Abstract

We have determined all the three independent components of quadratic nonlinear-optical coefficients of GaN by highly accurate Maker-fringe measurements on high-quality bulk samples combined with theoretical analysis taking account of the multiple-reflection effects in slightly misoriented optically anisotropic samples. Especially, the d33 coefficient is determined with sufficient accuracies for the first time, to our knowledge, by using (112¯0)-oriented samples. The obtained values of quadratic nonlinear-optical coefficients are d31=2.5±0.1  pm/V, d15=2.5±0.1  pm/V, and d33=3.8±0.2  pm/V at the fundamental wavelength of 1.064μm.

© 2010 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  27. J. Chen, Z. H. Levine, and J. W. Wilkins, “Calculated second-harmonic susceptibilities of BN, AlN, and GaN,” Appl. Phys. Lett. 66, 1129–1131 (1995).
    [CrossRef]
  28. J. L. P. Hughes, Y. Wang, and J. E. Sipe, “Calculation of linear and second-order optical response in wurtzite GaN and AlN,” Phys. Rev. B 55, 13630–13640 (1997).
    [CrossRef]
  29. N. Bloembergen and P. S. Pershan, “Light waves at the boundary of nonlinear media,” Phys. Rev. 128, 606–622 (1962).
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2009 (2)

H. Yoshida, M. Kuwabara, Y. Yamashita, Y. Takagi, K. Uchiyama, and H. Kan, “AlGaN-based laser diodes for the short-wavelength ultraviolet region,” New J. Phys. 11, 125013 (2009).
[CrossRef]

H. Sato, M. Abe, I. Shoji, J. Suda, and T. Kondo, “Accurate measurements of second-order nonlinear-optical coefficients of 6H and 4H silicon carbide,” J. Opt. Soc. Am. B 26, 1892–1896 (2009).
[CrossRef]

2008 (2)

S. Pezzagna, J. Brault, M. Leroux, J. Massies, and M. de Micheli, “Refractive indices and elasto-optic coefficients of GaN studied by optical waveguiding,” J. Appl. Phys. 103, 123112 (2008).
[CrossRef]

M. Abe, I. Shoji, J. Suda, and T. Kondo, “Comprehensive analysis of multiple-reflection effects on rotational Maker-fringe experiments,” J. Opt. Soc. Am. B 25, 1616–1624 (2008).
[CrossRef]

2006 (1)

F. Kawamura, H. Umeda, M. Morishita, M. Kawahara, M. Yoshimura, Y. Mori, T. Sasaki, and Y. Kitaoka, “Growth of a two-inch GaN single crystal substrate using the Na flux method,” Jpn. J. Appl. Phys., Part 2 45, L1136–L1138 (2006).
[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]

2004 (1)

D. Passeri, M. C. Larciprete, A. Belardini, S. Paoloni, A. Passaseo, C. Sibilia, and F. Michelotti, “Second harmonic generation in AlGaN, GaN and AlxGa1−xN/GaN multiple quantum well structures,” Appl. Phys. B 79, 611–615 (2004).
[CrossRef]

2003 (1)

N. A. Sanford, L. H. Robins, A. V. Davydov, A. Shapiro, D. V. Tsvetkov, A. V. Dmitriev, S. Keller, U. K. Mishra, and S. P. DenBaars, “Refractive index study of AlxGa1−xN films grown on sapphire substrates,” J. Appl. Phys. 94, 2980–2991 (2003).
[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)

Ü. Özgür, G. Webb-Wood, H. O. Everitt, F. Yun, and H. Morkoç, “Systematic measurement of AlxGa1−xN refractive indices,” Appl. Phys. Lett. 79, 4103–4105 (2001).
[CrossRef]

2000 (4)

B. Monemar and G. Pozina, “Group III-nitride based hetero and quantum structures,” Prog. Quantum Electron. 24, 239–290 (2000).
[CrossRef]

I. V. Kravetsky, I. M. Tiginyanu, R. Hildebrandt, G. Marowsky, D. Pavlidis, and A. Eisenbach, “Nonlinear optical response of GaN layers on sapphire: the impact of fundamental beam interface,” Appl. Phys. Lett. 76, 810–812 (2000).
[CrossRef]

T. Fujita, T. Hasegawa, M. Haraguchi, T. Okamoto, M. Fukui, and S. Nakamura, “Determination of second-order nonlinear optical susceptibility of GaN film on sapphire,” Jpn. J. Appl. Phys., Part 1 39, 2610–2613 (2000).
[CrossRef]

H. Iwanaga, A. Kunishige, and S. Takeuchi, “Anisotropic thermal expansion in wurtzite-type crystals,” J. Mater. Sci. 35, 2451–2454 (2000).
[CrossRef]

1999 (1)

M. J. Bergmann, Ü. Özgür, H. C. Casey, Jr., H. O. Everitt, and J. F. Muth, “Ordinary and extraordinary refractive indices for AlxGa1−xN epitaxial layers,” Appl. Phys. Lett. 75, 67–69 (1999).
[CrossRef]

1997 (3)

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]

F. Bernardini, V. Fiorentini, and D. Vanderbilt, “Spontaneous polarization and piezoelectric constants of III-V nitrides,” Phys. Rev. B 56, R10024 (1997).
[CrossRef]

J. L. P. Hughes, Y. Wang, and J. E. Sipe, “Calculation of linear and second-order optical response in wurtzite GaN and AlN,” Phys. Rev. B 55, 13630–13640 (1997).
[CrossRef]

1996 (1)

H. Y. Zhang, X. H. He, Y. H. Shin, M. Schurman, Z. C. Feng, and R. A. Stall, “Study of nonlinear optical effects in GaN:Mg epitaxial film,” Appl. Phys. Lett. 69, 2953–2955 (1996).
[CrossRef]

1995 (3)

1993 (1)

1992 (2)

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]

T. Detchprohm, K. Hiramatsu, K. Itoh, and I. Akasaki, “Relaxation process of the thermal strain in the GaN/α-Al2O3 heterostructure and determination of the intrinsic lattice constants of GaN free from the strain,” Jpn. J. Appl. Phys., Part 2 31, L1454–L1456 (1992).
[CrossRef]

1980 (1)

T. Ishidate, K. Inoue, and M. Aoki, “Second harmonic generation of epitaxially-grown GaN crystal,” Jpn. J. Appl. Phys. 19, 1641–1645 (1980).
[CrossRef]

1974 (1)

B. F. Levine, “Origin of the unusual dependence of the nonlinear optical susceptibility on bond length for ionic ferroelectrics,” Phys. Rev. B 10, 1655–1664 (1974).
[CrossRef]

1973 (1)

B. F. Levine, “Bond-charge calculation of nonlinear optical susceptibilities for various crystal structures,” Phys. Rev. B 7, 2600–2626 (1973).
[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]

1968 (1)

F. N. H. Robinson, “Relation between the components of the non-linear polarisability tensor in cubic and hexagonal II-VI compounds,” Phys. Lett. 26A, 435–436 (1968).

