The Binocular Handbook: Function, Performance and Evaluation of Binoculars 3031444078, 9783031444074

Citation preview

Holger Merlitz

The Binocular Handbook Function, Performance and Evaluation of Binoculars

The Binocular Handbook

Holger Merlitz

The Binocular Handbook Function, Performance and Evaluation of Binoculars

Holger Merlitz Leibniz-Institute of Polymer Research Dresden Dresden, Germany

ISBN 978-3-031-44407-4 ISBN 978-3-031-44408-1 https://doi.org/10.1007/978-3-031-44408-1

(eBook)

Translation from the German language edition: “Handferngläser: Funktion, Leistung, Auswahl” by Holger Merlitz, © Holger Merlitz 2019. Published by Europa-Lehrmittel. All Rights Reserved. © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023, corrected publication 2024 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Paper in this product is recyclable.

Foreword

Wisdom resteth in the heart of him that hath understanding. Proverbs 14:33

It was while conducting research for my own book, Choosing & Using Binoculars: A Guide for Stargazers, Birders and Outdoor Enthusiasts (Springer Nature 2023), that I first came across the work of Holger Merlitz. I was attempting to flesh out the factors which affect depth of focus in binoculars, when I hit on a 2002 Cloudy Nights post by Holger, which provided just enough information for me to ‘reverse engineer’—albeit with a little algebraic manipulation—the main factor (magnification) involved. I’ve since learned that Dr Merlitz, a senior researcher of theoretical physics at the Leibniz-Institute of Polymer Research Dresden, Germany, is a towering figure in the world of binocular optics, having published a string of influential papers—both for the amateur and professional optics community—over many years that have received international approbation. Indeed, the optics giant, Zeiss, recently adopted Dr Merlitz’s theory with an eye to designing their new SFL binoculars, employing his ‘ideal distortion profiles’ to create a uniquely comfortable panning experience. The Binocular Handbook: Function, Performance and Evaluation of Binoculars, freshly translated into the English language, is the culmination of decades of work carried out by Holger, often in his spare time. In just nine chapters, Dr Merlitz walks the reader through the fascinating world of binoculars, their nuts and bolts, the optical principles underpinning their design, as well as the role of the human eye in engaging with the binocular image. In this volume, you’ll find the answers to all sorts of interesting questions. Can a binocular generate a brighter image than that seen with the naked eye? What factors govern depth perception? Does a binocular need to have a flat transmission profile across the visible spectrum to produce images rich in natural colours? How do phase correction coatings really work? Are roof prism binoculars better than Porro prism designs? Is argon really a better gas than nitrogen in rendering a binocular fog proof? The book is divided into three neat sections. Part I covers the basic optical principles including aberrations, glass and prism types and the principles of eyepiece design. Part II delves into the fascinating subject of human vision and how it responds in bright and dim light conditions, depth and colour perception, stereopsis v

vi

Foreword

and some interesting laws that govern these parameters. Finally, in Part III, Dr Merlitz discusses the interface between the human eye and the binocular, discussing concepts such as luminance, contrast and resolution as they apply to imaging through a binocular and how the interplay between these factors changes as the ambient light changes. In addition, Merlitz offers some of the best advice I’ve seen for field testing binoculars, doubtless informed by his own extensive handson experience with a plethora of contemporary and classic binocular models. One of the great strengths of this volume is that it’s grounded on well-established physical principles, which sets it apart from unbridled speculation too often experienced in online forum discussions. The unwary novice can spend months or even years going round and round trying to comprehend an optical concept without ever gaining much in the way of understanding. This work offers a powerful panacea for such wanderings in the desert. The diligent reader will be rewarded with expert analysis, but not in a way that quickly loses the attention of the student. Holger has a real gift at making difficult concepts much easier to understand using vivid analogies from the workaday world. These skills will greatly endear the work to a wide readership, including birders, hunters, amateur astronomers or indeed anyone who wants to understand and better enjoy what he/she is seeing though their binoculars. It will also help the prospective buyer to sift through the morass of advertising hype, thereby empowering the reader to make informed decisions on purchasing an instrument before parting with their hard-earned cash. Without a shadow of a doubt, this tour de force in binocular optics is not likely to go out of date any time soon—surely a hallmark of a classic work in optical science. Fintry, Scotland, UK June 20 2023

Dr Neil English

Preface

Binoculars are fascinating instruments: once placed in front of the eyes, they allow the user to immerse himself in an altered reality. First of all, distant objects appear to be closer. But binoculars do much more than a simple magnification: the image of a good binocular is so vivid that it seems to invite the viewer to participate in what is being observed. On the contrary, a conventional telescope excludes from what is seen in its image—the observer always seems to remain an external element. Hidden behind the magic of binocular observation is the human visual perception, which unfolds its full capacity only in the context of two-eyed vision. It would therefore be misleading to judge a binocular solely by its optical and mechanical components, since its performance in practical use emerges in close cooperation with visual perception: the eye and the fieldglass, man and machine, form a symbiosis, whose characteristic properties arise from the mutual interaction between the image forming and the image processing layer. This book attempts to treat binoculars in such a broader context. In the first two parts, the technical aspects of binoculars and the properties of human visual perception are developed. Subsequently, the synthesis is carried out and the performance limits, as experienced by the observer in a variety of external conditions, are derived. Finally, this book takes the reader outdoors, where he learns to evaluate the properties and limitations of his binoculars in the field, and to recognise possible problems that may be due to manufacturing errors or accidental damages. Thus, a level of knowledge is provided that will enable the reader to fully exploit the capacities of his binoculars and to avoid bad purchases. Dresden, Germany June 2023

Holger Merlitz

vii

Acknowledgements

This book has not been written in isolation. Large parts of the content are the result of countless discussions between the author and other binocular enthusiasts and experts. In particular, the Internet with its discussion boards has made communication between people of different backgrounds possible. The author is grateful to the participants and operators of the forums on www.juelich-bonn.com, forum.astronomie.de, www.cloudynights.com and www.birdforum.net. It would be impossible to name everybody who, with his ideas and suggestions, has contributed to the knowledge that is condensed in the present book. I want to explicitly thank Walter Besenmatter, Dominique Blach, Börries von Breitenbuch, Bill Cook, Jan van Daalen, Gerhard Eller, Stefan Emsel, Beat Fankhauser, Dale Forbes, Gijs van Ginkel, Stephen Green, Albrecht Köhler, Stefan Korth, Tobias Mennle, Mathias Metz, Klaus Müscher, David W.J. Norton, Arek Olech, Andreas Perger, John Russel, Walter Schwab, Hans Seeger, Barry Simon, Volker Tautz, Hans Weigum, Heiko Wilkens, Wolfgang Wimmer and Ed Zarenski, and apologise to those which I have forgotten to mention. I am particularly grateful to Christophe Zumstein, who helped with the translation and continuously encouraged me to forge ahead with the making of this edition, as well as to Dr Neil English, who wrote the foreword for this book and more than once offered his kind support whenever I needed assistance. Last but not least, I want to thank the editorial team of Springer Nature for their professional and patient efforts to make this book possible.

ix

Contents

Part I The Technical Aspects of Binoculars 1

Optical Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Light in a Vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The Law of Refraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Refractive Index and Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Optical Glasses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Ray-Tracing a Lens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 The Image Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Lens Aberrations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.1 Longitudinal Chromatic Aberration . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.2 Spherical Aberration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.3 Field Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.4 Astigmatism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.5 Coma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.6 Lateral Chromatic Aberration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Optical Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 4 7 8 11 14 18 19 19 20 23 24 25 26 27

2

The Telescope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The Telescope of Keplerian Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Functional Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Focal Ratio. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Magnification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.4 Angle of Field and Field of View (FOV) . . . . . . . . . . . . . . . . . . . . . 2.1.5 Distortion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Telescope of Galilean Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Virtual Image and Depth of Field (DOF) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Wave Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Huygens’ Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Diffraction-Limited Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29 29 29 31 31 32 33 35 36 39 39 40

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2.4.3 Anti-Reflective Coatings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42 44

3

Image Erecting Prisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Porro-Type Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Total Internal Reflection (TIR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Porro I System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Porro II System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.4 Perger Prism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Roof Prisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Schmidt–Pechan Prism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Abbe-König Prism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Uppendahl Prism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Mirror–Prism Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.5 Reflective Coatings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.6 Phase-Shift and Its Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45 45 45 47 49 51 54 54 57 59 60 62 63 66

4

The Anatomy of Binoculars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Objective Lenses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 The Achromatic Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 ED Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Apochromatic Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Eyepieces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Simple Eyepieces for Narrow Subjective Angles of View . . 4.2.2 Wide-Angle Eyepieces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Focus Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Three Different Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Observations at Close Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Dimensions of the Ray Bundle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Reduced Path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Dimensions of the Intermediate Image . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Prism Entrance Face Width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 8x42 and 10x42 Binocular . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 8x30 and 6x30 Wide-Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.3 7x50 (Super-) Wide-Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Aperture and Field Stop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Image Stabilisation Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Sealing and Purging of Binoculars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67 67 67 69 70 71 72 73 76 77 79 82 83 84 85 86 86 87 88 90 92 95 98

5

Report on a Self-made High-Performance Binocular . . . . . . . . . . . . . . . . . . . . 99 Gerhard Eller Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

Contents

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Part II Elements of Human Vision 6

The Eye. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 The Eye as an Optical Instrument. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 The Field of Vision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 The Pupil Width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 The Accommodation Width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Aberrations of the Eye. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 The Retinal Photoreceptors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Day and Night Vision. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8 Twilight Vision: The Stiles–Crawford Effect. . . . . . . . . . . . . . . . . . . . . . . . . 6.9 Retinal Data Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

107 107 108 109 111 112 114 116 117 118 121

7

The Visual Perception . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Laws of Visual Perception . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 The Laws of Ricco, Piper and Weber–Fechner . . . . . . . . . . . . . . 7.1.2 Berek’s Model of Target Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 The Perception of Colour. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Stereoscopic Depth Perception. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Further Mechanisms of Depth Perception. . . . . . . . . . . . . . . . . . . . 7.4 Saccadic Image Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 The Optical Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 The Visual Distortion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

123 123 124 126 130 132 134 135 136 137 140

Part III Binocular Performance and Its Evaluation 8

Eye and Binocular: The Man-Machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Luminous Flux and Magnification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Performance: Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Performance: Handheld vs. Mounted . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Performance: Target Sighting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Which Approach to Binocular Performance is Most Relevant? . . . . . 8.6 Night Sky Performance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.1 Stellar Magnitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.2 Astro Indices for Limiting Magnitudes. . . . . . . . . . . . . . . . . . . . . . . 8.6.3 Limiting Magnitudes in Berek’s Model . . . . . . . . . . . . . . . . . . . . . . 8.6.4 Comparison with Observation Data . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Atmospheric Scatter and Seeing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8 Visual Transmission, Colour Contrast and Saturation . . . . . . . . . . . . . . . 8.9 Depth of Field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.10 Depth Resolution and Cardboard Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.11 Distortion and Globe Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.12 The Search for the Ideal Distortion Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

143 143 146 149 151 154 155 155 156 158 159 162 165 168 171 174 176 180

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9

Contents

Binocular Evaluation and Field Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Laboratory Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 A Quick Test in- and Outside the Store . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 First Impression: Design, Ergonomics, Haptics . . . . . . . . . . . . . 9.2.2 Checking for Additional Rejection Criteria . . . . . . . . . . . . . . . . . . 9.2.3 Evaluating Optical Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Field Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Resistance to Stray Light. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 Ghost Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.3 Off-centre Sharpness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.4 Low-light Performance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.5 Chromatic Aberration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.6 Ease of View: The Unspeakable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.7 Ergonomics and Haptics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

181 181 184 184 185 189 191 191 195 198 200 201 202 205 206

Correction to: Report on a Self-made High-Performance Binocular . . . . . .

C1

A Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1 Technical Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Visual Perception . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3 Binoculars and Scopes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

207 207 208 208

B Picture Credits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

Part I

The Technical Aspects of Binoculars

Chapter 1

Optical Imaging

1.1 Light in a Vacuum Visible light is the term used to describe electromagnetic radiation in the wavelength range between about 400 nm and 700 nm. This visible range is actually only a very small part of a much broader electromagnetic spectrum, which extends into the short-wave regime, through the UV (ultraviolet) into the X-ray range, up to gamma radiation and into the long-wave regime via the IR (infrared) and microwaves up to radio waves. The classical theory of electromagnetic radiation was developed by Maxwell in 1864 in the form of the famous equations named after him, and its modern incarnation, quantum electrodynamics, dates from the late 1940s. According to both approaches, light in a vacuum exhibits no dispersion, i.e. the wavelength .λ is related to the frequency .ν via a constant factor c, λ·ν =c ,

.

(1.1)

known as the speed of light in vacuum, .c = 2.997 924 58 × 108 m s−1 . The visible spectrum encompasses the colour range between the short-wave violet to long-wave red. For the definition of selected wavelengths, one traditionally uses the Fraunhofer-lines, which the optician Fraunhofer1 observed in the spectral decomposition of sunlight for the first time around 1814, but which had been described before by Wollaston2 in 1802 (Fig. 1.1). These absorption lines are caused by various chemical elements, which exist in traces in the atmospheres of the sun and earth. If light passes through these atmospheres on its way into the observer’s spectrometer, these elements are excited at very specific frequencies at which the intensity of light appears weakened. The excited ions re-emit light of the same wavelengths, but in random directions, resulting in an observed dimming in the

1 Joseph

von Fraunhofer, 1787–1826. Hyde Wollaston, 1766–1828.

2 William

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Merlitz, The Binocular Handbook, https://doi.org/10.1007/978-3-031-44408-1_1

3

4

1 Optical Imaging

Fig. 1.1 Spectral decomposition of visible light with Fraunhofer lines

direction of the light source, i.e. the sun. Since these lines are easily generated in the laboratory, they are ideal for measurement purposes and for the calibration of instruments. The most common absorption lines are named with letters, for example the G-line of iron (430.790 nm), the D double line of sodium (588.997 nm and 589.594 nm) or the C-line of hydrogen (656.281 nm).

1.2 The Law of Refraction Geometrical optics is a field that deals with the behaviour of light in different media. It is called geometric, because it is based on the assumption that the propagation of light can be described as a ray, whose trajectory obeys certain geometric laws. This is remarkable, because light is a rather complex electromagnetic oscillation phenomenon, obeying Maxwell’s field equations. Geometrical optics nevertheless works amazingly well, which has to do with the fact that the wavelengths of visible light are usually much smaller than the optical components we have to deal with in everyday life. The core of geometrical optics is the law of refraction, which became popular in Europe through the work of the Dutchman Willebrord van Roijen Snell in the early seventeenth century, after whom it is nowadays named as Snell’s law of refraction.3 It provides the relationship between the angle of incidence and the exit angle of a light ray at the interface of two transparent media with different refractive indices n1 and n2 . Referring to Fig. 1.2 one obtains .

n2 sin α = , sin β n1

(1.2)

3 Apparently, the law was already known during the tenth century in the Persian area, as it appears on a handwritten text by Abu Sad al-Ala ibn Sahl (about 940–1000). See also the discussion in the book by Iqbal [1].

1.2 The Law of Refraction Fig. 1.2 A beam of light (red) at the interface between two transparent media of different optical densities. Refraction occurs (solid line) as well as reflection (dashed line)

5

n1

α α

n2 β

where the angles are taken to the perpendicular (or normal to the surface) of the interface. The refractive index of a vacuum is exactly n = 1, and that of air under normal conditions amounts to n = 1.0003, practically the vacuum value. Since optical glasses have far higher refractive indices than that, we will be able to set the refractive index of air to n = 1 without sacrificing the accuracy of the results. For reflection, the simple law of reflection applies: angle of incidence = angle of exit. That is about all is required—starting from the law of refraction and the law of reflection, we are in a position to derive the course of rays through most complicated optical systems by calculating the angular ratios of incoming and outgoing rays at each point of incidence on the respective interfaces. This procedure is called raytracing and was largely carried out manually until the 1940s. At that time, the optical calculating bureaus employed human ‘computers’ whose tedious task was to track ray bundles with pencil and ruler through optical arrays, supported by mechanical calculating devices. Only after the Second World War were program-controlled computers increasingly employed for ray-tracing tasks. In Germany, Ernst Leitz AG in 1953 put a Zuse Z5 machine into operation for the purpose of computer-aided lens design, followed by the East German Carl Zeiss Jena company (1955, an inhouse computer design by Wilhelm Kämmerer, named OPREMA) and Zeiss (West) in Oberkochen (1956, Zuse Z11) [2]. For illustration, we calculate the path of a light beam through a planar glass plate—a simple case of ray-tracing, which already reveals some of the typical properties of an optical element (Fig. 1.3). The glass plate has the refractive index n and a thickness d, and the light beam (red line) an incident angle α to the perpendicular (green line). The angle of refraction inside the optically denser glass then results from the law of refraction to ( ) sin α (1.3) .β = arcsin . n Inside the glass plate, the light beam covers a distance of h = d/ cos β. When the beam leaves the glass plate, it passes from the optically dense medium into the optically thinner air, and the exit angle is identical (α) to the one at which it had previously hit the plate. Beyond the glass plate, light rays thus propagate in the same direction as before, but they are offset by a certain amount. This parallel displacement p can be easily computed from the triangle enclosing the difference

6

1 Optical Imaging

Fig. 1.3 A glass plate of thickness d is traversed by a light beam. The angle of incidence to the perpendicular (green) is α, the parallel offset of the outgoing beam is p

angle α − β, whose hypotenuse h is the light path in the glass (d/ cos β) since sin(α − β) = p/ h. It then follows that p = h sin(α − β) = d

.

sin(α − β) . cos β

(1.4)

So let the thickness of the glass plate be 5 cm, the angle of incidence 30° and the refractive index n = 1.57, corresponding to a BaK4 glass which is commonly used in image erecting prisms. With these input data, we then obtain an angle β = 18, 57◦ and thus the parallel offset to p ≈ 1.05 cm. It has been common in technical optics to use approximations in order to simplify the resulting formulas. Trigonometrical functions can be expanded into power series, such as for sine .

sin x = x −

x3 x5 − ... , + 5! 3!

(1.5)

.

cos x = 1 −

x2 x4 − ... , + 4! 2!

(1.6)

or for cosine

where the angle must be entered in radians, i.e. the angle (in degrees) multiplied by π and divided by 180. In fact, each pocket calculator computes its builtin trigonometrical functions with the help of these series expansions, which in principle have an infinite number of terms, but which in practice are truncated after a couple of terms, once a sufficient degree of accuracy has been reached. During the times in which these calculators were not yet available, Gauss4 developed a 4 Carl

Friedrich Gauss, 1777–1855.

1.3 Refractive Index and Dispersion

7

computation scheme for his optical calculations which employed only the leading order (linear) terms of the expansions, e.g. sin x ≈ x and cos x ≈ 1. This framework, which is accurate only for small angles, is known as Gaussian optics. In the Gaussian optics approximation, the law of refraction simplifies to β = α/n, and the parallel offset to ) ( 1 , p ≈dα 1− n

.

(1.7)

which for our example (d = 5 cm, α = 30◦ and n = 1.57) yields 0.95 cm, quite far from the more accurate result of 1.05 cm, since the angle of incidence was not exactly small. Let us repeat the calculation with an angle of incidence of 4°, then we obtain the parallel offset to 0.1269 cm (exact calculation) and to 0.1267 cm (Gaussian approximation), which are almost identical. If Eq. (1.4) is expanded up to terms that include third powers of the angle α, the computation scheme results in what is known as Seidel’s error theory. The Seidel approximation allows an in-depth analysis of some of the higher-order aberrations, which are excluded from the simplified equations of the Gaussian theory. Obviously, in modern, computer-driven optical design these different levels of approximation have become obsolete—equations are number-crunched to any level of accuracy within fractions of seconds. Yet, traditional approximations are still applied in order to classify the primary sources of imaging errors and thus support the evaluation of the properties of a new optical design.

1.3 Refractive Index and Dispersion The above-mentioned refractive index n is a material property of glass. Light is an electromagnetic wave that stimulates the countless atoms of the glass to oscillate— to be more precise, it is the atomic shells, formed by electron clouds, which oscillate in the electromagnetic field. During this process, the electrons are not excited, i.e. not lifted to higher energy levels, since if that happened, the light would be absorbed and the glass would no longer remain transparent. Now the electron shells have their own preferred oscillation frequencies, called natural frequencies. Since the frequency range of the incident light does not coincide with this natural frequency, the atoms cannot oscillate exactly in phase with the electromagnetic wave, and through their own oscillations, they themselves generate a secondary electromagnetic field that is superimposed on that of the incident light. Thus, a new effective field is generated, which is associated with an altered, frequency-dependent phase velocity. This is the reason why light, when passing through a transparent medium, alters its phase velocity, which leads to the observed refractive index and to deflections, as described by Snell’s law. From what has been said, it is understandable why the exact value of the refractive index depends not only on the material, but also on the light: The phase shifts depend on the ratio between the natural frequency and on the (vacuum-)

8

1 Optical Imaging

frequency of the incident light. The dependence of the refractive index on the wavelength is called dispersion. In most cases, the refractive index increases with the frequency, but exceptions occur near the natural frequency: close to the resonance, the refractive index drops with increasing frequency, a phenomenon called abnormal dispersion, which is of great practical importance for the correction of chromatic aberrations in optical systems. How exactly then does the refractive index depend on the wavelength .λ of the light? This relationship is specific to each type of glass, but it is so complex in detail that there exists no general law derived from first principles (in other words: it is not yet fully understood). For this reason, technical optics relies on series expansions of the form n2 (λ) = A0 + A1 λ2 + A2 λ−2 + A3 λ−4

.

+ A4 λ−6 + A5 λ−8 .

(1.8)

When this formula is fitted to the measured dispersion curve of a particular glass sample, the empirical coefficients .A0 . . . A5 are determined and subsequently listed in the glass catalogues of the manufacturers.

1.4 Optical Glasses So if the refractive index depends on the wavelength, how can it be given for each individual type of glass? For this purpose, it has been agreed that the yellow d-line of (.λd = 587.6 nm) wavelength would serve as the basis of the so-called d-system. The refractive index .nd at this wavelength is defined as the principal refractive index.5 The principal dispersion is the difference between the refractive indices in the blue and in the red spectral range, more specifically between the F-line (.nF , 6 .λF = 486.1 nm) and the C-line (.nC , .λC = 656.3 nm). Finally Abbe introduced the quotient v=

.

nd − 1 , nF − nC

(1.9)

known as the Abbe number. Glasses with low dispersion (ED-glasses) then exhibit a high Abbe number. In the glass catalogues of the manufacturers, the different glass types are listed in two-dimensional diagrams, the axes of which being the Abbe number and the principal dispersion (Fig. 1.4).

exists a second convention, which is based on the e-line of mercury (.λe = 546.1 nm) and consequently called the e-system. 6 Ernst Karl Abbe, 1840–1905. 5 There

1.4 Optical Glasses

9

Fig. 1.4 Optical glasses of the Schott catalogue, listed after principal refractive index and Abbe number. Copyright (©) SCHOTT AG, Advanced Optics

In addition to these specifications, the catalogues also contain other, qualityrelated classifications. It is worth remembering that no glass block can be perfect, but has inhomogeneities, streaks, inclusions and suffers mechanical stresses. Glasses are classified according to different levels of refractive index homogeneities, according to deviations from the specified refractive index or according to stressinduced birefringence.7 Note that the catalogues also give details of the internal transmittance: it refers to the proportion of light leaving the glass sample relative to the proportion that enters the sample; reflection losses at the interfaces between different media are not taken into account. However, this tabulated information often refers to the violet 400 nm line—for an evaluation of the transmission in the visual range this information is unsuitable, because the eye is no longer sensitive inside this short-wavelength region. Instead, of relevance for visual instruments is the visualtransmittance class: the internal transmittance of a 10 cm glass path is averaged over the entire spectrum of visual light and the glass sample subsequently classified into one of 10 quality levels as shown in Table 1.1.8 It would therefore be incorrect to assume that each glass type comes with its specific visual transmission. The buyer of a glass blank of given type has the choice between these quality classes, whereby class 4 is considered standard, class 3

7 Birefringence is here an unwanted phenomenon, a dependence of the refractive index on the polarisation and the direction of light. 8 Thanks to Albrecht Köhler for these data.

10 Table 1.1 Visual-transmittance classes of optical glasses: guaranteed internal transmittance .τvi , averaged over the spectrum of visible light through a distance of 100 mm of glass path

1 Optical Imaging Quality class 9 8 7 6 5 4 3 2

.τvi .≤

0.859 0.860 0.880 0.900 0.920 0.940 0.960 0.980

Fig. 1.5 Internal transmittance: conventional BK7 and BaK4 glasses and the new HT-variants, converted to 10 cm glass path (visible light: 400–700 nm; data courtesy of SCHOTT AG, Advanced Optics)

is considered high quality and glasses of transmission class 2 have to be picked from carefully selected glass melts, are expensive and not always available. Hightransmission classes are of particular relevance for binoculars, as these incorporate image erecting prisms with typical glass path lengths of 10 cm or more. Progress is still being made in this field today, as the recently introduced HTglasses (HT for ‘high transmission’) from Schott suggest. Figure 1.5 shows a comparison of the internal transmittance of conventional BK7 and BaK4 glasses with their corresponding HT variants. The results are shown for the highest visual transmittance class that is available as standard, usually class 3 for the majority of conventional glass types. Within the interval 400 nm to 700 nm, there is an average advantage for BK7 HT over conventional BK7 of at least 1.2 % following a glass path of 10 cm. BaK4, which is a glass of high refractive index and used in most image erecting prisms, is somewhat less transparent in the blue spectral range—the new HT version achieves some improvements here, between 400–500 nm, but offers no further advantages within the rest of the visible spectrum. At least the somewhat flattened transmittance curve should provide an improved colour neutrality than its conventional BaK4 counterpart, which is known for its slightly warm (yellowish) colour tint. Whether the investment in expensive HT glasses is worthwhile has to be decided for each individual optical design. In the binocular industry, the use of special optical glasses, in particular for military optics, was not uncommon. These included glass types which achieved a high refractive index through the addition of lead, to be used for the image erecting

1.5 Ray-Tracing a Lens

11

prisms of wide-angle optics (Sect. 3.1.1), or exhibited an improved resistance to radioactivity through the addition of cerium oxides. Most of the rather toxic special glasses that were used in earlier times have since been replaced by environmentally friendly glasses. Of particular relevance for the optical design of camera lenses, telescopes and binoculars has been the gradual replacement of fluorite crystals (CaF.2 ), which are difficult to coat and to work mechanically (because they are brittle), by special glasses which include fluoride ions (commonly known as ‘ED glasses’, Sect. 4.1) with similar abnormal dispersion characteristics but superior mechanical properties. Unfortunately, modern-day marketing adds to the confusion among the technical terms. For example, some manufacturers advertise the use of ‘HD-glass’, where ‘HD’ stands for ’high definition’. Such types of glass do not exist in the databases of any glassworks—a high image definition is the result of a good optical design in combination with precise manufacturing.

1.5 Ray-Tracing a Lens Section 1.2 has described the ray-tracing through a planar glass plate, by which the light beam has been found to be offset but unchanged in its direction. In order to build optical systems that can focus light, we will need an optical element featuring a curved surface—a lens. We shall now apply our trigonometrical methods to compute the path of rays through a lens with one planar and one spherical surface. We construct the lens as shown in Fig. 1.6 by taking a sphere of glass with radius R and cutting off a cap of thickness d. Even though the figure shows a complete sphere for illustration, we assume that only the spherical cap on the left side is made of optical glass. We have thus made a plano-convex lens, one surface of which being planar, and the other, spherically curved outwards. Almost all lenses used in binocular design have spherical surfaces, because they can be ground easily and with high precision. Other, aspherical lens shapes are possible, but are rather difficult to shape and expensive (if made of glass) or of limited quality (if moulded from plastic). As the principal axis or optical axis we denote the straight line passing from left to right through the lens vertex and the centre of the sphere. Light rays that enter the lens along the principal axis are passing straight through without being deflected. We now assume that all light rays entering on the left are parallel to each other and to the principal axis, as would be the case with binoculars that focus on a distant point-like object (such as a star) in the centre of the image. A bundle of rays with these properties is referred to as the principal beam. The subset of rays which run very close along the principal axis (for which the Gaussian approximation is valid) are called paraxial rays. As an example, Fig. 1.6 shows a single ray (red) that hits the lens from the left at a height h above the principal axis. At this point it is deflected according to the law of refraction, passes through the lens along a straight path and exits the lens at the

12

1 Optical Imaging

Fig. 1.6 Plano-convex lens, a spherical cap of thickness d, cut-off from a glass sphere of radius R. We are looking for the distance between the planar lens surface and the focal point, .Fbf , known as the back focal length. The focal length F is the distance between the principal plane H and the focal point

planar surface, where it is deflected once again. We are interested in the distance from the planar lens surface at which it intersects the principal axis, a distance known as back focal length. The point of intersection is called principal focal point F, or, more precisely, the secondary principal focal point since there exists another (primary-) principal focal point to the left of the lens. First, we need the angle of incidence to the surface normal at the point of incidence—the latter runs along the radius through the centre of the sphere (green dashed line), and we can easily obtain from trigonometry ( ) h . .α = arcsin R

(1.10)

On this occasion, we also calculate the length of the blue triangle side y (see Fig. 1.7 for a detailed view) to be .y = R cos α. In the glass, the light beam is then deflected according to the law of refraction, and with the refractive index n, the new angle ( β = arcsin

.

sin α n

) (1.11)

1.5 Ray-Tracing a Lens

13

Fig. 1.7 Light path through the plano-convex lens

emerges, again with respect to the surface normal. The ray is then picking up a vertical offset, which we can calculate with the help of the red triangle: the acute angle .γ is equal to the difference .α − β, and the adjacent to .γ (the lower side of the triangle) has a length of .d − (R − y), hence the offset is v = (d − R + y) tan(α − β) .

.

(1.12)

At this point the light beam exits the glass, and we need the exit angle .δ, to which we again apply the the law of refraction, so that δ = arcsin[n sin γ ] = arcsin[n sin(α − β)] .

.

(1.13)

Now it is almost done: we only need the horizontal distance to the lens at which our light beam intersects the principal axis, or the vertical shift exactly offsets the incident height h. Since the vertical shift has already reached v at this intersection point, the remaining offset to be covered beyond the lens is .h − v and we obtain the back focal length Fbf =

.

h−v . tan δ

(1.14)

14

1 Optical Imaging

If we insert the previously computed quantities, we get Fbf =

.

h − [d − R(1 − cos α)] tan(α − β) , tan {arcsin [n sin(α − β)]}

(1.15)

where .α is shown in Eq. (1.10) and .β in Eq. (1.11). Thus we have a value for the focal point of a ray as a function of the height of incidence h. Obviously, the entire bundle of rays, when covering a wide range of incident heights, does not focus at a single distance .Fbf from the lens surface and there is no unique focal point. The reason for that is related to a group of imperfections of the imaging properties of a single spherical lens, commonly denoted as aberrations. We are going into the details behind these aberrations in the following sections. Since this equation looks rather complicated, let us first take the Gaussian approximation for paraxial rays—here the height of incidence is small, and so are all the angles of deflection. Therefore, all trigonometrical functions are expanded into the power series (1.5) and (1.6) and terminated after the linear term. The same is done with the inverse sine function: .arcsin x ≈ x. One obtains after a short calculation the Gaussian approximation Fbf ≈

.

d R − , n−1 n

(1.16)

in which the height of incidence no longer appears at all. In other words: for paraxial rays, the back focal length does not depend upon the height of incidence of the ray. The second term of this approximation still contains the thickness d of the lens. For this reason, Gauss had proposed to define the equivalent focal length, in short focal length, as the distance between an imaginary plane, shifted by .d/n to the left, and the focal point, such that the focal length becomes independent of the lens thickness. This proposal was indeed very sensible, the new, shifted reference plane is called the principal plane H (Fig. 1.6) of the lens, and by using these principal planes, the raytracing of complex optical systems in the Gaussian approximation is considerably simplified. The attentive reader may have already come to the conclusion that each lens should actually feature two principal planes—one referring to light that enters from the left, and another one for light that hits the lens from the right. This is correct, and in general both principal planes do not coincide. We will return back to that point shortly.

1.6 The Image Equation We now calculate the axis-parallel ray path of the plano-convex lens in the reverse direction (Fig. 1.8): the ray of light (red) hits the planar surface, where it is not refracted because it enters parallel to the normal, to intersect the principal axis after

1.6 The Image Equation

15

Fig. 1.8 Plano-convex lens in which a beam of light impinges parallel to the optical axis into the planar surface. In this case, the principal plane H lies on the lens vertex

exiting the lens at the back focal distance .Fbf . The details of the calculation, which are analogous to the one in the previous section, are left to the reader and lead to Fbf =

.

h − R(1 − cos α) , tan(β − α)

(1.17)

or, in paraxial approximation, to Fbf = F ≈

.

R , n−1

(1.18)

which is identical to the result from Eq. (1.16), after the offset of the principal plane had been accounted for. What we learn from this exercise: the principal plane now falls exactly onto the lens vertex, so that focal length and back focal distance coincide. When related to the respective principal plane, the focal length of any lens remains invariant when the lens is passed in reverse direction. This holds true as long as the lens is surrounded on both sides by the same medium (in this case: air). More important is the fact that the concept of principal planes simplifies the optical path through optical systems to such an extent that the exact nature of the lens or lens group is no longer of relevance. It suffices to know the positions of the principal planes and the focal length of each optical element to construct the beam path. Figure 1.9 demonstrates the principle: given a lens or lens group with its two principal planes H (object-side) and .H ' (image-side) and the corresponding focal points .Fp and .Fp' . The object-side focal length F is the distance between H and ' .Fp , identical in magnitude with .F and negative according to the sign convention in ' technical optics, so that .F = −F holds. Let an object be located at the distance z (which by convention is also negative) from the principal plane H , having a height y. The distance between object and focal point is called focal-point-related object distance .zF = z − F .

16

1 Optical Imaging

Fig. 1.9 Construction of the beam path with two construction rays: The arrowhead of the object (left) and its image point (right). The entire construction is fully symmetric, object and image are interchangeable

The image is constructed with two beams as follows: an axis-parallel ray from the left is refracted at the principal plane .H ' such that it passes through the focal point on the image side. A second ray, which passes through the focal point .Fp on the object side, is refracted at the object-sided principal plane H in such a way, that it thereafter continues parallel to the optical axis, which is a compelling consequence of the fact that the ray path could as well be constructed from the image towards the object. This fact is known as the law of reciprocity, the reversibility of ray paths, which in the framework of geometrical optics (though not in wave optics) is strictly satisfied. Now it follows on the object side due to the similarity of the triangles 1 and 2 that .y/y ' = zF /F , and on the image side accordingly .y ' /y = zF' /F ' , thus .

z' − F ' F z' = = −1, z−F F' F'

(1.19)

F' 1 F = =1− ' F z z 1 + z−F

(1.20)

which we rearrange to .

1.6 The Image Equation

17

With .F = −F ' (object and image are in the same optical medium) we then obtain the paraxial image equation .

1 1 1 − = ' , ' z z F

(1.21)

which relates the object distance z to the image distance .z' and the focal length .F ' . This equation is extremely useful during initial design stages of an optical system, because it offers an overview of the course of the optical path, possible bundle truncations as well as the mechanical dimensions of the optical assembly prior to the in depth ray-tracing analysis. Note that in the case of distant objects (a rather common situation in daily life binocular practice) the factor .1/z approaches zero and we obtain .z' ≈ F ' , and the image lies close to the principal focal plane of the lens. With the aid of the focal point-related object and image distances we derive from Eq. (1.19) the simple relation .

z' F = F' , zF F

(1.22)

which transforms into the Newtonian image equation9 zF zF' = F F '

.

(1.23)

and in case of .F = −F ' further simplifies to zF zF' = −F 2 .

.

(1.24)

In our example (Fig. 1.9), we have oriented ourselves to a converging lens, which maps any object that is located beyond the focal point .Fp , into its real image on the opposite side of the lens. If the object is closer than .Fp to the lens, the formation of a real image is no longer possible, because the light rays diverge after having passed through the lens. In this case, the construction rays have to be extended backwards in the direction of the object (to the left) to the point at which they intersect. At this point, the virtual image is formed. The image equations (1.21) and (1.24) remain valid for this case, provided that the sign convention is observed and distances to the left of the lens are given a negative sign. Furthermore, there exist diverging lenses which exclusively produce virtual images. Their focal lengths are determined with the help of incident construction rays parallel to the optical axis, which again are extended backwards onto the object side until they intersect. According to the sign convention, diverging lenses have negative focal lengths and thus are referred to as negative lenses.

9 Sir

Isaac Newton, 1643–1727.

18

1 Optical Imaging

As a first application of the image equation, we want to calculate the focal length of a binocular with a focusing mechanism at its eyepieces (see also Sect. 4.3). To do this, we first focus on a close object at the distance .z1 and measure, as accurately as possible, the position of the eyepiece in relation to some prominent point on the binocular body, e.g. the prism cover plate. We note its value, .x1 . Now we repeat the measurement with a distant object (whose distance .z2 is sufficiently large so that we can safely set it to infinity) and once again take the eyepiece position, with the result of .x2 . Its difference, the eyepiece travel .x1 − x2 = Δx, corresponds to the distance .z1' − z2' between the two image planes. With a little algebra, we obtain from the imaging equation (1.21) the focal length / F' =

.

(δx)2 δx − z1 δx − , 4 2

(1.25)

which we can simplify, without significant loss of precision, to F' ≈

.

/ δx −z1 δx − . 2

(1.26)

For an Opticron 8x30 binocular, when focusing on an object 3 m away (.z1 = −300 cm), I measure the eyepiece distance to the prism cover plate to be .x1 = 7.124 cm, and when focusing to infinity, .x2 = 6.688 cm, a focusing travel of ' .Δx ≈ 0.436 cm results, thus yielding the focal length .F ≈ 11.2 cm. While performing these calculations, we must recall that the framework of the image equation (1.21), being based on the Gaussian (paraxial) approximation, simplifies the properties of the optical elements. An analysis of the aberrations is not feasible on this level of approximation, which is why in the following sections we once again return to the ray-tracing procedures.

1.7 Lens Aberrations Light rays passing through a lens do not converge to a single focus. A careful analysis of this simple computational fact leads to a distinctive set of conditions, known as aberrations, which may be studied separately to better understand the origins of these imperfections. In the following sections, we are going to perform such an exercise for the plano-convex lens which we have ray-traced in Sect. 1.5. We do this in order to obtain an intuitive and qualitative understanding of how such a lens bends light—for a thorough treatment of aberrations, and what these aberrations do to the image of a point-like object, I refer the reader to the specialised literature [3].

1.7 Lens Aberrations

19

1.7.1 Longitudinal Chromatic Aberration Within the Gaussian approximation, the location of the focal point is a function of the refractive index n, and since every glass has a dispersion (n being a function of the wavelength), the focal points of different wavelengths are at different locations. Let us take as an example a standard glass, the SF2 flint glass from the Schott catalogue. The refractive index of the red C-line (656.281 nm) is given as .nC = 1.642 10, and for the violet h-line (410.175 nm) .nh = 1.682 33 holds. If we apply Eq. (1.18), assuming a radius of curvature of .R = 10 cm, then paraxial rays of the red C-line intersect with the optical axis at .FC = 15.58 cm, while those of the violet h-line have their intersection point at .Fh = 14.66 cm, and so a considerable difference exceeding 9 mm emerges. This difference in the focal length of paraxial rays is known as longitudinal chromatic aberration or colour aberration. The resulting colour spectrum is also called the primary spectrum of the lens, and objects in the central part of the field develop colourful fringes or halos. Next we take the FK56 glass from the same catalogue, a fluoride crown glass of particularly low dispersion, and find .nC = 1.432 85, .nh = 1.441 85, and consequently .FC = 23.10 cm and .Fh = 22.63 cm, though at a significantly increased focal length—a consequence of the reduced refractive index. In general, the focal length of a lens is set already at the initial stage of the layout, and in order to restore the desired focal length of 15 cm, the radius of curvature of the lens must be reduced to about .R = 6.7 cm. With these modified specifications, we obtain .FC = 15.48 cm and .Fh = 15.16 cm, i.e. a focal length difference of roughly 3 mm. Even with this glass material, the secondary spectrum would be prohibitive to form an image that remains well defined over the spectral range of visible light. To make matters worse, with the now smaller radius of curvature of our FK56 lens, the remaining aberrations would increase. Thus, in practice, there is no way to make objectives out of single lenses; compound lenses that are composed of several, well matching glass types are needed for visual optical instruments. Such achromatic lenses are the technical standard in binocular design and will be discussed in detail in Sect. 4.1.

1.7.2 Spherical Aberration As already mentioned, the focal length in the Gaussian approximation (1.16) does not depend on the height of incidence h, but that changes with the exact formula (1.15). It is thus obvious that for an assessment of the image quality, it is essential to go beyond the Gaussian approximation. The next approximation of higher order is that due to Seidel, in which all terms up to the third power of the angles

20

1 Optical Imaging

Fig. 1.10 Spherical aberration: the focal distance F as a function of the incidence height h (see also Fig. 1.6)

are considered. We are not going to go down that road, which had been paved centuries ago in order to simplify numerical calculations, but directly apply the formula (1.15), which today is evaluated conveniently by computer. In Fig. 1.10 the intersection distance F is shown as a function of the height of incidence h, for a monochromatic light ray, refractive index .n = 1.5, radius of curvature .R = 10 cm and lens thickness .d = 1 cm. The focal point changes with the height of incidence h, due to an aberration which is called spherical aberration or aperture error. An object in the field centre exhibits an out-of-focus halo, also called the circle of confusion, a phenomenon that can be reduced by stopping down the lens. Any spherical lens surface necessarily leads to spherical aberration, to be corrected by a suitable modification of the shape of the lens curve (aspherical lens). This significantly increases the production costs, and since several spherical lenses can be combined to eliminate the spherical aberration, aspherical surfaces are virtually absent in binocular design, with the exception of certain high-performance wideangle eyepieces.

1.7.3 Field Curvature Up to now, we have dealt with an axis-parallel incident ray bundle, e.g. from a star, located in the field centre. Since every optical instrument encompasses a certain angle of view, images of objects are not restricted to the centre, but scattered all about the field of view up to the field edge. A typical value for the objective halfangle of field amounts to 4°, and this corresponds to the inclination of the incident ray bundle (with respect to the optical axis) of an object that is seen near the edge of field. Let us call this angle of incidence .e. We are interested in the location at which two light rays from the same object intersect on the image side of the lens (Fig. 1.11). This is not going to happen on the optical axis, because the image of the star is no longer in the centre of field, but offset by the image height .h' . To calculate the image point, we assume that the two incident rays are incident on the lens at the heights h (above the optical axis) and .−h (below the optical axis).

1.7 Lens Aberrations

21

Fig. 1.11 Light rays strike the lens at an angle .e to the principal axis, so that the image is not formed on the axis, but at the image height ' .h

After passing through the lens they exit at heights .y1 and .y2 at the angles .δ1 and .δ2 to the optical axis, to finally meet again at the intersection point with coordinates x and .h' . In the following calculation it should be noted that both .y2 and .δ2 become negative since the ordinate of the incident ray (.−h) is also negative. This yields the intersection coordinate y1 − y2 , tan δ1 − tan δ2

(1.27)

h' = y1 − x tan δ1 ,

(1.28)

yi = hi − [d − R(1 − cos αi )] tan(αi − βi )

(1.29)

x=

.

at an image height of .

where .

for the two indices .i ∈ {1, 2}, and the angles (

hi .αi = arcsin R

) (1.30)

,

as well as ( βi = arcsin

sin(αi + e) n

) ,

(1.31)

δi = arcsin [n sin(αi − βi )] .

(1.32)

.

and .

22

1 Optical Imaging

Fig. 1.12 Field curvature: varying the angle of incidence ◦ .e (Fig. 1.11) between .+4 and .−4◦ makes the image point move along a curved image shell

Figure 1.12 shows a curve, actually a sequence of image points, which results when the angle of incidence .e is varied in small steps from .+4◦ to .−4◦ . Imagine that we are dealing with a star that is slowly moved from one edge of the field through the centre to the opposite edge. Lens parameters featured a radius of curvature of .R = 10, a lens thickness of .d = 1 and the heights of incidence of the two construction rays were set to .h1 = 2 and .h2 = −2, respectively. The images of the star do not fall onto a plane, but onto a shell. This phenomenon is called field curvature and as a consequence of this aberration, the field as a whole cannot simultaneously be brought into focus: when focused on a star close to the centre of field, the outlying stars remain out of focus, and after bringing the latter into focus, the centre is blurred. The field curvature is eliminated when the radius of curvature .rp of the image shell, sometimes called Petzval-shell10 becomes infinitely large. The latter can be calculated via the petzal sum E 1 1 = rp ni fi' k

.

(1.33)

i=1

which runs over all optical elements of the system with their respective refractive indices .ni and individual focal lengths .fi' . If the design succeeds in eliminating the Petzval sum (and the residual astigmatism is made sufficiently small), then the resulting image falls onto a plane. In binocular optics, a so-called field-flattening lens (or lens group) is employed for this purpose, also called Smyth lens (see also Fig. 4.9). This is a negative lens, placed in front of the real image produced by the objective, to contribute a negative term to the Petzval sum and thus to compensate for the positive contributions from eyepiece and objective. In all these considerations, for the sake of simplicity, we have cheated somewhat, since the heights of incidence .hi were kept constant. In reality, the light bundle

10 After

Josef Maximilian Petzval, 1807–1891.

1.7 Lens Aberrations

23

would cover the entire entrance aperture, and the field curvature would become superimposed with spherical aberration.

1.7.4 Astigmatism In the previous section, we had tacitly made some simplifications: Fig. 1.11 shows only a cross section of the lens, and the ray paths hit the lens on a twodimensional plane, which hosts both the object and the principal axis. This plane is called meridional plane or tangential plane and is shown in purple in Fig. 1.13. Additionally, there exists a plane of incidence perpendicular to the meridional plane, as shown in green in Fig. 1.13, which is known as sagittal plane. The latter does not lie on the principal axis once the object is off-centre. Three-dimensional light bundles that stem from point-like objects illuminate the entire lens surface, and the image shells generated by meridional and sagittal sub-bundles do not coincide. As a result, the object—if located away from the centre of field—is mapped into a complicated three-dimensional figure. The difference between the meridional and the sagittal image shell is called astigmatism and a delicate correction of that aberration leaves only a single image shell—the field curvature. In poorly corrected binoculars, astigmatism leads to strongly deformed star images near the edges of field, which—contrary to the blur caused by field curvature—cannot be brought into focus by turning the focus wheel.

Fig. 1.13 Astigmatism (schematic): the light bundle falls in the meridional (red) and sagittal directions (green) of the lens. The meridional plane lies on the principal axis, while the sagittal plane does not. The focal points (BM , BS ) of both bundle cross sections are located at different distances from the lens (the optical axis is shown in black). Source: Michael Schmid on Wikimedia Commons under CC BY-SA 3.0

24

1 Optical Imaging

1.7.5 Coma The coma is another aberration that occurs with oblique rays which hit the lens far from the optical axis. Let us once again return to Fig. 1.11: here the rays enter at an angle .e and a distance h to the principal axis. To obtain the image field curvature (Sect. 1.7.3), we had fixed the distance h and varied the angle of incidence .e, then used Eqs. (1.27) and (1.28) to find the respective image intersection points, which lay on the image shell. In a next step, we fix the angle of incidence while varying the heights of incidence of the rays. Figure 1.14 shows what happens: the corresponding image points now migrate along a straight line (green) that intersects the image shell (a section of this shell is shown as the black curve). Technically, coma is a superposition of two image errors, spherical aberration and field curvature. We recall that the rays change their intersection points when the incidence heights are varied (Fig. 1.10), which corresponds to a shift of the focus along the principal axis. Now, the simultaneous variation of the angle of incidence to the lens normal makes the image points move along the image shell. The superposition of both motions—a horizontal shift due to spherical aberration plus a migration along the image shell results in the green line shown in Fig. 1.14. As a result of coma, stars near the edge of field are deformed into drop-like shapes, or even developing comet-like tails, as their images consist of superpositions of several circles of confusion, the centres of which do not coincide. Conrady described coma as being the most unpleasant of all aberrations, since it generates an asymmetrical blur and therefore hampers the precise determination of the image location. Thus, the elimination of coma must be given the highest priority in optical design [4]. In 1873, Ernst Abbe succeeded in formulating a condition which—after prior correction of spherical aberration—safely eliminates coma. The sine condition can be formulated as .

