Abstract

We present a general phenomenological theory for optical second- and third-harmonic generation obtained in reflection from all cubic centrosymmetric single crystals whose faces have macroscopic C1υ symmetry, including vicinal faces with such symmetry. This extends earlier research [ Phys. Rev. B 35, 1129 ( 1987)] for crystals with low-index faces. As a test of the theory we offer experimental results for anisotropic harmonic generation from vicinal, single crystals of Si, with and without an oxide layer, for which we use 130-fs, λ = 765 nm pulses. Overall there is good agreement between the phenomenological theory and experimental results. It is demonstrated that third-harmonic generation, which is dominated by bulk electric dipole contributions, can be used as a diagnostic of crystal orientation and crystal miscut angle. The intensity and the anisotropy of second-harmonic generation (SHG), which arise from surface electric dipole and bulk electric quadrupole effects, are seen to be sensitive to surface steps, an effect that also varies with surface oxidation. Finally, the use of vicinal surfaces allows us to identify bulk and surface contributions to SHG separately.

© 1994 Optical Society of America

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  1. S. Janz, D. J. Bottomley, H. M. van Driel, and R. S. Timsit, “Influence of steps on second-harmonic generation from vicinal metal surfaces,” Phys. Rev. Lett. 66, 1201–1204 (1991).
    [Crossref] [PubMed]
  2. C. W. van Hasselt, M. A. Verheijen, and Th. Rasing, “Vicinal Si(111) surfaces studied by optical second-harmonic generation: step induced anisotropy and surface bulk discrimination,” Phys. Rev. B 42, 9263–9266 (1990); M. A. Verheijen, C. W. van Hasselt, and Th. Rasing, “Optical second harmonic generation study of vicinal Si(111) surfaces,” Surf. Sci. 251, 467–471 (1991).
    [Crossref]
  3. R. W. J. Hollering, D. Dijkkamp, and H. B. Elswijk, “Optical second-harmonic generation on a vicinal Si(111) surface,” Surf. Sci. 243, 121–126 (1991).
    [Crossref]
  4. G. Lüpke, D. J. Bottomley, and H. M. van Driel, “Si/SiO2interfacial structure on vicinal Si(100) studied with second-harmonic generation,” Phys. Rev. B 47, 10389–10394 (1993).
    [Crossref]
  5. Y. R. Shen, “Surfaces probed by second-harmonic and sum-frequency generation,” Nature (London) 337, 519–525 (1989).
    [Crossref]
  6. G. L. Richmond, J. M. Robinson, and V. L. Shannon, “Second harmonic generation studies of interfacial structure and dynamics,” Prog. Surf. Sci. 28, 1–70 (1988).
    [Crossref]
  7. J. E. Sipe, D. J. Moss, and H. M. van Driel, “Phenomenological theory of optical second- and third-harmonic generation from cubic centrosymmetric crystals,” Phys. Rev. B 35, 1129–1141 (1987), and references therein.
    [Crossref]
  8. G. Lüpke and G. Marowsky, “Third-order processes and their relation to structural symmetry,” Appl. Phys. B 53, 71–81 (1991).
    [Crossref]
  9. E. D. Williams and N. C. Bartelt, “Thermodynamics of surface morphology,” Science 251, 393–400 (1991).
    [Crossref] [PubMed]
  10. R. Kaplan, “LEED study of the stepped surfaces of vicinal Si(100),” Surf. Sci. 93, 145–158 (1980).
    [Crossref]
  11. A. Ourmazd, P. H. Fuoss, J. Bevk, and J. F. Morar, “The Si(001)/SiO2 interface,” Appl. Surf. Sci. 41/42, 365–371 (1989).
    [Crossref]
  12. R. S. Timsit, W. G. Waddington, C. J. Humphreys, and J. L. Hutchison, “Structure of the Al/Al2O3interface,” Appl. Phys. Lett. 46, 830–832 (1985).
    [Crossref]
  13. P. Guyot-Sionnest and Y. R. Shen, “Bulk contribution in surface second-harmonic generation,” Phys. Rev. B 38, 7985–7989 (1988); R. W. J. Hollering and M. Barmentlo, “Symmetry analysis of vicinal (111) surfaces by optical second-harmonic generation,” Opt. Commun. 88, 141–145 (1992).
    [Crossref]
  14. A. Liebsch and W. L. Schaich, “Second-harmonic generation at simple metal surfaces,” Phys. Rev. B 40, 5401–5410 (1989).
    [Crossref]
  15. S. V. Govorkov, N. I. Koroteev, G. I. Petrov, I. L. Shumay, and V. V. Yakovlev, “Laser nonlinear-optical probing of silicon/SiO2interfaces: surface stress formation and relaxation,” Appl. Phys. A 50, 439–443 (1990); L. L. Kulyuk, D. A. Shutov, E. E. Strumban, and O. A. Aktsipetrov, “Second-harmonic generation by an SiO2-Si interface: influence of the oxide layer,” J. Opt. Soc. Am. B 8, 1766–1769 (1991).
    [Crossref]
  16. D. J. Moss, H. M. van Driel, and J. E. Sipe, “Third harmonic generation as a structural diagnostic of ion-implanted amorphous and crystalline silicon,” Appl. Phys. Lett. 48, 1150–1152 (1986).
    [Crossref]
  17. One can derive the bulk angular functions Φn(αm) independently of the model developed here by expressing them as products of functions in sin(αm) and cos(αm) and considering where each angular function and its derivative vanish.
  18. R. J. Pressley, ed., Handbook of Lasers (CRC, Cleveland, Ohio, 1971), p. 491; M. D. Levenson, Introduction to Nonlinear Laser Spectroscopy (Academic, New York, 1982), p. 121.
  19. J. E. Sipe, V. Mizrahi, and G. I. Stegeman, “Fundamental difficulty in the use of second-harmonic generation as a strictly surface probe,” Phys. Rev. B 35, 9091–9094 (1987).
    [Crossref]
  20. G. Lüpke, G. Marowsky, R. Steinhoff, A. Friedrich, B. Pettinger, and D. M. Kolb, “Symmetry superposition studied by surface second-harmonic generation,” Phys. Rev. B 41, 6913–6919 (1990).
    [Crossref]
  21. O. L. Alerhand, A. N. Berker, J. D. Joannopoulos, D. Vanderbilt, R. J. Hamers, and J. E. Demuth, “Finite-temperature phase diagram of vicinal Si(100) surfaces,” Phys. Rev. Lett. 64, 2406–2409 (1990).
    [Crossref] [PubMed]
  22. The theory presented in this paper is insufficient to derive the orientation of a general cubic crystal face by the use of THG. This is because we consider only faces with C1υ symmetry, and not those with C1 symmetry. See D. J. Bottomley, G. Lüpke, J. G. Mihaychuk, and H. M. van Driel, “Determination of the crystallographic orientation of cubic media to high resolution using optical harmonic generation,” J. Appl. Phys. 74, 6072–6078 (1993).
    [Crossref]
  23. E. D. Palik, ed., Handbook of Optical Constants of Solids, (Academic, Orlando, Fla., 1985), p. 562.
  24. D. J. Moss, H. M. van Driel, and J. E. Sipe, “Dispersion in the anisotropy of optical third-harmonic generation in silicon,” Opt. Lett. 14, 57–59 (1989).
    [Crossref] [PubMed]
  25. H. W. K. Tom, T. F. Heinz, and Y. R. Shen, “Second-harmonic reflection from silicon surfaces and its relation to structural symmetry,” Phys. Rev. Lett. 51, 1983–1986 (1983).
    [Crossref]
  26. S. Janz, K. Pedersen, H. M. van Driel, and R. S. Timsit, “Structural transformations in adsorbed oxygen layers on Al surfaces observed using optical second-harmonic generation,” J. Vacuum Sci. Technol. A 9, 1506–1510 (1991).
    [Crossref]

