Abstract

The propagation of partially coherent Hermite-cosh-Gaussian (H-ChG) beams through atmospheric turbulence is studied in detail. The analytical expression for the angular spread of partially coherent H-ChG beams in turbulence is derived. It is shown that the angular spread of partially coherent H-ChG beams with smaller spatial correlation length σ0, smaller waist width w 0, smaller beam parameter Ω0, and larger beam orders m, n is less affected by turbulence than that of partially coherent H-ChG beams with larger σ0, w 0, Ω0, and smaller m, n. Under a certain condition partially coherent H-ChG beams may generate the same angular spread as a fully coherent Gaussian beam in free space and also in atmospheric turbulence. The angular spread of partially coherent Hermite-Gaussian (H-G), cosh-Gaussian (ChG), Gaussian Schell-model (GSM) beams, and fully coherent H-ChG, H-G, ChG, Gaussian beams is studied and treated as special cases of partially coherent H-ChG beams. The results are interpreted physically.

©2008 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Spreading and directionality of partially coherent Hermite-Gaussian beams propagating through atmospheric turbulence

Xiaoling Ji, Xiaowen Chen, and Baida Lü
J. Opt. Soc. Am. A 25(1) 21-28 (2008)

Comparative study of the beam-width spreading of partially coherent Hermite-sinh-Gaussian beams in atmospheric turbulence

Jinhong Li, Ailin Yang, and Baida Lü
J. Opt. Soc. Am. A 25(11) 2670-2679 (2008)

Angular spread and directionality of the Hermite-Gaussian array beam propagating through atmospheric turbulence

Xiaoqing Li and Xiaoling Ji
Appl. Opt. 48(22) 4338-4347 (2009)

