Angular spread of partially coherent Hermite-cosh-Gaussian beams propagating through atmospheric turbulence

Ailin Yang; Entao Zhang; Xiaoling Ji; Baida Lü

doi:10.1364/OE.16.008366

Angular spread of partially coherent Hermite-cosh-Gaussian beams propagating through atmospheric turbulence

Ailin Yang, Entao Zhang, Xiaoling Ji, and Baida Lü

Optics Express
Vol. 16,
Issue 12,
pp. 8366-8380
(2008)
•https://doi.org/10.1364/OE.16.008366

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Ailin Yang, Entao Zhang, Xiaoling Ji, and Baida Lü, "Angular spread of partially coherent Hermite-cosh-Gaussian beams propagating through atmospheric turbulence," Opt. Express 16, 8366-8380 (2008)

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Related Topics
Table of Contents Category
- Coherence and Statistical Optics
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Atmospheric turbulence (010.1330)
- Coherence (030.1640)
- Diffractive optics (050.1970)

About this Article
History
- Original Manuscript: February 21, 2008
- Revised Manuscript: April 29, 2008
- Manuscript Accepted: May 7, 2008
- Published: May 23, 2008

Accessible

Open Access

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.

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References

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Year
|
Author
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Publication

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]
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]
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]
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]
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]
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]
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]
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]
H. T. Eyyuboğlu, “Propagation of Hermite-cosh-Gaussian laser beams in turbulent atmosphere,” Opt. Commun. 245, 37–47 (2005).
[Crossref]
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]
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]
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]
Y. Cai and S. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express, 14, 1353–1367 (2006).
[Crossref] [PubMed]
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]
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]
G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19, 1592–1598 (2002).
[Crossref]
A. Dogariu and S. Amarande, “Propagation of partially coherent beams: turbulence-induced degradation,” Opt. Lett. 28, 10–12 (2003).
[Crossref] [PubMed]
M. Zahid and M. S. Zubairy, “Directionality of partially coherent Bessel-Gauss beams,” Opt. Commun. 70, 361–364 (1989).
[Crossref]
L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 1998).
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]
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]
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]
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]
C. Leader, “Atmospheric propagation of partially coherent radiation,” J. Opt. Soc. Am. 68, 175–178 (1978).
[Crossref]

2008 (1)

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]

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)

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]

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]

Y. Cai and S. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express, 14, 1353–1367 (2006).
[Crossref] [PubMed]

2005 (3)

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

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]

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]

2004 (1)

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]

2003 (2)

A. Dogariu and S. Amarande, “Propagation of partially coherent beams: turbulence-induced degradation,” Opt. Lett. 28, 10–12 (2003).
[Crossref] [PubMed]

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]

2002 (2)

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]

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

1998 (2)

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]

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]

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)

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

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]

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]

1978 (3)

C. Leader, “Atmospheric propagation of partially coherent radiation,” J. Opt. Soc. Am. 68, 175–178 (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]

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]

Andrews, L. C.

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

Banakh, V. A.

Baykal, Y.

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]

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. 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]

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]

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]

Cai, Y.

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]

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]

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]

Y. Cai and S. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express, 14, 1353–1367 (2006).
[Crossref] [PubMed]

Casperson, L. W.

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]

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]

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.

A. Dogariu and S. Amarande, “Propagation of partially coherent beams: turbulence-induced degradation,” Opt. Lett. 28, 10–12 (2003).
[Crossref] [PubMed]

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]

Eyyuboglu, H. T.

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

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

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]

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]

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]

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.

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

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.

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]

Y. Cai and S. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express, 14, 1353–1367 (2006).
[Crossref] [PubMed]

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]

Ji, X.

Leader, C.

C. Leader, “Atmospheric propagation of partially coherent radiation,” J. Opt. Soc. Am. 68, 175–178 (1978).
[Crossref]

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.

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]

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.

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]

Shirai, T.

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]

Tovar, A. A.

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]

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]

Wang, S.

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]

Wolf, E.

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]

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

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]

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.

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]

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)

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]

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]

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)

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]

C. Leader, “Atmospheric propagation of partially coherent radiation,” J. Opt. Soc. Am. 68, 175–178 (1978).
[Crossref]

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

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

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. 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]

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]

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]

Opt. Commun. (6)

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]

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]

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

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]

Y. Cai and S. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express, 14, 1353–1367 (2006).
[Crossref] [PubMed]

Opt. Lett. (6)

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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.