Fig. 1.14 Coma: varying the heights of incidence h at constant angle of incidence to the optical axis (here: ◦ .e = 20 , as depicted in Fig. 1.11), leads to a drift of the image point along a straight line (green), which intersects the image shell

h =C, sin δ

(1.34)

1.7 Lens Aberrations

25

Fig. 1.15 Sine condition: For each angle .δ of the light ray (red) to the optical axis, the distance between the imaginary spherical shell (blue, dashed) and the focal point F has to be a constant C. This is the continuation of the concept of the Gaussian principal plane H to large incidence heights h

where h is the height of incidence of the ray, .δ its angle to the optical axis on the way to its focal point, and C is a constant. Figure 1.15 sketches the geometry of the sine condition: the right-angled triangle (green) requires, for any choice of the angle .δ, a hypotenuse of given length C. This implies that the (imaginary) point at which the incident light ray is refracted must lie on a spherical shell with F at its centre. This is a generalisation of the concept of the principal plane, as previously introduced by Gauss, to large incidence heights h. According to Gauss, the effect of refraction of a lens group has, for paraxial rays, to be regarded as refraction by an (imaginary) plane H . Abbe then showed that a similar simplification holds for larger incidence heights, given that the principal plane is replaced by a spherical shell. Once that condition is realised, the resulting optical design is free of coma.

1.7.6 Lateral Chromatic Aberration The image height .h' , which we had calculated in Eq. (1.28) using the example of the plano-convex lens, contains various angles, which in turn depend on the refractive index n. The latter, as a result of dispersion, is a function of the wavelength, thus is the image height. This phenomenon is called lateral chromatic aberration. It produces colour fringes along edges of objects far from the centre of field. For illustration purposes, we repeat the calculations that led to Fig. 1.12, but this time using two different wavelengths. Just as in Sect. 1.7.1, we assume that we are dealing with SF2 flint glass from the Schott catalogue, the refractive index of the red C-line being .nC = 1.642 10, and that of the violet h line being .nh = 1.682 33. Figure 1.16 shows a blue image shell and a (dashed) red one, so that we are dealing with two image shells that are shifted along the optical axis—a consequence of the longitudinal chromatic aberration as discussed in Sect. 1.7.1. When superposing both image shells, it is noticeable that they also differ in size, the red one being somewhat larger than the violet one. Consequently, the image height of a single white object is spread out into a spectrum, as if each colour had its individual magnification. This leads to the already mentioned colour fringes along edges of

26

1 Optical Imaging

Fig. 1.16 Lateral chromatic aberration: image shells at two different wavelengths (red and blue), with size comparison

objects, which are particularly pronounced when this edge occupies the outer areas of the field. For hand-held binoculars and their wide fields at low powers, the effects of lateral chromatic aberration are generally far more an issue than the effects arising from longitudinal chromatic aberration. Colour fringes are primarily generated in the eyepieces, and a complete elimination of this aberration is virtually impossible. Longitudinal chromatic aberration, which leads to coloured halos about objects at the centre of field, becomes dominant at high powers and is an issue with spotting scopes or mounted binoculars for astronomy that are commonly used with high magnifications. In this case, a complete elimination of chromatic aberration is feasible with the help of apochromatic objective lenses.

1.8 Optical Design The preceding sections have provided a rough overview of the basic properties of different aberrations.11 The aim of optical design is to bring all these aberrations to a tolerable level, making use of the available degrees of freedom such as different types of glass, radii of curvature and spacing between individual lenses. Every optical design is the result of a compromise, a delicate balance of different aberrations, which are impossible to eliminate entirely. It should be noted that the aberrations generated by the objective and those arising in the eyepiece necessarily exhibit different characteristics: objective lenses of binoculars or—even more so—spotting scopes are hit by light rays at relatively small incident angles. An optical designer of the traditional school would then begin with a minimisation of longitudinal chromatic aberration (by choice of wellmatching glass types), then eliminate spherical aberration and coma, i.e. to construct an achromatic and aplanatic lens. In practice, a two-lens setup with an air gap is usually sufficient to achieve that goal. Astigmatism and field curvature remain 11 A

discussion of distortion is left out and carried out later in Sects. 2.1.3 and 8.11.

References

27

partially uncorrected in this case, but usually within tolerable limits, while lateral chromatic aberration and distortion remain negligible [5]. In case of an eyepiece, the designer has to deal with light bundles which fan out over angles exceeding 30°. Here astigmatism, lateral chromatic aberration, field curvature and distortion have a considerable impact. Well-corrected eyepieces therefore require a higher number of lens elements than objectives in order to perform well. Even the simplest eyepieces consist of three lens elements, while wide-angle eyepieces require the implementation of five lenses or more. Ray-tracing is performed by the designer with the support of powerful computers and the corresponding software [6]. He usually starts with an existing configuration from the programme database which resembles the optic to be designed. Important parameters such as focal ratio and magnification are specified, as well as the upper limits for the aberrations that are to be tolerated. The latter are compiled to form a merit-function in which each aberration is processed with appropriate weights to yield a single score, which is then optimised by iteratively varying the lens parameters and performing a full ray-tracing cycle. If the optimisation does not lead to the desired result, the designer intervenes by, for example, splitting a selected lens into a doublet of suitable glass types and repeating the entire optimisation cycle. Experience is required to keep the optical construction as simple as possible, while at the same time having an eye on the manufacturing tolerances, which must not be too tight for the subsequent series production. Finally, attention must be paid to material properties—the coefficients of thermal expansion of the glass types must be matching whenever lenses have to be cemented, otherwise mechanical deformations and thus new aberrations occur. Other boundary conditions such as weight or mechanical strength (e.g. the resulting thickness of the lens), machining efforts (grinding of brittle glass types may be slow and associated with scrap production), cost and availability of optical glasses on the world market and environmental compatibility (heavy metal additives in the glass) are to be considered for the optical design to result in a successful product.

References 1. Iqbal, M: New Perspectives on the History of Islamic Science, vol. 3, p. 274. Routledge, London and New York (2017) 2. Thiele, H.: Die ersten Computer für Optikrechnung. In: Legenden und Geschichten der Photoindustrie. Privatdruck, München (2006) 3. Suiter, H.R.: Star Testing Astronomical Telescopes. Willmann-Bell, Richmond (2008) 4. Conrady, A.E.: Applied Optics and Optical Design, vol. 1 (reprint), p. 367. Dover Publications, New York (1992) 5. Rutten, H., Venrooij, M.: Telescope Optics. Willmann-Bell, Richmond (2002) 6. Smith, W.J.: Modern Lens Design. McGraw-Hill, Boston, Burr Ridge, Dubuque, Madison, New York, San Francisco, St. Louis (2004)

Chapter 2

The Telescope

2.1 The Telescope of Keplerian Design 2.1.1 Functional Principles A convergent lens produces a real image of an object, and this is ideally located in the image plane. This image could be projected onto a screen or, in the case of a camera, onto a photographic film or digital sensor. If the screen is omitted and instead the image observed visually through a kind of magnifying glass, the result is a Keplerian telescope,1 where the magnifying glass is called an eyepiece or ocular. Figure 2.1 sketches the functional principle: there is the objective and the eyepiece. In this schematic sketch, the construction details of both assemblies remain hidden, only the positions of their principal planes (blue, see also Sect. 1.5) are shown. Additionally, aberrations are neglected so that all image points of distant objects fall onto the image plane at a distance F from the principal plane of the objective. Images of nearby objects would be shifted to the right, as defined by the imaging equation (1.21). Consider light rays arriving from three separate objects, such as three distant stars. The diameter of the complete bundle of rays is defined by the aperture stop (also called the entrance pupil), which could be the edge of the objective lens or alternately a downstream annular diaphragm. On the focal plane lies the field stop, which is usually a part of the eyepiece. In the Keplerian telescope, the real image produced by the objective is referred to as the intermediate image because it is located in between the objective and the eyepiece. Alternately, it is often referred to as the instrument’s primary focus. At a distance f from the intermediate image lies the eyepiece. If f is identical to the focal length of the eyepiece, then the rays from each star exit as a parallel bundle. To the observer, the emerging virtual image of the object then appears to lie at infinity again, and the instrument is thus functioning as

1 Named

after Johannes Kepler, 1571–1630.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Merlitz, The Binocular Handbook, https://doi.org/10.1007/978-3-031-44408-1_2

29

30

2 The Telescope

Fig. 2.1 Keplerian telescope: red rays arise from a star in the centre of the field of view, green and magenta rays from stars at the field edges. lXP stands for the longitudinal distance of the exit pupil to the vertex of the last eyepiece lens. The shaded areas form two similar triangles

an afocal device. In practice, telescopes or binoculars are rarely used in this afocal mode: the observer involuntarily adjusts the focus position in such a way that the virtual image lies at a comfortable accommodation distance of the order of a metre. Figure 2.1 contains three separate construction rays in red (star in the centre of the field), green (upper edge of the field) and magenta (lower edge of the field). The latter two bundles enter the lens at an angle .Φ/2 to the principal axis, and .Φ is called the objective or real angle of field of the telescope. The same rays leave the eyepiece at the angle of .φ/2 to the principal axis—.φ being the subjective or apparent angle of field. Behind the eyepiece, all ray beams intersect at the exit pupil (XP), which, in the absence of aberrations, lies on a plane perpendicular to the principal axis.2 The exit pupil is, technically speaking, the real image of the entrance pupil. The field stop defines the circular boundary of the field of view that the observer perceives through his instrument. It determines the diameter of the intermediate image and the objective angle of field of the instrument. The distance between the lens vertex of the ocular’s eye lens (which is the lens facing the eye) and the exit pupil is known as the eye relief of the eyepiece. Note that the sketch in Fig. 2.1 is inaccurate in the sense that the actual position of the eye lens is not drawn, instead the principal plane of the eyepiece. Only an explicit raytracing of the telescope, including the complete set of its lens elements, would yield an accurate number for .lXP , the eye relief of the instrument. For spectacle-wearers, a sufficiently long eye relief is of particular importance to prevent a loss of field of view if the eye pupil cannot be brought onto the level of the exit pupil.

2 Aberrations caused by the eyepiece are commonly known as pupil aberrations. In particular the spherical aberration of the eyepiece leads to a bending of the exit pupil, which may cause the swivelling eye to experience the volatile shadowing of image sectors known as kidney beaning.

2.1 The Telescope of Keplerian Design

31

2.1.2 Focal Ratio If the diameter of the entrance pupil is D, what is then the resulting diameter of the exit pupil, d? We note that the shaded areas in Fig. 2.1 form two similar triangles, since the red construction rays are passing through the intermediate image without deflection. We therefore derive the corresponding relations .

tan α =

f F = , D/2 d/2

(2.1)

from which it follows that d=D

.

D f = F m

(2.2)

for a magnification m, which we will discuss in detail in the following section. The exit pupil is therefore equal to the entrance pupil multiplied by the ratio of the eyepiece focal length to the objective focal length. An important characteristic of the telescope is its focal ratio or f-number N=

.

F , D

(2.3)

the ratio of the focal length to the diameter of the entrance pupil. It is rather common to write f-numbers preceded by f/, for example f/5 which stands for a focal ratio of 5. Its reciprocal value is known as the relative aperture or lens speed C=

.

D . F

(2.4)

In reference to common practice in photography, one speaks of a fast focal ratio when the f-number is small or the lens speed is high. In this case, the incident beam forms a rapidly converging, obtuse cone, which has a negative effect on the aberrations. Telescopes with slow focal ratios are easier to correct and also more tolerant against errors in collimation. In binocular design, high demands on the compactness of the device lead to f-numbers between f/3.5 and f/5, which often result in considerable residual aberrations.

2.1.3 Magnification The magnification of a telescope tells us by which factor the apparent ‘size’ of an object is enhanced when observed through the eyepiece. In order to evaluate this factor, we first recall that for ray paths, the principle of reciprocity applies: if the direction of the beam is reversed in Fig. 2.1, then the light entering through the

32

2 The Telescope

eyepiece produces an intermediate image at a distance f from the eyepiece, which is then imaged by the objective from a distance F . We return once again to the imaging equation as shown in Fig. 1.9: from the left, the light ray is incident with the object angle A. With the support of triangle 2 the relation .tan A = y ' /F results, where .y ' is the object size in primary focus. Now we take into account that this intermediate image is processed by the eyepiece in the same manner, where we simply replace the objective focal length F with the eyepiece focal length f and the objective angle A with the subjective angle a, obtaining the same triangular relation ' ' .tan a = y /f . This allows us to eliminate .y to obtain .

tan a D F = = =m f tan A d

(2.5)

where the second equality arises from Eq. (2.2). This is the definition of the magnification (’power’) m of the telescope. It is nothing else but the ratio of the tangent of the subjective angle a to the tangent of the objective angle A, both taken with respect to the centre of field. However, the ratio of the entrance pupil diameter D to the exit pupil diameter d is far easier measured with a binocular, and since it is independent of the degree of distortion, it is generally the superior way to define the magnification of a binocular.

2.1.4 Angle of Field and Field of View (FOV) We define the angle of field as the maximum angle covered by the instrument, stretching from one edge of the field through the centre to the opposite field edge. When observed through the eyepiece, we denote the subjective angle of field as .φ, being twice the maximum possible subjective angle of view, a, from the centre to the field edge. The corresponding objective angle of field is .Φ and twice the maximum possible objective angle of view, A. Thus, while the angles of view, A and a are variables that cover all possible values from the centre to the edge, the angles of field, .Φ and .φ are (twice) their respective extrema and set by the instrument’s field stop. The field of view (FOV) of a telescope is given in metres per kilometre and is strictly related to the objective angle of field via the trigonometrical relation FOV(in m) = 2000 · tan(Φ/2)

.

(2.6)

or, conversely, [

FOV(in m) .Φ = 2 · arctan 2000

] .

(2.7)

A straightforward calculation of the subjective angle of field from either the field of view or the objective angle of field is possible under the condition that rectilinear distortion (Sect. 2.1.5) is absent. If this condition is satisfied, then the equation

2.1 The Telescope of Keplerian Design

33

Fig. 2.2 Distortion: chequerboard pattern with pincushion distortion (left) and barrel distortion (right). Note that the diameters of the images (which correspond to the subjective field angles) are not identical

φ = 2 arctan [m tan(Φ/2)]

.

(2.8)

computes the subjective angle, or conversely [ Φ = 2 arctan

.

tan(φ/2) m

] (2.9)

computes the objective angle of field. These relations also correspond to the (rather misleading) official ISO 14132-1:2002 standard for the calculation of the subjective angle of field, which Nikon has adopted for its binocular specification-sheets. A photographic lens or eyepiece that obeys these relations for all radial angles is sometimes called rectilinear or orthoscopic. Unfortunately, hardly any binocular on the market is free of distortion, and therefore, the ISO 14132-1:2002 standard does not actually apply; instead, more sophisticated methods are required to compute the subjective angle of view, as will be discussed in the following section.

2.1.5 Distortion Distortion is a lens aberration which does not blur the image; instead, it alters its geometry (Fig. 2.2). As per definition, the image is free of rectilinear distortion if it satisfies the so-called tangent condition3

3 This is a translation of the German term ‘Tangensbedingung’; in English terminology, the term f-tan theta relation is also common for this condition.

34

2 The Telescope .

tan a = m tan A

(2.10)

for any objective angle A and its image a. Note that Eq. (2.8) is a special case of the tangent condition when applied to the field edge. A lens or visual optical instrument that satisfies the tangent condition images any straight line into a straight line. One then defines the relative rectilinear distortion at any given angle A or a as the relative deviation from the tangent condition: tan a −m tan A .Vr = . m

(2.11)

If this distortion, as a function of the subjective angle a, is increasing monotonically, then straight lines that intersect the field (but fail to pass through its centre) are bending inwards and one speaks of pincushion distortion. If, on the other hand, the distortion is monotonically decreasing with the subjective angle, then a case of barrel distortion is found. More complex distortion patterns are possible and not uncommon in binoculars—the popularly used term moustache distortion is a case in which pincushion distortion crosses over into a barrel distortion when the angle approaches the edge of field. With hand-held binoculars, a well-selected amount of pincushion distortion is often intentionally implemented to improve the panning behaviour of the device. We will return to this interesting topic in Sect. 8.11. A special case of lenses with pincushion distortion satisfies the equiangular condition in which for all radial angles the simple formula a = mA

.

(2.12)

holds—also known as the angle condition.4 The subjective angle of field is then identically related to the objective angle via φ = mΦ ,

.

(2.13)

a formula commonly used by manufacturers to compute approximate specifications for the subjective angle of field. An equiangular lens offers a constant angular magnification .m = a/A from the centre of field all the way to the edge, and deviations from the condition of equiangularity are sometimes called angular magnification distortion (AMD). From the defining equations, it is obvious that no lens could possibly be corrected for rectilinear distortion and AMD at the same time: the equiangular condition necessarily implies a significant degree of pincushion distortion. It was the Zeiss employee, August Sonnefeld, who was the first to understand the importance of AMD for visually applied instruments [1]. In Sect. 8.12, we are going to analyse in detail the interactions between instrumental distortion and optical perception. 4 A translation of the German term ‘Winkelbedingung’; in English terminology, the term f-theta relation is also commonly found.

2.2 Telescope of Galilean Design

35

By this point, it should be clear that neither Eq. (2.10) nor (2.12) is likely to be accurate for a given binocular—the precise characteristics of its distortion are a result of the optimisation process and usually described by a distortion curve which the manufacturers do not disclose.5 A precise calculation of the subjective angle from the objective angle requires knowledge of the relative distortion at the edge of field, .Vr (Φ/2), which then yields the subjective angle of field φ = 2 arctan [m (Vr (Φ/2) + 1) tan(Φ/2)] .

.

(2.14)

Meanwhile, some manufacturers have begun to specify precisely measured values of their product’s subjective angles of field, instead of approximate numbers taken from either the tangent- or the angle condition. This is a welcome improvement which also allows the user to calculate the relative distortion via Eq. (2.14).

2.2 Telescope of Galilean Design A second look at Fig. 2.1 reveals that the Keplerian telescope inverts the image: the light from a star above the line of sight (green beam) emerges from below at the eyepiece, and its image thus lies at the lower edge of the field. In astronomy, this inversion is of little relevance, but during daytime observation, an upright and laterally correct image cannot be dispensed with. As a remedy, it would be possible to insert an additional lens into the beam path, which would also generate an image erection. This principle of the terrestrial telescope is realised in so called ‘pirate telescopes’, but it is impractical for the construction of binoculars because of its lengthy tubes. Instead, erecting prisms are used to invert the image. Before we go into this approach in great detail in the following chapter, we will briefly discuss an alternative design, which no longer plays a significant role in binocular construction, but is still surviving in a market niche known as opera glasses. With the Galilean telescope,6 the creation of a real intermediate image is avoided by placing a negative lens group in front of the primary focus, which diverges the beam and generates a virtual image to be observed by the eye (Fig. 2.3). Just as the Keplerian telescope, this is an afocal construction—parallel incident beams emerge in parallel, but the resulting image is upright. Another advantage of this system lies in its short construction: the overall length does not, as in the Keplerian telescope, correspond to the sum of the focal lengths of objective and eyepiece, .F + f , but their difference, .F − f . The magnification is still calculated via .m = F /f . However, the Galilean telescope also has serious disadvantages: since the eyepiece is a diverging lens, the exit pupil turns virtual and is located in front (to the left) of the eyepiece—unreachable by the eye. As a result, the subjective angle of

5 Whereas, 6 Galileo

with camera lenses, the corresponding distortion curves are commonly published. Galilei, 1564–1642.

36

2 The Telescope

Fig. 2.3 The parallel beams entering the objective emerge from the eyepiece, which has a negative focal length, without creating a real intermediate image. The exit pupil is virtual, and its location in front of the eyepiece is a function of the angle of field

field in such a telescope decreases rapidly with its magnification, so that today, the Galilean telescope is used exclusively at low magnifications at which the diameter of the exit pupil by far exceeds that of the eye pupil. Figure 2.3 shows schematically the function of such a device, including the entrance pupil of the observer’s eye as defined by the iris: the iris now assumes the function of the aperture stop which effectively stops down the entrance pupil of diameter D to .D ' . The ray-fan also demonstrates how certain obliquely incident rays from objects at the edge of field (shown here in green and magenta) are partially blocked out by the iris, causing an increasing amount of vignetting until the image fades near its boundary. In the absence of a real intermediate image, no field stop exists, and thus, no crisp field edge is visible to the eye. Furthermore, certain oblique incident rays can contribute to the image even though they enter from the outside of the effective entrance pupil, which is why—in contrast to the Keplerian telescope—the maximum angle of view is in some cases limited by the diameter of the objective lens. The ease of view through a Galilean telescope is affected by its fuzzy, undefined field edge and by the absence of a real exit pupil. To increase the angle of field, the observer tends to push his eye as close as possible onto the eyepiece lens, which easily fogs up or smudges. Yet, the subjective angles remain disappointingly narrow because of the un-bridgeable distance to the exit pupil. Today, binoculars of the Galilean type are predominantly used as opera glasses, although a few products persist in a niche segment of binoculars for astronomy, where at very low magnifications, they offer particularly wide objective angles of field and the coverage of entire star constellations.

2.3 Virtual Image and Depth of Field (DOF) In what follows, we turn back to the Keplerian telescope, whereby our findings will be equally valid for the Galilean telescope. We first determine the distance of the virtual image, under which it appears to the eye through the eyepiece. Note that we now refer to the accommodation distance, i.e.

2.3 Virtual Image and Depth of Field (DOF)

37

Fig. 2.4 An object at large distance produces an afocal beam when the telescope is focused at infinity (red beam). An object at close range is imaged at a distance .zf' in front of the exit pupil (XP) (green beam), if the telescope’s focus remains being set to infinity. The locations of the principal planes are shown in blue

the distance the eye has to focus in order to perceive a sharp image. Figure 2.4 shows the imaging principle in a simplified sketch: an object is located at a large distance .zF along the optical axis. Its incident light beam (red construction ray) falls parallel to the optical axis onto the objective and leaves the eyepiece, again parallel to the optical axis, when the telescope is focused at infinity. This is the already-mentioned principle of afocal imaging. The virtual image of the object is then observed at an accommodation distance of infinity, because the rays emerge parallel from the eyepiece and enter the eye just as they would if the observed object were far away. Now we imagine a second object which is located in the foreground and represented by the green construction ray: being close to the observer, the beam of rays emitted by that object hits the lens divergently, and the primary focus (which is the real intermediate image of that object) is now shifted to the right by a certain offset .zF' . This image is no longer in the focus of the eyepiece (which remains unchanged at its infinity setting). Since the intermediate image of the object is now closer to the eyepiece than its focal length f , the bundle of rays that leaves the eyepiece is divergent, too. The distance of the virtual image is now determined by a backward extension of the rays leaving the eyepiece (green, dashed), by the point at which this extension intersects the optical axis, which happens at .zf' when measured from the exit pupil. We now calculate .zf' using Newton’s imaging equation (1.24). This is achieved in two steps: first, the object (distance .zF ) is imaged by the lens of focal length F into the intermediate image, which has a distance .zF' from the focal plane of the lens, zF' zF = −F 2 ,

.

(2.15)

38

2 The Telescope

which we solve to zF' = −

.

F2 . zF

(2.16)

This real intermediate image now serves as the object for the second imaging process, this time through the eyepiece. The offset of the intermediate image to the objective’s focal plane, .zF' , is identical with its offset from the focus of the eyepiece, ' .zf , because the eyepiece remains unchanged in its infinity position. Thus, .zf = z F applies, and for the resulting imaging process we have zf' zf = −f 2 ,

.

(2.17)

yielding zf' = −

.

f 2 zF' f2 = , zf F2

(2.18)

or, when applying .m = F /f for the magnification, zf' =

.

zF' . m2

(2.19)

Let us as an example take a telescope of magnification .m = 10, which is focused at infinity, and through which an object at distance .30 m is observed, then the virtual image is at the distance .30 m/100 = 30 cm. Note the enormous impact of the squared magnification. This reduction in the accommodation distance by the square of the magnification is also known as the depth scale7 and is not to be confused with the influence of the magnification on the perceived size of the object: the magnification is the factor by which the (paraxial) linear expansion of the object is multiplied, and it amounts to m, whereas the depth scales with its square, .m2 . Obviously, Eq. (2.19) has something to do with the depth of field of the telescope, i.e. with the range of distances the eye of the observer can accommodate. There is a minimum distance, below which the image would be too close so that the eye would remain out of focus. We will discuss this in Sect. 6.4, which deals with the accommodation width of the eye, but at this point, we can already state that the perceived depth of field scales by the reciprocal square of the magnification. Upon further investigations, we will find the effective pupil diameter as another, though less influential factor that affects the depth of field. As a result of the dominating role of the magnification, the depth of field diminishes rapidly with increasing power. For example, binoculars with 7x magnification already offer a .(10/7)2 ≈ 2 times deeper depth of field than binoculars with 10x magnification.

7 In

German ‘Tiefenmaßstab’.

2.4 Wave Optics

39

There have been repeated claims on Internet forums that the depth of field of telescopes or binoculars may—apart from their decisive parameters such as magnification and exit pupil diameter—be affected or optimised by a particularly clever optical construction. This is not the case: a modification of the focal length of the objective requires a corresponding change of the focal length of the eyepiece by the same factor to arrive at the desired magnification, and by this process, Eq. (2.19) would remain unchanged. Is there any possibility to modify the depth of field without affecting the magnification, perhaps by the choice of a particular eyepiece type or objective design? Different designs would in fact change the spacings between the principal planes of the respective optical components, be it the objective or the eyepiece. Figure 2.4 displays their positions (blue), but the latter do not enter Eq. (2.19) at all and leave the virtual image’s location unaffected.8 In real-life applications, an abundant amount of field curvature (Sect. 1.7.3) may under certain circumstances mimic an increased depth of field, if objects that are located in the foreground and set into lower areas of the image appear sharp, while being out of focus if shifted to the centre of field.9 This effect is similar to that of varifocal glasses and not to be confused with the actual depth of field of an instrument. Field curvature is an aberration that occurs off-centre, while the depth of field is calculated in the Gaussian approximation for paraxial rays that propagate near the centre of field. The question remains still left open as to which distance range the image appears sharp, if the telescope is focused to a finite distance. We will address this problem in Sect. 8.9 in which we will explicitly include the observer’s accommodation range into the equations.

2.4 Wave Optics 2.4.1 Huygens’ Principle All previous derivations of the imaging properties have been based on the approximation of geometrical optics, in which the wave properties of light propagating through media were neglected and replaced by rays which are traced according to geometric rules. This approximation is not always justified: if light from a point source is brought into focus by the objective lens, and its image is then substantially magnified through the eyepiece, the wave properties emerge in the form of a ring-like pattern—a result of diffraction at the aperture stop. The field of optics, which deals with the physical properties of light beyond geometrical optics, is also known as physical optics or wave optics. Diffraction is among those 8 The 9 As

author would like to thank Volker Tautz for this simple proof. pointed out by Börries von Breitenbuch.

40

2 The Telescope

Fig. 2.5 Huygens’ principle: the front of a plane wave hits a slit from the left. In addition to the component that passes the slit (red), spherical components (green) are generated at its edges, which are superimposed on the plane wave

classical wave properties based on Huygens’ principle:10 every single point of an illuminated object behaves like a light source of its own, emitting spherical waves in all directions, which then combine to form a new wave front. For illustration consider Fig. 2.5: the plane wave front incident from the left generates additional spherical waves at a slit, which subsequently illuminate those areas which would appear to be shadowed by the slit. However, this illumination does not happen uniformly: depending on the light path difference at a given time and coordinate, the wave crest of one wave may superimpose with wave crests of other wave fronts to generate a locally amplified field. This phenomenon is known as constructive interference. Vice versa, at other coordinates or times the phase difference may lead to a partial or complete cancellation of the field, known as destructive interference. Interference patterns composed of alternating bright and dark areas are omnipresent phenomena in wave optics, and they cannot be neglected when dealing with visual optical instruments.

2.4.2 Diffraction-Limited Resolution A completely analogous phenomenon arises when a single point is imaged by an optical system: the aperture stop is to be regarded as a spherical slit at which diffraction occurs. As a result, even with perfect optics, a point source of light such as a star is not imaged as a point, but exhibits a characteristic diffraction pattern, known as the Airy-disc.11 Figure 2.6 shows such a point image (left): a central diffraction disc is visible, which defines the intensity maximum, surrounded by a sequence of ring-shaped diffraction minima and maxima. Given the light source at the centre of field, the mathematical description of the intensity distribution is a spherical Bessel function .J1 (x), ( I (θ ) = I0

.

10 Christiaan 11 Sir

Huygens, 1629–1695. George Biddell Airy, 1801–1892.

2J1 (x) x

)2 ,

(2.20)

2.4 Wave Optics

41

Fig. 2.6 Left: Airy-disc, the perfect, diffraction-limited image of a point source. Right: Rayleigh criterion: two point sources that are just barely separable. Source: Wikimedia Commons (public domain, left; Geek3 under CC BY-SA 3.0, right)

where .I0 represents the intensity at the centre (maximum), .θ is the angle (to the principal axis) of the imaginary beam, connecting the centre of the entrance pupil with the image point in the focal plane, D the aperture, .λ the wavelength and .x = (π D sin θ)/λ. Without going into further details, it is sufficient to know the angular distance from the centre of this disc to the first diffraction minimum, which is approximated by θ ≈ 1.22

.

λ . D

(2.21)

According to the (empirically found) Rayleigh criterion,12 this angular distance corresponds to the minimum distance at which two point-like objects can still be distinguished from each other. This rule of thumb for the separation of point-like objects is only valid for a perfectly corrected optical system, in which all other aberrations hide behind the central diffraction disc. The critical angle is proportional to the reciprocal value of the diameter D of the optics and proportional to the wavelength of the light. For instruments used visually, it is common to specify the resolution limit at the standardised wavelength .λ = 550 nm. For linear, periodic patterns (see for example Fig. 8.5) the above limiting resolution applies after removing the pre-factor 1.22, i.e. linear structures remain separable when point-like objects are already indistinguishable.

12 Named

after John William Strutt, 3. Baron Rayleigh, 1842–1919.

42

2 The Telescope

In practice, resolution limits are approached e.g. in astronomical observations of narrow binary stars, or planets at very high magnifications. In the case of hand-held binoculars, the diffraction-limited resolution is hardly of any relevance, due to their low magnifications and limited resolution of the eye (see Sect. 6.5): the healthy eye itself is approximately diffraction-limited only in bright daylight at pupil widths below 2 mm. Exit pupils of binoculars are, however, usually wider than that, and a binocular of magnification m is with such a narrow eye-pupil stopped down to an effective entrance pupil of .D ' = m · (2 mm). It would thus make sense to require the optical design of such a binocular to be diffraction-limited while the entrance pupil is stopped down to .D ' , though no industrial norm exists which would encourage the implementation of such a criterion.

2.4.3 Anti-Reflective Coatings Another effect of the wavelike nature of light can be used to reduce disturbing reflections on the glass surfaces. In Sect. 1.2 we discussed the law of reflection in connection with the law of refraction: a ray of light is partially reflected at the interface between two media of different optical densities, in such a way that the angle of incidence and the angle of exit, measured to the perpendicular, are identical. But how large is the proportion of reflected light? This so-called reflectance .ρ can be determined via Fresnel’s equation13 which, when using the angles in Fig. 1.2 (Snell’s law of refraction), yields 1 .ρ = 2

[

sin2 (α − β)

tan2 (α − β) + sin2 (α + β) tan2 (α + β)

] .

(2.22)

If this formula is plotted as a function of the angle of incidence .α, it remains almost constant for small angles up to about 45°. So let us first apply the already-discussed approximations of the trigonometric functions for small angles, .sin x ≈ x, .tan x ≈ x, furthermore the Gaussian approximation of the law of refraction, .n1 α ≈ n2 β, then the formula simplifies to [

n2 − n1 .ρ ≈ n2 + n1

]2 ,

(2.23)

.

(2.24)

or, if the ray transits from air (.n1 = 1) to glass, [ ρ≈

.

13 Named

n2 − 1 n2 + 1

after Augustin Jean Fresnel, 1788–1827.

]2

2.4 Wave Optics

43

Fig. 2.7 Thin layer coating: reflected partial rays (red, dashed) emerge at the transitions from air to coating and coating to glass. With a suitable choice of coating thickness d, destructive interference diminishes their intensities

For flint glass with main refractive index of .n = 1.7, we obtain a reflectance of .ρ = 0.067, i.e. almost 7 %, and a BaK4 glass with .n = 1.57 would still yield .ρ = 0.049, about 5 %. Of course these values are also affected by the dispersion, which makes the refractive index a function of the wavelength, .n2 (λ). When calculating reflection losses in eyepieces, however, the approximation of small angles no longer remains valid, since particularly with wide-angle eyepieces, increasingly oblique angles of incidence occur so that the more accurate equation (2.22) has to be applied. If these reflection losses take place in complex optical systems with a large number of air–glass transitions, they could easily sum up to numbers exceeding 50 %, which not only would yield a dull, dark image, but also disturbing arrays of ghostly images of bright light sources as a result of multiple reflections. As a solution, a technique is applied that exploits the wave property of light and that was patented in 1935 by Alexander Smakula of Zeiss: after vapour deposition of a thin, transparent layer with refractive index .n1 onto a glass surface with refractive index .n2 , the incident light produces not a single, but a pair of reflected partial waves. The thickness of the layer is chosen in such a way that both reflected beams suffer a relative phase shift of half a wavelength (or an odd multiple thereof) so that the wave crest of the first reflection meets the wave trough of the second reflection and destructive interference occurs. While reflected light is suppressed, the transmission increases accordingly, because the conservation of energy prevents any losses of the total intensity, and thus, the coating has fulfilled its task (Fig. 2.7). The ideal layer thickness d, at which the optical path lengths of both reflected partial beams differ by just .λ/2, is exactly .kλ/4n1 for perpendicular incidence with odd values for k, and at oblique incidence we obtain these with the refraction law from Eq. (1.2) as [2] kλ d= / , 4 n21 − n20 sin2 α

.

(2.25)

44

2 The Telescope

where .n0 = 1 was again used for air. Since the thickness depends on the angle of incidence .α and the wavelength .λ, it is impossible to design an optimally effective single coating for a convergent (or divergent) beam, in particular when the incident light is polychromatic and covering wavelengths from the entire visible spectrum. Multiple layers (known as multi-coatings) are employed to address that shortcoming, but reflectance reduction can only ever be partial, which is why most surface coatings change their colour tone when viewed from different angles. The total reflectivity of a single optical element with optimum film thickness and at small angle of incidence .α is calculated according to Fresnel’s equation (2.23) from the square of the amplitude difference of the two partial beams, [

n2 − n1 n1 − n0 − .ρ ≈ n1 + n0 n2 + n1

]2 .

(2.26)

Here the first term stands for the reflected partial beam at the air–coating interface and the second for the same at the coating–glass interface. If the thickness of the thin layer were only increased by .λ/4n1 , then constructive interference would occur and both amplitudes would add up, so there would have to be a “+” between the two terms. As an example, we choose again the BaK4 glass (.n2 = 1.57) and a magnesium fluoride (MgF.2 ) coating (.n1 = 1.38), which, due to its high durability, has been used over decades as the standard recipe for an effective single layer coating. With .n0 = 1 for air we get .ρ = 0.009, i.e. a loss of less than 1 %, which is a considerable improvement over the 5 % we obtained for the uncoated glass. Modern multi-coatings consist of up to eight layers and reduce the reflectivity for visible light to 0.2 %, in some cases less than that.

References 1. Sonnefeld, A.: Über die Verzeichnung bei optischen Instrumenten, die in Verbindung mit dem blickenden Auge gebraucht werden. Deutsche Optische Wochenschrift 13, 97 (1949) 2. Gabler, H: Optische Hilfsmittel und Geräte zur Verbesserung der Schiffsführungsoptik. Deutsche Hydrografische Zeitschrift 5, 32 (1952)

Chapter 3

Image Erecting Prisms

3.1 Porro-Type Systems 3.1.1 Total Internal Reflection (TIR) From Snell’s law of refraction (Fig. 1.2), it follows that a ray of light, when passing from an optically denser (.n2 ) to an optically thinner medium (.n1 ), is refracted away from the plane of incidence. It is thus clear that there must exist a critical incident angle, at which the exit angle exceeds 90°, and the ray is reflected back into the optically denser medium. At this critical angle .αc , the sine of the exit angle amounts to exactly unity, so that we can replace  αc = arcsin

.

n1 n2

 .

(3.1)

Let us take as an optically dense medium, a BK7 glass with .n2 = 1.517, and a transition to air (.n1 = 1), then the critical angle yields .αc = 41, 2◦ . Once this incident angle is exceeded, a total internal reflection of the light beam back into the glass medium takes place, while not even a fraction of the beam passes into the optically thinner medium. Such an interface then behaves like a perfect mirror. This effect is exploited in a deflection prism (Fig. 3.1): a light ray (red) enters the prism (with two 45° angles, also called a Porro-prism.1 ) is totally reflected on both sloped faces and exits in the opposite direction. Since we assume that the ray of light is incident perpendicular to the prism base, it hits the sloped faces at an angle of 45°. If the prism is made of BK7 glass, then this angle of incidence exceeds the critical angle for TIR of 41.2°, and the ray is indeed reflected. We may now calculate how the critical angle for TIR translates into the critical angle of incidence .αg on the prism base. It is reached as soon as the exit angle .β

1 Named

after Ignazio Porro, 1801–1875.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Merlitz, The Binocular Handbook, https://doi.org/10.1007/978-3-031-44408-1_3

45

46

3 Image Erecting Prisms

Fig. 3.1 Total internal reflection (TIR): a light ray (red) enters a rectangular prism perpendicular at its base and passes through the prism as a result of TIR at its sloped faces. Another ray (blue) fails to reach the critical angle and leaves the prism prematurely

(shown with the blue ray in Fig. 3.1) at the sloped face exceeds 90°. The calculation is fairly simple: The beam that enters at angle .α to the normal of the base is refracted towards the perpendicular according to the refraction law and continues its path inside the prism at the new angle .α1 = arcsin(n−1 sin α), while at the sloped face, the angle of incidence is now .45◦ − α1 , whereupon the second refraction occurs. We then obtain for the sine of the exit angle 



sin α . sin β = n sin 45 − arcsin n ◦

 ,

(3.2)

and this is set to unity to obtain the critical angle of entry, with the result     1 αg = arcsin n sin 45◦ − arcsin . n

.

(3.3)

For BK7-glass (.n = 1.517) we obtain a critical angle of .αg = 5, 7◦ . The blue ray in Fig. 3.1 has an angle of incidence of .6, 6◦ and would therefore, as shown, leave the prism without total internal reflection. If we replace the BK7 by a glass of higher refractive index, such as BaK4-glass (.n = 1.57), then .αg = 8, 6◦ , and under these circumstances the same ray would pass the entire prism. This is also the reason why the manufacturer’s advertising slogans like to tout their use of Bak4 optical glass as the superior prism material. However, this is not generally so: even though the critical angle of 5.7° is frequently exceeded with wide-angle binoculars, which then indeed require the implementation of highindex glasses such as Bak4, the BK7 optical glass is often perfectly sufficient for binoculars of ordinary angles of field, it is cheaper than Bak4 and offers a superior internal transmittance at short wavelengths (Fig. 1.5). The ideal choice of the prism

3.1 Porro-Type Systems

47

Fig. 3.2 Simplified projection of a Porro I image erecting system: a ray is shown along the principal axis as it passes through the prism pair (left). In binoculars, a convergent ray cone is common. The bundle of rays enters the first prism with a diameter w, but exits with a reduced cross section (right), which often allows a reduction of the size of the downstream prism

Fig. 3.3 3D-representation of the beam path through a Porro I system (design: Stefan Emsel)

material is thus not merely a matter of alleged quality, but a simple consequence of the optical layout and the resulting ray-fans.

3.1.2 Porro I System A Porro prism has the shape of an isosceles triangle with a 90° angle, and two opposing Porro prisms form a Porro I erecting system. Figure 3.2 is somewhat simplified for clarity: actually, both prisms are rotated by 90° against each other, so that the lower prism protrudes from the paper plane (see the 3D-representation in Fig. 3.3). Only the ray along the optical axis is shown, whereas in practice the entire

48

3 Image Erecting Prisms

beam almost always forms a convergent cone, whose diameter at the prism entrance roughly coincides with the prism-entrance width w. The entrance width provides a length scale with which all the other dimensions of the prism may be conveniently scaled. Thus, it is possible to calculate the glass path length along the principal axis to .L = 4w, as demonstrated in Fig. 3.2 (numbers along the partial paths are given in units of w). We want to generalise this relation and therefore define the geometric glass-path length L = κw

.

(3.4)

with the geometric glass-path factor .κ, which assumes an individual value for each prism type, in this case .κ = 4. The assembly length of the prism block is .M = 2w + x, where x is the air gap size between the two prisms. This gap may equal zero in case of cemented prisms, which further offers the advantage of reducing the number of air-to-glass transitions and the resulting reflection losses. If two identical prisms are implemented, then the glass volume of the prism block amounts to .V = 2w 3 , and we may calculate its weight by multiplying this volume by the specific density of the optical glass. In the case of wide-angle binoculars, which require large prism widths w, the implementation of asymmetric prisms is often preferable. Figure 3.4 gives an example of such an asymmetric Porro prism: the prism entrance width of .w = 33 mm would, in the case of a symmetric prism, yield a volume of 71.8 cm3 , which in turn, with the specific density of BaK4 glass, 3.05 g/cm3 , yields a considerable mass of 219 g. However, the asymmetric prism shown here is a compound of two partial prisms, which weighs only 89.6 g, while the second, smaller prism weighs 41.9 g. Summing up, we obtain 131.5 g, roughly 60 % of the symmetric Porro prism. √ A symmetric Porro I system causes a lateral axis offset of . 2w between objective and eyepiece axes which increases the stereo baseline and thus improves the stereoscopic (three-dimensional) visual impression of its image (Fig. 3.5). The extended distance between the objectives also enables the designer to implement objective diameters that considerably exceed the value of 54 mm, generally used as the minimum inter-pupillary distance of the eyepieces. A disadvantage of the axis offset is an increase in the close-up range (at close ranges, the images from Fig. 3.4 Asymmetric prism block: at the prism entrance (bottom left, width: 33 mm) a black lens hood made of plastic is mounted. The first prism is composed of two cemented partial prisms, the downstream prism is smaller and the total assembly weighs only 60 % of a symmetric Porro prism

3.1 Porro-Type Systems

49

Fig. 3.5 The 10x50 Zeiss (Oberkochen, field of view: 130 m/1000 m) is a typical representative of a wide-angle binocular with Porro I prisms

widely separated objectives are impossible to match into a single image). The wider dimension of the binocular housing also adds to its weight. With certain compact pocket binoculars, the axial offset is therefore reversed to reduce the distance between the objectives (reverse Porro design), which leads to a reduction in weight and to a considerable shortening of the usable close-up distance (Fig. 4.15).

3.1.3 Porro II System Take a Porro I image erecting prism, cut one of the prisms in the middle and glue the two rider prisms in opposite orientations onto the base prism (see Fig. 3.6). The result is a Porro II prism, which Ernst Abbe once designed as a modification of the Porro I system. Weight and glass path length being identical to the Porro I, the Porro II has the advantage of a shorter assembly length, which amounts to just .M = w, leading to the characteristic short, can-shaped prism housings (see Fig. 3.7). The Porro II system generates a lateral axis offset of w, shorter than that of the Porro I system. Stereoscopic vision is therefore somewhat less impressive, but the slimmer shape of the housing allows for an overall compact design of moderate weight—an advantage in particular with wide-angle binoculars. Variations of the conventional symmetric design also exist. Figure 3.8 displays one example, used in the Zeiss 8x60 submarine commander’s glass. Here, the base prism was somewhat smaller in size in relation to the rider prisms in order to reduce volume and weight of the assembly. Note that this particular design consists of two cemented parts instead of three, with the vertical plane of symmetry forming the cemented surface. The reduction from initially two to a single cemented surface presumably had process advantages and is currently also used for the Perger prism (Sect. 3.1.4). Until the end of the Second World War, Porro II prisms were frequently found in state-of-the-art military binoculars. Among other things, they had the advantage that the field lens of the eyepiece could be cemented onto the surface of the prism exit to further reduce transmission losses, thus increasing the binocular’s low-light performance. This way, the moving parts of the eyepiece remained separated from

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Fig. 3.6 Ray-tracing through a Porro II system (design: Stefan Emsel) Fig. 3.7 The 10x50 Stepmur (field of view: 123 m/1000 m) of the British Ross company, a binocular with Porro II prisms

the often rather large field lens, which allowed for a reduction of the eyepiece sleeve diameter. The invention of anti-reflective coatings subsequently eliminated the advantage of the cemented field lens, and the increasing demand for lighter

3.1 Porro-Type Systems

51

Fig. 3.8 Porro II prism block from a vintage Zeiss 8x60 binocular (reproduction from [1] with kind permission from Hans Seeger)

(albeit less wide-angle) binoculars ensured that the Porro II design has now largely disappeared from the market. Among the few binoculars still existing of this prism type are some of the imagestabilised Canon IS models (see also Fig. 4.24). Here, the inter-pupillary distance is not adjusted by means of a bridge, but by rotating the eccentrically mounted prism housings. The same technical solution is found with some large mounted binoculars where the eyepiece distance adjustment cannot easily be achieved via a central bridge. Here, the compact can-shaped prism housings are conveniently rotated against each other, whereby the spacing between the objective lenses remains fixed.

3.1.4 Perger Prism A Contribution by Dr Andreas Perger The image erecting systems presented here have been built almost unchanged for several decades. Apart from minor optimisations of their geometries, it was, above all, improvements to the coatings on which the developers have focused recently. However, fundamentally new geometries have not been seen in binocular construction for quite some time. The commonly used prisms are based on inventions which, in part, date back more than a hundred years. This is all the more astonishing because the prisms that are currently used are by no means without disadvantages. In terms of their optical properties, closest to the ideal are the Porro prisms, which, however, due to their axis offset result in a rather bulky beam path and thus inevitably lead to a design that is widely regarded as antiquated. Binoculars with roof prisms, which are advertised as being the more sophisticated designs, offer no advantages apart from their compactness and the coaxial position of objective and eyepiece. On the contrary, the presence of a roof-edge,

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Fig. 3.9 Perger prism with a principal axis ray (red) and the cemented plane (dark blue)

the roof surfaces of which have to be P-coated,2 and in the case of most types also a surface that requires a reflective coating, and in some cases also an air space between prism-blocks, are obvious disadvantages, which the manufacturers largely, but not entirely, manage to compensate for. The objective in developing the Perger prism was to create a system which, like a Porro prism, requires only four beam deflections and no roof edge, no surface to be mirrored and no air space, but with a reduced lateral offset between the objective and eyepiece axes as compared to the classical Porro systems of types I or II (Fig. 3.9). The starting point for its design was the type II Porro prism. This can be made, as shown in Fig. 3.8, from two identical pieces of glass, which are cemented together on the surface that aligns with the main axis of symmetry. A tilt of this surface with respect to the path of rays passing through it offers the first possibility of reducing the beam offset somewhat. The offset can be reduced even further if the first and the last faces at which total internal reflection takes place are no longer tilted by 45° with respect to the optical axis, but by a little more. Then the beam is deflected by more than 90°, and the central part of the ray path, which passes through the cemented surface, is no longer perpendicular to the direction of the optical axis (i.e. in a plane normal to the objective and eyepiece main axes), but at an angle. This results in a lateral axial beam-offset of .0.7w or even less. The axial offset can be further reduced by rotating the first and last reflecting surfaces about the optical axes of objective and eyepiece, respectively, whereby the relevant beam sections no longer fall onto parallel planes. In this way, the offset may

2 Roof

prisms require a phase-correcting coating, see Sect. 3.2.6.

3.1 Porro-Type Systems

53

Fig. 3.10 Ray-tracing through a Perger prism (design: Stefan Emsel)

at least theoretically be reduced to zero. An illustrative representation can be found in the patent specification EP2463692A1. Other advantages of the Perger prism are its small volume of typically .1.37w 3 , a short glass path, and the fact that all reflection angles are greater than 45°, which allows for a higher speed of the objective, or alternatively the choice of a glass type with a lower refractive index. The short glass path length (typically .3.3w) allows for a high transmittance, with the disadvantage that the Perger prism does not reduce the overall length of the binocular as much as, for example, a Schmidt–Pechan system, since its assembly length is somewhat larger (typically .1.55w). These values are offered here as typical examples, because the designer is given a considerable degree of freedom through the choices of angles and the resulting geometrical properties of the prism (Fig. 3.10). Further, it should be noted that, depending on the layout, the maximum possible beam diameter is reduced at the central cemented surface and not quite matching the value w at its entrance. However, this is rarely a disadvantage because the most common case in binocular design is the one in which the ray-bundle diameter is widest at the prism entrance. In other prism designs, the beam is also stopped down inside the prism system, to reduce stray-light, as with the Schmidt–Pechan prism by means of a diaphragm that is inserted into the air gap. The Perger prism was also developed with regard to the usability in binoculars with an integrated rangefinder. Due to the inclination of the cemented surface with respect to the optical axis, a dichroic beam splitter can be inserted without the need for additional optical components. Such a dichroic coating enables light of a selective wavelength to be deflected into the optical path, while allowing other frequency ranges to pass through un-hindered. In this way, a display of a given colour can be superimposed onto the image, or the measuring beam, whose frequency is usually in the infrared range, and can be coupled into the beam

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path without any significant loss in the overall transmission. A similar type beam insertion into prisms is already described in the patent specification EP1069442A3. The first binocular which used the Perger prism is the third generation Leica Geovid. This binocular comes with an integrated rangefinder, in which the inclined cemented surface is employed for the implementation of the above-described measuring beam and display options.

