The effect of spherical aberration on the optical transfer function.
Parrent, George Burl
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This paper discusses the behavior of the optical transfer function for an aberrated lens on an axis at several focus settings. The aberration considered is spherical and the focus is varied in five steps from paraxial to marginal focus. The curves are then interpreted in terms of spurious resolution, band pass, and shift of focus with spatial frequency. The problem is formulated for circular apertures in both cartesian and polar coordinates and the difficulties associated with such a general approach are described. The equivalent one dimensional problem for a slit aperture is then formulated in detail. Evaluation of the transfer function is accomplished by one of three methods depending on the amount of spherical aberration. For the case of zero aberration the integral is evaluated analytically by direct integration. For a quarter wave of aberration the integrand is expanded in a Taylors series and integrated term by term. Finally for larger aberrations Simpson's method of numerical integration is utilized and the properties of the function which make it particularly adaptable to numerical integration are established. lt was possible t hereby to obtain the transfer function at five positions between the paraxial and marginal focus for four different amounts of spherical aberration. In order to demonstrate how refocusing can compensate for spherical aberration, a frunily of curves was plotted for each aberration vvith focal ratio as a parameter. It was possible after squaring these curves ordinate by ordinate to obtain the equivalent band pass by planimeter integration. To demonstrate that the focal setting for maximum contrast depends on the spatial frequency the following graphical method was utilized. For each aberration a family of curves was plotted, contrast versus focal setting, with frequency as a parameter. It was observed that the maxima of these curves tended to shift toward the paraxial then toward the marginal focus as the frequency continued to increase. Finally an evaluation of the methods used and results obtained is presented.
Thesis (M.A.)--Boston University