Abstract

A general method for calculating second-order nonlinear processes in crystals is presented. This approach includes beam depletion and diffraction while permitting a spatial variation of the indices of refraction. As a particular example, second-harmonic generation in a crystal is studied. In this case the spatial variation of the indices of refraction is due to local heating of the crystal caused by the partial absorption of the fundamental beam.

© 1990 Optical Society of America

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References

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  1. D. A. Kleinman, “Theory of second harmonic generation of light,” Phys. Rev. 128, 1761 (1962).
    [CrossRef]
  2. G. Boyd, A. Ashkin, J. Dziedzic, D. Kleinman, “Second-harmonic generation of light with double refraction,” Phys. Rev. 137, 1305 (1965).
    [CrossRef]
  3. D. Kleinman, A. Ashkin, G. Boyd, “Second harmonic generation of light by focused laser beams,” Phys. Rev. 145, 338 (1966).
    [CrossRef]
  4. D. Kleinman, R. Miller, “Dependence of second harmonic generation on the position of the focus,” Phys. Rev. 148, 302 (1966).
    [CrossRef]
  5. J. E. Bjorkholm, “Optical second harmonic generation,” Phys. Rev. 126, 1452 (1966).
  6. Y. S. Liu, D. Dentz, R. Belt, “High-average-power intracavity second-harmonic generation using KTiOPO4 in an acousto-optically Q-switched Nd:YAG laser oscillator at 5 kHz,” Opt. Lett. 9, 76 (1984).
    [CrossRef] [PubMed]
  7. R. M. Kogan, R. M. Pixton, T. G. Crow, “High-efficiency frequency doubling of Nd:YAG,” Opt. Eng. 1, 120 (1978).
  8. S. Sheng, A. E. Siegman, “Nonlinear-optical calculations using gast-transform methods: second-harmonic generation with depletion and diffraction,” Phys. Rev. A 21, 599 (1980).
    [CrossRef]
  9. J. A. Fleck, M. D. Feit, “Beam propagation in uniaxial anisotropic media,” J. Opt. Soc. Am. 73, 920 (1983).
    [CrossRef]
  10. J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129 (1976).
    [CrossRef]
  11. J. Van Roey, J. van der Donk, P. E. Lagasse, “Beam-propagation method: analysis and assessment,” J. Opt. Soc. Am. 71, 803 (1980).
    [CrossRef]
  12. M. Okada, S. Ieri, “Influence of self-induced thermal effects on second harmonic generation,” IEEE J. Quantum Electron. QE-7, 12 (1971).
  13. D. T. Hon, H. Brusselbach, “Beam shaping to suppress phase mismatch in high power second harmonic generation,” IEEE J. Quantum Electron. QE-16, 1356 (1979).
  14. J. F. Nye, Physical Properties of Crystals (Oxford, New York, 1986).
  15. G. Boyd, D. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39, 3597 (1968).
    [CrossRef]
  16. J. A. Armstrong, N. Bloembergen, J. Ducuing, P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918 (1962).
    [CrossRef]
  17. J. E. Midwinter, J. Warner, “The effects of phase matching method and of uniaxial crystal symmetry on the polar distribution on second-order nonlinear optical polarization,” Br. J. Appl. Phys. 16, 1135 (1965).
    [CrossRef]
  18. A. Yariv, Quantum Electronics (Wiley, New York, 1979).
  19. R. A. Phillips, “Temperature variation of the index of refraction of ADP, KDP, and deuterated KDP*,” J. Opt. Soc. Am. 56, 629 (1966).
    [CrossRef]

1984 (1)

1983 (1)

1980 (2)

J. Van Roey, J. van der Donk, P. E. Lagasse, “Beam-propagation method: analysis and assessment,” J. Opt. Soc. Am. 71, 803 (1980).
[CrossRef]

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

1979 (1)

D. T. Hon, H. Brusselbach, “Beam shaping to suppress phase mismatch in high power second harmonic generation,” IEEE J. Quantum Electron. QE-16, 1356 (1979).

