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Transport and distribution of light energy for illuminating engineering applications Whitehead, Lorne Arthur 1989

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TRANSPORT AND DISTRIBUTION OF LIGHT ENERGY FOR ILLUMINATING ENGINEERING APPLICATIONS by LORNE ARTHUR WHITEHEAD B. Sc., University of British Columbia, 1977 M. Sc., University of British Columbia, 1979 J A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES (Physics Department) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA April 1989 © Lome Arthur Whitehead, 1989 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. The University of British Columbia Vancouver, Canada Department DE-6 (2/88) ii ABSTRACT This thesis concerns the transport of light along a class of cylindrical hollow optical structures, with an aim to develop practical light transport and distribution systems for illuminating engineering applications. Based on intuition, it is postulated that it may be possible to guide light energy in a hollow cylindrical optical structure, by means of total internal reflection on the external surfaces. Such a development would allow light to be guided with the efficiency of optical fibres, but with reduced material costs. In order to assess this possibility, a new technique is presented for analytic ray tracing in general cylindrical structures. The technique makes it possible to trace ray paths in the cross sectional plane of the structure, with the motion in the dimension out of the cross sectional plane precisely taken into account with a simple correction. This technique greatly simplifies the ray tracing calculations which are necessary to study the light transport properties of the proposed structures. It is then shown with the aid of this technique that a certain class of prismatic structures do indeed have the capability of guiding light, and members of this class are termed prism light guides. In any real prism light guide, light is not conducted perfectly, but is lost as a result of a number of deviations from the ideal model. Of these, the one intrinsic loss mechanism, which distinguishes a prism light guide from optical fibres, is diffraction at the corners of the prismatic surfaces. Fortunately, diffraction effects are significantly smaller than losses from practical imperfections in prism light guides. iii A detailed study of the other types of loss mechanisms is then presented. These arise from imperfections of the optical material comprising the prism light guide (absorption and scatter), and imperfections in the shape of the prism light guide. The importance of these effects is discussed first in an approximate assessment, and they are then modelled precisely by means of computer ray tracing Monte Carlo techniques. These predictions are compared with experimental measurements of actual prism light guides, with substantial agreement. Consideration is then given to the use of a prism light guide in illuminating engineering applications. In these situations light is distributed along the length of a prism light guide, and is uniformly emitted from the surface. The performance of such a system is modelled with Monte Carlo computer ray tracing, and experimentally tested, with substantial agreement. This study concludes with a measurement of diffraction losses. These losses were enhanced by the use of longer wavelength radiation (3mm wavelength microwaves). The results confirm that diffraction losses are small relative to other loss mechanisms. In general, the results reported in this thesis demonstrate the existence of a useful class of light guidance structures which are particularly well suited to the requirements for distribution and transport of optical radiation for illuminating engineering applications. iv TABLE OF CONTENTS Abstract ii Table of Contents iv List of Tables vii List of Figures viii Acknowledgements xi 1 INTRODUCTION 1 2 DISCUSSION OF PREVIOUS LIGHT GUIDES . 4 2.1 Introduction 4 2.2 Hollow metal waveguides 4 2.3 Solid dielectric waveguides 5 2.4 Beam waveguides 8 2.5 Inability of simple hollow dielectric structures to guide light 10 3 RAY TRACING METHODS 12 3.1 Introduction 12 3.2 Computer simulation and the Monte Carlo method 12 3.3 A method of exact ray tracing in cylindrical systems 14 4 A NEW CLASS OF LIGHT GUIDANCE STRUCTURES 21 4.1 Introduction 21 4.2 Mathematical analysis of the proposed light guidance structures . . . . 21 4.3 Advantages and limitations of prism light guides 27 V 5 LOSS MECHANISMS IN IMPERFECT PRISM LIGHT GUIDES 28 5.1 Introduction 28 5.2 Diffraction . . 30 5.3 Bulk absorption 32 5.4 Large angle surface scatter 36 5.5 Small angle surface scatter 36 5.6 Large angle bulk scatter 39 5.7 Small angle bulk scatter 40 5.8 Edge imperfections 41 6 REALIZATION OF A LIGHT CONDUCTING CONCEPT 43 6.1 Introduction 43 6.2 Fabrication technique 44 6.3 Measurement of physical imperfection in moulded samples of prism light guide wall material 46 6.4 Measurement of light transport in a prism light guide 48 6.5 Monte Carlo modelling of a prism light guide 50 7 THE PRISM LIGHT GUIDE AS A DISTRIBUTOR OF LIGHT 57 7.1 Introduction 57 7.2 Implications on system design arising from intrinsic properties of prism light guide and light sources 57 7.3 Construction, testing and ray tracing analysis of a sample system 61 vi 8 MICROWAVE SIMULATION OF THE DIFFRACTION LIMITED PRISM LIGHT GUIDE 67 8.1 The importance of diffraction loss in prism light guides 67 8.2 Practical difficulties in direct measurement and theoretical evaluation of diffraction loss of optical radiation in a prism light guide 71 8.3 The microwave simulation approach 72 8.4 Experimental design requirements and how they were met 73 8.5 Experimental results 85 8.6 Interpretation of the diffraction data . 88 8.7 Extrapolation of ray tracing model to conditions in an optically perfect prism light guide 90 9 CONCLUSION AND SUGGESTIONS FOR FUTURE WORK 94 9.1 Conclusions 94 9.2 Suggestions for future work 95 REFERENCES 96 APPENDICES APPENDIX A l Optical measurement techniques 98 APPENDIX A2 Optical constraints on minimum aperture size for collimated light 109 APPENDIX A3 Measurements of bulk absorptivity of polytetraflorethylene to 3mm wavelength microwaves . . 113 APPENDIX A4 Interpretation of results from Monte Carlo ray tracing programs 115 APPENDIX A5 Calculation of average number of reflections undergone by light rays traveling down a square light guide 118 vi i LIST OF TABLES Table 5.1 Prism light guide loss mechanisms 29 Table 6.10 Monte Carlo predictions for attenuation in prism light guide vs. experimental results 56 Table 6.11 Magnitudes of various loss mechanisms in sample based on Monte Carlo program 56 Table 7.5 Comparison of computer model output to experimental data. . . 65 Table 8.1 Theoretical approaches to analysis of diffraction in prism light guide wall 70 vii i LIST OF FIGURES Figure 2.1 Cross section of hollow metal waveguide 6 Figure 2.2 Cross section of solid dielectric waveguide 6 Figure 2.3 Beam waveguide 9 Figure 2.4 A simple hollow dielectric structure cannot guide light 11 Figure 3.1 General cylindrical optical system 16 Figure 3.2 Graph of ratio of projection-corrected effective refractive indices vs. off-axis angle 19 Figure 4.1 Various linear prismatic structures 22 Figure 4.2 Examples of a new class of light guidance structures, having surfaces in "octature" 22 Figure 4.3 Cross section of prism light guide wall 24 Figure 5.2 The evanescent wave occurring with total internal reflection 31 Figure 5.3 Escape of evanescent wave flux at corner of prism 31 Figure 5.4 Definition of prism light guide wall dimensions 33 Figure 5.5 Transverse view of ray reflecting in PLG wall 35 Figure 5.6 Imperfect prism corners 38 Figure 5.7 Terminology for surface angular errors in cylindrical structure . 42 Figure 5.8 Loss mechanisms in prismatic wall 42 Figure 6.1 Photographs of molding apparatus 45 Figure 6.2 Photograph of molded prismatic panel 45 Figure 6.3 Conceptual diagram of measurement of large angle scatter in prism light guide molded wall sample 47 Figure 6.4 Electron micrograph of prism corner 49 Figure 6.5 Measurement of attenuation of luminous flux in prism light guide 49 Figure 6.6 Measured attenuation of luminous flux in prism light guide vs. distance 51 Figure 6.7 Pictorial view of light rays as traced by ray tracing program. 51 ix Figure 6.8 Definition of dimensions for Monte Carlo prism light guide simulation '. 54 Figure 6.9 Definition of terms in Eq. (33) 54 Figure 7.1 Attenuation per PLG wall interaction limits practical length of guide, as seen in graph of luminous flux vs. distance 60 Figure 7.2 The prism light guide illumination system (not to scale) 62 Figure 7.3 Cross section of prism light guide with cover (not to scale)... 62 Figure 7.4 PLG surface brightness: comparison of computer model to experimental data 63 Figure 7.6 Simplified light ray flow chart for the covered prism light guide 65 Figure 8.2 Microwave horn used for emission and detection of 3mm microwave radiation 74 Figure 8.3 Polar plot of relative gain vs. off-axis angle 74 Figure 8.4 Single panel detail of the PTFE prism light guide 74 Figure 8.5 Fictitious plot of log sampled intensity vs. distance along axis of prism light guide 76 Figure 8.6 Aluminum waveguide for spatial mixing of 3mm wavelength microwave 79 Figure 8.7 Example of phase-space distribution of intensity (in one dimension) at beginning and end of mixing guide, using dimensions shown in Figure 10.5 80 Figure 8.8 Microwave integrating structure 82 Figure 8.9 Orientation of emitting horn relative to input end of mixing guide 86 Figure 8.10 Graph of log intensity vs. distance of propagation down the PTFE prism light guide 87 Figure 8.11 Monte carlo simulation program results, compared to average of experimental results, for attenuation of microwave radiation propagating in a PTFE prism light guide 91 Figure A l . l The integrating sphere 99 Figure A1.2 Measuring bulk scatter and surface scatter 101 Figure A1.3 Graph of scattered light signal reading, V, vs. PMMA sample length, /. When the fullbeam was scattered in the integrating structure, the value of V was 1.322 V 101 Figure A1.4 Measuring sum of bulk absorption and bulk scatter 103 X Figure A1.5 Graph of logarithm of photomultiplier output voltage vs. PMMA sample length, /. The reference voltage V 0 = 2.65 V was measured with no sample in place 103 Figure A1.6 Measuring surface reflectivity of a diffuse surface. 104 Figure A1.7 Measuring transmissivity of a diffuse sample 104 Figure A1.8 Measuring sum of small surface angular errors and angular errors due to small refractive index inhomogeneities 105 Figure A1.9 Integrating light meter readings to obtain luminous flux from a source 107 Figure A2.1 Terminology for luminance theorem 110 Figure A2.2 Implication of luminance theorem for collimation of light from diffuse sources 112 Figure A3.1 Measurement of attenuation of 3mm microwaves passing through samples of PTFE 114 Figure A3.2 Measured transmission vs. frequency for various PTFE sample lengths 114 Figure A4.1 Graph of logarithm of fraction of rays absorbed in a sample Monte Carlo run, vs. logarithm of bulk absorption constant. Error bars represent probabilistic error due to Monte Carlo statistics 117 Figure A5.1 Determination of rate of reflection of light rays per unit length of guide 119 xi ACKNOWLEDGEMENTS A number of people have provided significant help in the wide range of activities described in this thesis and their contributions are truly appreciated. I thank John Berlinsky and Walter Hardy for providing a research environment which was conducive to the creation of the initial concept for this thesis, and I especially thank Roy Nodwell for his early and continued support for research into the prism light guide. In addition, I appreciate the help of Boye Ahlborn, Frank Curzon and Walter Hardy regarding experimental design and I also appreciate the guidance of the other members of my Ph.D. committee, Andrew Ng and Bill Unruh. Boye Ahlborn's help in planning and preparing this thesis was invaluable, and is especially appreciated. The experimental studies described would not have been possible without the dedicated assistance of research assistants and technicians at UBC, notably Phil Acres, Cobus Barnard, Jack Bosma, Dave Brown, Greg Chwelos, Rob Clark, Malcolm Greaves, Peter Haas, and Cy Sedger. Their efforts are very much appreciated. Additionally, significant contributions have been made by researchers at TIR Systems Ltd., and the 3M Corporation. I thank Barry Lee, Jon Scott and Brent York of TIR for their excellent research assistance, and additionally, Terri Orchard for her tremendous effort in preparation of this thesis. At the 3M Corporation, Roger Appeldorn, Sandy Cob, Einar Home, and Steve Saxe are to be thanked for their research, enthusiasm, and commercial commitment to the prism light guide. A number of individuals at other organizations have also been very supportive and helpful during this research endeavor. I thank Gary Albach, xii Dave Camm and Steve Richards of Vortek Industries Ltd.; Don Cox, Jack Lane, Ken Lund and Drummond McCorkidale of the National Research Council; Jack Cole and Bill McGarry of Energy Mines and Resources Canada; Steve Selkowitz and Mark Spitzglass of Lawrence Berkeley Labs; and, additionally, Bernard Crocker, Michel Duguay, David Eijadi, John Hawley, Newton Hockey, Roy Hughes, Shale Mahanty, Dave Miller and Chris Stephens. A number of organizations have provided either direct or indirect support for this research, for which I am very grateful. These are: The Canadian Electrical Association, Canadian Patents and Development Ltd., Energy Mines and Resources Canada, The Ernest C. Manning Awards Foundation, The Illuminating Engineering Society of North America, Lawrence Berkeley Laboratory, The 3M Corporation, The National Research Council, Natural Sciences and Engineering Research Council, and TIR Systems Ltd. Finally, I thank my parents, Mr. and Mrs. Arthur Whitehead, for my upbringing and their encouragement in this endeavor, and I especially thank my wife, Leah Whitehead, for her unwavering support and understanding. 1 Chapter 1 INTRODUCTION It is well known that there are many uses for devices which confine and transport electromagnetic radiation. With regard to the guiding of light, applications fall into two general categories - transport for the purpose of relaying information, and transport for the purpose of illumination. Although the optical principles are similar in the two areas, the practical requirements are so different that there is little overlap between the two fields. For example, the amount of light required in a typical illumination application may be one hundred thousand times greater than that required in a typical fibre optics communication link. To appreciate the differences between transporting bulk power and information, one need only remember the analogous situation for conduction of electricity. Although the conductivity of metals is the central theme, the optimization of performance leads to radically different solutions in the two cases. For example, a high voltage transmission line bears almost no resemblance physically to a television signal cable distribution system. In the former, the most important design considerations involved are resistance to dielectric breakdown, electrical conductivity, weight, tensile strength of materials, and atmospheric interactions. In contrast, in the latter the dominant concerns are precision of longitudinal symmetry, dielectric loss, efficient coupling, shielding, and frequency response. Thus, even though both areas of endeavor are concerned with maximizing the transfer of energy from one point to another, the fact that power and frequency ranges are very different leads to very different solutions. 2 With respect to light energy, it is interesting to note that while the field of transport of light signals for communication was the first to see large scale commercial development, transport for illumination purposes has a much longer history. There are a number of practical motivations for such light transport in illuminating engineering, which stem from limitations with conventional light sources. All sources of light produce significant quantities of waste heat, require maintenance, and are hazardous in certain circumstances. Furthermore, most practical efficient light sources are smaller and therefore brighter than would be desirable for illumination purposes. As a result of these facts, in the majority of situations where artificial light is required, it would be beneficial to have a practical way of transporting and/or distributing light from the source. The topic of this thesis is the study of the principles, limitations and opportunities that are associated with the transport and distribution of light energy for illuminating engineering applications. It will be shown that it is both possible and practical to transport large amounts of light energy in a type of hollow light guide. This conclusion is reached by theory, models, and measurements. Practical implications concerning feasible manufacturing methods are also considered. In detail: Chapter 2 discusses previous work on light guides for illumination purposes. Chapter 3 concerns ray tracing techniques, including a new ray tracing method which is useful in cylindrical systems. In Chapter 4, this method is employed to demonstrate the principle of the prism light guide - a novel device arising from this research, which combines the advantages of two previous forms of light guides. In Chapter 5 the various limitations of practical prism light guides are discussed, and in Chapter 6 these are studied by fabrication, testing, and modelling of this light conduction concept. Chapter 7 is concerned with the 3 integration of prism light guides into complete illumination systems and includes discussion of simulations and experimental testing of prototype lighting systems based on prism light guide. Chapter 8 concerns the ultimate potential transport efficiency for prism light guides, which is limited by diffraction effects. An experimental study of enhanced diffraction loss through the use of longer wavelength microwave radiation instead of light is described, and its implications are discussed. Finally, Chapter 9 contains general conclusions and suggestions for future work. 4 Chapter 2 DISCUSSION OF PREVIOUS LIGHT GUIDES 2.1 Introduction Since early history scientists and engineers have been concerned with illuminating our surroundings by delivering optical radiation to . people's place of work. Since it became possible to generate artificial light of high intensity, there have been attempts to build devices for guiding light for illumination purposes. Several principles have been employed, all of which have drawbacks which have precluded their widespread use. The following brief discussion about these guides is intended to provide relevant background information for the developments presented later in this thesis. 