## Measurement of Helmholtz wave fields

JOSA A, Vol. 17, Issue 7, pp. 1256-1264 (2000)

http://dx.doi.org/10.1364/JOSAA.17.001256

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### Abstract

A simple formalism is found for the measurement of wave fields that satisfy the Helmholtz equation in free space. This formalism turns out to be analogous to the well-known theory of measurements for quantum-mechanical wave functions: A measurement corresponds to the squared magnitude of the inner product (in a suitable Hilbert space) of the wave field and a field that is associated with the detector. The measurement can also be expressed as an overlap in phase space of a special form of the Wigner function that is tailored for Helmholtz wave fields.

© 2000 Optical Society of America

**OCIS Codes**

(030.1670) Coherence and statistical optics : Coherent optical effects

(030.5630) Coherence and statistical optics : Radiometry

(040.0040) Detectors : Detectors

(260.0260) Physical optics : Physical optics

(350.7420) Other areas of optics : Waves

**History**

Original Manuscript: August 19, 1999

Revised Manuscript: March 1, 2000

Manuscript Accepted: March 1, 2000

Published: July 1, 2000

**Citation**

Miguel A. Alonso, "Measurement of Helmholtz wave fields," J. Opt. Soc. Am. A **17**, 1256-1264 (2000)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-17-7-1256

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### References

- C. Cohen-Tannoudji, B. Diu, F. Laloë, Quantum Mechanics (Wiley, Paris, 1977), Vol. 1, pp. 214–227.
- E. Wigner, “On the correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932). [CrossRef]
- H. W. Lee, “Theory and application of the quantum phase-space distribution functions,” Phys. Rep. 259, 147–211 (1995). [CrossRef]
- See, for example, S. Stenholm, “The Wigner function. I. The physical interpretation,” Eur. J. Phys. 1, 244–248 (1980). [CrossRef]
- K. B. Wolf, M. A. Alonso, G. W. Forbes, “Wigner functions for Helmholtz wave fields,” J. Opt. Soc. Am. A 16, 2476–2487 (1999). [CrossRef]
- P. González-Casanova, K. B. Wolf, “Interpolation for solutions of the Helmholtz equation,” Numer. Methods Partial Differential Equations 11, 77–91 (1995). [CrossRef]
- A. Erdélyi, Asymptotic Expansions (Dover, New York, 1956), pp. 51–57.
- L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, 1st ed (Cambridge U. Press, New York, 1995), pp. 128–133.
- 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]
- See Ref. 1, p. 216.
- Yu. A. Kravtsov, Yu. A. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, Berlin, 1990), p. 23.
- Notice that, for example, the spectral function for the first measurement in Eq. (18b) can be written as μ(θ)=2 sin θρ(r, θ)=i2limL→0 ρ(x+L/2, z, θ)-ρ(x-L/2, z, θ)kL.The corresponding measurement is then achieved by a phase-locked pair of point detectors, aligned in the z direction with a separation much smaller than the wavelength, and with a phase mismatch of π. Notice that these two detectors are coupled (i.e., their relative weights including phase are fixed) and therefore do not belong to the class of composite detectors presented in Section 8. The remaining measurements in Eqs. (18b) and (18c) are achieved with similar detectors.
- See Ref. 1, pp. 221.
- A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 58, 1256–1259 (1968). [CrossRef]
- M. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979). [CrossRef]
- H. O. Bartelt, K. H. Brenner, A. W. Lohmann, “The Wigner function and its optical production,” Opt. Commun. 32, 32–37 (1980). [CrossRef]
- See Ref. 8, p. 169.

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