A filter function has been derived as a means of optimally weighting the wavefront estimates obtained in image-based phase retrieval performed at multiple points distributed over the field of view of a telescope or other optical system. When the data obtained in wavefront sensing and, more specifically, image-based phase retrieval, are used for controlling the shape of a deformable mirror or other optic used to correct the wavefront, the control law obtained by use of the filter function gives a more balanced optical performance over the field of view than does a wavefront- control law obtained by use of a wavefront estimate obtained from a single point in the field of view. (The terms "wavefront sensing," "image-based," and "phase retrieval" are defined in the immediately preceding article.)

In a conventional approach to sensing and control of wavefronts, optical phase errors are estimated from the image of a single star or equivalent point source of light at a specific single location on a focal-plane image sensor. In effect, a wavefront control law is derived from a small area surrounding a single field point and is subsequently used to correct the performance of the optical system over the entire field of view. The disadvantage of this approach is that the performance of the system at other field points can suffer additional degradation because the wavefront information obtainable at those field points can differ from that obtained at the chosen field point.

Mean Values of Corrected Root-Mean-Square Wavefront Error were computed for several field points and fitted with straight lines to show that errors can be reduced and/or distributed more evenly when multiple field points and the filter function are used.
A mathematically complete description of the filter function and its derivation would exceed the space available for this article; it must suffice to summarize The derivation of the filter function begins with the concept of an anisoplanatic function, defined as a phase function representative of the degree to which imaging performance varies over the field of view. The wavefront phase at a given field point is assumed to be given by the sum of the isoplanatic and anisoplanatic contributions. It is further assumed that an estimate of an isoplanatic phase function at a given field point can be modeled as a sum, over all other field points, of the convolutions of the filter function with the wavefront phase. Then the filter-function problem is formulated as one of choosing the filter coefficients to minimize the sum, over all field points, of the squares of the differences between the estimated and exact phase values of the isoplanatic phase function. To minimize this sum, one sets the partial derivatives of this sum with respect to the filter coefficients equal to zero. After some further algebraic manipulations, one obtains equations for the filter coefficients and an equation for the corrected wavefront generated by use of the filter function.

The filter function was tested in a computational simulation based on the optical design of the James Webb Space Telescope. Among the results, the variation of phase error over the field of view was 83 percent less in the case of a multiplefield- point/filter-function control law than in the case of a single-field-point control law (see figure).

This work was done by Bruce H. Dean of Goddard Space Flight Center. For more information, download the Technical Support Package (free white paper) at www.techbriefs.com/tsp under the Physical Sciences category.