Abstract

Temporally-stationary electromagnetic fields with arbitrary second-order coherence functions are simulated using standard statistical tools. In cases where the coherence function takes a commonly-used separable form, a computationally-efficient variation of the approach can be applied. This work provides a generalization of previous spatio-temporal simulators which model only scalar fields and require either restrictions on the coherence function or consider only two points in space. The simulation of a partially-polarized Gaussian Schell-model beam and a partially-radially-polarized beam are demonstrated.

© 2007 Optical Society of America

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References

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  1. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).
  2. J. W. Goodman, Statistical Optics (Wiley-Interscience, 1985).
  3. G. Gbur, "Simulating fields of arbitrary spatial and temporal coherence," Opt. Express 14, 7567-7578 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-17-7567.
    [CrossRef] [PubMed]
  4. E. Wolf, "Correlation-induced changes in the degree of polarization, the degree of coherence, and the spectrum of random electromagnetic beams on propagation," Opt. Lett. 28, 1078-1080 (2003).
    [CrossRef] [PubMed]
  5. P. Vahimaa and J. Turunen, "Finite-elementary-source model for partially coherent radiation," Opt. Express 14, 1376-1381 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-4-1376.
    [CrossRef] [PubMed]
  6. 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]
  7. E. Wolf, "Invariance of the spectrum of light on propagation," Phys. Rev. Lett. 56, 1370-1372 (1986).
    [CrossRef] [PubMed]
  8. G. Magyar and L. Mandel, "Interference fringes produced by superposition of two independent Maser light beams," Nature 198, 255-256 (1963).
    [CrossRef]
  9. B. Davis, E. Kim, and J. R. Piepmeier, "Stochastic modeling and generation of partially polarized or partially coherent electromagnetic waves," Radio Sci. 39, RS1001 (2004).
  10. R. Simon and N. Makunda, "Twisted Gaussian Schell-model beams," J. Opt. Soc. Am. A 10, 95-109 (1993).
    [CrossRef]
  11. E. Wolf and G. S. Agarwal, "Coherence theory of laser resonator modes," J. Opt. Soc. Am. A 1, 541-546 (1984).
    [CrossRef]
  12. Y. Cai and S. He, "Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere," Appl. Phys. Lett. 89, 041117 (2006).
    [CrossRef]
  13. E. Tervonen, J. Turunen, and A. T. Friberg, "Transverse Laser-mode structure determination from spatial coherence measurements: experimental results," Appl, Phys. B 49, 409-414 (1989).
    [CrossRef]
  14. L. D. A. Lundeberg, G. P. Lousberg, D. L. Boiko, and E. Kapon, "Spatial coherence measurements in arrays of coupled vertical cavity surface emitting lasers," Appl. Phys. Lett. 90, 121103 (2007).
    [CrossRef]
  15. D. F. V. James, "Change of polarization of light beams on propagation in free space," J. Opt. Soc. Am. A 11, 1641-1643 (1994).
    [CrossRef]
  16. O. Korotkova, M. Salem, and E. Wolf, "The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence," Opt. Commun. 233, 225-230 (2004).
    [CrossRef]
  17. G. H. Golub and C. F. Van Loan, Matrix Computations (Johns Hopkins University Press, 1996).
  18. J. H. Michels, P. K. Varshney, and D. D. Weiner, "Synthesis of correlated multichannel random processes," IEEE Trans. Signal Process. 42, 367-375 (1994).
    [CrossRef]
  19. A. Papoulis and S. U. Pillai, Probability, Random Variables and Stochastic Processes (McGraw-Hill, 2002).
  20. W. H. Carter and E. Wolf, "Coherence and radiometry with quasihomogeneous planar sources," J. Opt. Soc. Am. 67, 785-796 (1977).
    [CrossRef]
  21. R. Loudon, The Quantum Theory of Light (Oxford University Press, 2000).
  22. F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, "Partially polarized Gaussian Schellmodel beams," J. Opt. A 3, 1-9 (2001).
    [CrossRef]
  23. N. Hodgson and H. Weber, Laser Resonators and Beam Propagation: Fundamentals, Advanced Concepts and Applications (Springer, 2005).
  24. K. C. Toussaint, Jr., S. Park, J. E. Jureller, and N. F. Scherer, "Generation of optical vector beams with a diffractive optical element interferometer," Opt. Lett. 30, 2846-2848 (2005).
    [CrossRef]
  25. S. Quabis, R. Dorn, and G. Leuchs, "Generation of a radially polarized doughnut mode of high quality," Appl. Phys. B 81, 597-600 (2005).
    [CrossRef]
  26. P. M. Lurie and M. S. Goldberg, "An approximate method for sampling correlated random variables from partially-specified distributions," Manage. Sci. 44, 203-218 (1998).
    [CrossRef]