2.2 Hollow Metal Waveguides The concept of the hollow metal waveguide is perhaps the simplest form of light guide to understand. Its behavior is basically that of a standard electromagnetic waveguide, as is well known and well used in the guidance of microwave power. The only difference with a light guide is that the guide dimensions are very much larger than the wavelength of the electromagnetic radiation in question. Given this very large size, it is quite appropriate to view the operation of a hollow metal wave guide with the ray model for light, and Figure 5 2.1 shows the very straightforward manner in which light is guided down such a structure. The concept of using such a structure for guiding light for illumination purposes was actually patented in 18821, and has been studied more recently^, but this idea has never seen widespread commercial use. The reason for this fact is that, unfortunately, metallic reflectors have limited efficiencies. Two of the best materials for reflecting light are aluminum, with approximately 90% reflectivity averaged over the visible spectrum, and silver with a corresponding value of 95%. At first glance, these efficiencies seem quite good, however a typical light distribution system will involve more than ten reflections off such surfaces, so that the resulting efficiency is only 0.35 - 0.60, which is unacceptably low. Nevertheless, the hollow metal waveguide structure is truly intriguing because it involves so little material — the actual metallic reflector, for example, can be a film less than 1 micron thick, deposited on a thin, inexpensive substrate. 2.3 Solid Dielectric Waveguides Solid dielectric waveguides conduct light by a completely different principle, which is most often seen in the field of fibre optics. As shown in Figure 2.2, this waveguide is basically a solid linear structure of constant cross section, composed of substantially non-absorptive dielectric material^. Light is guided due to total internal reflection at the external surface of the conducting dielectric. It can be shown that for light rays within a certain critical collimation angle about the axis direction of the structure, total internal reflection will occur each time a given ray encounters the external surface of the conducting medium. 6 smooth metallic walls Figure 2.1 Cross Section of Hollow Metal Waveguide smooth outer surfaces solid transparent medium Figure 2.2 Cross Section of Solid Dielectric Waveguide 7 Such solid dielectric waveguides have the tremendous advantage of using a reflection phenomenon which is truly 100% efficient. However, they suffer the disadvantage of forcing the light energy to travel the entire length of the guide within the dielectric material, possibly leading to attenuation and scattering. Furthermore, the entire cross section of the structure must consist of such dielectric material. In the communications industry, where very small amounts of energy are transmitted, it is possible to employ very small diameter solid dielectric waveguides so that only a small amount of dielectric material is used, and this fact makes it practical to employ extremely low absorptivity dielectric materials for such communication fibres. This fact is partly responsible for the widespread success of solid dielectric waveguide structures in the form of optical fibres in the communication industry^. Unfortunately, the situation is quite different when it is necessary to transmit a large amount of light energy in a guidance structure having a large cross sectional area, as the actual cost of large amounts of dielectric material is then very significant. Thus, it is not even remotely feasible to use the high purity materials found in the communications industry to eliminate the problem of absorption of light which occurs because the light rays travel the entire length of the guide in the dielectric medium. Thus, while work has taken place on the subject of providing general illumination with light guided by solid dielectric waveguides^, and a number of specialized applications for this concept will exist in situations where cost and efficiency concerns are not too great, this approach is impractical for general illumination purposes. 8 2.4 Beam Waveguides Clearly, it is desirable to have a means of guiding light energy in such a way that the light spends most of its time in open air, to reduce bulk absorption losses without the use of metallic reflectors, which are necessarily absorptive. One possible solution is the class of beam waveguides6. Figure 2.3 is an example of a beam waveguide system which consists of a line of converging lenses positioned confocally. As shown by the example ray in the diagram, this array of lenses is capable of confining light rays which have a certain limited angular divergence about the central axis. There are a number of possible configurations for beam waveguides and many different types of lenses which can be used, but all employ the concept of refractive redirection and confining the rays about the central axis. While the beam waveguide concept does accomplish guiding of light rays predominantly in air without metallic reflection losses, the actual efficiency and cost effectiveness of these structures is somewhat limited. In order for a beam waveguide to be very efficient, the lenses employed must be free of aberrations and must be substantially free of surface reflection loss. The only lenses which can be inexpensively mass produced are molded polymeric Fresnel lenses, which exhibit very substantial aberrations and reflectivities. Lenses which would be desirable for beam waveguide operations are of photographic quality, and are thus too expensive for general use. Therefore, for the time being, the beam waveguide, while an extremely interesting optical concept, is not a solution to the challenge of guiding light for illumination purposes. 9 Figure 2.3 Beam Waveguide 10 2.5 Inability of Simple Hollow Dielectric Structures to Guide Light On considering the above, it would seem desirable to find a way to combine the efficiency of the total internal reflection phenomenon utilized in solid dielectric light guides with the cost effectiveness of open air transmission within hollow metal or beam wave guides. It is worthwhile to consider if a simple hollow pipe made of dielectric material, having an annular cross section and parallel walls, could achieve this desired effect. For example, one might wonder if light rays traveling in the air inside such a structure, subsequently entering the dielectric wall, and then encountering the external surface of the wall, might undergo total internal reflection under the right conditions. Unfortunately, it can be shown that this is generally not the case for simple pipes. The reason that total internal reflection would not occur for light rays propagating in such a structure is precisely the same as the reason that total internal reflection is not observed in planar windows. As long as the inner and outer surfaces of a refractive material are parallel, the refracted ray entering the material is always within the critical angle at the external interface and does not therefore undergo total internal reflection. As shown in Figure 2.4, in order for a ray to enter the medium from one side its internal angle Px will be smaller than its critical angle B c . For parallel surfaces, therefore, the incident angle p 2 = P i will also be smaller than 3 C and so that internal reflection cannot occur. Clearly then, a device which could cause total internal reflection to occur at the external surface of a hollow structure would require a different angular relationship between the internal and external surfaces. This train of thought is continued in Chapter 4, but it is beneficial in the development to first consider the various ray tracing techniques depicted in the next chapter. 11 Figure 2.4 A Simple Hollow Dielectric Structure Cannot Guide Light 12 Chapter 3 RAY TRACING METHODS 3.1 Introduction A complete understanding of any type of light guidance system would require a complete solution to the Maxwell equations. In practice, this is neither practical nor necessary. The basic principle of operation of any large light guide system can be understood in terms of the ray model of light, which assumes that the wavelength of the radiation is so much smaller than the characteristic dimensions of the structure that effects arising from this non-zero wavelength are insignificant. Later, we will consider much more carefully the accuracy of this assumption, with regard to specific cases. In this chapter various optical ray tracing techniques will be studied with regard to their applicability to light guides in general, and a particular ray tracing concept will be developed to simplify consideration of general cylindrical optical structures. 3.2 Computer Simulation and the Monte Carlo Method Mathematical ray tracing of many rays, with the help of a computer, is an ideal method for analyzing the behavior of complex optical systems, simply because there is in general no precise solution to the behavior of an arbitrary complex shaped optical structure. However, in a system in which light undergoes a large number of optical interactions, there is a fundamental drawback to ray tracing simulation, in that each single light ray actually becomes two light rays upon interacting with a surface (except in those cases where true total internal 13 reflection occurs). In typical light guides, it is not at all unusual for an optical path to involve about one hundred surface interactions, which means that a single initial ray splits up into about 106 rays in the course of these interactions. Since it is desirable to employ at least 104 input rays to achieve sufficient accuracy, and since the calculation of each interaction involves up to one hundred arithmetic calculations, the result is that the simulation of a single light guide would require at least 10 1 2 computations, which is a completely impractical number. There are two basic solutions for getting around this problem with computer ray tracing, neither of which is perfect. In one method the computer allows the number of rays stemming from a single original ray to grow to a certain point, and from that point on rejects the weakest of any subsequently created rays, allocating such minor rays' energy on a pro rata basis to the other rays in the group. This is a particularly sensible approach when the partially reflected rays are very weak, and therefore the partial reflections of partial reflections represent only a very minor effect. The disadvantage of this method is that it is intrinsically approximate, and there is always a danger that the ignored rays represent physically meaningful information. The alternative method which overcomes the unknown inaccuracies outlined above is known as Monte Carlo ray tracing^. In this method, a single ray is maintained throughout the entire calculation. At each interface which would result in both a transmitted and reflected ray, a calculation is performed to determine the fraction of the energy of the ray which is transmitted, reflected, and absorbed. The full ray energy is then assigned to one of these three categories, with a probability equal to the corresponding fractional energies as determined for that particular interaction. Thus a single ray is categorized as 14 absorbed, or as continuing on either the reflected or transmit path, and a similar calculation and decision is performed at the next interface. The beauty of the Monte Carlo method is that in the limit of an infinite number of randomly selected rays treated in this manner, the technique will perfectly simulate the exact optical performance of the device. Moreover, even with a finite number of rays, the probability distribution for the resulting statistical error is simple and precisely known**. Thus any desired level of accuracy can be obtained with confidence simply by using the correct number of input rays. A further benefit is that the design of the program is simple and requires no assumptions about what are and what are not important effects. 3.3 A Method of Exact Ray Tracing in Cylindrical Systems Of course, it is also useful to study ideal systems in which exact ray tracing is possible. Such analytic ray tracing is much more difficult in three dimensions than two, and the results of such analyses are generally difficult to understand intuitively. However, there is a type of optical problem in which one might expect some form of simplification, namely, where the optical system possesses translational symmetry in a direction Y , so that the shape in a cross-sectional plane perpendicular to Y is constant, as illustrated in Fig. 3.1. Examples of such general cylindrical systems include optical fibres, cylindrical lenses, and prisms. Since a cross-sectional diagram can completely describe such shapes, one is tempted to follow the path of a' ray as it would appear projected onto this cross-sectional plane (henceforth referred to as the transverse plane). It is easily shown that in this plane the angles formed at the intersection of the projected ray path and the optical interfaces do not obey Snell's law. In other words, the motion of the ray in the third dimension cannot 15 simply be ignored. However, as shown below, this motion can be taken into account by replacing the various refractive indices in the system, n[, with translation-corrected effective refractive indices, n'j, and then applying Snell's law to the projection of the ray on the transverse plane^. The values n'i are constant for any ray and are determined by the original ray direction and the values nj. Such an approach greatly simplifies ray tracing and provides an intuitively pleasing way to take the third dimension into account. Consider the general optical system in Fig. 3.1. The direction of translational symmetry is specified by the unit vector Y . The light ray direction in a region of refractive index nj is specified by the unit vector rj . nQ and Y 0 refer to the initial values for the ray. When a ray strikes an optical interface, the unit vector ^ will represent the normal to the interface at that point, directed toward the region the ray is entering. The assumed geometry requires that s -Y =0 (1) Snell's law can be written in a 3-D vector notation as ni( r i x s ) = nj( r j x s ). (2) The direction of the projected ray path on the transverse plane is represented by the unit vector Y-', where (r * t ) t r 1 " V l - ( r ( 3 ) If we let Is ' be the unit vector normal to an interface in the transverse plane, by virtue of Eq.(l) we have s'=s. (4) We can now consider an interface between a material of refractive index nj and refractive index nj. Let 0i be the angle between Y 'i and s '. In a truly 2-D system, these angles would be related by the following form of Snell's law: 16 symmetry ) Figure 3.1 General Cylindrical Optical System 17 ni sin0i = njsin0j (5) Let us evaluate sin0j in terms of our ray vectors. We have by definition and the fact that Y- ' and "s 'are perpendicular to Y that sin0 = r x s • t . . (6) Using Eqs.(3) and (4) to eliminate Y- ' and "s ' and again employing Eq.(l), we obtain after simplification r x s • t S l n 0 = -v/i - 2^ ( ? ) v l - ( r • t ) z Taking the dot product of (2) with Y , we obtain the relationship ni( r i x s • t ) = nj(r j x s • t ). (8) Solving Eq. (7) for (Y- x s • t) and substituting in Eq. (8), we obtain n f \ / l - ( Y i • Y ) 2 sin0i = nj-\/l-(r j • Y ) 2 sin0j (9) Now taking the cross product of Eq.(2) with Y and simplifying with the aid of Eq.(l), we obtain n i (r i • t ) = nj(r j • Y ). (10) Thus the number n(r • t ) is a constant for a given ray, and in any medium i will be given by ni( r i • t ) = n 0 (r 0 • t ), ( H ) where r o>no refer to the ray's initial conditions. Substituting Eq.(ll) into Eq.(9) and simplifying, we obtain - ^ n 2 . - n 2 Q ( r 0 • Y ) 2 sin0i - <\j n 2 ^ - n 2 Q ( r 0 * Y ) 2 sin0i (12) This is the same form as Eq.(5), if one replaces the refractive index n by the projection-corrected effective refractive index n', where 18 n' = A / n 2 - n 2 o ( V 0 . ? )2 (13) If we let 0 be the angle between r 0 and t , we can summarize with the generalized form of Snell's law for projections onto the transverse plane of 3-D cylindrical systems: n'jsin0i = n'jsin0j (14) where n" = - ^ n 2 - n 2 o c o s 2 ( 9 ) (15) The value of n' has the following physical significance: if we consider the ray path in space as normal to plane waves, the speed of apparent transverse motion of those waves in the transverse plane is easily shown to be c' = c/n' (16) in direct analogy to the usual 2-D situation. In media where n' is found to be imaginary, the entry of rays will be barred by total internal reflection. This is also the case when solving Eq.(14) yields a complex value for 0 in the medium the ray encounters. Equation 15 provides a new tool to analyze the propagation of light along general cylindrical structures. As an example of the insights it can provide, consider a cylindrical system consisting of two refractive indices, n 0 , and nj . Figure 3.2 is a graph of the ratio of the projection-corrected effective refractive indices, nj'/no', for various values of the ratio rii/nQ, as a function of the original ray angle 0. It is interesting to note that for light rays which are in the range of 60 to 90 degrees (that is, rays which are less than 30 degrees out of the perpendicular plane) the ratio of the projection-corrected effective refractive indices do not differ greatly from the ratio of the true refractive indices. In other words, the correction associated with the motion of the light rays out of the cross sectional plane is a minor one. This is the geometrical 1 9 Figure 3.2 Graph of ratio of projection-corrected effective refractive indices vs. off-axis angle. 