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Exploration of Ray Mapping Methodology in FreeformOptics Design for Non-Imaging Applications
Item Type text; Electronic Dissertation
Authors Ma, Donglin
Publisher The University of Arizona.
Rights Copyright © is held by the author. Digital access to this materialis made possible by the University Libraries, University of Arizona.Further transmission, reproduction or presentation (such aspublic display or performance) of protected items is prohibitedexcept with permission of the author.
Download date 04/08/2021 17:04:14
Link to Item http://hdl.handle.net/10150/594394
EXPLORATION OF RAY MAPPING METHODOLOGY IN FREEFORM
OPTICS DESIGN FOR NON-IMAGING APPLICATIONS
by
Donglin Ma
____________________________
A Dissertation Submitted to the Faculty of the
DEPARTMENT OF PHYSICS
In Partial Fulfillment of the Requirements
For the Degree of
DOCTOR OF PHILOSOPHY
In the Graduate College
THE UNIVERSITY OF ARIZONA
2015
2
THE UNIVERSITY OF ARIZONA
GRADUATE COLLEGE
As members of the Dissertation Committee, we certify that we have read the
dissertation prepared by Donglin Ma titled Exploration of Ray Mapping
Methodology in Freeform Optics Design for Illumination Engineering and
recommend that it be accepted as fulfilling the dissertation requirement for the
Degree of Doctor of Philosophy.
_______________________________________________________________________ Date: (08/01/2015)
Rongguang Liang
_______________________________________________________________________ Date: (08/01/2015)
Koen Visscher
_______________________________________________________________________ Date: (08/01/2015)
Charles A Stafford
_______________________________________________________________________ Date: (08/01/2015)
Alexander D Cronin
_______________________________________________________________________ Date: (08/01/2015)
Richard J Koshel
Final approval and acceptance of this dissertation is contingent upon the candidate’s
submission of the final copies of the dissertation to the Graduate College.
I hereby certify that I have read this dissertation prepared under my direction and
recommend that it be accepted as fulfilling the dissertation requirement.
________________________________________________ Date: (08/06/2015)
Dissertation Director: Rongguang Liang
________________________________________________ Date: (08/06/2015)
Dissertation Director: Koen Visscher
3
STATEMENT BY AUTHOR
This dissertation has been submitted in partial fulfillment of the
requirements for an advanced degree at the University of Arizona and is deposited
in the University Library to be made available to borrowers under rules of the
Library.
Brief quotations from this dissertation are allowable without special
permission, provided that an accurate acknowledgement of the source is made.
Requests for permission for extended quotation from or reproduction of this
manuscript in whole or in part may be granted by the head of the major
department or the Dean of the Graduate College when in his or her judgment the
proposed use of the material is in the interests of scholarship. In all other
instances, however, permission must be obtained from the author.
.
SIGNED: Donglin Ma
4
ACKNOWLEDGEMENTS
I am deeply thankful to my advisor Professor Rongguang Liang for his valuable
advice for my career development and numbers of guidance during my whole Ph.D.
program.
I have to pay my greatest gratefulness to Dr. Zexin Feng for his continuous
guidance and abundant discussion for my Ph.D. research program, and especially
for his technique support during my Ph.D. research program.
I really appreciate other dissertation committee members, Professor Charles A
Stafford, Professor Richard J Koshel, Professor Koen Visscher, and Professor
Alexander D Cronin, for spending their time on reviewing my dissertation
manuscript and providing their valuable feedbacks.
I especially express my great thanks to Professor Richard J Koshel, who also serves
as a member of my dissertation committee, for his original enlightenment on the
research in illumination engineering and valuable technique support on the
illumination design.
I am gratefully acknowledge Professor Dr. Koen Visscher for his sincere encourage
and supervision for my completion of my Ph.D. program.
I want to thank Professor Russell A. Chipman for his inspiration and discussion on
the topic of “double pole” coordinate system in his classes.
I would like to express my sincere gratitude to Prof. Matthew A. Kupinski, Dr.
Roger Haar, Dr. Daewook Kim, Dr. Shawn Jackson, Dr. Drew Milsom, Professor
Michael Shupe, Dr. Brokk Toggerson, and Professor Koen Visscher, for their
valuable support and working as the supervisors for my teaching assistant position
in University of Arizona.
I am grateful to all of my friends for their friendship in my life. Especially thanks
to Zheng Cai, Yitong Wang, Yan Mao, Xingzhi Xu, and Zhuopei Li for their
accompany in Tucson. Great acknowledgement is due to Chengliang Wang, Nan
Zhu, Shaun Pacheco, and Zhenyue Chen for their discussion in lab.
Finally, I want to pay my earnest gratitude to my mother, Xilian Yang, who is the
best mother in the world, for her selfless love and support to me during her whole
life, regardless of little return.
5
DEDICATION
This dissertation is dedicated to the lasting memory of my dear father Mr. Ma
Kangwu (马康武, 1954-2013), who always loved me, believed in me and
inspired me.
6
TABLE OF CONTENTS
LIST OF FIGURES ········································································ 9
LIST OF TABLES·········································································13
ABSTRACT················································································14
CHAPTER 1 INTRODUCTION························································16
1.1 OVERVIEW OF ILLUMINATION ENGINEERING························16
1.2 OVERVIEW OF FREEFORM OPTICS········································18
1.3 INTRODUCTION TO THE BASIS OF GEOMETRICAL OPTICS·······20
1.3.1 GEOMETRICAL RAY TRACING·······································21
1.3.2 ETENDUE AND TRANSFER LIMIT····································23
1.3.3 SKEW INVARIANT AND TRANSFER LIMIT························24
1.4 OVERVIEW OF TRADITIONAL DESIGN METHODS FOR FREEFORM
OPTICS IN LED ILLUMINATION···········································27
1.4.1 ELLIPTICAL MONGE-AMPERE METHOD··························27
1.4.2 SMS APPROCHES··························································29
1.4.3 METHJODS OF SUPPORTING ELLIPSOIDS·························30
1.4.4 GEOMETRICAL RAY MAPPING APPROACH ······················31
1.5 OUTLINE OF THIS DISSERTATION·········································33
CHAPTER 2 GEOMETRICAL RAY MAPPING····································36
2.1 DESIGN SPECIFICATIONS AND EVALUATION·························36
2.2 INTRODUCTION TO θ-φ RAY MAPPING METHOD·····················39
2.2.1 DESIGN PRINCIPLE·······················································39
2.2.2 SURFACE CONSTRUCTION ············································43
2.2.3 DESIGN EXAMPLE························································45
2.3 u-v RAY MAPPING METHOD IN FREEFORM OPTICS DESIGN·······48
2.3.1 DESIGN PRINCIPLE·······················································49
2.3.2 SURFACE CONSTRUCTION ············································51
2.3.3 DESIGN EXAMPLE OF FREEFORM LENS FOR RECTANGULAR
ILLUMINATION ····························································52
2.4 x-y RAY MAPPING METHOD·················································54
2.4.1 RAY MAPPING PRINCIPLE ·············································55
2.4.2 DESIGN EXAMPLE FOR x-y RAY MAPPING – FREEFORM LENS
FOR RECTANGULAR ILLUMINATION·······························57
2.5 LIMITATION TO RAY MAPPING METHOD·······························60
2.6 CONCLUSIONS AND DISCUSSIONS········································64
CHAPTER 3 DOUBLE POLE RAY MAPPING METHOD························66
3.1 DOUBLE POLE RAY MAPPING METHOD WITH UNIFORM
SAMPLING···········································································66
3.1.1 DOUBLE-POLE RAY MAPPING METHANISM·····················66
3.1.2 DESIGN EXAMPLE························································71
3.2 DOUBLE POLE RAY MAPPING METHOD WITH NON-UNIFORM
SAMPLING ·········································································75
7
3.2.1 RAY MAPPING MECHANISM··········································75
3.2.2 DESIGN EXAMPLE AND SIMULATION·····························79
3.2.2.1 FREEFORM LENS DESIGN FOR LED SOURCE···············79
3.2.2.2 FREEFORM LENS DESIGN FOR LASER SOURCE············82
3.3 DOUBLE POLE COORDINATE SYSTEM FOR FREEFORM
REFLECTOR CONSTRUCTION················································83
3.3.1 DESIGN METHOD·························································85
3.3.2 FREEFORM REFLECTIVE SURFACE CONSTRUCTION·········88
3.3.3 DESIGN EXAMPLE························································89
3.3.3.1 FREEFORM REFLECTORS DESIGN WITH UNIFORM
SAMPLING ·······························································90
3.3.3.2 SMOOTH REFLECTOR DESIGN WITH NON-UNIFORM
SAMPLING ·······························································93
3.4 CONCLUSION AND DISCUSSION···········································95
CHAPTER 4 RAY MAPPING METHODOLOGY’S APPLICATION IN
FREEFORM TOTAL INTERNAL REFLECTIVE (TIR) OPTICS·········96
4.1 TIR LENS IN LED ILLUMINATION ·········································97
4.2 COMPOSITE RAY MAPPING IN DESIGNING FREEFORM TIR
LENS ·················································································98
4.2.1 COMPOSITE RAY MAPPING PRINCIPLE··························100
4.2.2 FREEFORM SURFACE DESIGN·······································103
4.2.3 DESIGN EXAMPLE······················································108
4.2.4 ROBUSTNESS ANALYSIS ·············································111
4.3 MULTI-xy RAY MAPPING IN TIR LENS DESIGN······················115
4.3.1 COMPOSITE RAY MAPPING METHOD WITH MULTIPLE x-y
RAY MAPPING MECHANISM··········································115
4.3.2 FREEFORM SURFACE CONSTRUCTION··························117
4.3.3 DESGIN EXAMPLES AND RESULT ANALYSIS··················120
4.4 DESIGN FREEFORM TIR LENS (ARRAY) USING INTEGRAL RAY
MAPPING METHOD·······················································121
4.4.1 FRESNEL TIR COLLIMATOR DESIGN······························122
4.4.2 FREEFORM LENS ARRAY DESIGN·································124
4.4.3 DESIGN EXAMPLE OF FRESNEL TIR FREEFORM LENS FOR
SQUARE ILLUMINATION ···············································125
4.4.4 FREEFORM TIR LENS ARRAYS FOR OTHER PATTERN······128
4.5 CONCLUSIONS AND DISCUSSIONS······································130
CHAPTER 5 SUBSEQUENT DESIGN METHODS FOR EXTENDED
SOURCES·········································································131
5.1 OPTIMIZATION·································································132
5.1.1 GENERAL DESIGN PROCEDURES FOR OPTIMIZATION
METHOD·····································································132
5.1.2 DESCRIPTION OF FREEFORM OPTICAL SURFACES ··········133
5.1.2.1 XY POLYNOMIAL SURFACE····································134
5.1.2.2 ZERNIKE POLYNOMIAL SURFACE ···························134
5.1.2.3 NURBS SURFACE ··················································135
5.2 FEEDBACK MODIFICATION METHOD··································135
8
5.2.1 FEEDBACK MODIFICATION PRINCIPLE··························136
5.2.2 SIMPLE EXAMPLE WITH FEEDBACK MODIFICATION·······136
5.3 DECONVOLUTION METHOD···············································137
5.3.1 MECHANISM OF DECONVOLUTION METHOD·················138
5.3.2 FREEFORM LENS ARRAY DESIGN AND ANALYSIS···········141
5.4 DISCUSSION ····································································145
CHAPTER 6 SUMMARY AND FUTURE WORK·································147
6.1 SUMMARY·······································································147
6.2 FUTURE WORK·································································148
REFERENCES ···········································································150
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LIST OF FIGURES
Figure 1.1 Diagram for illumination design of the transfer function T that takes the
light from the source (on the left) and efficiently transfers it to the target (right).
[4] ····························································································17
Figure 1.2 Typical applications of freeform optics: (a) freeform prism eyepiece for head-
mounted display; (b) solar concentrator; (c) rear-view mirror with no blind-spot; (d) laser
beam shaper. [41-44]·················································································20
Figure 1.3 Fermat’s principle [1] ···································································21
Figure 1.4 Diagram for the law of reflection [1] ·······································22
Figure 1.5 Vector formulation of refraction [1] ·················································23
Figure 1.6 Geometry of ray emission from differential-area patch on surface of
axisymmetric source [1] ··································································25
Figure 1.7 Example of skewness distribution mismatch between source and target [1, 47]
···········································································································26
Figure 1.8 Geometrical layout of collimated beam shaper design [59]·············· 28
Figure 1.9 Diagram of SMS method generating two surfaces that transform two input
congruences into two output ones [4] ·····························································30
Figure 1.10 Ellipsoids reflect all rays emitted from one focus to the other focus
···········································································································31
Figure 1.11 Single-surface setup with a collimated source [67] ·····················32
Figure 2.1 Design parameters·····························································37
Figure 2.2 Flow diagram of the design process [68] ············································40
Figure 2.3 Topological mapping form source (bottom) to a rectangular target (above)
based on the (θ, φ) ray mapping method···························································41
Figure 2.4 Geometrical layout of freeform lens design by θ-φ ray mapping method [58]
···········································································································43
Figure 2.5 Diagram for freeform surface construction in (θ, φ) coordinate system········44
Figure 2.6 (a) Lens model; (b) related contour plot of lens surface···························46
Figure 2.7 Simulation results for θ-φ ray mapping: (a) irradiance distribution; (b) x profile;
(c) y profile. ···························································································48
Figure 2.8 Topological ray mapping mechanism from source to target based on (u, v)
spherical coordinate system. ········································································50
Figure 2.9 Surface construction diagram in (u, v) ray mapping [73] ·························52
Figure 2.10 Freeform lens designed in u-v coordinate system: (a) lens model; (b) lens
surface contour. ······················································································53
Figure 2.11. Simulation results for u-v ray mapping: (a) irradiance distribution; (b) x
profile; (c) y profile. ··················································································54
Figure 2.12 Ray mapping mechanism for x-y ray mapping method··························57
Figure 2.13 Geometrical layout of freeform lens design by x-y ray mapping method·····58
Figure 2.14 Freeform lens designed in Cartesian coordinate system: (a) lens model; (b)
lens surface contour. ·················································································59
10
Figure 2.15 Simulation results for u-v ray mapping: (a) irradiance distribution; (b) x profile;
(c) y profile. ···························································································60
Figure 2.16 (ith, jth) grid on the freeform surface·················································62
Figure 2.17 RSE map for first quadrant of freeform lens designed by (a) θ-φ ray mapping,
(b) u-v ray mapping, and (c) x-y ray mapping····················································64
Figure 3.1 Diagram for double pole coordinate system: (a) perspective view; (b) top view;
(c) bottom view. ······················································································68
Figure 3.2 Mapping mechanism from (α, β) to (x, y). ···········································70
Figure 3.3 Irradiance distribution for freeform lenses designed in (a) (θ, φ) spherical
coordinate system, (b) (u, v) spherical coordinate system, and (c) double pole coordinate
system. ·································································································73
Figure 3.4 RSE map for first quadrant of freeform lens designed in (a) (θ, φ) spherical
coordinate system, (b) (u, v) spherical coordinate system, and (c) double pole coordinate
system. ·································································································74
Figure 3.5 Traditional methods to sample source intensity distribution in double coordinate
system: (a) emission region subtended by sampled grids (b) sampled grids subtended by
the source emission. ·················································································76
Figure 3.6 Non-uniform sampled grids in first quadrant of modified double pole coordinate
system and related mapping mechanism between source and target·························77
Figure 3.7 (a) Freeform lens contour and (b) NURBS surface entity for the LED freeform
lens······································································································80
Figure 3.8 Simulation results in FRED software: (a) irradiance distribution, (b) X profile
plot, and (c) Y profile plot. ··········································································81
Figure 3.9 (a) Freeform lens contour and (b) NURBS surface entity for the fiber laser
source freeform lens··················································································82
Figure 3.10 Simulation results of freeform lens for fiber laser illumination: (a) irradiance
distribution and (b) X and Y profile plot. ··························································83
Figure 3.11 (a) θφ coordinate system; (b) uv coordinate system; (c) double pole coordinate
system··································································································84
Figure 3.12 (a) Determinant of the Jacobian in double pole coordinates. The red lines is
the boundary governed by Equation (4.18). The green dot is the intersection of a line at 45˚
and the boundary. (b) The source mapping in α-β space. (c) Source map in Cartesian
coordinates. (d) Target map for double pole coordinate system. ······························86
Figure 3.13 Ray trace diagram for the (ith, jth) grid on the freeform reflector················89
Figure 3.14 Diagram of reflector for two different source-target mappings: (a) uncrossed
mapping; (b) crossed mapping ·····································································91
Figure 3.15 Freeform lens model designed under uniform sampling in double pole space
and corresponding irradiance distribution: (a) (e) uncrossed design with FOV of 120°; (b)
(f) uncrossed design with FOV of 140°; (c) (g) crossed design with FOV of 120°; (d) (h)
crossed design with FOV of 140°.·································································92
Figure 3.16 Freeform lens model designed under non-uniform sampling in double pole
space and corresponding irradiance distribution: (a) (e) uncrossed design with FOV of
120°; (b) (f) uncrossed design with FOV of 140°; (c) (g) crossed design with FOV of 120°;
(d) (h) crossed design with FOV of 140°.·························································94
Figure 4.1 (a) TIR lens with outside flat top surface, (b) TIR lens with inside flat top
surface; (c) TIR lens with faceted (Fresnel) TIR surface [88] ·································98
11
Figure 4.2 The topological mapping from source to target based on the composite mapping
method································································································101
Figure 4.3 Ray trace of TIR freeform lens in 2D diagram····································105
Figure 4.4 Diagram for freeform surface construction: (a) central refractive surface; (b)
peripheral TIR surface··············································································107
Figure 4.5 TIR freeform lens model······························································108
Figure 4.6 Simulation result (illumination distribution and line chart) for TIR freeform
lens: (a) trace rays through the central freeform surface; (b) trace rays through the TIR
surface; (c) trace all rays through the total TIR freeform lens. ··················110
Figure 4.7 Effect of LED’s shift on RSD and collection efficiency for: (a) regular refractive
freeform lens; (b) TIR freeform lens. ····························································112
Figure 4.8 (a) Actual LED’s intensity distributions based various power index m; (b) the
effect of variations in power index m on the illumination performance for both TIR
freeform lens (TIR) and general freeform lens (general) with regards to RSD and collection
efficiency. ···························································································114
Figure 4.9 Ray mapping mechanism for multi-xy mapping method························115
Figure 4.10 Ray trace of freeform TIR lens design in multi-xy ray mapping method
·········································································································117
Figure 4.11 Ray trace diagram and freeform surface construction: (a) central freeform
refractive surface; (b) peripheral freeform TIR surface. ······································118
Figure 4.12 (a) TIR lens model based on composite ray mapping; and (b) simulation result
·········································································································121
Figure 4.13 Structure and geometrical layout of Fresnel TIR collimator···················123
Figure 4.14 Schematic diagram for the working process of single micro lenslet ·········124
Figure 4.15 Schematic of ray mapping mechanism for single lenslet [95] ················125
Figure 4.16 Diagrams for Fresnel TIR collimator: (a) bottom view of Fresnel TIR surface
of the collimator; (b) wireframe side view; (c) ray tracing diagram. ························126
Figure 4.17 Fresnel TIR freeform lens array model: (a) rendered view in Rhino; (b) ghost
view in Rhino.·······················································································127 Figure 4.18 Simulation results for the Fresnel TIR freeform lens array for square illumination pattern: (a) irradiance distribution on the target surface; (b) irradiance distribution cross- section profile on both direction. ···········································128 Figure 4.19 Fresnel TIR lens with hexagon lens array: (a) lens model; (b) related irradiance
distribution. ··························································································128
Figure 4.20 Fresnel TIR lens with cross-hair lens array: (a) diagram for single lenslet; (b)
diagram for TIR lens combined with cross-hair lens array; (c) illumination pattern for the
TIR lens array system.··············································································129
Figure 5.1 Flow chart of optimization method [98] ···········································133
Figure 5.2 Feedback modification for the designed laser beam shaper. ····················137
Figure 5.3 The geometrical layout of freeform lens array for structured light illumination.
