UNIVERSITY OF LONDON Imperial College of Science and Technology Applied Optics Section THE VISUAL MACH EFFECT IN MICROSCOPY by Bohdan Maciej WATRASIEWICZ Thesis submitted for the Ph.D. degree 1964 2. ABSTRACT An eye presented with a transition between regions of different luminance levels may perceive bands or fringes parallel to the edge which are purely subjective, i.e. they are not present in the light flux distribution as measured by, say, a photoelectric cell. This phenomenon is called the Mach effect and the bands are called Mach bands. In this thesis we give a historical review of work on the Mach effect in the unaided eye and related topics in the study of the eye itself. Almost no work has been reported on the Mach effect in vision through instruments up to the present time but, as we show, there are fundamental differences from the effects seen with the unaided eye; in particular some remarkable effects are found with the microscope. We describe measurements of the positions and intensities of the Mach bands with the following varying parameters: degree of coherence, direction of polarization, wavelength and retinal illumination level. Also the effect of the object contrast is investigated. Finally, theoretical treatments of the Mach effect are reviewed and theoretical predictions are compared with the experimental measurements. 2a. PREFACE As this thesis incorporates some measurements made by other observers, I should like to state explicitly which items were originated and carried out by myself. They are: 1. A method for the preparation of the edge. 2. The visual photometer for intensity measurements on the Mach bands.(resign and construction.). 3. The improvement in stability of the scanning apparatus to enable scans extending over 20 minutes to be made. 4. Physical measurements. a) of _edge profiles at high numerical apeture. b) of edge profiles in the polarised light. c)discovery of hitherto unreported effect of "turned up" fringes due to defocusing in the interferograms of the mi:.Ircscope objective on Twyman—Green interferometer. 5. Subjective measurements of the Mach effect, a)in partially coherent light (at various degrees of coherence). b) at different levels of retinal illumination. 2b. c)variation of the Mach bands with wavelength. d) the effect of the object contrast on the bmnds. e) the effect of polarised light on the widths and positions of the Mach bands. 6. Realisation of the r6le of diffraction inflections in the Mach effect. 7. the investigation of theoretical models of the Mach effect by numerical computation. 3. CONTENTS Page No. ABSTRACT 2 2REFACE 2a CHAPTER 1: INTRODUCTION 1.1. Introduction 7 1.2. The retina 7 1.3. The Mach effect 14 1.4. Discussion of current theories 33 1.5. The Mach effect in microscopy 35 1.6. Notes on terminology 37 CHAPTER 2: APPARATUS AND TEST OBJECT 2.1. General introduction 41 2.2. The -nhotoelectric microscope 42 2.3. Additions to the apparatus 53 2.4. The photometric arrangement for visual matching 56 2.5. Test object preparation 60 2.6. Preparation of the neutral density wedge ' 63 2.7. Choice of the microscope objective 66 CHAPTER 3: PHYSICAL IMAGES OF A STRAIGHT EDGE AND OF CIRCULAR APERTURES 3.1. Introduction 73 3.2. Method of scanning 74 3.3. Luminance distribution along the line perpendicular to the edge 75 3.4. Images of a straight edge in polarized light 80 3.5. Diffraction images of apertures in opaque films 85 3.6. Conclusions 86 CHAPTER 4: THE DEPENDENCE OF THE MACH EFFECT ON LEVEL OF RETINAL ILLUMINATION AND ON COHERENCE 4.1. Introduction 90 4. Page No. 4.2. Method of measurement 91 4.3. Measurements at different S—values 95 4.4. Mach bands as a function of retinal illumination 107 4.5. Calculation and measurement of retinal illumination 108 4.6. Conclusions 117 CHAPTER 5: THE EFFECT OF WAVELENGTH ON THE MACH BANDS 5.1. Introduction 119 5.2. Method and results 119 5.3. Conclusions 121 CHAPTER 6: THE EFFECT OF EYEPIECE POWER, OBJECT CONTRAST AND POLARIZATION ON THE APPEARANCE OF THE MACH BANDS 6.1. Introduction 134 6.2. The effect of the eyepiece power 135 6.3. The effect of the objeot contrast on the appearance of Mach bands 138 6.4. The effect of the plane of polarization of the illuminating beam on the Mach bands 146 5 . Page No. CHAPTER 7: THEORETICAL TREATMENTS OF THE MACH EFFECT PARTS 7.1. Introduction 148 7.2. Mach's equation 149 7.3. Fry's approach 150 7.4. Treatments of Hartline et al. and Taylor 153 7.5. Approximate conditions for positive roots of equation (4) 158 7.6. Integral equations corresponding to the previous models 162 7.7. Comparison of various mathematical models 163 1. Introduction 163 2. Comparison of descrete models 164 3. Comparison of continuous models 165 4. Two—dimensional models 167 7.8. The solution of the alternate form of the simplified Hartline et al.equation 174 7.9. Solution of Taylor's equation 176 PART II: NUMERICAL CALCULATIONS 7.10. The numerical solution of tl,.o model of L.rtline et al. 181 7.11. The numerical solution of Taylor's equation 184 6 Page No . 