Abstract

A theoretical model was presented to deal with optical frequency conversion in mixed nonlinear crystals with a small random variation of the composition ratio. For the pump of collimated Gaussian beams, the output powers of sum and difference frequency generations were deduced upon the assumption that the randomness is subject to the stationary Gaussian stochastic process. The induced efficiency reduction relies on the variance and correlation scales of a given varying composition ratio, which widens the phase matching bandwidth as well. The result was generalized to combined mixed crystals involving the gradual index variation. Acceptable composition ratios in different cases were also developed, as they could be applied more easily in practice. The study also includes an analysis of an intentional sinusoidal compositional modulation, which provokes potential applications in the frequency conversions of multiband lasers. The model was also exemplified by applying it to AgGa1xInxSe2.

© 2007 Optical Society of America

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  1. J. C. Mikkelson, Jr. and H. Kildal, "Phase studies, crystal growth, and optical properties of CdGe(As1−xPx)2 and AgGa(Se1−xSx)2 solid solutions," J. Appl. Phys. 49, 426-431 (1978).
    [CrossRef]
  2. K. Kato, E. Takaoka, N. Umemura, and T. Chonan, "Temperature-tuned type-2 90° phase-matched SHG of CO2 laser radiation at 9.2714-10.5910 μm in CdGe(As1−xPx)2," in Proceedings of International Conference on Lasers and Electro-Optics Europe--Technical Digest (IEEE, 2000), paper CThE7, p. 295.
  3. G. C. Bhar, S. Das, D. V. Satyanarayan, P. K. Datta, U. Nundy, and Yu. M. Andreev, "Efficient generation of mid-infrared radiation in an AgGaxIn1−xSe2 crystal," Opt. Lett. 20, 2057-2059 (1995).
    [CrossRef] [PubMed]
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    [CrossRef]
  5. P. G. Schunemann, S. D. Setzler, and T. M. Pollak, "Phase-matched crystal growth of AgGaSe2 and AgGa1−xInxSe2," J. Cryst. Growth 211, 257-264 (2000).
    [CrossRef]
  6. V. Petrov and F. Rotermund, "Application of the solid solution CdxHg1−xGa2S4 as a nonlinear optical crystal," Opt. Lett. 27, 1705-1707 (2002).
    [CrossRef]
  7. H. Jin-Zhe, R. De-Ming, Hu Xiao-Yong, Qu Yan-Chen, Y. Andreev, P. Geiko, V. Badikov, and G. Lanskii, "Nonlinear optical properties of mixed Cd0.35Hg0.65Ga2S4 crystal," Acta Phys. Sin. 53, 3761-3765 (2004).
  8. V. Petrov, V. Badikov, V. Panyutin, G. Shevyrdyaeva, S. Sheina, and F. Rotermund, "Mid-IR optical parametric amplification with femtosecond puping near 800 nm using CdxHg1−xGa2S4," Opt. Commun. 235, 219-226 (2004).
    [CrossRef]
  9. L. Isaenko, A. Yelisseyev, S. Lobanov, A. Titov, V. Petrov, J.-J. Zondy, P. Krinitsin, A. Merkulov, V. Vedenyapin, and J. Smirnova, "Growth and properties of LiGaX2(X=S,Se,Te) single crystals for nonlinear optical applications in the mid-IR," Cryst. Res. Technol. 38, 379-387 (2003).
    [CrossRef]
  10. L. Isaenko, I. Vasilyeva, A. Merkulov, A. Yelisseyev, A. Yelisseyev, and S. Lobanov, "Growth of new nonlinear crystals LiMX2 (M=Al,In,Ga; X=S,Se,Te) for the mid-IR optics," J. Cryst. Growth 275, 217-223 (2005).
    [CrossRef]
  11. Yu. M. Andreev, V. V. Atuchin, G. V. Lanskii, N. V. Pervukhina, V. V. Popov, and N. C. Trocenco, "Linear optical properties of LiIn(S1−xSex)2 crystals and tuning of phase matching conditions," Solid State Sci. 7, 1188-1193 (2005).
    [CrossRef]
  12. J.-J. Huang, V. V. Atuchin, Yu. M. Andreev, G. V. Lanskii, and N. V. Pervukhina, "Potentials of LiGa(S1−xSex)2 mixed crystals for optical frequency conversion," J. Cryst. Growth 292, 500-504 (2006).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  18. C. A. Wang, D. Carlson, S. Motakef, M. Wiegel, and M. J. Wargo, "Research on macro- and microsegregation in semiconductor crystals grown from the melt under the direction of August F. Witt at the Massachusetts Institute of Technology," J. Cryst. Growth 264, 565-577 (2004).
    [CrossRef]
  19. J. J. Huang, G. Jü Ji, T. Shen, Yu M. Andreev, A. V. Shaiduko, G. V. Lanskii, and U. Chatterjee, "Influence of composition ratio variation on optical frequency conversion in mixed crystals. I. Gradual variation of composition ratio," J. Opt. Soc. Am. B 24, 2443-2453 (2007).
    [CrossRef]
  20. J. J. Huang, Y. M. Andreev, G. V. Lanskii, A. V. Shaiduko, S. Das, and U. Chatterjee, "Acceptable composition ratio variations of a mixed crystal for nonlinear laser device applications," Appl. Opt. 44, 7644-7650 (2005).
    [CrossRef] [PubMed]
  21. A. Papoulis and S. U. Pillai, Probability, Random Variables and Stochastic Processes (McGraw-Hill, 2002).
  22. H. J. Scheel, "Theoretical and technological solutions of the striation problem," J. Cryst. Growth 287, 214-223 (2006).
    [CrossRef]
  23. S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, "Optimization by simulated annealing," Science 220, 671-680 (1983).
    [CrossRef] [PubMed]
  24. P. C. Li, X. F. Chen, Y. P. Chen, and Y. X. Xia, "Pulse compression during second-harmonic generation in engineered aperiodic quasi-phase-matching gratings," Opt. Express 13, 6807-6814 (2005).
    [CrossRef] [PubMed]
  25. B. Y. Gu, Y. Zhang, and B. Z. Dong, "Optimization design of aperiodic optical supperlattices for harmonic generations," in Nonlinear Materials, Devices and Applications, J.W.Pierce, eds., Proc. SPIE 3828, 115-123 (2000).
  26. M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999), Chap. 3.

