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I had a question regarding the spacing of a zone plate, which is a way to focus light without a traditional lens.

Background

Zone Plate Spacing

The spacing between the fringes is governed by the criterion that:

$$ \dfrac{(n-1)\lambda}{2} < l - f < \dfrac{n\lambda}{2} $$

where n=1,3,5,..., $f$ is the focal length, $l$ is the distance from a point on the zone plate to the focal length, and $\lambda$ is the wavelength. By solving the distances:

$$ l = \sqrt{f^2 + r_n^2} $$

Substituting this in:

$$ l-f = \dfrac{n\lambda}{2} $$ $$ \sqrt{f^2 + r_n^2} - f = \dfrac{n\lambda}{2} $$

We can solve for the beginning radius of each ring:

$$r_n^2 = n\lambda \left( f + \dfrac{n\lambda}{4} \right)$$

The thinking is that, by blocking the areas where destructive interference occurs, it is possible to create an area where only constructive interference occurs.

The Problem

If the radius refers to the boundary between the light and dark rings, it seems to include some destructive interference.

If the radius refers to the center of the dark rings, the size of the rings look different from any of the other pictures of zone plates I've seen, such as the picture above. Especially the center ring. However, this assumption only seems to include constructive interference, which makes me think it is correct.

I made a video to visualize the two approaches. https://www.youtube.com/watch?v=N4K1anaX-3Q

The Question

Does this radius $r_n$ refer to the boundary between the light and dark rings, or to the center of the dark rings?

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For those of you wondering, I believe the radius $r_n$ refers to the middle of each zone, not the boundaries.

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