,” J. Appl. Phys. 41, 1667-1681 (1970).
[CrossRef]

Jiang, M. H.

S. X. Dou, M. H. Jiang, Z. S. Shao, and X. T. Tao, “Maker fringes in biaxial crystals and the nonlinear optical coefficients of thiosemicarbazide cadmium chloride monohydrate,” Appl. Phys. Lett. 54, 1101-1103 (1989).
[CrossRef]

Kaneda, Y.

R. Morita, Y. Kaneda, A. Sugihashi, T. Kondo, N. Ogasawara, S. Umegaki, and R. Ito, “Multiple-reflection effects in optical second-harmonic generation,” Jpn. J. Appl. Phys., Part 2 27, L1134-L1136 (1988).
[CrossRef]

Kano, S. S.

M. Ohashi, T. Kondo, K. Kumata, S. Fukatsu, S. S. Kano, Y. Shiraki, and R. Ito, “Determination of quadratic nonlinear optical coefficient of AlxGa1−xAs system by the method of reflected second harmonics,” J. Appl. Phys. 74, 596-601 (1993).
[CrossRef]

Keller, S.

N. A. Sanford, A. V. Davydov, D. V. Tsvetkov, A. V. Dmitriev, S. Keller, U. K. Mishra, S. P. DenBaars, S. S. Park, J. Y. Han, and R. J. Molnar, “Measurement of second order susceptibilities of GaN and AlGaN,” J. Appl. Phys. 97, 053512 (2005).
[CrossRef]

Ketterson, J. B.

P. M. Lundquist, W. P. Lin, G. K. Wong, M. Razeghi, and J. B. Ketterson, “Second harmonic generation in hexagonal silicon carbide,” Appl. Phys. Lett. 66, 1883-1885 (1995).
[CrossRef]

Kitamoto, A.

Kitamura, K.

V. Gopalan, N. A. Sanford, J. A. Aust, K. Kitamura, and Y. Furukawa, in Handbook of Advanced Electronic and Photonic Materials and Devices: Ferroelectrics and DielectricsH.S.Nalwa, ed. (Academic, 2001), Vol. 4, p. 57.
[CrossRef]

Kondo, T.

I. Shoji, T. Kondo, and R. Ito, “Second-order nonlinear susceptibilities of various dielectric and semiconductor materials,” Opt. Quantum Electron. 34, 797-833 (2002).
[CrossRef]

I. Shoji, T. Kondo, A. Kitamoto, M. Shirane, and R. Ito, “Absolute scale of second-order nonlinear-optical coefficients,” J. Opt. Soc. Am. B 14, 2268-2294 (1997).
[CrossRef]

N. Hashizume, M. Ohashi, T. Kondo, and R. Ito, “Optical harmonic generation in multilayered structures: a comprehensive analysis,” J. Opt. Soc. Am. B 12, 1894-1904 (1995).
[CrossRef]

M. Ohashi, T. Kondo, K. Kumata, S. Fukatsu, S. S. Kano, Y. Shiraki, and R. Ito, “Determination of quadratic nonlinear optical coefficient of AlxGa1−xAs system by the method of reflected second harmonics,” J. Appl. Phys. 74, 596-601 (1993).
[CrossRef]

R. Morita, Y. Kaneda, A. Sugihashi, T. Kondo, N. Ogasawara, S. Umegaki, and R. Ito, “Multiple-reflection effects in optical second-harmonic generation,” Jpn. J. Appl. Phys., Part 2 27, L1134-L1136 (1988).
[CrossRef]

Kumata, K.

M. Ohashi, T. Kondo, K. Kumata, S. Fukatsu, S. S. Kano, Y. Shiraki, and R. Ito, “Determination of quadratic nonlinear optical coefficient of AlxGa1−xAs system by the method of reflected second harmonics,” J. Appl. Phys. 74, 596-601 (1993).
[CrossRef]

Kurtz, S. K.

J. Jerphagnon and S. K. Kurtz, “Maker fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals,” J. Appl. Phys. 41, 1667-1681 (1970).
[CrossRef]

Lin, W. P.