20 situation which is common in lens design, in which such minor effects are "aberrations" from the ideal. In great contrast, and intuitively rather surprising, we see the divergent behavior near 9 = 0 . Since most light guidance applications will involve light rays in the range of 0 to 30 degrees, it is clear that the concept of the projection-corrected effective refractive index cannot be viewed simply as a small correction. These concepts will be of great benefit for the analysis of the new light guiding structures discussed in the next chapter. 21 Chapter 4 A NEW CLASS OF LIGHT GUIDANCE STRUCTURES 4.1 Introduction The problem at hand is to, find a practical configuration to conduct and distribute large amounts of light for illumination purposes. As indicated in section 2.5, simple hollow tubular structures cannot carry light by total internal reflection, and thus at the very least it is necessary to have some difference in angle between the inside and outside surfaces of any proposed cylindrical structure. Figure 4.1 shows various linear prismatic structures having this property. Of the infinite variety of such shapes, it is natural to consider a restricted class in which the outer corners form a 90° angle, due to the retro-reflective property of 90° corners. In particular, symmetry considerations suggest that the case in which the outer corners form 45° angles with respect to the inner surface might be particularly advantageous. This chapter concerns analysis of such optical structures. 4.2 Mathematical Analysis of the Proposed Light Guidance Structures Figure 4.2 shows three examples of the proposed class of light guidance structures, all of which have the following angular relationships: The inner surfaces are either parallel or perpendicular to one another.. Similarly, the outer surfaces are either parallel or perpendicular to one another. Finally, the outer surfaces are inclined at 45° relative to the inner surfaces. Of the three examples shown, it is anticipated that the regular shape in the third example, in 22 Figure 4.2 Examples of a New Class of Light Guidance Structures, Having Surfaces in "Octature" 23 which most of the cross section of the structure is air, would likely be the most practical design, but the foregoing mathematical analysis is completely general to this entire class of structures. Our hope is that structures of this type can guide light, primarily in the interior air space, by means of total internal reflection at the external guide surface. In this way the efficiency of total internal reflection can be combined with the low cost and efficiency of air transmission of light. This is most easily shown by determining the optical properties of the guide for rays in the transverse plane and then generalizing to three dimensions, using the projection-corrected effective refractive index procedure of equation 15. In Fig. 4.3, consider a ray in the transverse plane which has just left the interior space with refractive index ni and entered the wall material of refractive index n2- At any* point in the subsequent trajectory of the ray in the wall material, two angles a and 6 can be defined, a is the angle the ray makes with any inner surface of the pipe or with the perpendicular to that inner face, whichever is less. Similarly, B is the angle the ray makes with any outer face or the perpendicular to that face, whichever is less. It is easily shown that the values of a and 8 are constant for a ray and for all total and partial reflections of that ray within the wall of the guide. In addition, it can be seen that fi = (JI/4) - a (17) These two facts are a direct result of the prescribed geometry of the guide. There are two requirements for satisfactory operation of the guide. First, it must be possible for a ray which has entered the wall to reenter the interior space rather than undergoing total internal reflection at all inner surfaces. This necessitates that a<sin-l(4U (18) 24 medium 1 Figure 4.3 Cross Section of Prism Light Guide Wall 25 Second, it must be impossible for a ray to escape through an outer face; that is, it must always undergo total internal reflection at all external faces. This necessitates that 6 - s i n (jrij ( 1 9 ) Now the values of a and 8 obviously depend on the angle of a ray in the inner n i space when it strikes the wall, and on — . Providing that n 2 as will always be the case in practice, it is always true that a " s i n l^nfj ( 2 1 ) by virtue of Snell's law, and thus the first requirement (18) is satisfied automatically. Substituting (21) in Eq.(17), we obtain ^ t - ^ 1 ^ ) ( 2 2 ) To ensure the second requirement, (19), it must, therefore, be true that f - s i n - l ( ^ s i n - l ( ^ - ) (23) which leads to It is interesting that the solution to this 2-D problem shows that the device requires a wall material of unrealistically high refractive refractive index to ensure that total internal reflection will always occur at the outer surfaces. We can now generalize to three dimensions by considering an original ray in the inner space to make an angle 0 with the guide axis, determine the effective refractive indices n'j and n'2, and see what values of 0 will allow satisfaction of (24). Employing Eq.(15), we obtain 26 (25) n - j ' ^ n 2 1 - n 2 l C os2(0) n 2 ^n^2 ' n 2 i c o s ^ ( 0 ) Replacing ni/n2 in Eq.(24) by the above ratio of effective refractive indices and solving for 0 we obtain 0 < cos - 1 f n 2 1 - n 2 2sin2(7t/8) >l n 2 . n 2 sin2(jr/8) 1/2 (26) Providing n2 >ni , there is always a value of 0 small enough to satisfy (26), and the device thus behaves as described for rays which in the inner space are originally collimated to within an angle 0 m ax of the guide axis, where ©max is given by -1 ©max - COS" ' n 2 1 - n 2 2 sin2(jr/8) > n2j - n2jSin2(ji/8) (27) For a pipe with an air interior and an acrylic wall with n2=1.5, 0 m a x is -27.6° , a readily achievable degree of collimation for light guidance applications. Thus the class of structures shown by example in Figure 4.2 do in fact guide light, predominantly in the interior air space, by means of total internal reflection of the external surface. Henceforth, structures belonging to this class will be termed prism light guides. 27 4.3 Advantages and Limitations of Prism Light Guides It is worthwhile at this point to consider the advantages and limitations of prism light guides. To keep these in perspective, it is important to recall from Chapter 2 that all other light guidance structures suffer intrinsic loss mechanisms at optical interfaces, or require that the optical radiation spends all of its time within a dielectric structure. In contrast, from the perspective of the ray model of light, the mathematically ideal prism light guide is an optically perfect device — within the collimation angle accepted by the guide, all radiation undergoes total internal reflection in the wall of the guide. Moreover, light rays spend the vast majority of their time in the airspace within their guide, and therefore absorption in dielectric material can be very small. Thus, the prism light guide has tremendous potential with respect to the lighting requirements anticipated for distribution of light for illumination purposes. However, these conclusions are made for a precise theoretical optical design, which is free of construction inaccuracies and suffers neither dielectric imperfections nor diffraction, and it is of course impossible to construct a light conductor which is completely free of these real effects. Thus, it is critical to consider how sensitive the optical behavior of a prism light guide is to imperfections in the optical shape and optical materials qualities, and diffraction effects. These are the subjects of the next chapter. 28 Chapter 5 LOSS MECHANISMS IN IMPERFECT PRISM LIGHT GUIDES 5.1 Introduction In a prism light guide as described in Chapter 4, various types of losses may occur, attenuating the beam as it travels through the guide. One of these losses is of a principal nature, and the others are due to imperfections of present day materials and fabrication techniques. A practical prism light guide will always deviate to some extent from the ideal mathematical shape because of unavoidable errors in the fabrication process. In general, such a practical device will have an indefinably complex shape, but it is possible to characterize the nature of the shape in terms of a model which incorporates numerically defined deviations from the ideal shape, and from ideal optical material performance. Table 5.1 lists all imperfections concerned. The first of these loss mechanisms, diffraction, exists even in perfect prism light guides. Fortunately, as described first in section 5.2 and in more detail in chapter 8, it turns out that diffraction at this time is the least serious loss mechanisms in practical structures of this type. After considering diffraction, the rest of this chapter is devoted to losses that dominate the practical prism light guides due to the state of the art in material impurities and surface fabrication techniques. 29 Loss Type Loss Mechanism Diffraction Wavelength non-zero Absorption Imaginary component of dielectric constants Scatter Large surface angular errors in small areas Small surface angular error in large areas Bulk large angle scatter sites Bulk inhomogencity of dielectric constant Imperfect prism edges Table 5.1 Prism Light Guide Loss Mechanisms 30 5.2 Diffraction In order to estimate the effects of non-zero wavelength, it is worthwhile to review in more detail the concept of total internal reflection. The exact electromagnetic solution for total internal reflection for plane wave is well known. Within the incident dielectric medium, the solution is simply the sum of the incident plane wave and a reflected plane wave, corresponding to the incident and reflected ray in the ray model of light. In the external medium the solution consists of an "evanescent wave" corresponding to a flow of electromagnetic radiation parallel to the surface and also parallel to the plane containing the incident and reflected directions. If the incident wave power flux is I 0 , and the incident angle is 0, then the intensity of the energy flux in the evanescent wave is given by I e = 2Iosin0e \le J where z is the distance from the interface in the external medium 1 ^- 1 1 ' 1 2 . The value of l e depends on 0, ni and n2, but is always of order X. This information is shown pictorially in Figure 5.2. One way of thinking of the flux in the evanescent wave is to consider that a portion of the energy flux being transferred from the incident to the reflected wave must actually flow through the external medium as total internal reflection occurs. This "evanescent portion" is an infinitesimally small fraction of the incident flux for an infinite plane wave, and for a plane wave striking a region of width d as shown in Figure 5.2, this fraction will be of the order of XI d. Considering now that prism light guide wall as shown in Figure 5.3, in which the width d corresponds to the prism width, it seems plausible that the X "evanescent fraction" -r would escape at the edge of the prism, as shown pictorially. This ratio can be thought of as the ratio of light spilling out at the corner (via a beam of width X, with an average intensity I 0 for a power loss of 1(A). to the total reflected power of approximately Iod. Of course, this argument is only 31 Figure 5.2 The Evanescent Wave Occurring With Total Internal Reflection incident beam portion which interacts ^ with one prism surface Figure 5.3 Escape of Evanescent Wave Flux at Corner of Prism 32 a very rough plausibility discussion, and certainly does not represent an exact solution of the Maxwell equations for this very complex electromagnetic problem. However, this is taken to be a good estimate of the magnitude of this effect, for comparison with other possible loss mechanisms. We will see later that the nature of the other losses associated with practical imperfections in real devices will dictate prism sizes in the range of .5 mm to 5 mm, and that in these situations we therefore expect diffraction losses to be roughly in the range of .1% to .01%. This is too small to be noticeable compared to the other losses of today's best prism light guides. However, for completeness, in Chapter 9 diffraction effects will be more carefully addressed for an ideal future embodiment that is free of other imperfections. 5.3 Bulk Absorption One of the primary benefits of hollow light guides of the prism light guide type is that at any point in the conductor most of the electromagnetic energy is traveling through the air in the interior of the guide, rather than in the dielectric material of the wall, so that the effect of absorption in the bulk material comprising the wall is far less serious than that would be the case in a solid dielectric light guide of comparable size. Nevertheless bulk absorption is an extremely significant effect in the performance of prism light guide, in part because it is the only mechanism in the device which actually causes true loss of electromagnetic energy. (All the other loss mechanisms simply cause deviations of the propagation direction of rays which allow energy to escape from the device, so that this energy could still be used for lighting.) As shown in Fig. 5.4, the portion of the ray which enters the dielectric section travels an average cross-sectional distance / +2t within the dielectric, I 33 Figure 5.4 Definition of Prism Light Guide Wall Dimensions 34 being the length of the prism hypotenuse. However, before entering the dielectric, the ray is also moving in the direction of the guide axis with an angular deviation 8 less than 6m ax> as shown in Fig 5.5. The true average distance traveled in the dielectric will therefore be / + 2t 1 + 2t / + 2t cos 4> V , . c o s (Q ) V l - l / n ; n 2 (since cos9 ~ 1) (28) where n is the refractive index of the medium. The dielectric attenuation per reflection through the medium will therefore be: . V l - 1/n 2 assuming k d « l , as will be the case, where k is the attenuation per unit length in the dielectric. For a prism light guide made of polymethylmethacrylate with typical absorption^3 k=0.15m-1, and prism dimension 3mm, this yields an absorption of approximately 0.12%. While this loss will, in most circumstances, be larger than diffraction, it is nevertheless very small, particularly when compared to absorption losses of metallic surfaces, which are typically two orders of magnitude greater! 35 Figure 5.5 Transverse View of Ray Reflecting in Prism Light Guide Wall 36 5.4 Large Angle Surface Scatter Generally, it is assumed that the surface is quite similar to the ideal prismatic shape, and under these conditions, it is felt that there are two important types of surface shape imperfections to consider. Surface large angle scatter is an effect generally caused by imperfections which comprise a small fraction of the surface, but represent very large angular errors in those small areas. Physical examples are scratches and dents in the surface. The simplest way to model this type of imperfection, is to simply define the probability that a given light ray on a given interaction with the surface will in fact strike an imperfection region. The magnitude of wide angle scatter probability can be very low in high quality optical systems. In glass lenses, for example, its magnitude can be significantly less than one tenth of one percent, while in relatively low quality plastic optics it may be between .1 and 1 percent. Even with plastic optical surfaces it is possible with great care to reduce this probability to below Since a typical interaction with a prism light guide wall will involve three surface interactions, this loss can reasonably be expected to be of order .3% per interaction with the wall. 5.5 Small Angle Surface Scatter A light ray on a given interaction with the surface has a small defined probability of a large angular error occurring, but in addition it would be expected that for all interactions there would be some kind of probability distribution of small angular errors corresponding to extremely slight deviations from the ideal optical shape. Examples of causes of this type of error would be 37 inaccuracies in the fabrication process, changes in shape due to thermal effects, aging, etc. Obviously, the probability distribution for such angular error may be quite complex, and is, of course, a two dimensional distribution in solid angle. Particularly in view of the fact that a given light ray will undergo many surface interactions in the course of its travel down the guide, it would seem to be a reasonable simplification to model the probability distribution as a Gaussian, since the effect of convoluting several distributions as a ray progresses down the guide will be a rapid approach to a Gaussian distribution anyways. At this point it is worthwhile to define a vocabulary for surface angular errors, which is useful for three dimensional cylindrical systems. Figure 5.6 shows for a point on a particular surface in a cylindrical three dimensional system, the direction of the ideal normal vector nj, (which would be perpendicular to the surface if there were no angular error), and the actual normal vector n a . The angular deviation of n a relative to n i can be described by two quantities, A z and A p , defined mathematically as follows: A z = (ni x na) • t (30) where t is the axial direction of the cylindrical system, and A p = (ni x na) • (nj x t). (31) Intuitively, A z can be thought of as surface angular errors corresponding to rotations of the normal vector about the symmetry axis t, and Ap can be thought of similarly as rotations of the normal vector in a perpendicular direction. Thus, A z corresponds to errors in cross sectional shape, whereas A p corresponds to errors associated with change of the cross sectional dimensions down the length of the guide. Although it is tempting to do so, it would be an over-simplification to make the probability of an angular error in the surface normal vector a Gaussian Figure 5.6 Imperfect Prism Corners 39 function of the absolute value of the angular error, ( A P 2 + A z 2 )1 /2 . The reason for this is that the physical effect of angular error of the normal vector corresponding to A Z may be very different than that of A P (this will be seen to be the case in the next chapter). Therefore we will represent the probability of a small angular deviation as follows: -rp4v Pill P ( A P , A Z ) a e V V V J \a^JJ (32) where a p and cxz are the standard deviation for errors in the two directions. With glass optical systems, it is quite feasible to achieve angular deviations which are so small that the deviation from the ideal shape is significantly less than a wavelength of light throughout. With plastic optical systems, this is much more difficult. Nevertheless, as will be seen later, it is quite feasible to achieve angular errors less than 10" 3 radians, and in some cases as low as 10"4 radians. With respect to the impact on the efficiency of a prism light guide, the situation is extremely complex, but a reasonable first guess would be that the fraction of light lost per reflection will on the average be comparable to the relative magnitude of the angular errors. 