Design and development of compact optical systems = Diseño y desarrollo de sistemas ópticos compactos

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(1)UNIVERSIDAD POLITÉCNICA DE MADRID. ESCUELA TÉCNICA SUPERIOR DE INGENIEROS DE TELECOMUNICACIÓN. TESIS DOCTORAL. DESIGN AND DEVELOPMENT OF COMPACT OPTICAL SYSTEMS. JIAYAO LIU Ingeniero en Electrónica 2015.

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(3) UNIVERSIDAD POLITÉCNICA DE MADRID Instituto de Energía Solar Departamento de Electrónica Física Escuela Técnica Superior de Ingenieros de Telecomunicación. TESIS DOCTORAL. DESIGN AND DEVELOPMENT OF COMPACT OPTICAL SYSTEMS DISEÑO Y DESARROLLO DE SYSTEMAS ÓPTICOS COMPACTOS. AUTOR: D. Jiayao Liu Ingeniero en Electrónica DIRECTOR: D. Juan Carlos Miñano Domínguez Doctor en Ingeniería de Telecomunicación 2015.

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(5) Tribunal nombrado por el Magfco. y Excmo. Sr. Rector de la Universidad Politécnica de Madrid. PRESIDENTE:. VOCALES:. SECRETARIO:. SUPLENTES:. Realizado el acto de defensa y lectura de la Tesis en Madrid, el día ___ de ______ de 20___. Calificación:. EL PRESIDENTE. LOS VOCALES. EL SECRETARIO.

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(7) 献给我亲爱的父亲、母亲和我的爱人感谢你们无私的爱与支持 Dedicado a mis padres y mi esposa por su amor y apoyo. “天行健，君子以自强不息地势坤，君子以厚德载物”.

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(9) AGRADECIMIENTOS En esta tesis se ha resumido el trabajo de varios años en el grupo de óptica del CeDInt. No hubiese sido posible la realización de la misma sin Juan Carlos Miñano y Pablo Benítez. Gracias por darme la oportunidad de formar parte del grupo de óptica. A ellos les agradezco toda la confianza ofrecida en mi formación. Su dirección, consejos y guía han sido fundamentales para el desarrollo de mi tesis. Asimismo, quería agradecer a Wang Lin, por ayudarme conseguir la oportunidad de unirme a este grupo líder mundial en óptica y la ayuda de integración en un mundo para mí desconocido. Quería agradecer a mi compañera del despacho, Marina Buljan, por su amistad y su ayuda en el trabajo. Quería agradecer a Juan Carlos Gonzalez y Dejan Grabovickic por las ayudas en el laboratorio. Quería agradecer a Juan Vilaplana por la auyda en la medida del multichannel RXI. También quería agradecer al resto del grupo de óptica: Pablo Zamora, José Infante, Milena Nikolic, Rengmao Wu, Joao Mendes Lopes, Bharathwaj Narasimhan, Cristina Cocho, Guillermo Biot, Jesus López y a todo el personal del CeDInt por hacer el ambiente de trabajo más agradable. A Bharathwaj Narasimhan le agradezco la corrección de inglés de la tesis y los artículos. A Pablo Zamora le agradezco también la traducción del resumen de la tesis. Quería agradecer a otros investigadores de LPI con los que he trabajado directamente en el desarrollo de la tesis, Julio Chaves, Aleksandra Cvetković, José Blen Flores, Rubén Mohedano y Maikel Hernández. Quería agradecer especialmente a mis padres y mi esposa por todo su amor, por apoyarme durante tantos años..

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(11) RESUMEN Esta tesis considera dos tipos de aplicaciones del diseño óptico: óptica formadora de imagen por un lado, y óptica anidólica (nonimaging) o no formadora de imagen, por otro. Las ópticas formadoras de imagen tienen como objetivo la obtención de imágenes de puntos del objeto en el plano de la imagen. Por su parte, la óptica anidólica, surgida del desarrollo de aplicaciones de concentración e iluminación, se centra en la transferencia de energía en forma de luz de forma eficiente. En general, son preferibles los diseños ópticos que den como resultado sistemas compactos, para ambos tipos de ópticas (formadora de imagen y anidólica). En el caso de los sistemas anidólicos, una óptica compacta permite tener costes de producción reducidos. Hay dos razones: (1) una óptica compacta presenta volúmenes reducidos, lo que significa que se necesita menos material para la producción en masa; (2) una óptica compacta es pequeña y ligera, lo que ahorra costes en el transporte. Para los sistemas ópticos de formación de imagen, además de las ventajas anteriores, una óptica compacta aumenta la portabilidad de los dispositivos, que es una gran ventaja en tecnologías de visualización portátiles, tales como cascos de realidad virtual (HMD del inglés Head Mounted Display). Esta tesis se centra por tanto en nuevos enfoques de diseño de sistemas ópticos compactos para aplicaciones tanto de formación de imagen, como anidólicas. Los colimadores son uno de los diseños clásicos dentro la óptica anidólica, y se pueden utilizar en aplicaciones fotovoltaicas y de iluminación. Hay varios enfoques a la hora de diseñar estos colimadores. Los diseños convencionales tienen una relación de aspecto mayor que 0.5. Con el fin de reducir la altura del colimador manteniendo el área de iluminación, esta tesis presenta un diseño de un colimador multicanal. En óptica formadora de imagen, las superficies asféricas y las superficies sin simetría de revolución (o freeform) son de gran utilidad de cara al control de las aberraciones de la imagen y para reducir el número y tamaño de los elementos ópticos. Debido al rápido desarrollo de sistemas de computación digital, los trazados de rayos se pueden realizar de forma rápida y sencilla para evaluar el rendimiento del sistema óptico analizado. Esto ha llevado a los diseños ópticos modernos a ser generados mediante el uso de diferentes técnicas de optimización multi-paramétricas. Estas técnicas requieren un buen diseño I.

(12) inicial como punto de partida para el diseño final, que será obtenido tras un proceso de optimización. Este proceso precisa un método de diseño directo para superficies asféricas y freeform que den como resultado un diseño cercano al óptimo. Un método de diseño basado en ecuaciones diferenciales se presenta en esta tesis para obtener un diseño óptico formado por una superficie freeform y dos superficies asféricas. Esta tesis consta de cinco capítulos. En Capítulo 1, se presentan los conceptos básicos de la óptica formadora de imagen y de la óptica anidólica, y se introducen las técnicas clásicas del diseño de las mismas. El Capítulo 2 describe el diseño de un colimador ultra-compacto. La relación de aspecto ultra-baja de este colimador se logra mediante el uso de una estructura multicanal. Se presentará su procedimiento de diseño, así como un prototipo fabricado y la caracterización del mismo. El Capítulo 3 describe los conceptos principales de la optimización de los sistemas ópticos: función de mérito y método de mínimos cuadrados amortiguados. La importancia de un buen punto de partida se demuestra mediante la presentación de un mismo ejemplo visto a través de diferentes enfoques de diseño. El método de las ecuaciones diferenciales se presenta como una herramienta ideal para obtener un buen punto de partida para la solución final. Además, diferentes técnicas de interpolación y representación de superficies asféricas y freeform se presentan para el procedimiento de optimización. El Capítulo 4 describe la aplicación del método de las ecuaciones diferenciales para un diseño de un sistema óptico de una sola superficie freeform. Algunos conceptos básicos de geometría diferencial son presentados para una mejor comprensión de la derivación de las ecuaciones diferenciales parciales. También se presenta un procedimiento de solución numérica. La condición inicial está elegida como un grado de libertad adicional para controlar la superficie donde se forma la imagen. Basado en este enfoque, un diseño anastigmático se puede obtener fácilmente y se utiliza como punto de partida para un ejemplo de diseño de un HMD con una única superficie reflectante. Después de la optimización, dicho diseño muestra mejor rendimiento. El Capítulo 5 describe el método de las ecuaciones diferenciales ampliado para diseños de dos superficies asféricas. Para diseños ópticos de una superficie, ni la superficie de imagen ni la correspondencia entre puntos del objeto y la imagen pueden ser prescritas. Con esta superficie adicional, la superficie de la imagen se puede prescribir. Esto conduce a un conjunto de tres ecuaciones diferenciales ordinarias implícitas. La solución numérica se puede obtener a través de cualquier software de cálculo numérico. II.

