Abstract

The angular dependence of the sum-frequency signal radiated in a total internal reflection geometry is studied both theoretically and experimentally for a monolayer of octadecyl trichlorosilane chemisorbed upon CaF2. A predicted signal enhancement of 2 orders of magnitude is verified experimentally. The line amplitudes of the CH-stretching modes of the terminal methyl group depend strongly on the polarization geometry, the molecular hyperpolarizability, and the angles of incidence of the fundamental waves. By exploiting both the angular and the polarization dependence of the line amplitudes, one can obtain the molecular properties unambiguously. The hyperpolarizability tensor elements of the terminal methyl group are determined directly from the sum-frequency spectra.

© 1997 Optical Society of America

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1997 (1)

K. Wolfrum, J. Löbau, W. Birkhölzer, and A. Laubereau, Quantum Semiclassic. Opt. 9, 257 (1997).
[CrossRef]

1996 (4)

R. P. Chin, J. Y. Huang, Y. R. Shen, T. J. Chuang, and H. Seki, Phys. Rev. B 54, 8243 (1996).
[CrossRef]

J. Löbau, A. Rumphorst, K. Galla, S. Seeger, and K. Wolfrum, Thin Solid Films 289, 272 (1996).
[CrossRef]

K. B. Eisenthal, Chem. Rev. 96, 1343 (1996), and references therein.
[CrossRef] [PubMed]

J. C. Conboy, M. C. Messmer, and G. L. Richmond, J. Phys. Chem. 100, 7617 (1996).
[CrossRef]

1995 (4)

F. Geiger, R. Stolle, G. Marowsky, M. Palenberg, and B. U. Felderhof, Appl. Phys. B 61, 135 (1995).
[CrossRef]

F. Lindenberger, R. Stöckl, R. Laenen, and A. Laubereau, Opt. Commun. 117, 268 (1995).
[CrossRef]

A. Peremans, A. Tadjeddine, and P. Guyot-Sionnest, Chem. Phys. Lett. 247, 243 (1995).
[CrossRef]

M. E. Schmidt and P. Guyot-Sionnest, J. Chem. Phys. 104, 2438 (1995).
[CrossRef]

1994 (4)

Y. R. Shen, Appl. Phys. A 59, 541 (1994), and references therein.
[CrossRef]

J. C. Conboy, J. L. Daschbach, and G. L. Richmond, Appl. Phys. A 59, 623 (1994).
[CrossRef]

J. C. Conboy, J. L. Daschbach, and G. L. Richmond, J. Phys. Chem. 98, 9688 (1994).
[CrossRef]

K. Wolfrum and A. Laubereau, Chem. Phys. Lett. 228, 83 (1994).
[CrossRef]

1993 (2)

K. Wolfrum, H. Graener, and A. Laubereau, Chem. Phys. Lett. 213, 41 (1993).
[CrossRef]

K. Wolfrum, R. Laenen, and A. Laubereau, Opt. Commun. 97, 41 (1993).
[CrossRef]

1992 (5)

M. Morin, P. Jakob, N. J. Levinos, Y. J. Chabal, and A. L. Harris, J. Chem. Phys. 96, 6203 (1992).
[CrossRef]

M. Morin, N. J. Levinos, and A. L. Harris, J. Chem. Phys. 96, 3950 (1992).
[CrossRef]

C. Hirose, N. Akamatsu, and K. Domen, J. Chem. Phys. 96, 997 (1992).
[CrossRef]

C. Hirose, N. Akamatsu, and K. Domen, Appl. Spectrosc. 46, 1051 (1992).
[CrossRef]

S. R. Hatch, R. S. Polizzotti, S. Dougal, and P. Rabinowitz, Chem. Phys. Lett. 196, 97 (1992).
[CrossRef]

1991 (2)

P. Guyot-Sionnest, Phys. Rev. Lett. 66, 1489 (1991).
[CrossRef] [PubMed]

R. Superfine, J. Y. Huang, and Y. R. Shen, Phys. Rev. Lett. 66, 1066 (1991).
[CrossRef] [PubMed]

1988 (1)