2007

L. D. A. Lundeberg, G. P. Lousberg, D. L. Boiko, and E. Kapon, "Spatial coherence measurements in arrays of coupled vertical cavity surface emitting lasers," Appl. Phys. Lett. 90, 121103 (2007).
[CrossRef]

2006

2005

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]

K. C. Toussaint, Jr., S. Park, J. E. Jureller, and N. F. Scherer, "Generation of optical vector beams with a diffractive optical element interferometer," Opt. Lett. 30, 2846-2848 (2005).
[CrossRef]

S. Quabis, R. Dorn, and G. Leuchs, "Generation of a radially polarized doughnut mode of high quality," Appl. Phys. B 81, 597-600 (2005).
[CrossRef]

2004

O. Korotkova, M. Salem, and E. Wolf, "The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence," Opt. Commun. 233, 225-230 (2004).
[CrossRef]

B. Davis, E. Kim, and J. R. Piepmeier, "Stochastic modeling and generation of partially polarized or partially coherent electromagnetic waves," Radio Sci. 39, RS1001 (2004).

2003

2001

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, "Partially polarized Gaussian Schellmodel beams," J. Opt. A 3, 1-9 (2001).
[CrossRef]

1998

P. M. Lurie and M. S. Goldberg, "An approximate method for sampling correlated random variables from partially-specified distributions," Manage. Sci. 44, 203-218 (1998).
[CrossRef]

1994

J. H. Michels, P. K. Varshney, and D. D. Weiner, "Synthesis of correlated multichannel random processes," IEEE Trans. Signal Process. 42, 367-375 (1994).
[CrossRef]

D. F. V. James, "Change of polarization of light beams on propagation in free space," J. Opt. Soc. Am. A 11, 1641-1643 (1994).
[CrossRef]

1993

1989

E. Tervonen, J. Turunen, and A. T. Friberg, "Transverse Laser-mode structure determination from spatial coherence measurements: experimental results," Appl, Phys. B 49, 409-414 (1989).
[CrossRef]

1986

E. Wolf, "Invariance of the spectrum of light on propagation," Phys. Rev. Lett. 56, 1370-1372 (1986).
[CrossRef] [PubMed]

1984

1977

1963

G. Magyar and L. Mandel, "Interference fringes produced by superposition of two independent Maser light beams," Nature 198, 255-256 (1963).
[CrossRef]

Agarwal, G. S.

Boiko, D. L.

L. D. A. Lundeberg, G. P. Lousberg, D. L. Boiko, and E. Kapon, "Spatial coherence measurements in arrays of coupled vertical cavity surface emitting lasers," Appl. Phys. Lett. 90, 121103 (2007).
[CrossRef]

Borghi, R.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, "Partially polarized Gaussian Schellmodel beams," J. Opt. A 3, 1-9 (2001).
[CrossRef]

Cai, Y.

Y. Cai and S. He, "Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere," Appl. Phys. Lett. 89, 041117 (2006).
[CrossRef]

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]

Carter, W. H.

Davis, B.

B. Davis, E. Kim, and J. R. Piepmeier, "Stochastic modeling and generation of partially polarized or partially coherent electromagnetic waves," Radio Sci. 39, RS1001 (2004).

Dorn, R.

