Abstract

The propagation of a Lorentz-Gauss beam in turbulent atmosphere is investigated. Based on the extended Huygens-Fresnel integral and the Hermite-Gaussian expansion of a Lorentz function, analytical formulae for the average intensity and the effective beam size of a Lorentz-Gauss beam are derived in turbulent atmosphere. The average intensity distribution and the spreading properties of a Lorentz-Gauss beam in turbulent atmosphere are numerically demonstrated. The influences of the beam parameters and the structure constant of the atmospheric turbulence on the propagation of a Lorentz-Gauss beam in turbulent atmosphere are also discussed in detail.

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References

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    [CrossRef]
  8. G. Zhou, “Nonparaxial propagation of a Lorentz-Gauss beam,” J. Opt. Soc. Am. B 26(1), 141–147 (2009).
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2009

2008

Y. Cai, O. Korotkova, H. T. Eyyuboğlu, and Y. Baykal, “Active laser radar systems with stochastic electromagnetic beams in turbulent atmosphere,” Opt. Express 16(20), 15834–15846 (2008).
[CrossRef] [PubMed]

Y. Zhu, D. Zhao, and X. Du, “Propagation of stochastic Gaussian-Schell model array beams in turbulent atmosphere,” Opt. Express 16(22), 18437–18442 (2008).
[CrossRef] [PubMed]

G. Zhou, “Focal shift of focused truncated Lorentz-Gauss beam,” J. Opt. Soc. Am. A 25(10), 2594–2599 (2008).
[CrossRef]

G. Zhou, J. Zheng, and Y. Xu, “Investigation in the far field characteristics of Lorentz beam from the vectorial structure,” J. Mod. Opt. 55(6), 993–1002 (2008).
[CrossRef]

G. Zhou, “Analytical vectorial structure of a Lorentz-Gauss beam in the far field,” Appl. Phys. B 93(4), 891–899 (2008).
[CrossRef]

G. Zhou, “Propagation of vectorial Lorentz beam beyond the paraxial approximation,” J. Mod. Opt. 55(21), 3573–3577 (2008).
[CrossRef]

J. Yang, T. Chen, G. Ding, and X. Yuan, “Focusing of diode laser beams: a partially coherent Lorentz model,” Proc. SPIE 6824, 68240A (2008).
[CrossRef]

2007

2006

Y. Cai and S. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express 14(4), 1353–1367 (2006).
[CrossRef] [PubMed]

O. E. Gawhary and S. Severini, “Lorentz beams and symmetry properties in paraxial optics,” J. Opt. A, Pure Appl. Opt. 8(5), 409–414 (2006).
[CrossRef]

2005

2004

1990

1976

P. P. Schmidt, “A method for the convolution of lineshapes which involve the Lorentz distribution,” J. Phys. B 9(13), 2331–2339 (1976).
[CrossRef]

1975

W. P. Dumke, “The angular beam divergence in double-heterojunction lasers with very thin active regions,” IEEE J. Quantum Electron. 11(7), 400–402 (1975).
[CrossRef]

Baykal, Y.

Cai, Y.

Chen, T.

J. Yang, T. Chen, G. Ding, and X. Yuan, “Focusing of diode laser beams: a partially coherent Lorentz model,” Proc. SPIE 6824, 68240A (2008).
[CrossRef]

Chen, Y.

Chu, X.

Ding, G.

J. Yang, T. Chen, G. Ding, and X. Yuan, “Focusing of diode laser beams: a partially coherent Lorentz model,” Proc. SPIE 6824, 68240A (2008).
[CrossRef]

Du, X.

Dumke, W. P.

W. P. Dumke, “The angular beam divergence in double-heterojunction lasers with very thin active regions,” IEEE J. Quantum Electron. 11(7), 400–402 (1975).
[CrossRef]

Durst, F.

Elgawhary, O.

O. Elgawhary and S. Severini, “Lorentz beams as a basis for a new class of rectangular symmetric optical fields,” Opt. Commun. 269(2), 274–284 (2007).
[CrossRef]

Eyyuboglu, H. T.

Gawhary, O. E.

O. E. Gawhary and S. Severini, “Lorentz beams and symmetry properties in paraxial optics,” J. Opt. A, Pure Appl. Opt. 8(5), 409–414 (2006).
[CrossRef]

He, S.

