Abstract

We present a numerical model to deal with optical nonlinear processes at a strong focusing situation where general methods possibly fail to work. We find a special curvilinear coordinate system through which a new paraxial wave equation set is developed and its eigenmode transmission solutions are derived. On the basis of the wave equations, a detailed algorithm for second-harmonic generation is proposed with the Fourier-space method. Numerical results in non-walk-off and walk-off cases show that the new model can handle wave interactions on the strong focusing Lb1 condition within a relative small sampling grid, with which traditional methods will lose efficiency, and can provide unique features in comparison with the model in a Cartesian coordinate system. Concrete assessment of the new model is rendered by error analysis.

© 2006 Optical Society of America

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  1. D. A. Kleinman, "Theory of second harmonic generation of light," Phys. Rev. 128, 1761-1775 (1962).
    [CrossRef]
  2. J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, "Interactions between light waves in a nonlinear dielectric," Phys. Rev. 127, 1918-1939 (1962).
    [CrossRef]
  3. G. D. Boyd, A. Ashkin, J. Dziedzic, and D. Kleinman, "Second harmonic generation of light with double refraction," Phys. Rev. 137, A1305-A1320 (1965).
    [CrossRef]
  4. D. Kleinman, A. Ashkin, and G. Boyd, "Second harmonic generation of light by focused laser beams," Phys. Rev. 145, 338-379 (1966).
    [CrossRef]
  5. G. D. Boyd and D. A. Kleinman, "Parametric interaction of focused Gaussian light beams," J. Appl. Phys. 39, 3597-3641 (1968).
    [CrossRef]
  6. D. Fluck, "Theory on phase-matched second-harmonic generation in biaxial planar waveguides," IEEE J. Quantum Electron. 35, 53-59 (1999).
    [CrossRef]
  7. J.-J. Zondy, "Comparative theory of walkoff-limited type-II versus type-I second harmonic generation with Gaussian beams," Opt. Commun. 81, 427-440 (1991).
    [CrossRef]
  8. J.-J. Zondy, "The effects of focusing in type-I and type-II difference-frequency generations," Opt. Commun. 149, 181-206 (1998).
    [CrossRef]
  9. P. Tidemand-Lichtenberg and P. Buchhave, "Generalised theory describing non-linear interaction of focused Gaussian waves of arbitrary polarisation states," Opt. Commun. 180, 159-165 (2000).
    [CrossRef]
  10. Y. Jiang, L. Haowen, X. Shengwu, Y. Xuelin, and X. Yuxing, "Walkoff effects in type-II second-harmonic generation of Gaussian beams," Acta Photonica Sin. 29, 322-326 (2000) (in Chinese).
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    [CrossRef]
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    [CrossRef]
  13. G. Arisholm, "General numerical methods for simulating second-order nonlinear interactions in birefringent media," J. Opt. Soc. Am. B 14, 2543-2549 (1997).
    [CrossRef]
  14. P. Pliszka and P. P. Banerjee, "Nonlinear transverse effects in second-harmonic generation," J. Opt. Soc. Am. B 10, 1810-1819 (1993).
    [CrossRef]
  15. J. A. Fleck, Jr. and M. D. Feit, "Beam propagation in uniaxial anisotropic media," J. Opt. Soc. Am. 73, 920-926 (1983).
    [CrossRef]
  16. D. Eimerl, J. M. Auerbach, and P. W. Milonni, "Paraxial wave theory of second and third harmonic generation in uniaxial crystals. I. Narrowband pump fields," J. Mod. Opt. 42, 1037-1067 (1995).
    [CrossRef]
  17. M. A. Dreger and J. K. McIver, "Second-harmonic generation in a nonlinear, anisotropic medium with diffraction and depletion," J. Opt. Soc. Am. B 7, 776-784 (1990).
    [CrossRef]
  18. J. E. Durnin, J. J. Miceli, and J. H. Eberly, "Experiments with nondiffracting needle beams," in International Quantum Electronics Conference, Vol. 21 of OSA Technical Digest Series (Optical Society of America, 1987), p. 208.
  19. R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, 1965).
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  21. E. Lalor, "The angular spectrum representation of electromagnetic fields in crystals. II. Biaxial crystals," J. Math. Phys. 13, 443-449 (1972).
    [CrossRef]

2000 (2)

P. Tidemand-Lichtenberg and P. Buchhave, "Generalised theory describing non-linear interaction of focused Gaussian waves of arbitrary polarisation states," Opt. Commun. 180, 159-165 (2000).
[CrossRef]

Y. Jiang, L. Haowen, X. Shengwu, Y. Xuelin, and X. Yuxing, "Walkoff effects in type-II second-harmonic generation of Gaussian beams," Acta Photonica Sin. 29, 322-326 (2000) (in Chinese).

