Abstract

We predict the frequency-dependent bulk quadrupole contribution to second harmonic generation in silicon quantitatively from the linear susceptibility by means of a generalized classical anharmonic oscillator model and the simplified bond hyperpolarizability model. We show that in single-beam setups the main contribution is found for the silicon (111) surface, and only a minor contribution for the (001) and (011) facets. The dipole contribution obtained from our model is compared to literature values for SiC, AlAs and GaAs to demonstrate the viability of the method.

© 2017 Optical Society of America

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Simplified bond-hyperpolarizability model of second harmonic generation, group theory, and Neumann’s principle

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  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]
  2. X. D. Zhu, H. Suhr, and Y. R. Shen, “Surface vibrational spectroscopy by infrared-visible sum frequency generation,” Phys. Rev. B 35, 3047–3050 (1987).
    [Crossref]
  3. F. Brown, R. E. Parks, and A. M. Sleeper, “Nonlinear optical reflection from a metallic boundary,” Phys. Rev. Lett. 14, 1029–1031 (1965).
    [Crossref]
  4. N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, “Optical second-harmonic generation in reflection from media with inversion symmetry,” Phys. Rev. 174, 813–822 (1968).
    [Crossref]
  5. Y. R. Shen, “Surface studies by optical second harmonic generation: An overview,” J. Vac. Sci. & Technol. B 3, 1464 (1985).
    [Crossref]
  6. H. W. K. Tom and G. D. Aumiller, “Observation of rotational anisotropy in the second-harmonic generation from a metal surface,” Phys. Rev. B 33, 8818–8821 (1986).
    [Crossref]
  7. Y. R. Shen, “Surface properties probed by second-harmonic and sum-frequency generation,” Nat. 337, 519–525 (1989).
    [Crossref]
  8. J. F. McGilp, “A review of optical second-harmonic and sum-frequency generation at surfaces and interfaces,” J. Phys. D: Appl. Phys. 29, 1812–1821 (1996).
    [Crossref]
  9. K. B. Eisenthal, “Liquid interfaces probed by second-harmonic and sum-frequency spectroscopy,” Chem. Rev. 96, 1343–1360 (1996).
    [Crossref] [PubMed]
  10. Y. R. Shen, “Surface nonlinear optics [Invited],” J. Opt. Soc. Am. B 28, A56 (2011).
    [Crossref]
  11. M. Buck and M. Himmelhaus, “Vibrational spectroscopy of interfaces by infrared–visible sum frequency generation,” J. Vac. Sci. & Technol. A 19, 2717 (2001).
    [Crossref]
  12. G. L. Richmond, “Molecular bonding and interactions at aqueous surfaces as probed by vibrational sum frequency spectroscopy,” Chem. Rev. 102, 2693–2724 (2002).
    [Crossref] [PubMed]
  13. F. Vidal and A. Tadjeddine, “Sum-frequency generation spectroscopy of interfaces,” Reports on Prog. Phys. 68, 1095–1127 (2005).
    [Crossref]
  14. T. F. Heinz, M. M. T. Loy, and W. A. Thompson, “Study of symmetry and disordering of Si(111)-7×7 surfaces by optical second harmonic generation,” J. Vac. Sci. & Technol. B 3, 1467 (1985).
    [Crossref]
  15. H. Tom, X. Zhu, Y. Shen, and G. Somorjai, “Investigation of the Si(111)-(7×7) surface by second-harmonic generation: Oxidation and the effects of surface phosphorus,” Surf. Sci. Lett. 167, A76 (1986).
    [Crossref]
  16. H. W. K. Tom, C. M. Mate, X. D. Zhu, J. E. Crowell, T. F. Heinz, G. A. Somorjai, and Y. R. Shen, “Surface studies by optical second-harmonic generation: The adsorption of O2, CO, and sodium on the Rh(111) surface,” Phys. Rev. Lett. 52, 348–351 (1984).
    [Crossref]
  17. P. Guyot-Sionnest and Y. R. Shen, “Local and nonlocal surface nonlinearities for surface optical second-harmonic generation,” Phys. Rev. B 35, 4420–4426 (1987).
    [Crossref]
  18. G. D. Powell, J.-F. T. Wang, and D. E. Aspnes, “Simplified bond-hyperpolarizability model of second harmonic generation,” Phys. Rev. B 65, 205320 (2002).
    [Crossref]
  19. H. Held, A. I. Lvovsky, X. Wei, and Y. R. Shen, “Bulk contribution from isotropic media in surface sum-frequency generation,” Phys. Rev. B 66, 205110 (2002).
    [Crossref]
  20. C. Reitböck, D. Stifter, K. Hingerl, A. Alejo-Molina, and H. Hardhienata, “Bulk quadrupole and interface dipole contribution for second harmonic generation in Si(111),” J. Opt. 18, 1–7 (2016).
    [Crossref]
  21. A. Alejo-Molina, H. Hardhienata, and K. Hingerl, “Simplified bond-hyperpolarizability model of second harmonic generation, group theory, and Neumann’s principle,” J. Opt. Soc. Am. B 31, 526 (2014).
    [Crossref]
  22. J. Kwon, M. C. Downer, and B. S. Mendoza, “Second-harmonic and reflectance-anisotropy spectroscopy of vicinal Si (001) Si O2 interfaces: Experiment and simplified microscopic model,” Phys. Rev. B 73, 1–12 (2006).
    [Crossref]
  23. E. J. Adles and D. E. Aspnes, “Application of the anisotropic bond model to second-harmonic generation from amorphous media,” Phys. Rev. B 8202, 10 (2008).
  24. D. E. Aspnes, “Bond models in linear and nonlinear optics,” Phys. Status Solidi B 247, 1873–1880 (2010).
    [Crossref]
  25. E. J. Adles and D. E. Aspnes, “The anisotropic bond model of nonlinear optics,” Phys. Status Solidi A 205, 728–731 (2008).
    [Crossref]
  26. G. Lefkidis and W. Hübner, “Ab initio treatment of optical second harmonic generation in NiO,” Phys. Rev. Lett. 95, 2–5 (2005).
    [Crossref]
  27. E. Adler, “Nonlinear optical frequency polarization in a dielectric,” Phys. Rev. 134, 728–733 (1964).
    [Crossref]
  28. R. W. Boyd, Nonlinear optics (Elsevier Science, 2003).
  29. K.-D. Bauer, M. Panholzer, and K. Hingerl, “Bulk quadrupole contribution to second harmonic generation from a microscopic response function,” Phys. Status Solidi B 253, 234–240 (2016).
    [Crossref]
  30. R. C. Miller, “Optical second harmonic generation in piezoelectric crystals,” Appl. Phys. Lett. 5, 17–19 (1964).
    [Crossref]
  31. V. I. Gavrilenko, R. Q. Wu, M. C. Downer, J. G. Ekerdt, D. Lim, and P. Parkinson, “Optical second harmonic spectra of silicon-adatom surfaces: Theory and experiment,” Thin Solid Films 364, 1–5 (2000).
    [Crossref]
  32. M. Gajdoš, K. Hummer, G. Kresse, J. Furthmüller, and F. Bechstedt, “Linear optical properties in the projector-augmented wave methodology,” Phys. Rev. B 73, 045112 (2006).
    [Crossref]
  33. G. Kresse and J. Hafner, “Ab initio molecular dynamics for liquid metals,” Phys. Rev. B 47, 558–561 (1993).
    [Crossref]
  34. G. Kresse and J. Hafner, “Norm-conserving and ultrasoft pseudopotentials for first-row and transition elements,” J. Phys. Condens. Matter 6, 8245–8257 (1994).
    [Crossref]
  35. G. Kresse and J. Furthmüller, “Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set,” Comput. Mater. Sci. 6, 15–50 (1996).
    [Crossref]
  36. G. Kresse and J. Furthmüller, “Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set,” Phys. Rev. B 54, 11169–11186 (1996).
    [Crossref]
  37. G. Kresse and D. Joubert, “From ultrasoft pseudopotentials to the projector augmented-wave method,” Phys. Rev. B 59, 1758–1775 (1999).
    [Crossref]
  38. E. Luppi, H. Hübener, and V. Véniard, “Ab initio second-order nonlinear optics in solids: Second-harmonic generation spectroscopy from time-dependent density-functional theory,” Phys. Rev. B 82, 1–15 (2010).
    [Crossref]
  39. D. E. Aspnes and A. A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27, 985–1009 (1983).
    [Crossref]
  40. P. Yu and M. Cardona, Fundamentals of semiconductors: Physics and materials properties, Graduate Texts in Physics (SpringerBerlin Heidelberg, 2010).
    [Crossref]
  41. A. Meurer, C. P. Smith, M. Paprocki, O. Čertík, S. B. Kirpichev, M. Rocklin, A. Kumar, S. Ivanov, J. K. Moore, S. Singh, T. Rathnayake, S. Vig, B. E. Granger, R. P. Muller, F. Bonazzi, H. Gupta, S. Vats, F. Johansson, F. Pedregosa, M. J. Curry, A. R. Terrel, v. Roučka, A. Saboo, I. Fernando, S. Kulal, R. Cimrman, and A. Scopatz, “Sympy: Symbolic computing in python,” PeerJ Comput. Sci. 3, e103 (2017).
    [Crossref]