1962 (1)

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

Abe, M.

Akasaki, I.

T. Detchprohm, K. Hiramatsu, K. Itoh, and I. Akasaki, “Relaxation process of the thermal strain in the GaN/α-Al2O3 heterostructure and determination of the intrinsic lattice constants of GaN free from the strain,” Jpn. J. Appl. Phys., Part 2 31, L1454–L1456 (1992).
[CrossRef]

Aoki, M.

T. Ishidate, K. Inoue, and M. Aoki, “Second harmonic generation of epitaxially-grown GaN crystal,” Jpn. J. Appl. Phys. 19, 1641–1645 (1980).
[CrossRef]

Belardini, A.

D. Passeri, M. C. Larciprete, A. Belardini, S. Paoloni, A. Passaseo, C. Sibilia, and F. Michelotti, “Second harmonic generation in AlGaN, GaN and AlxGa1−xN/GaN multiple quantum well structures,” Appl. Phys. B 79, 611–615 (2004).
[CrossRef]

Bergmann, M. J.

M. J. Bergmann, Ü. Özgür, H. C. Casey, Jr., H. O. Everitt, and J. F. Muth, “Ordinary and extraordinary refractive indices for AlxGa1−xN epitaxial layers,” Appl. Phys. Lett. 75, 67–69 (1999).
[CrossRef]

Bernardini, F.

F. Bernardini, V. Fiorentini, and D. Vanderbilt, “Spontaneous polarization and piezoelectric constants of III-V nitrides,” Phys. Rev. B 56, R10024 (1997).
[CrossRef]

Bloembergen, N.

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

Brault, J.

S. Pezzagna, J. Brault, M. Leroux, J. Massies, and M. de Micheli, “Refractive indices and elasto-optic coefficients of GaN studied by optical waveguiding,” J. Appl. Phys. 103, 123112 (2008).
[CrossRef]

Bryden, W. A.

Casey, H. C.

M. J. Bergmann, Ü. Özgür, H. C. Casey, Jr., H. O. Everitt, and J. F. Muth, “Ordinary and extraordinary refractive indices for AlxGa1−xN epitaxial layers,” Appl. Phys. Lett. 75, 67–69 (1999).
[CrossRef]

Chen, J.

J. Chen, Z. H. Levine, and J. W. Wilkins, “Calculated second-harmonic susceptibilities of BN, AlN, and GaN,” Appl. Phys. Lett. 66, 1129–1131 (1995).
[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]

N. A. Sanford, L. H. Robins, A. V. Davydov, A. Shapiro, D. V. Tsvetkov, A. V. Dmitriev, S. Keller, U. K. Mishra, and S. P. DenBaars, “Refractive index study of AlxGa1−xN films grown on sapphire substrates,” J. Appl. Phys. 94, 2980–2991 (2003).
[CrossRef]

de Micheli, M.

S. Pezzagna, J. Brault, M. Leroux, J. Massies, and M. de Micheli, “Refractive indices and elasto-optic coefficients of GaN studied by optical waveguiding,” J. Appl. Phys. 103, 123112 (2008).
[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]

N. A. Sanford, L. H. Robins, A. V. Davydov, A. Shapiro, D. V. Tsvetkov, A. V. Dmitriev, S. Keller, U. K. Mishra, and S. P. DenBaars, “Refractive index study of AlxGa1−xN films grown on sapphire substrates,” J. Appl. Phys. 94, 2980–2991 (2003).
[CrossRef]

Detchprohm, T.

T. Detchprohm, K. Hiramatsu, K. Itoh, and I. Akasaki, “Relaxation process of the thermal strain in the GaN/α-Al2O3 heterostructure and determination of the intrinsic lattice constants of GaN free from the strain,” Jpn. J. Appl. Phys., Part 2 31, L1454–L1456 (1992).
[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]

N. A. Sanford, L. H. Robins, A. V. Davydov, A. Shapiro, D. V. Tsvetkov, A. V. Dmitriev, S. Keller, U. K. Mishra, and S. P. DenBaars, “Refractive index study of AlxGa1−xN films grown on sapphire substrates,” J. Appl. Phys. 94, 2980–2991 (2003).
[CrossRef]

Eisenbach, A.

I. V. Kravetsky, I. M. Tiginyanu, R. Hildebrandt, G. Marowsky, D. Pavlidis, and A. Eisenbach, “Nonlinear optical response of GaN layers on sapphire: the impact of fundamental beam interface,” Appl. Phys. Lett. 76, 810–812 (2000).
[CrossRef]

Everitt, H. O.

Ü. Özgür, G. Webb-Wood, H. O. Everitt, F. Yun, and H. Morkoç, “Systematic measurement of AlxGa1−xN refractive indices,” Appl. Phys. Lett. 79, 4103–4105 (2001).
[CrossRef]

M. J. Bergmann, Ü. Özgür, H. C. Casey, Jr., H. O. Everitt, and J. F. Muth, “Ordinary and extraordinary refractive indices for AlxGa1−xN epitaxial layers,” Appl. Phys. Lett. 75, 67–69 (1999).
[CrossRef]

Feng, Z. C.

H. Y. Zhang, X. H. He, Y. H. Shin, M. Schurman, Z. C. Feng, and R. A. Stall, “Study of nonlinear optical effects in GaN:Mg epitaxial film,” Appl. Phys. Lett. 69, 2953–2955 (1996).
[CrossRef]

Fiorentini, V.

F. Bernardini, V. Fiorentini, and D. Vanderbilt, “Spontaneous polarization and piezoelectric constants of III-V nitrides,” Phys. Rev. B 56, R10024 (1997).
[CrossRef]

Fujita, T.

T. Fujita, T. Hasegawa, M. Haraguchi, T. Okamoto, M. Fukui, and S. Nakamura, “Determination of second-order nonlinear optical susceptibility of GaN film on sapphire,” Jpn. J. Appl. Phys., Part 1 39, 2610–2613 (2000).
[CrossRef]

Fukui, M.

T. Fujita, T. Hasegawa, M. Haraguchi, T. Okamoto, M. Fukui, and S. Nakamura, “Determination of second-order nonlinear optical susceptibility of GaN film on sapphire,” Jpn. J. Appl. Phys., Part 1 39, 2610–2613 (2000).
[CrossRef]

Hagimoto, K.

