Abstract

A theoretical model is presented to deal with optical frequency conversion in a mixed nonlinear crystal with a small linear variation in composition ratio. Using the quasi-geometrical optics method, what we believe to be new diffraction-free coupling equations are developed to describe sum and difference frequency generations. With these new frequencies, we find that an optimal crystal length exists like that in the plane-wave model. Furthermore, the optimization of generated powers with absorption, transverse index modulation, and walk-off effect are studied in detail. According to different efficiency reductions, the tolerance and acceptance of composition ratios along and vertical to the beam propagating direction are presented. Intended gradual index crystals are also discussed for their possible applications in the frequency conversion of wideband lasers and in generating pulse compressed second harmonic ultrashort pulses.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |

  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 (Optical Society of America, 2000), pp. 295.
  3. G. C. Bhar, S. Das, U. Chatterjee, and Yu. M. Andreev, "Efficient generation of mid-infrared radiation in an AgGaxIn1−xSe2 crystal," Appl. Phys. Lett. 63, 1316-1318 (1993).
    [CrossRef]
  4. E. Takaoka and K. Kato, "90° phase-matched third-harmonic generation of CO2 laser frequencies in AgGa1−xInxSe2," Opt. Lett. 24, 902-904 (1999).
    [CrossRef]
  5. Yu. M. Andreev, V. V. Badikov, V. G. Voevodin, L. G. Geiko, P. P. Geiko, M. V. Ivashchenko, A. I. Karapuzikov, and I. V. Sherstov, "Radiation resistance of nonlinear crystals at a wavelength of 9.55 μm," Quantum Electron. 31, 1075-1078 (2001).
    [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. 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]
  8. L. Isaenko, I. Vasilyeva, A. Merkulov, 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]
  9. 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]
  10. 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]
  11. 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]
  12. 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]
  13. 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]
  14. M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).
  15. M. A. Arbore, O. Marco, and M. M. Fejer, "Pulse compression during second-harmonic generation in aperiodic quasi-phase-matching gratings," Opt. Lett. 17, 865-867 (1997).
    [CrossRef]
  16. G. Imeshev, M. A. Arbore, S. Kasriel, and M. M. Fejer, "Pulse shaping and compression by second-harmonic generation with quasi-phase-matching gratings in the presence of arbitrary dispersion," J. Opt. Soc. Am. B 17, 1420-1437 (2000).
    [CrossRef]
  17. 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]

2006

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]

2005

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, 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

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

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

2001

Yu. M. Andreev, V. V. Badikov, V. G. Voevodin, L. G. Geiko, P. P. Geiko, M. V. Ivashchenko, A. I. Karapuzikov, and I. V. Sherstov, "Radiation resistance of nonlinear crystals at a wavelength of 9.55 μm," Quantum Electron. 31, 1075-1078 (2001).
[CrossRef]

2000

1999

1997

1993

G. C. Bhar, S. Das, U. Chatterjee, and Yu. M. Andreev, "Efficient generation of mid-infrared radiation in an AgGaxIn1−xSe2 crystal," Appl. Phys. Lett. 63, 1316-1318 (1993).
[CrossRef]

1978

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]

Appl. Opt.

Appl. Phys. Lett.

G. C. Bhar, S. Das, U. Chatterjee, and Yu. M. Andreev, "Efficient generation of mid-infrared radiation in an AgGaxIn1−xSe2 crystal," Appl. Phys. Lett. 63, 1316-1318 (1993).
[CrossRef]

Cryst. Res. Technol.

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.

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

L. Isaenko, I. Vasilyeva, A. Merkulov, 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]

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]

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]

J. Opt. Soc. Am. B

Opt. Lett.

Quantum Electron.

Yu. M. Andreev, V. V. Badikov, V. G. Voevodin, L. G. Geiko, P. P. Geiko, M. V. Ivashchenko, A. I. Karapuzikov, and I. V. Sherstov, "Radiation resistance of nonlinear crystals at a wavelength of 9.55 μm," Quantum Electron. 31, 1075-1078 (2001).
[CrossRef]

Solid State Sci.

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

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 (Optical Society of America, 2000), pp. 295.

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

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

Dephasing functions D g 1 , 2 , D t [Eqs. (35b, 56, 58a)] and G 3 ( τ cl , c 2 ) [Eq. (54a)] versus the scaled crystal lengths τ cl and τ t 2 [Eq. (58b)] with c 2 = 0 , 0.2 , 0.5 . The maximum value of G 3 ( τ cl , 0 ) ( τ 0 ) and the asymptotic limit of G 3 (0.5) are denoted by thin curves.