P. M. Lundquist, W. P. Lin, G. K. Wong, M. Razeghi, and J. B. Ketterson, “Second harmonic generation in hexagonal silicon carbide,” Appl. Phys. Lett. 66, 1883-1885 (1995).
[CrossRef]

Lundquist, P. M.

P. M. Lundquist, W. P. Lin, G. K. Wong, M. Razeghi, and J. B. Ketterson, “Second harmonic generation in hexagonal silicon carbide,” Appl. Phys. Lett. 66, 1883-1885 (1995).
[CrossRef]

Maker, P. D.

P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M. Savage, “Effects of dispersion and focusing on the production of optical harmonics,” Phys. Rev. Lett. 8, 21-22 (1962).
[CrossRef]

Mishra, U. K.

N. A. Sanford, A. V. Davydov, D. V. Tsvetkov, A. V. Dmitriev, S. Keller, U. K. Mishra, S. P. DenBaars, S. S. Park, J. Y. Han, and R. J. Molnar, “Measurement of second order susceptibilities of GaN and AlGaN,” J. Appl. Phys. 97, 053512 (2005).
[CrossRef]

Molnar, R. J.

N. A. Sanford, A. V. Davydov, D. V. Tsvetkov, A. V. Dmitriev, S. Keller, U. K. Mishra, S. P. DenBaars, S. S. Park, J. Y. Han, and R. J. Molnar, “Measurement of second order susceptibilities of GaN and AlGaN,” J. Appl. Phys. 97, 053512 (2005).
[CrossRef]

Morita, R.

R. Morita, Y. Kaneda, A. Sugihashi, T. Kondo, N. Ogasawara, S. Umegaki, and R. Ito, “Multiple-reflection effects in optical second-harmonic generation,” Jpn. J. Appl. Phys., Part 2 27, L1134-L1136 (1988).
[CrossRef]

Niedermeier, S.

S. Niedermeier, H. Schillinger, R. Sauerbrey, B. Adolph, and F. Bechstedt, “Second-harmonic generation in silicon carbide polytypes,” Appl. Phys. Lett. 75, 618-620 (1999).
[CrossRef]

Nisenoff, M.

P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M. Savage, “Effects of dispersion and focusing on the production of optical harmonics,” Phys. Rev. Lett. 8, 21-22 (1962).
[CrossRef]

Ogasawara, N.

R. Morita, Y. Kaneda, A. Sugihashi, T. Kondo, N. Ogasawara, S. Umegaki, and R. Ito, “Multiple-reflection effects in optical second-harmonic generation,” Jpn. J. Appl. Phys., Part 2 27, L1134-L1136 (1988).
[CrossRef]

Ohashi, M.

N. Hashizume, M. Ohashi, T. Kondo, and R. Ito, “Optical harmonic generation in multilayered structures: a comprehensive analysis,” J. Opt. Soc. Am. B 12, 1894-1904 (1995).
[CrossRef]

M. Ohashi, T. Kondo, K. Kumata, S. Fukatsu, S. S. Kano, Y. Shiraki, and R. Ito, “Determination of quadratic nonlinear optical coefficient of AlxGa1−xAs system by the method of reflected second harmonics,” J. Appl. Phys. 74, 596-601 (1993).
[CrossRef]

Park, S. S.

N. A. Sanford, A. V. Davydov, D. V. Tsvetkov, A. V. Dmitriev, S. Keller, U. K. Mishra, S. P. DenBaars, S. S. Park, J. Y. Han, and R. J. Molnar, “Measurement of second order susceptibilities of GaN and AlGaN,” J. Appl. Phys. 97, 053512 (2005).
[CrossRef]

Pershan, P. S.

N. Bloembergen and P. S. Pershan, “Light waves at the boundary of nonlinear media,” Phys. Rev. 128, 606-622 (1962).
[CrossRef]

Pikhtin, A. N.

A. N. Pikhtin and A. D. Yaskov, “Refractive index and birefringence of semiconductors with the wurtzite structure,” Sov. Phys. Semicond. 15, 8-10 (1981).

