Abstract

We discuss electric-field-induced second harmonic (EFISH) generation for silicon and zincblende facets (001), (011), and (111), employing the full fourth-rank tensor representation of the third-order susceptibility. Then we directly relate these 81 tensor elements with the contracted or Voigt matrix representation. Using group theory, we show that the number of independent elements is only two; however, at different facets, different linear combinations thereof appear. Also, specific expressions for the resulting s- and p-polarized second harmonic polarization are given for incident s- and p-polarizations, for the first time explaining the facet and angle of incidence dependence of EFISH. Finally, a classical oscillator model is used to explain the response of the electrons and the material combined with a direct physical interpretation of the breaking of the symmetry and thus the deformation of the electronic charge density along the bonds. Through this model we propose a connection between the strength parameter b for third harmonic generation and the second harmonic signal originated by EFISH mechanism.

© 2017 Optical Society of America

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  1. J. P. Gordon, H. J. Zeiger, and C. H. Townes, “The maser, new type of microwave amplifier, frequency, standard and spectrometer,” Phys. Rev. 99, 1264–1274 (1955).
    [Crossref]
  2. P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
    [Crossref]
  3. P. D. Maker and R. W. Terhune, “Study of optical effects due to an induced polarization third order in the electric field strength,” Phys. Rev. 137, A801–A818 (1965).
    [Crossref]
  4. J. F. Ward and G. H. C. New, “Optical third harmonic generation in gases by a focused laser beam,” Phys. Rev. 185, 57–72 (1969).
    [Crossref]
  5. R. W. Terhune, P. D. Maker, and C. M. Savage, “Optical harmonic generation in calcite,” Phys. Rev. Lett. 8, 404–406 (1962).
    [Crossref]
  6. Y.-S. Lee, M. H. Anderson, and M. C. Downer, “Fourth-harmonic generation at a crystalline GaAs(001) surface,” Opt. Lett. 22, 973–975 (1997).
    [Crossref]
  7. Y.-S. Lee and M. C. Downer, “Reflected fourth-harmonic radiation from a centrosymmetric crystal,” Opt. Lett. 23, 918–920 (1998).
    [Crossref]
  8. Y.-S. Lee and M. C. Downer, “Reflected optical fourth harmonic generation at crystalline surfaces,” Thin Solid Films 364, 80–85 (2000).
    [Crossref]
  9. J.-K. Hansen, H. J. Peng, and D. E. Aspnes, “Application of the simplified bond-hyperpolarizability model to fourth-harmonic generation,” J. Vac. Sci. Technol. B 21, 1798–1803 (2003).
    [Crossref]
  10. J. A. Giordmaine and R. C. Miller, “Tunable coherent parametric oscillation in LiNbO3 at optical frequencies,” Phys. Rev. Lett. 14, 973–976 (1965).
    [Crossref]
  11. J. A. Giordmaine and R. C. Miller, “Optical parametric oscillation in the visible spectrum,” Appl. Phys. Lett. 9, 298–300 (1966).
    [Crossref]
  12. R. W. Boyd and C. H. Townes, “An infrared upconverter for astronomical imaging,” Appl. Phys. Lett. 31, 440–442 (1977).
    [Crossref]
  13. R. L. Byer and R. L. Herbst, Tunable Infrared Generation, Y. R. Shen, ed. (Springer, 1977).
  14. C. H. Lee, R. K. Chang, and N. Bloembergen, “Nonlinear electroreflectance in silicon and silver,” Phys. Rev. Lett. 18, 167–170 (1967).
    [Crossref]
  15. S. Kielich, “DC electric field-induced second harmonic light generation in gases and liquids,” Acta Phys. Pol. A37, 205–219 (1970).
  16. C. G. Bethea, “Electric field induced second harmonic generation in glass,” Appl. Opt. 14, 2435–2437 (1975).
    [Crossref]
  17. O. A. Aktsipetrov, A. A. Fedyanin, V. N. Golovkina, and T. V. Murzina, “Optical second-harmonic generation induced by a dc electric field at the Si-SiO2 interface,” Opt. Lett. 19, 1450–1452 (1994).
    [Crossref]
  18. O. A. Aktsipetrov, A. A. Fedyanin, and A. V. Melnikov, “DC electric field induced second-harmonic generation spectroscopy of the Si(001)-SiO2 interface: separation of the bulk and surface non-linear contributions,” Thin Solid Films 294, 231–234 (1997).
    [Crossref]
  19. K. Kikuchi and K. Tada, “Theory of electric field-induced optical second harmonic generation in semiconductors,” Opt. Quantum Electron. 12, 199–205 (1980).
    [Crossref]
  20. C. Ohlhoff, G. Lüpke, C. Meyer, and H. Kurz, “Static and high-frequency electric fields in silicon MOS and MS structures probed by optical second-harmonic generation,” Phys. Rev. B 55, 4596–4606 (1997).
    [Crossref]
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    [Crossref]
  22. D. E. Aspnes, “Bond models in linear and nonlinear optics,” Phys. Status Solidi B 247, 1873–1880 (2010).
    [Crossref]
  23. D. E. Aspnes, “Bond models in linear and nonlinear optics,” Proc. SPIE 9584, 95840A (2015).
    [Crossref]
  24. A. Alejo-Molina, K. Hingerl, and H. Hardhienata, “Model of third harmonic generation and electric field induced optical second harmonic using simplified bond-hyperpolarizability model,” J. Opt. Soc. Am. B 32, 562–570 (2015).
    [Crossref]
  25. G. D. Powell, J. F. Wang, and D. E. Aspnes, “Simplified bond hyperpolarizability model of second harmonic generation,” Phys. Rev. B 65, 205320 (2002).
    [Crossref]
  26. J.-F. T. Wang, G. D. Powell, R. S. Johnson, G. Lucovsky, and D. E. Aspnes, “Simplified bond-hyperpolarizability model of second harmonic generation: application to Si-dielectric interfaces,” J. Vac. Sci. Technol. B 20, 1699–1705 (2002).
    [Crossref]
  27. E. Pavarini, “Crystal-field theory, tight-binding method, and Jahn-Teller effect,” in Correlated Electrons: From Models to Materials, E. Koch, F. Anders, and M. Jarrell, eds. (Forschungszentrum Jülich, 2012), pp. 6.1–6.39.
  28. R. C. Powell, Symmetry, Group Theory, and the Physical Properties of Crystals, Lecture Notes in Physics (Springer, 2010), Vol. 824.
  29. P. Y. Yu and M. Cardona, Fundamental of Semiconductors, Physics and Materials Properties (Springer, 2010).
  30. J. F. Nye, Physical Properties of Crystals, Their Representations by Tensors and Matrices (Clarendon, 1957).
  31. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1998).
  32. D. E. Aspnes and A. A. Studna, “Anisotropies in the above—band-gap optical spectra of cubic semiconductors,” Phys. Rev. Lett. 54, 1956–1959 (1985).
    [Crossref]
  33. R. W. Boyd, Nonlinear Optics, 2nd ed. (Academic, 2003).
  34. H. Hardhienata, A. Prylepa, D. Stifter, and K. Hingerl, “Simplified bond-hyperpolarizability model of second-harmonic generation in Si(111): theory and experiment,” J. Phys. 423, 012046 (2013).
    [Crossref]
  35. H. Hardhienata, A. Alejo-Molina, C. Reitböck, A. Prylepa, D. Stifter, and K. Hingerl, “Bulk dipolar contribution to second-harmonic generation in zincblende,” J. Opt. Soc. Am. B 33, 195–201 (2016).
    [Crossref]

