Abstract

We present a Monte Carlo-derived Green’s function for the propagation of partially spatially coherent fields. This Green’s function, which is derived by sampling Huygens–Fresnel wavelets, can be used to propagate fields through an optical system and to compute first- and second-order field statistics directly. The concept is illustrated for a cylindrical f/1 imaging system. A Gaussian copula is used to synthesize realizations of a Gaussian Schell-model field in the pupil plane. Physical optics and Monte Carlo predictions are made for the first- and second-order statistics of the field in the vicinity of the focal plane for a variety of source coherence conditions. Excellent agreement between the physical optics and Monte Carlo predictions is demonstrated in all cases. This formalism can be generally employed to treat the interaction of partially coherent fields with diffracting structures.

© 2009 Optical Society of America

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References

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  2. G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19, 1592-1598 (2002).
    [CrossRef]
  3. G. Yao and L. V. Wang, “Monte Carlo simulation of an optical coherence tomography signal in homogeneous turbid media,” Phys. Med. Biol. 44, 2307-2320 (1999).
    [CrossRef] [PubMed]
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    [CrossRef]
  6. F.Mayinger and O.Feldmann, eds., Optical Measurements: Techniques and Applications (Springer-Verlag, 2001).
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  7. J.Pawley, ed., Handbook of Biological Confocal Microscopy (Springer, 1995).
  8. M. Keijzer, S. L. Jacques, S. A. Prahl, and A. J. Welch, “Light distributions in artery tissue: Monte Carlo simulations for finite-diameter laser beams,” Lasers Surg. Med. 9, 148-154 (1989).
    [CrossRef] [PubMed]
  9. L. Wang, S. L. Jacques, and L. Zheng, “MCML-Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131-146 (1995).
    [CrossRef] [PubMed]
  10. S. Bartel and A. H. Hielscher, “Monte Carlo simulations of the diffuse backscattering Mueller matrix for highly scattering media,” Appl. Opt. 39, 1580-1588 (2000).
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    [CrossRef]
  15. F. Riechert, F. Dürr, U. Rohlfing, and U. Lemmer, “Ray-based simulation of the propagation of light with different degrees of coherence through complex optical systems,” Appl. Opt. 48, 1527-1534 (2009).
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  23. J. W. Goodman, Statistical Optics (Wiley, 1985).
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    [CrossRef]

2009 (1)

2008 (3)

2005 (1)

A. M. Zysk, P. S. Carney, and J. C. Schotland, “Eikonal method for calculation of coherence functions,” Phys. Rev. Lett. 95, 043904 (2005).
[CrossRef] [PubMed]

2004 (3)

2002 (1)

2000 (1)

1999 (1)

G. Yao and L. V. Wang, “Monte Carlo simulation of an optical coherence tomography signal in homogeneous turbid media,” Phys. Med. Biol. 44, 2307-2320 (1999).
[CrossRef] [PubMed]

1995 (2)

L. Wang, S. L. Jacques, and L. Zheng, “MCML-Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131-146 (1995).
[CrossRef] [PubMed]

N. G. Douglas, A. R. Jones, and F. J. van Hoesel, “Ray-based simulation of an optical interferometer,” J. Opt. Soc. Am. A 12, 124-131 (1995).
[CrossRef]

1989 (1)

M. Keijzer, S. L. Jacques, S. A. Prahl, and A. J. Welch, “Light distributions in artery tissue: Monte Carlo simulations for finite-diameter laser beams,” Lasers Surg. Med. 9, 148-154 (1989).
[CrossRef] [PubMed]

1949 (1)

N. Metropolis and S. Ulam, “The Monte Carlo method,” J. Am. Stat. Assoc. 44, 335-341 (1949).
[CrossRef] [PubMed]

Alonso, M. A.

Andrews, L. C.

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

Bartel, S.

Carney, P. S.

A. M. Zysk, P. S. Carney, and J. C. Schotland, “Eikonal method for calculation of coherence functions,” Phys. Rev. Lett. 95, 043904 (2005).
[CrossRef] [PubMed]

Dogariu, A.

Douglas, N. G.

