Stigmatic Optics. Rafael G González-Acuña

This treatise focuses on a particular concept of geometric optics, stigmatism. Stigmatism refers to the image property of an optical system that focuses a single point source in object space at a single point in image space. Two of these points are called a stigmatic pair of the optical system.

The treatise starts from the foundations of stigmatism: Maxwellʼs equations, the eikonal equation, the ray equation, the Fermat principle and Snellʼs law. Then we study the most important stigmatic optical systems without any paraxial or third order approximation or without any optimization process. These systems are the conical mirrors, the Cartesian ovals and the stigmatic lenses.

Conical mirrors are studied step by step with clear examples.

In the case of the Cartesian ovals, two paradigms are studied. The first, the Cartesian ovals are obtained by means of a polynomial series and the second by means of a general equation of the Cartesian oval.

For stigmatic lenses, the case is studied when the two refractive surfaces are Cartesian ovals. Then the general equation for stigmatic lenses is obtained.

Finally, the similarities of optical systems and their nature are studied.

It is recommended to read this treatise in order.

The Authors

Series Editor’s foreword

For over a millennium, scientists have attempted to create mirrors and lenses free of spherical aberration that is the only monochromatic axial aberration. Geometrical imaging of an axial point object to form a perfect axial point image is known as axial stigmatic imaging. When an optical system produces a perfect image over the entire field-of-view, it known as a stigmatic optical system. Over time, it was learned that spherical aberration is a constant aberration over the entirety of the image surface. For a long time, it has been known that certain forms of lenses provided axial stigmatic imaging. For example, when the object is at infinity, a geometrically-perfect point image can be formed by a lens having (i) an ellipsoidal front surface and a plane rear surface or (ii) a plane front surface and hyperbolic rear surface. A stigmatic image for finite conjugates (magnification < 0) can be formed by using a pair of plano-hyperbolic lenses (having focal length of f₁ and f₂) with the plane surface facing one another. The magnification is simply −f2/f1. An example of a mirror forming a stigmatic image is a parabola with the object at infinity. Although such lenses and mirrors suffer no spherical aberration, other aberrations such as coma and astigmatism can be bothersome. Indeed, a fast parabola can become useless for imaging due to coma.

It is generally understood that there is not a generalized closed-form solution to the design of a singlet lens that is axially stigmatic, i.e., one that is free of spherical aberration. The authors of this treatise, Stigmatic Optics, elegantly attack this challenge to develop such a generalized closed-form solution. They begin the book by presenting Maxwell’s equations describing the behavior of electromagnetic fields and develop the eikonal equation that provides the basis for geometric ray propagation equation and Snell’s Law. Next is provided the necessary mathematics needed to understand their development of the equations describing the surfaces of lenses having the property of axial stigmatic imaging. Optical systems

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