Potopowicz, J. R.

S. Singh, J. R. Potopowicz, L. G. Van Uitert, and S. H. Wemple, “Nonlinear optical properties of hexagonal silicon carbide,” Appl. Phys. Lett. 19, 53-56 (1971).
[CrossRef]

Razeghi, M.

P. M. Lundquist, W. P. Lin, G. K. Wong, M. Razeghi, and J. B. Ketterson, “Second harmonic generation in hexagonal silicon carbide,” Appl. Phys. Lett. 66, 1883-1885 (1995).
[CrossRef]

Roberts, D. A.

D. A. Roberts, “Simplified characterization of uniaxial and biaxial nonlinear optical crystals: a plea for standardization of nomenclature and conventions,” IEEE J. Quantum Electron. 28, 2057-2074 (1992).
[CrossRef]

Sanford, N. A.

N. A. Sanford, A. V. Davydov, D. V. Tsvetkov, A. V. Dmitriev, S. Keller, U. K. Mishra, S. P. DenBaars, S. S. Park, J. Y. Han, and R. J. Molnar, “Measurement of second order susceptibilities of GaN and AlGaN,” J. Appl. Phys. 97, 053512 (2005).
[CrossRef]

V. Gopalan, N. A. Sanford, J. A. Aust, K. Kitamura, and Y. Furukawa, in Handbook of Advanced Electronic and Photonic Materials and Devices: Ferroelectrics and DielectricsH.S.Nalwa, ed. (Academic, 2001), Vol. 4, p. 57.
[CrossRef]

N. A. Sanford and J. A. Aust, “Nonlinear optical characterization of LiNbO3. I. Theoretical analysis of Maker fringe patterns for x-cut wafers,” J. Opt. Soc. Am. B 15, 2885-2909 (1998).
[CrossRef]

Sauerbrey, R.

S. Niedermeier, H. Schillinger, R. Sauerbrey, B. Adolph, and F. Bechstedt, “Second-harmonic generation in silicon carbide polytypes,” Appl. Phys. Lett. 75, 618-620 (1999).
[CrossRef]

Savage, C. M.

P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M. Savage, “Effects of dispersion and focusing on the production of optical harmonics,” Phys. Rev. Lett. 8, 21-22 (1962).
[CrossRef]

Schillinger, H.

S. Niedermeier, H. Schillinger, R. Sauerbrey, B. Adolph, and F. Bechstedt, “Second-harmonic generation in silicon carbide polytypes,” Appl. Phys. Lett. 75, 618-620 (1999).
[CrossRef]

Shaffer, P. T. B.

Shao, Z. S.

S. X. Dou, M. H. Jiang, Z. S. Shao, and X. T. Tao, “Maker fringes in biaxial crystals and the nonlinear optical coefficients of thiosemicarbazide cadmium chloride monohydrate,” Appl. Phys. Lett. 54, 1101-1103 (1989).
[CrossRef]

Shiraki, Y.

M. Ohashi, T. Kondo, K. Kumata, S. Fukatsu, S. S. Kano, Y. Shiraki, and R. Ito, “Determination of quadratic nonlinear optical coefficient of AlxGa1−xAs system by the method of reflected second harmonics,” J. Appl. Phys. 74, 596-601 (1993).
[CrossRef]

Shirane, M.

Shoji, I.

I. Shoji, T. Kondo, and R. Ito, “Second-order nonlinear susceptibilities of various dielectric and semiconductor materials,” Opt. Quantum Electron. 34, 797-833 (2002).
[CrossRef]

I. Shoji, T. Kondo, A. Kitamoto, M. Shirane, and R. Ito, “Absolute scale of second-order nonlinear-optical coefficients,” J. Opt. Soc. Am. B 14, 2268-2294 (1997).
[CrossRef]

Singh, S.

S. Singh, J. R. Potopowicz, L. G. Van Uitert, and S. H. Wemple, “Nonlinear optical properties of hexagonal silicon carbide,” Appl. Phys. Lett. 19, 53-56 (1971).
[CrossRef]

Sipe, J. E.

