Examples

An example of how phase retrieval can improve computed PSFs is shown in Fig. 1. At this wavelength in the middle of the optical

band, the Tiny Tim PSF is in reasonably good qualitative agreement with the observations. However, there are some discrepancies between the observed PSF and the Tiny Tim PSF; note especially that the shape of the PSF core is not well modeled and the position and brightness of the bright ring are not exactly correct. The most likely explanation is that the mirror map and aberration model used by Tiny Tim is not accurate enough. Any discrepancies in the optical model become more obvious at shorter wavelengths.

The simplest phase retrieval method is to model the wavefront error using a low order polynomial in . For HST the appropriate polynomials are the Zernike polynomials given in the HST Optical Telescope Assembly Instrument Handbook (Burrows 1990). The various Zernike polynomial coefficients correspond to wavefront tilt (which shifts the position of the object), focus, spherical aberration, astigmatism, and so on.

The lower left panel of Fig. 1 shows the result of adjusting these coefficients to fit the observed PSF. A conjugate gradient optimization method was used to search for the set of Zernike coefficients that maximizes the Poisson log likelihood. The agreement with the observations is improved, though there are still differences. For example, note the bright knot in the observed PSF where the tendril trailing off towards 8 o'clock crosses the bright ring. Also note that the outer dark ring, seen easily in the Zernike PSF, is hardly visible at all in the observed PSF. Other differences are readily visible when the PSFs are blinked on an image display.

The agreement with the data can be improved further by solving for a map of wavefront errors across the pupil using the method described above. The results of 20 iterations of such a method are shown in the lower right panel of Fig. 1. Note the improvement in features such as the strength and position of the ``tendrils'' that extend downward from the center of the PSF.

Fig. 2 shows the wavefront error maps derived by the two

phase retrieval methods.