INVESTIGATION OF INJECTION MOLDING PROCESS FOR HIGH PRECISION
POLYMER LENS MANUFACTURING
DISSERTATION
Presented in Partial Fulfillment of the Requirements for
the Degree Doctor of Philosophy in the Graduate
School of The Ohio State University
By
Chunning Huang, M.S.
*****
The Ohio State University
2008
Dissertation Committee:
Approved by
Professor Allen Y. Yi, Adviser
Professor Jose M. Castro
Professor L. James Lee
Adviser
Industrial and Systems Engineering
Graduate Program
ABSTRACT
Injection molding polymer optical components have long been used for its high
volume, low cost and lightweight capability over traditional glass optics. However, the
process has not been readily accepted in precision optical fabrication industry because
several difficult issues such as geometry deviation, inhomogeneous index distribution,
birefringence and freeform fabrication have hindered the implementation of injection
molding process in high precision optical applications.
This dissertation research was an attempt to create a methodology for injection
molding process for high precision polymer lens manufacturing. The study included both
experimental approach and numerical modeling in order to identify the proper polymer
lens manufacturing processes. The scope of this research involved in both fundamental
and systematic investigation in optical design, mold and lens fabrication, as well as
optical metrology issues related to polymer lens manufacturing to obtain precision macro
and micro polymer freeform optics with accurate geometry and proper optical
performance by the state-of-the-art mold fabrication and molding technology.
With the aid of DOE (design of experiment) and DEA (data envelopment analysis)
methods, the critical process parameters were narrowed down and the optimal conditions
ii
were determined for lens geometry compensation. The mold compensation methodology
was developed based on advanced freeform measurement and data analysis technology
and STS (slow tool servo) freeform mold fabrication.
The effects of the process
parameters on optical performance such as birefringence, index distribution and surface
scattering were carefully studied by theoretical and empirical analysis.
Due to the
complexity of the injection molding process, single process condition cannot fulfill all the
requirements for lens quality, therefore balanced process parameters need to be selected
as a compromise for desired specifications. Moreover, fabrication of macro Alvarez lens,
micro Alvarez lens array, diffractive lens and Fresnel lens has proven that the advanced
mold fabrication and injection molding process can provide an easy and quick solution
for freeform optics.
In addition, simulation with Moldflow Plastic Insight 6.1 was
implemented to verify the experiment results and the prediction of the simulation results
was validated using experiment results. Experimental results also showed that injection
molding process is capable for precision optics manufacturing with accurate mold
compensation and process control.
iii
Dedicated to my parents
iv
ACKNOWLEDGMENTS
I would like to express my sincere gratitude to my advisor, Professor Allen Yi, for
his guidance and support throughout my PhD study and during the completion of this
dissertation. I have learned a great deal from his intellectual insight and knowledgeable
expertise. It was an invaluable opportunity to work with him and this experience will
enrich me for the rest of my life. I also would like to thank Professor Jose M. Castro,
Professor L. James Lee and Professor Rebecca B. Dupaix for their service and
suggestions on my doctoral committee. I also appreciate the assistance from Professor
Thomas Raasch on SHS (Shack-Hartmann Sensor) and Alvarez lens research. I sincerely
thank Dr. Nelson Claytor for his valuable discussions during a visit to his company,
Fresnel Technologies, Inc. and at CAPCE meetings and for his generous financial support.
I would like to thank the members of Professor Yi’s group, for their suggestions
and help to my dissertation research. Lei Li helped me in ultraprecision machining and
measurement setup. He never hesitated to share with me his invaluable experience. Dr.
Chunhe Zhang helped me with machining setup. Greg Firestone taught me how to use
CMM (Coordinate Measuring Machine) and thermocouples. Thanks also go to Dr.
Anurag Jain, Yang Chen and Lijuan Su for their advice to my research. Special thanks
v
go to Denia R. Coatney for our cooperation on geometry measurement study and
friendship.
I also want to thank for the help from the machine shop supervisors in Department
of Industrial, Welding and Systems Engineering. Bob Miller provided assistance in
setting up the injection molding process which is very important for my research. Mary
Hartzler taught me and allowed me use the machines in the basement.
The financial support from the graduate school and CAPCE of OSU is gracefully
appreciated.
Last, but not the least, I would like to thank my parents, Xuye and Xiuhua, and
my husband, Jianqing, for their encouragement and support. Without their support, I
would not have accomplished what I have.
vi
VITA
April 19, 1977…………………………
Born –Anshan, China
2000…………………………................ B.S. Precision Instruments, Measurement and
Control Technology,
Tsinghua University, Beijing, China
2002…………………………………… M.S. Optical Engineering,
Tsinghua University, Beijing, China
2002 – 2004…………………………… Engineer,
Nuctech Company Limited, Beijing, China
2004 – 2005…………………………… University Fellow,
The Ohio State University
2005 – 2008…………………………… Graduate Fellow,
Center for Advanced Polymer and Composite
Engineering, College of Engineering,
The Ohio State University
PUBLICATIONS
Research Publication
1. L. Li, A. Y. Yi, C. Huang, D. A. Grewell, A. Benatar, and Y. Chen, “Fabrication of
Diffractive Optics by Use of Slow Tool Servo Diamond Turning Process,” Optical
Engineering, Vol.45, No.11, 113401, November, 2006.
vii
2. A. Y. Yi, C. Huang, F. Klocke, C. Brecher, G. Pongs, M. Winterschladen, A.
Demmer, S. Lange, T. Bergs, M. Merz, and F. Niehaus, “Development of A
Compression Molding Process for Three-dimensional Tailored Free-form Glass
Optics,” Applied Optics, Vol.45, No.25, 6511-6518, September, 2006.
3. L. Li, C. Huang, and A. Y. Yi, “Fabrication of micro and diffractive optical devices
by use of slow tool servo diamond turning process,” ASPE Annual Meeting, Norfolk,
VA, October 9-14, 2005.
FIELDS OF STUDY
Major Field: Industrial and Systems Engineering
viii
TABLE OF CONTENTS
Page
ABSTRACT………………………………………………………………………… ii
DEDICATION………………………………………………………………………
iv
ACKNOWLEDGMENTS…………………………………………………………... v
VITA………………………………………………………………………………...
vii
LIST OF TABLES......................................................................................................
xiii
LIST OF FIGURES…………………………………………………………………. xiv
CHAPTER 1 INTRODUCTION............................................................................... 1
1.1 Research Motivation…………………………………………………………….
1
1.2 Literature Review……………………………………………………………….. 4
1.3 Theoretical models…………………………………………................................
6
1.4 Research Objective................................................................................................ 10
CHAPTER 2 PRECISION MOLD DESIGN AND FABRICATION …….............
14
2.1 Lens Design……………………………………………………………………... 15
2.2 Mold Inserts Fabrication………………………………………………………...
16
2.3 Injection Molding Experiments…………………………………………………. 19
CHAPTER 3 GEOMOETERY MEASUREMENT AND COMPENSATION…… 21
3.1 Basic Measurement……………………………………………………………...
21
3.2 Surface Geometry and Part Thickness…………………………………………..
24
ix
3.3 Mold Compensation……………………………………………………………..
29
3.4 Freeform Measurement………………………………………………………….
31
3.4.1 Surface Measurement……………………………………………………... 33
3.4.2 Image Reconstruction……………………………………………………... 38
CHAPTER 4 OPTICAL MEASUREMENT………………………………………
43
4.1 Birefringence (Residual Stress) Measurement………………………………….. 43
4.2 Refractive Index Measurement………………………………………………….
52
4.3 Optical Effects of Surface Finish………………………………………………..
57
4.3.1 Theoretical Analysis………………………………………………………. 59
4.3.1.1 Surface Characteristics of a Diamond Machined Surface………...
59
4.3.1.2 Scalar Method for Diffraction and Scattering Calculation………..
61
4.3.2 Experiment and Measurement…………………………………………….. 63
4.3.3 Results……………………………………………………………………..
66
4.3.3.1 Comparison of Surface Profile Measurement and Direct
Scattering Measurement………………………………………………….. 66
4.3.3.2 Relationship of Molded Surface Quality and Injection Molding
Process Conditions………………………………………………………... 71
CHAPTER 5 ALVAREZ LENS MANUFACTURING…………………………...
80
5.1 Alvarez Lens…………………………………………………………………….
80
5.1.1 Alvarez Lens Design………………………………………………………
80
5.1.2 Alvarez Lens Fabrication………………………………………………….
84
5.1.3 Alvarez Lens Measurement……………………………………………….. 86
5.1.3.1 Zernike Polynomials………………………………………………
86
5.1.3.2 Wavefront Aberration Measurement……………………………...
89
x
5.1.3.3 Surface Measurement……………………………………………..
94
5.2 Micro Alvarez Lens Array………………………………………………………
96
5.2.1 Mold Design and Fabrication……………………………………………... 97
5.2.2 Measurement………………………………………………………………
100
5.2.2.1 Microlens Array…………………………………………………... 100
5.2.2.2 Geometry Measurement…………………………………………... 101
5.2.2.3 Surface Roughness………………………………………………... 103
5.2.2.4 Adjustable Focal Length Measurement…………………………...
104
CHAPTER 6 DIFFRACTIVE LENS MANUFACTURING……………………… 108
6.1 Diffractive Lens…………………………………………………………………
108
6.1.1 Lens Design……………………………………………………………….. 108
6.1.2 DOEs Fabrication…………………………………………………………. 110
6.1.2.1 Polar Coordinate – Spiral Tool Path………………………………
112
6.1.2.2. Cartesian Coordinate – Broaching……………………………….. 113
6.1.3 Profile Measurement………………………………………………………
114
6.2 Fresnel Lens……………………………………………………………………..
119
6.2.1 Lens Design……………………………………………………………….. 119
6.2.2 Mold Fabrication………………………………………………………….. 120
6.2.3 Profile Measurement………………………………………………………
121
6.2.4 Optical Performance Simulation…………………………………………..
125
CHAPTER 7 CONCLUSION……………………………………………………...
129
CHAPTER 8 FUTURE WORK……………………………………………………
134
APPENDIX A SPECIFICATION OF MOLDING MATERIAL………………….. 136
APPENDIX B SH50M MAIN SPECIFICATION………………………………… 139
xi
APPENDIX C PROCESS CONDITIONS FOR FULL FRACTIONAL
FACTORIAL EXPERIMENTS………………………………..…………………… 141
APPENDIX D ANOVA RESULTS FOR FULL FRACTIONAL FACTORIAL
EXPERIMENTS………………………………..…………………………………... 144
APPENDIX E DEA RESULTS FOR FULL FRACTIONAL FACTORIAL
EXPERIMENTS………………………………..…………………………………... 146
REFERENCE……………………………………………………………………….. 149
xii
LIST OF TABLES
Table
Page
3.1
Thermal Properties of Mold insert materials………………………………. 26
3.2
Thickness Measurement Locations………………………………………...
27
5.1
Zernike Polynomials (up to 4th order)……………………………………...
88
B.1
Main specification of SH50M injection molding machine………………...
140
C.1
Process conditions for full fractional factorial experiments……………….. 142
D.1
ANOVA results for full fractional factorial experiments………………….. 145
E.1
DEA results for full fractional factorial experiments……………………… 147
xiii
LIST OF FIGURES
Figure
Page
1.1
Some applications of injection molding optics. (a) f-θ lenses for laser
scanner (b) Projection lenses for television (c) Domes for surveillance
camera…………………………………………………………………….... 3
1.2
Narrow gap geometry as analyzed by the Hele-Shaw approximation……..
6
1.3
Advanced Compensation procedure for quality lens injection molding…...
12
2.1
Nanotech 350FG ultra precision machine………………………………….
17
2.2
Schematic drawing of the ultraprecision machine and diamond machining
process (a) Ultraprecision machine (b) Close up view of diamond
machining process…………………………………………………………. 18
2.3
Sumitomo SH50M injection molding machine……………………………. 20
3.1
DEA method for total weight vs. standard deviation………………………
3.2
Thickness and surface measurement setup………………………………… 25
3.3
Thickness measurement comparison between the molded lenses from
nickel inserts and aluminum inserts……………………………………….. 28
3.4
Thickness distribution on the molded lens…………………………………
3.5
First round compensated mold insert surface……………………………… 30
3.6
Lens thickness measurement result………………………………………...
31
3.7
Schematic of illumination principle………………………………………..
32
3.8
Finished nickel mold……………………………………………………….
33
3.9
MicroGlider profilometer…………………………………………………..
34
xiv
23
29
3.10
Measurement coordinate system manipulation…………………………….
3.11
Measurement result of the freeform molded lens (a) Targeted design
surface (b) Molded lens surface (c) error between design and molded lens
surface………………………………………………………………............ 37
3.12
Snell’s Law………………………………………………………………… 38
3.13
Needed points for refractive ray calculation……………………………….
3.14
Image reconstruction using the CMM measurement………………………. 41
3.15
Image formed by the molded freeform optics……………………………...
41
4.1
Principle sketch of plane polariscope………………………………………
44
4.2
Retardation comparison with different packing pressure………………….. 47
4.3
Retardation comparison with different mold temperature…………………. 48
4.4
Retardation comparison with different melt temperature………………….. 49
4.5
3D model and birefringence simulation result from Moldflow……………. 50
4.6
Retardation simulation result comparison with different packing
pressure…………………………………………………………………….. 51
4.7
Calculation of the slope of the wavefront at individual lenslet…………….
53
4.8
Index measurement setup…………………………………………………..
54
4.9
Wavefront error of the molded lens under different mold temperature in
fluid (a) Lower mold temperature (b) Higher mold temperature………….. 55
4.10
Wavefront error of the molded lens under different packing pressure in
fluid (a) Higher packing pressure (b) Lower packing pressure………….… 56
4.11
Wavefront error of the molded lens under different packing pressure in
air................................................................................................................... 57
4.12
Profile of a typical diamond machined surface…………………………….
4.13
Specular reflection, high order diffraction and scattering from the
diamond machined surface in Figure 4.12………………………………… 60
4.14
Diffraction from a diamond machined surface…………………………….. 62
xv
35
39
59
4.15
Schematic of phase shift interferometry …………………………………..
64
4.16
Setup of the scattering measurement device……………………………….
65
4.17
Scattering measurement system……………………………………………
66
4.18
Comparison of the mold insert and molded lens (a) 3D surface profile of a
20 μm tool mark spacing mold insert, measured by Veeco white light
profilometer (b) 3D surface profile of the molded lens, measured by
Veeco white light profilometer (c) Calculated average 1D spectrum of the
same mold surface (d) Calculated average 1D spectrum of the same
molded lens surface (e) Directly measured surface scattering of the same
mold surface (f) Directly measured surface scattering of the same molded
lens surface……………………………….................................................... 70
4.19
Experimental results of the lens molded under different packing pressure
(a) First order diffraction intensity (b) Surface roughness measured by
Veeco (c) Measured tool mark depth……………………………………… 73
4.20
Experimental results of the lens molded under different mold temperature
(a) First order diffraction intensity (b) Surface roughness measured by
Veeco (c) Measured tool mark depth……………………………………… 76
4.21
Experimental results of the lens molded under different melt temperature
(a) First order diffraction intensity (b) Surface roughness measured by
Veeco (c) Measured tool mark depth……………………………………… 78
5.1
Schematic drawing of Alvarez lens pair…………………………………… 81
5.2
Alvarez lens mold insert and molded lens (a) Alvarez lens mold
(b) Molded freeform lenses………………………………………………... 85
5.3
Measurement Setup for Alvarez Lens……………………………………...
5.4
Low order Zernike coefficients of the molded Alvarez lens pair while the
relative x-axis translation………………………………………………….. 91
5.5
RMS value of the molded Alvarez lens pair while the relative x-axis
translation………………………………………………………………….. 92
5.6
RMS value of the molded Alvarez lens pair while the relative x-axis
translation under different packing pressure………………………………. 93
5.7
RMS value of the molded Alvarez lens pair while the relative x-axis
translation under different mold temperature……………………………… 93
xvi
90
5.8
Retardation of the molded Alvarez lens under different process
parameters………………………………………………………………….. 94
5.9
Alvarez lens geometry measurement………………………………………. 95
5.10
Schematic drawing of Alvarez lens array………………………………….. 98
5.11
Broaching CNC tool path…………………………………………………..
99
5.12
Machined micro Alvarez lens array mold insert…………………………...
101
5.13
Design and 3D measurement results (a) Design (b) Measurement result of
lenslet in the middle of the array (c) Difference between the lenslet in the
middle of the array and design (d) Difference between the lenslet in the
middle and at the edge on the molded microlens array……………………. 103
5.14
Test setup for measuring the focal length of a molded microlens array
pair…………………………………………………………………………. 105
5.15
Focal length measurement result…………………………………………...
6.1
General concept of a DOE’s function (amplitude type)…………………… 109
6.2
Design of 256 level DOE…………………………………………………..
110
6.3
SEM picture of the half-radius diamond tool………………………………
111
6.4
Spiral CNC tool path for DOE fabrication…………………………………
113
6.5
Broaching CNC tool path for DOE fabrication……………………………. 114
6.6
Sectional SEM scan of a 256-level DOE…………………………………..
116
6.7
Sectional AFM scan of the 256-level DOE design………………………...
117
6.8
Sectional AFM line scan of the 256-level DOE design……………………
118
6.9
Fresnel lens design……………………………………………………........
120
6.10
Fresnel lens mold insert and molded lens………………………………….. 121
6.11
Measurement result from SEM…………………………………………….
122
6.12
Feature comparison with different mold temperature……………………...
123
6.13
Feature comparison with different packing pressure………………………. 124
xvii
106
6.14
Feature comparison with different melt temperature………………………
125
6.15
Designed lens diffractive pattern distribution……………………………...