3.2 Roof Prisms Since the early days of the development of prism binoculars, attempts were made to invent alternative image erecting systems of lower weight and superior compactness rather than using Porro prisms. Mention should be made of the pentaprism system that Moritz Hensoldt incorporated into his first roof-prism binoculars at the end of the nineteenth century. Since that time, numerous different types of roof prisms have been invented, of which only the most important representatives will be discussed here. The common feature of these prisms is the presence of the roof edge, located at the junction of two roof faces to form a precise 90° angle, which splits the beam in half.

3.2.1 Schmidt–Pechan Prism By far the most widely used roof prism system is known as the Schmidt–Pechan prism (Fig. 3.11). It consists of two single prisms, which are separated by a narrow air space of only a fraction of a millimetre. This prism owes its popularity to its two main features: its enormous compactness and the very small axial offset, which is why it belongs to the class of direct-vision prisms. Both properties are recognisable from the sketch in Fig. 3.11:3 the axial assembly length is given by .M = 1.21w, the glass volume is only .V = 1.80w 3 , and the cross-section of the prism block remains rather compact in all dimensions. Yet, the geometric glass path amounts to .L = 4.62w, quite a bit longer when compared to a Porro system. With such a complex optical path, it is unavoidable that the critical angle for total internal reflection is missed at one of the prism faces, which therefore has to be coated with a reflective layer (blue). The illustration gives the impression that the same coating is also necessary on the green (dashed) face, but that is a deception, based on the fact that the 45° inclinations of the roof faces are invisible in this flat projection. For illustration, Fig. 3.12 shows the prism system as a 3Dsketch and Fig. 3.13 the photo of a mounted prism block.

3 The

dimensions used here are taken from the book by Yoder [2].

3.2 Roof Prisms

55

Fig. 3.11 Sketch of a Schmidt–Pechan prism. The principal axis ray is shown in red, the reflective face in blue, the roof edge in green. The top view onto the upper roof face is shown as a hatched square

Fig. 3.12 3D-ray-tracing through the Schmidt–Pechan prism (design: Stefan Emsel)

The air space between the two individual prism blocks is required to ensure that the first and fifth deflections provide total internal reflections, so that there is no option to cement the two blocks together. To minimise reflection losses, it would be sensible to apply an anti-reflective multi-coating onto the two surfaces that face the

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Fig. 3.13 Mounted Schmidt–Pechan prism, the roof edge is visible on the right-hand prism block, the entrance to the left

air gap, but this leads to undesirable complications [3]: the anti-reflective coating affects the efficiency of the total internal reflection at the surface that lies below the coating. We recall that total internal reflection requires a sudden jump of the refractive index at the interface between two media, but the anti-reflective coating mitigates such a jump, since the refractive index of the vapour-deposited layer has to lie in between those of glass and air (Fig. 2.7). With a convergent ray-bundle incident at a given angle, the critical angle of total internal reflection between glass and coating could be missed by some of the partial rays, whereupon these rays penetrate the coating, to be subsequently reflected at the coating-to-air interface, but now with a path-length difference and an associated shift of its dynamic phase, which eventually causes a loss of micro-contrast of the image. Since it was Swarovski’s optical designer Konrad Seil, who first reported this phenomenon, for which an official term appears to be absent, we shall here denote the impairment of total internal reflection by a simultaneously applied anti-reflective coating the Seil-effect. Seil further evaluated that the best compromise, in terms of the resolution of the binocular, would be the application of a single-layer on the two affected surfaces. In practice, however, manufacturers seem to ignore the implications of the Seil effect and nonetheless apply multi-coatings to avoid the danger of ghost images caused by multiple reflections from the surfaces which are facing the air gap. As for the appropriate choice of glass for a Schmidt–Pechan prism, the same rules apply as for the Porro I and II prisms due to the identical 45° inclination angles: for wide-angle binoculars, the use of a high-index glass is necessary to fulfil the conditions for total internal reflection. But the production of a high-performance Schmidt–Pechan prism is considerably more demanding than that of a comparable Porro prism, which is partly due to the additional reflective coating and the two phase-correcting coatings (see Sect. 3.2.6). Additionally, the 90° angle between the roof faces allows maximum deviations of the order of a few seconds of arc [4], while the angular tolerances of a Porro prism are higher by a factor of 10. Furthermore, it should be noted that the light beam touches a total of nine surfaces that are relevant for imaging (in the case of a cemented Porro system there would only be

3.2 Roof Prisms

57

Fig. 3.14 Sketch of the Abbe-König prism. The ray (red) enters along the principal axis, the roof edge is shown in green. The top view onto the upper roof face is shown as a hatched square

six surfaces, of which only two need to be coated). Common to all roof prisms is the problem of light diffraction at the roof edge, which generates spikes perpendicular to the roof edge on bright point-like objects. A very precise cut of the roof edge is required to minimise these disturbing artefacts. Despite these difficulties, modern manufacturing methods allow for the fabrication of Schmidt–Pechan prisms that almost match the performance of well-made Porro prism systems, though at higher costs.

3.2.2 Abbe-König Prism Another roof-prism system was initially suggested by Ernst Abbe and Albert König and later modified and implemented by the Hensoldt company, to be initially known as the Hensoldt prism. After Hensoldt had been taken over by Zeiss, the same prism, as shown in Fig. 3.14, was renamed to become the Abbe-König prism.4 It is almost of the direct-vision type, though some variants do feature a considerable lateral axis offset. It does not offer the compactness of a Schmidt–Pechan system (Fig. 3.15), but comes with another significant advantage: it does not require the deposition of any reflective coatings, because the critical angle for total internal reflection is never exceeded. Another advantage is that both prism blocks may be cemented together, which reduces the transmittance losses (Fig. 3.16). The geometric glass path is, with .L = 5.20w, somewhat longer compared to the Schmidt–Pechan and significantly longer than with the Porro system.5 The mechanical assembly length

4 Thanks

to Walter Schwab for a clarification of these historical events.

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Fig. 3.15 3D-plot of the functioning of an Abbe-König prism (design: Stefan Emsel) Fig. 3.16 Abbe-König prism of a vintage Zeiss binocular (reproduction from [5] with kind permission from Hans Seeger)

is a considerable .3.46w, which makes binoculars with Abbe-König prisms have their characteristically tall and slim shape (compare, for example, with Fig. 4.19). Moreover, the volume with .V = 3.72w 3 and thus, its mass are higher than those in comparable image erecting systems. For this reason, some designs come with a wide air space between the prism blocks, which helps in reducing both mass and assembly length,6 since for a given geometrical path length, the optical path length through air is longer than through glass. If such an air space is implemented, the involved surfaces may be multi-coated without being compromised by the Seil effect (see previous section).

5 The

dimensions are taken from Yoder [2]. communication, Andreas Perger.

6 Personal

3.2 Roof Prisms

59

The elongated prismatic system implies that the beam is reflected at shallow angles, which is why for these prisms, unlike the Porro or Schmidt–Pechan, highindex glasses are not necessarily required.7 It is therefore possible to choose from a wider range of optical glasses and to select one that offers a lower specific density, which at least partially compensates for its volume-related disadvantage: BaK4 glass has a specific density of 3.05 g/cm3 , while BK7 with 2.51 g/cm3 would yield a noticeably lighter prism. With a transmittance that is superior to the Schmidt–Pechan prism, the AbbeKönig prism has been the preferable choice for night-glasses, and the traditional 8x56 Dialyt hunting binoculars from Hensoldt (later: Zeiss) once enjoyed a legendary reputation. Since 2012, there exists the option to implement HT-glasses (either BK7 HT or similar types) of further improved transmittance (Fig. 1.5). In this way, it was possible to keep a certain transmittance advantage of the AbbeKönig system over the Schmidt–Pechan, even after technical improvements such as dielectric reflective coatings had rendered the latter design highly competitive. This is presumably the reason why the Abbe-König prism is, despite its bulky dimensions and relatively high weight, still found to this day in some high-performance binoculars.

3.2.3 Uppendahl Prism Another roof prism less often found is the Uppendahl design. It consists of three individual prisms which are cemented together as shown in the sketch in Fig. 3.17. Here, the distance b is a free parameter that has to be chosen so that the width of the roof face accommodates the entire light-cone. If the latter converges sufficiently Fig. 3.17 Sketch of an Uppendahl prism. The ray along the principal axis (red), the roof edge (green) and the location of the reflective coating (blue). A view onto the upper roof face is shown as a hatched square

7 Personal

communication, Walter Besenmatter.

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fast, then the choice .b = 0 is possible, which yields a fully symmetric prism. If instead we assume a ray-bundle of constant diameter w, then the projection of the roof face (shown as the hatched square) has to have a width of .w/2. Hardly any literature exists about the Uppendahl prism, so that we have to carry out our own calculations: a little elementary geometry yields √ 2 cos 22, 5◦ + − 1 ≈ 0.169 .b = 2 2

(3.5)

(in units of the prism entrance width w). The geometric glass-path length then adds up to L=3+

.

√ √ 2 + (1 + 2 2)b ≈ 5.06 ,

(3.6)

the axial assembly length amounts to M=

.

√ 2 + b ≈ 1.58 ,

(3.7)

and the volume is approximately .V ≈ 2, 25◦ w 3 . For the special case of the symmetrical Uppendahl prism (.b = 0) we get .L ≈ 4.41w, .M ≈ 1.41w with a volume of about .V ≈ 1.80w 3 , identical to the volume of the Schmidt-Pechan prism. When compared to the Schmidt–Pechan prism, the Uppendahl design has the advantage of being fully cemented, which reduces reflection-related losses (Fig. 3.18). It is somewhat less compact and protrudes far to one side, but the bulky part can be oriented in the direction of the bridge, which then allows for a particularly slim body shape, as shown in Fig. 4.14. On the down side, such a configuration excludes the possibility of an open bridge design. Further more, fabrication costs are high since they involve three glass blocks that have to be cemented to the highest precision. As the Schmidt–Pechan, the Uppendahl prism suffers from the Seil effect (Sect. 3.2.1). The Uppendahl prism was used in the Leitz Trinovid models of the 1960s to 1980s and later in the Leica Geovid models. The latter contain a rangefinder and one of the inclined surfaces at which the prism blocks are cemented, serves, after being coated with a dichroic layer, as a beam splitter at which the measuring beam is deflected into the optical path. Meanwhile, the Perger prism (Sect. 3.1.4) has replaced the Uppendahl in the Geovid models, so that—to the author’s best knowledge—none of the currently sold binocular models are making use of the Uppendahl prism.

3.2.4 Mirror–Prism Combinations Binocular technology knows of further, rather exotic image erecting systems, which consist partly or entirely of mirrors. It is obvious that a Porro I system, for instance,

3.2 Roof Prisms

61

Fig. 3.18 3D-representation of the functioning of an Uppendahl prism (design: Stefan Emsel) Fig. 3.19 Leitz 6x24 Amplivid of the 1950s (field of view: 212 m/1000 m)

could be replaced by four separate mirrors, with the advantage of the associated weight reduction. However, such a setup would come with disadvantages, too: it is rather difficult to adjust four separate mirrors and to ensure the stability of collimation under harsh conditions; after all, binoculars do not stay stationary on an optical bench, but are exposed to mechanical and thermal stresses. Furthermore, every reflective layer comes along with reflection losses, which is why prism blocks which employ total internal reflection are so useful. For this reason, combined mirror–prism erecting systems had a certain experimental character, as in the case of the Leitz 6x24 Amplivid (Fig. 3.19), which offered a particularly compact and lightweight body despite of its impressive field of view

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of 212 m/1000 m. The sophisticated design by Ludewig and Schade was abandoned by Leitz after only a short production phase in favour of the Uppendahl design. Problems arose due to the poor collimation stability and the short life of the mirror coating, which lost much of its reflectivity after only a couple of years. Another example was the 6x18 Klein-DF compact binocular made by Zeiss Jena during the 1970s and 1980s, which also contained a mirror–prism reversal system. By that time, the reflective mirror coatings could be sealed sufficiently so that they remain functional to date. The 6x18 Klein-DF not only offered a particularly low weight of only 170 g, but also an internal focusing mechanism and a body made of macrolon, an impact-resistant polycarbonate, from which the housings of professional cameras are still made of today. In the author’s opinion, such hybrid erecting systems are likely to offer new opportunities in modern binocular design, because after the invention of dielectric coating technology, almost perfect and durable mirrors are available. It remains to be seen whether binocular manufacturers will once again go into these hybrid erecting systems, especially since the demands for compactness and further weight reduction are becoming increasingly pressing in modern binocular technology.

3.2.5 Reflective Coatings As mentioned in the previous section, most of the roof prisms used in hand-held binoculars today, with the exception of the Abbe-König system, contain a surface at which total internal reflection fails and which therefore requires a reflective coating. In the past, thin metal layers were vapour-deposited onto the corresponding prism surface and subsequently sealed against corrosion. Smooth metal surfaces serve as mirrors, because their freely moving electrons are excited by the incident light to oscillate, in the process building up an electromagnetic field that superposes with the incident light and thus causes the reflection. Conversely, light cannot penetrate the metal since it is almost perfectly shielded. Thin metallic layers achieve reflectivities in visible light of typically 92 % (aluminium) or 96 % (silver). A disadvantage comes along with the frequency dependence of the reflectivity: the electrons cannot respond arbitrarily fast to an external field, because every metal has its specific plasma frequency—the natural frequency at which the electrons would oscillate freely against the ion lattice. This plasma frequency lies for most metals in the near UV frequency range, and the short wavelength end of visible light is already approaching dangerously close to this resonance condition. As a result, blue to violet light is able to penetrate deeper into the metal coating and is thereby partially lost. Gold is an extreme case in which violet light is almost entirely absorbed, which gives the metal its characteristic metallic-yellow sheen. Accordingly, metal-coated prisms display a loss of transmittance near the short wavelength end of the visible spectrum, as shown in Fig. 3.20, and the images of the corresponding binoculars tend to exhibit a somewhat warm tint. In recent years,

3.2 Roof Prisms

63

Fig. 3.20 Reflectivity: dielectric multilayer and silver coating, at an angle of incidence of 24°

the invention of dielectric reflective coatings has led to significant improvements in terms of both total transmittance and colour fidelity of the prism. Their functional principle is—similar to anti-reflective coatings—based on destructive interference (Fig. 2.7), but with the opposite effect: a thin, transparent layer of refractive index higher than that of the glass substrate is applied such that the transmitted rays interfere destructively; conservation of energy then leads to a corresponding increase of reflectivity. A single thin layer, however, would be limited in its effect, as it is only optimised for a single wavelength and angle of incidence. The performance of dielectric layers is dramatically improved by vapour deposition of several layers, designed after a carefully chosen sequence of materials and refractive indices. These multi-layer dielectric coatings consist of layers that are arranged in pairs, always so that the outer layer features the higher refractive index than its adjacent inner layer [6]. Highperformance dielectric mirrors consist of more than 30 layers and achieve an average reflectivity that exceeds 99 % over the entire visible spectrum. Since the deposition of these high-end coatings is a time-consuming process that requires continuous supervision, the resulting prisms are expensive and only available with binoculars of the high-end market sector. Yet, simpler and insignificantly less effective dielectric coatings are nowadays found in lower-to-mid-range roof-prism binoculars, whereas metallic mirror coatings remain restricted to the low budget binocular sector, in which Porro prisms would certainly be the preferable choice.

3.2.6 Phase-Shift and Its Correction A peculiar effect, first described by G. Joos in 1943 and, independently, in 1945 by A. I. Mahan [7, 8], emerges in roof prisms: word had got around among the manufacturers that along with the application of roof prisms, a loss of image resolution of a degree not expected with well-corrected optics, had been observed. In the direction perpendicular to the roof edge the resolution turned out to be significantly affected, since from a point-like light source, multiple images emerged,

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Fig. 3.21 Disturbed point image in a roof binocular without phase correction. Reproduction from [9] with kind permission from the DOZ-Verlag

producing a spike-like artefact as shown in Fig. 3.21, a reproduction from an article by Weyrauch [9]. At first, this circumstance was attributed to manufacturing defects since the tolerances of these prisms were tight, and it could not be ensured that the 90° angle could be cut to within requirements. However, the theoretical studies mentioned above led to the shocking conclusion that even mathematically perfect prisms would necessarily suffer from the observed loss of resolution. In order to understand the causes of such an imaging error, we must once again refer to the wave properties of light: In the case of total reflection, a partial polarisation occurs. Light is said to be polarised when it oscillates preferentially in one (or several) plane(s). In everyday life these kinds of polarisation effects show up whenever light is reflected (e.g. on a water surface) or scattered (Rayleigh scattering leads to a partial polarisation of sunlight, see also Sect. 8.7). A photographer uses polarising filters to block light of a selected plane of oscillation in order to reduce unwanted reflections or to make the sky appear darker. In each prism, a partial polarisation of the light beam appears as a result of total internal reflection. What is special about a roof prism is that the light-bundle is split up, because by reflection at the roof faces, the upper and the lower halves of the beam are forced into different directions. Down the optical train, the partial beams reunite in the primary focus of the binocular. What happens between splitting and reunification of the individual partial rays can be explained with the help of an illustrative example: Figure 3.22 shows, strongly schematised, the flight of an aeroplane, which starts from the North Pole and flies along a line of longitude to the equator, turns to the left, continues along the equator a quarter of the circumference, and finally turns again to return to its starting point. On the way, an attentive passenger notes two precise 90° turns to the left. Otherwise, the trajectory always seems to be straight ahead. When he reaches the airport of departure, he expects to arrive at the runway (after two 90° turns) from the opposite direction, yet to his surprise, he crosses the runway at a right angle. The reason behind his confusion is, of course, the fact that the trajectory was not flat, but forming a three-dimensional curve. Along this curve, the airplane experienced an additional turn—invisible to the passenger—as a result of the Earth’s curvature. In physics, this additional displacement of direction is known as a geometric phase. Similarly, an analogous process occurs in a roof prism: the partial polarisation of light results in a polarisation vector, which, like the nose of an aeroplane, defines a certain preferred direction. The beam of rays is split at the roof edge, and both

3.2 Roof Prisms

65

Fig. 3.22 Geometric phase: an aircraft (green) takes off from the North Pole for a round trip and, after two 90° turns, arrives at its airport of departure 270° displaced

partial beams pass through the prism on different paths. When the two partial beams are joined, their polarisation vectors point in different directions because they have picked up different geometric phases along their respective paths. The phase-shifted beams interfere with each other and cause the loss of image resolution in the primary focus. More specifically, the geometric phase of the polarisation vector is called the Pancharatnam phase [10], and M.V. Berry recognised in the 1980s that it is an example for a more fundamental phenomenon, associated with the phase shift of quantum mechanical states in adiabatic cyclic processes [11]. In technical optics, this effect is simply called phase shift in roof prisms, and its correction is consequently called the phase-correction or P-correction. The first phase-corrected roof prisms were developed at Zeiss in 1988 by A. Weyrauch. In the process, a thin dielectric coating is applied to both roof surfaces, called a P-coating, by which the partial polarisation, which would otherwise occur during total internal reflection, is largely eliminated (Fig. 3.23). In the absence of a polarisation vector, its subsequent phase shift becomes irrelevant, and the problem of interference of differently polarised partial beams is thus solved. As always with thin layer coatings, such a correction works only for certain angles of incidence and wave lengths, a problem that can be mitigated with complex multiple coatings [12]. With the introduction of the P-coating towards the end of the 1980s, manufacturers succeeded for the first time in bringing the premium segment of their roof-prism binoculars to a level comparable to that of Porro binoculars. The remaining differences were diminished in the following two decades with further refinements such as dielectric reflective coatings and continuing improvements of

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Fig. 3.23 Phase correction: the roof edges are coated with a dielectric layer which largely eliminates the partial polarisation of light during reflection

the P-coatings. Today, it can be assumed that a high quality Schmidt–Pechan prism has become almost as efficient as a Porro erecting system and that a modern AbbeKönig prism is even on a par with Porro in all practical respects. The manufacturing cost of a good roof prism, however, is partly responsible for the high price of a premium binocular. With the invention of the Perger prism, it has become possible to combine the advantages of several traditional image erecting systems, and it remains to be seen whether manufacturers will make more use of this opportunity in the near future, or whether, with the advent of digital binoculars, the erecting prism is going to lose its function altogether in modern binocular design.

References 1. Seeger, H.T.: Zeiss Feldstecher. Zeiss Handferngläser 1919–1946. Dr. Hans T. Seeger, Hamburg (2015) 2. Yoder Jr., P.R.: Mounting Optics in Optical Instruments. SPIE Press Bellingham, Washington (2008) 3. Seil, K.: Progress in binocular design. Proc. SPIE 1533, 48 (1992) 4. Rosendahl, G.R.: Tolerances for Roof Prisms. J. Opt. Soc. Am. 49, 830 (1959) 5. Seeger, H.T.: Zeiss Feldstecher. Handferngläser 1894–1919. Dr. Hans T. Seeger, Hamburg (2010) 6. Dohlus, R.: Photonik: Physikalisch-Technische Grundlagen der Lichtquellen, der Optik und des Lasers. Oldenbourg Wissenschaftsverlag GmbH, München (2010) 7. Joos, G.: Die Bildverschlechterung durch Dachprismen und ihre Behebung. Zeiss Nachrichten 4, 9 (1943) 8. Mahan, A.I.: Focal plane anomalies in roof prisms. J. Opt. Soc. Am. 35, 623 (1945) 9. Weyrauch, A., Dörband, B.: P-Belag: Verbesserte Abbildung bei Ferngläsern durch phasenkorrigierte Dachprismen. Deutsche Optikerzeitung 4, (1988) 10. Pancharatnam, S.: Generalized theory of interference, and its applications. Part I. Coherent pencils. Proc. Indian Acad. Sci. Sect. A 44, 247 (1956) 11. Berry, M.V.: The adiabatic phase and pancharatnam’s phase for polarized light. J. Mod. Opt. 34, 1401 (1987) 12. Mauer, P.: Phase compensation of total internal reflection. J. Opt. Soc. Am. 56, 1219 (1966)

Chapter 4

The Anatomy of Binoculars

4.1 Objective Lenses In Sect. 1.5, we discussed the imaging properties of a single plano-convex lens (Fig. 1.6). Such single lenses were used in the seventeenth and eighteenth century as the objectives of refractors which, with their huge aberrations, could only produce a usable image because they had very slow focal ratios of about f/40. Since the second half of the eighteenth century, objectives with two lens elements became the new standard; the Abbe numbers of both elements differed in such a way that an acceptable low level of chromatic aberration resulted. Such objectives were called achromatic, indicating a low level of colour fringing in their images.

4.1.1 The Achromatic Objective Figure 4.1 shows an achromatic objective made of two lens elements separated by a wide air space—a rather popular design with modern binoculars [1]. The front element is a convex lens, made of crown glass of low refractive index (.n1 = 1.5168), followed by a concave lens of flint glass of high index (.n2 = 1.7283). Note the construction of that drawing: since all lens surfaces are spherical, the sectional view can be drawn using circles with their radii and positions optimised in the process of the optical design. The two construction rays (red) are principal rays which enter the objective on-axis and which, when crossing the assembly of lenses, get refracted in line with the laws of refraction. Tracing the converging rays backwards from the focal point F (the straight blue dotted lines), they intersect with the incoming principal rays at the position of the principal plane. It is interesting to note that the principal plane is not located somewhere inside the objective assembly, as in case of the plano-convex lens of Fig. 1.6, but rather in front of it. This enables the designer to keep the overall length of the instrument shorter than the focal length of its objective—a consequence of the air space between the © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Merlitz, The Binocular Handbook, https://doi.org/10.1007/978-3-031-44408-1_4

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Fig. 4.1 Achromatic objective with wide air space (tele-lens): The principal plane (H) is situated outside the objective cell, which shortens the physical length of the instrument

lens elements: the convex lens in front has a short focal length which lets the cone of the light beam converge rapidly within the air space. The concave lens then reduces that slope and slows down the focal ratio to about .f ≈ 1/4, a typical magnitude for binoculars. An objective with this configuration is called a tele-lens. The tolerances of manufacturing and alignment of the two lens elements are rather tight so that the tele-lens principle is found only in binoculars of elevated price sectors. Achromatic objectives may also be built without an air space, using two cemented lens elements that are sealed together with a transparent putty such as Canada balsam. Such a lens assembly is rather resistant against misalignment and, since the cemented surfaces do not need to be coated, can be produced at lower costs. However, the optical design is then restricted in two ways: the radii of the two cemented lens elements have to be identical, and the function of the air space as an air lens is lost. In the past, without the benefits of advanced coating technologies, the main advantages of a cemented objective were its superior transmittance and reduced stray light. Today, cemented objectives are mainly implemented in binoculars in which high optical performance or compact build is secondary to competitive pricing. An objective, designed with a narrow air space between the convex crown lens and the concave flint lens is known as a Fraunhofer objective; if this configuration is reversed—flint glass to the front, followed by crown glass—then the objective is called a Steinheil objective.1 Both configurations offer additional degrees of 1 Carl

August von Steinheil, 1801–1870.

4.1 Objective Lenses

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Fig. 4.2 The 8.5x44 Audubon ED (field of view: 137 m/1000 m) of the well-known US distributor SWIFT, produced in Japan, was a rare example of a modern Porro binocular that features ED objectives

freedom to balance the aberrations and thus deliver an imaging quality that is superior to cemented doublets. If both coma and spherical aberration are fully corrected for a given wavelength, then the objective is called aplanatic.

4.1.2 ED Objectives After replacing the first lens element in Fig. 4.1 with a glass of high Abbe number, in combination with a second element of crown instead of flint glass, the chromatic aberration of the extra-low dispersion (ED) objective may turn out to be substantially reduced (Fig. 4.2). Yet, attention has to be paid to the fact that ED glass mixed with fluoride ions, e.g. the so-called FPL-53 of the Japanese manufacturer Ohara, is somewhat brittle and therefore harder to figure than crown glass. It further comes with a high thermal expansion coefficient, which in cases of rapid temperature changes, may generate problematic deformations of the lens, temporarily affecting its optical performance. That is why in practice, the order of the lens elements in a doublet with ED glass is often reversed to have the more rugged concave element of crown glass in front (turning it into a Steinheil objective); this set-up efficiently protects the ED element against mechanical damage or thermal effects [2]. The advantage of ED objectives—with their low dispersion levels—has to be paid for with a low refractive index. To yield a pre-described focal length, the curvatures of the lens surfaces then have to be increased, which leads to higherorder residual aberrations and compromises the overall image quality. This fact and the general trend toward compact instruments with faster focal ratios have driven the manufacturers of high-end binoculars to implement objectives with three lens elements, known as triplet objectives. Usually, these objectives are composed of a cemented doublet, separated by an air space and followed by a third element. This set-up with its increased number of degrees of freedom allows the optical designer to build compact binoculars with well-corrected aberrations.

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Fig. 4.3 Achromatism and apochromatism (schematic): A single lens (black line) intersects the optimum image plane (dotted line) at one point. With an achromatic lens (blue), there exist two, with an apochromatic lens (red) three points of intersection

4.1.3 Apochromatic Objectives An achromatic objective lens is composed of two lenses of different Abbe numbers, such as flint and crown glass, and allows for the elimination of longitudinal chromatic aberration at two well-separated wavelengths as shown in Fig. 4.3: the image plane is crossed at two (widely separated) wavelengths at which the longitudinal aberration vanishes. The remaining deviations from the (paraxial) focal plane at all other wavelengths are known as the secondary spectrum of the lens. This secondary spectrum determines the remaining colour-related deviations of an image from the focal plane and has to be sufficiently suppressed, even at fast focal ratios of .f = 1/4 that are common with binoculars. Since magnifications are low with hand-held binoculars, effects of the secondary spectrum, i.e. colour fringes about objects near the centre of the field, remain often invisible. On the contrary, spotting scopes or mounted astro-binoculars are commonly used at far higher magnifications than hand-held binoculars. Bright objects in the centre of field may then, even when in optimum focus, be overlaid by a range of out-of-focus images of different colours and thus losing definition. Depending on the focal ratio, an ED objective may further reduce the secondary spectrum, but still fail to yield a close-to-perfect image rendition. Therefore, while no hand-held binocular is built with apochromatic objectives, some spotting scopes and a few astro-related giant binoculars are in fact apochromatic. To define apochromatism, the position of the focal point of a paraxial beam as a function of the wavelength may be inspected (see Fig. 4.3): if the location of the focus intersects the (ideal) focal plane at three sufficiently separate points within the visible spectrum, then the objective is called apochromatic. In contrast, an achromatic objective would intersect the focal plane twice, a single chromatic lens only once. Such a definition, however, is not sufficiently precise to fully describe the quantity of residual colour of an optical system. Achromatic objectives, which produce a low-level secondary spectrum, may perform better than a poorly designed apochromatic lens. For this reason, additional criteria have been introduced in the technical literature to define apochromatism. Ernst Abbe requested the correction

4.2 Eyepieces

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Fig. 4.4 Apochromatic re-design of the objective shown in Fig. 4.1, with a crown glass in front and a cemented doublet made of two (leaded) flint glass elements

of spherical aberration and of coma for two sufficiently separated wavelengths as additional necessary criteria for apochromasie. Using glass with anomalous partial dispersion, e.g. fluoride glass, even relatively simple objectives in the form of air-spaced doublets can be designed to be apochromatic as long as their focal ratio is sufficiently slow. However, triplet apochromats which allow a superior correction of several aberrations concurrently, especially in instruments with fast focal ratios, are more commonly employed. Figure 4.4 shows a classic apochromatic objective developed in the 1950s by Conradi and Köhler [1]; it included a heavily leaded flint lens with anomalous dispersion characteristics. Manufacturer’s advertisements sometimes spread vocabulary which is not well defined in a technical sense, such as the attribute ’semi-apochromatic’. It is supposed to refer to an optical layout which, though not fully matching the criteria for apochromatism, performs sufficiently well to qualify as ‘semi-’. In the absence of any generally accepted technical standards, this term belongs to the realm of advertising lyrics and may not be taken as a particular qualification of the instrument. Likewise, the implementation of ED glasses into the objectives does not guarantee the absence or even the reduction of colour fringes on edges far off the centre of field, since the latter are predominantly generated inside the eyepiece rather than the objective. Without an eyepiece that is properly matched to the objective and to the prism, no optical instrument would be able to operate close to its maximum performance.

4.2 Eyepieces An eyepiece may be regarded as a magnifying glass used to transform the real aerial image produced by the objective into a magnified virtual image which can be viewed with the eye. Numerous eyepiece designs exist, each of them with their own advantages and disadvantages. In binoculars, eyepieces always come in pairs with a lateral distance determined by the inter-pupillary distance of the observer’s eyes, commonly starting from 56 mm and thus limiting the dimensions of the eye-cups.

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Fig. 4.5 Kellner eyepiece, suitable for subjective angles of field up to 50°. The focal length in this example amounts to 29 mm, XP is the location of the exit pupil

This is contrary to telescopes or spotting scopes, in which the eyepiece diameter may exceed 60 mm without causing any steric restrictions posed by the nose.

4.2.1 Simple Eyepieces for Narrow Subjective Angles of View Eyepieces that are installed in most classic binoculars are often variations of just a few basic designs, the Kellner and the Erfle eyepiece being their most prominent representatives. Figure 4.5 shows a later variation of the Kellner design;2 it combines three lenses, of which the last two form a cemented doublet. Such a set-up is called a 1-2 assembly, as the first element is just a single lens, and the second element is a group composed of two lens elements. The lens positioned next to the intermediate image of the binocular is generally called the field lens, whereas the element closest to the eye of the observer is the eye lens. The Kellner eyepiece is the adequate choice for binoculars with a subjective angle of field limited to about 50°. In this eyepiece, the light beam passes four air–glass/glass– air transitions, which implies that losses caused by reflections remain low. Kellner eyepieces have therefore traditionally been implemented into binoculars for twilight and night observations, e.g. 7x50 or 8x56 night glasses; as a result of restrictions imposed on the dimensions of their prisms, these binoculars do commonly feature narrow subjective angles of field. Figure 6.5 shows two marginal rays which have already passed through the objective in Fig. 4.1, crossed at the focal point F , to continue their propagation through the eyepiece. A focal length of 29 mm was chosen for the Kellner eyepiece, which would be a typical value for a 7x50 binocular. With a focal ratio of 1/4, as assumed in Fig. 4.1, the focal length of the objective would then amount to 200 mm. The resulting eye relief of 17 mm is already sufficient for observations with glasses. Applying the angle condition, the objective angle of field of such a 7x50 binocular (with a subjective angle of field of .φ = 50◦ ) yields .O = φ/m = 7, 1◦ , or a field of view of 124 m/1000 m. In eyepiece designs during earlier times, the Kellner eyepiece had a distinct disadvantage, being rather susceptible to internal reflections (ghost images). This

2 Originally invented around 1849 by Carl Kellner, 1826–1855. The technical data for the Kellner, orthoscopic and Erfle eyepieces presented here are from König et al. [1].

4.2 Eyepieces

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Fig. 4.6 Orthoscopic eyepiece, with very low distortion, moderate eye relief, and capable of apparent angles of field up to about 50°. The focal length in this example is 17 mm

has since been significantly improved thanks to the development of effective anti-reflective coatings. A popular variation of the Kellner design exists, initially patented by the Edmund Scientific Corporation, which features a reversed 2-1 order of the lens groups and just a narrow space between them; this design is known as the reversed Kellner eyepiece or Rank Kellner Eyepiece (RKE). The reversed Kellner has a lower Petzval sum and therefore less field curvature than the classical Kellner eyepiece, and its design may be stretched to somewhat wider angles of field. Another eyepiece that deserves attention is the orthoscopic eyepiece (Fig. 4.6). It was originally designed in 1880 as an eyepiece for microscopes by Ernst Abbe, but subsequently widely used in telescopes, too. The design contains four lenses in a characteristic 3-1 array and is practically free of rectilinear distortion (or orthoscopic, a feature that is the origin of its name). With three of its four lenses being cemented together, the orthoscopic eyepiece suffers minimum losses in transmission, just like the Kellner eyepiece, while its stray-light resistance and general imaging performance are generally superior. In addition, its eye relief— in relation to its focal length—is fairly long, which makes the orthoscopic eyepiece an ideal choice for short focal lengths or high magnifications. Although the orthoscopic eyepiece in its original form was rarely fitted into handheld binoculars, various modifications based on this design led to improvements to its original narrow angle of field. As early as 1932, Zeiss implemented an orthoscopic eyepiece in reverse order, including an aspherical lens, into their 8x30 Deltrintem binocular—these modifications allowed them to expand the subjective angle of field to impressive 70° [3]. Similar designs were also used in the 8x40 Deltarem with a subjective angle of field that almost reached 90°. Initially, the aspherical lens elements had to be shaped manually with considerable effort, while nowadays they are sometimes made of plastic and used even in cheap wide-angle binoculars.

4.2.2 Wide-Angle Eyepieces Binoculars with wide subjective angles of field usually employ rather complex eyepieces, among which the variations of the Erfle design (Fig. 4.7) have traditionally

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Fig. 4.7 Wide-angle eyepiece of the Erfle type for subjective angles of field up to about 70°. The figure shows a design with a focal length of 20 mm. Note the short eye relief of only 12.3 mm

been most widely used.3 The design presented here is made of five lens elements grouped in a 2-1-2 array and is suitable for subjective angles of field up to about 70°. Numerous variations exist with different performances based on this design, including those made of six lens elements. Apart from being used in wide-angle binoculars, Erfle eyepieces are also built into instruments with moderate angles of field of about 60° where they deliver excellent images with reasonable peripheral sharpness. The eyepiece shown in Fig. 4.7 is drawn to scale to yield a focal length of 20 mm; in combination with the focal length of the objective of Fig. 4.1 it produces a magnification of 10x. If we assume an unchanged objective angle of field of ◦ .O = 7, 1 and apply the angle condition, a subjective angle of field of 71° would result. It is therefore common for manufacturers to use one and the same binocular body, including objectives and prisms, to be matched with eyepieces of different focal lengths, yielding a series of binoculars with different magnifications; within such a series, the models with the higher magnifications are often the ones which offer wider subjective (though equally wide objective) angles of field. It can be gathered from Fig. 4.7 that the eye relief .lXP of the eyepiece, when assuming a focal length of 20 mm, measures only 12.3 mm—not sufficient for observers wearing glasses. This is a generic problem which is linked, among other factors, to the diameter of the eye lens. As shown in Fig. 2.1, the diameter of the bundle of chief rays that pass through the lens is equal to the diameter of the exit pupil d. The entire ray fan, however, diverges, when traced back from the exit pupil towards the eyepiece, at an angle corresponding to the subjective half-angle of field, ◦ .φ/2 = 35 (compare with the triangle in Fig. 4.7). The diameter of the rear eyepiece lens thus yields del = d + 2lXP tan

.

( ) φ . 2

(4.1)

Keeping everything true to scale, a diameter of the eye lens of .del = 22.2 mm results. Taking another eyepiece with a spectacle-friendly eye relief of .lXP = 18 mm, it would require a lens diameter of more than 30 mm, and since the eyepiece would

3 Developed

around 1917 by Heinrich Erfle, 1884–1923.

4.2 Eyepieces

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Fig. 4.8 Ultra wide-angle eyepiece of the Nagler type, 2-1-2-1-2-array, suitable for subjective angles of field up to about 90°. A Smyth lens with negative refractive power is placed in front of the image plane (red dotted line). Data from US Patent Nr. 4747675, May 31, 1988)

need to be mounted and fitted with eye cups, it is obvious that the dimensions of such an assembly might turn problematic at narrow inter-pupillary distances if the observer’s nose has to fit between both eyepieces. A concave shape for the last lens surface relaxes the requirements, because it enables the designer to somewhat reduce its diameter. Striving for a longer eye relief, a significant improvement can be scored with the introduction of the so-called Smyth lens.4 First of all, this lens of negative power leads to a reduction of the Petzval sum and thus the field curvature (Sect. 1.7.3). It commonly serves as a field-flattening lens and improves the sharpness in the peripheral areas of the image. Figure 4.8 displays a sketch of a Nagler high performance astronomical eyepiece: the Smyth lens is shown as a concave lens between prism exit and intermediate image. Precisely like the popular Barlow lens, the Smyth lens actually increases the effective focal length of the objective and thus slows down its focal ratio, but since the widening of the ray fan occurs just shortly before the eyepiece proper, the overall length of the optical system remains almost unaffected. In order to keep the magnification of the entire optical system unchanged, an eyepiece with a longer focal length has to be fitted, and the eye relief is scaled up accordingly. As a result, the entire eyepiece becomes somewhat larger and heavier, but the increase in eye relief makes it accessible to spectacle wearers. The Nagler eyepiece shown in Fig. 4.8 is designed for astronomical telescopes and would be too bulky to serve as a pair of eyepieces in binoculars. Yet, modern high-performance binoculars apply variations of its design idea, sometimes tailored around narrower viewing angles. Figure 4.9 shows such a modification: a Smyth lens in the form of an air-spaced doublet, then the eyepiece in a 1-2-1 assembly, which looks almost like a Nagler eyepiece after the removal of its two final elements.

4 Charles

Piazzi Smyth, 1819–1900.

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Fig. 4.9 Cutaway model of a Swarovski 10x42 EL WB. From the left, the first element is a triplet objective in a 1-2 array, followed by the movable focusing lens, then the Schmidt–Pechan prism. At its exit lies a Smyth lens, composed of two elements, finally the eyepiece, a 1-2-1 array (with kind permission: Jan van Daalen)

Thanks to powerful computer-aided ray-tracing techniques, new highly specialised glass types, and a coating technology which allows the designer to add lens elements almost at will, modern eyepiece designs have little in common with the classic designs of the past century. They have led to binoculars with close to perfect imaging characteristics at amazing subjective angles of field exceeding 70°.

4.3 Focus Mechanisms When observing an object at close range, its intermediate image is not on the same focal plane as the image of a distant object, but shifted somewhat towards the eyepiece (Fig. 2.4). This is so because the light fan that emerges from the object and hits the objective lens is somewhat divergent when the object is close. On the condition that the binocular remains focused on infinity, this shift allows a calculation of the depth of field, as discussed in Sect. 2.3. However, if we observe objects in our proximity, we will usually focus our binoculars in line with the object distance. The image distance .z' as a function of the object distance then follows from the image equation (1.21)

4.3 Focus Mechanisms

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Fig. 4.10 Image distance as a function of the (negative) object distance, assuming a focal length of 20 cm. For objects in close proximity, the travel of the focus wheel increases rapidly

z' =

.

[

1 1 + F' z

]−1 .

(4.2)

z represents the object distance (being negative according to the sign convention) and .F ' is the focal length of the objective. As an example, if we set .F ' = 20 cm, then the image distance of a remote object is 20 cm, whereas an object at 2 m distance is focused at .z' = 22.2 cm. Figure 4.10 illustrates how the image distance varies with object distance. The shift of the image plane has to be compensated for with a focus mechanism, to match the focal plane of the eyepiece, which has to transform the intermediate image into a virtual image, the latter being at a comfortable distance for accommodation.

4.3.1 Three Different Approaches Quite generally, focus mechanisms in binoculars are based on three alternative technical concepts: firstly, a modification of the distance between the objective and the eyepiece; secondly, a shift of the image plane of the intermediate (aerial) image; and finally, a shift of the principal plane of the eyepiece. The traditional focusing method is based on a variation of the eyepiece positions and is still the standard focus mechanism for binoculars with Porro prisms. The shift may be achieved either through the individual eyepiece focusing (IF), in which each eyepiece is travelling on its individual thread and focus ring, or as centre focus (CF) mechanism, in which both eyepieces are connected to a movable bridge that is supported at its centre by a threaded rod in the central hinge. Turning a central focus wheel on the rod causes the bridge to move, thus shifting both eyepieces in or out. The IF mechanism tends to be rather robust and is easily sealed against water, which is the reason why it is the standard solution found with military binoculars. The main drawback is its somewhat cumbersome and slow operation, which requires the setting of both eyepieces whenever the object distance varies; it is therefore inefficient in combination with close and mobile objects.

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Fig. 4.11 Internal focusing (schematic): sketch of the beam when focused on infinity (top) and on an object at close distance (bottom), with h representing the focus travel

By far the most common focusing technique, implemented in modern roof-prism binoculars, employs an internal focusing lens which travels along the optical axis in the space between the objective and the prism entrance. This principle is called the internal focusing mechanism. Figure 4.11 shows how it works: most often, a concave lens of low refractive power is mounted on a movable shuttle. When focusing on infinity, this lens is close to the objective (top part of figure). Viewing an object close by, the incoming beam of rays does not stay parallel, but turns slightly divergent so that the focal point moves to the right. To compensate for this, the focusing lens is shifted to the right over a well-defined distance, the focus travel h (bottom part of figure). In addition, to allow myopic observers to use their binoculars without glasses, a certain quantity of excess travel beyond infinity has to be available. A sketch of the Zeiss Victory 8x32 FL, Fig. 4.12, illustrates the mechanism of the setup. Since the focusing lens is usually of low refractive power, it produces only minor aberrations, and thus, it is often made of a single lens. To achieve maximum image quality at different positions, it has become rather common with high-end binoculars to design the focusing lens as an achromatic doublet. A focusing mechanism also applied, although far less commonly, is one in which the objectives are shifted (Fig. 4.13). However, making this design waterproof is not an easy task. Moreover, moving objectives may compromise an optimised setup of baffles and thus promote the generation of stray-light. Such a mechanism is currently used in the image stabilised binoculars made by Canon (Fig. 4.24), where some models are also sealed with an additional planar glass plate, placed in front of the objectives. Finally, another focusing mechanism is based on a movable group of lenses inside the eyepiece. This somewhat exotic design has been implemented in the early Leitz Trinovid series (Fig. 4.14). Its advantage is the short focus travel, its drawback is a change in both magnification and image distortion when focusing on targets at different distances. This design is therefore not recommended for binoculars which have to feature a particularly short close focus distance. As the eyesight of both eyes is commonly not equally strong, binoculars with a centre focus mechanism have to feature a dioptre setting that allows tuning the focus of both optical paths independently. Once done for a single object distance, the centre focus may then be applied to focus on objects at any distance, until the binocular is handed over to another observer who needs an adjustment of the dioptre

4.3 Focus Mechanisms

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Fig. 4.12 Sketch of a Zeiss 8x32 Victory FL: a 1-2 triplet objective, followed by a negative focusing lens, then a Schmidt–Pechan prism, and finally a 2-2-1 eyepiece (graphic ZEISS)

setting. This setting was traditionally realised by rotating the right eyepiece, which was mounted on a screw thread. With modern roof prism binoculars, it has become rather common to have the dioptre adjustment wheel associated with the centralfocus wheel, so that setting the dioptre does not require the user to lose a firm hold of the instrument.

4.3.2 Observations at Close Range Figure 4.10 reveals an important aspect concerning observations of objects at close distance: in this regime, a variation of the object distance has a substantial impact on the position of the focal plane. If these observation distances are common, e.g. during studies of birds at close range, which vary their distances continuously, then a fast focus train is required. During recent decades, as the binocular market has been increasingly targeting the birding community, it may not come as a surprise to find that focusing speeds have generally increased. Quality binoculars, including those of the premium class, now often require only a 2/3 turn of the focus wheel to change focus between 3 m and infinity; in some instances, it is hardly more than half a turn. But this speedup has come along with some unwelcome consequences: finding the optimum focus point with a single stride is no longer guaranteed, and in certain instances mechanical shortcomings, which years ago have only been observed in cheap binoculars, have made their way up into the high-end ranges.

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Fig. 4.13 Sectional view of a Zeiss 8x30 B Dialyt (field of view: 130 m/1000 m) of the 1970s, using movable objectives as the focusing mechanism: a cemented doublet objective, followed by a Schmidt–Pechan prism, then the 1-2-1 eyepiece (graphic ZEISS)

Apparently, some manufacturers have experimented with focus mechanisms which feature non-linear gear ratios, so that close ranges were bridged faster than far ranges. However, the construction of such a mechanism is not only technically demanding, the observers also had to get familiar with the nonlinear response behaviour of the gear train, and these designs never seemed to gain popularity. Apart from these mechanical hurdles, the short range is plagued with difficulties concerning the optical design: it can be formally proven that the image formed by an objective can be optimised only for a single object distance [4]. A workaround consists of floating lens elements, which effectively alter the optical layout along with the focusing distance. Since binoculars are expected to produce sharp images over an extended range of distances, the designer is forced to compromise image quality, and rather seriously so when the focusing range is extended. This implies that a binocular with a close focus distance below 2 m, which may be a good choice for insect watching, may not necessarily be an ideal performer under the night sky where the objects are at infinity. Observing at close range with binoculars also poses a challenge for visual perception. To avoid seeing double images, the lines of sight of the two eyes have to meet at the observed object. This is achieved by squinting, also called

4.3 Focus Mechanisms

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Fig. 4.14 Sketch of a Leitz 7x42 B Trinovid (field of view: 140 m/1000 m) of the 1970s, with an internal eyepiece focusing mechanism: A cemented doublet objective is followed by an Uppendahl prism, then by the 2-1-2 Erfle eyepiece, in which a group of three lens elements is movable (graphic courtesy of Leica Camera AG / Leica Archive)

vergence; it happens involuntarily and poses no particular problem in practice. However, observing through the binoculars increases the stereoscopic disparity of image points by the factor of the magnification, increasing the angle of vergence accordingly. To keep both eye pupils properly centred on the exit pupils, it is now necessary to slightly reduce the inter-pupillary distance of the eyepieces. A further impediment to comfortable observation stems from the fact that due to the parallax, the two—still fully parallel—tubes do not show exactly the same section of the image; only the central part of the image overlaps in both tubes. In Porrotype binoculars with their increased lateral offset of the objectives (Sect. 3.1.2), these effects become even more severe than in roof-prism binoculars, so that the latter are generally preferable for observations at close ranges. An exception is binoculars with reverse Porro prisms, in which the axis offset is oriented such that it reduces the distance between the two objectives. This design has been successfully adopted to satisfy the specific requirements for observations at short ranges. The Pentax Papilio 6.5x21 (Fig. 4.15) is an example of such a binocular. A mechanism called focus-coupled vergence correction, which tilts the two optical axes so that

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Fig. 4.15 Pentax 6.5x21 Papilio (Field of view: 131 m/1000 m) with reverse Porro design and focus-coupled vergence correction, used for observations at very close ranges down to 50 cm and particularly popular with children. Photo by RICOH IMAGING DEUTSCHLAND GmbH

they intersect approximately at the distance of the observed object, allows for comfortable observations even at ranges down to 50 cm. When changing the focus distance, the observer should always start with a close range setting and then turn the focus wheel toward infinity, not the other way around. This ensures that the virtual image approaches from ‘beyond infinity’ and prevents the eye from accommodating to a particularly close distance, by which considerable eye strain is involved. In some binoculars, modifications of the distance setting also alter magnification and field of view to some extent. During observations at close range, magnification is not defined as the ratio of the objective focal length and the eyepiece focal length, but instead as the ratio of image distance and eyepiece focal length. In those focus mechanisms, where either the eyepieces or the objectives are moved, the image distance increases at close ranges, and so does the magnification. To demonstrate this, we consider the 8x30 Opticron from Sect. 1.6: when focused on an object at a distance of 3 m, the image distance is 4.36 mm larger than the focal length of 110 mm, and the magnification increased by about 4 %. The situation is different in the case of internal focus mechanisms: there, the magnification varies with the position of the principal plane, which is shifted during movements of the focusing lens. The precise value of the magnification can therefore only be established via ray-tracing, in which the characteristics of the combined system of objective and focusing lens are determined. To avoid incorrect results, magnification and field of view of a binocular should always be measured on distant objects.

