Abstract

The average intensity and spreading of cosh-Gaussian laser beams in the turbulent atmosphere are examined. Our research is based principally on formulating the average-intensity profile at the receiver plane for cosh-Gaussian excitation. The limiting cases of our formulation for the average intensity are found to reduce correctly to the existing Gaussian beam wave result in turbulence and the cosh-Gaussian beam result in free space (in the absence of turbulence). The average intensity and the broadening of the cosh-Gaussian beam wave after it propagates in the turbulent atmosphere are numerically evaluated versus source size, beam displacement, link length, structure constant, and two wavelengths of 0.85 and 1.55 μm, which are most widely used in currently employed free-space-optical links. Results indicate that in turbulence the beam is widened beyond its free-space diffraction values. At the receiver plane, analogous to the case of free space, this diffraction eventually leads to transformation of the cosh-Gaussian beam into an oscillatory average-intensity profile with a Gaussian envelope.

© 2005 Optical Society of America

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References

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    [CrossRef]
  3. C. Y. Young, Y. V. Gilchrest, B. R. Macon, “Turbulence-induced beam spreading of higher-order mode optical waves,” Opt. Eng. 41, 1097–1103 (2002).
    [CrossRef]
  4. T. Shirai, A. Dogariu, E. Wolf, “Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 20, 1094–1102 (2003).
    [CrossRef]
  5. S. A. Ponomarenko, J. J. Greffet, E. Wolf, “The diffusion of partially coherent beams in turbulent media,” Opt. Commun. 208, 1–8 (2002).
    [CrossRef]
  6. A. Dogariu, S. Amarande, “Propagation of partially coherent beams: turbulence-induced degradation,” Opt. Lett. 28, 10–12 (2003).
    [CrossRef] [PubMed]
  7. C. C. Davis, I. I. Smolyaninov, S. D. Milner, “Flexible optical wireless links and networks,” IEEE Commun. Mag. 41, 51–57 (2003).
    [CrossRef]
  8. H. A. Willebrand, B. S. Ghuman, “Fiber optics without fiber,” IEEE Spectrum 38, 40–45 (2001).
    [CrossRef]
  9. X. Zhu, J. M. Kahn, J. Wang, “Mitigation of turbulence-induced scintillation noise in free-space optical links using temporal-domain detection techniques,” IEEE Photonics Technol. Lett. 15, 623–625 (2003).
    [CrossRef]
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    [CrossRef]
  12. B. Lü, H. Ma, B. Zhang, “Propagation properties of cosh-Gaussian beams,” Opt. Commun. 164, 165–170 (1999).
    [CrossRef]
  13. B. Lü, S. R. Luo, “Beam propagation factor of hard-edge diffracted cosh-Gaussian beams,” Opt. Commun. 178, 275–281 (2000).
    [CrossRef]
  14. S. Yu, H. Guo, X. Q. Fu, W. Hu, “Propagation properties of elegant Hermite-cosh-Gaussian laser beams,” Opt. Commun. 204, 59–66 (2002).
    [CrossRef]
  15. I. S. Gradysteyn, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1980).

2004

2003

A. Dogariu, S. Amarande, “Propagation of partially coherent beams: turbulence-induced degradation,” Opt. Lett. 28, 10–12 (2003).
[CrossRef] [PubMed]

T. Shirai, A. Dogariu, E. Wolf, “Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 20, 1094–1102 (2003).
[CrossRef]

C. C. Davis, I. I. Smolyaninov, S. D. Milner, “Flexible optical wireless links and networks,” IEEE Commun. Mag. 41, 51–57 (2003).
[CrossRef]

X. Zhu, J. M. Kahn, J. Wang, “Mitigation of turbulence-induced scintillation noise in free-space optical links using temporal-domain detection techniques,” IEEE Photonics Technol. Lett. 15, 623–625 (2003).
[CrossRef]

2002

S. A. Ponomarenko, J. J. Greffet, E. Wolf, “The diffusion of partially coherent beams in turbulent media,” Opt. Commun. 208, 1–8 (2002).
[CrossRef]

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

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

2001

H. A. Willebrand, B. S. Ghuman, “Fiber optics without fiber,” IEEE Spectrum 38, 40–45 (2001).
[CrossRef]

2000

B. Lü, S. R. Luo, “Beam propagation factor of hard-edge diffracted cosh-Gaussian beams,” Opt. Commun. 178, 275–281 (2000).
[CrossRef]

1999

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

1998

1979

1967

Z. I. Feizulin, Y. A. Kravtsov, “Broadening of a laser beam in a turbulent medium,” Radiophys. Quantum Electron. 10, 33–35 (1967).
[CrossRef]

Amarande, S.