2016 (1)

2015 (2)

2013 (1)

H. Hardhienata, A. Prylepa, D. Stifter, and K. Hingerl, “Simplified bond-hyperpolarizability model of second-harmonic generation in Si(111): theory and experiment,” J. Phys. 423, 012046 (2013).
[Crossref]

2010 (1)

D. E. Aspnes, “Bond models in linear and nonlinear optics,” Phys. Status Solidi B 247, 1873–1880 (2010).
[Crossref]

2003 (1)

J.-K. Hansen, H. J. Peng, and D. E. Aspnes, “Application of the simplified bond-hyperpolarizability model to fourth-harmonic generation,” J. Vac. Sci. Technol. B 21, 1798–1803 (2003).
[Crossref]

2002 (2)

G. D. Powell, J. F. Wang, and D. E. Aspnes, “Simplified bond hyperpolarizability model of second harmonic generation,” Phys. Rev. B 65, 205320 (2002).
[Crossref]

J.-F. T. Wang, G. D. Powell, R. S. Johnson, G. Lucovsky, and D. E. Aspnes, “Simplified bond-hyperpolarizability model of second harmonic generation: application to Si-dielectric interfaces,” J. Vac. Sci. Technol. B 20, 1699–1705 (2002).
[Crossref]

2000 (1)