2017 (1)

A. Meurer, C. P. Smith, M. Paprocki, O. Čertík, S. B. Kirpichev, M. Rocklin, A. Kumar, S. Ivanov, J. K. Moore, S. Singh, T. Rathnayake, S. Vig, B. E. Granger, R. P. Muller, F. Bonazzi, H. Gupta, S. Vats, F. Johansson, F. Pedregosa, M. J. Curry, A. R. Terrel, v. Roučka, A. Saboo, I. Fernando, S. Kulal, R. Cimrman, and A. Scopatz, “Sympy: Symbolic computing in python,” PeerJ Comput. Sci. 3, e103 (2017).
[Crossref]

2016 (2)

C. Reitböck, D. Stifter, K. Hingerl, A. Alejo-Molina, and H. Hardhienata, “Bulk quadrupole and interface dipole contribution for second harmonic generation in Si(111),” J. Opt. 18, 1–7 (2016).
[Crossref]

K.-D. Bauer, M. Panholzer, and K. Hingerl, “Bulk quadrupole contribution to second harmonic generation from a microscopic response function,” Phys. Status Solidi B 253, 234–240 (2016).
[Crossref]

2014 (1)

2011 (1)

2010 (2)

E. Luppi, H. Hübener, and V. Véniard, “Ab initio second-order nonlinear optics in solids: Second-harmonic generation spectroscopy from time-dependent density-functional theory,” Phys. Rev. B 82, 1–15 (2010).
[Crossref]

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

2008 (2)

E. J. Adles and D. E. Aspnes, “The anisotropic bond model of nonlinear optics,” Phys. Status Solidi A 205, 728–731 (2008).
[Crossref]

E. J. Adles and D. E. Aspnes, “Application of the anisotropic bond model to second-harmonic generation from amorphous media,” Phys. Rev. B 8202, 10 (2008).

2006 (2)

M. Gajdoš, K. Hummer, G. Kresse, J. Furthmüller, and F. Bechstedt, “Linear optical properties in the projector-augmented wave methodology,” Phys. Rev. B 73, 045112 (2006).
[Crossref]

J. Kwon, M. C. Downer, and B. S. Mendoza, “Second-harmonic and reflectance-anisotropy spectroscopy of vicinal Si (001) Si O2 interfaces: Experiment and simplified microscopic model,” Phys. Rev. B 73, 1–12 (2006).
[Crossref]

2005 (2)

G. Lefkidis and W. Hübner, “Ab initio treatment of optical second harmonic generation in NiO,” Phys. Rev. Lett. 95, 2–5 (2005).
[Crossref]

F. Vidal and A. Tadjeddine, “Sum-frequency generation spectroscopy of interfaces,” Reports on Prog. Phys. 68, 1095–1127 (2005).
[Crossref]

2002 (3)

G. L. Richmond, “Molecular bonding and interactions at aqueous surfaces as probed by vibrational sum frequency spectroscopy,” Chem. Rev. 102, 2693–2724 (2002).
[Crossref] [PubMed]

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

H. Held, A. I. Lvovsky, X. Wei, and Y. R. Shen, “Bulk contribution from isotropic media in surface sum-frequency generation,” Phys. Rev. B 66, 205110 (2002).
[Crossref]

2001 (1)

M. Buck and M. Himmelhaus, “Vibrational spectroscopy of interfaces by infrared–visible sum frequency generation,” J. Vac. Sci. & Technol. A 19, 2717 (2001).
[Crossref]

2000 (1)

V. I. Gavrilenko, R. Q. Wu, M. C. Downer, J. G. Ekerdt, D. Lim, and P. Parkinson, “Optical second harmonic spectra of silicon-adatom surfaces: Theory and experiment,” Thin Solid Films 364, 1–5 (2000).
[Crossref]

1999 (1)

G. Kresse and D. Joubert, “From ultrasoft pseudopotentials to the projector augmented-wave method,” Phys. Rev. B 59, 1758–1775 (1999).
[Crossref]

1996 (4)

G. Kresse and J. Furthmüller, “Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set,” Comput. Mater. Sci. 6, 15–50 (1996).
[Crossref]

G. Kresse and J. Furthmüller, “Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set,” Phys. Rev. B 54, 11169–11186 (1996).
[Crossref]

J. F. McGilp, “A review of optical second-harmonic and sum-frequency generation at surfaces and interfaces,” J. Phys. D: Appl. Phys. 29, 1812–1821 (1996).
[Crossref]

K. B. Eisenthal, “Liquid interfaces probed by second-harmonic and sum-frequency spectroscopy,” Chem. Rev. 96, 1343–1360 (1996).
[Crossref] [PubMed]

1994 (1)

G. Kresse and J. Hafner, “Norm-conserving and ultrasoft pseudopotentials for first-row and transition elements,” J. Phys. Condens. Matter 6, 8245–8257 (1994).
[Crossref]

1993 (1)

G. Kresse and J. Hafner, “Ab initio molecular dynamics for liquid metals,” Phys. Rev. B 47, 558–561 (1993).
[Crossref]

1989 (1)

Y. R. Shen, “Surface properties probed by second-harmonic and sum-frequency generation,” Nat. 337, 519–525 (1989).
[Crossref]

1987 (2)

P. Guyot-Sionnest and Y. R. Shen, “Local and nonlocal surface nonlinearities for surface optical second-harmonic generation,” Phys. Rev. B 35, 4420–4426 (1987).
[Crossref]

X. D. Zhu, H. Suhr, and Y. R. Shen, “Surface vibrational spectroscopy by infrared-visible sum frequency generation,” Phys. Rev. B 35, 3047–3050 (1987).
[Crossref]

1986 (2)

H. W. K. Tom and G. D. Aumiller, “Observation of rotational anisotropy in the second-harmonic generation from a metal surface,” Phys. Rev. B 33, 8818–8821 (1986).
[Crossref]

H. Tom, X. Zhu, Y. Shen, and G. Somorjai, “Investigation of the Si(111)-(7×7) surface by second-harmonic generation: Oxidation and the effects of surface phosphorus,” Surf. Sci. Lett. 167, A76 (1986).
[Crossref]

1985 (2)

T. F. Heinz, M. M. T. Loy, and W. A. Thompson, “Study of symmetry and disordering of Si(111)-7×7 surfaces by optical second harmonic generation,” J. Vac. Sci. & Technol. B 3, 1467 (1985).
[Crossref]

Y. R. Shen, “Surface studies by optical second harmonic generation: An overview,” J. Vac. Sci. & Technol. B 3, 1464 (1985).
[Crossref]

1984 (1)

H. W. K. Tom, C. M. Mate, X. D. Zhu, J. E. Crowell, T. F. Heinz, G. A. Somorjai, and Y. R. Shen, “Surface studies by optical second-harmonic generation: The adsorption of O2, CO, and sodium on the Rh(111) surface,” Phys. Rev. Lett. 52, 348–351 (1984).
[Crossref]

1983 (1)

D. E. Aspnes and A. A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27, 985–1009 (1983).
[Crossref]

1968 (1)

N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, “Optical second-harmonic generation in reflection from media with inversion symmetry,” Phys. Rev. 174, 813–822 (1968).
[Crossref]

1965 (1)

F. Brown, R. E. Parks, and A. M. Sleeper, “Nonlinear optical reflection from a metallic boundary,” Phys. Rev. Lett. 14, 1029–1031 (1965).
[Crossref]

1964 (2)

R. C. Miller, “Optical second harmonic generation in piezoelectric crystals,” Appl. Phys. Lett. 5, 17–19 (1964).
[Crossref]

E. Adler, “Nonlinear optical frequency polarization in a dielectric,” Phys. Rev. 134, 728–733 (1964).
[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]

Adler, E.

E. Adler, “Nonlinear optical frequency polarization in a dielectric,” Phys. Rev. 134, 728–733 (1964).
[Crossref]

Adles, E. J.

E. J. Adles and D. E. Aspnes, “Application of the anisotropic bond model to second-harmonic generation from amorphous media,” Phys. Rev. B 8202, 10 (2008).

E. J. Adles and D. E. Aspnes, “The anisotropic bond model of nonlinear optics,” Phys. Status Solidi A 205, 728–731 (2008).
[Crossref]

Alejo-Molina, A.

C. Reitböck, D. Stifter, K. Hingerl, A. Alejo-Molina, and H. Hardhienata, “Bulk quadrupole and interface dipole contribution for second harmonic generation in Si(111),” J. Opt. 18, 1–7 (2016).
[Crossref]

A. Alejo-Molina, H. Hardhienata, and K. Hingerl, “Simplified bond-hyperpolarizability model of second harmonic generation, group theory, and Neumann’s principle,” J. Opt. Soc. Am. B 31, 526 (2014).
[Crossref]

Aspnes, D. E.

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

E. J. Adles and D. E. Aspnes, “The anisotropic bond model of nonlinear optics,” Phys. Status Solidi A 205, 728–731 (2008).
[Crossref]

E. J. Adles and D. E. Aspnes, “Application of the anisotropic bond model to second-harmonic generation from amorphous media,” Phys. Rev. B 8202, 10 (2008).

