Abstract

Calculations of the nonlinear-optical behavior are developed for model composites of nanospheres with a metallic core and nonlinear shell or with a nonlinear core and metallic shell suspended in a nonlinear medium. Optical phase conjugation is shown to be enhanced from each nonlinear region because the optical field can be concentrated in both the interior and the exterior neighborhoods of the particle and magnified at the surface-mediated plasmon resonance. For the model composite with a metallic core, a limited range of resonance tunability can be achieved by adjustment of shell thickness; the frequency range is dependent on the dielectric dispersion of the metal. For the composite with a metallic shell instead of a metallic core, this restriction is reduced so that tunability from ultraviolet to infrared can be attained. Enhancement of the phase-conjugate signal is calculated for the electrostrictive mechanism dominant in the microsecond time scale and for the electronic mechanism dominant in the picosecond time scale. Calculations based on the dielectric functions for gold and for aluminum indicate that phase-conjugate reflectivity enhancements of 108 can be achieved. The imaginary components of the composite dielectric functions are shown to limit the magnitude of the field enhancement at the surface-plasmon resonance and determine the absorption and figure of merit of the composite.

© 1989 Optical Society of America

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References

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  1. R. K. Jain and R. C. Lind, J. Opt. Soc. Am. 73, 647 (1983).
    [Crossref]
  2. P. Roussignol, D. Ricard, J. Lukasik, and C. Flytzanis, J. Opt. Soc. Am. B 4, 5 (1987).
    [Crossref]
  3. M. A. Kramer, W. R. Tompkin, and R. W. Boyd, Phys. Rev. A 34, 2026 (1986).
    [Crossref] [PubMed]
  4. P. W. Smith, A. Ashkin, and W. J. Tomlinson, Opt. Lett. 6, 284 (1981).
    [Crossref] [PubMed]
  5. D. Rogovin and S. O. Sari, Phys. Rev. A 31, 2375 (1985).
    [Crossref] [PubMed]
  6. A. E. Neeves and M. H. Birnboim, J. Opt. Soc. Am. B 5, 701 (1988).
    [Crossref]
  7. D. Rogovin, Opt. News 12(9), 138 (1986).
  8. K. M. Leung, Opt. Lett. 7, 347 (1985).
    [Crossref]
  9. D. Ricard, P. Roussignol, and C. Flytzanis, Opt. Lett. 10, 511 (1985).
    [Crossref] [PubMed]
  10. D. Ricard, in Nonlinear Optics: Materials and Devices, C. Flytzanis and J. L. Oudar, eds. (Springer-Verlag, Berlin, 1986).
  11. F. Hache, D. Ricard, and C. Flytzanis, J. Opt. Soc. Am. B 3, 1647 (1986).
    [Crossref]
  12. A. Wokaun, Solid State Phys. 38, 223 (1984).
    [Crossref]
  13. J. W. Haus, R. Inguva, and C. M. Bowden, “Effective medium theory of nonlinear ellipsoidal composites,” submitted to Phys. Rev. A.
  14. A. E. Neeves and M. H. Birnboim, in Digest of Second Topical Meeting on Microphysics of Surfaces, Beams, and Adsorbates (Optical Society of America, Washington, D.C., 1988), p. 219.
  15. A. E. Neeves and M. H. Birnboim, Opt. Lett. 13, 1087 (1988).
    [Crossref] [PubMed]
  16. A. E. Neeves and M. H. Birnboim, “Composite structures for enhancement of nonlinear-optical materials over a wide frequency range. II. Metallic shell model,” submitted to Opt. Lett.
  17. J. A. A. J. Perenboom, P. Wyder, and F. Meier, Phys. Rev. B 7, 173 (1981).
  18. C. Kittel, Introduction to Solid State Physics (Wiley, New York, 1976).
  19. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), p. 258.
  20. E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, New York, 1985).
  21. P. B. Johnson and R. W. Christy, Phys. Rev. B 6, 4370 (1972).
    [Crossref]
  22. C. G. Granqvist, in Electrical Transport and Optical Properties of Inhomogeneous Media, J. C. Garland and D. B. Tanner, eds. (American Institute of Physics, New York1978).
  23. R. W. Hellwart, Prog. Quantum Electron. 5, 1 (1977).
    [Crossref]
  24. D. M. Pepper and A. Yariv, in Optical Phase Conjugation, R. A. Fisher, ed. (Academic, New York, 1983).
  25. J. W. Haus, N. Kalyaniwalla, R. Inguva, M. Bloemer, and C. M. Bowden, J. Opt. Soc. Am. B 6, 797 (1989).
    [Crossref]
  26. J. Haus, N. Kalyaniwalla, R. Ingawa, and C. M. Bowden, Opt. News 13(9), 116 (1987).