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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.

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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.

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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.

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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.

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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.

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Tables (2)

Table 1. Beam parameters relating to Fig. 5.

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Table 2. Beam parameters relating to Fig. 6.

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Equations (82)

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U (ρ, z = 0) = H_{m} (\frac{\sqrt{2}}{w_{0}} ρ_{x}) H_{n} (\frac{\sqrt{2}}{w_{0}} ρ_{y}) \exp (- \frac{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} (\frac{\sqrt{2}}{w_{0}} ρ_{1 x}) H_{n} (\frac{\sqrt{2}}{w_{0}} ρ_{1 y}) \exp (- \frac{ρ_{1 x}^{2} + ρ_{1 y}^{2}}{w_{0}^{2}})

\times \cosh (Ω_{0} ρ_{1 x} + Ω_{0} ρ_{1 y}) \exp [- \frac{{(ρ_{1 x} - ρ_{2 x})}^{2}}{2 {σ_{0}}^{2}}]

\times H_{m} (\frac{\sqrt{2}}{w_{0}} ρ_{2 x}) H_{n} (\frac{\sqrt{2}}{w_{0}} ρ_{2 y}) \exp (- \frac{ρ_{2 x}^{2} + ρ_{2 y}^{2}}{w_{0}^{2}})

\times \cosh (Ω_{0} ρ_{2 x} + Ω_{0} ρ_{2 y}) \exp [- \frac{{(ρ_{1 y} - ρ_{2 y})}^{2}}{2 {σ_{0}}^{2}}],

W (ρ_{1}^{'}, ρ_{2}^{'}, z) = {(\frac{k}{2 π z})}^{2} \iint d^{2} ρ_{1} \iint d^{2} ρ_{2} W (ρ_{1}, ρ_{2}, z = 0)

\times \exp {\frac{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}

\approx \exp {- \frac{1}{ρ_{0}^{2}} [{(ρ_{1} - ρ_{2})}^{2} + (ρ_{1} - ρ_{2}) \cdot (ρ_{1}^{'} - ρ_{2}^{'}) + {(ρ_{1}^{'} - ρ_{2}^{'})}^{2}]},

ρ_{0} = {(0.545 C_{n}^{2} k^{2} z)}^{\frac{- 3}{5}},

u = \frac{ρ_{2} + ρ_{1}}{2}, v = ρ_{2} - ρ_{1} .

I (ρ', z) = W (ρ', ρ', z)

= {(\frac{k}{4 π z})}^{2} \iint d^{2} u \iint d^{2} v \exp (- \frac{2 u^{2}}{w_{0}^{2}}) \exp (- \frac{v^{2}}{ε^{2}}) \exp (- \frac{i k}{z} u \cdot v) \exp (\frac{i k}{z} ρ' \cdot v)

\times H_{m} [\frac{\sqrt{2}}{w_{0}} (u_{x} - \frac{v_{x}}{2})] H_{m} [\frac{\sqrt{2}}{w_{0}} (u_{x} + \frac{v_{x}}{2})] H_{n} [\frac{\sqrt{2}}{w_{0}} (u_{y} - \frac{v_{y}}{2})] H_{n} [\frac{\sqrt{2}}{w_{0}} (u_{y} + \frac{v_{y}}{2})]

\times {\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})]}

\frac{1}{ε^{2}} = \frac{1}{2 w_{0}^{2}} + \frac{1}{2 σ_{0}^{2}} + \frac{1}{ρ_{0}^{2}} .

w (z) = \sqrt{\frac{\int ρ'^{2} I (ρ', z) d^{2} ρ'}{\int I (ρ', z) d^{2} ρ'}} .

w (z) = \sqrt{P + \frac{Q}{k^{2}} z^{2} + 4 {(0.545 C_{n}^{2} k^{\frac{1}{3}})}^{\frac{6}{5}} z^{\frac{16}{5}}},

P = \frac{R_{1}}{R_{0}},

Q = \frac{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}) (\frac{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}) + \frac{1 + 2 w_{0}^{2} Ω_{0}^{2}}{2}

