Evaluation of a packaged intervention for treating selective mutism: application in a school setting

Date of Award


Degree Name

Ed.S. in School Psychology


School of Education and Health Sciences


Advisor: Elana R. Bernstein


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 separation 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 so that partially coherent illumination with a small enough system bandwidth could relax OPD requirements for the sparse aperture array.


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Copyright © 2015, author