## Effects of astigmatism on spectra, coherence and polarization of stochastic electromagnetic beams passing through an astigmatic optical system

Optics Express, Vol. 17, Issue 9, pp. 7310-7321 (2009)

http://dx.doi.org/10.1364/OE.17.007310

Acrobat PDF (425 KB)

### 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

## 1. Introduction

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]

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]

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]

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]

## 2. Theoretical analysis

*z*=0, with its axis along the

*z*direction, whose cross-spectral density matrix [17

17. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A **312**, 263–267 (2003). [CrossRef]

*z*=0 is given by

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]

*x*,

*y*) can be calculated from the general formula [17

17. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A **312**, 263–267 (2003). [CrossRef]

*x*

_{1},

*y*

_{1}) and (

*x*

_{2},

*y*

_{2}) is defined by the formula [17

17. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A **312**, 263–267 (2003). [CrossRef]

*x*,

*y*) is given by the expression [17

**312**, 263–267 (2003). [CrossRef]

**312**, 263–267 (2003). [CrossRef]

*W*↔ (

*x*

_{1},

*y*

_{1},

*x*

_{2},

*y*

_{2},

*z*,

*ω*) of EGSM beams through an astigmatic optical system at two points (

*x*

_{1},

*y*

_{1}) and (

*x*

_{2},

*y*

_{2}) in a transverse plane

*z*can be written as

*k*=

*ω*/

*c*denotes the wave number,

*c*is the velocity of light in vacuum.

*A*,

*B*and

*D*are elements of the transfer

*ABCD*matrix. In Eq.(6) the astigmatism in the

*ABCD*system is considered, and characterized by a term exp[-

*ikC*

_{6}(

*x*́

^{2}-

*y*́

^{2})],

*C*

_{6}being the astigmatic coefficient [19], but other phase aberrations can be neglected.

*W*↔ (

*x*

_{1},

*y*

_{1},

*x*

_{2},

*y*

_{2},

*z*,

*ω*) of EGSM beams at two points (

*x*

_{1},

*y*

_{1}) and (

*x*

_{2},

*y*

_{2}) at the

*z*plane is given by

*z*=0 and the separation between the lens and observation

*z*plane, we have

## 3. Numerical calculations and discussions

*A*=

_{x}*A*=

_{y}*A*,

*B*=1,

_{xx}*B*=B,

_{yy}*B*=

_{xy}*B*=0. On substituting from Eqs.(1) and (2) into Eq.(5), we readily obtain the expression for the polarization at the plane

_{yx}*z*=0 as

*P*

^{(0)}=0.5 (i.e., the case of a partially polarized beam) and present some numerical examples in Figs.1-10 to show the influence of astigmatic coefficient

*C*

_{6}on the spectral degree of polarization, spectra and coherence of EGSM beams, respectively.

*x*-

*z*plane. It can be seen from Fig.1 that the astigmatism affects the spectral degree of polarization, and the distribution of spectral degree of polarization is different for different values of

*C*

_{6}. In Fig.2 we give the spectral degree of polarization of the EGSM beam in the

*y*-

*z*plane. The corresponding results for the aberration-free case of

*C*

_{6}=0 the distribution of spectral degree of polarization in the

*y*-

*z*plane depicted in Fig. 2(a) is same behavior in the

*x*-

*z*plane in Fig. 1(a). For the cases of

*C*

_{6}≠ 0, the distribution of the spectral degree of polarization in the

*x*-

*z*plane is different from that in the

*y*-

*z*plane, and the distribution of spectral degree of polarization in the

*y*-

*z*plane is nearly

*z*axis as

*z*>400mm. The main reason for this phenomenon is the astigmatism of lens. In addition, from Fig.2 we see that there is a narrow band of higher degree of polarization close to

*z*=0.32m for

*C*

_{6}=0.3×10

^{-3}mm

^{-1}in Fig. 2(c) and

*z*=0.3m for

*C*

_{6}=0.5×10

^{-3}mm

^{-1}in Fig. 2(d), and numerical results show that the strongly elliptical plots appear in the

*x*-

*y*plane. The results can be interpreted as follows. In the

*y*-

*z*plane, for a fixed value

*y*near

*z*-axis, there is a maximum of the spectral degree of polarization close

*z*=0.3m at larger the astigmatism of lens for

*C*

_{6}=0.3×10

^{-3}mm

^{-1}and 0.5×10

^{-3}mm

^{-1}in Figs. 2(c) and 2(d).

*x*-

*y*plane for the aberration-free case of

*C*

_{6}=0 and aberration case of

*C*

_{6}=0.3×10

^{-3}mm

^{-1}, respectively. In Fig.3, for the aberration-free case of

*C*

_{6}=0 the distribution of spectral degree of polarization in the

*x*-

*y*plane is of circular symmetry. In Fig.4, for the aberration case of

*C*

_{6}=0.3×10

^{-3}mm

^{-1}the distribution of spectral degree of polarization is of elliptical distributions at

*z*=300mm, 500mm and 600mm(see Figs.4(a), 4(c) and 4(d)) because of the astigmatism of lens. In Fig.5, the EGSM source is changed with the correlation length

*δ*increased from 0.2 to 0.4mm.

