## Wigner functions for Helmholtz wave fields

JOSA A, Vol. 16, Issue 10, pp. 2476-2487 (1999)

http://dx.doi.org/10.1364/JOSAA.16.002476

Enhanced HTML Acrobat PDF (1392 KB)

### Abstract

We investigate a general form of the Wigner function for wave fields that satisfy the Helmholtz equation in two-dimensional free space. The momentum moment of this Wigner function is shown to correspond to the flux of the wave field. For a forward-propagating wave field, the negative regions of the Wigner function are seen to be associated with small regions of backward flux in the field. We also study different projections of the Wigner function, each corresponding to a distribution in a reduced phase space that fully characterizes the wave field. One of these projections is the standard Wigner function of the field at a screen. Another projection introduced by us has the added property of being conserved along rays and is better suited to the description of nonparaxial wave fields.

© 1999 Optical Society of America

**OCIS Codes**

(000.3860) General : Mathematical methods in physics

(030.5630) Coherence and statistical optics : Radiometry

(070.2590) Fourier optics and signal processing : ABCD transforms

(350.6980) Other areas of optics : Transforms

(350.7420) Other areas of optics : Waves

**History**

Original Manuscript: December 1, 1998

Revised Manuscript: April 16, 1999

Manuscript Accepted: April 16, 1999

Published: October 1, 1999

**Citation**

Kurt Bernardo Wolf, Miguel Angel Alonso, and Gregory W. Forbes, "Wigner functions for Helmholtz wave fields," J. Opt. Soc. Am. A **16**, 2476-2487 (1999)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-16-10-2476

Sort: Year | Journal | Reset

### References

- E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932);H.-W. Lee, “Theory and application of the quantum phase-space distribution functions,” Phys. Rep. 259, 147–211 (1995). [CrossRef]
- A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 58, 1256–1259 (1968). [CrossRef]
- A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 63, 1622–1623 (1973). [CrossRef]
- R. G. Littlejohn, R. Winston, “Corrections to classical radiometry,” J. Opt. Soc. Am. A 10, 2024–2037 (1993). [CrossRef]
- Yu. A. Kravtsov, Yu. A. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, Berlin, 1990), p. 23.
- The existence of an invertible Fourier transform [Eq. (3)] of a (twice-differentiable) Helmholtz wave field Ψ(q) is ensured by the conditions of the Fourier integral theorem only when it is absolutely integrable and of bounded variation. Plane waves are in this sense limit functions for which Fourier analysis can be extended as usual with Dirac δ’s in the p plane and on the unit circle [Eq. (4)]. However, evanescent solutions (i.e., of exponential growth in the plane) do not possess a Fourier transform.
- P. González-Casanova, K. B. Wolf, “Interpolation for solutions of the Helmholtz equation,” Num. Meth. Partial Diff. Eqs. 11, 77–91 (1995). [CrossRef]
- For other references about flux vortices see, for example, M. V. Berry, “Evanescent and real waves in quantum billiards and Gaussian beams,” J. Phys. A Nucl. Math. Gen. 27, L391–L398 (1994). [CrossRef]
- A. Lohmann, “The Wigner function and its optical production,” Opt. Commun. 42, 32–37 (1980).
- K. B. Wolf, A. L. Rivera, “Holographic information in the Wigner function,” Opt. Commun. 144, 36–42 (1997). [CrossRef]
- M. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978);“Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979);“The Wigner distribution function and Hamilton’s characteristics of a geometric-optical system,” Opt. Commun. 30, 321–326 (1979). [CrossRef]
- S. Steinberg, K. B. Wolf, “Invariant inner products on spaces of solutions of the Klein–Gordon and Helmholtz equations,” J. Math. Phys. 22, 1660–1663 (1981). [CrossRef]
- L. M. Nieto, N. M. Atakishiyev, S. M. Chumakov, K. B. Wolf, “Wigner distribution function for finite systems,” J. Phys. A Nucl. Math. Gen. 31, 3875–3895 (1998). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.