Diffraction, Aperture Size and Focal Length

Started Nov 13, 2017 | Questions thread
Re: Edge diffraction
1

Joe Pineapples II wrote:

Not really to answer your question, but just to show some pictures to reinforce a comment by Alan Robinson, that the diffraction is caused by the restriction of the transverse extent of the field, rather than being generated by the edge.

The reason that people talk about the edge as a source of diffraction is that mathematically it is convenient to split up the field at an edge into two parts: a "geometrical optics" field (simple ray tracing) that has a perfect shadow, and a "diffraction" field that seems to have its source at the edge. The actual field is the sum of these two parts, and neither part, on its own, is a real, physical field. The splitting is just a mathematical convenience. Anyway, here are the pictures showing field intensity for a plane wave striking an edge:

When you add those two parts together you get the actual, physical field. The sharp boundaries cancel out, but you see a nice smooth wave field with the diffraction into the shadow region:

Total (GO + diffraction) field

I hope this helps, and doesn't make things more complicated.

It probably makes more sense if you have read (or in my case, glanced at) Pauli's 1938 paper. https://journals.aps.org/pr/abstract/10.1103/PhysRev.54.924

You said : "the generalisation of Pauli's 1938 paper that is commonly referred to as the uniform theory of diffraction"

Did you extend Pauli's asymptotic expansions (equation 35) to apply for off-normal incidence, or is this simply a numerical solution to the two-dimensional scalar wave equation, with the boundary conditions used by Pauli to satisfy Maxwell's equations for a perfectly reflecting surface?

The point you correctly stated, but which may not have sunk in, is that both geometric optics and correction fields are discontinuous and non-physical. Neither is a solution to the wave equation. However, their sum is a solution of the scalar wave equation.

Like the GO wave, the correction wave has a step discontinuity in field amplitude at the boundary of the geometric shadow, which persists out to arbitrary distances from the edge. This is partially masked by plotting intensity, rather than amplitude, but the field is quite unphysical. Even if one somehow precisely matched the exact time-varying electrical and magnetic field components along the boundary, the propagated field would not look like this.

Although tempting to think of this as a scattered wave, centred on the edge of the aperture, this view does not withstand close scrutiny.  As you say, it is a mathematical convenience to derive an a analytic approximation.

Personally, I prefer the Cornu spiral visualisation of diffraction at an edge, but that is probably down to familiarity.  From a swift dip into Google:

http://mysite.du.edu/~etuttle/optics/ywave.htm#Cor

Less detailed (but more colourful): http://www.mike-willis.com/Tutorial/PF7.htm

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Alan Robinson

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