Which photons pass through a circular annulus?

[jimgraber]

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.

 

[jfizzix]

I couldn't say about the polarization of the light, but if you have light passing through a ring shaped aperture, it will look like a cylindrical Bessel function in the far field. To see what the diffraction pattern due to an elliptical ring aperture would look like, you could do a sort of change of coordinates from the circular case. That circular Bessel function would be squashed in the direction parallel to the long axis of the ellipse, and stretched in the perpendicular direction.

 

[jfizzix]

A cylindrical Bessel function would look like a circular wave, with a brightest spot in the middle, and rings outward slowly dying out.

 

[jimgraber]

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.