11.4.7 Case 7: virtual finite object—real finite image
11.4.8 Case 8: virtual finite object—virtual finite image
11.4.9 Case 9: real infinite object—real infinite image
Preface
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 utilizing Cartesian ovals are comprehensively discussed and numerous examples are provided. Subsequently, they meticulously develop the general equations for designing axial stigmatic lenes. The resultant surface shapes can be described by a combination of a conventional shape and an aspheric shape, or both surfaces being aspheric. These surfaces, in general, cannot be described by the well-known polynomial aspheric equation commonly used in lens design computer programs. The authors skillfully explain the development of their new aspheric equations and provide numerous examples. Throughout the book, the authors have richly included graphics that aid in clarification of their discussions. Readers will likely appreciate that the authors included computer code for algorithms useful in computing axial stigmatic designs.
Stigmatic Optics is an excellent book to gain an understanding of the basics of optical imaging and the formulation of the new aspheric shapes for achieving axial stigmatic imaging. To continue learning about this topic, readers are encouraged to read Analytical Lens Design by these authors, along with Julio C Gutiérrez-Vega, which also explores the development of aspherical-shaped surface(s) that create lenses which are aplanatic, i.e., free of both spherical aberration and linear coma. One might ask the question: ‘what is the value of this rather esoteric approach to lens design using closed-form solutions?’. My answer is that exploring closed-form solutions can provide further insight into creating better optical systems, although closed-form solutions of complex optical systems seems improbable. Also, recent advances in manufacturing free-form surfaces makes possible the creation of lenses incorporating the authors’ new aspheric shapes.
R Barry Johnson, FInstP, FOSA, FSPIE, HonSPIE
Series Editor, Emerging Technologies in Optics and Photonics
Huntsville, Alabama
Acknowledgements
Acknowledgements of Rafael G González-Acuña
Almighty God, creator of the Universe, this book aims to honor your glory. Thank you Lord for giving me the intelligence, the desire, the faith and the means to complete it. Lord, please help me to be a faithful servant of your will. Help me to deserve the promises of your son Jesus Christ, give me a sign, and show me the way …
I want to thank my family.
I want to thank my mother Carmen Leticia Acuña Medellín and to my father Rogelio González Cantú to my brothers Rolando and Rogelio,
Héctor Alejandro Chaparro-Romo, Comrade, once again! we did it comrade! time rewards!
To Professor R Barry Johnson for your support, patience and fruitful discussions.
To Ashley Gasque and Robert Trevelyan for all the support!
I would also like to thank Yoshio Catillejos, Israel Meléndez, Gustavo Medina, Daniel Lomas, Roberto Martinez, César López, Roberto Vera, Ileana Paulette Zambrano, Eliel Guadarrama, Miguel Rojas, Joel Guerra, Homero Pérez, Mauricio Arroyo, Esteban Lankenau, Alejandra Guajardo, Adad Yepiz, Erick Patiño, Alberto Silva, Michelle C Rocha, Adrian Lozano, Luis Garza, Roberto Acuña, Rogelio Acuña, Adriana Mabel Serrano, Maria Isabel González Villarreal, Sonia Villarreal, Dr Job Mendoza, Dr Dorilian López, Dr Servando López, Dr Raúl Aranda, Dr Benjamin Perez-Garcia, Dr Maximino Avendaño, Dr Genaro Zavala, Dr Carlos Hinojosa, Dr Francisco Cuevas, Dr Rafael Torres, Dr Blas Manuel Rodríguez Lara, Professor Reinhard Klette, Professor Alois Herkommer, Professor Russell Chipman, Professor Simon Thibault and Stephen Wolfram.
To Professor Julio C Gutiérrez-Vega and Dr Bernardino Barrientos García, my advisors during the PhD and masters degree, respectively.
I would also like to thank several institutions: Institute of Physics, Conacyt, Instituto
