Abstract

Optical coherence theory typically deals with the average properties of randomly fluctuating fields. However, in some circumstances the averaging process can mask important physical aspects of the field propagation. We derive a new method of simulating partially coherent fields of nearly arbitrary spatial and temporal coherence. These simulations produce the expected coherence properties when averaged over sufficently long time intervals. Examples of numerous fields are given, and an analytic formula for the intensity fluctuations of the field is given. The method is applied to the propagation of partially coherent fields through random phase screens.

© 2006 Optical Society of America

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References

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  1. E Verdet, Lecons d’Optique Physique, (L’Imprimierie Imperiale, Paris, 1869), vol. 1.
  2. M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, 1999, 7th (expanded) edition).
  3. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995).
  4. J. Wu, "Propagation of a Gaussian-Schell beam through turbulent media," J. Mod. Opt. 37, 671-684 (1990).
    [CrossRef]
  5. G. Gbur and E. Wolf, "Spreading of partially coherent beams in random media," J. Opt. Soc. Am. A 19, 1592-1598 (2002).
    [CrossRef]
  6. J. Wu and A. D. Boardman, "Coherence length of a Gaussian Schell-model beam and atmospheric turbulence," J. Mod. Opt. 38, 1355-1363 (1991).
    [CrossRef]
  7. J. C. Ricklin and F. M. Davidson, "Atmospheric turbulence effects on a partially coherent Gaussian Beam: implications for free-space laser communication," J. Opt. Soc. Am. A 19, 1794-1802 (2002).
    [CrossRef]
  8. O. Korotkova, L. C. Andrews, R. L. Phillips, "A Model for a Partially Coherent Gaussian Beam in Atmospheric Turbulence with Application in Lasercom," Opt. Eng. 43, 330-341 (2004).
    [CrossRef]
  9. L.C. Andrews and R.L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, Bellingham, Washington, 1998).
  10. M.S. Soskin and M.V. Vasnetsov, Singular Optics, in Progress in Optics, ed. E.Wolf (Elsevier, Amsterdam, 2001), vol. 42, p. 219-276.
  11. G. Gbur, T.D. Visser and E. Wolf, "’Hidden’ singularities in partially coherent wavefields", J. Opt. A 6, S239-S242 (2004).
    [CrossRef]
  12. G. Gbur and T.D. Visser, "Phase singularities and coherence vortices in linear optical systems," Opt. Commun. 259, 428-435 (2006).
    [CrossRef]
  13. W. Wang and E. Wolf, "Invariance properties of random pulses and of other random fields in dispersive media," Phys. Rev. E 52, 5532-5539 (1995).
    [CrossRef]
  14. S.O. Rice, Mathematical analysis of random noise, in Selected Papers on Noise and Stochastic Processes, ed. N. Wax (Dover, NY, 1954).
  15. L.C. Andrew and R.L. Phillips, Laser Beam Scintillation with Applications (SPIE Press, Bellingham, Washington, 2001).
    [CrossRef]
  16. R. Loudon, The Quantum Theory of Light (Oxford University Press, Oxford, 1983, 2nd edition).
  17. F.T. Arecchi, E. Gatti and A. Sona, "Time distribution of photons from coherent and Gaussian sources," Phys. Lett. 20, 27-29 (1966).
    [CrossRef]
  18. D.L. Knepp, "Multiple phase-screen calculation of the temporal behavior of stochastic waves," Proc. IEEE 71, 722-737 (1983).
    [CrossRef]

2006

G. Gbur and T.D. Visser, "Phase singularities and coherence vortices in linear optical systems," Opt. Commun. 259, 428-435 (2006).
[CrossRef]

2004

O. Korotkova, L. C. Andrews, R. L. Phillips, "A Model for a Partially Coherent Gaussian Beam in Atmospheric Turbulence with Application in Lasercom," Opt. Eng. 43, 330-341 (2004).
[CrossRef]