Baykal, Y.

Casperson, L. W.

Davis, C. C.

C. C. Davis, I. I. Smolyaninov, S. D. Milner, “Flexible optical wireless links and networks,” IEEE Commun. Mag. 41, 51–57 (2003).
[CrossRef]

Dogariu, A.

Feizulin, Z. I.

Z. I. Feizulin, Y. A. Kravtsov, “Broadening of a laser beam in a turbulent medium,” Radiophys. Quantum Electron. 10, 33–35 (1967).
[CrossRef]

Fu, X. Q.

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

Ghuman, B. S.

H. A. Willebrand, B. S. Ghuman, “Fiber optics without fiber,” IEEE Spectrum 38, 40–45 (2001).
[CrossRef]

Gilchrest, Y. V.

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

Gradysteyn, I. S.

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

Greffet, J. J.

S. A. Ponomarenko, J. J. Greffet, E. Wolf, “The diffusion of partially coherent beams in turbulent media,” Opt. Commun. 208, 1–8 (2002).
[CrossRef]

Guo, H.

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

Hu, W.

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

Kahn, J. M.

X. Zhu, J. M. Kahn, J. Wang, “Mitigation of turbulence-induced scintillation noise in free-space optical links using temporal-domain detection techniques,” IEEE Photonics Technol. Lett. 15, 623–625 (2003).
[CrossRef]

Kravtsov, Y. A.

Z. I. Feizulin, Y. A. Kravtsov, “Broadening of a laser beam in a turbulent medium,” Radiophys. Quantum Electron. 10, 33–35 (1967).
[CrossRef]

Lü, B.

B. Lü, S. R. Luo, “Beam propagation factor of hard-edge diffracted cosh-Gaussian beams,” Opt. Commun. 178, 275–281 (2000).
[CrossRef]

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

Luo, S. R.

B. Lü, S. R. Luo, “Beam propagation factor of hard-edge diffracted cosh-Gaussian beams,” Opt. Commun. 178, 275–281 (2000).
[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. 41, 1097–1103 (2002).
[CrossRef]

Milner, S. D.

C. C. Davis, I. I. Smolyaninov, S. D. Milner, “Flexible optical wireless links and networks,” IEEE Commun. Mag. 41, 51–57 (2003).
[CrossRef]

Plonus, M. A.

Ponomarenko, S. A.

S. A. Ponomarenko, J. J. Greffet, E. Wolf, “The diffusion of partially coherent beams in turbulent media,” Opt. Commun. 208, 1–8 (2002).
[CrossRef]

Ryzhik, I. M.

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

Shirai, T.

Smolyaninov, I. I.

C. C. Davis, I. I. Smolyaninov, S. D. Milner, “Flexible optical wireless links and networks,” IEEE Commun. Mag. 41, 51–57 (2003).
[CrossRef]

Tovar, A. A.

Wang, J.

X. Zhu, J. M. Kahn, J. Wang, “Mitigation of turbulence-induced scintillation noise in free-space optical links using temporal-domain detection techniques,” IEEE Photonics Technol. Lett. 15, 623–625 (2003).
[CrossRef]

Wang, S. C. H.

Willebrand, H. A.

H. A. Willebrand, B. S. Ghuman, “Fiber optics without fiber,” IEEE Spectrum 38, 40–45 (2001).
[CrossRef]

Wolf, E.

T. Shirai, A. Dogariu, E. Wolf, “Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 20, 1094–1102 (2003).
[CrossRef]

S. A. Ponomarenko, J. J. Greffet, E. Wolf, “The diffusion of partially coherent beams in turbulent media,” Opt. Commun. 208, 1–8 (2002).
[CrossRef]

Young, C. Y.

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

Yu, S.

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

Zhang, B.

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

Zhu, X.

X. Zhu, J. M. Kahn, J. Wang, “Mitigation of turbulence-induced scintillation noise in free-space optical links using temporal-domain detection techniques,” IEEE Photonics Technol. Lett. 15, 623–625 (2003).
[CrossRef]

IEEE Commun. Mag.