R. J. Gehr, G. L. Fischer, R. W. Boyd, and J. E. Sipe, “Nonlinear optical response of layered composite materials,” Phys. Rev. A 53, 2792-2798 (1996).
[CrossRef]

J. E. Sipe, “New Green-function formalism for surface optics,” J. Opt. Soc. Am. B 4, 481-489 (1987).
[CrossRef]

Sugihashi, A.

R. Morita, Y. Kaneda, A. Sugihashi, T. Kondo, N. Ogasawara, S. Umegaki, and R. Ito, “Multiple-reflection effects in optical second-harmonic generation,” Jpn. J. Appl. Phys., Part 2 27, L1134-L1136 (1988).
[CrossRef]

Tao, X. T.

S. X. Dou, M. H. Jiang, Z. S. Shao, and X. T. Tao, “Maker fringes in biaxial crystals and the nonlinear optical coefficients of thiosemicarbazide cadmium chloride monohydrate,” Appl. Phys. Lett. 54, 1101-1103 (1989).
[CrossRef]

Terhune, R. W.

P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M. Savage, “Effects of dispersion and focusing on the production of optical harmonics,” Phys. Rev. Lett. 8, 21-22 (1962).
[CrossRef]

Tsvetkov, D. V.

N. A. Sanford, A. V. Davydov, D. V. Tsvetkov, A. V. Dmitriev, S. Keller, U. K. Mishra, S. P. DenBaars, S. S. Park, J. Y. Han, and R. J. Molnar, “Measurement of second order susceptibilities of GaN and AlGaN,” J. Appl. Phys. 97, 053512 (2005).
[CrossRef]

Umegaki, S.

R. Morita, Y. Kaneda, A. Sugihashi, T. Kondo, N. Ogasawara, S. Umegaki, and R. Ito, “Multiple-reflection effects in optical second-harmonic generation,” Jpn. J. Appl. Phys., Part 2 27, L1134-L1136 (1988).
[CrossRef]

Van Uitert, L. G.

S. Singh, J. R. Potopowicz, L. G. Van Uitert, and S. H. Wemple, “Nonlinear optical properties of hexagonal silicon carbide,” Appl. Phys. Lett. 19, 53-56 (1971).
[CrossRef]

Wemple, S. H.

S. Singh, J. R. Potopowicz, L. G. Van Uitert, and S. H. Wemple, “Nonlinear optical properties of hexagonal silicon carbide,” Appl. Phys. Lett. 19, 53-56 (1971).
[CrossRef]

Wong, G. K.

P. M. Lundquist, W. P. Lin, G. K. Wong, M. Razeghi, and J. B. Ketterson, “Second harmonic generation in hexagonal silicon carbide,” Appl. Phys. Lett. 66, 1883-1885 (1995).
[CrossRef]

Yaskov, A. D.

A. N. Pikhtin and A. D. Yaskov, “Refractive index and birefringence of semiconductors with the wurtzite structure,” Sov. Phys. Semicond. 15, 8-10 (1981).

Appl. Opt. (1)

Appl. Phys. (1)

P. S. Bechthold and S. Haussühl, “Nonlinear optical properties of orthorhombic barium formate and magnesium barium fluoride,” Appl. Phys. 14, 403-410 (1977).
[CrossRef]

Appl. Phys. Lett. (4)

S. X. Dou, M. H. Jiang, Z. S. Shao, and X. T. Tao, “Maker fringes in biaxial crystals and the nonlinear optical coefficients of thiosemicarbazide cadmium chloride monohydrate,” Appl. Phys. Lett. 54, 1101-1103 (1989).
[CrossRef]

S. Singh, J. R. Potopowicz, L. G. Van Uitert, and S. H. Wemple, “Nonlinear optical properties of hexagonal silicon carbide,” Appl. Phys. Lett. 19, 53-56 (1971).
[CrossRef]