1993 (2)

G. Lüpke, D. J. Bottomley, and H. M. van Driel, “Si/SiO2interfacial structure on vicinal Si(100) studied with second-harmonic generation,” Phys. Rev. B 47, 10389–10394 (1993).
[Crossref]

The theory presented in this paper is insufficient to derive the orientation of a general cubic crystal face by the use of THG. This is because we consider only faces with C1υ symmetry, and not those with C1 symmetry. See D. J. Bottomley, G. Lüpke, J. G. Mihaychuk, and H. M. van Driel, “Determination of the crystallographic orientation of cubic media to high resolution using optical harmonic generation,” J. Appl. Phys. 74, 6072–6078 (1993).
[Crossref]

1991 (5)

S. Janz, K. Pedersen, H. M. van Driel, and R. S. Timsit, “Structural transformations in adsorbed oxygen layers on Al surfaces observed using optical second-harmonic generation,” J. Vacuum Sci. Technol. A 9, 1506–1510 (1991).
[Crossref]

S. Janz, D. J. Bottomley, H. M. van Driel, and R. S. Timsit, “Influence of steps on second-harmonic generation from vicinal metal surfaces,” Phys. Rev. Lett. 66, 1201–1204 (1991).
[Crossref] [PubMed]

G. Lüpke and G. Marowsky, “Third-order processes and their relation to structural symmetry,” Appl. Phys. B 53, 71–81 (1991).
[Crossref]

E. D. Williams and N. C. Bartelt, “Thermodynamics of surface morphology,” Science 251, 393–400 (1991).
[Crossref] [PubMed]

R. W. J. Hollering, D. Dijkkamp, and H. B. Elswijk, “Optical second-harmonic generation on a vicinal Si(111) surface,” Surf. Sci. 243, 121–126 (1991).
[Crossref]

1990 (4)

S. V. Govorkov, N. I. Koroteev, G. I. Petrov, I. L. Shumay, and V. V. Yakovlev, “Laser nonlinear-optical probing of silicon/SiO2interfaces: surface stress formation and relaxation,” Appl. Phys. A 50, 439–443 (1990); L. L. Kulyuk, D. A. Shutov, E. E. Strumban, and O. A. Aktsipetrov, “Second-harmonic generation by an SiO2-Si interface: influence of the oxide layer,” J. Opt. Soc. Am. B 8, 1766–1769 (1991).
[Crossref]

C. W. van Hasselt, M. A. Verheijen, and Th. Rasing, “Vicinal Si(111) surfaces studied by optical second-harmonic generation: step induced anisotropy and surface bulk discrimination,” Phys. Rev. B 42, 9263–9266 (1990); M. A. Verheijen, C. W. van Hasselt, and Th. Rasing, “Optical second harmonic generation study of vicinal Si(111) surfaces,” Surf. Sci. 251, 467–471 (1991).
[Crossref]

G. Lüpke, G. Marowsky, R. Steinhoff, A. Friedrich, B. Pettinger, and D. M. Kolb, “Symmetry superposition studied by surface second-harmonic generation,” Phys. Rev. B 41, 6913–6919 (1990).
[Crossref]

O. L. Alerhand, A. N. Berker, J. D. Joannopoulos, D. Vanderbilt, R. J. Hamers, and J. E. Demuth, “Finite-temperature phase diagram of vicinal Si(100) surfaces,” Phys. Rev. Lett. 64, 2406–2409 (1990).
[Crossref] [PubMed]

1989 (4)

D. J. Moss, H. M. van Driel, and J. E. Sipe, “Dispersion in the anisotropy of optical third-harmonic generation in silicon,” Opt. Lett. 14, 57–59 (1989).
[Crossref] [PubMed]

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

A. Liebsch and W. L. Schaich, “Second-harmonic generation at simple metal surfaces,” Phys. Rev. B 40, 5401–5410 (1989).
[Crossref]

A. Ourmazd, P. H. Fuoss, J. Bevk, and J. F. Morar, “The Si(001)/SiO2 interface,” Appl. Surf. Sci. 41/42, 365–371 (1989).
[Crossref]

1988 (2)

P. Guyot-Sionnest and Y. R. Shen, “Bulk contribution in surface second-harmonic generation,” Phys. Rev. B 38, 7985–7989 (1988); R. W. J. Hollering and M. Barmentlo, “Symmetry analysis of vicinal (111) surfaces by optical second-harmonic generation,” Opt. Commun. 88, 141–145 (1992).
[Crossref]

G. L. Richmond, J. M. Robinson, and V. L. Shannon, “Second harmonic generation studies of interfacial structure and dynamics,” Prog. Surf. Sci. 28, 1–70 (1988).
[Crossref]

1987 (2)

J. E. Sipe, D. J. Moss, and H. M. van Driel, “Phenomenological theory of optical second- and third-harmonic generation from cubic centrosymmetric crystals,” Phys. Rev. B 35, 1129–1141 (1987), and references therein.
[Crossref]

J. E. Sipe, V. Mizrahi, and G. I. Stegeman, “Fundamental difficulty in the use of second-harmonic generation as a strictly surface probe,” Phys. Rev. B 35, 9091–9094 (1987).
[Crossref]