Fig. 2
Fig. 2

Optimal mismatch δ ¯ opt versus the scaled crystal length τ cl with different wave losses of the product parameter c 1 τ cl . The dotted ones are of the approximation given by Eq. (42), while, the strict results come from Eq. (40).

Fig. 3
Fig. 3

Optimal length shift divided by the total absorption coefficient: δ L c Σ versus the latter parameter c Σ with different absorption ratios. The limit line is for Eq. (48), “A” for Eq. (47b), and other curves for the strict results of Eq. (46).

Fig. 4
Fig. 4

Optimal length shift δ L and scaled output power G 3 versus c 2 τ opt 2 at the optimal length τ opt .

Equations (104)

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

x = x 0 + β g ̂ r ,
n 2 ( x , κ ̂ ) = n 0 2 ( κ ̂ ) [ 1 + β ¯ ( κ ̂ ) ( x x 0 ) + o ( β ¯ ) ] .
n 2 ( r , κ ̂ ) = n 0 2 ( κ ̂ ) [ 1 + ε g ̂ r + o ( ε L ) ] ,
× × E ( r ) ω 2 ϵ E ( r ) c 2 = 0 ,
E ( r ) = 1 2 E ̂ ( r ) A ( r ) exp [ i k 0 S ( r ) ] + c.c.
E ( r ) = ν = 0 E ( ν ) ( r ) = ν = 0 E ̂ ( ν ) ( r ) A ( ν ) ( r ) ( i k 0 ) ν .
ν = 0 : Re ( ϵ ) E ̂ ( 0 ) + S × ( S × E ̂ ( 0 ) ) = 0 ,
ν = 1 : S × A ( 0 ) × E ̂ ( 0 ) k 0 Im ( ϵ ) E ( 0 ) + × ( A ( 0 ) S × E ̂ ( 0 ) ) = 0 .
S 2 = [ Re ( ϵ ) E ̂ ( 0 ) ] ( E ̂ ( 0 ) ) 1 = n 2 ( r , κ ̂ ) ,
I ( r 0 ) = I ( r ) .
x j = x z tan ρ j 1 ε j z 2 g 1 4 ,
y j = y z tan ρ j 2 ε j z 2 g 2 4 ,
z j = z , g 1 = g ̂ e ̂ x , g 2 = g ̂ e ̂ y ,
E j ( r j ) = e ̂ j E j 2 [ n j 0 n j ( r j ) ] 1 2 exp [ i k j 0 S j ( r j ) ] exp ( x j 2 + y j 2 w j 2 α j z 2 ) + c.c. ,
S 3 × A 3 × e ̂ 3 + × ( A 3 S 3 ) × e ̂ 3 k 30 A 3 Im ( ϵ 3 ) e ̂ 3 + i k 30 ε 0 P 3 NL exp ( i k 30 S 3 ) = 0 ,
E j ( r ) = 1 2 e ̂ j A j ( r ) exp [ i k j 0 S j ( r ) ] + c.c. ,
P 3 NL ( r ) = 1 2 [ 2 ε 0 χ : e ̂ 1 e ̂ 2 A 1 ( r ) A 2 ( r ) ] exp { i [ k 10 S 1 ( r ) + k 20 S 2 ( r ) ] } + c.c. ,
n j 2 ( r ) = n j 0 2 ( 1 + ε j g ̂ r ) , k j 0 = ω j c ,
2 ( e ̂ 3 S 3 A 3 e ̂ 3 S 3 A 3 ) + A 3 [ e ̂ 3 e ̂ 3 : S 3 Δ S 3 k 30 e ̂ 3 Im ( ϵ 3 ) e ̂ 3 ] + 2 i k 30 χ eff A 1 A 2 exp ( i δ p ) = 0 ,