More Recommended Articles

References

  • View by:
  • Article Order
  • |
  • Year
  • |
  • Author
  • |
  • Publication
  1. E. Collett and E. Wolf, “Is complete spatial coherence necessary for the generation of highly directional light beams?” Opt. Lett. 2, 27–29 (1978).
    [Crossref] [PubMed]
  2. E. Wolf and E. Collett, “Partially coherent sources which produce the same far-field intensity distribution as a laser,” Opt. Commun. 25, 293–296 (1978).
    [Crossref]
  3. P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29, 256–260 (1979).
    [Crossref]
  4. J. D. Farina, L. M. Narducci, and E. Collett, “Generation of highly directional beams from a globally incoherent source,” Opt. Commun. 32, 203–208 (1980).
    [Crossref]
  5. T. Shirai, A. Dogariu, and E. Wolf, “Directionality of Gaussian Schell-model beams propagating in atmospheric turbulence,” Opt. Lett. 28, 610–612 (2003).
    [Crossref] [PubMed]
  6. X. Ji, X. Chen, and B. Lü, “Spreading and directionality of partially coherent Hermite-Gaussian beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 25, 21–28 (2008).
    [Crossref]
  7. L. W. Casperson and A. A. Tovar, “Hermite-sinusoidal-Gaussian beams in complex optical systems,” J. Opt. Soc. Am. A 15, 954–961 (1998).
    [Crossref]
  8. A. A. Tovar and L. W. Casperson, “Production and propagation of Hermite-sinusoidal-Gaussian laser beams,” J. Opt. Soc. Am. A 15, 2425–2432 (1998).
    [Crossref]
  9. C. Y. Young, Y. V. Gilchrest, and B. R. Macon. “Turbulence induced beam spreading of higher order mode optical waves,” Opt. Eng. 41, 1097–1103 (2002).
    [Crossref]
  10. H. T. Eyyuboğlu and Y. Baykal. “Analysis of reciprocity of cos-Gaussian and cosh-Gaussian laser beams in a turbulent atmosphere,” Opt. Express 12, 4659–4674 (2004).
    [Crossref] [PubMed]
  11. H. T. Eyyuboğlu, “Propagation of Hermite-cosh-Gaussian laser beams in turbulent atmosphere,” Opt. Commun. 245, 37–47 (2005).
    [Crossref]
  12. H. T. Eyyuboğlu and Y. Baykal, “Average intensity and spreading of cosh-Gaussian laser beams in the turbulent atmosphere,” Appl. Opt. 44, 976–983 (2005).
    [Crossref] [PubMed]
  13. H. T. Eyyuboğlu and Y. Baykal. “Hermite-sine-Gaussian and Hermite-sinh-Gaussian laser beams in turbulent atmosphere,” J. Opt. Soc. Am. A 22, 2709–2718 (2005).
    [Crossref]
  14. Y. Cai and S. He. “Average intensity and spreading of an elliptical Gaussian beam propagating in a turbulent atmosphere,” Opt. Lett. 31, 568–570 (2006).
    [Crossref] [PubMed]
  15. Y. Cai and S. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express,  14, 1353–1367 (2006).
    [Crossref] [PubMed]
  16. H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Complex degree of coherence for partially coherent general beams in atmospheric turbulence,” J. Opt. Soc. Am. A 24, 2891–2900 (2007).
    [Crossref]
  17. H. T. Eyyubolu, Y. Baykal, and Y. Cai, “Degree of polarization for partially coherent general beams in turbulent atmosphere,” Appl. Phys. B: Lasers Opt. 89, 91–97 (2007).
    [Crossref]
  18. H. T. Eyyuboğlu and Y. Baykal, “Transmittance of partially coherent cosh-Gaussian, cos-Gaussian and annular beams in turbulence,” Opt. Commun. 278, 17–22 (2007).
    [Crossref]
  19. Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89, 041117 (2006).
    [Crossref]
  20. G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19, 1592–1598 (2002).
    [Crossref]
  21. A. Dogariu and S. Amarande, “Propagation of partially coherent beams: turbulence-induced degradation,” Opt. Lett. 28, 10–12 (2003).
    [Crossref] [PubMed]
  22. M. Zahid and M. S. Zubairy, “Directionality of partially coherent Bessel-Gauss beams,” Opt. Commun. 70, 361–364 (1989).
    [Crossref]
  23. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 1998).
  24. S. Wang, C. Ouyang, and M. A. Plonus, “Optical beam propagation for a partially coherent source in the turbulent atmosphere,” J. Opt. Soc. Am. 69, 1297–1304 (1979).
    [Crossref]
  25. H. T. Yura, “Mutual coherence function of a finite cross section optical beam propagating in a turbulent medium,” Appl. Opt. 11, 1399–1406 (1972).
    [Crossref] [PubMed]
  26. V. A. Banakh and V. L. Mironov, “Phase approximation of the Huygens-Kirhhoff method in problems of laser beam propagation in the turbulent atmosphere,” Opt. Lett. 1, 172–174 (1977).
    [Crossref] [PubMed]
  27. V. A. Banakh and V. L. Mironov, “Phase approximation of the Huygens-Kirhhoff method in problems of spase-limited optical-beam propagation in the turbulent atmosphere,” Opt. Lett. 4, 259–261 (1979).
    [Crossref] [PubMed]
  28. C. Leader, “Atmospheric propagation of partially coherent radiation,” J. Opt. Soc. Am. 68, 175–178 (1978).
    [Crossref]

2008 (1)

2007 (3)

H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Complex degree of coherence for partially coherent general beams in atmospheric turbulence,” J. Opt. Soc. Am. A 24, 2891–2900 (2007).
[Crossref]

H. T. Eyyubolu, Y. Baykal, and Y. Cai, “Degree of polarization for partially coherent general beams in turbulent atmosphere,” Appl. Phys. B: Lasers Opt. 89, 91–97 (2007).
[Crossref]

H. T. Eyyuboğlu and Y. Baykal, “Transmittance of partially coherent cosh-Gaussian, cos-Gaussian and annular beams in turbulence,” Opt. Commun. 278, 17–22 (2007).
[Crossref]

2006 (3)

2005 (3)

2004 (1)

2003 (2)

2002 (2)

G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19, 1592–1598 (2002).
[Crossref]

C. Y. Young, Y. V. Gilchrest, and B. R. Macon. “Turbulence induced beam spreading of higher order mode optical waves,” Opt. Eng. 41, 1097–1103 (2002).
[Crossref]

1998 (2)

1989 (1)

M. Zahid and M. S. Zubairy, “Directionality of partially coherent Bessel-Gauss beams,” Opt. Commun. 70, 361–364 (1989).
[Crossref]

1980 (1)

J. D. Farina, L. M. Narducci, and E. Collett, “Generation of highly directional beams from a globally incoherent source,” Opt. Commun. 32, 203–208 (1980).
[Crossref]

1979 (3)

1978 (3)

1977 (1)

1972 (1)

Amarande, S.

Andrews, L. C.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 1998).

Banakh, V. A.

Baykal, Y.