_{yy}*C*

_{6}=0, 0.1×10

^{-3}mm

^{-1}, 0.3×10

^{-3}mm

^{-1}and 0.5×10

^{-3}mm

^{-1}. We find that maxima of the spectral degree of polarization along the

*z*axis become smaller with larger values of

*C*

_{6}. In particular, the distributions of spectral degree of polarization are split into two lines. In Fig.6(b), the EGSM source is changed with the correlation length

*δ*increased from 0.2 to 0.4mm. Comparing with

_{yy}*δ*=0.2mm, we find that maxima of the spectral degree of polarization along the

_{yy}*z*axis become smaller with larger value of

*δ*=0.4mm.

_{yy}*C*

_{6}=0 and three aberration case of

*C*

_{6}=0.1×10

^{-3}mm

^{-1}, 0.3×10

^{-3}mm

^{-1}and 0.5×10

^{-3}mm

^{-1}. The variation of the on-axis spectral density with the three different coherence lengths

*δ*and

_{xx}*δ*is also illustrated in Fig.7(b). As can be seen, the maxima of axial spectral density change and the position of maximum of axial spectral density moves toward the lens, depending on

_{yy}*C*

_{6},

*δ*and

_{xx}*δ*. From Fig.7(a), it is shown that as

_{yy}*C*

_{6}increases, the maxima of the spectral density along the

*z*axis become smaller and the position of the maximum of axial spectral density moves toward the lens. From Fig.7(b), we find that the maxima of the spectral density along the

*z*axis become larger with larger values of

*δ*and

_{xx}*δ*.

_{yy}*μ*(0,0,

*x*,0,

*z*,

*ω*)| at a pair of points (0,0) and (

*x*,0) and at a pair of points (0,0) and (0,

*y*) in the plane

*z*=300mm, 400mm, 500mm and 600mm for different values of

*C*

_{6}=0, 0.1×10

^{-3}mm

^{-1}, 0.3×10

^{-3}mm

^{-1}and 0.5×10

^{-3}mm

^{-1}, respectively. It is shown that for aberration-free case of

*C*

_{6}=0 and smaller aberration case of

*C*

_{6}=0.1×10

^{-3}mm

^{-1}, the coherence width (the width of the spectral degree of coherence curve) decreases with propagation distance increasing from

*z*=300mm to 400mm, and the coherence width increases with propagation distance increasing from

*z*=400mm to 600mm. For larger aberration case of

*C*

_{6}=0.3×10

^{-3}mm

^{-1}and 0.5×10

^{-3}mm

^{-1}, coherence width increases with increment of propagation distance

*z*on the

*x*axis, but coherence width decreases with increment of propagation distance

*z*on the

*y*axis. We further investigate the behavior of spectral degree of coherence with different values of

*δ*and

_{xx}*δ*in the geometrical focal plane

_{yy}*z*=400mm (see Fig. 10). It is seen that the coherence width along the

*x*axis and along the

*y*axis change not obviously for larger values of

*δ*and

_{xx}*δ*, and the coherence width along the

_{yy}*x*axis is noticeably larger than that along the

*y*axis for smaller values of

*δ*=0.6mm and

_{xx}*δ*=0.2mm.

_{yy}## 4. Conclusions

## Acknowledgments

## References and links

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. |

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 |

3. | O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. |

4. | H. Wang, X, Wang, A. Zeng, and K. Yang, “Effects of coherence on anisotropic electromagnetic Gaussian-Schell model beams on propagation,” Opt. Lett. |

5. | J. Pu, O. Korotkova, and E. Wolf, “Polarization-induced spectral changes on propagation of stochastic electromagnetic beams,” Phys. Rev. E |

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. |

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. |

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 |

9. | Y. Cai, O. Korotkova, H. T. Eyyuboglu, and Y. Baykal, “Active laser radar systems with stochastic electromagnetic beams in turbulent atmosphere,” Opt. Express |

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 |

11. | X. Du and D. Zhao, “Propagation of random electromagnetic beams through axially nonsymmetrical optical systems,” Opt. Commun. |

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. |

13. | G. Zhang and J. Pu, “Stochastic electromagnetic beams focused by a bifocal lens,” J. Opt. Soc. Am. A |

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 |

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 |

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 |

17. | E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A |

18. | L. Mandel and E. Wolf, |

**OCIS Codes**

(030.1640) Coherence and statistical optics : Coherence

(260.0260) Physical optics : Physical optics

(260.5430) Physical optics : Polarization

**ToC Category:**

Coherence and Statistical Optics

**History**

Original Manuscript: February 25, 2009

Revised Manuscript: April 6, 2009

Manuscript Accepted: April 6, 2009

Published: April 17, 2009

**Citation**

Liuzhan Pan, Mengle Sun, Chaoliang Ding, Zhiguo Zhao, and Baida Lu, "Effects of astigmatism on spectra, coherence and polarization of stochastic electromagnetic beams passing through an astigmatic optical system," Opt. Express **17**, 7310-7321 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-9-7310

Sort: Year | Journal | Reset

### References

- 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]
- 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]
- 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. 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]
- 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, 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]
- 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]
- 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]
- 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]
- 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]
- X. Du and D. Zhao, "Propagation of random electromagnetic beams through axially nonsymmetrical optical systems," Opt. Commun. 281, 2711-2715 (2008). [CrossRef]
- 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]
- G. Zhang and J. Pu, "Stochastic electromagnetic beams focused by a bifocal lens," J. Opt. Soc. Am. A 25, 1710-1715 (2008). [CrossRef]
- 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]
- 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]
- 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]
- E. Wolf, "Unified theory of coherence and polarization of random electromagnetic beams," Phys. Lett. A 312, 263-267 (2003). [CrossRef]
- L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, Cambridge, 1995)

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.