·········································································································139
Figure 5.4 Ray tracing of the freeform lens array. ·············································142
Figure 5.5 (a) Target sinusoidal pattern, (b) the blur response of the extended light source,
and (c) the target response of the point source. ·················································142
Figure 5.6 Ray tracing of the freeform lens array to generate sinusoidal fringe pattern.
·········································································································144
12
Figure 5.7 Simulation results for a 1x1 mm LED chip. (a) The irradiance pattern of the
freeform lens array which is designed using deconvolution approach, (b) the irradiance
pattern of the design without deconvolution, and (c) normalized line profiles for the
irradiance patterns of both designs.·······························································145
13
LIST OF TABLES
Table 2.1 Design parameters (Unit: mm) ··············································46
Table 2.2 Comparison for three freeform lenses ······································63
Table 3.1 Design parameters·····························································72
Table 3.2 Comparison for three freeform lenses ······································74
Table 3.3 Design parameters of the freeform lens for LED source (Unit: mm)···80
Table 3.4 Design parameters of freeform lens for Laser source (Unit: mm) ······82
Table 4.1 Design parameters for the freeform TIR lens construction (Unit: mm) ··
······························································································120
Table 5.1. Design parameters (unit: mm)·············································142
14
ABSTRACT
This dissertation investigates various design metrologies on designing
freeform surfaces for LED illumination applications.
The major goal of this dissertation is to study designing freeform optical
surfaces to redistribute the radiance (which can be simplified as intensity
distribution for point source) of LED sources for various applications. Nowadays
many applications, such as road lighting systems, automotive headlights, projection
displays and medical illuminators, require an accurate control of the intensity
distribution. Freeform optical lens is commonly used in illumination system
because there are more freedoms in controlling the ray direction.
Design methods for systems with rotational and translational symmetry
were well discussed in the 1930’s. However, designing freeform optical lenses or
reflectors required to illuminate targets without such symmetries have been proved
to be much more challenging. For the simplest case when the source is an ideal
point source, the determination of the freeform surface in a rigorous manner usually
leads to the tedious Monge-Ampère second order nonlinear partial different
equation, which cannot be solved with standard numerical integration techniques.
Instead of solving the differential equation, ray mapping is an easier and more
efficient method in controlling one or more freeform surfaces for prescribed
15
irradiance patterns. In this dissertation, we investigate the ray mapping metrologies
in different coordinate systems to meet the integrability condition for generating
smooth and continuous freeform surfaces.
To improve the illumination efficiency and uniformity, we propose a
composite ray mapping method for designing the total internal reflective (TIR)
freeform lens for non-rotational illumination. Another method called “double pole”
ray mapping method is also proposed to improve system performance. The ray
mapping designs developed for the point source do not work well for extended
sources, we have investigated different design methodologies including
optimization method, deconvolution method and feedback modification method to
design freeform optical surfaces for extended sources.
16
CHAPTER 1
INTRODUCTION
The purpose of this dissertation is to explore new ray mapping mechanisms
in freeform optics design for light emitting diode (LED) illumination. The major
objective is to design freeform lenses or reflectors to generate predefined
illumination patterns or structured illumination patterns for both point sources and
extended sources.
This chapter consists of 5 sections. Section 1 briefly overviews the
background of illumination engineering; Section 2 introduces the concept of
freeform optics and discusses its applications; Section 3 discusses the background
of geometrical optics related to the ray tracing, étendue and skewness; Section 4
summaries traditional freeform optics design methods in LED illumination
engineering; and the last section layouts the content of this dissertation.
1.1 OVERVIEW OF ILLUMINATION ENGINEERING
Illumination engineering has been lasting for thousands of years since the
first man-made fire by drilling wood. Recently, with the development of light
emitting diodes (LEDs), the illumination industry is booming due to the increasing
requirement on illumination quality and energy saving [1-3]. In brief, illumination
17
engineering is to transfer the light or radiation from the source(s) to target(s) [4],
the design of illumination systems to realize the transfer process, as shown in Figure
1, is one of the key subjects in illumination engineering. Based on optical principles,
general illumination systems can be divided into two different categories. The first
one is imaging based illumination system, such as Köhler illumination systems and
Abbe illumination systems [5-6]. The other type of illumination system is non-
imaging based system, typically including solar concentrators [7-11], automotive
headlight [12-15], LCD backlit illumination [16-18], road lighting [19-22] and so
on.
Figure 1.1 Diagram of a typical illumination system: the transfer function T that takes the
light from the source (on the left) and efficiently transfers it to the target (right). [4]
The development of illumination engineering has always been accompanied
by the invention of new lighting sources. With the advent of incandescent lamps
developed by Joseph Swan in 1878 and then improved by Edison in 1789, the
energy efficiency became 18 lm/W and greatly expanded the development of
illumination engineering [23-24]. The invention of fluorescent lamps brought
another revolution to the field of illumination engineering in 1930’s [25].
Nowadays, LEDs have been put into applications in almost every field because of
T(r; r0)
18
their great advantages compared to traditional light sources. These advantages
include compactness, high energy efficiency, environment friendliness, and long
life time [26]. With the broad applications of LEDs, the illumination engineering
has become one of the primary research foci in modern optical engineering. The
major task of this dissertation is focused on the illumination system design for LED
illumination and fiber coupled laser.
1.2 OVERVIEW OF FREEFORM OPTICS
With the development of modern precision fabrication technologies such as
diamond turning, injection molding and electron beam fabrication, it becomes
possible for people to manufacture freeform optical elements with various shapes
rather than the traditional rotationally symmetric optics [27-29]. The so-called
freeform optics can be defined as surfaces without any axis of rotational invariance
and freeform surface can have arbitrary shapes, and regular or irregular structures
[28]. Freeform optics offers more freedom in developing novel optics with better
performance and more compact structures. For example, a single freeform prism
can be used to project the images of a head mounted display (HMD) onto the human
eyes with high resolution [30-31], and the freeform reflector can make the near-
wall projector to be feasible [32-34].
Due to the incomparable advantages of freeform optics over traditional
optics, freeform optics has very broad applications in the high performance imaging
systems [35, 36], illumination systems [37, 38], solar concentrators [39, 40], and
19
some other fields. Some typical applications for the freeform optics are as shown
in Figure 1.2. In most of the non-traditional optical elements such as HMD system
for current daily applications, freeform optics plays the key role in the performance
of these optical systems. With regards to the non-imaging optics design, up to now
freeform lens has been put into applications in many different fields such as
designing road illuminators [19] and automotive headlights [12]. For the field of
solar concentration, different kinds of freeform reflectors or lenses have been
proposed using simultaneous multiple surface (SMS) method to increase the
acceptance angle and approximate to the concentration limit [40].
Undoubtedly, freeform optics design is a hot topic for current scientific
research regarding to the field of optics engineering. It opens a new door to the
future research era of traditional optics engineering. In this thesis, we will focus on
developing new design methods of freeform optics generation for its application in
LED’s non-rotational illumination.
20
Figure 1.2 Typical applications of freeform optics: (a) freeform prism eyepiece for head-
mounted display; (b) solar concentrator; (c) rear-view mirror with no blind-spot; (d) laser
beam shaper. [41-44]
1.3 INTRODUCTION TO THE BASIS OF GEOMETRICAL
OPTICS
Generally speaking, geometrical optics is the fundamental tool for
designing both imaging systems and non-imaging systems. In geometrical optics,
we take the light as a bundle of rays whose behavior obeys some basic principles
and does not consider the diffraction effect. By tracing the rays geometrically,
optical surfaces with relatively large scale could be generated by building the
mapping relationship between the source and target. In this section, we will provide
a simple description of some basic concepts and principles in geometrical optics.
21
1.3.1 GEOMETRICAL RAY TRACING
The direction of a ray of light obeys Fermat’s principle, which is also called
of the principle of least time [1, 45]. As shown in Figure 1.3, according to Fermat’s
principle, the optical path between point A and B will be the path that can be
traversed in the least time. If we define the optical path length of a ray from point
A to point B as follows:
B
A
S nds (1.1)
where n is the continuously varying refractive index of the material between point
A and point B. According to Fermat’s principle, the value related to the specific
optical ray path expressed by Equation (1.1) is an extremum.
Figure 1.3 Fermat’s principle [1]
Based on Fermat’s principle, it is convenient to derive the formula for the
laws of reflection and refraction separately. As shown in Figure 1.4, the unit vector
r represents the incident ray and the unit vector r’ is reflected ray and n is the unit
normal vector pointing into the reflecting surface. Using Equation (1.1), the law of
reflection can be expressed by the vector equation as follows
A B
22
2( ) r r n r n' (1.2)
Figure 1.4 Diagram for the law of reflection [1]
The derivation for the refraction law is similar to the law of reflection. By
using the Fermat’s principle, the law of refraction (also called Snell’s law) can be
stated in the form
'sin ' sinn I n I (1.3)
where I and I’ are the angles of incidence and refraction as shown in Figure 1.5, n
and n’ are the refractive indices of materials in incident space and exit space
separately. Through rearrangement of the equation (1.3), the vector formulation of
the Snell’s law can be expressed as
' ' ( ' ' )n n n n r r r n r n n (1.4)
where r is the unit incident vector, r’ is the unit refraction vector, and n is the unit
normal vector.
r
n
r'
23
Figure 1.5 Vector formulation of refraction [1]
1.3.2 ETENDUE AND TRANSFER LIMIT
Etendue is one of the most basic, yet important concepts in the design of
both non-imaging optical systems and imaging systems [1, 4]. Etendue
characterizes the geometrical properties of flux propagation in an optical system.
The étendue of a system is defined as
2 cos s
pupil
n dA d ， (1.5)
where n is the index of refraction, As is the area of source and Ω is the projected
solid angle of flux beam onto the source surface.
For any lossless optical systems, the étendue is always conserved. The
conservation of étendue can be derived based on the geometrical properties of
general optical systems through geometrical ray tracing [2, 46]. Besides, to be more
fundamental, the conservation of étendue is derived by combining first law of
I
I’
(n)
(n’)Surface
Refracted ray
n
r
r'
Normal
Incident ray
24
thermodynamics with second law of thermodynamics [2]. For the non-lossless
optical systems, we can use the étendue to analyze the transfer efficiency or
concentration efficiency. If we define εtr as the maximum transferred étendue to the
target and εsrc as the total source étendue, then the maximum transfer or
concentration efficiency of the optical system can be derived as:
max .tr
src
(1.6)
1.3.3 SKEW INVARIANT AND TRANSFER LIMIT
As we have shown above, the conservation of étendue will limit the
maximum transfer efficiency to optical systems. However, skew invariant can
provide a more stringent limitation to the flux transfer efficiency between the
source and target. The skew invariant, or skewness, of the light ray is defined as
follows [1]:
ˆ( ),s r k a (1.7)
where a is a unit vector along the optical axis, k is a vector along the ray direction
with magnitude of the refractive index, r is an arbitrary vector that connects the
optical axis and the light ray. Physically, the skewness represents the distance of
light ray and the optical axis of the optical system. Obviously, the skewness is an
invariant for rotationally symmetric optical systems. As a result, it is impossible to
couple all of the light from a source with specific structure to a target with a
different topological structure because of the inability to alter the skewness of light
rays.
25
Figure 1.6 Geometry of ray emission from differential-area patch on surface of
axisymmetric source [1]
In fact, we can use the mismatch of skewness distribution between source
and target to characterize this kind of loss, which is also called dilution. To specify
the skewness distribution of source, we consider a differential source patch of
source surface area dA as depicted in Figure 1.6, which is located a distance r form
the symmetry axis. By combining the definitions of étendue and skewness, we can
derive the skewness distribution of optical systems as a function of skewness:
max
min
2
2 2
( )1 sin d dA,
S
d s n s
ds r n r
(1.8)
where S represents the region of the source surface area over which the integrant is
defined, and ϕmin and ϕmax are the minimum and maximum values of the azimuthal
angle ϕ.
x
z
y
α
ϕθ
ϕ
dA
26
Figure 1.7 Example of skewness distribution mismatch between source and target [1, 47]
Figure 1.7 shows an example of skewness distribution mismatch between
source and target, which is specified in Ref. [1, 47]. If we take ε1 as the étendue of
source and ε2 as the étendue of target, then the maximum étendue that can be
transferred from the source to the target can be derived as:
1 2max
( ) ( )min , .
d s d sds
ds ds
(1.9)
As a result, the upper limit of the transfer efficiency is changed to be:
maxmax .
src
(1.10)
where ɛsrc is the total étendue of light source. Obviously, the shaded region in
Figure 1.7 evaluates the dilution loss of the optical system due to the skewness
distribution mismatch between the source and target.
Skewness s
dε/
ds
Dilution
loss
Source
Target
-2
0
2
4
6
8
10
12
14
-1.0 -0.5 0.0 0.5 1.0
27
1.4 OVERVIEW OF TRADITIONAL DESIGN METHODS FOR
FREEFORM OPTICS IN LED ILLUMINATION
With the broad application of freeform optics in the field of LED
illumination engineering, various kinds of methods have been developed to design
freeform lens or reflectors based on different cases. These methodologies include
supporting ellipsoids [48-52], trial and error approaches such as optimization [53,
54], simultaneous multiple surfaces (SMS) method [55, 56], solving corresponding
Monge-Ampere equation [57-59], and geometrical ray mapping method [60-62]. In
this section, we will provide brief comparison and analysis of selected design
methods.
1.4.1 ELLIPTICAL MONGE-AMPERE METHOD
For an ideal point source, designing a freeform optical surface to
redistribute the source radiance/intensity distribution onto a specific target will
always fall to an optimal transport problem, which leads to solving the nonlinear
partial differential equations (PDEs) - Monge-Ampère equation [63]. Wu has
designed single freeform optical surface for the collimated beam shaping illustrated
in Figure 1.8 by solving the Monge-Ampère equation [59]. By mapping the
collimated source to the target surface using energy conservation principle, the
derivation of the freeform surface could be reduced to solving the following general
form of Monge-Ampère equation:
2
1 2 3 4 5( ) 0,xx yy xy xx yy xyA z z z A z A z A z A (1.11)
28
where A1, A2, A3, A4 and A5 are functions of zx and zy. Here, zx is the first-order
partial differential of variable of x. Similarly zxx is the second-order partial
differential for variable x, and same as for zy, zyy and zxy.
Figure 1.8 Geometrical layout of collimated beam shaper design [59]
To build a simple one to one ray mapping between the source and target,
we need to guarantee that the derived surface is continuous. By solving the tedious
partial differential equation, a strict solution for a freeform optical surface can be
derived to optimally transfer flux from an ideal collimated source (or a point source
in some other cases) to a specific target. Unlike most of other design methods, the
integrability condition of the derived freeform surface is difficult to meet, the strict
solution of Monge-Ampère equation could lead to the optimal mapping between
source and target. Recently, Fournier etc. have successfully applied the Monge-
Ampère method to design freeform reflectors for extended sources. However, the
coefficients in Equation (1.11) are extremely specific and the extreme effort is
29
needed to derive the expressions for these coefficients in each case. Of course, it is
not easy to solve these strongly nonlinear PDEs, which require very careful
schemes of numerical samplings in both source space and target space.