5.6 Large Angle Bulk Scatter This type of imperfection corresponds to bubbles, and other scattering centres in the bulk material comprising the wall of the prism light guide. This effect is quantified in terms of a probability per unit length that a ray will interact with the scattering centre and undergo a large angle random angular displacement. It is expected that the effect of this type of imperfection will be virtually identical to that of large surface scatter, but that the intensity of the 40 effect will depend on the average path length in the bulk material, in the same way that bulk absorption does.' In anything other than perfect single crystals, there will be a lower limit to bulk large angle scatter resulting from Raleigh scattering from the solid mater ia l ly . i n general, however, this effect is smaller than other sources of scatter such as particulate impurities in the bulk material, trapped microscopic air bubbles, etc. With commercial grade acrylic plastics, for example, this scatter rate is approximately 0.1 m"l, which would result in a loss of .2%^ for a prism light guide wall interaction having a typical path length of 2 cm. Techniques for measuring scatter and other dielectric properties are addressed in Appendix 1. 5.7 Small Angle Bulk Scatter This effect results from inhomogeneities of the refractive index of bulk material, such that a ray undergoes random small changes in angle while passing through the medium. Because of the "random walk" aspect of these angular errors as they accumulate in passing through the bulk materials, the root mean square angular error generated after traversing a distance d is given by the expression A = V a d where the coefficient a is a characteristic of the material for this type of scattering. As might be expected, there are large variations of the magnitude of a with various materials, ranging from 10"2 nr 1 for certain treated acrylic plastics used in optical fabrication^, to virtually immeasurably low values for properly prepared samples of optical grade glass^. As a final note with respect to bulk angular scatter of the two types described in this chapter, it is expected that the nature of their impact will be no different than that of angular error at the surface, since it is not until a ray 41 reaches the surface that its angular error has any impact on the prism light guide. This fact will be employed in simplifying the computer ray tracing studies in chapter six. 5.8 Edge Imperfections One of the challenges in fabricating the prism light guide is that it is in general a difficult task to produce optically perfect sharp edges. Therefore it is reasonable to expect that within a corner region of size c, as shown in Fig. 5.7, the average surface angular error will be certain to be very large. This is a very simple model of the nature of the actual edge, but corresponds very well to observations of typical optical corners with a scanning electron microscope1^. It is common to find a very smooth, well defined planar surface up to a certain distance from the theoretical corner, beyond which the surface shape is very poorly defined. Again, there is extreme variation in the parameter c in various fabrication systems. One extreme example is commercial injection molded plastic articles, for which c may exceed 50 microns^, with the other extreme being precision prisms in optical instruments, where it is possible for c to be of order .1 micron or less19. In terms of the effect on prism light guide, it is virtually certain that the impact of imperfect corners will cause a loss per reflection of the order of the ratio of c to the edge separation distance, /, which for a sample case of 5 micron edges and prisms with d = 3mm would correspond to about 0.12%. Figure 5.8 is a pictorial drawing displaying all the loss mechanisms which have been described in this chapter. 42 Figure 5.7 Terminology for Surface Angular Errors in Cylindrical Structure (\) bulk small angle (5) bulk large angle (§) corner defect (4) surface small angle (|) diffraction © surface large angle (3) bulk absorption correctly reflected ray Figure 5.8 Loss Mechanisms in Prismatic Wall 43 Chapter 6 REALIZATIONS OF A LIGHT CONDUCTING CONCEPT 6.1 Introduction To summarize the results of the previous chapters, a practical method for fabricating prismatic light guides must be capable of producing the required optical shape with a minimum of physical imperfections. Particularly challenging are the requirements that the surface finish of the prismatic materials be optically smooth and flat, with the corners very sharp. Such properties are difficult to achieve by optical polishing, and are also difficult and expensive to achieve by direct machining, even with the highest level of optical machine tool technology available today. A further requirement for a practical system is that the production technique itself should be relatively inexpensive and should involve reasonably priced optical materials with very low bulk scatter and absorptivity. Upon investigation of these requirements, the only practical fabrication technique involves the molding of optical quality polymeric materials such as polymethylmethacralate (PMMA). However, conventional techniques of extrusion or injection moulding are not particularly good at producing flat, smooth surfaces with sharp features. Therefore production of the prism light guide, even in test quantities, was quite challenging. This chapter summarizes the experimental work involved in producing a prototype prism light guide, optical testing of its transport qualities, and ray tracing simulation of the imperfections in its optical behavior. 44 6.2 Fabrication Technique The first attempt to produce prism light guide walls of sufficient quality was done with the help of a heated molding press. In this technique, a pre-fabricated sheet of thermoplastic (in this case PMMA) is placed between two surfaces having the optical details required, with virtually no pressure applied at first. The assembly of the two mold surfaces and the polymeric sheet are then heated to a temperature which softens the sheet but does not melt it (approximately 200°C). In this softened state a moderate pressure is applied to the mold surfaces by a hydraulic jacks (approximately 106 Newtons per square meter), which forces the polymeric material to very accurately adopt the shape of the mold. Without removing the pressure, the entire assembly is then cooled to a temperature at which the acrylic is very stiff (approximately 80°C). During this cooling, the acrylic material shrinks volumetrically by approximately 5%, but is prevented from losing precision contact with the mold surfaces by the extreme applied pressure. (This is critical to ensuring that precise replication of the surface is maintained as the material hardens.) Because of the volumetric shrinking, the acrylic is under considerable tensile stress while cooling. Providing the shrinkage is not too great, this stress simply causes the plastic to pull itself off the prismatic surfaces, when the pressure is removed. Production of the actual mold surfaces was done by the machining and optical polishing of stainless steel elements which when bolted together formed the desired shape. Figure 6.1 shows the molding apparatus used to produce the samples and Figure 6.2 is a photograph of the prismatic materials produced. Figure 6.2 Photograph of Molded Prismatic Panel 46 6.3 Measurement of Physical Imperfection in Molded Samples of Prism Light  Guide Wall Material In order to develop an understanding of the loss mechanisms in the prism light guide, a series of measurements were made regarding the optical quality of both the bulk PMMA and the molded surfaces. The bulk absorption constant was determined from measurements of the sum of bulk scatter and absorption, and of bulk scatter alone, as described in Appendix 1, with a resulting value of 0.15 ± 0.05 m"1 . The bulk absorption depends on the chemical make up of the particular commercial brand of PMMA employed, and for this reason it was deemed most appropriate to measure it rather than using published information with a greater level of uncertainty. The measured value we obtained was also consistent with general observations of attenuations found by personnel at the solid dielectric waveguide construction facility of the TRIUMF nuclear accelerator in Vancouver, who generally employ the same material for guiding the light output from scintillators to photomultiplier tubes20. The surface scatter of the molded material was more difficult to measure, as the actual surface scatter fraction was quite small. The measurement method employed was that shown in Figure 6.3, in which a beam of light from a helium-neon laser was directed through a sample of the material which was mounted in an integrating sphere. With the arrangement shown, only the scattered light would be detected by the sphere and this could be compared to the total beam intensity by closing the exit aperture shown. Furthermore, due to the short optical path length in the PMMA, the contribution from bulk scatter was negligible. The result was a total scatter of 6.0 ± 0.5% from the three surfaces interacting with the beam in Figure 6.3, corresponding to 2% scatter per surface. 47 Figure 6.3 Conceptual Diagram of Measurement of Large Angle Scatter in Prism Light Guide Molded Wall Sample 48 It should be noted that this level of scattering is substantially higher than that which could be obtained with higher quality moulding surfaces. The size of the defect region associated with the corners was determined by observing the corners with a scanning . electron microscope, as shown in Figure 6.4. As can be seen, the comers had a defect size of 2pm, which is extremely good compared to the typical value of 50pm obtained by commercial injection molding techniques^. Finally, the angular accuracy of the material was determined by observing angular deflection of a laser beam passing through the prismatic material, as described in Appendix 1. Based on these deflections, it was determined that the average angular error of the surfaces corresponding to rotations about the longitudinal axis of the guide was .005 ± .001 radians, and that the average angular variation corresponding to deflections in the perpendicular direction was .001 ± .0003 radians. Based on these results, it was felt that the samples were of sufficient quantity to obtain meaningful measurements of the behavior of an actual prism light guide formed of this material. 6.4 Measurements of Light Transport in a Prism Light Guide The press moulded PMMA was used to construct a number of lengths of prism light guide, each 1.1 m long and 130 x 130 mm in cross section. The guides were constructed by bonding together the four wall sections with lens cement21. The guides were tested by butting six 1.1 m long sections end to end and shining into one end white light collimated to within 15° of the axial direction. Figure 6.5 shows the experimental arrangement used to measure the luminous 49 Figure 6.4 Electron Micrograph of Prism Corner prism light 9 u ' d e integrating cube light frosted » front ; optical fibre bundle white interior photo multiplier Figure 6.5 Measurement of Attenuation of Luminous Flux in Prism Light Guide 50 flux arriving at various points along the guide. An incoherent quartz fiber-optic bundle cable was used to sample light from the integrating cube. By illuminating various portions of the cube input surface with a He-Ne laser beam, it had been verified experimentally that this light sample weighed all regions of the front of the cube equally to within a few percent, so that a photomultiplier connected to the cable would give an accurate proportional reading of total light flux in the guide's air space at the point of measurement. Figure 6.6 is a graph of the natural logarithm of the light flux level as a function of distance down the guide. In the next section, this behavior is simulated with a ray tracing model. 6.5 Monte Carlo Modelling of a Prism Light Guide The Monte Carlo ray tracing method described in Section 3.2 was used to model the prototype prism light guide 2 2 . The program makes use of the large amount of repetition and symmetry in the prismatic structure to simplify the calculation. The program models a square guide with an arbitrarily large number of prisms, but the actual computation occurs within one half of one prism. The actual optical calculation begins with a determination of energy loss as a ray travels through the bulk material (given by I = I0e"k<^ where k is the bulk absorption constant and d is the distance traveled to that point), and a more complex calculation of the nature of the refracted and reflected ray at an interface. The latter calculation is of course based on the Fresnel equations, and takes into account both the direction of the ray relative to the normal vector to the surface at the point in the question, and the polarization of the ray (generally elliptical). 51 0 1 2 3 4 5 D (metres) Figure 6.6 Measured Attenuation of Luminous Flux in Prism Light Guide vs. Distance typically 104 rays per run inner air core T absorbed transmitted to end of guide typically L/w = 30 T Figure 6.7 Pictorial view of Light Rays as Traced by Ray Tracing Program 52 Each ray is given a random initial position, direction, and polarization at the input end of the guide, subject to the constraint that the angular deviation from the axial direction be less than an angle 0 C , where 9 C is the collimation angle of the incident light. The program then traces the ray's path through the system, as shown pictorially in Figure 6.7. At an interface the incident ray divides into reflected ray and transmitted ray. As described in Section 3.2, it would be prohibitively expensive to follow an exponentially growing number of rays. Therefore, at each interface the path of only one of the rays created is followed to the next interface, with the choice being made by the Monte Carlo selection process in which the probability of selecting the reflected or transmitted ray is equal to the corresponding reflection or transmission coefficients which are determined by the Fresnel equations. Angular errors in the wall surfaces are modelled by assigning a 2-D probability distribution for the direction of the normal vector (see Fig. 5.6) to the surface, with the distribution being different in two types of regions—the jagged corner regions of width c as discussed in Section 5.8 being the first and all other regions being the second. In the corner regions, over a distance c/2 from the apex, the normal vector probability distribution is modelled to be uniform over all possible angles. In other regions, the distribution is modelled to be Gaussian, with standard deviation a z for angular variation of the normal vector in the transverse plane and O p for variation perpendicular to that plane, as described in Section 5.5. In addition, a small fraction of probability s is assigned uniformly over all possible angles. In this way, the Gaussian part of the distribution models the effect of slight errors in shape in the guide wall, while the uniform part models the effect of small surface scratches and marks which cause wide-angle scattering, as described in Section 5.4. 53 Absorptivity of the wall material is taken into account by assigning a probability per unit length k of complete annihilation of a ray, which is equal to the bulk absorptivity. Small and wide angle scattering of the light ray in the bulk material was not modelled because these effects were very small relative to the analogous surface scattering and would be similar in effect. For sufficiently small values of c,a z,o"p and s, the loss mechanisms contribute independently to the attenuation of light. With the pipe dimensions specified by / and w in Fig. 6.8 and assuming that G c , the collimation angle, is small, the program results for attenuation per unit length (see Appendix 4) can be summarized in the approximate empirical formula: d lnl c6c lk0 c o79 c a_0c s8c ^ j ^ s (0.55)7-*- + (2.9)—~+ (2.0)—+ (20)-E-1+ (1.7)— (33) dz 1 w w w w w In this equation, the first term corresponds to corner loss, the second to absorption loss in the bulk material, the third to light escape due to angular error A z , the fourth to light escape due to angular error A p , and the fifth to light escape due to surface scatter. The numerical coefficients for each parameter were determined by setting the other parameters to zero, and varying the parameter in question as described in Appendix 4. Table 6.9 summarizes the definitions of the parameters in equation (33). A more convenient way to describe this attenuation is in terms of the loss of light per reflection off the guide wall. The mean number of reflections per unit length is shown in Appendix 5 to be d i N l _ £ A ( 3 4 ) dz "37CW K I * > and the fraction of light lost per reflection L r can then be determined from Eq. (33) to be L r = (0.65) j + (3.4)lk + (2.3) o z + (24) a p + (2.0)s (35) Figure 6.8 Definition of Dimensions for Monte Carlo Prism Light Guide Simulation I - number of rays at point in question e c - maximum angular deviation of rays from axis c corner dimension (See Fig. 5.7) 1 - prism width (See Fig. 5.4) w - guide width (See Fig. 6.8) k - bulk absorptivity "parallel" r.m.s. surface angular error (See Fig. 5.6) <*? -s "perpendicular" r.m.s. surface angular error (See Fig. 5.6) probability of random large angle surface scatter Table 6.9 Definition of terms in Eq. (33) 55 It is interesting that the coefficient of a p is about ten times larger than that of a z, indicating that angular variations stemming from waviness down the length of the guide are correspondingly more deleterious than variations of the angles in the cross-sectional plane. For parameters resulting in values of L r of more than a few percent, Eqs. (33) and (35) will become inaccurate, because of interactions between various types of loss. Angular errors in the surfaces, for example, tend to increase the time a ray spends in the wall, and this changes the importance of the bulk absorptivity of the wall material. For such cases it is simplest to just run the program for the parameters of the guide to be studied. Such a run was performed using the measured values from Section 6.3 for the loss mechanisms of the prototype prism light guide, with the resulting predictions for transmission shown in Figure 6.6 and Table 6.10. The errors stated for the Monte Carlo runs are the predicted standard deviations due to statistical fluctuations8, and the estimates in the experimental measurements were, simply an estimated reasonable upper limit, which exceeded the limits of reproducibility of the apparatus. Table 6.11 depicts the magnitudes of the various loss mechanisms, as predicted by this program, as well as the ratio X/d, which relates to diffraction effects. To summarize, the measurements of attenuation agree quite well with the Monte Carlo model, which suggests that the model captures the important loss mechanisms for the light rays in this structure. In other words, from the comparison of the measured loss rate to that in the Monte Carlo calculation, which does not include diffraction effects, one concludes that diffraction does not play a significant role for prism light guides of the present construction. 