Abstract

We demonstrate how the phenomenological constants usually used to describe the nonlinear surface and bulk polarization associated with optical second-harmonic generation from cubic [(100) crystal face] and isotropic centrosymmetric metals can be determined by combining the results of experiments in which the p-polarized second-harmonic intensity is determined as a function of (i) the angle of incidence, (ii) the (linear) polarization direction of the fundamental beam, and for single crystals (iii) the orientation of the crystal. Experimental results for polycrystalline Cu and Al and for single-crystalline Al with a (100) face are reported. Experimental results for hexagonal single-crystalline Zn giving the intensity of the p-polarized second-harmonic generation as a function of (i) the state of polarization of the incident light and (ii) the rotational angle of the crystal, which has the c axis in the surface plane, are presented.

© 1989 Optical Society of America

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References

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  1. T. A. Driscoll and D. Guidotti, “Symmetry analysis of second-harmonic generation in silicon,” Phys. Rev. B28, 1171–1173 (1983).
  2. 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]
  3. J. A. Litwin, J. E. Sipe, and H. M. van Driel, “Picosecond and nanosecond laser-induced second-harmonic generation from centrosymmetric semiconductors,” Phys. Rev. B 31, 5543–5546 (1985).
    [Crossref]
  4. 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]
  5. 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]
  6. P. S. Pershan, “Nonlinear optical properties of solids: energy considerations,” Phys. Rev. 130, 919–929 (1963).
    [Crossref]
  7. F. Brown, R. E. Parks, and A. M. Sleeper, “Nonlinear optical reflection from a metallic boundary,” Phys. Rev. Lett. 14, 1029–1031 (1965).
    [Crossref]
  8. F. Brown and R. E. Parks, “Magnetic-dipole contribution to optical harmonics in silver,” Phys. Rev. Lett. 16, 507–509 (1966).
    [Crossref]
  9. N. Bloembergen, R. K. Chang, and C. H. Lee, “Second-harmonic generation of light in reflection from media with inversion symmetry,” Phys. Rev. Lett. 16, 986–989 (1966).
    [Crossref]
  10. H. Sonnenberg and H. Heffner, “Experimental study of optical second-harmonic generation in silver,” J. Opt. Soc. Am. 58, 209–212 (1968).
    [Crossref]
  11. 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).
    [Crossref]
  12. A. Liebsch, “Second-harmonic generation of simple metal surfaces,” Phys. Rev. Lett. 61, 1233–1236 (1988).
    [Crossref] [PubMed]
  13. F. Forstmann and R. R. Gerhardts, Metal Optics near the Plasma Frequency, Vol. 109 of Springer Tracts in Modern Physics, G. Höhler, ed. (Springer-Verlag, Berlin, 1986).
    [Crossref]
  14. P. Guyot-Sionnest, W. Chen, and Y. R. Shen, “General considerations on optical second-harmonic generation from surfaces and interfaces,” Phys. Rev. B 33, 8254–8263 (1986).
    [Crossref]
  15. O. Keller, “Moment expansion of the optical second-harmonic response tensor of condensed media,” Phys. Status Solidi B (to be published).
  16. E. Palik, ed., Handbook of Optical Constants of Solids (Academic, New York, 1985).
  17. M. Weber and A. Liebsch, “Density-functional approach to second-harmonic generation at metal surfaces,” Phys. Rev. B 35, 7411–7416 (1987).
    [Crossref]
  18. J. C. Quail and H. J. Simon, “Second-harmonic generation from silver and aluminum films in total internal reflection,” Phys. Rev. B 31, 4900–4905 (1985).
    [Crossref]
  19. K. Pedersen and O. Keller, “Photoelastic properties of metals measured by off-null ellipsometry,” J. Appl. Opt. 25, 226–234 (1986).
    [Crossref]

1988 (1)

A. Liebsch, “Second-harmonic generation of simple metal surfaces,” Phys. Rev. Lett. 61, 1233–1236 (1988).
[Crossref] [PubMed]

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).
[Crossref]