2007 (1)

2006 (3)

J.-J. Huang, V. V. Atuchin, Yu. M. Andreev, G. V. Lanskii, and N. V. Pervukhina, "Potentials of LiGa(S1−xSex)2 mixed crystals for optical frequency conversion," J. Cryst. Growth 292, 500-504 (2006).
[CrossRef]

P. G. Schunemann, K. T. Zawilski, and T. M. Pollak, "Horizontal gradient freeze growth of AgGaGeS4 and AgGaGe5Se12," J. Cryst. Growth 287, 248-251 (2006).
[CrossRef]

H. J. Scheel, "Theoretical and technological solutions of the striation problem," J. Cryst. Growth 287, 214-223 (2006).
[CrossRef]

2005 (4)

P. C. Li, X. F. Chen, Y. P. Chen, and Y. X. Xia, "Pulse compression during second-harmonic generation in engineered aperiodic quasi-phase-matching gratings," Opt. Express 13, 6807-6814 (2005).
[CrossRef] [PubMed]

J. J. Huang, Y. M. Andreev, G. V. Lanskii, A. V. Shaiduko, S. Das, and U. Chatterjee, "Acceptable composition ratio variations of a mixed crystal for nonlinear laser device applications," Appl. Opt. 44, 7644-7650 (2005).
[CrossRef] [PubMed]

L. Isaenko, I. Vasilyeva, A. Merkulov, A. Yelisseyev, A. Yelisseyev, and S. Lobanov, "Growth of new nonlinear crystals LiMX2 (M=Al,In,Ga; X=S,Se,Te) for the mid-IR optics," J. Cryst. Growth 275, 217-223 (2005).
[CrossRef]

Yu. M. Andreev, V. V. Atuchin, G. V. Lanskii, N. V. Pervukhina, V. V. Popov, and N. C. Trocenco, "Linear optical properties of LiIn(S1−xSex)2 crystals and tuning of phase matching conditions," Solid State Sci. 7, 1188-1193 (2005).
[CrossRef]

2004 (5)

H. Jin-Zhe, R. De-Ming, Hu Xiao-Yong, Qu Yan-Chen, Y. Andreev, P. Geiko, V. Badikov, and G. Lanskii, "Nonlinear optical properties of mixed Cd0.35Hg0.65Ga2S4 crystal," Acta Phys. Sin. 53, 3761-3765 (2004).