A 3 z + ν = 1 2 ( tan ρ 3 ν + g ν ε 3 n 30 z 2 n 3 ) A 3 x ¯ ν + ( ε 3 n 30 g 4 n 3 + α 3 2 ) A 3 i k 30 χ eff A 1 A 2 n 3 cos ρ 3 exp ( i δ p ) ,
A 1 A 2 exp ( i δ p ) A 1 A 2 * exp ( i δ p ) , ω 2 ω 2 .
n j 0 n j ( r j ) 1 .
A 3 z 3 + α 3 2 A 3 i b A 1 A 2 exp ( i δ p ) , b = k 30 χ eff n 30 cos ρ 3 .
E j e ̂ j E j A ̃ j 2 exp ( i k j 0 S j α j z 2 ) + c.c. ,
A j = E j A ̃ j exp ( α j z 2 ) .
A ̃ 3 z 3 i b ¯ A ̃ 1 A ̃ 2 exp [ i δ p ( r ) δ α z 3 ] ,
δ α = ( α 1 + α 2 α 3 ) 2 , b ¯ = b E 1 E 2 ,
A 3 = A ̃ 3 exp ( α 3 z 2 ) .
Δ k 0 = ν = 1 3 k ¯ ν 0 ξ ν , k ¯ j 0 = k j 0 n j 0 , ξ = ( 1 , 1 , 1 ) ,
w 3 2 = w 1 2 + w 2 2 , ρ ¯ j l = tan ρ j l tan ρ 3 l ,
δ ρ 1 = 2 w 3 2 ν , ν = 1 2 ρ ¯ ν ν w ν 2 e ̂ ν , δ ρ 2 2 = w 3 2 ν , ν = 1 2 ρ ¯ ν ν 2 w ν 2 ,
δ ε = w 3 2 2 ν = 1 2 ε ν ε 3 w ν 2 , δ k = 1 2 g ν = 1 3 k ¯ ν 0 ε ν ξ ν ,
δ k = 1 2 ν = 1 3 k ¯ ν 0 ε ν ξ ν ( g + 2 ν = 1 2 g ν tan ρ 3 ν ) .
A ̃ 1 A ̃ 2 exp [ ( δ ε g r 3 δ ρ 2 2 ) z 3 2 w 3 2 ] exp [ ( δ ρ 1 r 3 r 3 2 ) w 3 2 ] ,
δ p ( r 3 ) ( Δ k 0 + δ k r 3 ) z 3 + δ k z 3 2 2 .
Δ k ( r ) = δ p ( r ) , Δ k ( z ) e ̂ z Δ k ( r ) ,
A ̃ 3 = i b ¯ exp ( r 3 2 w 3 2 ) 0 L exp [ i δ p ( r 3 ) δ ρ 2 2 ( z 3 w 3 ) 2 ] × exp [ ( δ ρ 1 r 3 w 3 2 δ α ) z 3 ] d z 3 ,
I 3 1 2 c ε 0 n 30 A 3 2 , P 3 = Ξ I 3 ( r 3 ) d r 3 ,
P 3 = 4 P 0 L NL 2 Ω 2 exp ( α 3 L ) 0 τ cl 0 τ cl exp { 2 π i [ ( δ ¯ τ cl ) ( τ τ ) ± ( τ 2 τ 2 ) ] } exp { 2 π [ c 1 ( τ + τ ) c 2 ( τ τ ) 2 + c 3 ( τ + τ ) 2 ] } exp [ 2 π c 4 ( τ 2 + τ 2 ) ] d τ d τ .
P j = π 4 c ε 0 n j 0 w j 2 E j 2 , P 0 = 2 P 1 P 2 ( P 1 + P 2 ) ,
L NL = w 1 w 2 w 3 [ 8 π χ eff 2 ( P 1 + P 2 ) n 10 n 20 n 30 c ε 0 λ 3 2 cos 2 ( ρ 3 2 ) ] 1 2 ,
Ω = 1 2 π ( 2 δ k + δ k δ ρ 1 ) 1 2 , τ cl = Ω L 2 ,
τ = Ω z 3 2 , τ = Ω z 3 2 , c 1 = 1 π Ω δ α ,
c 2 = w 3 2 4 π Ω 2 δ k 2 , c 3 = 1 4 π Ω 2 w 3 2 δ ρ 1 2 ,
c 4 = 2 δ ρ 2 2 π Ω 2 w 3 2 , δ ¯ = Δ k 0 π Ω ± τ cl ,
Ω 2 = δ k π = 2 L j = 1 3 n j ( L ) n j ( 0 ) λ j ξ j 2 δ x L j = 1 3 n j λ j x ξ j x = x 0 β g j = 1 3 n j 0 λ j β ¯ j ξ j ,
