NAOMI Performance

The PSF (point spread function) yielded by an adaptive-optics system depends on:

• the natural seeing
• the magnitude of the guide star (NGS = natural guide star, or LGS = laser guide star)
• the angular distance between guide star and science target
• the wavelength of observation

and, for LGS observations:

the magnitude of the tip-tilt guide star, and the angular distance from the science target

The PSF also depends on AO-system parameters such as the exposure time on the wavefront sensor, the number of elements into which the wavefront is divided, and the loop gain.

AO performance can be characterised in terms of achieved FWHM, FWHE (full width including half the energy) or Strehl (ratio between the peak heights of corrected and diffraction-limited PSFs). In the optical, the NAOMI-corrected PSFs are near-gaussian in form, so FWHM is a particularly useful metric. It can be misleading if the PSF has a narrow core (e.g. in the IR) superimposed on a plateau of uncorrected seeing, with the latter including most of the light.

Measurements and predictions of NAOMI's performance are given below. The first section deals with the PSF at the position of the guide star, the second section with degradation of PSF as a function of distance from the guide star.

Performance with the laser guide star, GLAS, is still being characterised, but one may reasonably assume that it will be similar to that obtained with a bright natural guide star (mag ~ 9.5).
When observing with the laser guide star, a natural guide star is still required, to effect the tip-tilt correction, but this can be much fainter, V < 17, and up to 60 arcsec from the science target.

Below are summarised measured on-sky corrected FWHM in different values of natural seeing (median seeing on La Palma is 0.7 arcsec in V band). The Z J H K measurements are from INGRID, the remainder from OASIS (with a 50% dichroic in the beam).

Natural seeing 0.7 arcsec (in the relevant band):

Guide star mag	B/V (0.5 mic)	R (0.65 mic)	I (0.82 mic)	Z (1.05 mic)	J (1.25 mic)	H (1.65 mic)	Ks (2.15 mic)
<= 8	0.3	0.3	0.3	0.3	0.25	0.15	0.15
9
10	0.5	0.4	0.3
11	0.4	0.4	0.35		0.25	0.2	0.2
12	0.5	0.5	0.5		0.3	0.3	0.3
13					0.4	0.4	0.4
14	0.5	0.6	0.5		0.4	0.4	0.4
15	0.5		0.5		0.4	0.4	0.4
16			0.5

Natural seeing 0.5 arcsec (in the relevant band):

Guide star mag	B/V (0.5 mic)	R (0.65 mic)	I (0.82 mic)	Z (1.05 mic)	J (1.25 mic)	H (1.65 mic)	Ks (2.15 mic)
<= 8	0.2	0.2	0.15		0.2	0.15	0.15
9	0.25	0.2	0.2
10
11	0.40		0.30			0.2	0.2
12						0.2	0.2
13						0.3	0.25
14						0.4
15						0.4

For comparison, the theoretical WHT diffraction limits in J, H and Ks bands correspond to FWHM 0.07, 0.08 and 0.12 arcsec.

NAOMI was not designed to deliver useful performance in seeing > 1.2 arcsec, and there has been no systematic characterisation in such seeing, but it does deliver some improvement e.g. PSF 0.4 arcsec in H band, in natural seeing ~ 1.5 arcsec, in Sep 2006.

To convert seeing in J, H and K bands to seeing in V band, multiply by 1.2, 1.2 and 1.3 respectively.

Cloud doesn't necessarily preclude good AO correction - on one occasion, a bright star was observed through ~ 3 mags of cloud, but the FWHM in Ks band was still improved from 0.6 -> 0.2 arcsec.

The above measurements were all made using an 8*8 lenslet array in the NGS Shack-Hartmann wavefront sensor. A 4*4 lenslet is available, and should give higher S:N for faint stars, V > 14, at the expense of poorer spatial sampling, but in practice this mode has never delivered a useful performance improvement on-sky. Each 1-mag increase in the guide-star limit corresponds to a factor of 3 increase in sky coverage.

Theoretical predictions of the Strehl delivered by NAOMI (in the standard 8*8-lenslet mode), as a function of seeing, band and guid-star mag, have been provided by Ron Humphries and Richard Wilson, Durham:

	J			H			K
Guide-star mag	1.0"	0.7"	0.5"	1.0"	0.7"	0.5"	1.0"	0.7"	0.5"
11	0.15	0.29	0.38	0.29	0.45	0.54	0.46	0.62	0.69
12	0.08	0.21	0.31	0.19	0.36	0.46	0.34	0.51	0.61
13	0.03	0.12	0.22	0.08	0.24	0.35	0.19	0.38	0.50
14	0.01	0.04	0.11	0.02	0.10	0.21	0.06	0.21	0.35

Seeing 1.0, 0.7 and 0.5 arcsec correspond to Fried parameter r₀ = 10, 15 and 20 cm respectively.

For Strehl ratios of more than a few tenths, the bulk of the light will be concentrated in a diffraction-limited core (FWHM ~ 0.15 arcsec in K), with the remainder of the light distributed over the (uncorrected) natural-seeing disk. With poorer AO correction, there will still be a diffraction-limited core, but it will contain only a small fraction of the light. The delivered FWHM is typically close to either the diffraction limit or the tip-tilt-corrected seeing diameter.

The above performance predictions do not take into account wavefront errors induced by the telescope on spatial scales too small (< 50 cm) to be sampled by the spots of the wavefront sensor, although experience suggests that these are not large.

Even with no atmosphere, and with perfect optical surfaces, NAOMI would not deliver a neat, circularly-symmetrical, diffraction-limited PSF. The segmentation of the primary mirror, stopping down of pupil, and support vanes for central-obscuration stops all degrade the PSF.

Strehl ratio is predicted to fall to half its on-axis value at distances 13, 17 and 23 arcsec from the guide star in J, H and K respectively (the size of the isoplanatic patch scales as wavelength^6/5). That is, for a given degree of correction, the guide star needs to be closer to the target for J and H imaging than for K. The sky coverage, i.e. the fraction of sky falling within the anisoplanatic patch of a bright star, depends on required Strehl, and on galactic latitude. In K band, for delivered Strehl 0.3, predicted sky coverage is ~ 10% at galactic latitude 30 deg.

So far, few measurements have been made of the deterioration of PSF with distance from the guide star. These few measurements indicate that the isoplanatic patch is larger than given above, perhaps because the seeing is dominated by turbulent layers at a lower altitude than expected.

Predictions of the decline in delivered Strehl with separation between target and guide star, for J, H and K bands are shown below. The 3 curves are for median and for 25 and 75 percentile seeing conditions, with a bright guide star, V ~ 12. These predictions were provided by Richard Wilson and the Durham team.

Below are the predicted variation of FWHM (median, and 25 and 75 percentile) with separation from the guide star:

and of 50%-included-energy diameter (FWHE):

The horizontal dotted lines in the figures indicate the median uncorrected values. Median predicted FWHM is < 0.2 arcsec for guide-star distance = 30, 40 and 50 arcsec in J, H and K. At the same distances, the 50% encircled-energy diameters are about 0.4 arscec in each band. I.e. the FWHM can be close to the diffraction limit even when there is < 50% of the total energy in the central peak.