4.4 Dimensions of the Ray Bundle The analysis of the dimensions of ray bundles is among the main tasks in the design of an optical system, since it defines the dimensions of field stops, prisms and eyepieces. This might appear to be a formidable task, when considering the complicated ray paths within prisms (see e.g. Fig. 3.11). Fortunately, there exist ways to simplify this analysis significantly.

4.4 Dimensions of the Ray Bundle

83

Fig. 4.16 Objective and prism: the prism shifts the image plane by an amount ΔF to the right. The reduced path L0 corresponds to an imaginary prism with a refractive index of n = 1

4.4.1 Reduced Path In geometrical optics, the number of times a ray is being reflected is actually irrelevant. It is therefore possible to unfold the folded ray path through a prism and to represent the prism in terms of a single, thick glass plate. Figure 4.16 shows the resulting tunnel diagram schematically: in the absence of the prism, the two construction rays would follow the dotted lines (assuming a focal ratio of .f = 1/4.5). As a result of its refractive index of .n = 1.57, the same bundle converges rather slowly while passing through the prism (blue quadrangle), and the resulting focal plane is thus shifted to the right by an amount of .ΔF . To calculate this value, reference is made to Fig. 1.3 and the respective formula: assuming an angle of incidence .α, the angle .β follows from Snell’s law. The thickness of the glass plate amounts to L, so that the shift of the image plane equals ΔF = L

.

n−1 tan α − tan β ≈L . tan α n

(4.3)

The right-hand side of the equation corresponds to the paraxial (Gaussian) approximation. Let’s consider the following, even simpler way to look at this process: the ray bundle has a defined diameter when exiting the prism (green line in Fig. 4.16). We now assume an imaginary prism with the refractive index of air and a thickness of .L0 which makes the diameter of the bundle identical to the one in the original glass prism. Such an ‘air prism’ would be shorter by a defined amount, and a quick calculation of this reduced path results in L0 = L

.

L tan β ≈ , tan α n

(4.4)

with the paraxial approximation applied again on the right-hand side of the equation. In this context, the question may arise: why introduce a prism at all if it vanishes into thin air anyway with the introduction of the reduced path? Well, this approach comes with the considerable advantage that all of the geometrical limitations which the ray bundle is subjected to are still being taken into account, but without the complication of having to fold the bundle within the prism; the resulting reduced tunnel diagram is as simple as it can get. Obviously, reduced path

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diagrams do no longer display the true longitudinal dimensions of the set-up, and the aberrations induced by the prism are ignored. For a simple analysis of bundle dimensions, however, the simplified reduced picture is fully sufficient. All further details, including the influence of aberrations, would have to be studied in a later stage of the optical design process in the course of the usual ray-tracing analysis. To illustrate the above with an example, consider a binocular with an objective diameter of .D = 42 mm and a focal ratio of .f = 1/4.5, implying a focal length of .F = 189 mm. The incident bundle of rays is stopped down by the aperture of the objective, which defines the entrance pupil, and the two incoming construction rays are refracted by an angle of ( α = arctan

.

D 2F

) = arctan

( ) f = 6, 34◦ . 2

(4.5)

This angle equals the incident angle at the prism entrance face. We assume the prism to be made of BaK4 glass with a refractive index of .n = 1.57, and inside the prism, the convergence angle amounts to .β = 4, 03◦ . The geometrical path within the prism can then be calculated by multiplying the open width w with a numeric factor .κ, where .κ accounts for the prism geometry that is specific for the type of prism.5 Choosing a Schmidt-Pechan prism with a width of .w = 35 mm, the resulting light path length is therefore .L = 35 · 4.62 mm = 162 mm. Using the Eq. (4.3) we obtain a shift of the image plane by .ΔF = 59.2 mm (or, when applying the Gaussian paraxial approximation, .ΔF = 58.7 mm). The reduced path follows from Eq. (4.4) and measures .L0 = 103 mm (identical with the Gaussian approximation). The errors resulting from the paraxial approximation are thus in the range of typically 1% which is sufficiently accurate for our purpose.

4.4.2 Dimensions of the Intermediate Image Figure 4.17 shows the geometrical conditions of our model binocular in a schematic way. We assume, for the purpose of constructing the ray bundle, that we want to design a 10x42 binocular with an objective angle of field of .O = 6, 4◦ . A generously dimensioned prism with an open width of .w = 32 mm is positioned in such a way that it takes the entire ray fan without vignetting, i.e. a loss of peripheral rays due to undersized prisms or baffles. To simplify things, we leave aside the explicit modelling of the focusing lens—instead, assuming that it is fixed at the infinity setting and absorbed into the design of the objective. Care has to be taken to reserve sufficient space for the focus travel of that lens between the objective and the prism. Since the diagram employs a reduced path prism, there is no need to account for

= 4.62 for Schmidt–Pechan prisms, .κ = 4 for the (symmetrical) Porro prism, .κ = 3.3 for the Perger prism, .κ = 5.20 for the Abbe-König system, and .κ = 5.06 for the Uppendahl.

5 See Chap. 3: .κ

4.4 Dimensions of the Ray Bundle

85

Fig. 4.17 Binocular with unfolded and reduced ray path: The ray fan enters the objective (entrance pupil diameter: D, focal length F ) with an objective angle of view of O, then enters the prism (width w), after which it forms the intermediate (aerial) image (iI) of diameter Z, finally proceeding to form the exit pupil (XP) of diameter d—the real image of the entrance pupil

the refraction of the ray bundle inside the prism. We may now easily calculate the diameter of the intermediate image as if the prism were absent, using the triangle 2 in Fig. 1.9 to calculate .y ' . Since .Z = 2y ' , we obtain ( Z = 2F tan

.

O 2

) (4.6)

.

Using Eq. (2.6), we may alternately derive the diameter of the intermediate image as a function of the field of view, yielding ( Z=F ·

.

F OV (in m) 1000

) .

(4.7)

Interestingly, these two relations indicate that Z depends critically on the field of view. Since the intermediate image is positioned at the primary focus behind the prism, and since in almost every layout the light cone converges toward the primary focus, the prism entrance width w has to amount to at least the diameter of the intermediate image.6

4.4.3 Prism Entrance Face Width We shall now estimate the minimum prism entrance width w which is required to allow for the passage of the ray bundle without being clipped (vignetted). To simplify the calculation, we assume that the prism exit coincides precisely with the primary focus (i.e., the right edge of the blue rectangle in Fig. 4.17 would be positioned on the intermediate image, iI). This would not be the case in a real-life application, since some clearance for the eyepiece is required, e.g. for dioptre adjustments. Moreover, each little speckle of dust on the prism exit face 6 We

ignore the theoretically possible case of .Z > D, as it does rarely occur in hand-held binoculars.

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would be imaged in focus, with a negative impact on the image quality. Yet, the above approximation, which leads to a lower limit of the minimum entrance width w, is sufficiently close to realistic designs in which the prisms are routinely be somewhat under dimensioned, to reduce the overall weight of the instrument. With the application of some basic geometry, we arrive at the relation w>D−

.

(D − Z)L0 . F

(4.8)

Reminding that the reduced path .L0 = L/n = wκ/n itself is a function of the prism width w, we obtain the final equation w>

.

DnF . nF + κ(D − Z)

(4.9)

In our example (.F = 189 mm, objective angle of field 6.4°), the intermediate image diameter is .Z = 21.1 mm, resulting in a minimum prism entrance width of .w = 31.7 mm. It is further instructive to replace Z and F and verify that w increases linearly with D, and thus the prism volume with .D 3 , given that the focal ratio is left unchanged. The next element behind the primary focus is the eyepiece. Please note that Fig. 4.17 shows only its principal plane and ignores all details about its construction. In fact, the drawing implies that the eyepiece diameter at its principal plane corresponds to the diameter of the ray bundle, but this is a simplification: the (convex) field lens is often located in front of the eyepiece’s principal plane and rather close to the image plane, to funnel the diverging ray bundle into the eyepiece. Therefore, looking for the minimum diameter of the eyepiece, we may conclude that its field lens diameter has to exceed the diameter of the intermediate image. In addition to that, for the size of the rear eyepiece lens (the eye lens), the condition discussed in the context of Eq. (4.1) still applies. With these two estimates for the field lens and the eye lens, the diameter of the eyepiece barrel is largely defined.

4.5 Examples 4.5.1 8x42 and 10x42 Binocular Consider a manufacturer who would like to produce a 8x42 version of his 10x42 binoculars already on the market. To save cost, he could select the general design shown in Fig. 4.17, but replace the eyepiece with another one of focal length .f = F /m = 23.6 mm. Naturally, he would aim at a subjective angle of field that is similar to that of the 10x42 model, about 64°. The objective angle of field of the new 8x42 binocular would then, using the angle condition, be computed as .O = 64◦ /8 = 8◦ , and, with Eq. (4.6), he obtains a diameter of the intermediate image of .Z = 26.4 mm. Obviously, an eyepiece of increased diameter would be needed,

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87

which is not a problem per se. However, since the ray bundle has to pass the prism before reaching the eyepiece, the prism entrance width has to be wider, too. With Eq. (4.9) we obtain a minimum of .w = 33.8 mm, so that the current value of 32 mm would lead to a vignetting of the ray bundle. In fact, manufacturers routinely tolerate such a clipping of peripheral rays, which results in a reduction of peripheral image luminance. In daily life applications, a luminance loss of 30 %–50 % towards the peripheral areas of the field remains usually unnoticed, though during night time observations, the eye is rather sensitive, and thus, vignetting should be applied with care for the design of high-performance night glasses. It is crucial that the vignetting does not obstruct the principal ray bundle (shown as red lines in Fig. 4.17), since otherwise the exit pupil d would be clipped and the objective diameter would then not be fully utilised. In practice, though, this should not happen anyway, as may be demonstrated as follows: to avoid stopping down the field of view, the prism exit must have a minimal width of the intermediate image diameter, Z; then, its entrance width should be of the same size. We then have to check under which conditions the diameter of the principal ray bundle exceeds Z at the location of the prism entrance. This leads to the triangle relation .

Z D =C, = L0 F

(4.10)

with the relative aperture (or lens speed) C and the reduced length of the prism L0 =

.

κZ , n

(4.11)

n . κ

(4.12)

yielding the inequality C
24.1 mm. Such a Porro prism would weigh as little as 87 g in the complete absence of vignetting. It is therefore quite easy to design and build compact and lightweight 8x30 binoculars with wide subjective angles of field. Of course, there exists a potential market for such a binocular with a lower magnifying power, e.g. 6x. Such an instrument would have larger exit pupils and therefore remain, contrary to the 8x30, usable under twilight. If we had to design such a 6x30 binocular in a similar wide-angle set-up, with a subjective angle of field of .φ = 65◦ , then an objective angle of .O = 65◦ /6 = 10, 8◦ would follow. Using identical 30mm objectives (.F = 126 mm), the diameter of the intermediate image would now become .Z = 23.8 mm and the prism entrance width .w > 26.2 mm, resulting in prisms which are 40 % heavier than in 8x30 binoculars of similar subjective angle. Moreover, the larger prisms require a larger chassis, the wider intermediate image asks for wider eyepieces, and the result is an instrument that is altogether quite a bit bulkier—a feature most unwelcome to the customer who is looking for a compact travel companion. Figure 4.18 shows the difference in size between two 30 mm binoculars with Porro I prisms and almost identical subjective angles of field (about 70°): the Russian Kronos BPWC 6x30 and the Zeiss Jena 8x30 Jenoptem. The considerable difference in eyepiece diameter and size of prism case is obvious. The weight of the 8x30 is about 550 g, whereas the 6x30 weighs a considerable 750 g.

4.5.3 7x50 (Super-) Wide-Angle Why do virtually no 7x50 wide-angle binoculars exist? To answer that question, we analyse the recently introduced Nikon 7x50 IF WX model (Fig. 4.19): this binocular has a formidable objective angle of field of .O = 10, 7◦ , or a field of view of 187 m/1000 m. With the angle condition, this would yield a subjective angle of field

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89

Fig. 4.19 The Nikon 7x50 IF WX (field of view: 187 m/1000 m) belongs to the elite club of wide-angle 7x50 binoculars. It weighs a considerable 2.4 kg and should be mounted during long-term observations

Fig. 4.20 Mass of a single prism cluster for a 7x50 binocular as a function of field of view, when assuming a focal ratio of f/4 and no vignetting. Data are based on the approximate formula equation (4.9)

of .φ = mO = 74, 9◦ , but since its distortion is rather low, a value of .φ ≈ 70◦ appears to be closer to reality. Despite these impressive parameters, the image is sharp virtually from edge to edge. The focal length has not been officially disclosed, but the patent [5] indicates a focal ratio about .f/4 and thus a focal length about .F = 200 mm. The prism is of the Abbe-König type. Using Eq. (4.6), we first evaluate the diameter Z of the intermediate image and obtain a giant size of .Z ≈ 37 mm. With Eq. (4.9), a prism entrance width of .w ≈ 41 mm follows. Here, we have used a refractive index of .n = 1.52 (assuming BK7 glass) and the light-path factor .κ = 5.2, which applies to the Abbe-König prism. To reduce the prism size somewhat, the designer might choose a prism entrance diameter of .w = Z ≈ 38 mm, allowing for a minor amount of vignetting. With a density of 2.51 g/cm3 , and a volume of .V = 3.72 · w 3 , such a prism would weigh slightly above 500 g, and a pair of them a considerable kilogram (Fig. 4.20)! To be added are two huge spectacle-friendly eyepieces of diameters exceeding the aerial image size of 38 mm, each of them perhaps adding another 400 g to the instrument, which weighs 2.4 kg and is therefore right at the upper limit of instruments that can reasonably be called hand-held. Had Nikon implemented Schmidt–Pechan prisms instead, made of BaK4 glass of index .n = 1.57 and density 3.1 g/cm3 , then the resulting prism would have a volume

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4 The Anatomy of Binoculars

Fig. 4.21 The Miyauchi 7x50 Binon (field of view: 166 m/1000 m) featured an extremely fast focal ratio of .f/3.2 and Porro II prisms to achieve a moderate weight of 1.3 kg

of .V = 1.8 · w3 (with .w ≈ 42 mm, or .≈ 38 mm when allowing for vignetting) and a weight slightly above 300 g, which would reduce the instrument’s total weight by 400 g. Nikon has claimed that the superior imaging of the Abbe-König prism, being a total internal reflection prism, as well as a lower degree of vignetting compared to a Schmidt–Pechan prism would justify the additional weight and bulk of the AbbeKönig design. On a historical side note, the US manufacturer Bausch & Lomb had, during the 1940s, developed the Mark 41 binocular. It is considered the best vintage 7x50 wideangle binocular in history, offering an impressive objective angle of field of 10°, which corresponds to a field of view of 175 m/1000 m, and, while equipped with Porro prisms, weighed just slightly above 1.6 kg. In the early 2000s, the Japanese manufacturer Miyauchi offered its 7x50 Binon, which featured Porro II prisms, a field of 166 m/1000 m at a moderate weight of 1.3 kg (Fig. 4.21). To achieve its compact layout, the focal ratio had to be increased to a very fast .f/3.2. Both the Mark 41 and the Miyauchi Binon were compromised in terms of aberration control and suffered a significant image blur towards the edges.

4.6 Aperture and Field Stop Aperture and field stops are crucial for the proper functioning of an optical system and form an essential part of the optical design work. To illustrate this, a shift of the aperture stop does not only alter the position of the exit pupil, but also affects the balance of aberrations such as coma or distortion. A proper choice for the position of the field stop may not only avoid colour fringes at the field stop (’ring of fire’), but also guarantees a sharp and crisp image boundary. In fact, two fundamentally different types of stops exist in optical systems (see Fig. 4.22): the purpose of the first type is to stop down the ray-bundle, while the second type is implemented to suppress stray-light. The aperture stop is located in

4.6 Aperture and Field Stop

91

Fig. 4.22 Schematic view of the system of stops in a binocular

proximity of the principal plane of the objective. In case of an objective of the telelens type, which has its principal plane located in front of the lens, the lens mount often serves as aperture stop; otherwise, the edge of the lens itself or a diaphragm positioned directly behind the objective may serve that function. The aperture stop defines the size of the entrance pupil, i.e. the maximum diameter of the principal ray bundle and the diameter of the exit pupil. The field stop is located close to the plane of the intermediate image; it defines the edge of field of the binocular. Often, the field stop is integrated into the eyepiece and positioned in front of (or behind) its field lens, depending on the eyepiece design. If the field stop is not located sufficiently close to the image plane, the boundary of the field of view is blurred, with a negative impact on the ease of view. The field curvature has to be considered to obtain a perfectly sharp image of the aperture stop. Whenever field curvature exists, the field stop—instead of being positioned at the point of the paraxial focus—is shifted in such a way that it intersects the curved image shell at its periphery. Most military binoculars include reticles engraved on glass plates; for the pattern to be properly visible to the observer, the glass plate has to be located close to the image plane, in which case its edge functions as a field stop. Care needs to be taken to keep any optical element in close proximity to the intermediate image as clean as possible, since each mark of dirt or speck of dust at this location will inevitably be seen in focus as well. In the case of the ray bundle being clipped at any other position along the beam path, vignetting occurs, in which certain peripheral rays are affected by the clipping. As discussed in Sect. 4.5.1, vignetting often occurs at the entrance of prisms which are commonly somewhat undersized to save weight. In addition to aperture and field stops, which clip the bundle of rays, additional glare stops or light baffles are implemented to reduce stray light. This is a difficult task during the design process. These stops and baffles do not clip the imageforming beam, but are supposed to trap any irregular light which emerges after reflections on housings, lens cells or poorly coated glass surfaces. In addition, the complicated folding of light paths inside image erecting prisms may create opportunities for parasitical partial bundles to enter the ordinary beam via unexpected

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4 The Anatomy of Binoculars

Fig. 4.23 A Porro prism with a grooved hypotenuse catches a vagabonding light beam which had escaped the glare stops inside the objective tube. The dotted line marks the path of the light in absence of the groove

shortcuts. For this reason, Porro prisms should be equipped with a groove cut across each hypotenuse (Fig. 4.23). Such a groove is, of course, just another example of a glare stop. Numerous technical solutions exist to reduce stray-light formation in the space between the objectives and the prisms, such as tapered or threaded baffles inside the objective tubes, or a cylindrical baffle placed around the prism entrance. Prisms made of several disconnected elements, e.g. those of the Schmidt–Pechan design, can be fitted with ring-shaped diaphragms, located inside the air space between the prism elements. Additionally, lens edges, in particular those in proximity to the image-forming ray bundle, should be painted matt black to prevent the formation of scattered stray light. A precise analysis of stray-light formation during the system design process, by means of ray-tracing techniques, is still a formidable task and so far only partially successful. It thus requires the special skills of experienced optical designers, in combination with repeated laboratory measurements and field tests to root out all effective sources of stray light.

4.7 Image Stabilisation Techniques It sounds like a dream that has finally come true: the image stability of a mounted binocular in combination with the flexibility and dynamics of handheld observations. This is the promise made by image stabilisation technology. At first glance, it seems strange that not every single high-quality binocular is already equipped with image stabilisation, but there are reasons for that. Image stabilisation requires complex mechanics and/or electronics, and it is no trivial task to make such binoculars sufficiently robust for everyday observation. They are rather more susceptible than conventional binoculars to impact-related damages; are rather expensive, heavier and chunkier, and they—with few exceptions—require power supplies.

4.7 Image Stabilisation Techniques

93

First image-stabilised binoculars were developed for the military in the late 1970s by the US-based manufacturer Fraser-Volpe [6]. Independent developments were made by the Belarusian (then Soviet) company Peleng and by the Japanese Fujinon, which around 1980 presented its Stabiscope. This first generation of image stabilisation was based on prisms, which could be decoupled from the movement of the chassis and were stabilised by fast rotating gyroscopes. Due to the conservation of angular momentum, such a gyroscope resists external constraints, and several gyroscopes, rotating in different planes, are coupled to the prism, keeping its orientation stable within the given mechanical leeway. Such a gyro stabilisation was expensive and also heavy, and until the collapse of the Soviet Union, when the first Peleng models surfaced on the second-hand markets, they were hardly available to users who were not related to the military. In 1990, Zeiss introduced an entirely novel approach to image stabilisation: Inside the 20x60 S, both prisms are firmly connected to each other and coupled to the chassis via a cardanic gimbal suspension. If the binocular moves, the prism system reacts after a delay due to its inertia, whereas the eddy-current damped suspension acts like a low-pass filter, suppressing vibrations within certain frequency bands. This inertial stabilisation mechanism requires no power source and is used in the Zeiss 20x60 S binoculars and monoculars. A variation of this design, in which unfortunately the light beam was clipped by under-sized prisms, was soon also available from Russian production. Possibly, the 7x50 model No. 16 by Elgeet Optical in Rochester may have been an even earlier prototype of inertia-stabilised handheld binoculars, which, however, never made it into series production. Another prototype, designed by Leica, is presented in the publicly accessible product collection in Wetzlar, Germany. It was only with the integration of modern microelectronics that stabilised binoculars became lighter, cheaper and thus interesting for the non-professionally oriented market sector. Several approaches to mechanisms of stabilisation have been realised: Fujinon’s Techno-Stabi uses piezo and gyro acceleration sensors that register every change in the orientation of the actuators, coupled to the erecting prisms, and compensate for these movements within fractions of a second and up to amplitudes of .±6◦ . In case of the Vari-angle mechanism, which Canon has been using in its IS models since the mid-1990s, the light beam passes through an oil-filled cavity, consisting of two planar optical glass plates, which are enclosed by a flexible bellow that prevents the oil from leaking out. In its zero position, the plates are parallel to each other, so that the component behaves like a single, plane-parallel plate. Electromagnets are used to tilt the plates against each other, whereby they now form a kind of prism which shifts the light beam sideways as it passes through. Electronic acceleration sensors trigger the actions of the magnets to compensate for the trembling movements of the binocular. This method of stabilisation reacts very quickly, but is limited to small compensation angles of .±0, 8◦ (as specified for the Canon 12x36 IS, Fig. 4.24). The Vari-angle principle is therefore ideal for applications in which the observer stands on solid ground, while the Techno-Stabi mechanism, also offered by Nikon, can be applied on moving vehicles or ships.

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4 The Anatomy of Binoculars

Fig. 4.24 Canon 12x36 IS II (field of view: 87 m/1000 m, currently available in 3’rd generation), a fairly compact image-stabilised binocular with Porro II prisms and still a reasonable weight of 700 g (with batteries)

Particularly high demands are placed on the stabilisation mechanism when observations on board a helicopter are necessary, since the rotor generates vibrations of high frequency. Here the servomotors of the Techno-Stabi mechanism would be overtaxed due to their inherent speed limitation, so that the classic gyro stabilisation system has to be used.7 The Fujinon Stabiscope or the Fraser–Volpe Aviator, examples of modern, gyro-stabilised binoculars, are still heavy with 2 kg at objective diameters of just 40 mm. It should further be noted that their gyroscopes cause a non-negligible background noise during operation. This form of stabilisation is therefore less suitable for nature lovers or hunters. A working group led by Jürgen Nolting from Aalen University of Applied Sciences conducted measurements in which they compared the effectiveness of different stabilisation systems [7]. The binoculars were hand-held, while only one of the eyepieces was used for observation, and a small video camera registered trembling movements through the second eyepiece. They compared the 14x40 Techno-Stabi from Fujinon with the 18x50 IS from Canon and the 20x60 S from Zeiss. General advantages of the stabilisation mechanisms over unaided hand-held observations were clearly demonstrated. Here, the electronic stabilisation proved generally superior to the mechanical stabilisation: with the purely mechanically stabilised Zeiss, certain frequency ranges exist, about 4.5 Hz, 11 Hz and 17 Hz, which are very effectively suppressed, while at 7 Hz, 14 Hz and 21 Hz, the stabilisation remains largely ineffective. These are typical characteristics of mechanically damped systems with their unavoidable resonances. With the electronically stabilised binoculars, the damping at frequencies beyond 5 Hz was close to perfect. The Fujinon showed advantages over the Canon in the presence of large amplitude vibrations, while the Canon had the advantage of a faster response after pressing the Stabi-button. The last word on image stabilisation in hand-held binoculars is far from over. In recent years, smaller and lighter binoculars with image stabilisation have surfaced,

7 Stefan

Korth, personal communication.

4.8 Sealing and Purging of Binoculars

95

though the market sector of semi-professional sports-optics binoculars largely remains untouched by these developments. Investments would be considerable, and it remains an open question how the fairly conservative group of high-end customers would react to the introduction of image-stabilised techniques into their favourite instruments.

4.8 Sealing and Purging of Binoculars The sealing of binoculars to prevent the penetration of water or dust into the instrument is among the most demanding tasks in binocular manufacturing. Delicate optical elements have to be safely shielded, preferably over decades, from the environment, and such a shield has to survive mechanical impacts and thermal exposures which are inevitable in the field. The focusing mechanism with its movable parts poses a particular difficulty in this respect. This is the reason why waterproof binoculars (mostly for military use) traditionally avoided the implementation of central focusing units and relied on individually focused eyepieces instead. If carefully machined and then lubricated, their threads can be made fully waterproof under the most demanding of conditions. Until after WWII, cover plates were often sealed with bituminous pitch, causing trouble to the binocular repair workshops, which have to open and service such instruments. Today, plastic seals are implemented that have a long lifetime and are easily replaced during a clean and repair job. Lenses are sealed either with the help of tight rubber o-rings or—what is unfortunately getting rather common—using water resistant glue. Binoculars for professional service are sometimes fitted with silica desiccant cartridges which are refillable and ensure dry optics even in the case that the seals might fail (Fig. 4.25). The sealing of central focusing mechanisms became much easier with the arrival of roof prism binoculars with an internal-focusing mechanism, in which case only a thin rotating axis has to penetrate the instrument body. In Porro binoculars, a conventional internal focusing is more difficult to build due to the wide separation of the two objectives. Instead, a central focusing with movable eyepieces is common and sealed with rubber o-rings. These, however, slide over a counter face and are Fig. 4.25 Chinese 8x30 military binoculars ‘Type 62’ (field of view: 145 m/1000 m) with desiccant cartridges integrated into the prism body

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4 The Anatomy of Binoculars

Fig. 4.26 Zeiss 8x30 Porro binocular (field of view: 150 m/1000 m). Seals are shown in red. Note the telelens objective with wide air space and the sophisticated 6-lens eyepiece of 2-1-1-2, an Erfle-variation designed by Horst Köhler (graphic ZEISS)

therefore subject to friction or take in dust. The better solution is using a rolling diaphragm; it has the advantage of less frictional resistance and does not become stiff at low temperatures. This technology has been used since the 1950s to make Zeiss Porro binoculars splash-proof (Fig. 4.26). Rolling diaphragms used to be made of rubber and had a limited lifetime; nowadays, very durable synthetic polymer materials have become the standard. An alternative solution consists of soldering metal bellows to the eyepiece tubes.8 Another solution which avoids the highfrictional resistance in waterproof Porro instruments is the less common internal eyepiece focusing mechanism as shown in Fig. 4.14. Water resistance is defined in a couple of industrial standard norms, ranging from splash-proof (which allows using an instrument in rainy weather) up to the resistance to water at several different levels, measured in submersion depth (e.g. 1 m to 5 m) of different durations. Yet, manufacturers do not routinely test for tightness by submerging their binoculars in a water tank, rather by application of pressure: by means of a connecting piece, excess pressure in line with a standard norm is created and the time over which this pressure is maintained in the binocular body is monitored. After the test, the connecting piece is removed and the binocular case permanently sealed. Prior to the final sealing, the binoculars are purged with a dry and chemically inert gas, usually industrial nitrogen which is easily available. Advertising texts commonly explain such purging with the aim to prevent internal fogging. However, fogging caused by condensation of water vapour inside the instrument after a drop in temperature could as well be prevented by purging the binoculars with ordinary dry air. Therefore, a more likely reason behind the application of nitrogen is the fact that dry pressurised nitrogen is cheap, and since it is inert and does not react

8 This was the case in several Huet binoculars of the French Navy; information provided by Hans Weigum.

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97

with lubricants inside the instrument, it likely reduces the generation of oxidised by-products which might outgas and condense onto internal glass surfaces. The close proximity of lubricants and optical surfaces, confined inside a sealed cavity, poses a rather general problem to the manufacturers of optical instruments. Industrial oils and fats are made of long-chained molecules which are polydisperse, i.e. of varying molecular weights. Even at room temperature, the lightest ones may tend to evaporate, though slowly, and condense on the optical glass. The heavier parts of the fats remain in the lubricating film, which gradually turns more viscous and finally resinous. Under normal conditions, this condensation process, which is accompanied by a further polymerisation of the molecules, takes years or decades, but at high temperatures, inside a car or behind the window of a shop that is exposed to the sun, the process may accelerate considerably. High-quality synthetic lubricants help in reducing the impact of these processes, and some manufacturers, such as Leica, dispense with the application of oils altogether and instead rely on lubricant-free bearings. In the recent past, some manufacturers have touted the application of the noble gas argon as a purging agent, claiming it to be superior to nitrogen. Argon (40 g/mol) does in fact have a higher molecular mass than nitrogen molecules (28 g/mol), and its diffusion rate in open space is therefore lower. Based on this fact, it has been claimed that argon would remain longer inside the binocular body. This argument is flawed, however, as demonstrated with the diffusion rate of a gas through a solid body (e.g. a thin dividing wall): R = PA

.

p 1 − p2 . d

(4.13)

R stands for the diffusion rate which determines how fast the diffusion progresses; P is the permeation coefficient—this is a factor, the value of which depends on the material used for the seals and the diffusing gas; A is the area of the surface through which diffusion takes place; d the wall thickness and .p1 − p2 is the difference of the partial pressures of the gas: inside vs. outside, i.e. the difference in concentration. Ambient air consists of 78 % nitrogen, but less than 1 % argon. If binoculars are in fact more permeable to nitrogen than to argon, as being claimed, then nitrogen diffuses into a binocular which is filled with argon—and this process happens rather rapidly because its partial pressure difference initially amounts to a substantial 0.78 bar. In such a case, osmotic overpressure inside the binocular case would result, adding strain on the seals. Looking at a binocular filled with nitrogen instead, the partial pressure difference (inside being one bar, outside 0.78 bar) would yield merely 0.22 bar, which implies that nitrogen will diffuse out of the binocular at a far slower rate than it would enter a binocular filled with argon. On the other hand, the enormous partial pressure difference of a binocular filled with pure argon contributes a large factor .p1 − p2 , which might as well compensate for its smaller permeation coefficient. In summary, nitrogen appears to be the preferred choice, and the application of argon may as well be interpreted as a clever marketing strategy of some manufacturers.

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References 1. König, A., Köhler, H.: Die Fernrohre und Entfernungsmesser. Springer, Berlin (1959) 2. Smith, G.H., Ceragioli, R., Berry, R.: Telescopes, Eyepieces, Astrographs. Willmann-Bell, Richmond (2012) 3. Seeger, H.T.: Zeiss Feldstecher. Zeiss Handferngläser 1919–1946. Dr. Hans T. Seeger, Hamburg (2015) 4. Conrady, A.E.: Applied Optics and Optical Design, vol. 1 (reprint), p. 367. Dover Publications, New York (1992) 5. US-patent: US11054609B2 (2017) https://patents.google.com/patent/US11054609B2/en 6. Parker, G.: The Story of Stabilised Binoculars, New Scientist, 5441, Nov. (1978) 7. Nolting, J., Kiesel, M.: Verwackelt? Bestimmung der Sichtlinienstabilität stabilisierter Ferngläser. DOZ 7, 34 (2004)

Chapter 5

Report on a Self-made High-Performance Binocular Gerhard Eller

Abstract In this guest contribution, Gerhard Eller (Nidderau, [email protected]) describes the making of his high-performance .12 × 62 binocular.

From early childhood, when he first came into contact with binoculars, the author was fascinated by binoculars. My uncle had brought home a small Zeiss binocular. If I remember correctly, it was a .6 × 24, which I proudly carried out on our walks together. These binoculars left their mark on me, so to speak, and my enthusiasm for telescopic vision has not only remained alive over the years, but only intensified through the use of other optics. My grandfather’s Galilean binoculars made little impression on me, since they never gave me the visual experience that the small Zeiss binocular did, and I always found reasons to seek my uncle’s company on nature walks to avoid grandfather’s ‘Galilei’ when I went out hiking. My memories of these excursions from the 1950s still remain vivid. With the wonderful viewing experiences I also learned to distinguish good from not so good and to enjoy the particular pleasure of an outstanding telescope image. Everyone will probably have this kind of key experience and sooner or later will learn to appreciate good binoculars and thereby build up a positive relationship with visual optical instruments in general. However, life is more than just contemplative telescopic observations! The world of work had other priorities and the interest in binoculars and optics laid dormant for quite some time, only to reawaken later. Amateur astronomy had grabbed me! A successfully completed apprenticeship as a precision mechanic allowed me to build a telescope myself with the aid of a manual. Together with a colleague I also built two parallactic mounts along with the telescope. He used an existing Quelle refractor with 60 mm objective diameter and 910 mm focal length. My refractor had a Lichtenknecker AK objective of 125 mm aperture and 1300 mm focal length. This telescope was my first self-made optical instrument. The original version of the chapter “Report on a Self-made High-Performance Binocular” was previously published with an incorrect corresponding author. It has now been changed to “Gerhard Eller”. A correction to this chapter can be found at https://doi.org/10.1007/978-3-031-44408-1_10. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023, corrected publication 2024 H. Merlitz, The Binocular Handbook, https://doi.org/10.1007/978-3-031-44408-1_5

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After completing my mechanical engineering studies, I was able to expand my technical and mechanical skills considerably. With regard to precision, materials and machining possibilities, I approached new horizons. After the construction of various astronomical telescopes, not only did my knowledge and experience grow, but also the ambition to further my optical education in order to create something even better. Now, monocular vision is good – but binocular vision through a good pair of eyepieces is something else. The one-eyed man is king among the blind, but not among those who see with both eyes. Thus, binoculars and hardly anything else turned into my field of vision. While collecting binoculars of different types and from many manufacturers, I never again had the same ‘wow-effect’ I had experienced in my younger years. Recent binoculars do have larger fields of view than my uncle’s old Zeiss binoculars, but their ease of view appears rather mediocre and not superior in principle to the old binoculars. I had binoculars from all manufacturers in my collection, but none of them gave me what I felt as a boy when looking through that small Zeiss binocular. The reader may object that the first sight through very good binoculars – like many experiences from memory – turns more and more beautiful in retrospect and cannot be displaced from memory by something superior. Of course, the optical industry has improved their binoculars over the course of time and incorporated a couple of innovations. A ’quantum leap’ – perhaps not the right term here – has, in my opinion, never been realised. The impression that one of those binoculars added something completely new or reached a new dimension in terms of the viewing experience has never once occurred to me with commercially available binoculars. I have never been able to notice any real perceptible progress with handheld binoculars. I was never really satisfied, and I even assumed that the optical companies made high-quality measuring instruments rather than products of daily use that must also give pleasure. By chance I got to know people who are interested in historical binoculars, especially military models from the Second World War. This gave me the opportunity to look through these instruments, and all the sudden it was there again, the woweffect: the visual experience of a very special kind. The telescopic image of these binoculars is simply phenomenal. These models included the Zeiss .10 × 80 20°, .10 × 80 80°, the Busch .10 × 80 45° and the .8 × 60 Deck-mounted by Carl Zeiss Jena and Carl Zeiss Oberkochen.1 There are certainly a few other glasses that could be mentioned, but here I would like to highlight one particular glass, the .8 × 60 Deck-mounted by Carl Zeiss Oberkochen, built in the 1960s for the German Navy.2 I was taken with the technical data that characterise this glass, but there was more. The visual impression of the enormous field of view is fantastic, even unbelievable. The very long eye relief is another positive feature of this glass. Though the .8 × magnification is rather low, due to its mass of 6.5 kg, nothing can be achieved without mounting the device onto a stable tripod. After all, it was designed to be 1 The

angles here refer to the angled eyepieces with respect to the viewing direction. descriptions and photos are provided in the book by Seeger [1], an absolute must for the binocular enthusiast.

2 Detailed

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mounted on the ship’s side or onto a torpedo-targeting column for use on board. One can no longer buy such binoculars, and if one did, one would have to spend at least the equivalent of about 42,000 DM, which was, around 1960, the purchase price the German Navy had to pay for such an Oberkochen .8 × 60 Deck-mounted binocular. My activities as a collector led me to a pair of eyepieces from these binoculars – a step in the right direction. Two Porro II prisms from another model and two APO lenses from Leica spotting scopes were also at my disposal to give shape to my idea of a new super binocular. From these optical components, I have constructed a binocular that is unparallelled. It is a .12 × 62 APO binocular, offering an ease of view that lies beyond the range of commercial optics and immediately produces that wow-effect of the kind I had been seeking for. Since the completion of this binocular, I no longer have any unfulfilled wishes for a high-performance binocular – I can experience my woweffect whenever I wish (Figs. 5.1 – 5.8). These binoculars were presented for the first time to the public in 2010 at the meeting of amateur astronomers (ITV), in Vogelsberg. All observers were highly impressed, even overwhelmed. ‘I have never seen anything like this, it cannot be real’ were the first comments. Fig. 5.1 Self-made: A Porro II wide-angle binocular (field of view: 113 m/1000 m); magnification: .11.7 × ; subjective angle of field: 74°; exit-pupil diameter: 5.3 mm; eye-relief: 23 mm; close-focus distance: 9.8 m; weight: 2.7 kg (optical elements only: 1.4 kg)

.12 × 62

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Fig. 5.2 Objective lenses: Two apochromatic objectives from Leica 62 mm APO spotting scopes; focal length: 352 mm

Fig. 5.3 Eyepieces: a pair of oculars of the Carl Zeiss .8 × 60 (Deck-mounted) of the 1950s; focal length: 30 mm

Fig. 5.4 Porro II prisms, as used with the Asembi and Asiola models of Carl Zeiss Jena. The entrance diameter amounts to 30 mm, the glass path to 131 mm. The objective lenses are optimised for a glass path of 100 mm, which is sufficiently close at the magnification used here

What seemed important to me was not only the praise itself, but also who was giving it. Amateur astronomers are not average observers, but experts who know how to recognise a good telescope image and outstanding eyepiece properties. In the meantime, some representatives from the optical industry have become aware of this .12 × 62 APO and have expressed more than a passing interest. At the ATT 2010 in Essen, some experts were again full of praise, and I not only heard words like ‘incredible, wonderful, absolutely top’, but there were also more concrete enquiries: Where to buy such a glass? How much would it cost? I do not have answers to these questions, which are asked over and over again. It was a

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Fig. 5.5 Further components of the .12 × 62 (clockwise from upper-left): objective tubes, prism cases, prism mounts and parts of the central hinge Fig. 5.6 View into the binocular tubes with mounted stray-light baffles

lucky coincidence that an eyepiece that formerly was in serial production, the prism system that I have chosen and the objective lens altogether form a perfect unit, which in terms of field of view, distortion and absence of chromatic aberration is almost flawless, and additionally offering that very special and overwhelming visual impression. It is a visual impression that cannot be matched by any commercially available binoculars, no matter how expensive they may be. When I consider not only the value of the optical components but also the mechanical work based on a minimum hourly wage, then I would have invested about 3500 Euro into the construction of this binocular. One does not calculate one’s working time according to Euro and cent, because all efforts are soon forgotten when the optimum result is finally achieved. This figure is just theoretical and would

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Fig. 5.7 View onto the mounted prism-cluster Fig. 5.8 Prism cover plates with mounted eyepieces

become even lower for the next specimen if only the optical components were still being made in considerable numbers. It remains to be seen whether this new/old binocular will remain a single specimen or whether the first contacts with the industry are going to pay off. It would be nice if we had optics again, with which observation becomes something special – an experience and an event, even a particular pleasure!

Reference 1. Seeger, H.T: Military Binoculars and Telescopes for Land, Air and Sea Service, 2nd edn. Hans T. Seeger, Hamburg (2002)

Part II

Elements of Human Vision

Chapter 6

The Eye

6.1 The Eye as an Optical Instrument The eye is a highly complex organ, the function of which must be approximated by means of optical models. Figure 6.1 shows such a model layout, made of a couple of simplified components. All optical surfaces are spherical, with radii of curvature adapted to the average human eye [1]. Such an eye model is actually an important ingredient of the computer-aided design procedure of a visual instrument, to which it is added during the ray-tracing procedure in order to optimise the interface between optical device and human eye. In such a set-up, various image errors, resulting from an imperfect centring of the eye on the exit pupil (when the eye swivels), as well as the effects of stray light, can be analysed. In Fig. 6.1, an incident ray bundle first hits the cornea of thickness 0.6 mm, and with the two radii of curvature .R1 = 7.8 mm and .R2 = 6.4 mm. A principal refractive index of .n = 1.377 is assumed here. The light then passes through a chamber filled with water (.n = 1.337) and the iris, which serves as a variable aperture stop, before it enters the lens. The lens is flexible and can be tightened or loosened by strings (zonula fibres), which are attached to the ring-shaped ciliary muscle and are responsible for accommodation. When under tension, the surface curvature of the lens reduces and thus its focal length increases. Figure 6.1 shows the situation in the infinity setting. The resulting radii of curvature of both lens surfaces are .R1 = 10.1 mm and .R2 = −6.1 mm, respectively, with a refractive index of approximately .n = 1.411. If instead the eye focuses on an object at close range (about 25 cm), the zonula fibres relax, the lens becomes rounder due to its intrinsic elasticity, and the corresponding radii of curvature are approximately .R1 = 5.95 mm and .R2 = −4.5 mm. Needless to say, this lens elasticity diminishes with age so that older people need reading glasses to focus on objects at close distance. The rest of

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Merlitz, The Binocular Handbook, https://doi.org/10.1007/978-3-031-44408-1_6

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Fig. 6.1 Simplified model of the human eye, containing spherical optical components (length-unit in mm), with an incident ray-bundle

the ray path once again leads through a water-like medium, until the light hits the retina about 17 mm behind the lens, where it forms the real image. For the sake of completeness, the figure also contains the fovea, which represents the central area of sharpest vision, whereas additional anatomical details, such as the laterally offset blind spot, at which the optic nerve attaches, are omitted.

6.2 The Field of Vision The field of vision is the angular range which the eye can see when looking straight ahead. Each individual eye overlooks an asymmetrical sector, the measurement of which is a part of the field of perimetry and subject to individual variations. Nasally (in the direction of the nose), the fields of the individual eyes are somewhat restricted (Fig. 6.2). The visual fields of both eyes overlap in a wide (binocular) sector, which covers a half-angle of at least 50.◦ in all directions. However, the perception of colours is not uniformly well developed over the entire angular range. When moving away from the centre of the field, first the colour saturation and then also colour fidelity decrease, and in the outermost periphery only a light/dark distinction remains – a consequence of the uneven distribution of cones in the retina (see also Sect. 6.7). The eye is a true wide-angle instrument, and the use of binoculars, which commonly cover subjective angles of field far below 90.◦ , implements a significant restriction of vision.

6.3 The Pupil Width

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Fig. 6.2 The field of binocular (meaning of ‘binocular’ here: when using both eyes) vision (yellow). Numbers indicate the angles of view with respect to the centre

6.3 The Pupil Width The pupil width is adaptive and changes with the light conditions, more precisely with the luminance L of the environment, whose base unit is the candela per square metre (1 cd/m2 ).1 This luminance depends on two factors: the intensity of the ambient lighting and the albedo of the object under consideration. The albedo quantifies the proportion of the light reflected by an object (with a dull, non-glossy surface) and reaches, for fresh snow, values near 0.9, but inside a forest only values about 0.1. Table 6.1 contains some examples of luminances of a white sheet of paper (albedo close to 1) under very different illumination conditions. The pupil width and its variation as a function of luminance and age has been the subject of numerous studies. In a publication by Watson and Yellott, the parameters that affect the pupil width have been compiled into a convenient, empirically tested formula [2]. After rounding the numerical factors to four significant digits, this formula reads da ≈

.

18.52 + 0.1222f − 0.1056y + 1.386 · 10−4 fy . 2 + 0.06306f

(6.1)

Here y represents the age of the person (in years), which must be at least 20 for the formula to be valid (a corresponding adjustment for younger subjects is also explained in the paper). The factor f is defined as f = (La)0.41

.

(6.2)

1 The candela is the SI unit of luminosity, physically defined as the luminous flux per unit of solid angle, and is roughly equivalent to the luminosity of a household candle, whereas e.g. a 100 W light bulb provides about 120 cd.

110 Table 6.1 Luminance (in cd/m2 ) of a white sheet of paper under different illumination conditions

6 The Eye Luminance 0.000001 0.001 0.1 1 10 100 1000 10000 100000

White paper sheet. . . Perceptional threshold Under starlight In moonlight Room with running TV-set Streetlight Room-light Daylight, rainy day Bright daylight In direct sunlight

Fig. 6.3 Maximum pupil widths of subjects of different ages (averaged)

with the luminance L in cd/m2 and the angular area a of the visual field. With the unaided eye, half the field of vision reaches 50.◦ , yielding an angular area of ◦ 2 .a = π(50 ) ≈ 7900 (square degrees). On a dark night, with an ambient luminance of 0.0001 cd/m2 , the pupil reaches its maximum width, which is shown in Fig. 6.3 as a function of age. However, as with all results that are derived from the calculation formula (6.1), it should be noted that these are mean values and that the deviations among individual subjects are considerable. The authors mention standard deviations of individual pupil diameters of up to 1 mm, which means that about one-third of all subjects have pupil diameters that differ by more than 1 mm from the calculated mean values. Figure 6.4 shows the pupil widths of people of different ages (20, 40, 60 and 80 years) as a function of the ambient luminance. For better orientation, dividing lines have been drawn here at luminances of 0.01 cd/m2 and 10 cd/m2 , which (somewhat arbitrarily) separate the ranges of night vision from twilight vision and day vision. It is easy to see that the pupil is dilated in the dark and contracts in brightness. This pupillary adaptation represents a mechanism, by which the eye reacts to different lighting conditions by regulating the flow of light to the retina. Once again, pupil widths are age-dependent and subject to individual variation. An interesting side aspect is the dependence of pupil width on the angle of vision: the dashed curve in Fig. 6.4 is the result obtained by Eq. (6.1) for a 20 year old subject whose field of vision is limited to 50.◦ (implying half-angles of 25.◦ ). This situation occurs when pupil adaptation takes place during observation

6.4 The Accommodation Width

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Fig. 6.4 Mean pupil widths of test subjects of different age as a function of ambient luminance. If the angle of vision is narrowed, for example during observation with binoculars, which have a subjective angle of field of 50.◦ , then the pupil becomes additionally dilated (dashed curve)

through binoculars with a subjective angle of field of 50.◦ . The pupil then dilates even further – a circumstance, which is not without relevance for the choice of a binocular. For example, consider a daylight observation inside a forest, in which the ambient luminance reaches levels of a couple of 100 cd/m2 . In order to fully exploit the available light, it would be advantageous if the exit pupil of the binocular had a diameter of at least 3 mm. Typical compact binoculars often come with pupils that are smaller than that, quickly reaching their limits with a dull, low-contrast image. As some animals are crepuscular (i.e. active in twilight), their observation requires exit pupil diameters of at least 5 mm, in case of young observers even 6 mm–7 mm, to avoid a possible loss of light due to the stopped down eye pupil. Anyone who regularly observes at dusk, or even at night, may consider having his pupil diameter measured by an ophthalmologist, since, as mentioned above, individual numbers may deviate significantly from the mean values shown in the diagrams.

6.4 The Accommodation Width In order to focus on an object at a finite distance, the eye must accommodate, that is, change its refractive power. This accommodation is traditionally expressed in dioptre (dpt) as the reciprocal of the distance in metres. To measure an object

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Fig. 6.5 Average accommodation width as a function of the subject’s age, as given by A. Duane [3]

in focus at a distance of 5 m, the eye must accommodate (relative to infinity) by 1/5m−1 , which corresponds to 1/5dpt. An object in 20 cm distance thus requires an accommodation of 5 dpt. Assuming that a person sees sharply at infinity and at close ranges down to a near point of 20 cm, then one speaks of an accommodation width of .δakk = 5 dpt. Since the eyelens thickens with age and loses flexibility, the average accommodation width of the eye is strongly dependent on age. Figure 6.5 shows the course of the mean accommodation width as a function of age. On average, a young child has an accommodation width of .δakk ≈ 14 dpt. This value then decreases at first almost linearly with age, and later – from 40 onwards – more rapidly, to saturate beyond 55 years of age. Here, too, exists an individual dispersion, with a standard deviation of 2 dpt in young people and 1 dpt in older people. Many individuals above the age of 45 need reading glasses to access a text at a comfortable reading distance of 30 cm.