Y.-S. Lee and M. C. Downer, “Reflected optical fourth harmonic generation at crystalline surfaces,” Thin Solid Films 364, 80–85 (2000).
[Crossref]

1998 (1)

1997 (3)

Y.-S. Lee, M. H. Anderson, and M. C. Downer, “Fourth-harmonic generation at a crystalline GaAs(001) surface,” Opt. Lett. 22, 973–975 (1997).
[Crossref]

O. A. Aktsipetrov, A. A. Fedyanin, and A. V. Melnikov, “DC electric field induced second-harmonic generation spectroscopy of the Si(001)-SiO2 interface: separation of the bulk and surface non-linear contributions,” Thin Solid Films 294, 231–234 (1997).
[Crossref]

C. Ohlhoff, G. Lüpke, C. Meyer, and H. Kurz, “Static and high-frequency electric fields in silicon MOS and MS structures probed by optical second-harmonic generation,” Phys. Rev. B 55, 4596–4606 (1997).
[Crossref]

1994 (1)

1987 (1)

1985 (1)

D. E. Aspnes and A. A. Studna, “Anisotropies in the above—band-gap optical spectra of cubic semiconductors,” Phys. Rev. Lett. 54, 1956–1959 (1985).
[Crossref]

1980 (1)

K. Kikuchi and K. Tada, “Theory of electric field-induced optical second harmonic generation in semiconductors,” Opt. Quantum Electron. 12, 199–205 (1980).
[Crossref]

1977 (1)

R. W. Boyd and C. H. Townes, “An infrared upconverter for astronomical imaging,” Appl. Phys. Lett. 31, 440–442 (1977).
[Crossref]

1975 (1)

1970 (1)

S. Kielich, “DC electric field-induced second harmonic light generation in gases and liquids,” Acta Phys. Pol. A37, 205–219 (1970).

1969 (1)

J. F. Ward and G. H. C. New, “Optical third harmonic generation in gases by a focused laser beam,” Phys. Rev. 185, 57–72 (1969).
[Crossref]

1967 (1)

C. H. Lee, R. K. Chang, and N. Bloembergen, “Nonlinear electroreflectance in silicon and silver,” Phys. Rev. Lett. 18, 167–170 (1967).
[Crossref]

1966 (1)

J. A. Giordmaine and R. C. Miller, “Optical parametric oscillation in the visible spectrum,” Appl. Phys. Lett. 9, 298–300 (1966).
[Crossref]

1965 (2)

J. A. Giordmaine and R. C. Miller, “Tunable coherent parametric oscillation in LiNbO3 at optical frequencies,” Phys. Rev. Lett. 14, 973–976 (1965).
[Crossref]

P. D. Maker and R. W. Terhune, “Study of optical effects due to an induced polarization third order in the electric field strength,” Phys. Rev. 137, A801–A818 (1965).
[Crossref]

1962 (1)

R. W. Terhune, P. D. Maker, and C. M. Savage, “Optical harmonic generation in calcite,” Phys. Rev. Lett. 8, 404–406 (1962).
[Crossref]

1961 (1)

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[Crossref]

1955 (1)

J. P. Gordon, H. J. Zeiger, and C. H. Townes, “The maser, new type of microwave amplifier, frequency, standard and spectrometer,” Phys. Rev. 99, 1264–1274 (1955).
[Crossref]

Aktsipetrov, O. A.

O. A. Aktsipetrov, A. A. Fedyanin, and A. V. Melnikov, “DC electric field induced second-harmonic generation spectroscopy of the Si(001)-SiO2 interface: separation of the bulk and surface non-linear contributions,” Thin Solid Films 294, 231–234 (1997).
[Crossref]

O. A. Aktsipetrov, A. A. Fedyanin, V. N. Golovkina, and T. V. Murzina, “Optical second-harmonic generation induced by a dc electric field at the Si-SiO2 interface,” Opt. Lett. 19, 1450–1452 (1994).
[Crossref]

Alejo-Molina, A.

Anderson, M. H.

Aspnes, D. E.

