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Applied Optics

Applied Optics


  • Vol. 40, Iss. 2 — Jan. 10, 2001
  • pp: 249–256

Wigner formulation of optical processing with light of arbitrary coherence

Avi Pe’er, Dayong Wang, Adolf W. Lohmann, and Asher A. Friesem  »View Author Affiliations

Applied Optics, Vol. 40, Issue 2, pp. 249-256 (2001)

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A unified mathematical formulation for designing and analyzing even the most general optical processor is presented. It exploits the Wigner distribution function to characterize the illumination, the input, the inherent filter, and the output results. To characterize the propagation of the light through the optical processor setup, we exploit the Wigner matrix formalism, which is appealing because it allows simple geometric analysis. The Wigner distribution function was extended to include illumination of arbitrary coherence so that processors using either coherent light or partially coherent light can be designed and analyzed with the same Wigner formalism. The basic principles, design, and analysis of the imaging and Fourier-transform operations and use of the Wigner formalism to evaluate the performance and tolerances of optical processors are presented.

© 2001 Optical Society of America

OCIS Codes
(030.1640) Coherence and statistical optics : Coherence
(030.4280) Coherence and statistical optics : Noise in imaging systems
(070.2580) Fourier optics and signal processing : Paraxial wave optics
(070.4550) Fourier optics and signal processing : Correlators
(080.2730) Geometric optics : Matrix methods in paraxial optics

Original Manuscript: July 18, 2000
Revised Manuscript: September 21, 2000
Published: January 10, 2001

Avi Pe’er, Dayong Wang, Adolf W. Lohmann, and Asher A. Friesem, "Wigner formulation of optical processing with light of arbitrary coherence," Appl. Opt. 40, 249-256 (2001)

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  1. E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932). [CrossRef]
  2. T. Alieva, M. J. Bastiaans, “Self-affinity in phase space,” J. Opt. Soc. Am. A 17, 756–761 (2000). [CrossRef]
  3. D. Peris, V. C. Georgopoulos, “Wigner distribution representation and analysis of audio signals: an illustrated tutorial review,” J. Audio Eng. Soc. 47, 1043–1053 (1999).
  4. K. Banaszek, K. Wodkiewicz, “Non-locality of the Einstein–Podolsky–Rosen state in the Wigner representation,” Phys. Rev. A 58, 4345–4347 (1998). [CrossRef]
  5. T. A. C. M. Classen, W. F. G. Mecklenbrauker, “The Wigner distribution—a tool for time-frequency signal analysis: Part I. Continuous time signals”; “Part II. Discrete time signals”; “Part III. “Relations with other time-frequency signal transformations,” Philips J. Res. 35, 217–250, 276–300, 372–389 (1980).
  6. M. J. Bastiaans, “Application of the Wigner distribution function to partially coherent light,” J. Opt. Soc. Am. A 3, 1227–1238 (1986). [CrossRef]
  7. M. J. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979). [CrossRef]
  8. M. J. Bastiaans, “New class of uncertainty relations for partially coherent light,” J. Opt. Soc. Am. A 1, 711–715 (1984). [CrossRef]
  9. G. A. Deschamps, “Ray techniques in electromagnetism,” Proc. IEEE 60, 1022–1035 (1972). [CrossRef]
  10. R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, Calif., 1966).
  11. A. W. Lohmann, D. Wang, A. Pe’er, A. A. Friesem, “Design of an achromatic Fourier system by means of Wigner algebra,” in 18th Congress of the International Commission for Optics, A. J. Glass, J. W. Goodman, M. Chang, A. H. Guenther, T. Asakura, eds., Proc. SPIE3749, 6–7 (1999).
  12. R. H. Katyl, “Compensating optical systems. III. Achromatic Fourier transformation,” Appl. Opt. 11, 1255–1260 (1972). [CrossRef] [PubMed]
  13. S. Leon, E. N. Leith, “Optical processing and holography with polychromatic point source illumination,” Appl. Opt. 24, 3638–3642 (1985). [CrossRef] [PubMed]
  14. P. Andres, J. Lancis, W. D. Furlan, “White-light Fourier transformer with low chromatic aberration,” Appl. Opt. 31, 4682–4687 (1992). [CrossRef] [PubMed]
  15. E. Tajahuerce, J. Lancis, V. Climent, P. Andres, “Hybrid (refractive-diffractive) Fourier processor: a novel optical architecture for achromatic processing with broadband point-source illumination,” Opt. Commun. 151, 86–92 (1998). [CrossRef]
  16. G. M. Morris, “Diffraction theory for an achromatic Fourier transformation,” Appl. Opt. 20, 2017–2025 (1981). [CrossRef] [PubMed]
  17. P. Andrés, V. Climent, J. Lancis, G. Mínguez-Vega, E. Tajahuerce, A. W. Lohmann, “All-incoherent dispersion-compensated optical correlator,” Opt. Lett. 24, 1331–1333 (1999). [CrossRef]
  18. A. Pe’er, D. Wang, A. W. Lohmann, A. A. Friesem, “Optical correlation with totally incoherent light,” Opt. Lett. 24, 1469–1471 (1999). [CrossRef]
  19. D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, “Graded-index fibers, Wigner-distribution functions, and the fractional Fourier transform,” Appl. Opt. 33, 6188–6193 (1994). [CrossRef] [PubMed]
  20. H. M. Ozaktas, D. Mendlovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743–751 (1995). [CrossRef]
  21. Y. Bitran, Z. Zalevsky, D. Mendlovic, R. G. Dorsch, “Fractional correlation operation: performance analysis,” Appl. Opt. 35, 297–303 (1996). [CrossRef] [PubMed]

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