1989 (1)

1988 (2)

1987 (2)

P. Roussignol, D. Ricard, J. Lukasik, and C. Flytzanis, J. Opt. Soc. Am. B 4, 5 (1987).
[Crossref]

J. Haus, N. Kalyaniwalla, R. Ingawa, and C. M. Bowden, Opt. News 13(9), 116 (1987).

1986 (3)

F. Hache, D. Ricard, and C. Flytzanis, J. Opt. Soc. Am. B 3, 1647 (1986).
[Crossref]

M. A. Kramer, W. R. Tompkin, and R. W. Boyd, Phys. Rev. A 34, 2026 (1986).
[Crossref] [PubMed]

D. Rogovin, Opt. News 12(9), 138 (1986).

1985 (3)

1984 (1)

A. Wokaun, Solid State Phys. 38, 223 (1984).
[Crossref]

1983 (1)

1981 (2)

P. W. Smith, A. Ashkin, and W. J. Tomlinson, Opt. Lett. 6, 284 (1981).
[Crossref] [PubMed]

J. A. A. J. Perenboom, P. Wyder, and F. Meier, Phys. Rev. B 7, 173 (1981).

1977 (1)

R. W. Hellwart, Prog. Quantum Electron. 5, 1 (1977).
[Crossref]

1972 (1)

P. B. Johnson and R. W. Christy, Phys. Rev. B 6, 4370 (1972).
[Crossref]

Ashkin, A.

Birnboim, M. H.

A. E. Neeves and M. H. Birnboim, J. Opt. Soc. Am. B 5, 701 (1988).
[Crossref]

A. E. Neeves and M. H. Birnboim, Opt. Lett. 13, 1087 (1988).
[Crossref] [PubMed]

A. E. Neeves and M. H. Birnboim, “Composite structures for enhancement of nonlinear-optical materials over a wide frequency range. II. Metallic shell model,” submitted to Opt. Lett.

A. E. Neeves and M. H. Birnboim, in Digest of Second Topical Meeting on Microphysics of Surfaces, Beams, and Adsorbates (Optical Society of America, Washington, D.C., 1988), p. 219.

Bloemer, M.

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), p. 258.

Bowden, C. M.

J. W. Haus, N. Kalyaniwalla, R. Inguva, M. Bloemer, and C. M. Bowden, J. Opt. Soc. Am. B 6, 797 (1989).
[Crossref]

J. Haus, N. Kalyaniwalla, R. Ingawa, and C. M. Bowden, Opt. News 13(9), 116 (1987).

J. W. Haus, R. Inguva, and C. M. Bowden, “Effective medium theory of nonlinear ellipsoidal composites,” submitted to Phys. Rev. A.

Boyd, R. W.

M. A. Kramer, W. R. Tompkin, and R. W. Boyd, Phys. Rev. A 34, 2026 (1986).
[Crossref] [PubMed]

Christy, R. W.

P. B. Johnson and R. W. Christy, Phys. Rev. B 6, 4370 (1972).
[Crossref]

Flytzanis, C.

Granqvist, C. G.

C. G. Granqvist, in Electrical Transport and Optical Properties of Inhomogeneous Media, J. C. Garland and D. B. Tanner, eds. (American Institute of Physics, New York1978).

Hache, F.

Haus, J.

J. Haus, N. Kalyaniwalla, R. Ingawa, and C. M. Bowden, Opt. News 13(9), 116 (1987).

Haus, J. W.

J. W. Haus, N. Kalyaniwalla, R. Inguva, M. Bloemer, and C. M. Bowden, J. Opt. Soc. Am. B 6, 797 (1989).
[Crossref]

J. W. Haus, R. Inguva, and C. M. Bowden, “Effective medium theory of nonlinear ellipsoidal composites,” submitted to Phys. Rev. A.

