Abstract

A numerical wave optics approach for simulating a partial spatially coherent beam is presented. The approach involves the application of a sequence of random phase screens to an initial beam field and the summation of the intensity results after propagation. The relationship between the screen parameters and the spatial coherence function for the beam is developed and the approach is verified by comparing results with analytic formulations for a Gaussian Schell-model beam. The approach can be used for modeling applications such as free space optical laser links that utilize partially coherent beams.

© 2006 Optical Society of America

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References

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  1. S. C. H. Wang and M. A. Plonus, "Optical beam propagation for a partially coherent source in the turbulent atmosphere," J. Opt. Soc. Am. 69, 1297-1304 (1979).
    [CrossRef]
  2. Y. Baykal, M. A. Plonus, and S. J. Wang, "The scintillations for a weak atmospheric turbulence using a partially coherent beam," Radio Sci. 18,551-556 (1983).
    [CrossRef]
  3. J. C. Ricklin and F. M. Davidson, "Atmospheric turbulence effects on a partially coherent Gaussian beam: implication for free-space laser communication," J. Opt. Soc. Am. A 19, 1794-1802 (2002).
    [CrossRef]
  4. W. Martienssen and E. Spiller, "Coherence and fluctuations in light beams," Am. J. Phys. 32, 919-926 (1964).
    [CrossRef]
  5. H. Arsenault and S. Lowenthal, "Partial coherence of an object illuminated with laser light through a moving diffuser," Opt. Commun. 1, 451-453 (1970).
    [CrossRef]
  6. E. Tervonen, A. T. Friberg, and J. Turunen, "Gaussian Schell-model beams generated with synthetic acousto-optic holograms," J. Opt. Soc. Am. A 9, 796-803 (1992).
    [CrossRef]
  7. D. G. Voelz and K. J. Fitzhenry, "Pseudo-partially coherent beam for free-space laser communication," in Free-Space Laser Communications IV, J. C. Ricklin, D. G. Voelz, eds., Proc. SPIE 5550, 218-224 (2004).
    [CrossRef]
  8. X. Xiao and D. G. Voelz, "Wave optics simulation of partially coherent beams," in Free-Space Laser Communications V, D. G. Voelz, J. C. Ricklin, eds., Proc. SPIE 5892, 219-227 (2005).
  9. D. G. Voelz, K. A. Bush, and P. S. Idell, "Illumination coherence effects in laser-speckle imaging: modeling and experimental demonstration," Appl. Opt. 36, 1781-1788 (1997).
    [CrossRef] [PubMed]
  10. J. W. Goodman, Statistical Optics, (John Wiley & Sons, 1985).
  11. L. Mandel and E. Wolf, "Radiation from sources of any state of coherence," in Optical Coherence and Quantum Optics, (Cambridge University, 1995), pp. 229-337.
  12. H. Stark and J. W. Woods, Probability and Random Process with Applications to Signal Processing, (Prentice Hall, 2002), Chap. 7.

2002

1997

1992

1983

Y. Baykal, M. A. Plonus, and S. J. Wang, "The scintillations for a weak atmospheric turbulence using a partially coherent beam," Radio Sci. 18,551-556 (1983).
[CrossRef]

1979

1970

H. Arsenault and S. Lowenthal, "Partial coherence of an object illuminated with laser light through a moving diffuser," Opt. Commun. 1, 451-453 (1970).
[CrossRef]

1964

W. Martienssen and E. Spiller, "Coherence and fluctuations in light beams," Am. J. Phys. 32, 919-926 (1964).
[CrossRef]

Arsenault, H.

H. Arsenault and S. Lowenthal, "Partial coherence of an object illuminated with laser light through a moving diffuser," Opt. Commun. 1, 451-453 (1970).
[CrossRef]

Baykal, Y.

Y. Baykal, M. A. Plonus, and S. J. Wang, "The scintillations for a weak atmospheric turbulence using a partially coherent beam," Radio Sci. 18,551-556 (1983).
[CrossRef]

Bush, K. A.

Davidson, F. M.

Friberg, A. T.

Idell, P. S.

Lowenthal, S.