1986 (1)

D. J. Moss, H. M. van Driel, and J. E. Sipe, “Third harmonic generation as a structural diagnostic of ion-implanted amorphous and crystalline silicon,” Appl. Phys. Lett. 48, 1150–1152 (1986).
[Crossref]

1985 (1)

R. S. Timsit, W. G. Waddington, C. J. Humphreys, and J. L. Hutchison, “Structure of the Al/Al2O3interface,” Appl. Phys. Lett. 46, 830–832 (1985).
[Crossref]

1983 (1)

H. W. K. Tom, T. F. Heinz, and Y. R. Shen, “Second-harmonic reflection from silicon surfaces and its relation to structural symmetry,” Phys. Rev. Lett. 51, 1983–1986 (1983).
[Crossref]

1980 (1)

R. Kaplan, “LEED study of the stepped surfaces of vicinal Si(100),” Surf. Sci. 93, 145–158 (1980).
[Crossref]

Alerhand, O. L.

O. L. Alerhand, A. N. Berker, J. D. Joannopoulos, D. Vanderbilt, R. J. Hamers, and J. E. Demuth, “Finite-temperature phase diagram of vicinal Si(100) surfaces,” Phys. Rev. Lett. 64, 2406–2409 (1990).
[Crossref] [PubMed]

Bartelt, N. C.

E. D. Williams and N. C. Bartelt, “Thermodynamics of surface morphology,” Science 251, 393–400 (1991).
[Crossref] [PubMed]

Berker, A. N.

O. L. Alerhand, A. N. Berker, J. D. Joannopoulos, D. Vanderbilt, R. J. Hamers, and J. E. Demuth, “Finite-temperature phase diagram of vicinal Si(100) surfaces,” Phys. Rev. Lett. 64, 2406–2409 (1990).
[Crossref] [PubMed]

Bevk, J.

A. Ourmazd, P. H. Fuoss, J. Bevk, and J. F. Morar, “The Si(001)/SiO2 interface,” Appl. Surf. Sci. 41/42, 365–371 (1989).
[Crossref]

Bottomley, D. J.

G. Lüpke, D. J. Bottomley, and H. M. van Driel, “Si/SiO2interfacial structure on vicinal Si(100) studied with second-harmonic generation,” Phys. Rev. B 47, 10389–10394 (1993).
[Crossref]

The theory presented in this paper is insufficient to derive the orientation of a general cubic crystal face by the use of THG. This is because we consider only faces with C1υ symmetry, and not those with C1 symmetry. See D. J. Bottomley, G. Lüpke, J. G. Mihaychuk, and H. M. van Driel, “Determination of the crystallographic orientation of cubic media to high resolution using optical harmonic generation,” J. Appl. Phys. 74, 6072–6078 (1993).
[Crossref]

S. Janz, D. J. Bottomley, H. M. van Driel, and R. S. Timsit, “Influence of steps on second-harmonic generation from vicinal metal surfaces,” Phys. Rev. Lett. 66, 1201–1204 (1991).
[Crossref] [PubMed]

Demuth, J. E.

O. L. Alerhand, A. N. Berker, J. D. Joannopoulos, D. Vanderbilt, R. J. Hamers, and J. E. Demuth, “Finite-temperature phase diagram of vicinal Si(100) surfaces,” Phys. Rev. Lett. 64, 2406–2409 (1990).
[Crossref] [PubMed]

Dijkkamp, D.

R. W. J. Hollering, D. Dijkkamp, and H. B. Elswijk, “Optical second-harmonic generation on a vicinal Si(111) surface,” Surf. Sci. 243, 121–126 (1991).
[Crossref]

Elswijk, H. B.

R. W. J. Hollering, D. Dijkkamp, and H. B. Elswijk, “Optical second-harmonic generation on a vicinal Si(111) surface,” Surf. Sci. 243, 121–126 (1991).
[Crossref]

Friedrich, A.

G. Lüpke, G. Marowsky, R. Steinhoff, A. Friedrich, B. Pettinger, and D. M. Kolb, “Symmetry superposition studied by surface second-harmonic generation,” Phys. Rev. B 41, 6913–6919 (1990).
[Crossref]

Fuoss, P. H.

A. Ourmazd, P. H. Fuoss, J. Bevk, and J. F. Morar, “The Si(001)/SiO2 interface,” Appl. Surf. Sci. 41/42, 365–371 (1989).
[Crossref]

Govorkov, S. V.

S. V. Govorkov, N. I. Koroteev, G. I. Petrov, I. L. Shumay, and V. V. Yakovlev, “Laser nonlinear-optical probing of silicon/SiO2interfaces: surface stress formation and relaxation,” Appl. Phys. A 50, 439–443 (1990); L. L. Kulyuk, D. A. Shutov, E. E. Strumban, and O. A. Aktsipetrov, “Second-harmonic generation by an SiO2-Si interface: influence of the oxide layer,” J. Opt. Soc. Am. B 8, 1766–1769 (1991).
[Crossref]

Guyot-Sionnest, P.

P. Guyot-Sionnest and Y. R. Shen, “Bulk contribution in surface second-harmonic generation,” Phys. Rev. B 38, 7985–7989 (1988); R. W. J. Hollering and M. Barmentlo, “Symmetry analysis of vicinal (111) surfaces by optical second-harmonic generation,” Opt. Commun. 88, 141–145 (1992).
[Crossref]

Hamers, R. J.

O. L. Alerhand, A. N. Berker, J. D. Joannopoulos, D. Vanderbilt, R. J. Hamers, and J. E. Demuth, “Finite-temperature phase diagram of vicinal Si(100) surfaces,” Phys. Rev. Lett. 64, 2406–2409 (1990).
[Crossref] [PubMed]

Heinz, T. F.

H. W. K. Tom, T. F. Heinz, and Y. R. Shen, “Second-harmonic reflection from silicon surfaces and its relation to structural symmetry,” Phys. Rev. Lett. 51, 1983–1986 (1983).
[Crossref]

Hollering, R. W. J.

R. W. J. Hollering, D. Dijkkamp, and H. B. Elswijk, “Optical second-harmonic generation on a vicinal Si(111) surface,” Surf. Sci. 243, 121–126 (1991).
[Crossref]

Humphreys, C. J.

R. S. Timsit, W. G. Waddington, C. J. Humphreys, and J. L. Hutchison, “Structure of the Al/Al2O3interface,” Appl. Phys. Lett. 46, 830–832 (1985).
[Crossref]

Hutchison, J. L.