56 Distance -In I/Io -In I/Io (m) Monte Carlo from measurements 1.1 .15 ± .007 .11 ± .03 2.2 .29 ± .010 .31 ± .03 3.3 .45 ± .014 .44 ± .03 4.4 .55 ± .016 .56 ± .03 5.5 .71 ± .019 .72 ± .03 Table 6.10 Monte Carlo Predictions for Attenuation in Prism Light Guide vs. Experimental Results Loss Type Loss Parameter Input to Program Loss per Reflection Corner Defects c = 2 um .00015 Bulk Absorption k = 0 .15 n r 1 .0043 Cross Section Angle Error CTz = .005 .012 Perpendicular Angle Error a D = .001 .024 Surface Scatter s = .02 .040 Total .080 X/d .000083 Table 6.11 Magnitudes of Various Loss Mechanisms in Sample Based on Monte Carlo Program 57 Chapter 7 THE PRISM LIGHT GUIDE AS A DISTRIBUTOR OF LIGHT 7.1 Introduction As was seen in the previous chapter, practical prism light guides have inherent loss mechanisms which limit the efficiency of conducting light from one place to another. However, it is an interesting fact that for illumination purposes, light should not only be ducted from one place to another, but it must also be spread out and distributed. Since most of the loss in a practical prism light guide takes the form of light escaping from the guide, rather than being absorbed by the dielectric material, it is therefore possible to utilize this loss as a means of providing light. As a result, the fraction of the light which enters the prism light guide and which is subsequently utilized for illumination purposes can be very high, even over lengths for which the prism light guide is not an efficient light conductor. This chapter explores constraints on the optical design of such light distribution systems, which arise from the optical properties of prism light guide and of available light sources. This is followed by a discussion of experimental study and computer ray tracing simulation of a sample prism light guide light distribution system. 7.2 Implications on System Design Arising From Intrinsic Properties of Prism Light Guide and Light Sources The most common illumination design requirement calls for a uniform distribution of light over a given area. Therefore, it would generally be desirable to have a prism light guide lighting system in which light is emitted uniformly 58 from the prism light guide along its length, rather than coming out from the end opposite to the light source. As discussed previously, a certain amount of light intrinsically leaks from practical prism light guides. In addition, this leakage rate can be increased by modifying the optical quality of transmission of the light at any point in the guide. Thus, for illumination purposes, the challenge is not to cause light to exit from the prism light guide, but to cause it to exit uniformly along the length, and with a reasonably high level of efficiency. Obviously, there will be many means of accomplishing this goal, but at the same time there will be certain overall constraints which arise purely from the properties of the prism light guide materials and the light sources available today23. The first such constraint concerns the physical dimensions of the light guide system. Since the prism light guide can only guide light rays which have an off-axis angle of less than 27.6° (for refractive index 1.5), it is essential to modify the light output from ordinary light sources (which radiate in all directions), to match this input requirement. Such modification is simple for point sources of light, but unfortunately, light sources which have reasonable efficiency and light output have a significant physical size. (For example, one of the most useful light sources available today is a 400 Watt metal halide arc, available from most lamp manufacturers and in general use in industrial lighting, which has a cylindrical light emitting region which is approximately 4 cm long and 1 cm in diameter.) In order for a reflector to redirect the light output of such a light source into a given angular range, it is necessary for the reflector to have a certain physical size. As described in more detail in Appendix 2, it is impossible to concentrate light energy in the phase space consisting of spatial and angular distributions. Thus, if a light source which emits in all directions is to be converted to a confined angular range, this can only be done at the expense of 59 increasing the physical cross section of the light beam. Intuitively, another way to understand this constraint is to realize that at any point on a reflector surface, the light reflecting off that point of the reflector must necessarily have an angular range which is comparable to the angular size of the light source as seen from that point, suggesting that the reflector must have a shape which results in all points of the reflector being at least a certain minimum distance from the light source. The result of these considerations is that useful reflector designs for such lamps will typically be 15cm to 20cm in diameter. Another physical constraint is that there will be a limitation in the maximum length of a prism light guide system over which it is possible to obtain a uniform distribution of light. The simplest way to think of this is that since the intrinsic loss in the guide will cause a certain percentage of the light to escape with each reflection off the wall, there is a limit to the number of reflections which can occur and still leave enough light in the guide to produce the desired level of surface brightness. This is shown pictorially in Figure 7.1. For a prism light guide having a loss of 5% per reflection, for example, it would be desirable to have at the most 10-20 bounces along the length, which with a typical reflector design would limit the overall length of light guide to 30-60 times the diameter of the guide. A further design requirement stems from the fact that most lighting systems require light to be emitted from a ceiling mounted fixture in the downward direction whereas light escaping from a prism light guide radiates in all directions. This can be dealt with by placing a diffuse reflective material on the sides and top of a prism light guide. Escaping light which reflects off these materials can quite efficiently pass through the prism light guide, and radiate out of the bottom surface, as the. wall of the prism light guide is transmissive to most angles of light. 60 dmax distance Figure 7.1 Attenuation per PLG Wall Interaction Limits Practical Length of Guide, as Seen in Graph of Luminous Flux vs. Distance 61 The remaining design features are all concerned with creating a uniform rate of light escape along the length of the guide, and ensuring that most of the light does in fact escape. For example, it could be helpful to reflect light which reaches the far end of the light guide back toward the source, perhaps with an increased angular divergence to enhance escape at the far end. A further adjustment of the emission of light can be achieved by selectively degrading certain portions of the interior of the guide. In order to assess the practicality of such a method of illumination, a sample system was constructed, tested and analyzed by ray tracing as described in the following section. 7.3 Construction. Testing, and Ray Tracing Analysis of a Sample System A light guide system was constructed as shown in Figures 7.2 and 7.3. The system utilized a 250 watt high pressure sodium light source mounted within a reflector, which caused all emitting light rays sent into prism light guide to be within the desired angular acceptance range. The light guide consisted of acrylic panels moulded from 3 mm thick acrylic plastic sheets, with prisms having an edge dimension of 3 mm. The end mirror consisted of 95% reflective silvered mylar and the prism light guide was surrounded on the sides and top with 92% reflective cellulose mat and on the bottom surface with 75% transmissive acrylic prismatic diffuser material as commonly is employed on the bottom surface of fluorescent light fixtures. Appendix 1 describes the measurement techniques employed to determine such material properties. Figure 7.4 shows the light emission from the bottom surface of the prism light guide system as a function of position, as measured by an illuminance 62 REFLECTOR INPUT /l/ WINDOW PRISM LIGHT GUIDE \ LAMP END MIRROR fS LIGHT EMITTED FROM BOTTOM SURFACE Figure 7.2 The Prism Light Guide Illumination System (not to scale) PROTECTIVE COVER DIFFUSE REFLECTING MATERIAL (CELLULOSE MAT) PRISMATIC WALL MATERIAL DIFFUSING LENS 3 MM PATTERN 12 STD. Figure 7.3 Cross Section of Prism Light Guide With Cover (not to scale) COMPUTER MODEL EXPERIMENTAL DATA AVERAGE 1200 Ul o I 600 co e> 400 ^  a! CO o 200 g QQ / 2 3 DISTANCE FROM PLG INPUT WINDOW IN METERS 4 Figure 7.4 PLG Surface Brighincss: Comparison of Computer Model lo Experimental Data 64 meter. By means of integrating a large number of illuminance measurements on a plane receiving illumination from light sources (see Appendix 1), it is possible to quantify the actual number of lumens of light radiating from any given source. Using this method the light output of the lamp, the reflector units, and the entire system, was measured. Using the Monte Carlo ray tracing method, the entire system was also modelled by computer, using the known values of reflectivities and absorptivities of various materials. The only unknowns in the analysis were the transmissivity per reflection, and absorption per reflection, of the prism light guide wall. While it was possible to estimate these parameters with measurements of single reflections off the surface, such measurements are very difficult to do accurately. In fact, it was found that a greater constraint on these values was the fact that, when incorporated in the computer model, all of the resulting model predictions must agree with the measured photometric results. As shown in the computer data included in Figure 7.4, it was possible to get very good agreement with the measured brightness along the guide, using an absorption per reflection of 1 ± .5% and the transmissivity per reflection of 8 ± 1%, both of which were consistent with the known material properties in the wall. Table 7.5 compares the computer prediction and measured values for various overall efficiencies, again showing good agreement. Based on these results, the computer model gives more detailed information about the various types of light loss in the device, as depicted in Figure 7.6. These results were very encouraging for several reasons. First, the good agreement between the experimental tests and the computer simulation showed that the behavior of the prism light guide system was consistent with our model of the types of light interactions which occurred. Second, the fraction of flux from the lamp which reaches the work plane, 52%, is comparable to that 65 COMPUTER MODEL EXPERIMENTAL DATA FRACTION OF LAMP OUTPUT INCIDENT ON PLG INPUT 83% ± 2% 84% ± 4% FRACTION OF LAMP OUTPUT EMITTED FROM BOTTOM SURFACE OF PLG 52% ± 3% 50% ± 4% BRIGHTNESS VARIATION FROM THE AVERAGE ± 12% ± 14% Table 7.5 Comparison of Computer Model Output to Experimental Data Absorbed by reflector 10% Lost in reflector lamp cut-out 7% Absorbed by input window glass 2% Returned to reflector 7% Absorbed by PLG walls 6% Absorbed by PLG end mirror 3% V Absorbed by reflective cover around PLG, and by diffusing emitting surface 13% Illuminating area below 52% Figure 7.6 Simplified Light Ray Flow Chart for the Covered Prism Light Guide 66 obtained in conventional fluorescent light fixtures, implying that the system does indeed have potential as a practical method of providing distributed light using small intense light sources. The next chapter considers in more detail what the ultimate limitations might be for the reflectivity of prism light guide wall material. Such limitations relate to other possible applications in which it is desirable to simply carry light from one place to another, rather than to distribute it. 67 Chapter 8 MICROWAVE SIMULATION OF THE DIFFRACTION LIMITED PRISM LIGHT GUIDE 8.1 The Importance of Diffraction Loss in Prism Light Guide In Section 4.2 it was shown that a prism light guide formed perfectly from non-absorptive material would be a loss free propagator of optical radiation, if diffraction effects were not present. In other words, a ray tracing analysis, properly including all partial and total reflections, shows that perfect propagation would occur. In Section 5.2 it was estimated that diffraction effects should be negligible compared to other loss mechanisms in prism light guides of the highest quality which can be made today. This estimate is consistent with the observation that measured losses in prism light guides agree well with theoretical loss predictions which do not include diffraction effects. Thus while the subject of diffraction effects is unimportant from a practical point of view today, it is most interesting from a theoretical perspective as it is the only truly unavoidable loss mechanism in the prism light guide, and additionally it is possible that in the future prism light guides will be made with such high quality that they will be limited by only diffraction effects. The prospects of such prism light guides is intriguing, because such devices could be extremely useful. For example, such devices could pipe sunlight great distances for illumination in underground buildings, mines and tunnels. To quantify matters, consider that the estimate we presented in section 5.2 for the magnitude of diffraction losses is that a fraction of order X/d of the radiation striking the prism light guide wall at any point would be expected to be 68 lost to diffraction. If this estimate is correct, then a diffraction limited prism light guide, made of low cost, thin walled optical material, could transmit light from conventional light sources with an attenuation length in the range of 1000 diameters. Such transmission capabilities would represent an important advance over the state of the art. In contrast, however, if the actual loss is roughly ten times greater than this value, this attenuation length would be only 100 diameters, which would limit the applications of prism light guide to light distribution applications as studied in chapter 7. Therefore, it is desirable to better estimate what the magnitude of diffraction losses actually will be. In section 8.2 various methods of assessing diffraction loss are described, and the selected method is further elaborated on in section 8.3. This is followed with a discussion of the experimental design, and experimental results are presented. Section 8.6 draws on the simulation procedures developed in chapter 6 to interpret the experimental results, in order to generalize them to prism light guides with different prism dimensions and radiation of different wavelengths. The implications of this work to high efficiency prism light guides is then considered and avenues of further study are suggested. 8.2 Practical Difficulties in Direct Measurement and Theoretical Evaluation  of Diffraction Loss of Optical Radiation in a Prism Light Guide On first consideration two straightforward approaches might be attempted for evaluating the magnitude of diffraction loss of optical radiation in a prism light guide. One obvious approach would be to fabricate a prism light guide so perfectly that it truly was diffraction limited, and then to simply measure the 69 propagation loss. While an appropriate method of fabrication is not known at this time, it is possible that a means of fabricating such a guide could be developed. Unfortunately, the process would entail an expenditure of hundreds of thousands of dollars, and no funds were available to do this work. The obvious alternative method was a theoretical analysis of the loss. This alternative breaks down into two categories - a full treatment, and a simplified treatment which retains diffraction effects but employs scalar radiation to simplify the analysis. As depicted in Table 8.1, each of these theoretical approaches can be further subdivided into an analytic treatment, and a computer simulation. A brief review of the literature on the subject and discussions with other researchers in electromagnetic fields led the author to believe that the analytical approach, even for the scalar problems, would likely require years of effort, with the very real possibility of never obtaining a reliable result 2^. The computer simulation approach to the full problem appeared also to involve a tremendous effort, both in time and money, and it was similarly rejected. The most intriguing of these four theoretical approaches, therefore, was the computer simulation of the propagation of scalar radiation. This approach is conceptually fairly simple and would at least provide an estimate for what the actual diffraction losses would be with optical radiation. Unfortunately this estimate was felt to be of no greater accuracy than the intuitive estimates already employed, and could even be completely wrong. The author therefore sought a method of estimating diffraction loss, which involved a full treatment of the problem, either experimentally or theoretically, which had a reasonable chance of success, and which also could be performed in a reasonable period of time and for a reasonable cost. The next section describes such an approach. 