G. Gbur, T.D. Visser and E. Wolf, "’Hidden’ singularities in partially coherent wavefields", J. Opt. A 6, S239-S242 (2004).
[CrossRef]

2002

1995

W. Wang and E. Wolf, "Invariance properties of random pulses and of other random fields in dispersive media," Phys. Rev. E 52, 5532-5539 (1995).
[CrossRef]

1991

J. Wu and A. D. Boardman, "Coherence length of a Gaussian Schell-model beam and atmospheric turbulence," J. Mod. Opt. 38, 1355-1363 (1991).
[CrossRef]

1990

J. Wu, "Propagation of a Gaussian-Schell beam through turbulent media," J. Mod. Opt. 37, 671-684 (1990).
[CrossRef]

1983

D.L. Knepp, "Multiple phase-screen calculation of the temporal behavior of stochastic waves," Proc. IEEE 71, 722-737 (1983).
[CrossRef]

1966

F.T. Arecchi, E. Gatti and A. Sona, "Time distribution of photons from coherent and Gaussian sources," Phys. Lett. 20, 27-29 (1966).
[CrossRef]

Andrews, L. C.

O. Korotkova, L. C. Andrews, R. L. Phillips, "A Model for a Partially Coherent Gaussian Beam in Atmospheric Turbulence with Application in Lasercom," Opt. Eng. 43, 330-341 (2004).
[CrossRef]

Arecchi, F.T.

F.T. Arecchi, E. Gatti and A. Sona, "Time distribution of photons from coherent and Gaussian sources," Phys. Lett. 20, 27-29 (1966).
[CrossRef]

Boardman, A. D.

J. Wu and A. D. Boardman, "Coherence length of a Gaussian Schell-model beam and atmospheric turbulence," J. Mod. Opt. 38, 1355-1363 (1991).
[CrossRef]

Davidson, F. M.

Gatti, E.

F.T. Arecchi, E. Gatti and A. Sona, "Time distribution of photons from coherent and Gaussian sources," Phys. Lett. 20, 27-29 (1966).
[CrossRef]

Gbur, G.

G. Gbur and T.D. Visser, "Phase singularities and coherence vortices in linear optical systems," Opt. Commun. 259, 428-435 (2006).
[CrossRef]

G. Gbur, T.D. Visser and E. Wolf, "’Hidden’ singularities in partially coherent wavefields", J. Opt. A 6, S239-S242 (2004).
[CrossRef]

G. Gbur and E. Wolf, "Spreading of partially coherent beams in random media," J. Opt. Soc. Am. A 19, 1592-1598 (2002).
[CrossRef]

Knepp, D.L.

D.L. Knepp, "Multiple phase-screen calculation of the temporal behavior of stochastic waves," Proc. IEEE 71, 722-737 (1983).
[CrossRef]

Korotkova, O.

O. Korotkova, L. C. Andrews, R. L. Phillips, "A Model for a Partially Coherent Gaussian Beam in Atmospheric Turbulence with Application in Lasercom," Opt. Eng. 43, 330-341 (2004).
[CrossRef]

Phillips, R. L.

O. Korotkova, L. C. Andrews, R. L. Phillips, "A Model for a Partially Coherent Gaussian Beam in Atmospheric Turbulence with Application in Lasercom," Opt. Eng. 43, 330-341 (2004).
[CrossRef]

Ricklin, J. C.

Sona, A.

F.T. Arecchi, E. Gatti and A. Sona, "Time distribution of photons from coherent and Gaussian sources," Phys. Lett. 20, 27-29 (1966).
[CrossRef]

Visser, T.D.

G. Gbur and T.D. Visser, "Phase singularities and coherence vortices in linear optical systems," Opt. Commun. 259, 428-435 (2006).
[CrossRef]

G. Gbur, T.D. Visser and E. Wolf, "’Hidden’ singularities in partially coherent wavefields", J. Opt. A 6, S239-S242 (2004).
[CrossRef]

Wang, W.