P. Guyot-Sionnest, R. Superfine, J. H. Hunt, and Y. R. Shen, Chem. Phys. Lett. 144, 1 (1988).
[CrossRef]

1987 (4)

P. Guyot-Sionnest and Y. R. Shen, Phys. Rev. B 35, 4420 (1987).
[CrossRef]

B. U. Felderhof and G. Marowsky, Appl. Phys. B 43, 161 (1987).
[CrossRef]

B. U. Felderhof and G. Marowsky, Appl. Phys. B 44, 11 (1987).
[CrossRef]

P. Guyot-Sionnest, J. H. Hunt, and Y. R. Shen, Phys. Rev. Lett. 59, 1597 (1987).
[CrossRef] [PubMed]

1983 (1)

P. Ye and Y. R. Shen, Phys. Rev. B 28, 4288 (1983).
[CrossRef]

1980 (2)

F. J. Boerio and S. L. Chen, J. Colloid Interface Sci. 73, 176 (1980).
[CrossRef]

J. Sagiv, J. Am. Chem. Soc. 102, 92 (1980).
[CrossRef]

1972 (1)

M. J. Colles and J. E. Griffith, J. Chem. Phys. 56, 3384 (1972).
[CrossRef]

1963 (1)

J. H. Schachtschneider and R. G. Snyder, Spectrochim. Acta 19, 117 (1963).
[CrossRef]

1962 (1)

N. Bloembergen and P. S. Pershan, Phys. Rev. 128, 606 (1962).
[CrossRef]

Akamatsu, N.

C. Hirose, N. Akamatsu, and K. Domen, Appl. Spectrosc. 46, 1051 (1992).
[CrossRef]

C. Hirose, N. Akamatsu, and K. Domen, J. Chem. Phys. 96, 997 (1992).
[CrossRef]

Birkhölzer, W.

K. Wolfrum, J. Löbau, W. Birkhölzer, and A. Laubereau, Quantum Semiclassic. Opt. 9, 257 (1997).
[CrossRef]

Bloembergen, N.

N. Bloembergen and P. S. Pershan, Phys. Rev. 128, 606 (1962).
[CrossRef]

Boerio, F. J.

F. J. Boerio and S. L. Chen, J. Colloid Interface Sci. 73, 176 (1980).
[CrossRef]

Chabal, Y. J.

M. Morin, P. Jakob, N. J. Levinos, Y. J. Chabal, and A. L. Harris, J. Chem. Phys. 96, 6203 (1992).
[CrossRef]

Chen, S. L.

F. J. Boerio and S. L. Chen, J. Colloid Interface Sci. 73, 176 (1980).
[CrossRef]

Chin, R. P.

R. P. Chin, J. Y. Huang, Y. R. Shen, T. J. Chuang, and H. Seki, Phys. Rev. B 54, 8243 (1996).
[CrossRef]

Chuang, T. J.

R. P. Chin, J. Y. Huang, Y. R. Shen, T. J. Chuang, and H. Seki, Phys. Rev. B 54, 8243 (1996).
[CrossRef]

Colles, M. J.

M. J. Colles and J. E. Griffith, J. Chem. Phys. 56, 3384 (1972).
[CrossRef]

Conboy, J. C.

J. C. Conboy, M. C. Messmer, and G. L. Richmond, J. Phys. Chem. 100, 7617 (1996).
[CrossRef]

J. C. Conboy, J. L. Daschbach, and G. L. Richmond, Appl. Phys. A 59, 623 (1994).
[CrossRef]

J. C. Conboy, J. L. Daschbach, and G. L. Richmond, J. Phys. Chem. 98, 9688 (1994).
[CrossRef]

Daschbach, J. L.

J. C. Conboy, J. L. Daschbach, and G. L. Richmond, J. Phys. Chem. 98, 9688 (1994).
[CrossRef]

J. C. Conboy, J. L. Daschbach, and G. L. Richmond, Appl. Phys. A 59, 623 (1994).
[CrossRef]

Domen, K.