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

D. E. Aspnes and A. A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27, 985–1009 (1983).
[Crossref]

Aumiller, G. D.

H. W. K. Tom and G. D. Aumiller, “Observation of rotational anisotropy in the second-harmonic generation from a metal surface,” Phys. Rev. B 33, 8818–8821 (1986).
[Crossref]

Bauer, K.-D.

K.-D. Bauer, M. Panholzer, and K. Hingerl, “Bulk quadrupole contribution to second harmonic generation from a microscopic response function,” Phys. Status Solidi B 253, 234–240 (2016).
[Crossref]

Bechstedt, F.

M. Gajdoš, K. Hummer, G. Kresse, J. Furthmüller, and F. Bechstedt, “Linear optical properties in the projector-augmented wave methodology,” Phys. Rev. B 73, 045112 (2006).
[Crossref]

Bloembergen, N.

N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, “Optical second-harmonic generation in reflection from media with inversion symmetry,” Phys. Rev. 174, 813–822 (1968).
[Crossref]

Bonazzi, F.

A. Meurer, C. P. Smith, M. Paprocki, O. Čertík, S. B. Kirpichev, M. Rocklin, A. Kumar, S. Ivanov, J. K. Moore, S. Singh, T. Rathnayake, S. Vig, B. E. Granger, R. P. Muller, F. Bonazzi, H. Gupta, S. Vats, F. Johansson, F. Pedregosa, M. J. Curry, A. R. Terrel, v. Roučka, A. Saboo, I. Fernando, S. Kulal, R. Cimrman, and A. Scopatz, “Sympy: Symbolic computing in python,” PeerJ Comput. Sci. 3, e103 (2017).
[Crossref]

Boyd, R. W.

R. W. Boyd, Nonlinear optics (Elsevier Science, 2003).

Brown, F.

F. Brown, R. E. Parks, and A. M. Sleeper, “Nonlinear optical reflection from a metallic boundary,” Phys. Rev. Lett. 14, 1029–1031 (1965).
[Crossref]

Buck, M.

M. Buck and M. Himmelhaus, “Vibrational spectroscopy of interfaces by infrared–visible sum frequency generation,” J. Vac. Sci. & Technol. A 19, 2717 (2001).
[Crossref]

Cardona, M.

P. Yu and M. Cardona, Fundamentals of semiconductors: Physics and materials properties, Graduate Texts in Physics (SpringerBerlin Heidelberg, 2010).
[Crossref]

Certík, O.

A. Meurer, C. P. Smith, M. Paprocki, O. Čertík, S. B. Kirpichev, M. Rocklin, A. Kumar, S. Ivanov, J. K. Moore, S. Singh, T. Rathnayake, S. Vig, B. E. Granger, R. P. Muller, F. Bonazzi, H. Gupta, S. Vats, F. Johansson, F. Pedregosa, M. J. Curry, A. R. Terrel, v. Roučka, A. Saboo, I. Fernando, S. Kulal, R. Cimrman, and A. Scopatz, “Sympy: Symbolic computing in python,” PeerJ Comput. Sci. 3, e103 (2017).
[Crossref]

Chang, R. K.

N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, “Optical second-harmonic generation in reflection from media with inversion symmetry,” Phys. Rev. 174, 813–822 (1968).
[Crossref]

Cimrman, R.

A. Meurer, C. P. Smith, M. Paprocki, O. Čertík, S. B. Kirpichev, M. Rocklin, A. Kumar, S. Ivanov, J. K. Moore, S. Singh, T. Rathnayake, S. Vig, B. E. Granger, R. P. Muller, F. Bonazzi, H. Gupta, S. Vats, F. Johansson, F. Pedregosa, M. J. Curry, A. R. Terrel, v. Roučka, A. Saboo, I. Fernando, S. Kulal, R. Cimrman, and A. Scopatz, “Sympy: Symbolic computing in python,” PeerJ Comput. Sci. 3, e103 (2017).
[Crossref]

Crowell, J. E.

H. W. K. Tom, C. M. Mate, X. D. Zhu, J. E. Crowell, T. F. Heinz, G. A. Somorjai, and Y. R. Shen, “Surface studies by optical second-harmonic generation: The adsorption of O2, CO, and sodium on the Rh(111) surface,” Phys. Rev. Lett. 52, 348–351 (1984).
[Crossref]

Curry, M. J.

A. Meurer, C. P. Smith, M. Paprocki, O. Čertík, S. B. Kirpichev, M. Rocklin, A. Kumar, S. Ivanov, J. K. Moore, S. Singh, T. Rathnayake, S. Vig, B. E. Granger, R. P. Muller, F. Bonazzi, H. Gupta, S. Vats, F. Johansson, F. Pedregosa, M. J. Curry, A. R. Terrel, v. Roučka, A. Saboo, I. Fernando, S. Kulal, R. Cimrman, and A. Scopatz, “Sympy: Symbolic computing in python,” PeerJ Comput. Sci. 3, e103 (2017).
[Crossref]

Downer, M. C.

J. Kwon, M. C. Downer, and B. S. Mendoza, “Second-harmonic and reflectance-anisotropy spectroscopy of vicinal Si (001) Si O2 interfaces: Experiment and simplified microscopic model,” Phys. Rev. B 73, 1–12 (2006).
[Crossref]

V. I. Gavrilenko, R. Q. Wu, M. C. Downer, J. G. Ekerdt, D. Lim, and P. Parkinson, “Optical second harmonic spectra of silicon-adatom surfaces: Theory and experiment,” Thin Solid Films 364, 1–5 (2000).
[Crossref]

Eisenthal, K. B.

K. B. Eisenthal, “Liquid interfaces probed by second-harmonic and sum-frequency spectroscopy,” Chem. Rev. 96, 1343–1360 (1996).
[Crossref] [PubMed]

Ekerdt, J. G.

V. I. Gavrilenko, R. Q. Wu, M. C. Downer, J. G. Ekerdt, D. Lim, and P. Parkinson, “Optical second harmonic spectra of silicon-adatom surfaces: Theory and experiment,” Thin Solid Films 364, 1–5 (2000).
[Crossref]

Fernando, I.

A. Meurer, C. P. Smith, M. Paprocki, O. Čertík, S. B. Kirpichev, M. Rocklin, A. Kumar, S. Ivanov, J. K. Moore, S. Singh, T. Rathnayake, S. Vig, B. E. Granger, R. P. Muller, F. Bonazzi, H. Gupta, S. Vats, F. Johansson, F. Pedregosa, M. J. Curry, A. R. Terrel, v. Roučka, A. Saboo, I. Fernando, S. Kulal, R. Cimrman, and A. Scopatz, “Sympy: Symbolic computing in python,” PeerJ Comput. Sci. 3, e103 (2017).
[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]

Furthmüller, J.

M. Gajdoš, K. Hummer, G. Kresse, J. Furthmüller, and F. Bechstedt, “Linear optical properties in the projector-augmented wave methodology,” Phys. Rev. B 73, 045112 (2006).
[Crossref]

G. Kresse and J. Furthmüller, “Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set,” Comput. Mater. Sci. 6, 15–50 (1996).
[Crossref]

G. Kresse and J. Furthmüller, “Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set,” Phys. Rev. B 54, 11169–11186 (1996).
[Crossref]

Gajdoš, M.

M. Gajdoš, K. Hummer, G. Kresse, J. Furthmüller, and F. Bechstedt, “Linear optical properties in the projector-augmented wave methodology,” Phys. Rev. B 73, 045112 (2006).
[Crossref]

Gavrilenko, V. I.

V. I. Gavrilenko, R. Q. Wu, M. C. Downer, J. G. Ekerdt, D. Lim, and P. Parkinson, “Optical second harmonic spectra of silicon-adatom surfaces: Theory and experiment,” Thin Solid Films 364, 1–5 (2000).
[Crossref]

Granger, B. E.

A. Meurer, C. P. Smith, M. Paprocki, O. Čertík, S. B. Kirpichev, M. Rocklin, A. Kumar, S. Ivanov, J. K. Moore, S. Singh, T. Rathnayake, S. Vig, B. E. Granger, R. P. Muller, F. Bonazzi, H. Gupta, S. Vats, F. Johansson, F. Pedregosa, M. J. Curry, A. R. Terrel, v. Roučka, A. Saboo, I. Fernando, S. Kulal, R. Cimrman, and A. Scopatz, “Sympy: Symbolic computing in python,” PeerJ Comput. Sci. 3, e103 (2017).
[Crossref]

Gupta, H.

A. Meurer, C. P. Smith, M. Paprocki, O. Čertík, S. B. Kirpichev, M. Rocklin, A. Kumar, S. Ivanov, J. K. Moore, S. Singh, T. Rathnayake, S. Vig, B. E. Granger, R. P. Muller, F. Bonazzi, H. Gupta, S. Vats, F. Johansson, F. Pedregosa, M. J. Curry, A. R. Terrel, v. Roučka, A. Saboo, I. Fernando, S. Kulal, R. Cimrman, and A. Scopatz, “Sympy: Symbolic computing in python,” PeerJ Comput. Sci. 3, e103 (2017).
[Crossref]

Guyot-Sionnest, P.