1999 (1)

D. Fluck, "Theory on phase-matched second-harmonic generation in biaxial planar waveguides," IEEE J. Quantum Electron. 35, 53-59 (1999).
[CrossRef]

1998 (1)

J.-J. Zondy, "The effects of focusing in type-I and type-II difference-frequency generations," Opt. Commun. 149, 181-206 (1998).
[CrossRef]

1997 (2)

1995 (1)

D. Eimerl, J. M. Auerbach, and P. W. Milonni, "Paraxial wave theory of second and third harmonic generation in uniaxial crystals. I. Narrowband pump fields," J. Mod. Opt. 42, 1037-1067 (1995).
[CrossRef]

1993 (1)

1991 (1)

J.-J. Zondy, "Comparative theory of walkoff-limited type-II versus type-I second harmonic generation with Gaussian beams," Opt. Commun. 81, 427-440 (1991).
[CrossRef]

1990 (1)

1983 (1)

1980 (1)

S.-C. Sheng and A. E. Siegman, "Nonlinear-optical calculations using fast-transform methods: second-harmonic generation with depletion and diffraction," Phys. Rev. A 21, 599-606 (1980).
[CrossRef]

1972 (1)

E. Lalor, "The angular spectrum representation of electromagnetic fields in crystals. II. Biaxial crystals," J. Math. Phys. 13, 443-449 (1972).
[CrossRef]

1968 (1)

G. D. Boyd and D. A. Kleinman, "Parametric interaction of focused Gaussian light beams," J. Appl. Phys. 39, 3597-3641 (1968).
[CrossRef]

1966 (1)

D. Kleinman, A. Ashkin, and G. Boyd, "Second harmonic generation of light by focused laser beams," Phys. Rev. 145, 338-379 (1966).
[CrossRef]

1965 (1)

G. D. Boyd, A. Ashkin, J. Dziedzic, and D. Kleinman, "Second harmonic generation of light with double refraction," Phys. Rev. 137, A1305-A1320 (1965).
[CrossRef]

1962 (2)

D. A. Kleinman, "Theory of second harmonic generation of light," Phys. Rev. 128, 1761-1775 (1962).
[CrossRef]

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, "Interactions between light waves in a nonlinear dielectric," Phys. Rev. 127, 1918-1939 (1962).
[CrossRef]

Arisholm, G.

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, "Interactions between light waves in a nonlinear dielectric," Phys. Rev. 127, 1918-1939 (1962).
[CrossRef]

Ashkin, A.

D. Kleinman, A. Ashkin, and G. Boyd, "Second harmonic generation of light by focused laser beams," Phys. Rev. 145, 338-379 (1966).
[CrossRef]

G. D. Boyd, A. Ashkin, J. Dziedzic, and D. Kleinman, "Second harmonic generation of light with double refraction," Phys. Rev. 137, A1305-A1320 (1965).
[CrossRef]

Auerbach, J. M.

D. Eimerl, J. M. Auerbach, and P. W. Milonni, "Paraxial wave theory of second and third harmonic generation in uniaxial crystals. I. Narrowband pump fields," J. Mod. Opt. 42, 1037-1067 (1995).
[CrossRef]

Banerjee, P. P.

Bloembergen, N.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, "Interactions between light waves in a nonlinear dielectric," Phys. Rev. 127, 1918-1939 (1962).
[CrossRef]

Boyd, G.

D. Kleinman, A. Ashkin, and G. Boyd, "Second harmonic generation of light by focused laser beams," Phys. Rev. 145, 338-379 (1966).
[CrossRef]

Boyd, G. D.