W. Wang and E. Wolf, "Invariance properties of random pulses and of other random fields in dispersive media," Phys. Rev. E 52, 5532-5539 (1995).
[CrossRef]

Wolf, E.

G. Gbur, T.D. Visser and E. Wolf, "’Hidden’ singularities in partially coherent wavefields", J. Opt. A 6, S239-S242 (2004).
[CrossRef]

G. Gbur and E. Wolf, "Spreading of partially coherent beams in random media," J. Opt. Soc. Am. A 19, 1592-1598 (2002).
[CrossRef]

W. Wang and E. Wolf, "Invariance properties of random pulses and of other random fields in dispersive media," Phys. Rev. E 52, 5532-5539 (1995).
[CrossRef]

Wu, J.

J. Wu and A. D. Boardman, "Coherence length of a Gaussian Schell-model beam and atmospheric turbulence," J. Mod. Opt. 38, 1355-1363 (1991).
[CrossRef]

J. Wu, "Propagation of a Gaussian-Schell beam through turbulent media," J. Mod. Opt. 37, 671-684 (1990).
[CrossRef]

J. Mod. Opt.

J. Wu, "Propagation of a Gaussian-Schell beam through turbulent media," J. Mod. Opt. 37, 671-684 (1990).
[CrossRef]

J. Wu and A. D. Boardman, "Coherence length of a Gaussian Schell-model beam and atmospheric turbulence," J. Mod. Opt. 38, 1355-1363 (1991).
[CrossRef]

J. Opt. A

G. Gbur, T.D. Visser and E. Wolf, "’Hidden’ singularities in partially coherent wavefields", J. Opt. A 6, S239-S242 (2004).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

G. Gbur and T.D. Visser, "Phase singularities and coherence vortices in linear optical systems," Opt. Commun. 259, 428-435 (2006).
[CrossRef]

Opt. Eng.

O. Korotkova, L. C. Andrews, R. L. Phillips, "A Model for a Partially Coherent Gaussian Beam in Atmospheric Turbulence with Application in Lasercom," Opt. Eng. 43, 330-341 (2004).
[CrossRef]

Phys. Lett.

F.T. Arecchi, E. Gatti and A. Sona, "Time distribution of photons from coherent and Gaussian sources," Phys. Lett. 20, 27-29 (1966).
[CrossRef]

Phys. Rev. E

W. Wang and E. Wolf, "Invariance properties of random pulses and of other random fields in dispersive media," Phys. Rev. E 52, 5532-5539 (1995).
[CrossRef]

Proc. IEEE

D.L. Knepp, "Multiple phase-screen calculation of the temporal behavior of stochastic waves," Proc. IEEE 71, 722-737 (1983).
[CrossRef]

Other

E Verdet, Lecons d’Optique Physique, (L’Imprimierie Imperiale, Paris, 1869), vol. 1.

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, 1999, 7th (expanded) edition).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995).

S.O. Rice, Mathematical analysis of random noise, in Selected Papers on Noise and Stochastic Processes, ed. N. Wax (Dover, NY, 1954).

L.C. Andrew and R.L. Phillips, Laser Beam Scintillation with Applications (SPIE Press, Bellingham, Washington, 2001).
[CrossRef]

R. Loudon, The Quantum Theory of Light (Oxford University Press, Oxford, 1983, 2nd edition).

L.C. Andrews and R.L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, Bellingham, Washington, 1998).

M.S. Soskin and M.V. Vasnetsov, Singular Optics, in Progress in Optics, ed. E.Wolf (Elsevier, Amsterdam, 2001), vol. 42, p. 219-276.

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Figures (7)

Fig. 1.
Fig. 1.

Illustrating several realizations of the intensity of the field generated by the method with σI = 2cm, σg = 1cm, and Gaussian spectrum of center frequency 1×1015 Hz and 1% bandwidth. The pictures show the gradual evolution of the field in time; the frames are each separated by 5 periods at the center frequency. The window size is 10cm on a side.

Fig. 2.
Fig. 2.