P. Guyot-Sionnest and Y. R. Shen, “Local and nonlocal surface nonlinearities for surface optical second-harmonic generation,” Phys. Rev. B 35, 4420–4426 (1987).
[Crossref]

Hafner, J.

G. Kresse and J. Hafner, “Norm-conserving and ultrasoft pseudopotentials for first-row and transition elements,” J. Phys. Condens. Matter 6, 8245–8257 (1994).
[Crossref]

G. Kresse and J. Hafner, “Ab initio molecular dynamics for liquid metals,” Phys. Rev. B 47, 558–561 (1993).
[Crossref]

Hardhienata, H.

C. Reitböck, D. Stifter, K. Hingerl, A. Alejo-Molina, and H. Hardhienata, “Bulk quadrupole and interface dipole contribution for second harmonic generation in Si(111),” J. Opt. 18, 1–7 (2016).
[Crossref]

A. Alejo-Molina, H. Hardhienata, and K. Hingerl, “Simplified bond-hyperpolarizability model of second harmonic generation, group theory, and Neumann’s principle,” J. Opt. Soc. Am. B 31, 526 (2014).
[Crossref]

Heinz, T. F.

T. F. Heinz, M. M. T. Loy, and W. A. Thompson, “Study of symmetry and disordering of Si(111)-7×7 surfaces by optical second harmonic generation,” J. Vac. Sci. & Technol. B 3, 1467 (1985).
[Crossref]

H. W. K. Tom, C. M. Mate, X. D. Zhu, J. E. Crowell, T. F. Heinz, G. A. Somorjai, and Y. R. Shen, “Surface studies by optical second-harmonic generation: The adsorption of O2, CO, and sodium on the Rh(111) surface,” Phys. Rev. Lett. 52, 348–351 (1984).
[Crossref]

Held, H.

H. Held, A. I. Lvovsky, X. Wei, and Y. R. Shen, “Bulk contribution from isotropic media in surface sum-frequency generation,” Phys. Rev. B 66, 205110 (2002).
[Crossref]

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]

Himmelhaus, M.

M. Buck and M. Himmelhaus, “Vibrational spectroscopy of interfaces by infrared–visible sum frequency generation,” J. Vac. Sci. & Technol. A 19, 2717 (2001).
[Crossref]

Hingerl, K.

C. Reitböck, D. Stifter, K. Hingerl, A. Alejo-Molina, and H. Hardhienata, “Bulk quadrupole and interface dipole contribution for second harmonic generation in Si(111),” J. Opt. 18, 1–7 (2016).
[Crossref]

K.-D. Bauer, M. Panholzer, and K. Hingerl, “Bulk quadrupole contribution to second harmonic generation from a microscopic response function,” Phys. Status Solidi B 253, 234–240 (2016).
[Crossref]

A. Alejo-Molina, H. Hardhienata, and K. Hingerl, “Simplified bond-hyperpolarizability model of second harmonic generation, group theory, and Neumann’s principle,” J. Opt. Soc. Am. B 31, 526 (2014).
[Crossref]

Hübener, H.

E. Luppi, H. Hübener, and V. Véniard, “Ab initio second-order nonlinear optics in solids: Second-harmonic generation spectroscopy from time-dependent density-functional theory,” Phys. Rev. B 82, 1–15 (2010).
[Crossref]

Hübner, W.

G. Lefkidis and W. Hübner, “Ab initio treatment of optical second harmonic generation in NiO,” Phys. Rev. Lett. 95, 2–5 (2005).
[Crossref]

Hummer, K.

M. Gajdoš, K. Hummer, G. Kresse, J. Furthmüller, and F. Bechstedt, “Linear optical properties in the projector-augmented wave methodology,” Phys. Rev. B 73, 045112 (2006).
[Crossref]

Ivanov, S.

A. Meurer, C. P. Smith, M. Paprocki, O. Čertík, S. B. Kirpichev, M. Rocklin, A. Kumar, S. Ivanov, J. K. Moore, S. Singh, T. Rathnayake, S. Vig, B. E. Granger, R. P. Muller, F. Bonazzi, H. Gupta, S. Vats, F. Johansson, F. Pedregosa, M. J. Curry, A. R. Terrel, v. Roučka, A. Saboo, I. Fernando, S. Kulal, R. Cimrman, and A. Scopatz, “Sympy: Symbolic computing in python,” PeerJ Comput. Sci. 3, e103 (2017).
[Crossref]

Jha, S. S.

N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, “Optical second-harmonic generation in reflection from media with inversion symmetry,” Phys. Rev. 174, 813–822 (1968).
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Johansson, F.

A. Meurer, C. P. Smith, M. Paprocki, O. Čertík, S. B. Kirpichev, M. Rocklin, A. Kumar, S. Ivanov, J. K. Moore, S. Singh, T. Rathnayake, S. Vig, B. E. Granger, R. P. Muller, F. Bonazzi, H. Gupta, S. Vats, F. Johansson, F. Pedregosa, M. J. Curry, A. R. Terrel, v. Roučka, A. Saboo, I. Fernando, S. Kulal, R. Cimrman, and A. Scopatz, “Sympy: Symbolic computing in python,” PeerJ Comput. Sci. 3, e103 (2017).
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Joubert, D.

G. Kresse and D. Joubert, “From ultrasoft pseudopotentials to the projector augmented-wave method,” Phys. Rev. B 59, 1758–1775 (1999).
[Crossref]

Kirpichev, S. B.

A. Meurer, C. P. Smith, M. Paprocki, O. Čertík, S. B. Kirpichev, M. Rocklin, A. Kumar, S. Ivanov, J. K. Moore, S. Singh, T. Rathnayake, S. Vig, B. E. Granger, R. P. Muller, F. Bonazzi, H. Gupta, S. Vats, F. Johansson, F. Pedregosa, M. J. Curry, A. R. Terrel, v. Roučka, A. Saboo, I. Fernando, S. Kulal, R. Cimrman, and A. Scopatz, “Sympy: Symbolic computing in python,” PeerJ Comput. Sci. 3, e103 (2017).
[Crossref]

Kresse, G.

M. Gajdoš, K. Hummer, G. Kresse, J. Furthmüller, and F. Bechstedt, “Linear optical properties in the projector-augmented wave methodology,” Phys. Rev. B 73, 045112 (2006).
[Crossref]

G. Kresse and D. Joubert, “From ultrasoft pseudopotentials to the projector augmented-wave method,” Phys. Rev. B 59, 1758–1775 (1999).
[Crossref]

G. Kresse and J. Furthmüller, “Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set,” Comput. Mater. Sci. 6, 15–50 (1996).
[Crossref]

G. Kresse and J. Furthmüller, “Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set,” Phys. Rev. B 54, 11169–11186 (1996).
[Crossref]

G. Kresse and J. Hafner, “Norm-conserving and ultrasoft pseudopotentials for first-row and transition elements,” J. Phys. Condens. Matter 6, 8245–8257 (1994).
[Crossref]

G. Kresse and J. Hafner, “Ab initio molecular dynamics for liquid metals,” Phys. Rev. B 47, 558–561 (1993).
[Crossref]

Kulal, S.

A. Meurer, C. P. Smith, M. Paprocki, O. Čertík, S. B. Kirpichev, M. Rocklin, A. Kumar, S. Ivanov, J. K. Moore, S. Singh, T. Rathnayake, S. Vig, B. E. Granger, R. P. Muller, F. Bonazzi, H. Gupta, S. Vats, F. Johansson, F. Pedregosa, M. J. Curry, A. R. Terrel, v. Roučka, A. Saboo, I. Fernando, S. Kulal, R. Cimrman, and A. Scopatz, “Sympy: Symbolic computing in python,” PeerJ Comput. Sci. 3, e103 (2017).
[Crossref]

Kumar, A.

A. Meurer, C. P. Smith, M. Paprocki, O. Čertík, S. B. Kirpichev, M. Rocklin, A. Kumar, S. Ivanov, J. K. Moore, S. Singh, T. Rathnayake, S. Vig, B. E. Granger, R. P. Muller, F. Bonazzi, H. Gupta, S. Vats, F. Johansson, F. Pedregosa, M. J. Curry, A. R. Terrel, v. Roučka, A. Saboo, I. Fernando, S. Kulal, R. Cimrman, and A. Scopatz, “Sympy: Symbolic computing in python,” PeerJ Comput. Sci. 3, e103 (2017).
[Crossref]

Kwon, J.

J. Kwon, M. C. Downer, and B. S. Mendoza, “Second-harmonic and reflectance-anisotropy spectroscopy of vicinal Si (001) Si O2 interfaces: Experiment and simplified microscopic model,” Phys. Rev. B 73, 1–12 (2006).
[Crossref]

Lee, C. H.

N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, “Optical second-harmonic generation in reflection from media with inversion symmetry,” Phys. Rev. 174, 813–822 (1968).
[Crossref]

Lefkidis, G.

G. Lefkidis and W. Hübner, “Ab initio treatment of optical second harmonic generation in NiO,” Phys. Rev. Lett. 95, 2–5 (2005).
[Crossref]

Lim, D.

