Expand this Topic clickable element to expand a topic
Skip to content
Optica Publishing Group

Analysis of reciprocity of cos-Gaussian and cosh-Gaussian laser beams in a turbulent atmosphere

Open Access Open Access

Abstract

In a turbulent atmosphere, starting with a cos-Gaussian excitation at the source plane, the average intensity profile at the receiver plane is formulated. This average intensity profile is evaluated against the variations of link lengths, turbulence levels, two frequently used free-space optics wavelengths, and beam displacement parameters. We show that a cos-Gaussian beam, following a natural diffraction, is eventually transformed into a cosh-Gaussian beam. Combining our earlier results with the current findings, we conclude that cos-Gaussian and cosh-Gaussian beams act in a reciprocal manner after propagation in turbulence. The rates (paces) of conversion in the two directions are not the same. Although the conversion of cos-Gaussian beams to cosh-Gaussian beams can happen over a wide range of turbulence levels (low to moderate to high), the conversion of cosh-Gaussian beams to cos-Gaussian beams is pronounced under relatively stronger turbulence conditions. Source and propagation parameters that affect this reciprocity have been analyzed.

©2004 Optical Society of America

Full Article  |  PDF Article
More Like This
Average intensity and spreading of cosh-Gaussian laser beams in the turbulent atmosphere

Halil T. Eyyuboğlu and Yahya Baykal
Appl. Opt. 44(6) 976-983 (2005)

Propagation of a higher-order cosh-Gaussian beam in turbulent atmosphere

Guoquan Zhou
Opt. Express 19(5) 3945-3951 (2011)

Hermite-sine-Gaussian and Hermite-sinh-Gaussian laser beams in turbulent atmosphere

Halil Tanyer Eyyuboğlu and Yahya Baykal
J. Opt. Soc. Am. A 22(12) 2709-2718 (2005)

Cited By

Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1.
Fig. 1. Propagation geometry.
Fig. 2.
Fig. 2. (a) Normalized intensity of a cos-Gaussian beam at the source plane and (b) contour plots of the same cos-Gaussian beam.
Fig. 3.
Fig. 3. (a) Normalized intensity at the source plane and the normalized average intensity at the receiver plane for a typical cos-Gaussian beam and (b) contour plots for the same cos-Gaussian beam.
Fig. 4.
Fig. 4. Dependence of normalized average intensity at the receiver plane on link length.
Fig. 5.
Fig. 5. Dependence of normalized average intensity at the receiver plane on the real part of a complex displacement parameter.
Fig. 6.
Fig. 6. Dependence of normalized average intensity at the receiver plane on turbulence level and wavelength of operation.
Fig. 7.
Fig. 7. Dependence of normalized average intensity at the receiver plane on link length (cosh-Gaussian source excitation case).
Fig. 8.
Fig. 8. Dependence of normalized average intensity at the receiver plane on the real part of a complex displacement parameter (cosh-Gaussian source excitation case).
Fig. 9.
Fig. 9. Dependence of normalized average intensity at the receiver plane on turbulence level and wavelength of operation (cosh-Gaussian source excitation case).

Equations (39)