V. Petrov, V. Badikov, V. Panyutin, G. Shevyrdyaeva, S. Sheina, and F. Rotermund, "Mid-IR optical parametric amplification with femtosecond puping near 800 nm using CdxHg1−xGa2S4," Opt. Commun. 235, 219-226 (2004).
[CrossRef]

R. De-ming, Qu Yan-chen, H. Jin-zhe, Hu Xiao-yong, Y. Andreev, Geiko Pavel, V. Badikov, and A. Shaiduko, "Optical properties and frequency conversion with AgGaGeS4 crystal," Chin. Phys. 13, 1468-1473 (2004).
[CrossRef]

V. Petrov, F. Noack, V. Badikov, G. Shevyrdyaeva, V. Panyutin, and V. Chizhikov, "Phase-matching and femtosecond difference-frequency generation in the quaternary semiconductor AgGaGe5Se12," Appl. Opt. 43, 4590-4597 (2004).
[CrossRef] [PubMed]

C. A. Wang, D. Carlson, S. Motakef, M. Wiegel, and M. J. Wargo, "Research on macro- and microsegregation in semiconductor crystals grown from the melt under the direction of August F. Witt at the Massachusetts Institute of Technology," J. Cryst. Growth 264, 565-577 (2004).
[CrossRef]

2003 (1)

L. Isaenko, A. Yelisseyev, S. Lobanov, A. Titov, V. Petrov, J.-J. Zondy, P. Krinitsin, A. Merkulov, V. Vedenyapin, and J. Smirnova, "Growth and properties of LiGaX2(X=S,Se,Te) single crystals for nonlinear optical applications in the mid-IR," Cryst. Res. Technol. 38, 379-387 (2003).
[CrossRef]

2002 (1)

2001 (1)

X. W. Xu, T. C. Chong, G. Y. Zhang, and H. Kumagai, "Influence of [K]/[Li] and [Li]/[Nb] ratios in melts on the TSSG growth and SHG characteristics of potassium lithium niobate crystals," J. Cryst. Growth 225, 458-464 (2001).
[CrossRef]

2000 (1)

P. G. Schunemann, S. D. Setzler, and T. M. Pollak, "Phase-matched crystal growth of AgGaSe2 and AgGa1−xInxSe2," J. Cryst. Growth 211, 257-264 (2000).
[CrossRef]

1999 (1)

1995 (1)

1993 (1)

J. J. E. Reid, "Resonantly enhanced, frequency doubling of an 820 nm GaAlAs diode laser in a potassium lithium niobate crystal," Appl. Phys. Lett. 62, 19-21 (1993).
[CrossRef]

1983 (1)

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, "Optimization by simulated annealing," Science 220, 671-680 (1983).
[CrossRef] [PubMed]

1978 (1)

J. C. Mikkelson, Jr. and H. Kildal, "Phase studies, crystal growth, and optical properties of CdGe(As1−xPx)2 and AgGa(Se1−xSx)2 solid solutions," J. Appl. Phys. 49, 426-431 (1978).
[CrossRef]

Acta Phys. Sin. (1)

H. Jin-Zhe, R. De-Ming, Hu Xiao-Yong, Qu Yan-Chen, Y. Andreev, P. Geiko, V. Badikov, and G. Lanskii, "Nonlinear optical properties of mixed Cd0.35Hg0.65Ga2S4 crystal," Acta Phys. Sin. 53, 3761-3765 (2004).

Appl. Opt. (2)

Appl. Phys. Lett. (1)

J. J. E. Reid, "Resonantly enhanced, frequency doubling of an 820 nm GaAlAs diode laser in a potassium lithium niobate crystal," Appl. Phys. Lett. 62, 19-21 (1993).
[CrossRef]

Chin. Phys. (1)

R. De-ming, Qu Yan-chen, H. Jin-zhe, Hu Xiao-yong, Y. Andreev, Geiko Pavel, V. Badikov, and A. Shaiduko, "Optical properties and frequency conversion with AgGaGeS4 crystal," Chin. Phys. 13, 1468-1473 (2004).
[CrossRef]

Cryst. Res. Technol. (1)