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]

Haraguchi, M.

T. Fujita, T. Hasegawa, M. Haraguchi, T. Okamoto, M. Fukui, and S. Nakamura, “Determination of second-order nonlinear optical susceptibility of GaN film on sapphire,” Jpn. J. Appl. Phys., Part 1 39, 2610–2613 (2000).
[CrossRef]

Hasegawa, T.

T. Fujita, T. Hasegawa, M. Haraguchi, T. Okamoto, M. Fukui, and S. Nakamura, “Determination of second-order nonlinear optical susceptibility of GaN film on sapphire,” Jpn. J. Appl. Phys., Part 1 39, 2610–2613 (2000).
[CrossRef]

Hayden, L. M.

He, X. H.

H. Y. Zhang, X. H. He, Y. H. Shin, M. Schurman, Z. C. Feng, and R. A. Stall, “Study of nonlinear optical effects in GaN:Mg epitaxial film,” Appl. Phys. Lett. 69, 2953–2955 (1996).
[CrossRef]

Herman, W. N.

Hildebrandt, R.

I. V. Kravetsky, I. M. Tiginyanu, R. Hildebrandt, G. Marowsky, D. Pavlidis, and A. Eisenbach, “Nonlinear optical response of GaN layers on sapphire: the impact of fundamental beam interface,” Appl. Phys. Lett. 76, 810–812 (2000).
[CrossRef]

Hiramatsu, K.

T. Detchprohm, K. Hiramatsu, K. Itoh, and I. Akasaki, “Relaxation process of the thermal strain in the GaN/α-Al2O3 heterostructure and determination of the intrinsic lattice constants of GaN free from the strain,” Jpn. J. Appl. Phys., Part 2 31, L1454–L1456 (1992).
[CrossRef]

Hughes, J. L. P.

J. L. P. Hughes, Y. Wang, and J. E. Sipe, “Calculation of linear and second-order optical response in wurtzite GaN and AlN,” Phys. Rev. B 55, 13630–13640 (1997).
[CrossRef]

Inoue, K.

T. Ishidate, K. Inoue, and M. Aoki, “Second harmonic generation of epitaxially-grown GaN crystal,” Jpn. J. Appl. Phys. 19, 1641–1645 (1980).
[CrossRef]

Ishidate, T.

T. Ishidate, K. Inoue, and M. Aoki, “Second harmonic generation of epitaxially-grown GaN crystal,” Jpn. J. Appl. Phys. 19, 1641–1645 (1980).
[CrossRef]

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]

Itoh, K.

T. Detchprohm, K. Hiramatsu, K. Itoh, and I. Akasaki, “Relaxation process of the thermal strain in the GaN/α-Al2O3 heterostructure and determination of the intrinsic lattice constants of GaN free from the strain,” Jpn. J. Appl. Phys., Part 2 31, L1454–L1456 (1992).
[CrossRef]

Iwanaga, H.

H. Iwanaga, A. Kunishige, and S. Takeuchi, “Anisotropic thermal expansion in wurtzite-type crystals,” J. Mater. Sci. 35, 2451–2454 (2000).
[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]

Kan, H.

H. Yoshida, M. Kuwabara, Y. Yamashita, Y. Takagi, K. Uchiyama, and H. Kan, “AlGaN-based laser diodes for the short-wavelength ultraviolet region,” New J. Phys. 11, 125013 (2009).
[CrossRef]

Kawahara, M.

F. Kawamura, H. Umeda, M. Morishita, M. Kawahara, M. Yoshimura, Y. Mori, T. Sasaki, and Y. Kitaoka, “Growth of a two-inch GaN single crystal substrate using the Na flux method,” Jpn. J. Appl. Phys., Part 2 45, L1136–L1138 (2006).
[CrossRef]

Kawamura, F.

F. Kawamura, H. Umeda, M. Morishita, M. Kawahara, M. Yoshimura, Y. Mori, T. Sasaki, and Y. Kitaoka, “Growth of a two-inch GaN single crystal substrate using the Na flux method,” Jpn. J. Appl. Phys., Part 2 45, L1136–L1138 (2006).
[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]

N. A. Sanford, L. H. Robins, A. V. Davydov, A. Shapiro, D. V. Tsvetkov, A. V. Dmitriev, S. Keller, U. K. Mishra, and S. P. DenBaars, “Refractive index study of AlxGa1−xN films grown on sapphire substrates,” J. Appl. Phys. 94, 2980–2991 (2003).
[CrossRef]

Kistenmacher, T. J.

Kitamoto, A.

Kitaoka, Y.

F. Kawamura, H. Umeda, M. Morishita, M. Kawahara, M. Yoshimura, Y. Mori, T. Sasaki, and Y. Kitaoka, “Growth of a two-inch GaN single crystal substrate using the Na flux method,” Jpn. J. Appl. Phys., Part 2 45, L1136–L1138 (2006).
[CrossRef]

Kondo, T.

Kravetsky, I. V.

I. V. Kravetsky, I. M. Tiginyanu, R. Hildebrandt, G. Marowsky, D. Pavlidis, and A. Eisenbach, “Nonlinear optical response of GaN layers on sapphire: the impact of fundamental beam interface,” Appl. Phys. Lett. 76, 810–812 (2000).
[CrossRef]

Kunishige, A.

H. Iwanaga, A. Kunishige, and S. Takeuchi, “Anisotropic thermal expansion in wurtzite-type crystals,” J. Mater. Sci. 35, 2451–2454 (2000).
[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]

Kuwabara, M.

H. Yoshida, M. Kuwabara, Y. Yamashita, Y. Takagi, K. Uchiyama, and H. Kan, “AlGaN-based laser diodes for the short-wavelength ultraviolet region,” New J. Phys. 11, 125013 (2009).
[CrossRef]

Larciprete, M. C.

D. Passeri, M. C. Larciprete, A. Belardini, S. Paoloni, A. Passaseo, C. Sibilia, and F. Michelotti, “Second harmonic generation in AlGaN, GaN and AlxGa1−xN/GaN multiple quantum well structures,” Appl. Phys. B 79, 611–615 (2004).
[CrossRef]

Leroux, M.

S. Pezzagna, J. Brault, M. Leroux, J. Massies, and M. de Micheli, “Refractive indices and elasto-optic coefficients of GaN studied by optical waveguiding,” J. Appl. Phys. 103, 123112 (2008).
[CrossRef]

Levine, B. F.

