Abstract

We present a new formulation of Maker fringes in parallel-surface films, using self-consistent boundary conditions for reflections and allowing for any degree of refractive-index dispersion. This treatment of the second-harmonic reflections and dispersion, unlike a number of previous derivations, leads correctly to the expected form for the effective second-harmonic d coefficients. Complete expressions with physically meaningful factors are given for the generated second-harmonic power for either absorbing or birefringent films including reflections for the case of no pump depletion. A comparison with the isotropic approximation is given, and practical considerations in the use of these expressions for the fitting of experimental data are discussed.

© 1995 Optical Society of America

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  1. P. D. Maker, R. W. Terhune, M. Nisenhoff, and C. M. Savage, Phys. Rev. Lett. 8, 21 (1962).
    [CrossRef]
  2. J. Jerphagnon and S. K. Kurtz, "Maker fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals," J. Appl. Phys. 41, 1667 (1970).
    [CrossRef]
  3. N. Okamoto, Y. Hirano, and O. Sugihara, "Precise estimation of nonlinear-optical coefficients for anisotropic nonlinear films with C∞v symmetry," J. Opt. Soc. Am. B 9, 2083 (1992).
    [CrossRef]
  4. M. Eich, B. Reck, D. Y. Yoon, C. G. Willson, and G. C. Bjorklund, "Novel second-order nonlinear optical polymers via chemical cross-linking-induced vitrification under electric field," J. Appl. Phys. 66, 3241 (1989).
    [CrossRef]
  5. C. H. Wang and H. W. Guan, "Second harmonic generation and optical anisotropy of a spin cast polymer film," J. Polym. Sci. Part B 31, 1983 (1993).
    [CrossRef]
  6. M. G. Kuzyk, K. D. Singer, H. E. Zahn, and L. A. King, "Second-order nonlinear-optical tensor properties of poled films under stress," J. Opt. Soc. Am. B 6, 742 (1989).
    [CrossRef]
  7. N. Bloembergen and P. S. Pershan, "Light waves at the boundary of nonlinear media," Phys. Rev. 128, 606 (1962).
    [CrossRef]
  8. D. Chemla and P. Kupecek, "Analyse des expériences de génération de second harmonique," Rev. Phys. Appl. 6, 31 (1971).
    [CrossRef]
  9. L. M. Hayden, G. F. Sauter, F. R. Ore, P. L. Pasillas, J. M. Hoover, G. A. Lindsay, and R. A. Henry, "Second-order nonlinear optical measurements in guest-host and side-chain polymers," J. Appl. Phys. 68, 456 (1990).
    [CrossRef]
  10. D. J. Williams, "Nonlinear optical properties of guest-host polymer structures," in Nonlinear Optical Properties of Organic Molecules and Crystals, D. S. Chemla and J. Zyss, eds. (Academic, Orlando, Fla., 1987), p. 405.
    [CrossRef]
  11. J. L. Oudar, "Optical nonlinearities of conjugated molecules. Stilbene derivatives and highly polar aromatic compounds," J. Chem. Phys. 67, 446 (1977).
    [CrossRef]
  12. The anisotropic bound wave is also discussed in Ref. 8.
  13. In Ref. 3 Eq. (18) must be corrected so that the denominator divides only the second term in the numerator.
  14. A. Nahata, J. Shan, J. T. Yardley, and C. Wu, "Electro-optic determination of the nonlinear-optical properties of a covalently functionalized Disperse Red 1 copolymer," J. Opt. Soc. Am. B 10, 1553 (1993). [The same material with a different dye content, Poly(DR1-MMA) CAS #119989-05-8, can be purchased from IBM Almaden Research Center, San Jose, Calif.; attn.: Dan Dawson, phone (408) 927-1617.]
    [CrossRef]
  15. D. Jungbauer, I. Teraoka, D. Y. Yoon, B. Reck, J. D. Swalen, R. Tweig, and C. G. Willson, "Second-order nonlinear optical properties and relaxation characteristics of poled linear epoxy polymers with tolane chromophores," J. Appl. Phys. 69, 8011 (1991).
    [CrossRef]
  16. See, for example, M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1975), Sec. 14.2.
  17. N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1965), Chap. 4.
  18. See Sec. 14.6 of Ref. 16.
  19. D. Neher, A. Wolf, C. Bubeck, and G. Wegner, "Third-harmonic generation in polyphenylacetylene: exact determination of nonlinear optical susceptibilities in ultrathin films," Chem. Phys. Lett. 163, 116 (1989).
    [CrossRef]

1993

1992

1991

D. Jungbauer, I. Teraoka, D. Y. Yoon, B. Reck, J. D. Swalen, R. Tweig, and C. G. Willson, "Second-order nonlinear optical properties and relaxation characteristics of poled linear epoxy polymers with tolane chromophores," J. Appl. Phys. 69, 8011 (1991).
[CrossRef]

