In Part I of this book, we discussed imaging based on ray geometrical optics. The aberration-free image of an object according to it is the exact replica of the object, except for its magnification. The aberration-free image of a point object is also a point. In reality, however, the image obtained is not a point. Because of diffraction of the object wave at the aperture stop, or equivalently, the exit pupil of the imaging system, the actual aberration-free image for a circular exit pupil is a light patch surrounded by dark and light rings. Geometrical optics is still assumed to hold from the object to the exit pupil in that rays are traced through the system to determine the shape of the pupil and the aberration across it. The determination of the characteristics of the diffraction image of an object formed by an aberrated system is the subject discussed in Part II. The emphasis of this part is on the numerical results on the effects of aberrations on a diffraction image and not on the formalism, exposition, or a critique of the variety of diffraction theories proposed over the years. It is a compilation of the works of masters like Nijboer, Hopkins, Barakat, and Fried with a sprinkling of my own work.

In Chapter 1, the diffraction theory of image formation is discussed. Starting with a brief account of the Rayleigh-Sommerfeld theory from a Fourier-transform standpoint, we derive the Huygens-Fresnel principle from it. We show that the diffraction image of a point object, called the point-spread function (PSF), is proportional to the modulus square of the Fourier transform of the complex amplitude across the exit pupil. It is shown that the diffraction image of an isoplanatic incoherent object is equal to the convolution of its Gaussian image and the diffraction PSF. The optical transfer function (OTF) of an imaging system, which is the Fourier transform of its PSF, is also discussed. The spatial frequency spectrum of the image of an isoplanatic object is shown to be equal to the product of the spectrum of the object and the OTF. The OTF based on geometrical optics is also considered and an approximate expression valid for low spatial frequencies is given. A brief comparison of imaging based on diffraction and geometrical optics is given in terms of both the PSF and the OTF. The line of sight of a system is identified with the centroid of its PSF which, in turn, is obtained in terms of the OTF or the aberration function. The line- and edge-spread functions are introduced and obtained in terms of the PSF or the OTF. Expressions for Strehl, Hopkins, and Struve ratios, useful for obtaining aberration tolerances, are derived. Finally, imaging of coherently illuminated objects is discussed. It is shown that the image of an isoplanatic coherent object is equal to the convolution of its Gaussian amplitude image and the coherent PSF of the imaging system. This chapter forms the foundation of Part II in that the fundamental relations derived in it, and stated as theorems, are used in the succeeding chapters to obtain some practical results for imaging systems with circular, annular, and Gaussian pupils. However, a reader need not read all of this chapter before attempting to read others.

Chapter 2 on systems with circular pupils starts with the aberration-free PSF and its encircled and ensquared powers. The effects of primary aberrations on its Strehl ratio are discussed and aberration tolerances are obtained. Balanced primary aberrations are considered and identified with Zernike circle polynomials. Focused and collimated beams are discussed and the concept of near- and far-field distances is introduced. Aberrated PSFs and their symmetry properties are discussed, and a brief comparison is made with their counterparts in geometrical optics. The line of sight of an aberrated system, identified with the centroid of the PSF, is determined for primary aberrations. The OTF for these aberrations is also discussed, phase contrast reversal is explained, and aberration tolerances for a certain value of Hopkins ratio are given. Expressions for the geometrical OTF for primary aberrations are also given. Both incoherent and coherent line- and edge-spread functions are discussed, and aberration tolerances for a certain value of the Struve ratio are given. A brief comparison of incoherent and coherent imaging is given with special reference to the Rayleigh criterion of resolution. The Fourier-transforming property of wave propagation is illustrated in altering the image of an object by spatial filtering in the Fourier-transform plane.

Systems with annular pupils are given a cursory look at best in books where imaging is discussed. Our Chapter 3 is written in a manner similar to Chapter 2 where, for example, the balanced aberrations are identified with the Zernike annular polynomials. Although the propagation of Gaussian beams is discussed in books on lasers, their treatment is generally limited to the weakly truncated aberration-free beams. In Chapter 4, we consider the effect of arbitrary truncation of beams with and without aberrations. The balanced aberrations in this case are identified with the Zernike-Gauss polynomials. It is shown, for example, that the pupil radius must be at least three times the beam radius in order to neglect beam truncation without significant error.

Finally, random aberrations are considered in Chapter 5. The effect of random image motion is considered first, and expressions and numerical values of time-averaged Strehl ratio, PSF, and encircled power are given for systems with circular and annular pupils. The random aberrations introduced by atmospheric turbulence when a wave propagates through it, as in astronomical observations by a ground-based telescope, are discussed; and expressions for time-averaged PSF and OTF are obtained. The aberration function for Kolmogorov turbulence is expanded in terms of the Zernike polynomials, and auto-correlation and cross-correlation of the expansion coefficients are given. The atmospheric coherence length is defined, and it is shown that the resolution of a telescope cannot exceed that of an aberration-free telescope of this diameter. Both the short- and long-exposure images are considered.

As in Part I, each chapter ends with a set of problems. It is hoped that they will acquaint the reader with application of the theory in terms of some practical examples. References for addtional reading are given after the Bibliography. These references represent the author's collection as the editor of Milestone Series 74 entitled *Effects of Aberrations in Optical Imaging*, published by SPIE Press in 1993.

© 2001 Society of Photo-Optical Instrumentation Engineers