1978 (1)

R. M. Kogan, R. M. Pixton, T. G. Crow, “High-efficiency frequency doubling of Nd:YAG,” Opt. Eng. 1, 120 (1978).

1976 (1)

J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129 (1976).
[CrossRef]

1971 (1)

M. Okada, S. Ieri, “Influence of self-induced thermal effects on second harmonic generation,” IEEE J. Quantum Electron. QE-7, 12 (1971).

1968 (1)

G. Boyd, D. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39, 3597 (1968).
[CrossRef]

1966 (4)

R. A. Phillips, “Temperature variation of the index of refraction of ADP, KDP, and deuterated KDP*,” J. Opt. Soc. Am. 56, 629 (1966).
[CrossRef]

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

D. Kleinman, R. Miller, “Dependence of second harmonic generation on the position of the focus,” Phys. Rev. 148, 302 (1966).
[CrossRef]

J. E. Bjorkholm, “Optical second harmonic generation,” Phys. Rev. 126, 1452 (1966).

1965 (2)

G. Boyd, A. Ashkin, J. Dziedzic, D. Kleinman, “Second-harmonic generation of light with double refraction,” Phys. Rev. 137, 1305 (1965).
[CrossRef]

J. E. Midwinter, J. Warner, “The effects of phase matching method and of uniaxial crystal symmetry on the polar distribution on second-order nonlinear optical polarization,” Br. J. Appl. Phys. 16, 1135 (1965).
[CrossRef]

1962 (2)

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

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

Armstrong, J. A.

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

Ashkin, A.

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

G. Boyd, A. Ashkin, J. Dziedzic, D. Kleinman, “Second-harmonic generation of light with double refraction,” Phys. Rev. 137, 1305 (1965).
[CrossRef]

Belt, R.

Bjorkholm, J. E.

J. E. Bjorkholm, “Optical second harmonic generation,” Phys. Rev. 126, 1452 (1966).

Bloembergen, N.

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

Boyd, G.

G. Boyd, D. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39, 3597 (1968).
[CrossRef]

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

G. Boyd, A. Ashkin, J. Dziedzic, D. Kleinman, “Second-harmonic generation of light with double refraction,” Phys. Rev. 137, 1305 (1965).
[CrossRef]

Brusselbach, H.

D. T. Hon, H. Brusselbach, “Beam shaping to suppress phase mismatch in high power second harmonic generation,” IEEE J. Quantum Electron. QE-16, 1356 (1979).

Crow, T. G.

R. M. Kogan, R. M. Pixton, T. G. Crow, “High-efficiency frequency doubling of Nd:YAG,” Opt. Eng. 1, 120 (1978).

Dentz, D.

Ducuing, J.

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

Dziedzic, J.

G. Boyd, A. Ashkin, J. Dziedzic, D. Kleinman, “Second-harmonic generation of light with double refraction,” Phys. Rev. 137, 1305 (1965).
[CrossRef]

Feit, M. D.

J. A. Fleck, M. D. Feit, “Beam propagation in uniaxial anisotropic media,” J. Opt. Soc. Am. 73, 920 (1983).
[CrossRef]

J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129 (1976).
[CrossRef]

Fleck, J. A.

J. A. Fleck, M. D. Feit, “Beam propagation in uniaxial anisotropic media,” J. Opt. Soc. Am. 73, 920 (1983).
[CrossRef]

J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129 (1976).
[CrossRef]

Hon, D. T.

D. T. Hon, H. Brusselbach, “Beam shaping to suppress phase mismatch in high power second harmonic generation,” IEEE J. Quantum Electron. QE-16, 1356 (1979).

Ieri, S.

M. Okada, S. Ieri, “Influence of self-induced thermal effects on second harmonic generation,” IEEE J. Quantum Electron. QE-7, 12 (1971).

Kleinman, D.