L. Isaenko, A. Yelisseyev, S. Lobanov, A. Titov, V. Petrov, J.-J. Zondy, P. Krinitsin, A. Merkulov, V. Vedenyapin, and J. Smirnova, "Growth and properties of LiGaX2(X=S,Se,Te) single crystals for nonlinear optical applications in the mid-IR," Cryst. Res. Technol. 38, 379-387 (2003).
[CrossRef]

J. Appl. Phys. (1)

J. C. Mikkelson, Jr. and H. Kildal, "Phase studies, crystal growth, and optical properties of CdGe(As1−xPx)2 and AgGa(Se1−xSx)2 solid solutions," J. Appl. Phys. 49, 426-431 (1978).
[CrossRef]

J. Cryst. Growth (7)

J.-J. Huang, V. V. Atuchin, Yu. M. Andreev, G. V. Lanskii, and N. V. Pervukhina, "Potentials of LiGa(S1−xSex)2 mixed crystals for optical frequency conversion," J. Cryst. Growth 292, 500-504 (2006).
[CrossRef]

P. G. Schunemann, K. T. Zawilski, and T. M. Pollak, "Horizontal gradient freeze growth of AgGaGeS4 and AgGaGe5Se12," J. Cryst. Growth 287, 248-251 (2006).
[CrossRef]

L. Isaenko, I. Vasilyeva, A. Merkulov, A. Yelisseyev, A. Yelisseyev, and S. Lobanov, "Growth of new nonlinear crystals LiMX2 (M=Al,In,Ga; X=S,Se,Te) for the mid-IR optics," J. Cryst. Growth 275, 217-223 (2005).
[CrossRef]

X. W. Xu, T. C. Chong, G. Y. Zhang, and H. Kumagai, "Influence of [K]/[Li] and [Li]/[Nb] ratios in melts on the TSSG growth and SHG characteristics of potassium lithium niobate crystals," J. Cryst. Growth 225, 458-464 (2001).
[CrossRef]

C. A. Wang, D. Carlson, S. Motakef, M. Wiegel, and M. J. Wargo, "Research on macro- and microsegregation in semiconductor crystals grown from the melt under the direction of August F. Witt at the Massachusetts Institute of Technology," J. Cryst. Growth 264, 565-577 (2004).
[CrossRef]

H. J. Scheel, "Theoretical and technological solutions of the striation problem," J. Cryst. Growth 287, 214-223 (2006).
[CrossRef]

P. G. Schunemann, S. D. Setzler, and T. M. Pollak, "Phase-matched crystal growth of AgGaSe2 and AgGa1−xInxSe2," J. Cryst. Growth 211, 257-264 (2000).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Commun. (1)

V. Petrov, V. Badikov, V. Panyutin, G. Shevyrdyaeva, S. Sheina, and F. Rotermund, "Mid-IR optical parametric amplification with femtosecond puping near 800 nm using CdxHg1−xGa2S4," Opt. Commun. 235, 219-226 (2004).
[CrossRef]

Opt. Express (1)

Opt. Lett. (3)

Science (1)

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, "Optimization by simulated annealing," Science 220, 671-680 (1983).
[CrossRef] [PubMed]

Solid State Sci. (1)

Yu. M. Andreev, V. V. Atuchin, G. V. Lanskii, N. V. Pervukhina, V. V. Popov, and N. C. Trocenco, "Linear optical properties of LiIn(S1−xSex)2 crystals and tuning of phase matching conditions," Solid State Sci. 7, 1188-1193 (2005).
[CrossRef]

Other (4)

K. Kato, E. Takaoka, N. Umemura, and T. Chonan, "Temperature-tuned type-2 90° phase-matched SHG of CO2 laser radiation at 9.2714-10.5910 μm in CdGe(As1−xPx)2," in Proceedings of International Conference on Lasers and Electro-Optics Europe--Technical Digest (IEEE, 2000), paper CThE7, p. 295.

B. Y. Gu, Y. Zhang, and B. Z. Dong, "Optimization design of aperiodic optical supperlattices for harmonic generations," in Nonlinear Materials, Devices and Applications, J.W.Pierce, eds., Proc. SPIE 3828, 115-123 (2000).

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999), Chap. 3.

