# Calculating diffraction-limited resolution for a lens setup

Supposed a lens arrangement is prepared where light from an object is collimated, focused and recollimated etc. before entering a CCD array. Given that we can calculate the diffraction-limited resolution for each lens in the system, how do we measure the diffraction limited resolution for the whole setup?

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I want to briefly clarify what exactly is meant when we talk about being "diffraction limited." As light is focused, it will reach some minimum spot size before it begins to expand again. The size of this spot depends on how much the light beam is distorted.

A perfectly collimated beam (with perfectly planar wavefront) passing through a perfect lens would come out of the lens with perfectly spherical wavefront, and all of the rays in the beam would be converging to a single point. In this case, the spot size is determined solely by the angle occupied by the converging cone of light*. This is what we call "diffraction limited."

If the beam is abberated, for example by a poorly manufactured lens, then the beam will not have perfectly spherical converging wavefronts, and the resulting focal spot will be spread out over a larger area. The magnitude of these abberations is what determines the resolution of an optical system when it is not diffraction limited.

The size of the diffraction limited spot is a function of the f-number at the image plane. So, if you know the beam diameter after the last lens element, and the back focal distance, you can compute the diffraction limited spot size just like you would for any other lens.

*this is the case only assuming that the beam is always the same shape. In practice most beams are circular, so all we need to worry about is its diameter. If the beam is a different shape, then its diffraction limited spot size (and shape!) will change.

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I'm confused why diffraction by other lens in the system does not affect the resolution of the image. Should the presence of additional lens not compound the effects of diffraction? – flamearchon Jun 16 '11 at 6:58
@flamearchon: "Diffraction" is simply a term for the way in which light (and waves in general) propagates. We tend to think of it as a bad thing, as being that which limits our optical systems; but really light is always diffracting. When light goes through a lens, the lens alters the phase of the optical field such that it it comes to a focus as it diffracts beyond the lens. As such, diffraction cannot be thought of like noise which builds up in a signal as it goes through a chain of amplifiers. It is not some sort of cumulative effect that builds up and cannot be removed. – Colin K Jun 16 '11 at 15:13
Of course, if there is a lens in the system which distorts the beam, those distortions carry through the system and affect the focal spot, but if you are asking about the diffraction limited spot, that means you are talking about the situation where none of those aberrations are significant. That is the very meaning of "diffraction limited" – Colin K Jun 16 '11 at 15:16
I want to add a couple things to this nice answer: First, "the f-number at the image plane" is NOT the same as the f-number of the last lens of the system. I don't want Colin to be misinterpreted! This is just another way to discuss the "angle of the converging cone of light". Second, for the exact relation that Colin alluded to between convergence/divergence angle and diffraction-limited spot size (a.k.a. "beam waste"), you can use the Gaussian beam approximation. :-) – Steve B Jul 30 '11 at 2:01

In the spot diagrams of optic designs the ray aberrations are always compared to the size of diffraction limited spot. In order to do this you have to find the smallest aperture in your system. This can be a deliberately placed mechanical aperture but it can also be the circumference of a lens.

I copied the following spot diagram from http://www.astronomy.net/articles/17/. The black ring is the diffraction limited spot size (3.3 $\mu$m if you convert from inch) the red dots are the spot with ray aberrations. You can see a strong Coma with $1\mu$m spread in x.

I read a nice discussion about this topic a while ago but can't remember which book it was in. I tried to implement it in a raytracer. It involved an iteration over all apertures in the optical train. One had to trace all apertures into the image and chose which one produced the smallest opening angle.

I think any decent book on lens design should contain a chapter on the topic. See for example chapter 6 in Warren J. Smith: "Modern optical engineering: the design of optical systems".

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