# Do point spread functions from large single telescopes using adaptive optics still look like Airy functions for narrow-band filters?

...the possibility of attaining near diffraction-limited images using adaptive optics.

The most familliar example of near-diffraction-limited optics is the Hubble Space telescope which has been operating for about 30 years above Earth's atmosphere. By extensive analysis of the resulting images of a single stars at several out-of-focus positions using phase retrieval it has been possible to determine a static phase error map for the system and to use this to generate simulated point spread functions which can then be used to deconvolute intensity images, sharpening them in a science-based way to see dim features nearby much brighter stars for example.

The images below are from Krist, Hook & Stoehr (2011) 20 years of Hubble Space Telescope optical modeling using Tiny Tim (paywalled, also researchgate.

I should also cite J. D. Rhodes et al (2007) The Stability of the Point-Spread Function of the Advanced Camera for Surveys on the Hubble Space Telescope and Implications for Weak Gravitational Lensing (also arXiv and Caltech) which discussed (among other things) the interaction between drizzling and PSF models.

Question: Do point spread functions from large single1 telescopes using adaptive optics still look like Airy functions for narrow-band filters?

While the phase surfaces of the Hubble space telescope are relatively static, the whole point of adaptive optics is to dynamically modulate the phase map of the aperture to compensate for wavefront distortion in the atmosphere. Since this can not be done perfectly, the resulting point spread function may differ from the familiar diffraction pattern of a circular aperture occluded by a central obstruction and spider vanes.

For a narrow-band filter, what do they look like? Do they still show an Airy disk?

Here's a cropped, monochromed, ROI:

Further stretched in contrast and size:

Figure 2. Map of the combined primary and secondary mirror surface errors left from polishing obtained using phase retrieval on highly defocused star images. Shown between ±30 nm surface error. The HST and WFPC2 obscuration patterns are superposed.

Figure 3. Spherically-aberrated observed and simulated PSFs from the Faint Object Camera using (top) filter F253M (253 nm central wavelength) and (bottom) filter F486N. The models were generated using the old (pre-launch) optical surface error maps, the new maps from phase retrieval, and no surface error maps. Each PSF is approximately 6 arcseconds in diameter. The three lobes are shadows from the primary mirror restraining pads.

Figure 9. (Left) Long exposure image of the XZ Tauri binary system using WFPC2 PC and filter F675W. An outflow from the system is seen extending to the upper right, but the glare of the PSFs interfere with details closer to the stars. (Right) The image after subtraction of two Tiny Tim PSFs matched to the stars. The diffraction spike residuals have been masked.

• – uhoh
Jun 24, 2021 at 1:06
• I'm a bit confused as to why there's so much discussion of the HST PSF when your question is nominally about ground-based AO... Jun 24, 2021 at 20:27
• @PeterErwin 2nd question first: I link to What is the cause of all of these sharp, concentric rings around bright stars in this HST image? Scaling with $\lambda/d$ diffraction is chromatic so a narrow band filter can show a dozen or more distinct rings. Just like soap bubbles; if you view them through a narrow band filter you can see a larger number of interference bands in much thicker films, the same reason we use a low pressure mercury or sodium lamp to view fringes with an optical flat.
– uhoh
Jun 24, 2021 at 22:26
• @PeterErwin 1st question: so one reason I introduce what the PSF of Hubble looks like is just that; to show why the narrow band part of my question is important. I also do it to provide a contrast to whatever appears in answers; I explained that the HST's phase error map across the aperture is mostly static (doesn't change with time, but varies with position and drizzling) while for an adaptive optics system on the ground the phase error map is of course dynamic. Thus I set the stage for what I expect to be a very different time-integrated PSF. Maybe I'm wrong; I can't wait to find out!
– uhoh
Jun 24, 2021 at 22:31
• Ah, OK, I understand now. Thank you for the clear explanations. (I have to admit that my extensive experience with HST images has virtually always been with broad-band imaging, and so I was never actually made to see the real difference between narrow-band PSFs and broad-band PSFs. So I have learned something new -- thanks!) Jun 25, 2021 at 9:49

Yes, narrow-band images taken with adaptive optics on ground-based telescopes produce point-spread functions resembling Airy disks.

To answer this, I went fishing for some data, and randomly caught a great fish on the first try!

I went to the Keck Observatory Archive, and selected the NIRC2 camera only, specifying a 1-month date range far enough in the past that there should be plenty of publicly-available data. I scrolled through the search results until I found a narrow-band filter, "H2O_ice." The wavlength range listed for this filter in the table is 2.99-3.14 µm, which is a 5% bandpass, commonly considered narrow-band.

Here is what an image (N2.20150112.58609.fits) in this filter looks like with a square-root stretch to show the diffraction pattern:

I can make out the first and second Airy rings. They are lumpy because the AO correction is not perfect (we know it is an AO image because AODMSTAT and other keywords say the AO loops are closed), and the rings are hexagonal rather than circular due to the shape of the primary mirror segments.

It makes sense that narrow-band AO PSFs should resemble Airy functions. Here are the first 3 figures in the current Wiki article on Airy disk:

Fig. 1 - Usually basic calculations like these are for a single wavelength of light, and you do see the rings.

Fig. 2 visually shows the $$\lambda/D$$ dependence of the Airy function on wavelength. If the filter is broad enough, this chromatic dispersion would cause the ring to blend into the core, and eventually the photometric curve would look more like a Gaussian. So it's actually BROAD-band filters that show more poorly defined Airy rings than narrow-band.

You might be able to see this effect in Fig. 3 of the original question. The two rows are for filters F253M (top) and F486N (bottom). For HST filters "M" means medium bandwidth and "N" means narrow band. It looks like there is more fine structure in the bottom row, compared to a more blurred radial distribution in the top row, but these PSFs are very complex, and the image is stretched to show the outer halo of the PSF. You can't see the first Airy ring very well at this stretch and magnification.

Fig. 3 - You can clearly see the diffraction pattern from laser illumination, the ultimate real-life narrow-band light!

• Beautiful! I see, I guess "square-root stretch" makes the plots like absolute value of amplitude.
– uhoh
Feb 14, 2022 at 2:37
• Oh wow, I never made that connection but yes. I just thought of it as a way of showing the low-level variation and squashing the high-level variation. Feb 14, 2022 at 2:47