G. D. Boyd and D. A. Kleinman, "Parametric interaction of focused Gaussian light beams," J. Appl. Phys. 39, 3597-3641 (1968).
[CrossRef]

G. D. Boyd, A. Ashkin, J. Dziedzic, and D. Kleinman, "Second harmonic generation of light with double refraction," Phys. Rev. 137, A1305-A1320 (1965).
[CrossRef]

Buchhave, P.

P. Tidemand-Lichtenberg and P. Buchhave, "Generalised theory describing non-linear interaction of focused Gaussian waves of arbitrary polarisation states," Opt. Commun. 180, 159-165 (2000).
[CrossRef]

Coutts, J.

Dmitriev, V. G.

G. G. Gurzadyan, V. G. Dmitriev, and D. N. Nikogosyan, Handbook on Nonlinear Optical Crystals (Springer-Verlag, 1999).

Dreger, M. A.

Ducuing, J.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, "Interactions between light waves in a nonlinear dielectric," Phys. Rev. 127, 1918-1939 (1962).
[CrossRef]

Durnin, J. E.

J. E. Durnin, J. J. Miceli, and J. H. Eberly, "Experiments with nondiffracting needle beams," in International Quantum Electronics Conference, Vol. 21 of OSA Technical Digest Series (Optical Society of America, 1987), p. 208.

Dziedzic, J.

G. D. Boyd, A. Ashkin, J. Dziedzic, and D. Kleinman, "Second harmonic generation of light with double refraction," Phys. Rev. 137, A1305-A1320 (1965).
[CrossRef]

Eberly, J. H.

J. E. Durnin, J. J. Miceli, and J. H. Eberly, "Experiments with nondiffracting needle beams," in International Quantum Electronics Conference, Vol. 21 of OSA Technical Digest Series (Optical Society of America, 1987), p. 208.

Eimerl, D.

D. Eimerl, J. M. Auerbach, and P. W. Milonni, "Paraxial wave theory of second and third harmonic generation in uniaxial crystals. I. Narrowband pump fields," J. Mod. Opt. 42, 1037-1067 (1995).
[CrossRef]

Feit, M. D.

Feynman, R. P.

R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, 1965).

Fleck, J. A.

Fluck, D.

D. Fluck, "Theory on phase-matched second-harmonic generation in biaxial planar waveguides," IEEE J. Quantum Electron. 35, 53-59 (1999).
[CrossRef]

Freegarde, T.

Gurzadyan, G. G.

G. G. Gurzadyan, V. G. Dmitriev, and D. N. Nikogosyan, Handbook on Nonlinear Optical Crystals (Springer-Verlag, 1999).

Haowen, L.

Y. Jiang, L. Haowen, X. Shengwu, Y. Xuelin, and X. Yuxing, "Walkoff effects in type-II second-harmonic generation of Gaussian beams," Acta Photonica Sin. 29, 322-326 (2000) (in Chinese).

Hibbs, A. R.

R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, 1965).

Jiang, Y.

Y. Jiang, L. Haowen, X. Shengwu, Y. Xuelin, and X. Yuxing, "Walkoff effects in type-II second-harmonic generation of Gaussian beams," Acta Photonica Sin. 29, 322-326 (2000) (in Chinese).

Kleinman, D.

D. Kleinman, A. Ashkin, and G. Boyd, "Second harmonic generation of light by focused laser beams," Phys. Rev. 145, 338-379 (1966).
[CrossRef]

G. D. Boyd, A. Ashkin, J. Dziedzic, and D. Kleinman, "Second harmonic generation of light with double refraction," Phys. Rev. 137, A1305-A1320 (1965).
[CrossRef]

Kleinman, D. A.

G. D. Boyd and D. A. Kleinman, "Parametric interaction of focused Gaussian light beams," J. Appl. Phys. 39, 3597-3641 (1968).
[CrossRef]

D. A. Kleinman, "Theory of second harmonic generation of light," Phys. Rev. 128, 1761-1775 (1962).
[CrossRef]

Lalor, E.

E. Lalor, "The angular spectrum representation of electromagnetic fields in crystals. II. Biaxial crystals," J. Math. Phys. 13, 443-449 (1972).
[CrossRef]

McIver, J. K.