Hellwart, R. W.

R. W. Hellwart, Prog. Quantum Electron. 5, 1 (1977).
[Crossref]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), p. 258.

Ingawa, R.

J. Haus, N. Kalyaniwalla, R. Ingawa, and C. M. Bowden, Opt. News 13(9), 116 (1987).

Inguva, R.

J. W. Haus, N. Kalyaniwalla, R. Inguva, M. Bloemer, and C. M. Bowden, J. Opt. Soc. Am. B 6, 797 (1989).
[Crossref]

J. W. Haus, R. Inguva, and C. M. Bowden, “Effective medium theory of nonlinear ellipsoidal composites,” submitted to Phys. Rev. A.

Jain, R. K.

Johnson, P. B.

P. B. Johnson and R. W. Christy, Phys. Rev. B 6, 4370 (1972).
[Crossref]

Kalyaniwalla, N.

J. W. Haus, N. Kalyaniwalla, R. Inguva, M. Bloemer, and C. M. Bowden, J. Opt. Soc. Am. B 6, 797 (1989).
[Crossref]

J. Haus, N. Kalyaniwalla, R. Ingawa, and C. M. Bowden, Opt. News 13(9), 116 (1987).

Kittel, C.

C. Kittel, Introduction to Solid State Physics (Wiley, New York, 1976).

Kramer, M. A.

M. A. Kramer, W. R. Tompkin, and R. W. Boyd, Phys. Rev. A 34, 2026 (1986).
[Crossref] [PubMed]

Leung, K. M.

Lind, R. C.

Lukasik, J.

Meier, F.

J. A. A. J. Perenboom, P. Wyder, and F. Meier, Phys. Rev. B 7, 173 (1981).

Neeves, A. E.

A. E. Neeves and M. H. Birnboim, Opt. Lett. 13, 1087 (1988).
[Crossref] [PubMed]

A. E. Neeves and M. H. Birnboim, J. Opt. Soc. Am. B 5, 701 (1988).
[Crossref]

A. E. Neeves and M. H. Birnboim, “Composite structures for enhancement of nonlinear-optical materials over a wide frequency range. II. Metallic shell model,” submitted to Opt. Lett.

A. E. Neeves and M. H. Birnboim, in Digest of Second Topical Meeting on Microphysics of Surfaces, Beams, and Adsorbates (Optical Society of America, Washington, D.C., 1988), p. 219.

Pepper, D. M.

D. M. Pepper and A. Yariv, in Optical Phase Conjugation, R. A. Fisher, ed. (Academic, New York, 1983).

Perenboom, J. A. A. J.

J. A. A. J. Perenboom, P. Wyder, and F. Meier, Phys. Rev. B 7, 173 (1981).

Ricard, D.

Rogovin, D.

D. Rogovin, Opt. News 12(9), 138 (1986).

D. Rogovin and S. O. Sari, Phys. Rev. A 31, 2375 (1985).
[Crossref] [PubMed]

Roussignol, P.

Sari, S. O.

D. Rogovin and S. O. Sari, Phys. Rev. A 31, 2375 (1985).
[Crossref] [PubMed]

Smith, P. W.

Tomlinson, W. J.

Tompkin, W. R.

M. A. Kramer, W. R. Tompkin, and R. W. Boyd, Phys. Rev. A 34, 2026 (1986).
[Crossref] [PubMed]

Wokaun, A.

A. Wokaun, Solid State Phys. 38, 223 (1984).
[Crossref]

Wyder, P.

J. A. A. J. Perenboom, P. Wyder, and F. Meier, Phys. Rev. B 7, 173 (1981).

Yariv, A.