B. F. Levine, “Origin of the unusual dependence of the nonlinear optical susceptibility on bond length for ionic ferroelectrics,” Phys. Rev. B 10, 1655–1664 (1974).
[CrossRef]

B. F. Levine, “Bond-charge calculation of nonlinear optical susceptibilities for various crystal structures,” Phys. Rev. B 7, 2600–2626 (1973).
[CrossRef]

Levine, Z. H.

J. Chen, Z. H. Levine, and J. W. Wilkins, “Calculated second-harmonic susceptibilities of BN, AlN, and GaN,” Appl. Phys. Lett. 66, 1129–1131 (1995).
[CrossRef]

Marowsky, G.

I. V. Kravetsky, I. M. Tiginyanu, R. Hildebrandt, G. Marowsky, D. Pavlidis, and A. Eisenbach, “Nonlinear optical response of GaN layers on sapphire: the impact of fundamental beam interface,” Appl. Phys. Lett. 76, 810–812 (2000).
[CrossRef]

Massies, J.

S. Pezzagna, J. Brault, M. Leroux, J. Massies, and M. de Micheli, “Refractive indices and elasto-optic coefficients of GaN studied by optical waveguiding,” J. Appl. Phys. 103, 123112 (2008).
[CrossRef]

Michelotti, F.

D. Passeri, M. C. Larciprete, A. Belardini, S. Paoloni, A. Passaseo, C. Sibilia, and F. Michelotti, “Second harmonic generation in AlGaN, GaN and AlxGa1−xN/GaN multiple quantum well structures,” Appl. Phys. B 79, 611–615 (2004).
[CrossRef]

Miragliotta, J.

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]

N. A. Sanford, L. H. Robins, A. V. Davydov, A. Shapiro, D. V. Tsvetkov, A. V. Dmitriev, S. Keller, U. K. Mishra, and S. P. DenBaars, “Refractive index study of AlxGa1−xN films grown on sapphire substrates,” J. Appl. Phys. 94, 2980–2991 (2003).
[CrossRef]

Mito, A.

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]

Monemar, B.

B. Monemar and G. Pozina, “Group III-nitride based hetero and quantum structures,” Prog. Quantum Electron. 24, 239–290 (2000).
[CrossRef]

Mori, Y.

F. Kawamura, H. Umeda, M. Morishita, M. Kawahara, M. Yoshimura, Y. Mori, T. Sasaki, and Y. Kitaoka, “Growth of a two-inch GaN single crystal substrate using the Na flux method,” Jpn. J. Appl. Phys., Part 2 45, L1136–L1138 (2006).
[CrossRef]

Morishita, M.

F. Kawamura, H. Umeda, M. Morishita, M. Kawahara, M. Yoshimura, Y. Mori, T. Sasaki, and Y. Kitaoka, “Growth of a two-inch GaN single crystal substrate using the Na flux method,” Jpn. J. Appl. Phys., Part 2 45, L1136–L1138 (2006).
[CrossRef]

Morkoç, H.

Ü. Özgür, G. Webb-Wood, H. O. Everitt, F. Yun, and H. Morkoç, “Systematic measurement of AlxGa1−xN refractive indices,” Appl. Phys. Lett. 79, 4103–4105 (2001).
[CrossRef]

Muth, J. F.

M. J. Bergmann, Ü. Özgür, H. C. Casey, Jr., H. O. Everitt, and J. F. Muth, “Ordinary and extraordinary refractive indices for AlxGa1−xN epitaxial layers,” Appl. Phys. Lett. 75, 67–69 (1999).
[CrossRef]

Nakamura, S.

T. Fujita, T. Hasegawa, M. Haraguchi, T. Okamoto, M. Fukui, and S. Nakamura, “Determination of second-order nonlinear optical susceptibility of GaN film on sapphire,” Jpn. J. Appl. Phys., Part 1 39, 2610–2613 (2000).
[CrossRef]

Okamoto, T.

T. Fujita, T. Hasegawa, M. Haraguchi, T. Okamoto, M. Fukui, and S. Nakamura, “Determination of second-order nonlinear optical susceptibility of GaN film on sapphire,” Jpn. J. Appl. Phys., Part 1 39, 2610–2613 (2000).
[CrossRef]

Özgür, Ü.

Ü. Özgür, G. Webb-Wood, H. O. Everitt, F. Yun, and H. Morkoç, “Systematic measurement of AlxGa1−xN refractive indices,” Appl. Phys. Lett. 79, 4103–4105 (2001).
[CrossRef]

M. J. Bergmann, Ü. Özgür, H. C. Casey, Jr., H. O. Everitt, and J. F. Muth, “Ordinary and extraordinary refractive indices for AlxGa1−xN epitaxial layers,” Appl. Phys. Lett. 75, 67–69 (1999).
[CrossRef]

Paoloni, S.

D. Passeri, M. C. Larciprete, A. Belardini, S. Paoloni, A. Passaseo, C. Sibilia, and F. Michelotti, “Second harmonic generation in AlGaN, GaN and AlxGa1−xN/GaN multiple quantum well structures,” Appl. Phys. B 79, 611–615 (2004).
[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]

Passaseo, A.

D. Passeri, M. C. Larciprete, A. Belardini, S. Paoloni, A. Passaseo, C. Sibilia, and F. Michelotti, “Second harmonic generation in AlGaN, GaN and AlxGa1−xN/GaN multiple quantum well structures,” Appl. Phys. B 79, 611–615 (2004).
[CrossRef]

Passeri, D.

D. Passeri, M. C. Larciprete, A. Belardini, S. Paoloni, A. Passaseo, C. Sibilia, and F. Michelotti, “Second harmonic generation in AlGaN, GaN and AlxGa1−xN/GaN multiple quantum well structures,” Appl. Phys. B 79, 611–615 (2004).
[CrossRef]

Pavlidis, D.

I. V. Kravetsky, I. M. Tiginyanu, R. Hildebrandt, G. Marowsky, D. Pavlidis, and A. Eisenbach, “Nonlinear optical response of GaN layers on sapphire: the impact of fundamental beam interface,” Appl. Phys. Lett. 76, 810–812 (2000).
[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]

Pezzagna, S.

S. Pezzagna, J. Brault, M. Leroux, J. Massies, and M. de Micheli, “Refractive indices and elasto-optic coefficients of GaN studied by optical waveguiding,” J. Appl. Phys. 103, 123112 (2008).
[CrossRef]

Pozina, G.

B. Monemar and G. Pozina, “Group III-nitride based hetero and quantum structures,” Prog. Quantum Electron. 24, 239–290 (2000).
[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]

Robins, L. H.

