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In this chapter, we consider optical systems with Gaussian apodization or Gaussian pupils, i.e., those with a Gaussian amplitude across the wavefront at their exit pupils, which may be circular or annular. The discussion in this chapter is equally applicable to imaging systems with a Gaussian transmission (obtained, for example, by placing a Gaussian filter at its exit pupil) as well as laser transmitters in which the laser beam has a Gaussian distribution at its exit pupil. It is evident that whereas a Gaussian function extends to infinity, the pupil of an optical system can only have a finite diameter. The net effect is that the finite size of the pupil truncates the infinite-extent Gaussian function. If the Gaussian function is very narrow (i.e., its standard deviation is very small) compared to the radius of the pupil, it is said to be weakly truncated. In such cases, the truncation can be neglected, and the pupil can be assumed to be infinitely wide. The aberration-free image for a system with a Gaussian pupil shows that the Gaussian illumination reduces the central value, broadens the central bright spot, but reduces the power in the diffraction rings compared to a uniform pupil. Correspondingly, the OTF for a Gaussian pupil is higher for low spatial frequencies, and lower for the high. In these respects, the effect of a Gaussian illumination is opposite to that of a central obscuration in an annular pupil. The diffraction rings practically disappear when the pupil radius is twice the Gaussian radius, and the beam propagates as a Gaussian everywhere. The OTF in this case is also described by a Gaussian function. The standard deviation of a primary aberration over a Gaussian pupil is calculated and shown to be smaller than its corresponding value for a uniform pupil. This is due to the fact that the wave amplitude decreases as a function of the radial distance from the center of the pupil while the aberration increases, i.e., the amplitude is smaller where the aberration is larger. Accordingly, the Strehl ratio for a Gaussian pupil for a given amount of a primary aberration is higher than that for a uniform pupil, or the aberration tolerance for a given Strehl ratio is higher for a Gaussian pupil. The balanced primary aberrations with minimum variance are also obtained, and the diffraction focus for various values of the truncation ratio are given. The Gaussian polynomials orthonormal over a Gaussian pupil are obtained by orthogonalizing the circle polynomials over such a pupil. As expected, the Gaussian polynomials for primary aberrations represent balanced aberrations. Similarly, the orthonormal Gaussian annular polynomials are obtained by orthogonalizing the annular polynomials over a Gaussian pupil. Again, the primary Gaussian annular polynomials represent the balanced aberrations for a Gaussian annular pupil. The isometric, interferometric, and imaging characteristics of the Gaussian circular and annular polynomial aberrations are not discussed because of their similarity with those of the corresponding circle or annular polynomial aberrations for uniform pupils.
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