1.4.2 SMS APPROCHES
The Simultaneous Multiple Surfaces (SMS) method can help construct
optical systems which couple a pair of incoming wavefront and corresponding
outgoing wavefront as shown in Figure 1.9. The SMS method is a procedure for
designing two optical surfaces such that two given normal congruencies Wi1 and
Wi2 are transformed (by a combination of refractions and/or reflections at these
surfaces) into another two given normal congruencies Wo1 and Wo2 [4, 64]. It can
also be applied to design both rotational symmetric optics and non-rotational
freeform optics. SMS method is one of the few methods in non-imaging optics
which have the ability to control multiple surfaces at the same time. Since SMS
method is aimed to control the wavefronts, it has the ability to design optical
systems to generate prescribed illumination patterns directly for extended sources
rather than by time consuming interactive trial and error approaches such as
optimization and feedback modification methods.
Of course, there are still some limitations to SMS method. Only pairs of
surfaces can be computed in this approach. As a result, the SMS method cannot
help to generate optical systems with only one single optical surface, such as
freeform reflectors [65].
30
Figure 1.9 Diagram of SMS method generating two surfaces that transform two input
congruencies into two output ones [4]
1.4.3 METHODS OF SUPPORTING ELLIPSOIDS
The two foci of an ellipse in the plane have the following property: the light
emitted from on focus and then reflected by the interior of the ellipse will be
focused at the other focus as shown in Figure 1.10. The property is also correct in
3D for an ellipsoid. By applying this special property of ellipsoids, the methods of
supporting ellipsoids for designing freeform optics proposed by Kochengin and
Oliker [48, 49] can be described as follows: each point on the illumination target
surface that needs to be illuminated defines an ellipsoid whose one focus is located
at the point of source and the other focus is at the target point. By combining these
31
ellipsoids together through interpolation algorithms, researchers can finally build a
freeform optical surface which can realize the prescribed illumination pattern.
Figure 1.10 Ellipsoid reflects all rays emitted from one focus to the other focus
The problem to this supporting ellipsoids method is that there are infinitely
numbers of ellipsoids with two common foci but varying in diameters. As the
starting point, the diameter of the ellipsoid has to be defined by an initial guess.
However, the initial guess cannot guarantee the satisfied result, therefore it needs a
lot of iterations to modify the results until the design convergences. The process is
just as tedious as trial and error approaches since each step needs a numerical
integration over the emission solid angle of the source along with a huge number
of optimization steps to guarantee the smoothness of the surface.
1.4.4 GEOMETRICAL RAY MAPPING APPROACH
Geometrical ray mapping approach is another efficient freeform optics
design method for LED illumination [66]. Figure 1.11 shows an example of
mapping a collimated source to the target using a single freeform surface [67]. Each
32
source ray is uniquely associated with a point in a 2D plane Ω0 perpendicular to the
optical axis via a suitable projection, creating a flux density μ0 in this plane. The
target irradiance is equally projected in a consistent way onto a target plane Ω1 with
density μ1. The task of geometrical ray mapping method is finding a
diffeomorphism (“ray mapping”) so that the transformed irradiance distribution
matches the target distribution:
0 1 0 1: , , , .x yu x y t t with Du u (1.12)
Figure 1.11 Single-surface setup with a collimated source [67]
The ray mapping approach can help handle multiple optical surfaces for
prescribed illumination pattern simultaneously. The method is very fast and
efficient in designing optical surfaces for zero-étendue source (point-like source or
collimated source). However, the major problem of the ray mapping method is that
33
it probably fails to satisfy the integrability condition and thus it is hard to realize
the prescribed illumination target [62, 67]. This dissertation is aimed to explore new
ray mapping methodologies to design optical surfaces satisfying the integrability
condition to a better degree. More details will be introduced in the following
chapters.
1.5 OUTLINE OF THIS DISSERTATION
This dissertation consists of six chapters. Chapter 1 is the introduction part.
In this chapter, we provide an overview of illumination engineering and related
freeform optics in general. In addition, the fundamentals of geometrical optics and
freeform optics design methods are introduced in this chapter. Chapter 2 presents
two traditional ray mapping design methods of freeform optics in LED illumination
and investigates the limitations in actual design practices. Chapter 3 proposes a new
ray mapping method based on the so called “double pole” coordinate systems and
its applications in freeform lens and reflectors design for LED illumination.
Chapter 4 introduces several composite ray mapping mechanisms for general
freeform total internal reflection (TIR) lens design. Chapter 5 studies subsequent
design methodologies for extended sources such as optimization by commercial
software, feedback modification methods and deconvolution mechanism for shift
invariant systems. In chapter 6, the dissertation concludes with a summary of the
current research and future perspectives.
34
This dissertation investigates various design metrologies on designing
freeform surfaces for LED illumination applications.
The major goal of this dissertation is to study the ways of designing
freeform optical surfaces to redistribute the radiance distribution (which can be
simplified as intensity distribution for point source) of LED sources for various
actual applications. Nowadays many application fields such as road lighting
systems, automotive headlights, projection displays and medical illuminators
require an accurate control of the intensity distribution for different lighting systems.
Among various solutions, freeform optical lens can be a good way to provide
enough freedoms for the non-imaging optics design.
Design methods for systems with rotational and translational symmetry had
been well discussed in the 1930’s. However, the freeform optical lenses or
reflectors required to illuminate targets with no such symmetries have been proved
to be much more challenging to design. For the simplest case when the source is
assumed to be an ideal point source, the determination of the freeform surface in a
rigorous manner usually leads to the tedious Monge-Ampère second order
nonlinear partial different equation, which cannot be solved with standard
numerical integration techniques. Instead of solving the differential equation, ray
mapping is an easier and more efficient method in controlling one or more freeform
surfaces for prescribed irradiance patterns. In this dissertation, one of the major
tasks is to investigate the ray mapping metrologies in different coordinate system
to guarantee the integrability condition for the generation of smooth and continuous
freeform surfaces. To improve the design efficiency, we propose a composite ray
35
mapping method to design the total internal reflective (TIR) freeform lens for non-
rotational illumination. Another method called “double pole” ray mapping method
is also proposed to help escape solving the tedious Monge-Ampère equation.
While the pure ray mapping design does not work that well for extended
sources, another major goal of this dissertation is to explore different design
methodologies including optimization method, deconvolution method and
feedback modification method in designing freeform optical surfaces for extended
sources.
36
CHAPTER 2
GEOMETRICAL RAY MAPPIMG
This chapter focuses on several traditional ray mapping approaches for
freeform optics design in illumination engineering. Section 2.1 introduces the
design principle for traditional θ-φ ray mapping method and some examples.
Section 2.2 explores the design principle of another design method called “u-v” ray
mapping method and then introduces some specific examples. While in Section 2.3,
another ray mapping mechanism between source and target is discussed, where we
sample the source intensity distribution in the Cartesian coordinate system. In
Section 2.4, the limitations to these traditional ray mapping method will be
discussed and related factors leading to the limitations will also be explored in this
section. In the end, Section 2.5 summarizes these three traditional ray mapping
methods and discusses the potential improvement.
2.1 DESIGN SPECIFICATIONS AND EVALUATION
The main goal of this dissertation research is to develop design method of
freeform optical systems for the generation of pre-defined illumination based on
compact light sources. The specifications for the design are illustrated in Figure 2.1.
The source is a 1 mm x 1 mm LED chip, it has a Lambertian emission property
37
with a half divergence angle of 90°. The lens material is PMMA with refractive
index of 1.49. The task of the non-imaging optics design is to generate optical
surfaces that redistribute the radiance distribution or intensity distribution of light
on the target surface.
Figure 2.1 Design parameters
Lambertian source is defined as the source whose radiance is constant:
2
,cos
eLA
(2.1)
where Φe is radiant flux emitted by the light source, Ω is the solid angle, and Acosθ
is the projected area. The intensity of light source is defined as the radiant flux per
unit solid angle:
.eI
(2.2)
H
h1
h2
PMMA
2X
2Y
1 m
m
LE
D
38
For a Lambertian source with constant radiance distribution, we can derive its
intensity distribution by combining Eqs. (2.1) and (2.2):
0 cos ,I I (2.3)
where I0 is the intensity of light source at the normal direction. In order to evaluate
the performance of the optics design, we have to analyze the irradiance distribution
on the target surface, which is defined as radiant flux received by a surface per unit
area and thus expressed by:
.eeE
A
(2.4)
In most cases of illumination engineering, the goal of the optics design is to realize
a homogeneous irradiance distribution on the target surface. In this dissertation, the
uniformity of the specific irradiance distribution is defined as:
min
ave
,E
UniformityE
(2.5)
where Emin is the minimum irradiance on the target surface and Eave is the average
irradiance value of the whole target surface. A more comprehensive term to
evaluate the system performance is relative standard deviation (RSD), which is
defined as:
2
std 0
1ave 0
( ) ( )1,
( )
pN
S
ip
E E i E iRSD
E N E i
(2.6)
where Estd is the standard deviation of the illuminance distribution and Eave is the
average value of illuminance inside the prescribed illumination target. Np is the total
number of sampled points on the illumination target surface, ES(i) is the simulated
illuminance value at the i-th sampled point and E0(i) is the desired illumination
39
value at the i-th sampled point [68]. Generally, for a given irradiance distribution
on the target surface, higher uniformity will lead to smaller RSD value. In this
dissertation, RSD is used to evaluate the performance of freeform optics in
achieving the desired uniformity.
2.2 INTRODUCTION TO θ-φ RAY MAPPING METHOD
2.2.1 DESIGN PRINCIPLE
In order to simplify the design procedures, we firstly specify the ray
mapping mechanism between the source and target [66-68]. The design process for
the ray mapping method is illustrated in Figure 2.2. Ray mapping builds the
relationship between ray emitting direction out of the source and ray position on
the target surface using energy conservation principle. Then we can find the surface
slope using Snell’s law. With the slope at each point of the freeform surface, we
can construct the surface by integrating them together. Due to the spherically
symmetric intensity distribution of the point-like source, the traditional (θ, φ)
spherical coordinate system is often used to specify the ray emitting direction and
then establish the mapping relationships [69].
40
Figure 2.2 Flow diagram of the design process [68]
41
Figure 2.3 Topological mapping form source (bottom) to a rectangular target (above)
based on the (θ, φ) ray mapping method
The mapping mechanism for (θ, φ) ray mapping method for a rectangular
illumination is shown in Figure 2.3, where only one quadrant of the mapping is
plotted. If we let I(θ, φ) denote the intensity of the source at direction of (θ, φ) and
E(xt, yt) represent the irradiance at (xt, yt) on the target surface, thus by using the
energy conservation principle for lossless systems we can derive the following
expression:
, , , , ,t t t t t t
D
I J d d E x y J x y dx dy
(2.7)
Intensity
distribution
of source
θ1
θi
θi+1
θM
φ1 φjφj+1
φN
Target plane
x
x
y
y
I(θ, φ)
E(x, y)
42
where Ω is the total solid angle of source’s emission, D is the total illuminated area
on the target surface, and J is the corresponding Jacobi matrix. The absolute value
of the Jacobian determinant at given point gives us the factor or weight that decides
the topological volume of the uniformly sampled grid in the new coordinate system.
Specifically, Ω and D can be mathematically expressed as follows:
max 2 , 0
:
: 0,
,t tD X x X Y y Y
(2.8)
where θmax is the maximum polar angle that the source emission can cover, 2X and
2Y are the size of the rectangular target. By integrating Eq. (2.7) under the condition
expressed in Eq. (2.8) and separating variables on both sides, we can derive the
mapping relations in the first quadrant between the ray direction (θ, φ) and its target
position (xt, yt) as follows:
( , , , )
,( , , , )
t t
t t
f x y X Y
g x y X Y
(2.9)
where f, g cannot be expressed explicitly since the intensity distribution of the
source is not specified. For a Lambertian source and the homogeneous illumination
on the target surface, Eq. (2.9) can be derived as:
max
max
sin; 0
4 sin 4,
sin;
4 4 sin 4 2
tt
tt
yx X
Y
X xy Y
X
(2.10)
where θmax is the maximum half divergence angle of source emission. The
expression in Eq. (2.10) obviously reflects the discontinuous mapping along the
diagonal region.
43
Figure 2.4 Geometrical layout of freeform lens design by θ-φ ray mapping method [58]
2.2.2 SURFACE CONSTRUCTION
In the previous section, we have discussed the method to calculate the
normal vector field of the proposed freeform surface. Based on the calculated
normal vector field, we will how to reconstruct the freeform surface in this section.
Assume that the entrance surface is a spherical surface and the small source
is located at the center of the inner spherical surface. The exit surface is freeform
surface and the freeform lens redistributes the intensity distribution of the small
source onto a rectangular illumination target as shown in Figure 2.4. Assume the
incident ray intersects the freeform surface at P(x, y, z) and the ray is refracted to
the target plane at T(xt, yt, H), the normal vector N on the freeform surface can be
calculated as follows:
ϕ
x
y
z
S
A
BH
C
P(x,y, z)
In
Out
N
Target
Plane
Freeform
surface
Spherical surface
θ
T(xt, yt, z)
h1
h2
44
2 1/2[1 2 ( )]n n n , Out In N Out In (2.11)
where In, Out are the unit incident vector and unit output vector separately, and
they can be expressed by
/ ,In SP SP (2.12)
/ .Out PT PT (2.13)
Now with the information of the normal vector field, we can build the whole
freeform surface. The exact calculation procedures will be introduced in the next
section.
Figure 2.5 Diagram for freeform surface construction in (θ, φ) coordinate system
Pi, j+1Pi, j
N
S
x
y
zTarget plane
Ti, j
Ti, j+1
Points on
freeform surface
C(j)
C(j+1)
45
As shown in Figure 2.5, both the target plane and the source intensity space
are sampled into a number of grids and each grid of source intensity is projected to
the specific grid on the target plane. If we sample the intensity distribution of the
source equidistantly, the corresponding target position for the sampled points of
source intensity distribution can be calculated according to Equation (2.9) and the
normal vector field at point Pi, j can be derived according to Equation (2.11).
Assuming Pi,j on curve C(j) is a known quantity, the neighboring point Pi,j+1 on next
curve C(j+1) can be calculated from the following equation:
, 1 ,( ) 0.i j i j P P N (2.14)
This is the simplified iteration method (Newton’s method) to reconstruct the
freeform surface based on the normal vector field. For more precise requirement,
we usually adopt a more accurate modified Euler iteration method to calculate the
points on the freeform surface [70].
Once we have the initial curve C(1), we can then obtain all points on the
freeform surface using the above algorithms. The initial curve C(1) can also be
derived based on assuming an initial point’s coordinate (for example, the central
point). With the initial point position, the next point on the curve could also be
obtained since the collection between these two points should also satisfy Equation
(2.14). With all of these curves calculated, a smooth freeform surface can be finally
generated by integrating all the curves [71].
2.2.3 DESIGN EXAMPLE
46
Now we design a freeform lens by θ-φ ray mapping method for a rectangular
illumination and the design parameters (illustrated in Figure 2.1) are listed in Table 2.1.
The lens is supposed to collect all rays that have a polar angle less than 45°. The designed
lens model and the related lens surface contour plot are as shown in Figure 2.6 with
dimensions labelled.
Table 2.1 Design parameters (Unit: mm)
H h1 h2 X Y
1000 25 6 800 400
H is the distance of target to the source, h1 is the height of apex on the outer surface
of the lens, h2 is the height of the apex on the inner surface, and X and Y are the half
sizes of the illumination target.
Figure 2.6 (a) Lens model; (b) related contour plot of lens surface.
To reduce the statistical error, we trace one million rays in FRED software
[72]. The irradiance distribution and related cross-section profiles on both x and y
directions are shown in Figure 2.7. The RSD on the illumination target is 0.388,
relatively low for general performance. There are three reasons for the poor
47
performance of (θ, φ) ray mapping method in designing freeform surface. The first
one is that the surface error is relatively large because the sampling in the center is
too dense due to the existence of singular point effect. The distribution of the
surface error will be quantified in the next few sections. The second reason is the
discontinuity of mapping at the diagonal region caused by the topological break,
which leads to hot spots or peaks along the diagonal region of rectangular
illumination target. Most important, the skewness distribution of the (θ, φ) sampled
grids does not match with the corresponding sampled grids on illumination target
plane and thus the transfer efficiency from source to target is greatly reduced for
each grid.
48
Figure 2.7 Simulation results for θ-φ ray mapping: (a) irradiance distribution; (b) x
profile; (c) y profile.
2.3 u-v RAY MAPPING METHOD IN FREEFORM OPTICS
DESIGN
In order to solve these issues related to (θ, φ) ray mapping, (u, v) ray
mapping method has been proposed to generate freeform optics for non-imaging
applications [73]. This method moves the singular point to the side region, where
the intensity of source is approximated to zero.
49
2.3.1 DESIGN PRINCIPLE
Figure 2.8 shows the mapping mechanism of (u, v) → (x, y) ray mapping
method, where u is the angle between the light ray and the x axis, and v is the angle
of the plane containing the light ray that the x axis forms with the z axis. This
mapping method has relatively better performance compared to (θ, φ) ray mapping
method because it moves the dense sampling region from the center to the two polar
regions, where general LED sources have minimum energy density distribution.
Besides, the sampled grids in the u-v coordinate system also have better topological
match with the rectangular girds on the illumination target, which will further
reduce the surface error of the freeform optics.
50
Figure 2.8 Topological ray mapping mechanism from source to target based on (u, v)
spherical coordinate system
Now we employ the (u, v) coordinate system to represent the ray emitting
direction with uniform step of u or v and parametrize the target plane in Cartesian
coordinate system. Here we let I(u, v) denote the source intensity at direction of (u,
v) and E(x, y) represent the irradiance distribution on the illumination target plane.
Based on the energy conservation principle, we can have the following expression:
, , , , .D
I u v J u v dudv E x y J x y dxdy
(2.15)
where Ω is the solid angle of source emission pattern and D is the prescribed
illumination target. They can be expressed as follows:
ui
ui+1
uM
u1v1
vj+1
vj
vN
x
y
z
x
y
yj yj+1 yM
xi
xi+1
xN
Intensity
distribution
of source
Target plane
I(u, v)
E(x, y)
51
0 0 0
: ,
: ,,
min maxu u u v v v
D X x X Y y Y
(2.16)
where umin, umax, v0 and -v0 define the edge of the central region as shown in Figure
2.8. Through integrating two variables of the 2D integrals separately, we can build
the relationships of the source ray emitting direction (u, v) and the prescribed
position (x, y) on the target plane for the first quadrant, which can be expressed by
0
0
min
0
00
/2
0
0 00
0 00
( , ) ( , )
( , ) ( , ).