H. Arsenault and S. Lowenthal, "Partial coherence of an object illuminated with laser light through a moving diffuser," Opt. Commun. 1, 451-453 (1970).
[CrossRef]

Martienssen, W.

W. Martienssen and E. Spiller, "Coherence and fluctuations in light beams," Am. J. Phys. 32, 919-926 (1964).
[CrossRef]

Plonus, M. A.

Y. Baykal, M. A. Plonus, and S. J. Wang, "The scintillations for a weak atmospheric turbulence using a partially coherent beam," Radio Sci. 18,551-556 (1983).
[CrossRef]

S. C. H. Wang and M. A. Plonus, "Optical beam propagation for a partially coherent source in the turbulent atmosphere," J. Opt. Soc. Am. 69, 1297-1304 (1979).
[CrossRef]

Ricklin, J. C.

Spiller, E.

W. Martienssen and E. Spiller, "Coherence and fluctuations in light beams," Am. J. Phys. 32, 919-926 (1964).
[CrossRef]

Tervonen, E.

Turunen, J.

Voelz, D. G.

Wang, S. C. H.

Wang, S. J.

Y. Baykal, M. A. Plonus, and S. J. Wang, "The scintillations for a weak atmospheric turbulence using a partially coherent beam," Radio Sci. 18,551-556 (1983).
[CrossRef]

Am. J. Phys.

W. Martienssen and E. Spiller, "Coherence and fluctuations in light beams," Am. J. Phys. 32, 919-926 (1964).
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

H. Arsenault and S. Lowenthal, "Partial coherence of an object illuminated with laser light through a moving diffuser," Opt. Commun. 1, 451-453 (1970).
[CrossRef]

Radio Sci.

Y. Baykal, M. A. Plonus, and S. J. Wang, "The scintillations for a weak atmospheric turbulence using a partially coherent beam," Radio Sci. 18,551-556 (1983).
[CrossRef]

Other

D. G. Voelz and K. J. Fitzhenry, "Pseudo-partially coherent beam for free-space laser communication," in Free-Space Laser Communications IV, J. C. Ricklin, D. G. Voelz, eds., Proc. SPIE 5550, 218-224 (2004).
[CrossRef]

X. Xiao and D. G. Voelz, "Wave optics simulation of partially coherent beams," in Free-Space Laser Communications V, D. G. Voelz, J. C. Ricklin, eds., Proc. SPIE 5892, 219-227 (2005).

J. W. Goodman, Statistical Optics, (John Wiley & Sons, 1985).

L. Mandel and E. Wolf, "Radiation from sources of any state of coherence," in Optical Coherence and Quantum Optics, (Cambridge University, 1995), pp. 229-337.

H. Stark and J. W. Woods, Probability and Random Process with Applications to Signal Processing, (Prentice Hall, 2002), Chap. 7.

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Figures (2)

Fig. 1.
Fig. 1.

GSM beam intensity patterns for two sets of parameters. (a), (b), and (c) correspond to w 0 = 8 cm, λ = 0.785 μm, σr = 181.4 cm σf = 4 cm, (σg = 0.4422 cm), z = 1 km; (d), (e), and (f) correspond to w 0 = 2.5 cm, λ = 1.50 μm, σr = 181.4 cm σf = 2 cm, (σg = 0.1106 cm), z = 2 km. Wave optics simulation intensity patterns are shown in (a) and (d); analytic patterns are shown in (b) and (e). Intensity profiles of the wave optics (red) and analytic results (blue) are overlaid in (c) and (f) for comparison. Plot units in (c) and (f) are intensity (vertical) and meters (horizontal).

Fig. 2.
Fig. 2.

Fringe profiles for different partially coherent beams propagated through two pinholes with parameters w 0 = 8.0 cm, λ = 0.785 μm, Δ = 0, Δ = 1 cm, d = 3 mm, z = 1 km, σr = 181.4 cm, and σf is (a) 4.3 cm, (b) 5.0 cm, and (c) 6.0 cm. The cases correspond to the following values of μ: (a) 0.15, (b) 0.35, and (c) 0.6. Simulation (red) and analytical results (blue) are overlaid for comparison. Plot units are intensity (vertical) and meters (horizontal).