D. M. Pepper and A. Yariv, in Optical Phase Conjugation, R. A. Fisher, ed. (Academic, New York, 1983).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (4)

Opt. Lett. (4)

Opt. News (2)

D. Rogovin, Opt. News 12(9), 138 (1986).

J. Haus, N. Kalyaniwalla, R. Ingawa, and C. M. Bowden, Opt. News 13(9), 116 (1987).

Phys. Rev. A (2)

D. Rogovin and S. O. Sari, Phys. Rev. A 31, 2375 (1985).
[Crossref] [PubMed]

M. A. Kramer, W. R. Tompkin, and R. W. Boyd, Phys. Rev. A 34, 2026 (1986).
[Crossref] [PubMed]

Phys. Rev. B (2)

J. A. A. J. Perenboom, P. Wyder, and F. Meier, Phys. Rev. B 7, 173 (1981).

P. B. Johnson and R. W. Christy, Phys. Rev. B 6, 4370 (1972).
[Crossref]

Prog. Quantum Electron. (1)

R. W. Hellwart, Prog. Quantum Electron. 5, 1 (1977).
[Crossref]

Solid State Phys. (1)

A. Wokaun, Solid State Phys. 38, 223 (1984).
[Crossref]

Other (9)

J. W. Haus, R. Inguva, and C. M. Bowden, “Effective medium theory of nonlinear ellipsoidal composites,” submitted to Phys. Rev. A.

A. E. Neeves and M. H. Birnboim, in Digest of Second Topical Meeting on Microphysics of Surfaces, Beams, and Adsorbates (Optical Society of America, Washington, D.C., 1988), p. 219.

C. Kittel, Introduction to Solid State Physics (Wiley, New York, 1976).

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), p. 258.

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

D. Ricard, in Nonlinear Optics: Materials and Devices, C. Flytzanis and J. L. Oudar, eds. (Springer-Verlag, Berlin, 1986).

D. M. Pepper and A. Yariv, in Optical Phase Conjugation, R. A. Fisher, ed. (Academic, New York, 1983).

C. G. Granqvist, in Electrical Transport and Optical Properties of Inhomogeneous Media, J. C. Garland and D. B. Tanner, eds. (American Institute of Physics, New York1978).

A. E. Neeves and M. H. Birnboim, “Composite structures for enhancement of nonlinear-optical materials over a wide frequency range. II. Metallic shell model,” submitted to Opt. Lett.

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Figures (5)

Fig. 1
Fig. 1

Metallic dielectric functions. (a) Bulk Au. Experimental data of Johnson and Christy,21 and theory is for the combined Drude–Lorentz model fit with parameters ωpf = 1.3 × 1016 sec−1, τf = 9.3 fsec, ωpb = 1.3 × 1016 sec−1, ω0 = 7.0 × 1015 sec−1, and τb = 0.2 fsec. (b) Bulk Al. Experimental data are from Ref. 20, and theory is for the Drude model fit with parameters ωpf = 2.28 × 1016 sec−1, τf = 6.9 fsec.

Fig. 2
Fig. 2

Magnitude of the electric-field ratio E1/E0 in the core region as a function of frequency. The cladding thickness parameter r1/r2 varies from 0.1 to 1.0: (a) Au core, (b) Al core, (c) Au shell, (d) Al shell. The metallic dielectric functions for Au and for Al are fitted to the Drude–Lorentz and Drude models, respectively, with the same parameters as in Fig. 1. For the dielectric shell or core, /0 = 2.52 and 3/0 = 1.76.

Fig. 3
Fig. 3

Spatial dependence of the electric-field ratio in a composite at resonance. The solid curves are for the θ = 0 (parallel to E0), and the dashed curves are for the θ = 90° (perpendicular to E0) components of the field. (a) Au core with r1/r2 = 0.5, 1/0 = −5.3 + i2.5, 2/0 = 2.52, 3/0 = 1.76, ωR = 4.05 × 1015 sec−1. (b) Al core with r1/r2 = 0.5, 1/0 = −2.5 + i0.042, 2/0 = 1.21, 3/0 = 1.76, ωR = 12.14 × 1015 sec−1. (c) Au shell with r1/r2 = 0.83, 2/0 = −16.77 + i1.98, 1/0 = 2.52, 3/0 = 1.76, ωR = 2.8 × 1015 sec−1. (d) Al shell with r1/r2 = 0.955, 2/0 = −65.1 + i3.4, 1/0 = 2.52, 3/0 = 1.76, ωR = 2.8 × 1015 sec−1.

