Provided by the author(s) and NUI Galway in accordance with publisher policies. Please cite the published version when available. Title Chromatic Aberrations in Optical Systems: Prediction and Correction Author(s) Bahrami, Mehdi Publication Date 2012-12-19 Item record http://hdl.handle.net/10379/3463 Downloaded 2021-09-26T11:41:24Z Some rights reserved. For more information, please see the item record link above. Chromatic Aberrations in Optical Systems: Prediction and Correction by Mehdi Bahrami Supervisor: Dr. Alexander V. Goncharov A thesis submitted in partial fulfilment of the requirements for the degree of Doctor of Philosophy, School of Physics, Science Faculty, National University of Ireland, Galway December 2012 Abstract The variation of refractive index with wavelength, known as dispersion, was what redirected Isaac Newton from refractive telescope designs toward a reflective one, since he found the chromatic effect incurable. Later it was shown that different ma- terials demonstrate different chromatic characteristics and refractive optical elements can compensate each other’s chromatic contributions. This promoted the simple dis- persion effect to the field of formulizing and categorizing chromatic aberrations and their corrections. In this study an introduction to this process is provided, where the historical aspects are followed by the mathematical derivation and description of dif- ferent kinds of chromatic aberrations accompanied with a variety of approaches to correct the chromatic effects in different levels. The provided mathematical basis is employed in studying three distinctive topics. In the first one the flexibility of refrac- tive elements is used to provide a middle-sized catadioptric telescope design with all-spherical surfaces. Employing a new combination of chromatic lens correctors, the image quality can be improved so that it becomes comparable to an equivalent aspheric Ritchey-Chrétien telescope design. As the second topic the atmospheric dis- persion and its effect in extremely large telescopes are discussed, where a new atmo- spheric dispersion corrector design is proposed. In the third task the chromatic be- havior in an inhomogeneous medium is considered. A new gradient refractive index lens model for the crystalline lens of the eye is established and a different approach in characterizing its chromatic effects is developed. These three research topics are underpinning the main goal of the theses, that is the role of chromatic aberration in image formation in various optical systems. i List of publications Paper I: Mehdi Bahrami and Alexander V. Goncharov: All-spherical catadioptric telescope design for wide-field imaging Applied Optics, Vol. 49, Issue 30, pp. 5705-5712 (2010) Paper II: Mehdi Bahrami and Alexander V. Goncharov: The achromatic design of an atmospheric dispersion corrector for extremely large telescopes Optics Express, Vol. 19, Issue 18, pp. 17099-17113 (2011) Paper III: Mehdi Bahrami and Alexander V. Goncharov: Geometry-invariant gradient refractive index lens: analytical ray tracing Journal of Biomedical Optics, Vol. 17, Issue 5, pp. 55001-55009 (2012) Paper IV: Mehdi Bahrami and Alexander V. Goncharov: Geometry-invariant GRIN lens: iso-dispersive contours Biomedical Optics Express, Vol. 3, Issue 7, pp. 1684-1700 (2012) ii Acknowledgements This research was supported by Science Foundation Ireland under grant number 07/IN.1/1906. I express my sincerest gratitude to my supervisor, Dr. Alexander Goncharov, who has supported me throughout my research with his kindness, patience and knowledge whilst allowing me the freedom to work in my own way. I would like to thank Prof. Chris Dainty for providing a marvelous atmosphere and environment in the Applied Optics Group and for his valuable comments and ad- vices. The greatest appreciations to Dr. Hamid Fallah, who I am indebted to forever, who was the most help in the worst days. Thanks to my best friends Meisam, Javad, Hamed, Vahid, Maysam, Hossein, Reza, Vahid, and Marwan. Most of all, thanks to my parents, my brother, my sister, and my supportive wife for everything they have done for me. I will not forget all the love and the support that you have given me throughout my life and education. Thank you all. iii Acronyms ADC Atmospheric Dispersion Corrector AO Adaptive Optics ELT Extremely Large Telescope E-ELT The European Extremely Large Telescope GIGL Geometry-Invariant Gradient Refractive Index Lens GMT Giant Magellan Telescope GRIN Gradient Refractive Index LADC Linear Atmospheric Dispersion Corrector LGS Laser Guide Star OPD Optical Path Difference OPL Optical Path Length OT Optical Transmittance OT∗ Normalized Optical Transmittance RADC Rotating Atmospheric Dispersion Corrector RC Ritchey-Chrétien TMT Thirty Meter Telescope iv Contents Abstract i List of publications ii Acknowledgements iii Acronyms iv 1 Chromatic Aberrations 1 1.1 Historical aspects . .1 1.2 Axial color and achromats . .2 1.2.1 Axial chromatic aberration coefficients . .4 1.2.2 Achromats . .6 1.3 Secondary spectrum and apochromats . 