1990

L. M. Hayden, G. F. Sauter, F. R. Ore, P. L. Pasillas, J. M. Hoover, G. A. Lindsay, and R. A. Henry, "Second-order nonlinear optical measurements in guest-host and side-chain polymers," J. Appl. Phys. 68, 456 (1990).
[CrossRef]

1989

M. Eich, B. Reck, D. Y. Yoon, C. G. Willson, and G. C. Bjorklund, "Novel second-order nonlinear optical polymers via chemical cross-linking-induced vitrification under electric field," J. Appl. Phys. 66, 3241 (1989).
[CrossRef]

M. G. Kuzyk, K. D. Singer, H. E. Zahn, and L. A. King, "Second-order nonlinear-optical tensor properties of poled films under stress," J. Opt. Soc. Am. B 6, 742 (1989).
[CrossRef]

D. Neher, A. Wolf, C. Bubeck, and G. Wegner, "Third-harmonic generation in polyphenylacetylene: exact determination of nonlinear optical susceptibilities in ultrathin films," Chem. Phys. Lett. 163, 116 (1989).
[CrossRef]

1977

J. L. Oudar, "Optical nonlinearities of conjugated molecules. Stilbene derivatives and highly polar aromatic compounds," J. Chem. Phys. 67, 446 (1977).
[CrossRef]

1971

D. Chemla and P. Kupecek, "Analyse des expériences de génération de second harmonique," Rev. Phys. Appl. 6, 31 (1971).
[CrossRef]

1970

J. Jerphagnon and S. K. Kurtz, "Maker fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals," J. Appl. Phys. 41, 1667 (1970).
[CrossRef]

1962

N. Bloembergen and P. S. Pershan, "Light waves at the boundary of nonlinear media," Phys. Rev. 128, 606 (1962).
[CrossRef]

Bjorklund, G. C.

M. Eich, B. Reck, D. Y. Yoon, C. G. Willson, and G. C. Bjorklund, "Novel second-order nonlinear optical polymers via chemical cross-linking-induced vitrification under electric field," J. Appl. Phys. 66, 3241 (1989).
[CrossRef]

Bloembergen, N.

N. Bloembergen and P. S. Pershan, "Light waves at the boundary of nonlinear media," Phys. Rev. 128, 606 (1962).
[CrossRef]

N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1965), Chap. 4.

Born, M.

See, for example, M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1975), Sec. 14.2.

Bubeck, C.

D. Neher, A. Wolf, C. Bubeck, and G. Wegner, "Third-harmonic generation in polyphenylacetylene: exact determination of nonlinear optical susceptibilities in ultrathin films," Chem. Phys. Lett. 163, 116 (1989).
[CrossRef]

Chemla, D.

D. Chemla and P. Kupecek, "Analyse des expériences de génération de second harmonique," Rev. Phys. Appl. 6, 31 (1971).
[CrossRef]

Eich, M.

M. Eich, B. Reck, D. Y. Yoon, C. G. Willson, and G. C. Bjorklund, "Novel second-order nonlinear optical polymers via chemical cross-linking-induced vitrification under electric field," J. Appl. Phys. 66, 3241 (1989).
[CrossRef]

Guan, H. W.

C. H. Wang and H. W. Guan, "Second harmonic generation and optical anisotropy of a spin cast polymer film," J. Polym. Sci. Part B 31, 1983 (1993).
[CrossRef]

Hayden, L. M.

L. M. Hayden, G. F. Sauter, F. R. Ore, P. L. Pasillas, J. M. Hoover, G. A. Lindsay, and R. A. Henry, "Second-order nonlinear optical measurements in guest-host and side-chain polymers," J. Appl. Phys. 68, 456 (1990).
[CrossRef]

Henry, R. A.

L. M. Hayden, G. F. Sauter, F. R. Ore, P. L. Pasillas, J. M. Hoover, G. A. Lindsay, and R. A. Henry, "Second-order nonlinear optical measurements in guest-host and side-chain polymers," J. Appl. Phys. 68, 456 (1990).
[CrossRef]

Hirano, Y.

Hoover, J. M.

L. M. Hayden, G. F. Sauter, F. R. Ore, P. L. Pasillas, J. M. Hoover, G. A. Lindsay, and R. A. Henry, "Second-order nonlinear optical measurements in guest-host and side-chain polymers," J. Appl. Phys. 68, 456 (1990).
[CrossRef]

Jerphagnon, J.

J. Jerphagnon and S. K. Kurtz, "Maker fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals," J. Appl. Phys. 41, 1667 (1970).
[CrossRef]

Jungbauer, D.