V. I. Gavrilenko, R. Q. Wu, M. C. Downer, J. G. Ekerdt, D. Lim, and P. Parkinson, “Optical second harmonic spectra of silicon-adatom surfaces: Theory and experiment,” Thin Solid Films 364, 1–5 (2000).
[Crossref]

Loy, M. M. T.

T. F. Heinz, M. M. T. Loy, and W. A. Thompson, “Study of symmetry and disordering of Si(111)-7×7 surfaces by optical second harmonic generation,” J. Vac. Sci. & Technol. B 3, 1467 (1985).
[Crossref]

Luppi, E.

E. Luppi, H. Hübener, and V. Véniard, “Ab initio second-order nonlinear optics in solids: Second-harmonic generation spectroscopy from time-dependent density-functional theory,” Phys. Rev. B 82, 1–15 (2010).
[Crossref]

Lvovsky, A. I.

H. Held, A. I. Lvovsky, X. Wei, and Y. R. Shen, “Bulk contribution from isotropic media in surface sum-frequency generation,” Phys. Rev. B 66, 205110 (2002).
[Crossref]

Mate, C. M.

H. W. K. Tom, C. M. Mate, X. D. Zhu, J. E. Crowell, T. F. Heinz, G. A. Somorjai, and Y. R. Shen, “Surface studies by optical second-harmonic generation: The adsorption of O2, CO, and sodium on the Rh(111) surface,” Phys. Rev. Lett. 52, 348–351 (1984).
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McGilp, J. F.

J. F. McGilp, “A review of optical second-harmonic and sum-frequency generation at surfaces and interfaces,” J. Phys. D: Appl. Phys. 29, 1812–1821 (1996).
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Mendoza, B. S.

J. Kwon, M. C. Downer, and B. S. Mendoza, “Second-harmonic and reflectance-anisotropy spectroscopy of vicinal Si (001) Si O2 interfaces: Experiment and simplified microscopic model,” Phys. Rev. B 73, 1–12 (2006).
[Crossref]

Meurer, A.

A. Meurer, C. P. Smith, M. Paprocki, O. Čertík, S. B. Kirpichev, M. Rocklin, A. Kumar, S. Ivanov, J. K. Moore, S. Singh, T. Rathnayake, S. Vig, B. E. Granger, R. P. Muller, F. Bonazzi, H. Gupta, S. Vats, F. Johansson, F. Pedregosa, M. J. Curry, A. R. Terrel, v. Roučka, A. Saboo, I. Fernando, S. Kulal, R. Cimrman, and A. Scopatz, “Sympy: Symbolic computing in python,” PeerJ Comput. Sci. 3, e103 (2017).
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R. C. Miller, “Optical second harmonic generation in piezoelectric crystals,” Appl. Phys. Lett. 5, 17–19 (1964).
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Moore, J. K.

A. Meurer, C. P. Smith, M. Paprocki, O. Čertík, S. B. Kirpichev, M. Rocklin, A. Kumar, S. Ivanov, J. K. Moore, S. Singh, T. Rathnayake, S. Vig, B. E. Granger, R. P. Muller, F. Bonazzi, H. Gupta, S. Vats, F. Johansson, F. Pedregosa, M. J. Curry, A. R. Terrel, v. Roučka, A. Saboo, I. Fernando, S. Kulal, R. Cimrman, and A. Scopatz, “Sympy: Symbolic computing in python,” PeerJ Comput. Sci. 3, e103 (2017).
[Crossref]

Muller, R. P.

A. Meurer, C. P. Smith, M. Paprocki, O. Čertík, S. B. Kirpichev, M. Rocklin, A. Kumar, S. Ivanov, J. K. Moore, S. Singh, T. Rathnayake, S. Vig, B. E. Granger, R. P. Muller, F. Bonazzi, H. Gupta, S. Vats, F. Johansson, F. Pedregosa, M. J. Curry, A. R. Terrel, v. Roučka, A. Saboo, I. Fernando, S. Kulal, R. Cimrman, and A. Scopatz, “Sympy: Symbolic computing in python,” PeerJ Comput. Sci. 3, e103 (2017).
[Crossref]

Panholzer, M.

K.-D. Bauer, M. Panholzer, and K. Hingerl, “Bulk quadrupole contribution to second harmonic generation from a microscopic response function,” Phys. Status Solidi B 253, 234–240 (2016).
[Crossref]

Paprocki, M.

A. Meurer, C. P. Smith, M. Paprocki, O. Čertík, S. B. Kirpichev, M. Rocklin, A. Kumar, S. Ivanov, J. K. Moore, S. Singh, T. Rathnayake, S. Vig, B. E. Granger, R. P. Muller, F. Bonazzi, H. Gupta, S. Vats, F. Johansson, F. Pedregosa, M. J. Curry, A. R. Terrel, v. Roučka, A. Saboo, I. Fernando, S. Kulal, R. Cimrman, and A. Scopatz, “Sympy: Symbolic computing in python,” PeerJ Comput. Sci. 3, e103 (2017).
[Crossref]

Parkinson, P.

V. I. Gavrilenko, R. Q. Wu, M. C. Downer, J. G. Ekerdt, D. Lim, and P. Parkinson, “Optical second harmonic spectra of silicon-adatom surfaces: Theory and experiment,” Thin Solid Films 364, 1–5 (2000).
[Crossref]

Parks, R. E.

F. Brown, R. E. Parks, and A. M. Sleeper, “Nonlinear optical reflection from a metallic boundary,” Phys. Rev. Lett. 14, 1029–1031 (1965).
[Crossref]

Pedregosa, F.

A. Meurer, C. P. Smith, M. Paprocki, O. Čertík, S. B. Kirpichev, M. Rocklin, A. Kumar, S. Ivanov, J. K. Moore, S. Singh, T. Rathnayake, S. Vig, B. E. Granger, R. P. Muller, F. Bonazzi, H. Gupta, S. Vats, F. Johansson, F. Pedregosa, M. J. Curry, A. R. Terrel, v. Roučka, A. Saboo, I. Fernando, S. Kulal, R. Cimrman, and A. Scopatz, “Sympy: Symbolic computing in python,” PeerJ Comput. Sci. 3, e103 (2017).
[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. T. Wang, and D. E. Aspnes, “Simplified bond-hyperpolarizability model of second harmonic generation,” Phys. Rev. B 65, 205320 (2002).
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Rathnayake, T.

A. Meurer, C. P. Smith, M. Paprocki, O. Čertík, S. B. Kirpichev, M. Rocklin, A. Kumar, S. Ivanov, J. K. Moore, S. Singh, T. Rathnayake, S. Vig, B. E. Granger, R. P. Muller, F. Bonazzi, H. Gupta, S. Vats, F. Johansson, F. Pedregosa, M. J. Curry, A. R. Terrel, v. Roučka, A. Saboo, I. Fernando, S. Kulal, R. Cimrman, and A. Scopatz, “Sympy: Symbolic computing in python,” PeerJ Comput. Sci. 3, e103 (2017).
[Crossref]

Reitböck, C.

C. Reitböck, D. Stifter, K. Hingerl, A. Alejo-Molina, and H. Hardhienata, “Bulk quadrupole and interface dipole contribution for second harmonic generation in Si(111),” J. Opt. 18, 1–7 (2016).
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Richmond, G. L.

G. L. Richmond, “Molecular bonding and interactions at aqueous surfaces as probed by vibrational sum frequency spectroscopy,” Chem. Rev. 102, 2693–2724 (2002).
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Rocklin, M.

A. Meurer, C. P. Smith, M. Paprocki, O. Čertík, S. B. Kirpichev, M. Rocklin, A. Kumar, S. Ivanov, J. K. Moore, S. Singh, T. Rathnayake, S. Vig, B. E. Granger, R. P. Muller, F. Bonazzi, H. Gupta, S. Vats, F. Johansson, F. Pedregosa, M. J. Curry, A. R. Terrel, v. Roučka, A. Saboo, I. Fernando, S. Kulal, R. Cimrman, and A. Scopatz, “Sympy: Symbolic computing in python,” PeerJ Comput. Sci. 3, e103 (2017).
[Crossref]

Roucka, v.

A. Meurer, C. P. Smith, M. Paprocki, O. Čertík, S. B. Kirpichev, M. Rocklin, A. Kumar, S. Ivanov, J. K. Moore, S. Singh, T. Rathnayake, S. Vig, B. E. Granger, R. P. Muller, F. Bonazzi, H. Gupta, S. Vats, F. Johansson, F. Pedregosa, M. J. Curry, A. R. Terrel, v. Roučka, A. Saboo, I. Fernando, S. Kulal, R. Cimrman, and A. Scopatz, “Sympy: Symbolic computing in python,” PeerJ Comput. Sci. 3, e103 (2017).
[Crossref]

Saboo, A.