A. Papoulis and S. U. Pillai, Probability, Random Variables and Stochastic Processes (McGraw-Hill, 2002).

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Figures (3)

Fig. 1
Fig. 1

Dephasing functions denoted by D 0 , D g 1 , D r 1 , and D r 2 , which is illustrated in the inset versus ϕ with different values of ψ. Besides D r 3 at δ α = α 3 = 0.10 cm 1 , η = 2.1 cm 1 is also shown with an estimated point at L = 40.7 mm , as is defined in Section 6.

Fig. 2
Fig. 2

Approximated relative errors of P 3 versus the parameter ψ with different values of t r [Eq. (A12)] where the maximal limit at t r = 0 is calculated by Eq. (A16), whereas the truncation is denoted by a truncation in Eq. (A14) with ν < 4 .

Fig. 3
Fig. 3

Dephasing function of the SHG in a 1 cm AGISe crystal with a sinusoidally varying CR centered at x = 0.324 , where the peaks are labeled as corresponding fundamental wavelengths.

Equations (74)

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n j ( z ) = n ̃ j ( z ) + δ n j ( z ) , j = 1 , 2 , 3 ,
δ n j ( z ) = n j ( z , x ) x x = x ¯ δ x ( z ) .
d S ( κ ) d κ = n ( κ ) ,
S j ( z ) = S ̃ j ( z ) + 0 z δ n j ( z ) d z , S ̃ j ( z ) = 0 κ n ̃ j ( τ ) d τ ,
δ p ( z ) δ p ̃ ( z ) + k 30 δ S ( z ) , δ S ( z ) = 0 z δ n ( z ) d z ,
δ p ̃ ( z ) = ν = 1 3 k ν 0 S ̃ ν ( z ) ξ ν ,
δ n ( z ) = k 30 1 ν = 1 3 k ν 0 δ n ν ( z ) ξ ν ,
ξ = ( 1 , 1 , 1 ) ,
F x ( δ x ) = 1 σ x 2 π exp ( δ x 2 2 σ x 2 ) ,
F n ( δ n ) = 1 σ n 2 π exp ( δ n 2 2 σ n 2 ) .
ϱ 1 ( z z ) = exp ( z z μ 3 ) ,
R ( δ S , δ S ) E [ δ S δ S ] = 0 z 0 z E [ δ n ( z 1 ) δ n ( z 2 ) ] d z 1 d z 2 = σ n 2 0 z 0 z exp ( z 1 z 2 μ 3 ) d z 1 d z 2 = μ 3 { 2 z μ 3 [ 1 exp ( z μ 3 ) ] × [ 1 + exp ( z z μ 3 ) ] } σ n 2 .
σ s 2 ( z ) = 2 μ 3 [ z μ 3 + μ 3 exp ( z μ 3 ) ] σ n 2 2 μ 3 z σ n 2 ,
ϱ 2 ( z , z ) = R ( δ S , δ S ) σ s ( z ) σ s ( z ) ( z z ) 1 2 .
A 3 ( r ) = U 0 L exp { i [ δ p ̃ ( z ) + k 30 δ S ( z ) ] δ α z } d z ,
U = U ( x , y ) = i b ¯ exp ( α 3 L 2 x 2 + y 2 w 3 2 ) ,
P 3 = U ̃ ( x , y ) 2 A ̃ 3 A ̃ 3 * d x d y ,
P ¯ 3 E [ P 3 ] = U ̃ ( x , y ) 2 E [ A 3 A 3 * ] d x d y .