D. Jungbauer, I. Teraoka, D. Y. Yoon, B. Reck, J. D. Swalen, R. Tweig, and C. G. Willson, "Second-order nonlinear optical properties and relaxation characteristics of poled linear epoxy polymers with tolane chromophores," J. Appl. Phys. 69, 8011 (1991).
[CrossRef]

King, L. A.

Kupecek, P.

D. Chemla and P. Kupecek, "Analyse des expériences de génération de second harmonique," Rev. Phys. Appl. 6, 31 (1971).
[CrossRef]

Kurtz, S. K.

J. Jerphagnon and S. K. Kurtz, "Maker fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals," J. Appl. Phys. 41, 1667 (1970).
[CrossRef]

Kuzyk, M. G.

Lindsay, G. A.

L. M. Hayden, G. F. Sauter, F. R. Ore, P. L. Pasillas, J. M. Hoover, G. A. Lindsay, and R. A. Henry, "Second-order nonlinear optical measurements in guest-host and side-chain polymers," J. Appl. Phys. 68, 456 (1990).
[CrossRef]

Maker, P. D.

P. D. Maker, R. W. Terhune, M. Nisenhoff, and C. M. Savage, Phys. Rev. Lett. 8, 21 (1962).
[CrossRef]

Nahata, A.

Neher, D.

D. Neher, A. Wolf, C. Bubeck, and G. Wegner, "Third-harmonic generation in polyphenylacetylene: exact determination of nonlinear optical susceptibilities in ultrathin films," Chem. Phys. Lett. 163, 116 (1989).
[CrossRef]

Nisenhoff, M.

P. D. Maker, R. W. Terhune, M. Nisenhoff, and C. M. Savage, Phys. Rev. Lett. 8, 21 (1962).
[CrossRef]

Okamoto, N.

Ore, F. R.

L. M. Hayden, G. F. Sauter, F. R. Ore, P. L. Pasillas, J. M. Hoover, G. A. Lindsay, and R. A. Henry, "Second-order nonlinear optical measurements in guest-host and side-chain polymers," J. Appl. Phys. 68, 456 (1990).
[CrossRef]

Oudar, J. L.

J. L. Oudar, "Optical nonlinearities of conjugated molecules. Stilbene derivatives and highly polar aromatic compounds," J. Chem. Phys. 67, 446 (1977).
[CrossRef]

Pasillas, P. L.

L. M. Hayden, G. F. Sauter, F. R. Ore, P. L. Pasillas, J. M. Hoover, G. A. Lindsay, and R. A. Henry, "Second-order nonlinear optical measurements in guest-host and side-chain polymers," J. Appl. Phys. 68, 456 (1990).
[CrossRef]

Pershan, P. S.

N. Bloembergen and P. S. Pershan, "Light waves at the boundary of nonlinear media," Phys. Rev. 128, 606 (1962).
[CrossRef]

Reck, B.

D. Jungbauer, I. Teraoka, D. Y. Yoon, B. Reck, J. D. Swalen, R. Tweig, and C. G. Willson, "Second-order nonlinear optical properties and relaxation characteristics of poled linear epoxy polymers with tolane chromophores," J. Appl. Phys. 69, 8011 (1991).
[CrossRef]

M. Eich, B. Reck, D. Y. Yoon, C. G. Willson, and G. C. Bjorklund, "Novel second-order nonlinear optical polymers via chemical cross-linking-induced vitrification under electric field," J. Appl. Phys. 66, 3241 (1989).
[CrossRef]

Sauter, G. F.

L. M. Hayden, G. F. Sauter, F. R. Ore, P. L. Pasillas, J. M. Hoover, G. A. Lindsay, and R. A. Henry, "Second-order nonlinear optical measurements in guest-host and side-chain polymers," J. Appl. Phys. 68, 456 (1990).
[CrossRef]

Savage, C. M.

P. D. Maker, R. W. Terhune, M. Nisenhoff, and C. M. Savage, Phys. Rev. Lett. 8, 21 (1962).
[CrossRef]

Shan, J.

Singer, K. D.

Sugihara, O.

Swalen, J. D.

D. Jungbauer, I. Teraoka, D. Y. Yoon, B. Reck, J. D. Swalen, R. Tweig, and C. G. Willson, "Second-order nonlinear optical properties and relaxation characteristics of poled linear epoxy polymers with tolane chromophores," J. Appl. Phys. 69, 8011 (1991).
[CrossRef]

Teraoka, I.

D. Jungbauer, I. Teraoka, D. Y. Yoon, B. Reck, J. D. Swalen, R. Tweig, and C. G. Willson, "Second-order nonlinear optical properties and relaxation characteristics of poled linear epoxy polymers with tolane chromophores," J. Appl. Phys. 69, 8011 (1991).
[CrossRef]

Terhune, R. W.