Cai, Y.

Casperson, L. W.

Chen, X.

Collett, E.

J. D. Farina, L. M. Narducci, and E. Collett, “Generation of highly directional beams from a globally incoherent source,” Opt. Commun. 32, 203–208 (1980).
[Crossref]

E. Wolf and E. Collett, “Partially coherent sources which produce the same far-field intensity distribution as a laser,” Opt. Commun. 25, 293–296 (1978).
[Crossref]

E. Collett and E. Wolf, “Is complete spatial coherence necessary for the generation of highly directional light beams?” Opt. Lett. 2, 27–29 (1978).
[Crossref] [PubMed]

De Santis, P.

P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29, 256–260 (1979).
[Crossref]

Dogariu, A.

Eyyuboglu, H. T.

Eyyubolu, H. T.

H. T. Eyyubolu, Y. Baykal, and Y. Cai, “Degree of polarization for partially coherent general beams in turbulent atmosphere,” Appl. Phys. B: Lasers Opt. 89, 91–97 (2007).
[Crossref]

Farina, J. D.

J. D. Farina, L. M. Narducci, and E. Collett, “Generation of highly directional beams from a globally incoherent source,” Opt. Commun. 32, 203–208 (1980).
[Crossref]

Gbur, G.

Gilchrest, Y. V.

C. Y. Young, Y. V. Gilchrest, and B. R. Macon. “Turbulence induced beam spreading of higher order mode optical waves,” Opt. Eng. 41, 1097–1103 (2002).
[Crossref]

Gori, F.

P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29, 256–260 (1979).
[Crossref]

Guattari, G.

P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29, 256–260 (1979).
[Crossref]

He, S.

Ji, X.

Leader, C.

Lü, B.

Macon, B. R.

C. Y. Young, Y. V. Gilchrest, and B. R. Macon. “Turbulence induced beam spreading of higher order mode optical waves,” Opt. Eng. 41, 1097–1103 (2002).
[Crossref]

Mironov, V. L.

Narducci, L. M.

J. D. Farina, L. M. Narducci, and E. Collett, “Generation of highly directional beams from a globally incoherent source,” Opt. Commun. 32, 203–208 (1980).
[Crossref]

Ouyang, C.

Palma, C.

P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29, 256–260 (1979).
[Crossref]

Phillips, R. L.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 1998).

Plonus, M. A.

Shirai, T.

Tovar, A. A.

Wang, S.

Wolf, E.

Young, C. Y.

C. Y. Young, Y. V. Gilchrest, and B. R. Macon. “Turbulence induced beam spreading of higher order mode optical waves,” Opt. Eng. 41, 1097–1103 (2002).
[Crossref]

Yura, H. T.

Zahid, M.

M. Zahid and M. S. Zubairy, “Directionality of partially coherent Bessel-Gauss beams,” Opt. Commun. 70, 361–364 (1989).
[Crossref]

Zubairy, M. S.

M. Zahid and M. S. Zubairy, “Directionality of partially coherent Bessel-Gauss beams,” Opt. Commun. 70, 361–364 (1989).
[Crossref]

Appl. Opt. (2)

Appl. Phys. B: Lasers Opt. (1)

H. T. Eyyubolu, Y. Baykal, and Y. Cai, “Degree of polarization for partially coherent general beams in turbulent atmosphere,” Appl. Phys. B: Lasers Opt. 89, 91–97 (2007).
[Crossref]

Appl. Phys. Lett. (1)

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89, 041117 (2006).
[Crossref]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (6)

Opt. Commun. (6)

M. Zahid and M. S. Zubairy, “Directionality of partially coherent Bessel-Gauss beams,” Opt. Commun. 70, 361–364 (1989).
[Crossref]

H. T. Eyyuboğlu and Y. Baykal, “Transmittance of partially coherent cosh-Gaussian, cos-Gaussian and annular beams in turbulence,” Opt. Commun. 278, 17–22 (2007).
[Crossref]

E. Wolf and E. Collett, “Partially coherent sources which produce the same far-field intensity distribution as a laser,” Opt. Commun. 25, 293–296 (1978).
[Crossref]

P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29, 256–260 (1979).
[Crossref]

J. D. Farina, L. M. Narducci, and E. Collett, “Generation of highly directional beams from a globally incoherent source,” Opt. Commun. 32, 203–208 (1980).
[Crossref]

H. T. Eyyuboğlu, “Propagation of Hermite-cosh-Gaussian laser beams in turbulent atmosphere,” Opt. Commun. 245, 37–47 (2005).
[Crossref]

Opt. Eng. (1)

C. Y. Young, Y. V. Gilchrest, and B. R. Macon. “Turbulence induced beam spreading of higher order mode optical waves,” Opt. Eng. 41, 1097–1103 (2002).
[Crossref]

Opt. Express (2)

Opt. Lett. (6)

Other (1)

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 1998).

