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Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Editor: Franco Gori
  • Vol. 30, Iss. 12 — Dec. 1, 2013
  • pp: 2611–2617

Elementary-field analysis of partially coherent beam shaping

Manisha Singh, Jani Tervo, and Jari Turunen  »View Author Affiliations

JOSA A, Vol. 30, Issue 12, pp. 2611-2617 (2013)

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We consider spatial shaping of partially coherent fields in two types of optical systems: a 2F Fourier-transforming system with the beam shaping element in the input plane and a 4F imaging system with the element in the intermediate Fourier plane. Different representations of the spatially partially coherent field in terms of fully coherent fields are examined to permit reduction of the dimensionality of the propagation integrals. The standard Mercer-type coherent-mode representation of the incident cross-spectral density (CSD) function is compared to expansions of CSD in either spatially or angularly shifted elementary field modes, all sharing the same spatial profile. In Fourier-transforming systems, the angular elementary-field representation proves computationally superior, while in imaging systems the spatially shifted elementary-field expansion is the best choice. Considering the Fourier-plane element as a generalized pupil, the latter leads to the concept’s generalized amplitude associated with the elementary field and to a generalized transfer function of the system. These concepts reduce to the standard point spread function and the optical transfer function in the limit of spatial incoherence at the object plane. Examples of the effects of partial coherence in spatial beam shaping are given.

© 2013 Optical Society of America

OCIS Codes
(030.1640) Coherence and statistical optics : Coherence
(110.0110) Imaging systems : Imaging systems
(110.4850) Imaging systems : Optical transfer functions
(140.3300) Lasers and laser optics : Laser beam shaping

ToC Category:
Imaging Systems

Original Manuscript: October 9, 2013
Manuscript Accepted: October 31, 2013
Published: November 25, 2013

Manisha Singh, Jani Tervo, and Jari Turunen, "Elementary-field analysis of partially coherent beam shaping," J. Opt. Soc. Am. A 30, 2611-2617 (2013)

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  1. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  2. H. H. Hopkins, “On the diffraction theory of optical images,” Proc. R. Soc. Lond. Ser. A 217, 408–432 (1953). [CrossRef]
  3. J. W. Goodman, Statistical Optics (Wiley, 2000), Chap. 7.
  4. F. Gori, “Collett-Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980). [CrossRef]
  5. E. Wolf, “New theory of partial coherence in the space-frequency domain. Part I: spectra and cross spectra of steady-state sources,” J. Opt. Soc. Am. 72, 343–351 (1982). [CrossRef]
  6. F. Gori, M. Santarsiero, R. Simon, G. Piquero, R. Borghi, and G. Guattari, “Coherent-mode decomposition of partially polarized, partially coherent sources,” J. Opt. Soc. Am. A 20, 78–84 (2003). [CrossRef]
  7. J. Tervo, T. Setälä, and A. T. Friberg, “Theory of partially coherent electromagnetic fields in the space–frequency domain,” J. Opt. Soc. Am. A 21, 2205–2215 (2004). [CrossRef]
  8. A. Starikov and E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and of their radiation fields,” J. Opt. Soc. Am. 72, 923–928 (1982). [CrossRef]
  9. A. Starikov, “Effective number of degrees of freedom of partially coherent sources,” J. Opt. Soc. Am. 72, 1538–1544 (1982). [CrossRef]
  10. F. Gori and C. Palma, “Partially coherent sources which give rise to highly directional laser beams,” Opt. Commun. 27, 185–188 (1978). [CrossRef]
  11. F. Gori, “Directionality and partial coherence,” Opt. Acta 27, 1025–1034 (1980). [CrossRef]
  12. F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32, 3531–3533 (2007). [CrossRef]
  13. F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A 11, 085706 (2009). [CrossRef]
  14. R. Martínez-Herrero and P. M. Mejías, “Elementary-field expansions of genuine cross-spectral density matrices,” Opt. Lett. 34, 2303–2305 (2009). [CrossRef]
  15. J. Turunen, “Elementary-field representations in partially coherent optics,” J. Mod. Opt. 58, 509–527 (2011). [CrossRef]
  16. J. Turunen, P. Pääkkönen, M. Kuittinen, P. Laakkonen, J. Simonen, T. Kajava, and M. Kaivola, “Diffractive shaping of excimer laser beams,” J. Mod. Opt. 47, 2467–2475 (2000). [CrossRef]
  17. H. Partanen, J. Tervo, and J. Turunen, “Spatial coherence of broad-area laser diodes,” Appl. Opt. 52, 3221–3228 (2013). [CrossRef]
  18. A. C. Schell, “The multiple plate antenna,” Ph.D. dissertation (Massachusetts Institute of Technology, 1961).
  19. A. C. Schell, “A technique for the determination of the radiation pattern of a partially coherent aperture,” IEEE Trans. Antennas Propag. 15, 187–188 (1967). [CrossRef]
  20. O. Bryngdahl, “Geometrical transformations in optics,” J. Opt. Soc. Am. 64, 1092–1099 (1974). [CrossRef]
  21. S. R. Seshadri, “Partially coherent Gaussian Schell-model electromagnetic beams,” J. Opt. Soc. Am. A 16, 1373–1380 (1999). [CrossRef]
  22. K. Miyamoto, “The phase Fresnel lens,” J. Opt. Soc. Am. 51, 17–20 (1961). [CrossRef]
  23. P. Laakkonen, J. Lautanen, V. Kettunen, J. Turunen, and M. Schirmer, “Analog diffractive elements in SiO2 by electron beam lithography and proportional etching with negative resist,” J. Mod. Opt. 46, 1295–1307 (1999). [CrossRef]
  24. W.-H. Lee, “Computer-generated holograms: techniques and applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, 1978), Vol. 16, Chap. 3, pp. 119–223.

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