P. D. Maker, R. W. Terhune, M. Nisenhoff, and C. M. Savage, Phys. Rev. Lett. 8, 21 (1962).
[CrossRef]

Tweig, R.

D. Jungbauer, I. Teraoka, D. Y. Yoon, B. Reck, J. D. Swalen, R. Tweig, and C. G. Willson, "Second-order nonlinear optical properties and relaxation characteristics of poled linear epoxy polymers with tolane chromophores," J. Appl. Phys. 69, 8011 (1991).
[CrossRef]

Wang, C. H.

C. H. Wang and H. W. Guan, "Second harmonic generation and optical anisotropy of a spin cast polymer film," J. Polym. Sci. Part B 31, 1983 (1993).
[CrossRef]

Wegner, G.

D. Neher, A. Wolf, C. Bubeck, and G. Wegner, "Third-harmonic generation in polyphenylacetylene: exact determination of nonlinear optical susceptibilities in ultrathin films," Chem. Phys. Lett. 163, 116 (1989).
[CrossRef]

Williams, D. J.

D. J. Williams, "Nonlinear optical properties of guest-host polymer structures," in Nonlinear Optical Properties of Organic Molecules and Crystals, D. S. Chemla and J. Zyss, eds. (Academic, Orlando, Fla., 1987), p. 405.
[CrossRef]

Willson, C. G.

D. Jungbauer, I. Teraoka, D. Y. Yoon, B. Reck, J. D. Swalen, R. Tweig, and C. G. Willson, "Second-order nonlinear optical properties and relaxation characteristics of poled linear epoxy polymers with tolane chromophores," J. Appl. Phys. 69, 8011 (1991).
[CrossRef]

M. Eich, B. Reck, D. Y. Yoon, C. G. Willson, and G. C. Bjorklund, "Novel second-order nonlinear optical polymers via chemical cross-linking-induced vitrification under electric field," J. Appl. Phys. 66, 3241 (1989).
[CrossRef]

Wolf, A.

D. Neher, A. Wolf, C. Bubeck, and G. Wegner, "Third-harmonic generation in polyphenylacetylene: exact determination of nonlinear optical susceptibilities in ultrathin films," Chem. Phys. Lett. 163, 116 (1989).
[CrossRef]

Wolf, E.

See, for example, M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1975), Sec. 14.2.

Wu, C.

Yardley, J. T.

Yoon, D. Y.

D. Jungbauer, I. Teraoka, D. Y. Yoon, B. Reck, J. D. Swalen, R. Tweig, and C. G. Willson, "Second-order nonlinear optical properties and relaxation characteristics of poled linear epoxy polymers with tolane chromophores," J. Appl. Phys. 69, 8011 (1991).
[CrossRef]

M. Eich, B. Reck, D. Y. Yoon, C. G. Willson, and G. C. Bjorklund, "Novel second-order nonlinear optical polymers via chemical cross-linking-induced vitrification under electric field," J. Appl. Phys. 66, 3241 (1989).
[CrossRef]

Zahn, H. E.

Chem. Phys. Lett.

D. Neher, A. Wolf, C. Bubeck, and G. Wegner, "Third-harmonic generation in polyphenylacetylene: exact determination of nonlinear optical susceptibilities in ultrathin films," Chem. Phys. Lett. 163, 116 (1989).
[CrossRef]

J. Appl. Phys.

J. Jerphagnon and S. K. Kurtz, "Maker fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals," J. Appl. Phys. 41, 1667 (1970).
[CrossRef]

M. Eich, B. Reck, D. Y. Yoon, C. G. Willson, and G. C. Bjorklund, "Novel second-order nonlinear optical polymers via chemical cross-linking-induced vitrification under electric field," J. Appl. Phys. 66, 3241 (1989).
[CrossRef]

L. M. Hayden, G. F. Sauter, F. R. Ore, P. L. Pasillas, J. M. Hoover, G. A. Lindsay, and R. A. Henry, "Second-order nonlinear optical measurements in guest-host and side-chain polymers," J. Appl. Phys. 68, 456 (1990).
[CrossRef]

D. Jungbauer, I. Teraoka, D. Y. Yoon, B. Reck, J. D. Swalen, R. Tweig, and C. G. Willson, "Second-order nonlinear optical properties and relaxation characteristics of poled linear epoxy polymers with tolane chromophores," J. Appl. Phys. 69, 8011 (1991).
[CrossRef]

J. Chem. Phys.

J. L. Oudar, "Optical nonlinearities of conjugated molecules. Stilbene derivatives and highly polar aromatic compounds," J. Chem. Phys. 67, 446 (1977).
[CrossRef]

J. Opt. Soc. Am. B

J. Polym. Sci. Part B

C. H. Wang and H. W. Guan, "Second harmonic generation and optical anisotropy of a spin cast polymer film," J. Polym. Sci. Part B 31, 1983 (1993).
[CrossRef]

Phys. Rev.