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 (6)

Fig. 1.
Fig. 1. Angular spread θ sp of a partially coherent H-ChG beam versus the spatial correlation length σ0. The calculation parameters are z=10km, m=n=1, w 0=3cm, Ω0=100m-1.

Download Full Size | PPT Slide | PDF

Fig. 2.
Fig. 2. Angular spread θ sp of a partially coherent H-ChG beam versus the waist width w 0. The calculation parameters are z=10km, m=n=1, σ0=1.732cm, Ω0=100m-1, C 2 n =10-14m-2/3.

Download Full Size | PPT Slide | PDF

Fig. 3.
Fig. 3. Angular spread θ sp of a partially coherent H-ChG beam versus the beam parameter Ω0. The calculation parameters are z=10km, m=n=1, w 0=3cm, σ0=4cm, C 2 n =10-15m-2/3.

Download Full Size | PPT Slide | PDF

Fig. 4.
Fig. 4. Angular spread θ sp of a partially coherent H-ChG beam versus the beam order m, n. The calculation parameters are z=10km, w 0=1cm, σ0=2 cm, Ω0=300m-1, C 2 n =10-14m-2/3.

Download Full Size | PPT Slide | PDF

Fig. 5.
Fig. 5. Normalized rms width w(z) of the four equivalent partially coherent beams and the corresponding fully coherent Gaussian beam versus propagation distance z in free space and in turbulence. a: the corresponding fully coherent Gaussian beam; b: the equivalent GSM beam; c: the equivalent partially coherent H-G beam: d: the equivalent partially coherent H-ChG beam; e: the equivalent partially coherent ChG beam. The calculation parameters are listed in Table 1, and the other parameters are λ=1.06µm, C 2 n =10-14m-2/3.

Download Full Size | PPT Slide | PDF

Fig. 6.
Fig. 6. Normalized rms width w(z) of the three fully coherent beams and the corresponding fully coherent Gaussian beam versus propagation distance z. a: the corresponding fully coherent Gaussian beam; b: the equivalent fully coherent ChG beam; c: the equivalent fully coherent H-ChG beam; d: the equivalent fully coherent H-G beams. The calculation parameters are listed in Table 2, and the other parameters are λ=1.06µm, C 2 n =10-14m-2/3.

Download Full Size | PPT Slide | PDF

Tables (2)

Tables Icon

Table 1. Beam parameters relating to Fig. 5.

View Table | View all tables in this article
Tables Icon

Table 2. Beam parameters relating to Fig. 6.

View Table | View all tables in this article

Equations (82)