D. E. Aspnes, “Bond models in linear and nonlinear optics,” Proc. SPIE 9584, 95840A (2015).
[Crossref]

D. E. Aspnes, “Bond models in linear and nonlinear optics,” Phys. Status Solidi B 247, 1873–1880 (2010).
[Crossref]

J.-K. Hansen, H. J. Peng, and D. E. Aspnes, “Application of the simplified bond-hyperpolarizability model to fourth-harmonic generation,” J. Vac. Sci. Technol. B 21, 1798–1803 (2003).
[Crossref]

G. D. Powell, J. F. Wang, and D. E. Aspnes, “Simplified bond hyperpolarizability model of second harmonic generation,” Phys. Rev. B 65, 205320 (2002).
[Crossref]

J.-F. T. Wang, G. D. Powell, R. S. Johnson, G. Lucovsky, and D. E. Aspnes, “Simplified bond-hyperpolarizability model of second harmonic generation: application to Si-dielectric interfaces,” J. Vac. Sci. Technol. B 20, 1699–1705 (2002).
[Crossref]

D. E. Aspnes and A. A. Studna, “Anisotropies in the above—band-gap optical spectra of cubic semiconductors,” Phys. Rev. Lett. 54, 1956–1959 (1985).
[Crossref]

Bethea, C. G.

Bloembergen, N.

C. H. Lee, R. K. Chang, and N. Bloembergen, “Nonlinear electroreflectance in silicon and silver,” Phys. Rev. Lett. 18, 167–170 (1967).
[Crossref]

Boyd, R. W.

R. W. Boyd and C. H. Townes, “An infrared upconverter for astronomical imaging,” Appl. Phys. Lett. 31, 440–442 (1977).
[Crossref]

R. W. Boyd, Nonlinear Optics, 2nd ed. (Academic, 2003).

Byer, R. L.

R. L. Byer and R. L. Herbst, Tunable Infrared Generation, Y. R. Shen, ed. (Springer, 1977).

Cardona, M.

P. Y. Yu and M. Cardona, Fundamental of Semiconductors, Physics and Materials Properties (Springer, 2010).

Chang, R. K.

C. H. Lee, R. K. Chang, and N. Bloembergen, “Nonlinear electroreflectance in silicon and silver,” Phys. Rev. Lett. 18, 167–170 (1967).
[Crossref]

Downer, M. C.

Fedyanin, A. A.

O. A. Aktsipetrov, A. A. Fedyanin, and A. V. Melnikov, “DC electric field induced second-harmonic generation spectroscopy of the Si(001)-SiO2 interface: separation of the bulk and surface non-linear contributions,” Thin Solid Films 294, 231–234 (1997).
[Crossref]

O. A. Aktsipetrov, A. A. Fedyanin, V. N. Golovkina, and T. V. Murzina, “Optical second-harmonic generation induced by a dc electric field at the Si-SiO2 interface,” Opt. Lett. 19, 1450–1452 (1994).
[Crossref]

Franken, P. A.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[Crossref]

Giordmaine, J. A.

J. A. Giordmaine and R. C. Miller, “Optical parametric oscillation in the visible spectrum,” Appl. Phys. Lett. 9, 298–300 (1966).
[Crossref]

J. A. Giordmaine and R. C. Miller, “Tunable coherent parametric oscillation in LiNbO3 at optical frequencies,” Phys. Rev. Lett. 14, 973–976 (1965).
[Crossref]

Golovkina, V. N.

Gordon, J. P.

J. P. Gordon, H. J. Zeiger, and C. H. Townes, “The maser, new type of microwave amplifier, frequency, standard and spectrometer,” Phys. Rev. 99, 1264–1274 (1955).
[Crossref]

Hansen, J.-K.

J.-K. Hansen, H. J. Peng, and D. E. Aspnes, “Application of the simplified bond-hyperpolarizability model to fourth-harmonic generation,” J. Vac. Sci. Technol. B 21, 1798–1803 (2003).
[Crossref]

Hardhienata, H.

Herbst, R. L.

R. L. Byer and R. L. Herbst, Tunable Infrared Generation, Y. R. Shen, ed. (Springer, 1977).

Hill, A. E.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[Crossref]

Hingerl, K.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1998).

Johnson, R. S.

J.-F. T. Wang, G. D. Powell, R. S. Johnson, G. Lucovsky, and D. E. Aspnes, “Simplified bond-hyperpolarizability model of second harmonic generation: application to Si-dielectric interfaces,” J. Vac. Sci. Technol. B 20, 1699–1705 (2002).
[Crossref]

Kielich, S.

S. Kielich, “DC electric field-induced second harmonic light generation in gases and liquids,” Acta Phys. Pol. A37, 205–219 (1970).

Kikuchi, K.

K. Kikuchi and K. Tada, “Theory of electric field-induced optical second harmonic generation in semiconductors,” Opt. Quantum Electron. 12, 199–205 (1980).
[Crossref]

Kurz, H.