P. M. Lundquist, W. P. Lin, G. K. Wong, M. Razeghi, and J. B. Ketterson, “Second harmonic generation in hexagonal silicon carbide,” Appl. Phys. Lett. 66, 1883-1885 (1995).
[CrossRef]

S. Niedermeier, H. Schillinger, R. Sauerbrey, B. Adolph, and F. Bechstedt, “Second-harmonic generation in silicon carbide polytypes,” Appl. Phys. Lett. 75, 618-620 (1999).
[CrossRef]

IEEE J. Quantum Electron. (1)

D. A. Roberts, “Simplified characterization of uniaxial and biaxial nonlinear optical crystals: a plea for standardization of nomenclature and conventions,” IEEE J. Quantum Electron. 28, 2057-2074 (1992).
[CrossRef]

J. Appl. Phys. (3)

N. A. Sanford, A. V. Davydov, D. V. Tsvetkov, A. V. Dmitriev, S. Keller, U. K. Mishra, S. P. DenBaars, S. S. Park, J. Y. Han, and R. J. Molnar, “Measurement of second order susceptibilities of GaN and AlGaN,” J. Appl. Phys. 97, 053512 (2005).
[CrossRef]

J. Jerphagnon and S. K. Kurtz, “Maker fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals,” J. Appl. Phys. 41, 1667-1681 (1970).
[CrossRef]

M. Ohashi, T. Kondo, K. Kumata, S. Fukatsu, S. S. Kano, Y. Shiraki, and R. Ito, “Determination of quadratic nonlinear optical coefficient of AlxGa1−xAs system by the method of reflected second harmonics,” J. Appl. Phys. 74, 596-601 (1993).
[CrossRef]

J. Opt. Soc. Am. B (5)

Jpn. J. Appl. Phys., Part 2 (1)

R. Morita, Y. Kaneda, A. Sugihashi, T. Kondo, N. Ogasawara, S. Umegaki, and R. Ito, “Multiple-reflection effects in optical second-harmonic generation,” Jpn. J. Appl. Phys., Part 2 27, L1134-L1136 (1988).
[CrossRef]

Opt. Commun. (1)

H. Hellwig and L. Bohatý, “Multiple reflections and Fabry-Perot interference corrections in Maker fringe experiments,” Opt. Commun. 161, 51-56 (1999).
[CrossRef]

Opt. Quantum Electron. (1)

I. Shoji, T. Kondo, and R. Ito, “Second-order nonlinear susceptibilities of various dielectric and semiconductor materials,” Opt. Quantum Electron. 34, 797-833 (2002).
[CrossRef]

Phys. Rev. (1)

N. Bloembergen and P. S. Pershan, “Light waves at the boundary of nonlinear media,” Phys. Rev. 128, 606-622 (1962).
[CrossRef]

Phys. Rev. A (1)

R. J. Gehr, G. L. Fischer, R. W. Boyd, and J. E. Sipe, “Nonlinear optical response of layered composite materials,” Phys. Rev. A 53, 2792-2798 (1996).
[CrossRef]

Phys. Rev. Lett. (1)

P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M. Savage, “Effects of dispersion and focusing on the production of optical harmonics,” Phys. Rev. Lett. 8, 21-22 (1962).
[CrossRef]

Sov. Phys. Semicond. (1)

A. N. Pikhtin and A. D. Yaskov, “Refractive index and birefringence of semiconductors with the wurtzite structure,” Sov. Phys. Semicond. 15, 8-10 (1981).

Other (2)

J. A. Aust, “Maker-fringe analysis and electric-field poling of lithium niobate,” Ph.D. dissertation (University of Colorado, 1999).

V. Gopalan, N. A. Sanford, J. A. Aust, K. Kitamura, and Y. Furukawa, in Handbook of Advanced Electronic and Photonic Materials and Devices: Ferroelectrics and DielectricsH.S.Nalwa, ed. (Academic, 2001), Vol. 4, p. 57.
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Incident, transmitted, and reflected waves at the two interfaces ( z = 0 , z = L ), and their associated wave and unit polarization vectors.