R. S. Timsit, W. G. Waddington, C. J. Humphreys, and J. L. Hutchison, “Structure of the Al/Al2O3interface,” Appl. Phys. Lett. 46, 830–832 (1985).
[Crossref]

Janz, S.

S. Janz, D. J. Bottomley, H. M. van Driel, and R. S. Timsit, “Influence of steps on second-harmonic generation from vicinal metal surfaces,” Phys. Rev. Lett. 66, 1201–1204 (1991).
[Crossref] [PubMed]

S. Janz, K. Pedersen, H. M. van Driel, and R. S. Timsit, “Structural transformations in adsorbed oxygen layers on Al surfaces observed using optical second-harmonic generation,” J. Vacuum Sci. Technol. A 9, 1506–1510 (1991).
[Crossref]

Joannopoulos, J. D.

O. L. Alerhand, A. N. Berker, J. D. Joannopoulos, D. Vanderbilt, R. J. Hamers, and J. E. Demuth, “Finite-temperature phase diagram of vicinal Si(100) surfaces,” Phys. Rev. Lett. 64, 2406–2409 (1990).
[Crossref] [PubMed]

Kaplan, R.

R. Kaplan, “LEED study of the stepped surfaces of vicinal Si(100),” Surf. Sci. 93, 145–158 (1980).
[Crossref]

Kolb, D. M.

G. Lüpke, G. Marowsky, R. Steinhoff, A. Friedrich, B. Pettinger, and D. M. Kolb, “Symmetry superposition studied by surface second-harmonic generation,” Phys. Rev. B 41, 6913–6919 (1990).
[Crossref]

Koroteev, N. I.

S. V. Govorkov, N. I. Koroteev, G. I. Petrov, I. L. Shumay, and V. V. Yakovlev, “Laser nonlinear-optical probing of silicon/SiO2interfaces: surface stress formation and relaxation,” Appl. Phys. A 50, 439–443 (1990); L. L. Kulyuk, D. A. Shutov, E. E. Strumban, and O. A. Aktsipetrov, “Second-harmonic generation by an SiO2-Si interface: influence of the oxide layer,” J. Opt. Soc. Am. B 8, 1766–1769 (1991).
[Crossref]

Liebsch, A.

A. Liebsch and W. L. Schaich, “Second-harmonic generation at simple metal surfaces,” Phys. Rev. B 40, 5401–5410 (1989).
[Crossref]

Lüpke, G.

G. Lüpke, D. J. Bottomley, and H. M. van Driel, “Si/SiO2interfacial structure on vicinal Si(100) studied with second-harmonic generation,” Phys. Rev. B 47, 10389–10394 (1993).
[Crossref]

The theory presented in this paper is insufficient to derive the orientation of a general cubic crystal face by the use of THG. This is because we consider only faces with C1υ symmetry, and not those with C1 symmetry. See D. J. Bottomley, G. Lüpke, J. G. Mihaychuk, and H. M. van Driel, “Determination of the crystallographic orientation of cubic media to high resolution using optical harmonic generation,” J. Appl. Phys. 74, 6072–6078 (1993).
[Crossref]

G. Lüpke and G. Marowsky, “Third-order processes and their relation to structural symmetry,” Appl. Phys. B 53, 71–81 (1991).
[Crossref]

G. Lüpke, G. Marowsky, R. Steinhoff, A. Friedrich, B. Pettinger, and D. M. Kolb, “Symmetry superposition studied by surface second-harmonic generation,” Phys. Rev. B 41, 6913–6919 (1990).
[Crossref]

Marowsky, G.

G. Lüpke and G. Marowsky, “Third-order processes and their relation to structural symmetry,” Appl. Phys. B 53, 71–81 (1991).
[Crossref]

G. Lüpke, G. Marowsky, R. Steinhoff, A. Friedrich, B. Pettinger, and D. M. Kolb, “Symmetry superposition studied by surface second-harmonic generation,” Phys. Rev. B 41, 6913–6919 (1990).
[Crossref]

Mihaychuk, J. G.

The theory presented in this paper is insufficient to derive the orientation of a general cubic crystal face by the use of THG. This is because we consider only faces with C1υ symmetry, and not those with C1 symmetry. See D. J. Bottomley, G. Lüpke, J. G. Mihaychuk, and H. M. van Driel, “Determination of the crystallographic orientation of cubic media to high resolution using optical harmonic generation,” J. Appl. Phys. 74, 6072–6078 (1993).
[Crossref]

Mizrahi, V.

J. E. Sipe, V. Mizrahi, and G. I. Stegeman, “Fundamental difficulty in the use of second-harmonic generation as a strictly surface probe,” Phys. Rev. B 35, 9091–9094 (1987).
[Crossref]

Morar, J. F.

A. Ourmazd, P. H. Fuoss, J. Bevk, and J. F. Morar, “The Si(001)/SiO2 interface,” Appl. Surf. Sci. 41/42, 365–371 (1989).
[Crossref]

Moss, D. J.

D. J. Moss, H. M. van Driel, and J. E. Sipe, “Dispersion in the anisotropy of optical third-harmonic generation in silicon,” Opt. Lett. 14, 57–59 (1989).
[Crossref] [PubMed]

J. E. Sipe, D. J. Moss, and H. M. van Driel, “Phenomenological theory of optical second- and third-harmonic generation from cubic centrosymmetric crystals,” Phys. Rev. B 35, 1129–1141 (1987), and references therein.
[Crossref]

D. J. Moss, H. M. van Driel, and J. E. Sipe, “Third harmonic generation as a structural diagnostic of ion-implanted amorphous and crystalline silicon,” Appl. Phys. Lett. 48, 1150–1152 (1986).
[Crossref]

Ourmazd, A.

A. Ourmazd, P. H. Fuoss, J. Bevk, and J. F. Morar, “The Si(001)/SiO2 interface,” Appl. Surf. Sci. 41/42, 365–371 (1989).
[Crossref]

Pedersen, K.

S. Janz, K. Pedersen, H. M. van Driel, and R. S. Timsit, “Structural transformations in adsorbed oxygen layers on Al surfaces observed using optical second-harmonic generation,” J. Vacuum Sci. Technol. A 9, 1506–1510 (1991).
[Crossref]

Petrov, G. I.