Fig. 4
Fig. 4

Concentration dependence of the enhancement factors f4q for the phase-conjugate amplitude for the core, shell, and outer regions. The solid curves are for the clad particles, and the dashed curves are for the unclad (r1/r2 = 1.0) particles, with the same parameters as in Fig. 3: (a) Au core (unclad ωR = 4.3 × 1015 sec−1), (b) Al core (unclad ωR = 10.7 × 1015 sec−1), (c) Au shell (no unclad ωR), (d) Al shell (no unclad ωR).

Fig. 5
Fig. 5

Figure of merit as a function of frequency is proportional to the ratio of the enhancement factor f4q to molecular absorption coefficient γm. (a) Al core, 2/0 = 2.5 (solid curve), 2/0 = 2.5 + i0.25 (dashed curve), 3/0 = 1.76. (b) Al shell, 1/0 = 2.5 (solid curve), 1/0 = 2.5 + i0.25 (dashed curve), 3/0 = 1.76.

Tables (2)

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Table 1 Composite Model Calculations for Electrostrictive Mechanism in Polystyrene Compositea

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Table 2 Composite Model Calculations for Electronic Mechanism

Equations (56)

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E 1 = 9 2 3 2 a + 2 3 b E 0 [ cos ( Θ ) e ^ r - sin ( Θ ) e ^ θ ] .
E 2 = 3 3 2 a + 2 3 b [ ( 1 + 2 2 ) + 2 ( 1 - 2 ) ( r 1 / r ) 3 ] E 0 cos ( Θ ) e ^ r - 3 3 2 a + 2 3 b [ ( 1 + 2 2 ) - ( 1 - 2 ) ( r 1 / r ) 3 ] E 0 sin ( Θ ) e ^ θ ,
E 3 = ( 2 2 a - 3 b 2 a + 2 3 b r 2 3 r 3 + 1 ) E 0 cos ( Θ ) e ^ r + ( 2 a - 3 b 2 a + 2 3 b r 2 3 r 3 - 1 ) E 0 sin ( Θ ) e ^ θ ,
a 1 ( 3 - 2 P ) + 2 2 P ,
b 1 P + 2 ( 3 - P ) ,
P 1 - ( r 1 / r 2 ) 3 .
p = α E 0 ,
α 4 π 3 r 2 3 [ ( 2 a - 3 b ) / ( 2 a + 2 3 b ) ] .
Re ( 2 a + 2 3 b ) = 0
Re ( 2 a + 2 3 b ) = ( 1 2 - 1 2 ) ( 3 - 2 P ) + 2 P 1 3 + 2 P ( 2 2 - 2 2 ) + 2 2 3 ( 3 - P ) ,
Im ( 2 a + 2 3 b ) = 1 [ 3 2 - 2 P ( 2 - 3 ) ] + 2 [ 3 ( 1 + 2 3 ) + 4 P 2 - 2 P ( 1 + 3 ) ] ,
2 P 2 2 + 2 [ 3 ( 1 + 2 3 ) - 2 P ( 1 + 3 ) ] - 3 1 2 + 2 P ( 1 3 + 1 2 - 2 2 ) = 0.
1 = - 2 3             for P = 0
1 = - 2 2             for P = 1.
P 2 = - 3 / 2 ( 1 + 2 3 )             for P = 0
2 = - 2 3             for P = 1.
f ( ω ) = 0 ( 1 - ω pf 2 1 ω 2 + i ω γ f ) ,
