# Which photons pass through a circular annulus?

Passing light through a circular sieve:
Well, actually, let’s think about radar or microwaves with a wavelength of order a centimeter or two, so you can tailor your aperture, say by etching a silver screen on glass. If you have a reflective metal screen, and you cut a long narrow rectangle in it, it will pass (some) photons of the properly oriented linear polarization of wavelength shorter than the length of the rectangle.
What if you cut a narrow circular annulus into your screen? Would it pass circularly polarized radiation of the proper wavelength? Bonus Points: What about an elliptical annulus? Please ignore photons of wavelength shorter than or comparable to the narrow dimension of the slit.

You can make beams of light that have orbital as well spin angular momentum and they can go through an annular aperture.

So circular versus linear isn't enough. Linear has the phase advance orthogonal to the advancement with the polarization at a fixed angle. Circular has the polarization rotate but with an annular filter you can give the wavefront a twist where it can have an integer number of twists as it goes around the angle so you allow orbital angular momentum.

Electrons can do the same thing.

One way to produce such beams is via holography. A defraction grating with a point that has multiple lines coming from the top to be one line going down can produce a beam with orbital angular momentum with the amount related to how many lines came into the point. So the lines look like a lower case l or the number 1 but then there is a capital Y in the middle or maybe a Y and an I and so forth.

As for the polarization being linear in a vertical or a horizontal slit. If you had a rope going through a vertical slit it might be obvious that it is easier to shake it up and down. But an electric field is located at all the points along the lone it propagates and the vector points in a direction. The size of the field is measured in V/m and so isn't directly related to how tall the vertical slit it. So unlike the rope, the amplitude of the vertical electric waves isn't directly limited by the height of the vertical slit. Its limited by things like dielectric breakdown of the air and vacuum polarization and whether it would destroy the material you hope will block it.

But there is a general difficulty of passing a wavelength through a slit that is smaller in width than the wavelength. This is more a diffraction effect and isn't obvious at all.

But you asked about polarization. Of s material can pass both vertical and horizontal polarization through, and pass them both through with the same phase then it passes both kinds of circular polarization as well.

• Points for considering orbital angular momentum – Jim Graber Aug 28 '15 at 14:55

I am not totally sure of this answer, which is why I asked the question.
However, I think the answer is that only relatively short wavelengths can pass through an annular aperture.

Specifically, I think that if the outer radius of the annulus is R and the width is W, where W << R, the maximum wavelength that passes through is approximately 2R times the square root of 2W/R, or more exactly 2R Sin(ArcCos(1-W/R)). No Wavelengths comparable to R pass through this annular filter unless W is itself comparable to R.