A. Meurer, C. P. Smith, M. Paprocki, O. Čertík, S. B. Kirpichev, M. Rocklin, A. Kumar, S. Ivanov, J. K. Moore, S. Singh, T. Rathnayake, S. Vig, B. E. Granger, R. P. Muller, F. Bonazzi, H. Gupta, S. Vats, F. Johansson, F. Pedregosa, M. J. Curry, A. R. Terrel, v. Roučka, A. Saboo, I. Fernando, S. Kulal, R. Cimrman, and A. Scopatz, “Sympy: Symbolic computing in python,” PeerJ Comput. Sci. 3, e103 (2017).
[Crossref]

Scopatz, A.

A. Meurer, C. P. Smith, M. Paprocki, O. Čertík, S. B. Kirpichev, M. Rocklin, A. Kumar, S. Ivanov, J. K. Moore, S. Singh, T. Rathnayake, S. Vig, B. E. Granger, R. P. Muller, F. Bonazzi, H. Gupta, S. Vats, F. Johansson, F. Pedregosa, M. J. Curry, A. R. Terrel, v. Roučka, A. Saboo, I. Fernando, S. Kulal, R. Cimrman, and A. Scopatz, “Sympy: Symbolic computing in python,” PeerJ Comput. Sci. 3, e103 (2017).
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Shen, Y.

H. Tom, X. Zhu, Y. Shen, and G. Somorjai, “Investigation of the Si(111)-(7×7) surface by second-harmonic generation: Oxidation and the effects of surface phosphorus,” Surf. Sci. Lett. 167, A76 (1986).
[Crossref]

Shen, Y. R.

Y. R. Shen, “Surface nonlinear optics [Invited],” J. Opt. Soc. Am. B 28, A56 (2011).
[Crossref]

H. Held, A. I. Lvovsky, X. Wei, and Y. R. Shen, “Bulk contribution from isotropic media in surface sum-frequency generation,” Phys. Rev. B 66, 205110 (2002).
[Crossref]

Y. R. Shen, “Surface properties probed by second-harmonic and sum-frequency generation,” Nat. 337, 519–525 (1989).
[Crossref]

X. D. Zhu, H. Suhr, and Y. R. Shen, “Surface vibrational spectroscopy by infrared-visible sum frequency generation,” Phys. Rev. B 35, 3047–3050 (1987).
[Crossref]

P. Guyot-Sionnest and Y. R. Shen, “Local and nonlocal surface nonlinearities for surface optical second-harmonic generation,” Phys. Rev. B 35, 4420–4426 (1987).
[Crossref]

Y. R. Shen, “Surface studies by optical second harmonic generation: An overview,” J. Vac. Sci. & Technol. B 3, 1464 (1985).
[Crossref]

H. W. K. Tom, C. M. Mate, X. D. Zhu, J. E. Crowell, T. F. Heinz, G. A. Somorjai, and Y. R. Shen, “Surface studies by optical second-harmonic generation: The adsorption of O2, CO, and sodium on the Rh(111) surface,” Phys. Rev. Lett. 52, 348–351 (1984).
[Crossref]

Singh, S.

A. Meurer, C. P. Smith, M. Paprocki, O. Čertík, S. B. Kirpichev, M. Rocklin, A. Kumar, S. Ivanov, J. K. Moore, S. Singh, T. Rathnayake, S. Vig, B. E. Granger, R. P. Muller, F. Bonazzi, H. Gupta, S. Vats, F. Johansson, F. Pedregosa, M. J. Curry, A. R. Terrel, v. Roučka, A. Saboo, I. Fernando, S. Kulal, R. Cimrman, and A. Scopatz, “Sympy: Symbolic computing in python,” PeerJ Comput. Sci. 3, e103 (2017).
[Crossref]

Sleeper, A. M.

F. Brown, R. E. Parks, and A. M. Sleeper, “Nonlinear optical reflection from a metallic boundary,” Phys. Rev. Lett. 14, 1029–1031 (1965).
[Crossref]

Smith, C. P.

A. Meurer, C. P. Smith, M. Paprocki, O. Čertík, S. B. Kirpichev, M. Rocklin, A. Kumar, S. Ivanov, J. K. Moore, S. Singh, T. Rathnayake, S. Vig, B. E. Granger, R. P. Muller, F. Bonazzi, H. Gupta, S. Vats, F. Johansson, F. Pedregosa, M. J. Curry, A. R. Terrel, v. Roučka, A. Saboo, I. Fernando, S. Kulal, R. Cimrman, and A. Scopatz, “Sympy: Symbolic computing in python,” PeerJ Comput. Sci. 3, e103 (2017).
[Crossref]

Somorjai, G.

H. Tom, X. Zhu, Y. Shen, and G. Somorjai, “Investigation of the Si(111)-(7×7) surface by second-harmonic generation: Oxidation and the effects of surface phosphorus,” Surf. Sci. Lett. 167, A76 (1986).
[Crossref]

Somorjai, G. A.

H. W. K. Tom, C. M. Mate, X. D. Zhu, J. E. Crowell, T. F. Heinz, G. A. Somorjai, and Y. R. Shen, “Surface studies by optical second-harmonic generation: The adsorption of O2, CO, and sodium on the Rh(111) surface,” Phys. Rev. Lett. 52, 348–351 (1984).
[Crossref]

Stifter, D.

C. Reitböck, D. Stifter, K. Hingerl, A. Alejo-Molina, and H. Hardhienata, “Bulk quadrupole and interface dipole contribution for second harmonic generation in Si(111),” J. Opt. 18, 1–7 (2016).
[Crossref]

Studna, A. A.

D. E. Aspnes and A. A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27, 985–1009 (1983).
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Suhr, H.

X. D. Zhu, H. Suhr, and Y. R. Shen, “Surface vibrational spectroscopy by infrared-visible sum frequency generation,” Phys. Rev. B 35, 3047–3050 (1987).
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Tadjeddine, A.

F. Vidal and A. Tadjeddine, “Sum-frequency generation spectroscopy of interfaces,” Reports on Prog. Phys. 68, 1095–1127 (2005).
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Terrel, A. R.

A. Meurer, C. P. Smith, M. Paprocki, O. Čertík, S. B. Kirpichev, M. Rocklin, A. Kumar, S. Ivanov, J. K. Moore, S. Singh, T. Rathnayake, S. Vig, B. E. Granger, R. P. Muller, F. Bonazzi, H. Gupta, S. Vats, F. Johansson, F. Pedregosa, M. J. Curry, A. R. Terrel, v. Roučka, A. Saboo, I. Fernando, S. Kulal, R. Cimrman, and A. Scopatz, “Sympy: Symbolic computing in python,” PeerJ Comput. Sci. 3, e103 (2017).
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Thompson, W. A.

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[Crossref]

Tom, H.

H. Tom, X. Zhu, Y. Shen, and G. Somorjai, “Investigation of the Si(111)-(7×7) surface by second-harmonic generation: Oxidation and the effects of surface phosphorus,” Surf. Sci. Lett. 167, A76 (1986).
[Crossref]

Tom, H. W. K.

H. W. K. Tom and G. D. Aumiller, “Observation of rotational anisotropy in the second-harmonic generation from a metal surface,” Phys. Rev. B 33, 8818–8821 (1986).
[Crossref]

H. W. K. Tom, C. M. Mate, X. D. Zhu, J. E. Crowell, T. F. Heinz, G. A. Somorjai, and Y. R. Shen, “Surface studies by optical second-harmonic generation: The adsorption of O2, CO, and sodium on the Rh(111) surface,” Phys. Rev. Lett. 52, 348–351 (1984).
[Crossref]

Vats, S.

A. Meurer, C. P. Smith, M. Paprocki, O. Čertík, S. B. Kirpichev, M. Rocklin, A. Kumar, S. Ivanov, J. K. Moore, S. Singh, T. Rathnayake, S. Vig, B. E. Granger, R. P. Muller, F. Bonazzi, H. Gupta, S. Vats, F. Johansson, F. Pedregosa, M. J. Curry, A. R. Terrel, v. Roučka, A. Saboo, I. Fernando, S. Kulal, R. Cimrman, and A. Scopatz, “Sympy: Symbolic computing in python,” PeerJ Comput. Sci. 3, e103 (2017).
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Véniard, V.

E. Luppi, H. Hübener, and V. Véniard, “Ab initio second-order nonlinear optics in solids: Second-harmonic generation spectroscopy from time-dependent density-functional theory,” Phys. Rev. B 82, 1–15 (2010).
[Crossref]

Vidal, F.

F. Vidal and A. Tadjeddine, “Sum-frequency generation spectroscopy of interfaces,” Reports on Prog. Phys. 68, 1095–1127 (2005).
[Crossref]

Vig, S.

A. Meurer, C. P. Smith, M. Paprocki, O. Čertík, S. B. Kirpichev, M. Rocklin, A. Kumar, S. Ivanov, J. K. Moore, S. Singh, T. Rathnayake, S. Vig, B. E. Granger, R. P. Muller, F. Bonazzi, H. Gupta, S. Vats, F. Johansson, F. Pedregosa, M. J. Curry, A. R. Terrel, v. Roučka, A. Saboo, I. Fernando, S. Kulal, R. Cimrman, and A. Scopatz, “Sympy: Symbolic computing in python,” PeerJ Comput. Sci. 3, e103 (2017).
[Crossref]

Wang, J.-F. T.

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

Wei, X.