ω pf 2 = N f e 2 / e 0 m 0 ,
1 / τ f = 1 / τ 0 + 2 V f / d ,
V f 2 = ( h / 2 π m ) 2 ( 3 π 2 N f ) 2 / 3 ,
b ( ω ) = 0 ( 1 + ω pb 2 1 ω 0 2 - ω 2 - i ω γ b ) ,
ω pb 2 = N b e 2 / 0 m 0 ,
( ω ) = 0 ( 1 - ω pf 2 1 ω 2 + i ω γ f + ω pb 2 1 ω 0 2 - ω 2 + i ω γ b ) .
˜ = 3 + 3 ρ 3 2 a - 3 b 2 a + 2 3 b .
γ m = ( 2 π / λ 0 n ˜ ) ˜ / 0 .
γ m = ( 6 π n ˜ / λ 0 ) ρ Re ( 2 a - 3 b ) Im ( 2 a + 2 3 b ) ,
γ m = ( 6 π n ˜ / λ 0 ) ρ × ( 1 2 - 1 2 ) ( 3 - 2 P ) + 2 ( 2 2 - 2 2 ) P - 3 [ 1 P + 2 ( 3 - P ) ] 1 [ 2 ( 3 - 2 P ) + 2 3 P ] + 2 [ 1 ( 3 - 2 P ) + 4 2 P + 2 3 ( 3 - P ) ] .
γ m = ( 6 π n ˜ / λ 0 ) ρ 9 1 / ( 1 + 2 P ) .
γ s = 32 π 4 n ˜ 4 λ 0 4 ρ r 2 3 | 2 a - 3 b 2 a + 2 3 b | 2 .
γ s = 32 π 4 n ˜ 4 λ 0 4 ρ r 2 3 ( { ( 1 2 - 1 2 ) ( 3 - 2 P ) + 2 ( 2 2 - 2 2 ) P - 3 [ 1 P + 2 ( 3 - P ) ] 1 [ 2 ( 3 - 2 P ) + 2 3 P ] + 2 [ 1 ( 3 - 2 P ) + 4 2 P + 2 3 ( 3 - P ) ] } 2 +     { ( 1 2 + 1 2 ) ( 3 - 2 P ) + 4 2 2 - 3 [ 1 P + 2 ( 3 - P ) ] 1 [ 2 ( 3 - 2 P ) + 2 3 P ] + 2 [ 1 ( 3 - 2 P ) + 4 2 P + 2 3 ( 3 - P ) ] } 2 ) .
γ = γ m + γ s .
E 1 / E 0 = 3 3 / 1 ,
E 1 / E 0 = 9 2 3 / [ 1 ( 1 + 2 3 ) ] ,
E 1 / E 0 = 3 e [ ( 2 / 2 ) 2 + 1 ] 1 / 2 / ( 1 + 2 3 ) ,
E 1 / E 0 = 3 3 / 2 ,
χ eff ( 3 ) = 12 π 3 k b T ρ r 2 3 ( 2 a - 3 b 2 a + 2 3 b ) 2 ,
D = ˜ ( E ) E = 0 E + P ( E ) ,
P ( E ) = 0 χ ( 1 ) E + 0 χ ( 2 ) E 2 + 0 χ ( 3 ) E 3 ,
˜ ( E ) = 0 + 0 χ ( 1 ) + 0 χ ( 2 ) E + 0 χ ( 3 ) E 2 ,
P NL = δ ˜ ( E ) E ,
δ ˜ ( E ) = 0 χ ( 2 ) E + 0 χ ( 3 ) E 2 .
δ ˜ ( E ) = ˜ q δ q ( E ) ,
δ q ( E ) = 0 χ q ( 3 ) E q 2             for q = 1 , 2 , 3 ,
P NL = 0 q = 1 3 f 1 q 2 ¯ f 2 q f 3 ¯ χ q ( 3 ) E 0 3 ,
E q f 1 q ( r ) E 0 ,
f 2 q = ˜ q ,
E ˜ f ¯ 3 E 0 .
˜ 1 = 3 ρ 3 2 2 2 9 - 9 P + P 2 ( 2 a + 2 3 b ) 2 ,
˜ 2 = 3 ρ 3 2 P 1 2 ( 3 - 2 P ) + 2 1 2 P + 2 2 ( 3 - P ) ( 2 a + 2 3 b ) 2 ,
˜ 3 = 1 + ρ 2 2 a 2 - 2 3 2 b 2 - 2 2 3 a b ( 2 a + 2 3 b ) 2 .
P NL = 0 q f 1 q 2 ¯ f 2 q f 3 ¯ χ q ( 3 ) E f E b E p * .
P NL = 0 χ eff ( 3 ) E f E b E p * ,
f 4 q f 1 q 2 ¯ f 2 q f 3 ¯ ,
χ eff ( 3 ) f 4 q χ q ( 3 ) .
η η 2 / γ = 12 π / n 0 χ ( 3 ) / γ .
R = 0.131 η 2 ( I 2 / n 2 λ 2 ) ,

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