Illustrating the average intensity of the field (a) in the source plane and (b) through a cross-section of the source plane. The dots indicate the numerically calculated result; the solid line indicates the expected result of Eq. (41). The average is taken over 50 instantaneous values of the field, each separated by 20 periods at the center frequency. All other parameters are as in Fig. 1.

Fig. 3.
Fig. 3.

Illustrating the spectral degree of coherence of the field as calculated using 50 instantaneous values of the field, each separated by 20 periods at the center frequency. For (a), σg = 1cm, while for (b), σg = 2cm. The dashed lines indicate the expected result of Eq. (41).

Fig. 4.
Fig. 4.

Illustrating the temporal coherence function calculated by time averaging. The solid lines indicate the analytic prediction. For (a), the function is Gaussian, while for (b), the function is Lorentzian.

Fig. 5.
Fig. 5.

Illustrating (a) the instantaneous intensity and (b) the average intensity of a spatially coherent field after propagating through a Gaussian random phase screen at z = 1.0km, with the detector plane at z = 2km. Here L 0 = 15cm and ϕ02 = 1.

Fig. 6.
Fig. 6.

Illustrating (a) the instantaneous intensity and (b) the average intensity of a partially coherent field with σg = 1cm after propagating through a Gaussian random phase screen at z = 1.0km, with the detector plane at z = 2km. Here L 0 = 15cm and ϕ02 = 1.

Fig. 7.
Fig. 7.

A qualitative explanation of the behavior of partially coherent beams in turbulence. A coherent laser essentially propagates its energy through a single coherent mode, which is subject to distortion on propagation through the inhomogeneous medium. The partially coherent beam sends its energy through multiple (independent) modes, each of which propagates differently in the turbulent medium.

Equations (70)

