## Uncertainty products for nonparaxial wave fields

JOSA A, Vol. 17, Issue 12, pp. 2391-2402 (2000)

http://dx.doi.org/10.1364/JOSAA.17.002391

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### Abstract

Although maximal localization is a basic notion in the consideration of phase-space representations of fields, it has not yet been pursued for general wave fields. We develop measures of spatial and directional spreads for nonparaxial waves in free space. These measures are invariant under translation and rotation and are shown to reduce to the conventional ones when applied to paraxial fields. The associated uncertainty relation sets limits to joint localization in coordinate and frequency space. This relation provides a basis for the definition of a joint localization measure that is analogous to the beam propagation factor (i.e.,

© 2000 Optical Society of America

**OCIS Codes**

(030.1670) Coherence and statistical optics : Coherent optical effects

(070.2590) Fourier optics and signal processing : ABCD transforms

(260.2160) Physical optics : Energy transfer

(350.5730) Other areas of optics : Resolution

(350.7420) Other areas of optics : Waves

**History**

Original Manuscript: March 20, 2000

Revised Manuscript: July 7, 2000

Manuscript Accepted: June 6, 2000

Published: December 1, 2000

**Citation**

M. A. Alonso and G. W. Forbes, "Uncertainty products for nonparaxial wave fields," J. Opt. Soc. Am. A **17**, 2391-2402 (2000)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-17-12-2391

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### References

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- This form follows simply upon consideration of the process of wrapping a Gaussian distribution around a ring and summing the multiple values at each location. The Fourier coefficients of the result are then given as an integral over all θ of a Gaussian times exp(imθ), which is once again a Gaussian. The form of Eq. (9.4) then follows.
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