Prescription-Based Optical Modeling

An ideal model of the image formation process would integrate Maxwell's equations directly to solve for the fields in the telescope. This is quite a formidable task. Finite-element solutions of Maxwell's equations work for simple instrument geometries, but require very large amounts of computation, and do not provide a practical aproach for Hubble. Fortunately, simpler theories are available that are quite good.

Optical designers have long relied on geometric optics, or ray-tracing, to predict performance of optical instruments. Geometric optics makes the assumption that the wavelength of the light is infinitesimally small compared to the diameter of the beam. This turns out to be a very good assumption while the beam is away from a focus. Geometric optics provides an accurate means of determining both the wavefront phase and vignetting patterns in the expanded beam of a telescope, even in highly aberrated systems. It does not require simplification of the geometry of the instrument, and works directly from the optical prescription. It enables accurate computation of the effects of figure errors, induced aberrations and misalignments. Ray tracing alone is not capable of reproducing images in the far field with any accuracy, however.

For modeling images and highly diffracted beams (such as laser cavity modes) physical optics theories offer an alternative. Fresnel physical optics makes a different set of assumptions, such as time-invariance and slowly-varying phase, to provide a simplified set of propagation integrals that can be solved rapidly using Fourier transform techniques. As commonly implemented, physical optics codes unfold an optical system into a simplified linear sequence of ``thin lenses,'' which collapse the phase effects of optical elements into a plane. Single-plane PO models then lump all of the phase effects onto a single plane at the exit pupil of the telescope; images are created by propagating the exit pupil irradiance to the far field, at the detector. Multiple-plane physical optics models propagate from planar element to planar element in the near field, and then to the far field at the detector. Neither approach allows for accurate prediction of induced aberrations without phase retrieval. Multiple-plane models do allow for multiple planes of obscuration, which can be important in capturing details of the point spread function.

We have implemented a hybrid approach, which exploits the advantages of both geometric and physical optics. We use ray-tracing to track the phase and determine vignetting in the expanded beam (Fig. 1). This information is used

to drive Fresnel near- and far-field diffraction propagators, which compute the beam complex amplitude and, ultimately, the image or PSF. This approach accurately determines the effects of vignetting, aberrations and induced aberrations, based on the actual optical instrument geometry and prescription. It also enables us to model multiple planes of obscuration and other important diffraction effects.

This hybrid approach is realized in the Controlled Optics Modeling Package (COMP) (Lee et al. 1990), which is available from the NASA COSMIC software library (COSMIC can be reached at (706)542-3265, or by e-mail at service@cossak.cosmic.uga.edu). COMP is a full-featured optical modeling code that can be used either as a stand-alone application, or, through a subroutine interface, to provide optics functions in-line to other programs (Table 1). The subroutine version is called SCOMP. The point-spread function generator we are writing incorporates SCOMP, providing a simplified user interface and reduced function set specialized to the HST cameras.

COMP associates a physical optics complex-amplitude matrix with a geometric optics ray grid (Fig. 1), so that each ray acts as a phase and obscuration probe for a corresponding diffraction cell in the beam. Ray-tracing determines the wavefront aberrations and the vignetting effects of obscurations based on the instrument physical configuration and field angle. Phase errors due to figure errors, misalignments or other optical aberrations are introduced into the modeled system in the same way they occur in the actual system. Fresnel near- and far-field diffraction algorithms use the optical path length information to propagate the complex amplitude matrix through the system to the focal plane to generate the PSFs. Multiple obscuring surfaces can be specified. COMP incorporates various features to ensure correct sampling for the diffraction calculations and to avoid aliasing. Polarization effects can be included using polarization ray-trace and vector diffraction functions.

Using COMP to generate PSFs for image restoration allows us to avoid some of the limitations of a pure physical optics approach. Prescription-based PSF generation uses a fundamental parameterization that is capable of predicting accurate PSFs across the entire operational range of the instrument. Equivalent performance using a pure physical optics approach would probably require iterative data-matching or prescription retrieval at each field point. The ability to accurately predict PSFs is especially crucial for restoration of extended-object images, where objects for matching are not available.