6.5 Aberrations of the Eye Like any optical instrument, the eye is subject to imaging errors, which, even in the case of a healthy person, affect the resolving power of optical vision. Well studied is the chromatic aberration, whose variation in refractive power (in dioptre) is shown in Fig. 6.6. The values demonstrate that, within the visible spectrum, the eye has to deal with a substantial refractive variation that exceeds 2 dpt. While the wavelength of 590 nm (yellow-green) falls on the image plane, the eye is slightly far-sighted (hyperopic) in the red region and clearly short-sighted (myopic) in the blue or even violet region. In view of these values, it is astonishing that the eye is able to focus on a colourful motif at all. The reason for this daily-life experience is to be found in the retina, which masks chromatic aberration in normal daytime vision via various mechanisms. However, Fig. 6.6 provides an explanation for the fact that many people perceive a distant blue neon sign blurred at night (when the eye pupils are wide open and the resulting depth of field is shallow), while, at the same distance, red illuminated

6.5 Aberrations of the Eye

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Fig. 6.6 Chromatic refractive index difference of the healthy eye as a function of wavelength. Here, 590 nm was set as the reference wavelength, following the accommodative behaviour of the average eye. Data from Atchison et al. [4]

signs are readable clearly and distinctly. The advertising industry follows a rule not to use red lettering on blue background (or vice versa), since otherwise the eye’s accommodation is forced to jump back and forth between the two colours, so that reading becomes strenuous and the viewer tired. As already mentioned by Newton, colour fringes are observable at edges that separate areas of different brightness, once the eye pupil is asymmetrically stopped down [5]. In his experiments, he held a sheet of paper close to the eye so that parts of the eye pupil were obscured. This is of relevance when observing with optical instruments whose exit pupils are not centred accurately. Often the eye pupils are partially clipped when looking towards the edge of the field, by which the observer gets the impression of a lateral chromatic aberration of the instrument, although in reality, the colour fringes are not caused by the instrument, but due to the asymmetric trimming of the light bundle. This transversal chromatic aberration of the eye is caused by a circumstance, which, in simplified models such as the one shown in Fig. 6.1, is neglected: the optical axis of the eye is not identical with the line of vision, so that its point of intersection on the retina, relative to the fovea, is slightly displaced towards the nose. As a result, the focal points of different colours are not only longitudinally displaced, but also slightly transversely. In binocular vision, this can lead to an optical illusion, as the lateral offset of the images on the retina is wrongly interpreted as of having a stereoscopic origin. The resulting colour depth effect is described in Sect. 7.3. The healthy eye also exhibits a spherical aberration, which can vary individually between 1 and 2mm. This aberration is responsible, among other things, for the fact that bright point-light sources appear to be radiant and show starry spikes [5]. The cause of these spikes is found in the suspension of the lens of the eye on several zonula fibres, which are attached to the ring-shaped ciliary muscle and are responsible for accommodation. These fibres produce minor deformations of the gelatinous lens and thereby create the observable anisotropic spherical aberration. This fact is of relevance for the popular star test, which is used to analyse the imaging quality of an optical instrument (Sect. 9.3.3). The luminosity of the pointlight source used for such a test should be chosen with care, so that the aberrations of the optical system remain visible, but, at the same time, the eye-related aberrations do not outshine the image of the star.

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Almost every healthy eye also has an astigmatism, triggered by an anisotropic deformation of the cornea. However, its image-degrading power usually becomes significant in the peripheral areas of the lens and is particularly visible under lowlight conditions when the iris is wide open. Amateur astronomers are familiar with the effect that – with small exit pupil diameters – star images often look ‘cleaner’ than through instruments with very large exit pupils, since the outer areas of the image-forming eyelens are stopped down. The aberrations mentioned here are responsible for the fact that in the vast majority of situations, the eye does not have a diffraction-limited resolving power. In tests, under good lighting conditions, optimal resolution values have been found to reach about 1 minute of arc for point-light sources and pupil diameters between 2 mm and 2.5 mm. For larger pupils, the resolution was diminished by the aberrations, and when stopped down even further, diffraction became the dominating factor in image degradation. In addition to the aberrations of the healthy eye, there are also a number of vision impairments, which are corrected with vision aids (glasses, contact lenses) or may even be surgically removed. As long as this defective vision remains restricted to either short-sightedness (myopia) or long-sightedness (hyperopia), binoculars can be used without the need for any additional vision aids, since the correction is accounted for by the focusing mechanism. Cornual astigmatism, however, cannot be corrected by focusing.

6.6 The Retinal Photoreceptors In the photosensitive sensory cells of the retina, light is eventually converted into electrical impulses via a mechanism called photochemical transduction. It is based on the excitation of a light-sensitive visual pigment, a complex molecule, which changes its conformation when it absorbs a photon of a specific spectral range. Then, through a cascade of enzymatic reactions, an action potential is generated. The action potentials provided by a group of sensory cells are encoded in the downstream ganglion cells, and the resulting signals are then transmitted to the brain via the optic nerve. In the case of photosensitive cells, a distinction is made between two basic types, the cones and the rods. The cones are responsible for vision in bright daylight, which is also called photopic vision. There are three different types of cones, each with different spectral sensitivities, which make colour vision possible. The greensensitive M-cones have a sensitivity maximum at 534 nm, the red-sensitive L-cones have their maximum at 564 nm, and the blue-sensitive S cones at 420 nm. In reality, all three cones have broad absorption curves, with a substantial overlap in the case of the M and L cones. From the relative signal strengths of these cones our visual system assigns a colour tone to each object – a process known as trichromatic colour vision. Figure 6.7 shows the combined spectral sensitivity of the three cones (green), whose maximum is at 555 nm.

6.6 The Retinal Photoreceptors

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Fig. 6.7 Spectral stimulus response (‘sensitivity’) of the rods and cones, normalised to their respective maxima. These curves are also referred to in the literature as V-lambda curves

The L and M cones are anatomically indistinguishable from each other, but contain chemically slightly different visual pigments that respond to light quanta of characteristic wavelengths. It is no surprise that such a photoreceptor can also saturate: when exposed to an intense light, the concentration of pigment that is still receptive to light decreases and the cell temporarily loses its sensitivity. At low luminosity, the visual pigment regenerates within about 3 minutes (cones) or about 30 minutes (rods). Besides the pupil-size adaptation, this is the second mechanism that the eye uses to adapt to different lighting conditions. The rods are about 100 times more sensitive to light than the cones and are largely saturated during daytime vision. As twilight progresses, they regain their sensitivity and facilitate the night vision (also called scotopic vision). Since there is only one type of rod, no colour discrimination is possible during night vision. The transition from day vision to night vision is gradual and extends over a broad range of ambient luminance between about 10 cd/m2 and 0.01 cd/m2 . Vision under such twilight conditions is referred to as mesopic vision. In this phase, with gradually diminishing ambient luminance, the colours become paler, with the L cones for red perception being the fist sensor type to pass out: under a full moon, red flowers appear colourless and dark, since the rods are insensitive to light of long visual wavelengths (compare the mutually displaced V-lambda curves in Fig. 6.7). This shift in colour sensitivity, as twilight progresses, is called the Purkinje effect2 and the result of an evolutionary adaptation to the shift in spectral distribution of the ambient light after sunset, when Rayleigh-scattered photons of short wavelengths are abundant (see also Sect. 8.7). Astronomers take advantage of this peculiarity of night vision when using red light, which is perceived exclusively by cone vision and does not compromise the dark adaptation of the rods, to read out their star charts. In the next section, we will go into more detail about how the structure of the retina shapes the properties of visual perception.

2 Johann

Evangelist Purkinje, 1787–1869.

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6.7 Day and Night Vision The retina consists of a central area called the fovea, and its periphery, the extrafoveal area. The angular diameter of foveal vision comprises only about 2.◦ – this is the part of our visual field, in which the eye reaches its optimal visual acuity in daylight. The fovea predominantly consists of cones of the M and L variety, while the short-wavelength sensitive S variety cones are rare. The reason for the lack of blue vision in the fovea is unknown, but it is assumed that there is a connection with the eye’s significant chromatic aberration in the short-wave range, which impairs a sharp image in that spectral range (see Fig. 6.6). Similarly, the role of the macula – the yellow spot – is to be interpreted. It refers to an area which contains a yellow pigment and which covers the fovea and the central 6.◦ angle of vision. The macula acts as a yellow filter, attenuating the short-wave portion of the incoming light, and thus reduces the deteriorating effects of chromatic aberration [6]. However, in this light the question arises as to how a blue or violet object acquires a proper colour saturation when placed in the centre of the field. It has been speculated that, while the perception of the object’s contours is effected via the M- and L cones, its colour information is subsequently overridden by additional signals from individual S cones. Thus, the blue/violet colour is, in foveal vision, not perceived directly, but rather reconstructed in the brain. The cone density reaches its maximum of 150,000 mm.−2 in the fovea and decreases rapidly towards the extrafoveal region, in which the rod density rises steeply, reaching its maximum of 150,000 mm.−2 about 15.◦ away from the centre of the visual field. Further out, the cone density decreases continuously towards the edge of the visual field. In the retina, there exist altogether 120 million rods and 6 million cones. During the day, the eye’s resolving power is determined by the distribution of cones in the retina. Figure 6.8 shows a linear approximation of the resolving power in daytime vision, which reaches approximately 1 arc minute in the foveal area and drops to 15 minutes of arc at about 30.◦ away from the centre of vision. While daytime vision is dominated by its maximum resolution in the foveal area, night vision is rather extrafoveal and most efficient in the region of highest Fig. 6.8 Resolution of the eye during day vision as a function of the angle of view (linear approximation). Data from Pelli et al. [7]

6.8 Twilight Vision: The Stiles–Crawford Effect

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Fig. 6.9 Maximum visual resolution as a function of ambient luminance. Data from Yoder et al. [9]

rod density. Astronomers know the trick of detecting a faint nebula through the eyepiece using indirect vision, in which, instead of aiming directly at the object, the direction of view is shifted sideways by about 15.◦ . In frontal vision, the pale nebula disappears because of the small number of rods in the fovea. Tests revealed that the dark-adapted light sensitivity of the retina increases steeply between the centre of the visual field and 12.◦ extrafoveal, then remains almost constant until about 32.◦ , to decrease slowly towards the edge of the field [8]. At the same time, the extrafoveal resolving power falls continuously towards the edge of the field. In summary, Fig. 6.9 shows the maximum resolution of the eye as a function of ambient luminance. In daylight (luminance .L > 10 cd/m2 ), high resolution values are achieved with foveal vision. In the twilight phase (.0.01 cd/m2 < L < 10 cd/m2 ), the transition from cone-dominated foveal vision to rod-dominated extrafoveal vision takes place, whereby the resolution decreases progressively. At night (.L < 0.01 cd/m2 ), the luminance becomes so low that the signals of an entire group of rods are added to improve the signal-to-noise ratio, which takes place at the cost of resolving power. We will look at this retinal data processing in Sect. 6.9.

6.8 Twilight Vision: The Stiles–Crawford Effect Twilight (mesopic) vision offers another peculiarity, which is known as the Stiles– Crawford effect: the light sensitivity of the cones (much more than that of the rods) depends to a high degree on the direction of incidence of the light. As a consequence, the photochemical transduction of a light beam that enters the retina through a large pupil and that contains portions with shallow angles of incidence, is only partially efficient. In purely arithmetical terms, this corresponds to an entrance pupil, which loses transparency towards the edge. This phenomenon is irrelevant in daylight, when the diameter of the eye pupil is small, and at night, at which the cones no longer contribute to visual perception. The Stiles–Crawford effect can be formally described as a reduced effective pupil aperture, and on the basis of a series of experiments, the following conversion formula has been derived [10]:

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Fig. 6.10 The Stiles–Crawford effect: the effective pupil aperture as a function of the real pupil aperture during photopic (cone-dominated) vision

da' =

.

/

da2 − 1.06 × 10−2 da4 + 4.17 × 10−5 da6 .

(6.3)

Here, .da represents the actual pupil aperture, and .da' the corresponding effective aperture. Figure 6.10 shows this relationship in graphical form: due to the diminishing receptor efficiency, the luminous efficacy is reduced in such a way that the effective pupil aperture increasingly stays behind the actual pupil aperture. A pupil aperture of .da = 3 mm leads to a luminous efficacy corresponding to ' .da = 2.9 mm of effective pupil aperture, so that no impairment of the light yield occurs during the day. However, with .da = 4 mm, the corresponding effective diameter is .da' = 3.7 mm, and with .da = 6 mm, only .da' = 4.9 mm of effective opening remain. In the context of observations with binoculars, the Stiles–Crawford effect implies that the benefit of a large exit pupil is somewhat reduced at twilight. In Fig. 6.4 we see that during an intermediate phase of twilight (ambient luminances around 1 cd/m2 ), the pupil widths already exceed .da = 5 mm, and approach even 6 mm in young observers, so that a certain impairment of the light-yield by the Stiles– Crawford effect is to be expected. For the selection of binoculars that are to be used at such ambient luminosities, this implies that exit pupils beyond 5 mm do not provide as much a benefit as would be expected from luminous flux calculations. But as soon as the observations are extended into the night, at which wide exit pupils once again become fully effective, the Stiles–Crawford effect loses its relevance.

6.9 Retinal Data Processing The retina contains a total of 126 million photoreceptors, but the optic nerve, which transmits signals to the brain, is a bundle of about 1 million nerve fibres. This implies a data reduction which is called convergence. Figure 6.11 shows the underlying principle in a simplified sketch: multiple photoreceptors are connected via intermediate bipolar cells, whose signals are then transmitted via a reduced

6.9 Retinal Data Processing

119

Fig. 6.11 Convergence (schematic): the excitations of several photoreceptors are bundled via the convergent interconnections of the bipolar cells. There are significantly fewer ganglion cells than photoreceptors

number of ganglion cells to the optic nerve. Each one of the approximately 1 million ganglion cells contributes one nerve fibre to the optic nerve. The necessity of this data reduction in the retina is also related to the limited capacity of the brain: the retina may be compared to a light-sensitive digital sensor, consisting of 126 megapixels – a lot of information to be transmitted in real time to the brain and processed there. But even among the already reduced data, a further selection takes place in the visual cortex. It is interesting to realise that this enormous data reduction, which, apart from foveal vision, also affects the resolution of the image, is not perceived as an impairment in everyday life. This is because visual perception is not based on individual snapshots of a complete motif, but rather on the principle of a gazing eye, which focuses on various objects in quick succession (called saccades). The brain integrates the individual pieces of information into an overall picture of the environment and ensures that important information is processed immediately, while less significant details are ignored and never enter the viewer’s consciousness. In Sect. 7.4 on the visual image construction we will return to this mechanism. A straightforward convergence in connection with a signal addition, as shown in Fig. 6.11, occurs in the extrafoveal circuits of the rods. Those photoreceptors, which are connected to a single ganglion cell, belong to a group called the receptive field. Such a structure serves to increase the signal-to-noise ratio of the receptor: at luminance levels below 0.001 cd/m2 , the perception threshold of a cone is already undercut. The rods are about 100 times more sensitive, and if operating as individual receptors, they would detect 0.00001 cd/m.2 , while below that threshold, they would lose their function. Nevertheless, the perception threshold of the dark-adapted eye is again lower by a factor of 10 (see Table 6.1) – a consequence of the signal addition by receptive fields. Digital cameras use the same technique to increase their light sensitivity by bundling pixels while sacrificing resolution. It is primarily the rods

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Fig. 6.12 Lateral inhibition (schematic): if photoreceptors are stimulated (left), lateral connections via inhibitory synapses reduce the activity of neighbouring neurons, thereby amplifying a contrast in stimulation

of the extrafoveal retina that are grouped into convergent receptive fields, whereas cones in the fovea are commonly assigned to individual ganglion cells. This explains the high degree of resolution of the eye in the central area of the visual field during daytime vision. However, the interconnections of nerve cells in the retina can perform far more complex tasks. Figure 6.12 shows the example of lateral inhibition: photoreceptors are interconnected horizontally, via inhibitory synapses.3 In the section to the lefthand side, the receptors are exposed to a stimulus, which leads to an excitation of the strength 10, whereas in the right section, a weaker stimulus generates an excitation of the strength 2. Via horizontal interconnections, a fraction of the excitation is passed on to neighbouring cells as inhibition, where it is subtracted from the intrinsic excitation of that receptor. The processed signal strengths, which are finally passed on to the ganglion cells in the direction of the brain, are shown in the lower row. 3 Synapses represent the electrochemical connections between nerve cells. In the case of an inhibitory synapse, the signal has an inhibitory effect on the excitation.

References

121

Fig. 6.13 Mach bands to demonstrate the effect of lateral inhibition: each individual band is perfectly uniform in brightness, the apparent bright lines at the band’s boundaries are optical illusions

The process results in a contrast enhancement, which supports the recognition of outlines and shapes in low-contrast motifs. The Mach bands (Fig. 6.13) are a popular demonstration of an optical illusion that originates in lateral inhibition:4 at the brightness transitions, there seem to exist thin lines of enhanced intensity, which are absent in the actual motif – the effect of the lateral inhibition is unmistakable. The lateral inhibition is no recent invention of evolution. It has been found in the lateral eyes of the limulus (also known as ‘horseshoe crab’), a living fossil with an extremely simple neuroanatomy, which has already been raving about the good old times to the dinosaurs. In experiments with cats, far more complex receptive fields were found, serving tasks such as recognising orientations (horizontal/vertical) or movements. Receptive fields with similar tasks have also been proven to exist in the human retina in experiments on perception. The general properties of retinal information processing can be summarised as follows: the complexity of the neural network increases extrafoveally towards the periphery, and primarily the rods are grouped into receptive fields, which take over numerous complex tasks of information pre-processing in the retina. These insights make it clear that scotopic vision (night vision) in particular, in which the cones of the fovea are hardly involved, is highly dependent on neuronal data processing in the retina. This is not without relevance for the appropriate choice of binoculars: binoculars that are primarily used under low-light conditions should offer wide (subjective) angles of field and engage the peripheral areas of the retina. Unfortunately, such night glasses need large and heavy prisms in order to offer wide views with a low degree of vignetting, are heavy and bulky and therefore hard to sell.

References 1. Walker, B.H.: Optical design for visual systems. SPIE PRESS, Bellingham, Washington (2000) 2. Watson, A.B., Yellott, J.I.: A unified formula for light-adapted pupil size. J. Vis. 12, 1 (2012)

4 Ernst

Mach, 1838–1916.

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3. Duane, A.: Studies in monocular and binocular accommodation with their clinical applications. Am. J. Ophthalmol. Ser. 3 5, 865 (1922) 4. Atchison, D.A., Smith, G.: Chromatic Dispersions of the ocular media of human eyes. J. Opt. Soc. Am. A 22, 29 (2005) 5. König, A., Köhler, H.: Die Fernrohre und Entfernungsmesser. Springer, Berlin (1959) 6. Reading, V.M, Weale, R.A.: Macular pigment and chromatic aberration. J. Opt. Soc. Am. 64, 231 (1974) 7. Pelli, D.G., Farell, B.: Psychophysical methods. In: Bass, M. (ed.) Handbook of Optics, Volume III: Vision and Vision Optics. McGraw-Hill Companies (2010) 8. Riopelle, A.J., Bevan Jr., W.: The distribution of scotopic sensitivity in human vision. Am. J. Psychol. 66, 73 (1953) 9. Yoder, Jr., P.R., Vukobratovich, D.: Field Guide to Binoculars and Scopes, SPIE PRESS, Bellingham, Washington (2011) 10. Moon, P., Spencer, D.E.: On the Stiles-Crawford effect. J. Opt. Soc. Am. 34, 319 (1944)

Chapter 7

The Visual Perception

7.1 Laws of Visual Perception The theme of perception – in the context of long-range visual optical instruments – is concerned first and foremost with the sighting and identification of distant objects. The answer to the question of whether an object is perceptible depends on three basic factors [1]: on the object size, on the ambient luminance and on the contrast between object and background. There exist different models with different emphases and demands, with the help of which the attempt is made to simulate the performance of object reconnaissance under various lighting conditions. Despite the diversity of these models, each individual approach has to fulfil a number of prerequisites, which have been established over several decades in the form of basic laws of visual perception. Relatively easy to quantify is the visual acuity – every reader of this book has probably had the experience of an eye test, during which either Landolt rings (Fig. 7.1) or similar motifs, placed in different sizes and orientations on test cards, had to be deciphered. This is an experimental set-up that tests the ability to perceive fine details on high-contrast motifs. Such types of tests of visual acuity produce results as shown e.g. in the Figs. 6.8 and 6.9. Even if visual acuity is a good measure of the performance of perception in daylight – after all, foveal vision, apart from colour perception, is primarily optimised for maximum resolution – it is nevertheless not the only factor that defines the recognition of objects. Some objects are recognisable even though they cannot be resolved: distant high-voltage cables or the countless stars of the night sky are among the most common examples of objects that can be perceived without being resolvable. On the other hand, there are objects that could easily be resolved, but are nevertheless not perceptible due to their low contrast to the environment (camouflage). Especially in twilight and at night, contrast plays a major role in deciding whether an object can still be seen or not.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Merlitz, The Binocular Handbook, https://doi.org/10.1007/978-3-031-44408-1_7

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Fig. 7.1 The Landolt ring for determining visual acuity (resolving power)

7.1.1 The Laws of Ricco, Piper and Weber–Fechner We consider the question about the perceptual threshold of an object. For this purpose, we imagine a circular test target that is presented to the observer at an angular diameter of .σ and with a surface luminance of .Lt . This test disc is put in front of an unstructured background with a homogeneous luminance .Lb , and the observer’s eye is adapted to an adaptation luminance .La , which would usually lie in between .Lt and .Lb . The contrast between test disc and environment is defined as1 C=

.

L t − Lb , Lb

(7.1)

and we assume here for simplicity that it is positive, so that the object is brighter than its surroundings. If both the test target and the surroundings are non-illuminating motifs, which only passively reflect the incident light, then the albedo (i.e. the diffuse reflectivity) of both motifs may replace the corresponding luminance in Eq. (7.1). For motifs in nature, the albedo usually spans a range from about 0.1 (forest) to 0.9 (snow), and a white, non-luminous target in front of a forest line should therefore have a contrast not exceeding .C ≈ 8. The question therefore arises as to the location of the perception threshold of the target, i.e. suitable combinations of contrast, object size and ambient luminance, at which the disc is just barely visible. For simplification, we first make the

specifically, this defines the Weber contrast, while an alternative definition, .C = (Lt − Lb )/(Lt + Lb ), is known as the Michelson contrast.

1 More

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125

(usually realistic) assumption that adaptation luminance and ambient luminance are identical, i.e. the eye has adapted to the ambient luminance. Three laws can then be applied to determine the perception threshold, sorted according to object size [2]: under conditions such as those that occur in twilight, and for test disks of small angular diameter (up to about 24 arc minutes), Ricco’s law applies, which states that at the threshold of perception the product of contrast, object area and ambient luminance is a constant, i.e. Cσ 2 Lb = const .

.

(7.2)

The larger the disc, the weaker its contrast may be, so that it can stand out visibly from the background; a large target of low contrast is just as easily seen as a small object of correspondingly higher contrast. The unknown constant has to be determined experimentally in a series of tests. Figure 7.2 schematically illustrates the principle: the four disks differ in area and albedo, but at a certain level of ambient luminance and corresponding adaptation, they are similarly hard to detect. Physiologically, Ricco’s law can be explained with receptive fields of the retina, which, by means of convergence, bundle the signals of several sensors (Sect. 6.9). For this reason, its validity in daylight is limited to rather small angular target sizes, because convergence occurs mainly during twilight- and night vision in the extrafoveal areas of the retina. The larger the receptive fields, which are involved in the target detection process, the broader the range of application of Ricco’s law. For targets of larger angular diameters, the perception threshold is determined by Piper’s law Cσ Lb = const ,

.

Fig. 7.2 Ricco’s law: the smaller the object, the higher the minimum contrast that permits a sighting

(7.3)

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to which not the area, but the diameter of the disc contributes. The constant, however, is not identical with that of Ricco’s law. The range of validity of Piper’s law is limited to angular diameters between roughly 36 minutes of arc and several degrees. With even larger objects, the dependence on the angular diameter is lost. This is expressed in terms of a special case of the Weber–Fechner law, which states that the just perceptible difference in luminance between the object and the environment depends on the logarithm of the ambient luminance (but not on the apparent size of the object), yielding .

CLb = const . log Lb

(7.4)

If the ambient luminance, .Lb , is set to a given value, then the contrast that just permits detection of the target is called the threshold contrast. In bright daylight, the contrast between object and surroundings may be very low, and still a perception is possible. If it is dimly lit, a significantly higher contrast is required, since otherwise the target may be overlooked.

7.1.2 Berek’s Model of Target Detection In 1943, Max Berek of the Ernst Leitz AG established a mathematical model of perception, which compiled the then existing rules on perceptual thresholds into a consistent formalism [3]. The resulting equations allowed, for the first time, to estimate the perceptual thresholds of targets of any diameter under all conceivable illumination conditions and degrees of adaptation of the eye. The main equation for the threshold contrast is / / √ b(La ) 1 φ(La ) . C = + , (7.5) σ La La where .La stands for the adaptation luminance, .σ for the angular diameter of the object and C once more for the threshold contrast, the minimum contrast required for target sighting. The characteristic luminous flux function .φ(La ) and the characteristic luminance function .b(La ) contain the respective constants from the laws (7.2)–(7.4), which were determined and summarised in series of experiments with volunteers. In Berek’s original work, these luminance functions are recorded in cumbersome tables, which makes their use in computer programs difficult. However, they can be mathematically interpolated, as shown in Fig. 7.3, and the resulting smoothed functions parameterised as follows [4]: .

log10 b = −1.77 + 0.824 x ,

(7.6)

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Fig. 7.3 The luminance functions b and .φ from Berek’s original work (black circles) and their mathematical interpolations (red)

Table 7.1 Coefficients needed for Eq. (7.7)

= 0.421 46 = 0.071 90 .a4 = −0.000 795 9 .a0

.a1

.a2

.a3

= 0.395 57 = 0.010 21

where .x = log10 (La ) is the decadic logarithm of the adaptation luminance. Furthermore we obtain for the function .φ .

log10 φ = a0 + a1 x + a2 x 2 + a3 x 3 + a4 x 4 ,

(7.7)

with coefficients .ai that are summarised in Table 7.1. With the parameterisations of the luminance functions, it is now easy to automate the calculations of the perceptual equations and to obtain practical perceptual thresholds for many situations and under a variety of conditions. Figure 7.4 shows, as a first application, the threshold contrast as a function of the object’s angular diameter under three different lighting conditions. Here it is assumed that the eye is always adapted to the background luminance, .Lb . It is easy to see how the continuous transition from Ricco’s law (the slope of the function −2 is indicated by the dashed line in the upper left corner) to the Weber– .C ∼ σ Fechner law (.C = const for large object diameters) occurs. In this sense, Piper’s law stands less for an independent scaling law, but rather for a broad crossover regime between Ricco’s and the Weber–Fechner law. It is also clearly visible how the range of validity of Ricco’s law diminishes when the ambient luminance increases. Next, we consider a white disc, set up in front of a forest line, with a contrast of .C = 8 (dashed horizontal line in Fig. 7.4). The disc falls below the perception threshold at night, once its angular diameter is smaller than 15 minutes of arc,

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Fig. 7.4 Threshold contrast as a function of the target’s angular diameter .σ in daylight (.Lb = 1000 cd/m2 ), in twilight (.Lb = 0.1 cd/m2 ) and at night (.Lb = 0.0001 cd/m2 )

since then the threshold contrast is always above the dashed line. In twilight, the corresponding perception threshold is about 1.5 minutes of arc. Here, a comparison with Fig. 6.9 is interesting, which shows the resolving power of the eye at different levels of brightness. To obtain the corresponding data, presumably Landolt rings (Fig. 7.1) served as test targets, which lead to results that are consistent with Berek’s law in terms of their magnitudes. This consistency offers an important general insight into the nature of visual acuity: during the day, acuity is determined by the maximal resolving power of the eye and the retina. However, in twilight and at night, the contrast becomes the limiting factor for the recognition of fine details. With the perceptual model as represented in Eq. (7.5), interesting scenarios can be played out, such as estimating the the threshold detection range of an object with a given size and contrast, as a function of ambient luminance. It is derived from Eq. (7.5) to yield √ )}−1 { ( φ(La )/La π R = s 2 tan , · √ √ 1802 C − b(La )/La

.

(7.8)

where s is the object diameter in metre.2 Let us take as an example a wild boar in front of a forest edge: its contrast would be rather low, perhaps .C = 0.2, its diameter roughly 1 m (let us allow here to approximate the pig as a circular disc), 2 Note that the numerical factor .π/1802

into radians.

arises from the conversion of the apparent target half-angle

7.1 Laws of Visual Perception

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Fig. 7.5 Detection range of a wild boar, a white rabbit and a candle flame in front of a forest edge, as a function of the ambient luminance, determined with Eq. (7.8)

and for comparison a white dwarf rabbit (contrast: .C = 4 and diameter 15 cm). We calculate the maximum distance at which the object in question is still perceptible. The result is shown in Fig. 7.5: deep in the night this distance amounts to less than 9 m for the well-camouflaged boar – one almost steps on its feet before it becomes visible. The rabbit, though much smaller, is still visible at a distance of 20 m. In the twilight at ambient luminance of 0.05 cd/m2 , a point of intersection exists, at which the threshold distances of both rabbit and boar are equal at about 200 m. From then on, with increasing light levels, the larger pig is visible from a greater distance than the rabbit. It is interesting to note that in daylight the rabbit is still detectable at almost 2 km distance, after its angular diameter has dropped below the eye’s resolution limit, which lies at a much closer distance of 500 m. Thus, what is seen between 500 m and 2 km is not the rabbit itself, but its diffraction image as a tiny white spot. The behaviour of self-luminous objects is also covert by Berek’s perception model: consider a candle flame, whose diameter is approximately 1 cm, with a surface area of roughly .π cm2 and a luminous intensity of .1 cd. The surface luminance then becomes .π −1 cd/cm2 = 104 π −1 cd/m2 . Since the flame’s luminance is independent of the daytime (assuming that it doesn’t reflect ambient light), its contrast to the forest edge depends only on the background luminance: in broad daylight, the forest (with an albedo of 0.1) has a luminance of 1000 cd/m2 and the contrast to the candle flame is only .C ≈ 2.2. At night (luminance of the forest in this example: .3.2 × 10−5 cd/m2 ), the same contrast reaches a considerable 8 .C ≈ 10 . From these specifications, the law of perception calculates a visibility of the nocturnal candle flame up to beyond 7 km, whereas the same flame can only be seen up to a maximum distance of 90 m during the day (Fig. 7.5). If the sighting of a candle from a distance of 7 km sounds incredible, the reader may be assured that it is by no means a misjudgement: every soldier learns that in deep darkness, a smouldering cigarette can be seen over distances of several kilometres, and uses the technique of concealing the cigarette in his cupped hand. Berek’s model of perception is also applicable in combination with a visual optical instrument, including telescopes or binoculars, and then serves to calculate its performance, as will be discussed in Sect. 8.4. There exist also more modern

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perceptual models of higher complexity, which include further elements such as the duration of sighting or target movement – factors that are relevant to military applications. The interested reader is referred here to the specialised literature [5].

7.2 The Perception of Colour Every ornithologist will point out the enormous importance of colour perception for a reliable identification of certain bird species. Colour fidelity of an optic and the ability to differentiate colour are considered by naturalists to be an indispensable prerequisite for an accurate classification of living nature. The colour perception serves as a classic example of the complexity of data processing in visual perception. As a result of the trichromatic mechanism (based on three cone types) of colour perception, the experience of a given colour is not necessarily linked to the physical wavelength: the colour impression ‘green’ can either be derived from light of wavelength about 550 nm or from a light source with a broad colour spectrum whose red component has been suppressed. Conversely, the impression of a colour-neutral illumination must not lead to the erroneous conclusion that the light is composed of a continuous spectrum. Fluorescent tubes provide a discontinuous spectrum of individual lines, which only through additive colour mixing create the impression of a colour-neutral light source. A quick look at the colour spectrum (Fig. 1.1), in which the wavelength is continuously tuned, reveals that the visual colour impression correlates only partially with the physical properties of light: our ‘colour detector’ is non-linear and produces sectors of different widths, which we are used to call ‘colours’. It is therefore no surprise to learn that Goethe,3 within the framework of his ‘Farbenlehre’ (means: ‘theory of colours’), was able to develop a theory of light that was, from the point of view of physical sciences, completely wrong, yet consistent from a perceptualphysiological point of view. Our perception of colour has more peculiar features to offer. Figure 7.6 shows typical spectral distributions of daylight, once under direct sunlight and then also under overcast sky. The illumination is warmer (shifted towards yellow) when the sun is shining, while under overcast skies, the cooler colours dominate. Nevertheless, daylight always appears neutral to us, regardless of the weather. A white sheet of paper appears in all situations white and not coloured. Obviously, some form of ‘white balance’ takes place under variable lighting conditions. This phenomenon is known as colour constancy and is of astonishing robustness and experienced even in artificial room lighting, where the colour composition can deviate considerably from that of daylight. The power of this colour constancy becomes particularly obvious whenever photographs are taken under different lighting conditions while the automatic white balance is deactivated.

3 Johann

Wolfgang von Goethe, 1749–1832.

7.2 The Perception of Colour

131

Fig. 7.6 Spectral intensity distribution of daylight in sunshine and under overcast skies. Data from Pokorny et al. [6]

Fig. 7.7 Simultaneous contrast: the colours of a bird appear more intense against dark branches than on a green meadow

The colour constancy comes along with the ability of perception to perform a partial colour adaptation: well known is a simple experiment, in which a red square is stared at for 30 seconds, after which its replica appears on a white sheet of paper in the complementary colour green. This effect is easily interpreted as a temporary fading of the activated colour receptors in the retina. Rather complicated is the phenomenon of simultaneous contrast (Fig. 7.7): colour intensities, but also colour tones of an object are perceived differently when the object is placed in front of different backgrounds. Here, a far more complex information processing takes place in the visual cortex, the details of which are still largely unknown. There exist several attempts to model the trichromatic properties of colour vision in order to simulate the colour impression produced by coloured light sources, the threshold values for the perceivable colour contrasts and their dependencies on the lighting conditions or the state of adaptation of the eye. So far, a self-consistent law of colour perception, analogous to Berek’s model for the perception of brightness (Sect. 7.1.2), has not been convincingly formulated due to the high dimensionality

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of the colour space. Such a task is further complicated by the fact that the individual variability in colour perception, especially among the male population, is by far greater than is the case with brightness perception. As a result, experimental set-ups with test subjects rarely lead to fully consistent conclusions. Fortunately, many of these theoretical details are of secondary importance to observational practice: each observer may perceive the colour composition of a bird in his own individual way, but this fact has no influence on his ability to identify the bird’s species, as long as the conditions for colour constancy are fulfilled, i.e. the observer’s white balance remains effective. An optical instrument can support the observer in this task by providing a spectral transmission curve which is as flat as possible to provide images of true colour under all conceivable lighting conditions. The influence of the instrument’s transmission curve on brightness and colour saturation of the image is discussed in Sect. 8.8.

7.3 Stereoscopic Depth Perception One advantage of binocular observation is the feature of stereoscopic depth perception: if two objects are at close range, they appear to both eyes at slightly different angles due to the parallax. Different angles of incidence then lead to a lateral shift of the retinal images of both eyes, known as cross-disparity. From the quantity of the lateral shift, the brain can calculate a spatial impression of depth, which, at close range, contributes significantly to a rapid orientation. Evolutionary biologists suggest that the human’s two eyes – just as is the case with other higher primates – are located on the front of the head and not on the side, because in this way the overlap of the two visual fields (Fig. 6.2) and thus the effectivity of stereoscopic vision could be maximised. Figure 7.8 clarifies the principle: both observation points (L and R) are separated from each other with the distance of the base length b. Objects are observed at the distance E and .E − δE, which appear to the right eye at the two angles .α1 and .α2 . We apply the trigonometrical equations .tan α1 = b/E and .tan α2 = b/(E − AE), Fig. 7.8 Parallax: two objects of distance E and .(E − AE) appear to the two eyes (L and R) at different angles, and the disparity depends on the base length b

7.3 Stereoscopic Depth Perception

133

assuming that the angles are sufficiently small so that we approximate the tangent of the angle by the angle itself (in radians). For the angular difference, we then get α2 − α1 = Aα =

.

bAE . E(E − AE)

(7.9)

There is a threshold of perception for the minimal disparity that the brain can interpret as depth information, and this translates to a minimal angular difference, .(Aα)min . Precise measurements of this minimum transverse disparity can be made with random point stereograms: such a stereogram is constructed by first creating a random dot pattern. Then, an exact copy of this pattern is placed next to the first pattern. In the copy, a section of the dot pattern is shifted sideways by a minimal amount. The two images next to each other appear identical at first glance, and only in stereoscopic vision, when the images are viewed with both eyes separately but simultaneously, the lateral shift is recognisable. The previously shifted section of the image then appears to separate vertically from the background. In measurements, test subjects were able to perceive minimum angular differences .(Aα)min in the range of 10–30 arc seconds, which is well below the foveal resolving power of each individual eye. Such an over-accuracy of perception indicates a massive data processing of both retinal images in the brain, as a result of which the limitations of the individual sensors are overcome. An observation seemingly independent of this question supports the hypothesis of a complex data post-processing: the cumulative brightness perception of both eyes is not additive. Purely arithmetically, the perceptual threshold should binocularly – due √ to the improved signal-to-noise ratio – be lower by a factor of . 2 than in monocular observation. However, such an improvement has not been observed in experiments, which suggests that the signals from both eyes are not simply added together, but used for more complex operations, which may include some sort of interferometry [7]. A minimum angle .(Aα)min corresponds to a minimum distance difference, i.e. a depth resolution .AE of two objects, which an observer can achieve by stereoscopic means. The calculation, in which the angles are in radians, yields AE =

.

E 2 (Aα)min , b + E(Aα)min

(7.10)

and the result is shown in Fig. 7.9. Here, an inter-pupillary distance of .b = 65 mm was taken as the base length, and a transverse disparity threshold .(Aα)min of 30 arc seconds. At 22 m distance, two objects with distances that differ by 1 m can be spatially resolved, at a distance of 70 m, the subjects require a minimum separation of 10 m, which increases to 100 m at a distance of 270 m (of the far object). Note that stereoscopic depth perception does not require any additional cues to be effective. It thus functions perfectly well in low light, despite the lack of textural features and various other details that support depth perception in daylight. For this

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Fig. 7.9 Depth resolution: Minimum range difference .AE of two objects that an observer can still perceive stereoscopically, as a function of the distance E of the far object

reason, binoculars, which are primarily used under low-light conditions, should have as wide a stereo base (i.e. separation between its objectives) as possible, to optimally support the perception of depth in the dark (see also Sect. 8.10).

7.3.1 Further Mechanisms of Depth Perception The stereoscopically induced depth perception is not the only way in which our brain generates three-dimensional vision. There exist cues, which create the impression of depth in monocular vision, too. Examples are texture gradients on the surface, perspective, lighting and shading, as well as the phenomenon of atmospheric perspective, i.e. a loss of contrast that increases with distance. Over the centuries, painters have optimised their techniques, to make best use of these effects, and thereby created impressive paintings with stunning impressions of depth. Additionally, there are dynamical cues: by changing the point of view, the observer is able to assign distances to motifs on the basis of the resulting motion parallax. Due to an optical illusion, there exists yet another phenomenon of (an incorrect) depth perception, known as colour depth effect or colour stereoscopy: as a result of the transverse chromatic aberration of the eye (see Sect. 6.5), objects of different colours are laterally displaced on the retina. In binocular vision, the perception interprets this chromatic transverse disparity as a stereoscopic impression of depth, even when the objects are simply spots of colour on a poster. This phenomenon becomes particularly striking at night with coloured neon signs, and the impression of depth is further intensified by binoculars. By varying the inter-ocular eyepiece distance, it is even possible to reverse the depth impression of coloured illuminated lettering [8]: when the binocular is set to the normal inter-ocular distance, at which exit- and eye pupils are concentric, then the red letters appear to be in the foreground and in front of the blue or violet letters. The same impression becomes more pronounced when the eyepiece separation is slightly increased. If, however, the interocular distance is narrowed down, then the blue letters seem to float in front

7.4 Saccadic Image Construction

135

of the red letters, i.e. the colour depth effect is reversed. The reason behind this stunning reversal of depth perception is the function of the exit pupil as an aperture stop: By varying the inter-ocular distance, different sections of the light bundles are excluded from entering the eye, which also affects the character of the retinal transverse chromatic shifts and hence, the interpretation of depth.

7.4 Saccadic Image Construction In the first half of the 20th century, the perceptual image construction, i.e. the process in which the observer generates a mental map of the surrounding scene, was still compared to that of a (film) camera. It was believed that the eye captured an image of the environment in the form of a snapshot, which was then somehow passed to the brain for further interpretation. This process would have to be constantly repeated, about 20 times per second, in order to give the impression of a flowing sequence of movements as the motif changes. Today, it is known that such an approach to image construction would far exceed the available capacities for data transmission and data processing. Instead, the perceptual psychology has identified a number of methods and techniques that enable our perception to form a picture of the environment in a far more efficient way, through the mechanism of selective attention: in daytime observations, the eye only provides a small angular range of about 2.◦ of sharp, foveal vision (see Sect. 6.7), which operates at highest acuity and on which the maximum perceptual attention is directed. The extrafoveal regions of the retina see along, at far lower resolution, and serve mainly to identify peripheral motifs, particularly moving objects, which might be of interest or danger. Several times per second, the eyes then move around to fixate other objects, and during this involuntary process, known as saccadic movements, the brain gradually constructs a mental map of the environment. Figure 7.10 schematically illustrates such a scanning process: attention is first focused on the person in the foreground. Then the gaze falls to other, Fig. 7.10 Possible course of eye movement when looking at a motif selection with person and landscape. The lines show the saccadic eye movements during the initial few seconds

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striking features in the background. The brain integrates the information gained during such a scan and combines it to form a virtual picture of the surroundings. Such a mental image is far from being complete – in order to clarify certain questions of detail, the viewer will have to repeatedly shift attention to additional motifs in order to complete his picture of the surroundings. During the process, the viewer succumbs to the illusion of having the entire scene in view and not having overlooked any detail. The fact that this is not the case is proven by the popular picture puzzles in leisure magazines, the task of which is to discover a predefined number of differences in two almost identical pictures. This search for differences, even in ridiculously simple drawings, can take several minutes, because our perception, due to selective attention, does not have the capacity to take in a picture with all its details instantaneously. Famous painters of the past centuries instinctively knew how to use the peculiarities of selective perception by attracting the viewer’s eye by means of carefully placed elements. In this way, they succeeded in creating very realistic and vivid paintings with a limited number of details. Perceptual psychologists employ eye-trackers – cameras that follow the eye movement of test subjects – to uncover the secrets of selective attention. The potential return on investment from the advertising industry is obvious.

7.5 The Optical Flow The perception of movement is based on the dynamic change of positions, whereby several situations can be distinguished: (1) a single object moves in front of an otherwise immobile backdrop; (2) the object stands motionless in front of a dynamic background, such as a radio mast with clouds moving behind it; (3) both object and background perform movements. Think of a cornfield with ears of corn swaying in the wind, while at the same time a deer hops through the scene. Our visual perception is trained to separate movements from each other and to identify potentially interesting or even dangerous objects against moving backgrounds. Yet another variant is the coherent movement of the entire landscape, such as that which occurs on a train ride, and this case is of particular relevance for binocular observation, because it occurs when the binocular is panned. Here, each image point moves on its individual trajectory, and the bundle of all trajectories forms a velocity field. In the psychology of perception, one is interested in the effect of such a velocity field – in this context referred to as optical flow – on depth perception. The connection with animated computer graphics is obvious, in the computer games industry, but also in serious applications such as navigation aids for pilots. It is amazingly easy to create the impression of being in motion with only a few pixels (Fig. 7.11): if the point objects emerge from a fixed point and then move towards the observer, he has the impression of gliding in the direction of this vanishing point. Conversely, if the pixels converge towards a common point, the observer believes to move backwards. Figure 7.12 shows another example: a

7.6 The Visual Distortion

137

Fig. 7.11 Optical flow: if point objects move from a fixed point towards the observer, he has the impression of gliding in the direction of this vanishing point

Fig. 7.12 Divergence creates the impression of an object that is approaching, convergence that of an object moving away

group of pixels, approaching from one side of the image, diverges and increases in velocity, passes through the centre, then converges and decelerates while moving towards the other side of the image. The result is the impression of an orbit, distorted by perspective, as if a rotating merry-go-round were viewed from a distance of a few metres. In this example, the observer concludes that he himself remains stationary while observing a rotating or an orbiting motif. During the panning of a binocular, which has a non-vanishing rectilinear distortion (Sect. 2.1.5), some of the effects of optical flow described here are of relevance. Astonishingly, it has been demonstrated that such a distortion does not only occur in the optical instrument, but also within the optical perceptual process. This insight will be discussed in the next section, and it will eventually provide us with important clues about the ideal quantity of instrumental distortion, which has to be implemented in order to compensate for some of the less desirable features of our perception.

7.6 The Visual Distortion In the 19th century, Hermann v. Helmholtz made an interesting observation: a chequerboard pattern with a pincushion distortion can appear regular (i.e. free of distortion) to the eye if it is viewed at close range, so that it fills a sufficiently wide angular area of the optical field [9]. Figure 7.13 shows such a distorted chessboard pattern: according to Helmholtz’s instructions, this pattern has to be viewed from a distance the length of the black bar (at the bottom of the figure). The gaze should be kept fixed on the centre of the chequerboard, and at the same time the contour lines should be observed by indirect peripheral vision. The thus (to some observers)

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Fig. 7.13 Chequerboard with pincushion distortion according to Helmholtz: when viewed from a close range of the length of the black bar (bottom), it appears undistorted to some observers

visible ‘straightening’ of the contours can be interpreted as an intrinsic perceptual barrel distortion, which compensates for the pincushion-shaped distortion of the chequerboard pattern. Recent computer-based and systematic studies by the working group around A. H. J. Oomes have confirmed Helmholtz’s assumption that visual space commonly contains a certain amount of barrel distortion. They also found that there are significant individual differences in the quantity of this visual distortion [10]. In fact, it was found to be in average about half as strong as originally postulated by Helmholtz. Formally, an instrumental distortion can be parameterised after a modification of the tangent condition (Eq. (2.10)), which is is free of rectilinear distortion according to Eq. (2.11). For that, an new distortion parameter k is introduced: .

tan(ka) = m tan(kA) .

(7.11)

Here again, A means the objective angle, a the subjective angle of view and m the magnification of the optics. In the case .k = 1, we retain the tangent condition. In the limiting case .k → 0, we first expand the tangents, taking only the linear term, ka ≈ mkA ,

.

(7.12)

and then eliminate k, whereupon we obtain the angle condition (Eq. (2.12)), which involves a pincushion distortion, as is easily found after insertion into Eq. (2.11). As demonstrated by H. Slevogt, the Helmholtz chequerboard pattern corresponds to the specific parameter choice .k = 0.5, and he coined the term circle condition for the resulting relation [11]:

7.6 The Visual Distortion

139

Fig. 7.14 Relative frequency of subjects who perceived a chequerboard, previously generated using the distortion parameter k, as regular. As a consequence, the average visual distortion would be about .kv ≈ 0.75

.

tan(a/2) = m tan(A/2) .

(7.13)

The author created a gallery of chequerboards with varying degrees of distortions and asked volunteers via the Internet to repeat the Helmholtz experiment [12]. The aim was to identify the board that appeared, to the individual observer, closest to being regular when viewed from a prescribed distance. Figure 7.14 shows the results in terms of a smoothed, interpolated graph: displayed is a wide distribution of answers, ranging from .k = 1 (distortion-free board) to the circle condition (.k = 0.5), largely consistent with the earlier results of the Oomes study. Some (albeit few) subjects perceived even strongly distorted patterns, made according to the angle condition (.k = 0), as regular. This experiment seems to verify that the visual perception of a significant proportion of test persons involves a non-negligible amount of barrel distortion. In the framework of perceptual psychology, it is rather common to define the concept of visual space, which is then implemented in terms of a suitable mathematical modelling. Visual distortion can equivalently be interpreted as a curvature of the visual space, and such non-Euclidean visual spaces have been investigated for more than half a century [13]. A barrel distortion of the visual space follows from the relation y=

.

1 tan (kv A) , kv

(7.14)

where .kv is the visual distortion parameter and y is the radial coordinate of the image in visual space, measured from the centre of the visual field [14]. The statistical distribution of this parameter .kv is then identical to the distribution of the parameter k in Fig. 7.14, since for that experiment, it was the barrel distortion – represented by .kv – that had to eliminate the pincushion distortion of the board, which had previously been generated with the parameter k. Then, the average visual distortion among the test subjects would amount to .kv ≈ 0.75. In Sect. 8.12, we are going to work out the mathematical details behind this mutual compensation of distortions.