C. Ohlhoff, G. Lüpke, C. Meyer, and H. Kurz, “Static and high-frequency electric fields in silicon MOS and MS structures probed by optical second-harmonic generation,” Phys. Rev. B 55, 4596–4606 (1997).
[Crossref]

Lee, C. H.

C. H. Lee, R. K. Chang, and N. Bloembergen, “Nonlinear electroreflectance in silicon and silver,” Phys. Rev. Lett. 18, 167–170 (1967).
[Crossref]

Lee, Y.-S.

Lucovsky, G.

J.-F. T. Wang, G. D. Powell, R. S. Johnson, G. Lucovsky, and D. E. Aspnes, “Simplified bond-hyperpolarizability model of second harmonic generation: application to Si-dielectric interfaces,” J. Vac. Sci. Technol. B 20, 1699–1705 (2002).
[Crossref]

Lüpke, G.

C. Ohlhoff, G. Lüpke, C. Meyer, and H. Kurz, “Static and high-frequency electric fields in silicon MOS and MS structures probed by optical second-harmonic generation,” Phys. Rev. B 55, 4596–4606 (1997).
[Crossref]

Maker, P. D.

P. D. Maker and R. W. Terhune, “Study of optical effects due to an induced polarization third order in the electric field strength,” Phys. Rev. 137, A801–A818 (1965).
[Crossref]

R. W. Terhune, P. D. Maker, and C. M. Savage, “Optical harmonic generation in calcite,” Phys. Rev. Lett. 8, 404–406 (1962).
[Crossref]

Melnikov, A. V.

O. A. Aktsipetrov, A. A. Fedyanin, and A. V. Melnikov, “DC electric field induced second-harmonic generation spectroscopy of the Si(001)-SiO2 interface: separation of the bulk and surface non-linear contributions,” Thin Solid Films 294, 231–234 (1997).
[Crossref]

Meyer, C.

C. Ohlhoff, G. Lüpke, C. Meyer, and H. Kurz, “Static and high-frequency electric fields in silicon MOS and MS structures probed by optical second-harmonic generation,” Phys. Rev. B 55, 4596–4606 (1997).
[Crossref]

Miller, R. C.

J. A. Giordmaine and R. C. Miller, “Optical parametric oscillation in the visible spectrum,” Appl. Phys. Lett. 9, 298–300 (1966).
[Crossref]

J. A. Giordmaine and R. C. Miller, “Tunable coherent parametric oscillation in LiNbO3 at optical frequencies,” Phys. Rev. Lett. 14, 973–976 (1965).
[Crossref]

Murzina, T. V.

New, G. H. C.

J. F. Ward and G. H. C. New, “Optical third harmonic generation in gases by a focused laser beam,” Phys. Rev. 185, 57–72 (1969).
[Crossref]

Nye, J. F.

J. F. Nye, Physical Properties of Crystals, Their Representations by Tensors and Matrices (Clarendon, 1957).

Ohlhoff, C.

C. Ohlhoff, G. Lüpke, C. Meyer, and H. Kurz, “Static and high-frequency electric fields in silicon MOS and MS structures probed by optical second-harmonic generation,” Phys. Rev. B 55, 4596–4606 (1997).
[Crossref]

Pavarini, E.

E. Pavarini, “Crystal-field theory, tight-binding method, and Jahn-Teller effect,” in Correlated Electrons: From Models to Materials, E. Koch, F. Anders, and M. Jarrell, eds. (Forschungszentrum Jülich, 2012), pp. 6.1–6.39.

Peng, H. J.

J.-K. Hansen, H. J. Peng, and D. E. Aspnes, “Application of the simplified bond-hyperpolarizability model to fourth-harmonic generation,” J. Vac. Sci. Technol. B 21, 1798–1803 (2003).
[Crossref]

Peters, C. W.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[Crossref]

Powell, G. D.

G. D. Powell, J. F. Wang, and D. E. Aspnes, “Simplified bond hyperpolarizability model of second harmonic generation,” Phys. Rev. B 65, 205320 (2002).
[Crossref]

J.-F. T. Wang, G. D. Powell, R. S. Johnson, G. Lucovsky, and D. E. Aspnes, “Simplified bond-hyperpolarizability model of second harmonic generation: application to Si-dielectric interfaces,” J. Vac. Sci. Technol. B 20, 1699–1705 (2002).
[Crossref]

Powell, R. C.

