In Chapter 8 we have considered optical systems with circular exit pupils. Now we consider systems with annular pupils, for example, a Cassegrain telescope in which its secondary mirror obscures the central portion of its primary mirror. As in the case of a system with a circular pupil, we discuss the aberration-free PSF, axial irradiance, and the Strehl ratio of a system with an annular pupil. We show that the radius of the central bright spot of the PSF decreases, its principal or central maximum decreases in its value, and the secondary maxima increase in their values as the obscuration increases. However, the tolerance for a given Strehl ratio increases or decreases depending on the type of the aberration. The aberrated PSFs and OTFs are not discussed. Optical systems with circular pupils and Gaussian illumination across them are also considered along similar lines. For these systems, it is shown that the tolerance for an aberration increases compared to the corresponding tolerance for a system with a uniformly illuminated circular pupil. Finally, systems with weakly truncated Gaussian pupils, i.e., those having a very wide pupil compared to the width or the radius of the Gaussian illumination, are considered. In this case, the tolerance for a primary aberration is obtained in terms of its peak value at the Gaussian radius rather than at the edge of the pupil.
9.2. Annular Pupils
In this section, we discuss the imaging characteristics of systems with annular pupils. The aberration-free PSF, encircled power, axial irradiance, and Strehl ratio are discussed for increasing value of the obscuration of the pupil. The results obtained are compared with the corresponding results for systems with circular pupils.
9.2.1 Aberration-Free PSF
Consider a system with an annular exit pupil having inner and outer radii of Ïµa and a, where Ïµ is called its obscuration ratio. The PSF of the system, i.e., the irradiance distribution of the image of a point object formed by it, is given by Eq. (8-1) except that now the lower limit in the radial integration is Ïµ instead of zero. The aberration-free PSF thus obtained is given by I(r;Ïµ)=1 (1âÏµ 2 ) 2 [2J 1 (Ïr) Ïr âÏµ 2 2J 1 (ÏÏµr) ÏÏµr ] 2 , where J 1 (â
) is the first-order Bessel function of the first kind. It is normalized to unity at the center r=0 by the central irradiance PS p âÎ» 2 R 2 , where P is the total power transmitted by the annular pupil and S p =Ïa 2 (1âÏµ 2 ) is its clear area, Î» is the wavelength of object radiation and R is the distance between the pupil plane and the image plane.
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