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

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  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.

1982 (1)

1981 (2)

M. J. Bastiaans, "The Wigner distribution function of partially coherent light," Opt. Acta 28, 1215–1224 (1981).

A. J. E. M. Janssen, "Positivity of weighted Wigner distributions," SIAM J. Math. Anal. 12, 752-758 (1981).

1977 (1)

M. J. Bastiaans, "A frequency-domain treatment of partial coherence," Opt. Acta 24, 261–274 (1977).

1976 (1)

1964 (1)

H. Gamo, "Thermodynamic entropy of partially coherent light beams," J. Phys. Soc. Jpn. 19, 1955–1961 (1964).

1932 (1)

E. Wigner, "On the quantum correction for thermodynamic equilibrium," Phys. Rev. 40, 749–759 (1932).

Bastiaans, M. J.

M. J. Bastiaans, "An uncertainty principle for spatially quasistationary, partially coherent light," J. Opt. Soc. Am. 72, 1441–1443 (1982).

M. J. Bastiaans, "The Wigner distribution function of partially coherent light," Opt. Acta 28, 1215–1224 (1981).

M. J. Bastiaans, "A frequency-domain treatment of partial coherence," Opt. Acta 24, 261–274 (1977).

Courant, R.

R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. 1 (Interscience, New York, 1953).

Gamo, H.

H. Gamo, "Thermodynamic entropy of partially coherent light beams," J. Phys. Soc. Jpn. 19, 1955–1961 (1964).

H. Gamo, "Matrix treatment of partial coherence," in Progress in Optics, Vol. 3, E. Wolf, ed. (North-Holland, Amsterdam, 1964), pp. 187–332.

Hilbert, D.

R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. 1 (Interscience, New York, 1953).

Janssen, A. J. E. M.

A. J. E. M. Janssen, "Positivity of weighted Wigner distributions," SIAM J. Math. Anal. 12, 752-758 (1981).

Mandel, L.

Marcus, M.

M. Marcus and H. Minc, Introduction to Linear Algebra (Macmillan, New York, 1965), Sec. 3.5, Theorem 5.9.

Minc, H.

M. Marcus and H. Minc, Introduction to Linear Algebra (Macmillan, New York, 1965), Sec. 3.5, Theorem 5.9.

Papoulis, A.

A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1977).

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

Riesz, F.

F. Riesz and B. Sz.-Nagy, Functional Analysis (Ungar, New York, 1955).

Sz-Nagy, B.

F. Riesz and B. Sz.-Nagy, Functional Analysis (Ungar, New York, 1955).

Wigner, E.

E. Wigner, "On the quantum correction for thermodynamic equilibrium," Phys. Rev. 40, 749–759 (1932).

Wolf, E.

J. Opt. Soc. Am. (2)

J. Phys. Soc. Jpn. (1)

H. Gamo, "Thermodynamic entropy of partially coherent light beams," J. Phys. Soc. Jpn. 19, 1955–1961 (1964).

Opt. Acta (2)

M. J. Bastiaans, "The Wigner distribution function of partially coherent light," Opt. Acta 28, 1215–1224 (1981).

M. J. Bastiaans, "A frequency-domain treatment of partial coherence," Opt. Acta 24, 261–274 (1977).

Phys. Rev. (1)

E. Wigner, "On the quantum correction for thermodynamic equilibrium," Phys. Rev. 40, 749–759 (1932).

SIAM J. Math. Anal. (1)

A. J. E. M. Janssen, "Positivity of weighted Wigner distributions," SIAM J. Math. Anal. 12, 752-758 (1981).

Other (8)

The proof in Appendix A is due to M. L. J. Hautus, Technische Hogeschool Eindhoven, Eindhoven, The Netherlands (personal communication).

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.

M. Marcus and H. Minc, Introduction to Linear Algebra (Macmillan, New York, 1965), Sec. 3.5, Theorem 5.9.

A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1977).

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

H. Gamo, "Matrix treatment of partial coherence," in Progress in Optics, Vol. 3, E. Wolf, ed. (North-Holland, Amsterdam, 1964), pp. 187–332.

R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. 1 (Interscience, New York, 1953).

F. Riesz and B. Sz.-Nagy, Functional Analysis (Ungar, New York, 1955).

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