127
6.16
Lens 1 which is under higher packing pressure diffractive pattern
distribution…………………………………………………………………. 127
6.17
Lens 2 which is under lower packing pressure diffractive pattern
distribution…………………………………………………………………. 128
A.1
Product data sheet for Plexiglas® V825……………………………………
A.2
Product data sheet for Plexiglas® V825 (Figure A.1 continued)…………... 138
xviii
137
CHAPTER 1
INTRODUCTION
1.1 Research Motivation
A lens is an optical device that transmits or refracts light to either concentrate or
diverge. It is usually formed from a piece of shaped high purity glass or plastic. A high
precision lens is manufactured with very high tolerances, and a slight defect in the lens
can cause it to focus the light beam improperly, making it completely ineffective for its
intended purpose. The optical aberrations can result from geometry deviation, surface
roughness, sub-surface defects from fabrication process, physical and mechanical
properties of the optical material and optical conditions, etc. Applications that use high
precision lens include medical and military equipment, collision-avoidance devices for
the transportation industry, and scientific testing devices. High precision lenses are
generally made of glass and require high shape accuracy (a few microns or less) and
smooth surfaces (Ra ~ 2 ~ 20 nanometers) and a minimum subsurface damage (< 50 nm)
[Fahnle, 1988]. The performance of a lens therefore largely depends upon the fabrication
process which needs to be carefully designed in order to meet optical requirements.
1
Due to some theoretical limitation of the traditional symmetrical optical elements,
freeform elements are now beginning to be used in more applications, such as photograph,
illumination, optometry and many others. Freeform optical surfaces are defined as those
that do not have rotational symmetry and sometimes cover those that have rotational
symmetry but with aspheric surface. The lens arrays are also included among freeform
optics, since they face the same problems in fabrication, alignment and metrology as the
general freeform optics. This category of optics has its obvious advantages in reduction
of optical aberrations, system components and favorable positioning of optical elements.
This means that freeform optics can replace some of the spherical optics if they can be
improved with respect to geometry accuracy and ease of production at a comparable cost.
However, the design, fabrication (including direct fabrication and molding) and
metrology for freeform elements remain a difficult, case by case and complex task.
The conventional production of precision lenses is by all means a complicated
process and involves progressive material removal from a raw glass blank by grinding,
lapping and polishing operations to obtain a finished optical component. This process is
more suitable for manufacturing spherical glass lenses because of their simple geometry.
However, the grinding and polishing process makes it difficult to produce freeform
surface shapes economically other than sphere or flat using glass materials. As compared
to glass optics fabrication, the injection molding process makes it feasible to
economically produce more complicated optical shapes such as aspheric lenses,
diffractive lenses and freeform lenses in plastic when the optical mold is well designed
(compensated) and fabricated. Moreover, injection molding process can be used for high
2
volume production, thus the unit cost can be very low. For these apparent advantages,
injection molded polymer optics become alternative components in many applications.
For example, the pickup lenses for DVD (digital video disk) or CD (compact disk)
players and micro lenses for cellular phones are injection-molded of plastics. Figure 1.1
shows some examples of injection molded optics. These optics include f-θ lenses used in
scanner, projection lenses for rear projection television and plastic domes for camera
systems.
(a)
(b)
(c)
Figure 1.1: Some applications of injection molding optics. (a) f-θ lenses for laser scanner
(b) Projection lenses for television (c) Domes for surveillance camera
Although injection molding polymer optics is increasingly used in industry for
many applications, the requirements for image quality are not demanding. However, for
precision optical components, optical functionality is always the most important concern
for producers and consumers. Recently more requirements in product quality for the
injection molding optics are expected.
For example, at Videolarm corporate
(www.videolarm.com), improvements in domes for surveillance camera are needed for
3
both geometry accuracy and residual stress level to match the high resolution cameras
that are being introduced to the system.
Some of the drawbacks for injection molded optics include geometry deviation
from the original mold design and inhomogeneous index distribution during
manufacturing. The geometry deviation resulted from volume shrinkage and warpage are
strongly dependent on process conditions.
The inhomogeneous index distribution
resulted from the residual stresses and non-uniform molecular orientation in the injection
molded parts. These are the main reasons that injection molded polymer optics are not
suited for high precision applications. Therefore, investigation in injection molding
process for high precision polymer lenses is critical to solving the technical issues
associated with surface conformance to design and ultimately providing an affordable
high precision manufacturing process for satisfactory optical performance.
This dissertation research is focused on precision polymer optics fabrication by
injection molding.
The study involves both experimental approach and numerical
modeling in order to identify the proper polymer lens manufacturing processes. The
scope of this research includes investigation in optical design, mold and lens fabrication,
as well as optical metrology related to polymer lens manufacturing.
1.2 Literature Review
In previous research involving in polymer injection molding, most of the work
was focused on determining of process parameters in order to optimize part quality.
4
Many approaches, including mathematical modeling, numerical simulation, process
windows, design of experiment, expert systems, artificial neural networks, case based
reasoning, genetic algorithms, and evolutionary strategies, have been tested [Isayev, 1987;
Mok, 1999; Kwak, 2005; Shen, 2004; Tan, 1997; Kumar, 2002; Lu, 2001]. With an ever
increasing demand on molded part quality, more sophisticated studies were carried out.
Shape deformation including shrinkage and warpage, residual stress distribution,
molecular orientation, and cooling system were performed by many researchers [Young,
2004; Choi, 1999; Wimberger-Friedl, 1995; Kang, 1998; Liou, 1989].
The above
mentioned research activities were conducted with great details but did not address the
issues concerning mold compensation for high precision polymer lenses.
This
dissertation research will demonstrate our efforts to modify the mold design and
fabrication in order to compensate the geometrical and optical deviation from design.
Our investigation will be focused on study of the effects of the process parameters and on
development of the process and methodology to fabricate the freeform lenses with high
accuracy and efficiency.
On a different note, for high precision optical systems, freeform optics can
provide a practical solution for some design and manufacturing problems. Notably,
microlens arrays or diffractive optical elements can be injection molded in high volume
at a low cost. Numerous publications highlighted the contributions to this field such as
the effects of the process variables and size of the micro features for the molded parts
[Gale, 1997; Sha, 2007].
However the success of the process also relies on the
fabrication of the mold inserts. Fewer articles discussed the advanced mold fabrication
5
issue. This dissertation research will develop a methodology that is different from the
traditional fabrication processes in the sense that not only macro size but micro lenses
mold was also simultaneously machined using STS (slow tool servo) process. In addition,
contact and non-contact measurement and data analysis methods will be developed for
freeform polymer lens replication technology in this dissertation research.
1.3 Theoretical models
Because
most
injection
molded
polymer
products
have
asymmetrical
configurations and the rheological response of polymer melt is generally non-Newtonian
and non-isothermal, it is difficult to analyze the filling process without simplifications.
The GHS (generalized Hele-Shaw) flow model is the most common approximation that
provides simplified governing equations for non-isothermal, non-Newtonian and inelastic
flows in a thin cavity as shown in Figure 1.2 which is recreated from [Dantzig, 2001].
z
y
x
Vin or Pin
Polymer melt
2b
Figure 1.2: Narrow gap geometry as analyzed by the Hele-Shaw approximation
6
The assumptions [Su, 2004] of the GHS flow model are:
(1) The thickness of the cavity is much smaller than the other dimensions.
(2) The velocity component in the direction of thickness is neglected, and pressure is a
function of x and y only.
(3) The flow regions are considered to be fully developed Hele-Shaw flows in which
inertia and gravitational forces are much smaller than viscous forces.
(4) The flow kinematics is shear-dominated and the shear viscosity is taken to be both
temperature and shear rate dependent.
The detailed derivation has been developed by Hieber and Shen [Hieber, 1980].
In view of these assumptions and neglecting compressibility during the filling stages, the
momentum equation in the Cartesian coordinate system reduces to:
0=
∂ ⎡ ∂υ x ⎤ ∂P
−
η
∂z ⎢⎣ ∂z ⎥⎦ ∂x
(1-1)
0=
∂ ⎡ ∂υ y ⎤ ∂P
⎢η
⎥−
∂z ⎣ ∂z ⎦ ∂y
(1-2)
Where υ x and υ y are velocity components in the x and y directions, respectively; P(x, y) is
the pressure, η (γ&, T ) is the shear viscosity, γ& is the shear rate and T is temperature.
Under the present assumptions, γ& is given by
7
1/ 2
⎧⎪ ⎡ ∂υ ⎤ 2 ⎡ ∂υ y ⎤ 2 ⎫⎪
γ& = ⎨ ⎢ x ⎥ + ⎢
⎥ ⎬
⎪⎩ ⎣ ∂z ⎦ ⎣ ∂z ⎦ ⎪⎭
(1-3)
Because of the temperature difference between mold and polymer melt and the
viscous heating inside the flow, the filling process should be treated as a non-isothermal
case. Heat conduction in the direction of flow is neglected based on the assumption that
the thickness 2b is much smaller than the other two dimensions. The energy equation in
the melt region becomes
⎡ ∂T
∂T
∂T ⎤
∂ 2T
+υx
+υy
=
k
+ ηγ& 2
2
⎥
∂x
∂y ⎦
∂z
⎣ ∂t
ρc p ⎢
(1-4)
Where the ηγ& 2 is the viscous heating term, and ρ , c p and k are density, specific
heat and thermal conductivity, respectively.
For simplicity, it is assumed that the
velocities of polymer melt on the mold surfaces are zero and the temperature of mold
remains at Tw during filling. The boundary conditions are given by
υ x = υ y = 0 at z = b
T = Tw
∂v x ∂v y
=
= 0 at z = 0
∂z
∂z
∂T
at z = ±b
= 0 at z = 0
∂z
(1-5)
Applying the lubrication approximation, the thickness-averaged continuity
equation results in
8
∂ (bυ x ) ∂ (bυ y )
+
=0
∂x
∂y
(1-6)
Where υ x and υ y are averaged velocities over z, and b is half of the thickness.
The velocities and shear rate can be obtained as
υx = Λ x ∫
b
z
~
b ~
z ~
z
dz , υ y = Λ y ∫ d~
z
η
z
(1-7)
η
zΛ
γ& =
(1-8)
η
Where
Λx = −
∂P
,
∂x
Λy = −
∂P
∂y
and
[
Λ = Λ2x + Λ2y
]
1/ 2
(1-9)
In addition, the gapwise-averaged velocities are obtained as:
υ x = (Λ x / b) S , υ y = (Λ y / b) S
(1-10)
Where S is the flow conductance which is defined as
b
z2
0
η
S=∫
dz
Hence, substituting (1-10) into (1-6) gives:
9
(1-11)
∂ ⎡ ∂P ⎤ ∂ ⎡ ∂P ⎤
S
S
+
=0
∂x ⎢⎣ ∂x ⎥⎦ ∂y ⎢⎣ ∂y ⎥⎦
(1-12)
As can be seen, the equations of this model are nonlinear and coupled. It is
difficult to solve them analytically. In this dissertation research, simulation software
Moldflow Plastic Insight 6.1 will be used to simulate the process and experiments will be
conducted to verify the theory and simulation results.
1.4 Research Objective
Modeling and optimization of injection molding process for polymer optics have
been studied extensively for a long time. Previous studies were focused on the effects of
the process variables and material properties to obtain the optimal condition and improve
the part quality. However, only a few publications showed efforts in modifying mold
design and fabrication to compensate the geometry and optical deviation from design.
None provided a general strategy for low cost, high precision lens manufacturing. Also
fabrication and measurement of macro and micro freeform polymer optics were not
systematically studied before. Thus the overall objective of the dissertation research is to
develop a methodology to obtain high precision macro and micro polymer freeform
optics with accurate geometry and proper optical performance by the state-of-the-art
mold fabrication technology.
The polymer optics fabricated by injection molding are usually not suitable for
high precision applications due to issues related to geometry deviation, inhomogeneous
index distribution, birefringence and thermal instability of molded polymer lenses. The
10
geometrical deviation of the molded lenses will be used for mold compensation in this
dissertation research. In an optical assembly, optical path length is equal to the product
of the physical dimension of the medium and the refractive index. Therefore, the index
deviation should also be included for mold compensation.
By obtaining the index
variation in the molded lens under specific process conditions, the modified mold inserts
can be designed and fabricated by combining the surface and thickness measurement
results and index distribution. The residual stresses and surface scattering will also be
optimized under the same process condition.
With the modified mold (generally a
freeform shape), the molded lens will have improved optical performance. The advanced
iteration compensation procedure is shown in Figure 1.3.
11
Set optimal process condition
Freeform Mold design and fabrication
Lens molding process
Lens OPD (optical path difference) test
Lens optical performance test
(birefringence and optical scattering)
Acceptable?
Yes
Quality lens collection
No
Geometry measurement
Index variation measurement
Figure 1.3: Advanced compensation procedure for quality lens injection molding
The specific objectives of this dissertation research can be summarized as:
•
Investigate the feasibility of using injection molding process to manufacture high
precision polymer lenses by performing experiments (both axisymmetrical and
freeform lenses) and evaluating surface geometry and optical performance.
12
•
Explore the effects of process variables and material property to perform process
optimization for specific objective function (surface shape deviation, birefringence,
optical retardation, optical scattering).
•
Improve current measurement method to obtain real freeform surface shape, part
thickness and optical performance.
•
Develop a methodology to design and fabricate modified mold inserts to compensate
geometry error and optical aberration for the molded optics.
•
Design and fabricate multiple freeform mold inserts and obtain functional injection
molded freeform optics including compensated lens, Alvarez lens, micro Alvarez lens
arrays and diffractive lenses.
13
CHAPTER 2
PRECISION MOLD DESIGN AND FABRICATION
Injection molding polymer optical components are used for its high volume and
lightweight capability over traditional glass optics. Injection molding is an inherent
freeform process thus complex geometry (including aspherical and freeform) can be
readily manufactured. However several difficult issues associated with the injection
molded optics have hindered the implementation of injection molding process in wider
applications.
These issues include geometry deviation and inhomogeneous index
distribution due to thermal shrinkage; birefringence incurred during the molding process
also limited the adoption of polymer optics in certain polarization sensitive optical
systems; thermal instability of molded polymer lenses can also render the optics less
effective in application where temperature changes become large and frequent (such as
optics designed for out door use or high temperature applications).
In this research, our goal is to establish a high precision polymer lens
manufacturing protocol based on the state-of-the-art ultraprecision machining
technologies. Specifically, two focused research subjects were studied: injection molding
of macro (imaging optics) and micro optics (including microlens array and diffractive
14
optics).
In a departure from previous approaches where modifications of process
conditions or material properties were the first choice, our aim was to utilize the newly
acquired freeform optical fabrication capability using ultraprecision machining process to
compensate for optical performance degradation due to injection molding process
variability. By precisely measuring the optical retardation and surface deviation resulted
from molding process variations, accurate surface geometry of a freeform mold can be
constructed.
To obtain an injection molded lens for optical applications, three steps need to be
completed in sequence.
2.1 Lens Design
In this part of the proposed research, two types of optical lenses will be studied.
The first type includes precision imaging optics and ophthalmic lenses. The second type
is micro optics, specifically issues related to design and fabrication of microlens array
and diffractive optics will be studied.
For regular lenses, commercial optical design software such as Zemax®
(www.zemax.com) and Code V® (www.opticalres.com) are often used to obtain the
surface profile and other dimension information. In this research dissertation, a plano
lens is chosen for its simple characteristic since plano lens will simplify the shape
measurement, surface diffraction, residual stress and birefringence measurement and
15
index measurement. The methodology developed based on the plano lens can be then
implemented in other applications without loss of generality.
In addition to the plano lenses, lenses with a non axisymmetrical surface profile
will also be molded using modified molds in this research. Since traditional fabrication
method for freeform elements is difficult or costly, or time consuming, freeform lens
manufacturing process has not been used for high volume and low cost production. For
this dissertation research, advanced fabrication methods will be developed and precision
freeform optics will be fabricated.
2.2 Mold Inserts Fabrication
Mold inserts for polymer optics must have optical quality. The inserts used in this
research were fabricated on the Moore Nanotech 350FG machine, a state-of-the-art 5-axis
ultraprecision diamond machine.
The machine is shown in Figure 2.1.
Typical
applications for this machine include axisymmetric machining of aspheric and toroidal
surfaces, raster flycutting of freeform, linear diffractive, and micro-prismatic optical
structures,
as
well
as
slow
tool
servo
(www.nanotechsys.com).
16
machining
of
freeform
surface
Figure 2.1: Nanotech 350FG ultra precision machine
The 350FG (Freeform Generation) ultraprecision machine used in this study was
built by Moore Nanotechnology, Inc. It has three linear axes that are equipped with
linear laser-scales capable of resolving 8.6 nm at a maximum speed of 1800 mm/min.
The straightness on all slides is less than 250 nm over the entire travel up to 350 mm.
The work spindle is capable of reaching 6,000 rpm while maintaining axial and radial
error motion of less than 25 nm. The work spindle can also maintain angular position to
less than 0.5 arc sec in a modulated mode. The main specifications of the ultraprecision
machine were detailed elsewhere [Tomhe, 2003]. The C axis was fixed during freeform
broaching process while during slow tool servo machining process, the C axis rotated
with accurate control. The diamond tool was located on Z axis. Figure 2.2 illustrates the
machine operation and the details of the freeform machining process. Arrows in Figure
2.2 (a) indicate positive directions of the linear axes.
17
Y
Diamond tool
Z(X,Y)
Freeform optic
X
(a)
(b)
Figure 2.2: Schematic drawing of the ultraprecision machine and diamond machining
process (a) Ultraprecision machine (b) Close up view of diamond machining process
With this machine, the inserts can be fabricated with very low surface roughness
(Normally Ra is in several nanometers), therefore no post machining polishing is needed.
For plano lens mold inserts, the initial inserts can be fabricated by traditional
SPDT (single point diamond turning) process. The following modified mold inserts are
fabricated by slow tool servo process to create the nonsymmetrical surface profile. The
slow tool servo process makes the freeform inserts with accurate geometry and optical
finish in one single operation. After the mold inserts are diamond machined, non contact
measurement is preferred to measure the surface geometry to protect the optical surface
finish.
