Abstract

Hermite-sine-Gaussian and Hermite-sinh-Gaussian laser beam intensities in a turbulent atmosphere are investigated. The received intensity is formulated by applying the extended Huygens–Fresnel principle to generalized Hermite-hyperbolic-Gaussian and Hermite-sinusoidal-Gaussian beam incidences. From this result, the association to different types of Hermite-hyperbolic-Gaussian and Hermite-sinusoidal-Gaussian beams are defined. The average receiver intensity expressions for Hermite-sine-Gaussian and Hermite-sinh-Gaussian laser beams are evaluated and plotted against the variations in source parameters and propagation conditions. It is observed that the propagation of Hermite-sine-Gaussian and Hermite-sinh-Gaussian laser beams in turbulence have many similarities to their counterparts, Hermite-cosine-Gaussian and Hermite-cosh-Gaussian laser beams, that are examined earlier. It is further observed that under certain conditions the main features of the previously established reciprocity concept between cosine-Gaussian and cosh-Gaussian beams are mostly applicable to Hermite-sine-Gaussian and Hermite-sinh-Gaussian laser beams.

© 2005 Optical Society of America

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References

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  1. L. C. Andrews, R. L. Phillips, “Free space optical communication link and atmospheric effects: single aperture and arrays,” in Proceedings of Free-Space Laser Communication Technologies XVI, C. Y. Young and J. S. Stryjewski, eds., Proc. SPIE5338, 265–275 (2004).
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    [CrossRef]
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    [CrossRef]
  4. X. Zhu, J. M. Kahn, “Performance bounds for coded free-space optical communications through atmospheric turbulence channels,” IEEE Trans. Commun. 51, 1233–1239 (2003).
    [CrossRef]
  5. C. Y. Young, Y. V. Gilchrest, B. R. Macon, “Turbulence induced beam spreading of higher order mode optical waves,” Opt. Eng. (Bellingham) 41, 1097–1103 (2002).
    [CrossRef]
  6. Y. Baykal, “Correlation and structure functions of Hermite-sinusoidal-Gaussian laser beams in a turbulent atmosphere,” J. Opt. Soc. Am. A 21, 1290–1299 (2004).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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  11. B. Lü, H. Ma, B. Zhang, “Propagation properties of cosh-Gaussian beams,” Opt. Commun. 164, 165–170 (1999).
    [CrossRef]
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    [CrossRef]
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2005 (2)

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

H. T. Eyyuboğlu, Y. Baykal, “Average intensity and spreading of cosh-Gaussian laser beams in the turbulent atmosphere,” Appl. Opt. 44, 976–983 (2005).
[CrossRef] [PubMed]

2004 (3)

2003 (2)

X. Zhu, J. M. Kahn, “Performance bounds for coded free-space optical communications through atmospheric turbulence channels,” IEEE Trans. Commun. 51, 1233–1239 (2003).
[CrossRef]

J. C. Ricklin, F. M. Davidson, “Atmospheric optical communication with a Gaussian Schell beam,” J. Opt. Soc. Am. A 20, 856–866 (2003).
[CrossRef]

2002 (2)

R. K. Tyson, “Bit-error rate for free-space adaptive optics laser communications,” J. Opt. Soc. Am. A 19, 753–758 (2002).
[CrossRef]

C. Y. Young, Y. V. Gilchrest, B. R. Macon, “Turbulence induced beam spreading of higher order mode optical waves,” Opt. Eng. (Bellingham) 41, 1097–1103 (2002).
[CrossRef]

1999 (1)

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

1998 (2)

1979 (1)

1977 (1)

Andrews, L. C.

L. C. Andrews, R. L. Phillips, “Free space optical communication link and atmospheric effects: single aperture and arrays,” in Proceedings of Free-Space Laser Communication Technologies XVI, C. Y. Young and J. S. Stryjewski, eds., Proc. SPIE5338, 265–275 (2004).

Baykal, Y.

Baykal, Y. K.

Casperson, L. W.

Davidson, F. M.

Eyyuboglu, H. T.

Gilchrest, Y. V.

C. Y. Young, Y. V. Gilchrest, B. R. Macon, “Turbulence induced beam spreading of higher order mode optical waves,” Opt. Eng. (Bellingham) 41, 1097–1103 (2002).
[CrossRef]

Gradysteyn, I. S.

I. S. Gradysteyn, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, 2000).

Ishimaru, A.

Kahn, J. M.

X. Zhu, J. M. Kahn, “Performance bounds for coded free-space optical communications through atmospheric turbulence channels,” IEEE Trans. Commun. 51, 1233–1239 (2003).
[CrossRef]

Liu, H.

D. Zhao, H. Mao, H. Liu, “Propagation of off-axial Hermite-cosh-Gaussian laser beams,” J. Opt. A, Pure Appl. Opt. 6, 77–83 (2004).
[CrossRef]

Lü, B.

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

Ma, H.

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

Macon, B. R.

C. Y. Young, Y. V. Gilchrest, B. R. Macon, “Turbulence induced beam spreading of higher order mode optical waves,” Opt. Eng. (Bellingham) 41, 1097–1103 (2002).
[CrossRef]

Mao, H.