\times [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}

\times [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})]) + \frac{m + n + 1}{2}},

R_{2} = \exp (w_{0}^{2} Ω_{0}^{2}) {2 (\frac{1}{w_{0}^{2}} + \frac{1}{σ_{0}^{2}}) L_{m}^{0} (- w_{0}^{2} Ω_{0}^{2}) L_{n}^{0} ({- w}_{0}^{2} Ω_{0}^{2}) + \frac{2}{w_{0}^{2}} [L_{m - 1}^{1} (- w_{0}^{2} Ω_{0}^{2})

\times 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 (\frac{1}{w_{0}^{2}} + \frac{1}{σ_{0}^{2}} - Ω_{0}^{2}) + \frac{2 (m + n)}{w_{0}^{2}}

θ_{sp} (z) = \frac{w (z)}{z} |_{z \to \infty} = \sqrt{\frac{Q}{k^{2}} + {4 (0.545 C_{n}^{2} k^{\frac{1}{3}})}^{\frac{6}{5}} z^{\frac{6}{5}}} .

θ_{sp} (z) = \sqrt{\frac{Q_{1}}{k^{2}} + {4 (0.545 C_{n}^{2} k^{\frac{1}{3}})}^{\frac{6}{5}} z^{\frac{6}{5}}},

Q_{1} = 2 (\frac{m + n + 1}{w_{0}^{2}} + \frac{1}{σ_{0}^{2}}) .

θ_{sp} (z) = \sqrt{\frac{Q_{2}}{k^{2}} + {4 (0.545 C_{n}^{2} k^{\frac{1}{3}})}^{\frac{6}{5}} z^{\frac{6}{5}}},

Q_{2} = 2 [(\frac{1}{w_{0}^{2}} + \frac{1}{σ_{0}^{2}}) - \frac{Ω_{0}^{2}}{1 + \exp (w_{0}^{2} Ω_{0}^{2})}] .

θ_{sp} (z) = \sqrt{\frac{Q_{3}}{k^{2}} + {4 (0.545 C_{n}^{2} k^{\frac{1}{3}})}^{\frac{6}{5}} z^{\frac{6}{5}}},

Q_{3} = 2 (\frac{1}{w_{0}^{2}} + \frac{1}{σ_{0}^{2}}) .

θ_{sp} (z) = \sqrt{\frac{Q_{4}}{k^{2}} + {4 (0.545 C_{n}^{2} k^{\frac{1}{3}})}^{\frac{6}{5}} z^{\frac{6}{5}}},

Q_{4} = \exp (w_{0}^{2} Ω_{0}^{2}) [\frac{2}{w_{0}^{2}} L_{m}^{0} (- w_{0}^{2} Ω_{0}^{2}) L_{n}^{0} (- w_{0}^{2} Ω_{0}^{2}) + \frac{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 (\frac{1}{w_{0}^{2}} - Ω_{0}^{2}) + \frac{2 (m + n)}{w_{0}^{2}} .

θ_{sp} (z) = \sqrt{\frac{Q_{5}}{k^{2}} + 4 {(0.545 C_{n}^{2} k^{\frac{1}{3}})}^{\frac{6}{5}} z^{\frac{6}{5}}},

Q_{5} = 2 (\frac{m + n + 1}{w_{0}^{2}}) .

θ_{sp} (z) = \sqrt{\frac{Q_{6}}{k^{2}} + {4 (0.545 C_{n}^{2} k^{\frac{1}{3}})}^{\frac{6}{5}} z^{\frac{6}{5}}},

Q_{6} = 2 [(\frac{1}{w_{0}^{2}} - \frac{Ω_{0}^{2}}{1 + \exp (w_{0}^{2} Ω_{0}^{2})}] .

θ_{sp} (z) = \sqrt{\frac{Q_{7}}{k^{2}} + {4 (. 0545 C_{n}^{2} k^{\frac{1}{3}})}^{\frac{6}{5}} z^{\frac{6}{5}}},

Q_{7} = \frac{2}{w_{0}^{2}} .

Q = Q_{1} = Q_{2} = Q_{3} = Q_{7}

Q_{4} = Q_{5} = Q_{6} = Q_{7}

w (z) = \sqrt{\frac{F}{F_{0}}},

F_{0} = \int I (ρ', z) d^{2} ρ',

F_{0} = \int ρ'^{2} I (ρ', z) d^{2} ρ' .