S. V. Govorkov, N. I. Koroteev, G. I. Petrov, I. L. Shumay, and V. V. Yakovlev, “Laser nonlinear-optical probing of silicon/SiO2interfaces: surface stress formation and relaxation,” Appl. Phys. A 50, 439–443 (1990); L. L. Kulyuk, D. A. Shutov, E. E. Strumban, and O. A. Aktsipetrov, “Second-harmonic generation by an SiO2-Si interface: influence of the oxide layer,” J. Opt. Soc. Am. B 8, 1766–1769 (1991).
[Crossref]

Pettinger, B.

G. Lüpke, G. Marowsky, R. Steinhoff, A. Friedrich, B. Pettinger, and D. M. Kolb, “Symmetry superposition studied by surface second-harmonic generation,” Phys. Rev. B 41, 6913–6919 (1990).
[Crossref]

Rasing, Th.

C. W. van Hasselt, M. A. Verheijen, and Th. Rasing, “Vicinal Si(111) surfaces studied by optical second-harmonic generation: step induced anisotropy and surface bulk discrimination,” Phys. Rev. B 42, 9263–9266 (1990); M. A. Verheijen, C. W. van Hasselt, and Th. Rasing, “Optical second harmonic generation study of vicinal Si(111) surfaces,” Surf. Sci. 251, 467–471 (1991).
[Crossref]

Richmond, G. L.

G. L. Richmond, J. M. Robinson, and V. L. Shannon, “Second harmonic generation studies of interfacial structure and dynamics,” Prog. Surf. Sci. 28, 1–70 (1988).
[Crossref]

Robinson, J. M.

G. L. Richmond, J. M. Robinson, and V. L. Shannon, “Second harmonic generation studies of interfacial structure and dynamics,” Prog. Surf. Sci. 28, 1–70 (1988).
[Crossref]

Schaich, W. L.

A. Liebsch and W. L. Schaich, “Second-harmonic generation at simple metal surfaces,” Phys. Rev. B 40, 5401–5410 (1989).
[Crossref]

Shannon, V. L.

G. L. Richmond, J. M. Robinson, and V. L. Shannon, “Second harmonic generation studies of interfacial structure and dynamics,” Prog. Surf. Sci. 28, 1–70 (1988).
[Crossref]

Shen, Y. R.

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

P. Guyot-Sionnest and Y. R. Shen, “Bulk contribution in surface second-harmonic generation,” Phys. Rev. B 38, 7985–7989 (1988); R. W. J. Hollering and M. Barmentlo, “Symmetry analysis of vicinal (111) surfaces by optical second-harmonic generation,” Opt. Commun. 88, 141–145 (1992).
[Crossref]

H. W. K. Tom, T. F. Heinz, and Y. R. Shen, “Second-harmonic reflection from silicon surfaces and its relation to structural symmetry,” Phys. Rev. Lett. 51, 1983–1986 (1983).
[Crossref]

Shumay, I. L.

S. V. Govorkov, N. I. Koroteev, G. I. Petrov, I. L. Shumay, and V. V. Yakovlev, “Laser nonlinear-optical probing of silicon/SiO2interfaces: surface stress formation and relaxation,” Appl. Phys. A 50, 439–443 (1990); L. L. Kulyuk, D. A. Shutov, E. E. Strumban, and O. A. Aktsipetrov, “Second-harmonic generation by an SiO2-Si interface: influence of the oxide layer,” J. Opt. Soc. Am. B 8, 1766–1769 (1991).
[Crossref]

Sipe, J. E.

D. J. Moss, H. M. van Driel, and J. E. Sipe, “Dispersion in the anisotropy of optical third-harmonic generation in silicon,” Opt. Lett. 14, 57–59 (1989).
[Crossref] [PubMed]

J. E. Sipe, V. Mizrahi, and G. I. Stegeman, “Fundamental difficulty in the use of second-harmonic generation as a strictly surface probe,” Phys. Rev. B 35, 9091–9094 (1987).
[Crossref]

J. E. Sipe, D. J. Moss, and H. M. van Driel, “Phenomenological theory of optical second- and third-harmonic generation from cubic centrosymmetric crystals,” Phys. Rev. B 35, 1129–1141 (1987), and references therein.
[Crossref]

D. J. Moss, H. M. van Driel, and J. E. Sipe, “Third harmonic generation as a structural diagnostic of ion-implanted amorphous and crystalline silicon,” Appl. Phys. Lett. 48, 1150–1152 (1986).
[Crossref]

Stegeman, G. I.

J. E. Sipe, V. Mizrahi, and G. I. Stegeman, “Fundamental difficulty in the use of second-harmonic generation as a strictly surface probe,” Phys. Rev. B 35, 9091–9094 (1987).
[Crossref]

Steinhoff, R.

G. Lüpke, G. Marowsky, R. Steinhoff, A. Friedrich, B. Pettinger, and D. M. Kolb, “Symmetry superposition studied by surface second-harmonic generation,” Phys. Rev. B 41, 6913–6919 (1990).
[Crossref]

Timsit, R. S.

S. Janz, K. Pedersen, H. M. van Driel, and R. S. Timsit, “Structural transformations in adsorbed oxygen layers on Al surfaces observed using optical second-harmonic generation,” J. Vacuum Sci. Technol. A 9, 1506–1510 (1991).
[Crossref]

S. Janz, D. J. Bottomley, H. M. van Driel, and R. S. Timsit, “Influence of steps on second-harmonic generation from vicinal metal surfaces,” Phys. Rev. Lett. 66, 1201–1204 (1991).
[Crossref] [PubMed]

R. S. Timsit, W. G. Waddington, C. J. Humphreys, and J. L. Hutchison, “Structure of the Al/Al2O3interface,” Appl. Phys. Lett. 46, 830–832 (1985).
[Crossref]

Tom, H. W. K.

H. W. K. Tom, T. F. Heinz, and Y. R. Shen, “Second-harmonic reflection from silicon surfaces and its relation to structural symmetry,” Phys. Rev. Lett. 51, 1983–1986 (1983).
[Crossref]

van Driel, H. M.

