Abstract

Analytical formulas for the cross-spectral density matrix of stochastic electromagnetic Gaussian Schell-model (EGSM) beams passing through an astigmatic optical system are derived. We show both analytically and by numerical examples the effects of astigmatism on spectra, coherence and polarization of stochastic electromagnetic EGSM beams propagating through an astigmatic lens. A comparison with the aberration-free case is made, and shows that the astigmatism has significant effect on the spectra, coherence and polarization.

© 2009 Optical Society of America

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  1. 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]
  2. T. Shirai and E. Wolf, "Coherence and polarization of electromagnetic beams modulated by random phase screens and their changes on propagation in free space," J. Opt. Soc. Am. A 21, 1907-1916 (2004).
    [CrossRef]
  3. O. Korotkova and E. Wolf, "Changes in the state of polarization of a random electromagnetic beam on propagation," Opt. Commun. 246, 35-43 (2005).
    [CrossRef]
  4. H. Wang, X, Wang, A. Zeng, and K. Yang, "Effects of coherence on anisotropic electromagnetic Gaussian-Schell model beams on propagation," Opt. Lett. 32, 2215-2217 (2007).
    [CrossRef] [PubMed]
  5. J. Pu, O. Korotkova, and E. Wolf, "Polarization-induced spectral changes on propagation of stochastic electromagnetic beams," Phys. Rev. E 75, 056610-1 (2007).
    [CrossRef]
  6. 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]
  7. H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, "Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere," J. Mod. Opt. 52, 1611-1618 (2005).
    [CrossRef]
  8. X. Du, D. Zhao, and O. Korotkova, "Changes in the statistical properties of stochastic anisotropic electromagnetic beams on propagation in the turbulent atmosphere," Opt. Express 15, 16909-16915 (2007),http://www.opticsexpress.org/abstract.cfm?URI=OPEX-15-25-16909.
    [CrossRef] [PubMed]
  9. Y. Cai, O. Korotkova, H. T. Eyyuboglu, and Y. Baykal, "Active laser radar systems with stochastic electromagnetic beams in turbulent atmosphere," Opt. Express 16, 15834-15846 (2008), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-16-20-5834.
    [CrossRef] [PubMed]
  10. H. Roychowdhury, G. P. Agrawal, and E. Wolf, "Changes in the spectrum, in the spectral degree of polarization, and in the spectral degree of coherence of a partially coherent beam propagating through a gradient-index fiber," J. Opt. Soc. Am. A 23, 940-948 (2006).
    [CrossRef]
  11. X. Du and D. Zhao, "Propagation of random electromagnetic beams through axially nonsymmetrical optical systems," Opt. Commun. 281, 2711-2715 (2008).
    [CrossRef]
  12. M. Yao, Y. Cai, H. T. Eyyuboglu, Y. Baykal, and O. Korotkova, "Evolution of the degree of polarization of an electromagnetic Gaussian Schell-model beam in a Gaussian cavity," Opt. Lett. 33, 2266-2268 (2008).
    [CrossRef] [PubMed]
  13. G. Zhang and J. Pu, "Stochastic electromagnetic beams focused by a bifocal lens," J. Opt. Soc. Am. A 25, 1710-1715 (2008).
    [CrossRef]
  14. Y. Zhu and D. Zhao, "Generalized Stokes parameters of a stochastic electromagnetic beam propagating through a paraxial ABCD optical system," J. Opt. Soc. Am. A 25, 1944-1948 (2008).
    [CrossRef]
  15. S. G. Hanson, W. Wang, M. L. Jakobsen, and M. Takeda, "Coherence and polarization of electromagnetic beams modulated by random phase screens and their changes through complex ABCD optical systems," J. Opt. Soc. Am. A 25, 2338-2346 (2008).
    [CrossRef]
  16. X. Du and D. Zhao, "Changes in the polarization and coherence of a random electromagnetic beam propagating through a misaligned optical system," J. Opt. Soc. Am. A 25, 773-779 (2008).
    [CrossRef]
  17. E. Wolf, "Unified theory of coherence and polarization of random electromagnetic beams," Phys. Lett. A 312, 263-267 (2003).
    [CrossRef]
  18. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, Cambridge, 1995)