F_{0} = \iint W^{(0)} (ρ, ρ, z = 0) d^{2} ρ

= \frac{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} = \int ρ'^{2} I_{1} (ρ', z) d^{2} ρ',

F_{2} = \int ρ'^{2} I_{2} (ρ', z) d^{2} ρ',

F_{3} = \int ρ'^{2} I_{3} (ρ', z) d^{2} ρ',

F_{4} = \int ρ'^{2} I_{4} (ρ', z) d^{2} ρ',

I_{1} (ρ', z) = \frac{1}{4} {(\frac{k}{2 π z})}^{2} \iint d^{2} u \iint d^{2} v

\times H_{m} [\frac{\sqrt{2}}{w_{0}} (u_{x} - \frac{v_{x}}{2})] H_{n} [\frac{\sqrt{2}}{w_{0}} (u_{y} - \frac{v_{y}}{2})] H_{m} [\frac{\sqrt{2}}{w_{0}} (u_{x} + \frac{v_{x}}{2})] H_{n} [\frac{\sqrt{2}}{w_{0}} (u_{y} + \frac{v_{y}}{2})]

\times \exp (- \frac{2 u^{2}}{w_{0}^{2}}) \exp (- \frac{v^{2}}{ε^{2}}) \exp (- \frac{i k}{z} u \cdot v) \exp (\frac{i k}{z} ρ' \cdot v) \exp [2 Ω_{0} (u_{x} + u_{y})],

I_{2} (ρ', z) = \frac{1}{4} {(\frac{k}{2 π z})}^{2} \iint d^{2} u \iint d^{2} v

\times H_{m} [\frac{\sqrt{2}}{w_{0}} (u_{x} - \frac{v_{x}}{2})] H_{n} [\frac{\sqrt{2}}{w_{0}} (u_{y} - \frac{v_{y}}{2})] H_{m} [\frac{\sqrt{2}}{w_{0}} (u_{x} + \frac{v_{x}}{2})] H_{n} [\frac{\sqrt{2}}{w_{0}} (u_{y} + \frac{v_{y}}{2})]

\times \exp (- \frac{2 u^{2}}{w_{0}^{2}}) \exp (- \frac{v^{2}}{ε^{2}}) \exp (- \frac{i k}{z} u \cdot v) \exp (\frac{i k}{z} ρ' \cdot v) \exp [Ω_{0} (v_{x} + v_{y})] .

\int x^{2} \exp (- i 2 π x s) d x = - \frac{1}{{(2 π)}^{2}} δ ″ (s),

F_{1} = F_{11} + F_{12},

F_{11} = - \frac{1}{4} {(\frac{z}{k})}^{2} \iint d^{2} u \iint d^{2} v

\times H_{m} [\frac{\sqrt{2}}{w_{0}} (u_{x} - \frac{v_{x}}{2})] H_{n} [\frac{\sqrt{2}}{w_{0}} (u_{y} - \frac{v_{y}}{2})] H_{m} [\frac{\sqrt{2}}{w_{0}} (u_{x} + \frac{v_{x}}{2})] H_{n} [\frac{\sqrt{2}}{w_{0}} (u_{y} + \frac{v_{y}}{2})]

\times \exp (- \frac{2 u^{2}}{w_{0}^{2}}) \exp (- \frac{v^{2}}{ε^{2}}) \exp (- \frac{i k}{z} u \cdot v) \exp [2 Ω_{0} (u_{x} + u_{y})] δ ″ (v_{x}) δ (v_{y}),

F_{12} = - \frac{1}{4} {(\frac{z}{k})}^{2} \iint d^{2} u \iint d^{2} v

\times H_{m} [\frac{\sqrt{2}}{w_{0}} (u_{x} - \frac{v_{x}}{2})] H_{n} [\frac{\sqrt{2}}{w_{0}} (u_{y} - \frac{v_{y}}{2})] H_{m} [\frac{\sqrt{2}}{w_{0}} (u_{x} + \frac{v_{x}}{2})] H_{n} [\frac{\sqrt{2}}{w_{0}} (u_{y} + \frac{v_{y}}{2})]