Miceli, J. J.

J. E. Durnin, J. J. Miceli, and J. H. Eberly, "Experiments with nondiffracting needle beams," in International Quantum Electronics Conference, Vol. 21 of OSA Technical Digest Series (Optical Society of America, 1987), p. 208.

Milonni, P. W.

D. Eimerl, J. M. Auerbach, and P. W. Milonni, "Paraxial wave theory of second and third harmonic generation in uniaxial crystals. I. Narrowband pump fields," J. Mod. Opt. 42, 1037-1067 (1995).
[CrossRef]

Nikogosyan, D. N.

G. G. Gurzadyan, V. G. Dmitriev, and D. N. Nikogosyan, Handbook on Nonlinear Optical Crystals (Springer-Verlag, 1999).

Pershan, P. S.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, "Interactions between light waves in a nonlinear dielectric," Phys. Rev. 127, 1918-1939 (1962).
[CrossRef]

Pliszka, P.

Sheng, S.-C.

S.-C. Sheng and A. E. Siegman, "Nonlinear-optical calculations using fast-transform methods: second-harmonic generation with depletion and diffraction," Phys. Rev. A 21, 599-606 (1980).
[CrossRef]

Shengwu, X.

Y. Jiang, L. Haowen, X. Shengwu, Y. Xuelin, and X. Yuxing, "Walkoff effects in type-II second-harmonic generation of Gaussian beams," Acta Photonica Sin. 29, 322-326 (2000) (in Chinese).

Siegman, A. E.

S.-C. Sheng and A. E. Siegman, "Nonlinear-optical calculations using fast-transform methods: second-harmonic generation with depletion and diffraction," Phys. Rev. A 21, 599-606 (1980).
[CrossRef]

Tidemand-Lichtenberg, P.

P. Tidemand-Lichtenberg and P. Buchhave, "Generalised theory describing non-linear interaction of focused Gaussian waves of arbitrary polarisation states," Opt. Commun. 180, 159-165 (2000).
[CrossRef]

Xuelin, Y.

Y. Jiang, L. Haowen, X. Shengwu, Y. Xuelin, and X. Yuxing, "Walkoff effects in type-II second-harmonic generation of Gaussian beams," Acta Photonica Sin. 29, 322-326 (2000) (in Chinese).

Yuxing, X.

Y. Jiang, L. Haowen, X. Shengwu, Y. Xuelin, and X. Yuxing, "Walkoff effects in type-II second-harmonic generation of Gaussian beams," Acta Photonica Sin. 29, 322-326 (2000) (in Chinese).

Zondy, J.-J.

J.-J. Zondy, "The effects of focusing in type-I and type-II difference-frequency generations," Opt. Commun. 149, 181-206 (1998).
[CrossRef]

J.-J. Zondy, "Comparative theory of walkoff-limited type-II versus type-I second harmonic generation with Gaussian beams," Opt. Commun. 81, 427-440 (1991).
[CrossRef]

Acta Photonica Sin. (1)

Y. Jiang, L. Haowen, X. Shengwu, Y. Xuelin, and X. Yuxing, "Walkoff effects in type-II second-harmonic generation of Gaussian beams," Acta Photonica Sin. 29, 322-326 (2000) (in Chinese).

IEEE J. Quantum Electron. (1)

D. Fluck, "Theory on phase-matched second-harmonic generation in biaxial planar waveguides," IEEE J. Quantum Electron. 35, 53-59 (1999).
[CrossRef]

J. Appl. Phys. (1)

G. D. Boyd and D. A. Kleinman, "Parametric interaction of focused Gaussian light beams," J. Appl. Phys. 39, 3597-3641 (1968).
[CrossRef]

J. Math. Phys. (1)

E. Lalor, "The angular spectrum representation of electromagnetic fields in crystals. II. Biaxial crystals," J. Math. Phys. 13, 443-449 (1972).
[CrossRef]

J. Mod. Opt. (1)

D. Eimerl, J. M. Auerbach, and P. W. Milonni, "Paraxial wave theory of second and third harmonic generation in uniaxial crystals. I. Narrowband pump fields," J. Mod. Opt. 42, 1037-1067 (1995).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Commun. (3)