( , ) ( , )
( , ) ( , )
v
tv
u
v
t v
I u v J u v dvdu
x X
I J dvdu
I u v J u v dvy Y
I u v J u v dv
(2.17)
2.3.2 SURFACE CONSTRUCTION
The ray tracing diagram in u-v spherical coordinate is exactly same as
shown in Figure 2.4. Figure 2.9 shows the diagram to construct the freeform surface.
The procedure is completely same as what we have done above to construct the
freeform surface in (θ, φ) ray mapping approach. Firstly based on the ray mapping
mechanism, the corresponding target point can be obtained. Then the normal vector
N can be derived based on the Snell’s law. Using the Newton or Euler’s iteration
algorithm, the neighboring point’s coordinate can also be derived using Equation
(2.14). After all points and lines are calculated, we can finally integrate all points
52
and lines together to reconstruct the whole freeform surface by using the spline
interpolation algorithm.
Figure 2.9 Surface construction diagram in (u, v) ray mapping [73]
2.3.3 DESIGN EXAMPLE OF FREEFORM LENS FOR
RECTANGULAR ILLUMINATION
In order to demonstrate the performance of (u, v) ray mapping method on
the freeform optics generation for LED rectangular illumination, we design a
freeform lens with exactly same specifications on the design target requirement and
Pi, j+1
N
S
x
y
zTarget plane
Ti, j Ti, j+1
Points on
freeform surface
C(j) C(j+1)
Pi, j
53
same lens dimensions (assuming same lens height). The designed lens model and
related lens contour plot are illustrated in Figure 2.10.
Figure 2.10 Freeform lens designed in u-v coordinate system: (a) lens model; (b) lens
surface contour.
We build the CAD model of freeform lens in Rhino software [74] and
import it into the FRED software. The irradiance distribution on the target surface
and related cross-section profiles are shown in Figure 2.11. RSD for this lens
system is about 0.19.
54
Figure 2.11. Simulation results for (u, v) ray mapping: (a) irradiance distribution; (b) x
profile; (c) y profile.
2.4 x-y RAY MAPPING METHOD
The key task of ray mapping method is to find a way of sampling the source
intensity distribution to make the sampled grids topologically similar to the
sampled grids of the illumination target. As we can see from the design examples
using the previous two methods, both can be used to design freeform lens to
generate rectangular illumination. Relatively, u-v sampled grids are more
55
topologically similar to the rectangular grids, and thus it has better performance.
However, with the increase of collection angle in u-v ray mapping method, the
skewness distribution mismatch between source grids and target grids is becoming
more significant. In this section, we will introduce x-y ray mapping to design the
freeform lens for rectangular illumination. In this method, we sample the source
intensity distribution on a projected x-y plane and sample it into square grids in the
plane. As a result, this sampled source grids have the best ability to match the
topological structure of the rectangular target grids.
2.4.1 RAY MAPPING PRINCIPLE
The x-y ray mapping mechanism is shown in Figure 2.12. In the design, the
projected x-y plane for intensity distribution sampling can be any virtual plane
above the light source or the actual collection surface of freeform lens. Assume the
distance between the source and the projection plane is h0. In this case, the direction
of each light ray for first quadrant can be specified by the intersection position at
the projection plane as follows:
2 2
1
0
1
tan
.
tan
s s
s
s
x y
h
y
x
(2.18)
As a result, a new Jacobi function J(xs, ys) can be specified by combining Eqs. (2.18)
and (2.7) to satisfy the following equation:
( , ) ( , ) .s s s sJ x y dx dy J d d (2.19)
56
Plugging Eq. (2.19) into the previous ray mapping equation, we can represent the
energy conservation principle in the x-y ray mapping methodology as follows:
, , , , ,s s s s s s
D
I x y J x y dx dy E x y J x y dxdy
(2.20)
where Ω is the solid angle subtended by the projection plane and D is the
illumination target. Similarly, by integrating the both sides of energy conservation
equation, we can easily obtain the ray mapping mechanism in x-y ray mapping
method as shown in Equation (2.21).
0
0 0
0
0 0
00
,
00
0 00
,
0 00
( , ) ( , )
|
( , ) ( , ),
( , ) ( , )|
( , ) ( , )
m
s s mm
s s m
xy
s s s s s s
t x x y y xy
s s s s s s
y
s s s
t x x y y y
s s s
I x y J x y dx dy
x X
I x y J x y dx dy
I x y J x y dyy Y
I x y J x y dy
(2.21)
where xm and ym define the size of the sampling plane.
57
Figure 2.12 Ray mapping mechanism for x-y ray mapping method
2.4.2 DESIGN EXAMPLE FOR x-y RAY MAPPING – FREEFORM
LENS FOR RECTANGULAR ILLUMINATION
In this section, we will apply the x-y ray mapping method to design a
freeform lens for rectangular illumination. Figure 2.13 shows the geometrical
layout for the design of the freeform lens, where we have a real flat plane to sample
the source intensity. The sampling surface can also be a virtual plane with a
spherical surface as the collection surface, in which there is no refraction. Assume
the ray intersects with the flat surface at point Ri,j and with the freeform surface at
point Pi,j, finally reach the target at point Ti,j. Then the Snell’s law in Eq. (2.11) can
be applied to the ray refraction at the incident flat surface and the freeform surface
separately. For the refraction at the flat surface, the ray tracing vector can be
expressed as follows:
xs
ys
I(xs, ys)
xt
yt
E(xt, yt)
Source
intensity
distribution
Target plane
Light source
58
1
1
1
/ .
/
i, j i, j
i, j i, j i, j i, j
N z
In SR SR
Out R P R P
(2.22)
While for the refraction in freeform surface, the unit incident vector and exit vector can
be expressed by
/
./
i, j i, j i, j i, j
i, j i, j i, j i, j
In R P R P
Out P T P T (2.23)
Figure 2.13 Geometrical layout of freeform lens design by x-y ray mapping method
For the case that we use spherical surface as the collection surface, the incident
vector is just the ray emitting vector as we state in the previous two design methods.
By plugging the above equations into the Snell’s law, we can obtain the normal
Pi, j+1
N
S
x
y
zTarget plane
Ti, j Ti, j+1
Points on freeform
surface
Pi, j
Ri, j
Ri, j+1 Incident flat
surface
2Y
2Y
h1
h2
H
2ym
2xm
59
vector at each point on the freeform surface. Then we can reconstruct the whole
freeform surface as shown in Figure 2.13.
We employ the x-y ray mapping method to design the freeform lens for
generating a rectangular illumination with exactly same design requirements and
constraints as the designs for u-v ray mapping and θ-φ ray mapping. The designed
lens model and related lens surface contour are illustrated in Figure 2.14, where we
choose spherical surface as the collection surface in order to stay consistent with
the previous designs. After 1 million rays are traced in FRED software, the
irradiance distribution for the optical system is as shown in Figure 2.15. The RSD
for this system is about 15.4%, which is consistent with our statement that x-y ray
mapping can provide the best topological similarity between source and target in
our design examples.
Figure 2.14 Freeform lens designed in Cartesian coordinate system: (a) lens model; (b)
lens surface contour.
60
Figure 2.15 Simulation results for u-v ray mapping: (a) irradiance distribution; (b) x
profile; (c) y profile.
2.5 LIMITATIONS TO RAY MAPPING METHODS
We have introduced several ray mapping algorithms for freeform optics
designs to generate rectangular illumination targets. For each method, there are
three common tasks during the design process:
(1) Building the ray mapping between the source intensity distribution and
the output irradiance patterns;
61
(2) Deriving the surface normal vector field from the ray mapping;
(3) Generating the freeform optics by integrating the surface slope or
normal vector field.
As a result, the key factor in ray mapping algorithms is to find a diffeomorphism
(“ray mapping”) so that the prescribed irradiance distribution and source intensity
distribution could be well coupled together. Once the ray mapping is built, each
source ray will be uniquely associated with a point on the target plane and thus the
normal vector or surface slope of the freeform surface at the point where the ray
intersects with the freeform surface can be specified by using Snell’s law.
However, it is not guaranteed that the ray mapping mechanism is the
optimal transport of energy from source to target. The normal vector field derived
from ray mapping may not satisfy the integrability condition of surface [62, 67]. As
a result, surface discontinuities or surface errors may happen. Generally, the
constraint of integrability condition can be written as
0,C
dl N (2.24)
where C is an arbitrary small close loop on the freeform surface, N is the normal
vector of the surface, and dl is a differential displacement vector. By using Stokes’
theorem, Eq. (2.24) can be also written as
0,C S
d ds N l N N (2.25)
where S is the surface contoured by C, and ds is a differential surface area.
Therefore, to guarantee the derived surface is continuous with no surface error, the
normal field of surface at any point must satisfy the following constraint:
62
0. N N = (2.26)
Any failure to satisfy this integrability condition in Eqs. (2.24) or (2.26) can lead
to surface error or step discontinuity when constructing the surface. For simplicity, we use
the normalized residue of the integral in Eq. (2.27) to evaluate the surface error. For the
shaded (ith, jth) grid on the freeform surface in Figure 2.16, we can calculate the residual
surface error (RSE) for the grid as follows:
,
,
,
.i j
i j
i j
d
dl
N l (2.27)
Figure 2.16 (ith, jth) grid on the freeform surface
The numerator term and denominator term in Eq. (2.27) can be separately expressed
by:
, 1, 1, 1, 1
,
1, 1 , 1 , 1 ,
2 2
,2 2
i j i j i j i jAB BC
i j
i j i j i j i jCD DA
d d d
d d
N N N N
N l
N N N N (2.28)
A B
CD
i i+1 i+2
j
j+1
j+2
63
,
,AB BC CD DAi j
d d d d d l (2.29)
where Ni,j is the normalized normal vector at the (ith, jth) sampling point of the
freeform surface. Obviously, RSE is a dimensionless quantity which evaluates the
relative surface error for small area with unit circumference.
Now we can implement the above algorithm to evaluate the amount of surface
error in all three different ray mapping mechanisms we have introduced. The three
freeform lenses are designed with same parameters. RSE maps for first quadrant of these
three freeform lenses are plotted in (θ, φ), (u, v) and (x, y) spaces separately in Figures 2.17
(a), 2.17 (b) and 2.17 (c) separately. The ray mapping method in (θ, φ) spherical coordinate
is significantly affected by the discontinuity at diagonal region where φ equals to π/4, as
shown in Figure 2.17 (a). RSE of the freeform lens designed in x-y ray mapping method is
one magnitude smaller than that designed in (u, v) coordinate system, whose RSE is only
1/3 of the freeform surface designed in (θ, φ) coordinate system. The RSD of irradiance on
target for the lens designed in x-y coordinate system is also much smaller than other
methods.
Table 2.2 Comparison for three freeform lenses
RSD Maximum RSE RMS of RSE
(θ, φ) Spherical 42.4% 2.2e-3 1.1e-3
(u, v) Spherical 19.2% 6.3e-4 3.9e-4
(x, y) Cartesian 15.4% 5.8e-4 3.4e-5
64
Figure 2.17 RSE map for first quadrant of freeform lens designed by (a) θ-φ ray
mapping, (b) u-v ray mapping, and (c) x-y ray mapping
As shown in Table 2.2, the x-y ray mapping method has significant advantages
over the other two traditional ray mapping methods because its sampling grids of source
intensity distribution are most topologically similar to those of the target irradiance
distribution. This topological similarity is usually quantified by the skewness distribution
of the optics surface. The ray mapping method with best match of skewness distribution
between source and target has the best irradiance uniformity as well as the smallest surface
error. The other two methods suffer from the effect of central singular point, two side
singular points, source-target skewness distribution mismatch and so on.
2.6 CONCLUSIONS AND DISCUSSIONS
0 5 10 15 200
5
10
15
x/mm
(a)
y/m
m
0 5 10 150
5
10
15
20
x/mm
(b)
y/m
m0 5 10 15
0
5
10
15
20
x/mm
(c)
y/m
m
0.5
1
1.5
2
x 10-3
65
In this chapter, we have discussed different ray mapping methods that sample
source intensity distribution in three completely different coordinate systems. Due to the
topological structure of sampled grids, they have very different degree of match with the
rectangular target and thus they have very different performance in generating freeform
optics for rectangular illumination. Compared to θ-φ ray mapping and u-v ray mapping,
the x-y ray mapping method has the best performance for the generatiing rectangular
illumination pattern from LED light source as well as the least surface error. The θ-φ ray
mapping method has the least ability to generate rectangular illumination pattern since
sampled source intensity grids have the worst topological match with the rectangular grids
on the target surface.
66
CHAPTER 3
DOUBLE-POLE RAY MAPPING METHOD
In the previous chapter, three different ray mapping methods based on
various coordinate systems are introduced to design freeform optics for non-
rotational LED illumination. In this chapter, another completely new method,
double-pole ray mapping method, will be introduced for the design of freeform
optics. This ray mapping method has the best topological similarity to rectangular
target grids. Section 3.1 discusses the principle of the double-pole ray mapping
method, Section 3.2 explores the application of double-pole ray mapping method
in freeform lens design, and Section 3.3 demonstrates its application in freeform
reflector design for non-rotationally symmetric illumination.
3.1 DOUBLE-POLE RAY MAPPING METHOD WITH
UNIFORM SAMPLING
3.1.1 DOUBLE-POLE RAY MAPPING MECHANISM
Based on symmetric properties of point-like sources, it is preferable to use
normal (θ, φ) spherical coordinate system mapping method to sample the intensity
distribution of the light source [69]. However, the polar region in the center will
definitely lead to unstable solutions for the surface construction. Another ray
67
mapping method samples source intensity distribution in (u, v) spherical coordinate
system [73], it shifts the central singular point to the two peripheral polar regions,
causing large surface error when extending the collection angle to the whole
hemisphere. The double-pole coordinate system is proposed to completely avoid
the effect of poles by moving the two poles of the spherical coordinate system to the
southernmost point of the sphere and overlapping them together as shown in Figure 3.1. In
this special coordinate system, there is no singular point over the whole hemisphere above
the LED source. The double-pole coordinate system has been used to describe a linearly
polarized grid of rays on a spherical wavefront [75], and it has also been brought into
applications in neuroscience [76]. The double-pole coordinate system is similar to the
stereographic projection coordinate introduced in [57]. The stereographic projection
coordinate was used to sample the intensity distribution of source to derive the Monge-
Ampère second order nonlinear partial different equation for freeform reflector
design [57]. In this paper, we will develop ray mapping method in double-pole
coordinate system to design freeform surface for LED illumination.
68
Figure 3.1 Diagram for double pole coordinate system: (a) perspective view; (b) top
view; (c) bottom view.
In double-pole coordinate system, we locate the source in the origin O and let (R,
α, β) represent the coordinate of any specific point A on the sphere, R is the radius of the
sphere, α is the angle between y-z plane and the shaded surface containing OA and Oy,
O
O’
xy
x’ y’
z
A
R
(a)
(b) (c)-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
69
and β is the angle between x-z plane and the shaded surface containing OA and Ox.
Therefore, the point (x, y, z) on sphere with a radius of R can be expressed by
2 2 2
1
1
tan ( ) .
tan ( )
R x y z
x
z R
y
z R
(3.1)
As a result, the transformation from the modified double pole coordinates to
Cartesian coordinates can be obtained as:
2 2
2 2
2 2
2 2
2 2
2 2
1 tan tan(1 ) tan
1 tan tan
1 tan tan(1 ) tan .
1 tan tan
1 tan tan
1 tan tan
x R
y R
z R
(3.2)
If we let J(α, β) denote the Jacobi matrix of the transformation between modified
double pole coordinate system and Cartesian coordinate system, J(α, β) can be
derived by
( , ) .
x x x
R
y y yJ
R
z z z
R
(3.3)
By plugging Eq. (3.2) into the above matrix, we can obtain the determinant of the
above Jacobi matrix as:
2 2
22 2
4sec sec.
1 tan tan( , )J
(3.4)
70
By using the energy conservation principle, we can get the following equation:
0, , , , ,D
I J d d E x y J x y dxdy
(3.5)
where Ω defines the boundary of source intensity distribution, 𝐷 is the whole
prescribed rectangular illumination target, I is the intensity distribution of the
source, and E0 is the irradiance on the target plane. Ω and D are defined as
0 0 0 0: ;
.: ;t t
A A B B
D X x X Y y Y
(3.6)
where A0 and B0 define the edges of source intensity distribution as shown in Figure
3.2, X and Y define the half sizes of the prescribed illumination target.
Figure 3.2 Mapping mechanism from (α, β) to (x, y).
The mapping mechanism in Figure 3.2 only shows the first quadrant of the
target plane. Through integrating two variables of the 2D integrals in Eq. (3.5)
separately, we can establish the relationships of the emitting ray direction (α, β)
from the light source and the prescribed ray position (xt, yt) on the target plane,
I( , )E(x, y)
x
y
xy
z
X
YIntensity
distribution grids
Illumination target grids
71
0
0 0
0
0 0
0 0
0 00
0 00
( , ) ( , )
( , ) ( , ),
( , ) ( , )
( , ) ( , )
B
t A B
t B
I J d dx X
I J d d
I J dy Y
I J d
(3.7)
where I(α, β) is the intensity distribution of source in (α, β) space, and α0 is any specific
value for α. Here represents different ray mapping mechanisms, we only take “+” for
freeform lens design in order to minimize the ray deviation angle during the refraction and
thus minimize the surface error. With this ray mapping mechanism, we can construct the
one-to-one correspondence relationship between the source and target and thus reconstruct
the freeform surface.