G. Boyd, D. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39, 3597 (1968).
[CrossRef]

D. Kleinman, R. Miller, “Dependence of second harmonic generation on the position of the focus,” Phys. Rev. 148, 302 (1966).
[CrossRef]

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

G. Boyd, A. Ashkin, J. Dziedzic, D. Kleinman, “Second-harmonic generation of light with double refraction,” Phys. Rev. 137, 1305 (1965).
[CrossRef]

Kleinman, D. A.

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

Kogan, R. M.

R. M. Kogan, R. M. Pixton, T. G. Crow, “High-efficiency frequency doubling of Nd:YAG,” Opt. Eng. 1, 120 (1978).

Lagasse, P. E.

Liu, Y. S.

Midwinter, J. E.

J. E. Midwinter, J. Warner, “The effects of phase matching method and of uniaxial crystal symmetry on the polar distribution on second-order nonlinear optical polarization,” Br. J. Appl. Phys. 16, 1135 (1965).
[CrossRef]

Miller, R.

D. Kleinman, R. Miller, “Dependence of second harmonic generation on the position of the focus,” Phys. Rev. 148, 302 (1966).
[CrossRef]

Morris, J. R.

J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129 (1976).
[CrossRef]

Nye, J. F.

J. F. Nye, Physical Properties of Crystals (Oxford, New York, 1986).

Okada, M.

M. Okada, S. Ieri, “Influence of self-induced thermal effects on second harmonic generation,” IEEE J. Quantum Electron. QE-7, 12 (1971).

Pershan, P. S.

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

Phillips, R. A.

Pixton, R. M.

R. M. Kogan, R. M. Pixton, T. G. Crow, “High-efficiency frequency doubling of Nd:YAG,” Opt. Eng. 1, 120 (1978).

Sheng, S.

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

Siegman, A. E.

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

van der Donk, J.

Van Roey, J.

Warner, J.

J. E. Midwinter, J. Warner, “The effects of phase matching method and of uniaxial crystal symmetry on the polar distribution on second-order nonlinear optical polarization,” Br. J. Appl. Phys. 16, 1135 (1965).
[CrossRef]

Yariv, A.

A. Yariv, Quantum Electronics (Wiley, New York, 1979).

Appl. Phys. (1)

J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129 (1976).
[CrossRef]

Br. J. Appl. Phys. (1)

J. E. Midwinter, J. Warner, “The effects of phase matching method and of uniaxial crystal symmetry on the polar distribution on second-order nonlinear optical polarization,” Br. J. Appl. Phys. 16, 1135 (1965).
[CrossRef]

IEEE J. Quantum Electron. (2)

M. Okada, S. Ieri, “Influence of self-induced thermal effects on second harmonic generation,” IEEE J. Quantum Electron. QE-7, 12 (1971).

D. T. Hon, H. Brusselbach, “Beam shaping to suppress phase mismatch in high power second harmonic generation,” IEEE J. Quantum Electron. QE-16, 1356 (1979).

J. Appl. Phys. (1)

G. Boyd, D. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39, 3597 (1968).
[CrossRef]

J. Opt. Soc. Am. (3)

Opt. Eng. (1)

R. M. Kogan, R. M. Pixton, T. G. Crow, “High-efficiency frequency doubling of Nd:YAG,” Opt. Eng. 1, 120 (1978).

Opt. Lett. (1)

Phys. Rev. (6)

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

G. Boyd, A. Ashkin, J. Dziedzic, D. Kleinman, “Second-harmonic generation of light with double refraction,” Phys. Rev. 137, 1305 (1965).
[CrossRef]

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

D. Kleinman, R. Miller, “Dependence of second harmonic generation on the position of the focus,” Phys. Rev. 148, 302 (1966).
[CrossRef]

J. E. Bjorkholm, “Optical second harmonic generation,” Phys. Rev. 126, 1452 (1966).

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

Phys. Rev. A (1)

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

Other (2)

A. Yariv, Quantum Electronics (Wiley, New York, 1979).

J. F. Nye, Physical Properties of Crystals (Oxford, New York, 1986).

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Figures (7)

Fig. 1
Fig. 1

Crystallographic coordinates (x′, y′, z′) and laboratory coordinates (x, y, z). The optic axis lies along z′; η is the walk-off angle

Fig. 2
Fig. 2

Conversion efficiency versus z/zI for type I phase matching.