D. Zhao, H. Mao, H. Liu, “Propagation of off-axial Hermite-cosh-Gaussian laser beams,” J. Opt. A, Pure Appl. Opt. 6, 77–83 (2004).
[CrossRef]

Phillips, R. L.

L. C. Andrews, R. L. Phillips, “Free space optical communication link and atmospheric effects: single aperture and arrays,” in Proceedings of Free-Space Laser Communication Technologies XVI, C. Y. Young and J. S. Stryjewski, eds., Proc. SPIE5338, 265–275 (2004).

Plonus, M. A.

Ricklin, J. C.

Ryzhik, I. M.

I. S. Gradysteyn, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, 2000).

Tovar, A. A.

Tyson, R. K.

Wang, S. C. H.

Young, C. Y.

C. Y. Young, Y. V. Gilchrest, B. R. Macon, “Turbulence induced beam spreading of higher order mode optical waves,” Opt. Eng. (Bellingham) 41, 1097–1103 (2002).
[CrossRef]

Zhang, B.

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

Zhao, D.

D. Zhao, H. Mao, H. Liu, “Propagation of off-axial Hermite-cosh-Gaussian laser beams,” J. Opt. A, Pure Appl. Opt. 6, 77–83 (2004).
[CrossRef]

Zhu, X.

X. Zhu, J. M. Kahn, “Performance bounds for coded free-space optical communications through atmospheric turbulence channels,” IEEE Trans. Commun. 51, 1233–1239 (2003).
[CrossRef]

Appl. Opt. (2)

IEEE Trans. Commun. (1)

X. Zhu, J. M. Kahn, “Performance bounds for coded free-space optical communications through atmospheric turbulence channels,” IEEE Trans. Commun. 51, 1233–1239 (2003).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

D. Zhao, H. Mao, H. Liu, “Propagation of off-axial Hermite-cosh-Gaussian laser beams,” J. Opt. A, Pure Appl. Opt. 6, 77–83 (2004).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (5)

Opt. Commun. (2)

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

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

Opt. Eng. (Bellingham) (1)

C. Y. Young, Y. V. Gilchrest, B. R. Macon, “Turbulence induced beam spreading of higher order mode optical waves,” Opt. Eng. (Bellingham) 41, 1097–1103 (2002).
[CrossRef]

Opt. Express (1)

Other (2)

L. C. Andrews, R. L. Phillips, “Free space optical communication link and atmospheric effects: single aperture and arrays,” in Proceedings of Free-Space Laser Communication Technologies XVI, C. Y. Young and J. S. Stryjewski, eds., Proc. SPIE5338, 265–275 (2004).

I. S. Gradysteyn, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, 2000).

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Figures (10)

Fig. 1
Fig. 1

Three-dimensional intensity profile of two Hermite-sine-Gaussian beams for α s x = 5 cm , V x r = 50 m 1 , a x = 20 m 1 , and b x = 0 . Mode indices are n = 2 , m = 0 for the upper plot and n = 0 , m = 0 for the lower plot.

Fig. 2
Fig. 2

Contour plots of two Hermite-sine-Gaussian beams, the upper plot with n = 2 , m = 0 and the lower plot with n = 1 , m = 1 , retaining the rest of the parameter settings as in Fig. 1.

Fig. 3
Fig. 3

Contour plots of two Hermite-sinh-Gaussian beams for α s x = 3 cm , a x = 33.3 m 1 , and b x = 0 , one with n = 2 , m = 0 and V x i = V y i = 10 m 1 and the other with n = 2 , m = 1 , and V x i = V y i = 50 m 1 .

Fig. 4
Fig. 4

Two superimposed contour plots of Hermite-cosh-Gaussian beams that have the same parameter values as in Fig. 3.

Fig. 5
Fig. 5

Intensity distribution of a Hermite-sinh-Gaussian beam with n = 1 , m = 0 at L = 0 , 2, 5, and 20 km .

Fig. 6
Fig. 6

Intensity distribution of a Hermite-cosh-Gaussian beam with n = 1 , m = 0 at L = 0 , 2, 5, and 20 km .

Fig. 7
Fig. 7

Intensity distribution of a Hermite-sinh-Gaussian beam with n = 2 , m = 1 at L = 0 , 0.05, 1, and 250 km .

Fig. 8
Fig. 8

Intensity distribution of a Hermite-sinh-Gaussian beam with n = 2 , m = 0 at L = 0 , 0.05, 1, and 250 km .

Fig. 9
Fig. 9

Intensity distribution of a Hermite-sine-Gaussian beam with n = 1 , m = 0 at L = 0 , 0.4, 2, and 20 km .

Fig. 10
Fig. 10

Intensity distribution of a Hermite-cosine-Gaussian beam with n = 1 , m = 1 at L = 0 , 0.4, 2, and 50 km .