R. C. Powell, Symmetry, Group Theory, and the Physical Properties of Crystals, Lecture Notes in Physics (Springer, 2010), Vol. 824.

Prylepa, A.

H. Hardhienata, A. Alejo-Molina, C. Reitböck, A. Prylepa, D. Stifter, and K. Hingerl, “Bulk dipolar contribution to second-harmonic generation in zincblende,” J. Opt. Soc. Am. B 33, 195–201 (2016).
[Crossref]

H. Hardhienata, A. Prylepa, D. Stifter, and K. Hingerl, “Simplified bond-hyperpolarizability model of second-harmonic generation in Si(111): theory and experiment,” J. Phys. 423, 012046 (2013).
[Crossref]

Reitböck, C.

Savage, C. M.

R. W. Terhune, P. D. Maker, and C. M. Savage, “Optical harmonic generation in calcite,” Phys. Rev. Lett. 8, 404–406 (1962).
[Crossref]

Sipe, J. E.

Stifter, D.

H. Hardhienata, A. Alejo-Molina, C. Reitböck, A. Prylepa, D. Stifter, and K. Hingerl, “Bulk dipolar contribution to second-harmonic generation in zincblende,” J. Opt. Soc. Am. B 33, 195–201 (2016).
[Crossref]

H. Hardhienata, A. Prylepa, D. Stifter, and K. Hingerl, “Simplified bond-hyperpolarizability model of second-harmonic generation in Si(111): theory and experiment,” J. Phys. 423, 012046 (2013).
[Crossref]

Studna, A. A.

D. E. Aspnes and A. A. Studna, “Anisotropies in the above—band-gap optical spectra of cubic semiconductors,” Phys. Rev. Lett. 54, 1956–1959 (1985).
[Crossref]

Tada, K.

K. Kikuchi and K. Tada, “Theory of electric field-induced optical second harmonic generation in semiconductors,” Opt. Quantum Electron. 12, 199–205 (1980).
[Crossref]

Terhune, R. W.

P. D. Maker and R. W. Terhune, “Study of optical effects due to an induced polarization third order in the electric field strength,” Phys. Rev. 137, A801–A818 (1965).
[Crossref]

R. W. Terhune, P. D. Maker, and C. M. Savage, “Optical harmonic generation in calcite,” Phys. Rev. Lett. 8, 404–406 (1962).
[Crossref]

Townes, C. H.

R. W. Boyd and C. H. Townes, “An infrared upconverter for astronomical imaging,” Appl. Phys. Lett. 31, 440–442 (1977).
[Crossref]

J. P. Gordon, H. J. Zeiger, and C. H. Townes, “The maser, new type of microwave amplifier, frequency, standard and spectrometer,” Phys. Rev. 99, 1264–1274 (1955).
[Crossref]

Wang, J. F.

G. D. Powell, J. F. Wang, and D. E. Aspnes, “Simplified bond hyperpolarizability model of second harmonic generation,” Phys. Rev. B 65, 205320 (2002).
[Crossref]

Wang, J.-F. T.

J.-F. T. Wang, G. D. Powell, R. S. Johnson, G. Lucovsky, and D. E. Aspnes, “Simplified bond-hyperpolarizability model of second harmonic generation: application to Si-dielectric interfaces,” J. Vac. Sci. Technol. B 20, 1699–1705 (2002).
[Crossref]

Ward, J. F.

J. F. Ward and G. H. C. New, “Optical third harmonic generation in gases by a focused laser beam,” Phys. Rev. 185, 57–72 (1969).
[Crossref]

Weinreich, G.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[Crossref]

Yu, P. Y.

P. Y. Yu and M. Cardona, Fundamental of Semiconductors, Physics and Materials Properties (Springer, 2010).

Zeiger, H. J.

J. P. Gordon, H. J. Zeiger, and C. H. Townes, “The maser, new type of microwave amplifier, frequency, standard and spectrometer,” Phys. Rev. 99, 1264–1274 (1955).
[Crossref]

Acta Phys. Pol. (1)

S. Kielich, “DC electric field-induced second harmonic light generation in gases and liquids,” Acta Phys. Pol. A37, 205–219 (1970).