C. Hirose, N. Akamatsu, and K. Domen, J. Chem. Phys. 96, 997 (1992).
[CrossRef]

C. Hirose, N. Akamatsu, and K. Domen, Appl. Spectrosc. 46, 1051 (1992).
[CrossRef]

Dougal, S.

S. R. Hatch, R. S. Polizzotti, S. Dougal, and P. Rabinowitz, Chem. Phys. Lett. 196, 97 (1992).
[CrossRef]

Eisenthal, K. B.

K. B. Eisenthal, Chem. Rev. 96, 1343 (1996), and references therein.
[CrossRef] [PubMed]

Felderhof, B. U.

F. Geiger, R. Stolle, G. Marowsky, M. Palenberg, and B. U. Felderhof, Appl. Phys. B 61, 135 (1995).
[CrossRef]

B. U. Felderhof and G. Marowsky, Appl. Phys. B 43, 161 (1987).
[CrossRef]

B. U. Felderhof and G. Marowsky, Appl. Phys. B 44, 11 (1987).
[CrossRef]

Galla, K.

J. Löbau, A. Rumphorst, K. Galla, S. Seeger, and K. Wolfrum, Thin Solid Films 289, 272 (1996).
[CrossRef]

Geiger, F.

F. Geiger, R. Stolle, G. Marowsky, M. Palenberg, and B. U. Felderhof, Appl. Phys. B 61, 135 (1995).
[CrossRef]

Graener, H.

K. Wolfrum, H. Graener, and A. Laubereau, Chem. Phys. Lett. 213, 41 (1993).
[CrossRef]

Griffith, J. E.

M. J. Colles and J. E. Griffith, J. Chem. Phys. 56, 3384 (1972).
[CrossRef]

Guyot-Sionnest, P.

M. E. Schmidt and P. Guyot-Sionnest, J. Chem. Phys. 104, 2438 (1995).
[CrossRef]

A. Peremans, A. Tadjeddine, and P. Guyot-Sionnest, Chem. Phys. Lett. 247, 243 (1995).
[CrossRef]

P. Guyot-Sionnest, Phys. Rev. Lett. 66, 1489 (1991).
[CrossRef] [PubMed]

P. Guyot-Sionnest, R. Superfine, J. H. Hunt, and Y. R. Shen, Chem. Phys. Lett. 144, 1 (1988).
[CrossRef]

P. Guyot-Sionnest, J. H. Hunt, and Y. R. Shen, Phys. Rev. Lett. 59, 1597 (1987).
[CrossRef] [PubMed]

P. Guyot-Sionnest and Y. R. Shen, Phys. Rev. B 35, 4420 (1987).
[CrossRef]

Harris, A. L.

M. Morin, P. Jakob, N. J. Levinos, Y. J. Chabal, and A. L. Harris, J. Chem. Phys. 96, 6203 (1992).
[CrossRef]

M. Morin, N. J. Levinos, and A. L. Harris, J. Chem. Phys. 96, 3950 (1992).
[CrossRef]

Hatch, S. R.

S. R. Hatch, R. S. Polizzotti, S. Dougal, and P. Rabinowitz, Chem. Phys. Lett. 196, 97 (1992).
[CrossRef]

Hirose, C.

C. Hirose, N. Akamatsu, and K. Domen, Appl. Spectrosc. 46, 1051 (1992).
[CrossRef]

C. Hirose, N. Akamatsu, and K. Domen, J. Chem. Phys. 96, 997 (1992).
[CrossRef]

Huang, J. Y.

R. P. Chin, J. Y. Huang, Y. R. Shen, T. J. Chuang, and H. Seki, Phys. Rev. B 54, 8243 (1996).
[CrossRef]

R. Superfine, J. Y. Huang, and Y. R. Shen, Phys. Rev. Lett. 66, 1066 (1991).
[CrossRef] [PubMed]

Hunt, J. H.