J.-J. Zondy, "Comparative theory of walkoff-limited type-II versus type-I second harmonic generation with Gaussian beams," Opt. Commun. 81, 427-440 (1991).
[CrossRef]

J.-J. Zondy, "The effects of focusing in type-I and type-II difference-frequency generations," Opt. Commun. 149, 181-206 (1998).
[CrossRef]

P. Tidemand-Lichtenberg and P. Buchhave, "Generalised theory describing non-linear interaction of focused Gaussian waves of arbitrary polarisation states," Opt. Commun. 180, 159-165 (2000).
[CrossRef]

Phys. Rev. (4)

D. A. Kleinman, "Theory of second harmonic generation of light," Phys. Rev. 128, 1761-1775 (1962).
[CrossRef]

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, "Interactions between light waves in a nonlinear dielectric," Phys. Rev. 127, 1918-1939 (1962).
[CrossRef]

G. D. Boyd, A. Ashkin, J. Dziedzic, and D. Kleinman, "Second harmonic generation of light with double refraction," Phys. Rev. 137, A1305-A1320 (1965).
[CrossRef]

D. Kleinman, A. Ashkin, and G. Boyd, "Second harmonic generation of light by focused laser beams," Phys. Rev. 145, 338-379 (1966).
[CrossRef]

Phys. Rev. A (1)

S.-C. Sheng and A. E. Siegman, "Nonlinear-optical calculations using fast-transform methods: second-harmonic generation with depletion and diffraction," Phys. Rev. A 21, 599-606 (1980).
[CrossRef]

Other (3)

J. E. Durnin, J. J. Miceli, and J. H. Eberly, "Experiments with nondiffracting needle beams," in International Quantum Electronics Conference, Vol. 21 of OSA Technical Digest Series (Optical Society of America, 1987), p. 208.

R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, 1965).

G. G. Gurzadyan, V. G. Dmitriev, and D. N. Nikogosyan, Handbook on Nonlinear Optical Crystals (Springer-Verlag, 1999).

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Figures (4)

Fig. 1
Fig. 1

Relationship between the laboratory coordinate system x , y , z and the principal dielectric axes X , Y , Z of a uniaxial anisotropic crystal, where S denotes the Poynting vector of an extraordinary wave.

Fig. 2
Fig. 2

(a) Power and sampling errors (PE, SE) versus the focusing parameter L b with the error tolerance of 10 4 and (b) run times versus error tolerance with different focusing parameters 0.1, 1.0, 4.1 (critical point), and 20 in the non-walk-off case with pump strength σ = 5.0 and a centered focus.

Fig. 3
Fig. 3

(a) Number of steps versus the focusing parameter L b with error tolerance of 10 4 but different walk-off angles 0.005 and 0.023 versus (b) error tolerance with different focusing parameters 0.1 and 2.0 in the walk-off case with the pump strength σ = 2.0 .

Fig. 4
Fig. 4

Evolution of the harmonic (H) and fundamental (F) waves in the type I SHG process, simulated by the non-walk-off CURM, where the fundamental beam is initialized with the fourth-order super-Gaussian profile.

Equations (86)

Equations on this page are rendered with MathJax. Learn more.