M. Weber and A. Liebsch, “Density-functional approach to second-harmonic generation at metal surfaces,” Phys. Rev. B 35, 7411–7416 (1987).
[Crossref]

1986 (3)

K. Pedersen and O. Keller, “Photoelastic properties of metals measured by off-null ellipsometry,” J. Appl. Opt. 25, 226–234 (1986).
[Crossref]

P. Guyot-Sionnest, W. Chen, and Y. R. Shen, “General considerations on optical second-harmonic generation from surfaces and interfaces,” Phys. Rev. B 33, 8254–8263 (1986).
[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]

1985 (2)

J. A. Litwin, J. E. Sipe, and H. M. van Driel, “Picosecond and nanosecond laser-induced second-harmonic generation from centrosymmetric semiconductors,” Phys. Rev. B 31, 5543–5546 (1985).
[Crossref]

J. C. Quail and H. J. Simon, “Second-harmonic generation from silver and aluminum films in total internal reflection,” Phys. Rev. B 31, 4900–4905 (1985).
[Crossref]

1983 (2)

T. A. Driscoll and D. Guidotti, “Symmetry analysis of second-harmonic generation in silicon,” Phys. Rev. B28, 1171–1173 (1983).

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]

1968 (2)

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]

H. Sonnenberg and H. Heffner, “Experimental study of optical second-harmonic generation in silver,” J. Opt. Soc. Am. 58, 209–212 (1968).
[Crossref]

1966 (2)

F. Brown and R. E. Parks, “Magnetic-dipole contribution to optical harmonics in silver,” Phys. Rev. Lett. 16, 507–509 (1966).
[Crossref]

N. Bloembergen, R. K. Chang, and C. H. Lee, “Second-harmonic generation of light in reflection from media with inversion symmetry,” Phys. Rev. Lett. 16, 986–989 (1966).
[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]

1963 (1)

P. S. Pershan, “Nonlinear optical properties of solids: energy considerations,” Phys. Rev. 130, 919–929 (1963).
[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]

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]

N. Bloembergen, R. K. Chang, and C. H. Lee, “Second-harmonic generation of light in reflection from media with inversion symmetry,” Phys. Rev. Lett. 16, 986–989 (1966).
[Crossref]

Brown, F.

F. Brown and R. E. Parks, “Magnetic-dipole contribution to optical harmonics in silver,” Phys. Rev. Lett. 16, 507–509 (1966).
[Crossref]

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

N. Bloembergen, R. K. Chang, and C. H. Lee, “Second-harmonic generation of light in reflection from media with inversion symmetry,” Phys. Rev. Lett. 16, 986–989 (1966).
[Crossref]

Chen, W.

P. Guyot-Sionnest, W. Chen, and Y. R. Shen, “General considerations on optical second-harmonic generation from surfaces and interfaces,” Phys. Rev. B 33, 8254–8263 (1986).
[Crossref]

Driscoll, T. A.

T. A. Driscoll and D. Guidotti, “Symmetry analysis of second-harmonic generation in silicon,” Phys. Rev. B28, 1171–1173 (1983).

Forstmann, F.

F. Forstmann and R. R. Gerhardts, Metal Optics near the Plasma Frequency, Vol. 109 of Springer Tracts in Modern Physics, G. Höhler, ed. (Springer-Verlag, Berlin, 1986).
[Crossref]

Gerhardts, R. R.

F. Forstmann and R. R. Gerhardts, Metal Optics near the Plasma Frequency, Vol. 109 of Springer Tracts in Modern Physics, G. Höhler, ed. (Springer-Verlag, Berlin, 1986).
[Crossref]

Guidotti, D.

T. A. Driscoll and D. Guidotti, “Symmetry analysis of second-harmonic generation in silicon,” Phys. Rev. B28, 1171–1173 (1983).

Guyot-Sionnest, P.

P. Guyot-Sionnest, W. Chen, and Y. R. Shen, “General considerations on optical second-harmonic generation from surfaces and interfaces,” Phys. Rev. B 33, 8254–8263 (1986).
[Crossref]

Heffner, H.

