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Journal of the Optical Society of America A

Journal of the Optical Society of America A

| OPTICS, IMAGE SCIENCE, AND VISION

  • Vol. 1, Iss. 7 — Jul. 1, 1984
  • pp: 711–715

New class of uncertainty relations for partially coherent light

Martin J. Bastiaans  »View Author Affiliations


JOSA A, Vol. 1, Issue 7, pp. 711-715 (1984)
http://dx.doi.org/10.1364/JOSAA.1.000711


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Abstract

A class of uncertainty relations for partially coherent light is derived; the uncertainty relations in this class express the fact that the product of the effective widths of the space-domain intensity and the spatial-frequency-domain intensity of the light has a lower bound and that this lower bound is inversely proportional to the overall degree of coherence. The different members of this class of uncertainty relations correspond to different choices of the quantity that measures the overall degree of coherence. One particular uncertainty relation has the property of being compatible with the ordinary uncertainty principle; the corresponding overall degree of coherence might therefore be considered the best possible choice of all the quantities that measure the overall degree of coherence of the light.

© 1984 Optical Society of America

History
Original Manuscript: June 21, 1983
Manuscript Accepted: February 17, 1984
Published: July 1, 1984

Citation
Martin J. Bastiaans, "New class of uncertainty relations for partially coherent light," J. Opt. Soc. Am. A 1, 711-715 (1984)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-1-7-711


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References

  1. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).
  2. M. J. Bastiaans, “Uncertainty principle for partially coherent light,” J. Opt. Soc. Am. 73, 251–255 (1983). [CrossRef]
  3. M. J. Bastiaans, “A frequency-domain treatment of partial coherence,” Opt. Acta 24, 261–274 (1977). [CrossRef]
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  15. The proof in Appendix A is due to M. L. J. Hautus, Technische Hogeschool Eindhoven, Eindhoven, The Netherlands (personal communication).
  16. A. Starikov, “Effective number of degrees of freedom of partially coherent sources,” J. Opt. Soc. Am. 72, 1538–1544 (1982). [CrossRef]
  17. M. J. Bastiaans, “Lower bound in the uncertainty principle for partially coherent light,” J. Opt. Soc. Am. 73, 1320–1324 (1983). [CrossRef]
  18. The nonincreasing behavior of μq can be proved by showing that its derivative is nonpositive; indeed, with f(t) = t log t(t≥ 0) a convex function, we haveq2(∑m=0∞λmq)1-1/qddq(∑m=0∞λmq)1/q=∑m=0∞f(λmq)-f(∑m=0∞λmq),which is nonpositive by means of Ref. 20, Sec. 1.4.7, Eq. (1).
  19. M. Marcus, H. Minc, Introduction to Linear Algebra (Macmillan, New York, 1965), Sec. 3.5, Theorem 5.9.
  20. D. S. Mitrinović, Analytic Inequalities (Springer-Verlag, Berlin, 1970).
  21. The proof in Appendix C is partly due to J. Boersma, Technische Hogeschool Eindhoven, Eindhoven, The Netherlands (personal communication).

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