2008

2007

2006

2005

O. Korotkova and E. Wolf, "Changes in the state of polarization of a random electromagnetic beam on propagation," Opt. Commun. 246, 35-43 (2005).
[CrossRef]

H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, "Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere," J. Mod. Opt. 52, 1611-1618 (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]

T. Shirai and E. Wolf, "Coherence and polarization of electromagnetic beams modulated by random phase screens and their changes on propagation in free space," J. Opt. Soc. Am. A 21, 1907-1916 (2004).
[CrossRef]

2003

Agrawal, G. P.

Baykal, Y.

Cai, Y.

Du, X.

Eyyuboglu, H. T.

Hanson, S. G.

Jakobsen, M. L.

Korotkova, O.

M. Yao, Y. Cai, H. T. Eyyuboglu, Y. Baykal, and O. Korotkova, "Evolution of the degree of polarization of an electromagnetic Gaussian Schell-model beam in a Gaussian cavity," Opt. Lett. 33, 2266-2268 (2008).
[CrossRef] [PubMed]

X. Du, D. Zhao, and O. Korotkova, "Changes in the statistical properties of stochastic anisotropic electromagnetic beams on propagation in the turbulent atmosphere," Opt. Express 15, 16909-16915 (2007),http://www.opticsexpress.org/abstract.cfm?URI=OPEX-15-25-16909.
[CrossRef] [PubMed]

J. Pu, O. Korotkova, and E. Wolf, "Polarization-induced spectral changes on propagation of stochastic electromagnetic beams," Phys. Rev. E 75, 056610-1 (2007).
[CrossRef]

O. Korotkova and E. Wolf, "Changes in the state of polarization of a random electromagnetic beam on propagation," Opt. Commun. 246, 35-43 (2005).
[CrossRef]

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]

Ponomarenko, S. A.

H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, "Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere," J. Mod. Opt. 52, 1611-1618 (2005).
[CrossRef]

Pu, J.

G. Zhang and J. Pu, "Stochastic electromagnetic beams focused by a bifocal lens," J. Opt. Soc. Am. A 25, 1710-1715 (2008).
[CrossRef]

J. Pu, O. Korotkova, and E. Wolf, "Polarization-induced spectral changes on propagation of stochastic electromagnetic beams," Phys. Rev. E 75, 056610-1 (2007).
[CrossRef]

Roychowdhury, H.

H. Roychowdhury, G. P. Agrawal, and E. Wolf, "Changes in the spectrum, in the spectral degree of polarization, and in the spectral degree of coherence of a partially coherent beam propagating through a gradient-index fiber," J. Opt. Soc. Am. A 23, 940-948 (2006).
[CrossRef]

H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, "Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere," J. Mod. Opt. 52, 1611-1618 (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]

Shirai, T.

Takeda, M.

Wang, H.

Wang, W.

Wolf, E.

J. Pu, O. Korotkova, and E. Wolf, "Polarization-induced spectral changes on propagation of stochastic electromagnetic beams," Phys. Rev. E 75, 056610-1 (2007).
[CrossRef]

H. Roychowdhury, G. P. Agrawal, and E. Wolf, "Changes in the spectrum, in the spectral degree of polarization, and in the spectral degree of coherence of a partially coherent beam propagating through a gradient-index fiber," J. Opt. Soc. Am. A 23, 940-948 (2006).
[CrossRef]

O. Korotkova and E. Wolf, "Changes in the state of polarization of a random electromagnetic beam on propagation," Opt. Commun. 246, 35-43 (2005).
[CrossRef]

H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, "Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere," J. Mod. Opt. 52, 1611-1618 (2005).
[CrossRef]

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]

T. Shirai and E. Wolf, "Coherence and polarization of electromagnetic beams modulated by random phase screens and their changes on propagation in free space," J. Opt. Soc. Am. A 21, 1907-1916 (2004).
[CrossRef]

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]

E. Wolf, "Unified theory of coherence and polarization of random electromagnetic beams," Phys. Lett. A 312, 263-267 (2003).
[CrossRef]

Yao, M.