Fig. 3
Fig. 3

Fundamental and harmonic beam profiles for type I phase matching: a, z/zI = 0; b, z/zI = 1/3; c, z/zI = 2/3; d, z/zI = 1.

Fig. 4
Fig. 4

Conversion efficiency versus z/zI for type II phase matching.

Fig. 5
Fig. 5

Fundamental ordinary, fundamental extraordinary, and harmonic beam profiles for type II phase matching: a, z/zI = 0; b, z/ zI = 1/3; c, z/zI = 2/3; d, z/zI = 1. The fundamental beams are indicated by solid curves, whereas dashed curves represent the doubled beam.

Fig. 6
Fig. 6

Loss of conversion efficiency as a function of crystal heating.

Fig. 7
Fig. 7

Effect on conversion efficiency of an intentional phase mismatch of a Gaussian beam in a nonuniformly heated crystal.

Tables (4)

Tables Icon

Table 1 Angle Phase-Matching Schemes for Uniaxial Crystals

Tables Icon

Table 2 Refractive Indices of KDP

Tables Icon

Table 3 Nonlinear Coupling for KDP17,18

Tables Icon

Table 4 Temperature Dependence19 of Refractive Indices of KDP at 298 K

Equations (84)

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× × E ( r , t ) + μ 0 2 t 2 D ( r , t ) = 0 ,
E ( r , t ) = E ( r , ω ) e i ω t d ω .
× × E ( i ) ( r ) μ 0 ω i 2 D ( i ) ( r ) = 0 ,
D ( r ) = D L ( r ) + P ( r ) .
2 E ( 1 ) ( r ) [ E ( 1 ) ( r ) ] + k 0 2 0 D ( 1 ) L ( r ) = k 0 0 P ( 1 ) ( r ) ,
2 E ( 2 ) ( r ) [ E ( 2 ) ( r ) ] + 4 k 0 2 0 D ( 2 ) L ( r ) = 4 k 0 2 0 P ( 2 ) ( r ) ,
= 0 [ n o 2 + δ o ( r ) 0 0 0 n o 2 + δ o ( r ) 0 0 0 n o 2 + δ e ( r ) ] ,
E ( x , y , z ) = E ( x , y , z ) exp ( i k 0 n z ) ,
| , x | k 0 n | |
| , z | k 0 n | |
| , z z | k 0 n | , z | .
= x ̂ y + ŷ ( cos θ x + sin θ z ) + ( sin θ z + cos θ z ) .
D L = D P = E ,
D L = 0 n 0 2 [ E + ( γ 2 1 ) E 3 ] + 0 ( x ̂ δ o E 1 + ŷ δ o E 2 + δ e E 3 ) ,
D = 0 n o 2 [ E + ( γ 2 1 ) ( sin θ E 3 , x + cos θ E 3 , z ) ] + 0 [ ( δ o E i ) , y cos θ ( δ o E 2 ) , x + sin θ ( δ o E 2 ) , z + sin θ ( δ e E 3 ) , x + cos θ ( δ e E 3 ) , z + P = 0 .
E = ( γ 2 1 ) ( sin θ E 3 , x + cos θ E 3 , z ) 1 0 n o 2 P sin θ E 2 , z δ o + cos θ E 3 , z δ e n o 2 .