N. A. Sanford, L. H. Robins, A. V. Davydov, A. Shapiro, D. V. Tsvetkov, A. V. Dmitriev, S. Keller, U. K. Mishra, and S. P. DenBaars, “Refractive index study of AlxGa1−xN films grown on sapphire substrates,” J. Appl. Phys. 94, 2980–2991 (2003).
[CrossRef]

Robinson, F. N. H.

F. N. H. Robinson, “Relation between the components of the non-linear polarisability tensor in cubic and hexagonal II-VI compounds,” Phys. Lett. 26A, 435–436 (1968).

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]

N. A. Sanford, L. H. Robins, A. V. Davydov, A. Shapiro, D. V. Tsvetkov, A. V. Dmitriev, S. Keller, U. K. Mishra, and S. P. DenBaars, “Refractive index study of AlxGa1−xN films grown on sapphire substrates,” J. Appl. Phys. 94, 2980–2991 (2003).
[CrossRef]

Sasaki, T.

F. Kawamura, H. Umeda, M. Morishita, M. Kawahara, M. Yoshimura, Y. Mori, T. Sasaki, and Y. Kitaoka, “Growth of a two-inch GaN single crystal substrate using the Na flux method,” Jpn. J. Appl. Phys., Part 2 45, L1136–L1138 (2006).
[CrossRef]

Sato, H.

Schurman, M.

H. Y. Zhang, X. H. He, Y. H. Shin, M. Schurman, Z. C. Feng, and R. A. Stall, “Study of nonlinear optical effects in GaN:Mg epitaxial film,” Appl. Phys. Lett. 69, 2953–2955 (1996).
[CrossRef]

Shapiro, A.

N. A. Sanford, L. H. Robins, A. V. Davydov, A. Shapiro, D. V. Tsvetkov, A. V. Dmitriev, S. Keller, U. K. Mishra, and S. P. DenBaars, “Refractive index study of AlxGa1−xN films grown on sapphire substrates,” J. Appl. Phys. 94, 2980–2991 (2003).
[CrossRef]

Shin, Y. H.

H. Y. Zhang, X. H. He, Y. H. Shin, M. Schurman, Z. C. Feng, and R. A. Stall, “Study of nonlinear optical effects in GaN:Mg epitaxial film,” Appl. Phys. Lett. 69, 2953–2955 (1996).
[CrossRef]

Shirane, M.

Shoji, I.

Sibilia, C.

D. Passeri, M. C. Larciprete, A. Belardini, S. Paoloni, A. Passaseo, C. Sibilia, and F. Michelotti, “Second harmonic generation in AlGaN, GaN and AlxGa1−xN/GaN multiple quantum well structures,” Appl. Phys. B 79, 611–615 (2004).
[CrossRef]

Sipe, J. E.

J. L. P. Hughes, Y. Wang, and J. E. Sipe, “Calculation of linear and second-order optical response in wurtzite GaN and AlN,” Phys. Rev. B 55, 13630–13640 (1997).
[CrossRef]

Stall, R. A.

H. Y. Zhang, X. H. He, Y. H. Shin, M. Schurman, Z. C. Feng, and R. A. Stall, “Study of nonlinear optical effects in GaN:Mg epitaxial film,” Appl. Phys. Lett. 69, 2953–2955 (1996).
[CrossRef]

Suda, J.

Takagi, Y.

H. Yoshida, M. Kuwabara, Y. Yamashita, Y. Takagi, K. Uchiyama, and H. Kan, “AlGaN-based laser diodes for the short-wavelength ultraviolet region,” New J. Phys. 11, 125013 (2009).
[CrossRef]

Takeuchi, S.

H. Iwanaga, A. Kunishige, and S. Takeuchi, “Anisotropic thermal expansion in wurtzite-type crystals,” J. Mater. Sci. 35, 2451–2454 (2000).
[CrossRef]

Tiginyanu, I. M.

I. V. Kravetsky, I. M. Tiginyanu, R. Hildebrandt, G. Marowsky, D. Pavlidis, and A. Eisenbach, “Nonlinear optical response of GaN layers on sapphire: the impact of fundamental beam interface,” Appl. Phys. Lett. 76, 810–812 (2000).
[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]

N. A. Sanford, L. H. Robins, A. V. Davydov, A. Shapiro, D. V. Tsvetkov, A. V. Dmitriev, S. Keller, U. K. Mishra, and S. P. DenBaars, “Refractive index study of AlxGa1−xN films grown on sapphire substrates,” J. Appl. Phys. 94, 2980–2991 (2003).
[CrossRef]

Uchiyama, K.

H. Yoshida, M. Kuwabara, Y. Yamashita, Y. Takagi, K. Uchiyama, and H. Kan, “AlGaN-based laser diodes for the short-wavelength ultraviolet region,” New J. Phys. 11, 125013 (2009).
[CrossRef]

Umeda, H.

F. Kawamura, H. Umeda, M. Morishita, M. Kawahara, M. Yoshimura, Y. Mori, T. Sasaki, and Y. Kitaoka, “Growth of a two-inch GaN single crystal substrate using the Na flux method,” Jpn. J. Appl. Phys., Part 2 45, L1136–L1138 (2006).
[CrossRef]

Vanderbilt, D.

F. Bernardini, V. Fiorentini, and D. Vanderbilt, “Spontaneous polarization and piezoelectric constants of III-V nitrides,” Phys. Rev. B 56, R10024 (1997).
[CrossRef]

Wang, Y.

J. L. P. Hughes, Y. Wang, and J. E. Sipe, “Calculation of linear and second-order optical response in wurtzite GaN and AlN,” Phys. Rev. B 55, 13630–13640 (1997).
[CrossRef]

Webb-Wood, G.

Ü. Özgür, G. Webb-Wood, H. O. Everitt, F. Yun, and H. Morkoç, “Systematic measurement of AlxGa1−xN refractive indices,” Appl. Phys. Lett. 79, 4103–4105 (2001).
[CrossRef]

Wickenden, D. K.

Wilkins, J. W.

J. Chen, Z. H. Levine, and J. W. Wilkins, “Calculated second-harmonic susceptibilities of BN, AlN, and GaN,” Appl. Phys. Lett. 66, 1129–1131 (1995).
[CrossRef]

Yamashita, Y.

H. Yoshida, M. Kuwabara, Y. Yamashita, Y. Takagi, K. Uchiyama, and H. Kan, “AlGaN-based laser diodes for the short-wavelength ultraviolet region,” New J. Phys. 11, 125013 (2009).
[CrossRef]

Yoshida, H.

H. Yoshida, M. Kuwabara, Y. Yamashita, Y. Takagi, K. Uchiyama, and H. Kan, “AlGaN-based laser diodes for the short-wavelength ultraviolet region,” New J. Phys. 11, 125013 (2009).
[CrossRef]

Yoshimura, M.