(13) Dicho procedimiento también se explica en este capítulo. Este método de diseño da como resultado una lente anastigmática, que se comparará con una lente aplanática. El diseño anastigmático converge mucho más rápido en la optimización y la solución final muestra un mejor rendimiento.. III.

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(15) ABSTRACT We will consider optical design from two points of view: imaging optics and nonimaging optics. Imaging optics focuses on the imaging of the points of the object. Nonimaging optics arose from the development of concentrators and illuminators, focuses on the transfer of light energy, and has wide applications in illumination and concentration photovoltaics. In general, compact optical systems are necessary for both imaging and nonimaging designs. For nonimaging optical systems, compact optics use to be important for reducing cost. The reasons are twofold: (1) compact optics is small in volume, which means less material is needed for mass-production; (2) compact optics is small in size and light in weight, which saves cost in transportation. For imaging optical systems, in addition to the above advantages, compact optics increases portability of devices as well, which contributes a lot to wearable display technologies such as Head Mounted Displays (HMD). This thesis presents novel design approaches of compact optical systems for both imaging and nonimaging applications. Collimator is a typical application of nonimaging optics in illumination, and can be used in concentration photovoltaics as well due to the reciprocity of light. There are several approaches for collimator designs. In general, all of these approaches have an aperture diameter to collimator height not greater than 2. In order to reduce the height of the collimator while maintaining the illumination area, a multichannel design is presented in this thesis. In imaging optics, aspheric and freeform surfaces are useful in controlling image aberrations and reducing the number and size of optical elements. Due to the rapid development of digital computing systems, ray tracing can be easily performed to evaluate the performance of optical system. This has led to the modern optical designs created by using different multi-parametric optimization techniques. These techniques require a good initial design to be a starting point so that the final design after optimization procedure can reach the optimum solution. This requires a direct design method for aspheric and freeform surface close to the optimum. A differential equation based design method is presented in this thesis to obtain single freeform and double aspheric surfaces. V.

(16) The thesis comprises of five chapters. In Chapter 1, basic concepts of imaging and nonimaging optics are presented and typical design techniques are introduced. Readers can obtain an understanding for the following chapters. Chapter 2 describes the design of ultra-compact collimator. The ultra-low aspect ratio of this collimator is achieved by using a multichannel structure. Its design procedure is presented together with a prototype and its evaluation. The ultra-compactness of the device has been approved. Chapter 3 describes the main concepts of optimizing optical systems: merit function and Damped Least-Squares method. The importance of a good starting point is demonstrated by presenting an example through different design approaches. The differential equation method is introduced as an ideal tool to obtain a good starting point for the final solution. Additionally, different interpolation and representation techniques for aspheric and freeform surface are presented for optimization procedure. Chapter 4 describes the application of differential equation method in the design of single freeform surface optical system. Basic concepts of differential geometry are presented for understanding the derivation of partial differential equations. A numerical solution procedure is also presented. The initial condition is chosen as an additional freedom to control the image surface. Based on this approach, anastigmatic designs can be readily obtained and is used as starting point for a single reflective surface HMD design example. After optimization, the evaluation shows better MTF. Chapter 5 describes the differential equation method extended to double aspheric surface designs. For single optical surface designs, neither image surface nor the mapping from object to image can be prescribed. With one more surface added, the image surface can be prescribed. This leads to a set of three implicit ordinary differential equations. Numerical solution can be obtained by MATLAB and its procedure is also explained. An anastigmatic lens is derived from this design method and compared with an aplanatic lens. The anastigmatic design converges much faster in optimization and the final solution shows better performance.. VI.

(17) TABLE OF CONTENTS. CHAPTER 1 FUNDAMENTALS .............................................................................. 1 1.1 INTRODUCTION TO GEOMETRICAL OPTICS ............................................................. 1 1.2 NONIMAGING OPTICS ............................................................................................ 2 1.2.1 Design problem in nonimaging optics .......................................................... 3 1.2.2 Edge-ray principle......................................................................................... 4 1.2.3 Non-onimagingimaging design methods ...................................................... 4 1.3 IMAGING OPTICS.................................................................................................... 5 1.3.1 The Generalized Cartesian Oval ................................................................... 5 1.3.2 Differential equation method ........................................................................ 6 1.3.3 Wave and transverse ray aberrations............................................................. 7 1.4 IMAGE EVALUATION .............................................................................................. 8 1.4.1 Ray spot diagram .......................................................................................... 9 1.4.2 Ray aberration curves ................................................................................... 9 1.4.3 The Modulation Transfer Function ............................................................. 10 REFERENCE........................................................................................................... 12 CHAPTER 2 DESIGN OF ULTRA-COMPACT MULTICHANNEL COLLIMATOR ......................................................................................................... 13 2.1 INTRODUCTION ................................................................................................... 13 2.1.1 Solid state lighting ...................................................................................... 13 2.1.2 Ultra-compact collimator design ................................................................ 14 2.2 MULTICHANNEL RXI DESIGN .............................................................................. 14 2.2.1 Aplanatic design .......................................................................................... 17 2.2.2 Distributor design ....................................................................................... 17 2.2.3 Channel mirror and channel lens design ..................................................... 21 2.2.4 Central part design ...................................................................................... 28 2.3 MULTICHANNEL RXI DESIGN EXAMPLE .............................................................. 31 2.3.1 Example for comparison ............................................................................. 31. VII.

(18) 2.3.2 Example for prototype ................................................................................ 32 2.4 MULTICHANNEL RXI RAY TRACING .................................................................... 35 2.5 MULTICHANNEL RXI PROTOTYPE ....................................................................... 37 2.5.1 Plastic material ........................................................................................... 37 2.5.2 Prototype of multichannel RXI .................................................................. 39 2.5.3 Prototype characterization .......................................................................... 40 2.5.4 Analysis ...................................................................................................... 45 2.6 CONCLUSION AND FUTURE WORK ....................................................................... 47 REFERENCE .......................................................................................................... 47 CHAPTER 3 OPTIMIZATION OF OPTICAL SYSTEMS ................................. 51 3.1 INTRODUCTION ................................................................................................... 51 3.2 THE MERIT FUNCTION ......................................................................................... 51 3.3 DAMPED LEAST-SQUARES METHOD (DLS) ........................................................ 53 3.4 STARTING POINT FOR THE OPTIMIZATION ............................................................ 55 3.4.1 Compound Cartesian oval .......................................................................... 57 3.4.2 Modified SMS 2D ...................................................................................... 58 3.4.3 Differential equation method...................................................................... 60 3.4.4 Comparison of the three design methods ................................................... 62 3.5 CURVE AND SURFACE FITTING FOR OPTIMIZATION OF OPTICAL SURFACES ........... 65 3.5.1 2D Curve fitting ......................................................................................... 65 3.5.2 3D surface fitting ........................................................................................ 66 3.6 CONCLUSION ...................................................................................................... 66 REFERENCE .......................................................................................................... 67 CHAPTER 4 SINGLE SURFACE IMAGING SYSTEM DESIGN ..................... 69 4.1 INTRODUCTION ................................................................................................... 69 4.2 PROBLEM STATEMENT ......................................................................................... 71 4.3 3D ASPHERIC SURFACE DESIGN ........................................................................... 72 4.4 3D FREEFORM SURFACE DESIGN ......................................................................... 82 4.4.1 Elementary concepts of differential geometry ........................................... 83 4.4.2 Differential equations for freeform optical surface .................................... 85 4.4.3 Numerical solution to differential equation................................................ 88 4.5 3D FREEFORM DESIGN EXAMPLE......................................................................... 92. VIII.