Equations (24)

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U ( p , q ; t ) = U 0 ( p , q ) t A ( p , q ; t ) ,
t A ( p , q ; t ) = exp ( j ξ ( p , q ; t ) ) ,
I ( x , y ) = 1 ( λ z ) 2 U ( p , q ; t ) exp ( j 2 π λ z ( px + qy ) ) dpdq 2 t ,
I ( x , y ) = 1 ( λ z ) 2 U ( p 1 , q 1 ; t ) U * ( p 2 , q 2 ; t ) t exp [ j 2 π λ z ( ( p 1 p 2 ) x + ( q 1 q 2 ) y ) ] d p 1 d q 1 d p 2 d q 2 .
I ( x , y ) = 1 ( λ z ) 2 u 0 ( Δ p , Δ q ) R ( Δ p , Δ q ; t ) t exp [ j 2 π λ z ( Δ px + Δ qy ) ] d Δ pd Δ q ,
u 0 ( Δ p , Δ q ) = U 0 ( p ¯ + Δ p 2 , q ¯ + Δ q 2 ) U 0 * ( p ¯ Δ p 2 , q ¯ Δ q 2 ) d p ¯ d q ¯ .
ξ p q t = r ( p , q ; t ) f ( p , q ) ,
f ( p , q ) = 1 2 π σ f 2 exp ( p 2 + q 2 2 σ f 2 ) .
R ( Δ p , Δ q ) = exp ( j ξ ( p 1 , q 1 ; t ) ) exp ( j ξ ( p 2 , q 2 ; t ) ) t = exp ( 1 2 σ Δ ξ 2 ) ,
σ Δ ξ 2 = ( ξ ( p 2 , q 2 ; t ) ξ ( p 1 , q 1 ; t ) ) 2 t = 2 ( Γ ξ ( 0 , 0 ) Γ ξ ( Δ p , Δ q ) ) ,
Γ ξ ( Δ p , Δ q ) = R rr ( Δ p , Δ q ) [ f ( Δ p , Δ q ) f * ( Δ p , Δ q ) ] .
R rr ( Δ p , Δ q ) = σ r 2 δ ( Δ p , Δ q ) ,
f ( Δ p , Δ q ) f * ( Δ p , Δq ) = 1 4 π σ f 2 exp ( ( Δ p + Δ q ) 2 4 σ f 2 ) .
Γ ξ ( Δ p , Δ q ) = σ r 2 4 π σ f 2 exp ( Δ p 2 + Δ q 2 4 σ f 2 ) .
R ( Δ p , Δ q ) = exp { σ r 2 4 π σ f 2 [ 1 exp ( Δ p 2 + Δ q 2 4 σ f 2 ) ] } .
R ( Δ p , Δ q ) exp [ ( Δ p 2 + Δ q 2 ) 2 σ g 2 ] ,
σ g 2 = 8 π σ f 4 σ r 2 .
f ( n , m ) = 1 2 π σ f 2 exp ( ( n Δ n ) 2 + ( m Δ m ) 2 2 σ f 2 ) ; n [ N 2 , N 2 ] , m [ M 2 , M 2 ] ,
r ( n , m ) = rand ( n , m ) ( Δ n Δ m ) 1 2 ,
I x y z = 1 ( Δ ( z ) ) 2 exp [ 2 ( x 2 + y 2 ) w 0 2 ( Δ ( z ) ) 2 ] ,
Δ ( z ) = [ 1 + ( 2 z k w 0 2 ) 2 ( 1 + w 0 2 σ g 2 ) ] 1 2 ,
I x y z = 1 8 λ 2 z 2 [ π d 2 exp ( Δ p 2 + Δ q 2 4 w 0 2 ) ] 2 [ J 1 ( k d 2 z x 2 + y 2 ) ( k d 2 z x 2 + y 2 ) ] 2
{ 1 + μ cos [ k z ( Δ p x + Δ q y ) ] } ,
μ = R ( Δ p , Δ q ) R ( 0 , 0 ) ,

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