Apart from imaging optical lenses, although individual optical elements in a
microlens array or diffractive components may have an axisymmetric curve, multiple
18
micro lenses arranged in a matrix format can be treated as a freeform surface and
therefore fabrication method used to produce freeform surfaces can be employed to
generate the array. The method is different from traditional fabrications processes in the
sense that the micro lenses were simultaneously machined using slow tool servo process.
A rapid optical manufacturing process from mold making to completed polymer optics
based on STS (slow tool servo) will be developed.
2.3 Injection Molding Experiments
In the injection molding experiments, PMMA (Polymethyl methacrylate), code
named Plexiglas® V825, is selected. The specification of this polymer material is shown
in Appendix A. Injection molding process is very complicated since more than two
hundred variables are involved in the whole process [Greis, 1983]. However, under the
conditions that are crucial to our experiments, only several parameters are important to
the part quality therefore will be the focus in our study. These parameters include mold
temperature, polymer melt temperature, packing pressure, packing time and cooling time.
These parameters will be set to different levels to complete a full fractional factorial
experiment to optimize the process condition. To evaluate the part quality under each
process condition, the collected parts need to be consistent in the concerned specification.
For our experiments, the initial ten trial parts will be made and discarded then five or ten
parts will be collected for measurement. The room temperature and humidity are also
important for the part quality, so any selected experiments will be conducted in one
single day to keep the environmental condition consistent. All the experiments in this
19
research dissertation were conducted on Sumitomo SH50M injection molding machine
shown in Figure 2.3. The specification of SH50M is listed in Appendix B.
Figure 2.3: Sumitomo SH50M injection molding machine
20
CHAPTER 3
GEOMOETERY MEASUREMENT AND COMPENSATION
Although optics fabricated by injection molding are increasingly used in industry,
requirements for image quality in many applications are not necessarily demanding.
However, for precision optical components, optical functionality is the most important
factor for producers and consumers. The geometry deviation resulted from volume
shrinkage and warpage is one of the drawbacks for injection molded optics. This is the
main reason that prevents injection molded optics from being used in high precision
applications. Therefore, investigation in injection molding process for high precision
polymer lenses is critical to solving the technical issues associated with surface
conformance to design and ultimately providing an affordable high precision
manufacturing process for satisfactory optical performance.
3.1 Basic Measurement
The quality of the injection molded components is strongly dependent on process
conditions. For this research, the optimal process conditions were obtained with the aid
21
of DOE (Design of Experiments) and DEA (Data Envelopment Analysis) methods
according to basic measurement results.
The mold inserts were made of copper nickel C715 (www.farmerscopper.com)
and the design was a plano lens with diameter of 50 millimeters and thickness of 3
millimeters. After initial tuning of the process, five parameters in different levels were
set up for a full factorial design of experiments. Seventy-two experiment conditions were
listed in Appendix C.
For the plano lens, both the diameter and the thickness of the injection molded
part were measured. A micrometer was opted for diameter measurement and a precision
indicator was used for thickness measurement on specific position. A precision scale was
used for part weight measurement.
Every part was measured and the average and
standard deviations were calculated for each group under the same process condition.
The measurement data (total weight and its standard deviation) were processed
using ANOVA (analysis of variance) built in MINITAB and DEA to obtain the effects of
the process variables and the optimal conditions (Refer to Appendix D and Appendix E).
From ANOVA results, melt temperature, mold temperature, packing pressure were found
to be the most important variables and cooling time, packing time were less important.
The conclusion from ANOVA results just narrowed down the critical process parameters
for the following study. From DEA results, four process conditions (Condition 12, 27, 32
and 36) were obtained which means they were better choices for designated objective
functions. As optimal process condition, Condition 12 (melt temperature 210 °C (450 °F),
22
mold temperature 65 °C (150 °F), cooling time 40 sec, packing pressure 76.3 MPa (35%)
and packing time 7.5 sec) was chosen for the following study. The result for total weight
vs. standard deviation by DEA is shown below and four optimal conditions are marked
on Figure 3.1.
Total Weight vs Standard Deviation
0.0400
Standard Deviation
0.0350
0.0300
0.0250
0.0200
0.0150
0.0100
0.0050
0.0000
20.1
20.2
20.3
20.4
20.5
20.6
20.7
20.8
20.9
21
Total Weight (grams)
Figure 3.1: DEA method for total weight vs. standard deviation
These basic measurement results provide qualitative analysis and show the
process consistency and comparison of the rough dimension of the molded lens to the
design. However, they cannot be used to assess the part optical performance due to the
rough measurement and different merit functions. The main benefit from these results is
23
to narrow down the critical process parameters and choose one optimal condition for
following studies.
3.2 Surface Geometry and Part Thickness
The surface geometry accuracy on a lens is critical to its optical performance.
The deviation of the molded surface with the design surface will introduce unwanted
aberrations in an optical assembly. Each single surface geometry and part thickness need
to be measured accurately and the aberration from the geometry can be estimated for the
following compensation scheme.
The surface geometry and part thickness can be
measured by two LVDTs (linear variable difference transformers) mounted on the 350
FG machine (Figure 3.2(b) as viewed in the Z direction). The axial movement accuracy
of the 350 FG machine is only several nanometers, much higher than the molded surface
geometry deviation. The two LVDTs are coaxially mounted and the molded part surface
is perpendicular to the direction between two LVDT tips.
accuracy for thickness measurement as shown in Figure 3.2.
24
This setup provides the
(a) 350 Machine Frame
(b) LVDT setup
Figure 3.2: Thickness and surface measurement setup
The molded lens is held on the machine main spindle and the LVDT setup is
installed on the Z slide (as shown in Figure 3.2 (b)). Since the valid measurement range
of LVDT is only ±100 µm, the spindle and Z slide moving position for each measurement
point should be preset on the estimated spot to prevent probe from over-traveling. The
geometry of each surface can be obtained from single surface LVDT measurement data
on the same side after removing the tilt error. The thickness can be obtained from both
surface LVDT measurement data for the corresponding pair of points. By modifying the
part holder and keeping a constant environment condition (temperature, noise etc), the
measurement repeatability were maintained to less than 0.4 µm.
Only selected non-ferrous materials can be machined by diamond turning process
to optical quality without polishing, so in this dissertation research aluminum and copper
25
nickel alloy were chosen to fabricate mold inserts. The thermal properties of the mold
insert materials are listed in Table 3.1.
CTE*, linear 250 °C
Specific Heat Capacity
Thermal Conductivity
Copper Nickel C715
16.2 µm/m-°C
0.380 J/g-°C
29.0 W/m-K
Aluminum 6061 T6
25.2 µm/m-°C
0.896 J/g-°C
167 W/m-K
* Coefficient of Thermal Expansion
Table 3.1: Thermal properties of mold insert materials
Due to the different material thermal properties, even under same process
condition, the molded parts by using different mold insert materials also will be different
in final shape such as part thickness P-V (peak to valley) value. The thickness of the
molded plano lenses from nickel inserts and aluminum inserts but under same process
condition was measured in the locations listed in Table 3.2. The center of the molded
lens was set as origin of the coordinate system and Y axis and Z axis were in the same
direction as the machine coordinate system. The measurement area for surface geometry
and thickness was limited by the dimensions of the lens holder and LVDT probes, so the
area that is close to the edge of the molded lenses could not be measured by current
measurement setup. For this study, the radius of the measurement area was 12 mm to
avoid the interference between the lens holder and the probes.
26
Location
1
2
3
4
5
6
7
8
9
10
11
12
13
Y (mm)
0
-6
0
6
-12
-6
0
6
12
-6
0
6
0
Z (mm)
-12
-6
-6
-6
0
0
0
0
0
6
6
6
12
Table 3.2: Thickness measurement locations
The measurement results from nickel inserts and aluminum inserts are shown in
Figure 3.3. The thickness P-V value of the molded plano lenses with copper nickel C715
inserts was about 20% less than that with aluminum 6061 inserts. Since aluminum is
easier to machine, aluminum was again chosen as main mold insert material for the
following experiments in this dissertation research.
The measurement results from
different mold materials also show the same tendency for the thickness distribution on the
molded lens, therefore, the conclusions from the following experiments which were
based on aluminum inserts can be applied for the molded lenses with copper nickel
inserts.
27
Figure 3.3: Thickness measurement comparison between the molded lenses from nickel
inserts and aluminum inserts
With the aluminum flat mold inserts, the P-V value of the thickness deviation of
the molded plano lens is about 7 µm from the first experiment round under condition 12
(melt temperature 210 °C (450°F), mold temperature 65 °C (150 °F), cooling time 40 sec,
packing pressure 76.3 MPa (35%) and packing time 7.5 sec) on all measurement
locations. Figure 3.4 shows the thickness measurement results. From the figure, it can
be seen that the square area was measured due to the constraints for measurement point
selection. The lens compensation scheme that would be implemented was based on this
measurement result.
28
Figure3.4: Thickness distribution on the molded lens
3.3 Mold Compensation
When lens geometry and thickness measurement are performed, the amount of
mold compensation can be determined under the same process condition. First, the
difference between the measured molded lens surfaces and design surface profile can be
obtained. Second, the surface geometry can be fitted by Zernike polynomials (details
will be explained later), which is a convenient tool for wavefront description. The
compensated mold surface will be a complex freeform surface. Finally, slow tool servo
machining will be used to fabricate the compensation mold with optical surface quality.
With this mold insert, the molded lens should have better geometry and optical
29
performance that can be quantified using the techniques described in this dissertation
research.
The first round compensation is based on the molded lens thickness measurement
results from aluminum flat mold inserts. The modified mold insert surface is shown in
Figure 3.5. The analytical expression for the fitted surface is second order Zernike
polynomials. The P-V value for the modified mold surface with 40 mm in diameter is
about 11 µm to compensate the uneven thickness of the molded lens with flat mold insert.
The first round compensated mold insert surface is also fabricated with Aluminum 6061.
Figure 3.5: First round compensated mold insert surface
The lens thickness measurement results are shown in Figure 3.6. The blue line is
the molded lens thickness measurement result from original flat mold insert. The red and
30
black lines are two molded lenses from the first round compensated mold insert. The
measurement locations can be referred in Table 3.2.
2.938
Thickness (mm)
2.936
flat mold
2.934
modified mold
modified mold
2.932
2.93
2.928
2.926
0
2
4
6
8
10
12
14
Location
Figure 3.6: Lens thickness measurement result
From the measurement shown in Figure 3.6, it can be seen there is more than 50%
improvement on the thickness variation for the molded lens after compensation
comparing with the original lens.
The compensation method for lens quality
improvement has been proven to be effective. The further work will be focused on
improvement in compensation results.
3.4 Freeform Measurement
Benefiting from continuing research and development, freeform optical surface
are now becoming a practical solution to many optomechanical designs. However,
because of the asymmetrical geometry of the freeform optics, it is difficult to obtain
31
accurate surface curvature information of the freeform components. In this dissertation
research, a methodology for freeform measurement and data analysis was introduced.
The optical lens used in this research has a 3D tailored free-form surface. Three
dimensional tailoring is a constructive method for the design of freeform illumination
optics [Ries, 2002]. Light from a point source is transmitted by the freeform lens and
redirected to cast a prescribed illumination distribution on an image surface. The exact
shape of the lens surface is calculated by solving a set of differential equations that
describe a piecewise smooth surface, the desired trimming, and the redirection of
radiation defined by the slope and the curvature of the surface. The second surface of the
glass lens is flat. Figure 3.7 shows the 3D tailored free-form lens that refracts the light
rays from a point light source to form the Fraunhofer Institute for Production Technology
(IPT) logo as bright lines on a flat screen (the image plane). The finished lens has a
diameter of 20 mm, and the image has a size of 20 mm×20 mm.
Projection
Molded
freeform lens
20 mm
(Point)
Light source
20 mm
Figure 3.7: Schematic of illumination principle
32
Due to the complexity of the current freeform surface design, an FTS (fast tool
servo) designed with aerostatic ways was chosen to machine the free-form lens mold in
this research. The fast tool servo unit was developed at Fraunhofer IPT [Weck, 1999]. In
Figure 3.8, a finished nickel alloy electrolytically plated on the stainless steel substrate
freeform mold is shown.
Figure 3.8: Finished nickel mold
The low Tg freeform lens was fabricated by a Toshiba precision glass molding
press series 211V.
After molding, the finished lenses were cleaned and thermally
saturated in the metrology room where temperature is controlled at 20 ± 0.05 °C for at
least 24 hours before curve and surface roughness measurements were performed.
3.4.1 Surface Measurement
Molded lenses were measured using the MicroGlider profilometer (Fries Research
and Technology GmbH, Friedrich-Ebert-Strasse, D-51429 Bergisch Gladbach, Germany,
33
shown in Figure 3.9). The lateral accuracy (both x and y) is 2 μm, and the vertical
accuracy is ± 0.1 μm. The vertical axis resolution is 2 nm.
Figure 3.9: MicroGlider profilometer
To generate the error map, the design values of the freeform optical surface need
to be compared to the corresponding measured molded lens surface. However, since
there are no fiducial marks on the functional freeform lens surface, during the
measurement, the molded lens cannot be positioned with the same orientation and
leverage. To obtain the actual error between the design and molded lens, the measured
surface need be manipulated in all three linear translations and three angular rotations
shown in Figure 3.10 until the minimal error was reached [Li, 2004].
34
Z
Z’
X
X’
Y
Y’
Design Coordinate System
Measurement Coordinate System
Figure 3.10: Measurement coordinate system manipulation
The rotation transformations can be expressed as the following matrices:
0
⎡1
⎢0 cos θ
R x (θ ) = ⎢
⎢0 sin θ
⎢
0
⎣0
⎡cos θ
⎢ sin θ
R z (θ ) = ⎢
⎢ 0
⎢
⎣ 0
⎡1
⎢0
T (r ) = ⎢
⎢0
⎢
⎣0
[
v = vx
vy
0
1
0
0
0
− sin θ
cos θ
0
− sin θ
cos θ
0
0
0
0
1
0
0⎤
0⎥⎥
0⎥
⎥
1⎦
⎡ cos θ
⎢ 0
R y (θ ) = ⎢
⎢− sin θ
⎢
⎣ 0
,
0 sin θ
1
0
0 cos θ
0
0
0⎤
0⎥⎥
0⎥
⎥
1⎦
and
0⎤
0⎥⎥
. The translation transformation can be expressed as
0⎥
⎥
1⎦
0 rx ⎤
0 ry ⎥⎥
. The 3D position vector v was replaced with its 4D version
1 rz ⎥
⎥
0 1⎦
]
v z 1 . The homogeneous transformation matrix is obtained by combining
a sequence of rotation and translation transformations ( M = R x ⋅ R y ⋅ R z ⋅ T ) and the new
35
position vectors can be obtained by left multiplying M with the original 4D position
vector. The objective function is the difference between the corresponding points on the
design surface and molded lens surface. The minimal error was obtained by optimizing
the objective function with optimal axial rotation angles and translations using NelderMead simplex (direct search) method built in Matlab fminsearch function.
Figure 3.11 shows the 3D plots of the lens surface design, molded lens surface
and error between these two surfaces after data manipulation described above. After
optimizing the three direction translations and three axis rotations, the error between the
design and the molded lens surface is around ± 3.5 μm. Since the maximum vertical
measurement was limited to 300 μm on the model that was used, stitching was utilized to
create the measurements over the entire optical surface (which corresponded to the
vertical deviation of approximately 1.28 mm). The stitching created slight bumps on the
measurement as seen in Figure 3.11 (c).
(a) Targeted design surface
36
(b) Molded lens surface
(c) Error between design and molded lens surface
Figure 3.11: Measurement result of the freeform molded lens
37
This section described the steps for freeform surface measurement and data
analysis which can be applied in freeform component fabrication industry as a
methodology and quantify the geometry accuracy.
3.4.2 Image Reconstruction
The freeform lens surface was re-measured on a Sheffield Cordax coordinate
measuring machine (CMM). The measurement results can be used to reconstruct the
image using ray trace method based on Snell’s law shown in Figure 3.12.
A
N
θ1
n
n’
sin θ1 n'
=
sin θ 2 n
θ2
A’
Figure 3.12: Snell’s Law
Snell’s law gives the relationship between angles of incidence and refraction for a
wave impinging on an interface between two media with different indices of refractions.
According to reference [Yin, 1996], Snell’s law can be expressed as vector format:
38
nA × N = n' A'× N and in this formula A, N, A’ are unit vectors. The refractive vector can
be obtained as: A' =
n
[A − N ( A ⋅ N )] + N 1 − ( n ) 2 + ( n ) 2 ( A ⋅ N ) 2 . So for refractive ray
n'
n'
n'
calculation, incident ray vector, normal vector and refractive index are needed. However,
due to the irregularity of the freeform surface, the normal vector on the object point need
be calculated by multi-cross product method [Lin, 2003] with its four nearest points. The
coordinates of the objects point and its four nearest points are shown in Figure 3.13.
P2(xi,yi+1,z2)
P3(xi-1,yi,z3)
P1(xi+1,yi,z1)
P(xi,yi,z)
P4(xi,yi-1,z4)
Figure 3.13: Needed points for refractive ray calculation
The normal on the object point can be calculated as the following steps: (1)
Calculate the four tangential vectors as: V1=P1-P, V2=P2-P, V3=P3-P, V4=P4-P and find
out the four normal vectors of the point P as: N1=V1×V2, N2=V2×V3, N3=V3×V4,
39
N4=V4×V1; (2) Obtain the unit vectors of the four normal vectors as: eN1, eN2, eN3, eN4 and
the average normal vector of the compensated point P as Nm= eN1+eN2+eN3+eN4; (3)
Calculate the angle θi between vector Nm and Ni by the dot product of the two vectors:
Ni ⋅ Nm
, i=1, 2, 3, 4; (4) The normal vector of the compensated point P is
Ni ⋅ Nm
cos θ i =
4
N ww = ∑ e Ni (cos θ i ) 2 .
i =1
With the aid of Snell’s law and the multi-cross product method, the reconstructed
image was obtained shown in Figure 3.14. Interestingly, very light circular patterns on
the image plane can be observed which was not presented in the direct measurements
shown in Figure 3.11. This might have been the result of the circular motion by the fast
tool servo path, since the CMM probe that mapped the lens surface followed a
meandering trajectory. The edges were not fully reconstructed because of the probing tip
radius that prevents the measurements from being taken too close to the edge as the lens
was secured in a lens holder. This information is a demonstration that the molded lenses
retained the design geometry.