Zhang, G.

Zhao, D.

Zhu, Y.

J. Mod. Opt.

H. Roychowdhury, S. A. Ponomarenko, and E. Wolf, "Change in the polarization of partially coherent electromagnetic beams propagating through the turbulent atmosphere," J. Mod. Opt. 52, 1611-1618 (2005).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

X. Du and D. Zhao, "Propagation of random electromagnetic beams through axially nonsymmetrical optical systems," Opt. Commun. 281, 2711-2715 (2008).
[CrossRef]

O. Korotkova and E. Wolf, "Changes in the state of polarization of a random electromagnetic beam on propagation," Opt. Commun. 246, 35-43 (2005).
[CrossRef]

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. Lett. A

E. Wolf, "Unified theory of coherence and polarization of random electromagnetic beams," Phys. Lett. A 312, 263-267 (2003).
[CrossRef]

Phys. Rev. E

J. Pu, O. Korotkova, and E. Wolf, "Polarization-induced spectral changes on propagation of stochastic electromagnetic beams," Phys. Rev. E 75, 056610-1 (2007).
[CrossRef]

Other

Y. Cai, O. Korotkova, H. T. Eyyuboglu, and Y. Baykal, "Active laser radar systems with stochastic electromagnetic beams in turbulent atmosphere," Opt. Express 16, 15834-15846 (2008), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-16-20-5834.
[CrossRef] [PubMed]

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

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Figures (10)

Fig. 1.
Fig. 1.

Distribution of the spectral degree of polarization in the x-z plane. (a) C 6=0, (b) C 6=0.1 × 10-3mm-1, (c) C 6=0.3×10-3mm-1, and (d)C 6=0.5×10-3mm-1. The parameters are f=400mm, ω=3×1015 rad/s, σ=1 mm, δxx =0.6mm, δyy =0.2mm.

Fig. 2.
Fig. 2.

Distribution of the spectral degree of polarization in the y-z plane. (a)C 6=0, (b)C 6=0.1×10-3mm-1, (c)C 6=0.3×10-3mm-1, and (d)C 6=0.5×10-3mm-1. The other parameters are the same as in the Fig.1.

Fig. 3.
Fig. 3.

Distribution of the spectral degree of polarization in the x-y plane for the aberration-free case of C 6=0. (a)z=300mm, (b)z=400mm, (c)z=500mm, and (d)z=600mm. The other parameters are the same as in the Fig.1.

Fig. 4.
Fig. 4.

Distribution of the spectral degree of polarization in the x-y plane for the aberration case of C 6=0.3×10-3mm-1. (a)z=300mm, (b)z=400mm, (c)z=500mm, and (d)z=600mm. The other parameters are the same as in the Fig.1.

Fig. 5.
Fig. 5.

As Fig.4, but δyy =0.4mm.

Fig. 6.
Fig. 6.

Spectral degree of polarization on the z axis. (a)δyy =0.2mm and (b)δyy =0.4mm. The other parameters are the same as in the Fig.1.

Fig.7.
Fig.7.

Spectral density on the z axis. The curves in (a) are associated with different values of the astigmatic coefficient C 6. The curves in (b) are associated with different values of the parameters δxx and δyy . The other parameters are the same as in the Fig.1.

Fig. 8.
Fig. 8.

Spectral degree of coherence along the x axis for different values of C 6. (a)z=300mm, (b)z=400mm, (c)z=500mm, and (d)z=600mm. The other parameters are the same as in the Fig.1.