(19) 4.5.1 Initial curves ............................................................................................... 92 4.5.2 Optical surface and image surface .............................................................. 94 4.5.3 Optimization and evaluation ....................................................................... 98 4.6 CONCLUSION ..................................................................................................... 102 REFERENCE......................................................................................................... 103 CHAPTER 5 DOUBLE SURFACE IMAGING SYSTEM DESIGN ................. 105 5.1 INTRODUCTION ................................................................................................. 105 5.2 PROBLEM STATEMENT ....................................................................................... 105 5.3 DIFFERENTIAL EQUATIONS FOR OPTICAL SURFACE PROFILE ............................... 107 5.3.1 Derivation of differential equations .......................................................... 107 5.3.2 Numerical solution using MATLAB......................................................... 111 5.4 DESIGN EXAMPLE .............................................................................................. 114 5.4.1 Anastigmatic design .................................................................................. 114 5.4.2 Aplanatic design for comparison .............................................................. 117 5.4.3 Optimization and evaluation ..................................................................... 120 5.5 CONCLUSION AND FUTURE WORK ...................................................................... 124 REFERENCE......................................................................................................... 124 CONCLUSIONS ...................................................................................................... 127 PUBLICATIONS ..................................................................................................... 131. IX.

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(21) Fundamentals. Chapter 1 FUNDAMENTALS 1.1 Introduction to geometrical optics Geometrical optics is a convenient fundamental tool in most optical designs, imaging and nonimaging. It is obtained from Maxwell´s equations with the condition that the wavelength is short enough compared to the spatial variation of the electromagnetic field. Thus the rays can be defined as a line-like path along which the energy propagates. According to the theory of Geometric Optics, one of the following can happen when a ray intercepts an optical surface: Reflection, Refraction (transmission) and Scattering. When it is transmitted /refracted, the direction in which the light travels will change according to the law of refraction: the famous Snell’s law. It states that the ratio of the sines of the incidence and refraction angles is inversely proportional to the ratio of the indices of refraction of the two mediums. In case that the ray reflects off a smooth surface (reflecting), it follows the law of reflection, which states that the incident and reflected rays make the same angle with normal to the surface. In both cases, all three directions are coplanar. Fermat’s principle is one of several equivalent formulations of geometrical optics. It is based on the concepts of optical path length, which is mathematically defined by the expression [1, 2]: B. L   n  x, y, z  dl ,. (1.1.1). A. where n(x, y, z) is the refractive index of the medium at the point (x, y, z) and dl is the differential length along the light’s path between points A and B. Suppose we have a point source O emitting a number of light rays in the medium, then we can mark off points at the same optical path length from O along all the rays using the Equation(1.1.1). These points can be joined to form a surface called wavefront. Fermat’s principle is a way of predicting the path of a light ray from point A to point B in a medium by stating that a physically possible ray path is one for which the optical path length along it from A to B is a minimum as compared to other neighboring paths. Hamiltonian formulation is another important formulation in geometrical optics. 1.

(22) Jiayao Liu Suppose that we have a ray represented as a 6-dimensional vector (x, y, z, p, q, r), where (x, y, z) is the point in the space and (p, q, r) are optical direction cosines, which are cosines of the angle between the ray direction and the axis x, y, z multiplied by the refractive index. The Hamiltonian formulation states that the trajectories of the rays are given as the solution to the following system of first-order ordinary differential equations: dx  Hp, dt dy  Hq , dt dz  Hr , dt. dp  H x , dt dq  H y , dt dr  H z , dt. (1.1.2). where H = n2 (x, y, z) - p2 - q2 - r2 and t is a parameter without physical significance. Any solution of the preceding Hamiltonian system fulfills H = constant, since dH/dt=0 (note dH. that. dt.  Hx. dx dt.  Hy. dy dt.  Hz. dz dt.  Hp. dp dt. +Hq. dq dt. +Hr. dr dt. and this is zero in view of Hamilton. equations. Rays are only the solutions of the system (1.1.2) contained in H = 0, as can be deduced from the definition of (p, q, r). When n is known and one of the direction cosines is positive, a ray can be defined by a point and two direction cosines as a five-parameter entity. This five-dimension space is called Extended Phase Space. A ray-bundle M4D (or ray manifold) is a four-parameter entity, a closed set of points in the extended phase space, with each point representing a different ray (i.e., two different points cannot correspond to the same ray at two different instants). Often, a ray manifold M4D is defined at its intersection with a reference surface. R, which must observe the condition of intersecting only the trajectories of the rays belonging to M4D. This reference surface defines a four-parameter manifold called Phase Space. For instance, if the reference surface is a plane z = 0, then the phase space is (x, y, p, q). In 2D geometry, all these concepts can be defined similarly. For example, the extended phase space is the three-dimensional sub-manifold, defined by p2 + q2 = n2 (x, y), in the four-dimensional space of coordinates x, y, p and q.. 1.2 Nonimaging optics Nonimaging optics was informally founded in the early sixties with invention of the 2.

(23) Fundamentals compound parabolic concentrator (CPC), and has been rapidly developing ever since [3]. Nonimaging optical systems have the goal of efficiently transferring luminous power from the source to the receiver without the need for image formation. Nonetheless, the design of nonimaging optical systems does not necessarily imply that image formation never occurs. A collimator is an optical system that provides sharper intensity pattern at the receiver than the one at its entry aperture. Since the ray trajectories are reversible, a collimator can be used in the opposite directions as a concentrator.. Figure 1.1 The difference between imaging and nonimaging optical systems is that a specific pointto-point correspondence between the source and the receiver being required.. Classic imaging systems are appropriate solutions for some paraxial nonimaging problems, i.e., those in which the transmitted rays at no time form large angles with the axis of the optical system. When the design problem is non-paraxial, as often occurs in the design of collimators or concentrators, the restriction imposed by image formation are usually quite inconvenient, but more importantly unnecessary in most illumination situations. Imaging optics has the prime goal of preserving spatial contrasts, while illumination engineering usually wants a contrast-free distribution of light upon a surface. On the contrary, in nonimaging optics blurring is welcome, since it needs only boundaries of the source in order to do the projection. All these factors make nonimaging optics a distinct field.. 1.2.1 Design problem in nonimaging optics There are two main groups of design problems in nonimaging optics, bundle3.

(24) Jiayao Liu coupling (such as the case of a collimator or a concentrator) and prescribed irradiance (such as the case of backlight or automotive lighting). A bundle of rays impinging on the surface of the entry aperture of the nonimaging device is called the input bundle, and is denoted by Mi. The bundle of rays that links the surface of the exit aperture of the device with the receiver is the exit bundle Mo. The set of the rays common to Mi and Mo is called the collected bundle Mc. The input and exit bundles are coupled by the action of the device. In the first group, bundle-coupling, the design problem is to specify the bundles Mi and Mo, and the goal is to design the nonimaging device to couple the two bundles, i.e., making Mi = Mo = Mc. The good example is the multichannel collimator presented in Chapter 2. For the second group of design problems, prescribed-irradiance, it is only specified that one bundle must be included in the other, for example, Mi in Mo (so that Mi and Mc coincide), with the additional condition that the bundle Mc produces a prescribed irradiance distribution on one target surface at the output side. As Mc is not fully specified, this problem is less restrictive than the bundle-coupling one.. 1.2.2 Edge-ray principle The edge-ray theorem is a fundamental tool in nonimaging optics design. This theorem states that for an optical system to couple two ray bundles Mi and Mo it suffices to couple bundles Mi and Mo, where Mi and Mo are the edge-ray subsets of bundle Mi and Mo (and have one dimension fewer). A perfect matching between bundles Mi and Mo implies the coupling of their edge-rays. This theorem was proven by Miñano [4] in the mid-eighties, and Benítez [5] extended this demonstration in the late nineties. The edge-ray principle is the design key in most nonimaging devices, and shows the benefits that arise from the elimination of the imaging requirement.. 1.2.3 Nonimaging design methods There are several nonimaging optical design methods: the flow-line, the Simultaneous Multiple Surface method in two dimensions (SMS2D) and in three dimensions (SMS3D), as well as some others. The flow-line method is also called the. 4.