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

U ( ρ , z = 0 ) = H m ( 2 w 0 ρ x ) H n ( 2 w 0 ρ y ) exp ⁡ ( − ρ x 2 + ρ y 2 w 0 2 ) cosh ⁡ ( Ω 0 ρ x + Ω 0 ρ y ) ,
W ( ρ 1 , ρ 2 , z = 0 ) = H m ( 2 w 0 ρ 1 x ) H n ( 2 w 0 ρ 1 y ) exp ⁡ ( − ρ 1 x 2 + ρ 1 y 2 w 0 2 )
× cosh ( Ω 0 ρ 1 x + Ω 0 ρ 1 y ) exp ⁡ [ − ( ρ 1 x − ρ 2 x ) 2 2 σ 0 2 ]
× H m ( 2 w 0 ρ 2 x ) H n ( 2 w 0 ρ 2 y ) exp ⁡ ( − ρ 2 x 2 + ρ 2 y 2 w 0 2 )
× cosh ( Ω 0 ρ 2 x + Ω 0 ρ 2 y ) exp ⁡ [ − ( ρ 1 y − ρ 2 y ) 2 2 σ 0 2 ] ,
W ( ρ 1 ′ , ρ 2 ′ , z ) = ( k 2 π z ) 2 ∬ d 2 ρ 1 ∬ d 2 ρ 2 W ( ρ 1 , ρ 2 , z = 0 )
× exp ⁡ { i k 2 z [ ( ρ 1 ′ − ρ 1 ) 2 − ( ρ 2 ′ − ρ 2 ) 2 ] } 〈 exp ⁡ [ ψ ( ρ 1 , ρ 1 ′ ) + ψ * ( ρ 2 , ρ 2 ′ ) ] 〉 m ,
〈 exp ⁡ [ ψ ( ρ 1 , ρ 1 ′ ) + ψ * ( ρ 2 , ρ 2 ′ ) ] 〉 m
≈ exp { − 1 ρ 0 2 [ ( ρ 1 − ρ 2 ) 2 + ( ρ 1 − ρ 2 ) · ( ρ 1 ′ − ρ 2 ′ ) + ( ρ 1 ′ − ρ 2 ′ ) 2 ] } ,
ρ 0 = ( 0.545 C n 2 k 2 z ) − 3 5 ,
u = ρ 2 + ρ 1 2 , v = ρ 2 − ρ 1 .
I ( ρ ′ , z ) = W ( ρ ′ , ρ ′ , z )
  = ( k 4 π z ) 2 ∬ d 2 u ∬ d 2 v exp ( − 2 u 2 w 0 2 ) exp ( − v 2 ε 2 ) exp ( − i k z u · v ) exp ( i k z ρ ′ · v )
  × H m [ 2 w 0 ( u x − v x 2 ) ] H m [ 2 w 0 ( u x + v x 2 ) ] H n [ 2 w 0 ( u y − v y 2 ) ] H n [ 2 w 0 ( u y + v y 2 ) ]
× { exp [ 2 Ω 0 ( u x + u y ) ] + exp [ − Ω 0 ( v x + v y ) ]
+ exp [ Ω 0 ( v x + v y ) ] + exp [ − 2 Ω 0 ( u x + u y ) ] }
1 ε 2 = 1 2 w 0 2 + 1 2 σ 0 2 + 1 ρ 0 2 .
w ( z ) = ∫ ρ ′ 2 I ( ρ ′ , z ) d 2 ρ ′ ∫ I ( ρ ′ , z ) d 2 ρ ′ .
w ( z ) = P + Q k 2 z 2 + 4 ( 0.545 C n 2 k 1 3 ) 6 5 z 16 5 ,
P = R 1 R 0 ,
Q = R 2 R 0 ,
R 0 = exp ⁡ ( w 0 2 Ω 0 2 ) L m 0 ( − w 0 2 Ω 0 2 ) L n 0 ( − w 0 2 Ω 0 2 ) + 1 ,
R 1 = w 0 2 { exp ⁡ ( w 0 2 Ω 0 2 ) ( 1 + w 0 2 Ω 0 2 2 L m 0 ( − w 0 2 Ω 0 2 ) L n 0 ( − w 0 2 Ω 0 2 ) + 1 + 2 w 0 2 Ω 0 2 2
× [ L n 0 ( − w 0 2 Ω 0 2 ) L m − 1 1 ( − w 0 2 Ω 0 2 ) + L m 0 ( − w 0 2 Ω 0 2 ) L n − 1 1 ( − w 0 2 Ω 0 2 ) ] + w 0 2 Ω 0 2
× [ L n 0 ( − w 0 2 Ω 0 2 ) L m − 2 2 ( − w 0 2 Ω 0 2 ) + L m 0 ( − w 0 2 Ω 0 2 ) L n − 2 2 ( − w 0 2 Ω 0 2 ) ] ) + m + n + 1 2 } ,