2 E + ( γ 2 1 ) ( sin θ E 3 , x + cos θ E 3 , z ) + k o 2 n o 2 ( x ̂ E 1 + ŷ E 2 + γ 2 E 3 ) = S + I ,
I = k 0 2 ( x ̂ δ o E 1 + ŷ ̂ δ o E 2 + δ e E 3 ) 1 n o 2 ( sin θ E 2 , z δ o + cos θ E 3 , z δ e )
S = 1 0 [ k 0 2 P + 1 n o 2 ( P ) ] .
S = k 0 2 0 ( x ̂ P o + ŷ cos θ P e sin θ P e ) ,
E = 0 exp ( i k 0 n z )
k 0 2 n 2 1 0 + k 0 2 n o 2 1 0 = 0 ,
k 0 2 n 2 2 0 k 0 2 n 2 ( γ 2 1 ) sin θ cos θ 3 0 + k 0 2 n o 2 2 0 = 0 ,
k 0 2 n 2 2 0 k 0 2 n 2 ( γ 2 1 ) cos 2 θ 3 0 + k 0 2 n e 2 3 0 = 0 ,
k 0 6 ( n 0 2 n 2 ) 2 [ n e 2 n 2 ( γ 2 cos 2 θ + sin 2 θ ) ] = 0 .
n = n o ,
2 0 = 3 0 = 0 ,
e 0 = k 0 = 0 .
n = n e / β ,
1 0 = 0
2 0 = γ 2 cot θ 2 0 ,
o 0 = 0 , k 0 = τ β 2 e 0 ,
1 n 2 = sin 2 θ n e 2 + cos 2 θ n o 2 .
U ( 0 ) = [ 0 1 0 ]
U ( e ) = [ 1 0 τ / β 2 ] [ 1 0 0 ] ,
2 i k 0 n o o , z + o , x x + o , y y = ( S 1 + I 1 ) exp ( i k 0 n o z ) ,
I 1 = k 0 2 δ o 1 exp ( i k 0 n o z ) .
S 1 = k 0 2 0 P o .
3 = e sin θ / β 2 .
2 i k 0 n e β e , z + α 2 e , x x + e , y y 2 i k 0 τ n e β e , x = β 2 sin θ ( S 3 + I 3 ) exp ( i k 0 n o z / β ) ,
I 3 = k 0 δ e 2 E 3 1 n o 2 × ( sin θ x + cos θ z ) ( cos θ δ e E 3 , z + sin θ δ o E 2 , z ) .
I 3 = k 0 2 sin θ β 4 ( sin 2 θ δ e + cos 2 θ δ o γ 4 ) e exp ( i k 0 n e z / β ) ,
( x , y , z ) = 1 2 π d k x d k y ¯ ( k x , k y , z ) × exp [ i ( k x x + k y y ) ] ,
¯ ( k x , k y , z ) = d x d y ( x , y , z ) exp [ i ( k x x + k y y ) ] .
F [ f ( x , y , z ) g ( x , y , z ) ] = f ( k x , k y , z ) * g ( k x , k y , z ) ,
f * g = d k x d k y f ( k x , k y ) g ( k x k x , k y k y ) .
i σ o ¯ o + ¯ o , z = i k 0 2 0 n o P ¯ o exp ( i k 0 n o z ) + H o * o ,
H o i k 0 2 n o δ o
σ o = ( k x 2 + k y 2 ) / 2 k 0 n o .
i σ e ¯ e + ¯ e , z = i 2 k 0 n e P e exp ( i k 0 n e z / β ) + H e * ¯ e ,
H e i k 0 2 n e β 3 ( cos 2 θ δ o γ 4 + sin 2 θ δ e )
σ e = α 2 k k x 2 + k x 2 + 2 τ k 0 k x n e / β 2 k 0 n e β .
P i ( 1 ) = 2 d i j l E j ( 2 ) * E l ( 1 ) ,
P i ( 2 ) = d i j l E j ( 1 ) E l ( 1 ) ,