(25) Fundamentals Winston-Welford design method. Let´s consider that we have two wavefronts, wA and wB, propagating through an optical system. After propagation, wA1 is transformed to wA2 and wB1 to wB2. The flow line bisects the rays, at each point, the rays coming from wA1 and wB1. It also bisects the rays going to wA2 and wB2. At each point, the optical path lengths to the wavefronts satisfy the following equations:. SA1  SB1  const,. (1.2.1). SA2  SB2  const.. (1.2.2). The flow-line method allows a high efficient control of light confinement inside a light guide while conserving étendue [6]. The SMS method is a very powerful tool, capable of designing simultaneously several surfaces at once. It was originally developed for two dimensional designs, and later extended to three dimensions as SMS3D, so that the surfaces without symmetry (free-form surfaces) could be designed. This method generates two surfaces at the same time (refractive or reflective) [7, 8, 9, 10].. 1.3 Imaging optics Imaging optical systems usually have three main components: the object (instead of the light source), the optic and the image (instead of the receiver) it forms. The light or part of it from each point on the object is collected by the optical system and converged to the corresponding point on the image. Therefore, the correspondence between the points on the objects and those on the image are required.. 1.3.1 The Generalized Cartesian Oval From wavefront’s point of view, the optical path length from object point to image point is the same along all rays to achieve “perfect” images. To solve this problem, the Generalized Cartesian Ovals are used. They are refractive or reflective surfaces that transform one input wavefront w1 from a point A into another output wavefront w2, which converges to another point B, as shown in Figure 1.2.. 5.

(26) Jiayao Liu. Figure 1.2 A refractive Cartesian Oval transforms the spherical wavefront from point A into the spherical wavefront converges to point B. The condition that the optical surface must fulfill is. n1  AP  n2  PB  const,. (1.3.1). where P is any point on the Cartesian Oval.. 1.3.2 Differential equation method The generalized ray tracing equations link the curvatures of the optical surfaces, the ones of the input and output wavefronts and the twist of the principal lines of curvature caused by refraction. Considering the plane of incidence and the plane perpendicular to the incident plane which contains the normal to the surface as well, the Coddington equations are the ones giving the propagation of the parameters of the wavefronts as [11]:.  no cos2  o ni cos2 i no cos  o  ni cos i      qs qo qi   no n n cos  o  ni cos i  i  o ,     po pi ps   n cos  n cos i no cos  o  ni cos i o  o  i  o i s . (1.3.2). where suffix q and p refer to the plane of incidence and the perpendicular one respectively, n denotes the refractive index, ρ denotes the radius of curvature, α denotes the angle between the ray and the normal of the surface and 1/ denotes the torsion of the geodesic 6.

(27) Fundamentals curve. Suffix i, o and s refer to input, output and surface respectively. The local properties of optical surfaces and wavefronts such as slope and radius of curvature can be expressed by the concepts of differential geometry. Thus, series of differential equations can be derived to describe the optical system. Although in most cases, the acquisition of analytical solutions are impossible, there are already existing methods to obtain numerical solutions.. 1.3.3 Wave and transverse ray aberrations [12, 13] As we can see, the Cartesian Oval can provide “perfect” imaging for a point object. But in real applications, the object is usually extended, which means there are infinite points in place of only one object point. Therefore, there will be infinite spherical wavefronts coming from the object and crossing the optical system. They will be transformed into other wavefronts that are not spherical in general, in other words, they are aberrated from the spherical wavefronts that center at the corresponding image points.. Figure 1.3 The spherical wavefronts from the point object is converted to aberrated wavefront by the optical system. As shown in Figure 1.3, the dash curve S refers to the spherical wavefront that centers at the image point P. The solid curve W refers to the aberrated wavefront. A general ray GR intersects the spherical wavefront, the aberrated wavefront and the image plane at Q, Q' and P' respectively. The geometrical deviation QQ’ times the refractive index of the medium of the wavefront from a spherical wavefront that centers at the image point is 7.

(28) Jiayao Liu called the wave aberration. It can be presented as a power series expansion:  A2 rh cos . Magnification.   . 1st order.  B1r 4. Spherical aberration   Coma   Astigmatism   Petzval   Distortion . 3rd order. A1r 2. Defocus.  B2 r h cos  3. W  r ,  , h    B3 r 2 h 2 cos 2   B4 r h 2. 2.  B5 rh 3 cos . .  etc. (1.3.3). high order. The definition of r, h and  are shown in Figure 1.4.. Figure 1.4 A ray from the object point y=h, x=0 in the object passes through the optical system aperture at a point defined by its polar coordinates (r, ), and intersects with the image surface (x’, y’, z’).. The displacement PP’ of the two image points formed by the aberrated wavefront and the spherical wavefront is called the transverse ray aberrations.. 1.4 Image evaluation [12, 13] There are many criteria to evaluate the performance of an optical system from different aspect. Some of them give information about which aberrations an optical system has and the amount of the certain aberrations; some of them provide information about the capability of an optical system to resolve object. The main criteria used along this thesis are ray spot diagram, ray aberration curves and modulation transfer function (MTF). 8.

(29) Fundamentals. 1.4.1 Ray spot diagram If the entrance pupil of an optical system is uniformly divided into a large number of equal areas and through the center of each, a ray is traced from the point object, the intersection of the rays with the image plane will form a representation of the geometrical image. The more rays traced, the more accurate the representation of the image will be. This ray intercept plot is called the ray spot diagram. A convenient metric for the quality of the image, called the RMS spot size, can be derived from the ray spot diagram. The “center of gravity” of the image spot is determined as the average coordinates of the ray intersections with the image plane. Then the RMS spot size is defined as RMS . Ri2 n. (1.4.1). where RMS stands for root mean square, Ri is the radial distance of the ith spot from the “center of gravity” and n is the number of the rays traced. The RMS spot size gives intuitive information how the image of the point object expands.. 1.4.2 Ray aberration curves The ray aberration curves are another useful criteria for image evaluation. It helps designers to know not only the type of the aberrations an optical system has, but also the amount of the image blurring. The ray aberration curves represent the behavior of a fan of rays in tangential plane and sagittal plane respectively. The rays are traced from the object point through y (for tangential) and x (for sagittal) axis on the aperture of the optical system shown in Figure 1.4. The normalized radius of the intersection point on the aperture will be the horizontal axis of the ray aberration curves. The distance between the intersection points of the rays traced and the chief ray, which passes through the center of the aperture, will be the vertical axis. Figure 1.5 shows typical ray aberration curves [14].. 9.

(30) Jiayao Liu. Figure 1.5 Ray aberration curves in tangential and sagittal plane respectively. 1.4.3 The Modulation Transfer Function The Modulation Transfer Function (MTF) is another commonly used metric for image evaluation. A series of alternating light and dark bars of equal width is used as the test pattern and is shown in Figure 1.6 together with its brightness level plot.. Figure 1.6 (a) The test pattern of N bars per unit length. (b) The brightness level plot of squarewave form.. The modulation is defined as the contrast of the pattern and given by the equation. Modulation . I max  I min . I max  I min. (1.4.2). When the pattern is imaged by an optical system, each bar will be imaged as a blurred 10.