40
Figure 3.14: Image reconstruction using the CMM measurement
The optical performance of the molded lens was studied using a molded lens in a
similar setup as shown in Figure 3.7. As illustrated in Figure 3.15, when illuminated by a
point light source (at 30 mm above the lens but outside of the figure), the clear image of
the IPT logo is projected to a screen placed 30 mm below the freeform lens.
Figure 3.15: Image formed by the molded freeform optics
41
The results in this section proved that the ray trace method and multi-cross
product method for normal vector determination are effective and accurate for freeform
optics image reconstruction; also the method can be used for contact iterative
measurement with less error from the variable surface slopes and probe radius.
42
CHAPTER 4
OPTICAL MEASUREMENT
The optical performance of the injection molded optical components is influenced
by the change of the process parameters. To obtain high precision and low cost injection
molding optical elements, it is very important to systematically investigate the
relationship between the process conditions and the optical performance of the molded
optics. Typical optical performance indicators include birefringence, refractive index,
and surface scattering.
4.1 Birefringence (Residual Stress) Measurement
When the injection molded parts are ejected from the mold cavity, residual
stresses due to the molding process will remain in the molded parts and additional
stresses will occur during cooling. The residual stresses are mainly from three sources:
flow induced stress due to filling, thermally induced stress due to cooling, and frozen-in
stress due to packing pressure. The thermal induced stresses are much higher than the
flow induced stresses so the latter can be omitted sometime. It is well known that the
molded-in residual stresses result in birefringence in an injection molded optical lens and
43
thus affect its optical quality (refractive index variation, unwanted light path deviation as
well as intensity change that can all result in image quality deterioration). Therefore the
residual stresses and birefringence measurement are very important for optical system
quality assessment.
Figure 4.1: Principle sketch of plane polariscope
The birefringence is usually measured by polarimeter. The principles of a plane
polarimeter are schematically illustrated in Figure 4.1.
A typical plane polarimeter
includes three major components, an illuminator or light source, a polarizer, and an
analyzer. The polarizer and analyzer are two identical plane polarizers. The light
intensity (which can be displayed on a screen or viewed directly) behind the analyzer can
be described using the following equation [Aben, 1993]:
I = I 0 sin 2 2ϕ sin 2
44
Δ
2
(4-1)
where φ is the inclination angle between the principal stress and axis of polarization for
the analyzer. The phase difference Δ is related to the wavelength λ of the light wave by:
Δ=
2π
2π
δ
2π =
(n2 − n3 )d =
C (σ 2 − σ 3 )d
λ
λ
λ
(4-2)
where C is a material property called the stress-optic constant, ni are the refractive indices
along the principal axis and σi are the principal stresses.
The birefringence is measured by PS-100-SF plane polarimeter (Strainoptics Inc,
www.strainoptics.com).
On the PS-100-SF polarimeter, the analyzer can be rotated
around the central axis to adjust the fringe color of the point of interest. To measure the
residual stresses, the molded lens was first placed in the optical system and rotated
around its central axis until the point of interest was in the brightest region. The analyzer
was then rotated to a position when the neighboring fringe appeared at the point of
interest on the sample. The neighboring fringe appeared when the analyzer introduced an
equal amount of retardation at the point of interest on the sample. The reading off the
marks on the analyzer then provides quantitative information of the optical retardation of
that point.
Most of the injection molded lens area has a relatively low level of residual
stresses. Thus it is difficult to quantitatively read the neighboring fringe that passed the
point of interest. To obtain more accurate measurement results, five lenses under the
same process condition with the same orientation are packing together during the
45
measurement. The value of the residual stress for the interested point is then only onefifth of the measurement value.
The measurement results are shown below. The unit of vertical axis is nanometer
which stands for the optical retardation from the molded lens and the unit of horizontal
axis is millimeter, representing the distance between the measured point and the lens
center. The flow direction is the inverse direction of +X.
In Figure 4.2, packing pressure was adjusted at eight levels as 5% (10.9 MPa),
15% (32.7 MPa), 25% (54.5 MPa), 27% (58.9 MPa), 29% (63.2 MPa), 31% (67.6 MPa),
33% (71.9 MPa) and 35% (76.3 MPa) of the maximal machine injection pressure (218
MPa) while all the other process parameters were remained unchanged. From the figure,
it can be concluded that the higher the packing pressure, the higher the birefringence
value in the part. However, with increasing distance from the gate, the dependence of
retardation on packing pressure became weak. Also the nearer to the gate location, the
larger the retardation value is because flow-induced residual stress is more concentrated
around the gate area. Moreover, the retardation value is almost the same for the parts
with packing pressure less than 25%. If the uniform retardation for the optical lens is
needed, the lower packing pressure is a better option.
46
Figure 4.2: Retardation comparison with different packing pressure
In Figure 4.3, mold temperature was adjusted at three levels as 150 °F (65.5 °C),
170 °F (76.6 °C) and 190 °F (87.7 °C) while all other process parameters were kept the
same.
With increasing of mold temperature, the thickness of solidified layer is
decreasing as well as the associated stresses. In the figure, it can be seen that the higher
the mold temperature, the lower the birefringence in the part.
47
Figure 4.3: Retardation comparison with different mold temperature
In Figure 4.4, different polymer melt temperature was tested at two levels as
450 °F (232.2 °C) and 470 °F (243.3 °C) while all the other process parameters were the
same to each other. More flow induced stresses and frozen-in stresses will occur under
lower melt temperature. As a result, in the same figure, higher melt temperature would
result in lower birefringence in most of the part.
48
Figure 4.4: Retardation comparison with different melt temperature
Moldflow Plastic Insight 6.1 (www.moldflow.com) can simulate optical
birefringence. 3D model of the molded part includes two symmetric lens cavity, gates,
runner and sprue for accurate simulation. The 3D model and simulation result under
different packing pressure were shown in Figure 4.5 and Figure 4.6 which show the same
tendency as the result from the polarimeter for points at the center part of the molded lens.
With the increasing of the distance from the gate, the retardation is decreased and with
the increasing the packing pressure, the retardation increases in the molded lenses.
However, since the material used in this dissertation research (PMMA Plexiglas® V825)
is not included in the birefringence material database of MPI 6.1 and the substitute in the
simulation is PMMA Sumipex HT55X from Sumitomo Chemical Company which may
be different from the experiment material in optical, mechanical and thermal properties.
In addition, simulation on the lens edge is not accurate (even in MPI 6.1’s birefringence
49
analysis examples, the retardation value on the edge is as high as tens of microns), the
simulation results are as high as times of the measurement results.
Figure 4.5: 3D model and birefringence simulation result from Moldflow
50
Figure 4.6: Retardation simulation result comparison with different packing pressure
From the MPI 6.1 simulation and measurement results, it is clear that lower
packing pressure, higher mold and melt temperature will result in smaller birefringence
(residual stresses). The process conditions can be optimized for lower birefringence,
however this may lead to more geometry deviation at the same time without proper
compensation scheme (such as a freeform lens mold). Base on the measurement results,
the magnitude of the maximal residual stress in the 30 mm diameter range is less than
100 nm. It is believed that such a low level birefringence can be neglected when
comparing with the geometry error (minimal level about 3 to 10 µm). However the
birefringence may play a bigger role after the lens geometry has been compensated (i.e.,
the geometry error has been reduced).
51
4.2 Refractive Index Measurement
In scientific terms, a wavefront may be defined as “the surface over which an
optical disturbance has constant phase” or “the surface which joins individual points on
rays which have the same optical path length”. The optical path length is simply the
distance a ray travels, multiplied by the refractive index of the material travels in. For an
ideal plano lens, the exiting wavefront surface should be a plane which means the exiting
wavefront error should be zero. However, injection molding process can introduce
inhomogeneous index distribution across the part. If the part geometry is perfectly flat on
both side and two sides are parallel to each other, the index change in the lens can be
obtained from the measurement of wavefront change.
In reality, lenses with perfect geometry are hard to come by for index
measurement thus we need to remove the effect of geometry error. One solution for the
surface geometry compensation is to submerge the molded lenses into an index matching
fluid, a special fluid with nominal refractive index that matches that of PMMA. The
index variation can be measured by a wavefront measuring system, such as a Shack
Hartmann sensor (SHS). SHS has been widely used in both precision optical and vision
science research with object to sample various points on the emerging wave and derive
the shape of the wavefront. In essence, a Shack-Hartmann plate is a series of microlenses
arranged in a linear fashion. Each lenslet focuses a view of the point source through
various points of the entrance pupil. As such, the SHS determines the shape of the
wavefront on the exiting pupil.
The slope of the wavefront is calculated with the
52
displacement and the focal length of the lenslet as shown in Figure 4.7 [Trusit, 2004].
After examining the slope at each micro lenslet in the x and y meridian, the entire
wavefront can be plotted in 3D format. The wavefront error which describes the optical
path difference between the measured wavefront and the reference wavefront is derived
mathematically from the reconstructed wavefront.
CCD device
Single micro lenslet
Position of laser spot for
calibrated plane wavefront
θ
Optical axis of lenslet
θ
∆y
Measured wavefront
Position of laser spot for
measured wavefront
Plane wavefront
Focal length of micro lenslet
Figure 4.7: Calculation of the slope of the wavefront at individual lenslet
A typical SHS based measurement setup is shown in Figure 4.8. The filter in the
optical path is used to adjust the intensity of the laser to avoid saturation on the sensor.
The aperture of the Hartmann sensor is 6.4 mm × 4.8 mm of a rectangular shape. If large
size samples are measured, beam reducer is needed in the measurement setup.
53
Because of different slope, each section of the wavefront will be imaged at
different position on the CCD (Coupled Charge Device) in the SHS system. The system
error will be nullified prior to taking each measurement. With the aid of the matching
fluid, index variations can be mapped for the entire molded lens. The different index
variation for the lenses molded under different process conditions can be transferred into
geometry changes in the optical system.
This information is useful for mold
compensation.
He-Ne Laser Filter
Matching Fluid
Beam Reducer Hartmann Sensor
Test lens
Microlens CCD
Array
camera
Figure 4.8: Index measurement setup
The measurement results are shown as follows. The unit for the color bar is
micron. The dimension of the molded lens is in millimeter.
54
In Figure 4.9 (a), the mold temperature was 150 °F while in Figure 4.9 (b), it was
190 °F. In this experiment, only mold temperature was adjusted, all other process
parameters were kept unchanged. The flow direction is from left to right. From the
measurement results, it can be seen that higher mold temperature will bring smaller index
deviation distribution.
(a) Lower mold temperature
(b) Higher mold temperature
Figure 4.9: Wavefront error of the molded lens under different mold temperature in fluid
In Figure 4.10 (a), the packing pressure was 35 % and 27 % in Figure 4.10 (b). In
this experiment, only the packing pressure was adjusted, all other process parameters
were kept the same. From the measurement results, it can be seen that lower packing
pressure will also introduce smaller index variation in the lens.
55
(a) Higher packing pressure
(b) Lower packing pressure
Figure 4.10: Wavefront error of the molded lens under different packing pressure in fluid
With the aid of the optical matching fluid, we can determine the average index
distribution in the part. While for the real optical system, the optical elements cannot be
immersed into the matching fluid. When the lens is used in the air or other media, the
wavefront aberration from the part is not only caused by inhomogeneous index but also
by geometry error. Sometimes the geometry error may compensate for some index
change in the part, so the RMS (root mean square) value for the wavefront may actually
be smaller than that in the matching fluid.
The same lenses as in Figure 4.10 (a) and (b) were also measured in the air. The
measurement results are shown in Figure 4.11.
In Figure 4.11, the measurement was obtained using the same setup in Figure 4.8
except the lens sample was in the air. From the surface measurements, we discovered
that the surface geometry accuracy of the lens molded using 35% packing pressure is
56
much better than that of the lens molded using 27% packing pressure, so the
measurement result in the air does make sense. In this case, the surface geometry of the
lens in the lower packing pressure brings more aberration to the wavefront error.
(a) Higher packing pressure
(b) Lower packing pressure
Figure 4.11: Wavefront error of the molded lens under different packing pressure in air
The index deviation measurements can be used for mold compensation.
Combining the surface and thickness measurement results and index distribution, the
modified mold inserts can be designed and fabricated. With the modified freeform mold,
the molded lens will bring improved optical performance.
4.3 Optical Effects of Surface Finish
Ultraprecision single point diamond turning process has been used extensively for
direct optical mold fabrication in injection molding. With this process, optical surfaces
can be fabricated without post machining polishing. Although ultraprecision diamond
57
machining process can create optics with surface roughness down to tens of nanometers
or even lower, the diamond machined surfaces still have characteristic periodic tool
marks and defects [Ikawa, 1991; Hocheng, 2004; Cheung, 2000; Khanfir, 2006]. In
optical applications, these defects result in reduced optical performance, such as
scattering and distortion, which can negatively affect efficiency and imaging quality of
the optical surface. Due to the transcribility of the injection molding process, the molded
optical elements will be affected by the mold quality. Injection molding process can
reduce the optical effects from the tool marks comparing to the mold insert surface. In
addition to mold surface quality, process conditions also influence the surface finish and
optical behavior of transparent plastics [Gunes, 2006; Gunes, 2007] in injection molded
plastic lenses.
To improve optical performance of the injection molded lenses, in this
dissertation research, the following studies were performed: 1) Study the characteristics
of the diamond machined mold surface and molded lens surface; interpret the surface
profile based on their optical effects. 2) Analytically correlate surface characteristics to
optical effects such as high order diffractions and background scattering; quantify the
surface effects on the optical performance, specifically the effects due to tool marks both
on the mold and lens surface.
3) Experimentally identify the relationship between
process conditions (including packing pressure, mold temperature and melt temperature)
and the surface characteristics (optical performance).
Develop a methodology to
determine the optimal conditions for an injection molded lens.
58
4.3.1 Theoretical Analysis
4.3.1.1 Surface Characteristics of a Diamond Machined Surface
Generally speaking, ultraprecision diamond machined surfaces have these
characteristics: 1) Periodic tool marks along the feed direction. 2) Low frequency spatial
variation. 3) High frequency and other random vibration. The periodic tool marks, low
frequency surface variation and high frequency vibration can appear on a diamond
machined surface. The causes of these surface characteristics are complicated. It may
include machine tool vibration, material induced vibration, servo control, temperature
variation, tool wear, and chips scratches. Figure 4.12 shows a typical two-dimensional
(2D) diamond machined surface profile.
Figure 4.12: Profile of a typical diamond machined surface
59
A diamond machined surface can be considered as the combination of these three
components. In terms of optical performance, these components pose different effects as
shown in Figure 4.13.
For example, the periodic tool marks introduce high order
diffraction in addition to the specular reflection.
The position of the high order
diffraction is decided by the amount of tool mark spacing. The diffraction intensity, on
the other hand, is related to the tool mark depth. The periodic tool marks induced high
order diffraction will affect surface reflectivity and imaging quality. For lower frequency
variation, one potential effect is the sideband diffraction peak alongside the high order
diffraction peaks as in Figure 4.13. The background scattering is related to random
surface roughness component.
1.0E+07
Sensor C urrent (am p)
1.0E+06
1.0E+05
Specular Reflection
High order diffraction
1.0E+04
1.0E+03
Sideband diffraction peak
1.0E+02
1.0E+01
Background scattering
1.0E+00
1.0E-01
1.0E-02
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95
Angle (Degree)
Figure 4.13: Specular reflection, high order diffraction and scattering from the diamond
machined surface in Figure 4.12
60
The injection molding process can duplicate the mold surface to molded lens
surface so the characteristics on the molded lens surface will be similar to the diamond
machined mold surface.
In the following section, the scattering from the diamond
machined mold surface and molded lens surface will be analyzed using scalar method
[Stover, 1995]. The specular reflectivity and imaging quality of the mold and molded
surfaces will also be analyzed.
4.3.1.2 Scalar Method for Diffraction and Scattering Calculation
The distribution of light diffracted off a diamond-machined surface can be
calculated using two-dimensional Discrete Fourier Transform (2D DFT), which is based
on the scalar diffraction theory.
In Figure 4.14, the electrical field E1(x1, y1,z1 ) at point B(x1, y1,z1 ) on the
observation plane X1 − Y1 is the result of light diffracting from object plane X 0 − Y0 and
can be expressed as [Collins, 2002]:
E1 (x1, y1, z1 ) =
jk ∞ ∞
e− jkr
dx0dy0
∫ ∫ E0 (x0 , y0 , z0 ) cosθ
2π −∞ −∞
r
(3-3)
where r is the length of line AB, E 0 ( x 0 , y 0 , z 0 ) is the output electrical field from
plane X 0 − Y0 at point A(x 0 , y 0 ,z 0 ) , θ is the included angle between line AB and Z axis.
61
A( x 0 , y 0 , z 0 )
X0
Y0
b
a
r
Y1
X1
θ
B ( x1 , y1 , z1 )
Z
Ei
Figure 4.14: Diffraction from a diamond machined surface
Under the far field diffraction condition, the electrical field on the observation
plane can be written as:
1 1
jk z1 ab − jkr1
E1 (u1 , v1 , z1 ) =
e
E0 ( x0 , y 0 , z 0 )e j 2π ( u0u1 +v0v1 ) du 0 dv0
2
∫
∫
2π r1
0 0
where u 0 =
(3-4)
x0
y
x a
y b
, v0 = 0 , u1 = 1 , v1 = 1
, and r1 = x12 + y12 + z12 , a, b are the
a
b
λ r1
λ r1
dimensions of the sample in X and Y direction.