Korotkova, O.

Naqwi, A.

Qu, J.

Schmidt, P. P.

P. P. Schmidt, “A method for the convolution of lineshapes which involve the Lorentz distribution,” J. Phys. B 9(13), 2331–2339 (1976).
[CrossRef]

Severini, S.

O. Elgawhary and S. Severini, “Lorentz beams as a basis for a new class of rectangular symmetric optical fields,” Opt. Commun. 269(2), 274–284 (2007).
[CrossRef]

O. E. Gawhary and S. Severini, “Lorentz beams and symmetry properties in paraxial optics,” J. Opt. A, Pure Appl. Opt. 8(5), 409–414 (2006).
[CrossRef]

Xu, Y.

G. Zhou, J. Zheng, and Y. Xu, “Investigation in the far field characteristics of Lorentz beam from the vectorial structure,” J. Mod. Opt. 55(6), 993–1002 (2008).
[CrossRef]

Yang, J.

J. Yang, T. Chen, G. Ding, and X. Yuan, “Focusing of diode laser beams: a partially coherent Lorentz model,” Proc. SPIE 6824, 68240A (2008).
[CrossRef]

Yuan, X.

J. Yang, T. Chen, G. Ding, and X. Yuan, “Focusing of diode laser beams: a partially coherent Lorentz model,” Proc. SPIE 6824, 68240A (2008).
[CrossRef]

Yuan, Y.

Zhao, D.

Zheng, J.

G. Zhou, J. Zheng, and Y. Xu, “Investigation in the far field characteristics of Lorentz beam from the vectorial structure,” J. Mod. Opt. 55(6), 993–1002 (2008).
[CrossRef]

Zhou, G.

G. Zhou, “Nonparaxial propagation of a Lorentz-Gauss beam,” J. Opt. Soc. Am. B 26(1), 141–147 (2009).
[CrossRef]

G. Zhou, “Fractional Fourier transform of Lorentz-Gauss beams,” J. Opt. Soc. Am. A 26(2), 350–355 (2009).
[CrossRef]

G. Zhou, “The beam propagation factors and the kurtosis parameters of a Lorentz beam,” Opt. Laser Technol. 41(8), 953–955 (2009).
[CrossRef]

G. Zhou, “Beam propagation factors of a Lorentz-Gauss beam,” Appl. Phys. B 96(1), 149–153 (2009).
[CrossRef]

G. Zhou, “Focal shift of focused truncated Lorentz-Gauss beam,” J. Opt. Soc. Am. A 25(10), 2594–2599 (2008).
[CrossRef]

G. Zhou, “Analytical vectorial structure of a Lorentz-Gauss beam in the far field,” Appl. Phys. B 93(4), 891–899 (2008).
[CrossRef]

G. Zhou, “Propagation of vectorial Lorentz beam beyond the paraxial approximation,” J. Mod. Opt. 55(21), 3573–3577 (2008).
[CrossRef]

G. Zhou, J. Zheng, and Y. Xu, “Investigation in the far field characteristics of Lorentz beam from the vectorial structure,” J. Mod. Opt. 55(6), 993–1002 (2008).
[CrossRef]

Zhu, Y.

Appl. Opt.

Appl. Phys. B

G. Zhou, “Analytical vectorial structure of a Lorentz-Gauss beam in the far field,” Appl. Phys. B 93(4), 891–899 (2008).
[CrossRef]

G. Zhou, “Beam propagation factors of a Lorentz-Gauss beam,” Appl. Phys. B 96(1), 149–153 (2009).
[CrossRef]

IEEE J. Quantum Electron.

W. P. Dumke, “The angular beam divergence in double-heterojunction lasers with very thin active regions,” IEEE J. Quantum Electron. 11(7), 400–402 (1975).
[CrossRef]

J. Mod. Opt.

G. Zhou, “Propagation of vectorial Lorentz beam beyond the paraxial approximation,” J. Mod. Opt. 55(21), 3573–3577 (2008).
[CrossRef]

G. Zhou, J. Zheng, and Y. Xu, “Investigation in the far field characteristics of Lorentz beam from the vectorial structure,” J. Mod. Opt. 55(6), 993–1002 (2008).
[CrossRef]

J. Opt. A, Pure Appl. Opt.

O. E. Gawhary and S. Severini, “Lorentz beams and symmetry properties in paraxial optics,” J. Opt. A, Pure Appl. Opt. 8(5), 409–414 (2006).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

J. Phys. B

P. P. Schmidt, “A method for the convolution of lineshapes which involve the Lorentz distribution,” J. Phys. B 9(13), 2331–2339 (1976).
[CrossRef]

Opt. Commun.

