The position and the size of the Gaussian image of an object formed by an optical imaging system is determined by using its Gaussian imaging equations. The aperture stop of the system limits the amount of light entering it the most. Its entrance pupil determines the amount of light from an object that enters it, and the exit pupil determines how that light is distributed in the image. The Gaussian image is an exact replica of the object, except for its magnification. The diffraction image of an isoplanatic incoherent object is given by the convolution of the Gaussian image and the diffraction image of a point object, called the point-spread function (PSF). In the spatial frequency domain, the spectrum of the image is correspondingly given by the product of the optical transfer function (OTF), which is the Fourier transform of the PSF, and the spectrum of the Gaussian image. The image is obtained by inverse Fourier transforming its spectrum. We define a pupil function, representing the complex amplitude at the exit pupil, and give equations for obtaining the PSF and the OTF.
The aberrations of the system determine the quality of an image. An important measure of the quality of an image is its Strehl ratio, which represents the ratio of the central irradiances of the PSF with and without the aberration. This ratio is discussed and simple but approximate expressions for it are derived for small aberrations in terms of the variance of the aberration at the exit pupil. Since the Strehl ratio is higher for a smaller variance, we discuss aberration balancing in which an aberration of a higher order is balanced with one or more aberrations of lower order to minimize its variance and thereby maximize the Strehl ratio. We discuss some general results on the effects of nonuniform amplitude, called apodization, and nonuniform phase, called aberration, at the exit pupil on the irradiance at the center of the reference sphere with respect to which the aberration is defined. For a given total power in the pupil and, therefore, in the image of a point object, maximum central irradiance is obtained for a system with an unapodized and unaberrated pupil. Moreover, the peak value of an unaberrated image lies at the center of curvature of the reference sphere regardless of the apodization of the pupil. Generally, the effect of even large amplitude variations across the pupil is relatively small compared to that of even small aberrations.
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