N. Bloembergen and P. S. Pershan, "Light waves at the boundary of nonlinear media," Phys. Rev. 128, 606 (1962).
[CrossRef]

Rev. Phys. Appl.

D. Chemla and P. Kupecek, "Analyse des expériences de génération de second harmonique," Rev. Phys. Appl. 6, 31 (1971).
[CrossRef]

Other

D. J. Williams, "Nonlinear optical properties of guest-host polymer structures," in Nonlinear Optical Properties of Organic Molecules and Crystals, D. S. Chemla and J. Zyss, eds. (Academic, Orlando, Fla., 1987), p. 405.
[CrossRef]

P. D. Maker, R. W. Terhune, M. Nisenhoff, and C. M. Savage, Phys. Rev. Lett. 8, 21 (1962).
[CrossRef]

The anisotropic bound wave is also discussed in Ref. 8.

In Ref. 3 Eq. (18) must be corrected so that the denominator divides only the second term in the numerator.

See, for example, M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1975), Sec. 14.2.

N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1965), Chap. 4.

See Sec. 14.6 of Ref. 16.

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Figures (11)

Fig. 1
Fig. 1

Three-layer slab geometry for SHG that is due to a nonlinear layer in region II with the origin at the center of the nonlinear film.

Fig. 2
Fig. 2

Maker fringe pattern predicted by JK and HH for a 300-μm-thick piece of X-cut quartz. Rotation is about the z axis.

Fig. 3
Fig. 3

Ratio of the nonlinear coefficients as obtained by the JK and HH methods.

Fig. 4
Fig. 4

Effect of dispersion on the ratio d31/d33 as calculated with the JK method. (d31/d33)HH is assumed to be 1/3.

Fig. 5
Fig. 5

Maker fringe patterns for a representative absorbing NLO poled polymer.

Fig. 6
Fig. 6

Effect of absorption on the determination of d33 in a NLO polymer. n1 = 1.566, n2 = 1.780, coherence length 1.24 μm.

Fig. 7
Fig. 7

Ratio of the predicted d33 and of the predicted film thickness for the cases of absorption and no absorption. The results are obtained from fits to Eq. (8) (with α2 = 0) for d33 and the thickness.

Fig. 8
Fig. 8

Maker fringe plots for 1000-μm-thick X-cut quartz.

Fig. 9
Fig. 9

Effect of birefringence on the calculation of d33 in a poled polymer. Extraordinary indices were used for the isotropic case.

Fig. 10
Fig. 10

Ratio of the predicted d33 and film thickness for the isotropic and the birefringent cases. The results are obtained from fits to Eq. (12) for d33 (crosses) and d33 and the thickness (filled diamonds and open boxes).

Fig. 11
Fig. 11

Definition of the walk-off angle γ in a birefringent medium.

Equations (55)

