The amplitude distribution at the aperture of an optical system for off-axis object points due to the introduction of a non-uniform transmission filter depends on the transmission function and on the longitudinal filter position. The effects of several filters on the axial and extra-axial point-spread function (PSF) and modulation-transfer function (MTF) of an optical system for different filter positions are studied. The filter transmission function and the filter position are obtained by the optimization of the axial and extra-axial Strehl ratios. The axial and extra-axial effects of the filters on the PSF and on the MTF are analogous. There are some positions of the filter for which the overall performance is significantly improved. The optimum longitudinal positioning'of the apodizing filter is achieved in a process which is analogous to the stop shift effect in conventional optical design. The lack of radial symmetry in the off-axis wavefront aberration function at any particular field angle is convolved with the apodizing function and this leads to the anticipated improvement. © 1989 Taylor & Francis Ltd.
Yzuel, M. J., & Campos, J. (1989). Aberrated optical systems with optimum apodizers for the strehl ratio: Axial and extra-axial point-spread function and modulation-transfer function. Journal of Modern Optics, 36(11), 1513-1526. https://doi.org/10.1080/09500348914551561