C. C. Davis, I. I. Smolyaninov, S. D. Milner, “Flexible optical wireless links and networks,” IEEE Commun. Mag. 41, 51–57 (2003).
[CrossRef]

IEEE Photonics Technol. Lett.

X. Zhu, J. M. Kahn, J. Wang, “Mitigation of turbulence-induced scintillation noise in free-space optical links using temporal-domain detection techniques,” IEEE Photonics Technol. Lett. 15, 623–625 (2003).
[CrossRef]

IEEE Spectrum

H. A. Willebrand, B. S. Ghuman, “Fiber optics without fiber,” IEEE Spectrum 38, 40–45 (2001).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

S. A. Ponomarenko, J. J. Greffet, E. Wolf, “The diffusion of partially coherent beams in turbulent media,” Opt. Commun. 208, 1–8 (2002).
[CrossRef]

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

B. Lü, S. R. Luo, “Beam propagation factor of hard-edge diffracted cosh-Gaussian beams,” Opt. Commun. 178, 275–281 (2000).
[CrossRef]

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

Opt. Eng.

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

Opt. Lett.

Radiophys. Quantum Electron.

Z. I. Feizulin, Y. A. Kravtsov, “Broadening of a laser beam in a turbulent medium,” Radiophys. Quantum Electron. 10, 33–35 (1967).
[CrossRef]

Other

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

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Figures (11)

Fig. 1
Fig. 1

Propagation geometry.

Fig. 2
Fig. 2

Normalized intensity of a cosh-Gaussian beam at the source plane.

Fig. 3
Fig. 3

Contour plot at the source plane of the same cosh-Gaussian beam as in Fig. 2.

Fig. 4
Fig. 4

Normalized intensity at the source plane and the normalized average intensity at the receiver plane for a cosh-Gaussian beam.

Fig. 5
Fig. 5

Contour plot of the normalized intensity at the source plane and the normalized average intensity at the receiver plane for the same cosh-Gaussian beam of Fig. 4.

Fig. 6
Fig. 6

Difference in beam sizes at the receiver and the source planes in the x direction Δαxbs versus link length in turbulence and in free space for a fixed source size and at FSO wavelengths of 0.85 and 1.55 μm.

Fig. 7
Fig. 7

Difference in beam sizes at the receiver and the source planes in the x direction Δαxbs versus the source size in turbulence and in free space for a fixed link length and at FSO wavelengths of 0.85 and 1.55 μm.

Fig. 8
Fig. 8

Difference in beam sizes at the receiver and the source planes in the x direction Δαxbs versus the structure constant Cn2 evaluated at source size αsx = 3 cm, Vxi = 55 cm−1 and link length L = 2 km and for three FSO wavelengths of 0.85, 1.3, and 1.55 μm.

Fig. 9
Fig. 9

Normalized average intensity at the receiver transverse plane (z = L) at various link lengths.

Fig. 10
Fig. 10

Average intensity at the receiver transverse plane (z = L) normalized with respect to the maximum value of the free-space-received intensity evaluated at various structure constants Cn2.

Fig. 11
Fig. 11

Ir0(px, py, z = L) for different Vxi values at the fixed source and link parameters of αsx = 0.1 cm, L = 1 km, λ = 1.55 μm, Cn2 = 1 × 10−15 m−2/3.

Equations (27)