P. Guyot-Sionnest, R. Superfine, J. H. Hunt, and Y. R. Shen, Chem. Phys. Lett. 144, 1 (1988).
[CrossRef]

P. Guyot-Sionnest, J. H. Hunt, and Y. R. Shen, Phys. Rev. Lett. 59, 1597 (1987).
[CrossRef] [PubMed]

Jakob, P.

M. Morin, P. Jakob, N. J. Levinos, Y. J. Chabal, and A. L. Harris, J. Chem. Phys. 96, 6203 (1992).
[CrossRef]

Laenen, R.

F. Lindenberger, R. Stöckl, R. Laenen, and A. Laubereau, Opt. Commun. 117, 268 (1995).
[CrossRef]

K. Wolfrum, R. Laenen, and A. Laubereau, Opt. Commun. 97, 41 (1993).
[CrossRef]

Laubereau, A.

K. Wolfrum, J. Löbau, W. Birkhölzer, and A. Laubereau, Quantum Semiclassic. Opt. 9, 257 (1997).
[CrossRef]

F. Lindenberger, R. Stöckl, R. Laenen, and A. Laubereau, Opt. Commun. 117, 268 (1995).
[CrossRef]

K. Wolfrum and A. Laubereau, Chem. Phys. Lett. 228, 83 (1994).
[CrossRef]

K. Wolfrum, R. Laenen, and A. Laubereau, Opt. Commun. 97, 41 (1993).
[CrossRef]

K. Wolfrum, H. Graener, and A. Laubereau, Chem. Phys. Lett. 213, 41 (1993).
[CrossRef]

Levinos, N. J.

M. Morin, N. J. Levinos, and A. L. Harris, J. Chem. Phys. 96, 3950 (1992).
[CrossRef]

M. Morin, P. Jakob, N. J. Levinos, Y. J. Chabal, and A. L. Harris, J. Chem. Phys. 96, 6203 (1992).
[CrossRef]

Lindenberger, F.

F. Lindenberger, R. Stöckl, R. Laenen, and A. Laubereau, Opt. Commun. 117, 268 (1995).
[CrossRef]

Löbau, J.

K. Wolfrum, J. Löbau, W. Birkhölzer, and A. Laubereau, Quantum Semiclassic. Opt. 9, 257 (1997).
[CrossRef]

J. Löbau, A. Rumphorst, K. Galla, S. Seeger, and K. Wolfrum, Thin Solid Films 289, 272 (1996).
[CrossRef]

Marowsky, G.

F. Geiger, R. Stolle, G. Marowsky, M. Palenberg, and B. U. Felderhof, Appl. Phys. B 61, 135 (1995).
[CrossRef]

B. U. Felderhof and G. Marowsky, Appl. Phys. B 43, 161 (1987).
[CrossRef]

B. U. Felderhof and G. Marowsky, Appl. Phys. B 44, 11 (1987).
[CrossRef]

Messmer, M. C.

J. C. Conboy, M. C. Messmer, and G. L. Richmond, J. Phys. Chem. 100, 7617 (1996).
[CrossRef]

Morin, M.

M. Morin, N. J. Levinos, and A. L. Harris, J. Chem. Phys. 96, 3950 (1992).
[CrossRef]

M. Morin, P. Jakob, N. J. Levinos, Y. J. Chabal, and A. L. Harris, J. Chem. Phys. 96, 6203 (1992).
[CrossRef]

Palenberg, M.

F. Geiger, R. Stolle, G. Marowsky, M. Palenberg, and B. U. Felderhof, Appl. Phys. B 61, 135 (1995).
[CrossRef]

Peremans, A.

A. Peremans, A. Tadjeddine, and P. Guyot-Sionnest, Chem. Phys. Lett. 247, 243 (1995).
[CrossRef]

Pershan, P. S.

N. Bloembergen and P. S. Pershan, Phys. Rev. 128, 606 (1962).
[CrossRef]

Polizzotti, R. S.