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E 1 υ = ê υ E 1 exp [ i ( q 1 r ω t ) ] ,
Region I : E 2 = R ê p r exp { i q 2 r [ r + ( L / 2 ) ] } , H 2 = ( i c / 2 ω ) × E = R ê s r exp { i q 2 r [ r + ( L / 2 ) ] } ,
Region II : E 2 = e b exp ( i 2 k 1 r ) + A ê 2 p exp ( i k 2 r ) + B ê 2 p r exp ( i k 2 r r ) , H 2 = h b exp ( i 2 k 1 r ) + n 2 A ê 2 s × exp ( i k 2 r ) + n 2 B ê 2 s r exp ( i k 2 r r ) ,
Region III : E 2 = T ê 2 p sub exp [ i k 2 sub ( r L 2 ) ] , H 2 = n 2 s T ê 2 s sub × exp [ i k 2 sub ( r L 2 ) ] ,
e b = 4 π n 1 2 n 2 2 [ P NL k b ( k b P NL ) | k 2 | 2 ] , k b = 2 k 1 . h b = n 1 k ˆ 1 × e b , P NL | E 1 | 2 d : ê 1 ê 1 .
P 2 ω ( γ p ) = 128 π 3 c A [ t af ( 1 γ ) ] 4 [ t fs ( 2 p ) ] 2 [ t sa ( 2 p ) ] 2 n 2 2 c 2 2 P ω 2 ( 2 π L λ ) 2 d eff 2 ( sin 2 Ψ Ψ 2 + [ r af ( 2 p ) ] 2 R 2 sin 2 Φ Φ 2 2 r af ( 2 p ) R sin Ψ Ψ sin Φ Φ cos 2 ϕ 2 ) ( 1 + [ r af ( 2 p ) r fs ( 2 p ) ] 2 + 2 r af ( 2 p ) r fs ( 2 p ) cos 4 ϕ 2 ) ,
( d 31 d 33 ) JK = [ 2 c 1 ( n 2 c 2 n 1 c 1 ) n 1 s 1 2 + ( d 33 d 31 ) HH ] 1 .
p 2 ω ( r p ) = 128 π c A [ t af ( 1 γ ) ] 4 [ t fs ( 2 p ) ] 2 [ t sa ( 2 p ) ] 2 n 2 2 c 2 2 P ω 2 ( 2 π L λ ) 2 d eff 2 × exp [ 2 ( δ 1 + δ 2 ) ] sin 2 Ψ + sinh 2 χ Ψ 2 + χ 2 ,
i = [ n io 2 0 0 0 n io 2 0 0 0 n ie 2 ] , i = 1 , 2 ,
e b = 4 π n 2 2 ( θ 1 ) n 1 2 ( θ 1 ) n 2 2 ( θ 1 ) [ ( ) 1 P NL k b ( k b P NL ) ( 2 ω c n 2 o n 2 e ) 2 ] ,
n i ( θ i ) = ( cos 2 θ i n io 2 + sin 2 θ i n ie 2 ) 1 / 2 , i = 1 , 2 , n 2 ( θ 1 ) = ( cos 2 θ 1 n 2 o 2 + sin 2 θ 1 n 2 e 2 ) 1 / 2 .
P 2 ω ( γ p ) = 128 π 3 c A [ t af ( 1 γ ) ] 4 [ t fs ( 2 p ) ] 2 [ t sa ( 2 p ) ] 2 n 2 2 ( θ 2 ) cos 2 γ 2 cos 2 ( θ 2 γ 2 ) P ω 2 ( 2 π L λ ) 2 × d eff 2 [ n 2 ( θ 1 ) n 2 o ] 4 [ n 1 2 ( θ 1 ) n 2 2 ( θ 2 ) n 1 2 ( θ 1 ) n 2 2 ( θ 1 ) ] 2 sin 2 Ψ bi Ψ bi 2 ,
n io = n iu σ i , n ie = n iu + 2 σ i ,
× × E 2 ( 2 ω c ) 2 2 E 2 = ( 2 ω c ) 2 4 π P NL exp ( 2 i k 1 r ) ,
k m = m ω n m ( θ m ) c ( s m , 0 , c m ) , m = 1 , 2 , k 2 r = 2 ω n 2 ( θ 2 ) c ( s 2 , 0 , c 2 ) ,
ê m = [ cos ( θ m γ m ) , 0 , sin ( θ m γ m ) ] , m = 1 , 2 ê 2 r = [ cos ( θ 2 γ 2 ) , 0 , sin ( θ 2 γ 2 ) ] .
cos( θ m γ m ) = [ n m ( θ m ) n mo ] 2 cos γ m cos θ m , ( sin θ m γ m ) = [ n m ( θ m ) n me ] 2 cos γ m sin θ m ,
tan γ m = 1 2 n m ( θ m ) 2 sin 2 θ m ( 1 n mo 2 1 n me 2 ) .
cos γ m = n mo n me n m ( θ m ) [ n mo 2 + n me 2 n m ( θ m ) 2 ] 1 / 2 .
sin θ m = n me sin θ [ ( n mo n me ) 2 + ( n me 2 n mo 2 ) sin 2 θ ] 1 / 2 .