Assume that the incident light is a plane wave, then Ei = Ee
− j ( x0κ ix + y0κ iy + z0κ iz )
, where
κ ix , κ iy ,κ iz are the unit vectors of the incident light. The surface profile of the sample is
known as d = d(x 0 , y 0 ) . For the diamond machined sample that has surface roughness
much less than wavelength, surface profile variation will introduce a phase delay to the
62
incident
light.
E0 ( x0 , y 0 , z 0 ) = Ee
At
Z0 = 0
the
− j ( x0κ ix + y0κ iy + 2 dκ iz )
plane,
the
reflected
wave
is
. Substitute E0 into Eq. (3-4) and write it in discrete
form:
E1 ( n2 , m2 , z1 ) =
N −1 M −1
j 2π (
jk z1ab − jkr1
− jκ z 2 d ( n0 , m0 )
e
E
e
00 ∑ ∑ e
2
2π r1 MN
0
0
where n0 = u 0 N , m0 = v0 M , n2 = N (
n0 n2 m0 m2
+
)
N
M
(3-5)
κy 1 κy
x1 κ x
y
1 κ
− ) /( − x ), m2 = M ( 1 − ) /( − )
r1λ 2π λ 2π
r1λ 2π λ 2π
Eq. (3-5) has the form of 2D DFT. By Eq. (3-5), the field and intensity of the diffraction
light can be calculated if the surface roughness data are known.
4.3.2 Experiment and Measurement
The mold inserts were fabricated on the Nanotech 350 FG Freeform Generator
with Al 6061. Mold inserts used in this research were diamond turned with a normal size
diamond tool (tool nose radius is 3.048 mm, rake angle is 0o and clearance angle is 8o).
The spindle speed was kept constant (1,000 rpm) while the feed rate was at 20 mm/min.
As a result, the tool mark spacing was 20 μm. The nominal depth of cut was 3 μm.
Based on the conclusion from basic measurement and ANOVA results, only a few
process parameters are crucial to our experiments including mold temperature, polymer
melt temperature and packing pressure. These parameters were set at different levels for
the experiments. To evaluate the part quality under each process condition, the collected
63
parts need to be consistent in the specifications of interest. For each process condition in
our experiments, ten trial parts were made and discarded then five parts were collected
for measurement. Room temperature and humidity were also important for the part
quality, so all selected experiments were conducted in one single day to keep the
environmental conditions consistent.
Both surface profile and surface scattering measurements were taken on the mold
and molded lens surfaces. The surface profiles of these samples were also measured on
the Veeco NT 3300 Profilometer.
The Veeco profilometer is built on phase shift
interferometry and its schematic is shown in Figure 4.15 (recreated from James C.
Wyant’s class notes [Wyant, 2000]).
TEST SAMPLE
PZT
DRIVING
MIRROR
LASER
BS
IMAGING LENS
DETECTOR
ARRAY
DIGITIZER
PZT
CONTROLLER
COMPUTER
Figure 4.15: Schematic of phase shift interferometry
64
The scattering from the sample surface was measured by a home built device
shown in Figure 4.16. A silicon photo diode detector (Edmund, 53-371) was connected
to an Aerotech ADRT-200 rotary stage that rotates around the sample and measures the
intensity of the scattering light from the mold or molded lens surface. The sample to be
measured was mounted on the Aerotech stage that is capable of 4-DOF (degree of
freedom) adjustment, thus can align the sample surface right above the rotational center
of the rotary stage. The incident angle of the laser beam on the sample was fixed but the
reception angle on the photodiode was adjusted on the rotary stage. In this research, the
incident angle was 5 degree from normal and the reception angle on the photo diode was
varied from 2 degree to 90 degree at 0.2 degree increment. The measurement system was
fully computerized and capable of high precision automatic scanning and measuring
(Figure 4.17).
Sample
Aerotech Rotary
Stage
Incident Angle
Adjust Stage
He-Ne Laser
Detector
Figure 4.16: Setup of the scattering measurement device
65
Incident light
Sample
Reception angle θ
Detector
To Rotary
Stage
Amplifier
Drive
Computer
Figure 4.17: Scattering measurement system
4.3.3 Results
4.3.3.1 Comparison of Surface Profile Measurement and Direct Scattering
Measurement
Several methods can be employed to measure surface roughness. In this research,
both Veeco surface profilometer measurement and direct scattering measurement were
used to evaluate the surface quality. Figure 4.18 (a) and 4.18 (b) are 3D surface profiles
of mold insert and molded lens surface off this insert measured by Veeco.
The
magnification used was 10 × 2, which had a 0.402768 μm × 0.469896 μm resolution and
296.0345 μm × 225.0802 μm measurement scope in X and Y direction respectively. By
using 2D FFT method, Figure 4.18 (c) and 4.18 (d) are the calculated average onedimensional spectrum along Y direction of the samples, which are shown in Figure 4.18
66
(a) and 4.18 (b). Figure 4.18 (e) and 4.18 (f) are the measured one-dimensional surface
scattering along Y direction of the samples from the same mold and molded lens surface
in Figure 4.18 (a) and 4.18 (b) respectively. In the scattering measurement, a He-Ne laser
(wavelength 632.8 nm) was used. From the comparison between Figure 4.18 (c) and
Figure 4.18 (e), both methods clearly were capable of revealing the existence of the
periodic tool marks.
On Figure 4.18 (e) and 4.18 (f), for clear demonstration, the
diffraction orders were not completely marked. The direct scattering measurement has a
better measurement bandwidth and it is more sensitive than Veeco surface profilometer
measurement so direct scattering measurement can provide more information for surface
quality evaluation.
(a) 3D surface profile of a 20 μm tool mark spacing mold insert, measured by Veeco
white light profilometer
67
(b) 3D surface profile of the molded lens, measured by Veeco white light profilometer
(c) Calculated average 1D spectrum of the same mold surface
68
(d) Calculated average 1D spectrum of the same molded lens surface
(e) Directly measured surface scattering of the same mold surface
69
(f) Directly measured surface scattering of the same molded lens surface
Figure 4.18: Comparison of the mold insert and molded lens
From the Veeco profilometer measurement result in Figure 4.18 (a), it shows that
diamond machined surface has characteristic periodic tool marks. These defects can
result in reduced optical performance, such as scattering and distortion. The molded lens
surface however is smoother than the mold insert surface which showed molding process
can average the roughness and improve the surface quality to some extent by comparison
of Figure 4.18 (a) and 4.18 (b). One dimension surface spectrum in Figure 4.18 (c) and
4.18 (d) shows the periodic tool marks are deeper and sharper on the mold surface than
the molded lens surface. The injection molding process can reduce the optical effect
when was compared to the mold insert surface, which are clearly shown in Figure 4.18 (e)
and 4.18 (f). The high order diffraction peaks can be clearly identified on the mold and
molded lens scattering results. Obviously the high order diffraction from the molded lens
70
surface is largely reduced comparing to the mold insert. The measurement results show
that it is possible to obtain optical quality injection molded elements with direct diamond
machined mold even the mold surface itself has tool marks and can generate higher
intensity high order diffraction just shown in Figure 4.18 (c) and 4.18 (e), which can
greatly reduce the production cost for mold fabrication and lens molding.
4.3.3.2 Relationship of Molded Surface Quality and Injection Molding Process
Conditions
For the injection molded optical lens in this research, it was designed to be used
as a refractive device. However, since the refraction scattering measurement will be
influenced by both surfaces of the lens, it is difficult to establish the relationship of the
molded surface quality and injection molding process conditions. Therefore, for the
injection molded lens, reflection scattering measurement was also used to evaluate the
molded lens surface optical quality. Since the first order diffraction has the highest
intensity among all diffraction orders and was the closest to specular reflection, therefore
first order diffraction was used to measure the surface diffraction effects. The surface
roughness and tool mark depth were used to evaluate the molded surface quality.
To investigate the relationship of the optical effects and the packing pressure, in
the molding experiments, the packing pressure was set at seven levels, 5% (10.9MPa),
10% (21.8MPa), 15% (32.7MPa), 20% (43.6MPa), 25% (54.5MPa), 30% (65.4MPa) and
35% (76.3MPa) of the maximal machine injection pressure (218MPa). Packing pressure
higher than 40% of the full capacity was considered over packing for this case study. In
71
addition to packing pressure, all other process parameters remained unchanged. Higher
packing pressure results in smaller shrinkage as well as surface quality. Figure 4.19 (a)
shows the relationship between first order diffraction intensity from the molded lens
surface reflection and the packing pressure.
The first order diffraction intensity
decreased as the packing pressure increased with approximate linear relationship. In
optical industry, the common method used to evaluate the surface quality is the root
means square value of the surface roughness (Ra). Figure 4.19 (b) shows the measured
surface roughness Ra of the molded lens surface under different packing pressure. Figure
4.19 (c) shows the measured tool mark depth of the molded lens surface. The surface
roughness and tool mark depth also decreased when packing pressure increased with
similar trend as the first order diffraction intensity. Based on the experiment results, to
obtain optical performance molded lenses especially for imaging purpose, higher packing
pressure should be selected to reduce the high order diffraction losses and ghost image.
(a) First order diffraction intensity
72
b) Surface roughness measured by Veeco
(c) Measured tool mark depth
Figure 4.19: Experimental results of the lens molded under different packing pressure
73
To investigate the relationship of the optical effects and the mold temperature, in
the molding experiments, the mold temperature was set at five levels, 110°F (43.3°C),
130°F (54.4°C), 150°F (65.5°C), 170°F (76.6°C) and 190°F (87.7°C). The mold heating
system controls and adjusts the mold temperature by controlling the temperature of the
circulating water system, so the temperature higher than water boiling point can not be
done for this case study.
In addition to the mold temperature, all other process
parameters remained unchanged. Figure 4.20 (a) shows the relationship between the first
order diffraction intensity from the molded lens surface reflection and the mold
temperature. The first order diffraction intensity increased as the mold temperature
increased. Figure 4.20 (b) and 4.20 (c) show the measured surface roughness Ra and the
tool mark depth of the molded lens surface under different mold temperature. The tool
mark depth also increased when mold temperature increased with similar trend as the first
order diffraction intensity. The surface roughness under different mold temperature was
close and had no obvious tendency as the mold temperature was changed. From the
experiment results, lower mold temperature was better for reducing the effect of high
order diffraction for the molded lens.
74
(a) First order diffraction intensity
(b) Surface roughness measured by Veeco
75
(c) Measured tool mark depth
Figure 4.20: Experimental results of the lens molded under different mold temperature
To investigate the relationship of the optical effects and the melt temperature, in
the molding experiments, the melt temperature was set at four levels, 430°F (221.1°C),
450°F (232.2°C), 470°F (243.3°C) and 490°F (254.4°C). The melt temperature from the
processing information provided by material vendor is 430°F. If the melt temperature
was raised to more than 500°F it will cause degradation in the material. In this part of the
study, all other process parameters remained unchanged. Figure 4.21 (a) shows the
relationship between first order diffraction intensity from the molded lens surface
reflection and the melt temperature. The first order diffraction intensity decreased as the
melt temperature was increased. Figure 4.21 (b) and 4.21 (c) show the measured surface
roughness Ra and the tool mark depth of the molded lens surface under different melt
temperature. The tool mark depth also decreased when melt temperature was increased
76
with similar trend as the first order diffraction intensity. The surface roughness has
different tendency with the first order diffraction intensity as the melt temperature
changes which means sometimes the optical surface quality cannot be evaluated by
surface roughness alone. From the experiment results, higher melt temperature was
better for reducing the effect of high order diffraction for the molded lenses.
(a) First order diffraction intensity
77
(b) Surface roughness measured by Veeco
(c) Measured tool mark depth
Figure 4.21: Experimental results of the lens molded under different melt temperature
78
The surface roughness Ra of the mold insert is 26.76 nm and the measured tool
mark depth of the mold insert is about 99 nm. Comparing with molded lenses, it can be
seen that injection molding process will reduce the surface roughness and smooth the
sharp tool marks duplicated from the mold insert surface which can enhance the optical
performance of the molded lens.
In this section, the optical effects of the diamond turned mold insert surface and
molded lens surface were studied. The optical scattering and lens surface profiles were
related by scalar analysis of the light reflected off the optical surfaces. The direct
scattering measurement is more sensitive than surface profile measurement. Diamond
machined surfaces have characteristic periodic tool marks which can cause degradation in
efficiency and imaging quality of the optical surface. Injection molding process has
smoothing effect that can reduce optical effects caused by tool marks on the molded lens
surface comparing to the mold insert surface itself. By conducting selected experiments
under different molding conditions, optical effects of the process conditions for molded
lens surface were investigated in this study for further optimization of the surface finish.
Specifically, three parameters, packing pressure, mold temperature and melt temperature,
are varied in numerous levels. For each setting, the characteristics of mold and molded
part surface and scattering were both analytically and experimentally studied. The results
showed that the appropriate process conditions will enhance the optical performance of
the molded lens, such as higher packing pressure, higher melt temperature and lower
mold temperature.
79
CHAPTER 5
ALVAREZ LENS MANUFACTURING
5.1 Alvarez Lens
5.1.1 Alvarez Lens Design
Alvarez lens is a unique optical device. An Alvarez lens pair [Alvarez, 1967]
consists of a pair of bicubic phase profile optics, one is the inverse shape of the other.
When these plates are placed in registration, the resulting phase profile is a null, owing to
the cancellation of the two phase profiles. However, if one of the plates is translated in
the plane of the phase profile, there is no longer perfect cancellation of the two phase
profiles. The residual phase variation is the differential of the cubic profiles, resulting in
a quadratic phase variation. This quadratic phase variation is equivalent to a lens of a
certain focal length, as determined by the steepness of the individual cubic phase profiles
and the translation distance. Variable spherical or astigmatic power can be produced by
varying the relative translation in the x and y directions for wavefront correction.
For ophthalmic use, Alvarez lens makes possible simple, inexpensive, thin and
attractive variable-power spectacles that can be focused quickly and easily for near and
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distant vision, and yet provides a sharp, substantially undistorted view throughout the
field of vision at each setting. The lens is also useful in cameras and other optical devices.
The versatility of the Alvarez lens allows for dynamic correction of arbitrary astigmatic
aberrations.
In order to precisely define the lens parameters, a rectangular coordinate system is
used. The optical axis of the lens system is taken to be z axis and the lens thickness t is
measured parallel to the optical axis. If x axis is the lens moving direction then y axis is
perpendicular to x axis and z axis. Figure 5.1 is a schematic drawing of Alvarez lens pair
which is recreated from [Alvarez, 1967].
Figure 5.1: Schematic drawing of Alvarez lens pair
For the designed elements, the lower lens (element 1) thickness equation can be
expressed as:
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1
t1 = A( xy 2 + x 3 ) + Dx + E
3
(5-1)
1
where A( xy 2 + x 3 ) is the characterizing terms, A is a constant, which determines the
3
rate at which the power of the lens varies with movement of the lens elements relative to
each other along the x axis. D may be selected to minimize the lens thickness.
The upper lens (element 2) thickness equation can be expressed as:
1
t 2 = − A( xy 2 + x 3 ) − Dx + E
3
(5-2)
which is the same as the lens equation of element 1, except for a reversal of the algebraic
sign of all terms of the lens equation except the constant term E.
In the neutral or zero-power position of the lens element shown in Figure 5.1, the
curved surface of element 2 exactly fits the curved top surface of element 1, so that the
two elements could perfectly fit together with no space between them. However, in
practice, a small space is left between the two lens elements to permit the movement of
one element relative to the other for adjusting the power of the lens. The space should be
as small as convenient practice to keep the validity of the thin-lens approximations.
The optical thickness tc of the composite lens at any point is equal to the sum of
the optical thicknesses of element 1 and 2 at that point. In the neutral position, the
82
composite optical thickness is obtained as t c = t1 + t 2 = 2 E . So for this neutral position,
the composite lens is optically equivalent to a flat plate of glass.
Points in the plane of the lens system may now be identified by a system of
coordinates X, Y that remain stationary while the lens elements move in opposite
directions along the X axis or Y axis. The optical axis passes through the point X=0, Y=0.
If both elements are moved by equal amounts (represented by d) in opposite directions
along X axis, the thickness equations for the two lens elements may be written:
t1 = A( X − d )Y 2 +
1
A( X − d ) 3 + D( X − d ) + E
3
t 2 = − A( X + d )Y 2 −
1
A( X + d ) 3 − D( X + d ) + E
3
(5-3)
(5-4)
The composite-lens optical thickness tc is obtained by adding t1 and t2, with the
following result:
t c = t1 + t 2 = −2 Ad ( X 2 + Y 2 ) −
2
Ad 3 − 2 Dd + 2 E
3
(5-5)
The term − 2 Ad ( X 2 + Y 2 ) describes a convex or converging spherical lens
having a power (the reciprocal of focal length) proportional to Ad. All of the other terms
are independent of X and Y, and therefore represent a uniform thickness over the whole
area of the lens. Thus the composite lens is, in thin lens approximation, a theoretically
perfect spherical lens of variable power, the power being linearly proportional to the
83
distance d that the lens elements are displaced from their neutral positions. The range of
power variation for a given displacement d is determined by the value of the coefficient A,
which is designed by purpose.
If both elements are moved by equal amounts (represented by d) in opposite
directions along Y axis, the thickness equations for the two lens elements may be written:
t1 = AX (Y − d ) 2 +
1
AX 3 + DX + E
3
t 2 = − AX (Y + d ) 2 −
1
AX 3 − DX + E
3
(5-6)
(5-7)
The composite-lens optical thickness tc is obtained by adding t1 and t2, with the
following result:
t c = t1 + t 2 = −2 AdXY + 2 E
(5-8)
The term -2AdXY represents a variable prism along the Y axis of the lens which
can be used to compensate for parallax.