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u s ( s x ,     s y ,     z = 0 ) = 0.5 A exp ( - i ϕ ) exp { - 0.5 [ ( s x / α s x ) 2 + ( s y / α s y ) 2 ] } [ exp ( V x i s x + V y i s y ) + exp ( - V x i s x - V y i s y ) ] ,
I s ( s x ,     s y ,     z = 0 ) = 0.5 A exp [ - ( s x 2 / α s x 2 + s y 2 / α s y 2 ) ] × { 1 + cosh [ 2 ( V x i s x + s y 2 / α s y 2 ) ] } = 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 ) ] .
- ( 2 s x p / α s x 2 ) cosh ( V x i s x p + V y i s y p ) + 2 V x i × sinh ( V x i s x p + V y i s y p ) = 0 ,
- ( 2 s y p / α s y 2 ) cosh ( V x i s x p + V y i s y p ) + 2 V y i × sinh ( V x i s x p + V y i s y p ) = 0 ,
cosh ( arg ) sinh ( arg ) 0.5 exp ( arg ) .
( 2 s x p / α s x 2 ) ± 2 V x i = 0 ,
( 2 s y p / α s y 2 ) ± 2 V y i = 0.
I o = I s ( s x = 0 ,     s y = 0 ,     z = 0 ) / I s ( s x = s x p ,     s y = s y p ,     z = 0 )
I s [ s x = α x b s ( z = 0 ) ,     s y = α y b s ( z = 0 ) ,     z = 0 ] / I s ( s x = s x p ,     s y = s y p ,     z = 0 ) = exp ( - 2 ) ,
α x b s ( z = 0 ) = α s x ( V x i α s x + 1 ) ,
α y b s ( z = 0 ) = α s y ( V y i α s y + 1 ) .
I s N ( s x ,     s y ,     z = 0 ) = I s ( s x ,     s y ,     z = 0 ) / I s ( s x = s x p ,     s y = s y p ,     z = 0 ) .
< u ( p , L , t ) > = k exp ( i k L ) / ( 2 π i L ) × - - 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 { 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 ( 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 [ - ( s 1 x 2 + s 2 x 2 ) / ( 2 α s x 2 ) - ( s 1 y 2 + s 2 y 2 ) / ( 2 α s y 2 ) ] { exp [ V x i ( s 1 x + s 2 x ) + V y i ( s 1 y + s 2 y ) ] + exp [ - V x i ( s 1 x + s 2 x ) - V y i ( s 1 y + s 2 y ) ] + exp [ V x i ( s 1 x - s 2 x ) + V y i ( s 1 y - s 2 y ) ] + exp [ - V x i ( s 1 x - s 2 x ) - V y i ( 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 i 2 ( ρ 0 2 a s x + 1 ) / D s x ) + V y i 2 ( ρ 0 2 a s y + 1 ) / D s y ] } × cosh [ ( 2 ρ 0 4 k 2 / L 2 ) ( V x i p x / D s x + V y i p y / D s y ) ] + exp { ρ 0 4 [ V x i 2 / ( α s x 2 D s x ) + V y i 2 / ( α s y 2 D s y ) ] } cos { ( 2 ρ 0 4 k / L ) × [ V x i p x / ( α s x 2 D s x ) + V y i p y / ( α s y 2 D s y ) ] } )
α x b s ( z = L ) = V x i α s x 2 + α s x L D s x 0.5 / ( ρ 0 2 k ) = V x i α s x 2 + ( k α s x ) - 1 [ L 2 + ( 4 α s x 2 L 2 / ρ 0 2 ) + k 2 α s x 4 ] 0.5 ,
α y b s ( z = L ) = V y i α s y 2 + α s y L D s y 0.5 / ( ρ 0 2 k ) = V y i α s y 2 + ( k α s y ) - 1 [ L 2 + ( 4 α s y 2 L 2 / ρ 0 2 ) + k 2 α s y 4 ] 0.5 ,
arg = 2 { [ V x i 2 α s x 2 / ( k - 2 L 2 α s x - 4 + 4 k - 2 L 2 α s x - 2 ρ 0 - 2 + 1 ) ] + [ V y i 2 α s y 2 / ( k - 2 L 2 α s y - 4 + 4 k - 2 L 2 α s y - 2 ρ 0 - 2 + 1 ) ] } .
< 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 2 ρ 0 2 ( ρ x 2 + p y 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 i 2 g x + α s y 2 V y i 2 g y ) ] { cosh [ 2 k 2 ( α s x 4 V x i g x p x + α s y 4 V y i g y p y ) ] + cos [ 2 k L ( α s x 2 V x i g x p x + α s y 2 V y i g y p y ) ] } ,
I r N ( p x ,     p y ,     z = L ) = < I ( p x ,     p y ,     z = L ) > / I s ( s x = s x p ,     s y = s y p ,     z = 0 ) .
Δ α x b s = α x b s ( z = L ) - α x b s ( z = 0 ) = ( k α s x ) - 1 [ L 2 + ( 4 α s x 2 L 2 / ρ 0 2 ) + k 2 α s x 4 ] 0.5 - α s x .
I r C ( p x ,     p y ,     z = L ) = < I ( p x ,     p y ,     z = L ) > / max [ < I ( p x ,     p y ,     z = L ) > ] C n 2 = 0 .
I r 0 ( p x ,     p y ,     z = L ) = < I ( p x ,     p y ,     z = L ) > / < I ( p x = 0 ,     p y = 0 ,     z = L ) > .

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