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Master’s Thesis Defense
Monday, June 25, 2015
10:00 AM in Fitz Hall Room 580
DIGITAL PHASE CORRECTION OF A PARTIALLY
COHERENT SPARSE APERTURE SYSTEM
Sarah Krug
University of Dayton
Abstract
Sparse aperture image synthesis requires proper phasing between sub-apertures. Phasing can be difficult
due to hardware misalignments, atmospheric turbulence, and many other causes of optical path differences
(OPD). Common synthesis techniques include incoherent and coherent methods. Incoherent methods
utilize passive illumination and adaptive optics while coherent methods rely on active illumination and
phase reconstruction approaches such as phase retrieval or spatial heterodyne. In this thesis, we present a
partially coherent technique with the capability to use either active or passive illumination to digitally
correct for piston phase errors. This technique requires an anamorphic pupil relay system and a piston
correction algorithm. The anamorphic pupil relay causes two closely spaced sub-apertures in the entrance
pupil to appear to be shifted further apart in the exit pupil. Analytic and numerical wave optics models
demonstrate the effectiveness of this relay system, matching with experimental results. An analytic model
shows that the higher frequency terms are equivalent to scaled cross-correlations of the two sub-apertures,
which are shifted due to the anamorphic separation. The constant shifts due to the separation are found
experimentally using a registration algorithm with a calibration target. The cross-correlations are
dependent on the piston phase errors between sub-apertures. We show that a piston correction algorithm
can be used to shift the cross-correlations to their original positions dictated by the entrance pupil, multiply
a cross-correlation with the complex conjugate of the auto-correlation, use the summation of this product
to calculate the piston, and correct the phase error in each cross-correlation before recombining them with
the auto-correlation. Examples show diffraction limited results for both simulated and experimental images
that are supported by analytical, numerical, and experimental analysis of the system’s modulation transfer
function (MTF). In addition, analysis of the piston for multiple wavelengths reveals two effects of
bandwidth on the system. First, the impact of a constant spatial shift resulting in different spatial frequency
shifts for different wavelengths, referred to as field dependent contrast (FDC), is addressed. Second, the
inverse relationship between the bandwidth of the system and OPD tolerances is shown. In future
research, diffraction gratings could eliminate the FDC and partially coherent illumination with a small
enough system bandwidth could relax OPD requirements for the sparse aperture array.