JOURNAL OF THE OPTICAL SOCIETY OF AMERICA VOLUME 64, NUMBER 9 SEPTEMBER 1974

Reply to Marchand and Wolf Adriaan Walther

Worcester Polytechnic Institute, Worcester, Massachusetts 01609 (Received 29 April 1974)

Index Headings: Coherence; Radiometry.

Marchand and Wolf1 restrict their discussion to the radiometric properties of a plane source, and are concerned with only the radiance in the surface of the source itself. If we go along with these restrictions, their arguments are correct but indecisive: either de­finition can be used.

In my papers, I address a second problem as well: What is the connection between the radiance in a beam of light, and the coherence properties of the beam? This type of problem allows more-general coordinate transformations than those used by Mar­chand and Wolf. Consider, for example, a searchlight placed well to the left of the plane z= 0, and radiating light towards the half space z>0. The allowable co­ordinate transformations now include rotations around the x axis and the y axis, over angles limited only by the requirement that the z plane does not intersect the searchlight aperture. This leads to the problem described in my second paper.

I appreciate the authors' observation that the general­ized radiance, radiant emittance and, I may add, the irradiance, can on occasion be negative. The reason is that the energy flow vector —∂u/∂t· gradu can very

well point to the left in fields that propagate to the right. For instance, in the Airy disk associated with an unapodized hemispherical wave, the field amplitude u can be shown to be proportional to

in which k is the wave number and ρ is the radial coordinate. For this wave, the z component of the field gradient in the Airy disk is proportional to

In the regions where the sine function and the Bessel function have opposite signs, the energy flow vector points backwards.

1 E. W. Marchand and E. Wolf, J. Opt. Soc. Am. 64, 1273 (1974).

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