Chapter 4 presented a simplistic introduction to some of the ideas behind diffraction and the Huygens-Fresnel principle. For the purpose of discussion, we assumed every point on the wavefront re-radiates the wave. That convenient fiction has some difficulties, however. If each point is allowed to radiate in every direction equally, the aperture would arbitrarily reflect back into space some of the energy that reached it.

Reflection at the aperture is not observed, nor is it seen at any point on the wavefront. Waves don't suddenly reverse direction unless they encounter a change in the medium. The wave sum also doesn't deliver the correct answer when integrated over a situation with no aperture, just a source of light and a receiver. The integral is over a complete sphere, and the radiators at right angles between the source and receiver have too strong a weight.

Fresnel realized that these were impediments, so he made certain approximations. His model was not derived from first principles, which probably helps explain the intense scrutiny by such worthies as Poisson. In the early 19th century, even the physical process that produces light was not yet understood.

Later research of Gustav Kirchhoff produced a version of the Huygens-Fresnel theory derived from first principles. His result, called the Fresnel-Kirchhoff diffraction formula, no longer requires that the rearward direction be ignored, though it demands a few conditions:

1. Light is modeled as a scalar wave. Polarization is not included. The model is oblivious to the vector nature of light.

2. The field values near the aperture are the same as they would be in the absence of the aperture (weighted by a simple trigonometric function).

3. The distance from the aperture is sufficient to ensure that no bound or evanescent fields are present.

One consequence of the second condition is the simultaneous specification of both the field values on the aperture in addition to their derivatives. The time-harmonic form of the wave equation1 is a second-order differential equation for which the normal method of solution is to set either the value of the field on the boundaries or the derivative of that field. It is unusual that Kirchhoff gave both.

With its overspecified boundaries, we must regard Kirchhoffs solution as an approximation. This approximation will not affect results if the aperture does not have a lot of structure. We may rightly suspect that the Fresnel-Kirchhoff equation will be less accurate for devices like high-resolution diffraction gratings. For gently sloped optics, with most of the area many wavelengths from all edges, the approximation does not cause much trouble. More detail concerning these criticisms is available in Baker and Copson (1950).

A modified form of the Fresnel-Kirchhoff formula (using variables de-

1 This formulation is called the Helmholtz equation.

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