E [ A 3 A 3 * ] 0 L 0 L exp { i [ δ p ̃ ( z ) δ p ̃ ( z ) ] δ α ( z + z ) } × exp ( k 30 2 μ 3 σ n 2 z z ) d z d z .
E [ A 3 A 3 * ] 2 η 2 [ ψ 1 + exp ( ψ ) ] ,
ψ = η L , η = k 30 2 μ 3 σ n 2 .
P ¯ 3 = P 0 L 2 L NL 2 2 ψ 1 + exp ( ψ ) ψ 2 P 0 L 2 L NL 2 D r 1 ( ψ ) ,
ϕ = 1.392 , τ cl = 1.318 , ψ = 2.557 ,
P ¯ 3 = P 0 L 2 L NL 2 D r 2 ( ψ ̃ ) , ψ ̃ = ψ + 2 i ϕ ,
D r 2 ( ψ ̃ ) ψ ̃ 1 + exp ( ψ ̃ ) ψ ̃ 2 + c.c.
D r 2 ( ψ ̃ ) D r 2 ( ψ ) ψ 2 ψ ̃ + ψ 2 ψ ̃ * = ψ 2 ψ ̃ ψ ̃ * = η 2 η 2 + Δ k 0 2 .
P ¯ 3 = P 0 L 2 L NL 2 D r 3 , D r 3 ( δ α , α 3 , η , L ) { 1 exp ( 2 δ α L ) δ α ( η δ α ) L 2 + 2 1 exp [ ( δ α + η ) L ] ( δ α 2 η 2 ) L 2 } exp ( α 3 L ) .
δ P 3 [ Q ( t r , θ ) ] 1 2 2 ψ 3 5 .
x ¯ = 1 N j = 1 N x j , σ x 2 = 1 f ( N ) j = 1 N ( δ x j ) 2 ,
ϱ ( Δ z m ) = 1 f ( N ) σ x 2 j = 1 N m δ x j δ x j + m , m N ,
P ¯ 3 = 2 P 0 L NL 2 Ω 2 exp ( 2 π c Σ τ cl ) τ cl τ cl d τ τ τ cl τ cl τ d τ + × exp { 2 π i [ ( δ ¯ τ cl ) τ ± ( τ + + τ cl ) τ ] } × exp [ 2 π ( c 1 τ + + c 2 τ 2 + η ¯ τ ) ] = P 0 L 2 L NL 2 D r g ,
D r g ( c Σ , c 2 , η ¯ , τ cl ) exp ( 2 π c Σ τ cl ) 2 π τ cl 2 τ cl τ cl exp ( 2 π η ¯ τ ) × sin [ 2 π ( τ cl τ ) ( τ ± i c 1 ) ] exp [ 2 π ( c 2 τ 2 i δ ¯ τ ) ] ( τ ± i c 1 ) d τ .
P ¯ 3 = P 0 L 2 L NL 2 exp [ ( 2 π c Σ τ cl ) ] × [ 1 2 π τ cl η ¯ 3 + π 2 τ cl 2 3 ( c 1 2 δ ¯ 2 + η ¯ 2 c 2 π ) π 2 τ cl 4 45 ( 1 6 c 2 2 4 π c 1 δ ¯ ) + ] ,
δ ¯ opt c 1 [ τ cl Re ( F 1 ) ( π η ¯ τ cl 1 ) + F 1 2 + 3 η ¯ Im ( F 1 ) ] [ τ cl Im ( F 1 ) ] 1 .
P ¯ 3 opt P 0 L 2 L NL 2 D rga ( c Σ , η ¯ , τ cl ) P 0 L 2 L NL 2 F 1 2 + ( 2 η ¯ π η ¯ 2 τ cl ) Im ( F 1 ) exp ( 2 π c Σ τ cl ) τ cl 2 ,
τ opt τ 0 ( 1.086 + 1.837 η ¯ ) c Σ + 0.5721 η ¯ 2 .
L opt 2.419 Ω ( 0.3457 Ω 2 + 0.1861 η Ω 3 ) ν = 1 3 α ν + 0.1159 η 2 Ω 3 .
Δ x ( L , x 0 ) π L Δ k x x = x 0 1 ,
Δ x ( L , x 0 ) 2.783 L Δ k x x = x 0 1 .
Δ x g δ x 2 τ cl = 1.318 3.924 Δ x .
k 30 2 μ 3 σ n 2 L = 2.557 .
Δ k = 1.960 k 30 σ n Δ k x x = x ¯ δ x r .
Δ x r δ x r ψ = 2.557 3.134 L Δ k x x = x ¯ 1 ( L μ 3 ) 1 2 1.126 ( L μ 3 ) 1 2 Δ x .
n j = n j 0 + n ¯ j sin ( 2 π z T + ϑ ) ,
P 3 = P 0 L 2 L NL 2 D s ( Δ k 0 , δ k ¯ , T , ϑ ) ,