O. Elgawhary and S. Severini, “Lorentz beams as a basis for a new class of rectangular symmetric optical fields,” Opt. Commun. 269(2), 274–284 (2007).
[CrossRef]

Opt. Express

Opt. Laser Technol.

G. Zhou, “The beam propagation factors and the kurtosis parameters of a Lorentz beam,” Opt. Laser Technol. 41(8), 953–955 (2009).
[CrossRef]

Opt. Lett.

Proc. SPIE

J. Yang, T. Chen, G. Ding, and X. Yuan, “Focusing of diode laser beams: a partially coherent Lorentz model,” Proc. SPIE 6824, 68240A (2008).
[CrossRef]

Other

I. S. Gradshteyn, and I. M. Ryzhik, Table of integrals, series, and products (Academic Press, New York, 1980).

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Figures (3)

Fig. 1
Fig. 1

Normalized average intensity distributions in the x-direction of a Lorentz-Gauss beam (the solid curve) and a Gaussian beam (the dotted curve) at different propagation distances in turbulent atmosphere. (a) z = 1km. (b) z = 3km. (c) z = 10km.

Fig. 2
Fig. 2

Three-dimensional normalized intensity distributions and the corresponding contour graphs of Lorentz-Gauss beams in turbulent atmosphere. z = 1km. (a) w 0 = 0.01m and w 0 x = w 0 y = 0.02m. (b) w 0 = w 0 x = w 0 y = 0.02m. (c) w 0 = 0.03m and w 0 x = w 0 y = 0.01m.

Fig. 3
Fig. 3

The effective beam size of a Lorentz-Gauss beam versus the propagation distance z in turbulent atmosphere. (a) w 0 = 0.02m and Cn 2 = 10−14m-2/3. (b) w 0 x = 0.02m and Cn 2 = 10−14m-2/3. (c) w 0 = 0.02m and Cn 2 = 10−15m-2/3.

Equations (18)