Fig. 9.
Fig. 9.

Spectral degree of coherence along the y axis for different values of C 6. (a)z=300mm, (b)z=400mm, (c)z=500mm, and (d)z=600mm. The other parameters are the same as in the Fig.1.

Fig. 10.
Fig. 10.

Spectral degree of coherence along the x axis and along the y axis in the geometrical focal plane for different values of the parameters δxx and δyy . The other parameters are the same as in the Fig.1.

Equations (21)

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

W ( 0 ) x 1 y 1 x 2 y 2 ω = ( W xx ( 0 ) x 1 y 1 x 2 y 2 ω W xy ( 0 ) x 1 y 1 x 2 y 2 ω W yx ( 0 ) x 1 y 1 x 2 y 2 ω W yy ( 0 ) x 1 y 1 x 2 y 2 ω )
W ij ( 0 ) x 1 y 1 x 2 y 2 ω = A i A j B ij exp [ x 1 2 + x 2 2 4 σ 2 y 1 2 + y 2 2 4 σ 2 ] exp [ ( x 1 x 2 ) 2 2 δ ij 2 ( y 1 y 2 ) 2 2 δ ij 2 ]
( i = x , y , j = x , y )
S x y ω = Tr W x y x y ω
μ x 1 y 1 x 2 y 2 ω = Tr W x 1 y 1 x 2 y 2 ω Tr W x 1 y 1 x 1 y 1 ω Tr W x 2 y 2 x 2 y 2 ω
P x y ω = 1 4 Det W x y x y ω [ Tr W x y x y ω ] 2
W ij x 1 y 1 x 2 y 2 z ω = ( k 2 πB ) 2 W ij ( 0 ) x 1 y 1 x 2 y 2 ω exp { ik C 6 [ ( x 1 2 x 2 2 ) ( y 1 2 y 2 2 ) ] }
× exp { ik 2 B [ ( A ( x 1 2 + y 1 2 ) 2 ( x 1 x 1 + y 1 y 1 ) + D ( x 1 2 + y 1 2 ) )
( A ( x 2 2 + y 2 2 ) 2 ( x 2 x 2 + y 2 y 2 ) + D ( x 2 2 + y 2 2 ) ) ] } d x 1 d y 1 d x 2 d y 2
W ij x 1 y 1 x 2 y 2 z ω = A i A j B ij Q x Q y exp [ 1 4 σ 2 ( x 1 2 + x 2 2 Q x + y 1 2 + y 2 2 Q y ) ]
× exp [ ( x 1 x 2 ) 2 2 δ ij 2 Q x ( y 1 y 2 ) 2 2 δ ij 2 Q y ] exp [ ( x 1 2 x 2 2 ) 2 c R x ( y 1 2 y 2 2 ) 2 c R y ]
Q x = ( A 2 B C 6 ) 2 + c 2 B 2 4 ω 2 σ 4 ( 1 + 4 σ 2 δ ij 2 )
Q y = ( A + 2 B C 6 ) 2 + c 2 B 2 4 ω 2 σ 4 ( 1 + 4 σ 2 δ ij 2 )
R x = B Q x D Q x A + 2 B C 6
R y = B Q x D Q y A 2 B C 6
( A B C D ) = ( 1 z 0 1 ) ( 1 0 1 f 1 ) = ( 1 z f z 1 f 1 )
Q x = ( 1 z f 2 z C 6 ) 2 + c 2 z 2 4 ω 2 σ 4 ( 1 + 4 σ 2 δ ij 2 )
Q y = ( 1 z f + 2 z C 6 ) 2 + c 2 z 2 4 ω 2 σ 4 ( 1 + 4 σ 2 δ ij 2 )
R x = z Q x Q x 1 + z f + 2 z C 6
R y = z Q x Q y 1 + z f 2 z C 6
P ( 0 ) = 1 B 1 + B

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