Appl. Opt. (1)

Appl. Phys. Lett. (2)

J. A. Giordmaine and R. C. Miller, “Optical parametric oscillation in the visible spectrum,” Appl. Phys. Lett. 9, 298–300 (1966).
[Crossref]

R. W. Boyd and C. H. Townes, “An infrared upconverter for astronomical imaging,” Appl. Phys. Lett. 31, 440–442 (1977).
[Crossref]

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

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Figures (3)

Fig. 1.
Fig. 1. Conventional silicon cell.
Fig. 2.
Fig. 2. (a) SHG, inside the high index nonlinear material, has a very small angle with the normal. There is a coherent contribution just for a certain depth d . (b) High index prims keeps a large angle with the normal inside the nonlinear material.
Fig. 3.
Fig. 3. Equilibrium position of the electric charge is shifted due to the DC field E Z . Facets (a) Si(001), (b) Si(110), and (c) Si(111).

Equations (23)

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P ( ω ) = ϵ 0 { χ ( 1 ) · E ( ω ) + χ ( 2 A ) · · [ E ( ω ) E ( 0 ) ] + } P ( 2 ω ) = ϵ 0 { χ ( 2 ) · · [ E ( ω ) E ( ω ) ] + χ ( 3 A ) [ E ( ω ) E ( ω ) E ( 0 ) ] + } P ( 3 ω ) = ϵ 0 { χ ( 3 ) [ E ( ω ) E ( ω ) E ( ω ) ] + } ,
P i = χ i j k l ( 3 ) E j ( ω ) E k ( ω ) E l ( 0 ) .
χ ( 3 ) = ( ( χ 1111 χ 1112 χ 1113 χ 1121 χ 1122 χ 1123 χ 1131 χ 1132 χ 1133 ) ( χ 1211 χ 1212 χ 1213 χ 1221 χ 1222 χ 1223 χ 1231 χ 1232 χ 1233 ) ( χ 1311 χ 1312 χ 1313 χ 1321 χ 1322 χ 1323 χ 1331 χ 1332 χ 1333 ) ( χ 2111 χ 2112 χ 2113 χ 2121 χ 2122 χ 2123 χ 2131 χ 2132 χ 2133 ) ( χ 2211 χ 2212 χ 2213 χ 2221 χ 2222 χ 2223 χ 2231 χ 2232 χ 2233 ) ( χ 2311 χ 2312 χ 2313 χ 2321 χ 2322 χ 2323 χ 2331 χ 2332 χ 2333 ) ( χ 3111 χ 3112 χ 3113 χ 3121 χ 3122 χ 3123 χ 3131 χ 3132 χ 3133 ) ( χ 3211 χ 3212 χ 3213 χ 3221 χ 3222 χ 3223 χ 3231 χ 3232 χ 3233 ) ( χ 3311 χ 3312 χ 3313 χ 3321 χ 3322 χ 3323 χ 3331 χ 3332 χ 3333 ) ) .
S = ( s 11 s 12 s 13 s 14 s 15 s 16 s 21 s 22 s 23 s 24 s 25 s 26 s 31 s 32 s 33 s 34 s 35 s 36 s 41 s 42 s 43 s 44 s 45 s 46 s 51 s 52 s 53 s 54 s 55 s 56 s 61 s 62 s 63 s 64 s 65 s 66 ) ,
χ i j k l = R i m R j n R k o R l p χ m n o p ,
χ G T ( 3 ) ( 001 ) = ( s 11 s 12 s 12 0 0 0 s 21 s 11 s 12 0 0 0 s 21 s 21 s 11 0 0 0 0 0 0 s 44 0 0 0 0 0 0 s 44 0 0 0 0 0 0 s 44 ) ,