H. Held, A. I. Lvovsky, X. Wei, and Y. R. Shen, “Bulk contribution from isotropic media in surface sum-frequency generation,” Phys. Rev. B 66, 205110 (2002).
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P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[Crossref]

Wu, R. Q.

V. I. Gavrilenko, R. Q. Wu, M. C. Downer, J. G. Ekerdt, D. Lim, and P. Parkinson, “Optical second harmonic spectra of silicon-adatom surfaces: Theory and experiment,” Thin Solid Films 364, 1–5 (2000).
[Crossref]

Yu, P.

P. Yu and M. Cardona, Fundamentals of semiconductors: Physics and materials properties, Graduate Texts in Physics (SpringerBerlin Heidelberg, 2010).
[Crossref]

Zhu, X.

H. Tom, X. Zhu, Y. Shen, and G. Somorjai, “Investigation of the Si(111)-(7×7) surface by second-harmonic generation: Oxidation and the effects of surface phosphorus,” Surf. Sci. Lett. 167, A76 (1986).
[Crossref]

Zhu, X. D.

X. D. Zhu, H. Suhr, and Y. R. Shen, “Surface vibrational spectroscopy by infrared-visible sum frequency generation,” Phys. Rev. B 35, 3047–3050 (1987).
[Crossref]

H. W. K. Tom, C. M. Mate, X. D. Zhu, J. E. Crowell, T. F. Heinz, G. A. Somorjai, and Y. R. Shen, “Surface studies by optical second-harmonic generation: The adsorption of O2, CO, and sodium on the Rh(111) surface,” Phys. Rev. Lett. 52, 348–351 (1984).
[Crossref]

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Chem. Rev. (2)

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[Crossref]

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C. Reitböck, D. Stifter, K. Hingerl, A. Alejo-Molina, and H. Hardhienata, “Bulk quadrupole and interface dipole contribution for second harmonic generation in Si(111),” J. Opt. 18, 1–7 (2016).
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M. Buck and M. Himmelhaus, “Vibrational spectroscopy of interfaces by infrared–visible sum frequency generation,” J. Vac. Sci. & Technol. A 19, 2717 (2001).
[Crossref]

J. Vac. Sci. & Technol. B (2)

T. F. Heinz, M. M. T. Loy, and W. A. Thompson, “Study of symmetry and disordering of Si(111)-7×7 surfaces by optical second harmonic generation,” J. Vac. Sci. & Technol. B 3, 1467 (1985).
[Crossref]

Y. R. Shen, “Surface studies by optical second harmonic generation: An overview,” J. Vac. Sci. & Technol. B 3, 1464 (1985).
[Crossref]

Nat. (1)

Y. R. Shen, “Surface properties probed by second-harmonic and sum-frequency generation,” Nat. 337, 519–525 (1989).
[Crossref]

PeerJ Comput. Sci. (1)

A. Meurer, C. P. Smith, M. Paprocki, O. Čertík, S. B. Kirpichev, M. Rocklin, A. Kumar, S. Ivanov, J. K. Moore, S. Singh, T. Rathnayake, S. Vig, B. E. Granger, R. P. Muller, F. Bonazzi, H. Gupta, S. Vats, F. Johansson, F. Pedregosa, M. J. Curry, A. R. Terrel, v. Roučka, A. Saboo, I. Fernando, S. Kulal, R. Cimrman, and A. Scopatz, “Sympy: Symbolic computing in python,” PeerJ Comput. Sci. 3, e103 (2017).
[Crossref]

Phys. Rev. (2)

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E. J. Adles and D. E. Aspnes, “Application of the anisotropic bond model to second-harmonic generation from amorphous media,” Phys. Rev. B 8202, 10 (2008).

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G. Kresse and D. Joubert, “From ultrasoft pseudopotentials to the projector augmented-wave method,” Phys. Rev. B 59, 1758–1775 (1999).
[Crossref]

E. Luppi, H. Hübener, and V. Véniard, “Ab initio second-order nonlinear optics in solids: Second-harmonic generation spectroscopy from time-dependent density-functional theory,” Phys. Rev. B 82, 1–15 (2010).
[Crossref]

D. E. Aspnes and A. A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27, 985–1009 (1983).
[Crossref]

X. D. Zhu, H. Suhr, and Y. R. Shen, “Surface vibrational spectroscopy by infrared-visible sum frequency generation,” Phys. Rev. B 35, 3047–3050 (1987).
[Crossref]

H. W. K. Tom and G. D. Aumiller, “Observation of rotational anisotropy in the second-harmonic generation from a metal surface,” Phys. Rev. B 33, 8818–8821 (1986).
[Crossref]

P. Guyot-Sionnest and Y. R. Shen, “Local and nonlocal surface nonlinearities for surface optical second-harmonic generation,” Phys. Rev. B 35, 4420–4426 (1987).
[Crossref]

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

H. Held, A. I. Lvovsky, X. Wei, and Y. R. Shen, “Bulk contribution from isotropic media in surface sum-frequency generation,” Phys. Rev. B 66, 205110 (2002).
[Crossref]

M. Gajdoš, K. Hummer, G. Kresse, J. Furthmüller, and F. Bechstedt, “Linear optical properties in the projector-augmented wave methodology,” Phys. Rev. B 73, 045112 (2006).
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[Crossref]

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

G. Lefkidis and W. Hübner, “Ab initio treatment of optical second harmonic generation in NiO,” Phys. Rev. Lett. 95, 2–5 (2005).
[Crossref]

H. W. K. Tom, C. M. Mate, X. D. Zhu, J. E. Crowell, T. F. Heinz, G. A. Somorjai, and Y. R. Shen, “Surface studies by optical second-harmonic generation: The adsorption of O2, CO, and sodium on the Rh(111) surface,” Phys. Rev. Lett. 52, 348–351 (1984).
[Crossref]

Phys. Status Solidi A (1)

E. J. Adles and D. E. Aspnes, “The anisotropic bond model of nonlinear optics,” Phys. Status Solidi A 205, 728–731 (2008).
[Crossref]

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D. E. Aspnes, “Bond models in linear and nonlinear optics,” Phys. Status Solidi B 247, 1873–1880 (2010).
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K.-D. Bauer, M. Panholzer, and K. Hingerl, “Bulk quadrupole contribution to second harmonic generation from a microscopic response function,” Phys. Status Solidi B 253, 234–240 (2016).
[Crossref]

Reports on Prog. Phys. (1)

F. Vidal and A. Tadjeddine, “Sum-frequency generation spectroscopy of interfaces,” Reports on Prog. Phys. 68, 1095–1127 (2005).
[Crossref]

Surf. Sci. Lett. (1)

H. Tom, X. Zhu, Y. Shen, and G. Somorjai, “Investigation of the Si(111)-(7×7) surface by second-harmonic generation: Oxidation and the effects of surface phosphorus,” Surf. Sci. Lett. 167, A76 (1986).
[Crossref]

Thin Solid Films (1)

V. I. Gavrilenko, R. Q. Wu, M. C. Downer, J. G. Ekerdt, D. Lim, and P. Parkinson, “Optical second harmonic spectra of silicon-adatom surfaces: Theory and experiment,” Thin Solid Films 364, 1–5 (2000).
[Crossref]

Other (2)

P. Yu and M. Cardona, Fundamentals of semiconductors: Physics and materials properties, Graduate Texts in Physics (SpringerBerlin Heidelberg, 2010).
[Crossref]

R. W. Boyd, Nonlinear optics (Elsevier Science, 2003).

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Figures (4)

Fig. 1
Fig. 1 The bond directions of a zincblende structure relative to one of the atom types. For our calculations we chose the representation b1 ∝ (1, 1, 1), b2 ∝ (−1, −1, 1), b3 ∝ (1, −1, 1), b4 ∝ (−1, 1, −1).
Fig. 2
Fig. 2 Non-linear spectra obtained with anharmonic oscillator model in SBHM from ab-initio linear response obtained with VASP, for SiC, GaAs, and AlAs. (top) Comparison of the obtained dipole response function to the shape of the absolute value |ε(2ω)| of the linear dielectric function and to the IPA ab-initio spectrum obtained by Luppi et al. [38], denoted “Ref.” in the plot, using fixed and frequency-dependent anharmonicity β as described in the text. (bottom) Predicted quadrupole response given as χ(2QD) = ik χ(2Q) as described in the text, and comparison to the shape of the dielectric function.
Fig. 3
Fig. 3 Quadrupole response prediction from SBHM using a simulated ab-initio dielectric function and the experimental dielectric function obtained by Aspnes et al. [39]. The corresponding plots are denoted as Si(sim) and Si(exp) respectively. (a) Simulated and experimental dielectric functions used in the calculation. (b) Predicted quadrupole response given as χ(2QD) = ik χ(2Q) as described in the text. The quadrupole response is compared to the dielectric function εr(ω) and εr(2ω) whose peeks are aligned with the peeks of the quadrupole response as expected. (c) For completeness the real and imaginary parts of χ(2QD) is given.
Fig. 4
Fig. 4 Structure of (left) zincblende and (right) diamond type lattices, represented in the conventional cubic unit cell.

