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Chapter 4:
Optical Systems with Gaussian Pupils
Abstract

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 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. However, the advantage of the reduced secondary maxima is lost when, for example, spherical aberration is present. Although a pupil transmits more light for a narrower Gaussian beam, the focal-point irradiance is correspondingly smaller becuase of the larger diffraction spread. Accordingly, an optimum radius of a Gaussan beam is defined that yields the maximum focal-point irradiance. Since the central obscuration of an annular pupil reduces the size of the central disc and increases the power in the diffraction rings, the difference between the diffraction effects of uniform and Gaussian beams decreases as the obscuration increases. As in the case of uniformly illuminated pupils, the principal maximum of the axial irradiance of a focused Gaussian beam with a small Fresnel number also lies at a point that is closer to the pupil plane than the geometrical focus. However, maximum central irradiance on a target at a fixed distance is still obtained when the beam is focused on it.