5.1.2 Alvarez Lens Fabrication
Although the Alvarez lens has a surface profile that can be described precisely
using an analytical formula, it has been largely an impractical task to create such an
optical surface because the cubic surfaces were difficult to fabricate using conventional
machining processes such as grinding and polishing. There have been a few fabrication
84
methods mentioned in reported recently [Wilhemsen, 1999] which are small tool
polishing and photolithography. Both methods are expensive and time-consuming.
Injection molding is a good choice for Alvarez lens fabrication for its high
volume and low cost with high precision. With the accurate mold insert and precision
molding condition control, Alvarez lens or lens array can be fabricated in very good
quality and reasonable cost.
With the aid of the high precision machine and the
innovative STS process, Alvarez lens mold inserts were fabricated first on the 350 FG
machine and then the Alvarez lenses were injection molded. The Alvarez lens mold
insert and some samples of molded lens are shown in Figure 5.2. The diameter of the
mold insert is 39.88 mm and the width of the Alvarez lens is 20 mm.
(a) Alvarez lens mold
(b) Molded freeform lenses
Figure 5.2: Alvarez lens mold insert and molded lens
85
5.1.3 Alvarez Lens Measurement
5.1.3.1 Zernike Polynomials
Optical system aberration is expressed as a weighted sum of power series terms
that are functions of the pupil coordinates. Each term is associated with a particular
aberration or mode, for example, spherical aberration, coma, astigmatism, field curvature,
distortion, and other higher order modes [Malacara, 1978].
Zernike polynomials form a complete set of functions or modes that are
orthogonal over a circle of unit radius and are convenient for serving as a set of basis
functions. This makes them suitable for accurately describing wavefront aberrations as
well as for data fitting. Zernike polynomials are usually expressed in polar coordinates,
and are readily convertible to Cartesian coordinates. These polynomials are mutually
orthogonal, and are therefore mathematically independent, making the variance of the
sum of modes equal to the sum of the variances of each individual mode. They can be
scaled so that non-zero order modes have zero mean and unit variance. This places all
modes in a common reference frame that enables meaningful relative comparison among
them. The wavefront may be described as [Trusit, 2004]:
W ( ρ , θ ) = C1−1 Z 1−1 + C11 Z 11 + C 2−2 Z 2−2 + C 20 Z 20 + C 22 Z 22 + C 3−3 Z 3−3 + C 3−1 Z 3−1 + C 31 Z 31 + ⋅ ⋅ ⋅ (5-9)
Each polynomial has three components: the normalization factor, a radially
dependent polynomial, and an azimuthally dependent sinusoid.
A double indexing
scheme is used where: n is the highest power or order of the radial polynomial and m is
86
the azimuthal or angular frequency of the sinusoidal component. An accompanying
single indexing scheme is also employed where the index j is used to represent the mode
number.
Normalization of each mode means that observation of the coefficients
immediately gives an indication of the level of influence that each type of aberration has
on the total wavefront error.
The Zernike polynomials are defined as [Thibos, 2002]:
Z nm ( ρ ,θ ) = N nm Rn ( ρ ) cos(mθ )
m
Z nm ( ρ ,θ ) = − N nm Rn ( ρ ) sin(mθ )
m
for m ≥ 0,0 ≤ ρ ≤ 1,0 ≤ θ ≤ 2π
for m < 0,0 ≤ ρ ≤ 1,0 ≤ θ ≤ 2π
(5-10)
(5-11)
For a given n, m can only take on values of -n, -n+2, -n+4, …, n
N nm is the normalization factor
N nm =
2(n + 1)
1 + δ m0
δ m 0 = 1 for m = 0, δ m 0 = 0 for m ≠ 0
(5-12)
Rn ( ρ ) is the radial polynomial
m
R (ρ ) =
m
n
( n− m ) / 2
∑
s =0
(−1) s (n − s )!
ρ n−2 s
s! ⎣0.5(n + m ) − s ⎦! ⎣0.5(n − m ) − s ⎦!
87
(5-13)
Table 5-1 contains a list of Zernike polynomials up to order 4 and their meanings
relative to the traditional Seidel or Primary aberrations [Mahajan, 1998]. The wavefront
aberration can be fitted as Zernike polynomials.
mode
j
order
n
Frequency
m
0
1
0
1
0
-1
Z nm ( ρ , θ )
1
2 ρ sin(θ )
2
1
1
2 ρ cos(θ )
3
2
-2
6 ρ 2 sin(2θ )
4
2
0
3 (2 ρ 2 − 1)
5
2
2
6 ρ 2 cos(2θ )
6
3
-3
8 ρ 3 sin(3θ )
7
3
-1
8 (3ρ 3 − 2 ρ ) sin(θ )
Coma along y-axis
8
3
1
8 (3ρ 3 − 2 ρ ) cos(θ )
Coma along x-axis
9
3
3
8 ρ 3 cos(3θ )
10
4
-4
10 ρ 4 sin(4θ )
11
4
-2
10 (4 ρ 4 − 3ρ 2 ) sin(2θ )
Secondary Astigmatism
12
4
0
5 (4 ρ 4 − 6 ρ 2 + 1)
13
4
2
10 (4 ρ 4 − 3ρ 2 ) cos(2θ )
Spherical Aberration,
Defocus
Secondary Astigmatism
14
4
4
10 ρ 4 cos(4θ )
Meaning
Constant term, or Piston
Tilt in y-direction,
Distortion
Tilt in x-direction,
Distortion
Astigmatism with axis at
± 45o
Focus shift
Astigmatism with axis at
0 o or 90 o
Table 5.1: Zernike Polynomials (up to 4th order)
The first order term, prism, is not relevant to the wavefront as they represent tilt
and are corrected using prism. The second order terms represent low order aberrations,
88
namely, defocus and astigmatism. Defocus represents the spherical component of the
wavefront. The astigmatic terms conversely describe the cylinder. Using these three
terms, any sphero-cylindrical lens can be described. Every mode after second order is a
high order aberration.
In order to summarize the wavefront error, numerical index were tried to describe
the wavefront error. At present, the mostly widely used means is the root mean square
(RMS) error. This term describes the weighted mean of the individual Zernike modes.
The RMS value describes the overall aberration. The RMS error can be calculated as:
RMS = (C 2−2 ) 2 + (C 20 ) 2 + (C 22 ) 2 + (C 3−3 ) 2 + (C 3−1 ) 2 + ⋅ ⋅ ⋅
(5-14)
5.1.3.2 Wavefront Aberration Measurement
According to the design principle, a pair of molded Alvarez lenses can be
measured together. The measurement system is shown in Figure 5.3. The wavefront
information can be obtained by the Shack-Hartmann sensor while the Alvarez lens pair
move to opposite direction either along the x-axis or the y-axis. In order to analyze the
wavefront quantitatively, the wavefront error needs to be expressed as Zernike
polynomials to fit the data in three dimensions.
89
Shack-Hartmann Sensor
Light Source
Alvarez Lens Pair
Figure 5.3: Measurement Setup for Alvarez Lens
As explained for Alvarez lens design in Section 5.1.1, when the relative
displacement of the Alvarez lens pair is in opposite direction along x-axis, the Alvarez
lens pair can be treated as a convex or converging spherical lens having a power
proportional to the relative displacement, so the spherical coefficient of the Alvarez lens
pair has linear relationship with the relative displacement. According to the design
principle in thickness expression Equation (5-5), the astigmatic coefficients should be
zero which conforms to the measurement results that the astigmatic coefficients are
almost zero.
Basic measurement results are shown below: The change in sphere power looks
fairly linear, and the astigmatism and higher order terms (up through 10th order) stay
fairly low. In Figure 5.4, the black line stands for the spherical coefficient (focus shift)
and the red line stands for the astigmatic coefficient with axis at 0° or 90° and green line
stands for the astigmatic coefficient with axis at ±45°.
90
Figure 5.4: Low order Zernike coefficients of the molded Alvarez lens pair while the
relative x-axis translation
In Figure 5.5, the blue line is the RMS value without spherical aberration and the
red line stands for the higher order terms only. The maximum RMS value is about 0.08
μm in the measurement range.
91
Figure 5.5: RMS value of the molded Alvarez lens pair while the relative x-axis
translation
Clinically, high order aberrations were found to have a mean RMS of
0.305±0.095 μm in 532 eyes across a 6 mm pupil [Trusit, 2004]. Comparing to the
measurement results, currently the molded Alvarez lens can fulfill the requirement of the
vision test.
The molded Alvarez lens quality will be affected by injection molding process
parameters. The molding experiments were conducted with different packing pressure
and different mold temperature. From the measurements of the pairs, the lens pair with
the lower RMS value is with a higher packing pressure and lower mold temperature. The
measurement results are shown in Figure 5.6 and Figure 5.7.
92
Figure 5.6: RMS value of the molded Alvarez lens pair while the relative x-axis
translation under different packing pressure
Figure 5.7: RMS value of the molded Alvarez lens pair while the relative x-axis
translation under different mold temperature
93
The residual stresses in the molded Alvarez lens also were measured by
polarimeter. The measurement result was shown in Figure 5.8.
Figure 5.8: Retardation of the molded Alvarez lens under different process parameters
Although the lower residual stresses (optical retardation) are obtained under lower
packing pressure, since the retardation of the molded Alvarez lens under higher packing
pressure (about 30%) is also in the acceptable range comparing to the common human
eyes aberration value, packing pressure 30% is a better choice to obtain functional
Alvarez lens.
5.1.3.3 Surface Measurement
Alvarez lens is difficult to fabricate because of its non axisymmetrical surface
with large deviation (or sag). For the same reason, it is also very difficult to measure the
94
molded Alvarez lens surface geometry.
Malacara and Cornejo used the method of
Newton’s fringes to determine the aspheric profile of a surface that deviates markedly
from a spherical surface [Malacara, 1970]. This method is useful if the aspheric deviates
from the nearest spherical by a few wavelength of light (10 to 20 λ). The principle of this
method can be used to measure the Alvarez lenses. The schematic of the metrology
system is shown in Figure 5.9.
CCD camera
Observing fringes
Partly reflecting
glass sheet
Laser head
Lens
Tiny gap
Mold insert
Figure 5.9: Alvarez lens geometry measurement
To do the measurement, the molded Alvarez lens and the null component (either
mold insert or machined plastic lens) are placed very close together until the
interferometric fringes appear. Fringes occur because the amplitude of each single light
95
ray from the source is divided by the lens surface and mold surface, one part of light is
reflected from the lens surface and the remaining light transmits the lens surface and then
reflected from the mold surface. The lights travel in different optical path. When the
light beams recombine, the interference may take place.
When mathematically
interpreting the interferograms, the deviation between the molded lens surface and mold
insert will be obtained. Using this method, freefrom optical elements such as a lens or a
mirror can be measured if a master (the null) is available.
With the success of the Alvarez lens fabrication, more and more lenses with
nonsymmetrical freeform surface profiles can be fabricated with high optical quality,
high volume and at a low cost.
5.2 Micro Alvarez Lens Array
Microlenses are important optical components that image, detect and couple light.
With
the
growing
demands
of
industrial
applications,
including
imaging,
telecommunication and detection systems, there are more requirements for microlens
arrays with higher geometry accuracy, more complicated surface profile and integrated
functions. However, for most microlens arrays, it is impossible to adjust their focal
length due to their fixed geometry. Alvarez lenses allow the focal length to be adjusted
by simply translating the lens pair along the normal direction to the light propagation.
Moreover, due to the asymmetrical surface profile, it is difficult to fabricate micro
Alvarez lens array by using conventional fabrication technology [Fritze, 1998; Keyworth,
1997; Mihailov, 1993; Popovic, 1988; Yu, 2003]. In this dissertation research, a freeform
96
surface with individual freeform lenslets was fabricated by ultraprecision machining
using slow tool servo and injection molding process to produce freeform microlenses
with high optical quality at a low cost. The methodology developed in this dissertation
research can be used for many other applications as well.
5.2.1 Mold Design and Fabrication
The micro Alvarez lens array design used in this dissertation research consists of
5×5 lens cavities as shown in Fig. 5.10. As mentioned, each micro Alvarez lenslet is a
bi-cubic phase profile optic. The dimension of each lenslet cavity is 1 mm × 0.5 mm and
the sagittal height (or sag) is 10 μm. The edge to edge interval distance of the optical
cavity is 0.4 mm and 0.2 mm, which is for lateral and vertical translation of the functional
Alvarez lens pair. A 5×5 lens array was machined on aluminum 6061 substrate to
demonstrate the machine capability although other patterns on non-ferrous materials can
be also easily obtained using this method. This approach has been adapted to making
almost any shapes for other optical applications, such as diffractive optical elements [Li,
2006].
97
Figure 5.10: Schematic drawing of Alvarez lens array
The mold inserts were fabricated by broaching process with C axis fixed on
FG350 ultraprecision CNC machine. The CNC tool path is nearly identical to the profile
of the Alvarez lens surface shape, differing only in diamond tool radius compensation. In
this setup, the Alvarez lens array mold was held on the vacuum chuck in C-axis mode,
i.e., the main work spindle was fixed without rotating during machining process. The
workpiece was first moved in X direction at a fixed step distance.
It was then
continuously moved vertically while the diamond tool was fed in Z direction based on the
analytical expression of the Alvarez lens surface. This process continued until the entire
expected surface was completely machined. Figure 5.11 shows the perspective three
dimensional (3D) view of the tool path generated for this 5×5 Alvarez lens array mold.
In the figure, the number of steps in X direction was reduced to show the tool path. For
each point on the calculated tool path, the Cartesian coordinates were decided by the step
98
size, the point position on its lens cavity, slope of the surface profile curvature on the
contact point between the diamond tool and the machined surface, and the cutter radius.
Figure 5.11: Broaching CNC tool path
The Alvarez lens surface equation for this study is t = a( xy 2 + 1 / 3x 3 ) + bx + c
where the parameters a, b and c are 0.32, -0.026 and 0 respectively. To reduce sag of the
entire surface, 10 µm limit was set for the range in Z direction. Z coordinates on the lens
surface were calculated from the surface equation or were set as either 0 or -10 µm when
the z value was out of range. For the points between the lens cavities, Z coordinates were
also set to zero.
To machine the Alvarez lens array mold insert surface, the workpiece was
mounted on the main spindle (C axis) with its angular position fixed. The movements of
99
three linear axes (X, Y and Z axis) were simultaneously controlled to preset positions in
sequence based on the surface equation. The diamond tool feed rate was 100 mm/min.
The depth of cut was 3 μm. The feed step size in X direction was 10 μm for rough cut
and 0.5 μm for finishing cut; in Y direction it was 10 μm for rough cut and 2 μm for
finishing cut. Tool nose radius of the diamond cutter is 250 μm in this study. After the
finishing cut, the optical surface finish was confirmed first by visual inspection and later
by using the Veeco optical profilometer.
5.2.2 Measurement
Alvarez lenses work in pairs. Broaching can provide prototype but is not suitable
for mass production due to its expensive and time-consuming nature. Injection molding
is an inherent freeform process, which is another reason for freeform optics fabrication
where complex geometries can be readily manufactured. After the optical molds were
constructed using single point diamond broaching process, micro Alvarez lens arrays
were injection molded.
5.2.2.1 Microlens Array
The mold inserts were fabricated using 6061 aluminum alloy as shown in Figure
5.12. There were no visible tool marks or other surface defects on the broached mold
surfaces. After molding, no visible cosmetic difference among the 25 lenslets was
observed. The diameter of the mold insert was 39.88 mm. For research purpose, on the
same mold insert another 4×4 Alvarez lens array (2 mm × 1 mm for each Alvarez lens
100
cavity, as shown in the upper part of the mold surface in the same figure) was machined
by the same broaching process.
Figure 5.12: Machined micro Alvarez lens array mold insert
5.2.2.2 Geometry Measurement
Since contact method is not suitable for surface and geometry measurement of
optical components, machined surfaces and injection molded lens surfaces were
measured using Veeco NT 3300 white light profilometer.
Alvarez lens surface is
asymmetrical, so 2D profile comparison is not enough to verify the fabrication quality.
Due to the limitations of the measurement range on the Veeco NT3300 profilometer, the
entire micro Alvarez lens array can not be measured in a single scan so each Alvarez lens
cavity had to be measured separately. Moreover, the steep edge on the lens surface can
not be resolved from Veeco profilometer measurements either. After removing tilt,
translation and rotation in the valid measurement region, 3D measurement results of a
101
single lenslet can be obtained. Figure 5.13 shows the design and measurement results. A
single lenslet design is shown in Figure 5.13 (a); measurement result of the lenslet in the
middle of the molded microlens array is shown in Figure 5.13 (b); the difference between
the lenslet in the middle of the microlens array and design is shown in Figure 5.13 (c)
where the maximal deviation was around 0.2 μm, indicating that the fabrication accuracy.
In addition, the difference between the molded lenslet in the middle and at the edge is
shown in Figure 5.13 (d) where the maximal deviation was again around 0.2 μm. After
removing noise, the average deviation was less than 100 nm, indicating that all lenslets in
the microlens array met high quality optical requirements, a demonstration of
repeatability by this process.
(a)
(b)
102
(c)
(d)
Figure 5.13: Design and 3D measurement results (a) Design (b) Measurement result of
lenslet in the middle of the array (c) Difference between the lenslet in the middle of the
array and design (d) Difference between the lenslet in the middle and at the edge on the
molded microlens array
5.2.2.3 Surface Roughness
The mold and molded lens surface roughness were also measured on the Veeco
profilometer. After removing tilt and curve from the surface measurement results, the
surface roughness can be obtained. For the broached mold surface, the Ra value is 37.25
nm and for molded lens surface, Ra value is 24.48 nm at the same area. The Ra values
103
met the requirements for optical applications and the fact that surface roughness of the
molded lens was improved over the mold surface indicating that injection molding
process has a smoothing effect. For the optical molds fabricated in this research, no post
polishing was needed. It is concluded from this research that broaching and injection
molding process provide a practical solution to producing freeform optical elements.