(31) Fundamentals one and the brightness level will be smoothened as shown in Figure 1.7.. Figure 1.7 (a) The blurred pattern. (b) The smoothened brightness level plot.. The modulation transfer function is the ratio of the modulation in the image to that in the object as a function of frequency of the sine-wave pattern instead of the squarewave pattern shown previously. MTF   . Modulationimage Modulationobject. (1.4.3). where frequency is defined as cycles per unit length, which indicates the number of lines per unit length of the test pattern. The MTF is used to evaluate the capability of an optical system of resolving object at different frequencies. See Figure 1.8 for example, system A has superior performance at lower frequency while system B has higher limiting resolution.. Figure 1.8 System A has superior performance at low frequency while system B has higher limiting resolution. 11.

(32) Jiayao Liu. REFERENCE 1. R. Winston, J.C. Miñano, P. Benítez, “Nonimaging Optics”, Elsevier, Academic Press, (2004) 2. Julio Chaves, “Introduction to Nonimaging Optics”, CRC Press, 2008 3. W.T. Welford, R. Winston. “High Collection Nonimaging Optics”, Academic Press, New York, 1989 4. J.C. Miñano, “Two-dimensional nonimaging concentrators with inhomogeneous media: a new look”, J. Opt. Soc. Am. A 2(11), pp. 1826-1831, (1985) 5. P. Benitez, “Conceptos avanzados de óptica anidólica: diseño y fabricación”, Thesis Doctoral, E.T.S.I.Telecomunicación, Madrid (1998) 6. Julio Chaves, “Introduction to Nonimaging Optics”, CRC Press, 2008 7. P. Benítez, R. Mohedano, J.C. Miñano. “Design in 3D geometry with the Simultaneous Multiple Surface design method of Nonimaging Optics”, in Nonimaging Optics: Maximum Efficiency Light Transfer V, Roland Winston, Editor, SPIE, Denver (1999) 8. P. Benítez, J.C. Miñano, et al, “Simultaneous multiple surface optical design method in three dimensions”, Opt. Eng, 43(7) 1489-1502, (2004) 9. O. Dross, P. Benitez, J.C. Miñano et al, “Review of SMS Design Methods and Real World Applications”, SPIE, San Diego (2004) 10. US patent “Three- Dimensional Simultaneous Multiple Surface Method and FreeForm Illumination Optics Designed Therefrom” Inventors Pablo Benitez, Juan.C 11. Orestes N. Stavroudis, “The Mathematics of Geometrical and Physical Optics: The k-function and its Ramifications”, Wiley-VCH, Berlin, 97-114 (2006) 12. Warren Smith, “Modern Optical Engineering, 4th Ed”, McGraw-Hill Professional, 2007 13. V. N. Mahajan, “Aberration Theory Made Simple, 2º Ed”, SPIE PRESS, 2011 14. Synopsis®, Introduction Course to CODEV notes, 2013. 12.

(33) Design of ultra-compact multichannel collimator. Chapter 2 DESIGN OF ULTRA-COMPACT MULTICHANNEL COLLIMATOR 2.1 Introduction. 2.1.1 Solid state lighting One important application of nonimaging optics is illumination system design, especially for solid state lighting, which uses Light Emitting Diode (LED) as its source. With the advantages such as high efficiency, long life time, stability and environmental friendly, solid state lighting has become the latest topic of interest illumination technology. In March of 2014, Cree announced a new efficiency record of white-light production with a laboratory demonstration of 303 lumens per watt at a correlated color temperature of 5150 K and 350 mA [1]. The rapid improvement in LED efficiency shows great potential in the role it can play in Solid State Lighting (SSL).. Figure 2.1 White-light LED package efficacy projections for commercial product [2]. (pc-LED stands for phosphor-converted LED, Qual Data stands for qualitative data). 13.

(34) Jiayao Liu LEDs are more suitable for compact optics than incandescent lamps owing to the fact that their operating temperature allows low cost optics, and also their higher radiance comparing with fluorescent lamps makes them more suitable for collimating applications.. 2.1.2 Ultra-compact collimator design Collimator is an important application of LED lamps used in spot lighting in home, retail and advertisement sectors. The design of LED collimator has to meet several requirements like high efficiency and on axis intensity, good beam control, low cost, and compactness. Compact collimators not only allow sufficient space for electronic and cooling systems, but also lead to weight and volume reduction, which reduces material and shipping costs. In a quite similar application to LED optics, Concentrating Photovoltaics (CPV) has also explored the world of compact optics: Recently, several companies in CPV have proposed very thin concentrators [3, 4]. In general, the compactness can be achieved by low aspect ratio (Aspect ratio=height/diameter) architectures, namely Fresnel lenses, parabolic or aspheric reflectors and TIR lenses. All these architectures share the idea of dividing the light flow into different “channels” each one of them having its own optics whose surfaces are not shared with neighboring channels. At the turn of this century, multichannel devices called Stepped flow-line optics (SFL) were developed [5, 6, 7]. They are designed with the flowline method [8] and soon enough, found applications in the design of backlights [9, 10], optics for combining light sources or efficiently distributing light to several targets [11, 12, 13, 14, 15]. The main disadvantage of these devices lies in their manufacturability which is in general complex because the flow-line mirrors, derived from the method, used to have complex shapes and large areas (relative to the aperture area). In the next section, a novel multichannel design approach for an ultra-compact collimator for a Lambertian LED located on the optical axis to solve aforementioned disadvantage of flow-line based designs is illustrated.. 2.2 Multichannel RXI design The multichannel collimator makes use of reflection (denoted by X), total internal reflection (denoted by I) and refraction (denoted by R) to accomplish collimation. The 14.

(35) Design of ultra-compact multichannel collimator collimator is designed with rotational symmetry, i.e., it’s a 2D design in the sense that only the degrees of freedom of a 2 dimensional design are allowed. The multichannel RXI is characterized by having an optical surface called distributor d which collects light from the source (an LED in this thesis) and distributes it over a given support surface m. The optics will be confined between m and another support surface l. In Figure 2.2, the cross sections of both support surfaces are straight lines. The flow lines of the input and output beam are also shown in Figure 2.2. The input beam is formed by the rays emitted from the LED and the output beam is the collimated beam that we want to have at the exit aperture of the system.. Figure 2.2 The design of the reflective distributor. The optics is confined between the two lines l and m.. Besides the distributor, the multichannel RXI is formed by several optical “channels” (each one of them formed by a lens along the line l and a mirror along the line m) that work in parallel. Figure 2.3 shows how the ray bundles are separated by the channel mirrors and collimated by the channel lenses.. 15.

(36) Jiayao Liu. axis. channel lens. distributor d: reflector. channel mirror. Figure 2.3 The ray bundles are chopped by channel mirrors and collimated by channel lenses. The design procedure is schematically shown in Figure 2.4.. Figure 2.4 Design procedure of multichannel RXI. The design starts from the selection of the position of the source S, the support line l and the support line m (Figure 2.4(a)). The distributor is designed so that the rays from the source S after reflection on it intersect with the support line m with coordinate x such that x=f sin (Figure 2.4(b)). The mirrors along the support line m and the lenses along the support line l are designed so that the rays are emitting vertically from the top of the multichannel RXI (Figure 2.4(c)). Figure 2.4(d) shows the ray trajectories.. 16.