Δ L E 1 ϵ 0 c 2 D L t 2 ( E ) = 1 ϵ 0 c 2 P NL t 2 ,
( D L + P NL ) = 0 ,
Δ L E X j ( 1 γ 2 ) 2 E Z X j Z + ω 2 c 2 ϵ X j X j E X j = ω 2 ϵ 0 c 2 P X j NL , j = 1 , 2 , 3 ,
Δ L E y + k o 2 E y = ω o 2 ϵ 0 c 2 P y NL , k o = k 1 = ω o c n 1 .
( j = 1 3 k X j 2 ω e 2 c 2 ϵ X m X m ) E X m = ( 1 γ 2 ) k X m k Z E Z , m = 1 , 2 , 3 .
k X 2 + k Y 2 + γ 2 k Z 2 = k 2 2 = ω e 2 c 2 n 2 2 ,
k X 2 + k Y 2 + k Z 2 = k e 2 = ω e 2 n e 2 ( θ ) c 2 ,
n e ( θ ) = n 1 n 2 ( n 1 2 sin 2 θ + n 2 2 cos 2 θ ) 1 2 .
E Z = E x sin θ β 2 , E z = ρ E x ,
α 2 = cos 2 θ + γ 2 sin 2 θ ,
β 2 = sin 2 θ + γ 2 cos 2 θ ,
ρ = sin θ cos θ ( γ 2 1 ) β 2 .
[ Δ L + ( γ 2 1 ) 2 Z 2 ] E x + k 2 2 E x = ω e 2 ϵ 0 c 2 P x NL .
( 2 x 2 + 2 y 2 + 2 z 2 ) E o + k o 2 E o = ω o 2 ϵ 0 c 2 P o NL ,
( α 2 2 x 2 + 2 y 2 + β 2 2 z 2 + 2 β 2 ρ 2 x z ) E e + k 2 2 E e = ω e 2 ϵ 0 c 2 P e NL .
Ψ ( x , y , z ) = E 0 1 + i τ exp [ i k z x 2 + y 2 ( 1 + i τ ) w 0 2 ] , τ ( z ) = 2 ( z z f ) b ,
F [ Ψ ( x , y , z ) ] = E 0 w 0 2 2 exp [ ( k ̃ x 2 + k ̃ y 2 ) w 0 2 ( 1 + i τ ) 4 ] exp ( i k z ) ,
ξ x w 0 ( 1 + τ 2 ) 1 2 = x w ( z ) ,
η y w 0 ( 1 + τ 2 ) 1 2 = y w ( z ) ,
ζ 1 w 0 [ z + τ b x 2 + y 2 1 + τ 2 ] = 1 w 0 [ z + x 2 + y 2 2 R ( z ) ] .
Ψ ( ξ , η , ζ ) = E 0 1 + i τ exp [ 2 i κ ζ ( ξ 2 + η 2 ) ] E 0 1 + i τ exp [ 2 i κ ζ ( ξ 2 + η 2 ) ] = E 0 M exp [ 2 i κ ζ ( ξ 2 + η 2 ) i tan ( τ ) ] ,
τ = ( ζ ζ f ) κ , M = 1 ( 1 + τ 2 ) 1 2 ,
κ = k w 0 2 = 1 δ 0 , ζ f = z f w 0 , ζ 0 = z 0 w 0 .
E o = E in A o exp ( 2 i k o ζ ) , E e = E in A e exp ( 2 i κ e ζ ) ,
κ o = κ = k w 0 2 , κ e = k e w 0 2 ,
2 A o , e ζ 2 2 κ o , e A o , e ζ 4 κ o , e 2 A o , e ,
A o ζ M 2 κ [ τ + i ( ξ 2 + η 2 ) i 4 ( 2 ξ 2 + 2 η 2 ) ] A o + i κ P o NL ϵ 0 n o 2 E in exp ( 2 i κ ζ ) ,
A e ζ M 2 κ [ τ ( 1 + α 2 ) 2 β 2 + ρ κ M ( 2 i t τ ξ + ξ ) + τ ( δ 1 ξ ξ + δ 2 η η ) + i t ( 1 + δ 1 τ 2 ) ξ 2 + i t ( 1 + δ 2 τ 2 ) η 2 i 4 β 2 t ( α 2 2 ξ 2 + 2 η 2 ) ] A e + i κ e P e NL ϵ 0 n 2 2 E in exp ( 2 i κ e ζ ) ,
δ 1 = α 2 β 2 1 , δ 2 = 1 β 2 1 , t = κ e κ .
x w [ x ρ ( z z 0 ) ] α , y w y , z w z β ,