70 A n a l y t i c C o m p u t e r S i m u l a t i o n Vector Wave extremely difficult, perhaps impossible very complex and expensive Scalar Wave difficult, perhaps impossible, dubious accuracy dubuious accuracy . but possible Table 8.1 Theoretical Approaches to Analysis of Diffraction in Prism Light Guide Wall 71 8.3 The Microwave Simulation Approach Since the basic problem in experimentally measuring the diffraction effect of optical radiation was that the effect was smaller than the other loss mechanisms, it was decided to investigate the possibility of enhancing the diffraction loss so that it would dominate the loss. Since it was believed that the diffraction loss would be proportional to X/d, the goal was to increase the magnitude of this ratio relative to the other loss mechanisms. Two approaches were considered - decreasing d and increasing X. A problem encountered in the most obvious approach of decreasing d, is that one of the other loss mechanism - loss from imperfect corners - also is inversely proportional to d. (One way of looking at this is to think of the diffraction effect as corresponding to an intrinsic corner defect having a size of order X. The problem was that it was not possible to manufacture corners which would have defects significantly smaller than X.) The alternative of increasing X had constraints with regard to transmissivity of optical materials, and availability of sources and detectors of radiation. Looking at increasing wavelengths it is found first that significant absorption bands precluded the use of near infrared radiation, and that availability of sources and detectors further reduced the possibility of far infrared radiation. The first wavelength for which it was possible to identify a transparent material, and a suitable source and detector, was approximately 3mm microwave radiation. At this wavelength, there appeared to be substantial opportunities to measure diffraction effects. It was established that a prism light guide could be manufactured from polytetrafluoroethyline (PTFE), with a value of d of 6mm, and 72 dimensional errors not exceeding 50 microns, essentially perfect relative to the wavelength of the radiation. It was possible to obtain published values 2^ for the dielectric constant, e" e', and loss angle a where tan a = — , where e" is the imaginary component of the e dielectric constant, for microwave radiation at a wavelength of 12mm. Assuming the value of E' of 2.08 and tan a of 6 X 10"4 are the same at a wavelength of 3mm, the value of the refractive index would be n = 1.44 and the absorption constant would be k = 1.26 nv 1 . As a further check, approximate measurements were made of attenuation of the 3mm radiation traveling through a 1" diameter copper pipe containing various lengths of P T F E ranging from 3" to 21", as described in Appendix 3. The observed absorption constant of 1.03 ± .33 n r 1 was consistent with the previous estimate. Referring to section 7.4, it was estimated that the total loss per prism light guide wall reflection due to absorption, would be approximately 15%, which is smaller than the value X/d, 50%. We therefore proceeded with this experimental concept as a means of observing the diffraction effect. 8.4 Experimental Design Requirements and How Thev Were Met The requirements for this experiment to have predictive value fell into 3 categories. First, it would be necessary to be able to measure total radiant flux of microwave radiation as a function of distance down the prism light guide. Second, we would have to establish that spurious effects were not responsible for, or distorting the observed flux decay. Third, it would be highly desirable to find a method of modelling the observed decay with the Monte Carlo ray tracing model developed in chapter 6, in order to make it possible to estimate how the observed 73 diffraction losses might change for different values of X and d. These three design criteria are described below. 8.4.1 Flux Measurement The source of 3mm microwave radiation was a Siemens 756110GHz Backward Wave Oscillator having a frequency range of 75 to 140 GHz, with the radiation emanating from a horn emitter as depicted in Figure 8.2, with an approximate radiation pattern also as shown in Figure 8.3. The detector consisted of a Hughes diode detector mounted on the same make of horn. The detector output was observed on an oscilloscope and measured by a digital volt meter using the oscilloscope amplifier as a preamplifier. It was easily demonstrated that zero offset errors of this apparatus were negligible and manufacturer's data on the detector established that the digital volt meter reading would be very accurately proportional to the power level of the microwave radiation being detected. The PTFE prism light guide had a square cross section composed of 4 panels with dimensions as shown in Figure 8.4. The simplest experimental approach would be to direct the output of the emitting horn into one end of the prism light guide and to simply sample the radiation at various locations down the length of the prism light guide with the detector horn. If a much longer prism light guide had been available for study, this very crude approach might have been acceptable by itself. The obvious error with this approach is that the detector measures the intensity of the radiation at a given point with a particular angular sensitivity, rather than measuring the total flux of radiation in the prism light guide, and the ratio of these two quantities could vary as a function of distance down the guide. Nevertheless, since this ratio would likely be confined to a limited range, over sufficiently long length the attenuation of measured Full scale Figure 8.2 Microwave Horn Used for Emission and Detection of 3mm Microwave Radiation Figure 8.4 Single Panel Detail of the PTFE Prism Light Guide 75 intensity would be dominated by true loss of flux rather than variations in this "sampling ratio" and a meaningful result would be obtained. Although this approach was not taken because of the limited length of the prism light guide, it is a worthwhile introduction to consider in more detail how one would attempt to interpret the flux decay rate from measurements of the decay rate of sampled intensities. First, it is important to realize that there are two quite distinct effects which can cause the ratio of intensity of the detector signal to total flux to vary as a function of position. The first such effect is the analog to laser "speckle". Since the microwave radiation is coherent, the entire prism light guide is filled with a standing diffraction pattern, so that the detector will register a wide variety of intensity readings and at some places will actually measure zero intensity despite the fact that there is significant radiation in the guide. The second factor is purely geometrical - the actual intensity profile of the radiant flux across the guide is not necessarily uniform and the distribution of intensity across the guide will therefore almost certainly vary as a function of length down the guide. Fortunately, both of these effects can be expected to be random in nature, and the random variations occurring to the ratio of intensity to flux can be expected to have fairly well defined spatial frequencies. In the case of speckle, the inverse of the spatial frequency was of order several millimeters, and in the case of geometrical effects, one expects and indeed observes that the spatial frequency is approximately the inverse of the width of the guide, which is the length required for significant change due to transverse motion of the components of the flux distribution. Figure 8.5 is a; fictitious graph of the log of intensity versus distance which, displays how both of these features might appear. The slope of the dashed line shown can be taken as a good representation of the rate of flux decay in the 76 Figure 8.5 Fictitious Plot of Log Sampled Intensity vs. Distance Along Axis Prism Light Guide 77 device, for the following reasons: First, we expect that the flux is monotonically decreasing as a function of length, and it is reasonable to expect that this decrease is fairly smooth. Second, it is clear from the graph that the very high spatial frequency speckle effects simply cause periodic random dips below the envelope represented by the dotted curve and are therefore easily ignored, and thirdly, the magnitude of the longer spatial frequency geometric effects appear to be constant but random and hence the dashed line which is centered on the dotted curve probably gives a good summary of the actual flux decay. Of course we must estimate the error in the slope of the dashed line. A reasonable estimate would be that the error in graphically estimating the slope would be approximately equal to one half of the magnitude of the lower spatial frequency variations divided by the length of the guide. With this approach in mind it should be possible to reasonably interpret the attenuation rate of flux down the prism light guide by simply sampling detector intensity down the prism light guide and employing this graphical procedure, and a criteria for this procedure being useful is that the magnitude of the lower spatial frequency variations in intensity should be small compared to the total intensity attenuation down the guide. The purpose of the consideration of this graphical technique depicted above is to establish a criteria for flux measurement, as any attempted measurement of flux is relying on a measurement of intensity in the detector. Since the very simple approach discussed above would yield unsatisfactory errors relative to the attenuation over the short length of prism light guide actually available, it was desirable to reduce as much as possible variations in the ratio of detector signal to total flux. Since geometrical variations were the prime problem, a method was adapted to reduce these variations as much as possible. Considering the input end, it was recognized that to whatever extent the intensity 78 of the radiation as a function of position entering the Prism Light guide could be made uniform, geometrical variations would be reduced. With this purpose in mind, the output of the emitting horn was modified by a mixing device as shown in Figure 8.6. This device is simply a square cross section metallic wave guide whose purpose is to transform a relatively non uniform profile of input radiation into a much more uniform profile of output radiation. Its operation is easiest to understand from a ray tracing perspective which is quite appropriate given that the wavelength of the input radiation is very substantially smaller than all of the guide dimensions. Imagine a ray entering the guide at one end at a certain position and with a certain angle. It will reflect down this mixing guide and will exit at a second cross sectional position and with an angle that may be different, but it will have the same angular deviation from the axis as it originally had. (This results from the specular reflectivity of the metal and the constant cross section of the guide.) If now the ray angle, 0, is changed by A0, the output position will correspondingly change approximately /A0, where / is the length of the guide. It is thus apparent that if the input radiation has a range of angles significantly greater than d/l where d is the width of the guide, the output radiation profile will be very well "mixed". This can be understood more graphically by referring to Figure 8.7. The first diagram in that figure shows an example of a non-uniform distribution of intensity as a function of one physical dimension x, and one direction of angle 0 X . The second diagram shows how this distribution would look at the end of the guide. As can be seen, there is a much greater degree of uniformity of the radiation at the end of the mixing guide. A further method of reducing geometrical variations in the ratio of intensity readings to total flux is to sample the intensity at a number of points in 79 support leg Isometric view 1/20 actual size Figure 8.6 Aluminum Waveguide for Spatial Mixing of 3mm Wavelength Microwave 80 0x BEGINNING OF GUIDE END OF GUIDE Figure 8.7 Example of Phase-space Distribution of Intensity (in one dimension) at Beginning and End of Mixing Guide, Using Dimensions Shown in Figure 10.5 81 the cross section of the guide, at each measurement location down the length of the guide. Since only one detector was available, and since it was physically difficult to move the detector to various orientations for measurement, it was decided to build an "integrating structure" which would sample radiation from a number of different points across the cross section and direct a portion of the sampled radiation toward the detector. Figure 8.8 shows the dimensions of the device employed. At the input surface, an array of nine 1cm diameter holes admit radiation into the cubic structure. The interior of the structure was lined with aluminum foil which was textured with wrinkles having spatial frequency approximately 2 cm"* so that they would reflect the microwave radiation approximately randomly. The concept of this integrating structure is essentially that of an Ulbricht sphere: the radiation entering the structure randomly reflects many times around the interior of the structure before being absorbed, exiting through one of the entrance holes or entering the detector. The aim was to have the fraction of the energy which entered each hole and which was subsequently detected by the detector approximately independent of the angle of that radiation, and also independent of the location of the entrance point. The baffle in the centre of the structure prevents the detector from having a direct view of the central hole, which was expected to further reduce variations in intensity readings. It is critical to understand that the only true means of assessing the success of these attempts to remove geometrical variations is graphical consideration of the decay of intensity versus length, which yields a direct measure of the magnitude of these geometrical effects. As will be seen in section 8.5, these measures did result in a suitably low level of error due to geometrical variation. The next section considers other potential measurement problems and their elimination. Figure 8.8 Microwave Integrating Structure 83 8.4.2 Other Potential Experimental Errors One of the most obvious concerns in the experimental design is that the detector may receive radiation from a route other than transmission down the prism light guide. In all measurements, it was established that blocking the input aperture of the mixing guide with a metallic cover reduced the signal to below noise. A more subtle concern involved the housing of the prism light guide itself. Since it was not possible to bond the individual walls of the PTFE guide together, it was necessary to clamp them at the corners with thin strips of material, and it was recognized that these strips would be exposed to evanescent radiation from the guide. Aluminum was selected as the material for these strips as it was felt that a metallic material would perturb the behaviour less than a dielectric due to its reflectivity. The fraction of the outer cross sectional area of the guide so affected was kept to a minimum (-3%) and given that the aluminum was not very absorptive, and was smooth along the length, it was felt that any effect from this aluminum strip would be extremely small. As already mentioned, there was good reason to believe that any losses due to absorption and scattering in the PTFE would be very small relative to the anticipated diffraction loss. As a further check that the observed loss did indeed correspond to radiation escaping from the prism light guide, the detector horn was moved to the outside of the guide and levels of radiation were detected outside the PTFE at a level which seemed reasonable in view of the loss which was occurring. Again, it was confirmed that these readings resulted only from radiation which had originated within the prism light guide. The polarization of the input horn was periodically varied without substantial effect, and as a final safeguard that all the effects were real, the wavelength of the radiation was varied slightly in different runs to make sure that there were no effects being 84 observed that were extremely wavelength dependent. Unfortunately, the source was not capable of substantial variation of frequency,' so it was not possible to measure the wavelength dependence of the diffraction loss. Of course, there is always the possibility that some unanticipated effect was present, and it is important to note that if this were the case, the effect on the final experimental result would almost certainly be an increase in loss rate beyond that caused purely by diffraction. Since the primary goal of the experiment was to put an upper limit on the magnitude of diffraction losses, such a possibility is much better than the opposite alternative. 8.4.3 Designing the Experiment to Correspond to the Rav Tracing Computer  Model As will be described in section 8.6, it was desirable to be able to adapt the Monte Carlo ray tracing computer model of a prism light guide to attempt to model the experimental design. The only constraint this imposed on the experiment was that it was desirable to have an input of radiation to the prism light guide which corresponded to that employed in the computer model. In the computer model the input radiation is uniformly distributed across the cross section of the guide, and a given ray has a randomly selected angle within a certain defined range of angular deviation from the axial direction. Fortunately, the mixing guide which was employed to reduce geometrical variations also provided this distribution, and all that was required was to ensure that the range of off axis input angles from the horn emitting radiation into the input of the mixing guide was appropriate. The range of angles most suitable for study are those which are large enough to involve at least one bounce as the radiation passes down the guide, and small enough to be within the acceptance angle of the 85 prism light guide, with an overall range wide enough to obtain substantial uniformity of the radiation at the end of the mixing guide. With these constraints in mind, the selected range of off axis angles was 10° to 25°, which was simply achieved by orienting the emitting horn relative to the input of the guide as shown in the diagram in Figure 8.9. With this final adjustment, the experiment was ready to proceed. 