by M. Isshiki
Designing a lens is like searching an ore in the crust of the earth. Even if one has found a good vein, he would always think that there must be better ones at some other locations. The situation is the same with lens design. If a designer finds a solution by using some optimization technique, the design thus found is only a local minimum of the merit function located near the starting design. What makes the design problem more complex is the fact that solutions are distributed in a multi-dimensional parameter space while the crust of the earth extends only in a three-dimensional space.
The problem of finding the global solution has not been solved yet. Our "Global Explorer" (GE) method finds multiple local minima automatically, so that the designer can choose the most appropriate one among them. The good points of the method are:
2. Optimization in lens design
In designing a lens, we have, first of all, to set up a starting design parameters, such as number of lens elements, arrangement of them, materials used and so on. To find a good starting design is so important that it almost decides the quality of the solution reached. Once a good starting point is given, computer can easily lead the design to an excellent solution automatically.
However, there is no standard or systematic approach to find a promising initial design. This is a serious defect of the Damped Least Squares (DLS) method for optimization. Furthermore, there are other defects; the changes below can never be made without the intervention of designers.
3. Error functions
An error function shows some character of an optical system. For example, it may be the lateral aberration i.e. deviation of a point at which an image-forming ray crosses the object plane from its ideal position. These error functions are shown as
fi ( i=1, 2,,.., m)
Design parameters are denoted by
xj (j=1, 2, ...,n)
They are surface curvatures, axial distances between surfaces, refractive indices of glasses and so on. Image errors are functions of xj’s shown as
fi(xj ) ( i=1, 2,,.., m)
Time consuming ray trace calculation is indispensable to get error functions. In order to save the calculation time for image evaluation, there are several ideas, some of which are described in references (1 - 3).
4. Local optimization (Damped Least Squares method)
Designing a lens is to find a set of xj’s that makes error functions as small as possible. The computer tries to minimize the value of the merit function f defined by
where fi’s are error functions to be controlled, fiT ’s are their target values (zeros in most cases), and wi's are weights for fi's. In the following description, weights and target values are included in fi's, namely wi(fi - fi T ) 2so that the error function can simply be written as
The conditions of minimum f are given by
. (j=1,2,…,n)
The design problem is to find the values for xj's that satisfy these equations, but that is impossible because f is composed of errors fi's that are very complicated functions of parameter xj's; these functions can only be numerically calculated with ray tracing.
In order to avoid this complication, the fi's are linearly approximated in the vicinity of the starting point xj0 as
where aij’s are,
Using these the definition of the error function, we have an approximated merit function
which is a second order polynomial equation with regard to xj. The values for xj that minimize this approximated merit function can be obtained by solving simultaneous linear equations below.
(j=1,...,n)
The solution thus obtained is no more than an approximated solution. This is regarded as the starting point for the next iteration of the above process; it is expected that these repeated processes will lead the design to converge to a real local minimum.
In order to assure that the design converges, a damping factor D is introduced. Namely, the term
is added to the merit function so that
is to be minimized. In doing so, the design shift distance or the step size
remains within a reasonable range in which the approximation error is not so serious. If D is small, convergence to a solution is not always guaranteed; if too large, the step size of design is too small and it will take too much time to reach a solution. Therefore, value of D should be decided to fit the geographical features of merit function in each iteration run. Since automatic control of D had been successfully found(8), this method, called "Damped Least Squares" or DLS, has become the standard optimization technique.
The solution thus obtained is no more than a local solution or local minimum that happens to be near the starting design. Once the design is trapped there, it is impossible to get out of that place, because damping factor D becomes too large around the local minimum and this prevents the design from jumping out of the trap. This is one of the most serious defects of the DLS method.
5. Global optimization with escape function (4-5)
The following figure illustrates a model of merit function f having a design parameter xj. When the design falls into a local minimum at xjL, an escape function fE is set up there which is to be added to the error function. The escape function is defined by
where
xjL: Local minimum from which the design is to escape.
m j: Weights for design parameters.
