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Chapter 4:
Optical Systems with Gaussian Pupils
In Chapters 2 and 3, we have considered optical systems with uniformly illuminated (i.e., with uniform amplitude across) circular and annular pupils, respectively. Systems with nonuniform amplitude across their exit pupils are referred to as apodized systems. Often, the transmission of a system is made nonuniform by placing an absorbing filter at its entrance or exit pupil in order to reduce the secondary maxima of its PSF. The word apodization in Greek means “without feet” implying without or at least reduced secondary maxima. The purpose in reducing the secondary maxima is to improve the resolution of the system. 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 of this chapter is applicable equally 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. In the case of an imaging system, we are interested in its PSF, i.e., the irradiance distribution of the image of a point object. In the case of a laser transmitter focusing a beam on a target, we are interested in the irradiance distribution in the target plane or the focal plane of the beam. 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. It is shown that the Gaussian illumination of the pupil broadens the central disc and reduces the secondary maxima of the Airy pattern obtained for a uniform pupil. The corresponding OTF is higher for low spatial frequencies and lower for the high. As in the case of uniformly illuminated pupils, here too the principal maximum of axial irradiance of a focused beam with a small Fresnel number lies at a point that is inside and not at the geometrical focus. However, maximum central irradiance on a target at a fixed distance is still obtained when the beam is focused on it. 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 essence, the pupil can be assumed to be also infinitely wide with the result that a Gaussian beam exiting from the pupil remains Gaussian as it propagates. The standard deviation of an aberration for a Gaussian pupil is smaller compared to that 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 but 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.
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