Duncan, D. D.

Dürr, F.

Fischer, D. G.

Gan, X.

Gbur, G.

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, 1985).

Goodman, W.

W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J.C.Dainty, ed. (Springer-Verlag, 1975), pp. 9-75.
[CrossRef]

Gu, M.

Hielscher, A. H.

Jacques, S. L.

L. Wang, S. L. Jacques, and L. Zheng, “MCML-Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131-146 (1995).
[CrossRef] [PubMed]

M. Keijzer, S. L. Jacques, S. A. Prahl, and A. J. Welch, “Light distributions in artery tissue: Monte Carlo simulations for finite-diameter laser beams,” Lasers Surg. Med. 9, 148-154 (1989).
[CrossRef] [PubMed]

Jones, A. R.

Keijzer, M.

M. Keijzer, S. L. Jacques, S. A. Prahl, and A. J. Welch, “Light distributions in artery tissue: Monte Carlo simulations for finite-diameter laser beams,” Lasers Surg. Med. 9, 148-154 (1989).
[CrossRef] [PubMed]

Kirkpatrick, S. J.

Lemmer, U.

Lu, Q.

Luo, Q.

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, (Cambridge U. Press, 1999).

Metropolis, N.

N. Metropolis and S. Ulam, “The Monte Carlo method,” J. Am. Stat. Assoc. 44, 335-341 (1949).
[CrossRef] [PubMed]

Mujat, C.

Nelson, R. B.

R. B. Nelson, An Introduction to Copulas (Springer-Verlag, 1999).

Papoulis, A.

A. Papoulis and S. U. Pillai, Probability, Random Variables, and Stochastic Processes, 4th ed. (McGraw-Hill, 2002).

Petruccelli, J. C.

Phillips, R. L.

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

Pillai, S. U.

A. Papoulis and S. U. Pillai, Probability, Random Variables, and Stochastic Processes, 4th ed. (McGraw-Hill, 2002).

Prahl, S. A.

D. G. Fischer, S. A. Prahl, and D. D. Duncan, “Monte Carlo modeling of spatial coherence: free-space diffraction,” J. Opt. Soc. Am. A 25, 2571-2581 (2008).
[CrossRef]

M. Keijzer, S. L. Jacques, S. A. Prahl, and A. J. Welch, “Light distributions in artery tissue: Monte Carlo simulations for finite-diameter laser beams,” Lasers Surg. Med. 9, 148-154 (1989).
[CrossRef] [PubMed]

Riechert, F.

Rohlfing, U.

Schotland, J. C.

A. M. Zysk, P. S. Carney, and J. C. Schotland, “Eikonal method for calculation of coherence functions,” Phys. Rev. Lett. 95, 043904 (2005).
[CrossRef] [PubMed]

Stamnes, J. J.

J. J. Stamnes, Waves in Focal Regions: Propagation, Diffraction, and Focusing of Light, Sound, and Water Waves (Adam Hilger, 1986).

Ulam, S.

N. Metropolis and S. Ulam, “The Monte Carlo method,” J. Am. Stat. Assoc. 44, 335-341 (1949).
[CrossRef] [PubMed]

van Hoesel, F. J.

Visser, T. D.

Wang, L.

L. Wang, S. L. Jacques, and L. Zheng, “MCML-Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131-146 (1995).
[CrossRef] [PubMed]

Wang, L. V.

G. Yao and L. V. Wang, “Monte Carlo simulation of an optical coherence tomography signal in homogeneous turbid media,” Phys. Med. Biol. 44, 2307-2320 (1999).
[CrossRef] [PubMed]

Welch, A. J.

M. Keijzer, S. L. Jacques, S. A. Prahl, and A. J. Welch, “Light distributions in artery tissue: Monte Carlo simulations for finite-diameter laser beams,” Lasers Surg. Med. 9, 148-154 (1989).
[CrossRef] [PubMed]

Wolf, E.

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

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, (Cambridge U. Press, 1999).

Yao, G.

G. Yao and L. V. Wang, “Monte Carlo simulation of an optical coherence tomography signal in homogeneous turbid media,” Phys. Med. Biol. 44, 2307-2320 (1999).
[CrossRef] [PubMed]

Zheng, L.