Fig. 2
Fig. 2

Field distribution of the fundamental wave inside the sample is composed of the multiply reflected beams. The forward propagating fundamental beam distribution function B + ( x , z ) is schematically shown.

Fig. 3
Fig. 3

Schematic of the calculation of second-harmonic output. Second-harmonic field emitted from a point on the output surface can be obtained by integrating the second-harmonic output along the second-harmonic beam path.

Fig. 4
Fig. 4

Numerical simulation results calculated with the fundamental beam radius a = 400 μ m . The second-harmonic output powers are plotted as functions of the fundamental incident angle θ ω from 0° to 60° in (a). The black solid curve, black dashed curve, and gray solid curve represent single path, two round trip, and five round trip results, respectively. (b) and (c) show expanded plots from 20° to 30° and from 50° to 60°, respectively.

Fig. 5
Fig. 5

Numerical simulation results under the five round trips condition. The black and gray solid curves represent the results calculated with beam radii of 400 and 4000 μ m , respectively. The second-harmonic output powers are plotted as functions of the fundamental incident angle θ ω from 0° to 60° in (a). (b) and (c) show expanded plots from 20° to 30° and from 50° to 60°, respectively.

Fig. 6
Fig. 6

Measured Maker fringes of a (0001) 6H-SiC plate obtained in the s p configuration. The output second-harmonic intensities are normalized with respect to the envelope amplitude of the Maker fringes of quartz ( d 11 ) . The inset shows a portion of the Maker-fringe data around + 20 ° plotted on an expanded horizontal scale.

Fig. 7
Fig. 7

Maker fringes of a (0001) 6H-SiC plate obtained in the s p configuration. The open circles are experimental data, and the solid curves show the theoretical fitting, which are plotted only in the positive angle regions in three separated parts on expanded horizontal scales.

Fig. 8
Fig. 8

Maker fringes of a (0001) 6H-SiC plate obtained in the 45° s configuration. The open circles are experimental data and the solid curves show the theoretical fitting.

Fig. 9
Fig. 9

Maker fringes of a (0001) 6H-SiC plate obtained in the p p configuration. The open circles are experimental data and the solid curves show the theoretical fitting.

Equations (55)