\times \exp (- \frac{2 u^{2}}{w_{0}^{2}}) \exp (- \frac{v^{2}}{ε^{2}}) \exp (- \frac{i k}{z} u \cdot v) \exp [2 Ω_{0} (u_{x} + u_{y})] δ (v_{x}) δ ″ (v_{y}),

\int f (x) δ (x) d x = f (0),

\int \exp [- {(x - y)}^{2}] H_{m} (x) H_{n} (x) d x = 2^{n} \sqrt{π} m! y^{n - m} L_{n}^{n - m} (- 2 y^{2}),

F_{11} = - \frac{1}{4} {(\frac{z}{k})}^{2} 2^{n} n! \sqrt{π} \frac{w_{0}}{\sqrt{2}} \exp (\frac{w_{0}^{2} Ω_{0}^{2}}{2}) L_{n}^{0} (- w_{0}^{2} Ω_{0}^{2}) \iint d v_{x} d u_{x}

\times H_{m} [\frac{\sqrt{2}}{w_{0}} (u_{x} - \frac{v_{x}}{2})] H_{m} [\frac{\sqrt{2}}{w_{0}} (u_{x} + \frac{v_{x}}{2})] \exp (\frac{2 u_{x}^{2}}{w_{0}^{2}})

\times \exp (- \frac{v_{x}^{2}}{ε^{2}}) \exp (- \frac{i k}{z} u_{x} v_{x}) \exp (2 Ω_{0} u_{x}) δ ″ (v_{x}) .

\int \exp ({- x}^{2}) H_{m} (x + y) H_{n} (x + z) d x = 2^{n} \sqrt{π} m! y^{n - m} z^{n - m} L_{n}^{n - m} (- 2 y z),

\int f (x) δ ″ (x) d x = f ″ (0),

F_{11} = - \frac{1}{4} {(\frac{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})

\times [(- \frac{k^{2} w_{0}^{2}}{4 z^{2}} - \frac{2}{ε^{2}}) L_{m}^{0} (- w_{0}^{2} Ω_{0}^{2})] - \frac{k^{2} w_{0}^{4} Ω_{0}^{2}}{4 z^{2}} L_{m}^{0} (- w_{0}^{2} Ω_{0}^{2}) - \frac{k^{2} w_{0}^{4} Ω_{0}^{2}}{z^{2}} L_{m - 2}^{2} (- w_{0}^{2} Ω_{0}^{2})

- (\frac{2}{w_{0}^{2}} + \frac{k^{2} w_{0}^{2}}{2 z^{0}}) L_{m - 1}^{1} (- w_{0}^{2} Ω_{0}^{2}) - \frac{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} = - \frac{1}{4} {(\frac{z}{k})}^{2} 2^{m + n - 1} m! n! π w_{0}^{2} [(- \frac{k^{2} w_{0}^{2}}{4 z^{2}} - \frac{2}{ε^{2}}) + Ω_{0}^{2} - m (\frac{2}{w_{0}^{2}} + \frac{k^{2} w_{0}^{2}}{2 z^{2}})] .

w (z) = \sqrt{P + \frac{Q}{k^{2}} z^{2} + 4 {(0.545 C_{n}^{2} k^{\frac{1}{3}})}^{\frac{6}{5}} z^{\frac{16}{5}}} .

	m	n	w ₀(cm)	σ₀(cm)	Ω₀(m^-1)
a	0	0	1	∞	0
b	0	0	2.8	1.07	0
c	2	1	3.5	1.22	0
d	2	2	2	1.23	150
e	0	0	3	1.07	100

	m	n	w ₀(cm)	Ω₀(m^-1)
a	0	0	1.01	0
b	0	0	1.00	300
c	2	1	1.20	200
d	3	2	2.47	0

	m	n	w ₀(cm)	σ₀(cm)	Ω₀(m^-1)
a	0	0	1	∞	0
b	0	0	2.8	1.07	0
c	2	1	3.5	1.22	0
d	2	2	2	1.23	150
e	0	0	3	1.07	100

	m	n	w ₀(cm)	Ω₀(m^-1)
a	0	0	1.01	0
b	0	0	1.00	300
c	2	1	1.20	200
d	3	2	2.47	0

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