L. Wang, S. L. Jacques, and L. Zheng, “MCML-Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131-146 (1995).
[CrossRef] [PubMed]

Zysk, A. M.

A. M. Zysk, P. S. Carney, and J. C. Schotland, “Eikonal method for calculation of coherence functions,” Phys. Rev. Lett. 95, 043904 (2005).
[CrossRef] [PubMed]

Appl. Opt. (3)

Comput. Methods Programs Biomed. (1)

L. Wang, S. L. Jacques, and L. Zheng, “MCML-Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131-146 (1995).
[CrossRef] [PubMed]

J. Am. Stat. Assoc. (1)

N. Metropolis and S. Ulam, “The Monte Carlo method,” J. Am. Stat. Assoc. 44, 335-341 (1949).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (7)

Lasers Surg. Med. (1)

M. Keijzer, S. L. Jacques, S. A. Prahl, and A. J. Welch, “Light distributions in artery tissue: Monte Carlo simulations for finite-diameter laser beams,” Lasers Surg. Med. 9, 148-154 (1989).
[CrossRef] [PubMed]

Phys. Med. Biol. (1)

G. Yao and L. V. Wang, “Monte Carlo simulation of an optical coherence tomography signal in homogeneous turbid media,” Phys. Med. Biol. 44, 2307-2320 (1999).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

A. M. Zysk, P. S. Carney, and J. C. Schotland, “Eikonal method for calculation of coherence functions,” Phys. Rev. Lett. 95, 043904 (2005).
[CrossRef] [PubMed]

Other (9)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, (Cambridge U. Press, 1999).

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

F.Mayinger and O.Feldmann, eds., Optical Measurements: Techniques and Applications (Springer-Verlag, 2001).
[CrossRef]

J.Pawley, ed., Handbook of Biological Confocal Microscopy (Springer, 1995).

A. Papoulis and S. U. Pillai, Probability, Random Variables, and Stochastic Processes, 4th ed. (McGraw-Hill, 2002).

W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J.C.Dainty, ed. (Springer-Verlag, 1975), pp. 9-75.
[CrossRef]

J. W. Goodman, Statistical Optics (Wiley, 1985).

J. J. Stamnes, Waves in Focal Regions: Propagation, Diffraction, and Focusing of Light, Sound, and Water Waves (Adam Hilger, 1986).

R. B. Nelson, An Introduction to Copulas (Springer-Verlag, 1999).

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Figures (12)

Fig. 1
Fig. 1

Illustration of geometrical configuration.

Fig. 2
Fig. 2

Normalized modulus of Green’s function from numerical evaluation of Eq. (3) (left) and of MC ray trace (right). For the MC Green’s function, a total of 10 8 rays were traced. For purposes of this comparison, a measurement function (see Subsection 3D) was applied to the PO Green’s function.

Fig. 3
Fig. 3

Phase in radians of Green’s function from numerical evaluation of Eq. (9) (left) and of MC ray trace (right). These displays are relative to the zero phase value at (0, 0). For purposes of this comparison, a measurement function (see Subsection 3D) was applied to the PO Green’s function.

Fig. 4
Fig. 4

Normalized moduli of cross-spectral density matrices for source σ g a = 0.6 (left) and focal plane (right).

Fig. 5
Fig. 5

Axial cross-spectral density matrix in the vicinity of the focal plane for the case σ g a = 0.6 .

Fig. 6
Fig. 6

Real part of spectral degree of coherence μ ( 0 , 0 , 0 ; x , 0 , 0 ) and normalized spectral density S ( x , 0 , 0 ) S ( 0 , 0 , 0 ) for various values of σ g a : (a) σ g a = 2.0 , (b) σ g a = 1.0 , (c) σ g a = 0.6 , (d) σ g a = 0.4 , (e) σ g a = 0.2 , (f) σ g a = 0.1 . MC and PO results are for propagation of 65,536 field realizations.