8.5 Experimental Results Apart from employing the safeguards described in the preceding sections, the actual operation of the experiment was straightforward. The integrating structure was positioned at a number of points down the prism light guide, and the intensity was read at each point. One minor additional feature in the experimental design was that the microwave source unit had the capability of slight frequency modulation, which had the effect of blurring out the laser speckle referred to earlier. While this was not a necessary part of the experimental design, it was employed as a means of greatly reducing the number of measurements required to graphically interpret the attenuation in flux as earlier described. This F M modulation speckle smoothing technique was therefore employed throughout. Figure 8.10 shows a graph of intensity versus position down the prism light guide for three runs having very slightly different wavelengths. (Since the source amplitude varied dramatically with the slight changes in wavelengths, the curves shown in Figure 8.10 have been multiplied by arbitrary constants so that they all begin with approximately the same intensity.) As expected there is variation in the detailed structure of these curves, presumably resulting from minor geometrical differences in the radiation pattern resulting from the slightly different wavelengths used, but the average 86 Figure 8.9 Orientation of Emitting Horn Relative to Input End of Mixing Guide 87 i—i r — i 1 — i — r • - R U N # I 0 20 40 60 80 100 DISTANCE DOWN PRISM LIGHT GUIDE (cm) Figure 8.10 Graph of Log Intensity vs. Distance of Propagation Down the PTFE Prism Light Guide 88 shape of these curves agree quite well. The dash line represents the best interpretation of fractional loss as a function of length down the prism light guide, based on this data. The magnitude of the low spatial frequency geometrical intensity variations can be seen to correspond to a variation in the natural logarithm of ±0.2 which is relatively small compared to the overall 2.0 reduction in logarithm of intensity over the length of the guide. It is therefore felt that to an accuracy of ±10% the decay depicted by the dashed curve in Figure 8.10 is representative of the true decay of flux as a function of length down the prism light guide. It is interesting to note that the shape of the dotted line depicting the attenuation of flux down the prism light guide in Figure 8.10 is not the straight line one would expect for simple exponential decay. This is not particularly surprising, because the radiation in the guide can be thought of as consisting of a large number of "subsets" of radiation having different angles, and it would be expected that the decay rate for any "subset" would be proportional to the rate of reflection off the walls, and would therefore depend on angle. Thus, the general decay curve one expects in a guide of this type carrying a distribution of angles of radiation is that of a sum of exponentials, which generally results in a positive curvature in a graph of log intensity versus distance, as observed. Intuitively, the simplest way to picture this is that wide angle, high loss radiation attenuates rapidly from the guide, causing a high loss rate in initial stages, and leaving a much lower loss rate for the residual radiation which is seen further down the guide. 89 8.6 Interpretation of the Diffraction Data The results of the preceding section are complete in and of themselves. Although it is an oversimplification, the results can be summarized in the following statement: 3mm microwave radiation transmitted by a prism light guide of the dimension selected will be attenuated by a factor of 7.4 ± 1.5, if the radiation consists of a range of angles from 10° to 25° off-axis. Based on the assumption that the loss rate per reflection is approximately independent of angle and geometrically estimating the number of reflections the radiation is undergoing as it travels down the guide, an alternative approximate statement would be that 3mm microwave radiation in the range of angles typically employed in prism light guides undergoes a diffraction loss of approximately 35% of the incident radiation in each encounter with prism light guide wall having 6mm prisms. In this section, we would like to carry the process one step further by attempting to estimate from this result what the diffraction loss would be with much smaller wavelength radiation and somewhat smaller prisms. The simplest estimating procedure would be simply to use the original estimate that diffraction loss is proportional to X/d. However, this approach is not very satisfactory because it is obvious that this relationship must break down as X approaches d and therefore it is desirable to employ an alternate extrapolation procedure which behaves more reasonably as X approaches d, and is based more closely on physical events in question. The procedure selected was to attempt to model the diffraction loss observed in the prism light guide with the Monte Carlo ray tracing program described in Chapter 6. The concept was basically this: Leaving all of the conventional loss parameters at zero, model the effect of diffraction of light as an 90 angular deviation of the rays in the ray tracing program. This procedure is basically an analogy to single slit • diffraction where parallel rays passing through a slit in effect become a bundle of rays having an angular width of approximately X/d. With this procedure in mind, the rays reflecting off the prismatic surfaces in the model were given an angular deviation having a probability distribution generated statistically, with the following properties: P(9)tx (1 + tKpLGYQ)] 2 )- 1 for -^<Q<j- (36) p (9) = 0 for 0 < j and 9 ^ f (37) This distribution is approximately the envelope of the intensity distribution of single slit diffraction, and the one free parameter, K P L G . which controls the width of the distribution, was to be adjusted in the hope that it could result in simulation data agreeing closely with both the magnitude and the detailed shape of the decay reported in the previous section. Figure 8.11 shows the experimental decay curve of the radiation flux from the previous section, and also the simulation results from the computer program which were obtained by setting the parameter K P L G = 1-174. The decay results were very sensitive to this parameter, and it was possible to determine its best value to within an accuracy of approximately 10%. As can be seen, there is substantial agreement with the shape of the decay curve, which is encouraging. 91 0.4 0 -1.2 -1.6 -2.0 -2.4 APPROXIMATE CURVE FITTED TO EXPERIMENTALLY MEASURED INTENSITY MONTE CARLO RESULTS WITH ERROR BAR BASED ON MONTE CARLO STATISTICS JL _L _L J_ J_ 0 20 40 60 80 100 DISTANCE DOWN PRISM LIGHT GUIDE (cm) Figure 8.11 Monte Carlo Simulation Program Results, Compared to Average of Experimental Results, for Attenuation of Microwave Radiation Propagating in a PTFE Prism Light Guide 92 8.7 Extrapolation of Rav Tracing Model to Conditions in an Optically Perfect  Prism Light Guide Using the approach discussed in the previous section, the wavelength was changed to approximately .5 microns, and a prediction was made for the diffraction loss in an optically perfect prism light guide of the same size and shape as that studied with microwaves, assuming that K P L G is independent of wavelength. The results are that such a guide, if truly optically perfect, would carry visible light approximately 1200m, with a 50% loss. From an economic point of view, it is desirable to consider thinner prism light guide materials, such as those now manufactured by 3M, as an outgrowth of this research, in the form Scotch Brand Optical Lighting Film, in which the prism width is approximately .18 mm. In this case, such a 10 cm diameter guide of the same size could still carry light 53 m with 50% loss, which would be extremely exciting. Considering that the typical number of bounces of a light ray traveling such a distance in a 10 cm square prism light guide is about 300 bounces, this corresponds to an escape rate of approximately .17% per reflection. This value is quite close to the ratio of the wavelength of light to the prism edge dimension for this material, a result which is the most easily remembered conclusion from this study. This study of microwave radiation travelling down a PTFE prism light guide has independently added support to the intuitively appealing estimate for the loss per reflection due to diffraction in . a prism light guide being approximately equal to the wavelength of the radiation divided by the prism edge dimension. This study is rather indirect, and in particular requires intuitive assumptions for extrapolation to visible light wavelengths. Nevertheless, it is the opinion of the author, based on this research, that the probability that prism 93 light guides can in fact achieve this level of performance is very high, assuming that the optical quality of the structure is comparable to that found elsewhere in high quality optical equipment (i.e. physical errors significantly smaller than the wavelength of light, and angular errors associated with transmission through bulk material are significantly smaller than the diffraction limited angular beam widths in the system). It is hoped that the economic significance of the possibility of "transport grade" prism light guide, combined with these encouraging results, will stimulate further work in this direction. The next steps could include economic study of possible applications of "transport grade" prism light guides, evaluation of optical fabrication techniques for making diffraction limited prism light guide, and further, more direct experimental tests. 94 Chapter 9 CONCLUSION AND SUGGESTIONS FOR FUTURE WORK 9.1 Summary It has been shown that there exists a class of cylindrically symmetrical hollow optical structures which are in principle capable of guiding light by means of total internal reflection, with very low loss. Based on the assumption that diffraction effects are negligibly small in practical prism light guide structures, a complete analysis of their practical behavior was presented, in the context of the geometrical optics model of light. It was seen that it is possible to fabricate prism light guide structures which transport light over lengths 50 to 100 times the guide width with very little absorption loss. Furthermore, this is achievable with commercial molding techniques, using polymethylmethacralate as the refractive material, which bodes well for possible commercial applications of the concept. Since one of the main motivations for this research was its possible applications in the field of illuminating engineering, an important result has been the demonstration that prism light guides can be used to efficiently distribute light from practical high intensity discharge light sources. Finally, the experimental confirmation that the magnitude of diffraction effects is indeed very small (with the loss per reflection roughly the wavelength divided by the prism size) is encouraging. As molding technologies substantially improve, we can be confident that diffraction will not impede corresponding improvements in the transport capabilities of prism light guides. Thus, in the future, this technology may find applications beyond today's conventional illuminating engineering needs, as it becomes possible to transport 95 light very efficiently over distances perhaps 1000 times greater than the diameter of the structure. 9.2 Suggestions For Further Work From a fundamental point of view, the most exciting area of further research could be the theoretical quantification of the fundamental diffraction limits of prism light guides. It is not known whether a precise analytic solution is possible, but even if not, there is the possibility that an approximate analytic treatment could yield meaningful results, or that an accurate computer simulation could prove feasible. At any rate, it would be most gratifying to determine precisely the ultimate diffraction limits. With regard to computer ray tracing, while this study's level of detail was sufficient for verifying the fundamental principles of the prism light guide, there is substantial room for improvement. Particularly in view of recent advances in sophisticated computer imaging in the cinemagraphic and illuminating engineering fields, there is the possibility of very advanced, detailed predictions of performance of prism light guide-based light distribution systems. Finally, with regard to such illumination systems generally, there is a need for improvement both in the quality of prism light guide materials themselves, and in the auxiliary optical materials which are used in conjunction. These include specular metallic reflectors for controlling light output from high intensity discharge lamps, and diffuse reflective materials used for redirecting the light output from prism light guides. It is anticipated that as commercial activity involving this technology increases, substantial improvements in these areas will be likely to occur. 96 REFERENCES 1 W. Wheeler, U.S. Patent #247229, September 20, 1881. 2 J.B. Aizenburg, C.B. Bukham and V . M . Pjatigorsky, "A new principle of lighting premises by means of the illuminating device with the slit lightguides", CIE Publication No. 36, Bureau Central de la CIE, Paris, 1976. 3 N.S. Kapany, Sci. Am., Nov. 1968, 72. 4 R. Kompfner, Applied Optics, 11, 2412 (1972). . 5 La Foret Engineering and Information Service, Himawari Bldg., Toranomoh 2-7-8, Minato-ku, Tokyo 105, Japan. 6 M.A. Duguay and R.M. Edgar, Appl. Opt. 16, 1444, 1977. 7 D. Stanger, "Monte Carlo Procedures in Lighting Design", J. IES 13, 368 (1984). 8 I.M. Sobol, "The Monte Carlo Method", University of Chicago Press, 1974. 9 L.A. Whitehead, Simplified ray tracing in cylindrical systems, Applied Optics, 21 (1982) 3536-3538. 10 E. Hecht, A. Zajac, Optics Section 4.3.4. 11 K. Drexhage, Sci. Am. 222, 108 (1970). 12 M. Lefrancois, F. Curzon, B. Ahlborn, Evanescent Waves and Mode Selection in Optical Fibres (submitted for publication from UBC Physics Dept., Vancouver, ' Canada). 13 N. Kapany, "Fiber Optics Principles and Applications", Page 326, Academic Press (1967). 14 Laboratory tests of optical samples using techniques described in Appendix A l . 15 R. Kompfner, Applied Optics, 11, P. 2418 specificially, (1972). 16 Elastomeric sub-optical microsphere-doped acrylic samples from S. Saxe, Optics Technology Centre, 3M St. Paul, Minnesota. 17 Observations by L . Whitehead at SEM facility, Department of Metallurgy, University of British Columbia. 18 Estimates obtained from a variety of commercial injection moulding firms, including Ian Howard, Columbia Plastics, Vancouver, BC. 19 E. Brown, "Optical Instruments", Chapter 27, Chemical Publishing Co., (1945). 20 Observations provided by Chris Stephens, TRIUMF, Vancouver, B.C. 21 NE 581 Optical Cement, manufactured by Nuclear Enterprises, available through Technical Marketing Associates, Richmond, BC. 22 Whitehead, L.A. , R.A. Nodwell and F.L. Curzon, A new efficient light guide for interior illumination, Applied Optics, 21 (1982) 2755. 97 23 IES Lighting Handbook, Reference Volume 2-5, 2-6 (1984). 24 Chang-Sung Joo, Jung-Woong RA, and Sang-Yung Shin, Scattering by Right Angle Dielectric Wedge, IEEE Transactions on Antennas and Propagation, Vol. AP-32, No. 1, January 1984. 25 Von Hippel, A, Tables of dielectric materials, 322. 26 Walsh, Q.J., "Photometry", Chapter VIII. 27 Winston, R. , "Light collection within the framework of geometrical optics", Journal of Optical Society of America, 60 (1970) 245-247. 