F. Kawamura, H. Umeda, M. Morishita, M. Kawahara, M. Yoshimura, Y. Mori, T. Sasaki, and Y. Kitaoka, “Growth of a two-inch GaN single crystal substrate using the Na flux method,” Jpn. J. Appl. Phys., Part 2 45, L1136–L1138 (2006).
[CrossRef]

Yun, F.

Ü. Özgür, G. Webb-Wood, H. O. Everitt, F. Yun, and H. Morkoç, “Systematic measurement of AlxGa1−xN refractive indices,” Appl. Phys. Lett. 79, 4103–4105 (2001).
[CrossRef]

Zhang, H. Y.

H. Y. Zhang, X. H. He, Y. H. Shin, M. Schurman, Z. C. Feng, and R. A. Stall, “Study of nonlinear optical effects in GaN:Mg epitaxial film,” Appl. Phys. Lett. 69, 2953–2955 (1996).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. B (1)

D. Passeri, M. C. Larciprete, A. Belardini, S. Paoloni, A. Passaseo, C. Sibilia, and F. Michelotti, “Second harmonic generation in AlGaN, GaN and AlxGa1−xN/GaN multiple quantum well structures,” Appl. Phys. B 79, 611–615 (2004).
[CrossRef]

Appl. Phys. Lett. (5)

H. Y. Zhang, X. H. He, Y. H. Shin, M. Schurman, Z. C. Feng, and R. A. Stall, “Study of nonlinear optical effects in GaN:Mg epitaxial film,” Appl. Phys. Lett. 69, 2953–2955 (1996).
[CrossRef]

I. V. Kravetsky, I. M. Tiginyanu, R. Hildebrandt, G. Marowsky, D. Pavlidis, and A. Eisenbach, “Nonlinear optical response of GaN layers on sapphire: the impact of fundamental beam interface,” Appl. Phys. Lett. 76, 810–812 (2000).
[CrossRef]

M. J. Bergmann, Ü. Özgür, H. C. Casey, Jr., H. O. Everitt, and J. F. Muth, “Ordinary and extraordinary refractive indices for AlxGa1−xN epitaxial layers,” Appl. Phys. Lett. 75, 67–69 (1999).
[CrossRef]

Ü. Özgür, G. Webb-Wood, H. O. Everitt, F. Yun, and H. Morkoç, “Systematic measurement of AlxGa1−xN refractive indices,” Appl. Phys. Lett. 79, 4103–4105 (2001).
[CrossRef]

J. Chen, Z. H. Levine, and J. W. Wilkins, “Calculated second-harmonic susceptibilities of BN, AlN, and GaN,” Appl. Phys. Lett. 66, 1129–1131 (1995).
[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. (4)

N. A. Sanford, L. H. Robins, A. V. Davydov, A. Shapiro, D. V. Tsvetkov, A. V. Dmitriev, S. Keller, U. K. Mishra, and S. P. DenBaars, “Refractive index study of AlxGa1−xN films grown on sapphire substrates,” J. Appl. Phys. 94, 2980–2991 (2003).
[CrossRef]

S. Pezzagna, J. Brault, M. Leroux, J. Massies, and M. de Micheli, “Refractive indices and elasto-optic coefficients of GaN studied by optical waveguiding,” J. Appl. Phys. 103, 123112 (2008).
[CrossRef]

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]

J. Mater. Sci. (1)

H. Iwanaga, A. Kunishige, and S. Takeuchi, “Anisotropic thermal expansion in wurtzite-type crystals,” J. Mater. Sci. 35, 2451–2454 (2000).
[CrossRef]

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

Jpn. J. Appl. Phys. (1)

T. Ishidate, K. Inoue, and M. Aoki, “Second harmonic generation of epitaxially-grown GaN crystal,” Jpn. J. Appl. Phys. 19, 1641–1645 (1980).
[CrossRef]

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

T. Fujita, T. Hasegawa, M. Haraguchi, T. Okamoto, M. Fukui, and S. Nakamura, “Determination of second-order nonlinear optical susceptibility of GaN film on sapphire,” Jpn. J. Appl. Phys., Part 1 39, 2610–2613 (2000).
[CrossRef]

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

F. Kawamura, H. Umeda, M. Morishita, M. Kawahara, M. Yoshimura, Y. Mori, T. Sasaki, and Y. Kitaoka, “Growth of a two-inch GaN single crystal substrate using the Na flux method,” Jpn. J. Appl. Phys., Part 2 45, L1136–L1138 (2006).
[CrossRef]

T. Detchprohm, K. Hiramatsu, K. Itoh, and I. Akasaki, “Relaxation process of the thermal strain in the GaN/α-Al2O3 heterostructure and determination of the intrinsic lattice constants of GaN free from the strain,” Jpn. J. Appl. Phys., Part 2 31, L1454–L1456 (1992).
[CrossRef]

New J. Phys. (1)

H. Yoshida, M. Kuwabara, Y. Yamashita, Y. Takagi, K. Uchiyama, and H. Kan, “AlGaN-based laser diodes for the short-wavelength ultraviolet region,” New J. Phys. 11, 125013 (2009).
[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. Lett. (1)

F. N. H. Robinson, “Relation between the components of the non-linear polarisability tensor in cubic and hexagonal II-VI compounds,” Phys. Lett. 26A, 435–436 (1968).

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. B (4)

B. F. Levine, “Bond-charge calculation of nonlinear optical susceptibilities for various crystal structures,” Phys. Rev. B 7, 2600–2626 (1973).
[CrossRef]

B. F. Levine, “Origin of the unusual dependence of the nonlinear optical susceptibility on bond length for ionic ferroelectrics,” Phys. Rev. B 10, 1655–1664 (1974).
[CrossRef]

J. L. P. Hughes, Y. Wang, and J. E. Sipe, “Calculation of linear and second-order optical response in wurtzite GaN and AlN,” Phys. Rev. B 55, 13630–13640 (1997).
[CrossRef]

F. Bernardini, V. Fiorentini, and D. Vanderbilt, “Spontaneous polarization and piezoelectric constants of III-V nitrides,” Phys. Rev. B 56, R10024 (1997).
[CrossRef]

Prog. Quantum Electron. (1)

B. Monemar and G. Pozina, “Group III-nitride based hetero and quantum structures,” Prog. Quantum Electron. 24, 239–290 (2000).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Maker fringes of a (0001) HVPE-grown GaN plate in the s - p configuration. The open squares are the experimental data, and the solid curves show the theoretical fitting curves.