Equations (16)

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u s ( s ) = u s ( s x , s y ) = A c H n ( a x s x + b x ) H m ( a y s y + b y ) exp [ 0.5 ( s x 2 α s x 2 + s y 2 α s y 2 ) ] { l 1 exp [ i ( V x s x + V y s y ) ] + l 2 exp [ i ( Y x s x + Y y s y ) ] } .
I s cosine ( s ) = I s ( s x , s y ) = H n 2 ( a x s x + b x ) H m 2 ( a y s y + b y ) 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 ) ] ,
I s cosh ( s ) = I s ( s x , s y ) = H n 2 ( a x s x + b x ) H m 2 ( a y s y + b y ) exp [ ( s x 2 α s x 2 + s y 2 α s y 2 ) ] cosh 2 [ ( V x i s x + V y i s y ) ] ,
I s sine ( s ) = I s ( s x , s y ) = H n 2 ( a x s x + b x ) H m 2 ( a y s y + b y ) exp [ ( s x 2 α s x 2 + s y 2 α s y 2 ) ] sin 2 [ ( V x r s x + V y r s y ) ] ,
I s sinh ( s ) = I s ( s x , s y ) = H n 2 ( a x s x + b x ) H m 2 ( a y s y + b y ) exp [ ( s x 2 α s x 2 + s y 2 α s y 2 ) ] sinh 2 [ ( V x i s x + V y i s y ) ] .
u r ( p , z = L , t ) = k exp ( i k L ) ( 2 i π L ) d 2 s u s ( s ) exp [ i k ( p s ) 2 ( 2 L ) + ψ ( s , p ) 2 i π f t ] ,
I r ( p , z = L ) = k 2 ( 2 π L ) 2 d 2 s 1 d 2 s 2 u s ( s 1 ) u s * ( s 2 ) exp { i k [ ( 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 , z = L ) = b 2 ρ 0 4 ( D s x D s y ) 1 2 exp { ρ 0 4 b 2 [ p x 2 ( α s x 2 D s x ) + p y 2 ( α s y 2 D s y ) ] } [ exp { 0.5 ρ 0 2 [ V x 2 ( ρ 0 2 a s x + 1 ) D s x + V y 2 ( ρ 0 2 a s y + 1 ) D s y ] } { B s 1 exp [ 2 i ρ 0 4 b 2 ( V x p x D s x + V y p y D s y ) ] S x 1 S y 1 + B s 2 exp [ 2 i ρ 0 4 b 2 ( V x p x D s x + V y p y D s y ) ] S x 2 S y 2 } + exp { 0.25 ρ 0 4 [ V x 2 ( α s x 2 D s x ) + V y 2 ( α s y 2 D s y ) ] } ( B s 3 exp { ρ 0 4 b [ V x p x ( α s x 2 D s x ) + V y p y ( α s y 2 D s y ) ] } S x 3 S y 3 + B s 4 exp { ρ 0 4 b [ V x p x ( α s x 2 D s x ) + V y p y ( α s y 2 D s y ) ] } S x 4 S y 4 ) ] ,
b = k 2 L , a s x = [ ( 0.5 α s x 2 ) + ( 1 ρ 0 2 ) ] ,
a s y = [ ( 0.5 α s y 2 ) + ( 1 ρ 0 2 ) ] ,
D s x = [ ρ 0 4 ( a s x 2 + b 2 ) 1 ] ,
D s y = [ ρ 0 4 ( a s y 2 + b 2 ) 1 ] ,
S x 1 = l x 1 = 0 [ n 2 ] l n x 1 = 0 n 2 l x 1 k x 1 = 0 [ l n x 1 2 ] l n x 11 = 0 l n x 1 2 k x 1 l x 2 = 0 [ n 2 ] l n x 2 = 0 n 2 l x 2 k x 2 = 0 [ ( l n x 11 + l n x 2 ) 2 ] ( 1 ) l x 1 + l x 2 2 2 n l x 1 l x 2 l n x 1 l n x 2 i l n x 1 + l n x 2 2 k x 1 2 k x 2 T l x 1 T l x 2 ( n 2 l x 1 ) ( n 2 l x 2 ) ( n 2 l x 1 l n x 1 ) ( l n x 1 2 k x 1 l n x 11 ) ( n 2 l x 2 l n x 2 ) l n x 1 ! ( l n x 1 2 k x 1 ) ! k x 1 ! ( l n x 11 + l n x 2 ) ! ( l n x 11 + l n x 2 2 k x 2 ) ! k x 2 ! × ( a x ) l n x 1 + l n x 2 ( b x ) 2 n 2 l x 1 2 l x 2 l n x 1 l n x 2 ( ρ 0 2 ) l n x 2 ( D s x ) k x 2 l n x 11 l n x 2 ( a s x i b ) l n x 1 + k x 1 + k x 2 ( V x 2 b p x ) l n x 1 2 k x 1 l n x 11 [ ρ 0 2 ( V x + 2 b p x ) ( a s x i b ) + ( V x 2 b p x ) ] l n x 11 + l n x 2 2 k x 2 .
I s N ( s x , s y ) = I s ( s x , s y ) max [ ( I s ( s x , s y ) ) ] ,
I r N ( p , z = L ) = I r ( p , z = L ) max [ ( I s ( s x , s y ) ) ] .

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