k 1 = ( ω / c ) n 1 o ( s 1 , 0 , c 1 ) , ê 1 = ( 0 , 1 , 0 ) , sin θ 1 = sin θ / n 1 o ,
[ n 1 2 ( θ 1 ) c 1 2 + n 2 o 2 ] e b x + n 1 2 ( θ 1 ) s 1 c 1 e b z = 4 π P x NL , n 1 2 ( θ 1 ) c 1 s 1 e b x + [ n 1 2 ( θ 1 ) s 1 2 + n 2 e 2 ] e b x = 4 π P z NL .
H 2 = h b exp ( i 2 k 1 r ) + n 2 ( θ 2 ) cos γ 2 [ A ê 2 s exp ( i k 2 r ) + B ê 2 s r exp ( i k 2 r r ) ] ,
cos ( θ ) R = e b x exp ( i ϕ 1 ) + cos ( θ 2 γ 2 ) × [ A exp ( i ϕ 2 ) B exp ( i ϕ 2 ) ] ,
R = h b y exp ( i ϕ 1 ) + n 2 ( θ 2 ) cos γ 2 × [ A exp ( i ϕ 2 ) + B exp ( i ϕ 2 ) ] ,
c 2 s T = e b x exp ( i ϕ 1 ) + cos ( θ 2 γ 2 ) × [ A exp ( i ϕ 2 ) B exp ( i ϕ 2 ) ] ,
n 2 s T = h b y exp ( i ϕ 1 ) + n 2 ( θ 2 ) cos γ 2 × [ A exp ( i ϕ 2 ) + B exp ( i ϕ 2 ) ] ,
T = ( 1 / Δ ) [ u a + υ + exp ( ϕ 1 2 ϕ 2 ) + u a υ exp ( ϕ 1 + 2 ϕ 2 ) + 2 n 2 ( θ 2 ) cos γ 2 cos ( θ 2 γ 2 ) ( h b y cos θ + e b x ) × exp ( i ϕ 1 ) ] ,
Δ = u a + u s + exp ( 2 i ϕ 2 ) + u a u s exp ( 2 i ϕ 2 ) ,
u a ± = cos ( θ 2 γ 2 ) ± n 2 ( θ 2 ) cos γ 2 cos θ , u s ± = n 2 s cos ( θ 2 γ 2 ) ± n 2 ( θ 2 ) c 2 s cos γ 2 , υ ± = h b y cos ( θ 2 γ 2 ) ± n 2 ( θ 2 ) e b x cos γ 2 .
exp i ( ϕ 1 ± 2 ϕ 2 ) = 2 i sin ( ϕ 1 ± ϕ 2 ) exp ( ± i ϕ 2 ) + exp ( i ϕ 1 )
υ ± = 4 π n 2 ( θ 1 ) 2 n 1 ( θ 1 ) 2 n 2 ( θ 1 ) 2 [ n 1 ( θ 1 ) c 1 ± n 2 ( θ 2 ) c 2 ] n 2 o 2 { ê 2 P NL ê 2 r P NL ,
T = 8 π i Δ [ n 2 ( θ 1 ) n 2 o ] 2 1 n 1 ( θ 1 ) 2 n 2 ( θ 2 ) 2 × { u a + [ n 1 ( θ 1 ) c 1 + n 2 ( θ 2 ) c 2 ] ê 2 P NL sin ( ϕ 1 ϕ 2 ) × exp ( i ϕ 2 ) + u a [ n 1 ( θ 1 ) c 1 n 2 ( θ 2 ) c 2 ] ê 2 r P NL × sin ( ϕ 1 + ϕ 2 ) exp ( i ϕ 2 ) } .
r af ( 2 p ) = u a / u a + , r fs ( 2 p ) = u s / u s + ,
t fs ( 2 p ) = 2 n 2 ( θ 2 ) cos γ 2 cos ( θ 2 γ 2 ) u s +
n 1 ( θ 1 ) 2 n 2 ( θ 2 ) 2 = [ n 1 ( θ 1 ) c 1 + n 2 ( θ 2 ) c 2 ] × [ n 1 ( θ 1 ) c 1 n 2 ( θ 2 ) c 2 ] ,
T = 4 π i t fs ( 2 p ) n 2 ( θ 2 ) cos γ 2 cos ( θ 2 γ 2 ) ( 2 π L λ ) [ n 2 ( θ 1 ) n 2 o ] 2 [ n 1 ( θ 1 ) 2 n 2 ( θ 2 ) 2 n 1 ( θ 1 ) 2 n 2 ( θ 1 ) 2 ] × [ ê 2 P NL sin Ψ bi Ψ bi exp ( i ϕ 2 ) r af ( 2 p ) ê 2 r P NL sin Φ bi Φ bi exp ( i ϕ 2 ) ] [ exp ( 2 i ϕ 2 ) + r af ( 2 p ) r fs ( 2 p ) exp ( 2 i ϕ 2 ) ] ,
Ψ bi = ϕ 1 ϕ 2 = 2 π L λ [ n 1 ( θ 1 ) c 1 n 2 ( θ 2 ) c 2 ] , Φ bi = ϕ 1 + ϕ 2 = 2 π L λ [ n 1 ( θ 1 ) c 1 + n 2 ( θ 2 ) c 2 ] .
ê 2 P NL = 8 π c d eff I 1 [ t af ( 1 γ ) ] 2 , ê 2 r P NL = 8 π c d eff r I 1 [ t af ( 1 γ ) ] 2 ,