The theory presented in this paper is insufficient to derive the orientation of a general cubic crystal face by the use of THG. This is because we consider only faces with C1υ symmetry, and not those with C1 symmetry. See D. J. Bottomley, G. Lüpke, J. G. Mihaychuk, and H. M. van Driel, “Determination of the crystallographic orientation of cubic media to high resolution using optical harmonic generation,” J. Appl. Phys. 74, 6072–6078 (1993).
[Crossref]

G. Lüpke, D. J. Bottomley, and H. M. van Driel, “Si/SiO2interfacial structure on vicinal Si(100) studied with second-harmonic generation,” Phys. Rev. B 47, 10389–10394 (1993).
[Crossref]

S. Janz, D. J. Bottomley, H. M. van Driel, and R. S. Timsit, “Influence of steps on second-harmonic generation from vicinal metal surfaces,” Phys. Rev. Lett. 66, 1201–1204 (1991).
[Crossref] [PubMed]

S. Janz, K. Pedersen, H. M. van Driel, and R. S. Timsit, “Structural transformations in adsorbed oxygen layers on Al surfaces observed using optical second-harmonic generation,” J. Vacuum Sci. Technol. A 9, 1506–1510 (1991).
[Crossref]

D. J. Moss, H. M. van Driel, and J. E. Sipe, “Dispersion in the anisotropy of optical third-harmonic generation in silicon,” Opt. Lett. 14, 57–59 (1989).
[Crossref] [PubMed]

J. E. Sipe, D. J. Moss, and H. M. van Driel, “Phenomenological theory of optical second- and third-harmonic generation from cubic centrosymmetric crystals,” Phys. Rev. B 35, 1129–1141 (1987), and references therein.
[Crossref]

D. J. Moss, H. M. van Driel, and J. E. Sipe, “Third harmonic generation as a structural diagnostic of ion-implanted amorphous and crystalline silicon,” Appl. Phys. Lett. 48, 1150–1152 (1986).
[Crossref]

van Hasselt, C. W.

C. W. van Hasselt, M. A. Verheijen, and Th. Rasing, “Vicinal Si(111) surfaces studied by optical second-harmonic generation: step induced anisotropy and surface bulk discrimination,” Phys. Rev. B 42, 9263–9266 (1990); M. A. Verheijen, C. W. van Hasselt, and Th. Rasing, “Optical second harmonic generation study of vicinal Si(111) surfaces,” Surf. Sci. 251, 467–471 (1991).
[Crossref]

Vanderbilt, D.

O. L. Alerhand, A. N. Berker, J. D. Joannopoulos, D. Vanderbilt, R. J. Hamers, and J. E. Demuth, “Finite-temperature phase diagram of vicinal Si(100) surfaces,” Phys. Rev. Lett. 64, 2406–2409 (1990).
[Crossref] [PubMed]

Verheijen, M. A.

C. W. van Hasselt, M. A. Verheijen, and Th. Rasing, “Vicinal Si(111) surfaces studied by optical second-harmonic generation: step induced anisotropy and surface bulk discrimination,” Phys. Rev. B 42, 9263–9266 (1990); M. A. Verheijen, C. W. van Hasselt, and Th. Rasing, “Optical second harmonic generation study of vicinal Si(111) surfaces,” Surf. Sci. 251, 467–471 (1991).
[Crossref]

Waddington, W. G.

R. S. Timsit, W. G. Waddington, C. J. Humphreys, and J. L. Hutchison, “Structure of the Al/Al2O3interface,” Appl. Phys. Lett. 46, 830–832 (1985).
[Crossref]

Williams, E. D.

E. D. Williams and N. C. Bartelt, “Thermodynamics of surface morphology,” Science 251, 393–400 (1991).
[Crossref] [PubMed]

Yakovlev, V. V.

S. V. Govorkov, N. I. Koroteev, G. I. Petrov, I. L. Shumay, and V. V. Yakovlev, “Laser nonlinear-optical probing of silicon/SiO2interfaces: surface stress formation and relaxation,” Appl. Phys. A 50, 439–443 (1990); L. L. Kulyuk, D. A. Shutov, E. E. Strumban, and O. A. Aktsipetrov, “Second-harmonic generation by an SiO2-Si interface: influence of the oxide layer,” J. Opt. Soc. Am. B 8, 1766–1769 (1991).
[Crossref]

Appl. Phys. A (1)

S. V. Govorkov, N. I. Koroteev, G. I. Petrov, I. L. Shumay, and V. V. Yakovlev, “Laser nonlinear-optical probing of silicon/SiO2interfaces: surface stress formation and relaxation,” Appl. Phys. A 50, 439–443 (1990); L. L. Kulyuk, D. A. Shutov, E. E. Strumban, and O. A. Aktsipetrov, “Second-harmonic generation by an SiO2-Si interface: influence of the oxide layer,” J. Opt. Soc. Am. B 8, 1766–1769 (1991).
[Crossref]

Appl. Phys. B (1)

G. Lüpke and G. Marowsky, “Third-order processes and their relation to structural symmetry,” Appl. Phys. B 53, 71–81 (1991).
[Crossref]

Appl. Phys. Lett. (2)

D. J. Moss, H. M. van Driel, and J. E. Sipe, “Third harmonic generation as a structural diagnostic of ion-implanted amorphous and crystalline silicon,” Appl. Phys. Lett. 48, 1150–1152 (1986).
[Crossref]

R. S. Timsit, W. G. Waddington, C. J. Humphreys, and J. L. Hutchison, “Structure of the Al/Al2O3interface,” Appl. Phys. Lett. 46, 830–832 (1985).
[Crossref]

Appl. Surf. Sci. (1)

A. Ourmazd, P. H. Fuoss, J. Bevk, and J. F. Morar, “The Si(001)/SiO2 interface,” Appl. Surf. Sci. 41/42, 365–371 (1989).
[Crossref]

J. Appl. Phys. (1)

The theory presented in this paper is insufficient to derive the orientation of a general cubic crystal face by the use of THG. This is because we consider only faces with C1υ symmetry, and not those with C1 symmetry. See D. J. Bottomley, G. Lüpke, J. G. Mihaychuk, and H. M. van Driel, “Determination of the crystallographic orientation of cubic media to high resolution using optical harmonic generation,” J. Appl. Phys. 74, 6072–6078 (1993).
[Crossref]

J. Vacuum Sci. Technol. A (1)

S. Janz, K. Pedersen, H. M. van Driel, and R. S. Timsit, “Structural transformations in adsorbed oxygen layers on Al surfaces observed using optical second-harmonic generation,” J. Vacuum Sci. Technol. A 9, 1506–1510 (1991).
[Crossref]

Nature (London) (1)

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

Opt. Lett. (1)

Phys. Rev. B (7)

J. E. Sipe, V. Mizrahi, and G. I. Stegeman, “Fundamental difficulty in the use of second-harmonic generation as a strictly surface probe,” Phys. Rev. B 35, 9091–9094 (1987).
[Crossref]

