Mathematical Methodology

The observed PSF is proportional to the power spectrum of the plane wave incident at the entrance pupil of the telescope (Goodman 1968). This can be written as:

where is the phase of the plane wave incident on the entrance pupil of the telescope and is a proportionality constant. Eq. (1) assumes that the effect of the telescope assembly can be expressed by a single equation and, hence, is an approximation in itself. However, Eq. (1) expresses the PSF as a function of the wave front phase at the entrance pupil. Hence, the observed PSF can be useful in reconstructing the wavefront at the entrance pupil of the telescope.

Zernike polynomials and their variants are known for better modeling of different aberrations of the phase function ( Zernike 1935). Analytical representation of the phase in terms of the Zernike polynomials is given by:

where is the Zernike coefficient corresponding to the Zernike polynomial and is the number of Zernike polynomials that are necessary to represent the analytical phase. Hence, the problem of analytic PSF generation reduces to estimating the Zernike polynomial coefficients that are necessary to represent the entrance pupil wave front.

For phase retrieval we used the iterative transform algorithm (ITA) of Fienup (1982), who based much of this ITA on the Gerchberg and Saxton (1972) algorithm, and a variation of the ITA. These two algorithms are shown in Fig. 1.

For Algorithm 1, which corresponds to annular Zernike polynomials, we used the telescope geometrical information such as the mirror sizes, mirror pads, spiders and the obscuration geometry in the pupil plane and the observed PSF in the Fourier transform domain. Algorithm 1 stagnated after a few thousand iterations. We used a robust QR algorithm (Golub and Van Loan 1992) to analytically represent the phase using the annular Zernike polynomials. When compared with the observed PSF, we noted that the simulated PSF did not have detailed structures such as the clover and pronounced wings that are so essential for the image restoration.

In an attempt to improve the phase retrieval results, we used Algorithm 2, which corresponds to the classical radial Zernike polynomials. Unlike Algorithm 1, Algorithm 2 does not impose any constraint in the pupil domain. This is equivalent to assuming that the wave front is continuous at the entrance pupil. We used a circular mask in the pupil domain region of interest and retrieved the phase. This algorithm also stagnated after a few thousand iterations. Since the classical radial Zernike polynomials are orthogonal over the circular pupil domain, the corresponding Zernike coefficients are obtained using a simple inner product. PSFs generated with Algorithm 2 led to better representation of the inner core structures than Algorithm 1, but still did not compare favorably to the observed PSF for better image restoration.

Fig. 2 presents the schematic diagram of the process involved in the PSF simulation. All the functions shown in the diagram are implemented on a massively parallel computer - the MasPar with 16,384 processors. Zernike polynomials are generated ``on-line'' during the computation and are not stored.