Constructive interference from a reflector

Question

Would it be possible to construct a reflector such that, for a given wavelength (perhaps part of the microwave spectrum?), the reflected wave interferes constructively with itself?

Ideally, this would work for any two arbitrary reflection points that are "close" to one another.

My question has two very interrelated ideas:

• Could such a shape even exist?
• How would you go about finding the defining curve?

Notes:

I initially assumed a parabolic reflector because one nicety of this thought experiment was to produce a collimated beam in addition to the interference pattern, and I remembered that parabolic dishes are widely used to roughly focus light into beams. In actuality I realize that if such an shape does exist, it probably wouldn't have such a nice defining equation.

I tried considering the problem from the $\Delta L =m\lambda$ standpoint, in two dimensions first to simplify things.

If the emitter is considered to be a point source at the focus of the parabola $y=ax^2$ then for any two rays with reflection points $(x,ax^2)$ and $(x+h,a(x+h)^2)$ then the path length difference is $\Delta L=\sqrt{x^2+(ax^2-\frac{1}{4a})^2}-\sqrt{(x+h)^2+(a(x+h)^2-\frac{a}{4a})^2}+a((x+h)^2-x^2)$, presuming I did my math right.

If I understand what I have correctly, it will give me the equation of the reflector where rays separated in the x-direction by h units interfere constructively. Ideally, however, there's a shape that will reflect a given wavelength with constructive interference for every h (within realistic constraints of course. Infinitely large reflectors can't exist, etc ).

Answer 1

One way to look at optics is that the reason the paraboloid focuses a plane wave onto a point is that there is construtive interference for reflections converging on that point and destructive interference for beams going any other direction. So the parabolic reflector is already using interference to do its job.

Beyond that, you could consider doing things like etching patterns on an optical reflector or making your microwave reflector from patterns of wire that produce some further diffractive effect in the reflected beam. You could also probably make a flat mirror that has the converging properties of a parabolic one, at least within some fairly narrow frequency band (and with some leakage into 2nd-order beams).