3.1.2 DESIGN EXAMPLE
To demonstrate the advantages of the proposed double pole coordinate system in
designing freeform illumination lens, we design three freeform lenses using normal (θ, φ)
spherical coordinate system as shown in Figure 2.3 [69], (u, v) spherical coordinate system
in Figure 2.8 [73], and the double-pole coordinate system as shown in Figure 3.2 [77]. For
all three methods, the first quadrants of freeform surfaces are tailored into 100 × 100 grids
for sampling source intensity distribution with equal (θ, φ), (u, v), and (α, β) step in each
grid. The design parameters are listed in Table 3.1. The LED chip is 1mm × 1mm and has
a Lambertian intensity distribution. All three lenses are designed to collect all light rays
emitted from the source and achieve a homogeneous irradiance distribution on the target.
72
Table 3.1 Design parameters
H h X Y
900 mm 25 mm 800 mm 400mm
H is the distance from the source to the target surface, h is the distance between the source
and the vertex of freeform surface, and X and Y are half sizes of the rectangular illumination
target.
We perform ray tracing with 1 million rays for all three lenses in Lighttools and
plot the irradiance distributions in Figure 3.3. Both freeform lenses designed in spherical
coordinate systems have hot spots as shown in Figures 3.3 (a) and 3.3 (b). The uniformity
of the freeform lens designed in double-pole coordinate system (Figure 3.3 (c)) is much
better than the other two methods. RSE maps for first quadrant of freeform lenses are
plotted in (θ, φ), (u, v) and (α, β) spaces separately in Figure 3.4. Traditional ray mapping
method in (θ, φ) spherical coordinate is significantly affected by the discontinuity at
diagonal region where φ equals to π/4, as shown in Figure 3.4 (a). RSE of the freeform lens
designed in double-pole coordinate system is one magnitude smaller than that designed in
(θ, φ) coordinate system and is only 1/3 of the freeform surface designed in (u, v) coordinate
system. The relative standard deviation (RSD) of irradiance on target for the lens designed
in double-pole coordinate system is also much smaller than other methods.
73
Figure 3.3 Irradiance distribution for freeform lenses designed in (a) (θ, φ) spherical
coordinate system, (b) (u, v) spherical coordinate system, and (c) double-pole coordinate
system.
As summarized in the Table 3.2, the ray mapping method in double-pole
coordinate system has significant advantages over the other two traditional ray mapping
methods. The proposed ray mapping method has the best irradiance uniformity as well as
the smallest surface error. The other two methods suffer from the effect of central singular
point, two side singular points, and source-target skewness distribution mismatch. In
contrast, the ray mapping method in double-pole coordinate system doesn’t have singular
point over the whole hemisphere.
(a) (b)
(c)
74
Figure 3.4 RSE map for first quadrant of freeform lens designed in (a) (θ, φ) spherical
coordinate system, (b) (u, v) spherical coordinate system, and (c) double-pole coordinate
system.
Table 3.2 Comparison for three freeform lenses
RSD Maximum RSE RMS of RSE
(θ, φ) Spherical 50.4% 2.5e-3 6.6e-4
(u, v) Spherical 15.6% 6.8e-4 2.1e-4
Double Pole 5.7% 1.9e-4 4.4e-5
75
3.2 DOUBLE-POLE RAY MAPPING METHOD WITH NON-
UNIFORM SAMPLING
3.2.1 RAY MAPPING MECHANISM
In our previous approach, we sample the source intensity distribution
uniformly in the double-pole coordinate space [77]. It works well if we do not take
the geometric shape and collection efficiency into consideration. However, in some
cases, light sources have a rotationally symmetric emission properties, but the
double-pole coordinate system is not rotationally symmetric with the uniform
sampled square or rectangular grids. As a result, there exists a mismatch between
the uniformly sampled grid in double-pole coordinate system and the actual light
source emission pattern. In addition, the freeform lens designed under the uniform
sampling method has a square or rectangular shape due to the structures of sampled
grids in the double-pole coordinate system, possibly making it more difficult to
fabricate and mount.
Two approaches have been developed to address this mismatch issue. As
shown in Figure 3.5(a), the sampled grid covers the whole emission region with the
assumption that the region outside the actual source emission region has zero
emission intensity. This sampling method covers the whole emission region and
can maximize the collection efficiency to 1. However, this mapping method
requires that the grids with nearly zero area at both left and right side are mapped
to the sampled grids on the target surface. This dimensional mismatch will
definitely lead to singular points at the edges, resulting some discontinuities in the
76
edge regions. The sampling approach in Figure 3.5(b) reduces the sampling region
to guarantee that all the sampled grids are well illuminated by the light source.
Obviously, this solution suffers the reduction of light collection efficiency and
results in some energy loss.
Figure 3.5 Traditional methods to sample source intensity distribution in double-pole
coordinate system: (a) emission region subtended by sampled grids (b) sampled grids
subtended by the source emission.
In order to overcome the shortcomings caused by the uniformly sampled
grids for freeform optics design, we propose to sample the light source intensity
distribution non-uniformly to exactly match its circular emission edge. Figure 3.6
shows the non-uniform sampling approach in double-pole coordinate system and
the mapping mechanism from non-uniform sampled grids to the rectangular grids
on the illumination target. The whole quadrant circular emission region is divided
into (M-1) × (N-1) grids with M × N grid points on the whole surface. If we assume
that the maximum emission zenith angle of light source is θM, then the coordinates
of 4 nominal corner points and edge grid points can be derived as follows:
I(α, β)=0
Source Emission Region
I(α, β)>0
(a) (b)
α α
ββ
77
1, ,1 ,1 1,
, ,
0, = / 2 1,2,..., ; 1,2,...,
,1 cos1arctan
1 cos2
j i M N M
M
M N M N
M
i M j N
(3.8)
The coordinates of grid points along the “③” edge are equally spaced while the
related coordinates are calculated to satisfy the boundary condition,
, 1, , 1,
2 2 2
, 0 , 0 0
1
1 1,2,..., .
arctan 1 (1 z ) tan / 1
i N N M N N
i N i N
i
M i M
z z
(3.9)
Figure 3.6 Non-uniform sampled grids in first quadrant of modified double-pole
coordinate system and related mapping mechanism between source and target
Illumination target grids
y
x
Y
X
E(x, y)
I( , )
Intensity
distribution
grids
(a0, b0)
(A0, 0)
(0, B0)
(0, 0)
( i,j, i,j)
②②
③③
④④
①
78
Similarly, the coordinates of the grid points along the “④” edge are sampled with
equal spacing in β direction and variable spacing in α direction to satisfy the
boundary conditions. Thus, the coordinates in edge “④” can be derived as:
, ,1 , ,1
2 2 2
, 0 , 0 0
1
1 1,2,..., .
arctan 1 (1 z ) tan / 1
M j M M N M
M j M j
j
N j N
z z
(3.10)
Now all the coordinates of grid points located on the emission edge in the double
pole coordinate system are defined. For all the other grid points, we take an equally
spacing on both directions, and the coordinates can be defined as follows:
, 1, , 1,
, ,1 , ,1
1
11,2,... ; 1,2,... .
1
1
i j j M j j
i j i i N i
i
Mi M j N
j
N
(3.11)
With all grid points defined on the surface, we firstly integrate the energy
distribution on the source space line by line in the direction, which corresponds
to the x coordinates of the lines on the illumination target. As a result, the x
coordinate of the target points on the kth line related with the (kth, sth) grid point on
the source space can be obtained as:
, , ,
2 2,
, , ,
2 2
0 1
( , ) ,,
Otherwise
( , ) ,
k N
i j i j i j
i jt k s
M N
i j i j i j
i j
k
E i j J dAx
X
E i j J dA
(3.12)
where dAi,j is the approximate area of the grid on (α, β) space and can be expressed
by
79
, , 1 1, 1, 1 , 1, , 1 1, 1
,2 2
i j i j i j i j i j i j i j i j
i jdA
. (3.13)
After each grid is mapped onto the target plane, by integrating the energy grid on
each line we can find the y coordinate of the target point corresponding to the (kth,
sth) grid point as follows:
, , ,
2,
, , ,
2
0 for 1
( , ) ,.
Otherwise
( , ) ,
s
i j i j i j
jt k s
N
i j i j i j
j
s
E i j J dAy
Y
E i j J dA
(3.14)
3.2.2 DESIGN EXAMPLE AND SIMULATION
To demonstrate the effectiveness of the ray mapping method with non-
uniformly sampled grids of source intensity in the modified double-pole coordinate
system, we have designed two freeform lenses with different configurations. One
freeform lens with spherical incident surface is designed for LED, whose emission
solid angle covers the whole hemisphere. The other one with flat incident surface
is designed for the beam shaping of a laser source, whose emission angle is
relatively small.
3.2.2.1 FREEFORM LENS DESIGN FOR LED SOURCE
The geometrical layout of the freeform lens design for LED source is shown
in Figure 2.4. Since LED source has a Lambertian intensity distribution, the lens is
80
designed with a spherical incident surface to collect all light rays inside the whole
hemisphere. The design parameters are listed in Table 3.3.
Table 3.3 Design parameters of the freeform lens for LED source (Unit: mm)
H h X Y r
1000 20 900 600 6
H is the distance from the source to the target surface, h is the
distance between the source and the vertex of the freeform lens, X
is the half of the length of the illumination target, Y is the half of
the width of the illumination target, and r the radius of the
collection surface.
The intensity distribution of the LED source can be expressed as follows:
2 2
0 0 2 2
1 tan tancos ,
1 tan tanI I I
(3.15)
By plugging Eq. (3.15) into the mapping equations, the one-to-one corresponding
relationship between the source and target can be derived. Using the algorithms and
design methods introduced above, the initial lens model and related contour plot of
lens surface for LED illumination are shown Figure 3.7.
Figure 3.7 (a) Freeform lens contour and (b) NURBS surface entity for the LED freeform
lens
14.1
mm
11.4 mm
(a) (b)
81
After the freeform lens is constructed in Rhino software, we simulate the
performance of the freeform lens for LED source in FRED software by tracing 10
million rays for smaller statistical error. The simulation result of the initial design
without feedback optimization is shown in Figure 3.8. The power of the LED is
normalized to be 1 Watt for simplicity. With Fresnel loss considered, the total
collection efficiency for the designed freeform lens is approximately 86%, and RSD
for the irradiance distribution on the target surface is about 15%. RSD can be further
improved by feedback optimization.
Figure 3.8 Simulation results in FRED software: (a) irradiance distribution, (b) X profile
plot, and (c) Y profile plot.
(a)
(b)
(c)
82
3.2.2.2 FREEFORM LENS DESIGN FOR LASER SOURCE
This lens is designed to generate a rectangular illumination pattern for the
laser light from a multimode fiber. The non-uniform sampling grid method is
implemented because of the rotational symmetric properties of laser light from the
fiber. The design parameters are listed in Table 3.4.
Table 3.4 Design parameters of the freeform lens for LED source (Unit: mm)
H h1 h2 X Y
350 4 7 70 35
H is the distance from the source to the target surface, h1 is the
distance between the source and the vertex of the freeform lens, h2
is the distance between the source and the vertex of the freeform
lens, X is the half of the length of the illumination target, and Y is
the half of the width of the illumination target.
Figure 3.9 (a) Freeform lens contour and (b) NURBS surface entity for the fiber laser
source freeform lens
The intensity distribution for the source can be expressed by the following
equation:
2 2
0= exp 2 / ,AI I (3.16)
(a) (b)
3 m
m
1.8 m
m
83
where θA is the numerical aperture of the multimode fiber, which is equal to 0.22
for this design. The diameter of the fiber is 105 μm. Similarly, by plugging Eq.
(3.16) into the mapping equations introduced above, the initial freeform lens model
and related contour plot of the lens surface are shown in Figure 3.9. As can be seen
from the lens model, the initial freeform lens has circular edge, much easier for
fabrication and mounting. The simulation results are shown in Figure 3.10. RSD
of irradiance distribution on the target surface is as low as 8.5%. The collection
efficiency when Fresnel loss considered is about 84% on the target surface.
Figure 3.10 Simulation results of freeform lens for fiber laser illumination: (a) irradiance
distribution and (b) X and Y profile plot.
3.3 DOUBLE POLE COORDINATE SYSTEM FOR FREEFORM
REFLECTOR CONSTRUCTION
(a)
Y/mm
X/m
m
(b)
84
A number of design methods for freeform reflectors have been developed.
These methodologies include supporting ellipsoids [48-52], trial and error
approaches [53, 54], simultaneous multiple surfaces (SMS) method [55, 56],
analytic solution of Monge-Ampere equations [57-59], and geometrical ray
mapping method [60-62]. Among these methods, the geometrical ray mapping
method is one of the promising design methods for non-rotational illumination. The
main drawback for the trial and error method is very time consuming due to the
costly ray tracing. The method of supporting ellipsoids also requires of a number
of optimization iterations for each ellipsoid, which is also very time consuming.
The SMS method usually requires at least one pair of surfaces and is not suitable to
design the reflector with only one single surface. For the Monge-Ampere method,
the derivation of the partial differential equation is very tedious and time consuming,
and it is hard to avoid the existence of singular value in solutions when numerically
solving the equation.
Figure 3.11 (a) θ-φ coordinate system; (b) u-v coordinate system; (c) double-pole
coordinate system
(a) (b) (c)
-1
-0.5
0
0.5
1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-0.8-0.6
-0.4-0.2
00.2
0.40.6
0.8
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
-0.8-0.6
-0.4-0.2
00.2
0.40.6
0.8
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
85
Traditionally, θ-φ spherical coordinate system in Figure 3.11(a) [69] and u-
v spherical coordinate system in Figure 3.11 (b) [73] are used to sample the source
intensity distribution in building the ray mapping relationship between source and
target. We propose a double-pole ray mapping method in Figure 3.11 (c) to design
a freeform lens with a total acceptance solid angle of π [77]. Besides, by taking
non-uniform sampling in double-pole coordinate system, we can generate freeform
lens with both smooth surface and more circular-like edge [78]. In next section, we
will apply this new ray mapping method to design the freeform reflectors with
acceptance solid angle of more than π for generating rectangular illumination
patterns.
3.3.1 DESIGN METHOD
The design process for general geometrical ray mapping method is illustrated in
Figure 2.2. The design phase has four procedures: source-target ray mapping
relationship establishment, freeform surface construction, ray trace simulation and
feedback modification. Ray mapping method determines the mapping efficiency
and system performance of initial design. Feedback modification is aimed to
optimize the system to satisfy the initial requirement and it also requires good initial
design. Good initial starting design can greatly reduce the optimization time for the
feedback modification algorithms.
86
Figure 3.12 (a) Determinant of the Jacobian in double pole coordinates. The red lines is
the boundary governed by Eq. (3.18). The green dot is the intersection of a line at 45˚ and
the boundary. (b) The source mapping in α-β space. (c) Source map in Cartesian
coordinates. (d) Target map for double pole coordinate system.
Typically, lenses or reflectors attempt to capture all flux from a source over
a full angle θ. Given a maximum angle θmax, the source captures the incident flux
for all z greater than
maxcosz R (3.17)
where R is the radius of a sphere . The boundary of the source in α-β space, given
a maximum θ, is:
22 2
2
1tan tan
1
z
z
(3.18)
87
The determinant of the Jacobian for the double-pole coordinate system is
shown in Figure 3.12 (a) when θmax=π/4. The boundary shown as a red line is
governed by Eq. (3.18). The goal of the non-uniform source mapping is to create
equal flux grids in α-β space. The source map should not contain any virtual
singularity caused by the mapping. One method to create a source map without any
virtual singularity is to use the 45˚ point (α45, β45) on the boundary of the Jacobian
as the end point for α and β grid lines.
To maintain rectangular grid lines, the grid lines along α start along β = [0,
βend] on the left and are connected to points along the boundary from β = [0, β45] on
the right of the source map. The grid lines along β start along α = [0, αend] on the
bottom and are connected to points along the boundary from α= [0, α45]. This grid
mapping is shown in Figure 3.12(b), the grid lines along α direction are in black
lines and the grid lines along β direction are in red lines. The corresponding
endpoints along the boundary of the source are in black and red points, respectively.
This non-uniform spacing along α and β prevent virtual singularities caused by the
source mapping. The grids are equally spaced along each line. The source map
obtained using the double-pole coordinate system in Cartesian coordinates is shown
in Figure 3.12(c).
The whole quadrant circular emission region is divided into (M-1) × (N-1)
grids with M × N grid points on the whole surface. Let (αi,j, βi,j) denote the
coordinates of (ith, jth) grid point in the double-pole space, the x coordinate of the
target points on the kth ribbon related to the (kth, sth) grid point on the source space
can be obtained as:
88
, , ,
2 2,
, , ,
2 2
0 1
( , ) ,,
Otherwise
( , ) ,
k N
i j i j i j
i jt k s
M N
i j i j i j
i j
k
E i j J dAx
X
E i j J dA
(3.19)
where dAi,j is the approximate area of the grid on (α, β) space and can be
approximately expressed by
, , 1 1, 1, 1 , 1, , 1 1, 1
,2 2
i j i j i j i j i j i j i j i j
i jdA
. (3.20)
After each ribbon is mapped onto the target plane, by integrating the energy grid
on each ribbon we can then find the y coordinate of the target point corresponding
to the (kth, sth) grid point as follows:
, , ,
2,
, , ,
2
0 for 1
( , ) ,.