τ w ( ζ w ζ f β ) κ , M w 1 ( 1 + τ w 2 ) 1 2 .
Δ ̃ L = [ ( 1 δ 3 ) 2 x w 2 + 2 y w 2 + 2 z w 2 ] ,
ξ w = γ 1 2 ξ ( x w , y w , z w ) , η w = γ 1 2 η ( x w , y w , z w ) , ζ w = ζ ( x w , y w , z w ) .
A o ζ M 2 κ [ τ + i ( ξ 2 + η 2 ) i 4 ( 2 ξ 2 + 2 η 2 ) ] A o + i κ P o NL ϵ 0 n o 2 E in exp ( 2 i κ ζ ) ,
A e ζ w M w 2 κ [ τ w + i r ( ξ w 2 + η w 2 ) i 4 r ( 2 ξ w 2 + 2 η w 2 ) ] A e + i κ 2 P e NL ϵ 0 n 2 2 E in exp ( 2 i κ 2 ζ w ) ,
E o = E in A o exp ( 2 i κ o ζ ) , E e = E in A e exp ( 2 i κ 2 ζ w )
q arctan ( τ ) , 0 q < π 2 ,
q w arctan ( τ w ) , 0 q w < π 2 ,
u A o ( ξ , η , q ) M ( ζ ) ,
v A e ( ξ w , η w , q w ) M ( ζ w ) .
i u q [ 1 4 ( 2 ξ 2 + 2 η 2 ) + ( ξ 2 + η 2 ) ] u ,
i v q w [ 1 4 r ( 2 ξ w 2 + 2 η w 2 ) + r ( ξ w 2 + η w 2 ) ] v .
u m , n = H m ( 2 1 2 ξ ) H n ( 2 1 2 η ) exp [ ( ξ 2 + η 2 ) ] exp [ i ( m + n + 1 ) q ] ,
v m , n = H m [ ( 2 r ) 1 2 ξ w ] H n [ ( 2 r ) 1 2 η w ] exp [ r ( ξ w 2 + η w 2 ) ] × exp [ ( i ( m + n + 1 ) q w ) ] ,
u ( ξ , η , q ) = 1 i π sin ( q q ) u ( ξ , η , q ) exp [ i ( ξ 2 + ξ 2 + η 2 + η 2 ) cot ( q q ) ] exp [ 2 i ( ξ ξ + η η ) sin ( q q ) ] d ξ d η ,
P o NL = 2 ϵ 0 d eff E in 2 A o * A e exp [ 2 i ( κ e κ ) ζ ] ,
P e NL = ϵ 0 d eff E in 2 A o 2 exp ( 4 i κ ζ ) ,
A e f = A e exp ( i t τ ξ 2 ) .
A o ζ M 2 κ [ τ + i ( ξ 2 + η 2 ) i 4 ( 2 ξ 2 + 2 η 2 ) ] A o + i σ o A o * A e f exp [ i ( 2 Δ κ ζ + t τ ξ 2 ) ] ,
A e f ζ M 2 κ [ τ ( 1 2 β 2 ξ ξ + δ 2 η η ) + ρ κ M ξ + i t ( 1 + δ 2 τ 2 ) η 2 i 4 β 2 t ( α 2 2 ξ 2 + 2 η 2 ) ] A e f + i σ e A o 2 exp [ i ( 2 Δ κ ζ + t τ ξ 2 ) ] ,
κ 2 = κ e β = 2 γ κ , Δ κ = 2 κ κ e ,
σ o = 2 κ d eff E in n 1 2 ( ω ) , σ e = κ 2 d eff E in n 2 2 ( 2 ω ) .
A ̃ o , e = F ( A o , e ) = A o , e ( ξ , η , ζ ) exp [ 2 π i ( s 1 ξ + s 2 η ) ] d ξ d η ,
A ̃ o ζ M 2 κ { τ A ̃ o + i F [ A o ( ξ 2 + η 2 ) ] + i π 2 ( s 1 2 + s 2 2 ) A ̃ o } + i σ o exp ( 2 i Δ κ ζ ) F [ A o * A e f exp ( i t τ ξ 2 ) ] ,
A ̃ e f ζ M 2 κ { [ 3 δ 2 2 τ + 2 π i s 1 ρ κ M ] A ̃ e f 2 π i τ F [ ( s 1 ξ δ 2 s 2 η ) A e f ] + i t ( 1 + δ 2 τ 2 ) F ( A e f η 2 ) } + i σ e F [ A o 2 exp ( i t τ ξ 2 ) ] exp ( 2 i Δ κ ζ ) .