11 1.3.1 Secondary spectrum . 11 1.3.2 Apochromats . 13 1.4 Lateral color . 15 1.5 Chromatic aberrations of a single surface . 18 1.6 Chromatic aberrations of a plane parallel plate . 20 1.7 Chromatic variations of Seidel aberrations . 21 v 2 Publications overview 23 2.1 Paper I . 23 2.2 Paper II . 25 2.3 Paper III . 28 2.4 Paper IV . 31 3 Conclusions, future work 33 Appendix 37 Bibliography 38 vi Chromatic Aberrations 1.1 Historical aspects The refractive index of any medium (other than vacuum) varies with wavelength. This means that all the optical properties of a medium are functions of wavelength. Traditionally, chromatic variations of paraxial properties such as image position and magnification have been considered as chromatic aberrations as well as variations of classical aberrations. These chromatic effects are usually the main problem in re- fractive systems. A well corrected refractive monochromatic imaging system could easily lose its acceptable optical performance even in a small range of wavelengths. Isaac Newton considered this problem intrinsically intractable and turned to reflec- tive systems. In fact, he thought that the chromatic aberrations of all lenses were proportional to their powers, with the same constant of proportionality even for dif- ferent glasses. Newton was the first to discuss chromatic aberrations in detail in his book, OPTICKS [1], which was released to the public in 1704. Newton’s conclusion was not the final word on this topic. Before Newton, it was a common belief that the human eye, because of containing different refractive me- dia, is optically corrected for chromatic aberration. James Gregory, the designer of the Gregorian telescope, had the same opinion in the 1660s. In addition to this, Christian Huygens had shown, in correspondence with Sir Christopher Wren in the 1660s, that a positive and a negative lens more or less in contact could correct spherical aberra- tion [2]. In 1729 Chester Moore Hall, a lawyer, followed on these two and could find a combination of positive and negative lenses made from different glasses, which pro- vided achromatic correction. He had such objectives made by opticians, and proved Newton wrong. In 1858 John Dollond, who expanded Hall’s work by experimenting with prisms, patented an achromatic lens [3]. From a technical point of view, manufacturing an acceptable reflective surface is much harder than a refractive element. The first practical reflective telescope was 1 1. Chromatic Aberrations built by Newton in 1668, 58 years after manufacturing the Galilean telescope (Al- though Zucchi made his reflective telescope in 1616, its image quality was not compa- rable to the refractive ones [4].) As manufactures approached 1 m objective doublets, in addition to being limited by mechanical difficulties of building large lenses, they faced another limitation related to difference in dispersion of the glasses used. In fact by, making larger and larger telescope doublet objectives they found that an achro- matic doublet was not quite free of chromatic aberration, and a residual error, known as secondary spectrum appeared. This led to the re-introduction of reflective objectives which are now universally used for apertures exceeding 1 m. From an optical designer’s point of view, the natural reaction to this new problem is adding one more lens to the doublet, thus making it an Apochromatic triplet. Imple- menting this solution was not simple because of the complexity of physics laws in dis- persion and the number of constraints which should be satisfied. On the other hand, the variety of glass combinations which can remove the residual chromatic aberra- tions is limited and sometimes expensive. Nevertheless this solution is implemented nowadays, and achromatic and apochromatic elements are used widely in optical sys- tems (e.g. small size telescope objectives, telescope correctors, photographic lenses, etc.). Before finishing this brief historical review, it is worth mentioning that a chromat- icly aberrated optical system or element is sometimes desired. For instance, when we need to compensate the chromatic aberration arising from the prior optical elements or media, for example the Earth’s atmosphere or human eye. 1.2 Axial color and achromats In the design and analysis of visual systems, it is common to choose F (blue, l = 486 nm), d (yellow, l = 588 nm), and C (red, l = 656 nm) spectral lines for the calculations. Fig. 1.1 shows that in a positive singlet, focus for F light is inside the paraxial focus for d light, while the C one lies on the outside. In fact, primary axial color is defined as the distance between the foci for F and C. To formulize this as a function of lens parameters one can start from a thin lens power equation [5], f = (n − 1)(C1 − C2), (1.1) 2 1. Chromatic Aberrations F d C Transverse axial color Primary Axial Color (Longitudinal axial color) Figure 1.1: F (blue), d (yellow), and C (red) spectral lines and their different foci. Primary axial color is the distance between the foci for F and C.