Otherwise
( , ) ,
s
i j i j i j
jt k s
N
i j i j i j
j
s
E i j J dAy
Y
E i j J dA
(3.21)
3.3.2 FREEFORM REFLECTIVE SURFACE CONSTRUCTION
The source intensity distribution is sampled into m×n grids in the double-
pole coordinate system, and the ray trace diagram for the (ith, jth) grid on the
freeform reflector is shown in Figure 3.13. Once the source intensity grid is
established, the incident vector is specified and the output vector can be obtained
by using the mapping relationships expressed in Eq. (3.7). Let In denote the unit
incident vector into the (ith, jth) grid on the reflector surface and Out denote the unit
89
output vector out of the grid. Using the law of reflection, the normal vector at the
(ith, jth) point on the reflector can be obtained as:
.
Out InN
Out In (3.22)
With the information of normal vector field at the specific point, the neighboring
points can be obtained using Newton’s iteration algorithm or Euler’s iteration
algorithm. Therefore, the whole surface can be reconstructed if we assume an initial
point on the freeform reflector. The exact calculation procedures can be found in
[70, 79].
Figure 3.13 Ray trace diagram for the (ith, jth) grid on the freeform reflector
3.3.3 DESIGN EXAMPLES
To show the feasibility of double-pole ray mapping method in designing
freeform reflectors with large acceptance solid angle, several freeform reflectors
90
with different mapping structures are designed for rectangular irradiance
distribution. For all designs, we set the distance between the LED source and the
target as 1000 mm (H) and the distance between the apex of reflector and source as
20 mm (h). The source is a sphere source with a radius of 1 mm and it has isotropic
intensity distribution. The acceptance solid angle of the freeform surfaces has a
zenith angle of 120° or 140°. The illumination target is 1 m × 0.5 m.
3.3.3.1 FREEFORM REFLECTORS DESIGN WITH UNIFORM
SAMPLING
Figure 3.14 shows two typical mapping structures: uncrossed mapping and
crossed mapping. For uncrossed mapping, we only take “+” for both mapping
relationships in Eq. (3.7), while for the crossed mapping structure, only “” is left.
In this section, we design and compare 4 different reflectors: uncrossed reflectors
with half field of view (HFOV) of 120° and 140°, and crossed reflectors with
HFOV of 120° and 140°. The lens models are as shown in Figures 3.15(a), 3.15(b),
3.15(c) and 3.15(d), they are not plotted in scale but with dimensions labelled in
the figure. The corresponding irradiance distributions for different reflectors after
5 million rays traced in FRED are shown in Figures 3.15(e), 3.15(f), 3.15(g), and
3.15(h), respectively.
91
Figure 3.14 Diagram of reflectors for two different source-target mappings: (a) uncrossed
mapping; (b) crossed mapping.
Obviously, reflectors with uncrossed mapping configurations are much less
compact compared to the designs with crossed mapping mechanisms. Besides, the
size of reflector increases dramatically with the increase of acceptance solid angle.
For these 4 different designs, the relative standard deviations (RSDs) are 12.8% for
the uncrossed design with HFOV of 120°, 15.1% for crossed design with HFOV of
120°, 18.4% for the uncrossed design with HFOV of 140°, and 25.9% for the
crossed design with HOFV of 140°.
92
Figure 3.15 Freeform lens models under uniform sampling in double-pole space and
corresponding irradiance distribution: (a) and (e) uncrossed design with FOV of 120°, (b)
and (f) uncrossed design with FOV of 140°, (c) and (g) crossed design with FOV of
120°, and (d) and (h) crossed design with FOV of 140°.
60
mm
100 mm
48
0 m
m
185mm
(a)
(b)
70 mm
80
mm
(c)
(d)
55
mm
52 mm
(e)
(f)
(g)
(h)
93
3.3.3.2 SMOOTH REFRLECTOR DESIGN WITH NON-UNIFORM
SAMPLING
In this subsection, we design another 4 reflectors with same design
parameters listed in the previous subsection by using the non-uniform sampling in
the source space. The designed lens models are shown in Figures 3.16(a), 3.16(b),
3.16(c) and 3.16(d). The corresponding irradiance distributions after 1 million rays
traced in FRED are as shown in Figures 3.16(e), 3.16(f), 3.16(g), and 3.16(h),
respectively. Compared to the designs with uniform sampling, the designs with
non-uniform sampling have much more smooth edge and relatively more compact
dimensions. More important is that we can also improve the irradiance uniformity
using non-uniform sampling. For example, the RSDs for these 4 different designs
are 9.5% for the uncrossed design with HFOV of 120°, 11.5% for crossed design
with HFOV of 120°, 12.6% for the uncrossed design with HFOV of 140°, and 20.0%
for the crossed design with HOFV of 140°, separately. In conclusion, the non-
uniform sampling design algorithm has been demonstrated to be much more
advantageous compared to the uniform sampling design algorithm when designing
the freeform reflectors with large collection solid angles.
94
Figure 3.16 Freeform lens model designed under non-uniform sampling in double-pole
space and corresponding irradiance distribution: (a) and (e) uncrossed design with FOV
of 120°, (b) and (f) uncrossed design with FOV of 140°, (c) and (g) crossed design with
FOV of 120°, (d) and (h) crossed design with FOV of 140°.
83 mm
60 m
m
170 mm
114 m
m60 mm
27 m
m
90 mm
35 m
m
(e)
(a)
(b)
(c)
(d)
(f)
(g)
(h)
95
3.4 CONCLUSION AND DISCUSSION
In this chapter, we firstly introduce a completely new coordinate system
double pole coordinate system, to freeform optics design for rectangular
illumination. Compared to traditional coordinate systems, the sampled grids in
double pole coordinate system can have the best topological match with the
rectangular sampled grids on the target surface. On the other hand, we take non-
uniform sampled grids on the double-pole space to achieve rotationally symmetric
edge of sampled grids, matching the radiation pattern of some light sources. Finally,
we demonstrate the proposed uniform and non-uniform sampling methods in
designing refractive freeform lenses and freeform reflectors in double-pole
coordinate system.
96
CHAPTER 4
RAY MAPPING METHODOLOGY’S APPLICATION
IN FREEFORM TOTAL INTERNAL REFLECTIVE
(TIR) OPTICS
To improve the light efficiency in LED illumination, total internal reflection
(TIR) lenses are often used because of the low light loss and compactness. For the
rotationally symmetric illumination target, TIR lenses have been well developed
[80-81]. However, there is a strong need of non-rotationally symmetric illumination
for better illumination efficiency, for example rectangular illumination in road
lighting [19, 38]. For the non-rotational illumination, the design of TIR freeform
surface is much more challenging. A few methods have been proposed to design
the TIR freeform lens for non-rotational illumination [82-84]. However, there are
some potential issues with current approaches, such as relatively large surface error
and hot spots in the illumination pattern. In this chapter, we will discuss new ray
mapping methods to design freeform TIR lenses for non-rotational illumination.
Section 4.1 proposes a composite ray mapping method by combining u-v ray
mapping and θ-φ ray mapping method to design the freeform TIR lens. Section 4.2
will introduce a multiple x-y ray mapping method to design the freeform TIR lens.
97
In Section 4.3, a combination of TIR collimating lens and freeform lens array is
implemented to generate various non-rotational illumination patterns.
4.1 TIR LENS IN LED ILLUMINATION
TIR lens has broad applications in non-imaging optics [85, 86]. Compared
to general refraction-only lenses or single-surface reflectors, TIR lens has a number
of advantages. For example, TIR lenses can achieve much larger bending angles
for light rays compared to traditional refractive optics, which can be easily more
than 90 degrees. TIR lenses can be able to collect all light emitted by source with
relatively compact structure. In addition, from the perspective of optics design, TIR
lenses have at least 3 degrees of freedom (one TIR surface and two refractive
surfaces) to control the light rays, while there are only 2 degrees of freedom for
general refraction-only lenses (two refractive surfaces) and only one degree of
freedom for single-surface reflectors (one reflective surface).
With their great advantages in controlling light rays compared to other
general lens or reflectors, TIR lenses have broad applications in LED illumination
[80], solar concentrators and daily life illumination such street and indoor
illumination [85]. TIR lens can also improve the color uniformity of multi-color
LEDs [86, 87].
Although TIR lenses have many different applications, there are only a few
types of TIR lenses in general due to their specific architectures. Figure 4.1 shows
three typical TIR lenses [88-91], and most TIR lenses for LED illumination can be
98
classified as one of these three types. The following sections will discuss different
design methods for freeform TIR lenses based on these three basic architectures.
Figure 4.1 (a) TIR lens with outside flat top surface, (b) TIR lens with inside flat top
surface; (c) TIR lens with faceted (Fresnel) TIR surface [88]
4.2 COMPOSITE RAY MAPPING IN DESIGNING FREEFORM
TIR LENS
LED
(c)
LED
(a)
LED
(b)
99
The ray mapping methods are commonly used to design TIR lenses because
they can simplify the design by separating the ray mapping and lens construction.
The problem of ray mapping methods is that they don’t fully satisfy the integrability
condition as discussed in previous chapters. The key to design TIR lenses for non-
symmetrical illumination target is how to achieve proper ray mappings for both the
refractive surfaces and the TIR surface. One convenient way is to partition the
source intensity distribution in traditional spherical (θ, φ) coordinates and partition
its corresponding target irradiance distribution into a grid as shown in Figure 2.3
[69]. However, one issue of this approach is that the surface error is large because
the sampling in the central region is too dense. As shown in Figure 2.17, due to the
discontinuity of mapping in the diagonal region of the target, the hot spots along
the diagonal region are unavoidable. Figure 2.8 shows another commonly used ray
mapping method (u, v) → (x, y), wherein u is the angle between the light ray and
the x axis, and v is the angle of the plane containing the light ray that the x axis
forms with the z axis [73, 79]. This mapping method moves the dense sampling
region from the center to the two polar regions. However, it results in hot spots at
the four corners of the target as we have proved through examples. Moreover, this
mapping method cannot be directly applied in designing TIR lens due to its dense
sampling at the polar regions, which will lead to large surface error to the TIR
surface.
In this section, we propose a composite ray mapping method [70] to design
TIR lens for non-symmetrical illumination. The central region of the source
intensity distribution is sampled in (u, v) coordinate to design refractive surface,
100
and the peripheral region is sampled in (θ, φ) coordinate to design TIR surfaces.
The key advantage of the proposed ray mapping method is that the illumination
uniformity is better and the lens is less sensitive to the errors in the manufacturing
and LED light distribution because this method superposes the light from the
central region and peripheral regions.
4.2.1 COMPOSITE RAY MAPPING PRINCIPLE
In the composite ray mapping method, we apply (u, v) coordinate and (θ, φ)
coordinate to specify the direction of LED rays in different regions, as shown in
Figure 4.2. We divide the emitted rays of LEDs into 5 different parts: one central
zone and four peripheral regions. The optical model is initially created based on a
point source by using a variable separation mapping method [73]. In the central
region, we employ the (u, v) coordinate to sample the source’s light distribution and
design the freeform refractive surface based on the ray mapping. In the four
peripheral regions, freeform TIR surfaces are designed by using (θ, φ) coordinate
to sample the source light distribution. As shown in Figure 4.2, since the edge θmin
in the θ direction for the peripheral region is dependent on a specific φ, the sampling
in the θ direction should be non-uniform in order to guarantee that the rectangular
grid sampling on the illumination target plane is same as the central zone. Based on
this mapping mechanism, source rays from three different parts are mapped to the
same part on the target surface. The overlapping mechanism is expected to improve
the illumination uniformity on the target surface. In addition, the lens is less
101
sensitive to the manufacturing errors and the errors in light distribution of the light
source due to the overlapping mechanism.
Figure 4.2 The topological mapping from source to target based on the composite
mapping method
For the central zone in Figure 4.2, let I(u, v) denote the intensity distribution
of the LED source and E0(x, y) be the illumination distribution contributed by the
Intensity
distribution
of source
Target plane
E(x, y)
I(θ, φ) I(u, v)
φmax
θmin(φ)
umax
umin
-v0
v0
I(θ, φ)
102
central part of the source on the prescribed illumination target. The energy
conservation of a lossless optical system for the central part can be expressed by
0
0, , , , .D
I u v J u v dudv E x y J x y dxdy
(4.1)
where Ω0 is the solid angle covered by the central region, and D is the prescribed
illumination target. They can be expressed by the following terms:
0 0 0
: ,
: ,.
min maxu u u v v v
D X x X Y y Y
(4.2)
where umin, umax, v0 and -v0 define the edge of the central region as shown in Figure
4.2. By integrating two variables of the 2D integrals separately, we can build the
relationships of the LED ray emitting direction (u, v) and the ray prescribed position
(x, y) on the target plane
0x f u (4.3)
00| |ux f u
y h v
(4.4)
For the four peripheral regions, we sample the source intensity distribution
in the spherical coordinate system. Let I(θ, φ) denote the intensity at the right-side
peripheral region and E1(x, y) be the illumination distribution contributed by the
right-side peripheral region on the prescribed illuminating target. The energy
conservation of a lossless optical system for the peripheral region can be shown as
1 1
1, , , , .D
I J d d E x y J x y dxdy
(4.5)
wherein Ω1 is the solid angle covered by the right-side peripheral region rendered
by light purple color in Figure 4.2, D1 is the right half of prescribed illumination
target. They can be expressed as
103
1 max mmin
1
in: , ( ) 2
: ,
.
0D x X Y y Y
(4.6)
wherein φmax, φmin, θmin(φ) define the edge of the right peripheral region as shown
in Figure 4.2 and θmin varies with φ. Similarly, by integrating Eq. (4.5) separately,
we can get the relationships between ray direction in spherical coordinate (θ, φ) and
its target position (x, y)
1y f (4.7)
11 |
y fhx
(4.8)
Because the ray angle in the central region is relatively narrow, a refractive
surface is sufficient to achieve uniform illumination over the illumination target. In
order to collect the light from peripheral regions, a TIR surface is most commonly
used, Figure 4.3 is a commonly used example of LED lens. The next two sections
will discuss how to design each surface using composite ray mapping method.
4.2.2 FREEFORM SURFACE DESIGN
Assume the incident position A(x, y, z) of the red ray in Figure 4.3 is known,
we can derive its target position C(xd, yd, H) using Eqs. (4.3) and (4.5). Its position
B at the plane surface can be determined by Fermat’s principle and Snell’s law.
According to Snell’s law at point B, we have:
ˆn k BC AB
zBC AB
(4.9)
104
where n is the refractive index of the freeform lens material, z is the normal vector
on the plane, and k is a constant. AB , BC , and z are located in the same plane
(the detailed proof for this equation is provided in [70]), which means that the
projections of the two vectors 𝐴𝐵 , 𝐵𝐶 onto the x-y plane is aligned on the same
straight line. Thus we can simplify Eq. (4.9) as follows
, ,d r d r r rx x y y t x x y y (4.10)
where t is a dimensionless scale factor. The optical path length (OPL) between point
A and C is:
OPL n AB BC (4.11)
By substituting Eq.(4.10) into Eq. (4.11), it is clear that OPL is a function of t.
According to Fermat’s principle, OPL has to be a minimum value, we have
0OPL
t
(4.12)
By solving the Eqs. (4.10), (4.11) and (4.12), we can derive the position of ray
intersection position B at the plane surface.
105
Figure 4.3 Ray trace of TIR freeform lens in 2D diagram
As shown in Figure 4.4, the target plane is sampled into a non-uniform
rectangular array, Ti, j is the (ith, jth) point on the target plane. The corresponding
point on the freeform surface is Pi, j (u(i), v(j), r(i, j)) and the intersection point at
the plane surface is Ri, j (xr, yr, h), where h is the distance between original point O
and the front plane surface of the freeform lens. If the coordinates of the point Pi, j
PMMA
Air
Air
Target Plane
OX-Axis
Z-A
xis 1 2
C(xd, yd, )
B(xr, yr, ℎ)
A( , , )
3
4
106
on curve C(j) are known, the normal vector N at point Pi, j can be calculated as
following steps:
| |In OA OA (4.13)
, , , , i j i j i j i j Out R P R P (4.14)
n n N Out In Out In (4.15)
where In is the unit incident vector into the freeform surface, Out is the unit
refractive vector out of the freeform surface, and n is the refractive index. The radial
coordinate r(i, j+1) of Pi, j+1 can be derived by following formula
, 1 ,( ) 0i j i j P P N (4.16)
This is the simplified iteration method (Euler method) to calculate ray positions
based on the normal vector field. We adopt a more accurate modified Euler method
to improve the construction accuracy (See Appendix II in [70]).
To derive the freeform surface using the above iterations, we will need an
initial curve C(1). This initial curve can be created with a start point P1,1. The other
points in the curve can be derived using Eq. (4.16). Using the same approach, we
can acquire all of the curves on the freeform surface. The smooth freeform surface
can be constructed by integrating all curves [71].
107
Figure 4.4 Diagram for freeform surface construction: (a) central refractive surface; (b)
peripheral TIR surface
Ti, j+1
Ti, j
z
xy
N
Ri, j+1
Ri, j
Pi, j+1
Pi, j
O
In
Out
C(j)
C(j+1)
Front plane surface of
freeform lens
Points on central freeform surface
Ri, j+1
Ri, j
Ti, j+1
Ti, j
Front plane surface
z
y
x
Pi, j+1
Pi, j
N
O
Target plane
Target plane
Qi, j+1
Qi, j
Inner plane
surface
Points on TIR
freeform surface
(a)
(b)
108
To calculate the TIR surface, we assume that the 4 inner surfaces (labeled
as 4 in Figure 4.3) are vertical planes for simplicity of calculation. As shown in
Figure 4.4 (b), the design procedure of the TIR surface (labeled as 2 in Figure 4.3)
is similar to the construction of central freeform refractive surface (labeled as 1 in
Figure 4.3). Because rays are reflected by the TIR surface, the refractive index in
Equation (4.15) is changed to 1. For a given ray through Pi, j on the TIR surface, the
intersection point between the ray and the inner surface is Qi, j. The unit incident
vector should be as follows:
, , , , i j i j i j i j In P Q P Q (4.17)
The coordinates of point Qi, j can be obtained through Snell’s law and Fermat’s
principle.