Fig. 2
Fig. 2

Maker fringes of a (0001) HVPE-grown GaN plate in the p - p configuration. The open squares are the experimental data, and the solid curves show the theoretical fitting curves.

Fig. 3
Fig. 3

Maker fringes of a (0001) HVPE-grown GaN plate in the 45 ° - s configuration. The open squares are the experimental data, and the solid curves show the theoretical fitting curves.

Fig. 4
Fig. 4

SH power versus sample thickness of the ( 11 2 ¯ 0 ) HVPE-grown GaN sample in the (a) d 31 , (b) d 15 , and (c) d 33 configurations. The open squares are the experimental data, and the solid curves represent the theoretical fitting curves.

Fig. 5
Fig. 5

SH power versus sample thickness of the ( 11 2 ¯ 0 ) LPE-grown GaN sample in the (a) d 31 , (b) d 15 , and (c) d 33 configurations. The open squares are the experimental data, and the solid curves represent the theoretical fitting curves.

Fig. 6
Fig. 6

Incident, transmitted, and reflected waves at the two interfaces, and their associated wave and unit polarization vectors in a plane-parallel sample with a misorientation ϕ.

Fig. 7
Fig. 7

Schematic of multiple reflection of the fundamental and SH waves in a plane-parallel plate with the misorientation angle of ϕ

Tables (1)

Tables Icon

Table 1 Measured Quadratic Nonlinear Optical Coefficients of GaN a

Equations (66)

