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

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References

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Applied Optics

A. Ishimaru, �??Phase fluctuations in a turbulent medium,�?? Applied Optics 16, 3190-3192 (1977).
[CrossRef] [PubMed]

J. Opt. A

D. Zhao, H. Mao, and H. Liu, �??Propagation of off-axial Hermite-cosh-Gaussian laser beams,�?? J. Opt. A 6, 77�??83 (2004).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

S. Yu, H. Guo, X. Fu, and W. Hu, �??Propagation properties of elegant Hermite-cosh-Gaussian laser beams,�?? Opt. Commun. 204, 59�??66 (2002).
[CrossRef]

S. Konar and J. Soumendu, �??Linear and nonlinear propagation of sinh-Gaussian pulses in dispersive media possessing Kerr nonlinearity,�?? Opt. Commun. 236, 7�??20 (2004).
[CrossRef]

N. Zhou and G. Zeng, �??Propagation properties of Hermite-cosine-Gaussian beams through a paraxial optical ABCD system with hard-edge aperture,�?? Opt. Commun. 232, 49�??59 (2004).
[CrossRef]

Y. Song. G. Wangyi, and G. Hong, �??Optical resonator with hyperbolic-cosine-Gaussian modes,�?? Opt. Commun. 221, 241�??247 (2003).
[CrossRef]

B. Lü, H. Ma, and B. Zhang, �??Propagation properties of cosh-Gaussian beams,�?? Opt. Commun. 164, 165�??170 (1999).
[CrossRef]

Opt. Eng.

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]

Radiophys. Quantum Electron.

Z. I. Feizulin and Y. Kravtsov, �??Broadening of a laser beam in a turbulent medium,�?? Radiophys. Quantum Electron. 10, 33�??35 (1967).
[CrossRef]

A. I. Kon and V. I. Tatarskii, �??On the theory of propagation of partially coherent light beams in a turbulent atmosphere,�?? Radiophys. Quantum Electron. 15, 1187�??1192 (1972).
[CrossRef]

Sov. J. Quantum Electron.

M. S. Belen�??kii, A. I. Kon, and V. L. Mironov, �??Turbulent distortions of the spatial coherence of a laser beam,�?? Sov. J. Quantum Electron. 7, 287�??290 (1977).
[CrossRef]

M. S. Belen�??kii and V. L. Mironov, �??Phase fluctuations of a multimode laser field in a turbulent atmosphere,�?? Sov. J. Quantum Electron. 12, 3�??6 (1982).
[CrossRef]

Other

H. T. Eyyubo�?lu and Y. Baykal, �??Average intensity and spreading of cosh-Gaussian laser beams in the turbulent atmosphere,�??Applied Optics, submitted for publication.

I. S. Gradysteyn, and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic Press, New York, 1980).

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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)

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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 )

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