Optimization in Lens Design

53 Pages·2018·3.52 MB·English

by  Akira Yabe

Optimization in
Lens Design
Akira Yabe
Optimization in Lens Design
by Akira Yabe
doi: http://dx.doi.org/10.1117/3.2322375
PDF ISBN: 9781510619838
epub ISBN: 9781510619845
mobi ISBN: 9781510619852
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Spotlight vol. SL36
Last updated: 25 April 2018
Table of Contents
Preface vi
1 Overview 1
1.1 Design requirements and independent variables 1
1.2 Problem of sensitivity and tolerances 1
2 Lens Design Optimization Principles 2
2.1 Merit function 2
2.2 Problem of nonlinear optimization 3
2.3 Metric in the parameter space 4
2.4 Control of boundary conditions 5
2.5 Local and global optimization 6
2.6 Damped least-squares method for local optimization 7
2.7 Effective use of the escape function for global optimization 7
3 Special Independent Variables 9
3.1 Fictitious glass model 9
3.2 Asphere model 10
3.3 Freeform model 12
3.4 Traveling asphere and traveling freeform model 15
4 Evaluation and Optimization 23
4.1 Ray aiming 23
4.2 Aberration control 24
4.3 Efficient method of MTF optimization 25
4.4 Sensitivity control 26
4.5 Control of the Monte Carlo simulation result 30
5 Design Process 30
5.1 Outline of the initial design of zoom lenses 30
5.2 Design example 31
6 Cost-Based Optimization of Tolerances 33
6.1 Concept 33
6.2 Minimization of tolerance cost 37
6.3 Tolerance optimization 40
iii
iv TableofContents
7 Future Development 44
7.1 Treatment of patents 44
7.2 Application of deep learning 44
7.3 Common framework of program development 44
References 44
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Preface
I started working in lens design in 1980, the same year that the International Lens
Design Conference was held in Oakland, California. The session titles included:
l Lens design using large computers,
l Lens design using small computers,
l Lens design using microcomputers,
l Optimization techniques,
l Optimization and aberration theory,
l Aberration theory and computational techniques, and so on.
The optimization techniques had been already established, and the race
among various commercial software companies had just started.
Over the past four decades, the improvement of the calculation speed and the
decrease in cost of computers have been remarkable. Calculation speed changed
frommegaflopstogigaflops.Thepriceofcomputerschangedfromamilliondollars
to a thousand dollars. Under these circumstances, the most important viewpoint of
the programmer was to make the best use of the improved calculation speed and
reduced cost. The purpose of a lens design program is to produce the best design
within the shortest time. My early work assumed that the intervention of the lens
designer needs to be reduced as much as possible to make the best use of the
improved calculation speed. Before optimization, lens designers needed to deter-
mine the pattern of the positive and negative elements. To improve the perfor-
mance, they repeated the small adjustment of targets and tried many different
starting points for the optimization. Many tasks, such as the choice of glasses,
the choice of surfaces to be aspherized, the reduction of the tolerance sensitivity,
and the control of manufacturing feasibility, needed to be determined and con-
trolled. The problems of the choice of glasses and the choice of aspheric surfaces
are discrete and combinatorial, and they seemed to be difficult to treat in the ordi-
nary optimization scheme. My design program was developed to control such
complicated tasks.
Thedevelopment ofglobaloptimization inthe1990screated adrastic change.
Global optimization found a lot of patterns of positive and negative elements
automatically. Many functions, intended to reduce the intervention of the lens
designer, were used to produce useful solutions.
vi
Preface vii
When I explained global optimization, a specialist of metrology asked me,
“If commercial software gives different solutions to the same problem, is it defec-
tive?” I did not immediately understand his question, but the result of the optimi-
zation strongly depends on the character of the software, contrary to software for
analysis or simulation. On the other hand, every time the level of control is raised,
a question comes to mind: “Are lens designers losing their roles?”
I wonder how many designers in the world are using their own personal
design codes. The implementation of raytracing and optimization algorithms is
not very difficult. It would be exciting if lens designers could use their own
code and be fully responsible for the design result without relying on
black-box commercial software. This Spotlight consists of the concepts that other
researchers established and I accepted, original design methods that I have pub-
lished, and previously unpublished concepts. I intend to offer guidelines to novice
lens designers who wish to develop their own design code.
Akira Yabe
March 2018
Yabe:OptimizationinLensDesign 1
1 Overview
1.1 Design requirements and independent variables
When lens data are given, various characteristics of the lens can be evaluated with