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

u s ( s x , s y , z = 0 ) = 0.5 A exp ( i ϕ ) exp [ 0.5 ( s x 2 α s x 2 + s y 2 α s y 2 ) ]
× { exp [ i ( V x s x + V y s y ) ] + exp [ i ( Y x s x + Y y s y ) ] }
I s ( s x , s y , z = 0 ) = exp [ ( s x 2 α s x 2 + s y 2 α s y 2 ) ] cos 2 ( V x r s x + V y r s y )
u ( p , L , t ) = k exp ( ikL ) ( 2 πiL ) d 2 s u s ( s )
× exp [ i k ( p s ) 2 ( 2 L ) + ψ ( s , p ) i 2 π f t ]
< I ( p , L ) > = k 2 ( 2 π L ) 2 d 2 s 1 d 2 s 2 u s ( s 1 ) u s * ( s 2 ) exp { ik [ ( p s 1 ) 2 ( p s 2 ) 2 ] ( 2 L ) }
× < exp [ ψ ( s 1 , p ) + ψ * ( s 2 , p ) ] >
< exp [ ψ ( s 1 , p ) + ψ * ( s 2 , p ) ] > = exp [ 0.5 D ψ ( s 1 s 2 ) ] = exp [ ρ 0 2 ( s 1 s 2 ) 2 ]
< I ( p , L ) > = 0.25 k 2 ( 2 π L ) 2 d s 1 x d s 1 y d s 2 x d s 2 y exp [ 0.5 ( s 1 x 2 + s 2 x 2 ) α s x 2 0.5 ( s 1 y 2 + s 2 y 2 ) α s y 2 ]
× { exp [ i V x r ( s 1 x + s 2 x ) + i V y r ( s 1 y + s 2 y ) ] + exp [ i V x r ( s 1 x + s 2 x ) i V y r ( s 1 y + s 2 y ) ]
+ exp [ i V x r ( s 1 x s 2 x ) + i V y r ( s 1 y s 2 y ) ] + exp [ i V x r ( s 1 x s 2 x ) i V y r ( s 1 y s 2 y ) ] }
× exp [ 0.5 ( i k L ) ( s 1 x 2 2 p x s 1 x s 2 x 2 + 2 p x s 2 x + s 1 y 2 2 p y s 1 y s 2 y 2 + 2 p y s 2 y ) ]
× exp [ ρ 0 2 ( s 1 x 2 2 s 1 x s 2 x + s 2 x 2 + s 1 y 2 2 s 1 y s 2 y + s 2 y 2 ) ]
< I ( p , L ) > = 0.5 ( k L ) 2 ρ 0 4 ( D s x D s y ) 1 2 exp { ( ρ 0 4 k 2 L 2 ) [ p x 2 ( α s x 2 D s x ) + p y 2 ( α s y 2 D s y ) ] }
× ( exp { 2 ρ 0 2 [ V x r 2 ( ρ 0 2 + 4 α s x 2 ) ( α s x 2 D s x ) + V y r 2 ( ρ 0 2 + 4 α s y 2 ) ( α s y 2 D s y ) ] }
× cos [ ( 2 ρ 0 4 k 2 L 2 ) ( V x r p x D s x + V s r p y D s y ) ]
+ exp { ρ 0 4 [ V x r 2 ( α s x 2 D s x ) + V y r 2 ( α s y 2 D s y ) ] }
× cosh { ( 2 ρ 0 4 k L ) [ ( V x r p x ( α s x 2 D s x ) + V y r p y ( α s y 2 D s y ) ) ] } )
I ( p , L ) = [ k 2 α s 4 ( L 2 + k 2 α s 4 ) ] exp { [ k 2 α s 2 ( L 2 + k 2 α s 4 ) ] ( p x 2 + p y 2 ) }
< I ( p , L ) > = [ k 2 α s 4 ρ 0 2 ( ρ 0 2 L 2 + 4 α s 2 L 2 + k 2 α s 4 ρ 0 2 ) ] exp [ k 2 α s 4 ρ 0 2 ( p x 2 + p x 2 ) ( ρ 0 2 L 2 + 4 α s 2 L 2 + k 2 α s 4 ρ 0 2 ) ]
I ( p , L ) = 0.5 k 2 α s x 2 α s y 2 g x 0.5 g y 0.5 exp { k 2 [ α s x 2 g x p x 2 + α s y 2 g y p y 2 ] } exp { L 2 [ α s x 2 V x r 2 g x + α s y 2 V y r 2 g y ] }
× ( cos { 2 k 2 [ α s x 4 V x r g x p x + α s y 4 V y r g y p y ] } + cosh { 2 k L [ α s x 2 V x r g x p x + α s y 2 V y r g y p y ] } )
< I ( p , L ) > = 0.5 ( k L ) 2 ρ 0 4 ( D s x D s y ) 1 2 exp { ( ρ 0 4 k 2 L 2 ) [ p x 2 ( α s x 2 D s x ) + p y 2 ( α s y 2 D s y ) ] }
× exp { ρ 0 4 [ V x r 2 ( α s x 2 D s x ) + V y r 2 ( α s y 2 D s y ) ] }
× cosh { ( 2 ρ 0 4 k L ) [ ( V x r p x ( α s x 2 D s x ) + V y r p y ( α s y 2 D s y ) ) ] }
I 0 = < I ( p x = p x p , p y = p y p , z = L ) > < I ( p x = 0 , p y = 0 , z = L ) >
ρ 0 4 V x r 2 α s x 2 L 2 ρ 0 4 L 2 + 4 ρ 0 2 α s x 2 L 2 + ρ 0 4 k 2 α s x 4 = 0.5 n ( 2 I 0 ) , ρ 0 4 V y r 2 α s y 2 L 2 ρ 0 4 L 2 + 4 ρ 0 2 α s y 2 L 2 + ρ 0 4 k 2 α s y 4 = 0.5 n ( 2 I 0 )
I s N ( s x , s y , z = 0 ) = I s ( s x , s y , z = 0 ) I s ( s x = s y = z = 0 )
I r N ( p x , p y , z = L ) = < I ( p x , p y , z = L ) > I s ( s x = s y = z = 0 )
I r 0 ( p x , p y , z = L ) = < I ( p x , p y , z = L ) > Max [ < I ( p x , p y , z = L ) > ]
I 1 x = d s 1 x d s 2 x exp [ 0.5 ( s 1 x 2 + s 2 x 2 ) α s x 2 ] exp [ i V x r ( s 1 x + s 2 x ) ]
× exp [ 0.5 ( i k L ) ( s 1 x 2 2 p x s 1 x s 2 x 2 + 2 p x s 2 x ) ] exp [ ( s 1 x 2 2 s 1 x s 2 x + s 2 x 2 ) ρ 0 2 ]
= d s 1 x exp { [ 0.5 α s x 2 + j k ( 2 L ) 1 ρ 0 2 ] s 1 x 2 + [ i V x r j k p x L + 2 s 2 x ] s 1 x }
× d s 2 x exp ( 0.5 s 2 x 2 α s x 2 ) exp ( i V x r s 2 x )
× exp [ 0.5 ( i k L ) ( s 2 x 2 + 2 p x s 2 x ) ] exp [ ( s 2 x 2 + s 1 y 2 2 s 1 y s 2 y + s 2 y 2 ) ρ 0 2 ]
d x exp ( p 2 x 2 q x ) = ( π 0.5 p ) exp [ q 2 ( 4 p 2 ) ]
I 1 x = π 0.5 [ 0.5 α s x 2 + 1 ρ 0 2 j k ( 2 L ) ] 0.5 d s 2 x
× exp ( { 0.5 α s x 1 + 1 ρ 0 2 + j k ( 2 L ) ρ 0 4 [ 0.5 α s x 2 + 1 ρ 0 2 j k ( 2 L ) ] } s 2 x 2 )
× exp ( { i V x r + i k p x L + ρ 0 2 ( i V x r i k p x L ) [ 0.5 α s x 2 + 1 ρ 0 2 j k ( 2 L ) ] } s 2 x )
Select as filters


Select Topics Cancel
© Copyright 2024 | Optica Publishing Group. All Rights Reserved