S. R. Hatch, R. S. Polizzotti, S. Dougal, and P. Rabinowitz, Chem. Phys. Lett. 196, 97 (1992).
[CrossRef]

Rabinowitz, P.

S. R. Hatch, R. S. Polizzotti, S. Dougal, and P. Rabinowitz, Chem. Phys. Lett. 196, 97 (1992).
[CrossRef]

Richmond, G. L.

J. C. Conboy, M. C. Messmer, and G. L. Richmond, J. Phys. Chem. 100, 7617 (1996).
[CrossRef]

J. C. Conboy, J. L. Daschbach, and G. L. Richmond, Appl. Phys. A 59, 623 (1994).
[CrossRef]

J. C. Conboy, J. L. Daschbach, and G. L. Richmond, J. Phys. Chem. 98, 9688 (1994).
[CrossRef]

Rumphorst, A.

J. Löbau, A. Rumphorst, K. Galla, S. Seeger, and K. Wolfrum, Thin Solid Films 289, 272 (1996).
[CrossRef]

Sagiv, J.

J. Sagiv, J. Am. Chem. Soc. 102, 92 (1980).
[CrossRef]

Schachtschneider, J. H.

J. H. Schachtschneider and R. G. Snyder, Spectrochim. Acta 19, 117 (1963).
[CrossRef]

Schmidt, M. E.

M. E. Schmidt and P. Guyot-Sionnest, J. Chem. Phys. 104, 2438 (1995).
[CrossRef]

Seeger, S.

J. Löbau, A. Rumphorst, K. Galla, S. Seeger, and K. Wolfrum, Thin Solid Films 289, 272 (1996).
[CrossRef]

Seki, H.

R. P. Chin, J. Y. Huang, Y. R. Shen, T. J. Chuang, and H. Seki, Phys. Rev. B 54, 8243 (1996).
[CrossRef]

Shen, Y. R.

R. P. Chin, J. Y. Huang, Y. R. Shen, T. J. Chuang, and H. Seki, Phys. Rev. B 54, 8243 (1996).
[CrossRef]

Y. R. Shen, Appl. Phys. A 59, 541 (1994), and references therein.
[CrossRef]

R. Superfine, J. Y. Huang, and Y. R. Shen, Phys. Rev. Lett. 66, 1066 (1991).
[CrossRef] [PubMed]

P. Guyot-Sionnest, R. Superfine, J. H. Hunt, and Y. R. Shen, Chem. Phys. Lett. 144, 1 (1988).
[CrossRef]

P. Guyot-Sionnest, J. H. Hunt, and Y. R. Shen, Phys. Rev. Lett. 59, 1597 (1987).
[CrossRef] [PubMed]

P. Guyot-Sionnest and Y. R. Shen, Phys. Rev. B 35, 4420 (1987).
[CrossRef]

P. Ye and Y. R. Shen, Phys. Rev. B 28, 4288 (1983).
[CrossRef]

Snyder, R. G.

J. H. Schachtschneider and R. G. Snyder, Spectrochim. Acta 19, 117 (1963).
[CrossRef]

Stöckl, R.

F. Lindenberger, R. Stöckl, R. Laenen, and A. Laubereau, Opt. Commun. 117, 268 (1995).
[CrossRef]

Stolle, R.

F. Geiger, R. Stolle, G. Marowsky, M. Palenberg, and B. U. Felderhof, Appl. Phys. B 61, 135 (1995).
[CrossRef]

Superfine, R.

R. Superfine, J. Y. Huang, and Y. R. Shen, Phys. Rev. Lett. 66, 1066 (1991).
[CrossRef] [PubMed]

P. Guyot-Sionnest, R. Superfine, J. H. Hunt, and Y. R. Shen, Chem. Phys. Lett. 144, 1 (1988).
[CrossRef]

Tadjeddine, A.

