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Journal of the Optical Society of America

Journal of the Optical Society of America

  • Vol. 73, Iss. 3 — Mar. 1, 1983
  • pp: 251–255

Uncertainty principle for partially coherent light

Martin J. Bastiaans  »View Author Affiliations


JOSA, Vol. 73, Issue 3, pp. 251-255 (1983)
http://dx.doi.org/10.1364/JOSA.73.000251


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Abstract

Two intensity functions are introduced for partially coherent light: one in a space domain and a second one in a spatial-frequency domain. Moreover, a quantity is defined that can be considered a measure of the overall degree of coherence of the partially coherent light. It is then shown that the following uncertainty principle can be formulated: the product of the effective widths of the two intensity functions has a lower bound, and this lower bound is inversely proportional to the overall degree of coherence.

© 1983 Optical Society of America

Citation
Martin J. Bastiaans, "Uncertainty principle for partially coherent light," J. Opt. Soc. Am. 73, 251-255 (1983)
http://www.opticsinfobase.org/josa/abstract.cfm?URI=josa-73-3-251


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References

  1. A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1977).
  2. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).
  3. M. J. Bastiaans, "The Wigner distribution function of partially coherent light," Opt. Acta 28, 1215–1224 (1981).
  4. M. J. Bastiaans, "An uncertainty principle for spatially quasistationary, partially coherent light," J. Opt. Soc. Am. 72, 1441–1443 (1982).
  5. M. J. Bastiaans, "A frequency-domain treatment of partial coherence," Opt. Acta 24, 261–274 (1977).
  6. L. Mandel and E. Wolf, "Spectral coherence and the concept of cross-spectral purity," J. Opt. Soc. Am. 66, 529–535 (1976).
  7. E. Wigner, "On the quantum correction for thermodynamic equilibrium," Phys. Rev. 40, 749–759 (1932).
  8. H. Gamo, "Matrix treatment of partial coherence," in Progress in Optics, Vol. 3, E. Wolf, ed. (North-Holland, Amsterdam, 1964), pp. 187–332.
  9. H. Gamo, "Thermodynamic entropy of partially coherent light beams," J. Phys. Soc. Jpn. 19, 1955–1961 (1964).
  10. R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. 1 (Interscience, New York, 1953).
  11. F. Riesz and B. Sz.-Nagy, Functional Analysis (Ungar, New York, 1955).
  12. A. J. E. M. Janssen, "Positivity of weighted Wigner distributions," SIAM J. Math. Anal. 12, 752-758 (1981).
  13. The proof in Appendix A is due to M. L. J. Hautus, Technische Hogeschool Eindhoven, Eindhoven, The Netherlands (personal communication).
  14. From Ref. 4, Eq. (10), one might conclude that the minimum value 8/9 can be reached even without going into a limit. Indeed, if the Wigner distribution function had a form as described by Eq. (10) of Ref. 4, this would be the case. However, a function having such a form cannot be a proper Wigner distribution function, i.e., the power spectrum that would correspond to this function is not nonnegative definite hermitian. Therefore the equality signs in Eqs. (8) and (9) of Ref. 4 are somewhat misleading.
  15. M. Marcus and H. Minc, Introduction to Linear Algebra (Macmillan, New York, 1965), Sec. 3.5, Theorem 5.9.

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