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Chapter 11:
Random Aberrations
11.1 Introduction 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. Although much of our discussion is on systems with circular pupils, systems with annular pupils are also considered. A brief discussion on the aberrations resulting from fabrication errors is also given. 11.2 Theory Consider an imaging system with a circular exit pupil of radius a imaging a point object radiating at a wavelength λ. Let Φ(r ⃗ p ) be the random phase aberration at a point r ⃗ p in the plane of the pupil. We assume that Φ(r ⃗ p ) is a stationary Gaussian random variable of zero mean, variance σ 2 Φ , and structure functionD Φ (∣r ⃗ p −r ⃗ p ′ ∣)=<[Φ(r ⃗ p )−Φ(r ⃗ p ′ )] 2 <=2[σ 2 Φ −R Φ (∣r ⃗ p −r ⃗ p ′ ∣)], where R Φ is the phase autocorrelation function given by R Φ (∣r ⃗ p −r ⃗ p ′ ∣)=<Φ(r ⃗ p )Φ(r ⃗ p ′ )<.
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