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I read the wiki article about angular resolution, but I struggle to understand the image sensors' role in telescopes. Will better image sensors can help go beyond the diffraction point? If not, how to find the largest pixel size of an image sensor that will not prevent the telescope from operating at the diffraction level?

Ilya Gazman
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  • Are you considering fun tricks like subpixel resolution like tricks where one makes assumptions about the scene which permit processing into a higher resolution product? Or are you just looking for equations about the size of the Airy Disk? – Cort Ammon Jan 08 '21 at 16:09
  • @CortAmmon I am looking for the equations, and I am always in for fun :). Ultimately I want to understand the connection to the image sensor. – Ilya Gazman Jan 08 '21 at 16:14
  • Does this answer your question? If not, what information do you miss? – A. P. Jan 09 '21 at 12:05
  • @A.P. This is what I am struggling to understand, the image sensor impact on the angular resolution. – Ilya Gazman Jan 09 '21 at 13:31
  • As long as the pixels of the detector are smaller than the point spread function of the imaging system they don't have an influence at all. – A. P. Jan 09 '21 at 15:11
  • @A.P. So you are saying that theoretically, you can rich any image resolution without being restricted by diffraction limit as long as you can generate very small pixels in the image sensor? Does it mean that image sensor affects the angular resolution of telescopes? – Ilya Gazman Jan 09 '21 at 15:43
  • No, it means that the Rayleigh diffraction limit is the best you can do. And to fully make use of it the pixel size must be small enough to resolve the point spread function. To make answering your question easier, could you add a few more details? It's easier to be helpful if the question is like "I've read this, but don't understand that." rather than asking for a full tutorial. – A. P. Jan 09 '21 at 15:53
  • @A.P. ok, I updated the question – Ilya Gazman Jan 09 '21 at 17:58

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The best possible resolution* which can be reached is given by the Rayleigh criterion $$\theta = 1.22 \frac{\lambda}{D} \text{,}$$ where $\theta$ is the angular resolution, $\lambda$ the wavelength of the used light and $D$ the diameter of the collecting lens. On the photodetector the image of the point spread function will have a diameter of $$d = \frac{\lambda}{2 \, \text{NA}}$$ with $\text{NA}$ being the numerical aperture of the light cone hitting the detector. If there are no abberations the point spread function for a circular aperture looks like this:

The pixel size of the detector should be smaller than the central spot, otherwise you lose resolution.

Imagine pixels which are 5 times larger than the points spread function. You would see 1 pixel with some intensity on it, but you can't tell where on the pixel it impinges.

Very small pixels don't help you improving the resolution. Imagine two point-like objects, each one resulting in a point-spread function on the detector:

The minimum distance at which you can tell them apart doesn't depend on how many pixels you use. For further information see Could Legolas actually see that far? and answers therein.

* Putting aside superresolution tricks, which usually have restrictions or requirements.

A. P.
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  • Can you please explain how to find the size of the central spot in PSF? For example, in the Hubble telescope, the pixel size is 10 micrometers. I want to check if it's indeed smaller than the central spot. – Ilya Gazman Jan 10 '21 at 00:10
  • @IlyaGazman Do you mean I should elaborate on the mentioned $\text{NA}$? – A. P. Jan 10 '21 at 00:13
  • Yes, please, how do I get to the calculation of the pixel size/central spot? – Ilya Gazman Jan 10 '21 at 00:25
  • @IlyaGazman Ok, I assume you don't know the divergence of the light cone between the last optical element and the detector, do you? If not, what do you know? – A. P. Jan 10 '21 at 00:30
  • I just saw that Wikipedia says that its $\text{NA}$ is $1 / 48$. So the main lobe of the PSF ideally would have a diameter of $24 \lambda$, which is 13 µm for green light. But it turns out that the main mirror has some spherical abberations. – A. P. Jan 10 '21 at 00:47
  • Awesome! I think I am starting to understand this now. My next question is, would it be possible to upgrade the Hubble telescope to increase NA? What is the upper limit that NA can get? – Ilya Gazman Jan 10 '21 at 01:37
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    @IlyaGazman Glad I could help! Feel free to ask this in a separate question, but keep in mind, that for increasing the resolution one shouldn't increase the $\text{NA}$ of the beam hitting the detector, as it would descrease the size of the PSF beyond the smallest technically possible pixel size at some point. Instead one needs to increase the size of the collecting lens (or mirror in the case of Hubble). – A. P. Jan 10 '21 at 10:53