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]

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).
[Crossref]

Keller, O.

K. Pedersen and O. Keller, “Photoelastic properties of metals measured by off-null ellipsometry,” J. Appl. Opt. 25, 226–234 (1986).
[Crossref]

O. Keller, “Moment expansion of the optical second-harmonic response tensor of condensed media,” Phys. Status Solidi B (to be published).

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]

N. Bloembergen, R. K. Chang, and C. H. Lee, “Second-harmonic generation of light in reflection from media with inversion symmetry,” Phys. Rev. Lett. 16, 986–989 (1966).
[Crossref]

Liebsch, A.

A. Liebsch, “Second-harmonic generation of simple metal surfaces,” Phys. Rev. Lett. 61, 1233–1236 (1988).
[Crossref] [PubMed]

M. Weber and A. Liebsch, “Density-functional approach to second-harmonic generation at metal surfaces,” Phys. Rev. B 35, 7411–7416 (1987).
[Crossref]

Litwin, J. A.

J. A. Litwin, J. E. Sipe, and H. M. van Driel, “Picosecond and nanosecond laser-induced second-harmonic generation from centrosymmetric semiconductors,” Phys. Rev. B 31, 5543–5546 (1985).
[Crossref]

Moss, D. J.

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).
[Crossref]

Parks, R. E.

F. Brown and R. E. Parks, “Magnetic-dipole contribution to optical harmonics in silver,” Phys. Rev. Lett. 16, 507–509 (1966).
[Crossref]

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

Pedersen, K.

K. Pedersen and O. Keller, “Photoelastic properties of metals measured by off-null ellipsometry,” J. Appl. Opt. 25, 226–234 (1986).
[Crossref]

Pershan, P. S.

P. S. Pershan, “Nonlinear optical properties of solids: energy considerations,” Phys. Rev. 130, 919–929 (1963).
[Crossref]

Quail, J. C.

J. C. Quail and H. J. Simon, “Second-harmonic generation from silver and aluminum films in total internal reflection,” Phys. Rev. B 31, 4900–4905 (1985).
[Crossref]

Shen, Y. R.

P. Guyot-Sionnest, W. Chen, and Y. R. Shen, “General considerations on optical second-harmonic generation from surfaces and interfaces,” Phys. Rev. B 33, 8254–8263 (1986).
[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]

Simon, H. J.

J. C. Quail and H. J. Simon, “Second-harmonic generation from silver and aluminum films in total internal reflection,” Phys. Rev. B 31, 4900–4905 (1985).
[Crossref]

Sipe, J. E.

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).
[Crossref]

J. A. Litwin, J. E. Sipe, and H. M. van Driel, “Picosecond and nanosecond laser-induced second-harmonic generation from centrosymmetric semiconductors,” Phys. Rev. B 31, 5543–5546 (1985).
[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]

Sonnenberg, H.

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, 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.

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).
[Crossref]

J. A. Litwin, J. E. Sipe, and H. M. van Driel, “Picosecond and nanosecond laser-induced second-harmonic generation from centrosymmetric semiconductors,” Phys. Rev. B 31, 5543–5546 (1985).
[Crossref]

Weber, M.

M. Weber and A. Liebsch, “Density-functional approach to second-harmonic generation at metal surfaces,” Phys. Rev. B 35, 7411–7416 (1987).
[Crossref]

J. Appl. Opt. (1)

K. Pedersen and O. Keller, “Photoelastic properties of metals measured by off-null ellipsometry,” J. Appl. Opt. 25, 226–234 (1986).
[Crossref]

J. Opt. Soc. Am. (1)

Phys. Rev. (2)

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]

P. S. Pershan, “Nonlinear optical properties of solids: energy considerations,” Phys. Rev. 130, 919–929 (1963).
[Crossref]

Phys. Rev. B (7)

T. A. Driscoll and D. Guidotti, “Symmetry analysis of second-harmonic generation in silicon,” Phys. Rev. B28, 1171–1173 (1983).