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E ( r 0 , 0 ) = 1 w 0 x w 0 y [ 1 + ( x 0 / w 0 x ) 2 ] [ 1 + ( y 0 / w 0 y ) 2 ] exp ( x 0 2 + y 0 2 w 0 2 ) ,
E ( r 0 , 0 ) = π 2 w 0 x w 0 y m = 0 N n = 0 N σ 2 m σ 2 n H 2 m ( x 0 w 0 x ) H 2 n ( y 0 w 0 y ) exp ( x 0 2 u x 2 y 0 2 u y 2 ) ,
1 u j 2 = 1 w 0 2 + 1 2 w 0 j 2 ,
E ( r , z ) = i k 2 π z E ( r 0 , 0 ) exp [ i k 2 z ( r 0 r ) 2 + ψ ( r 0 , r ) ] d x 0 d y 0 ,
< I ( r , z ) > = k 2 4 π 2 z 2 E ( r 01 , 0 ) E ( r 02 , 0 ) exp [ i k 2 z ( r 01 r ) 2 + i k 2 z ( r 02 r ) 2 ] × < exp [ ψ ( r 01 , r ) + ψ ( r 02 , r ) ] > d r 01 d r 02 ,
< exp [ ψ ( r 01 , r ) + ψ ( r 02 , r ) ] > = exp [ 0.5 D ψ ( r 01 r 02 ) ] = exp [ ( r 01 r 02 ) 2 ρ 0 2 ] ,
H 2 m ( x ) exp [ a ( x y ) 2 ] d x = π a ( 1 1 a ) m H 2 m [ y ( 1 1 a ) 1 / 2 ] ,
H 2 m ( x ) = l = 0 m ( 1 ) l ( 2 m ) ! l ! ( 2 m 2 l ) ! ( 2 x ) 2 m 2 l ,
x 2 n exp ( b x 2 + 2 c x ) d x = ( 2 n ) ! π b ( c b ) 2 n exp ( c 2 b ) s = 0 n 1 s ! ( 2 n 2 s ) ! ( b 4 c 2 ) s ,
< I ( r , z ) > = < I ( x , z ) > < I ( y , z ) > ,
< I ( j , z ) > = k π 4 z 1 α 1 j α 2 j exp ( β j 2 α 2 j k 2 w 0 j 2 j 2 4 α 1 j z 2 ) m = 0 N m = 0 N σ 2 m σ 2 m ( 1 1 α 1 j ) m l 1 = 0 m ( 1 ) l l ( 2 m ) ! l 1 ! ( 2 m 2 l 1 ) ! × l 2 = 0 m ( 1 ) l 2 ( 2 m ) ! l 2 ! ( 2 m 2 l 2 ) ! l 3 = 0 2 m 2 l 2 ( l 3 2 m 2 l 2 ) 2 2 ( m + m ) 2 l 1 2 l 2 γ j l 3 δ j 2 m 2 l 2 l 3 ( 2 m + l 3 2 l 1 ) ! × ( β j α 2 j ) 2 m + l 3 2 l 1 s = 0 [ m l 1 + l 3 / 2 ] 1 s ! ( 2 m + l 3 2 l 1 2 s ) ! ( α 2 j 4 β j 2 ) s ,
α 1 j = ( 1 u j 2 + 1 ρ 0 2 i k 2 z ) w 0 j 2 , α 2 j = ( 1 u j 2 + 1 ρ 0 2 + i k 2 z ) w 0 j 2 w 0 j 4 α 1 j ρ 0 4 , β j = i k w 0 j j 2 z i k w 0 j 3 j 2 z α 1 j ρ 0 2 ,
γ j = w 0 j 2 ( α 1 j 2 α 1 j ) 1 / 2 ρ 0 2 , δ j = k w 0 j j 2 i z ( α 1 j 2 α 1 j ) 1 / 2 .
W j z = 2 j 2 < I ( r , z ) > d x d y < I ( r , z ) > d x d y .
W j z = 2 A 1 j A 2 j ,
A 1 j = m = 0 N m = 0 N σ 2 m σ 2 m ( 1 1 α 1 j ) m l 1 = 0 m ( 1 ) l l ( 2 m ) ! l 1 ! ( 2 m 2 l 1 ) ! l 2 = 0 m ( 1 ) l 2 ( 2 m ) ! l 2 ! ( 2 m 2 l 2 ) ! l 3 = 0 2 m 2 l 2 ( l 3 2 m 2 l 2 ) × 2 2 ( m + m ) 2 l 1 2 l 2 γ j l 3 η j 2 m 2 l 2 l 3 ( 2 m + l 3 2 l 1 ) ! s = 0 [ m l 1 + l 3 / 2 ] 1 s ! ( 2 m + l 3 2 l 1 2 s ) ! 4 s × α 2 j 2 l 1 + s 2 m l 3 ξ j 2 m + l 3 2 l 1 2 s Γ ( m + m l 1 l 2 s + 3 2 ) p j l 1 + l 2 + s m m 3 / 2 ,
A 2 j = m = 0 N m = 0 N σ 2 m σ 2 m ( 1 1 α 1 j ) m l 1 = 0 m ( 1 ) l l ( 2 m ) ! l 1 ! ( 2 m 2 l 1 ) ! l 2 = 0 m ( 1 ) l 2 ( 2 m ) ! l 2 ! ( 2 m 2 l 2 ) ! l 3 = 0 2 m 2 l 2 ( l 3 2 m 2 l 2 ) × 2 2 ( m + m ) 2 l 1 2 l 2 γ j l 3 η j 2 m 2 l 2 l 3 ( 2 m + l 3 2 l 1 ) ! s = 0 [ m l 1 + l 3 / 2 ] 1 s ! ( 2 m + l 3 2 l 1 2 s ) ! 4 s × α 2 j 2 l 1 + s 2 m l 3 ξ j 2 m + l 3 2 l 1 2 s Γ ( m + m l 1 l 2 s + 1 2 ) p j l 1 + l 2 + s m m 1 / 2 ,
η j = k w 0 j 2 i z ( α 1 j 2 α 1 j ) 1 / 2 , ξ j = i k w 0 j 2 z i k w 0 j 3 2 z α 1 j ρ 0 2 , p j = k 2 w 0 j 2 4 α 1 j z 2 ξ j 2 α 2 j ,

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