(37) Design of ultra-compact multichannel collimator. 2.2.1 Aplanatic design Aplanatic designs are axisymmetric optical designs free from on-axis spherical aberration and linear coma. This entails two conditions: (1) the Abbe sine condition and (2) stigmatic on-axis image [16, 17, 18]. The Abbe sine condition x  f sin . (2.2.1). where f is the parameter to define the aperture size and  is the angle between the ray emanating from the source and the symmetric axis as indicated previously in the schematic design procedure. The collimated output rays and the source point ensure the stigmatic on-axis image. Two rotationally symmetric surfaces give enough degrees of freedom to meet both conditions, so that two surfaces (one reflective and one refractive) can produce an aplanat for the corresponding channel. The distributor surface doesn’t contribute to aplanatic conditions, but helps to reduce the thickness of the whole system.. 2.2.2 Distributor design According to the rotational symmetry, the design of the multichannel RXI is done in x-y plane in which the source point S is on the origin and the y axis is the axis of symmetry. The design starts from the line of the distributor, which is designed with the condition that a ray issuing from the source and reflected at the distributor must intersect with the “mirror supporting line” m with coordinate x fulfils Equation(2.2.1). S1 and S2 are used to indicate the y-coordinate of “lens supporting line” l and “mirror supporting line” m respectively, and A is used to indicate the distance from the initial point of the distributor to the source. The line of the distributor starts from the initial point P0 on the axis of symmetry, above the source point S, i.e., with the coordinate (0, A). The input ray R0 and the angle. 0=0º to the axis are derived from P0 and S. The point of intersection T0=(S1, 0) of the reflected ray and the mirror supporting line m is derived from the Equation(2.2.1). Thus the reflected ray R0' can be derived from T0 and P0. The normal N0 at the point P0 can be derived by the equation. N  unit  R  R . 17. (2.2.2).

(38) Jiayao Liu. N0. R0. R 0. Figure 2.5 The first point of the distributor is at (0, A). The intersection with the support line m and the normal of the point is calculated from the input ray and the reflected ray.. Then consider an input ray issuing from the source with a small increment of the angle , the intersection with the tangent line from P0 can approximate the second point P1 of the distributor. The corresponding point T1 on the support line m and the normal can be derived as described for P0 (Figure 2.6).. N1. R1. R 1. Figure 2.6 The second point of the distributor is the intersection of the input ray with a small increment of the angle  and the tangent line of the first point.. A series of points on the line of the distributor can be derived by iteration of the similar process (Figure 2.7).. 18.

(39) Design of ultra-compact multichannel collimator. Figure 2.7 A series of points of the distributor is derived by repeating the process.. If the increment of the angle  is small enough, every one of them can present the real point of the distributor at that place with enough accuracy. Thus the line of the distributor can be represented by these points. The distributor is affected by the parameters f, S1 and A simultaneously. But every parameter has a significant effect on the distributor. For example, A mainly defines the height of the distributor which more or less determines the thickness of the whole system. Figure 2.8 shows the distributor with different parameter A.. Figure 2.8 Distributors with different A. The parameter f simply defines the aperture of the optical system according to Equation(2.2.1). The parameter S1 mainly affects the slope of the distributor as shown in Figure 2.9. 19.

(40) Jiayao Liu. Figure 2.9 Distributors with different S1. As the parameter A affects the thickness of the system significantly, for designing an ultra-thin optical system, it is obvious to decrease its value. This method works perfectly under point source condition. But in real applications, 1mm×1mm LED as source for multichannel RXI, the size of the source cannot be ignored comparing to the dimension of the device. Figure 2.10 shows the effect of an extended source... Figure 2.10 The effect of an extended source.. As shown, with an extended source, every point on the distributor not only receives a single ray, but a bundle of rays. The reflected one will also be a bundle of rays. When the distance between the distributor and the LED is too small, there will be some rays that cannot be reflected to the supporting line m, thus cannot be reflected to the lens supporting line which leads to losses of energy. Thus the parameter A cannot be decreased without limit. 20.

(41) Design of ultra-compact multichannel collimator The distributor is designed to work under total internal reflection (TIR) in order to avoid energy loss. Thus it starts from the point that the ray from the worst position (right edge of LED) can fulfil the TIR condition. In Figure 2.11, the red part of the distributor can fulfil the TIR condition for every ray from the source.. Figure 2.11 TIR part of the distributor. 2.2.3 Channel mirror and channel lens design When the distributor is determined, the reflected rays from the distributor are distributed over the support line m with known x-coordinate and direction for different channels. Each channel consists of a mirror on the support line m and a lens on the support line l and works as an aplanatic system. To achieve the same irradiance (flow per unit surface) on both support lines, the collimated outgoing rays should have the same x-coordinate on the support line l as on the support line m, i.e., fulfils Equation(2.2.1). The design of the mirror starts from an initial point P0 on the support line m. From the x-coordinate of the point, the angle  of the ray, which reaches this point, when it issues from the source point can be derived. This ray will finally exit from the support line l as a collimated one, with the same x-coordinate, i.e., the ray becomes vertical after the reflection on the mirror supporting line and reaches the point P0' on the support line l. After the refraction, the ray exits vertically as shown in Figure 2.12.. 21.

(42) Jiayao Liu. Figure 2.12 Initial points on the support line m and support line l.. The normal of the point P0 and P0' can be derived from the three ray vectors. Now consider a ray emanating from the source with a small increment (or decrement) of the angle , after the reflection on the distributor, it will intersect with the support line m at a point near P0. If the increment is small enough, the intersection between the ray reflected by the distributor and the tangent line of the point P0 can represent the real point of the mirror. After reflection, this ray is expected to intersect with the tangent line of the point P0' at the point the x-coordinate of which fulfils Equation(2.2.1), then become the collimated output ray after refraction. The normal of both points can be derived from the ray vectors. The mirror and the lens keep extending until the lens reaches a threshold height. Thus a channel is formed by a pair of corresponding mirror and lens as shown in Figure 2.13.. Figure 2.13 A channel is formed by a pair of corresponding mirror and lens.. 22.

(43) Design of ultra-compact multichannel collimator Subsequent channels can be derived similarly when their initial points are chosen. But the channel obtained is very likely to be discontinuous with the previous one. Thus iteration has been applied to change the initial point according to the x-coordinate difference between the left edge of the current lens and the right edge of the previous one. The new pair of corresponding mirror and lens can be derived from this iteration so that the left edge of the new lens is just connected with the right edge of the former lens (see Figure 2.15), thus keeps the continuality of the lenses on support line l. Considering that the ray issuing from the source with an angle of  will have an x-coordinate which fulfills Equation(2.2.1), the continuality of the lenses guarantees the continuality of the flux.. Figure 2.14 The continuality of two pairs of corresponding mirror and lens.. Repeating this procedure, a series of corresponding mirrors and lenses are obtained to collimate the rays issuing from a point source. But as shown before, the LED is an extended source so that every point of the mirror not only receives one single light ray, but a bundle of rays. After the reflection on the mirror, the bundles of the rays go toward the lens. As shown in Figure 2.15, some of the rays are out of the range of the lens, which means these rays cannot be refracted by the corresponding lens, thus very probably being lost.. 23.

(44) Jiayao Liu. Figure 2.15 A pair of corresponding mirror and lens designed for point source cannot collect all the rays from an extended source.. An intuitive solution for eliminating this energy loss can be implemented by making all the rays fall within the range of the corresponding lens. This can be achieved either by enlarging the range of the lens or forcing the convergence of the rays reflected by the mirror. Since every lens is connected with each other, there is no room to enlarge it without affecting others; the second solution becomes reasonable and applicable. Forcing the convergence of the rays reflected by the mirror means changing the xcoordinate of the rays on the support line l. A factor scale has been introduced in the design to define the degree of convergence. Consider again a ray issuing from the source with an angle of , after the reflection on the mirror, it reaches the lens with x-coordinate no longer fulfills Equation(2.2.1), but the following new one:.  x  x0  scale   f  sin   x0  ( f  sin   x0 ) ,   x  x0  scale   x0  f  sin   ( f  sin   x0 ). (2.2.3). where x0 is the x-coordinate of the initial point. Figure 2.16 gives a better understanding.. 24.

