# Constructive interference from a reflector

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 ).

• If both of the questions I asked require entirely separate answers, then I'd be happy to ask them as separate questions. I asked them together because I assumed that the answer for one would probably rule out or heavily depend on the other. – CoilKid Dec 19 '17 at 19:12
• For reflectors check the Schwarzschild two-mirror system and similar structures; for lenses see the Holt-Mayer construction having a circle focus. – hyportnex Dec 19 '17 at 19:55
• Does this answer help at all? – Floris Dec 19 '17 at 20:08
• @Floris So to confirm: that should work in reverse too, and if you design a parabolic reflector of that form then you would have constructive interference for all reflected rays. Yes? If I understood correctly, then you've perfectly answered my questions. – CoilKid Dec 19 '17 at 20:40
• If the curve of the reflector is not a parabola, then it's not a parabolic reflector. – The Photon Dec 19 '17 at 20:54