5.2.2.4 Adjustable Focal Length Measurement
When moving the lenses in a micro Alvarez lens array pair in opposite directions,
the focal lengths of the lens array pair will be changed. To measure the adjustable focal
lengths of the micro Alvarez lens array pair, a measurement setup was employed in
Figure 5.14. The Alvarez lens array was enlarged for clarity. The detailed measurement
method and procedure were described earlier in a separate publication [Firestone, 2005].
The positive lenses used in the systems have focal lengths of 8 mm (lens 1), 50 mm (lens
2), 50 mm (lens 3) and 100 mm (lens 4). In the test setup, a He-Ne laser and a Hitachi
KP-D20BU CCD camera were used.
First, the micro Alvarez lens array pair was
manually moved along the optical axis (perpendicular to the lens surface) to a position so
that the focus was on the lens surface (top image of the embedded photos). Then the
Alvarez lens array pair was moved away from the CCD camera until sharp focused spots
(lower image of the embedded photos) were imaged on the CCD camera. The focal
lengths can be measured as the displacements from the lenslet surface to the positions of
the focused spots.
104
1
2
3
4
He-Ne Laser
(λ=633 nm)
CCD
Camera
25 μm
Pinhole
Monitor
Alvarez Lens
Array Pair
Figure 5.14: Test setup for measuring the focal length of a molded microlens array pair
When the microlens array pair was placed in registration, the composite lens is
optically equivalent to a flat plate and the measured focal length was at infinity. When
the lens pair was laterally translated, the focal length was changed simultaneously. The
equivalent spherical lens has a power (the reciprocal of focal length) proportional to the
translation distance that the lens elements are displaced from their neutral positions and
coefficient a in surface equation. Based on theoretical calculation, in this study, the focal
length from 6.4 mm to infinity can be obtained by translating this 5×5 Alvarez lens array
pair within 0.25 mm translation. The measurement result and design values are shown in
Figure 5.15. The error between the measurement and design was less than 10% and the
average error on the measurement positions was 5.2%.
105
Figure 5.15: Focal length measurement result
Through design, fabrication and measurement of a micro Alvarez lens array, the
capability for functional precision freeform microlens array manufacturing by combining
ultraprecision diamond turning machining using slow tool servo with injection molding
process was demonstrated.
Unlike conventional processes, this research provides a direct broaching process,
which allows the entire Alvarez lens array to be machined accurately in one single
operation. The machined mold inserts and injection molded lens arrays were measured to
ensure that surface geometry and roughness with optical quality were obtained. No post
machining and polishing are required which is important for complex optical surface
fabrication since current polishing process may compromise the shape accuracy. The
adjustable focal lengths were obtained by laterally translating the position of an Alvarez
lens array pair.
This research shows the possibility of fabricating many complex
106
(arbitrary) shape elements using the same methodology with optical quality with minimal
tooling and setup requirements.
Such a strategy would not have been practical or
possible if traditional fabrication processes were used.
107
CHAPTER 6
DIFFRACTIVE LENS MANUFACTURING
As special micro optical elements, diffractive lenses are selected for this
dissertation research. Two types of diffractive lens are described and the design and
fabrication processes involved are explained as follows.
6.1 Diffractive Lens
6.1.1 Lens Design
The basic concept of the DOEs (diffractive optical elements) relies on
constructive and destructive interference of spherical sources to produce the desired
illumination geometry. For example, in the simplest design, two parallel slits illuminated
by a collimated beam (planar wave front) such as a laser exhibit constructive and
destructive interference.
As the light propagates from the slits, both slits act as
independent light sources, and the light from each propagates in a spherical shape, as
shown in Figure 6.1. At some distance, the two spherical waves interact with each other.
In some regions, the interaction is constructive, producing “bright” regions, while at half
wavelengths from each of these regions, areas of destructive interference occur,
108
producing “dark” regions.
If a screen is placed at some distance from the slits,
alternating lines of dark and bright areas are produced.
Figure 6.1: General concept of a DOE’s function (amplitude type)
In order to produce two-dimensional (2D) patterns, the line pattern is replaced by
a 2D pattern, which appears to be very complicated. To design this pattern, an inverse
Fast Fourier Transform (iFFT) of complex arrays is performed [O’Shea, 2004] based on
the desired bitmap image. Because of limitations of lithography fabrication techniques,
the complex values of the iFFT are truncated to 2nd levels, depending on the number of
levels that will be fabricated in the final lens.
The efficiency of the final lens is
proportional to the number of levels. The design of 256 levels DOE to produce a circle is
shown in Figure 6.2.
109
Figure 6.2: Design of 256 level DOE
The pattern is a complex, non rotational layout. The size of each block is equal
and can be arbitrarily selected to achieve different divergence of the image. The center to
center distance of each block is 20 μm in these test. The smallest vertical step is
approximately 10 nm, same as the resolution of the ultraprecision machine.
6.1.2 DOEs Fabrication
In optical industry, lithography has been used for fabrication of diffractive optics
[Lee, 2003; Suleski, 1995; Ogura, 2001]. However, this process is expensive especially
for multilevel mask fabrication. In this dissertation research, STS process on 350 FG
machine was used to machine the diffractive DOEs simultaneously. Due to the limitation
of the tiny features on DOEs, a special diamond tool was selected which is shown in
110
Figure 6.3. The diamond tool tip was reduced to a half-radius tool with radius of 2.5μm
and one side of the cutting edge completely removed in the tool preparation stage. The
rake angle is 0 deg, and the clearance angle is 7 deg. This design allowed very straight
side walls to be machined while maintaining the smooth machined surface by the use of
the radius cutting edge.
Figure 6.3: SEM picture of the half-radius diamond tool
In this research, two different but similar approaches to micromachining of DOEs
were studied. Each of the two approaches can be effective depending on the optical
design. In this study, broaching produced slightly better results because the lens design
has a rectangular shape.
111
6.1.2.1 Polar Coordinate – Spiral Tool Path
The CNC tool path is nearly identical to the design surface shape, differing only
due to the tool radius compensation. In polar coordinates, the Y position for the tool was
fixed while the workpiece was rotating.
Figure 6.4 shows the perspective three
dimensional view of the tool path generated for the experiment, but the number of steps
was reduced for clarity. For each point on the tool path, the polar radius ρ and angle θ
were determined by the step size and arc length, and the tool height Z was determined
from the diffractive lens design for the corresponding the X and Y coordinates. The tool
nose radius can be compensated for either off-line method or using the onboard tool
compensation function. The angular position of the workpiece was controlled in real
time simultaneously with the three linear axes, resulting in a spiral cutting pattern as the
Z axis is modulated.
112
Figure 6.4: Spiral CNC tool path for DOE fabrication
6.1.2.2. Cartesian Coordinate – Broaching
The second method that was used in the fabrication process was broaching. In
this setup, the diffractive lens was held on a vacuum chuck.
During cutting, the
workpiece was first moved in the X direction at a fixed step distance (lateral, to the right
in Figure 6.5, the step size depends on the diamond tool size). Then it was continuously
moved vertically while the diamond tool was fed in the Z direction based on the
diffraction pattern.
This process continued until the entire surface was completely
machined. Figure 6.5 depicts the machine tool path for broaching where the number of
steps was reduced to show the straight tool travel paths and the return travel passes were
also removed for clarity.
113
Figure 6.5: Broaching CNC tool path for DOE fabrication
6.1.3 Profile Measurement
In reality, a 2 level or 4 level diffractive lens can be fabricated using lithography
with relative ease, although it still remains a lengthy process. However, a 256 level
diffractive lens is difficult and costly to fabricate using lithography technique due to the
multiple-exposures required for a multi-level diffractive design and the potential for
accumulated errors following each exposure and etching. This work demonstrated that
micromachining process using ultraprecision machine and a specially designed diamond
114
tool permits machining of multi-level micro features in a single operation without the
repetitive re-alignments.
For similar machining speed, the broaching approach produced smoother surface
especially on the edge of the features. This is due to the fact that the lens design was a
square pattern. If a circular pattern design was selected, it is expected that the spiral
machining process would produce better results.
Figure 6.6 shows an SEM photo of the 256 level DOE. The broaching diamond
machining process was selected to fabricate the device. Prior to machining of the lens,
diamond turning process was used to machine the sample flat on both sides. In the SEM
photo in Figure 6.6, the lens was tilted slightly to show the depth of the different levels
for only a portion of the machined surface. The SEM scan shows only a small section for
the machined DOE surface.
115
Figure 6.6: Sectional SEM scan of a 256-level DOE
AFM was used to study the topography of the DOEs.
Figure 6.7 shows a
sectional AFM scan of the 256 level DOE surface. To view the depth information, a line
scan was executed using the AFM (Atomic Force Microscopy) and the results are shown
in Fig. 6.8. As can be seen the draft angle in this case was 8.8 degree which is similar to
the results obtained by lithography technology and the surface finish of the scan area was
approximately 9.4 nm Ra (arithmetic average). The step features of the diffractive lens
have largely been replicated during the fabrication process.
116
Figure 6.7: Sectional AFM scan of the 256-level DOE design
117
Figure 6.8: Sectional AFM line scan of the 256-level DOE design
Surface measurement indicates that the direct machined DOEs surface meets the
requirements for precision optical applications.
The investigated strategy for DOE
fabrication has several inherited advantages over existing technologies. First, DOEs can
be machined directly on the substrate in a single turning operation with minimal tooling
and setup requirement and without the need for realignment. Second, the DOEs have
optical quality and finish that does not require postmachining polishing.
STS process is a powerful complementary tool for clean-room technology, where
mask-making and the lithography process can be very costly and time consuming. In this
118
dissertation research, the design from pattern calculation (using a self-written Matlab
program) to finished DOE can be completed in less than a day. This is ideal for
prototype fabrication, providing an easy and quick means to evaluate an optical design
before major investment is made for mass production. Another important feature of the
STS process is that multiple-level micro-/nano-scale features can be simultaneously
machined without the need of repeated alignment operations, as in a lithography method.
6.2 Fresnel Lens
6.2.1 Lens Design
The design of the Fresnel lens for this research is described as follows. The
center of the molding lens is a 50-zone Fresnel lens with focal length (f) of 100 mm. For
this design, wavelength λ is equal to 632.8 nm and refractive index of the plastic material
(Plexiglas® V825) n is equal to 1.49. The feature heights are equal to λ /(n − 1) which is
approximately 1.3 μm.
The transition location for each zone occurs at rp2 = 2 pλf
(where p = 0, 1, 2, …, 50). The design for the center part of the Fresnel lens is shown in
Figure 6.9 with only 5 zones shown for clear demonstration.
119
Figure 6.9: Fresnel lens design
6.2.2 Mold Fabrication
Since the Fresnel lens has an axisymmetrical design, the mold insert can be made
by traditional single point diamond turning process. The tool radius used for this Fresnel
lens mold insert fabrication is 2.5 µm. Since the tool radius can not be made infinitely
small (zero), the mold profile will be slightly different from the design which were
usually designed based on zero radius, this finite radius will be one of the sources of
errors affecting optical performance of a molded lens. A Fresnel lens mold insert and a
molded lens are shown in Figure 6.10. The mold insert is 6061 aluminum alloy and the
lenses were molded with Plexiglas® V825, same polymer as the other molded optical
components involved in this dissertation research. In addition to the diffractive lens
120
patterns in the center, selected groove designs were also fabricated on the insert for
testing the replication of micro features.
Figure 6.10: Fresnel lens mold insert and molded lens
6.2.3 Profile Measurement
For the tiny feature of a Fresnel lens as shown in an SEM (scanning electron
microscope) photo in Figure 6.11, it is difficult to measure the surface profile with
contact method. The non-contact optical profilometer Veeco NT 3300 was chosen to
perform the surface profile measurement.
121
Figure 6.11: Measurement result from SEM
In Figure 6.12, according to the Veeco profilometer measurement results, the
profiles of the mold insert and the lenses molded under different mold temperatures are
compared to each other. The mold temperature was set at five levels, 110°F (43.3°C),
130°F (54.4°C), 150°F (65.5°C), 170°F (76.6°C) and 190°F (87.7°C). In addition to the
different mold temperature, all the other process parameters remained unchanged.
122
Figure 6.12: Feature comparison with different mold temperature
Although the precision machine and machining process can be used to fabricate
the lens mold with high accuracy, due to the error from cutter path, tool geometry and
from machining process parameters such as feed rate and spindle speed, fabrication errors
between the mold insert profile and design exist. Moreover, because of shrinkage from
injection molding process, larger errors can occur between the molded lens profile and
design.
In Figure 6.13, the profiles of the mold insert and lenses molded under different
packing pressure are compared to each other. The packing pressure was set at six levels,
10% (21.8 MPa), 15% (32.7 MPa), 20% (43.6 MPa), 25% (54.5 MPa), 30% (65.4 MPa)
and 35% (76.3 MPa) of the maximal machine injection pressure (218 MPa). Except
different setting for packing pressure, all the other process parameters remained
unchanged.
123
Figure 6.13: Feature comparison with different packing pressure
In Figure 6.14, the profiles of the mold insert and lenses molded under different
melt temperature are compared to each other. In this experiment, the melt temperature
was set at three levels, 430°F (221.1°C), 450°F (232.2°C) and 470°F (243.3°C). In this
experiment, only the melt temperature was varied, all the other process parameters
remained unchanged.
124
Figure 6.14: Feature comparison with different melt temperature
The measurement results showed that more accurate lens profile will be obtained
under higher mold temperature, higher packing pressure and higher melt temperature.
The mold temperature makes obvious contribution to the lens geometry accuracy.
6.2.4 Optical Performance Simulation
The effect of the lens profile error is crucial to the optical performance for Fresnel
lenses. The analytical method described in Section 4.3.1 will be used to obtain the
diffractive pattern from the Veeco measurement results.
Normally the diffraction integral is used to derive simplified solutions. The
conventional approaches for approximation are near field approximation and far field
approximation. The criterion for near field approximation (Fresnel approximation) is
125
( z1 − z 0 ) nf ≥ (2.5a 4 / λ )1/ 3 where ( x 02 + y02 ) max = a 2 .
The criterion for far field
approximation (Fraunhofer approximation) is ( z1 − z 0 ) ff ≥ 10a 2 / λ . Unfortunately for
our design, neither far-field nor near-field approximation could be applied in this case so
we had to use the principle formula Equation (3-3) to calculate the diffraction distribution
of the design and the molded lens.
Figure 6.15, 6.16 and 6.17 are the simulation results for optical performance
based on design profile and Veeco profile measurements. For the molded lens, the
intensity of the first-order diffraction is much lower than the design value and the spread
of the central spot is larger than the desired value. Also the lenses molded under different
process variables will have different performance that can be simulated from the
measured lens profiles using Veeco profilometer. The difference between lens 1 and lens
2 is the packing pressure during the injection molding process. The packing pressure for
lens 1 is 35% (76.3 MPa) and for lens 2 is 10% (21.8 MPa).
126
Figure 6.15: Designed lens diffractive pattern distribution
Figure 6.16: Lens 1 which is under higher packing pressure diffractive pattern
distribution
127
Figure 6.17: Lens 2 which is under lower packing pressure diffractive pattern
distribution
From the simulation results above, it can be concluded the same conclusion as
directly from the profile measurement results of the molded Fresnel lenses that higher
packing pressure is suitable for more accurate feature replication.
128
CHAPTER 7
CONCLUSION
The motivation of this dissertation research was to investigate and develop a
methodology on precision polymer optics fabrication by injection molding that can be
used for high volume and low cost lens manufacturing. Injection molding polymer
optical components have long been used for its high volume, low cost and lightweight
capability over traditional glass optics. Injection molding is an inherent freeform process
thus complex geometry (including aspherical and freeform design) may be readily
manufactured. However, the process has not been readily accepted in precision optical
fabrication industry because several difficult issues related to the injection molded optics
have hindered the implementation of injection molding process in high precision
applications.
These issues include geometry deviation and inhomogeneous index
distribution due to thermal shrinkage; birefringence incurred during the molding process
also limited the adoption of polymer optics in certain polarization sensitive optical
systems; thermal instability of molded polymer lenses can also render the optics less
effective in application where temperature changes become large and frequent (such as
optics designed for out door use or high temperature applications). Currently, most of the
129
research involved in polymer injection molding was focused on the determination of
process parameters in order to optimize part quality but did not address the issues
concerning mold compensation for high precision polymer lenses. Also the optical
effects from process conditions of lens injection molding such as index distribution,
residual stress/birefringence and optical scattering were not studied systematically.
Furthermore, with the high precision requirement of the optical system, freeform optics
including microlens array and diffractive optics can provide a practical solution for some
design and manufacturing problems. The success of the process relies on the fabrication
of the mold inserts and measurement technology. Fewer articles discussed the advanced
mold fabrication and measurement issues. It is necessary for current researchers to make
efforts to improve the injection molding process on precision optical component
production.
This dissertation research involved fundamental and systematic study of precision
polymer optics fabrication by injection molding. The study included both experimental
approach and numerical modeling in order to identify the proper polymer lens
manufacturing processes. The scope of this research includes investigation in optical
design, mold and lens fabrication, as well as optical metrology issues related to polymer
lens manufacturing to obtain precision macro and micro polymer freeform optics with
accurate geometry and proper optical performance by state-of-the-art mold fabrication
and molding technology.
130
In Chapter 3, with the aid of DOE and DEA methods, the critical process
parameters including packing pressure, mold temperature and melt temperature were
narrowed down for other process and performance studies and the optimal condition was
found for compensation study both by the plano lens molding experiment and
measurement results. The mold compensation methodology was developed based on
advanced freeform measurement and data analysis technology and STS freeform mold
insert fabrication.
In Chapter 4, the effects of the process parameters to optical performance such as
birefringence, index distribution and surface scattering were carefully studied by
theoretical and empirical analysis. Lower packing pressure, higher mold temperature and
melt temperature were better setup for lens molding with lower birefringence. Lower
packing pressure and higher mold temperature were proven to be better for lens molding
with smaller index deviation. Higher packing pressure, lower mold temperature and
higher melt temperature were better for lens molding with lower optical scattering. Due
to the complexity of the injection molding process, single process condition cannot fulfill
all the requirements for lens quality requirements so process parameters need to be
selected as a compromise for desired specification.