(45) Design of ultra-compact multichannel collimator. Figure 2.16 The ray (cyan) fulfills Equation(2.2.1) and intersects the support line l with x=x1. The converged ray (purple) intersects the support line l with x=x2.. In Figure 2.16, the blue ray is the one which intersects with the initial point of the mirror, which doesn’t change whether the rays are converged or not. The cyan ray is the one from the design for a point source, the purple one is the converged ray. The relation between them is shown as:. x1  scale. x2. (2.2.4). For converging rays, the factor scale should be greater than 1. With the new Equation(2.2.4), a new pair of corresponding mirror and lens has been derived (Figure 2.17).. 25.

(46) Jiayao Liu. Figure 2.17 A pair of corresponding mirror and lens based on converged rays can collect more rays than the one designed for point source.. As shown in Figure 2.17 that the edge rays of the bundle from the endpoint of the mirror are all covered by the corresponding lens. According to the edge ray principle, if the edge rays of the bundle are covered by the lens, the whole bundle is covered by the lens. Thus all the rays reflected by the mirror can be collected by the lens. The extended source also has effect on the design of mirrors as shown in Figure 2.18. In the area pointed out by the red arrow, there are some rays that cannot reach the next mirror, which means they cannot be reflected to the corresponding lens, which leads to energy loss.. 26.

(47) Design of ultra-compact multichannel collimator. Figure 2.18 Energy losses between two mirrors. Obviously the mirror has to be extended in order to cover additional rays emitted by the extended source. As shown in Figure 2.19, for the channel on the right, solid ray refers to the ray that reaches the second mirror while dash rays refer to the rays chopped by the former mirror. Although some ray bundles are chopped by the previous mirror, the present channel is designed for the complete ray bundles.. Figure 2.19 Energy loss is eliminated by extending the mirror for complete ray bundles.. Repeating the above procedure, a series of channels consisting of corresponding mirrors and lenses has been derived (Figure 2.20).. 27.

(48) Jiayao Liu. Figure 2.20 A series of channels consisting of corresponding mirrors and lenses.. 2.2.4 Central part design The distributor is one of the main parts of the optical system and is always been placed. It blocks the trajectory of some of the rays reflected by the mirror. To avoid this problem, there are two solutions that can be used together or alternatively: 1. Eliminate the channels whose reflected rays are blocked by the distributor. This would eliminate the central channel shown in Figure 2.21 and partially its adjacent channel. When the central channels are deleted, the central part of the distributor becomes useless. This is important because this central part could not work by TIR. In most of these cases, the usable part of the distributor can work by TIR. Additionally, a lens can be placed in the deleted central part of the distributor to help the efficiency. This central part won’t have the same degree of collimation as the remaining parts of the multichannel RXI.. Lens to replace central channel. Figure 2.21 Channels blocked by the distributor are deleted and are replaced by a lens.. 28.

(49) Design of ultra-compact multichannel collimator 2. Create a thin layer of low refractive index at the position of the distributor so it works as a TIR mirror for most of the rays issuing from the source but is transparent for the incoming rays reflected from the mirror (in a similar way as the front reflector of the SMS 2D design called XX works). This is the case shown in Figure 2.22. Again, there is a central part that won’t work by TIR. This part can be metalized, which will produce some losses by light blocking (in general very small, <1%).. Figure 2.22 Multichannel RXI with a thin layer of low refractive index.. The first solution has been applied in this design. The central part is designed as a Cartesian oval for two wavefronts: the spherical wavefront centers at the point source and the plane output wavefront. The central part starts from the edge of the TIR part of the distributor. The optical path length can be calculated according to. L  n  l1  l2 , where. (2.2.5). n is the refractive index of the material, l1 is the geometrical length between the. source and the start point of the central part, which is the edge of the TIR part of the distributor, and. l2 is the geometrical length between the start point of the central part and. the plane output wavefront.. 29.

(50) Jiayao Liu. Figure 2.23 Start of the central part.. For any ray emanating from the source with an angle  to the axis within the central part, the corresponding point (x, y) on the central part can be derived by solving the following equations:.    n  . x  tan  y. (2.2.6).  x  x0    y  y0    yw  y   L 2. 2. where (x0, y0) is the coordinate of the source, yw is the y-coordinate of the plane output wavefront. The central lens is shown in Figure 2.24.. 30.

(51) Design of ultra-compact multichannel collimator. Figure 2.24 The central lens of multichannel RXI is a Cartesian oval.. 2.3 Multichannel RXI design example. 2.3.1 Example for comparison [19] An example of multichannel RXI has been presented in Figure 2.25 to compare with other collimators from different design approaches: Aplanatic lens, refractive Fresnel lens with teeth away from the source, TIR lens with curved exit surface, TIR Fresnel lens, RXI, grooved RXI, parabolic mirror, mirror with side emitting lens, slat collimator consisting of reflective parabolic segments with central refractive Fresnel lens, XX Fresnel. All the collimators have a dimensionless diameter of 10 with a source of diameter 1.. 31.

(52) Jiayao Liu. Figure 2.25 Design example of multichannel RXI for comparison with other approaches.. This multichannel RXI example has achieved a height of 1.49, which indicates an aspect ratio of 0.149. The comparison of the aspect ratio of different collimators is shown in Figure 2.26.. Aspect ratio XX Fresnel Slat collimator Mirror with side lens Parabolic mirror Grooved RXI RXI. Aspect ratio. TIR Fresnel lens TIR lens Refractive Fresnel lens Aplanatic lens Multichannel RXI 0. 0.2. 0.4. 0.6. 0.8. 1. 1.2. 1.4. Figure 2.26 Aspect ratio of collimators from different design approach.. It can be seen from the comparison that the multichannel RXI has significantly reduced the aspect ratio of the collimator, which dramatically increases the compactness of the optical device.. 2.3.2 Example for prototype Another multichannel RXI example has been developed for prototype manufacturing. The complete device of multichannel RXI is 98 mm in diameter, 8 mm in 32.

(53) Design of ultra-compact multichannel collimator thickness, which indicates an aspect ratio of 0.0816. The design was planned to use the LED LW W5SN from OSRAM as the source, which has a flat light emitting surface. To coincide with the LED, the entry aperture of the optical system is also designed as a flat surface and has a 50 micron air gap to the encapsulation of the LED.. Figure 2.27 LED LW W5SN from OSRAM is used as the source for multichannel RXI.. The multichannel RXI will be fabricated using a five-axis diamond turning machine, shown in Figure 2.28. This advanced machine can work in different modules depending on the form of the work piece.. Figure 2.28 Five-axis diamond turning machine. 33.

(54) Jiayao Liu Although the C0 continuity of the curves of multichannel RXI has been guaranteed, which means every part touches the adjacent ones at the point of intersection, the C1 continuity is not fulfilled, which means the two parts don’t share the same tangent direction at the point of intersection. This is shown in Figure 2.29 as sharp angles formed by two adjacent lenses. This kind of connection cannot be realized in real modeling neither by diamond turning or injection.. Figure 2.29 The joint points between lenses are sharp angles due to C1 discontinuity.. The solution is to smooth the angle, which is done by replacing the sharp angle with an arc line that is tangent to both lens curves as shown in Figure 2.30. This solution has also been applied for sharp angles on the mirror side, thus adding C1 continuity to multichannel RXI.. 34.

(55) Design of ultra-compact multichannel collimator. Figure 2.30 The sharp angle is replaced by an arc of radius of 50 µm.. In Figure 2.31, (a) the cross section of the whole multichannel RXI with the package of the LED is shown. (b) and (c) show the top view and the bottom view respectively.. lens. lens. distributor. LED mirror (a) Cross section of multichannel RXI. (b) Top view of multichannel RXI. (c) Bottom view of multichannel RXI. Figure 2.31 (a) Cross section of multichannel RXI. (b) Top view of multichannel RXI. (c) Bottom view of multichannel RXI.. 2.4 Multichannel RXI ray tracing The ray tracing simulation of the multichannel RXI has been done in Light Tools. The source is the LED model LW W5SN from OSRAM, which is used in the 35.