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In this context, the question arises as to why such a visual barrel distortion unveils itself under suitable experimental conditions, such as the Helmholtz experiment, but remains hidden in everyday life. After all it should bend straight lines, such as edges of houses or lampposts, if these lines intersect the peripheral areas of the visual field. An answer is provided by the already discussed mechanism of saccadic image formation (Sect. 7.4): an image of the environment is formed via the saccades, during which the attention is always selectively directed to the immediate vicinity of the targeted object. At the same time, it is a property of visual distortion, that it does not show up in the central, foveal region, but in the peripheral areas of the visual field. Whenever the eye directs its gaze to a straight edge, this edge intersects the centre of vision and thus remains straight. The brain composes its overall image of the environment from selectively recorded individual frames, which consistently contain straight edges, so that the presence of such a slight amount of barrel distortion remains hidden. Yet, the Helmholtz experiment is not the only situation in which the aforementioned barrel distortion of perception can become noticeable. When a binocular, which itself is designed to be free of rectilinear distortion, is panned, then the distortion of the visual space modifies the optical flow (Sect. 7.5) of the image points that are continuously moving in front of the observer’s eyes. As a result, the optical illusion of a curved image arises during the process of panning – a phenomenon, which is known as the globe effect and discussed in Sect. 8.11.

References 1. König, A., Köhler, H.: Die Fernrohre und Entfernungsmesser. Springer, Berlin (1959) 2. Kühl, A.: Zur visuellen Leistung von Fernrohren. Z. Phys. 51, 429 (1928) 3. Berek, M.: Zum physiologischen Grundgesetz der Wahrnehmung von Lichtreizen. Instrumentenkunde 63, 297 (1943) 4. Merlitz, H.: Berek’s model of target detection. J. Opt. Soc. Am. A 32, 101 (2015) 5. Matchko, R.M., Gerhart, G.R.: ABCs of foveal vision. Opt. Eng. 40, 2735 (2001) 6. Pokorny, J., Smith, V.C.: Colorimetry and color discrimination. In: Boff, K.R., Kaufman, L., Thomas, J.P. (eds.) Handbook of Perception and Human Performance, Volume I. Wiley, New York (1986) 7. Lauinger, N.: What the human eye tells the brain: a new approach towards a hardware-based modelling of mental functions. Proc. of SPIE 6764, 676406-1 (2007) 8. Kohler, I.: Experiments with Goggles. Sci. Am. 206, 62 (1962) 9. von Helmholtz, H.: Handbuch der physiologischen Optik. 2nd edn. Voss, Leipzig (1890) 10. Oomes, A.H.J., Koenderink, J.J., Doorn, A.J., de Ridder, H.: What are the uncurved lines in our visual field? A fresh look at Helmholtz’s checkerboard. Perception 38, 1284 (2009) 11. Slevogt, H.: Zur Definition der Verzeichnung bei optischen Instrumenten für den subjektiven Gebrauch. Optik 1, 358 (1946) 12. Merlitz, H.: Distortion of the visual field (2012). www.holgermerlitz.de/globe/test_distortion. html 13. Wagner, M.: The geometries of visual space. Lawrence Erlbaum Associates, Mahwah (2006) 14. Merlitz, H.: Distortion of binoculars revisited: does the sweet spot exist? J. Opt. Soc. Am. A 27, 50 (2010)

Part III

Binocular Performance and Its Evaluation

Chapter 8

Eye and Binocular: The Man-Machine

8.1 Luminous Flux and Magnification As previously exercised in Sect. 7.1.1, we imagine an observer, looking at a distant, circular target. That target has an apparent angular diameter of .σ (arc minutes), and a luminance of .Lt . The (uniform) background has a luminance of .Lb , yielding a contrast of C=

.

L t − Lb . Lb

(8.1)

The luminous flux from the target that enters the optical instrument is proportional to the target’s luminance, its apparent area, which is proportional to the square of its solid angle, and the area of the instrument’s entrance pupil, .∼ D 2 , so that Φ ∼ Lt σ 2 D 2 .

.

(8.2)

Ideally, the luminous flux that exits the instrument would be the same, if the loss of light due to reflections on the lens surfaces, absorption inside the glass elements and a vignetting of the ray fan due to baffling remains negligible. Modern binoculars are capable of transmitting more than 90 % of the incoming light (on axis, that is) to their exit pupils, which is commonly specified with the transmission coefficient .T > 0.9. The observer places her eye pupil (of diameter .de ) on the exit pupil of diameter .d = D/m, where m stands for the magnification. As long as .de ≥ d, the luminous flux exiting the instrument enters the eye without vignetting, otherwise the effective diameter of the light beam is stopped down to the diameter of the eye pupil. In that case, the effective exit pupil of the instrument is reduced to the eye pupil diameter .d ' = de , which in turn leads to a reduced effective entrance pupil (objective aperture) of .D ' = mde . In other words: whenever a telescope (or binocular) is placed

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Merlitz, The Binocular Handbook, https://doi.org/10.1007/978-3-031-44408-1_8

143

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8 Eye and Binocular: The Man-Machine

in front of the observer’s eye, and the eye pupil diameter is smaller than the exit pupil diameter, then a corresponding fraction of the instrument’s light collecting capacity is lost. Figure 8.1 sketches the situation in which both, exit pupil and eye pupil, have identical diameters. This situation is sometimes denoted as standard magnification and not solely a property of the instrument: depending on ambient light, the observer’s pupil diameter varies, and hence the validity of .de = d is subject to external conditions. Usually, spotting scopes have exchangeable eyepieces which allow a selection between different magnifications. The standard magnification would thus be significantly higher in daylight than in twilight or during the night. Binoculars, however, are usually equipped with permanently mounted eyepieces and – unless being zoom oculars – with fixed magnification values. In daylight, the eye pupil diameter is often smaller than the exit pupil, so that the instrument would actually require a higher magnification to yield matching eye- and exit pupil sizes. That is why the binocular is now used in under-magnification mode (Fig. 8.2). Given the magnification m, the image of the target’s solid angle is expanded by a factor m, and its area by a factor .m2 . Let .Φ be the luminous flux when the target is seen with the bare eye, and .Φ ' the flux that enters the eye after having passed through the instrument, then we obtain their ratio (once again neglecting transmission losses inside the instrument) as .

Fig. 8.1 Standard magnification: the exit pupil diameter d of the eyepiece (left) matches the eye pupil diameter .de

Fig. 8.2 Under-magnification: in daylight, the eye pupil is contracted and often narrower than the exit pupil. Apart from transmission losses, the object appears as bright as when seen with the bare eye

Lt σ 2 D '2 Φ' D '2 = = 2 = m2 . 2 2 Φ Lt σ de de

(8.3)

8.1 Luminous Flux and Magnification

145

Fig. 8.3 Over-magnification: in low light, the eye pupil diameter often exceeds the exit pupil. In this situation, the target appears enlarged, but less luminous than seen with the unaided eye

Since the target area scales with the same factor .m2 as does the luminous flux, its perceived luminance remains unchanged: during daytime, the target, seen through a (perfect) binocular, would be as bright and contrasty as if it were observed with the bare eye.1 This simple calculation also implies that no well-resolved object can ever appear brighter through a binocular than with the bare eye; it just appears larger and – obviously – with an increased level of detail. In twilight, or during the night, the eye pupil diameter .de is commonly wider than the exit pupil diameter d of the binocular (Fig. 8.3). In order to match both diameters, the instrument would have to reduce its magnification to the corresponding standard magnification .ms = D/de , and because its actual power is higher than that, the instrument is now used in over-magnification mode. In this case, the ratio of both luminous fluxes (instrumental vs. bare eye) becomes .

Φ' D2 = 2 = m2s . Φ de

(8.4)

The target area, however, is still expanded by a factor .m2 , so that its luminance drops by the factor .m2s /m2 , which makes its image appear darker than seen with the unaided eye. Note that its contrast to the background remains unchanged, since the latter is losing intensity by the same factor. At some point, the background appears black, because the ability of the eye to adapt to its luminance is exhausted, and from that point onwards, the perceived contrast as well is dropping with the magnification. Note that these considerations are valid only for well-resolved targets, which grow in size with the magnification. The perception of fixed stars under the night sky, which would remain point-like regardless of m, will be discussed in Sect. 8.6. An over-magnification may help discerning additional details on targets that are still sufficiently bright. As mentioned above, the eye reaches its maximum resolution at pupil diameters between 2 mm–2.5 mm (Sect. 6.5), and a sufficiently bright target reveals its maximum level of detail when the exit pupil diameter is of similar

1 Imperfections of the binocular, such as light loss, stray light and optical aberrations, are compromising that performance and will be addressed in Sect. 8.4.

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Fig. 8.4 Booster: the Nikon EII binocular, combined with a .6 × 30 monocular, yields a formidable .48 × magnification and an exit pupil diameter of just 0.625 mm

.8 × 30

dimension or even smaller than that in case of self-luminous targets such as the moon or the bright planets. Binoculars are significantly inferior – optically – to the best astronomical telescopes and approach a diffraction-limited resolution only in exceptional cases. This should be taken into account when these instruments are occasionally mounted and equipped with a booster – a monocular that is attached to one of the binocular’s eyepieces (Fig. 8.4). With such a configuration, the resulting magnification turns into the product .m1 · m2 of the magnifications of both instruments, and the exit pupil reduces to .D/(m1 · m2 ). Such a high magnification is often close to, if not beyond, the maximum usable power, and since the optical design of a binocular cannot be expected to be optimised to applications with boosters, the resulting image usually turns out to be far from being perfect.

8.2 Performance: Resolution The resolving power or resolution of an optical instrument generally refers to its capability of displaying fine structure. The test chart shown in Fig. 8.5 is a typical example for regular patterns of high contrast, which are employed to test the resolution limits of binoculars. The closer the bars, the higher the number of lines per unit length, or their spatial frequency. A magnification m of a visual instrument does not necessarily imply an identical increase of limiting resolution .R ' over the resolution R of the unaided eye. This is why the resolution-based efficiency .Er of the optical instrument is introduced and defined as the ratio Er =

.

R' , R

(8.5)

in which the resolutions .R ' , R are given in terms of their respective limiting spatial frequencies. With a perfect instrument and the absence of information loss due to atmospheric conditions, this efficiency could approach the magnification m, while, in real-life situations, it consistently remains below that limit.

8.2 Performance: Resolution

147

Fig. 8.5 USAF 1951 test chart: blocks of bars are reproduced at varying spatial frequencies to determine the limiting resolution of the instrument. Source: Alemily on Wikimedia Commons under CC BY-SA 2.5

During the 1940s, extensive studies with volunteers were conducted under the guidance of Zeiss opticians H. Köhler and R. Leinhos, which led to the following generalised approximation to the efficiency [1, 2]: ( Er = m

.

1−2x

D de

)2x Tx ,

(8.6)

in which D is the entrance pupil (usually the objective’s) diameter, .de the eye pupil diameter, T the light transmission of the instrument and x an empirically determined parameter. The test data were analysed following measurements on Landolt rings (Fig. 7.1), and the results suggested that the parameter x was a non-trivial function of the ambient luminance L. In terms of an approximative solution, three regimes were determined for that parameter: .x ≈ 0 during daylight and the brighter phases of twilight, as long as .L > 0.3 cd/m2 . Then, a crossover regime of .0.003 cd/m2 ≤ L ≤ 0.3 cd/m2 , in which .x ≈ 1/4 holds, and finally .x ≈ 1/2 in low-light conditions of .L < 0.003 cd/m2 . Equation (8.6) is based on another assumption: the eye pupil diameter .de is assumed to be as large (or larger) than the exit pupil diameter d of the binocular. This restriction is easily relaxed with the introduction of the effective exit pupil diameter ' ' .d : its value equals d, as long as .de ≥ d, and .de otherwise. In other words: .d is the smaller one of both pupils. The effective entrance pupil diameter is then easily

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8 Eye and Binocular: The Man-Machine

determined as .D ' = md ' . With this modification, we obtain the three luminance regimes Er ∼ m

.

(

' )1/2

Er ∼ m

d de

Er ∼ m

d ' 1/2 T de

T 1/4

daylight.

(8.7)

twilight.

(8.8)

night

(8.9)

In daylight, the binocular efficiency is proportional to the magnification m (assuming a securely mounted, flawless instrument and ideal atmospheric conditions). The transmission does not show up here – presumably, the eyes are capable of adapting to a moderate reduction of the luminous flux with a slight expansion of the pupil diameter, without any significant impact on resolution. In twilight (.0.003 cd/m2 ≤ L ≤ 0.3 cd/m2 ), the situation turns rather complex. Assuming a sufficiently wide eye pupil, i.e. .de > d, implying the effective exit pupil diameter of .d ' = d, and neglecting the transmission T , we arrive at ( Er ∼

.

mD de

)1/2 .

(8.10)

The eye pupil .de is a variable that belongs to the individual observer and may be omitted, too. This yields the well-known twilight index as defined by the industry norm DIN ISO 58386, Zt =

.

√ mD ,

(8.11)

in which the objective lens diameter D is given in millimetres. Most binocular manufacturers add this twilight index to their specification sheets, to indicate the performances of their glasses under twilight. The classical .8 × 42 then yields .Zt = 18.3, and the .10 × 42 reaches .Zt = 20.5. Should the exit pupil size exceed the eye pupil – a common situation with night glasses – then the twilight binocular efficiency changes to .Er ∼ mT 1/4 , which is, as in daylight, proportional to the magnification, with the addition of the transmission factor which no longer remains negligible. It is a well-known empirical fact that the optical instrument profits from a high transmission particularly when the ambient light levels turn low. Note that these relations do not explicitly account for optical vision effects like the Stiles–Crawford Effect (Sect. 6.8), but since they have been found empirically and hence gone through a validation procedure with test persons, these effects are possibly absorbed into the x parameter. At night (.L < 0.003 cd/m2 ), the eye pupils are fully dilated, usually beyond the exit pupil diameter, so that it is rather safe to set .d ' = d. This yields .Er ∼ DT 1/2 /de , and the efficiency now depends linearly on the objective diameter – ’aperture rules!’ is a popular battle cry among amateur astronomers. The term .de in the denominator

8.3 Performance: Handheld vs. Mounted

149

does not imply that the instrument’s performance drops with the eye pupil diameter. Instead, its performance gain, measured in relation to the unaided eye, is dropping when the latter increases. With a night glass, the opposite situation may √ arise, in which the exit pupil diameter exceeds the eye pupil. Here, .Er ∼ m T , the efficiency is proportional to the magnification m, just as in daylight, but augmented with a considerable dependence on the transmission factor. The twilight index asks for an in-depth analysis in the following sections of this book. During the 1950s, this index had been approved as an international industry norm, after a considerable promotion by Zeiss. Yet, its validity remained questionable ever since. After all, a .12 × 42 binocular, not particularly known for its low-light qualities, achieves .Zt = 22.5, while the full-fledged .8 × 56 night glass – highly popular among European hunters who observe from stationary hides during entire nights – scores a lower 21.2 points. This apparent contradiction will guide us to another, alternative approach to binocular performance, which will be derived in Sect. 8.4.

8.3 Performance: Handheld vs. Mounted Once an optical instrument is used hand-held, a performance loss occurs as a result of jitter. Image-stabilised binoculars have been designed to minimise efficiency losses (Sect. 4.7), but the majority of binoculars is still lacking such a functionality, which is expensive to implement and tends to compromise the ruggedness of the device. Unsteadiness of the image arises from two sources, which operate in separate frequency ranges: first, a muscular tremble, creating jitter within a frequency range of 6–12 Hz with an amplitude of typically 0.25°. Image-stabilising techniques are focusing on the elimination of this high-frequency shake. Second, a lowfrequency drift of 1–2 Hz and large amplitude, which arises from motions of the torso, continuously undertaken to keep the body in balance. The latter motions are effectively eliminated once the observer is sitting down or leaning against a solid support. D. Vukobratovich has studied the implications of jitter to the performance of handheld binoculars and compiled the following empirical formula [3]: .

Eh 1 . = Em 1 + 0.05m

(8.12)

Eh is the efficiency of the hand-held, and .Em of the mounted binocular. The accuracy of this relation was tested on available experimental data, taken with binoculars between 3.5.× and 18.×. His equation is plotted in Fig. 8.6: already at low powers of .6×, the resolving power of the handheld device has dropped to 77 %, and when approaching magnifications of .20×, the binocular performs rather like a mounted device of .10× power.

.

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8 Eye and Binocular: The Man-Machine

Fig. 8.6 Efficiency: handheld vs. mounted, as a function of magnification. The upper scale denotes an effective magnification, which accounts for the loss of resolution due to jitter

These are empirical facts, extracted from data sets that were obtained with test persons in the field. Yet, the experienced user may hesitate to acknowledge their validity in daily life: most binocular users seem to agree that binoculars of lower power, such as .7× or less, are conveniently hand-held without any perceived difficulties arising from jitter. On the other hand, the majority of observers regard binoculars of .12× magnification or above utterly useless for handheld applications. A sharp turn appears to take place at about .10×, separating the hand-held from the mounted regime of magnifications. We are going to hit discrepancies of this kind between synthetically derived test results and daily life experience. This is the case due to a fundamental feature of human perception: visual perception results from a tremendously complex body of data processing, and no simple laboratory setup is ever able to encompass the entire range of parameters that contribute to the impression of visual performance. Applications in the field are not restricted to the identification of letters on test charts, and it makes a difference whether an observation is carried out in a relaxed mood and at ease or in the framework of an experimental setup which requires the observer’s full attention and concentration. As a side note, investigations have been done to analyse how the weight of a binocular affects its hand-held efficiency. In fact, since it contributes inertia to the hand-arm chain, its mass is suppressing parts of the high-frequency jitter. Once the weight is passing over a certain threshold, however, the additional work is tiring the muscles and subsequently increasing their tremble. It is no surprise to learn that the optimum weight turns out to depend strongly on the physical fitness of the test subject, as much as on other factors like level of exhaustion, coffee consumption or altitude of the observation site. As a rule of thumb, the average binocular user may profit most from binoculars of roughly 800 g weight.2

2 Walter

Besenmatter, personal communication.

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151

8.4 Performance: Target Sighting The resolution-based efficiency .Er refers to observations of objects with finestructured textures – after all, it was designed around data sets drawn from observations of Landolt rings (Fig. 7.1). However, the nature enthusiast is often confronted with wildlife that is perfectly adapted to its environment, and the daily battle of survival has equipped wild animals with well-camouflaged outfits. Such targets do neither offer fine-structured details nor high levels of contrast that are part of the paradigms the resolution tests are based on. Instead of asking: what is the finest detail visible in the binocular?, the question turns into: is the object still visible under the available ambient luminance? The fundamental difference between these approaches has been addressed before in Sect. 7.1.2: object visibility is often not a matter of resolution, but rather of contrast. In low light, the wild boar, barely 10 m away, may remain undetected, while a fixed star, though grossly under-resolved, shines brightly above the landscape. It is therefore instructive to first analyse the threshold contrast C, at which a given target becomes visible to the unaided eye. In a second step, the corresponding threshold .C ' is determined, which applies to the image of this object, as seen through an optical instrument, and finally to define Ec =

.

C C'

(8.13)

as the contrast-based efficiency gain of that instrument.3 The evaluation of C is conducted with Berek’s theory of vision (Sect. 7.1.2). To evaluate .C ' , the same model is now applied to the virtual image of the object, which yields [4]: /

/ .

T ' 1 C = μ σ m'

/ φ(μLa ) + μLa

b(μLa ) . μLa

(8.14)

Once again, .σ stands for the objective angular diameter of the target (in arc minutes), and .La for the adaptive luminance of the unaided eye. We assume that the eye is well adapted to the scene luminance, and that the binocular modifies that luminance by the factor .μ, its visual light transmission. Berek writes .μ = T + ν, a composition of useful transmission T (the ‘signal’) and unwanted stray-light .ν (the ‘noise’). Instead of the magnification m, the effective magnification .m' = md ' /de is employed, scaled with the ratio of the effective exit pupil diameter .d ' (Sect. 8.2) and the eye pupil diameter. Note that .m' = m whenever .d ' > de , i.e. whenever the eye pupil diameter is smaller than the exit pupil. Berek’s empirically derived functions for the characteristic luminous flux .φ(L) and the characteristic luminance 3 Since with the aid of an instrument, the threshold contrast for object sighting is dropping, we have defined .Ec = C/C ' rather than .Ec = C ' /C, so that a high numerical value implies high efficiency.

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b(L) have been discussed in Sect. 7.1.2. Since the object is now observed through the instrument, they are functions of the adaptive luminance .La , multiplied with the light transmission .μ of that instrument. The implicit assumption is that the observer’s eye adapts to the modified flux that enters through the eyepiece. We assume that the target is sufficiently small and faint so that its light doesn’t blind the eyes and the adaptive luminance remains equal to the background luminance .Lb . After combining Eqs. (8.13), (7.5) and (8.14), we obtain the contrastrelated efficiency, defined as a gain in threshold contrast:

.

( Ec = T

.

√ 1√ σ φ(Lb ) + b(Lb )

√ 1 √ σ m' φ(μLb ) + b(μLb )

)2 .

(8.15)

To compute the effective magnification .m' , the eye pupil diameter is required, which we take from Eq. (6.1), assuming a young observer of age 30. The apparent angle of the (circular) target amounts to .σ = 1 arc minute, just barely resolvable to the unaided eye in bright daylight. Figure 8.7 displays the results for different binoculars with magnifications between .7× and .12×, and objective diameters from 20 mm to 56 mm. The vertical lines separate the ambient luminance regimes as defined by Köhler et al. (Sect. 8.2): daylight (.Lb > 0.3 cd/m2 ), night (.Lb < 0.003 cd/m2 ), and the twilight regime in between. For the computation of these curves, a light transmission of .μ = 0.9 was

Fig. 8.7 Threshold contrast gain for a young observer, when sighting a target of 1 arcmin apparent angle. The ambient luminance varies between almost complete darkness (left) and bright daylight (right). The .8 × 56 binocular (red) scores higher than the .12 × 42 (blue) under twilight conditions

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153

assumed for each binocular, stray light was neglected (by setting .ν = 0 and .μ = T ), and the subjective angle of field (which affects the eye pupil diameter) was set to 60°. At high luminance levels, the binocular performance is determined solely by its magnification, in close coincidence with the resolution-based efficiency of Eq. (8.7). In fact, this also follows from Ricco’s law (Sect. 7.1.1): the higher the power, the larger the target is seen through the ocular and the lower its threshold contrast. Since the eye pupils are well contracted, the exit pupil diameter consistently exceeds the eye pupil diameter, all binoculars are operating in under-magnification mode and the luminous flux that enters the eye from the target is always proportional to the square of the magnification. The situation changes at very low light levels: here, the eye pupil diameter commonly exceeds the exit pupil diameter and the binocular is operating in overmagnification mode. The luminance of the target, as seen through the instrument, is now proportional to the objective diameter. Hence, the threshold contrast gain is exclusively a function of D, once again in close agreement with previous results obtained for the resolution-based efficiency (Eq. (8.9)). Significant differences between both approaches to binocular performance emerge in the crossover regime between day and night. In twilight, the .8 × 56 (red curve) performs consistently higher than the .12 × 42 (blue curve), which dominates in daylight, but rapidly loses ground when the luminance levels drop. This appears to be in close agreement with empirical observations in the field. By contrast, the twilight index of Eq. (8.11) would suggest a higher performance of the .12 × 42 throughout the twilight regime. In Sect. 8.5 we are going to analyse the reasons behind this apparent contradiction. Figure 8.7 also confirms how poorly a compact .8 × 20 binocular performs, once the light level drops. With its small exit pupil of 2.5 mm, it is rapidly drifting into over-magnification mode, in which it produces a reduced contrast and a dull image. This reflects a general trend, observable also with larger instruments: as soon as the observer’s eye pupil begins to exceed the exit pupil diameter, the threshold contrast gain ceases to grow. The secret behind the performance of a night glass is its large exit pupil, which shifts this turnover point deep into the low-light regime. Figure 8.8 summarises the results obtained with the same binoculars as in Fig. 8.7, but for an observer of about 60 years. Since the eye pupils are no longer able to expand to sufficiently wide diameters, the performance of night glasses with wide exit pupils is not fully exploited. The .7 × 50 binocular performs precisely on the same level as the .7 × 42 (both curves are on top of each other), and the .8 × 56 (red curve) is clearly inferior to the .10 × 56 format, located even below the .10 × 50. A somewhat higher magnification may therefore be applied to compensate for the reduced eye pupil diameters. The ideal handheld binocular for low-light observations has now become the .10 × 50 or the .10 × 56, rather than the .8 × 56 which suits the younger observer. Berek’s theory of contrast-based binocular performance offers a variety of additional features: threshold-distances can be computed to determine (and compare) the maximum distance at which a target of given contrast remains visible. The

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Fig. 8.8 Threshold contrast gain obtained with the same instruments, but for an older observer (about 60 years of age). The .8 × 56 binocular (red) delivers a less stellar performance in twilight, and may be replaced with the .10 × 56 (blue) as a powerful low-light instrument

influence of coating technology (which affects light transmission and stray-light) on binocular performance can be studied, and in Sect. 8.6 we will demonstrate how this approach may be adapted to apply to astronomy and predict the limiting magnitudes of detectable stars. Yet, a couple of implicitly made assumptions, on which the performance model is based, should be called into attention: the thresholds of visibility of a target are evaluated while assuming that this target is already well in focus of a binocular, which itself is firmly mounted and pointing into the target’s direction. In real-life applications, neither the precise direction nor the distance of the object is usually known. Binoculars with wide angles of field and high depths of field (Sect. 8.9) would then support the observer’s task of sighting the target. Apart from that, any atmospheric seeing effects, which affect the contrast of a distant target, are neglected. These factors will be addressed in Sect. 8.7.

8.5 Which Approach to Binocular Performance is Most Relevant? For daylight and night, the resolution-based approach to binocular performance of Sect. 8.2 and the contrast-based efficiency model of Sect. 8.4 lead to similar results. In twilight, however, alarming discrepancies are visible: the resolution-

8.6 Night Sky Performance

155

based approach scores binoculars of higher magnifications higher, whereas the contrast-based approach prefers large objective diameters. To understand the origin of this apparent contradiction, we have to recall in which way the raw data, which led to the respective performance models, were obtained: Köhler and Leinhos made test persons identify fine detail on Landolt rings, while Berek’s visual data were obtained during field tests, in which test persons had to sight targets near the threshold contrast. As previously discussed in Sect. 6.6, human vision undergoes a transition from daylight vision (photopic vision, conducted by the cone cells) to low-light vision (scotopic vision of the rod cells), and the crossover regime during twilight is known as mesopic vision. Cone cells are particularly centred around the foveal area of the visual field, which leads to peak resolution values during daylight, while the rods occupy the extra-foveal areas of the retina and are often connected to provide higher sensitivity at the cost of resolution. Resolution-based visual experiments force the test person into a foveal vision mode, in which she attempts to discern the finest possible details. In such a situation, an over-magnification of the instrument (as it commonly occurs with most binoculars in twilight) would still generate a performance gain, as long as the contrast of the objects on the test chart remains sufficiently high. For the sighting of a target near its threshold contrast, however, mesopic vision may partially resort to extra-foveal vision to reap the higher sensitivity of the rod cells. In this case, when resolution becomes secondary to brightness, an increase of the aperture, and hence additional light, yields better results than an increase in magnification. It is therefore necessary to consider the context in which the observations are carried out: if a birder has to read out codes on ringed birds in low light, then she has to employ foveal vision to identify the numbers. In such a situation, the twilight index (Eq. 8.11) offers valuable guidance for the perfect choice of her instrument. If, on the contrary, hunters are about to locate a well-hidden deer between the underbrush or astronomers are seeking for a faint, diffuse comet after sunset, then the contrast-related binocular efficiency of Berek offers superior solutions. Binocular performance is therefore not a priori a well-defined quantity, since it requires the context, in which the observation takes place, to be set.

8.6 Night Sky Performance 8.6.1 Stellar Magnitudes The night sky is not entirely dark: in a moonless night, the background luminance of the sky typically varies between 3 × 10−4 cd/m2 and 3 × 10−3 cd/m2 , while for moonlit nights, higher values between 3 × 10−3 cd/m2 and 0.1 cd/m2 are common [5]. The deep sky astronomer, who is searching for faint diffuse nebula and galaxies, may apply the method of threshold contrasts, as discussed in Sect. 8.4, to evaluate the performance of her telescope or binocular on these objects. Quite generally, the performance reaches its maximum when the optical instrument operates

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at standard magnification, implying that the exit pupil diameter should be about the same as the eye pupil diameter. Occasionally, an over-magnification of such objects is suggested as a recipe to increase their contrast, and this claim is justified with the reduced background luminance of the sky at higher magnifications. However, the luminance of the object would be reduced by the same factor, leaving its contrast to the background invariant. What changes during extended, uninterrupted observation periods is the state of adaption of the eye, but this advantage is insignificant under properly dark skies, under which deep sky observations would commonly take place. For limiting magnitudes of stars, the situation is different. If we compare two randomly selected stars, then their difference in brightness is a result of different luminous fluxes .Φ1 and .Φ2 that enter the eye, as defined in Eq. (8.2). Following the peculiar properties of visual perception of light intensities, and in the tradition of an ancient Greek classification scheme, astronomers measure the brightness of an object on a logarithmic scale, in which the magnitude difference is defined as ( m2 − m1 = 1001/5 log

.

Φ2 Φ1

) (8.16)

and expressed in units of mag. The pre-factor .1001/5 ≈ 2.512 is defined in such a way that a magnitude-difference of 5 mag corresponds to a factor of 100 between the two luminous luxes, with higher stellar magnitudes indicating fainter stars. The absolute scale is calibrated such that the brightest stars have magnitudes about 1, and the faintest stars that are visible to the bare eye in sufficiently dark nights reach between 6–7 mag – a quantity also known as the visual limiting magnitude. Often, a poor transparency of the atmosphere, or the vicinity to cities restrict the limiting visual magnitudes to values below 5 mag. The naked-eye limiting magnitude is also a decisive parameter for the limiting magnitude achieved in combination with the optical instrument.

8.6.2 Astro Indices for Limiting Magnitudes During extensive field studies, Ed Zarenski has determined the limiting magnitudes of a considerably large set of binoculars [6] and summarised his findings in Fig. 8.9. The data suggest that it is not only the aperture that decides on the performance, but the magnification as well. The Adler index, proposed by Alan Adler and defined as the product of magnification and the square root of the objective diameter, IA = mD 1/2 ,

.

(Adler index)

(8.17)

is in close agreement with Zarenski’s limiting magnitudes. For example, a .10 × 50 binocular would yield .IA,1 = 70.7, while the larger .20 × 100 device would score

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157

Fig. 8.9 Limiting magnitudes of stars (in mag) with binoculars, as a function of the limiting magnitude of the unaided eye (figure courtesy of Ed Zarenski, CN Report: Limiting Magnitude in Binoculars, 2004, www. cloudynights.com)

IA,2 = 200, and thus a performance gain of .200/70.7 = 2.83. Translated into magnitudes, this performance gain would then amount to

.

log(IA,2 /IA,1 ) log(2.83) = = 1.13 mag . log(2.512) 0.4

A(mag) =

.

(8.18)

As Beat Fankhauser has pointed out, the scaling behaviour of the Adler index violates basic physical principles [7]: Obviously, the .20 × 100 binocular collects four times as much light as the .10 × 50 device. Assuming identical quality features such as transmission, that factor of four would apply to the flux of light exiting the binocular, too. Additionally, both binoculars have identical exit pupil diameters, so that on the perceptional side of the equation all other parameters would remain the same. Naturally, the performance gain of the .20 × 100 over the .10 × 50 binocular should exactly amount to four, and the gain in limiting magnitudes therefore 1.51 mag, rather than 1.13 mag, as predicted by the Adler index. Fankhauser correctly observed that every physically consistent performance index would have to satisfy the general scaling law I = (ma D b )2/(a+b) ,

.

(8.19)

with an arbitrary choice of the two exponents .(a, b), in order to conserve the energy flux that passes through the instrument. He proposed a choice of .a = 2 and .b = 1, to achieve a dominance of the magnification over the aperture, just as the Adler index, but this time with the correct scaling law IF = (m2 D)2/3 .

.

(Fankhauser index)

(8.20)

Another alternative approach, commonly known as visibility factor [8] and suggested by Roy Bishop, applies equal weights to both magnitude and aperture, with the choice .a = 1 and .b = 1, yielding IV = mD .

.

(Bishop index)

(8.21)

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But which of these indices would be most accurate? Is there a way to approach this problem from a rather scientific angle, based on models of human perception? As it turns out, Berek’s model may indeed be applied to compute limiting magnitudes of stars.

8.6.3 Limiting Magnitudes in Berek’s Model We begin with Eq. (7.5) of Sect. 7.1.2, and note that, because of the tiny apparent angle of the target, its first term clearly dominates over the second term, so that the latter may be omitted. After taking the square we obtain the threshold contrast C=

.

φ(La ) , σ 2 La

(8.22)

in which we set the adaptive luminance equal to the background luminance .Lsky of the night sky. The contrast between star and sky is then .C = (Lstar − Lsky )/Lsky , and the luminance of a barely detectable star amounts to Lstar =

.

φ(Lsky ) + Lsky . σ2

(8.23)

At this point, a few peculiarities that apply to the detection of stars have to be considered: the apparent angular size of any star disk is far below the resolution limit of both the unaided eye and the optical instrument. As a result, what is seen is never the star disk itself, but its diffraction disk, and when assuming a reasonably good imaging of the optical instrument, then the diameter of the star’s image on the retina is invariant of the magnification. It always corresponds to a circular object at the resolution limit of the bare eye, roughly 1 arc minute. Consequently, the target angle .σ in Eq. (8.23) will be set to unity in the remaining parts of this section. The luminance of a star seen through the eyepiece is proportional to the luminous flux that passes through the instrument, thus .∼ D 2 , and the area of its image is a fixed quantity. As long as the light enters the eye without losses, i.e. as long as the exit pupil diameter d remains smaller than the eye pupil diameter .de , the instrumental threshold luminance should scale as .L˜ star = Lstar /(D/de )2 , because the amount of light on the retina increases by the factor .(D/de )2 . If, on the other hand, the exit pupil diameter exceeds the eye pupil diameter, then some of the light collected by the instrument is wasted, and we once again use the effective magnification .m' , being identical to .m' = m if .d > de , or else .m' = D/da . With these preparations, we arrive at a preliminary solution, ? Lstar , L˜ star = (m' )2

.

(8.24)

hastingly adding that one contribution is still missing: contrary to the star image, the background luminance of the night sky is in fact a function of the magnification.

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159

Once the exit pupil is smaller than the eye pupil, the background turns increasingly dark as the exit pupil diameter shrinks. This leads to ˜ sky = Lsky .L

(

m' m

)2

( =

D da m

)2 .

(8.25)

The following facts have to be considered here: when compared to the unaided eye, the instrument collects an amount of background light that is higher by a factor 2 2 .(D/da ) . That light is then spread over an area that is larger by the factor .m , thus defining the instrumental luminance level of the background. If astronomers are patiently gazing into the eyepiece, then their adaptive luminances approach the further darkened background of luminance .L˜ sky , allowing her to discern stars of somewhat lower intensities. The correct threshold luminance then turns into L˜ star =

.

φ(L˜ sky ) + L˜ sky , m'2

(8.26)

and we define Berek’s astro index as the inverse of that threshold luminance, yielding IB = L˜ −1 star =

.

m'2 φ(L˜ sky ) + L˜ sky

.

(Berek index)

(8.27)

Next we compare the indices of two binoculars with objective diameters .D2 = 2D1 and magnifications .m2 = 2m1 . Both have identical exit pupils and thus yield identical background luminances to the observer’s eye (assuming identical transmission values). This leads to the ratio of their performance indices, .IB,2 /IB,1 = D22 /D12 = 4, consistent with the conservation of the luminous flux. Given the rather general case of non-identical exit pupil diameters, the situation is rather complex, since the evaluation of the specific luminance function .φ(L˜ sky ) is now required. This function is non-linear (Fig. 7.3) and does not yield any simple power-law function of D and m, as proposed in Eq. (8.19). The luminous flux is still conserved, but the perception of brightness differs with the level of retinal adaptation, so that the conservation law is no longer manifest in the general form of Berek’s astro index, Eq. (8.27). This is a striking example for a simple, physics-based performance law, Eq. (8.19), which is altered (and significantly complicated) by the features of human perception, once the observer is coupled into the image chain.

8.6.4 Comparison with Observation Data For a perfect .16 × 70 binocular, we obtain the Berek index of .IB = 139.2. Assuming the limiting magnitude of the bare eye to be 6 mag, we would gain .log 139.2/ log 2.152 mag = 5.36 mag with the instrument and arrive at the limiting magnitude of 11.36 mag. Taking Fig. 8.9, we find that Ed Zarenski reported a

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limiting magnitude of 11.18 mag with his .16 × 70 binocular, which seems sufficiently close to our theoretically derived value, when considering that no instrument is perfect in terms of transmission, stray light and absence of aberrations. We note that another assumption has entered our calculation, namely that the eye reaches its limiting magnitude of 6 mag in combination with the background luminance of 0.001 cd/m2 . To avoid pitfalls that may be encountered with these kinds of assumptions, it is advisable to normalise the set of observations to the data obtained with one particular instrument and to compare only relative performances, i.e. gains and losses with respect to that reference binocular. We choose the Fujinon .16 × 70 FMT-SX of Zarenski’s data set, which is likely to be the instrument of highest quality among the test samples. To evaluate the pupil size of the observer, we apply Eq. (6.1), assuming a young observer and a subjective angle of field of 60° for all instruments.4 In Fig. 8.10, the theoretical predictions are plotted over the observational data of Zarenski. It is interesting to note how closely the predictions of the different models agree with the observations and with one another: although deviations exist, they generally remain quite moderate, and it is hard to decide which of these astro indices actually performs best. This fact is all the more puzzling because we have already shown that the Adler index violates fundamental principles and thus should not be expected to remain accurate. The indices of Fankhauser and Bishop have different scaling exponents and thus should deviate in their predictions, too. An explanation for the surprising similarity of the various astro indices is found after the introduction of the exit pupil diameter .d = D/m into the general index of Eq. (8.19), yielding Fig. 8.10 Limiting magnitudes, derived with various theoretical models (normalised to the data point of the .16 × 70 Fujinon) and plotted over the observational data of Zarenski (Fig. 8.9) at a bare-eye limiting magnitude of 6 mag. The diagonal defines the identity of observation and theory

4 Differences in the angles of field would lead to tiny modifications of the results, which are conveniently neglected.

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161

I = m2 d 2b/(a+b) .

.

(8.28)

Expressed in this way, the entire set of possible indices, which is spanned over the parameter space of a and b, exclusively differs in their exponents of the exit pupil diameter: .2b/(a + b) = 1 (Bishop), or 2/3 (Fankhauser). We then derive (

I . log m2

)

( ) = mag(I) log (2.512) − log m2 ) ( = log d 2b/(a+b) ,

(8.29)

and the rescaled plot, .mag(I) · 0.4 versus .log d, then produces lines with the slopes 2b/(a + b). This plot is shown in Fig. 8.11: The magnitudes evaluated with the Bishop index (blue) are aligned along the curve of slope 1, the predictions of Fankhauser (green) of slope 2/3 and Berek’s predictions lie on a (non-linear) curve with an approximate slope of .≈1.25 (black curve). In this plot, the indices, due to their different slopes, would yield clearly distinguishable predictions, but once the observational data are added (red stars), it becomes clear, why no discrimination between the models is achieved: the data points scatter significantly, and hardly any well-defined slope can be extracted. A linear regression would yield the slope .≈1.18 ± 0.25 (dashed red line), including a significant uncertainty. The range of exit pupil diameters was limited to .3.75 mm ≤ d ≤ 7 mm, which, in combination with the scatter, is insufficient to derive a slope and thus a well-defined power law of the form .∼d x . The accuracy of the observational data would have to be increased significantly to allow for a clear discrimination between the different astro indices – a difficult task, considering the unavoidable quality differences between the samples. On the other hand, Fig. 8.10 indicates that, for practical purposes, the accuracy of both observations

.

Fig. 8.11 Scaled limiting magnitudes as a function of the exit pupil diameter. The wide scatter of the observation data (red stars; linear interpolation shown as dashed red line) prohibits a reliable discrimination between the various theoretical models

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and theoretical models is actually reasonably high, so that any one of the existing astro indices would be sufficiently accurate to estimate limiting magnitudes. In this light, the simplest approach, i.e. the Bishop index, might in fact be preferable.

8.7 Atmospheric Scatter and Seeing In optical design, it is possible to account for those parts of the optical path that lead through air by simply including the air’s refractive index (which is almost identical to .n = 1 of the vacuum) and ignoring any further effects of that medium. These simplifications are no longer possible when the optical path between the instrument and a distant object is concerned. Let us first recall the previous analysis of resolution-based efficiency of Eq. (8.6), which was based on a couple of simplifying assumptions: the instrument was assumed to be tightly mounted to avoid the information loss due to hand shake, as described in Eq. (8.12). Additionally, the test chart was supposed to be located sufficiently close to the observer, so that any effects of atmospheric haze (extinction) or seeing were negligible. Obviously, the latter conditions are impossible to satisfy once the target to be observed is located at far distances. Atmospheric extinction generates a loss in contrast which can be approximated with the Lambert-Beer law. We first introduce the visibility, the horizontal distance .Lh over which an object is visible to the bare eye. More specifically, the object is assumed to be extended and darker than the background (sky). The Lambert–Beer law then yields C(L) = C0 exp(−σ L) ,

.

(8.30)

where .σ is the damping or extinction coefficient, and .C0 the contrast in complete absence of extinction. The limiting visibility condition is defined as a contrast .C(Lh ) of 2 %. If we assume .C0 = −1 (a perfectly black target), which at the limiting visibility distance has to drop to .C(Lh ) = −0.02, then we obtain 0.02 = exp(−σ Lh )

(8.31)

3.91 ln 50 ≈ . Lh Lh

(8.32)

.

or σ =

.

On an average clear day, the visibility distance is typically .Lh ≈ 20 km, yielding an extinction coefficient of .σ ≈ 0.20 km−1 . Once the new, reduced contrast of the target against its background is evaluated, Berek’s theory of vision (Sect. 8.4) may be applied to evaluate the conditions of its visibility, with or without instrumental support.

8.7 Atmospheric Scatter and Seeing

163

So far we have assumed that the atmospheric extinction phenomena are indifferent to the spectral wavelength. This is approximately correct as long as water droplets (mist, fog) or pollutants such as aerosols (dust) are concerned. Yet, particularly on clear days, colourful phenomena are abundant: the sun is setting in bright orange or red colours, while the sky is blue. This is the result of light being scattered on atmospheric molecules, generally known as Rayleigh-scattering, which exhibits a strong dependence on the wavelength: photons of high frequency (blue) are subject to larger scattering angles than low-frequency photons on the red side of the spectrum. The spectral transmission of the atmosphere due to Rayleigh scattering is known as: ) ( 0.0084(μm)4 . .τ (λ) = exp − λ4 cos z

(8.33)

Here, z is the angle between the line of sight and the zenith, and .λ the wavelength. Figure 8.12 displays the spectral transmission curves for three different angles z. The transmission is highest when this angle is zero, because in this case the light path passing through the atmosphere is comparably short (roughly corresponding to 8 km air under normal pressure). The optical path increases with z, and so does the extinction. The dependence of Rayleigh scattering on the wavelength is significant: While red light (700 nm) is able to pass the atmosphere almost unhindered, blue (450 nm) and violet (400 nm) are scattered into all directions, thus creating the deep blue colour of the clear sky. For the same reason, the spectral composition of twilight differs considerably from its composition in daylight: It is shifted toward the short-wavelength end of the visible spectrum – with consequences for the spectral sensitivity of human vision (Fig. 6.7). Scattered light is not lost, but redirected. Visible light of short wavelengths thus contributes the bulk of the haze that limits long-distance visibility of objects on moderately clear days. This is why distant mountains appear to fade behind a bluish veil. Light-yellow filters, which truncate the transmitted spectrum at typically 420 nm and thereby eliminate the violet and a part of the blue light, are helpful tools, Fig. 8.12 Spectral transmission of the atmosphere due to Rayleigh-scattering, for different angles z between zenith and line of sight

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Fig. 8.13 Kowa .8.5 × 44 Genesis (Field of view: 122 m/1000 m) with a screw-in 46 mm camera filter

which somewhat increase the contrast of distant objects, and are thus among the standard gadgets of military binoculars. Threaded hoods in front of the objectives allow the attachment of camera filters, which are available in a plethora of variations (Fig. 8.13). Unfortunately, only a small subset of binocular makers are implementing these threads, which might alternately take polarisation filters (to reduce reflections during observations over water), deep sky filters for the astronomers, or detachable extended hoods for observations in difficult light conditions. Astronomers are well familiar with another atmospheric effect, commonly known as seeing: stars are twinkling, and at higher magnifications, their images are ‘dancing’ randomly about their average positions. The origins of this phenomenon are random fluctuations of the index of refraction in unsteady air. Local temperature gradients, but also high winds in the upper layers of the atmosphere, which lead to turbulences and density fluctuations, affect the optical density of the medium that lies between object and observer. A good overview of these effects is offered by Yoder et al. [9] and may be summarised as follows: we imagine taking a photo of the star with an exposure time of several seconds, during which the image is smeared out to form a seeing disk of diameter .ds . Its size is related to the Fried parameter .r0 , which, from a physical point of view, is the coherence length of the atmosphere, i.e. the average distance over which a wavefront propagates without being knocked out of phase. The Fried parameter is a function of the angular distance z from the zenith and the wavelength .λ, following r0 ∼ (λ cos z)6/5 ,

.

(8.34)

and in the zenith, it is related to the seeing disk via r0 =

.

0.98λ . ds

(8.35)

8.8 Visual Transmission, Colour Contrast and Saturation

165

Note that .ds is given in radians, and one arc second corresponds to .≈ 4.84 · 10−6 rad. If, at night and with good seeing, we would obtain a seeing disk of one arc second (while observing at .λ = 550 nm), then a Fried parameter of .r0 ≈ 11 cm would result. The Fried parameter leads to a rule of thumb for the maximum value of the aperture, that fully exploits the resolution of the optics: scopes with apertures D smaller than .r0 are potentially diffraction limited (or rather, aberration limited). The images of objects are scattering around mostly in lateral movements as long as the aperture remains below .D = 3.7r0 , and the visual observer may still be able to compensate for that motion just by tracing the position of that jumping point. However, once .D > 3.7r0 , then the image begins to move also in longitudinal direction along the light path and to jump out of the image plane, so that a blur emerges that can no longer be compensated for. Since during daytime the Fried parameter assumes values of 20 mm–40 mm, it hardly restricts the usability of handheld binoculars, though mounted spotting scopes of larger dimensions usually fail to reach their optimum performance under these conditions. Seeing conditions turn to the worse on sunny days, when objects have to be observed via a horizontal path that passes over heated surfaces. The turbulences above heated roofs or squares may reach intensities that limit the performance of even the smallest instruments. The only workaround would be to search for a better observation site, preferably at an elevated location, which increases the angle between the line of sight and the unsteady layer of air above a heated surface.

8.8 Visual Transmission, Colour Contrast and Saturation The colour contrast of an optical instrument is, just as the luminosity contrast, strongly affected by stray light. Anti-reflection coatings of highest quality and properly placed baffles, which protect the optical train against intruding reflections, are mandatory elements for a perfect colour contrast and colour saturation which the birder requires to distinguish between certain species. The perceived brightness of an image, as well as a possible colour bias, are a product of three equally important factors, the combined action of which eventually determines the character of the final image: the spectral composition of the ambient light (Fig. 7.6), the spectral response of the individual eye (V-lambda curve), and the spectral transmission of the optical instrument. The black curve in Fig. 8.14 displays the measured transmission of a military Zeiss (Jena) .7 × 40 EDF (Fig. 9.2) as a function of the wavelength. For comparison, average human V-lambda curves, valid in daylight (photopic vision, red) and in advanced twilight (scotopic vision, blue) are also plotted. The transmission maximum of the EDF is located at a wavelength of about 530 nm, indicating a compromise for a binocular that is supposed to be used during day and night. The maximum transmission of almost 85 % represents a score that is typical for a binocular of the 1980s with a three-layer multi-coating.

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Fig. 8.14 Spectral transmission of the Zeiss Jena .7 × 40 EDF (black), with the average human V-lambda curves of photopic vision (daytime, red) and scotopic vision (nighttime, blue). Transmission data with kind permission: Gijs v. Ginkel

Remarkable, however, is the significant drop of the transmission curve in the short-wavelength region below 480 nm. In deep violet, the curve dives even below the 50 % threshold. The V-lambda curves indicate that this drop occurs in a spectral range which adds little weight to the integrated visual transmission. With .T (λ) being the instrument’s spectral transmission curve, the visual transmission is defined as the weighted integral { Tv,i =

.