optical theory. The typical independent variables of lens design are
l total surface number,
l material between surfaces,
l curvature of surface,
l distance between surfaces,
l clear aperture of surface,
l surface number to be aspherized,
l shape of asphere,
l stop surface number,
l stop surface diameter, and so on.
The typical dependent variables of the lens design are
l effective focal length,
l back focal length,
l distance from the object surface to the image surface,
l imaging magnification,
l ray aberration,
l root mean square optical path difference (RMS OPD),
l modulation transfer function (MTF),
l distortion,
l relative illumination,
l ghost image intensity,
l total track length,
l maximum surface height,
l glass weight,
l transmittance,
l glass cost,
l tolerance sensitivity, and so on.
A major part of lens design is optimization, defined as the process of achiev-
ing the values of independent variables that realize the target values of dependent
variables.
1.2 Problem of sensitivity and tolerances
The problems of tolerance sensitivity and the determination of tolerances are
important in lens design. The as-built performance of a sensitive design is
low if the ordinary tolerances are applied. If tight tolerances are applied to get
a high as-built performance, then the cost is high. It might seem that the
2 Yabe:OptimizationinLensDesign
evaluation of the tolerance sensitivity takes a lot of time and cannot be included
in the optimization. This Spotlight will explain how to evaluate the tolerance
sensitivity within a short time and how to control it systematically during opti-
mization. The loss of nominal performance depends on the choice of the toler-
ances, which affects the manufacturing cost.
The goal of tolerance optimization is to choose the best tolerance. However,
this is not the same as the minimization of tolerance sensitivity. When the lens
design is finished, the tolerances need to be determined for manufacture. Tight
tolerances result in high performance and high cost. Loose tolerances cause
low performance and low cost. If a tolerance set is given, the performance of
the product can be estimated statistically and the production cost can be evalu-
ated. However, it seems very difficult to prove that a given tolerance set is opti-
mal and to find the optimal tolerance set. This Spotlight formulates this problem
as the optimization problem and proposes a simple and rapid method to solve it
(culminating in Section 6.3).
2 Lens Design Optimization Principles
2.1 Merit function
Generally speaking, if the number of dependent variables is larger than the
number of independent variables, then thetarget values of the dependent variables
cannot be fulfilled simultaneously. In this situation, the problem needs to be a
least-squares problem. The merit function Φ is defined as
X
n
Φ ¼ ½wðf −t Þ(cid:2)2, (1)
i i i
i¼1
where f is the value of the i’th-dependent variable, t is the target value, w is
i i i
the weight, and n is the number of dependent variables. There is a case where a
dependent variable f is allowed to be in an interval [c d]. In this case, the weight
i i i
w and the target value t change with the value f:
i i i
If f < c, then t ¼ c;
i i i i
If c ≤ f ≤ d , then w ¼ 0.0; and
i i i i
If d < f , then t ¼ d :
i i i i
The problem is to minimize the merit function Φ in the space of the independent
variables. The weight w is chosen to express the importance of each dependent
i
variable. The minimum point of Φ changes with the values of w changing the
i
residual values of f . The choice of w is critical to the performance of the design
i i
result.
Yabe:OptimizationinLensDesign 3
2.2 Problem of nonlinear optimization
The first step of optimization is to get the differential coefficients of the
dependent variables to the independent variables. The differential coefficients
are usually approximated with the finite difference
∂f f ðx þδxÞ−f ðxÞ
a ¼ i ≈ i j j i j , (2)
ij ∂x δx
j j
where x is the j’th-independent variable, and δx is a small increment of x.
j j j
The arguments other than x are omitted in this expression. The matrix (a ) is
j ij
sometimes called the variation table. If all dependent variables are linear to the
independent variables, then the merit function is written as
(cid:2) (cid:3) (cid:4)(cid:5)
X X
n m 2
Φ ¼ w f −t þ a Δx , (3)
i 0i i ij j
i¼1 j¼1
where f isthecurrent value off,Δx isthevariation ofx,and misthenumber of
0i i j j
independent variables. The differential coefficients of Φ to all independent varia-
bles are 0 at the minimum point
(cid:3) (cid:4)
X X
∂Φ n m
¼ 2w2 f −t þ a Δx a
∂x i 0i i ij j ik
k i¼1 j¼1
(4)
X X X
n m n
¼ 2 w2ðf −t Þa þ2 Δx w2a a ¼ 0:
i 0i i ik j i ij ik
i¼1 j¼1 i¼1
These equations can be written concisely as
At~yþAtAΔ~x ¼ 0, (5)
A ¼ wa , (6)
ij i ij
y ¼ wðf −t Þ: (7)
i i 0i i
The solution is written as
Δ~x ¼ −ðAtAÞ−1At~y: (8)
If the dependent variables are not linear to the independent variables, matrix A
does not express the behavior of dependent variables properly at the region far
from the starting point. In this case, the solution needs to be searched with
iterative procedures. This is called the nonlinear optimization. Nonlinear