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p ( N ) = N ̅ N N ! exp [ N ̅ ] .
V N r t = j = 1 N Λ r t t j exp [ i K j ∙r ] ,
Γ r 1 r 2 τ V * ( r 1 , t 1 ) V r 2 t 2 ,
V N * r 1 t 1 V N r 2 t 2 = i , j = 1 N Λ * ( r 1 , t 1 t i ) exp [ i K i r 1 ] Λ r 2 t 2 t j exp [ i K j r 2 ] .
Λ r t = 0 Λ ˜ r ω e iωt d ω .
V N * r 1 t 1 V N r 2 t 2 = i , j = 1 N 0 0 Λ ˜ * r 1 ω Λ ˜ r 2 ω exp [ i K i r 1 ] exp [ i K j r 1 ] × exp [ ( t 1 t i ) ] exp [ ( t 2 t j ) d ωd ω
F t i t j = 1 T 2 T 2 T 2 T 2 T 2 F t i t j d t i d t j .
exp [ i ( ωt i ω t j ) ] = { sin c [ ( ω ω ) T 2 ] f ( ω ω ) , i = j , sin c [ ωT 2 ] sin c [ ω T 2 ] g ( ω ) g ( ω ) , i j ,
exp [ i ωt i ] = sin c [ ωT 2 ] g ( ω ) ,
V N * ( r 1 , t 1 ) V N ( r 2 , t 2 ) = i = 1 N 0 0 Λ ˜ * ( r 1 , ω ) Λ ˜ ( r 2 , ω ' ) exp [ iK i ( r 1 r 2 ) ]
× exp [ i ( ωt 1 ω t 2 ) ] f ( ω ω ) d ω d ω
+ i j N 0 0 Λ ˜ * ( r 1 , ω ) Λ ˜ ( r 2 , ω ) exp [ i ( K i r 1 K j r 2 ) ]
× exp [ i ( ωt 1 ω t 2 ) ] g ( ω ) g ( ω ) d ω .
exp [ i K i ( r 1 r 2 ) ] = P ( K i ) exp [ iK i ( r 1 r 2 ) ] d 2 K i = ( 2 π ) 2 P ˜ ( r 2 r 1 ) ,
exp [ i ( K i r 1 K j r 2 ) ] = ( 2 π ) 4 P ˜ * ( r 1 ) P ˜ ( r 2 )
P ˜ ( r ) = 1 ( 2 π ) 2 P ( K ) exp [ i K∙r ] d 2 K .
Γ N ( r 1 , r 2 , τ ) = V N * ( r 1 , t 1 ) V N ( r 2 , t 2 )
= ( 2 π ) 2 N 0 0 Λ ˜ * ( r 1 , ω ) Λ ˜ ( r 2 , ω ) P ˜ ( r 2 r 1 )
× exp [ i ( ω t 1 ω t 2 ) ] f ( ω ω ) d ω d ω
+ ( 2 π ) 4 N ( N 1 ) 0 0 Λ ˜ * ( r 1 , ω ) Λ ˜ ( r 2 , ω ) P ˜ * ( r 1 ) P ˜ ( r 2 )
× exp [ i ( ω t 1 ω t 2 ) ] g ( ω ) g ( ω ) d ω d ω
Γ r 1 r 2 τ = N = 0 p ( N ) Γ N r 1 r 2 τ ,
N = 0 p ( N ) N = N ̅ ,
N = 0 p ( N ) N ( N 1 ) = N ̅ 2 .
Γ ( r 1 , r 2 , τ ) = ( 2 π ) 2 N ¯ 0 0 Λ ˜ * ( r 1 , ω ) Λ ˜ ( r 2 , ω ) P ˜ ( r 2 r 1 )
× exp [ i ( ω t 1 ω t 2 ) ] f ( ω ω ) d ω d ω
+ ( 2 π ) 4 N ¯ 2 0 0 Λ ˜ * ( r 1 , ω ) Λ ˜ ( r 2 , ω ) P ˜ * ( r 1 ) P ˜ ( r 2 )
× exp [ i ( ω t 1 ω t 2 ) ] g ( ω ) g ( ω ) d ω d ω .
Γ ( r 1 , r 2 , τ ) = ( 2 π ) 2 η 0 0 Λ ˜ * ( r 1 , ω ) Λ ˜ ( r 2 , ω ) P ˜ ( r 2 r 1 )
× exp [ i ( ω t 1 ω t 2 ) ] f ( ω ω ) T d ω d ω
+ ( 2 π ) 4 η 2 0 0 Λ ˜ * ( r 1 , ω ) Λ ˜ ( r 2 , ω ) P ˜ * ( r 1 ) P ˜ ( r 2 )
× exp [ i ( ω t 1 ω t 2 ) ] g ( ω ) T g ( ω ) T d ω d ω .
f ( ω ω ) T→ 2 πδ ( ω ω ) ,
g ( ω ) 4 π δ ( e ) ( ω ) ,
0 ε δ ( e ) ( ω ) d ω = 1 2 .