In Chapter 5, macro Alvarez lens and micro Alvarez lens array were fabricated.
The mold inserts were successfully machined using slow tool servo and broaching
process. The injection molded Alvarez lens can fulfill the requirement of vision test and
these lenses can be used for ophthalmic application. In addition, deviation between the
131
molded micro Alvarez lens array and design was around 0.2 μm with the P-V value of the
design at 10 μm. The average error of the adjustable focal length was only 5.2%.
In Chapter 6, diffractive lenses and Fresnel lenses were fabricated.
The
fabrication of the multilevel DOEs with STS has proven that STS can provide an easy
and quick solution without expensive and time-consuming mask making and lithography
in cleanroom. The measurement results of the Fresnel lens showed that more accurate
lens profile can be obtained under higher mold temperature, higher packing pressure and
higher melt temperature. The mold temperature is also critical to the lens geometry
accuracy. The same conclusion was drawn from optical performance simulation.
In addition, simulation using Moldflow was implemented to verify the experiment
results, for example, plano lens warpage and birefringence.
The tendency of the
simulation results was similar as the experiment results. However, accurate predictions
can not be easily obtained using commercial software in all cases.
This dissertation research was an attempt to create a methodology for injection
molding process for high precision polymer lens manufacturing. Experimental study and
process modeling were conducted to develop a fundamental understanding of the process.
The feasibility of lens compensation using freeform mold were fully tested.
Other
functional freeform optical elements were fabricated and numerical simulation was
utilized to predict the optical performance of the molded elements. The contributions of
this research are as follows:
132
•
Performed experiments (both axisymmetric lenses and freeform lenses) and evaluated
surface geometry and optical performance to investigate the feasibility of using
injection molding process to manufacture high precision polymer lenses.
•
Explored the effects of process variables and material property for specific objective
function (surface shape deviation, birefringence, optical retardation, optical scattering)
for lens performance optimization.
•
Utilized current measurement methods and developed freeform data analysis method
for real surface shape, part thickness and optical performance measurement.
•
Designed and fabricated multiple freeform mold inserts and obtain functional
injection molded freeform optics including compensated lens, Alvarez lens, micro
Alvarez lens arrays and diffractive lenses.
133
CHAPTER 8
FUTURE WORK
Current research focused on empirical study on the effects of process parameters
to the lens performance and the compensation practice relied on a trial and error approach,
in which initial molding was performed and the deviation on the lens was compensated
for on the mold inserts to obtain improved lens. This is a time consuming, labor intensive
and therefore expensive method. Currently, in computer simulation, geometry deviation,
structures in both macro and micro size can not be readily and properly modeled using
commercial software such as Moldflow Plastic Insight® 6.1. Some important information
for optical performance such as residual stresses of the molded lens is not available in the
software simulation results. Therefore, in the future, with the development of new and
reliable multi-scale approach and more powerful computational capability for injection
molding process simulation, the performance of the molded lens can be predicted and the
modified mold inserts and the optimal process conditions can be obtained by numerical
analysis for various mold materials, polymer materials and optical components without
performing the actual experimental work..
134
Future work will also be focused on analyses that may include study of errors in
mold fabrication, molding process and measurement. The CNC tool path for mold
fabrication is calculated based on an ideal tool geometry and accurate linear and radial
movement of the machine.
However, due to wear and measurement error of the
machining tools, the tool geometry data require modifications for proper tool path
compensation. In addition, the environmental factors including temperature, humidity
and vibration will also have an impact on machine accuracy. Long STS machining time
for steep freeform mold fabrication may also increase fabrication error. In the future, the
mold fabrication process can be improved with newer equipment. For example, using an
FTS (fast tool servo) to increase freeform mold fabrication rate can make this process
more robust and suitable for industrial applications. Finally, due to the uncertainties and
random factors involved in the lens and mold measurement procedure and measurement
equipment, analysis of mold fabrication error, molding process error and measurement
error will all be important for predicting the molded components performance.
135
APPENDIX A
SPECIFICATION OF MOLDING MATERIAL
136
Figure A.1: Product data sheet for Plexiglas® V825
137
Figure A.2: Product data sheet for Plexiglas® V825 (Figure A.1 continued)
138
APPENDIX B
SH50M MAIN SPECIFICATION
139
Clamping system
Clamping force
Opening force
Distance between tie-bars (H×V)
Overall size of platen
Mold space
Opening stroke
Daylight
Ejector type
fully hydraulic
50tf
3.2tf
325×325mm
470×467mm
Min.160mm
440mm
600mm
hydraulic and across multipoint ejection (5
points)
70mm
2.2tf
C160s (plasticizing unit)
28mm
2230kg/cm2 (approximately 218.54Mpa)
70cm3
67g (2.4oz)
(screw 37kg/h (400rpm)
Ejector stroke
Ejector force
Injection unit
Screw diameter
Injection pressure
Injection capacity
Injection weight (GPPS)
Plasticizing capacity (GPPS)
rotation speed)
Injection rate
Screw stroke
Max.injection speed
Screw driving system
Screw torque
Torque selector
Screw ratoation speed
No. of temperature control zone
Heater capacity (for open nozzle only)
Nozzle contact force
Injection unit displacement stroke
Hopper capacity
Electric and hydraulic:
Drive motor capacity
Pressure in hydraulic circuit
Oil reservoir capacity
Others:
Machine dimensions (L×W×H)
Machine weight
99cm3/s
114mm
160mm/s
hydraulic motor
36kgf⋅m
1
400rpm
4
4.6kw
4670kgf
245mm
15 l
11kw
155kgf/ cm2
110 l
3703×936×1635MM
2.2ton
Table B.1: Main specification of SH50M injection molding machine
140
APPENDIX C
PROCESS CONDITIONS FOR FULL FRACTIONAL FACTORIAL
EXPERIMENTS
141
RUNS
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
Tmelt
210°C (450°F)
210°C (450°F)
210°C (450°F)
210°C (450°F)
210°C (450°F)
210°C (450°F)
210°C (450°F)
210°C (450°F)
210°C (450°F)
210°C (450°F)
210°C (450°F)
210°C (450°F)
210°C (450°F)
210°C (450°F)
210°C (450°F)
210°C (450°F)
210°C (450°F)
210°C (450°F)
210°C (450°F)
210°C (450°F)
210°C (450°F)
210°C (450°F)
210°C (450°F)
210°C (450°F)
243°C (470°F)
243°C (470°F)
243°C (470°F)
243°C (470°F)
243°C (470°F)
243°C (470°F)
243°C (470°F)
243°C (470°F)
243°C (470°F)
243°C (470°F)
243°C (470°F)
Tmold
65°C (150°F)
65°C (150°F)
65°C (150°F)
65°C (150°F)
65°C (150°F)
65°C (150°F)
65°C (150°F)
65°C (150°F)
65°C (150°F)
65°C (150°F)
65°C (150°F)
65°C (150°F)
88°C (190°F)
88°C (190°F)
88°C (190°F)
88°C (190°F)
88°C (190°F)
88°C (190°F)
88°C (190°F)
88°C (190°F)
88°C (190°F)
88°C (190°F)
88°C (190°F)
88°C (190°F)
65°C (150°F)
65°C (150°F)
65°C (150°F)
65°C (150°F)
65°C (150°F)
65°C (150°F)
65°C (150°F)
65°C (150°F)
65°C (150°F)
65°C (150°F)
65°C (150°F)
tcool
30sec
30sec
30sec
30sec
30sec
30sec
40sec
40sec
40sec
40sec
40sec
40sec
30sec
30sec
30sec
30sec
30sec
30sec
40sec
40sec
40sec
40sec
40sec
40sec
30sec
30sec
30sec
30sec
30sec
30sec
40sec
40sec
40sec
40sec
40sec
Ppacking
65.4MPa (30%)
65.4MPa (30%)
65.4MPa (30%)
76.3MPa (35%)
76.3MPa (35%)
76.3MPa (35%)
65.4MPa (30%)
65.4MPa (30%)
65.4MPa (30%)
76.3MPa (35%)
76.3MPa (35%)
76.3MPa (35%)
65.4MPa (30%)
65.4MPa (30%)
65.4MPa (30%)
76.3MPa (35%)
76.3MPa (35%)
76.3MPa (35%)
65.4MPa (30%)
65.4MPa (30%)
65.4MPa (30%)
76.3MPa (35%)
76.3MPa (35%)
76.3MPa (35%)
65.4MPa (30%)
65.4MPa (30%)
65.4MPa (30%)
76.3MPa (35%)
76.3MPa (35%)
76.3MPa (35%)
65.4MPa (30%)
65.4MPa (30%)
65.4MPa (30%)
76.3MPa (35%)
76.3MPa (35%)
tpacking
6.5sec
7.0sec
7.5sec
6.5sec
7.0sec
7.5sec
6.5sec
7.0sec
7.5sec
6.5sec
7.0sec
7.5sec
6.5sec
7.0sec
7.5sec
6.5sec
7.0sec
7.5sec
6.5sec
7.0sec
7.5sec
6.5sec
7.0sec
7.5sec
6.5sec
7.0sec
7.5sec
6.5sec
7.0sec
7.5sec
6.5sec
7.0sec
7.5sec
6.5sec
7.0sec
Continued
Table C.1: Process conditions for full fractional factorial experiments
142
Table C.1 continued
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
243°C (470°F)
243°C (470°F)
243°C (470°F)
243°C (470°F)
243°C (470°F)
243°C (470°F)
243°C (470°F)
243°C (470°F)
243°C (470°F)
243°C (470°F)
243°C (470°F)
243°C (470°F)
243°C (470°F)
254°C (490°F)
254°C (490°F)
254°C (490°F)
254°C (490°F)
254°C (490°F)
254°C (490°F)
254°C (490°F)
254°C (490°F)
254°C (490°F)
254°C (490°F)
254°C (490°F)
254°C (490°F)
254°C (490°F)
254°C (490°F)
254°C (490°F)
254°C (490°F)
254°C (490°F)
254°C (490°F)
254°C (490°F)
254°C (490°F)
254°C (490°F)
254°C (490°F)
254°C (490°F)
254°C (490°F)
65°C (150°F)
88°C (190°F)
88°C (190°F)
88°C (190°F)
88°C (190°F)
88°C (190°F)
88°C (190°F)
88°C (190°F)
88°C (190°F)
88°C (190°F)
88°C (190°F)
88°C (190°F)
88°C (190°F)
65°C (150°F)
65°C (150°F)
65°C (150°F)
65°C (150°F)
65°C (150°F)
65°C (150°F)
65°C (150°F)
65°C (150°F)
65°C (150°F)
65°C (150°F)
65°C (150°F)
65°C (150°F)
88°C (190°F)
88°C (190°F)
88°C (190°F)
88°C (190°F)
88°C (190°F)
88°C (190°F)
88°C (190°F)
88°C (190°F)
88°C (190°F)
88°C (190°F)
88°C (190°F)
88°C (190°F)
40sec
30sec
30sec
30sec
30sec
30sec
30sec
40sec
40sec
40sec
40sec
40sec
40sec
30sec
30sec
30sec
30sec
30sec
30sec
40sec
40sec
40sec
40sec
40sec
40sec
30sec
30sec
30sec
30sec
30sec
30sec
40sec
40sec
40sec
40sec
40sec
40sec
143
76.3MPa (35%)
65.4MPa (30%)
65.4MPa (30%)
65.4MPa (30%)
76.3MPa (35%)
76.3MPa (35%)
76.3MPa (35%)
65.4MPa (30%)
65.4MPa (30%)
65.4MPa (30%)
76.3MPa (35%)
76.3MPa (35%)
76.3MPa (35%)
65.4MPa (30%)
65.4MPa (30%)
65.4MPa (30%)
76.3MPa (35%)
76.3MPa (35%)
76.3MPa (35%)
65.4MPa (30%)
65.4MPa (30%)
65.4MPa (30%)
76.3MPa (35%)
76.3MPa (35%)
76.3MPa (35%)
65.4MPa (30%)
65.4MPa (30%)
65.4MPa (30%)
76.3MPa (35%)
76.3MPa (35%)
76.3MPa (35%)
65.4MPa (30%)
65.4MPa (30%)
65.4MPa (30%)
76.3MPa (35%)
76.3MPa (35%)
76.3MPa (35%)
7.5sec
6.5sec
7.0sec
7.5sec
6.5sec
7.0sec
7.5sec
6.5sec
7.0sec
7.5sec
6.5sec
7.0sec
7.5sec
6.5sec
7.0sec
7.5sec
6.5sec
7.0sec
7.5sec
6.5sec
7.0sec
7.5sec
6.5sec
7.0sec
7.5sec
6.5sec
7.0sec
7.5sec
6.5sec
7.0sec
7.5sec
6.5sec
7.0sec
7.5sec
6.5sec
7.0sec
7.5sec
APPENDIX D
ANOVA RESULTS FOR FULL FRACTIONAL FACTORIAL EXPERIMENTS
144
Constant
Tmelt
Tmold
tcool
Ppack
tpack
Tmelt*Tmelt
tpack*tpack
Tmelt*Tmold
Tmelt*tcool
Tmelt*Ppack
Tmelt*tpack
Tmold*tcool
Tmold*Ppack
Tmold*tpack
tcool*Ppack
Tcool*tpack
Ppack*tpack
S
R-SQ
R-SQ adj
P
0.870
0.000
0.000
0.244
0.007
0.476
0.000
0.205
0.000
0.264
0.001
0.001
0.108
0.000
0.087
0.787
0.953
0.001
All Factors less than 0.05 are significant
0.7050
0.0000
0.0000
ELIMINATED IT
0.0050
0.4660
0.0000
0.2010
0.0000
ELIMINATED IT
0.0010
0.0010
ELIMINATED IT
0.0000
0.0850
ElIMINATED IT
ELIMINATED IT
0.0010
0.022
98.50%
98.10%
0.02197
98.40%
98.10%
Table D.1: ANOVA results for full fractional factorial experiments
145
APPENDIX E
DEA RESULTS FOR FULL FRACTIONAL FACTORIAL EXPERIMENTS
146
RUNS
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
Total Weight
20.6421
20.6997
20.7308
20.745
20.8202
20.871
20.6583
20.7024
20.7306
20.7486
20.8192
20.882
20.4877
20.5488
20.6022
20.5595
20.6379
20.7057
20.4979
20.5551
20.6046
20.5522
20.6378
20.7021
20.5527
20.6169
20.6854
20.63655556
20.7312
20.8221
20.5549
20.6342
20.6922
20.6345
20.7442
Standard Deviation
0.0112
0.0088
0.0146
0.0106
0.0139
0.0102
0.0068
0.0091
0.0343
0.0050
0.0140
0.0066
0.0153
0.0148
0.0159
0.0100
0.0049
0.0060
0.0081
0.0106
0.0045
0.0092
0.0062
0.0143
0.0044
0.0075
0.0033
0.0053
0.0064
0.0111
0.0091
0.0025
0.0082
0.0040
0.0092
Input
0.7885
0.8604
0.7579
0.8399
0.8058
0.9034
0.8908
0.8543
0.5236
0.9798
0.8032
1.0000
0.7126
0.7217
0.7034
0.8104
0.9331
0.9331
0.8510
0.7974
0.9408
0.8259
0.8979
0.7535
0.9451
0.8635
1.0000
0.9196
0.9338
0.8629
0.8281
1.0000
0.8703
0.9565
0.8704
Output
0.7243
0.7791
0.8004
0.8173
0.8973
0.9686
0.7483
0.7807
0.8002
0.9488
0.8961
1.0000
0.6188
0.6562
0.6929
0.6672
0.8447
0.8344
0.6325
0.6629
0.8364
0.6640
0.7577
0.7736
0.8025
0.7137
1.0000
0.8124
0.8291
0.9015
0.6661
1.0000
0.7738
0.8991
0.8205
Continued
Table E.1: DEA results for full fractional factorial experiments
147
Table E.1 continued
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
20.8961
20.4192
20.4653
20.539
20.432
20.5301
20.6181
20.3905
20.4583
20.5256
20.4217
20.5141
20.6007
20.4157
20.5097
20.5943
20.4723
20.5892
20.7047
20.4193
20.5071
20.5883
20.4644
20.5842
20.6966
20.2598
20.3166
20.4041
20.268
20.3527
20.4379
20.2331
20.3018
20.3768
20.2509
20.4317
20.3411
0.0117
0.0179
0.0133
0.0130
0.0146
0.0100
0.0155
0.0097
0.0136
0.0112
0.0106
0.0088
0.0113
0.0075
0.0068
0.0078
0.0059
0.0081
0.0105
0.0043
0.0053
0.0067
0.0042
0.0060
0.0082
0.0221
0.0208
0.0195
0.0142
0.0111
0.0066
0.0140
0.0058
0.0060
0.0073
0.0101
0.0086
148
1.0000
0.6736
0.7468
0.7525
0.7247
0.8099
0.7107
0.8151
0.7416
0.7854
0.7981
0.8345
0.7829
0.8640
0.8827
0.8582
0.9043
0.8499
0.8268
0.9466
0.9192
0.8851
0.9487
0.9016
0.8729
0.6182
0.6344
0.6519
0.7307
0.7873
0.8854
0.7348
0.9059
0.9011
0.8684
0.8077
0.8399
1.0000
0.5816
0.6061
0.6499
0.5882
0.6481
0.7046
0.5712
0.6023
0.6426
0.5852
0.6407
0.6927
0.5881
0.6425
0.6964
0.6593
0.6919
0.7793
0.7112
0.7135
0.6947
0.7446
0.7284
0.7778
0.5103
0.5336
0.5740
0.5135
0.5507
0.6016
0.5000
0.5729
0.5997
0.5144
0.5913
0.5501
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