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ε ν = ( ( n x ν ) 2 0 0 0 ( n y ν ) 2 0 0 0 ( n z ν ) 2 ) ,
k + = k 0 = ( k x , 0 , k z ) ,
k = ( k x , 0 , k z ) ,
k s + = ( k x , 0 , k z s ) = ( k x , 0 , n y 2 k 0 2 k x 2 ) ,
k s = ( k x , 0 , k z s ) ,
k p + = ( k x , 0 , k z p ) = ( k x , 0 , n x 2 k 0 2 ( n x 2 n z 2 ) k x 2 ) ,
k p = ( k x , 0 , k z p ) ,
s ̂ = ( 0 , 1 , 0 ) ,
p ̂ + = ( k z k 0 , 0 , k x k 0 ) ,
p ̂ = ( k z k 0 , 0 , k x k 0 ) ,
q ̂ + = ε 1 d ̂ + ε 1 d ̂ + ,
= 1 ε 1 d ̂ + n p k 0 ( k z p n x 2 , 0 , k x n z 2 ) ,
q ̂ = ε 1 d ̂ ε 1 d ̂ ,
= 1 ε 1 d ̂ n p k 0 ( k z p n x 2 , 0 , k x n z 2 ) ,
ε 1 d ̂ + = ε 1 d ̂ = [ k 0 2 n z 4 k x 2 ( n z 2 n x 2 ) k 0 2 n z 4 n x 4 + k x 2 n z 2 n x 2 ( n z 2 n x 2 ) ] 1 2 .
t 01 p = k 0 p x k 0 p x + k z p q x + p x k x q z + p x k 0 q x + ,
r 00 p = k 0 q x + k z p q x + p x + + k x q z + p x + k z p q x + p x k x q z + p x k 0 q x + ,
t 12 p = 2 k z p q x q x + + k x q z q x + k x q z + q x k 0 q x + k z p q x p x + + k x q z p x + ,
r 11 p = r 00 p ,
E ω , i ( x , y , z ) = ( E 0 ω , s s ̂ ω + E 0 ω , p p ̂ ω + ) exp [ i ( k x ω x + k z ω z ) ] exp [ { ( x cos θ ω z sin θ ω ) 2 + y 2 } a 2 ] ,
g 0 ( x ) = exp ( x 2 cos 2 θ ω a 2 ) .
E 0 + ω ( x , y , z ) = [ E 0 + ω , s ( x , z ) s ̂ ω + E 0 + ω , p ( x , z ) q ̂ ω + ] exp ( y 2 a 2 ) ,
= [ t 01 ω , s E 0 ω , s s ̂ ω g s + ( x , z ) exp { i ( k x ω x + k z ω , s z ) } + t 01 ω , p E 0 ω , p q ̂ ω + g p + ( x , z ) exp { i ( k x ω x + k z ω , p z ) } ] exp ( y 2 a 2 ) .
g s + ( x , z ) = g 0 ( x z tan θ ω , s ) ,
g p + ( x , z ) = g 0 ( x z tan θ ω , p ) ,
E 0 ω ( x , y , z ) = [ E 0 ω , s ( x , z ) s ̂ ω + E 0 ω , p ( x , z ) q ̂ ω ] exp ( y 2 a 2 ) , = [ t 01 ω , s r 11 ω , s E 0 ω , s s ̂ ω g s ( x , z ) exp ( i k z ω , s L ) exp { i ( k x ω x k z ω , s z ) } + t 01 ω , p r 11 ω , p E 0 ω , p q ̂ ω g p ( x , z ) exp ( i k z ω , p L ) exp { i ( k x ω x k z ω , p z ) } ] exp ( y 2 a 2 ) ,
g s ( x , z ) = g 0 ( x { 2 L z } tan θ ω , s ) ,
g p ( x , z ) = g 0 ( x { 2 L z } tan θ ω , p ) .
E ± ω , s ( x , z ) = t 01 ω , s E 0 ω , s B s ± ( x , z ) exp [ i ( k x ω x ± k z ω , s z ) ] ,
B s + ( x , z ) = m = 0 g s + ( x 2 m L tan θ ω , s , z ) ( r 11 ω , s ) 2 m exp ( 2 i m k z ω , s L ) ,
B s ( x , z ) = m = 0 g s ( x 2 m L tan θ ω , s , z ) ( r 11 ω , s ) 2 m + 1 exp [ i ( 2 m + 1 ) k z ω , s L ] .
E ± ω , p ( x , z ) = t 01 ω , p E 0 ω , p B p ± ( x , z ) exp [ i ( k x ω x ± k z ω , p z ) ] ,
B p + ( x , z ) = m = 0 g p + ( x 2 m L tan θ ω , p , z ) ( r 11 ω , p ) 2 m exp ( 2 i m k z ω , p L ) ,
