## Random sources for optical frames |

Optics Express, Vol. 22, Issue 9, pp. 10622-10633 (2014)

http://dx.doi.org/10.1364/OE.22.010622

Acrobat PDF (11813 KB)

### Abstract

Analytical models for random sources producing far fields with frame-like intensity profiles are introduced. The frames can have polar and Cartesian symmetry and adjustable sharpness of the inner and outer edges. The frames are shape invariant throughout the far zone but expand due to diffraction with growing distance from the source. The generalization to multiple nested frames is also discussed. The applications of the frames are envisioned in material surface processing and particle trapping.

© 2014 Optical Society of America

## 1. Introduction

11. S. Avramov-Zamurovic, O. Korotkova, C. Nelson, O. Korotkova, and R. Malek-Madani, “The dependence of the intensity PDF of a random beam propagating in the maritime atmosphere on source coherence,” Waves in Complex and Random Media. **24**(1), 69–82 (2014). [CrossRef]

12. F. Wang, X. Liu, Y. Yuan, and Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. **38**(11), 1814–1816 (2013). [CrossRef] [PubMed]

## 2. Modeling of the random frames

*z*axis, and the angular brackets denote an ensemble average. In what follows the angular frequency dependence of all the quantities of interest will be omitted but implied. A genuine cross-spectral density function is limited by the constraint of non-negative definiteness. This condition is fulfilled if the cross-spectral density can be written as a superposition integral of the form [13

13. F. Gori, V. R. Sanchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. **11**(8), 085706 (2009). [CrossRef]

*z*. Since

*μ*and

## 3. Optical frames with polar symmetry

*A*is discussed below. Equations (8) and (9) are slightly more general distributions than given in [6

6. S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. **37**(14), 2970–2972 (2012). [CrossRef] [PubMed]

*x*and

*y*directions. In order for

13. F. Gori, V. R. Sanchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. **11**(8), 085706 (2009). [CrossRef]

14. M. Santarsiero, G. Piquero, J. C. G. de Sande, and F. Gori, “Difference of cross-spectral densities,” Opt. Lett. **39**(7), 1713–1716 (2014). [CrossRef] [PubMed]

*p*varying with

*M*= 1, and Fig. 2 corresponds to

*M =*40. Here and in all subsequent figures the horizontal and vertical axes are given in meters for

^{−1}for

*p*. While for all figures

*M*= 1, the ones for

*M*= 40 are capable of having adjustable edge sharpness and adjustable frame thickness the latter being fully controllable by the difference in the r.m.s. correlation widths.

*M*= 40. It shows the spatial distribution of the absolute value of the source degree of coherence,

*p*varying with

*x*and

*y*directions it is possible to achieve practically any anisotropic features of the elliptic frames.

## 4. Optical frames with Cartesian symmetry

8. O. Korotkova, “Random sources for rectangular far fields,” Opt. Lett. **39**(1), 64–67 (2014). [CrossRef] [PubMed]

*p*(d)–(f) for the frames with Cartesian symmetry,

*M*= 40 and the same set of r.m.s. correlations as in Fig. 2: while for all figures

*M*, its size and thickness is controlled by the outer and inner correlation coefficients, respectively. We note that for thin frames the corners become somewhat highlighted.

*M =*40 and the same r.m.s. correlations as in Fig. 2, i.e.,

*x*and

*y*directions.

## 5. Frame combinations

*B*is introduced to meet the first of conditions (11),

## 6. Summary

## Acknowledgments

## References and links

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

2. | F. Gori, G. Guattari, and C. Padovani, “Modal expansion for J |

3. | Z. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. |

4. | Z. Mei and O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett. |

5. | H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. |

6. | S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. |

7. | O. Korotkova, S. Sahin, and E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A |

8. | O. Korotkova, “Random sources for rectangular far fields,” Opt. Lett. |

9. | O. Korotkova and E. Shchepakina, “Rectangular multi-Gaussian Schell-model beams in free space and atmospheric turbulence,” J. Opt. |

10. | C. Liang, F. Wang, X. Liu, Y. Cai, and O. Korotkova, “Experimental generation of Cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. |

11. | S. Avramov-Zamurovic, O. Korotkova, C. Nelson, O. Korotkova, and R. Malek-Madani, “The dependence of the intensity PDF of a random beam propagating in the maritime atmosphere on source coherence,” Waves in Complex and Random Media. |

12. | F. Wang, X. Liu, Y. Yuan, and Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. |

13. | F. Gori, V. R. Sanchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. |

14. | M. Santarsiero, G. Piquero, J. C. G. de Sande, and F. Gori, “Difference of cross-spectral densities,” Opt. Lett. |

**OCIS Codes**

(030.1640) Coherence and statistical optics : Coherence

(350.5500) Other areas of optics : Propagation

**ToC Category:**

Coherence and Statistical Optics

**History**

Original Manuscript: March 26, 2014

Manuscript Accepted: April 3, 2014

Published: April 24, 2014

**Citation**

Olga Korotkova and Elena Shchepakina, "Random sources for optical frames," Opt. Express **22**, 10622-10633 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-9-10622

Sort: Year | Journal | Reset

### References

- L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995).
- F. Gori, G. Guattari, C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64(4), 311–316 (1987). [CrossRef]
- Z. Mei, O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013). [CrossRef] [PubMed]
- Z. Mei, O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett. 38(14), 2578–2580 (2013). [CrossRef] [PubMed]
- H. Lajunen, T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011). [CrossRef] [PubMed]
- S. Sahin, O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012). [CrossRef] [PubMed]
- O. Korotkova, S. Sahin, E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A 29(10), 2159–2164 (2012). [CrossRef] [PubMed]
- O. Korotkova, “Random sources for rectangular far fields,” Opt. Lett. 39(1), 64–67 (2014). [CrossRef] [PubMed]
- O. Korotkova, E. Shchepakina, “Rectangular multi-Gaussian Schell-model beams in free space and atmospheric turbulence,” J. Opt. 16(4), 045704 (2014). [CrossRef]
- C. Liang, F. Wang, X. Liu, Y. Cai, O. Korotkova, “Experimental generation of Cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014). [CrossRef] [PubMed]
- S. Avramov-Zamurovic, O. Korotkova, C. Nelson, O. Korotkova, R. Malek-Madani, “The dependence of the intensity PDF of a random beam propagating in the maritime atmosphere on source coherence,” Waves in Complex and Random Media. 24(1), 69–82 (2014). [CrossRef]
- F. Wang, X. Liu, Y. Yuan, Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013). [CrossRef] [PubMed]
- F. Gori, V. R. Sanchez, M. Santarsiero, T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009). [CrossRef]
- M. Santarsiero, G. Piquero, J. C. G. de Sande, F. Gori, “Difference of cross-spectral densities,” Opt. Lett. 39(7), 1713–1716 (2014). [CrossRef] [PubMed]

## 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.