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So far we have considered deterministic aberrations such as those that are inherent in the design of an optical imaging system. These aberrations are deterministic in the sense that they are known or can be calculated, for example, by ray tracing the system. Now we consider the effects of aberrations that are random in nature on the quality of images. The aberration is random in the sense that it varies randomly with time for a given system, or it varies randomly from one sample of a system to another. An example of the first kind is the aberration introduced by atmospheric turbulence when an optical wave propagates through it, as in ground-based astronomical observations. An example of the second kind is the aberration introduced due to polishing errors of the optical elements of the system. The polishing errors of an element fabricated similarly in large quantities vary randomly from one sample to another. In either case, we cannot obtain the exact image unless the instantaneous aberration or the exact polishing errors are known. However, based on the statistics of the aberrations, we can obtain the time- or ensemble-averaged image. We discuss the effects of two types of random aberrations: random wavefront tilt causing random image motion, and random aberrations introduced by atmospheric turbulence. The time-averaged Strehl ratio, point-spread function (PSF), optical transfer function (OTF), and encircled power are discussed for the two types of aberrations. A coherence length of atmospheric turbulence is defined, which limits the resolution of an imaging system, regardless of how large its aperture is. Both long- and short-exposure images are discussed, and expressions for the aberration variance are given in both cases. These expressions can be used to define the requirements of a steering mirror for corrections of wavefront tilt and a deformable mirror for corrections of wavefront deformation or the aberrations. It is shown that in a severe turublence, a short-exposure image breaks up into speckles whose size is determined by the resolution of the system. Although much of our discussion is on systems with circular pupils, systems with annular pupils are also considered, and differences between the two types are outlined. In many optical imaging systems, especially those used in space, there is always some image motion during an exposure interval. The source of image motion may, for example, be vibration of optical elements and servo dither in the pointing system. In the case of beam transmitting systems, the beam itself may have some motion associated with it. Here, we obtain expressions for the time-averaged PSF, Strehl ratio, OTF, and encircled power for an imaging system with a circular exit pupil undergoing Gaussian random motion. A simple approximate model based on a Gaussian approximation of its motion-free PSF is also developed, and numerical results provided by it are compared with the exact results.
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