The material presented here has been gathered from my lectures at the Electrical Engineering-Electrophysics Department of the University of Southern California, where I have been teaching a graduate course on optical imaging and aberrations since 1984. My objective for this course has been to provide the students with an understanding of how aberrations arise in optical systems and how they affect optical wave propagation and imaging based on both geometrical and physical optics. This book has been written with the same objective in mind. The emphasis of the text is on concepts, physical insight, and mathematical simplicity. Figures and drawings are given wherever appropriate to facilitate understanding and make the book reader friendly. An abbreviated version called Aberration Theory Made Simple was published by the SPIE Press in 1991 in their Tutorial Text Series (Vol. TT6). The current detailed version is divided into two parts just like the abbreviated one. In Part I of this text, which contains the first seven chapters, ray geometrical optics is discussed. In Part II, wave diffraction optics is discussed.

In Part I, Chapter 1 begins with the foundations of geometrical optics. Fermat's principle, the laws of geometrical optics, the Malus-Dupin theorem, and Hamilton's point characteristic function are described. Starting with a brief outline of the sign convention for object and image distances, heights, and ray angles, Gaussian imaging by a spherical refracting surface, a thin lens, an afocal system, and a spherical reflecting surface (mirror) is discussed. The cardinal points of an imaging system are defined and a paraxial ray-tracing procedure to determine them is described. It is emphasized that the results for a reflecting surface can be obtained from those for a refracting surface by substituting the refractive index associated with the reflected rays equal to the negative of the refractive index associated with the incident rays. A two-lens system, a two-mirror system, and a catadioptric system consisting of a lens and a mirror are considered as examples of imaging systems for which the focal length is explicitly determined using the ray-tracing equations. Since the ray-tracing equations in Guassian optics are linear in ray heights and slopes, the whole imaging process can be represented by a 2 Ã 2 matrix. This is discussed in detail, especially for electrical engineers who seem to have a preference for the matrix approach.

The sign convention used throughout the book is the Cartesian sign convention of analytical geometry. It is different from the one used in the author's *Aberration Theory Made Simple* in some respects, which should be noted when making comparisons with the equations given there. A good understanding of Chapter 1 is essential for performing Gaussian (or first-order) design and analysis of an optical imaging system. Given the radii of curvature and the positions of the surfaces of an optical system, and the refractive indices of the media around them, one can determine its cardinal points, and, in turn, the position and size of the image for any position and size of the object. However, there is no discussion in this chapter on the intensity of the image of a point object in terms of the object intensity, or the irradiance distribution of the image of an extended object in terms of the object radiance distribution.

The concepts of aperture stop, entrance and exit pupils, chief and marginal rays, sizes of imaging elements, and vignetting of rays are introduced in Chapter 2. Radiometry of point and extended sources, and of point and extended object imaging, is discussed next. The origins and limitations of the cosine-cube law of image intensity for point objects, and the cosine-fourth law of image irradiance for extended objects, are discussed in detail. It is pointed out that because of pupil distortion, integration must be performed across the aperture stop to calculate the total flux entering the system from an object element. For integrating across a pupil, its distortions determined by detailed ray tracing must be taken into account. A brief discussion of photometry, which is a branch of radiometry involving the spectral response of the human eye, is also included.

Besides the position, size, and intensity or irradiance of an image, its quality, which depends on the aberrations of the system, is of paramount importance. In Chapter 3, the wave and ray aberrations are defined and a relationship between them is derived. Relationships between defocus wave aberration and longitudinal defocus, and wavefront tilt aberration and wavefront tilt angle, are described. The form of the aberration function of a rotationally symmetric system is derived, and its expansions in terms of a power series and Zernike circle polynomials are discussed. The relationships between the coefficients of the two expansions are given. It is shown that up to the fourth order in object and pupil coordinates, any system with an axis of rotational symmetry can have no more than five primary aberration terms, called Seidel aberrations. The form of the secondary (or Schwarzschild) and tertiary aberrations is discussed. How an aberration may be observed is described by discussing the interference patterns of the primary aberrations. The conditions under which an imaging system may form an aberration-free image are considered. In particular, the sine condition for coma-free imaging is discussed. It is not essential to understand all of the material in this chapter to understand the material in Chapters 5, 6, and 7, though it would be useful to read the first four sections and to know the form of the five primary aberrations of a rotationally symmetric system from Section 3.5.2.1.

In Chapter 4, the relationship between the ray and wave aberrations is utilized to discuss the geometrical point-spread functions and the ray spot diagrams for each of the five primary aberrations. The circle of least confusion is discussed for both spherical aberration and astigmatism, thereby introducing the concept of aberration balancing. The centroid, encircled power, and the standard deviation or sigma of an aberrated image spot are also discussed. The traditional examples of the image of a spoked wheel in the presence of astigmatism, and the image of a square grid in the presence of distortion, are explained. Thus, given the aberrations of a system, the quality of the image of a point object in terms of its size or the ray distribution can be determined using the material given in this chapter. Aberration tolerances can be obtained from the tolerable image spot sizes.