4.2.3 DESIGN EXAMPLE
Figure 4.5 TIR freeform lens model
109
To demonstrate the feasibility of the composite ray mapping method, we
have designed a TIR based freeform lens for a 550 mm x 550 mm square
illumination target. The distance between the LED source and the target is 800 mm
and the thickness of our freeform lens is 20mm. The LED chip is 1 mm 1 mm.
The Lambertian intensity distribution of LED can be expressed as
0 0cos cos sin ,I I I v u (4.18)
where θ, u, v are the polar coordinates as we have defined above. By substituting
Eq. (4.18) into Eqs. (4.1), (4.3), (4.4), (4.7), and (4.8), we can build the one-to-one
corresponding relationship between ray emitting direction (u, v) or (θ, φ) and ray
position on the target plane (xd, yd, H). The final design is shown in Figure 4.5. The
overall size is about 45mm × 45mm in transverse section. To evaluate the
performance we use RSD as defined in Eq.(2.6) to quantify the uniformity of the
irradiance distribution on the illumination plane.
110
Figure 4.6 Simulation result (illumination distribution and line chart) for TIR freeform
lens: (a) trace rays through the central freeform surface; (b) trace rays through the TIR
surface; (c) trace all rays through the total TIR freeform lens.
Figure 4.6 is the simulation result after 5 million rays are traced in
LightTools [92]. Figure 4.6 (a) is the simulation result when the light only from the
central region of the light source is traced through the central refractive surface,
and RSD is estimated as 11.4%. Figure 4.6 (b) is simulation result when only the
peripheral light from the light source is traced through TIR surfaces, and RSD is
Illu
min
ance/L
ux
0
1
-600 600Position/mm
0
1
-600 600-600
X/mm
Y/m
m
Illu
min
an
ce
0
1
-600 600
Position/mm0
1
-600 600
-600
600
X/mm
Y/m
m
Illu
min
an
ce
/Lu
x0
1
-600 600Position/mm
0
1
-600 600
-600
600
X/mm
Y/m
m
(a)
(b)
(c)
600
111
6.6%. When all rays from the light source is traced through the TIR freeform lens,
RSD is 8.0% as shown in Figure 4.6 (c). In the simulations, LED luminous flux is
100 lm and five million rays are traced to offset the effect of statistical error. The
total energy on the illumination target is approximately 90% (with Fresnel loss) of
the total energy emitted by the LED source.
4.2.4 ROBUSTNESS ANALYSIS
The current design has a number of advantages over the other freeform
illumination lenses because of the overlapping illumination mechanism. First, this
lens is less sensitive to the position of LED chip. Figure 4.7 studies the impact of
LED misalignment on the illumination performance for the TIR freeform in Figure
4.5 and regular refractive freeform lens [8]. Both RSD and light collection
efficiency are much better than the refractive freeform lens designed in [8], the
current lens still has reasonable performance even when LED chip is offset by ±1
mm as shown in Figure 4.7 (b). It can be expected that they are better than other
LED lenses without using the overlapping illumination mechanism. A LED lens
which is less insensitive to the LED chip position will improve the productivity, as
well as assemble time and cost.
112
Figure 4.7 Effect of LED’s shift on RSD and collection efficiency for: (a) regular
refractive freeform lens; (b) TIR freeform lens.
Most of LED lenses are designed with prescribed intensity distribution from
LEDs, for example Lambertian distribution where the intensity distribution is
proportional to cos(θ). However, the actual intensity distribution may be different
from the designed values. Figure 8 studies the sensitivity of the TIR freeform lens
to the intensity distribution. Both the current TIR freeform lens and regular
refractive lens are designed for the LED with Lambertian intensity distribution.
When the intensity distribution is not Lambertian, for example, cosm (θ) with m
(a)
(b)
113
ranging from 0.2 to 4.0, illumination uniformity and light efficiency are shown in
Figure 4.8. It is clear that the TIR freeform lens is much less sensitive to the
variation of LED intensity distribution. The light collection efficiency is almost
uniform and RSD only changes slightly with m from 0.2 to 4.0 as shown in Figure
4.8 (b).
Another key advantage of the proposed freeform TIR lens is that it is much
less sensitive to the manufacturing errors, for example the local slope, because each
point in the illumination target is illuminated by the light from three different
surfaces. For other LED lenses without overlapping mechanism, any defect in the
surface will be magnified in the illumination target.
114
Figure 4.8 (a) Actual LED’s intensity distributions based various power index m; (b) the
effect of variations in power index m on the illumination performance for both TIR
freeform lens (TIR) and general freeform lens (general) with regards to RSD and
collection efficiency.
(a)
(b)
115
4.3 MULTI-XY RAY MAPPING IN TIR LENS DESIGN
4.3.1 COMPOSITE RAY MAPPING METHOD WITH MULTIPLE X-Y
RAY MAPPING MECHANISM
Figure 4.9 Ray mapping mechanism for multi-xy mapping method
Obviously, the single x-y ray mapping method is unable to collect all rays
from a source whose emitting rays cover the whole hemisphere. A composite ray
mapping method with multiple x-y ray mapping is proposed to construct the
freeform optics. As shown in Figure 4.9, the source intensity distribution is divided
into five different regions with five collection surfaces: one upper surface S0, and
116
four side surfaces S1, S2, S3 and S4. To minimize the bending angle of light rays, the
light rays collected by central upper surface S0 are mapped to the whole target
surface, while the light rays from all other 4 side surfaces are mapped to half target
surface. As a result, each grid on the illumination target corresponds to three
different source intensity distribution grids. Let I0, I1, I2, I3, I4 represent the intensity
distribution of each collection surface in the source space, and Et0, Et1, Et2, Et3, Et4
denote the irradiance distribution on the illumination target when only one
corresponding collection surface is permitted to trace rays. According to the energy
conservation principle, we can have the following relationships:
, , , , , 0,1,2,3,4
i
i i i i i i i ti t t t t t t
D
I x y J x y dx dy E x y J x y dx dy i
(4.19)
where i is the solid angle subtended by the ith collection surface, D is the total
target surface. By integrating the surface on both sides and separating variables, we
can derive the mapping relationships between (xi, yi) and (xt, yt) based on Eq. (2.21)
separately.
The ray trace for the freeform TIR lens design is shown in Figure 4.10.
Obviously, ray trace and the surface construction in multi-xy ray mapping are
similar to the composite ray mapping method we have proposed in [70]. All
parameters for the freeform TIR lens are illustrated in Figure 4.10. The central
collection surface area is l2 × l2, and the corresponding illuminatin targt is l3 × l3.
The side collection surface area is l2 × l1, the corresponding illumination target is l3
× (l3/2). In our mapping scheme, the rays collected by the central surface are
redirected to cover the whole target surface, and the rays collected by the side
117
surface are redirected to cover the half of the target surface. As a result, rays from
three lens surfaces will overlap at the same point on the target surface.
Figure 4.10 Ray trace of freeform TIR lens design in multi-xy ray mapping method
4.3.2 FREEFORM SURFACE CONSTRUCTION
Based on our composite ray mapping mechanism between the source and
target, 5 freeform surfaces including 1 freeform refractive surface and 4 freeform
TIR surfaces can be constructed by geometrical method to generate the prescribed
irradiance pattern on the target surface.
118
Figure 4.11 Ray trace diagram and freeform surface construction: (a) central freeform
refractive surface; (b) peripheral freeform TIR surface.
Pi, j+1
N
S
x
y
zTarget plane
Ti, j Ti, j+1
Points on freeform
surface
Pi, j
Ri, j
Ri, j+1 Incident flat
surface
2X
2Y
h1
h3
H
l2
l2
S
x
y
zTarget plane
Ti, j Ti, j+1
2X
2Y
h1
H
l2
h2
Pi, j
Pi, j+1
Ri, j
Ri, j+1
Qi, j+1 Qi, j
Points on freeform
TIR surface
Incident flat
surface
Rim flat surface
N
(a)
(b)
119
The ray trace diagram for the rays through the central refractive surface is
shown in Figure 4.11(a). The unit incident vector In into the freeform refractive
surface is the normalized SRi,j, and the unit output vector Out away from the
freeform refractive surface is the normalization vector of Pi,jTi,j. By using the
Snell’s law, we can derive the normal vector N at the known point Pi,j as follows:
2 1/2[1 2 ( )]n n n , Out In N Out In (4.20)
where n is the refractive index of the lens material. Now the neighbor point Pi,j+1
can be derived based on the normal vector field and geometrical ray tracing through
the following equations:
, , 1 0.i j i j P P N (4.21)
, 1 , 1 , 1
, 1 , 1 , 1
.i j i j i j
i j i j i j
z zn
z z
SR R P
SR R P (4.22)
Detailed calculation procedures and more accurate iteration algorithms can be
found in [70, 79]. When all points are derived, the NURBS surface is used to fit the
point cloud for entity construction [71].
While for the TIR surface, each single ray experiences one reflection on the
TIR surface and two refractions between the source and target. As shown in Figure
4.11(b), the unit incident vector In into the TIR surface is specified by the
normalization vector of SRi,j, and the unit output vector Out away from the TIR
surface is evaluated by the normalization vector of Pi,jQi,j. According to the
reflection law, the normal vector N at the point Pi,j on the TIR surface can be
expressed as:
120
.
Out InN
Out In (4.23)
For the refraction on the incident flat surface and exit flat surface, the normal
vectors on these two surfaces are y and z separately, where we can continue to
trace the rays using Snell’s law through Eq. (4.22).
4.3.3 DESIGN EXAMPLES AND RESULT ANALYSIS
In this section, we discuss the design of a TIR based freeform lens for square
illumination target based on the composite x-y ray mapping method. The LED has
a dimension of 1mm 1mm and has a Lambertian intensity distribution. The other
design parameters for the lens dimension and illumination target requirements are
listed in Table 4.1. All the symbols representing the parameters are consistent with
what we have labeled in Figure 4.10 and Figure 4.11.
Table 4.1 Design parameters for the freeform TIR lens construction
(Unit: mm)
l1 l2 l3 h1 h2 h3 H
30 10 450 5 19 25 1000
Now by substituting the intensity distribution function into Eqs. (2.19),
(2.20), and (4.20) separately, we can find the one-to-one mapping relationship
between target position and source emitting direction for each region. With the
mapping mechanism, the freeform refractive surface and freeform TIR surface can
be derived following the algorithm introduced above. The designed freeform TIR
lens model is shown in Figure 4.12 (a).
121
Figure 4.12 (a) TIR lens model based on composite ray mapping; and (b) simulation
result
The irradiance pattern with 5 million rays traced are shown in Figure 4.12(b).
RSD of the irradiance distribution is about 0.15 and the collection efficiency with
Fresnel loss considered is about 93%, which is higher than most of freeform lenses
with only single refractive surface.
4.4 DESIGN FREEFORM TIR LENS (ARRAY) USING
INTEGRAL RAY MAPPING METHOD
In previous two sections, we have introduced two different mapping
mechanisms to design the freeform TIR lenses for its two different structures. As
we can see from the design example, the overlapping mechanism has significantly
improved the irradiance uniformity and the robustness to mounting tolerance and
manufacturing errors. To further improve the performance, we introduce integral
ray mapping method. In this approach, freeform lens array is used to sample the
22 mm
55 mm
(a) (b)
122
source emission. Then we redistribute the source rays incident to each single lenslet
onto the whole target surface. As a result, the final irradiance on the target surface
is the integrated illumination from all the lenslets of the lens array. As an example,
we use a Fresnel TIR collimator to redirect source rays into the sampling surface
with freeform lens array. In this section, we will apply the integral ray mapping
mechanism to design the freeform TIR lens with basic structure shown in Figure
4.1(c): firstly, a Fresnel TIR lens is designed to collimate the rays emitted from the
LED [93, 94], then a freeform lens array for the collimated light is designed to
generate various irradiance pattern on the illumination target [95].
4.4.1 FRESNEL TIR COLLIMATOR DESIGN
The basic cross-section of Fresnel TIR collimator is shown in Figure 4.13.
The lens is divided into two different parts: one is Fresnel TIR surface whose
collection zenith angle ranges from 30 to 90 and the other part is the general
refractive surface with collection zenith angle ranging from 0 to 30 . As seen from
the figure, all rays are directly refracted to a bundle of collimated rays by the
refraction surface. For a given point on the central refractive surface, using Snell’s
law we have
sin sinair glassn i n r (4.24)
RLi r (4.25)
r (4.26)
where α is the tangent angle at the given point, i is the incident angle, r is the
refractive angle and ωRL is the emission zenith angle of the light ray. With the value
123
of the slope, we can derive the neighboring points using Newton’s iteration
algorithm and then reconstruct the whole profile of the surface.
Figure 4.13 Structure and geometrical layout of Fresnel TIR collimator
For the TIR surface part, the rays are firstly refracted by a vertical cylinder
surface and then reflected to the collimation beam by the TIR surface. For the
refraction on the first cylinder surface, we can obtain using Snell’s law:
sin sin ,2 2
air TIR glassn n
(4.27)
where ωTIR is the emission zenith angle of the light ray incident to the TIR surface.
For the reflection at the TIR surface, we can obtain using geometrical relationships:
2 180 (4.28)
Then the slope angle of the TIR surface at that point is expressed as:
. (4.29)
Similarly, with the information of the slope of the specific point, we can construct
the whole TIR surface profile using Newton’s iteration algorithm.
εϕ
iωRL
ωTIR
θr
nair
nglassTangent Line
Horizontal Line
LED
h
α
124
4.4.2 FREEFORM LENS ARRAY DESIGN
As shown in Figure 4.14, each lenslet redistribute the collimated light
uniformly to the target surface. When the target surface is large enough, the
freeform lens is relatively small compared to the target surface, each lenslet can be
identical. The irradiance on the target surface is the integrated irradiance produced
by all the lenslets of the lens array.
Figure 4.14 Schematic diagram for the working process of single micro lenslet
The radiation transmitted through each lenslet can be sampled into numbers
of rectangular grids as shown in Figure 4.15. By using the x-y ray mapping
mechanism we have introduced in previous chapter, we can build the corresponding
relationship between the ray intersection position on the lenslet surface and the ray
position on the target surface. The only difference here is that the intensity
distribution projected to the x-y plane is closely constant. For each specific point on
125
the lenslet surface, the incident vector and output vector of light ray can be
expressed as follows:
(0 ,0 ,1 )In i j k (4.30)
i, j i, j
i, j i, j
P QOut
P Q (4.31)
The normal vector field can be obtained by Eq.(4.20). With normal vector field,
then the freeform lenslet surface can be constructed using Newton’s iteration
algorithm and Spline interpolation algorithm [70, 71].
Figure 4.15 Schematic of ray mapping mechanism for single lenslet [95]
4.4.3 DESIGN EXAMPLE OF FRESNEL TIR FREEFORM LENS FOR
SQUARE ILLUMINATION
126
To demonstrate the feasibility of the integral ray mapping method, we
design a lens array with Fresnel TIR freeform structure to generate square
illumination as a starting point. The target is set to be 1000 mm × 1000 mm, and is
1000 mm from the source. We first design a Fresnel TIR collimator to achieve a
collimated beam with relatively uniform irradiance distribution. The lens model for
the collimator with different views and related ray tracing diagrams for the
collimator are shown in Figure 4.16. The Fresnel TIR collimator has a good
performance in collimating the LED beams, the divergence of the light from this
collimator is less than 3°.
Figure 4.16 Diagrams for Fresnel TIR collimator: (a) bottom view of Fresnel TIR surface
of the collimator; (b) wireframe side view; (c) ray tracing diagram.
127
After the LED collimator is built, we design the freeform lens array to
redistribute the collimated beam into a square illumination pattern on the specific
target surface using the integral ray mapping method introduced above. Both the
“rendered” view and “ghost” view of the lens model are shown in Figure 4.17. Each
lenslet of the lens array will project all its incident rays onto the entire target. Thus
the final irradiance distribution on the target surface is the integrated irradiance
distribution of all lenslet. The irradiance distribution on the target surface is show
in Figure 4.18. As a result, this freeform illumination system can reach relatively
high irradiance uniformity, and RSD for the irradiance distribution is as low as 3%.
Figure 4.17 Fresnel TIR freeform lens array model: (a) rendered view in Rhino; (b) ghost
view in Rhino
128
Figure 4.18 Simulation results for the Fresnel TIR freeform lens array for square
illumination pattern: (a) irradiance distribution on the target surface; (b) irradiance
distribution cross-section profile on both direction.
4.4.4 FREEFORM TIR LENS ARRAYS FOR OTHER PATTERNS
Figure 4.19 Fresnel TIR lens with hexagon lens array: (a) lens model; (b) related
irradiance distribution
129
Using the same principle, we can generate uniform illumination for targets
with different shapes. Figure 4.19 shows the lens model of TIR lens combined with
a freeform hexagon lens array and related irradiance pattern on the target surface.
Another very interesting example is to tailor the lenslet into a composite
piece with cross-hair structure and square structure. The square part of the lenslet
provides the background illumination. A design example for generating the cross-
hair structure illumination pattern is as show in Figure 4.20.
Figure 4.20 Fresnel TIR lens with cross-hair lens array: (a) diagram for single lenslet; (b)
diagram for TIR lens combined with cross-hair lens array; (c) illumination pattern for the
TIR lens array system.
(a) (b)
(c)
130
4.5 CONCLUSIONS AND DISCUSSIONS
In this chapter, we have proposed composite ray mapping method to design
the freeform TIR lens or lens array system for non-rotational illumination. In this
method, we sample the source intensity distribution in multiple coordinate systems
and overlap together the rays from different source regions on the target surface.
With overlapping mechanism, we can greatly improve the irradiance uniformity
and robustness for fabrication error, lens mounting deviations, and source defects.