G. Lüpke, G. Marowsky, R. Steinhoff, A. Friedrich, B. Pettinger, and D. M. Kolb, “Symmetry superposition studied by surface second-harmonic generation,” Phys. Rev. B 41, 6913–6919 (1990).
[Crossref]

C. W. van Hasselt, M. A. Verheijen, and Th. Rasing, “Vicinal Si(111) surfaces studied by optical second-harmonic generation: step induced anisotropy and surface bulk discrimination,” Phys. Rev. B 42, 9263–9266 (1990); M. A. Verheijen, C. W. van Hasselt, and Th. Rasing, “Optical second harmonic generation study of vicinal Si(111) surfaces,” Surf. Sci. 251, 467–471 (1991).
[Crossref]

G. Lüpke, D. J. Bottomley, and H. M. van Driel, “Si/SiO2interfacial structure on vicinal Si(100) studied with second-harmonic generation,” Phys. Rev. B 47, 10389–10394 (1993).
[Crossref]

J. E. Sipe, D. J. Moss, and H. M. van Driel, “Phenomenological theory of optical second- and third-harmonic generation from cubic centrosymmetric crystals,” Phys. Rev. B 35, 1129–1141 (1987), and references therein.
[Crossref]

P. Guyot-Sionnest and Y. R. Shen, “Bulk contribution in surface second-harmonic generation,” Phys. Rev. B 38, 7985–7989 (1988); R. W. J. Hollering and M. Barmentlo, “Symmetry analysis of vicinal (111) surfaces by optical second-harmonic generation,” Opt. Commun. 88, 141–145 (1992).
[Crossref]

A. Liebsch and W. L. Schaich, “Second-harmonic generation at simple metal surfaces,” Phys. Rev. B 40, 5401–5410 (1989).
[Crossref]

Phys. Rev. Lett. (3)

S. Janz, D. J. Bottomley, H. M. van Driel, and R. S. Timsit, “Influence of steps on second-harmonic generation from vicinal metal surfaces,” Phys. Rev. Lett. 66, 1201–1204 (1991).
[Crossref] [PubMed]

O. L. Alerhand, A. N. Berker, J. D. Joannopoulos, D. Vanderbilt, R. J. Hamers, and J. E. Demuth, “Finite-temperature phase diagram of vicinal Si(100) surfaces,” Phys. Rev. Lett. 64, 2406–2409 (1990).
[Crossref] [PubMed]

H. W. K. Tom, T. F. Heinz, and Y. R. Shen, “Second-harmonic reflection from silicon surfaces and its relation to structural symmetry,” Phys. Rev. Lett. 51, 1983–1986 (1983).
[Crossref]

Prog. Surf. Sci. (1)

G. L. Richmond, J. M. Robinson, and V. L. Shannon, “Second harmonic generation studies of interfacial structure and dynamics,” Prog. Surf. Sci. 28, 1–70 (1988).
[Crossref]

Science (1)

E. D. Williams and N. C. Bartelt, “Thermodynamics of surface morphology,” Science 251, 393–400 (1991).
[Crossref] [PubMed]

Surf. Sci. (2)

R. Kaplan, “LEED study of the stepped surfaces of vicinal Si(100),” Surf. Sci. 93, 145–158 (1980).
[Crossref]

R. W. J. Hollering, D. Dijkkamp, and H. B. Elswijk, “Optical second-harmonic generation on a vicinal Si(111) surface,” Surf. Sci. 243, 121–126 (1991).
[Crossref]

Other (3)

One can derive the bulk angular functions Φn(αm) independently of the model developed here by expressing them as products of functions in sin(αm) and cos(αm) and considering where each angular function and its derivative vanish.

R. J. Pressley, ed., Handbook of Lasers (CRC, Cleveland, Ohio, 1971), p. 491; M. D. Levenson, Introduction to Nonlinear Laser Spectroscopy (Academic, New York, 1982), p. 121.

E. D. Palik, ed., Handbook of Optical Constants of Solids, (Academic, Orlando, Fla., 1985), p. 562.

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Figures (8)

Fig. 1
Fig. 1

Schematic of SHG and THG from the generic vicinal surface with C1υ symmetry. For details see text.

Fig. 2
Fig. 2

Diagram of the beam frame with respect to the cubic crystal’s coordinate system. For details see text.

Fig. 3
Fig. 3

Plots of the angular functions in Table 2 for the two distinct misorientations shown in Fig. 2.

Fig. 4
Fig. 4

I p , s ( 3 ω ) ( ψ ) at a fundamental wavelength of 765 nm for a, Si(111) misoriented by 5° toward [ 11 2 ¯ ]; b, Si(001) misoriented by 5° toward [110]. The points are data, and the curves are derived from theory.

Fig. 5
Fig. 5

I g , h ( 3 ω ) ( ψ ) at a fundamental wavelength of 765 nm from Si(111) misoriented by 5° toward [ 11 2 ¯ ] for the following polarization combinations: a, (p, p); b, (s, s); c, (s, p). The points are data, and the curves are derived from theory.

Fig. 6
Fig. 6

I g , h ( 2 ω ) ( ψ ) at a fundamental wavelength of 765 nm from Si(111) misoriented by 5° toward [ 11 2 ¯ ] for the following polarization combinations: a, (s, s); b, (p, p); c, (p, s); d, (s, p). The points are data, and the curves are best fits when complex Fourier coefficients up to fourfold in symmetry are used.

Fig. 7
Fig. 7

I g , p ( 2 ω ) ( ψ ) at a fundamental wavelength of 765 nm from Si(001) misoriented by 5° toward [110] for the following polarization combinations: a, (p, p); b, (s, p). The points are data, and the curves are best fits when Fourier coefficients up to fourfold in symmetry are used.

Fig. 8
Fig. 8

I g , p ( 2 ω ) ( ψ ) at a fundamental wavelength of 765 nm from clean Si(001) misoriented by 5° toward [110] for the following polarization combinations: a, (p, p); b, (s, p). The points are data, and the curves are best fits when Fourier coefficients up to fourfold in symmetry are used.