7.12. Application of frequency response methods tc visual. problems 189 Appendix I Appendix II 137 CHAPTER 8: DISCUSSION 199 REFERENCES ACKNOWLEDGMENT S 2 2 7. CHAPTER 1. INTRODUCTION 1.1. Introduction. The Mach effect may be briefly described as a subjec- tive enhancement of contrast in optical images which takes the following form: an edge between light and dark regions, in which the chageover of luminance may be sudden or gradual, appears to the eye to be sharper than it is and it may appear to have light and dark bands parallel to the direc- 2 tion of the edge. This effect, first noticed by Ernest Mach a century ago but it is only recently that physicists and psychologists have begun to study it in detail. Here we are concerned specifically with the Mach effect in relation to images seen in optical instruments, a field which so far does not appear to have been investigated at all. The Mach effect is certainly connected closely with the mechanism of vision and in this introductary chapter we survey briefly the rele- vant parts of the body of known anatomical and physiological knowledge. 1.2. The retina. The optical system of the eye,comprising the cornea, the crystalline lens, et.c., produces a real image of the external world on the light-sensitive screen at the back of the eye which is known as the retina; it is in the trans- 8. mission of signals from this image on the retina through the optic nerve to the brain and their interpretation there that some part of the explanation of the Mach effect may lie. A transverse section through the retina, fig.1.1.1 Shows a ten-layer structure consisting of five different types of cell ; they are: 1. the. receptors. Theae: cells constitute the outer layer of the retina i.e. the layer furthest from the lens et.c. They possess light sensitive portion, the so called rods or cones, which project through an external limiting membrane towards pigment epithelium (layer 1 fig.l.l. ). 2.Bipolar cells.These illustrate. the general structure of a nerve cell or "neuron"; it consists of a cell body with extended nerve fibres or "processes" of which one kind, the dendrites, receives impulses: from other neurons: and the other kind, axons transmit impulses to other neurons; the connections are called "synapses" and they may involve two or more neurons. 1,he bipolar cells (layer 6 in the figure) relay the signals from the receptors to the ganglion cells, the next stage, by synaptic couplings of a wide variety and complexity. They are called bipolar because they have two main processes,' one a dendrite and the other an axon. 3.Ganglion cells. These cells are multipolar and are responsible for relay of signals to the brain; they are 9 01•IfTiZA:k tteif:6MIA an • ...a . • on. •.. e Fig. I. 1 The Layers of the Retina6° I. Pigment Epithelium 6. Inner Nuclear Layer 2. Rod and Cone Layer 7. Inner Plexiform Layer 3. External Limiting Membrane 8. Ganglion Cell Layer 4. Outer Nuclear Layer 9. Nerve Fibre Layer 5. Outer Plexiform Layer 10. Internal Limiting Membrane 10. similar to cells found elsewhere in the nervous system. The short dendritic processes face outwards to the bipolars; their axons are prolonged forming a nerve fibre (layer nr.9) converging towards the optic nerve. 4.Association cells.Functionally these are thought to be responsible for intra—retinal interactions; they allow one part of the retina to modify the response in the adjacent part, or in the same part after a time interval ( spatial and temporal induction).For the most part they intermingle with the bipolars. There are three types: horizontal, amacrine cells and centrifugal bipolars. 5. -Dieuroglial cells. These cells are assumed to insulate and support the conduction cells i.e. reurons in general; in the retina they pass between the two limiting membranes and hold the retina together. The micro—anatomy of the retina, as briefly described above, is well established but there are still unsolved prob— lems, e.g. the details of the cellular interconnections are not well understood. The: physiological processes involved in vision are not understood in detail but briefly some process such as the following is thought to occur. A quantum of light falling on the retina may initiate a photochemical change in a rod or cone, since both these contain phototropic pigments.