B p ( x , z ) = m = 0 g p ( x 2 m L tan θ ω , p , z ) ( r 11 ω , p ) 2 m + 1 exp [ i ( 2 m + 1 ) k z ω , p L ] .
P NL ( x , y , z ) = P + NL ( x , y , z ) + P NL ( x , y , z ) ,
= ϵ 0 d : E + ω ( x , y , z ) E + ω ( x , y , z ) + ϵ 0 d : E ω ( x , y , z ) E ω ( x , y , z ) ,
2 E 2 ω ( r ) + ϵ 2 ω ( k 0 2 ω ) 2 E 2 ω ( r ) ( E 2 ω ( r ) ) = ( k 0 2 ω ) 2 ϵ 0 P NL ( r ) ,
P NL ( r ) = P ( z ) δ ( z z ) exp ( i k x 2 ω x ) ,
G ( z z ) = G + ( z z ) + G ( z z ) + G b ( z z ) ,
= i ( k 0 2 ω ) 2 2 ϵ 0 [ { s ̂ 2 ω s ̂ 2 ω k z 2 ω , s e i k z 2 ω , s ( z z ) + ( n x 2 ω ) 2 q ̂ 2 ω + q ̂ 2 ω + ( n p 2 ω ) 2 k z 2 ω , p e i k z 2 ω , p ( z z ) } θ ( z z ) + { s ̂ 2 ω s ̂ 2 ω k z 2 ω , s e i k z 2 ω , s ( z z ) + ( n x 2 ω ) 2 q ̂ 2 ω q ̂ 2 ω ( n p 2 ω ) 2 k z 2 ω , p e i k z 2 ω , p ( z z ) } θ ( z z ) ] z ̂ z ̂ ϵ 0 ( n z 2 ω ) 2 δ ( z z ) ,
E 2 ω ( r ) = 0 L G ( z z ) P ( z ) d z exp ( i k x 2 ω x ) .
E + 2 ω ( L ) = 0 L G + ( z z ) P + ( z ) d z ,
E 2 ω ( 0 ) = 0 L G ( z z ) P ( z ) d z ,
P NL ( x , y , z ) = [ P + ( x , z ) + P ( x , z ) ] exp ( i k x 2 ω x ) exp ( 2 y 2 a 2 ) ,
E + 2 ω , s ( x , L ) = 0 L d z i ( k 0 2 ω ) 2 2 ϵ 0 [ s ̂ 2 ω k z 2 ω , s s ̂ 2 ω P + ( x { L z } tan θ 2 ω , s , z ) e i k z 2 ω , s ( L z ) ] ,
E + 2 ω , p ( x , L ) = 0 L d z i ( k 0 2 ω ) 2 2 ϵ 0 [ ( n x 2 ω ) 2 q ̂ 2 ω + ( n p 2 ω ) 2 k z 2 ω , p q ̂ 2 ω + P + ( x { L z } tan θ 2 ω , p , z ) e i k z 2 ω , p ( L z ) ] ,
E 2 ω , s ( x , 0 ) = 0 L d z i ( k 0 2 ω ) 2 2 ϵ 0 [ s ̂ 2 ω k z 2 ω , s s ̂ 2 ω P ( x z tan θ 2 ω , s , z ) e i k z 2 ω , s z ] ,
E 2 ω , p ( x , 0 ) = 0 L d z i ( k 0 2 ω ) 2 2 ϵ 0 [ ( n x 2 ω ) 2 q ̂ 2 ω ( n p 2 ω ) 2 k z 2 ω , p q ̂ 2 ω P ( x z tan θ 2 ω , p , z ) e i k z 2 ω , p z ] .
E out 2 ω , s ( x , L ) = t 12 2 ω , s m = 0 { E + 2 ω , s ( x 2 m L tan θ 2 ω , s , L ) ( r 11 2 ω , s ) 2 m exp ( 2 i m k z 2 ω , s L ) + E 2 ω , s ( x { 2 m + 1 } L tan θ 2 ω , s , 0 ) ( r 11 2 ω , s ) 2 m + 1 exp [ i ( 2 m + 1 ) k z 2 ω , s L ] } ,
E out 2 ω , p ( x , L ) = t 12 2 ω , p m = 0 { E + 2 ω , p ( x 2 m L tan θ 2 ω , p , L ) ( r 11 2 ω , p ) 2 m exp ( 2 i m k z 2 ω , p L ) + E 2 ω , p ( x { 2 m + 1 } L tan θ 2 ω , p , 0 ) ( r 11 2 ω , p ) 2 m + 1 exp [ i ( 2 m + 1 ) k z 2 ω , p L ] } ,
P out 2 ω = ε 0 c cos θ ω 2 E 2 ω ( x , L ) out 2 exp ( 4 y 2 a 2 ) d x d y ,
= π ε 0 c a cos θ ω 4 E out 2 ω ( x , L ) 2 d x .
d 31 = d 32 = 6.7 ± 0.8 pm V ,
d 15 = d 24 = 6.4 ± 1.1 pm V ,
d 33 = 9.7 ± 8.0 pm V .

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