H and W: Escape parameters as shown in figure.
Illustration of local minima and an escape function
The shape of merit function f around its local minimum changes with the escape function (see figure); f is raised by an amount of fE2 when an escape function is added to the error function. This enables the design to escape from the trap. Repeating this process, you can automatically find a predetermined number of local minima. In the next section, the program named ‘Global Explorer’ is explained in detail.
From the above graph, it may look very difficult to select appropriate values for the two parameters H and W. However, in practical cases where the number of parameters is large enough, this problem is not so delicate; a rather crude choice of these two values would be acceptable in most cases. In the above figure, the number of parameters is only one, i. e. the picture shows a model in one dimensional parameter space. A model in two dimensional space is shown below.
Visualization of the escape function in 2D parameter space.
6. Design process
The process "Global Explorer" or GE is shown in the following a flow chart and also by the following description of steps.
Flow chart of program GE
In the step (4), there must be a criterion to judge whether the escape was successfully made or not. In this program, the distance of two solutions is defined as
where x and x¢ are positions of the local minima in the parameter space. If Dp is larger than a threshold value Dt, i.e.
these two solutions are regarded as independent. If this relation does not hold between a newly found solution and each of the already filed solutions, the escape is judged as a failure.
Solutions under a fine search; threshold value Dt=0.1. (a) is starting design, (b), (c), (d), and (e) are design No. 3, 6, 9, and 12 respectively. The lens types are rather similar.
Solutions under a rough search; threshold value Dt=10. (a) is starting design, (b), (c), (d), and (e) are design No. 3, 6, 9, and 12 respectively. The lens types are more independent
When threshold value Dt is set comparatively small, the design solutions tend to be similar in shape. For a large value of Dt, the solutions filed are more independent with each other. This example is for a lithographic projection lens, NA=0.3, image 10x10 mm, and wavelength of light 248.5 nm.
It might be good to start the global search with a comparatively large threshold value to find most promising area or areas. After that, fine search of the design could be performed with a smaller threshold value.
7. Default values for Global Explorer
In applying the Global Explorer to global optimization of lens design, following parameters should be assigned. They are
(1) m j: Weights for the j design variables. There is no rule now to determine these values, but recommended values are given in the following table.
| Parameter | Value | Description |
| ge_cvtypwgt | 1000 | Curvature weights |
| ge_thtypwgt | 1 | Thickness weights |
| ge_rntypwgt | 1 | Refractive index weights |
| ge_dntypwgt | 1 | Dispersion factor weights |
(2) H and W: Height and width of escape function’s contribution to merit function (fE2). As noted before, these values could be chosen rather arbitrarily, however, the recommended initial values are
ge_start_height = H0 = 0.1
ge_start_width = W0 = 0.5,
and if the escape is unsuccessful, these are changed in the following manner5).
Hj = ge_height_mult*Hj-1
Wj = ge_width_mult*Wj-1
The default value for ge_height_mult is 2.0 and the default value for ge_width_mult is 1.03.
(3) ge_min_diff = Dt : Threshold distance for identifying two solutions. Default value is 1.
8. Design examples
As an example, a lens having focal length 100 mm, f-number 2.5, and image height 21.6 mm. was designed. The distance of the first surface and the image plane was kept within 145 mm. The starting design was a plano-convex lens followed by five plane parallel plates. All the surface curvatures and axial separations between surfaces were taken as variables. The Global Explorer found 100 solutions including types shown in the figure below.
Typical solutions found with the software Global Explorer
Merit functions of those 100 solutions in the order of finding are shown in the figure below. Note that there is a kind of lower limit for those merit functions. This limiting value depends essentially on the complexity of lens structure. In order to reduce the merit function remarkably beyond this limit, it would not be practical to increase the number of filed designs – in this case more than 100. It is recommended to increase the complexity of the system, for example by introducing aspheric surfaces.
Relative Global Explorer merit functions in the order found for 100 solutions
References