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Flat topped beams and their characteristics in turbulent media

Halil Tanyer Eyyuboğlu, Çağlar Arpali, and Yahya Baykal

Open Access Open Access

Abstract

The source and receiver plane characteristics of flat topped (FT) beam propagating in turbulent atmosphere are investigated. To this end, source size, beam power and M 2 factor of source plane FT beam are derived. For a turbulent propagation medium, via Huygens Fresnel diffraction integral, the receiver plane intensity is found. Power captured within an area on the receiver plane is calculated. Kurtosis parameter and beam size variation along the propagation axis are formulated. Graphical outputs are provided displaying the variations of the derived source and receiver plane parameters against the order of flatness and propagation length. Analogous to free space behavior, when propagating in turbulence, the FT beam first will form a circular ring in the center. As the propagation length increases, the circumference of this ring will become narrower, giving rise to a downward peak emerging from the center of the beam, eventually turning the intensity profile into a pure Gaussian shape.

©2006 Optical Society of America

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Figures (11)
Fig. 1.
Fig. 1. FT source beam profiles at different flatness order.

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Fig. 2.
Fig. 2. Overlaid plots of four FT source beams belonging to Fig. 1 cut along the slanted axis.

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Fig. 3.
Fig. 3. Source beam size variation versus flatness order.

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Fig. 4.
Fig. 4. Source beam power variation versus flatness order.

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Fig. 5.
Fig. 5. Variation of Mx2 versus flatness order.

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Fig. 6.
Fig. 6. Propagation view of the FT beam for selected source and propagation parameters.

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Fig. 7.
Fig. 7. Overlaid plots of the FT beam belonging to Fig. 6 cut along the slanted axis.

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Fig. 8.
Fig. 8. Intensity distribution of an elliptical FT beam before and after propagation.

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Fig. 9.
Fig. 9. Receiver beam size variation versus flatness order.

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Fig. 10.
Fig. 10. Power in bucket variation versus flatness order.

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Fig. 11.
Fig. 11. Kurtosis parameter variation versus propagation length.

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Equations (20)

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u s ( s ) = u s ( s x , s y ) = 1 − [ 1 − exp ( − s x 2 ⁄ α sx 2 − s y 2 ⁄ α sy 2 ) ] N ,
α sxN = [ 2 ∫ − ∞ ∞ ∫ − ∞ ∞ s x 2 I s ( s ) d s x d s y ⁄ ∫ − ∞ ∞ ∫ − ∞ ∞ I s ( s ) d s x d s y ] 1 ⁄ 2 ,
α syN = [ 2 ∫ − ∞ ∞ ∫ − ∞ ∞ s y 2 I s ( s ) d s x d s y ⁄ ∫ − ∞ ∞ ∫ − ∞ ∞ I s ( s ) d s x d s y ] 1 ⁄ 2 ,
I s ( s ) = u s ( s ) u s * ( s ) .
α sxN = α sx [ 2 ∑ n = 1 N ∑ n 1 = 1 N ( − 1 ) n + n 1 ( N n ) ( N n 1 ) 1 ( n + n 1 ) 2 ⁄ ∑ n = 1 N ∑ n 1 = 1 N ( − 1 ) n + n 1 ( N n ) ( N n 1 ) 1 ( n + n 1 ) ] 2 ,
P sN = 2 π α sx α sy ∑ n = 1 N ∑ n 1 = 1 N ( − 1 ) n + n 1 ( N n ) ( N n 1 ) 1 ( n + n 1 ) .
M x 2 = 4 π ( σ sx 2 σ fx 2 ) 0.5 ,
M y 2 = 4 π ( σ sy 2 σ fy 2 ) 0.5 ,
σ sx 2 = ∫ − ∞ ∞ s x 2 I s ( s x ) d s x ∫ − ∞ ∞ I s ( s x ) d s x ,
σ fx 2 = ∫ − ∞ ∞ f x 2 I f ( f x ) d f x ∫ − ∞ ∞ I f ( f x ) d f x .
M x 2 = 2 [ ∑ n = 1 N ∑ n 1 = 1 N ( − 1 ) n + n 1 ( N n ) ( N n 1 ) ( ( n n 1 ) ( 1 + n 1 ) 3 ) ∑ n = 1 N ∑ n 1 = 1 N ( − 1 ) n + n 1 ( N n ) ( N n 1 ) 1 ( n + n 1 ) ] 1 ⁄ 2 .
< I r ( p ) > = < I r ( p x , p y ) > = k 2 ( 2 π L ) 2 ∫ − ∞ ∞ ∫ − ∞ ∞ ∫ − ∞ ∞ ∫ − ∞ ∞ d 2 s 1 d 2 s 2 u s ( s 1 ) u s ( s 2 ) exp { jk ( 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 r ( p ) > = 1 − ∑ n = 0 N ( − 1 ) n ( N n ) β α sx α sy ( ζ nx ζ ny ) 1 ⁄ 2 exp ( − β n p x 2 2 ζ nx − β n p y 2 2 ζ ny ) − ∑ n = 0 N ( − 1 ) n ( N n ) β α sx α sy ( ζ nx * ζ ny * ) 1 ⁄ 2 exp ( − β n p x 2 2 ζ nx * − β n p y 2 2 ζ ny * )
+ ∑ n = 0 N ∑ n 1 = 0 n ∑ n 2 = 0 n 1 ( − 1 ) n 1 ( N n ) ( n n 1 ) ( n 1 n 2 ) β α sx 2 α sy 2 ( γ n n 1 n 2 x γ n n 1 n 2 y ) 1 ⁄ 2 exp ( β n n 1 α sx 2 p x 2 2 γ n n 1 n 2 y + β n n 1 α sy 2 p y 2 2 γ n n 1 n 2 y ) ,
α pxN = [ 2 ∫ − ∞ ∞ ∫ − ∞ ∞ p x 2 I r ( p ) d p x d p y ⁄ P sN ] 1 ⁄ 2 .
K x = ∫ − ∞ ∞ ∫ − ∞ ∞ p x 4 I r ( p ) d p x d p y ⁄ P sN [ ∫ − ∞ ∞ ∫ − ∞ ∞ p x 2 I r ( p ) d p x d p y ⁄ P sN ] 2 ,
P αN = 2 π ∫ 0 α r r I r ( p ) dr ⁄ P sN .
Δ α pxN = ( α pxN − α sxN ) ⁄ α sxN .
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