S. Quabis, R. Dorn, and G. Leuchs, "Generation of a radially polarized doughnut mode of high quality," Appl. Phys. B 81, 597-600 (2005).
[CrossRef]

Friberg, A. T.

E. Tervonen, J. Turunen, and A. T. Friberg, "Transverse Laser-mode structure determination from spatial coherence measurements: experimental results," Appl, Phys. B 49, 409-414 (1989).
[CrossRef]

Gbur, G.

Goldberg, M. S.

P. M. Lurie and M. S. Goldberg, "An approximate method for sampling correlated random variables from partially-specified distributions," Manage. Sci. 44, 203-218 (1998).
[CrossRef]

Gori, F.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, "Partially polarized Gaussian Schellmodel beams," J. Opt. A 3, 1-9 (2001).
[CrossRef]

He, S.

Y. Cai and S. He, "Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere," Appl. Phys. Lett. 89, 041117 (2006).
[CrossRef]

James, D. F. V.

Jureller, J. E.

Kapon, E.

L. D. A. Lundeberg, G. P. Lousberg, D. L. Boiko, and E. Kapon, "Spatial coherence measurements in arrays of coupled vertical cavity surface emitting lasers," Appl. Phys. Lett. 90, 121103 (2007).
[CrossRef]

Kim, E.

B. Davis, E. Kim, and J. R. Piepmeier, "Stochastic modeling and generation of partially polarized or partially coherent electromagnetic waves," Radio Sci. 39, RS1001 (2004).

Korotkova, O.

O. Korotkova, M. Salem, and E. Wolf, "The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence," Opt. Commun. 233, 225-230 (2004).
[CrossRef]

Leuchs, G.

S. Quabis, R. Dorn, and G. Leuchs, "Generation of a radially polarized doughnut mode of high quality," Appl. Phys. B 81, 597-600 (2005).
[CrossRef]

Lousberg, G. P.

L. D. A. Lundeberg, G. P. Lousberg, D. L. Boiko, and E. Kapon, "Spatial coherence measurements in arrays of coupled vertical cavity surface emitting lasers," Appl. Phys. Lett. 90, 121103 (2007).
[CrossRef]

Lundeberg, L. D. A.

L. D. A. Lundeberg, G. P. Lousberg, D. L. Boiko, and E. Kapon, "Spatial coherence measurements in arrays of coupled vertical cavity surface emitting lasers," Appl. Phys. Lett. 90, 121103 (2007).
[CrossRef]

Lurie, P. M.

P. M. Lurie and M. S. Goldberg, "An approximate method for sampling correlated random variables from partially-specified distributions," Manage. Sci. 44, 203-218 (1998).
[CrossRef]

Magyar, G.

G. Magyar and L. Mandel, "Interference fringes produced by superposition of two independent Maser light beams," Nature 198, 255-256 (1963).
[CrossRef]

Makunda, N.

Mandel, L.

G. Magyar and L. Mandel, "Interference fringes produced by superposition of two independent Maser light beams," Nature 198, 255-256 (1963).
[CrossRef]

Michels, J. H.

J. H. Michels, P. K. Varshney, and D. D. Weiner, "Synthesis of correlated multichannel random processes," IEEE Trans. Signal Process. 42, 367-375 (1994).
[CrossRef]

Mondello, A.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, "Partially polarized Gaussian Schellmodel beams," J. Opt. A 3, 1-9 (2001).
[CrossRef]

Park, S.

Piepmeier, J. R.

B. Davis, E. Kim, and J. R. Piepmeier, "Stochastic modeling and generation of partially polarized or partially coherent electromagnetic waves," Radio Sci. 39, RS1001 (2004).

Piquero, G.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, "Partially polarized Gaussian Schellmodel beams," J. Opt. A 3, 1-9 (2001).
[CrossRef]

Quabis, S.

S. Quabis, R. Dorn, and G. Leuchs, "Generation of a radially polarized doughnut mode of high quality," Appl. Phys. B 81, 597-600 (2005).
[CrossRef]

Salem, M.