χ i j k l , G T ( 3 ) ( 001 ) = ( ( s 11 0 0 0 s 12 0 0 0 s 12 ) ( 0 s 44 4 0 s 44 4 0 0 0 0 0 ) ( 0 0 s 44 4 0 0 0 s 44 4 0 0 ) ( 0 s 44 4 0 s 44 4 0 0 0 0 0 ) ( s 21 0 0 0 s 11 0 0 0 s 12 ) ( 0 0 0 0 0 s 44 4 0 s 44 4 0 ) ( 0 0 s 44 4 0 0 0 s 44 4 0 0 ) ( 0 0 0 0 0 s 44 4 0 s 44 4 0 ) ( s 21 0 0 0 s 21 0 0 0 s 11 ) ) ,
χ i j k l , EFISH ( 3 ) ( 2 ω , ω , ω , 0 ) = 1 2 ( χ i j k l , EFISH ( 3 ) + χ i k j l , EFISH ( 3 ) ) ,
P s ( 2 ω ) = ( 0,0 , s 12 E y 2 E z ) .
P p ( 2 ω ) = E p 2 E z ( s 12 sin    2 θ 2 ( ω ) , 0 , s 11 sin 2 θ 2 ( ω ) + s 12 cos 2 θ 2 ( ω ) ) ,
n 1 ( ω ) sin    θ 1 ( ω ) = n 2 ( ω ) sin    θ 2 ( ω ) ,
n 2 ( 2 ω ) sin    θ 2 ( 2 ω ) = n 1 ( 2 ω ) sin    θ 1 ( 2 ω ) .
χ i j k l , EFISH ( 3 ) ( 111 ) = ( ( 1 2 ( s 11 + 3 s 12 ) 0 s 11 3 s 12 3 2 0 1 6 ( s 11 + 3 s 12 ) 0 s 11 3 s 12 3 2 0 s 11 3 ) ( 0 1 6 ( s 11 + 3 s 12 ) 0 1 6 ( s 11 + 3 s 12 ) 0 s 11 3 s 12 3 2 0 s 11 3 s 12 3 2 0 ) ( s 11 3 s 12 3 2 0 s 11 3 0 s 11 3 s 12 3 2 0 s 11 3 0 0 ) ( 0 1 6 ( s 11 + 3 s 12 ) 0 1 6 ( s 11 + 3 s 12 ) 0 s 11 3 s 12 3 2 0 s 11 3 s 12 3 2 0 ) ( 1 6 ( s 11 + 3 s 12 ) 0 s 11 3 s 12 3 2 0 1 2 ( s 11 + 3 s 12 ) 0 s 11 3 s 12 3 2 0 s 11 3 ) ( 0 s 11 3 s 12 3 2 0 s 11 3 s 12 3 2 0 s 11 3 0 s 11 3 0 ) ( s 11 3 s 12 3 2 0 s 11 3 0 s 11 3 s 12 3 2 0 s 11 3 0 0 ) ( 0 s 11 3 s 12 3 2 0 s 11 3 s 12 3 2 0 s 11 3 0 s 11 3 0 ) ( s 11 3 0 0 0 s 11 3 0 0 0 1 3 ( s 11 + 6 s 12 ) ) ) .
P s ( 2 ω ) = 1 3 E y 2 E z ( 2 2 [ s 11 3 s 12 ] , 0 , s 11 ) ,
P p ( 2 ω ) = 1 3 E p 2 E z ( 2 2 [ s 11 3 s 12 ] cos 2 θ 2 ( ω ) s 11 sin    2 θ 2 ( ω ) , 0 , s 11 + 6 s 12 sin 2 θ 2 ( ω ) ) ,
P ( 2 ω ) = ϵ 0 χ ( 2 ω ) E ( 2 ω ) ,
χ i j k l , EFISH ( 3 ) ( 110 ) = ( ( s 11 0 0 0 s 12 0 0 0 s 12 ) ( 0 s 12 0 s 12 0 0 0 0 0 ) ( 0 0 s 12 0 0 0 s 12 0 0 ) ( 0 s 12 0 s 12 0 0 0 0 0 ) ( s 12 0 0 0 s 11 + 3 s 12 2 0 0 0 s 11 s 12 2 ) ( 0 0 0 0 0 s 11 s 12 2 0 s 11 s 12 2 0 ) ( 0 0 s 12 0 0 0 s 12 0 0 ) ( 0 0 0 0 0 s 11 s 12 2 0 s 11 s 12 2 0 ) ( s 12 0 0 0 s 11 s 12 2 0 0 0 s 11 + 3 s 12 2 ) ) .
P s ( 2 ω ) = 1 2 E y 2 E z ( 0,0 , [ s 11 s 12 ] ) ,
P p ( 2 ω ) = 1 2 E p 2 E z ( 2 s 12 sin    2 θ 2 ( ω ) , 0,2 s 12 + [ s 11 + s 12 ] sin 2 θ 2 ( ω ) ) .
V ( r ) = 1 2 m ω 0 2 r 2 1 4 m b r 4 q sin    γ E DC r ,
r r 0 q sin    γ E DC m ω 0 2 = r 0 r eff ,
V ( r ) = 1 2 ( m ω 0 2 + 1 2 m b r eff 2 ) r eff 2 m b r eff 3 ( r r eff ) + 1 2 ( m ω 0 2 3    mb r eff 2 ) ( r r eff ) 2 m b r eff ( r r eff ) 3 + O [ r r eff ] 4 ,
a eff ( 2 ) ( 2 ω ) = 3 b q ω 0 2 sin    γ E DC .

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