(56) Jiayao Liu characterization of the prototype. This model of the source contains a ray file which presents the light distribution of the real model (Figure 2.32).. Figure 2.32 LED model LW W5SN in Light Tools.. A series of simulations have been done. Firstly, the theoretical model has been analyzed. This model includes the Fresnel losses and absorption of the material. Since the mirrors on the support line m don’t fulfill the condition of TIR, they are covered by a coating of silver which has a reflectivity of about 95% (Figure 2.33).. Figure 2.33 The bottom part of the model is covered with a coating of silver.. 36.

(57) Design of ultra-compact multichannel collimator The preliminary results show the angular distribution of luminous flux. Thus the percentage of flux within different angles can be derived (Table 1): Table 1 Percentage of flux within different angles. Angle(°). ±5. ±10. ±20. ±60. Percentage(%). 70.4. 75.2. 79.8. 92.46. The results also give intensity pattern shown as Figure 2.34, the maximum intensity as 635 cd/lm and the full width at half maximum (FWHM) as 2.4°.. Figure 2.34 Intensity pattern of the multichannel RXI. 2.5 Multichannel RXI prototype In previous sections, a detailed explanation of multichannel RXI design has been presented and a design example for prototype manufacturing has been made and simulated. In this section, the prototype of multichannel RXI fabricated by direct cut using a five axis diamond turning machine is presented and characterized.. 2.5.1 Plastic material The glasses have been used traditionally for optical manufacture, especially in optics 37.

(58) Jiayao Liu that requires high stability and thermal resistance. However, they are not suitable for mass production. On other hand, plastics are suitable for mass production, they are lighter than the glasses, but their thermal resistance is lower. Since multichannel RXI is designed as ultra-compact, light and cheap device, the plastic has been chosen to fabricate the device. The plastic materials mostly used in optics are Poly(methyl methacrylate) (PMMA) and Polycarbonate (PC) due to their high transparency. The transparency of a polymer is closely related to its molecular structure. Interatomic or intermolecular interactions of a polymer exposed to light produce absorption in the ultraviolet and visible region, due to electronic transitions or absorption in the infrared due to vibrational transitions. PMMA has higher transparency than PC, since in its molecular structure there are no absorption centers for visible light. Another important feature of polymers is their refractive index. The higher the index is, the larger the Fresnel losses are at the interfaces material-air and air-material. Polycarbonate has higher refractive index than PMMA, so the amount of transmitted light in this material is lower. Fresnel losses can be reduced when an anti-reflexive layer is placed on the optical surfaces, but it complicates the design. The following features are also important: birefringence (or double refraction) and heat resistance. Birefringence only occurs when a material is anisotropic, making the material behave as having different refractive indices for different polarizations of light. In situations where the temperature changes a lot, the plastics may not be appropriate material, due to huge refractive index variation which is 10-100 times higher than in the case of the glasses. PMMA characteristics include high transparency, low birefringence, high hardness and resistance to deterioration by light. The PMMA is used in large lenses, such as condensers, Fresnel lenses, automobile lenses or cameras. The PC features include: low level of impurities, good transparency, high refractive index (high light diffusion), resistance to heat. It is used for optical discs, optical lenses and optical films for liquid crystals. It has great impact resistance, so it is used in multiple applications. To obtain a high efficient system, the used material must have high transparency, so PMMA has been the proper choice.. 38.

(59) Design of ultra-compact multichannel collimator. 2.5.2 Prototype of multichannel RXI The fabricated prototype is shown in Figure 2.35.. Figure 2.35 Prototype of multichannel RXI.. The back of the prototype is metalized with a coating of silver by evaporation. To prevent the oxidation of the silver, a black painting has been added over the silver coating as shown in Figure 2.36.. Figure 2.36 Prototype of the multichannel RXI with black painting on the silver coating. 39.

(60) Jiayao Liu A demonstrator with LED on printed circuit and supportive structures has been made and shown in Figure 2.37.. LED under multichannel RXI. LED for comparison. selector Figure 2.37 Demonstrator of multichannel RXI and an LED for comparison.. The demonstrator contains two LEDs. One is under the multichannel RXI; the other one is beside for comparison. Both LEDs are soldered to the printed circuit on an aluminum board for obtaining a good dissipation of heat and each one is connected with a resistance of about 3Ω to prevent the current to exceed 500 mA. The two LEDs are controlled separately by a selector. The power source is 4.5V with 3 pillars.. 2.5.3 Prototype characterization The characterization of the prototype of multichannel RXI includes far field pattern and angular distribution of luminous flux. The far field pattern is derived by the system shown in Figure 2.38.. 40.

(61) Design of ultra-compact multichannel collimator. screen. fixture of multichannel RXI, LED and Fresnel lens. Figure 2.38 System for the measurement of far field pattern.. The screen is located in front of a Fresnel lens. The Fresnel lens is located in front of the multichannel RXI so that the multichannel RXI is at the focal of the lens. Thus the multichannel RXI will be imaged at infinity and the distribution on the screen will be the far field pattern. Figure 2.39 shows the detailed fixture of the multichannel RXI, LED and the Fresnel lens.. Fresnel lens LED. multichannel RXI. Figure 2.39 Fixture of multichannel RXI, LED and Fresnel lens. 41.

(62) Jiayao Liu The far field pattern is shown in Figure 2.40.. Figure 2.40 Far field pattern distribution.. It can be seen that most of the lights are concentrated to the center, which means they are collimated. There is a visible ring on the screen which is undesirable. The luminous flux is calculated by.    Id  ,. (2.5.1). where  stands for luminous flux, I stands for luminous intensity and  stands for the solid angle. The system for measuring its angular distribution is shown in Figure 2.41.. 42.

(63) Design of ultra-compact multichannel collimator. fixture of multichannel RXI, LED and goniometer light meter. Figure 2.41 System of the measurement of angular distribution of luminous flux.. The multichannel RXI and the LED are located on a goniometer and they are on its rotational axis (Figure 2.42 (a)), thus can rotate in 180˚. A light meter is fixed 3.674 m away on the same height facing the multichannel RXI and is covered by a black tube (Figure 2.42 (b)) which ensures that the light meter only receives the light of a certain angle from the multichannel RXI.. Figure 2.42 (a) Multichannel RXI and LED on goniometer. (b) Light meter with a dark tube.. 43.

(64) Jiayao Liu A schematic diagram is shown in Figure 2.43. During the measurement, the multichannel RXI will rotate together with the goniometer in  direction from 0˚ ~ 90˚ with a step size of 1˚. Suppose the area of the receiver of light meter is AR, it then will form a solid angle on the hemisphere centers at multichannel RXI as R . AR . r2. (2.5.2). receiver of light meter r = 3.674m.  multichannel RXI. . Figure 2.43 Schematic diagram of the measurement of angular distribution of luminous flux.. Thus the luminous intensity can be calculated by I. R R r 2  . R AR. (2.5.3). where R is the result from light meter, r and AR are constants. Thus the intensity can be quantified by the result of the light meter. Then the luminous flux can be derived from integrating over the desired area on the hemisphere as.    Id    I sin  d d  . R r 2 AR. sin  d d .. (2.5.4). Assuming that all the luminous flux is within ±90˚, then the percentage of flux within different angles have been derived from the measurement as Table 2 Percentage of flux within different angles. Angle(°). ±5. ±10. ±20. ±60. Percentage(%). 46.1. 49.1. 54.2. 91.4. 44.