O. Korotkova, M. Salem, and E. Wolf, "The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence," Opt. Commun. 233, 225-230 (2004).
[CrossRef]

Santarsiero, M.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, "Partially polarized Gaussian Schellmodel beams," J. Opt. A 3, 1-9 (2001).
[CrossRef]

Scherer, N. F.

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]

Simon, R.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, "Partially polarized Gaussian Schellmodel beams," J. Opt. A 3, 1-9 (2001).
[CrossRef]

R. Simon and N. Makunda, "Twisted Gaussian Schell-model beams," J. Opt. Soc. Am. A 10, 95-109 (1993).
[CrossRef]

Tervonen, E.

E. Tervonen, J. Turunen, and A. T. Friberg, "Transverse Laser-mode structure determination from spatial coherence measurements: experimental results," Appl, Phys. B 49, 409-414 (1989).
[CrossRef]

Toussaint, K. C.

Turunen, J.

P. Vahimaa and J. Turunen, "Finite-elementary-source model for partially coherent radiation," Opt. Express 14, 1376-1381 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-4-1376.
[CrossRef] [PubMed]

E. Tervonen, J. Turunen, and A. T. Friberg, "Transverse Laser-mode structure determination from spatial coherence measurements: experimental results," Appl, Phys. B 49, 409-414 (1989).
[CrossRef]

Vahimaa, P.

Varshney, P. K.

J. H. Michels, P. K. Varshney, and D. D. Weiner, "Synthesis of correlated multichannel random processes," IEEE Trans. Signal Process. 42, 367-375 (1994).
[CrossRef]

Weiner, D. D.

J. H. Michels, P. K. Varshney, and D. D. Weiner, "Synthesis of correlated multichannel random processes," IEEE Trans. Signal Process. 42, 367-375 (1994).
[CrossRef]

Wolf, E.

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, Phys. B

E. Tervonen, J. Turunen, and A. T. Friberg, "Transverse Laser-mode structure determination from spatial coherence measurements: experimental results," Appl, Phys. B 49, 409-414 (1989).
[CrossRef]

Appl. Phys. B

S. Quabis, R. Dorn, and G. Leuchs, "Generation of a radially polarized doughnut mode of high quality," Appl. Phys. B 81, 597-600 (2005).
[CrossRef]

Appl. Phys. Lett.

Y. Cai and S. He, "Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere," Appl. Phys. Lett. 89, 041117 (2006).
[CrossRef]

L. D. A. Lundeberg, G. P. Lousberg, D. L. Boiko, and E. Kapon, "Spatial coherence measurements in arrays of coupled vertical cavity surface emitting lasers," Appl. Phys. Lett. 90, 121103 (2007).
[CrossRef]

IEEE Trans. Signal Process.

J. H. Michels, P. K. Varshney, and D. D. Weiner, "Synthesis of correlated multichannel random processes," IEEE Trans. Signal Process. 42, 367-375 (1994).
[CrossRef]

J. Opt. A

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, "Partially polarized Gaussian Schellmodel beams," J. Opt. A 3, 1-9 (2001).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Manage. Sci.

P. M. Lurie and M. S. Goldberg, "An approximate method for sampling correlated random variables from partially-specified distributions," Manage. Sci. 44, 203-218 (1998).
[CrossRef]

Nature

G. Magyar and L. Mandel, "Interference fringes produced by superposition of two independent Maser light beams," Nature 198, 255-256 (1963).
[CrossRef]

Opt. Commun.

O. Korotkova, M. Salem, and E. Wolf, "The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence," Opt. Commun. 233, 225-230 (2004).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. Lett.

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]

E. Wolf, "Invariance of the spectrum of light on propagation," Phys. Rev. Lett. 56, 1370-1372 (1986).
[CrossRef] [PubMed]

RadioSci.

B. Davis, E. Kim, and J. R. Piepmeier, "Stochastic modeling and generation of partially polarized or partially coherent electromagnetic waves," Radio Sci. 39, RS1001 (2004).