In order to determine the quality of an image formed by a certain system, its aberrations must be known. The remainder of Part I discusses how to calculate the aberrations of an optical system given the radii of curvature and positions of its surfaces, and the refractive indices of the media surrounding them. Of course, the task of a lens designer is to choose these parameters in a way that is practical yet meets his/her image quality objectives. Chapter 5 describes an approach for calculating the primary aberrations of a multisurface optical system with an axis of rotational symmetry. The theory is developed by starting with the simplest problem, namely, the aberrations of a spherical refracting surface with its aperture stop located at the surface. An on-axis point object is considered first, so that the only aberration that arises is spherical aberration. An off-axis point object is considered next, and expressions for field aberrations (coma, astigmatism, field curvature, and distortion) are obtained with respect to the Petzval image point. These are generalized next to obtain the aberrations with respect to the Gaussian image point. Only field curvature and distortion terms change as the image point is changed from Petzval to Gaussian. This completes the derivation of primary aberrations of a spherical refracting surface. The Gaussian imaging equations are obtained as a by-product of this derivation. The primary aberrations of a spherical refracting surface with an arbitrary location of the aperture stop are considered next and its aplanatic points are determined. The aberrations of a conic refracting surface and finally a general aspheric surface are obtained. Instead of starting with a derivation for the most complex case, namely, a general aspheric surface with a remote aperture stop, a step-by-step derivation of increasing complexity is given so that physical insight on the differences between different steps is not lost.

How the results given for a single refracting surface can be extended to obtain the aberrations of a multisurface system is described. The changes in the aberration function as a result of a change in the position of the aperture stop are discussed next. The stop-shift equations relating the aberration coefficients for one position of the aperture stop to those for another are derived. The aberration function is also considered in terms of Seidel sums and Seidel coefficients of an optical system. As applications of the theory, the aberrations of a thin lens and a plane-parallel plate are derived and discussed. The aplanatic and field-flattening lenses are also considered. Next, the chromatic aberrations of a refracting system are discussed in terms of the wavelength dependence of the position and magnification of an image formed by it. Finally, pupil aberrations are considered and conjugate-shift equations are obtained that relate the aberrations of the image of one object in terms of those of another.

The primary aberrations of reflecting and catadioptric systems are discussed in Chapter 6. As in the case of imaging relations, the aberration expressions for a reflecting surface may be obtained from those for a corresponding refracting surface by substituting the refractive index associated with the reflected rays equal to the negative of the refractive index associated with the incident rays. As examples of reflecting systems, expressions are obtained for the primary aberrations of a spherical mirror, paraboloidal mirror, a beam expander consisting of two confocal paraboloidal mirrors, and two-mirror astronomical telescopes. Schmidt and Bouwers-Maksutov cameras and telescopes with aspheric plates are discussed as examples of catadioptric systems.

Even if a practical design of a system has been chosen, its elements must be fabricated and assembled into a system. In Chapter 7, the last chapter of Part I of the book, the primary aberrations due to perturbations such as a decenter, a tilt, or a despace of the surface of a system are considered. When one or more of the imaging elements is decentered and/or tilted, a system loses its rotational symmetry. Hence, new aberrations arise which have different dependence on the object height but the same dependence on pupil coordinates as the aberrations of the unperturbed system. The expressions derived for the primary aberrations produced when a perturbation is introduced into the system are used to obtain the aberrations of misaligned two-mirror telescopes. Finally, the relationships between the fabrication errors of the surfaces and the corresponding aberrations or wavefront errors introduced by them are derived for both the refracting and reflecting surfaces. The determination of system errors from fabrication or deformation errors and allocation of error tolerance are also described briefly.

Throughout the book, the primary aberrations of a system are emphasized since they are often the dominant aberrations in the early stages of the design of an optical system. Although expressions for higher-order aberrations have been given in the literature, their value in designing or analyzing optical systems has not been fully exploited or realized, mainly perhaps because of their complexity. The expressions for the primary aberrations of even simple systems such as a thin lens (made up of two surfaces with negligible thickness between them) or a two-mirror astronomical telescope are complex indeed. With the advent of computers and commercially available computerized ray-tracing and image-analysis programs (e.g., ZEMAX or CODE V), it is a simple matter to determine the aberrations of a system fully, not just its primary or secondary aberrations. However, it is this author's belief that it is essential to understand the primary aberrations of simple systems in order to be able to design systems that are more complex and provide high image quality. It is for this reason that full derivations and discussion of the expressions for the primary aberrations of simple systems are given. Key equations representing fundamental results are highlighted by putting a box around them. It is hoped that they will provide the reader with certain basic tools to develop new designs without endless surfing in a sea of potential designs.

Each chapter ends with a set of problems. These problems have been crafted carefully either as an extension of the theory given in the text, or, more often, as applications of the theory. They are an essential part of the book since only by working through such problems can the students appreciate the theory and validate their understanding of it.

In Part II, published by SPIE in 2001, imaging based on diffraction is discussed. It starts with an introduction of the diffraction point-spread function and optical transfer function of a general imaging system. An understanding of diffraction effects is essential since geometrical point-spread functions in terms of the spot diagrams give at best a qualitative understanding of the image quality aspects of an imaging system, especially for high-quality systems. Optical systems with circular, annular, and Gaussian pupils are considered and aberration-free as well as aberrated images are discussed. Aberration tolerances based on the Strehl and Hopkins ratios of an image are obtained. The effect of random aberrations such as those introduced by atmospheric turbulence or the fabrication errors on the image formed by a system is also discussed.

Virendra N. Mahajan

El Segundo, California

1998

© 1998 Society of Photo-Optical Instrumentation Engineers