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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 9 — May. 6, 2013
  • pp: 10511–10525

Wigner function measurement using a lenslet array

Lei Tian, Zhengyun Zhang, Jonathan C. Petruccelli, and George Barbastathis  »View Author Affiliations


Optics Express, Vol. 21, Issue 9, pp. 10511-10525 (2013)
http://dx.doi.org/10.1364/OE.21.010511


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Abstract

Geometrical–optical arguments have traditionally been used to explain how a lenslet array measures the distribution of light jointly over space and spatial frequency. Here, we rigorously derive the connection between the intensity measured by a lenslet array and wave–optical representations of such light distributions for partially coherent optical beams by using the Wigner distribution function (WDF). It is shown that the action of the lenslet array is to sample a smoothed version of the beam’s WDF (SWDF). We consider the effect of lenslet geometry and coherence properties of the beam on this measurement, and we derive an expression for cross–talk between lenslets that corrupts the measurement. Conditions for a high fidelity measurement of the SWDF and the discrepancies between the measured SWDF and the WDF are investigated for a Schell–model beam.

© 2013 OSA

OCIS Codes
(110.4980) Imaging systems : Partial coherence in imaging
(050.5082) Diffraction and gratings : Phase space in wave options

ToC Category:
Imaging Systems

History
Original Manuscript: March 8, 2013
Revised Manuscript: April 12, 2013
Manuscript Accepted: April 13, 2013
Published: April 23, 2013

Citation
Lei Tian, Zhengyun Zhang, Jonathan C. Petruccelli, and George Barbastathis, "Wigner function measurement using a lenslet array," Opt. Express 21, 10511-10525 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-9-10511


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References

  1. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  2. B. J. Thompson and E. Wolf, “Two-beam interference with partially coherent light,” J. Opt. Soc. Am.47, 895 (1957). [CrossRef]
  3. W. Tango and R. Twiss, “Michelson stellar interferometry,” Prog. Optics17, 239–277 (1980). [CrossRef]
  4. K. Itoh and Y. Ohtsuka, “Fourier-transform spectral imaging: retrieval of source information from three-dimensional spatial coherence,” J. Opt. Soc. Am. A3, 94–100 (1986). [CrossRef]
  5. D. L. Marks, R. A. Stack, and D. J. Brady, “Three-dimensional coherence imaging in the Fresnel domain,” Appl. Opt.38, 1332–1342 (1999). [CrossRef]
  6. M. G. Raymer, M. Beck, and D. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett.72, 1137–1140 (1994). [CrossRef]
  7. K. G. Larkin and C. J. R. Sheppard, “Direct method for phase retrieval from the intensity of cylindrical wave fronts,” J. Opt. Soc. Am. A16, 1838–1844 (1999). [CrossRef]
  8. D. M. Marks, R. A. Stack, and D. J. Brady, “Astigmatic coherence sensor for digital imaging,” Opt. Lett.25, 1726–1728 (2000). [CrossRef]
  9. S. Cho and M. A. Alonso, “Ambiguity function and phase-space tomography for nonparaxial fields,” J. Opt. Soc. Am. A28, 897–902 (2011). [CrossRef]
  10. L. Tian, J. Lee, S. B. Oh, and G. Barbastathis, “Experimental compressive phase space tomography,” Opt. Express20, 8296–8308 (2012). [CrossRef] [PubMed]
  11. L. Tian, S. Rehman, and G. Barbastathis, “Experimental 4D compressive phase space tomography,” in “Frontiers in Optics,” (Optical Society of America, 2012), p. FM4C.4.
  12. B. C. Platt and R. Shack, “History and principles of Shack-Hartmann wavefront sensing,” J. Refract. Surg.17(2001). [PubMed]
  13. G. Lippmann, “La photographie integrale,” Comptes-Rendus, Academie des Sciences146 (1908).
  14. A. Stern and B. Javidi, “Three-dimensional image sensing and reconstruction with time-division multiplexed computational integral imaging,” Appl. Opt.42, 7036–7042 (2003). [CrossRef] [PubMed]
  15. J.-H. Park, K. Hong, and B. Lee, “Recent progress in three-dimensional information processing based on integral imaging,” Appl. Opt.48, H77–H94 (2009). [CrossRef] [PubMed]
  16. E. H. Adelson and J. Y. A. Wang, “Single lens stereo with a plenoptic camera,” IEEE Trans. Pattern Anal. Mach. Intell.14, 99–106 (1992). [CrossRef]
  17. R. Ng, M. Levoy, M. Bredif, G. Duval, M. Horowitz, and P. Hanrahan, “Light field photography with a hand-held plenoptic camera,” Tech. Rep. CTSR 2005-02, Stanford (2005).
  18. A. T. Friberg, “On the existence of a radiance function for finite planar sources of arbitrary states of coherence,” J. Opt. Soc. Am.69, 192–198 (1979). [CrossRef]
  19. E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev.40, 0749–0759 (1932). [CrossRef]
  20. L. Dolin, “Beam description of weakly-inhomogeneous wave fields,” Izv. Vyssh. Uchebn. Zaved. Radiofiz7, 559–563 (1964).
  21. A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am.58, 1256–1259 (1968). [CrossRef]
  22. Z. Zhang and M. Levoy, “Wigner distributions and how they relate to the light field,” in “IEEE International Conference on Computational Photography (ICCP),” (IEEE, 2009), pp. 1–10. [CrossRef]
  23. H. Bartelt, K.-H. Brenner, and A. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun.32, 32–38 (1980). [CrossRef]
  24. A. Wax and J. E. Thomas, “Optical heterodyne imaging and Wigner phase space distributions,” Opt. Lett.21, 1427–1429 (1996). [CrossRef] [PubMed]
  25. H. N. Chapman, “Phase-retrieval X-ray microscopy by Wigner–distribution deconvolution,” Ultramicroscopy66, 153–172 (1996). [CrossRef]
  26. L. Waller, G. Situ, and J. Fleischer, “Phase–space measurement and coherence synthesis of optical beams,” Nat. Photonics6, 474–479 (2012). [CrossRef]
  27. H. Choi, S.-W. Min, S. Jung, J.-H. Park, and B. Lee, “Multiple-viewing-zone integral imaging using a dynamic barrier array for three-dimensional displays,” Opt. Express11, 927–932 (2003). [CrossRef] [PubMed]
  28. K. B. Wolf, M. A. Alonso, and G. W. Forbes, “Wigner functions for Helmholtz wave fields,” J. Opt. Soc. Am. A16, 2476–2487 (1999). [CrossRef]
  29. S. Cho, J. Petruccelli, and M. Alonso, “Wigner functions for paraxial and nonparaxial fields,” J. Mod. Optic.56, 1843–1852 (2009). [CrossRef]
  30. N. Lindlein, J. Pfund, and J. Schwider, “Algorithm for expanding the dynamic range of a shack-hartmann sensor by using a spatial light modulator array,” Opt. Eng.40, 837–840 (2001). [CrossRef]
  31. M. E. Gehm, S. T. McCain, N. P. Pitsianis, D. J. Brady, P. Potuluri, and M. E. Sullivan, “Static two-dimensional aperture coding for multimodal, multiplex spectroscopy,” Appl. Opt.45, 2965–2974 (2006). [CrossRef] [PubMed]
  32. Z. Zhang, Z. Chen, S. Rehman, and G. Barbastathis, “Factored form descent: a practical algorithm for coherence retrieval,” Opt. Express21, 5759–5780 (2013). [CrossRef] [PubMed]

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