T (λ) Vi (λ) dλ ,

(8.36)

where the index .i ∈ {p, s} stands for either the daylight (photopic) or the low light (scotopic) response curve. It is this visual transmission which determines the perceived brightness of the image. For daylight use, the binocular reaches a visual transmission of .Tv,p = 82.4%, and in low light, the integral yields .Tv,p = 81.6%. These are the two values which should be quoted by the manufacturer, rather than the popular, but less telling transmission maxima (here: 84.6 %). Although the drop of the transmission curve has no considerable impact on the brightness of the binocular, there is something that remains invisible in these numbers: the image of the EDF has a distinct yellow tint – a consequence of its low transmission inside the violet spectral regime. The lack of violet implies an over-representation of its complementary colour, and for humans that is yellow. This binocular is therefore rather incompatible with the birder’s requirements of a colourunbiased image. The reason behind its extraordinary transmission behaviour are cerium oxide additives to the SF3R flint optical glass, which improved the resistance of this military binocular against nuclear radiation damage. Figure 8.15 shows transmission spectra of two modern .8 × 42 binoculars, equipped not only with modern multi-coatings, but also with selected glass types of particularly high transmission. The Schmidt–Pechan prisms of the Leica Ultravid are coated with a dielectric mirror of high reflectivity (Sect. 3.2.5), while the Zeiss Victory contains Abbe-König prisms which exclusively employ total internal reflections (Sect. 3.2.2). An application of Eq. (8.36) to their transmission curves yields for the Ultravid .Tv,p = 89.4% (daylight) and .Tv,s = 86.5% (low-light transmission), and for the Victory .Tv,p = 93.7% and .Tv,s = 91.4%, respectively.

8.8 Visual Transmission, Colour Contrast and Saturation

167

Fig. 8.15 Spectral transmission curves of two modern binoculars (data with kind permission: A. Olech, www.Allbinos.com/ Optyczne.pl)

The high transmission values of the Zeiss are clearly discernible in real-life observations, especially in a direct comparison with the Leica. Despite the fact that the transmission curves in Fig. 8.15 are not flat, but raising towards higher wavelengths, the binoculars do not exhibit any obvious colour bias, which implies that a constant transmission throughout the visible spectrum is not among the necessary conditions for the perception of neutral colour. These facts are well known to the manufacturers of e.g. light tubes, the spectra of which are far from being flat, yet shining apparently white. Rather interesting to note is another phenomenon: most of the experienced binocular users seem to agree upon the impression that the Ultravid offers a superior colour saturation and thus a superior colour contrast. It may be speculated that the high transmission of the Leica binocular near the long-wavelength end of the spectrum is responsible for that feature. This section of the spectrum may help adding ‘punch’ to images of objects that predominantly contain warm colour tones like red, orange or brown.5 The question then arises whether or not a particular modulation of the transmission curve would boost the colour contrast of the perceived image even further, without introducing any colour bias. Carefully designed transmission filters might be placed in front of the objectives to achieve such a goal (Fig. 8.13). It has to be acknowledged that many issues regarding human colour perception still remain unsolved. Theoretical approaches such as Berek’s model of vision (which is restricted to luminosity contrasts) have not yet found their counterparts in the field of colour perception. One reason for this shortcoming is the parameter space, which is of considerably higher complexity, since all possible mixtures of colours would have to be covered. Another, perhaps equally serious problem is related to significant differences between individual observers, which prevents the scientists from drawing accurate and universal conclusions from their field tests. Eventually, the combined action of three colour sensors (cones), with partially overlapping spectral responses, and the subsequent data processing inside the visual cortex, with emergent phenomena such as the simultaneous colour contrast (Sect. 7.2), are the root of the difficulties in arriving at a quantitative understanding of human colour perception. 5 The

author thanks Stephen Green and Mathias Metz for insightful input to these considerations.

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8.9 Depth of Field The perceived depth of field of a binocular (or telescope) is determined by both the depth measure of its optics (Sect. 2.3) and the accommodation width of the observer (Sect. 6.4): the former determines the virtual distance, to which the eye has to accommodate in order to get the image into focus, and the latter decides whether such an accommodation is possible. We consider a binocular that is focused on infinity, and an observer with perfect long-distance vision (or, alternately, with her vision aids on), then the minimum distance of an object, which can still be accommodated to, amounts to Emin =

.

m2 . δacc

(8.37)

We remind that m is the magnification of the instrument, .m2 the corresponding depth measure, and .δacc stands for the accommodation width, being the inverse of the closest distance (in metres) at which the vision remains sharp. For example, a young observer may have an accommodation width of 12 dpt, implying that every object beyond 1/12 meter can be focused on. Through a binocular of magnification .m = 8, this minimum accommodation distance is moved away by the factor .m2 , yielding .64/12 dpt = 5.3 m. On the other hand, an older observer with .δakk = 1 dpt would only be able to get objects beyond .64 m in focus, while observing through the same binocular. Actually, this picture is somewhat simplified, since a (mathematically) perfect focus is not required to regard an image as being sharp. A certain degree of fuzziness, defined in terms of the circle of confusion, and expressed in units of arc minutes, is tolerable without considerable loss of perceived image quality. Albert König [2] has defined the maximum tolerable angular diameter of that circle of confusion to be 3.4 arc minutes.6 If for this choice of the circle of confusion we compute the focal tolerance, then an additional margin of .1/de dpt to the depth of field arises, on both sides, the close- and the far distance limit. Here, .de stands for the diameter (in millimetres, not metres!) of the observer’s eye pupil. The validity of this approximation is restricted to pupil diameters beyond 2 mm, since otherwise diffraction effects would become dominant. In combination with the optical instrument it is once again the effective exit pupil diameter .d ' , which is of relevance, and defined as the smaller one of both, the exit pupil and the eye pupil. To avoid any confusion between different length units, we here regard .d˜ as a unitless parameter, which assumes the numerical value of the effective exit pupil in millimetres. Due to the circle of confusion, the binocular doesn’t have to remain focused on infinity to get distant objects into focus. Instead, the hyperfocal distance Eh (in metres) = m2 d˜

.

(8.38)

6 Note that the maximum resolution of the unaided eye under ideal conditions is on the order of a single arc minute, considerably higher than 3.4 arc minutes.

8.9 Depth of Field

169

replaces the infinity setting and in this way extends the range of distances in which objects appear sharp: now that the binocular’s focus is set to a closer distance, the minimum accommodation distance approaches by yet another .1/d˜ dpt, and we obtain its new value as Emin =

.

m2 δacc + 2/d˜ dpt

.

(8.39)

For an observer with a particularly high visual acuity, the assumption of a 3.4 arc minutes wide circle of confusion may appear too lax, and a sharpening to half of its value would consequently halven the dioptric shift in Eq. (8.39) to .1/d˜ dpt instead of .2/d˜ dpt. Figure 8.16 displays the accommodation ranges of binoculars, which are set to the hyperfocal distance (dashed curve), as a function of magnification. Two observers of different ages and different accommodation ranges experience depths of field which differ in their minimum accommodation distances. For example, a 25-year old observer with 10 dpt accommodation width takes her .8 × 42 binocular, focuses on the hyperfocal distance of 192 m, and is able to accommodate to objects between 6 m and infinity. In contrast, an observer of 50 years of age and with

Fig. 8.16 In-focus areas of binoculars that are focused onto the hyperfocal distance (dashed curve), as a function of magnification. All objects between the minimum distance and infinity appear in focus, and the former depends on the accommodation width of the observer, as demonstrated for a young observer with .δakk = 10 dpt and an older observer with .δakk = 2 dpt. Further assumed is an effective exit pupil of three millimetres

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reduced 2 dpt accommodation range sets his .12 × 50 binocular to the hyperfocal distance of 432 m and gets every object between 54 m and infinity in focus. In all calculations we have assumed an effective exit pupil diameter of .d ' = 3mm, a typical value during daylight observations. At low ambient light, the observer’s pupils expand, and so does the effective exit pupil, which now adopts the value of the binocular’s exit pupil diameter. With .8 × 42, and .d ' = 5.25 mm, the young observer now experiences a depth of field between .Emin = 6.17 m and infinity, the older, equipped with his .12 × 50, a range between .Emin = 58 m and infinity. These ranges differ marginally from their daylight values, indicating that – contrary to the dominating influence of the magnification – the pupil size contributes little to the depth of field of visual optical instruments. Finally, we have to generalise these arguments to all distance settings of the binocular – the user will obviously focus onto objects at arbitrary distances, with consequences for the respective depths of field. Actually, this generalisation is easier than it seems: whenever a binocular’s focus is set to a certain distance, then the virtual images of objects at this particular distance are at infinity. Due to the tolerable circle of confusion, there exists a margin beyond this focus distance, in which objects would still appear sharp, and the accommodation width of the eye then adds a certain range of distances in front of that object that can be accommodated to. Using the depth scale .m2 , we then simply have to translate these virtual distances back into object space. This leads to the range of distances E of .

m2 m2 < E < , m2 1 1 m2 + dpt − dpt δacc + Efoc Efok d˜ d˜

(8.40)

in which objects appear in focus, once the binocular is focused on the distance .Efoc . The denominator of the left term contains a sum of three dioptre values: The virtual infinity, .m2 /Efok , the accommodation width, and the margin .1/d ' due to the circle of confusion. To the right, the denominator contains the same term, .m2 /Efok , while here the margin due to the circle of confusion is subtracted because it extends the depth of field to larger distances. Here, .δacc does not contribute because the eye would at .Efok arrive at its infinity setting without any possibility to accommodate beyond. Figure 8.17 displays the ranges of objects in focus, calculated for a magnification of .8× and an effective exit pupil of .d ' = 3 mm, as a function of .Efoc . At close distances, the depth of field gets very narrow, and once again, the minimum distance depends strongly on the accommodation range of the observer. The far distance (red curve) does not depend on .δacc , and runs towards infinity when the binocular’s distance setting approaches the hyperfocal distance. We may summarise our findings, pointing out that the perceived depth of field of a binocular is strongly dependent on its magnification and the observer’s accommodation width, but only marginally dependent on the pupil size diameter

8.10 Depth Resolution and Cardboard Effect

171

Fig. 8.17 Ranges of objects that appear sharp, as a function of the instrument’s distance setting, at magnification. The far distance limit (red curve) does not depend on the accommodation width of the observer, while the close distance point varies significantly with .δacc (i.e. observer’s age). The hyperfocal distance, beyond which the depth of field extends to infinity, is at .192 m

.8×

(or ambient light). We further remind the reader on what has been discussed before in Sect. 2.3: the depth of field does not depend on any additional factors, including the design of an instrument’s optical layout. If a manufacturer happens to praise the outstanding depth of field of his product, then he is either selling a binocular of particularly low magnification or trying to deceive the potential buyer.

8.10 Depth Resolution and Cardboard Effect In Sect. 7.3 we have discussed human depth resolution – the ability to distinguish between distances of objects by stereoscopic vision. In combination with the binocular, this ability to enjoy a three-dimensional view is improved considerably. First of all, the magnification enhances angular differences between incoming rays by a factor m. The lateral retinal disparities of images, and thus the ability of human vision to perceive depth, are enhanced by the same factor. Additionally, the increase of baseline separation – a result of an axis offset caused by the image erecting system that modifies the separation of the objectives – enhances stereoscopic√vision. This is particularly significant with prisms of Porro I type (axis offset: . 2 times the entrance width w), less so with Porro II systems (equal to w), and marginally so

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Fig. 8.18 Stereoscopic depth resolution of the Zeiss (Oberkochen) .10 × 50, an arbitrary .10 × 50 roof-prism binocular without axis-offset, and the .7 × 40 RISO-I (Fig. 8.19)

with certain variants of the Abbe-König prism. The stereoscopic depth perception of the unaided eye is thus boosted by the factor K=m

.

objective separation . eyepiece separation

(8.41)

As an example, the Zeiss .10 × 50 binocular of Fig. 3.5 has an objective separation (baseline) of .b' = 140 mm, when the eye distance is set to 65 mm, yielding a boost in stereoscopic depth resolution by a factor .K = 21.5, and the Porro II type Ross .10 × 50 of Fig. 3.7 yields .K = 16.9. Naturally, a roof-prism model without axis offset would contribute only its magnification to the corresponding boost factor. Figure 8.18 displays the resulting depth resolution, as computed by Eq. (7.10), with the following modification to accommodate the performance of the optical instrument: AE =

.

' E 2 αmin , ' b' + Eαmin

(8.42)

where E stands for the distance to the far object, and .AE for the minimum detectable separation between E and .E − AE, the distance of a second object in ' front. .αmin = αmin /m is the minimum detectable retinal disparity, after division by the magnification (angles in radian measure, average values amount to .αmin ≈ 1.45 · 10−4 rad). At a distance of 100 m, the Zeiss .10 × 50 with an objective separation of .b' = 0.140 m offers a depth resolution of 1 m, and at 1000 m distance, the resolution drops to 94 m. A .10 × 50 roof-prism binocular would – in absence of any axis offset and an objective separation of .b' = 0.065 m – yield a depth resolution that is inferior by about a factor of two. This is certainly among the reasons why the military has commonly relied on Porro-prism designs, despite of their disadvantage in terms of bulk. For naval military applications, binoculars with enhanced stereoscopic vision have occasionally been designed. Figure 8.19 shows the .7 × 40 RISO-I stereoscopic binocular, made in Japan and used by the US Navy during the Korean War. In this instrument, planar diagonal mirrors implement an axis offset of a factor

8.10 Depth Resolution and Cardboard Effect

173

Fig. 8.19 RISO-I .7 × 40 stereoscopic binocular with an objective baseline of 255 mm (at eye-distance setting of 65 mm (field of view: 133 m/1000 m). The eyepieces are located near the central upper part of the photo

of 3.9, so that the stereoscopic boost factor (Eq. (8.41)) over the unaided eye amounts to .K = 27. As Fig. 8.18 demonstrates, its gain in depth-resolution over a .10 × 50 Porro binocular remains moderate, due to its lower power of .7×. Mounted rangefinders with baselines of several meters and far superior depth resolution have routinely been employed by coast guards who had to measure precise distances to approaching vessels. Instruments with enhanced stereoscopic vision, including Porro-prism binoculars, generate an interesting optical illusion, which is generally known as the cardboard effect. Based on daily-life experience, human perception interprets an enhanced stereoscopic cue as a close distance to the motif. Through the binocular, the images of objects of distance E display a depth resolution that resembles objects at distances of .E/K. At the same time, the perspective proportions of those distant objects remain unaffected by the magnification, and consequently they appear strangely deformed. In particular, the extent in depth is disproportionate and apparently squeezed by a factor of m – an effect, which is well known to photographers when shooting with long tele-lenses. The combination of these perspective distortions with the enhanced depth cue leads to the cardboard effect, which makes the landscape appear like a collection of flat objects assembled from cardboard motifs. Another consequence of these mutually contradicting visual cues is the impression of objects being shrunk in size – after all, their distances seem reduced, and to fit into the field of view, they appear to have reduced their natural dimensions. The impression that a Porro-prism binocular seems to magnify less than a roofprism binocular of identical power, is sometimes attributed to that cardboard effect. Of course, magnification and hence resolution of details remain unaffected by stereoscopic vision, as is easily verified in comparative measurements. In summary, the enhancement of stereoscopic vision through beam-axis offsets comes along with both advantages and problems. The increased depth resolution is payed for with disproportionate landscape motifs, and the wide objective spacing prevents a close-focus distance that is required to observe insects or other small animals (Sect. 4.3.2). Yet, the particular impression of plasticity, which the Porro

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binocular offers, and which has been referred to as true ‘3D-imaging’, can be a pleasant experience, since it demonstrates the abilities of human binocular perception to form a spatial image of its surroundings.

8.11 Distortion and Globe Effect In 1827, G.B. Airy defined the condition under which an optical instrument was free of rectilinear distortion [10]. This tangent condition has been discussed before in Sect. 2.1.5; it relates the angle A of an object (with respect to the line of sight) to the angle a of its image (to the centre of field): .

tan a = m tan A ,

(8.43)

with m being the magnification. Before 1950, most binoculars were designed, whenever possible, closely according to this recipe, and their images thus were – as per definition – free of rectilinear distortion. Surprisingly, user complaints persisted about an unpleasant feature when the binoculars were panned. This globe effect, also known as rolling ball effect, makes the image appear rolling over a convex surface. Users have reported side effects of this optical illusion, ranging from minor dizziness to symptoms of motion sickness. In military applications, the emergence of unnatural motion patterns during panning was running the danger of masking true movements in the field that was under surveillance, and had to be addressed accordingly. The origin of this apparent rolling motion of an image, which – when steady – seemed distortion free, remained unclear. Zeiss designer H. Köhler reported about unnatural perspective shifts among motifs that were scattered over a threedimensional landscape [11], but he failed to notice that the same effect has also been observed under the night sky, where any effects of perspective shifts could be excluded. Only recently, the author of this book has pointed out a possible connection between the globe effect and the barrel distortion of the visual field (as discussed in Sect. 7.6): computer simulations have shown that the amount of visual barrel distortion would suffice to generate the impression of a globe effect with some individuals, in perfect absence of any instrumental distortion [12]. Figure 8.20 displays the optical flow field of image points, when barrel distortion enters the image chain, and in this case it is the visual perception, not the instrument, which is the source of that distortion. According to simulation results, a globe effect may arise once the visual distortion parameter, as introduced in Eq. (7.14), assumes values below .kv ≤ 0.8, and Fig. 7.14 suggests that slightly more than half of the population might be susceptible to this phenomenon, though the perceived intensity of this illusion would differ substantially.

8.11 Distortion and Globe Effect

175

Fig. 8.20 Optical flow during a horizontal panning motion, in the presence of barrel distortion (schematic)

With the static binocular, the globe effect remains invisible, for the same reason why the barrel distortion of the visual field remains unnoticed by the observer: although straight edges of objects bend outward, this distortion occurs in the peripheral part of the visual field and is virtually invisible. Its presence is verified, however, with the help of Helmholtz-chequerboards (Fig. 7.13). In dailylife situations, the image formed in the visual cortex is a superposition of numerous patches which are taken during saccadic scans of the environment, during which predominantly central retinal areas, which are free of distortion, are employed. As a result, the static image appears distortion free, and only the moving image, which rolls in front of the eye, reveals the global barrel distortion of the visual field. The question then arises why such a distortion remains invisible when turning the head. This is so because during motions of the head, the eyes remain fixated onto a certain motif for a short time, and then jump onto the next motif. This involuntary fixation during head movements is known as the vestibulo-ocular reflex, which prevents a continuous motion of the image and thus a perception of any curvature of the visual field. When observing through a binocular, then the speed of the moving image is increased by a factor of the magnification, and under these conditions the fixation of the line of sight on individual motifs is easily overcome, the image is flowing continuously in front of the eyes, and the optical flow of Fig. 8.20 – whenever such a visual distortion is present – becomes obvious. How can the globe effect be avoided? The short answer is: when the optical instrument includes a small amount of pincushion distortion, which compensates for the individual amount of visual barrel distortion, then the globe effect is eliminated. Unfortunately, Fig. 7.14 proves that barrel distortion is not a universal feature of human perception, but varying individually in its quantity. Yet, as will be discussed

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in the next section, reasonable compromises are possible, which effectively reduce the globe effect to such an extent that it remains unnoticed by the vast majority of binocular users.

8.12 The Search for the Ideal Distortion Curve The globe effect can be eliminated with the help of a pincushion distortion, which is deliberately implemented into the binocular. To prove that claim, we start with the generalised distortion relation, Eq. (7.11), .

tan(ka) = m tan(kA) ,

(8.44)

and solve for the angle in the virtual image, 1 arctan [m · tan (kA)] . k

a=

.

(8.45)

Once again, .k ∈ [0, 1] is the distortion parameter that defines the quantity of pincushion distortion: .k = 1 yields the tangent condition which is free of rectilinear distortion, and .k = 0 corresponds to the angle condition, which is free of angular magnification distortion (AMD) but comes with rectilinear pincushion distortion. Now we include the visual space of human perception (Sect. 7.6): in visual space, the image point is no longer perceived at an angle a, but it turns into a distance to the centre of field that has to be computed from a using Eq. (7.14), yielding y=

.

} { kv 1 arctan [m tan (kA)] . tan kv k

(8.46)

It is easily verified that in the special case of .k = kv , i.e. equality of instrumental pincushion distortion and visual barrel distortion, this equation simplifies to y=

.

m tan (kv A) , kv

(8.47)

and the angles of the objects, A, are usually small (less than 5° in most binoculars), so that we safely approximate y≈

.

m (kv A) = mA ≈ m tan A . kv

(8.48)

We remember that y is the radial coordinate of the perceived image point in visual space, and the relation above resembles the tangent condition, but in visual space, in which y has replaced the tangent of the half angle a of the original tangent condition. This resemblance implies that the perceived image is now free of (rectilinear)

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177

distortion, as long as the observer keeps fixating the centre of field. The absence of distortion in the visual field ensures a homogeneous (divergence-free) optical flow field during panning and hence an elimination of the globe effect. Note that we have used a small-angle approximation in Eq. (8.48). It can be shown numerically that a compensation requires .k < kv when the exact equation is applied, but .k ≈ kv remains sufficiently accurate for magnifications .m ≥ 7, which are common with binoculars (with a notable exception of opera glasses). It was the scientific work of August Sonnefeld in 1949, which made binocular designers question their design paradigms [13]. Since 1950, Zeiss began to add a certain quantity of pincushion distortion to their binoculars, on the order of the angle condition, to avoid the globe effect. Soon thereafter, most other manufacturers followed Zeiss’ example. However, in the early 21st century, some binocular makers, notably the Japanese Nikon and Kowa, and since about 2010 also parts of the European competition, have begun to offer binoculars with lower distortion levels. In particular the introduction of the .8.5 × 42 EL SV by Swarovski reanimated the discussions about distortion and its consequences on the panning behaviour. The question is how much of pincushion distortion is needed to suppress the globe effect to a negligible level, without allowing unwanted side effects of pincushion distortion to deteriorate the static image: whenever the observer fixates objects with straight edges in the peripheral areas of the field, then the pincushion distortion is bending their contours inwards. Note that a compensation of visual barrel distortion and instrumental pincushion distortion is possible only as long as the direction of view remains fixated on the centre of field, but binoculars, particularly if mounted, are also used with the eye swivelling around to observe offcentre objects. The relative distortion of the image as a function of the k parameter can be evaluated after solving Eq. (8.44) for A, and insertion into Eq. (2.11), which then yields Vr =

.

tan a ( )] − 1 . tan(ka) 1 arctan m tan k m [

(8.49)

This somewhat cumbersome expression may be simplified, since .tan(ka)/m is usually sufficiently small to allow the expansion of the trigonometric functions .arctan() and the outermost .tan() to first order, yielding Vr ≈

.

k tan a −1, tan(ka)

(8.50)

in which the magnification has vanished. The angle condition, .k → 0, then leads to .Vr ≈ (tan a)/a − 1, which contains a substantial amount of pincushion distortion. Computer simulations, conducted with different values of the visual distortion parameter, indicate that the choice .k ≈ 0.7 would be close to the ideal value to eliminate the globe effect for a vast majority of observers. Figure 8.21 shows that the residual amount of pincushion distortion would then be about half

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Fig. 8.21 Relative distortion as a function of the subjective half-angle, at different values of the distortion parameter k. The angle condition (black), frequently implemented since the 1950s, leads to distortion values roughly twice as high as the choice .k = 0.7 (red), which the author has proposed to be close to the ideal compromise. A moustache distortion (blue) yields an unpleasant panning behaviour Fig. 8.22 Soviet military binocular BPO .7 × 30, produced by KOMZ (Field of view: 150 m/1000 m). Its moustache distortion (as shown in Fig. 8.21, blue dashed curve) produces a particularly pronounced globe effect

as large as with the rather traditional choice of the angle condition. At least in parts, the optical industry has adopted this proposal as their approach to balance distortion and globe effect in visual optical instruments [14]. Of historical interest is further a suggestion that was based on theoretical studies by Helmholtz [15], and subsequently named circle condition by Slevogt [16]. The circle condition would correspond to a distortion parameter of .k = 0.5, containing a somewhat higher than necessary level of pincushion distortion. In real-life applications, in which the distortion curve .Vr (a) may show a rather complex behaviour, a parameterisation in terms of Eq. (8.44) is not always possible. Consider the distortion curve of the Soviet BPO .7 × 30 binocular of Fig. 8.22, as shown as the dashed blue curve in Fig. 8.21, which is an example for a moustache distortion. Here, .Vr (a) raises initially, following a pincushion distortion in the

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179

central part of the image, and then bends over toward the edge of field to produce a barrel distortion in the peripheral part of the image. The optical flow of the image points during panning alters with their velocities, which are proportional to the gradient of the distortion curve. Variations in the velocities are then interpreted by the visual perception as waves in the image plane, generating an unnatural panning experience. It is still unknown how such a distortion function would have to be shaped in order to optimise the perceived panning behaviour. Yet, the theoretical works by Helmholtz, Slevogt, and the experiments with Helmholtz chequerboards (Sect. 7.6) provide sufficient motivation to model the distortion curves closely around the function of Eq. (8.49). In fact, whenever binoculars of recent make exhibited problems during panning, then these problems could be attributed to either a lack of distortion, or to significant deviations of the distortion curve from the regular form of Eq. (8.49). Finally, there exist exceptions from the general rule of .k ≈ 0.7 for instruments that are predominantly used in a static mode, because they are mounted and rarely used for panning (e.g. instruments used to read out scores on targets at gun ranges). Those may be designed according to the tangent condition (.k = 1) and without any rectilinear distortion. On the contrary, binoculars for astronomy may feature a higher level of distortion, closely following the angle condition (.k = 0, Eq. (2.12)). For a proof, we take Eq. (8.46) in the limit .k → 0 and find y=

.

1 tan(kv a) . kv

(8.51)

This is now identical to Eq. (7.14) which we have derived for the unaided eye, with the sole difference, that the objective half-angle A is replaced with the subjective half-angle a. The view through the binocular then offers a night sky that exactly resembles the individual’s impression of the unaided eye: the observer has the impression of looking into a vault, the stars seem to surround him. This phenomenon is sometimes referred to as the spacewalk effect,7 and often implemented in wideangle eyepieces made for astronomical telescopes. It is important to understand that the angle condition offers an image that is free of AMD: a cluster of stars keeps precisely the same proportions, whether it is at the centre of field or shifted to the edge, as long as the observer’s eyes track its position. This is so, because when looking into a vault, the line of view pierces the image plane normally in any direction (see also Sect. 2.1.4). If instead the image is free of rectilinear distortion (as with orthoscopic eyepieces), star formations appear to be squeezed when the eye follows them toward the edge of field, since now the image plane is tilted against the line of sight. In some sense, the angle condition mimics the impression of a starry sky that is pinned onto the hollow side of a hemisphere, whereas the tangent condition makes the stars behave as if they were pinned onto a flat poster.

7 A term that was probably coined first in 1980 by eyepiece maker Al Nagler when he presented his 82° Nagler ocular.

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The impact of distortion on the properties of visual optical instruments is a complex topic. The fact that the individual response to cues related to distortion is far from being universal may motivate the optics designer to create eyepieces in which the amount of distortion is tunable. In a recent patent, Leica has proposed the idea of making a digital telescope with a processing unit that allows the user to choose the quantity of pincushion distortion in its image [17]. Further more, the instrument would also distinguish between static observation and panning mode and dynamically adjust the amount of pincushion distortion. Distortion levels could be lowered during an inspection of architecture or arts, tuned to moderate levels when observing a horse race while the binocular is panning, and turned high under the night sky, where AMD has to be eliminated.

References 1. Köhler, H. Leinhos, R.: Untersuchungen zu den Gesetzen des Fernrohrsehens. Opt. Acta 4, 88 (1957) 2. König, A., Köhler, H.: Die Fernrohre und Entfernungsmesser. Springer, Berlin (1959) 3. Vukobratovich, D.: Binocular performance and design. Proc. SPIE Int. Soc. Opt. Eng. 1168, 338 (1989) 4. Berek, M.: Die Nutzleistung binokularer Erdfernrohre, Z. Phys. A 125, 657 (1949) 5. Brandt, R., Müller, B., Splittgerber, E.: Himmelsbeobachtungen mit dem Fernglas. Johann Ambrosius Barth, Leipzig (1983) 6. Zarenski, E.: CN Report: Limiting Magnitude in Binoculars (2003). www.cloudynights.com/ 7. Fankhauser, B.: Eine neue Leistungsgröße für Ferngläser. ORION 387, 14 (2015) 8. Spix, L.: Fern-Seher. Oculum-Verlag GmbH, Erlangen (2009) 9. Yoder, Jr., P.R., Vukobratovich, D.: Field Guide to Binoculars and Scopes. SPIE Press Bellingham, Washington (2011) 10. Airy, G.B.: On the spherical aberration of the eyepieces of telescopes. Camb. Philos. Trans. 3, 1 (1827) 11. Köhler, H.: Grundsätzliches zum Fernrohrsehen. Deutsche Optische Wochenschrift 35, 41 (1949) 12. Merlitz, H.: Distortion of binoculars revisited: Does the sweet spot exist? J. Opt. Soc. Am. A 27, 50 (2010) 13. Sonnefeld, A.: Über die Verzeichnung bei optischen Instrumenten, die in Verbindung mit dem blickenden Auge gebraucht werden. Deutsche Optische Wochenschrift 13, 97 (1949) 14. Zeiss patent WO2022034231A2, p. 71 (2022) https://patents.google.com/patent/ WO2022034231A2/en 15. von Helmholtz, H.: Handbuch der physiologischen Optik. 2nd edn. Voss, Leipzig (1890) 16. Slevogt, H.: Zur Definition der Verzeichnung bei optischen Instrumenten für den subjektiven Gebrauch. Optik 1, 358 (1946) 17. Leica patent WO2021160195A1 (2021) https://patents.google.com/patent/ WO2021160195A1/en

Chapter 9

Binocular Evaluation and Field Testing

9.1 Laboratory Tests Let us begin this chapter with a short overview of laboratory tests which are routinely performed by manufacturers as part of their quality control procedures. Internationally agreed testing standards and norms exist, which may or may not be applied during these procedures. Some technical journals hire the service of manufacturer’s labs or independent institutions for their product reviews, in which binoculars are tested for optical performance and mechanical durability. On Albrecht Köhler’s website there exists a list of standards and norms that are used to test optical parameters [1]. They include measurements of the angular magnification, entrance pupil diameter, exit pupil diameter, angle of field, eye relief, suitability for spectacle wearers, dioptre adjustment and zero setting (all as per DIN ISO 14490-1), resolution (DIN ISO 14490-7), image quality (DIN ISO 93363), transmission (DIN ISO 14490-5), veiling glare index (DIN ISO 14490-6) and binocular alignment (DIN ISO 14490-2). Image quality is determined by measuring the contrast transfer function – a sophisticated and costly procedure. The veiling glare index is measured using a large photometer sphere, and to check the usability for spectacle wearers, a special adapter is attached to the eyepiece. Spectral transmission is measured with a spectrophotometer; however, before determining the integrated visual transmission data, the transmission curve needs to be weighted with the spectral stimulus response of the eye (Sect. 8.8), which implies that the finally recorded visual transmission values in daylight (photopic vision) and low light (scotopic vision) differ. The resolution is measured at full aperture of the entrance pupil. As previously discussed (Sect. 2.4.2), such a test should be repeated after the exit pupil has been stopped down to values below 2.5 mm, because the human eye performs best under these conditions, in which the binocular should therefore perform close to

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Merlitz, The Binocular Handbook, https://doi.org/10.1007/978-3-031-44408-1_9

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Fig. 9.1 Not recommended: testing for mechanical durability and water resistance may yield undesirable results (with kind permission: Barry Simon)

its diffraction limit.1 A majority of handheld binoculars have exit pupil diameters above 4 mm, which are exhausted only in low light in which the resolution of the eye remains well below the diffraction limit (Sect. 6.5). Thus, measurements taken at full aperture are often of limited relevance in the field – with the exception of applications in astronomy. Testing the quality of phase-correcting coatings (Sect. 3.2.6) requires two polarising filters, placed at both ends of the telescope tube with their transmission axes orientated parallel or perpendicular to each other [2]. The roof edge has to be aligned with the transmission axis of one of the polarisers. Narrow-band green light is used to assess the differences in brightness of the exit pupils, depending on the relative orientation of the polarising filters: without phase coating, the exit pupil is brighter with the parallel set-up of both polarisers, compared to the perpendicular set-up. It would be the opposite in the presence of an effective P-coating. Tests for mechanical ruggedness and water resistance are an essential ingredient of the design of any high-quality binocular (Fig. 9.1). The manufacturers follow here their own procedures, the details of which are not made public, but we do have the maintenance guide A050/1/501 of the East German NVA, which describes in detail the regular stress tests, to which their standard military binocular, the EDF .7 × 40 (Fig. 9.2), was subjected to: . tightness: implementation of an internal overpressure of 50.7 kPa. Requirement: pressure drop not permissible. . submersible water test: submersion depth: 1 m. Water temperature: 10°C to 15°C lower than unit temperature (resulting in a negative pressure in the unit). Duration: 1 h. Requirement: no water or fogging in the interior. . drop test: Drop height: 0.75 m. Direction: lying broadside with stretched central hinge, one fall.

1 Thanks

to David W.J. Norton for this important remark.

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Fig. 9.2 Zeiss (Jena) .7 × 40 EDF, the standard handheld binocular of the East German NVA (Nationale Volksarmee, Field of view: 131 m/1000 m)

. impact test:2 acceleration 15 g. Pulse duration: 5–10 ms. Direction: 250 strokes on objective standing, 150 strokes on broadside lying with stretched central hinge, 150 strokes lying on narrow side with stretched central hinge. . impact test: acceleration 120 g. Pulse duration: 1–5 ms. Direction: two strokes on objective standing, four strokes on broadside lying with stretched central hinge, four beats lying on narrow side with stretched central hinge. . vibration load: frequency range: 30–80 Hz. Acceleration: 6 g. Duration: 2 h standing on objective lens, 1 h lying on broadside with stretched central hinge, 1 h lying on narrow side with stretched central hinge. . cold resistance: −50°C, duration 2 h, after the equipment has reached the required temperature, rubber parts to −40°C. . heat resistance: 60°C, duration 2 h, after the units have reached the required temperature. Requirement: no grease leakage. . cyclic temperature test: upper temperature: 60°C, lower temperature: −50°C. Duration: 5 cycles per 2 h. Requirement: no grease leakage. . storage temperature: 80°C, duration 1 h, after the units have reached the required temperature. Requirement: no grease leakage. . resistance to sea mist: Temperature: 27°C. Duration: 168 h. Composition: sodium chloride 27 g/l, magnesium chloride 6 g/l, calcium chloride (anhydrous) 1 g/l, potassium chloride 1 g/l. Requirement: no corrosion. The German consumer association, Stiftung Warentest, revealed additional details in its testing brochure (September 2006), featuring tests for resistance to humidity, cold/heat and other environmental factors (DIN ISO 10109-4 and DIN ISO 9022), shock resistance (DIN ISO 58390) as well as abrasion resistance (DIN ISO 58196-4). 2 The unit g does not stand for gram here, but for the acceleration due to gravity. NASA astronauts were subjected to stress tests which included a maximum acceleration of 20 g.

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None of these test methods are usually available to the ordinary binocular user. If the intention is to use a new set of binoculars at very low temperatures, it certainly makes sense to leave the instrument inside a freezer for a night and to check whether the focusing mechanism and other mechanical parts remain operational. If not, unsuitable lubricants, which turn hard in the cold, were applied. This may lead to a temporary failure of the instrument in the field. Apart from these simple manipulations of the environmental conditions, the binocular purchaser does not have many options for testing ruggedness, and instead has to rely on the manufacturer’s tests and the warranties they are granting. Additionally, he may search the Internet for reports from long-term users. Not surprisingly, the purchase of a technical instrument such as a binocular also requires a certain degree of common sense: who demands minimum weight, may not expect the durability of a military device. It is also highly unlikely that a cheap branded binocular has ever enjoyed the treatment of a vibrating table as a part of its design procedure.

9.2 A Quick Test in- and Outside the Store Whoever is interested in purchasing binoculars may not necessarily have the opportunity to take several samples home and evaluate them against one other. It is therefore crucial to arrive at first conclusions, and perhaps to shortlist the lineup of interesting candidates, right in front of the counter of the store. With some experience, it is possible to learn a lot about the properties of a binocular within just ten minutes; this paragraph provides tips and hints about how to conduct quick tests in- or outside the shop.

9.2.1 First Impression: Design, Ergonomics, Haptics More often than not, the first impression sticks. This is a fact long known to advertising psychologists, even if the critical buyer would care to dispute it. ‘Gutfeeling’ prevails over the rational assessment of technical properties.3 A well designed binocular with a high quality appearance can leave a lasting impression on us, and we may thus overlook or ignore technical flaws and imperfections. The most important theme of the following quick tests is therefore to put function above beauty, since the latter will turn out rather irrelevant during practical applications in the field.

3 Consumption

creates subjective well-being and thus has to stay emotional. After: Hans Weigum.

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185

At the moment when the binocular is first picked up, it is advisable to pay attention to the following aspects: . Is it easy to adjust the settings of the instrument, i.e. . . . . . . . is the central hinge neither too lose nor too stiff? . . . . do the eyecups have a fold-down or twist-down mechanism, and do they allow bringing the eyes into a comfortable position behind the eyepieces so that the entire field of view can be seen? . is the field of view sufficiently wide, or does an impression of tunnel vision arise? . does the rim of the eyecup feel comfortable, or does it leave pressure marks on the skin that may turn uncomfortable after extended observations? . does the instrument rest comfortably in the hands, and does it remain well balanced while manipulating the focusing unit? . does the covering material offer a good grip so that the binocular is safely operated even with sweaty hands or gloves? In a next step, the focusing mechanism may be tried, after pointing the instrument towards a convenient target inside the store. Is it possible to operate the focusing mechanism without the need of changing the grip of the hands, so that the binocular, in particular one with higher magnification, remains in optimum position? The central focusing barrel, if present, should turn smoothly and precisely, without any jerk or play. The entire focus travel should be exploited to check whether the torque remains constant, or whether any glitches in the gears emerge. The dioptre setting is tested in the same way. To prevent undesired changes of its setting after an unintentional contact, it should turn with somewhat higher resistance or, alternatively, be lockable. Once a binocular has passed these initial tests, the next sample may be evaluated for the same criteria, or otherwise the second phase of testing as described below may follow.

9.2.2 Checking for Additional Rejection Criteria It is advisable to compile a list of rejection criteria prior to testing, which allows the reduction of the initial lineup of potential candidates at an early stage of the evaluation. If observations of insects at close ranges are required (Fig. 9.3), then those binoculars with a minimum focus distance of 3 m or above are quickly eliminated. Luckily, this can be tested even inside the smallest shop. It should be verified that a close target is observable in a relaxed state of accommodation and without any eye strain. Binoculars of the ordinary Porro type (i.e. not reverse Porro constructions) have a disadvantage here due to the wide separation of their objectives, which leads to a significant parallax when aiming at close objects (Sect. 4.3.2). The shortest focal distances of binoculars with roof prisms often reach 2 m or below, while with Porro prisms as much as 3 m may already become tedious.

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Fig. 9.3 Many users apply their binoculars to observations at close distances, at which the minimum close-focus range represents an important parameter

An observer, who is used to focusing clockwise from close to far distances, may be confused with a focusing mechanism that turns the opposite way. A small pocket lamp serves as an indispensable tool for the quick tester, even if the shop salesman may frown upon its application. It allows checking for visible damages like scratches on the lens- or prism surfaces. Once pointed from below into the eyepiece while simultaneously looking into the objective enables the detection of dust or impurities inside the glass material. It should be noted that a few dust particles or air bubbles inside the glass are rarely a serious issue; they turn relevant only if located in immediate proximity of the focal plane, which usually concerns the field lenses of the eyepieces, or reticles of military binoculars. At this point it is advisable to assess the quality of the anti-reflective coatings (Sect. 2.4.3). The various reflections caused by the flashlight are expected to be colourful and not too bright, i.e. of low intensity. In contrast, a bright white reflection would indicate an uncoated surface. When pointing the light downwards into the tube, shiny metallic parts which could cause stray-light should be absent, the prism entrance should ideally have a surrounding baffle, the flanks of Porro prisms should be covered with a non-reflective layer and their reflective surfaces covered with shields (Sect. 4.6). Ideally, the tubes are coated with matt varnish and supplemented with internal stray-light baffles (Fig. 9.4). Next, a distant target should be located to check for the state of collimation of the binocular. This may be difficult inside most stores, and a permission is required to take the instrument(s) outside and find some easy distant objects, e.g. a churchclock or roof antenna. Any object of distance beyond 1 km may safely be regarded as ‘infinitely far’ away. When adjusting focus, attention should be paid to the amount of extra focus travel remaining beyond the infinity setting. Such a reserve travel should remain to meet the needs of short-sighted observers who want to use the device without their spectacles. After careful adjustment and focusing, an easy and relaxed view onto the distant object should result. In case it is necessary to squint to properly see the object, or if a double image appears, then the two tubes are most likely not properly aligned and the binocular therefore out of consideration.

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Fig. 9.4 View into the objective tube. Left: clone of a military binocular with metallic reflections on the tube walls. Right: stray-light baffles of the Zeiss Jena .7 × 40 EDF (visible reflections are caused by the flashlight on the objective lenses)

If instead a minor degree of de-collimation is suspected, one may proceed as follows: one eye remains closed and the target is observed for a little while using monocular vision. Then, upon opening the second eye, a single image of the object should result instantly. If instead two images are visible, which rapidly merge together, then a sub-standard alignment of both binocular tubes is likely. Even though both eyes manage to find a superposition of both images, the application of such a binocular over an extended period will likely cause headaches or watery eyes. It should be noted that this procedure may not be applied by observers suffering from heterophoria, who would experience a double image even with a flawless instrument.4 An alternative test works as follows: a distant, almost point-like object is brought into focus and into the centre of field. Then the binocular is gradually shifted away from the eyes, carefully keeping the object firmly in sight. During this process, the discernible field of view rapidly shrinks, and when the binocular is 20–30 cm apart from the face, the object should simultaneously remain visible at the centres of both exit pupils. This method requires a steady hand and some experience – it actually works better if the tester moves back and away from a binocular mounted on a tripod. A quick inspection of the exit pupils does also reveal the vignetting of the light beam, which is frequently caused by undersized prisms (Sect. 4.5.1). Figure 9.5 displays how the test is carried out:5 when looking straight towards the eyepiece,

4 Thanks 5 Thanks

to Klaus Müscher. to Heiko Wilkens.

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Fig. 9.5 Vignetting of the edge pupil: the exit pupil is circular if seen face-on (left). But once seen at an angle, at which it touches the lens edge, it becomes partially vignetted (right)

the face-on exit pupil should be perfectly circular and of full size according to the specifications; otherwise, the principal ray fan would be partially obstructed and the instrument’s performance in low light was compromised. Next, the instrument is turned so that the exit pupil is seen from an angle, until it touches the perimeter of the eyepiece lens. This is the exit pupil as it appears close to the edge of field, also called edge pupil (German: Randpupille). Usually, the edge pupil is no longer circular, but obstructed to a cat-eye shape as a result of vignetting of the peripheral rays of the light cone. A moderate degree of vignetting does not significantly affect the performance of the binocular; in fact, it may even prove useful to the optical designer: the stopping-down of rays arriving off-axis hides their aberrations, in particular coma and astigmatism. As a result, the off-centre brightness of the image is somewhat reduced, but its sharpness is increased. Yet, the optical design of binoculars intended to be used in low light should apply vignetting sparingly, since at night, with widened eye pupils, the loss of light in the peripheral parts of the image becomes bothersome and affects the binocular’s performance (Sect. 6.9). Apart from that, a high degree of vignetting of the edge pupil creates negative side effects for the ease of view, because it promotes the perception of a partial shadowing of the image with a swivelling eye. Once a subset of binoculars has passed the selection criteria as described above, a couple of further tests may be performed to evaluate the imaging characteristics of their optics. During the following, one-on-one comparisons of the images of selected pairs will facilitate the selection of the highest performer among the remaining ones.

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9.2.3 Evaluating Optical Performance In order to arrive at definitive and final conclusions about the optical properties of the binocular, extended field tests are mandatory. However, with a certain degree of experience, a first and quite conclusive assessment is actually possible at a trade fair, in a department store or optics outlet, by making good use of the available targets. Perfect test objects are the lit billboards which are abundant at department stores. Focus carefully on such a board in the centre of the field, and then slowly pan the binocular, letting a selected letter or number move through the field of view, and observe how the sharpness varies between the centre and the edges of the image. The edge of the billboard will probably show some degree of colour fringing. While keeping the direction of view fixed on that edge, pan the binoculars again and let the edge move through the centre of field, at which the colour fringes should disappear, and toward the opposite edge, where the fringes will emerge again. These colour fringes are the result of lateral chromatic aberration (Sect. 1.7.6), which can never be entirely eliminated, but should not be too obtrusive. In the course of panning, you will possibly register the billboard edge bending: its two ends will seemingly bend away from the centre of the field when the image is moved towards the periphery of the field. This is caused by pincushion distortion, which is often implemented intentionally to reduce the globe effect (Sects. 2.1.3 and 8.12). If the billboard edge remains straight all over the field of view, then the image is said to be free of (rectilinear) distortion. The observer may then possibly perceive a distinct globe effect during panning. A minor quantity of such a distortion should therefore not generally be regarded as an imperfection of the binocular, but rather as the deliberate choice of the optical designer, who had to find a delicate balance between optical perfections of the static and the dynamic (panning) images. In many cases, a bright spotlight may be utilised. It should be used to test the optics for ghost images (multiple reflections on the glass surfaces), and hence the effectiveness of the anti-reflective coatings, while panning the spotlight slowly across the field of view. It is also instructive to position the light source just outside the field of view, then moving it clockwise along the edge of field – a test, which evaluates the quality of the stray-light baffles. During such a procedure, no glaring or diffuse loss in contrast should occur. In the following section, dealing with the test procedures outdoors in the field, further checks regarding the stray-light protection of binoculars will be carried out. The white paper test , as previously described by the technical journalist and camera expert Walter E. Schön, offers an easy way to assess natural colour rendition and transmittance of binoculars or camera lenses: a white sheet of paper is observed through the optical instrument, which, in the case of binoculars, is held upside down, i.e. while looking into the objective tube. It is then easy to see whether the light has picked up any colour tint or has been dimmed down considerably (Fig. 9.6). Two

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Fig. 9.6 Testing colour rendition and brightness with the paper test: a photo of a white sheet of paper is taken through the binocular-objective, and afterwards the brightness is scaled down by five f-stops. This reveals a slight tint of red for the Leica and Nikon, and a tint of green for both Zeiss instruments (with kind permission: Tobias Mennle)

different binoculars can easily be held in each hand for a comparison. The ambient light should be neutral, ideally daylight, but even artificial light works reasonably well. The precise character of such colour rendering depends primarily on the characteristic spectral transmission of the anti-reflective coatings, the absorption of the optical glass, and the spectral reflection characteristics of the reflective coating, if any, on the roof prisms (Sect. 8.8). A slight degree of colour bias is usually compensated by the colour adaptation of the eye, does not affect the observation and may therefore not be regarded as a flaw. When observing in daylight over long distances, a warm colour bias – a result of a reduced transmission in the short-wavelength range of the visible spectrum – may turn out to be an advantage, as it suppresses atmospheric scatter and increases contrast (Sect. 8.7). On the other hand, binoculars, which are optimised for lowlight observations, often exhibit a somewhat cool colour rendition as a result of an elevated transmission of the shorter wavelengths. This reflects the shift of the sensitivity of the human eye towards the blue end in situations, in which scotopic (night-) vision dominates (Sect. 6.6). At this stage, the optical and mechanical characteristics of the test candidates should be sufficiently developed to compile a shortlist of preferred candidates which

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191

may be taken home for additional, detailed field tests. In an ideal situation, the test procedure is now over, because a single instrument has been identified which performs well and best matches the requirements. The procedure may then be concluded by taking a look at the accessories supplied with the instrument. The strap should be mounted and its functionality checked. The protective caps should fit tightly, but also be removable in an instant when the binocular is needed in a hurry. If a bag comes with the instrument, it should allow the stowing or removal of the binocular without having to adjust the central hinge, and it should close safely and stay closed when on the move. In any case: if some of the accessories turn out to be less than perfect, it should not deter a buyer from purchasing an otherwise good instrument. Better and more adequate accessories can always be purchased afterwards from third party suppliers.