δ k π β j = 1 3 n j 0 λ j β ¯ j ξ j g .
P 3 = 4 P 0 L NL 2 Ω 2 0 τ cl exp { 2 π i [ ( δ ¯ τ cl ) τ ± τ 2 ] } d τ 2 .
F 0 ( τ cl , δ ¯ ) 0 τ cl exp { 2 π i [ ( δ ¯ τ cl ) τ ± τ 2 ] } d τ = 1 2 exp [ i π 2 ( δ ¯ τ cl ) 2 ] { C ( τ cl δ ¯ ) + C ( τ cl + δ ¯ ) ± i [ S ( τ cl δ ¯ ) + S ( τ cl + δ ¯ ) ] } ,
F 1 ( τ cl ) F 0 ( τ cl , 0 ) ,
P 3 = P 0 L 2 L NL 2 D g 1 ( τ cl ) P 0 L 2 L NL 2 [ 1 π 2 ( Ω L ) 4 45 16 + ] ,
D g 1 ( τ cl ) [ C 2 ( τ cl ) + S 2 ( τ cl ) ] τ cl 2 .
P 3 0.6158 P 0 ( L opt L NL ) 2 , L opt = 2 τ opt Ω 1 ,
Δ x tol = 4.838 L j = 1 3 n j 0 β ¯ j λ j ξ j 1 .
P 3 = 4 P 0 L NL 2 Ω 2 0 τ cl exp { 2 π i [ ( δ ¯ τ cl ) τ ± τ 2 ] 2 π c 1 τ } d τ 2 × exp ( α 3 L ) 4 P 0 exp ( α 3 L ) L NL 2 Ω 2 F 2 ( τ cl , δ ¯ , c 1 ) 2 .
F 2 ( τ cl , δ ¯ , c 1 ) = [ 1 ± π c 1 ( δ ¯ τ cl ) ] F 0 ( τ cl , δ ¯ ) c 1 exp ( i π δ ¯ τ cl ) sin ( π δ ¯ τ cl ) + ,
F 2 2 δ ¯ = 0 ,
F 0 2 δ ¯ ± 2 π F 1 2 c 1 π τ cl ( F 1 + F 1 * ) 0 .
δ ¯ = δ ¯ opt c 1 τ cl Re ( F 1 ) F 1 2 τ cl Im ( F 1 ) ,
= c 1 [ cos ( π τ cl 2 2 ) C ( τ cl ) + sin ( π τ cl 2 2 ) S ( τ cl ) C ( τ cl ) 2 + S ( τ cl ) 2 τ cl ] [ cos ( π τ cl 2 2 ) S ( τ cl ) sin ( π τ cl 2 2 ) C ( τ cl ) ] 1 .
δ ¯ opt ± 2 π c 1 τ cl 15 ( τ cl + π 2 τ cl 5 105 + ) .
P 3 4 P 0 L NL 2 Ω 2 exp ( α 3 L ) F 0 ( τ cl , δ ¯ ) 2 ( 1 2 π τ cl c 1 ) P 0 L 2 L NL 2 D g 1 ( τ cl ) exp ( 2 π τ cl c Σ ) ,
c Σ = ( 2 π Ω ) 1 ( α 1 + α 2 + α 3 ) .
P 3 δ ¯ = 0 , P 3 τ cl = 0 .
τ cl [ F 1 ( τ cl ) 2 exp ( 2 π τ cl c Σ ) ] = 0 ,
F 1 ( τ cl ) + F 1 * ( τ cl ) 2 π c Σ F 1 ( τ cl ) 2 = 0 .
δ L τ 1 2 π c Σ τ 0 τ 1 π 1 1.086 c Σ ,
L opt 2 Ω ( τ 0 1.086 ν = 1 3 α ν 2 π Ω ) 2.419 Ω 0.3457 Ω 2 ν = 1 3 α ν .
τ + = τ + τ τ cl , τ = τ τ .
P 3 = 2 P 0 L NL 2 Ω 2 τ cl τ cl d τ τ τ cl τ cl τ exp ( 2 π c 2 τ 2 ) × exp { 2 π i [ ( δ ¯ τ cl ) τ ± ( τ + + τ cl ) τ ] } d τ + ,
P 3 = 2 P 0 π L NL 2 Ω 2 τ cl τ cl exp [ 2 π ( i δ ¯ τ c 2 τ 2 ) ] × sin [ 2 π ( τ cl τ ) τ ] τ d τ P 0 L 2 L NL 2 G 2 ( δ ¯ , τ cl , c 2 ) τ cl 2 .
δ ¯ opt = 0 .
G 3 = 1 π 0 τ cl exp ( 2 π c 2 τ 2 ) sin [ 2 π ( τ cl τ ) τ ] τ d τ ,