Fig. 7
Fig. 7

Spectral degree of coherence μ ( 0 , 0 , 0 ; 0 , 0 , z ) and normalized spectral density S ( 0 , 0 , z ) S ( 0 , 0 , 0 ) for various values of σ g a : (a) σ g a = 2.0 , (b) σ g a = 1.0 , (c) σ g a = 0.6 , (d) σ g a = 0.4 , (e) σ g a = 0.2 , (f) σ g a = 0.1 . MC and PO results are for propagation of 65,536 field realizations.

Fig. 8
Fig. 8

Comparison of the native and corrected PO results for the transverse (a) and axial (b) spectral degrees of coherence for the case σ g a = 0.4 .

Fig. 9
Fig. 9

Distribution of standard deviations in Green’s matrix for various numbers of traced rays.

Fig. 10
Fig. 10

Mean relative standard deviation as a function of number of rays propagated. Also shown is the 1 N r asymptote.

Fig. 11
Fig. 11

PO–MC residuals versus model that is a function of number of rays traced and number of field realizations propagated. Solid line is the identity.

Fig. 12
Fig. 12

Illustration of phase wrapping from adjacent Riemann sheets into the fundamental interval.

Equations (24)

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U ( r , ω ) = i π λ a a U ( 0 ) ( r , ω ) exp ( i k f 2 + x 2 ) z s H 1 ( 1 ) ( k s ) d x ,
U ( r , ω ) = U ( 0 ) ( r , ω ) G ( r , r ) d x ,
G ( r , r ) = { i π λ exp ( i k f 2 + x 2 ) z s H 1 ( 1 ) ( k s ) , x < a 0 , else } .
W ( r 1 , r 2 , ω ) = U * ( r 1 , ω ) U ( r 2 , ω ) .
S ( r ) = W ( r , r ) ,
μ ( r 1 , r 2 ) = W ( r 1 , r 2 ) [ S ( r 1 ) S ( r 2 ) ] 1 2 ,
W ( r 1 , r 2 , ω ) = W ( 0 ) ( r 1 , r 2 , ω ) G * ( r 1 , r 1 ) G ( r 2 , r 2 ) d x 1 d x 2 ,
W ( 0 ) ( r , r ) = W ( 0 ) ( x , x ) = exp [ ( x x ) 2 2 σ g 2 ] ,
U = G U ( 0 ) .
θ = θ min + ξ ( θ max θ min ) .
Y 1 = 2 ln X 1 cos ( 2 π X 2 ) ,
Y 2 = 2 ln X 1 sin ( 2 π X 2 ) ,
[ Z 1 Z 2 ] = 1 2 [ 1 1 1 1 ] [ 1 + r 0 0 1 r ] [ Y 1 Y 2 ] ,
μ = exp ( 1 2 σ Δ ϕ 2 ) ,
W = ( 1 N f ) U U = ( 1 N f ) ( G U ( 0 ) ) ( G U ( 0 ) ) = G [ ( 1 N f ) U ( 0 ) U ( 0 ) ] G = G W ( 0 ) G ,
F ( r ) = 1 2 i k [ U ( r ) U * ( r ) U * ( r ) U ( r ) ] ,
F c ( r 1 , r 2 ) = 1 2 i k U * ( r 1 ) 2 U ( r 2 ) U ( r 2 ) 1 U * ( r 1 ) = 1 2 i k ( 2 1 ) W ( r 1 , r 2 ) .
M PO [ W PO ( r 1 , r 2 ) ] = [ F c ( r 1 , r 2 ) ] PO n ̂ D ,
s = f 2 + ( x x ) 2 ,
s = ( f z ) 2 + x 2 ,
σ i j 2 = 1 8 k = 1 9 [ G i j ( k ) G ¯ i j ] 2 ,
σ PO MC 2 = 1 N 1 i = 1 N [ S PO ( x i , 0 , 0 ) S PO ( 0 , 0 , 0 ) S MC ( x i , 0 , 0 ) S MC ( 0 , 0 , 0 ) ] 2 ,
σ ̂ PO MC 2 = ( 30.1 N r ) 2 + ( 0.895 N f ) 2 ,
U = 1 N k = 1 N exp ( i α k ) .

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