g o 1 G o τ = i σ o κ F [ A o * A e f exp ( i t τ ξ 2 ) ] exp ( 2 i Δ κ ζ ) i M 2 F [ ( ξ 2 + η 2 ) A o ] ,
g e 1 G e τ = i σ e κ F [ A o 2 exp ( i t τ ξ 2 ) ] exp ( 2 i Δ κ ζ ) + 2 π i τ M 2 [ s 1 F ( ξ A e f ) δ 2 s 2 F ( η A e f ) ] i t ( 1 + δ 2 τ 2 ) M 2 F [ A e f η 2 ] ,
G o = A ̃ o g o , G e = A ̃ e f g e ,
g o = M 1 exp [ i π 2 ( s 1 2 + s 2 2 ) q ] ,
g e = M ( 3 δ 2 ) 2 exp [ i π 2 β 2 t ( α 2 s 1 2 + s 2 2 ) q ] × exp [ 2 i π s 1 κ ρ arcsinh ( τ ) ] .
6 w 0 ( 1 + τ max 2 ) 1 2 = π N g w 0 8 .
τ max = [ ( π N g 48 ) 2 1 ] 1 2 ,
A o = E 0 exp [ ( x 2 + y 2 w 0 2 ) 2 ] , δ s = 2 3 4 λ w 0 π 5 4 .
Y i = Y i ( X 1 , X 2 , X 3 ) , X i = X i ( Y 1 , Y 2 , Y 3 ) ( i = 1 , 2 , 3 ) ,
d r = r Y i d Y i = e i d Y i ,
g i k = r Y i r Y k = X l Y i X l Y k = e i e k , G = det g .
h i = ( g i i ) 1 2 , e i , e j = e i e j ( e i 2 + e j 2 ) 1 2 .
d a i = e j × e k d Y j d Y k = ( g j j g k k g j k 2 ) 1 2 d Y j d Y k ,
d V = G 1 2 d Y 1 d Y 2 d Y 3 .
d a 3 [ 1 + τ 2 ( ζ ) ] d ξ d η = M 2 ( ζ ) d ξ d η
e 1 , e 3 = 2 ζ ξ 2 κ 2 + ζ 2 + o ( ξ 2 κ 2 ) ,
e 2 , e 3 = 2 ζ η 2 κ 2 + ζ 2 + o ( η 2 κ 2 ) ,
e 1 , e 2 = 0 .
Γ ( Ω , z ) = const 1 , Λ ( Ω , z ) = const 2
Γ ( Ω , z ) Λ ( Ω , z ) = 0 .
Λ Ω Λ z = Γ Ω Γ z = Ω z κ 2 + z 2 ,
d z d Ω = Ω z κ 2 + z 2 ,
C = z exp ( Ω 2 + z 2 2 κ 2 ) .
Π ( C ) = Π [ z exp ( Ω 2 + z 2 2 κ 2 ) ] .
Π ( z ) 1 = z exp ( z 2 2 κ 2 ) ,
Λ ( Ω , z ) = Π [ z exp ( Ω 2 + z 2 2 κ 2 ) ] .
Λ ( Ω , z ) = z [ 1 + Ω 2 2 ( κ 2 + z 2 ) ( z 2 κ 2 ) Ω 4 8 ( z 2 + κ 2 ) 3 ] + O [ ( Ω κ ) 6 ] .
ζ κ , 2 ζ 2 κ 2 ,
κ ρ , ζ ( κ 2 + ζ 2 ) 1 2 , ξ , η 1 ,
1 β 2 , α 2 β 2 , ρ δ 0 ,
w 0 2 Δ ̃ L w 0 2 ( α 2 2 x 2 + 2 y 2 + β 2 2 z 2 + 2 β 2 ρ 2 x z ) M 2 ( α 2 2 ξ 2 + 2 η 2 ) + M 2 δ 0 τ { 1 + α 2 + 2 [ ( α 2 β 2 ) ξ + ( 1 β 2 ) η ] } ζ + 2 M β 2 ρ 2 ξ ζ + β 2 M 2 δ 0 2 { 1 M 2 δ 0 2 + ( α 2 ξ 2 + η 2 ) τ 2 β 2 + 2 [ ρ ξ τ M δ 0 + ( ξ 2 + η 2 ) ( 1 τ 2 ) ] } 2 ζ 2 ,

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