G 3 1 π 0 τ cl ( 1 2 π c 2 τ 2 ) sin [ 2 π ( τ cl τ ) τ ] τ 1 d τ ,
G 3 = C 2 ( τ cl ) + S 2 ( τ cl ) ± c 2 τ cl Im ( F 1 ) ,
G 3 = τ cl 2 ( 1 c 2 π τ cl 2 3 π 2 τ cl 4 45 + ) .
1 π 0 τ cl sin [ 2 π ( τ cl τ ) τ ] τ d τ = C 2 ( τ cl ) + S 2 ( τ cl ) .
D g 2 ( τ cl , c 2 ) G 3 ( τ cl , c 2 ) τ cl 2 .
G 3 2 0 τ cl exp ( 2 π c 2 τ 2 ) ( τ cl τ ) d τ = τ cl 2 c 2 erf ( τ cl 2 π c 2 ) + 1 2 π c 2 [ exp ( 2 π c 2 τ cl 2 ) 1 ] .
D t ( τ t ) { π τ t erf ( τ t ) + 1 τ t 2 [ exp ( τ t 2 ) 1 ] } ,
P 3 = P 0 L 2 L NL 2 D t , Ω t 2 1 2 w 3 δ k , τ t Ω t L 2 .
A ̃ 3 = i b ̃ L exp ( r 3 2 w 3 2 ) exp ( i X ) sinc ( X ) ,
X = L ( Δ k 0 + δ k r 3 ) 2 .
Δ x t = 2.558 L t w 3 L j = 1 3 n j 0 λ j β ¯ j ξ j ,
w max = 2.558 L t δ x t L j = 1 3 n j 0 λ j β ¯ j ξ j .
F 0 ( τ cl , δ ¯ ) 2 = 1 4 [ C ( τ cl δ ¯ ) + C ( τ cl + δ ¯ ) ] 2 + 1 4 [ S ( τ cl δ ¯ ) + S ( τ cl + δ ¯ ) ] 2 .
F 0 ( τ cl ± δ ¯ ) 2 1 4 [ C ( τ cl ± δ ¯ ) + 0.5 ] 2 + 1 4 [ S ( τ cl ± δ ¯ ) + 0.5 ] 2 .
Δ k b π Δ δ ¯ Ω = π Ω ( Ω L 2 δ ¯ 0 ) .
t 1 = [ t 0 2 + ( C 1 t 0 ) 2 ] 1 2
t 2 = [ t 0 2 2 + 2 ( C 2 t 0 ) 2 ] 1 2 ,
C 2 = 1 2 [ C 1 + 2 π ( δ u Ω ) 2 sgn ( δ k ) ] ,
n ( r , κ ̂ ) = n ( r , κ ̂ ( r ) ) = n ( r ( κ ) ) = n ( κ ) = n ( s )
S ( r ) = ( κ ̂ ) S ( r ) = d S ( r ( κ ) ) d κ = n ( κ ) ,
S ( r ) n ( r ) = κ ̂ = s ̂ ( s ̂ κ ̂ ) = d r ( s ) d s ( s ̂ κ ̂ ) d r ( s ) d s ρ ,
d d s ( n d r d s ) d d s S + d d s ( n ρ ) = s ̂ S + s ̂ ( n ρ ) = κ ̂ S + ρ S + s ̂ ( n ρ ) = 1 2 n n 2 + ρ ( n κ ̂ ) + s ̂ ( n ρ ) .
d d t = n n 0 d d s ,
d 2 r d t 2 1 2 n 2 + n ρ ( n κ ̂ ) + n s ̂ ( n ρ ) .
n 2 = n d S d κ = n 0 d S d t d s d κ n 0 d S d t 1 cos ρ ,
S ( t ) = S 0 + cos ρ n 0 0 t n 2 ( r ( t ) ) d t ,
d 2 r d t 2 = 1 2 ε g ̂ , r t = 0 = r 0 , d r d t t = 0 = n ( r 0 ) n 0 ( e ̂ z + ρ ) ,
r ( t ) = ε t 2 4 g ̂ + n ( r 0 ) n 0 ( e ̂ z + ρ ) t + r 0 .
S ( t ) n 0 [ h 2 t + ε h 2 ( g + ρ g ̂ ) t 2 + ε 2 t 3 12 ] ,
h = [ 1 + ε ( r 0 g ̂ ) ] 1 2 .
S ( r ) = n 0 z [ 1 + ε 4 ( 2 g r + g z ) ] + O ( ε 2 z 3 ) + O ( ε ρ z 3 ) .

Metrics