t af ( 1 γ ) = { 2 cos θ cos ( θ 1 γ 1 ) + n 1 ( θ 1 ) cos γ 1 cos θ , γ = p 2 cos θ cos θ 1 + n 1 o cos θ , γ = s ,
d eff = ê 2 d : ê 1 ê 1 = { d 14 cos ( θ 2 γ 2 ) sin 2 ( θ 1 γ 1 ) + sin ( θ 2 γ 2 ) [ d 31 cos 2 ( θ 1 γ 1 ) + d 33 sin 2 ( θ 1 γ 1 ) ] p p d 31 sin ( θ 2 γ 2 ) , s p , d eff r = ê 2 r d : ê 1 ê 1 = { d 15 cos ( θ 2 γ 2 ) sin 2 ( θ 1 γ 1 ) + sin ( θ 2 γ 2 ) [ d 31 cos 2 ( θ 1 γ 1 ) + d 33 sin 2 ( θ 1 γ 1 ) ] , p p d 31 sin ( θ 2 γ 2 ) , s p .
t sa ( 2 p ) = 2 n 2 s c 2 s n 2 s cos θ + c 2 s .
P 2 ω γ p = 128 π 3 c A [ t sa ( 2 p ) ] 2 [ t fs ( 2 p ) ] 2 [ t af ( 1 γ ) ] 4 P ω 2 n 2 ( θ 2 ) 2 cos 2 γ 2 cos 2 ( θ 2 γ 2 ) ( 2 π L λ ) 2 [ n 2 ( θ 1 ) n 2 o ] 4 [ n 1 ( θ 1 ) 2 n 2 ( θ 2 ) 2 n 1 ( θ 1 ) 2 n 2 ( θ 1 ) 2 ] 2 × d eff 2 { sin 2 Ψ bi Ψ bi 2 + [ r af ( 2 p ) ] R 2 sin 2 Φ bi Φ bi 2 2 r af ( 2 p ) R sin Ψ bi Ψ bi sin Φ bi Φ bi cos 2 ϕ 2 } { 1 + [ r af ( 2 p ) r fs ( 2 p ) ] 2 + 2 r af ( 2 p ) r fs ( 2 p ) cos 4 ϕ 2 } ,
tanh θ i = κ tan θ r ,
cos 2 θ r = ( cos 2 θ R + κ 2 ) + [ ( cos 2 θ R κ 2 ) 2 + 4 κ 2 ] 1 / 2 2 ( 1 + κ 2 ) ,
ñ cos θ = n cos θ r cosh θ i + i n κ cos θ r cosh θ i .
ñ cos θ n cos θ R + i n κ / cos θ R .
T = 4 π i t fs ( 2 p ) n 2 c 2 ( 2 π L λ ) [ ê 2 P NL sin Ψ Ψ exp ( i ϕ 2 ) r af ( 2 p ) ê 2 r P NL sin Φ Φ exp ( i ϕ 2 ) ] [ exp ( 2 i ϕ 2 ) + r af ( 2 p ) r fs ( 2 p ) exp ( 2 i ϕ 2 ) ] ,
ϕ m = 2 π L λ ñ m c m = ϕ m + i δ m , δ m = 2 π L λ n m κ m c m ,
Ψ = ϕ 1 ϕ 2 = Ψ + i χ , χ = 2 π L λ ( n 1 κ 1 c 1 n 2 κ 2 c 2 ) ,
Φ = ϕ 1 + ϕ 2 = Φ + i Γ , Γ = 2 π L λ ( n 1 κ 1 c 1 + n 2 κ 2 c 2 ) ,
t fs ( 2 p ) = 2 n 2 c 2 n 2 s c 2 + n 2 c 2 s , t af ( 2 p ) = n 2 cos θ c 2 n 2 cos θ + c 2 , r fs ( 2 p ) = n 2 s c 2 n 2 c 2 s n 2 s c 2 + n 2 c 2 s .
P NL = | E 1 | 2 d : [ ê 1 exp ( δ 1 / 2 ) ] [ ê 1 exp ( δ 1 / 2 ) ] ,
P 2 ω ( γ p ) = 128 π 3 c A [ t sa ( 2 p ) ] 2 [ t fs ( 2 p ) ] 2 [ t af ( 1 γ ) ] 4 P ω 2 n 2 2 c 2 2 ( 2 π L λ ) 2 d eff 2 exp [ 2 ( δ 1 + δ 2 ) ] × { | sin Ψ / Ψ | 2 + [ r af ( 2 p ) ] 2 R 2 exp ( 4 δ 2 ) | sin Φ / Φ | 2 2 r af ( 2 p ) R exp ( 2 δ 2 ) | sin Ψ / Ψ | | sin Φ / Φ | cos ( 2 ϕ 2 θ x + θ y ) } { 1 + [ r af ( 2 p ) r fs ( 2 p ) ] 2 exp ( 8 δ 2 ) + 2 r af ( 2 p ) r fs ( 2 p ) cos 4 ϕ 2 exp ( 4 δ 2 ) } ,
| sin Ψ / Ψ | 2 = ( sin 2 Ψ + sinh 2 χ ) / ( Ψ 2 + χ 2 ) , | sin Φ / Φ | 2 = ( sin 2 Φ + sinh 2 Γ ) / ( Φ 2 + Γ 2 ) , tan θ x = Ψ tanh χ χ tan Ψ Ψ tan Ψ + χ tanh χ , tan θ y = Φ tanh Γ Γ tan Φ Φ tan Φ + Γ tanh Γ ,

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