D s L 2 0 L 0 L exp [ ± i Δ k 0 ( z z ) ] × exp [ ± i δ k ¯ sin ( 2 π z T + ϑ ) ] × exp [ i δ k ¯ sin ( 2 π z T + ϑ ) ] d z d z ,
δ k ¯ = ν = 1 3 k ν 0 n ¯ ν ξ ν ,
D r 3 ( δ α , α 3 , η , L ) 9.5 % ,
efficiency 50 % 6.92 W 11.8 W D r 3 2.8 % .
ϱ 3 ( r ¯ m r ¯ n ) = exp [ ( x ¯ m n 2 + y ¯ m n 2 μ t 2 + z ¯ m n 2 μ g 2 ) 1 2 ] ,
μ 1 = ( cos 2 θ μ t 2 + sin 2 θ μ g 2 ) 1 2 , μ 2 = μ t ,
μ 3 = ( sin 2 θ μ t 2 + cos 2 θ μ g 2 ) 1 2 ,
F ( δ S ) = Σ 1 2 ( 2 π ) 2 exp [ 1 2 ( δ S Σ 1 δ S ) ] ,
x ¯ m n = r m n cos θ cos φ z m n sin θ ,
y ¯ m n = r m n sin φ ,
z ¯ m n = z m n cos θ + r m n sin θ cos φ ,
ϱ 3 = exp { [ ( z m n Δ r ) 2 μ 3 2 + r 2 μ e 2 ] 1 2 } ,
exp ( z m n Δ r μ 3 ) exp ( r μ e ) ,
μ e 2 ( θ , φ ) μ ¯ e 4 ( θ ) μ t 2 μ ¯ e 4 ( θ ) sin 2 φ + μ t 4 cos 2 φ ,
μ ¯ e 2 ( θ ) μ t ( μ g 2 sin 2 θ + μ t 2 cos 2 θ ) 1 2 ,
Δ r ( θ , φ ) = r ( μ t 2 μ g 2 ) μ 3 2 sin 2 θ cos φ 2 μ g 2 μ t 2 ,
R ( δ S m , δ S n ) 2 μ 3 σ n 2 z [ m n ] exp ( r μ e ) 2 μ 3 σ n 2 z [ m n ] h ( r ) ,
Σ 2 μ 3 σ n 2 [ z 1 z [ 12 ] z [ 13 ] h z [ 14 ] h z [ 21 ] z 2 z [ 23 ] h z [ 24 ] h z [ 31 ] h z [ 32 ] h z 3 z [ 34 ] z [ 41 ] h z [ 42 ] h z [ 43 ] z 4 ] .
Δ P 3 = R ( P 3 , P 3 ) 4 P 0 2 L 4 π 2 w 3 4 L NL 4 V ( h ) × exp ( 2 r 2 + r 2 w 3 2 ) d r d r ,
V ( h , ψ ) = 1 L 4 0 L F ( δ S ) exp ( i k 30 C T δ S ) d 4 δ S d 4 z ( P ¯ 3 P 0 ) 2 ( L NL L ) 4 ,
V ( h , ψ ) = 1 L 4 0 L exp ( η 2 C T Σ C ) d 4 z ( P ¯ 3 L NL 2 P 0 L 2 ) 2 = 2 ψ 4 { exp [ 2 ( h 1 ) ψ ] ( 1 3 h + 2 h 2 ) 2 + exp [ 2 ( h + 1 ) ψ ] ( 1 + 3 h + 2 h 2 ) 2 2 h 2 [ 7 5 h 2 + 2 ψ ( h 2 1 ) ] ( h 2 1 ) 2 2 exp ( 2 ψ ) 32 h 2 exp ( ψ ) [ 1 + 4 h 2 + ψ ( 4 h 2 1 ) ] ( 4 h 2 1 ) 2 } .
Δ P 3 8 P 0 2 L 4 π w 3 2 L NL 4 0 2 π d φ 0 r exp ( 2 r 2 w 3 2 ) V ( h , ψ ) d r .
V ( h , ψ ) = 4 ν = 1 a ν ( ψ ) h 2 ν ,
a ν ( ψ ) ψ 4 ( 2 ν ) ! V ( h , ψ ) 4 2 ν h h = 0 .
Δ P 3 4 P 0 2 L 4 L NL 4 ν = 1 a ν ( ψ ) Q ν ( t r , θ ) ,
Q ν ( t r , θ ) 1 2 ν π 0 π 2 t exp ( ν 2 t 2 ) erfc ( ν t ) d φ ,
erfc ( x ) 1 erf ( x ) , t = 2 1 2 w 3 μ e ( θ , φ ) .
Δ P 3 P 0 2 L 4 L NL 4 V ( 1 , ψ ) .
a ν ( ψ ) 2 ψ ( 5 + 2 ν ) ψ 4 , ν 3 .

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