Other

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

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

G. H. Golub and C. F. Van Loan, Matrix Computations (Johns Hopkins University Press, 1996).

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

R. Loudon, The Quantum Theory of Light (Oxford University Press, 2000).

N. Hodgson and H. Weber, Laser Resonators and Beam Propagation: Fundamentals, Advanced Concepts and Applications (Springer, 2005).

Supplementary Material (2)

» Media 1: AVI (1469 KB)     
» Media 2: AVI (1469 KB)     

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Figures (2)

Fig. 1.
Fig. 1.

Realized and specified coherence properties for the Gaussian-Schell example. The specified intensity I(x, y) and field direction p(x,y) (in blue) are shown in (a) and the realized values in (b). The white bar indicates one spatial unit. An overlay of x-axis cross sections from (a) and (b) is shown in (c), where the dashed line gives the specified profile. Specified and realized images of ∣Γ xx (x,y,0,0,0)∣ are shown in (d) and (e) respectively, with an x-axis cross section in (f). A movie (1.4MB) showing 100 consecutive frames of the instantaneous intensity is given in (g). [Media 1] The realized degree of polarization P(x,y) is shown in (h) while the specified value was P = b = 0.5. A temporal correlation ∣Γ xx (0,0,0,0,t)∣ is shown in (i).

Fig. 2.
Fig. 2.

Realized and specified coherence properties for the radially-polarized example. The specified intensity I(x, y) and field direction p(x,y) (in blue) are shown in (a) and the realized values in (b). The white bar indicates one spatial unit. An overlay of x-axis cross sections from (a) and (b) is shown in (c), where the dashed line gives the specified profile. Specified and realized images of ∣Γ xx (x,y,3,0,0)∣ are shown in (d) and (e) respectively, with an x-axis cross section in (f). A movie (1.4MB) showing 100 consecutive frames of the instantaneous intensity is given in (g). [Media 2] The realized degree of polarization P(x,y) is shown in (h) while the specified value was P = 0.5. A temporal correlation ∣Γ xx (0,0,3,0,t)∣ is shown in (i).

Equations (16)

Equations on this page are rendered with MathJax. Learn more.

Γ jj′ x y x′ y′ τ = < 𝓔 j x y t 𝓔 j′ * x′ y′ t τ ) ,
< 𝒳 [ n , m ] 𝒳 * [ n , m ] > = δ [ n n ] [ m m′ ] ,
W [ n , n′ , ω ) = u Γ [ n , n′ , u ] e iωu .
W [ n , n′ , ω ) = r V [ n , r , ω ) V * [ n′ , r , ω ) .
h V [ n , r , m ] = 1 2 π π π V [ n , r , ω ) e iωm .
𝓔 [ n , m ] = r q h v [ n , r , m q ] 𝒳 [ r , q ] .
< 𝓔 [ n , m ] 𝓔 * [ n′ , m′ ] > = 1 2 π π π W [ n , n′ , ω ) e ( m m′ ) ,
= Γ [ n , n′ , m m′ ] .
Γ jj′ [ k , l , k′ , l′ m m′ ] = A j [ k , l ] A j′ * [ k′ , l′ ] B jj′ η [ k k′ , l l′ ] R [ m m′ ] .
< 𝒳 j [ k , l , m ] 𝒳 j′ [ k′ , l′ , m′ ] > = δ jj′ δ [ k k′ ] δ [ l l′ ] δ [ m m′ ] ,
H ̅ H ̅ = B ̅ = [ B xx B xy B yx B yy ] .
H ̅ = [ 1 0 b 1 b 2 ] .
R [ m ] = q h R [ q ] h R * [ q m ] ,
η [ k , l ] = o , p h η [ o , p ] h η * [ o k , p l ] .
𝓔 j [ k , l , m ] = A j [ k , l ] q h R [ m q ] o p h η [ k o , l p ] s = x , y H js 𝒳 s [ o , p , q ] ,
Γ jj′ x y x y , 0 = I ( x , y ) { [ 1 P ( x , y ) ] δ jj′ 2 + P ( x , y ) pj x y p j′ * x y } ,

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