Γ r 1 r 2 τ = ( 2 π ) 3 η 0 Λ ˜ * r 1 ω Λ ˜ r 2 ω P ˜ ( r 2 r 1 ) exp [ i ωτ ] d ω + ( 2 π ) 6 η 2 Λ ˜ * ( r 1 , 0 ) Λ ˜ r 2 0 P ˜ * ( r 1 ) P ˜ ( r 2 ) .
Λ r t = Θ ( r ) Φ ( t ) ,
Λ ˜ r ω = Θ ( r ) Φ ˜ ( ω ) .
Γ r 1 r 2 τ = η ( 2 π ) 3 Θ * ( r 1 ) Θ ( r 2 ) P ˜ ( r 2 r 1 ) 0 Φ ˜ ( ω ) 2 exp [ i ωτ ] d ω .
W r 1 r 2 ω = η ( 2 π ) 3 Φ ˜ ( ω ) 2 Θ * ( r 1 ) Θ ( r 2 ) P ˜ ( r 2 r 1 ) ,
W r 1 r 2 ω = 1 2 π Γ r 1 r 2 τ exp [ i ωτ ] .
σ I 2 ( r ) = I r t 2 I r t 2 1 ,
I N r t 2 = V N * r t V N * r t V N r t V N r t ,
I N ( r , t ) 2 = ijkl N 0 0 0 0 Λ ˜ * ( r , ω 1 ) Λ ˜ * ( r , ω 2 )
× Λ ˜ ( r , ω 3 ) Λ ˜ ( r , ω 4 ) exp [ i t ( ω 1 + ω 2 ω 3 ω 4 ) ]
× exp [ i ( ω 1 t i + ω 2 t j ω 3 t k ω 4 t l ) ]
× exp [ i r ( K i + K j K k K l ) ] d ω 1 d ω 2 d ω 3 d ω 4 .
i = j = k = l = α , N terms , α = i = k j = l = β N ( N 1 ) terms, α = i = l j = k = β N ( N 1 ) terms .
exp [ i ( ω 1 + ω 2 ω 3 ω 4 ) t α ] = 2 πδ ( ω 1 + ω 2 ω 3 ω 4 ) ,
exp [ i ( ω 1 ω 3 ) t α ] exp [ i ( ω 2 ω 4 ) t β ] = ( 2 π ) 2 δ ( ω 1 ω 3 ) δ ( ω 2 ω 4 ) ,
exp [ i ( ω 1 ω 4 ) t α ] exp [ i ( ω 2 ω 3 ) t β ] = ( 2 π ) 2 δ ( ω 1 ω 4 ) δ ( ω 2 ω 3 ) .
( 2 π ) 2 P ˜ ( 0 ) = 1 , i = j = k = l = α exp [ i ( K i + K j K k K l ) ·r ] = ( 2 π ) 4 ( P ˜ ( 0 ) ) 2 = 1 , α = i = k j = l = β , ( 2 π ) 4 ( P ˜ ( 0 ) ) 2 = 1 , α = i = l j = k = β
I r t 2 = 2 πη 0 0 0 Λ ˜ * r ω 2 + ω 3 + ω 4 Λ ˜ * r ω 2
× Λ ˜ r ω 3 Λ ˜ r ω 4 d ω 2 d ω 3 d ω 4
+ ( 2 π ) 2 2 η 2 2 0 Λ ˜ r ω 1 2 Λ ˜ r ω 2 2 d ω 1 d ω 2 .
I ( r , t ) 2 = η Θ ( r ) 4 Φ ( t ) 4 d t + 2 I r t 2 .
σ I 2 = 1 η Φ ( t ) 4 d t η Φ ( t ) 2 d t 2 + 1 .
Φ ( t ) = 1 π σ t exp [ t 2 σ t 2 ] .
σ I 2 = 1 π 1 η σ t + 1 .
Γ r 1 r 2 τ = γ ( τ ) I ( r 1 ) I ( r 2 ) μ ( r 2 r 1 ) ,
I ( r ) = exp [ r 2 2 σ I 2 ] ,
μ ( r 2 r 1 ) = exp [ ( r 2 r 1 ) 2 2 σ g 2 ] ,
V r t = z = 0 V 0 r t R c R d 2 r ,
V r t = exp [ i ω 0 t ] F 0 r t ,
V r t = exp [ i ω 0 t ] z = 0 exp [ i ω 0 R c ] F 0 ( r , t R c ) R d 2 r .
R = r r ' z + 1 2 ( x x ) 2 + ( y y ) 2 z .
V r t = exp [ i ω 0 t ] exp [ i ω 0 z c ] z z = 0 exp [ i ω 0 2 c ( x x ) 2 + ( y y ) 2 z ]
× F 0 [ r , t 1 c ( z + 1 2 ( x x ) 2 + ( y y ) 2 z ) ] d 2 r
V r t = exp [ 0 t ̂ ] z z = 0 exp [ i ω 0 2 c ( x x ) 2 + ( y y ) 2 z ] F 0 r t ̂ d 2 r ,
S ( K ) = ϕ 0 2 exp [ k 2 L 0 2 4 ] ,

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