A. Peremans, A. Tadjeddine, and P. Guyot-Sionnest, Chem. Phys. Lett. 247, 243 (1995).
[CrossRef]

Wolfrum, K.

K. Wolfrum, J. Löbau, W. Birkhölzer, and A. Laubereau, Quantum Semiclassic. Opt. 9, 257 (1997).
[CrossRef]

J. Löbau, A. Rumphorst, K. Galla, S. Seeger, and K. Wolfrum, Thin Solid Films 289, 272 (1996).
[CrossRef]

K. Wolfrum and A. Laubereau, Chem. Phys. Lett. 228, 83 (1994).
[CrossRef]

K. Wolfrum, R. Laenen, and A. Laubereau, Opt. Commun. 97, 41 (1993).
[CrossRef]

K. Wolfrum, H. Graener, and A. Laubereau, Chem. Phys. Lett. 213, 41 (1993).
[CrossRef]

Ye, P.

P. Ye and Y. R. Shen, Phys. Rev. B 28, 4288 (1983).
[CrossRef]

Appl. Phys. A (2)

J. C. Conboy, J. L. Daschbach, and G. L. Richmond, Appl. Phys. A 59, 623 (1994).
[CrossRef]

Y. R. Shen, Appl. Phys. A 59, 541 (1994), and references therein.
[CrossRef]

Appl. Phys. B (3)

B. U. Felderhof and G. Marowsky, Appl. Phys. B 43, 161 (1987).
[CrossRef]

B. U. Felderhof and G. Marowsky, Appl. Phys. B 44, 11 (1987).
[CrossRef]

F. Geiger, R. Stolle, G. Marowsky, M. Palenberg, and B. U. Felderhof, Appl. Phys. B 61, 135 (1995).
[CrossRef]

Appl. Spectrosc. (1)

Chem. Phys. Lett. (5)

A. Peremans, A. Tadjeddine, and P. Guyot-Sionnest, Chem. Phys. Lett. 247, 243 (1995).
[CrossRef]

S. R. Hatch, R. S. Polizzotti, S. Dougal, and P. Rabinowitz, Chem. Phys. Lett. 196, 97 (1992).
[CrossRef]

K. Wolfrum and A. Laubereau, Chem. Phys. Lett. 228, 83 (1994).
[CrossRef]

P. Guyot-Sionnest, R. Superfine, J. H. Hunt, and Y. R. Shen, Chem. Phys. Lett. 144, 1 (1988).
[CrossRef]

K. Wolfrum, H. Graener, and A. Laubereau, Chem. Phys. Lett. 213, 41 (1993).
[CrossRef]

Chem. Rev. (1)

K. B. Eisenthal, Chem. Rev. 96, 1343 (1996), and references therein.
[CrossRef] [PubMed]

J. Am. Chem. Soc. (1)

J. Sagiv, J. Am. Chem. Soc. 102, 92 (1980).
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J. Chem. Phys. (5)

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Figures (7)

Fig. 1
Fig. 1

Infinitely thin nonlinear polarization sheet located at z=0-, close to the boundary between two centrosymmetric media a and b but embedded in medium b; noncollinear planar geometry of incident fundamental waves with k1a and k2a, and k3a of the reflected SF signal.

Fig. 2
Fig. 2

Schematic representation (planar projection) of the CH2 chain of OTS. The symmetry axis of the terminal methyl group defines the z axis, which is tilted toward the molecular axis by an angle of approximately 35.5°. Θ denotes the angle between the methyl symmetry axis and the surface normal, which is oriented along the z axis.

Fig. 3
Fig. 3

Real parts (dashed curves), imaginary parts (dotted curves), and absolute values (solid curves) of the linear Fresnel factors (a) f1,xxba, (b) f1,yyba, and (c) f1,zzba as functions of the angle of incidence ϑ1a.

Fig. 4
Fig. 4

Real parts (dashed curves), imaginary parts (dotted curves), and absolute values (solid curves) of the nonlinear Fresnel factors Fx, Fy, and Fz as functions of the angle of reflection ϑ3a.