9.3 Field Tests The rather subtle character of a binocular comes to light once the instrument is taken out into the field. That is why – in the past – the design process for any new binocular used to include extensive field tests on prototypes by professionals or amateur testers. This strategy allowed the elimination of flaws or imperfections prior to the introduction of the final product onto the market. Unfortunately, cost pressure today often induces manufacturers to skip this important phase of field tests; instead, the optical engineer relies on the output of his computer simulations, which are restricted to idealised models of both the instrument and the environmental conditions. An accurate assessment of the instrument’s stray-light characteristics is virtually impossible, since it would have to encompass the reflective properties of the tiniest screw at any possible angle, or the reflectivities of all surfaces as a function of the angle of incidence. Simulations are thus complemented by laboratory tests which permit the measurement of some of the stray-light properties of the instrument, though hardly all of them. Standardised lab tests struggle to reproduce the tremendous diversity of lighting situations in nature, in combination with the delicate responses of the observer’s eyes (such as pupil diameter and eye placement) under real-life conditions (Fig. 9.7). Hence, its application in the field still provides the ultimate test for the instrument, and it is always interesting to observe how even premium binoculars do sometimes reveal their limitations.

9.3.1 Resistance to Stray Light Among the most characteristic, but also best hidden features of a binocular is its response to varying light situations. Particularly difficult to characterise is its susceptibility to generate stray light (glare). No single instrument exists which is

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Fig. 9.7 Nature offers a diversity of lighting situations which can only inadequately be simulated in a lab

Fig. 9.8 Exit pupils. Left: Hensoldt .10 × 50 Dialyt with brightly illuminated prism edges. Right: Zeiss (Oberkochen) .10 × 50, with well shielded prisms

absolutely immunised against this class of effects: too many options exist for a ray pencil to run astray and enter the optical path through an unwanted direction. Tests for the degree of resistance to stray light must therefore not be limited to a single test setup, but rather be conducted in the widest possible variety of circumstances, thus forming a permanent part of the entire testing procedure. As a matter of fact, stray light can originate from virtually any point in the optical path. Yet, it is often possible to determine its origin without the need to disassemble the binocular in question. Figure 9.8 presents an example, in which the exit pupils of two different binoculars are displayed. The vintage Hensoldt Dialyt to the left exhibits a bright frame-like structure around the exit pupil, caused by reflections from the edges of the Abbe-König prism which appears to be improperly shielded. In addition, at a clock angle of 4 o’clock, and very close to the edge of the exit pupil, a small bright spot is visible, a false exit pupil or prism-leak. False exit pupils are formed, when parasitic light enters the prism from outside the field of view and arrives at the eyepiece through an alternate optical path. The binoculars shown in

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the example are of size .10 × 50, i.e. their exit pupils measure 5 mm. In daylight, the eye pupil of the observer is always smaller than 5 mm, which implies that the binoculars show an impeccable image. However, at lower light levels during dawn or dusk, when the eye pupils are dilated, they are prone to take parasitic light from the peripheral regions of the exit pupils, leading to a visible drop of the image’s contrast: a typical case of a stray-light effect which turns apparent only under certain lighting conditions. Figure 9.8 shows on the right hand side the exit pupil of a vintage Zeiss .10 × 50 Porro glass with superior stray-light suppression. Despite the discernible illuminated areas outside the exit pupil, stray-light problems do not arise here, because those affected areas are sufficiently distant from the exit pupil: the eye pupil would have to widen up beyond 6 mm to get in touch with these structures, but that would only happen in situations of advanced darkness, in which potential sources of stray light are usually absent. An exceptional situation might arise in astronomy when areas in the vicinity of the moon are observed. Figure 9.9 shows an effective method to eliminate the stray-light effects mentioned above: a second inner tube, which is painted matte and which narrowly encloses the light cone, is placed inside the objective tube. This inner tube prevents light which, e.g., enters the objective from the lower left, to reach the upper Porro prism on a direct path, potentially causing stray light. As an additional measure to

Fig. 9.9 Fujinon .7 × 50 MTR cutaway model: a cemented doublet objective, a Porro prism and an eyepiece of type Kellner (with kind permission: William J. Cook)

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neutralise reflections on the inner wall of the stray-light tube, a short stray-light baffle could be placed directly in front of the prism entrance (Fig. 3.4). Such a stray-light tube is rather effective, but it does add to the weight of the binocular. Consequently, manufacturers often dispense with such baffling. Alternately, the sides of the upper prism could be painted black and the two reflecting flanks covered with small sheets of metal. The latter must not, however, touch the polished surface, otherwise it would affect its total internal reflection capability. In addition, prisms of the Porro type should have a small groove cut across their hypotenuse to block parasitic light, as discussed in Fig. 4.23. Prisms of the Schmidt-Pechan type should have a ring-shaped diaphragm placed into the narrow space between the two prisms in order to further reduce the impact of stray light (see also Sect. 4.6). Additionally, stray light may arise on the other side of the prism, in front (or even inside) the eyepiece. This is particularly relevant with binoculars that were originally designed to allow the passage of light cones which are wider than actually needed. As a common example, an .8 × 42 may be turned into a .10 × 42 after a replacement of its eyepieces. A careless manufacturer may not be willing to spend the effort to re-design the entire setup of baffles, in order to match the now reduced angle of view. As a result, the eyepiece is fully illuminated, and its field stop may not be sufficient to prevent unwanted skew-rays shining onto the inner walls of its barrel or onto the lens edges. The latter should therefore be blackened with a special paint, a procedure that has to be carried out manually, making it costly. A good opportunity to test for stray light arises during twilight (Fig. 9.10). In the direction of the sun (being below the horizon), the sky is bright, while large parts of the landscape lie in the shadow. When observing e.g. a forest edge, it is still possible to discern details of the trees, such as branches and leaves, but the illuminated sky right above the scene tends to throw a significant amount of unwanted light into the tubes. Under twilight, the eye pupils tend to be wide, thus allowing stray light to enter from the perimeters of the exit pupils – a phenomenon, which rarely occurs in daylight. As a result, the contrast Fig. 9.10 Checking for glare after sunset

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C=

.

L t − Lb Lb

(9.1)

is reduced. Here, .Lt is the luminance of an object (e.g. a deer) to be observed in front of a background with a luminance .Lb (e.g. tree trunks). Assuming that the entire field of view is filled with homogenous stray-light of luminance .Ls , then this amount has to be added to both luminance values, yielding a new contrast of C=

.

Lt − Lb . L b + Ls

(9.2)

Depending on the magnitude of .Ls , C may have dropped significantly and details are lost. Stray light originating close to or within the eyepiece often results in a diffuse veil which covers only the peripheral areas of the image. If the incoming stray light enters from the sky above the observed object it may exclusively shine into the lower parts of the barrel, then leading to an arc-like veil around the lower edges of the field. Air-spaced objectives are also prone to stray-light, in particular those with a less than perfect polish. In this case, however, the entire image suffers from an almost uniform loss of contrast (a ‘whiteout’). Mounting simple self-made stray-light hoods significantly reduces such a loss of contrast; manufacturers should therefore consider installing such pull-out or screw-in hoods, as they are commonly found on camera objectives. The stray-light performance of many binoculars could be improved considerably with the help of these – technically speaking – simple gadgets.

9.3.2 Ghost Images Ghost images are the result of multiple reflections on glass surfaces. If all optical elements of a binocular are coated with a sufficiently effective anti-reflective coating, then these ghost images do not show up. Strictly speaking, these reflections are nothing but yet another type of stray light, but since they have a particular cause and they offer an accurate assessment of the quality of coatings, they should be addressed in a separate test setup. Figure 4.9 may give an impression about how many times a light beam has to enter and exit glass elements within a binocular of moderate complexity. As a rule of thumb, an uncoated glass surface reflects roughly 5% of the incoming light. If the glass is treated with a single-layer coating, as patented by A. Smakula of Zeiss in the early 1930s, then the reflectivity reduces to values of about 1.5%. Modern multilayer coatings can reduce that loss to values below 0.3% over the entire visible spectrum of light and a wide range of incident angles (details are discussed in Sect. 2.4.3).

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The formation of a ghost image does always require an even number of reflections: two reflections cause a ghost image of the first order, four reflections result in a ghost image of the second order and so on. In binoculars with up-to-date anti-reflection coatings, only ghost images of the first order are of any relevance. Assuming a transmission of .T = 0.995 of an incoming light-ray, which is an average value for a multi-coated surface, a first order ghost image would have an intensity that is reduced by a factor of .(1 − T )2 ≈ 0.000025. This implies that such a ghost image may be potentially observable only at night and on a bright light source such as a street lantern. In astronomy, the reduction of a factor 0.000025 corresponds to a decrease of eleven magnitudes (Sect. 8.6.1), so that only the moon and none of the stars or planets would have sufficient brightness to generate a visible response. The situation is different with binoculars of vintage origin. Before 1978, only single-layer coatings were commonly applied to binoculars, and a light ray entering a single-coated glass surface would have a transmission of about .T = 0.985, yielding a reduction in brightness of .(1 − T )2 ≈ 0.00023 for the first order ghost image. This corresponds to nine stellar magnitudes, and under unfavourable conditions, the planet Venus may be able to cause a reflection. If glass surfaces exist, which, for cost reduction, remain entirely uncoated, then the situation turns out to be worse. It occurs with some of the cheaper imported products, in which the coating process of the prisms has been dispensed with. Note that Porro prisms are usually un-cemented, and the bases of both prism elements are set up face to face with a narrow air space. Ghost images of first order are now suppressed by the factor .(1 − 0.95)2 ≈ 0.0025, which corresponds to roughly six magnitudes. This implies that virtually all of the bright stars and planets are likely to cause irritating reflections, and the panoramic view onto a city centre at night creates flitting fireworks (Fig. 9.11). The facts mentioned above indicate that bright and ideally pinpoint light sources at night are ideal test objects to analyse the susceptibility of a binocular for ghost images, and hence the quality of its coatings (the lantern test): a light source, e.g. a bright street lantern at a distance of several 100 m, is inspected while slowly Fig. 9.11 Lights of different intensities enable the testing for ghost images

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panning, such that the light is repeatedly moved across the entire field of view. Reflections of that light may show up, and, depending on their origins, they may move at different rates. If the image stays entirely free of reflections, then another, brighter light source may be selected. Such a test is best done with more than one test sample, which simplifies a comparative rating of their coatings. The interpretation of the test results may turn tricky at times. Not solely the coating of the lenses, but also their mutual placements and curvatures may affect the intensities of the resulting ghost images. Common examples are field lenses of the eyepieces, which for some designs are located close to the focal plane, at which the image of the lantern is well focused. Naturally, the intensities are highest here, and the resulting ghost images may turn out to be particularly intense. Military binoculars, which require a reticle right in the focal plane of the instrument, are particularly prone to this problem. In some instruments, the ghost images may turn out highly diffuse (because far off-focus) and remain altogether undetected even with less than perfect coatings. Here the ghost image only adds to a general diffuse stray-light background that slightly reduces the contrast of the entire image. The locations of ghost images may yield information about the state of centring of the lens elements: in a perfect optical setup, all reflections on the lenses should exhibit a rotational symmetry about the centre of field. Hence, if several ghost images are visible, they should form a perfectly straight line which runs through the main image of the lantern. If the latter is placed into the centre of field, all ghost images have to be concentric. If this happens to be otherwise, then one or more lens elements must be de-centred or tilted. In some instances, experienced testers have been able to estimate the most probable eyepiece types of vintage binoculars, solely by examining the appearances and relative positions of their ghost images. With roof-prism binoculars, the lantern test exhibits another phenomenon of entirely different origin: since the roof-prism edge cuts through the light path like a very fine thread, the resulting diffraction of light may now become visible as a short, linear spike, not unlike the situation shown in Fig. 3.21. The orientation of that spike is perpendicular to that of the roof edge, and since the prisms in the left and the right barrels are usually orientated in different directions, the resulting diffraction effect resembles two crossing spikes. The intensity of that artefact is a function of the width of the roof edge, and it almost disappears with a well cut and polished prism. The lantern test thus assesses the quality of the roof prisms, too. In some instances – staying with the street lantern as the test object – a second, slightly blurred image can be observed next to the primary image, but this time, the position of the secondary image does not change in relation to the primary one, even when the instrument is panned and the image moved across the field of view. This effect is usually observable in one of the barrels only. The effect in question originates in a reflection, caused by a prism which has not been cut with adequate precision. It occurs at times with cheap roof-prism binoculars, in which the extremely narrow tolerances for the 90° angle between the two roof faces are not met (Sect. 3.2.1).

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The discussions of the present section provide convincing arguments in favour of the capabilities of the lantern test to yield abundant information about the characteristic properties and the condition of an optical instrument. Yet, a word of warning is adequate here: the lantern test is extremely sensitive – reflections are diagnosed which are sometimes ten thousand times weaker in intensity than the primary image. The detection of isolated ghost images does not necessarily imply a flaw in the instrument, nor a quality impairment that would necessarily affect the performance of that binocular in daily-life situations. When comparing highperformance devices, which commonly arrive with state of the art coatings, the lantern test may reveal certain minor differences that are of little to no relevance in the field. The test should rather be regarded as a sort of checkup on vintage or cheaper binoculars, since it enables the detection of uncoated glass surfaces, imprecisely cut roof edges, or optical surfaces that are insufficiently polished.

9.3.3 Off-centre Sharpness A star in the night sky represents a mathematically perfect pinpoint light source and is therefore an ideal object to test the image quality of an optical instrument [3]. Moreover, at night, the eye pupils are wider than during the day, and the entire light cone, which has passed the binocular, contributes to the image without being stopped down by the iris. Tests of the imaging properties of stars thus offer reliable results that are easier accessible than, yet similarly accurate as tests conducted in bright daylight. However, test results need to be analysed and interpreted with care: not only the binocular, but also the human eye is liable to aberrations. To most observers, a bright star appears surrounded by starry spikes instead of being pointlike – a consequence of the spherical aberration of the eye (Sect. 6.5). This is easily verified with the bare eye when observing a bright star: once tilting the head to the side, the rays emerging from the star image are rotating. These effects remain invisible with stars of lower – second or third – magnitude, which are therefore more suitable for our testing purposes than the very bright ones. Another caveat: observers with astigmatic vision need to put on their individually adjusted vision aids before testing the instrument. As a matter of fact, binoculars are usually incapable of producing diffractionlimited images, including the central (paraxial) areas of their field. But the aberrations should be sufficiently suppressed and visible only with the help of a booster (Fig. 8.4). A decent binocular should thus display stars in the centre of the field as pinpoint objects, and if that fails, then it may safely be assumed that the optics are flawed, so that any further testing would become unwarranted. Once the star is shifted off-centre towards the edge of the field of view, its image will turn increasingly blurred. If we now imagine a straight line cutting through the centre of the field, we may further imagine a linear scale which reads the percentage of the distance to the edges. The message: a star appears sharp within 60% of the angle and turns blurred further outside then defines the (relative) size of the so

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called sweet spot of that binocular. Naturally, each observer applies his individual standards when he has to decide from which angle onwards the image may be called ‘soft’ or ‘un-sharp’. Thus, a sweet-spot size is generally an individual measure, but independent observers are usually able to rank sets of different binoculars with reproducible and consistent results. The sweet-spot sizes of different instruments commonly vary between values of 60% and beyond 90%. Not surprisingly, vintage binoculars, in particular wideangle models, with fairly simple eyepiece constructions, perform poorly in this test. Historically, binoculars were primarily military instruments, used by young soldiers with wide accommodation ranges and thus capable of compensating the excessive amount of field curvature in these devices (Sect. 1.7.3). Instead, modern binoculars are often supplemented with field-flattening lenses (Sect. 4.2.2) and sometimes literally offering an edge-to-edge sharpness. Yet, a positive side of field curvature also exists: when observing a landscape, field curvature can effectively increase the perceived depth of field (Sect. 2.3). Further aberrations may contribute to the image blur. As long as the fuzzy star near the edge of field can be re-focused into a sharp image, field curvature was the exclusive origin of the blur. Otherwise, astigmatism, sometimes in combination with coma, affect the off-centre resolution. Additionally, lateral chromatic aberration (Sect. 1.7.6), responsible for colour fringes along contours near the periphery of the field, most certainly also has an impact here. The latter is easily detected on the moon edge, once the moon is shifted to the periphery of the field. In practice, the quantity of off-centre image blur is not always distributed isotropically. It is therefore instructive to examine the image blur in several different directions, starting from the centre. Since, in principle, an optical system has centrosymmetric properties, a non-isotropic distribution of image blur would be the result of an imperfect alignment. Sometimes, both tubes show exactly identical asymmetric distributions: more often than not, the lower half of the field appears sharper than the upper half, raising the suspicion that the manufacturer may have intentionally tuned the optics, e.g. by slightly tilting the prisms towards the optical axes, to cover a poor edge sharpness during terrestrial observations: when observing in daylight, the majority of objects are often scattered about the lower half of the field of view, the upper half being often occupied with structureless sky. Of course, this is not so in the case of astronomical observations, during which non-isotropic distributions of sharp and blurred areas in the image leave a rather unpleasant impression. To estimate the objective angle of field, it can be quite useful to observe star constellations in the night sky. A well-known constellation should be selected and a pair of stars located which just barely fit into the field of view. Afterwards, the angle between the two stars is determined using a star atlas or catalogue. Computer programs are also readily available for this purpose, and so are star maps on the Internet, created with these programs, which directly indicate angles between selected pairs of stars. The advantage of using stars to measure the field angle lies in their infinite distance: if instead the field of view were measured on a ruler placed at close range, then inaccurate specifications may result (Sect. 4.3.2).

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9.3.4 Low-light Performance The performance of a binocular should be assessed in a variety of different lighting situations. A particularly useful time for a tester is the twilight: within 1 – 2 hours, daylight conditions are gradually transforming into low-light conditions. Not only is this transition accompanied by a constant variation of illumination; but there also exists a highly complex modification of the way our visual perception functions. The increasingly dilated eye pupils begin to receive light from the peripheral areas of the exit pupils – we refer to Sect. 9.3.1 for a discussion on how this affects the stray-light performance. The mode of operation of our ‘sensor’, the retina, changes dramatically. During a first stage, the sensitivity of the retina is increased. Thereafter, a shift in colour perception occurs: the scotopic vision of the rods increasingly supersedes the photopic vision of cone cells, and the ability to distinguish colours diminishes during that process (Sect. 6.7). At the same time, resolution at the centre of the field declines and peripheral vision turns rather more significant. The ability to spot details of an object gradually gives way to the identification of object contours, and data pre-processing in the retina becomes increasingly complex (Sect. 6.9) (Fig. 9.12). In the common case, in which the transmission curve of the binocular is not entirely flat, the spectrum of the incoming light undergoes a modification. This may turn relevant under twilight conditions: instruments that have been optimised for observation in twilight tend to show particularly high transmission levels at the short-wave end of the spectrum, and may thus exhibit a somewhat cold colour rendition on sunny days. On the other hand, those binoculars, which excel in daylight with an excellent correction of chromatic aberration, lose that advantage in low light when any stark contrast fades. A high level of total visual transmission, which is not that important in daylight, can make a difference after sunset when every single photon contributes to the signal-to-noise ratio of hardly discernible objects. An observer testing the low-light performance may first assess how the binocular accurately displays colours of selected objects far into the twilight. Once again, such a test is most telling if several instruments are compared and rated against one Fig. 9.12 As dusk progresses, surface details diminish, and the detection of object contours and movements gains importance

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another. Additionally, the amount of details displayed on textured surfaces (e.g., the structure of bark, or individual leaves of bushes) should be analysed and compared through different binoculars. Then, after nightfall, it may be checked whether, and to what extent, the observation of a landscape through the binoculars still offers visual cues which enable general orientation: is it still possible to define one’s position relative to surrounding objects and determine their distances and relative proportions accurately? During observations under these conditions of highly suppressed visual details, wide angles of view are of advantage since they allow the relation of each object to its surroundings. In addition, a wide stereoscopic base helps to discern the three-dimensional arrangement of objects (Sect. 8.10). In this context, experimenting with several binoculars of different designs and formats can be very instructive. Care has to be taken to keep the eyes of the tester fully adapted to the respective lighting levels, and therefore, any artificial light source should be left out of sight during these test procedures. Pocket lights, used while writing the observation protocol, should be covered with a red filter (a coloured and transparent plastic foil will do). Such a dimmed light would not compromise dark adaptation of the retinal rods, which are entirely insensitive to red light (Fig. 6.7).

9.3.5 Chromatic Aberration Chromatic aberration appears particularly obtrusive wherever stark contrasts exist in the image. This is why e.g. overhead power lines and transmission towers are ideal test objects, which, when observed against the bright background skies, offer numerous contrast transitions throughout the entire field of view (Fig. 9.13). First, the centre of the field may be examined. Here, no visible colour fringes should be visible, since in a well-centred optical system only longitudinal chromatic aberration should exist near the centre, and these should not be perceptible at the typically low magnifications of handheld binoculars. In other words: unless mounted binoculars Fig. 9.13 Transmission tower: a perfect target to test for colour fringes

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are concerned, which are commonly used at very high magnifications, the centre of field should generally be free of colour fringes. Here it is important to have the eye pupils well aligned to the exit pupils, and thus the inter-pupillary distance has to be set accurately via the central hinge. An accidental misalignment may happen easily, especially with large exit pupils, when even poorly aligned eye pupils remain fully illuminated. Colour fringes do always show up if the eye pupils are partially clipped by the exit pupils, but these impairments are generated by the observer’s eye, not by the instrument (Sect. 6.5). If, after careful alignment of the instrument with the eyes, the colour fringes remain visible near the centre of field, then the binocular is probably out of collimation and in need for a repair. Note that, in such a situation, different results for both barrels should be expected. Once the centre of the field has been examined, the periphery of the image is inspected. Off-centre, even the best binoculars exhibit a certain amount of colour fringes. They are an implication of the lateral chromatic aberration, which cannot be fully corrected in visual instruments of reasonably wide fields of view (Sect. 1.7.6). Even if a designer were to succeed in eliminating it, colour fringes would still emerge in certain situations: Observations in the peripheral areas of the field require the eyes to swivel around in their sockets, so that a concentric alignment with the exit pupil becomes lost. Thus, a complete elimination of these effects is technically impossible, but colour fringes can be substantially reduced with the careful design of the eyepiece, and the purpose of this paragraph has been to determine how well the manufacturer has succeeded with his task.

9.3.6 Ease of View: The Unspeakable Vision is experienced in a new way and with an intensity that defies description. All other sensations are extinguished. Eye and binocular – the near and the far – fuse into a perfect and harmonious unity.

This is how Hans Seeger describes his experience with his vintage 8 × 60 Zeiss binocular, made in the 1940s [4]. The expression The eye and the binocular . . . fuse vividly describes a mode of vision which is free of any distracting side effects; nothing interferes with the act of natural and relaxed observation. Instruments, which allow, via the eyepiece as their interface, a seamless connection between eye and optical image, possess a distinct quality for which the German term ‘Einblickverhalten’ has been coined, and which is approximately translated with the expression ease of view. Several criteria contribute to the attribute of having an outstanding ease of view: firstly, the virtue of an instrument to allow the observer to easily see over the entire field of view right from the moment when the binocular is placed before the eyes. Secondly, the absence of any irritating reflexes or dark shadows, which would show up while the eye is swivelling around to look at objects in peripheral areas of the field, or while the binoculars are panned over a landscape.

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These criteria are all about tolerances, or, more precisely, about the deviations allowed for the relative placements of exit- and eye pupil, without any significant deterioration of the image quality. This is easier to achieve when the eye pupil, the exit pupil, or both, have large diameters, and a reason why many binoculars exhibit a superior ease of view in low-light situations, including the artificial lighting conditions in an exhibition hall or department store. Once taken out into broad daylight, the same instrument may then appear to lose some of its merits. The occurrence of dark shadows, blocking parts of the field of view, is commonly referred to as kidney beaning. The effect arises when, during panning or sudden leaps of the line of vision, parts of the ray-pencil are blocked off by the iris and prohibited from entering the eye. To minimise these effects, binoculars should exhibit as little as possible vignetting of the edge pupil.6 Since vignetting is often implemented to block off skew-rays, which suffer from insufficiently corrected astigmatism (Sect. 1.7.4), the optical designer then has to implement other, and costlier solutions to optimise the image. Furthermore, a proper correction of the spherical aberration of the exit pupil (Sect. 2.1.1) considerably improves the ease of view of the instrument. False exit pupils, or prism leaks (Fig. 9.8, left) may cause irritating flashes of whiteouts during eye movements and should be eliminated through sophisticated installations against stray light. Particular challenges await the optical designer who intends to ensure a pleasant ease of view to spectacle wearers, since that additionally requires a sufficiently high eye relief of the exit pupil. Besides the ease of view, additional factors exist that contribute to the ultimate viewing experience described by Dr. Seeger. An impeccable collimation and alignment, a wide subjective angle of field, and an image framed by a well-defined field stop are attributes that help to make the observation an unforgettable experience. Even the deliberate employment of carefully chosen quantities of aberrations into the optical design may contribute to the perceived performance of a binocular: a moderate degree of field curvature creates the impression of an increased depth of field (Sect. 2.3), and a well-chosen inclusion of pincushion distortion eliminates the globe effect when the binocular is panned (Sect. 8.12). Naturally, such aberrations are in conflict with the standards of exact optical image reproduction, and the definition of an adequate set of standards, which balance between perceived performance and optical bench quality, remains a challenging and even controversial task. As the binocular industry appears to approach a point of saturation with respect to traditional optical parameters, such a ‘human vision compatibility’, i.e. a set of design paradigms, which address the perceived satisfaction level during observation, may become an increasingly important factor for the competitiveness of the product. The Zeiss 8 × 60 binocular mentioned in the quote at the beginning of this section had a large exit pupil of 7.5 mm, a high eye-relief of 24 mm, as well as a wide subjective field of view of more than 70°; it was therefore well equipped to offer a competitive ease of view (Fig. 9.14). The SARD 6 × 42 Mark 43 of the US

6 Information

provided by Dale Forbes on birdforum.net.

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Fig. 9.14 Perhaps the most remarkable binoclars of the 20th and the (so far) 21st century: World War II era blc (Zeiss Jena) 8 × 60 submarine commander’s binocular (left; field of view: 158 m/1000 m, weight: 2.3 kg; property of Hans Seeger), and a modern Nikon 7 × 50 IF WX (field of view: 187 m/1000 m, weight: 2.4 kg)

Navy, used on anti-submarine aircrafts to spot submarines which surfaced during the nights to refill their air-tanks, had similar characteristics (Fig. 9.15). With its wide exit pupil of 7 mm and eye-relief of 20 mm, its very wide field of view could be scanned easily even from a shaky airplane. A number of binocular enthusiasts and experts believe that the best of those instruments, which were produced during the 1940s, still represent the overall state of the art of binocular design, and that similarly high standards in terms of ease of view and comfort have never been reached again since then. When considering today’s omnipresent trend towards sizeand weight reduction, modern binoculars with their considerably reduced prism sizes are unlikely to ever reach similarly generous specifications regarding field of view, exit pupil diameter and eye relief. After all, the SARD weighed 1.7 kg, and the Zeiss 8 × 60 scored 2.3 kg. Remarkably, with the recent arrival of the Nikon 7 × 50 and 10 × 50 WX models (Sect. 4.5.3), a line of currently made binoculars exists, which is capable of competing with these classic instruments in terms of ease of view, while, at the same time, delivering superior optical imaging. Obviously, as demonstrated in Fig. 9.14, this competitiveness includes the features of bulk and weight.

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Fig. 9.15 SARD 6 × 42 Mark 43 (Field of view: 193 m/1000 m) of the US Navy, with a respectable weight of 1720 g

9.3.7 Ergonomics and Haptics Inside the store, after the decision for the purchase of a binocular has been made, its ergonomic and haptic qualities have been checked and passed. However, surprises may be waiting for its new owner, once the instrument is taken outdoors into the fields for serious applications: all of a sudden, the high magnification binocular, which was so easily wielded at the shop counter, turns out to be hard to hold steady, particularly after a tiring up-hill tour. In cold air, the focus wheel suddenly turns much stiffer than before, and after the rain sets in, the initially tactile armor feels slippery. The carrying strap, so far considered an unimportant accessory, enters the stage and takes on a new and considerably less welcome role, by chafing the wanderer’s neck! It cannot be emphasised enough: a final assessment of the ergonomic and haptic qualities is not yet closed after a brief fumbling in the shop, and endless discussions on Internet forums are similarly unlikely to offer final answers. Different users employ their binoculars for different purposes, so that generalised advice is rarely useful. It is a common piece of wisdom that the selection of an ideal binocular is only possible out in the field. The requirements of different users as to the ergonomic and haptic properties do vary with the mode of application. When the instrument is used mostly as a stationary device, e.g. in a raised hide during hunting, on the bridge of a vessel, or on a checkpoint, then weight and ergonomics are less important than optical merits such as low-light performance or a wide field of view. Those who intend to carry the device around their neck during the larger part of the day and having to deal with varying weather conditions and opportunities arising suddenly, will appreciate binoculars which do not cause additional troubles. In fact, arriving at relevant conclusions about ergonomics is thus quite simple: the binocular should be tested under the same conditions which are likely to prevail in later applications. Problems or issues, which are unlikely to evaporate with gaining experience, should be carefully registered and evaluated when the test procedure

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is over. Unfortunately, certain aspects and issues may turn bothersome only after a considerable time, when it is already too late to return the instrument to the seller. Such experiences should, without rancour, be taken as an apprenticeship and are unavoidable for one who is gradually becoming an experienced binocular user. After all, there is no way around this simple fact: even the best possible binocular design and concept cannot relieve the user of this individual learning process. An optical instrument must be mastered, and only the persevering use of the instrument in practice will finally lead to a perfect symbiosis of man and machine. In the end, after achieving such mastery, some of the performance parameters, which have been listed on the specification sheets and regarded as the holy grails of binocular design during endless debates in the Internet’s discussion boards around the world, may suddenly be rendered irrelevant.

References 1. Köhler, A.: Prüfen von Fernrohren. www.akoehler.de 2. Weyrauch, A., Dörband, B.: P-Belag: Verbesserte Abbildung bei Ferngläsern durch phasenkorrigierte Dachprismen. Deutsche Optikerzeitung 4, (1988) 3. Suiter, H.R.: Star Testing Astronomical Telescopes. Willmann-Bell, Richmond (2008) 4. Seeger, H.T: Military Binoculars and Telescopes for Land, Air and Sea Service, 2nd edn. Hans T. Seeger, Hamburg (2002)

Correction to: Report on a Self-made High-Performance Binocular

Gerhard Eller

Correction to: Chapter 5 in: H. Merlitz, The Binocular Handbook, https://doi.org/10.1007/978-3-031-44408-1_5

The chapter “Report on a Self-made High-Performance Binocular” was previously published with an incorrect corresponding author. It has been changed to “Gerhard Eller”.

The original version of the chapter has been revised. A correction to this chapter/book can be found at https://doi.org/10.1007/978-3-031-44408-1_5 © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 H. Merlitz, The Binocular Handbook, https://doi.org/10.1007/978-3-031-44408-1_10

C1

Appendix A

Further Reading

A.1 Technical Optics • The Bureau of Naval Personnel: Basic Optics and Optical Instruments. Dover Publications Inc., New York, (1969) Mostly interesting for its description of maintenance procedures. Includes an introductory course on geometrical optics for Navy personnel. • Alexander E. Conrady and Rudolf Kingslake: Applied optics and optical design. Vol. 1&2 (reprint), Dover Publications, (2011) Very mathematical, but with true gems about classical approaches to optical design. • B.K. Johnson: Optics and Optical Instruments. Dover Publications Inc., New York, (1960) Concise and easy to read writeup covering geometrical optics and optical instruments. • Masud Mansuripur: Classical Optics and its Applications. Cambridge University Press, 2nd edition (2009) Undergraduate level textbook about wave optics and its applications. Harrie G.J. Rutten & Martin A.M. van Venrooij: Telescope Optics. WillmannBell, Richmond, (2002) A gem for the technically interested amateur astronomer. • George Smith and David A. Atchison: The eye and visual optical instruments. Cambridge University Press, (1997) A very comprehensive book about the design of optical instruments to be used in combination with the eye. • Gregory H. Smith, Roger Ceragioli and Richard Berry: Telescopes, Eyepieces and Astrographs: Design, Analysis and Performance of Modern Astronomical Optics. Willmann-Bell, (2012) Ray-tracing analysis of numerous optical instruments for the astronomer, including vintage eyepiece designs.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Merlitz, The Binocular Handbook, https://doi.org/10.1007/978-3-031-44408-1

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A Further Reading

• Warren J. Smith: Modern Lens Design. 2nd edition, McGraw Hill Professional, (2004) A professional-level selection of optical system layouts and their performances. • Bruce H. Walker: Optical Design for Visual Systems. SPIE PRESS, Bellingham, Washington, (2000) A concise writeup about the eye as an optical instrument in combination with visual optical instruments. • Paul R. Yoder, Jr.: Mounting Optics in Optical Instruments. SPIE PRESS, Bellingham, Washington, 2nd edition (2008) Covers the mechanical engineer’s aspects of optical instruments, including the layouts of numerous image-erecting prisms.

A.2 Visual Perception • Peter G.J. Barten: Contrast Sensitivity of the Human Eye and its Effects on Image Quality. SPIE PRESS, Bellingham, Washington, (1999). Contains data and mathematical models regarding the performance limits of the eye. • Michael Bass (chief editor): Handbook of Optics, Volume III: Vision and Vision Optics. McGraw-Hill Companies Inc., 3rd edition (2010) Very comprehensive book about human vision, with a plethora of data taken from recent studies. • K.R. Boff, L. Kaufman and J.P. Thomas (editors): Handbook of Perception and Human Performance. Volume I, Wiley & Sons Inc., New York (1986) Comprehensive collection of data about human visual performance from older studies. • Mark Wagner: The Geometries of Visual Space. Lawrence Erlbaum Associates Inc., USA, (2006) Contains numerous studies about the visual space and how its geometrical properties can be tested.

A.3 Binoculars and Scopes • Brin Best: Binoculars and people. Biosphere Publications, West Yorkshire, (2008) Interesting book about the interactions between binoculars and society. • William J. Cook: Binoculars: Fallacy & Fact. ISBN-13: 978-1548932190 (2017) An entertaining read, debunking common myths about binocular function, repair and collimation. • William J. Cook: Binocular Collimation. ISBN-13: 978-1790983780 (2018) Everything to know about binocular collimation.

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• Neil English: Choosing & Using Binoculars: A Guide for Stargazers, Birders and Outdoor Enthusiasts. Springer Nature (2023) The new standard book about the selection of binoculars and their different fields of application. To be published in 2023. • William Reid: ‘We’re Certainly not afraid of Zeiss’: Barr & Stroud binoculars and the Royal Navy. National Museums of Scotland, (2001) Interesting collection of historical material regarding Barr & Stroud binoculars. • Hans T. Seeger: Military Binoculars and Telescopes for Land, Air and Sea Service. Hans T. Seeger, Hamburg (1995) The reference book about military binoculars. Though the main text is written in German, all figure captions are dual language (and the book consists mainly of pictures). • J.W. Seyfried: Choosing, Using & Repairing Binoculars. University Optics Inc, Ann Arbor, 2nd edition (1995) Neat little book containing some basics about binoculars and their repair. • Harold Richard Suiter: Star Testing Astronomical Telescopes. Willmann-Bell, Richmond, 2nd edition (2013) The reference book about star testing astronomical optical instruments. • Stephen Tonkin: Binocular Astronomy. Springer Press, London, 2nd edition (2014) All about binoculars under the night sky. Contains an introduction into binocular function. • Paul R. Yoder, Jr. and Daniel Vukobratovich: Field Guide to Binoculars and Scopes. SPIE PRESS Bellingham, Washington, (2011) A concise summary of binocular and telescope function and performance.

Appendix B

Picture Credits

• Figure 1.1: Wikipedia: Fraunhoferlinien, Public Domain; • Figure 1.4: Copyright (©) SCHOTT AG, Advanced Optics; • Figure 1.13: Source: Michael Schmid on Wikimedia Commons under CC BY-SA 3.0; • Figure 2.6, (left): Wikipedia: Airy disk, Public Domain, (right): Wikipedia: Rayleigh-Kriterium, Geek3/CC BY-SA 3.0; • Figure 3.8: Reproduction from: Hans Seeger: Zeiss Feldstecher. Zeiss Handferngläser 1919–1946. Dr. Hans T. Seeger, Hamburg (2015); • Figure 3.16: Reproduction from: Hans Seeger: Zeiss Feldstecher. Handferngläser 1894–1919. Dr. Hans T. Seeger, Hamburg (2010); • Figure 3.21: (©) DOZ-Verlag Optische Fachveröffentlichung GmbH, reproduction from A. Weyrauch, B. Dörband, Deutsche Optikerzeitung 4, (1988); • Figure 4.12: Reproduction from the brochure HW 871/2: Handliche Brillanz: die neuen Victory 32 FL (2004). Copyright (©) Carl Zeiss Sports Optics GmbH. • Figure 4.13: Reproduction from the brochure CZO-F 488: Zeiss Ferngläser weltberühmt (1983). Copyright (©) Carl Zeiss Sports Optics GmbH; • Figure 4.14: Copyright (©) Leica Camera AG; • Figure 4.15: Copyright (©) RICOH IMAGING DEUTSCHLAND GmbH; • Figure 4.26: Reproduction from the brochure CZO-F 131: ZEISS-Feldstecher (1960). Copyright (©) Carl Zeiss Sports Optics GmbH; • Figure 8.5: Source: Alemily on Wikimedia Commons under CC BY-SA 2.5.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Merlitz, The Binocular Handbook, https://doi.org/10.1007/978-3-031-44408-1

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Index

A Abbe number, 8 Aberration, 14, 18 astigmatism, 23 coma, 24 lateral chromatic, 25 longitudinal chromatic, 19 spherical, 20 Accommodation, 111 Accommodation width, 112 Achromatic lens, 67 Achromatism, 70 Adler index, 156 Afocal, 30 Afocal imaging, 37 Air lens, 68 Airy-disc, 40 Albedo, 109, 124 Angle condition, 34 Angle of field, 32 Angular magnification distortion (AMD), 34, 179 Aperture error, 20 Aperture stop, 29, 90 Apochromatic objective, 70 Apochromatism, 70 Apparent angle of field, 30 Argon purging, 97 Astigmatism, 23 Astro-binoculars, 70 Axis offset Perger, 52 Porro I, 48 Porro II, 49

B Back focal length, 12 BaK4 (optical glass), 6 Base length, 132 Base prism (Porro II), 49 Beam splitter, 53 Bellows, 96 Berek index, 159 Binocular haptics, 205 laboratory tests, 181 night, 134 panning, 136 quick tests, 185 stress test, 182 Bipolar cells, 118 Bishop index, 157 BK7 (optical glass), 10 Booster, 146

C CaF2 (optical crystal), 11 Camouflage, 123 Candela, 109 Canon IS, 51 Canon 12x36 IS II, 94 Cardboard effect, 173 Centre focus, 77 Chequerboard (Helmholtz), 137 Chinese 8x30 (Typ 62), 95 Ciliary muscle, 107 Circle-condition, 139 Circle of confusion, 20, 168

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Merlitz, The Binocular Handbook, https://doi.org/10.1007/978-3-031-44408-1

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214 Colour aberration, 19 Colour adaptation, 131 Colour constancy, 130 Colour depth effect, 113, 134 Colour fringes, 25, 113, 201 Colour neutrality, 10 Colour perception, 130 Colour stereoscopy, 134 Colour tint, 189 Coma, 24 Cones, 114 Contrast, 124, 143, 194 Convergence, 118 Converging lens, 17 Critical angle, 45 Cross-disparity, 132

D Daylight spectral distribution, 130 Deflection prism, 45 Depth of field, 38, 168 Depth perception, 132 Depth resolution, 133 Depth scale, 38 Desiccant cartridge, 95 Diaphragm, 92 Dielectric coating, 63 Diffraction, 39 Digital telescope, 180 Dioptre, 111 Dioptre setting, 78 Direct-vision prism, 54 Dispersion, 3, 8 abnormal, 8 Distortion, 137 barrel, 34 parameter, 138, 176 perceptual, 138 pincushion, 34 rectilinear, 174 Diverging lens, 17

E Ease of view, 36, 202 ED (optical glass), 8, 11 Edge pupil, 188, 203 Effective entrance pupil, 143 Effective magnification, 158 Efficiency contrast based, 151 resolution based, 146

Index Entrance pupil, 29 Equivalent focal length, 14 Excess travel, 78 Exit pupil, 30 false, 192 formula, 31 spherical aberration, 203 Extinction, 162 Eye astigmatism, 114 cornea, 107 fovea, 108, 116 hyperopic, 112 imaging errors, 112 iris, 107 lens, 30, 72 longitudinal chromatic aberration, 112 macula, 116 myopic, 112 relief, 30 resolving power, 114 retina, 108 spherical aberration, 113 transversal chromatic aberration, 113 vision impairments, 114 Eyepiece diameter, 74 Erfle, 74 Kellner, 72 Nagler, 75 orthoscopic, 73 reversed Kellner (RKE), 73

F False exit pupil, 192 Fankhauser index, 157 Field curvature, 22, 39 Field-flattening lens, 22, 75 Field lens, 72 diameter, 86 Field of view formula, 32 Field of vision, 108 Field stop, 29, 91 FK56 (optical glass), 19 Fluoride glasses, 11 Fluorite (optical glass), 11 Focal length, 14 Focal ratio, 31 Focusing lens, 78 Focus travel, 78 Focus wheel gear ratio, 79

Index Fogging (lens), 97 Fovea, 108 F-ratio, 31 Fraunhofer-lines, 3 Frequency, 3 Fresnel’s equation, 42 Fried parameter, 164 F-tan theta relation, 33 F-theta relation, 34 Fujinon 7x50 MTR, 193

G Galilean telescope, 35 Ganglion cells, 119 Geometric phase, 64 Ghost images, 195 Glare stop, 91 Glass BaK4, 6, 10, 46 BaK4 HT, 10 BK7, 10, 45, 46 BK7 HT, 10, 59 CaF2 , 11 ED, 8, 11, 69 FK56, 19 HD, 11 SF2, 19 Glass-path factor, 48 Glass-path length Abbe-König prism, 57 Perger prism, 53 Porro I, 48 Porro II, 49 Schmidt-Pechan prism, 54 Uppendahl prism, 60 Glass plate, 5 Globe effect, 140, 174 Gyro stabilisation, 93

H HD (optical glass), 11 HT (optical glass), 10 Huygens’ principle, 40 Hyperfocal distance, 168

I Ideal layer thickness, 43 Image equation Newtonian, 17 paraxial, 17 Image plane, 29

215 shift, 76 Image stabilisation, 92 Imaging afocal, 37 Indirect vision, 117 Individual eyepiece focusing, 77 Inertial stabilisation, 93 Interference constructive, 40 destructive, 40 Intermediate image, 29 diameter, 85 Internal focusing, 78 eyepiece, 78 Internal transmittance, 9

K Kidney beaning, 30, 203 KOMZ 7x30 BPO, 178 Kowa 8.5x44 Genesis, 164

L Lambert-Beer law, 162 Landolt ring, 123 Lantern test, 196 Lateral inhibition, 120 Law of reciprocity, 16 Law of reflection, 5 Leica Geovid, 54 8x42 Ultravid, 166 Leitz 6x24 Amplivid, 61 7x42 B Trinovid, 81 Lens air, 68 aspherical, 11, 20, 73 plano-convex, 11 Smyth, 22 speed, 31 tele, 68 Light, 3 phase velocity, 7 Limiting magnitudes, 156 Limulus, 121 Low-light testing, 200 Lubricants, 97 Luminance, 109 Luminance function, 126 Luminous flux, 143 Luminous flux function, 126

216 M Macrolon, 62 Magnesium fluoride coating, 44 Magnification, 32 close range, 82 effective, 158 Meridional plane, 23 Merit-function, 27 Mesopic vision, 115 Miyauchi 7x50 Binon, 90 Monocular, 146 Motion parallax, 134 Moustache distortion, 34, 178 Multi-coating, 44 N Natural frequency, 7 Negative lens, 17 Nikon 7x50 IF WX, 89, 204 8x30 EII, 146 Nitrogen purging, 96 O Objective achromatic, 19, 67 angle of field, 30 aplanatic, 69 apochromatic, 70 cemented, 68 ED, 69 Fraunhofer, 68 semi-apochromatic, 71 Steinheil, 68 triplet, 69 Opera glasses, 35 Optical axis, 11 Optical flow, 136 Optical glass SF3R, 166 Optical illusion cardboard effect, 173 colour adaptation, 131 colour stereoscopy, 134 globe effect, 140, 174 Mach bands, 121 simultaneous contrast, 131 Optics Gaussian, 7 geometrical, 4 Orthoscopic, 33 Over-magnification, 145

Index P Pancharatnam phase, 65 Parallax, 132 Parallel displacement, 5 Paraxial rays, 11 P-coating, 65 test, 182 Pentax 6.5x21 Papilio, 81 Perception, 123 Perceptual threshold, 124 non-additivity, 133 Performance atmospheric effects, 162 contrast based, 151 depth of field, 168 depth resolution, 172 ease of view, 202 handheld, 149 limiting magnitudes, 156 resolution based, 146 visual transmission, 166 Perger prism, 52 Perimetry, 108 Petzval shell, 22 sum, 22 Photochemical transduction, 114 Photopic vision, 114 Physical optics, 39 Piper’s law, 125 Polarisation, 64 Power series, 6 Primary focus, 29 Primary spectrum, 19 Principal axis, 11 Principal beam, 11 Principal dispersion, 8 Principal focal point, 12 Principal plane, 14 Principal refractive index, 8 Prism Abbe-König, 57 Hensoldt, 57 leak, 192 porro, 45 Porro (reverse), 81 Porro I, 47 Porro II, 49 reflective coating, 62 reverse Porro, 49 Schmidt-Pechan, 54 Uppendahl, 59 Prism-entrance width, 48

Index Prism width formula, 86 Pupil aberration, 30 Pupillary adaptation, 110 Pupil width, 109 Purging argon, 97 nitrogen, 96 Purkinje effect, 115

R Random point stereogram, 133 Rangefinder, 53 Ray bundle, 82 Rayleigh criterion, 41 Rayleigh-scattering, 163 Ray-tracing, 5, 11, 27 Real angle of field, 30 Receptive field, 119, 125 Rectilinear, 33 Rectilinear distortion, 174 Reduced path, 83 Reflectance, 42 Reflectivity aluminium vs. silver, 62 Refraction law, Snell’s, 4 Refractive index, 4, 5 formula, 8 Relative aperture, 31 Relative distortion, 34 Resolution, 146 measurement, 181 Resolution limit stereoscopic, 133 Resolution test chart, 147 Reticle, 91 Retina, 114 cones, 114 contrast enhancement, 121 rods, 114 Ricco’s law, 125 Rider prism (Porro II), 49 RISO-I 7x40, 173 Rods, 114 Rolling diaphragm, 96 Roof prism phase-correction, 65 spikes, 57, 197 Ross 10x50 Stepmur, 50

217 S Saccades, 119, 135 Sagittal plane, 23 SARD 6x42 Mark 43, 205 Schmidt-Pechan prism optical glasses, 59 path-length difference, 56 Seil-effect, 56 Scotopic vision, 115 Secondary spectrum, 70 Seeing, 162, 164 Seidel’s error theory, 7 Selective attention, 135 Self-luminous objects, 129 SF2 (optical glass), 19 Side pupil, 203 Signal addition, 119 Simultaneous contrast, 131 Sine condition, 24 Smyth lens, 22, 75 Spacewalk effect, 179 Special optical glasses, 10 Spectrum, visible, 3 Speed of light, 3 Spikes, 197 Spotting scopes, 70 Standard magnification, 144 Star test, 113, 198 Stereoscopic binocular, 172 Stereoscopy, 132 Stiles-Crawford effect, 117 Subjective angle of field, 30 Swarovski 10x42 EL WB, 76 Sweet spot, 199 Swift 8.5x44 Audubon ED, 69

T Tangent condition, 33, 174 Tangential plane, 23 Tele-lens, 68 Telescope, 29 Galilean, 35 Keplerian, 29 terrestrial, 35 Threshold contrast, 126 Threshold detection range, 128 Total internal reflection, 45 Total reflectivity, 44

218 Transmission atmosphere, 163 curve, 132 visual, 166 Trichromatic colour vision, 114 Tunnel diagram, 83 reduced, 83 Twilight index, 148

U Under-magnification, 144 Uppendahl prism rangefinder, 60

V Vari-angle stabilisation, 93 Vergence, 81 Vergence correction, 81 Vestibulo-ocular reflex, 175 Vignetting, 36, 84, 87, 91, 187 Virtual image, 17, 29, 36 Visibility, 162 Visibility factor, 157 Visual acuity, 123, 128 Visual pigment, 114 Visual space, 139

Index Visual transmission, 166 Visual-transmittance class, 9 W Waterproof, 95 testing, 96 Wave optics, 39 Weber-Fechner law, 126 Whiteout, 195 White paper test, 189 Z Zeiss 8x30 (Porro), 96 8x30 B Dialyt, 80 8x32 Victory, 79 8x42 Victory, 166 10x50 (Porro), 49 20x60 S, 93 Zeiss (Jena) EDF 7x40, 165 7x40 EDF, 183 Transmission, 166 8x30 Jenoptem, 88 8x60 DF, 202, 204 ZOMZ 6x30 Kronos BPWC2, 88 Zonula fibres, 107