Tables (8)

Tables Icon

Table 1 A(g, h) and an(g, h) Combinations of Fresnel Factors for THG

Tables Icon

Table 2 Angular Functions Φn(αm)

Tables Icon

Table 3 A(g, h) and bn(g, h) Combinations of Fresnel Factors for Bulk SHG

Tables Icon

Table 4 s0,n(g, h) Combinations of Fresnel Factors for Surface SHG

Tables Icon

Table 5 s1,n(g, h) Additional Combinations of Fresnel Factors for Surface SHG

Tables Icon

Table 6 Normalized Complex Fourier Coefficients ci for NOS Si(111)5° Surfaces at λ = 765 nm

Tables Icon

Table 7 Deduced Nonlinear Optical Susceptibility Components at λ = 765 nm

Tables Icon

Table 8 Normalized Fourier Coefficients ci for NOS and Clean Si(001)5° Surfaces at λ = 765 nm

Equations (42)

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s ˆ = y ˆ ,
p ˆ = f s z ˆ + f c x ˆ ,
f s = sin θ 0 n ( ω ) ,
f c = ( 1 f s 2 ) 1 / 2
S ˆ = y ˆ ,
P ˆ = F s z ˆ F c x ˆ ,
F s = sin θ 0 n ( 2 ω ) ,
F c = ( 1 F s 2 ) 1 / 2 .
P i ( 3 ω ) = 0 χ i j k l ( 3 ) E j ( ω ) E k ( ω ) E l ( ω )
P i ( 2 ω ) = 0 Γ i j k l E j ( ω ) k E l ( ω ) .
P i ( 2 ω ) = 0 χ i j k ( 2 ) E j ( ω ) E k ( ω ) δ ( z + h ) .
P i ( 3 ω ) = 3 0 χ 1212 ( 3 ) E i ( ω ) | E ( ω ) | 2 + 0 [ χ 1111 ( 3 ) 3 χ 1212 ( 3 ) ] [ E i ( ω ) ] 3 .
χ 1111 ( 3 ) 3 χ 1212 ( 3 ) = σ χ 1111 ( 3 ) .
[ χ i j k l ( 3 ) ] = R i p ( α m ) R j q ( α m ) R k r ( α m ) R l s ( α m ) χ p q r s ( 3 ) ,
{ R i j ( α 1 ) } = [ cos α 1 0 sin α 1 0 1 0 sin α 1 0 cos α 1 ] .
{ R i j ( α 2 ) } = [ cos α 2 ( 2 ) 1 / 2 cos α 2 ( 2 ) 1 / 2 sin α 2 1 ( 2 ) 1 / 2 1 ( 2 ) 1 / 2 0 sin α 2 ( 2 ) 1 / 2 sin α 2 ( 2 ) 1 / 2 cos α 2 ] .
[ χ i j k l ( 3 ) ] = S i p ( ψ ) S j q ( ψ ) S k r ( ψ ) S l s ( ψ ) [ χ p q r s ( 3 ) ] ,
{ S i j ( ψ ) } = [ cos ψ sin ψ 0 sin ψ cos ψ 0 0 0 1 ] .
I g , g ( 3 ω ) ( ψ , α m ) | [ A ( g , g ) + n = 0 4 σ a n ( g , g ) Φ n ( α m ) cos ( n ψ ) ] [ E g ] 3 | 2 ,
I g , h ( 3 ω ) ( ψ , α m ) | n = 1 4 σ a n ( g , h ) Φ n ( α m ) sin ( n ψ ) [ E g ] 3 | 2 ,
P i ( 2 ω ) = 0 ( δ β 2 γ ) E j ( ω ) j E i ( ω ) + 0 β E i ( ω ) j E j ( ω ) + 0 γ i [ E j ( ω ) E j ( ω ) ] + 0 ζ E i ( ω ) i E i ( ω ) ,
δ = 2 Γ i i i i ,
β = 2 Γ i i j j ,
γ = Γ i j i j ,
ζ = δ β 2 γ 2 Γ i j j i .
E g , p bulk , ( 2 ω ) ( ψ , α m ) [ A ( g , p ) + K ζ n = 0 4 b n ( g , p ) Φ n ( α m ) cos ( n ψ ) ] [ E g ( ω ) ] 2 ,
E g , s bulk , ( 2 ω ) ( ψ , α m ) K ζ n = 1 4 b n ( g , s ) Φ n ( α m ) sin ( n ψ ) [ E g ( ω ) ] 2 ,
[ P x ( 2 ω ) P y ( 2 ω ) P z ( 2 ω ) ] = 0 [ 11 12 13 0 15 0 0 0 0 24 0 26 31 32 33 0 35 0 ] × [ E x ( ω ) 2 E y ( ω ) 2 E z ( ω ) 2 2 E y ( ω ) E z ( ω ) 2 E x ( ω ) E z ( ω ) 2 E x ( ω ) E y ( ω ) ] δ ( z + h ) ,
( 001 ) 15 = 24 , 31 = 32 , 33 ,
( 111 ) 11 = 12 = 26 , 15 = 24 , 31 = 32 , 33 ,
( 110 ) 15 , 24 , 31 , 32 , 33 .
[ χ i j k ( 2 ) ] = S i p ( ψ ) S j q ( ψ ) S k r ( ψ ) χ p q r ( 2 ) ,
E g , p surf , ( 2 ω ) ( ψ ) [ n = 0 3 s 0 , n ( g , p ) cos ( n ψ ) ] [ E g ( ω ) ] 2 ,
E g , s surf , ( 2 ω ) ( ψ ) [ n = 1 3 s 0 , n ( g , s ) sin ( n ψ ) ] [ E g ( ω ) ] 2 ,
11 ( 1 ) = 1 4 ( 3 11 + 12 + 2 26 ) ,
12 ( 1 ) = 1 4 ( 11 + 3 12 2 26 ) ,
11 ( 3 ) = 1 4 ( 11 12 2 26 ) .
I g , h ( 2 ω ) ( ψ , α m ) | E g , h bulk , ( 2 ω ) ( ψ , α m ) + E g , h surf , ( 2 ω ) ( ψ ) | 2 .
{ χ ˜ i j k ( 2 ) } = [ 11 ( 3 ) 11 ( 3 ) 0 0 15 0 0 0 0 24 0 11 ( 3 ) 31 32 33 0 0 0 ] .
[ χ ˜ i j k ( 2 ) ] = R i p ( α ) R j q ( α ) R k r ( α ) χ ˜ p q r ( 2 ) ,
E g , p surf , ( 2 ω ) ( ψ , α ) { n = 0 3 [ s 0 , n ( g , p ) + s 1 , n ( g , p ) tan ( α ) ] cos ( n ψ ) } [ E g ( ω ) ] 2 ,
E g , s s u r f , ( 2 ω ) ( ψ , α ) { n = 1 3 [ s 0 , n ( g , s ) + s 1 , n ( g , s ) tan ( α ) ] sin ( n ψ ) } [ E g ( ω ) ] 2 .

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