131
CHAPTER 5
SUBSEQUENT DESIGN METHODS FOR
EXTENDED SOURCES
We have proposed several ray mapping methods for designing freeform
optics for ideal point sources. However, most of the light sources are extended
sources with certain dimensions. The non-ideal source will degrade the illumination
performance predicted with the ideal point source, therefore it is necessary to
minimize the degradation in the final design stage. Generally, the commonly used
approach in the final design stage for extended sources is the trial and error
approach. In direct ray mapping method, feedback modification is often used to
improve the system performance. To improve the efficiency of feedback
modification, a number of optimization methods have been developed and
incorporated in ray tracing software. In this chapter, a brief introduction to feedback
modification and optimization approaches will be provided. Section 5.1 will
discuss the basic design procedures of optimization method, and Section 5.2 will
describe the method of using the feedback modification method to continuously
improve the designs with a simple example. In Section 5.3, a new design method
based on deconvolution to deal with the issue of extended sources will be
introduced.
132
5.1 OPTIMIZATION
Optimization is the most commonly used design method for illumination
systems with extended sources or complex design targets [96-98]. In this section,
we will systematically introduce the general procedures in using optimization
approach to design efficient illumination systems.
5.1.1 GENERAL DESIGN PROCEDURES FOR OPTIMIZATION
METHOD
The optimization method is aimed to improve the performance of final
designs. The first step is to generate a good initial design based on traditional
freeform optics design methods discussed in the previous chapters. The second step
is to establish the merit function based on the system requirement and set the
variables. The merit function for the illumination optics design is generally
expressed as:
1
1,
N
i i ti
i
MF w E EN
(5.1)
where Ei is the actual irradiance value at the ith grid on the target surface, Eti is the
target irradiance at the ith grid on the target surface and wi is the related weight
factor. The final step is to apply optimization algorithms to minimize the merit
function. There exist many optimization algorithms such as simplex algorithm [99,
100], Gauss-Newton algorithm [101, 102], and Least-Square algorithm [103]. The
design flow chart for the optimization method can be summarized in Figure 5.1.
133
Sometimes, it is necessary to modify merit function based on the intermediate
results.
Figure 5.1 Flow chart of optimization method [98]
5.1.2 DECRIPTION OF FREEFORM OPTICAL SURFACES
One of the key tasks in the optimization method is to parameterize the
freeform surfaces and thus determine the variables. A number of surface types, such
as XY polynomial surfaces [104], Zernike polynomial surfaces [105], and Non-
uniform rational B-spline (NURBS) surfaces [71], have been used to parameterize
freeform surfaces. This subsection will briefly introduce the above three commonly
used surface representations.
Specify design targets and
constraints
Output
Initial design
Surface fitting and
determine variables
Optimization
Desired result?
Y
N
134
5.1.2.1 XY POLYNOMIAL SURFACE
The mathematical definition of XY polynomial surface is expressed as [105]:
2
2 22
,1 1 (1 )
Nm n
i
i
crz A x y
k c r
(5.2)
where r2=x2+y2, k is the conic constant, c is the base surface curvature, Ai is the
coefficient of the monomial xmyn with i= [(m+n)2+m+3n]/2+1. The XY polynomial
surface is the most commonly used type of non-symmetric representation for
freeform surfaces.
5.1.2.2 ZERNIKE POLYNOMIAL SURFACE
The Zernike polynomial surface is another commonly used representation
to describe the freeform surfaces [106]. Zernike polynomials are a sequence of
polynomials that are orthogonal on the unit circle. The even Zernike polynomial
terms are defined as:
( , ) ( )cos( )m m
n nZ R m (5.3)
and the odd ones are defined as:
( , ) ( )sin( ),m m
n nZ R m (5.4)
where m and n are nonnegative integers with n ≥ m, φ is the azimuthal angle, ρ is
the radial distance 0≤ ρ ≤1, and Rmn are the radial polynomials. The Rm
n are defined
as follows:
2
2
0
( 1) ( )!( ) .
! ! !2 2
n m
km n k
n
k
n kR
n m n mk k k
(5.5)
135
5.1.2.3 NURBS SURFACE
NURBS is considered the most powerful and popular surface representation
in the CAD industry [107] and has broad applications in freeform optics [108, 109].
The mathematical description of NURBS surface is a vector valued function of two
parameters, u and v, and a NURBS surface with pth degree in the u direction and qth
degree in the v direction can be specified as:
,, ,
0 0
, ,
0 0
( ) ( )
( , ) (0 , 1),
( ) ( )
n m
i jij i p j q
i j
n m
ij i p j q
i j
w N u N v P
S u v u v
w N u N v
(5.6)
Here ,i jP are the control points which form the control polygon. Ni,p and Nj,q are
the pth-degree B-spline basis functions and qth-degree B-spline basis functions
defined on the knot vector 1 1
1 1
0,...0, ,..., ,1,...,1p r p
p p
U u u
and
1 1
1 1
0,...0, ,..., ,1,...,1q s q
q q
V v v
, where r=n+p+1 and s=m+q+1. The expression
for ith B-spline basis function of pth-degree is provided as follows:
1
,0
1
, , 1 1, 1
1 1
1( )
0.
( ) ( ) ( )
i i
i
i pii p i p i p
i p i i p i
u u uN u
otherwise
u uu uN u N u N u
u u u u
(5.7)
More detailed information on NURBS surfaces can be found in [71].
5.2 FEEDBACK MODIFICATION METHOD
136
5.2.1 FEEDBACK MODIFICATION PRINCIPLE
The freeform surfaces designed with ideal point sources and ray mapping
methods typically don’t satisfy the integrability condition strictly, and there are
significant deviations in performance for the extended sources. In order to minimize
the surface error as much as possible and realize the prescribed irradiance
distribution for extended sources, a feedback modification approach [110-112] is
often employed to finalize the freeform optical surface by manually modifying the
target irradiance distribution according to the simulated irradiance distribution. The
feedback function ( , )x y can be defined as
,0 , , 1, , / 1 , 1,2, , ; 1,2, , ,q
k s s k s ki j E x y p E p E i m j n (5.8)
where p (0≤p≤1) and q (q≥0) are the weight parameters, and Es,k is the kth simulated
irradiance distribution on the target surface. With t times of iterations, the modified
target irradiance distribution for next iteration of design can be expressed as
0
1
, , , , 1,2, , ; 1,2, , ,t
k l
l
E i j i j E i j i m j n
(5.9)
where E0(i, j) is also the prescribed irradiance distribution on the target surface [68].
5.2.2 A SIMPLE EXAMPLE WITH FEEDBACK MODIFICATION
The freeform reflector with uncrossed mapping and half FOV of 120° that
we have designed in Chapter 3 is used as an example to demonstrate the feasibility
of feedback modification method. As shown in the Figure 5.2, RSD of the
137
irradiance distribution on the target has been improved from 0.085 to 0.076 after
only 3 iterations of feedback modification to the design for a LED chip with size of
2mm × 2mm.
Figure 5.2 Feedback modification for the designed laser beam shaper.
5.3 DECONVOLUTION METHOD
The freeform reflectors has been developed to generate periodic illumination
patterns on the target surface [113]. The optical system with lens array can be approximated
as a source shift invariant optical system, therefore the pattern intensity can be estimated
as the superposition of the source intensity convolved with the lenslet intensity response
[114, 115]. For lens array systems design with ideal collimated incident beam, the non-
ideal collimated input beam blurs the sharpness of illumination pattern. For systems
designed with point source, the size of the real light source performs as the blur response
to the final illumination pattern. In this section, we will discuss how to design a freeform
RSD=0.085 RSD=0.081
RSD=0.076RSD=0.079
Initial Design Iteration One
Iteration Two Iteration Three
138
lens array to generate a sinusoidal illumination pattern using the blur response caused by
the source size. Since the illumination target is periodic fringe patterns, the freeform
illumination system can be approximated as a shift invariant system. We can use
convolution method to analyze the effect of extended source and use the deconvolution
approach to design the prescribed illumination patterns for extended light sources.
5.3.1 MECHANISM OF DECONVOLUTION METHOD
Figure 5.3 shows the layout of freeform lens array for structured light illumination.
The shift invariant approximation is assumed that the extended source is discretized to a
series of point sources and each point source has same response with shift in the
illumination plane. Under this approximation, the illumination pattern e(x, y) is the
superposition of the response p(x, y) of the point source along the extended source plane:
( , ) ( , ) ( )e x y p x y s x, y , (5.10)
where ⊗ represents the convolution operator and s(x, y) is the blur response of the extended
light source.
139
Figure 5.3 The geometrical layout of freeform lens array for structured light illumination.
Assume the shift of point source response profile in the illumination plane is –A/2
when the point source is shifted from the center to the edge with a distance of a/2 in x
direction, the blur response of the source size in the illumination plane is a rectangular
function with a window width of A. Based on the paraxial approximation, the width A is
estimated as follows:
3 2
1 2 1
,( ) /
H HA a
H H H n
(5.11)
where n is the refractive index of the freeform lens. The blur response of the
extended light source can be adjusted by changing at least one of parameters, the
source distance H1, thickness (H2-H1) of the lens array, and the illumination
distance (H3-H2).
When the freeform lens is designed with a point source to meet the
prescribed illumination profile, the final illumination pattern will be modified by
the blur response of the real extended light source. To design the freeform lens
which can achieve the prescribed illumination profile for the extended light source,
Extended
source
Freeform
lens array
Illu
min
atio
n
pla
ne
H1
H2
H3
a A
140
we reconstruct the response profile of the point source by taking the blur response
of the extended light source into consideration through deconvolution approach.
Based on Equation (5.10), the new point response function can be obtained as:
1( , ) ( , ) ( , ),p x y e x y s x y (5.12)
where ⊗-1 represents the deconvolution operator. The deconvolution operation can
be implemented in the frequency space. According to the convolution theorem of
Fourier transform, Equation (5.10) can be rewritten as follows:
( , ) ( , ) ( , ),E u v P u v S u v (5.13)
where E(u, v) is spatial frequency distribution for prescribed illumination pattern,
P(u, v) is the spatial frequency distribution for the theoretical point response of the
point source, and S(u, v) is the degradation function for non-zero étendue source
(extended light source). Therefore, the deconvolution in Fourier domain can be
simplified to calculate the spatial frequency distribution of point response pattern
according to the following formula:
( , )
( , ) .( , )
E u vP u v
S u v (5.14)
We can then reconstruct the point response function using the inverse Fourier
transform:
-1( , ) ( , ) ,p x y F P u v (5.15)
where F-1 is operator of inverse Fourier transform. This p(x, y) will be the actual
illumination target for designing the freeform lens with a point source to achieve
the prescribed illumination pattern e(x, y) for the extended source.
141
5.3.2 FREEFORM LENS ARRAY DESIGN AND ANALYSIS
To demonstrate the feasibility of the proposed deconvolution method, we
design a freeform lens array shown in Figure 5.4 to generate a sinusoidal fringe
pattern with the period of T based on a Lambertian LED source. The design
parameters are listed in Table 5.1, a is the size of LED chip， the lens material is
PMMA. The width of the blur response of the LED chip is set to be T/2, half of the
fringe period to avoid the overlapping between the adjacent fringes. The efficient
collection area on the bottom plane surface is rectangular with size of 2H1 × 2H1.
Using Equation (5.12), the target response of the point source in the illumination
plane is estimated and plotted in Figure 5.5 (c). Figures 5.5 (a) and 5.5 (b) are the
target sinusoidal illumination pattern and blur response of the extended light source.
142
Figure 5.4 Ray tracing of the freeform lens array.
Table 5.1. Design parameters (unit: mm)
H1 H2 H3 T a
20 40 450 25 1
Figure 5.5 (a) Target sinusoidal pattern, (b) the blur response of the extended light source,
and (c) the target response of the point source.
x
y
z
O
A(xs, ys, H1)
B(x, y, z)
C(xt, yt, H3)
N
Plane surface
Freeform
surface
Illumination
plane
x
y
H1
H2
H3
In
Out
=⊗-1
(a) (b) (c)
-12.5 -7.5 -2.5 2.5 7.5 12.5X/mm
-12.5 -7.5 -2.5 2.5 7.5 12.5X/mm
-12.5 -7.5 -2.5 2.5 7.5 12.5X/mm
143
With the target response of the point source, we will be able to design the freeform
lens array to achieve the prescribed sinusoidal fringe patterns as discussed above. We
design the freeform lens by tailoring the intensity distribution on the incident surface with
a rectangular border and map it to the rectangular target with periodic irradiance pattern as
shown in Figure 5.4. The design of freeform lens array also follows the energy
conservation principle of the mapping process:
0 , ,, ,
i i
s s
D
I d E x y J x y dxdyx y
(5.16)
where 𝛺𝑖 is the solid angle defined by the effective rectangular incident surface of
the ith ribbon of the freeform lens, Di is the rectangular target containing the ith
periodic fringes, and I(xs, ys) is the source’s intensity at any point A on the incident
surface. Through integrating two variables of the 2D integrals in Equation (5.16)
separately, we can establish the relationships of the emitting ray direction (xs, ys)
from the light source and the prescribed ray position (xt, yt) on the illumination
plane. Based on the mapping relationship between source and target, we can obtain
the ray position C(xt, yt, H3) for any specific ray direction. The freeform surface
shown in Figure 5.6 is constructed by using iteration algorithms and spline surface
fitting algorithms [70, 71]. For comparison, we also design another freeform lens
array using traditional x-y ray mapping method without considering the blur
response of extended sources. The second lens is completely same as the designed
one with deconvolution method. The only difference is that it is designed for a point
source with exact sinusoidal fringe pattern.
144
Figure 5.6 Ray tracing of the freeform lens array to generate sinusoidal fringe pattern.
We perform ray tracing with 20 million rays in LightTools for two designs
separately. Figure 5.7 (a) shows the irradiance pattern of the freeform lens designed with
deconvolution approach, while Figure 5.7 (b) is the pattern of the lens designed with
traditional ray mapping method. Figure 5.7 (c) shows the normalized line profiles of the
illumination patterns and the prescribed line profile, it is clear that the design using
deconvolution approach meets the design requirement and has a much higher fringe
contrast. The relative RMS for absolute difference between simulated profile and
prescribed theoretical sinusoidal profile is only 2.5%. The fringe contrast in Figure 5.7 (a)
is 97%, compared to the 62% in Figure 5.7 (b), demonstrating the proposed method is able
to design freeform lens array for precise structured light illumination with the consideration
of extended light source.
(a) (b)
145
Figure 5.7 Simulation results for a 1x1 mm LED chip. (a) The irradiance pattern of the
freeform lens array which is designed using deconvolution approach, (b) the irradiance
pattern of the design without deconvolution, and (c) normalized line profiles for the
irradiance patterns of both designs.
5.4 DISCUSSION
In this chapter, we have reviewed the two traditional design methods of
optimization and feedback modification to improve the illumination performance.
We also introduce the deconvolution method in designing freeform optics and
demonstrate this method in designing sinusoidal fringe patterns with freeform
(a)
-40 -20 0 20 400
0.5
1
X/mm
(c)
Irra
dia
nce
(b)
With Deconv
Prescribed
No Deconv
146
reflector. The method works well in designing optics for illumination pattern with
periodic properties and it can improve the contrast of structure illumination pattern.
However, the deconvolution method does not work well for uniform illumination
pattern because the deconvolved result of uniform function is a series of δ functions,
leading to number of discontinuities on the freeform surfaces.
147
CHAPTER 6
SUMMARY AND FUTURE WORK
6.1 SUMMARY
In this dissertation, we have developed several novel ray mapping methodologies
in designing freeform optical surfaces for non-imaging applications.
In Chapter 1, the background of the illumination engineering, freeform optics as
well as general non-imaging optics design methods, is presented. Besides, the introduction
to the basis of the geometrical optics principles in non-imaging optics is provided.
In Chapter 2, a review of several traditional ray mapping methods, such as θ-φ ray
mapping method, u-v ray mapping method, and x-y ray mapping method, are discussed
with detailed design procedures and related design examples. We have also reviewed the
limitations of general ray mapping methods and compared the performance of three
different ray mapping methods.
In Chapter 3, we propose a composite ray mapping method for designing the
freeform TIR lens systems. We have developed three different methods to design the TIR
lens based on the different structures. Both of the irradiance uniformity and robustness have
been improved greatly due to the overlapping mechanism.
In Chapter 4, we first introduce the double-pole coordinate system into the non-
imaging optics design. We compare the performance of θ-φ ray mapping, u-v ray mapping
and double-pole ray mapping in designing freeform lenses for rectangular illumination.
148
The proposed double-pole ray mapping method has a number of advantages, for example
better illumination uniformity and less surface errors. We also demonstrate the double-pole
ray mapping method in designing the freeform reflector with super large acceptance angle.
In Chapter 5, we first review two traditional design methods, optimization and
feedback modification method, for improving system performance for extended sources.
We also, for the first time, introduce the deconvolution method in the non-imaging optics
design and demonstrate its superior performance in generating accurate structured
illumination patterns.
6.2 FUTURE WORK
The dissertation has explored many different ray mapping methods in
designing freeform optics for non-imaging optics applications. Most of these
designs are based on the ideal point source and they have some limitations for
extended light sources. The future research should focus on developing efficient
design methods for extended light sources. For example, we will try to develop
more powerful feedback functions to take the feedback modification of designed
illumination systems with fewer iterations.
The major task of this dissertation is focused on the design freeform optics
for rectangular illumination pattern, which is the most basic example for non-
rotational illumination systems. In the future, we will try to apply our geometrical
ray mapping methods to design freeform optics for other freeform illumination
patterns, such as triangle illumination pattern, and other complicate patterns with
different features.
149
A lot of design algorithms for freeform optics generation have been
discussed in this dissertation. Most of the designs are based on theoretical models.
In the future, we will try to apply these design algorithms to design freeform optics
for actual applications. These applications can vary from automotive signal lighting
system design, automotive headlamp design, various types of street lighting design,
and so on.
150
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