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

d eff s - p = d 31   sin   θ 2 ω ,
d eff 45 ° - s = d 15   sin   θ ω ,
d eff p - p = d 15   cos   θ 2 ω   sin   2 θ ω + sin   θ 2 ω [ d 31 cos 2 θ ω + d 33 sin 2 θ ω ] ,
ε ν = ( ( n x ν ) 2 0 0 0 ( n y ν ) 2 0 0 0 ( n z ν ) 2 ) ,
k = ( k x , 0 , k z ) ,
k s ± = ( k x s ± , 0 , k z s ± ) ,
k p ± = ( k x p ± , 0 , k z p ± ) ,
k x = k x +   cos   2 ϕ + k z +   sin   2 ϕ ,
k z = k x +   sin   2 ϕ k z +   cos   2 ϕ ,
k x s ± = k x + cos 2 ϕ + k z +   sin   ϕ   cos   ϕ 1 2 sin   ϕ ( 2 n y 2 1 ) ( k x 2 + k z 2 ) 2 k x + k z +   sin   2 ϕ ( k x + k z + ) ( k x + + k z + ) cos   2 ϕ ,
k z s ± = k z + sin 2 ϕ + k x +   sin   ϕ   cos   ϕ ± 1 2 cos   ϕ ( 2 n y 2 1 ) [ ( k x + ) 2 + ( k z + ) 2 ] 2 k x + k z +   sin   2 ϕ ( k x + k z + ) ( k x + + k z + ) cos   2 ϕ ,
k x p ± = ( n z 2 k x 2 cos 2 ϕ + n z 2 k z 2   sin   ϕ   cos   ϕ n x n z   sin   ϕ { [ ( n z 2 1 ) ( k x + ) 2 + n z 2 ( k z + ) 2 ] cos 2 ϕ + [ n x 2 ( k x + ) 2 + ( n x 2 1 ) ( k z + ) 2 ] sin 2 ϕ 2 k x + k z +   sin   ϕ   cos   ϕ } 1 / 2 ) / ( n z 2 cos 2 ϕ + n x 2 sin 2 ϕ ) ,
k z p ± = ( n x 2 k z + sin 2 ϕ + n x 2 k x +   sin   ϕ   cos   ϕ ± n x n z   cos   ϕ { [ ( n z 2 1 ) ( k x + ) 2 + n z 2 ( k z + ) 2 ] cos 2 ϕ + [ n x 2 ( k x + ) 2 + ( n x 2 1 ) ( k z + ) 2 ] sin 2 ϕ 2 k x + k z +   sin   ϕ   cos   ϕ } 1 / 2 ) / ( n z 2 cos 2 ϕ + n x 2 sin 2 ϕ ) ,
s ̂ = ( 0 , 1 , 0 ) ,
p ̂ ± = ( k z ± / | k + | , 0 , k x ± / | k + | ) ,
q ̂ ± = ε 1 d ̂ ± | ε 1 d ̂ ± | = 1 | ε 1 d ̂ ± | | k p ± | ( k z p ± / n x 2 , 0 , k x p ± / n z 2 ) ,
t 01 = d 1 e 2 + d 2 e 1 d 2 e 3 d 3 e 2 ,
t 12 = d 3 e 4 d 4 e 3 d 1 e 4 d 4 e 1 ,
r 00 = d 1 e 3 + d 3 e 1 d 2 e 3 d 3 e 2 ,
r 11 = r 00 ,
d 1 = p x +   cos   ϕ p z +   sin   ϕ ,
d 2 = p x   cos   ϕ p z   sin   ϕ ,
d 3 = q x +   cos   ϕ q z +   sin   ϕ ,
d 4 = q x   cos   ϕ q z   sin   ϕ ,
e 1 = k z + p x + k x + p z + ,
e 2 = k z p x + k x + p z + ,
e 3 = k z p + q x + k x p + q z + ,
e 4 = k z p q x + k x p p z + ,
E α , β ( x , z , t ) = 1 2 u ̂ E α , β   exp [ i ( k x α , β x + k z α , β z ω t ) ] + c .c . ,
E s + = ( c 1 c 2 ) ( c 1 c 4 ) e i k z s L ( c 1 c 3 ) ( c 2 c 4 ) e i k z s + L ( c 1 c 4 ) ( c 2 c 3 ) e i k z s L E s , i ,
E s = ( c 1 c 2 ) ( c 1 c 3 ) e i k z s + L ( c 1 c 4 ) ( c 2 c 3 ) e i k z s L ( c 1 c 3 ) ( c 2 c 4 ) e i k z s + L E s , i ,
E p + = ( d 1 e 2 d 2 e 1 ) ( d 1 e 4 d 4 e 1 ) e i k z p L ( d 1 e 3 d 3 e 1 ) ( d 2 e 4 d 4 e 2 ) e i k z p + L ( d 1 e 4 d 4 e 1 ) ( d 2 e 3 d 3 e 2 ) e i k z p L E p , i ,
E p = ( d 1 e 2 d 2 e 1 ) ( d 1 e 3 d 3 e 1 ) e i k z p + L ( d 1 e 4 d 4 e 1 ) ( d 2 e 3 d 3 e 2 ) e i k z p L ( d 1 e 3 d 3 e 1 ) ( d 2 e 4 d 4 e 2 ) e i k z p + L E p , i ,
c 1 = k x +   sin   ϕ k z +   cos   ϕ ,
c 2 = k x   sin   ϕ k z   cos   ϕ ,
c 3 = k x s +   sin   ϕ k z s +   cos   ϕ ,
c 4 = k x s   sin   ϕ k z s   cos   ϕ .
E ω ( x , z ) = 1 2 s ̂ { E s +   exp [ i ( k x s + x + k z s + z ) ] + E s   exp [ i ( k x s x + k z s z ) ] } + 1 2 { q ̂ + E p +   exp [ i ( k x p + x + k z p + z ) ] + q ̂ E p   exp [ i ( k x p x + k z p z ) ] } + c .c .
P NL ( x , z ) i = 1 2 ε 0 d i j k E ω ( x , z ) j E ω ( x , z ) k + c .c . ,
P NL ( x , z ) = 1 2 P NL   exp [ i ( k x NL x + k z NL z ) ] + c .c .
× × E 2 ω ( x , z ) ε 2 ω ( k 0 2 ω ) 2 E 2 ω ( x , z ) = ( k 0 2 ω ) 2 ε 0 P NL ( x , z ) ,
E 2 ω ( x , z ) = 1 2 s ̂ 2 ω { E 2 ω s +   exp [ i ( k x 2 ω , s + x + k z 2 ω , s + z ) ] + E 2 ω s   exp [ i ( k x 2 ω , s x + k z 2 ω , s z ) ] } 1 2 { q ̂ 2 ω + E 2 ω p +   exp [ i ( k x 2 ω , p + x + k z 2 ω , p + z ) ] + q ̂ 2 ω E 2 ω p   exp [ i ( k x 2 ω , p x + k z 2 ω , p z ) ] } + 1 2 E NL   exp [ i ( k x NL x + k z NL z ) ] + c .c . ,
E x NL = [ ( n z 2 ω ) 2 ( k 0 2 ω ) 2 ( k x NL ) 2 ] P x NL k x NL k z NL P z NL ε 0 [ ( n x 2 ω ) 2 ( k x NL ) 2 + ( n z 2 ω ) 2 ( k z NL ) 2 ( n x 2 ω ) 2 ( n z 2 ω ) 2 ( k 0 2 ω ) 2 ] ,
E y NL = ( k 0 2 ω ) 2 P y NL ε 0 [ ( k x NL ) 2 + ( k z NL ) 2 ( n y 2 ω ) 2 ( k 0 2 ω ) 2 ] ,
E z NL = k x NL k z NL P x NL + [ ( n x 2 ω ) 2 ( k 0 2 ω ) 2 ( k z NL ) 2 ] P z NL ε 0 [ ( n x 2 ω ) 2 ( k x NL ) 2 + ( n z 2 ω ) 2 ( k z NL ) 2 ( n x 2 ω ) 2 ( n z 2 ω ) 2 ( k 0 2 ω ) 2 ] .
E 2 ω ( x , z ) = 1 2 s ̂ 2 ω E 2 ω s , r   exp [ i ( k x 2 ω , s , r x + k z 2 ω , s , r z ) ] + 1 2 p ̂ 2 ω E 2 ω p , r   exp [ i ( k x 2 ω , p , r x + k z 2 ω , p , r z ) ] + c .c . ,
E 2 ω ( x , z ) = 1 2 s ̂ 2 ω E 2 ω s , o   exp [ i ( k x 2 ω , s , o x + k z 2 ω , s , o z ) ] + 1 2 p ̂ 2 ω + E 2 ω p , o   exp [ i ( k x 2 ω , p , o x + k z 2 ω , p , o z ) ] + c .c . ,
I 2 ω = ε 0 c 2 ( | E 2 ω s , o | 2 + | E 2 ω p , o | 2 ) .
E 2 ω s , o = [ ( f 2 f 3 ) ( A 1 A 2 f 4 ) e i ( k z 2 ω , s + k z NL ) L ( f 2 f 4 ) ( A 1 A 2 f 3 ) e i ( k z 2 ω , s + + k z NL ) L + ( f 3 f 4 ) ( A 1 A 2 f 2 ) e i ( k z 2 ω , s + + k z 2 ω , s ) L ] / [ ( f 1 f 4 ) ( f 2 f 3 ) e i k z 2 ω , s L + ( f 3 f 1 ) ( f 2 f 4 ) e i k z 2 ω , s + L ] ,
E 2 ω p , o = [ ( g 3 h 4 g 4 h 3 ) ( g 2 A 4 h 2 A 3 ) e i ( k z 2 ω , p + + k z 2 ω , p ) L ( g 2 h 3 g 3 h 2 ) ( A 3 h 4 A 4 g 4 ) e i ( k z 2 ω , p + k z NL ) L + ( g 2 h 4 g 4 h 2 ) ( A 3 h 3 A 4 g 3 ) e i ( k z 2 ω , p + + k z NL ) L ] / [ ( g 3 h 1 g 1 h 3 ) ( g 4 h 2 g 2 h 4 ) e i k z 2 ω , p + L ( g 4 h 1 g 1 h 4 ) ( g 3 h 2 g 2 h 3 ) e i k z 2 ω , p L ] ,
f 1 = k x 2 ω +   sin   ϕ k z 2 ω +   cos   ϕ ,
f 2 = k x 2 ω   sin   ϕ k z 2 ω   cos   ϕ ,
f 3 = k x 2 ω , s +   sin   ϕ k z 2 ω , s +   cos   ϕ ,
f 4 = k x 2 ω , s   sin   ϕ k z 2 ω , s   cos   ϕ ,
g 1 = p x +   cos   ϕ p z +   sin   ϕ ,
g 2 = p x   cos   ϕ p z   sin   ϕ ,
g 3 = q x 2 ω +   cos   ϕ q z 2 ω +   sin   ϕ ,
g 4 = q x 2 ω   cos   ϕ q z 2 ω   sin   ϕ ,
h 1 = k z 2 ω + p x + k x 2 ω + p z + ,
h 2 = k z 2 ω p x k x 2 ω p z ,
h 3 = k z 2 ω , p + q x 2 ω + k x 2 ω , p + q z 2 ω + ,
h 4 = k z 2 ω , p q x 2 ω k x 2 ω , p q z 2 ω ,
A 1 = ( k x NL   sin   ϕ k z NL   cos   ϕ ) E y NL ,
A 2 = E y NL ,
A 3 = E x NL   cos   ϕ + E z NL   sin   ϕ ,
A 4 = k z NL E x NL k x NL E z NL .

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