R 2 = exp ( w 0 2 Ω 0 2 ) { 2 ( 1 w 0 2 + 1 σ 0 2 ) L m 0 ( − w 0 2 Ω 0 2 ) L n 0 ( − w 0 2 Ω 0 2 ) + 2 w 0 2 [ L m − 1 1 ( − w 0 2 Ω 0 2 )
× L n 0 ( − w 0 2 Ω 0 2 ) + L m 0 ( − w 0 2 Ω 0 2 ) L n − 1 1 ( − w 0 2 Ω 0 2 ) ] } + 2 ( 1 w 0 2 + 1 σ 0 2 − Ω 0 2 ) + 2 ( m + n ) w 0 2
θ sp ( z ) = w ( z ) z | z → ∞ = Q k 2 + 4 ( 0.545 C n 2 k 1 3 ) 6 5 z 6 5 .
θ sp ( z ) = Q 1 k 2 + 4 ( 0.545 C n 2 k 1 3 ) 6 5 z 6 5 ,
Q 1 = 2 ( m + n + 1 w 0 2 + 1 σ 0 2 ) .
θ sp ( z ) = Q 2 k 2 + 4 ( 0.545 C n 2 k 1 3 ) 6 5 z 6 5 ,
Q 2 = 2 [ ( 1 w 0 2 + 1 σ 0 2 ) − Ω 0 2 1 + exp ⁡ ( w 0 2 Ω 0 2 ) ] .
θ sp ( z ) = Q 3 k 2 + 4 ( 0.545 C n 2 k 1 3 ) 6 5 z 6 5 ,
Q 3 = 2 ( 1 w 0 2 + 1 σ 0 2 ) .
θ sp ( z ) = Q 4 k 2 + 4 ( 0.545 C n 2 k 1 3 ) 6 5 z 6 5 ,
Q 4 = exp ⁡ ( w 0 2 Ω 0 2 ) [ 2 w 0 2 L m 0 ( − w 0 2 Ω 0 2 ) L n 0 ( − w 0 2 Ω 0 2 ) + 2 w 0 2 [ L m − 1 1 ( − w 0 2 Ω 0 2 ) L n 0 ( − w 0 2 Ω 0 2 )
+ L m 0 ( − w 0 2 Ω 0 2 ) L n − 1 1 ( − w 0 2 Ω 0 2 ) ] + 2 ( 1 w 0 2 − Ω 0 2 ) + 2 ( m + n ) w 0 2 .
θ sp ( z ) = Q 5 k 2 + 4 ( 0.545 C n 2 k 1 3 ) 6 5 z 6 5 ,
Q 5 = 2 ( m + n + 1 w 0 2 ) .
θ sp ( z ) = Q 6 k 2 + 4 ( 0.545 C n 2 k 1 3 ) 6 5 z 6 5 ,
Q 6 = 2 [ ( 1 w 0 2 − Ω 0 2 1 + exp ⁡ ( w 0 2 Ω 0 2 ) ] .
θ sp ( z ) = Q 7 k 2 + 4 ( . 0545 C n 2 k 1 3 ) 6 5 z 6 5 ,
Q 7 = 2 w 0 2 .
Q = Q 1 = Q 2 = Q 3 = Q 7
Q 4 = Q 5 = Q 6 = Q 7
w ( z ) = F F 0 ,
F 0 = ∫ I ( ρ ′ , z ) d 2 ρ ′ ,
F 0 = ∫ ρ ′ 2 I ( ρ ′ , z ) d 2 ρ ′ .
F 0 = ∬ W ( 0 ) ( ρ , ρ , z = 0 ) d 2 ρ
= 1 2 2 m + n − 1 m ! n ! w 0 2 π [ exp ( w 0 2 Ω 0 2 ) L m 0 ( − w 0 2 Ω 0 2 ) L n 0 ( − w 0 2 Ω 0 2 ) + 1 ] .
F = F 1 + F 2 + F 3 + F 4 ,
F 1 = ∫ ρ ′ 2 I 1 ( ρ ′ , z ) d 2 ρ ′ ,
F 2 = ∫ ρ ′ 2 I 2 ( ρ ′ , z ) d 2 ρ ′ ,
F 3 = ∫ ρ ′ 2 I 3 ( ρ ′ , z ) d 2 ρ ′ ,
F 4 = ∫ ρ ′ 2 I 4 ( ρ ′ , z ) d 2 ρ ′ ,
I 1 ( ρ ′ , z ) = 1 4 ( k 2 π z ) 2 ∬ d 2 u ∬ d 2 v
× H m [ 2 w 0 ( u x − v x 2 ) ] H n [ 2 w 0 ( u y - v y 2 ) ] H m [ 2 w 0 ( u x + v x 2 ) ] H n [ 2 w 0 ( u y + v y 2 ) ]
  × exp ( − 2 u 2 w 0 2 ) exp ( − v 2 ε 2 ) exp ( − i k z u · v ) exp ( i k z ρ ′ · v ) exp [ 2 Ω 0 ( u x + u y ) ] ,
I 2 ( ρ ′ , z ) = 1 4 ( k 2 π z ) 2 ∬ d 2 u ∬ d 2 v