J. A. Litwin, J. E. Sipe, and H. M. van Driel, “Picosecond and nanosecond laser-induced second-harmonic generation from centrosymmetric semiconductors,” Phys. Rev. B 31, 5543–5546 (1985).
[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]

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).
[Crossref]

P. Guyot-Sionnest, W. Chen, and Y. R. Shen, “General considerations on optical second-harmonic generation from surfaces and interfaces,” Phys. Rev. B 33, 8254–8263 (1986).
[Crossref]

M. Weber and A. Liebsch, “Density-functional approach to second-harmonic generation at metal surfaces,” Phys. Rev. B 35, 7411–7416 (1987).
[Crossref]

J. C. Quail and H. J. Simon, “Second-harmonic generation from silver and aluminum films in total internal reflection,” Phys. Rev. B 31, 4900–4905 (1985).
[Crossref]

Phys. Rev. Lett. (5)

A. Liebsch, “Second-harmonic generation of simple metal surfaces,” Phys. Rev. Lett. 61, 1233–1236 (1988).
[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]

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

F. Brown and R. E. Parks, “Magnetic-dipole contribution to optical harmonics in silver,” Phys. Rev. Lett. 16, 507–509 (1966).
[Crossref]

N. Bloembergen, R. K. Chang, and C. H. Lee, “Second-harmonic generation of light in reflection from media with inversion symmetry,” Phys. Rev. Lett. 16, 986–989 (1966).
[Crossref]

Other (3)

F. Forstmann and R. R. Gerhardts, Metal Optics near the Plasma Frequency, Vol. 109 of Springer Tracts in Modern Physics, G. Höhler, ed. (Springer-Verlag, Berlin, 1986).
[Crossref]

O. Keller, “Moment expansion of the optical second-harmonic response tensor of condensed media,” Phys. Status Solidi B (to be published).

E. Palik, ed., Handbook of Optical Constants of Solids (Academic, New York, 1985).

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Figures (7)

Fig. 1
Fig. 1

Intensity of p-polarized SHG from polycrystalline Cu in arbitrary units as a function of the polarization direction (ϕ) of the incident light for the angles of incidence (a) θ = 45° and (b) θ = 60°. The results are normalized to the same height at ϕ = 0°. The plotted curves represent the ϕ dependence of Eq. (2.6) fitted to the experimental results using (a) |A|2 = 3.31, BR = 1.35, BI = −2.66 and (b) |A|2 = 2.25, BR = 0.77, BI = −5.57.

Fig. 2
Fig. 2

Fitting parameters |A|2 (arbitrary units), BR, and BI for polycrystalline Cu obtained at different angles of incidence (θ). The two solid curves represent the θ dependence of Eq. (2.8) based on the appropriate ratios of the phenomenological constants obtained when BI is assumed to be negative. The dashed curve is the θ dependence of BR of Eq. (2.8) obtained when BI is assumed positive.

Fig. 3
Fig. 3

Intensity of p-polarized SHG from (a) polycrystalline and (b) single-crystalline [(100) face] Al as a function of the direction of polarization (ϕ) of the incident light at an angle of incidence θ = 45°. The crystal is oriented so that a 〈100〉 direction falls in the plane of incidence. The intensities, given in arbitrary units, are normalized to the same height at ϕ = 0°. The plotted curves represent the ϕ dependence of (a) Eq. (3.16) with D = 12.3, E = 15.9 and (b) Eq. (2.6) with BR = −0.12, BI = −6.93.

Fig. 4
Fig. 4

Intensity of p-polarized SHG from an Al single crystal with a (100) surface as a function of the rotational angle (ψ) of the crystal obtained at an angle of incidence θ = 45° from (a) a p-polarized and (b) an s-polarized fundamental beam. For ψ = 0°, the 〈100〉 direction is parallel to the plane of incidence. The curves plotted represent (a) Eq. (2.10) with bp,p = 0.3 + i0.0 and (b) Eq. (2.11) with bs,p = −0.4 + i0.0.