Tables (1)

Tables Icon

Table 1 Numerical parameters used for the ab-initio calculation of the dielectric function. For the Brillouin zone sampling, Monkhorst-Pack grids excluding the Γ-point were used for faster convergence. a0 is the lattice constant of the equivalent conventional cell.

Equations (58)

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V ( r ) = m 2 ( ω 0 2 ) a b r a r b + m 3 β a b c r a r b r c
0 = r ¨ + 2 η r ˙ + ω 0 2 r + β : rr λ μ E ( r , t ) + λ μ ( B ( r , t ) × r ˙ )
P = χ ( 1 ) E + χ ( 2 D ) : EE + χ ( 2 Q ) : E / r E
χ ( 1 ) ( ω ) = + n e Γ ω μ
χ ( 2 D ) ( ω ) = n e Γ 2 ω β : ( Γ ω μ ) ( Γ ω μ )
χ ( 2 Q ) ( ω ) = + n e ( Γ 2 ω μ ) ( Γ ω μ )
χ ( 2 D ) ( ω ) = 1 ( n e ) 2 μ 1 χ ( 1 ) ( 2 ω ) β : χ ( 1 ) ( ω ) χ ( 1 ) ( ω )
χ ( 2 Q ) ( ω ) = 1 n e χ ( 1 ) ( 2 ω ) χ ( 1 ) ( ω )
P ( 2 MD ) ( t ) = + n e m Γ 2 ω μ i ω B 0 × ( Γ ω μ E m )
P ( 2 MD ) = m n ( χ ( 2 Q ) : E m i k m E n χ ( 2 Q ) : i k m E m E n ) ,
χ a b c ( 2 D ) = + β m e n v 2 e 0 3 χ ( 1 ) ( 2 ω ) [ χ ( 1 ) ( ω ) ] 2 | a b c |
χ a b c d ( 2 Q ) = 1 n v 2 e 0 χ ( 1 ) ( 2 ω ) χ ( 1 ) ( ω ) δ a b δ c d
Γ ω ( j ) = 1 ω 0 2 ω 2 2 i η ω b ^ j b ^ j γ ω b ^ b ^ .
χ ( 1 ) ( ω ) = + j n v 4 e μ γ ω b ^ j b ^ j = + 1 3 n v e μ γ ω 1
χ a b c ( 2 D ) ( ω ) = j n v 4 e μ 2 γ 2 ω γ ω 2 β ( b ^ j b ^ j b ^ j ) a b c = 1 27 n v e μ 2 γ 2 ω γ ω 2 β | a b c |
χ a b c d ( 2 Q ) = + j n v 4 e μ 2 γ 2 ω γ ω ( b ^ j b ^ j b ^ j b ^ j ) a b c d = + 1 9 n v e μ 2 γ 2 ω γ ω X a b c d
X a b c d = δ a b δ c d + δ a c δ b d + δ a b δ b c 2 δ a b δ b c δ c d
χ a b c ( 2 D ) ( ω ) = + 27 β m e n v 2 e 0 3 χ ( 1 ) ( 2 ω ) [ χ ( 1 ) ( ω ) ] 2 | a b c | χ ( 2 D ) ( ω ) | a b c |
χ a b c d ( 2 Q ) ( ω ) = 1 n v e 0 χ ( 1 ) ( 2 ω ) χ ( 1 ) ( ω ) X a b c d χ ( 2 Q ) ( ω ) X a b c d
| P ( 2 D ) | = 2 3 | χ ( 2 D ) ( ω ) | E 0 2
| P ( 2 Q ) | = 2 3 | χ ( 2 QD ) ( ω ) i k | E 0 2
| P ( 1 ) | 0.9 × 10 3 C / m 2
| P ( 2 D ) | 1.1 × 10 8 C / m 2
| P ( 2 Q ) | 1.3 × 10 10 C / m 2
χ ( 2 QD ) = i k χ ( 2 Q ) ,
r ( t ) = r 0 + λ r 1 e i ω t + λ 2 r 2 e 2 i ω t + 𝒪 ( λ 3 )
E ( r , t ) = ( E 0 + E 0 r ) e i ω t + 𝒪 ( k 2 )
0 = + λ 0 ( ω 0 2 r 0 + β : r 0 r 0 ) + λ 1 ( ω 2 r 1 2 i ω η r 1 + ω 0 2 r 1 μ E 0 ) e i ω t + λ 2 ( 4 ω 2 r 2 4 i ω η r 2 + ω 0 2 r 2 + β : r 1 r 1 μ E 0 r 1 ) e 2 i ω t + 𝒪 ( λ 3 )
r 1 = + Γ ω μ E 0
r 2 = Γ 2 ω β : r 1 r 1 + Γ 2 ω μ E 0 r 1 = Γ 2 ω β : ( Γ ω μ E 0 ) ( Γ ω μ E 0 ) + Γ 2 ω μ E 0 Γ ω μ E 0
P ( t ) = n e r ( t )
E ( r , t ) = m E m e i k m r i ω t
P ( 1 ) ( t ) = + n e Γ ω μ E 0 e i ω t
P ( 2 D ) ( t ) = n e Γ 2 ω β : ( Γ ω μ E 0 ) ( Γ ω μ E 0 ) e 2 i ω t
P ( 2 Q ) ( t ) = + n e m , n Γ 2 ω μ E m ( i k m Γ ω μ E n )
χ ( 1 ) ( ω ) = + n e Γ ω μ
χ ( 2 D ) ( ω ) = n e Γ 2 ω β : ( Γ ω μ ) ( Γ ω μ )
χ ( 2 Q ) ( ω ) = + n e ( Γ 2 ω μ ) ( Γ ω μ )
T ( k 1 ) T ( k 2 ) T ( k N ) ,
A i 1 , i R = T i 1 j 1 ( m ) T i R j R ( m ) A j 1 , . . j R T ( m )
( ( 0 0 0 ) ( 0 0 a ) ( 0 a 0 ) ( 0 0 a ) ( 0 0 0 ) ( a 0 0 ) ( 0 a 0 ) ( a 0 0 ) ( 0 0 0 ) )
( ( A 4 0 0 0 A 3 0 0 0 A 3 ) ( 0 A 2 0 A 1 0 0 0 0 0 ) ( 0 0 A 2 0 0 0 A 1 0 0 ) ( 0 A 1 0 A 2 0 0 0 0 0 ) ( A 3 0 0 0 A 4 0 0 0 A 3 ) ( 0 0 0 0 0 A 2 0 A 1 0 ) ( 0 0 A 1 0 0 0 A 2 0 0 ) ( 0 0 0 0 0 A 1 0 A 2 0 ) ( A 3 0 0 0 A 3 0 0 0 A 4 ) ) .
E ( r , t ) = E 0 e i kr i ω t
P ( r , t ) = P ( 1 ) e i kr i ω t + ( P ( 2 D ) + P ( 2 Q ) ) e 2 i kr 2 i ω t
k ^ [ 001 ] = e ^ z , E ^ [ 001 ] = cos φ e ^ x + sin φ e ^ y k ^ [ 011 ] = e ^ y + e ^ z 2 , E ^ [ 011 ] = cos φ e ^ x + sin φ e ^ y e ^ z 2 k ^ [ 111 ] = e ^ x + e ^ y + e ^ z 3 , E ^ [ 111 ] = cos φ 2 e ^ x e ^ y e ^ z 6 + sin φ e ^ y e ^ z 2
P ( 2 D ) = 2 χ ( 2 D ) ( E y E z E z E x E x E y )
P [ 001 ] ( 2 D ) / χ ( 2 D ) E 0 2 = k ^ sin 2 φ
P [ 001 ] ( 2 D ) / χ ( 2 D ) E 0 2 = k ^ sin 2 φ + E ^ 3 4 ( cos φ cos 3 φ )
P [ 111 ] ( 2 D ) / χ ( 2 D ) E 0 2 = k ^ 1 3 + E ^ 2 3 cos 3 φ B ^ 2 3 sin 3 φ
| P ( 2 D ) | | ( 1 k ^ k ^ ) P [ 111 ] ( 2 D ) | = 2 3 | χ ( 2 D ) | E 0 2
P ( 2 Q ) E ( k E ) = 0
P ( 2 Q ) = χ ( 2 Q ) ( i k E 0 2 2 a e ^ a i k a E 0 , a 2 )
P [ 001 ] ( 2 Q ) / χ ( 2 Q ) E 0 2 i k = k ^
P [ 011 ] ( 2 Q ) / χ ( 2 Q ) E 0 2 i k = k ^ 1 2 ( 1 + cos 2 φ )
P [ 111 ] ( 2 Q ) / χ ( 2 Q ) E 0 2 i k = k ^ 1 3 E ^ 2 3 cos 3 φ + B ^ 2 3 sin 3 φ
| P ( 2 Q ) | | ( 1 k ^ k ^ ) P [ 111 ] ( 2 Q ) | = 2 3 | χ ( 2 Q ) | i k E 0 2
P a ( 2 MD ) = χ ( 2 Q ) δ a b δ c d ( E b i k c E d i k b E c E d )
P ( 2 MD ) = χ ( 2 Q ) i k E 0 2

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