Fig. 5
Fig. 5

SF spectra of OTS chemisorbed upon CaF2 taken in TIR geometry for (a) ssp, (b) sps, and (c) ppp polarization. Points, experimental; curves, calculated. The maxima in the SF spectra are assigned to the following modes: symmetric methylene stretch at 2850 cm-1, antisymmetric methylene stretch at 2910 cm-1, symmetric methyl stretch at 2875 cm-1, Fermi resonance of the symmetric methyl stretch with the combination tone of the methyl bending mode at 2936 cm-1, and twofold degenerate antisymmetric methyl stretching vibration at 2964 cm-1. 24

Fig. 6
Fig. 6

Absolute value squared of the CH3 s-stretch SF amplitude (y axes) versus internal angle of incidence ϑ2a for (a) ssp and (b) ppp polarization. Points, experimental; curves, calculated (Table 1).

Fig. 7
Fig. 7

Absolute value squared of the CH3 d-stretch SF amplitude (y axes) versus internal angle of incidence ϑ2a for (a) ssp, (b) sps and (c) ppp polarization. Points, experimental; curves, calculated (Table 1).

Tables (1)

Tables Icon

Table 1 Summary of Molecular Parameters Determined from the Angular Dependences Shown in Figs. 6 and 7

Equations (27)

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P(ω3)=χ(2)(ω1):Eb(ω2)Eb(ω1).
Eb(ωi)=Lb(ωi)fba(ωi)Ea(ωi),
fi,xxba=2nia cos ϑibnib cos ϑia+nia cos ϑib,
fi,yyba=2nia cos ϑianib cos ϑib+nia cos ϑia,
fi,zzba=2(nia/nib)2nib cos ϑianib cos ϑia+nia cos ϑib,
sin ϑ3a=ω2ω3n2an3asin ϑ2a+ω1ω3n1an3asin ϑ1a.
χzzz(2),χyyz(2)=χxxz(2),χyzy(2)=χxzx(2)=χzyy(2)=χzxx(2).
Pyssp(ω3)=χyyz(2)Eyb(ω2)Ezb(ω1),
Pysps(ω3)=χyzy(2)Ezb(ω2)Eyb(ω1),
Pxppp(ω3)=χxxz(2)Exb(ω2)Ezb(ω1)+χxzx(2)Ezb(ω2)Exb(ω1),
Pzppp(ω3)=χzzz(2)Ezb(ω2)Ezb(ω1)+χzxx(2)Exb(ω2)Exb(ω1).
Fx=cos ϑ3bn3a cos ϑ3b+n3an3b cos ϑ3a,
Fy=1n3a cos ϑ3a+n3b cos ϑ3b,
Fz=sin ϑ3an3an3b cos ϑ3b+(n3b)2 cos ϑ3a.
Issp(ω3)=2πcn3aω32|FyLyybPyssp(ω3)|2,
Isps(ω3)=2πcn3aω32|FyLyybPysps(ω3)|2,
Ippp(ω3)=2πcn3aω32|FxLxxbPxppp(ω3)+FzLzzbPzppp(ω3)|2.
|Ea(ωi)|2U(ωi)cos ϑiaAiτinia.
U(ω3)I(ω3)cos ϑ1a.
χijk(2)=χijknr+νχijk,ν(2),χijk,ν(2)=Aijk,νων-ω1-iΓν/2.
Ayyz,s=βzzz2Nπ[(γ+1)cos Θ0+(γ-1)cos3 Θ0],
Ayzy,s=βzzz2Nπ(1-γ)(cos Θ0-cos3 Θ0),
Azzz,s=βzzzNπ[γ cos Θ0-(γ-1)cos3 Θ0];
γ=βxxzβzzz.
Ayyz,d=-βzxx2Nπ(cos Θ0-cos3 Θ0),
Ayzy,d=βzxx2Nπ cos3 Θ0,
Azzz,d=βzxxNπ(cos Θ0-cos3 Θ0).

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