Fig. 5
Fig. 5

Fitting parameters |A|2 (arbitrary units), BR, and BI for a single crystal of Al obtained at different angles of incidence (θ). The crystal has a (100) face and is oriented so that a 〈100〉 direction lies in the plane of incidence. The two solid curves represent the θ dependence of Eq. (2.8) based on the appropriate ratios of the phenomenological constants obtained when BI is assumed to be negative. The dashed curve is that obtained for BR when BI is assumed positive.

Fig. 6
Fig. 6

Intensity (in arbitrary units) of p-polarized SHG from a single crystal of Zn oriented with the c axis parallel to the surface as a function of the rotational angle (ψ) of the crystal. For ψ = 0°, the c axis is parallel to the plane of incidence. The results are obtained at an angle of incidence θ = 45° with (a) p-polarized and (b) s-polarized incident light.

Fig. 7
Fig. 7

Intensity of p-polarized SHG from Zn as a function of the direction of polarization (ϕ) of the incident light for three different orientations (ψ) of the crystal. For ψ = 0°, the c axis is parallel to the plane of incidence. The two curves plotted represent Eq. (2.6) fitted to the experimental results using BR = 0.66 and BI = 0.0 for ψ = 0° and BR = 1.9 and BI = 0.0 for ψ = 90°.

Tables (1)

Tables Icon

Table 1 Phenomenological Constant for the (100) Crystal Face of Al Determined Relative to γ + N231 at a Fundamental Wavelength 1.06 μm

Equations (17)

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P i ( 2 ω ) = ( δ β 2 γ ) ( E ) E i + β E i ( E ) + γ i ( E E ) + ξ E i i E i ,
P i S ( 2 ω ) = j , k Δ ijk E j E k δ ( z 0 ) ,
Δ ( 100 ) = [ 0 0 0 0 15 0 0 0 0 15 0 0 31 31 33 0 0 0 ] ,
E p , s ( 2 ω ) = α 1 p , s U 2 + α 2 p , s V 2 + α 3 p , s UV ,
E p ( 2 ω ) = A ( U 2 + B V 2 ) ,
| E p ( 2 ω ) | 2 = | A | 2 [ sin 4 ϕ + ( B R 2 + B I 2 ) cos 4 ϕ + 2 B R sin 2 ϕ cos 2 ϕ ] ,
A = 8 π i ( ω c 0 ) F s N cos θ + F c ( 2 cos θ cos θ + n f c ) 2 ( γ + N 2 31 ) E 0 2
B = ( cos θ + n f c n cos θ + f c ) 2 [ γ + N 2 31 f c 2 γ + N 2 31 + ξ γ + N 2 31 × n f s f c ( F c f c + F s f s ) 2 F s ( n f c + N F c ) + f s F s N 2 F s f s 33 2 F c f c 15 γ + N 2 31 ] ,
q = i ( R i | E p ( 2 ω ) ( ϕ i ) | 2 ) 2 ,
| E ( 2 ω ) ( p , p ) | 2 = | E p 2 K a p , p | 2 [ 1 + | b p , p | 2 cos 2 4 Ψ + 2 Re ( b p , p ) cos 4 Ψ ]
| E ( 2 ω ) ( s , p ) | 2 = | E s 2 K a s , p | 2 [ 1 + | b s , p | 2 cos 2 4 Ψ + 2 Re ( b s , p ) cos 4 Ψ ]
a p , p = ξ n f s ( 3 F c f c 2 + 4 F s f s f c ) 8 ( n f c + N F c ) + F s ( γ + N 2 31 ) + N 2 F s f s 2 ( 33 31 ) 2 f s f c F c 15 ,
a s , p = ξ n f s F c 8 ( n f c + N F c ) + F s ( γ + N 2 31 ) ,
b p , p = 1 a p , p ξ n f s F c f c 2 8 ( n f c + N F c ) ,
b s , p = 1 a s , p ξ n f s F c 8 ( n f c + N F c ) ,
K = 8 π i ω c 0 1 N cos θ + F c .
| E p ( 2 ω ) ( ϕ ) | 2 = C ( sin 4 ϕ + D cos 4 ϕ + E cos 2 ϕ sin 2 ϕ ) ,

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