98 Appendix A l  OPTICAL MEASUREMENT TECHNIQUES Throughout the course of the experimental studies described in this thesis, it was routinely necessary to make measurements of a large number of optical materials properties. The techniques which were used are well known to those skilled in the field of illuminating engineering, but it is worthwhile to briefly review them here. Most of the measurements were done with the aid of integrating structures of the form generally termed "integrating sphere" or "Ulbricht sphere", as shown diagrammatically in Figure A l . l 2 ^ . This device consists of an approximately spherical structure, which is coated on the interior with a reasonably high reflectivity, reasonably diffuse material. Ideally, the reflectivity should be in the range of .8 to .95 and the material should reflect approximately in a "Lambertian" manner, meaning that the intensity of the reflected light is proportional to the cosine of the angle from the normal direction. (Most diffusely reflective materials approximate this behavior.) The beauty of this Lambertian reflection, is that it can easily be shown that such reflected light uniformly illuminates the interior of the sphere. However, even approximately spherical shapes with approximately Lambertian reflectivity will result in very even illumination within the sphere, providing that the reflectivity is high enough that a typical light ray bounces several times before being absorbed. As a result, it is a fact that the illuminance within the sphere will be very uniform, and hence can be accurately sampled at one point by a linear response light detector (such as a photomultiplier tube or a silicon photodiode) and that the reading so taken will be very accurately proportional to the total amount of light flux entering, or originating within, the integrating sphere. As will be seen below, these properties enable a number of very important measurements to be made. 99 Figure A 1.1 The Integrating Sphere 100 As an example, Figure A1.2 shows a set up for measuring light scattered from a transparent sample, such as a sample of the acrylic material used to fabricate the prism light guide. • The basic idea is that the main light beam passes through the sphere without being detected by the light meter, but that the scattered light is substantially intercepted by the surface of the sphere, so that the light meter gives a reading proportional to the magnitude of the scattered light. For calibration, the main light beam can be caused to strike the interior of the sphere, to give a reading of its full intensity. The details shown in the diagram are examples of techniques used to deal with fairly low light intensities. Here the light output from the laser beam is chopped so as to cause the amplitude of the beam to rise and fall at a known phase and frequency. This phase and frequency is obtained electronically by a pickup coil which responds to the motion of the metal chopping wheel as a result of an ambient magnetic field in its vicinity. This electronic signal is used as the reference for a phase sensitive detector measuring the output from the photomultiplier tube, resulting in extremely good signal to noise. Such techniques are particularly useful when highly collimated light beams are used (necessitating low intensity), and when only a very small fraction of that light beam is actually measured. Figure A1.3 depicts a graphical interpretation of sample scatter measurements for various lengths of PMMA. As indicated on the graph, the "y-intercept" relates to surface reflection and the slope represents bulk scatter per unit length. Yet another important measurement is the attenuation of a light beam passing through a bulk material, which is a result of both bulk absorption and bulk scatter. The sum of these two effects can be measured by passing a light beam through samples of various lengths and then into an integrating sphere for 101 Figure A1.2 Measuring Bulk Scatter and Surface Scatter > a / (m) Figure .A1.3 Graph of scattered light signal reading, V, vs. PMMA sample length, /. When the fullbeam was scattered in the integrating structure, the value of V was 1.322 V. 102 measurement, as shown in Figure A 1.4. For calibration purposes, a reference intensity I 0 is obtained by measuring the intensity in the sphere with no sample present. Figure A 1.5 is a graph of sample measurements of attenuation of a laser beam passing through various lengths of PMMA. As shown, the intercept for zero length represents the loss in intensity resulting from surface reflection and surface scatter from the sample. The slope of the line is the negative of the sum of the bulk scatter and bulk absorption per unit length. Another important measurement involves determining the surface reflectivities of materials. In this case a well regulated light beam (such as the collimated output from a tungsten quartz-halogen lamp operating on a constant voltage power supply) is directed through the entrance aperture toward the other side of the integrating sphere, as shown in Figure A 1.6. At the point where the beam would normally intersect the surface of the sphere, a sample reflective material, or reference reflective material intercepts the beam. The ratio of the intensity with the sample in place to the intensity with the reference beam in place is then a measure of the surface reflectivities of the two materials. In this test it is assumed that the reflecting material is diffuse, so that only a negligible portion of the reflected light exits from the original input aperture. Another important optical measurement is the transmissivity of a diffuse material. In this case, a measurement is made by placing either the sample material, or nothing, immediately in front of the input aperture of the integrating sphere, as shown in Figure A 1.7. The ratio of the intensity with the sample with the intensity without the sample is then a measure of the transmissivity of the diffuse sample. It is often necessary to measure very slight angular errors in light traveling through samples having planar surfaces and ostensibly homogeneous refractive index. As shown in Figure A1.8, this can be done by passing a laser beam through the sample and observing the precise location of the beam at a 103 integrating sphere Figure A 1.4 Measuring Sum of Bulk Absorption and Bulk Scatter o * •0.1 •0.2 surface reflection and surface scatter \J slope =-0.25 m-1 Figure A 1.5 1 r 1 — — — r 0 0.1 a.2 0.3 0.4 Graph of logarithm of photomultiplier output voltage vs. PMMA sample length, /. The reference voltage V 0 = 2.65 V was measured with no sample in place. 104 integrating sphere Figure A1.6 Measuring Surface Reflectivity of a Diffuse Surface Figure Al.7 Measuring Transmissivity of a Diffuse Sample 105 Figure A 1.8 Measuring Sum of Small Surface Angular Errors and Angular Errors Due to Small Refractive Index Inhomogeneities 106 great distance. Typically, a screen is placed ten meters from the sample, and the beam location is measured. The sample is then moved around, so that the laser beam samples various regions of the surface, and various regions of the bulk material, and the RMS deviation of the spot on the screen is thus a good measure of the combination of the effects of small surface angular errors and bulk refractive index in homogeneities. For this measurement to be practical, these angles should be reasonably small, i.e. less than roughly 10~2 radians. Finally, Figure A1.9 depicts another method of measurement of total light flux leaving a source, namely the integration of a large number of illuminance meter measurements. This technique is used when a light source is too big to be placed inside an integrating structure. The basic tool is a calibrated illuminance meter, which by definition is cosine corrected so that it responds to a light beam with a sensitivity proportional to the cosine of the angle between the light beam and the surface normal of the meter. The reading of such a meter is, by definition, a reading of the luminous flux per square meter passing through space at the position of the light meter in the direction of the surface normal of the meter. As shown in the figure, the technique involves some arbitrary surface which receives all of the light from the source. For relatively collimated sources, this can be often be a plane surface, but if necessary it can be cylindrical, spherical, or any arbitrary shape that intercepts all the light from the source. The surface is divided up into a number of regions A n which are small enough so as to subtend a small angle from the source, and have substantially uniform illuminance. If A n is a measure of the area of each surface element, and I n is the corresponding illuminance reading, it is therefore the case that the total luminous flux from the source is given by the F = D s u r f a c e A n I n . Of course, it is important to know that the subdivision of the surface has been fine enough for an accurate measurement, and one way to check this is to subdivide to an extent greater than is thought necessary, and then group the area elements into larger groups, and perform the calculation using the illuminance . value from one light leaving source; luminous flux F arbitrary surface which receives all light from source normal of both surface element and p = A n I„ furface small surface element A n calibrated, cosine-corrected, illuminance meter, reading I n Figure A1.9 Integrating Light Meter Readings to Obtain Luminous Flux From Source 108 representative sample within each group and the aggregate area of the samples of the group. If the result is essentially unchanged, then one can be confident that a fine enough division of the area has been used. All of the above optical techniques, while straightforward, do require great care to ensure that the results are meaningful. For example, the overall sensitivity of integrating spheres can vary if test materials are placed inside the sphere, or near the input apertures and it is very important to watch out for such effects. Furthermore, the assumption that only a negligible amount of light leaves from the entrance aperture must also be checked and one must always be on guard for variations in the output intensity of light sources, which are notoriously unstable. Finally, when measuring very small scattered intensities or very small angular variations, absolute cleanliness of optical surfaces is essential, as the imperfections in question are so small as to be easily overpowered by surface oil contamination. Above all, checking for reproducibility of results is essential. 109 Appendix A2 OPTICAL CONSTRAINTS ON MINIMUM APERTURE SIZE FOR COLLIMATED LIGHT An important optical theorem involves a photometric concept termed luminance. Luminance is defined relative to a specific point in space, and a specific direction. For such a point and direction, the luminance is a measure of the density of luminous flux per unit area of a surface perpendicular to the direction in question, at the point in question, per unit solid angle at the direction in question. Thus the units of luminance are candela per square meter per steradian. Luminance is actually the most intuitively meaningful photometric term, as the human eye is basically a luminance meter which simultaneously samples a large number of directions. It is a fundamental theorem of optics27 that at any point along a given light ray, the luminance in the direction of the ray is constant, or decreasing as a result of loss mechanisms. (This assumes that all optical interactions are passive, i.e. no light amplification occurs.) Figure A2.1 shows, in a very general way, how this conservation of luminance can imply certain relationships between light passing through two surfaces. In this figure it is assumed that there is a certain light flux in question, all of which passes through first surface Si and second surface S2, with some kind of arbitrary intervening passive optical structure lying between the two surfaces. With the terms used as shown in the figure, it can be shown that the conservation of luminance for any given ray directly implies • the following general surface integral relationship; involving luminances L i and L2. C dfi2.dA2[cos(a2)L2(x2,y2.e2.02)]n (38) df2,dAi[cos(ai)Li(xi,y i,ei,0i,)]n = Si s2 (for all n) 110 arbitrary surface through arbitrary surface through which light also passes which light passes arbitrary intervening passive optical structure Figure A 2 . 1 Terminology for Luminance Theorem c I l l While equation 38 is in general complicated to use, it can be applied very easily to a very simple optical problem which directly relates to the collimation of light sources for use with light guides. In such a case, the light source is usually an object having a given surface area A \ , and having the property that light is emitted from the surface of the object in a diffuse manner, i.e. with a luminance which is independent of direction. It is usually desired to use an optical system to cause light to go through a second aperture of area A 2 , with the property that the light has a uniform, constant luminance for directions which are within an angle 9 C of the normal direction of the aperture area, and zero for angles beyond 9 C , as shown in Figure A2.2. Under these conditions, equation 38 implies that 7C A i = 7csin 20 cA2 which implies that A 9 — ^ -z s i n 2 0 c Of course in practice, there is nothing to say that this equality must hold. In fact, generally A 2 will be significantly larger than this value. However, this result is extremely useful in that there is no point in attempting to devise values of A 2 smaller than this — such attempts are equivalent to attempting to violate the second law of thermodynamics! 112 arbitrary passive optical structure Figure A2.2 Implication of Luminance Theorem for Collimation of Light From Diffuse Sources 113 Appendix A3 Measurements of Bulk Absorptivity of Polytetraflorethvlene  to 3mm Wavelength Microwaves. For purposes of interpreting the results of the experiment measuring the attenuation of microwaves traveling through a PTFE prism light guide, described in this thesis, it was necessary to measure the bulk absorptivity of PTFE to 3mm wavelength microwaves. Figure A3.1 diagrammatically depicts the set up used to measure such absorption. The same microwave source emitting horn and detecting horn were used as for the prism light guide experiment, but the path between the two horns was different. Radiation from the emitting horn was directed into one end of a 1" ID copper pipe measuring 30" long, from a distance approximately 12" away, along the axis. The other end of the copper pipe entered an approximately 1' diameter roughly textured aluminum foil integrating structure, which contained the detector horn at a direction well off axis. Thus the signal measured from the detection horn was a measure of the quantity of microwave radiation passing through the pipe. Since the sensitivity of the entire system was understandably wavelength dependent, all measurements were done by sweeping the microwave source through a range of frequency (approximately 80 GHz to 120 GHz wavelength) and the intensity output was plotted as a function of wavelength on a chart recorder, to ensure that spurious wavelength-dependent effects would not create misleading results. , The PTFE was then tested by putting various lengths of close fitting teflon plugs within the pipe. It was found that within this wavelength band the attenuation was small, and essentially independent of wavelength, with a calculated absorption constant of 1.03 ± 0.33 meters"1. Figure A3.2 shows the results from these transmission measurements for the various samples. 114 roughly textured aluminum detection horn Figure A3.1 Measurement of Attenuation of 3mm Microwaves Passing Through Samples of PTFE Microwave frequency GHz Figure A3.2 Measured Transmission vs. Frequency for Various PTFE Sample Lengths 115 Appendix A4 Interpretation' of Results From Monte Carlo Rav Tracing Programs In any Monte Carlo calculation, n t trials are performed, with each having a definite outcome. Generally, the range of possible outcomes is classified into two or more categories. For the sake of this discussion, let us assume that we are interested in one such category, and take as a sample case the outcome that a light ray is absorbed in an optical model of a prism light guide. Let us say that of a total of n t rays, n a are absorbed. Further, we assume that the Monte Carlo program in question can be manipulated by changing a relevant model parameter, such as, in this sample case, the bulk absorptivity, k. We have reason to believe that the probability of a given trial resulting in the outcome of absorption is proportional to k, for sufficiently small values of k. The ratio n a/nt is an approximate measure of the probability of absorption and the error in this measure of probability^ will be approximately V n e /nt when n e « n t. (The precise value is ( V n e /nt)^l - •) Figure A4.1 is a graph of sample results for the Monte Carlo ray tracing program used to model a prism light guide. (The axes are logarithmic to more easily show the wide range of data.) The error bars represent the range of probability for the outcome of escape, resulting from the Monte Carlo statistics. The straight line of unity slope shows the proportionality between absorption loss and bulk absorptivity for the selected range of values, and the intercept represents the constant of proportionality. The conclusion from the graph shown is that the probability of absorption P a , for this particular model, is given by P a = (.074m) x k, where the numerical coefficient is simply an approximate empirical result from this graphical data. 116 Of course, there is no need to use such an approximate empirical formula in practice, as it is always possible to run the Monte Carlo program for any desired value of k, but such an empirical formula may have value in gaining an intuitive understanding of the behavior of the program, or simply in saving computing costs. 117 -2 -7 —»—•—I—i—l—"—i—i—I -4 -3 -2 -1 0 1 In (k/lm-1) Figure A4.1 Graph of logarithm of fraction of rays absorbed in a sample Monte Carlo run, vs. logarithm of bulk absorption constant. Error bars represent probabilistic error due to Monte Carlo statistics. 118 Appendix A5 Calculation of Average Number of Reflections Undergone by Light Rays Traveling Down a Square Light Guide Physically, the behavior of hollow light guides can best be pictured in terms of the reflectivity of the hollow structure's internal surfaces. However, the most directly measured property of a light guide is its fractional loss per unit length, ~ [ ~ - This can be related to the loss per reflection L r in the following equation dz dz v ' In this equation the term "^p" is the average number of reflections per unit length undergone by the light rays. In general, this rate of reflection will be strongly dependent on the angle 0 which the light rays make with respect to the axial direction of the light guide, but will be only weakly dependent on the azimuthal angle. It is therefore worthwhile to consider the mean number of reflections per unit length experienced by light rays having a random azimuthal angle, but. a selected value of G for a given light guide cross sectional shape. This can be calculated in reference to Figure A5.1, which depicts a square cross section prism light guide and the array of mirror images effectively created by the reflective internal surfaces. The example ray in question has an azimuthal angle 0 and moves tanG units in this cross sectional plane, for every unit it moves in the longitudinal direction. The purpose of this "reflected matrix diagram" is to show that the rate of reflection of light rays off the horizontal surfaces shown, per unit length moved in the cross sectional plane, will be I sin0 I , and correspondingly the rate c 119 \ 1 \ i ! \! ' \ - _ — _ J r \ A V » • — 7 — — M Figure A5.1 Determination of Rate of Reflection of Light Rays per Unit Length of Guide 120 of reflection off the vertical surfaces shown will be I cos0 I Based on this idea, the mean number of reflections per unit length, averaged over 0, is 2K Till d(N) 1 f 1 - . , , K j tane 2 f N J tan9 4 , _ s - ^ = t a n 9 — J I s in 0 I + I cos0 I) d0 = — ~ J (sm0 + cos0) d0 = — (37) o o In practice, it will be the case that 9 is relatively small, and therefore a useful approximately that (38) dZ 7IW This is useful, as it is now of interest to consider the case where light rays have a random value of 9, up to a limiting value 0 C . In this case, it is necessary to work out the mean value of (38) over the solid angle, as follows: ec TC Jdee2 d £ N 2 = _ ± _ _ o = _ L 9 c dz jtw 9C 3JE W Tcjdee o This result is very useful in determining the average number of reflections occurring in a unit length of a square prism light guide of width w, for light rays having random angles up to a maximum off axis angle 9c. 

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