i σ e ( 2 ) o ( 1 ) + o , z ( 1 ) = i R o o ( 1 ) * * e ( 2 ) e i Δ k z + H o ( 1 ) * o ( 1 ) ,
i σ e ( 2 ) o ( 2 ) + e , z ( 2 ) = i R e o ( 1 ) * * o ( 1 ) e i Δ k z + H e ( 2 ) * e ( 2 ) ,
R o ( 1 ) = k 0 0 n o ( 1 ) d eff ,
R o ( 1 ) = k 0 0 n o ( 1 ) d eff ,
d eff = i j l d i j l U i ( 1 ) o u j ( 1 ) o U l ( 2 ) e ,
Δ k = 2 k 0 ( n e ( 2 ) n o ( 1 ) ) .
sin 2 θ ( n o ( 1 ) 2 n o ( 2 ) 2 ) / ( n e ( 2 ) 2 n o ( 2 ) 2 ) .
i σ o ( 1 ) o ( 1 ) + o , z ( 1 ) = i T o ( 1 ) e ( 1 ) * * e ( 2 ) e i Δ k z + H o ( 1 ) * o ( 1 ) ,
i σ e ( 1 ) e ( 1 ) + e , z ( 1 ) = i T e ( 1 ) o ( 1 ) * * e ( 2 ) e i Δ k z + H e ( 1 ) * e ( 1 ) ,
i σ e ( 2 ) e ( 2 ) + e , z ( 2 ) = i T e o ( 1 ) * * e ( 1 ) e i Δ k z + H e ( 2 ) * e ( 2 ) ,
T o ( 1 ) = k 0 0 n o ( 1 ) d eff ,
T e ( 1 ) = k 0 0 n e ( 1 ) d eff ,
T e ( 2 ) = 2 k 0 0 n e ( 2 ) d eff ,
d eff = i j l d i j l U i ( 1 ) o U j ( 1 ) e U l ( 2 ) e ,
Δ k = k 0 ( 2 n e ( 2 ) n o ( 1 ) n e ( 1 ) ) .
( 1 ) ( 1 ) + ( i R ( 1 ) * ( 2 ) e i Δ k z + H ( 1 ) ( 1 ) ) d z
e i σ d z
( 1 ) ( 1 ) ( i R ( 1 ) * * ( 2 ) e i Δ k z + i σ ( 1 ) ( 1 ) ) dz .
( k x , k y , 0 ) = 0 exp [ ( k x 0 + k y 2 ) w 0 2 / 4 ] ,
w ( z ) = w 0 ( 1 + z 2 / z R 2 ) 1 / 2 ,
z I = 0 d eff E 0 k 0 X { n o ( 1 ) ( type I ) [ n o ( 1 ) n e ( 1 ) ] 1 / 2 ( type II ) } .
T = 4 T m ( x 2 d 2 / 4 ) d 2 ,
Δ k = 2 k 0 n e ( 1 ) τ β ( 2 ) 3 / 2 Δ θ ,
[ e ̂ o ̂ k ̂ ] = [ cos θ cos ϕ cos θ sin ϕ sin θ sin ϕ cos ϕ 0 sin θ cos ϕ sin θ sin ϕ cos θ ] [ x ̂ ŷ ] ,
[ x ̂ ŷ ] = [ cos θ cos ϕ sin ϕ sin θ cos ϕ cos θ sin ϕ cos ϕ sin θ sin ϕ sin θ 0 cos θ ] [ e ̂ o ̂ k ̂ ] .
= x ̂ x + ŷ y + z ,
= x ̂ ( cos ϕ cos θ x + sin ϕ y + sin θ cos ϕ z ) + ŷ ( cos θ sin ϕ x + cos ϕ y + sin θ cos ϕ z ) + ( sin θ x + cos θ z ) .
D L = 0 n o 2 [ e ̂ ( α 2 E e + τ E k ) + o ̂ E o + k ̂ ( τ E e + β 2 E k ) ] ,
γ 2 = n e 2 / n o 2 , α 2 = cos 2 θ + γ 2 sin 2 θ , β 2 = γ 2 cos 2 θ + sin 2 θ , τ = sin θ cos θ ( γ 2 1 )
α 2 β 2 = γ 2 + τ 2 .

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