× H m [ 2 w 0 ( u x − v x 2 ) ] H n [ 2 w 0 ( u y - v y 2 ) ] H m [ 2 w 0 ( u x + v x 2 ) ] H n [ 2 w 0 ( u y + v y 2 ) ]
× exp ( − 2 u 2 w 0 2 ) exp ( − v 2 ε 2 ) exp ( − i k z u · v ) exp ( i k z ρ ′ · v ) exp [ Ω 0 ( v x + v y ) ] .
∫ x 2 exp ⁡ ( − i 2 π x s ) d x = − 1 ( 2 π ) 2 δ ″ ( s ) ,
F 1 = F 11 + F 12 ,
F 11 = − 1 4 ( z k ) 2 ∬ d 2 u ∬ d 2 v
× H m [ 2 w 0 ( u x − v x 2 ) ] H n [ 2 w 0 ( u y − v y 2 ) ] H m [ 2 w 0 ( u x + v x 2 ) ] H n [ 2 w 0 ( u y + v y 2 ) ]
× exp ⁡ ( − 2 u 2 w 0 2 ) exp ⁡ ( − v 2 ε 2 ) exp ⁡ ( − i k z u · v ) exp ⁡ [ 2 Ω 0 ( u x + u y ) ] δ ″ ( v x ) δ ( v y ) ,
F 12 = − 1 4 ( z k ) 2 ∬ d 2 u ∬ d 2 v
× H m [ 2 w 0 ( u x − v x 2 ) ] H n [ 2 w 0 ( u y − v y 2 ) ] H m [ 2 w 0 ( u x + v x 2 ) ] H n [ 2 w 0 ( u y + v y 2 ) ]
× exp ⁡ ( − 2 u 2 w 0 2 ) exp ⁡ ( − v 2 ε 2 ) exp ⁡ ( − i k z u · v ) exp ⁡ [ 2 Ω 0 ( u x + u y ) ] δ ( v x ) δ ″ ( v y ) ,
∫ f ( x ) δ ( x ) d x = f ( 0 ) ,
∫ exp ⁡ [ − ( x − y ) 2 ] H m ( x ) H n ( x ) d x = 2 n π m ! y n − m L n n − m ( − 2 y 2 ) ,
F 11 = − 1 4 ( z k ) 2 2 n n ! π w 0 2 exp ⁡ ( w 0 2 Ω 0 2 2 ) L n 0 ( − w 0 2 Ω 0 2 ) ∬ d v x d u x
× H m [ 2 w 0 ( u x − v x 2 ) ] H m [ 2 w 0 ( u x + v x 2 ) ] exp ⁡ ( 2 u x 2 w 0 2 )
× exp ⁡ ( − v x 2 ε 2 ) exp ⁡ ( − i k z u x v x ) exp ⁡ ( 2 Ω 0 u x ) δ ″ ( v x ) .
∫ exp ( − x 2 ) H m ( x + y ) H n ( x + z ) d x = 2 n π m ! y n − m z n − m L n n − m ( − 2 y z ) ,
∫ f ( x ) δ ″ ( x ) d x = f ″ ( 0 ) ,
F 11 = − 1 4 ( z k ) 2 2 m + n − 1 m ! n ! π w 0 2 exp ⁡ ( w 0 2 Ω 0 2 ) L n 0 ( − w 0 2 Ω 0 2 )
× [ ( − k 2 w 0 2 4 z 2 − 2 ε 2 ) L m 0 ( − w 0 2 Ω 0 2 ) ] − k 2 w 0 4 Ω 0 2 4 z 2 L m 0 ( − w 0 2 Ω 0 2 ) − k 2 w 0 4 Ω 0 2 z 2 L m − 2 2 ( − w 0 2 Ω 0 2 )
− ( 2 w 0 2 + k 2 w 0 2 2 z 0 ) L m − 1 1 ( − w 0 2 Ω 0 2 ) − k 2 w 0 4 Ω 0 2 z 2 L m − 1 1 ( − w 0 2 Ω 0 2 ) ] .
F 2 = F 21 + F 22 ,
F 21 = − 1 4 ( z k ) 2 2 m + n − 1 m ! n ! π w 0 2 [ ( − k 2 w 0 2 4 z 2 − 2 ε 2 ) + Ω 0 2 − m ( 2 w 0 2 + k 2 w 0 2 2 z 2 ) ] .
w ( z ) = P + Q k 2 z 2 + 4 ( 0.545 C n 2 k 1 3 ) 6 5 z 16 5 .

Metrics