Lens design: Global optimization of both performance and tolerance sensitivity. Masaki Isshiki*a, Douglas C. Sinclairb, Seiichi Kanekoc Isshiki Optics, 3-10-10 Terabun Kamakura 247-0064 Japan b Sinclair Optics, 3651 N. Grayhawk Loop, Lecanto, FL 34461 US c Chart Inc., 5-16-21 Shikahama Adachi-ku Tokyo 123-0864 Japan a

ABSTRACT We have previously developed a global optimization method using an escape function that finds multiple local solutions in the multidimensional parameter space of a lens. We have applied this method to various actual design problems and have been able to understand some topological features of the merit function in its parameter space. These experiences have led us to develop an improved global optimization method that takes into account the robustness of the lens with respect to manufacturing and assembly errors. The method uses a technique that we call ‘θ-segmentation’ to perform the escape function search in a smaller parameter space. After reaching an acceptable solution with the escape function, it often seems impossible to reduce the sensitivity in tolerance without degrading the image quality already acquired. Generally speaking, performance and robustness are in a trade-off relation, i.e. reduced sensitivity can be obtained only by increasing the merit function of the system. However, in reality, it is often possible to reduce sensitivity without substantial increase of the merit function. Furthermore, in many cases, it is possible to reduce the merit function and the sensitivity simultaneously by using our new method. Keywords: Lens Design, Global optimization, Tolerance

1. INTRODUCTION Optimization in lens design is a process to find a set of design parameters such as surface curvatures, axial separations between surfaces, refractive indices of lens materials, etc. Namely, it is a method to find a point in the multidimensional parameter space. First, a starting point in the parameter space should be given, its error functions are to be improved by an optimization algorithm. The process of improvement is expressed as a mathematical problem of minimizing the merit function that is composed of error functions showing the lens performance. This process is now mostly based on the “damped least squared method or DLS” which leads the design from its starting point to a local minimum of the merit function. Its simplicity and effectiveness is excellent, but the design thus obtained is no more than a local solution near the starting point. There can be better designs with DLS from different starting points; selection of the starting point is essentially important, but extremely difficult. The Escape function was introduced in 1995 for the purpose of finding multiple local solutions by adding an escape function to the group of the error functions.1,2 Strictly speaking, this is not real global optimization, so we call it “Global Explorer” instead of “Global Optimizer”. However, recently we come to the conclusion that Global Explorer can give us a good design, the performance of which is quite near to the optimum solution, provided the escape function parameters H and W are well controlled. Tolerance analysis of the lens is also important for manufacturing and assembling optical systems. It gives us statistical information on the sensitivities for parameters, but what is wanted more is to reduce those sensitivities in the design stage. It is desirable to reduce both image errors and sensitivity in the optimization process. Below is our proposal to achieve that purpose. _____________________________ *[email protected]

2. METHOD OF OPTIMIZATION 2.1 DLS (Damped least squares method) and GE (Global explorer) For optimization, the performance of an optical system is expressed by a single number called the merit function φ which is defined as m

φ = ∑w i f i 2 ,

(1)

i =1

where fi is an error function, and wi is its weight. It is a function of design parameters such as surface curvatures, axial surface separations, and refractive indices of materials. These parameters are denoted as xj (j =1,2,..,n), therefore

φ = φ (x 1, x 2 ,L, x n ) . DLS seeks a solution in the parameter space at which ∂φ =0 ∂x j

(2)

(3)

( j = 1, L, n )

In order to have the solution of Eq.3, error functions fi’s are linearly approximated, and the error due to this approximation is reduced by successively iterating the process of solving Eq.3, in the hope that the values xj converges to the real solution. A damping factor is introduced for the purpose of assuring the convergence and also to increase the speed of convergence. This process is widely used in the optimization as DLS ( Damped Least Squares) method, but it only gives a local minimum of the merit function near the starting design. This local optimization can be extended into a global optimization by adding an escape function fE to the group of error functions.

⎧

1 ⎩ 2W

f E = H exp⎨−

⎫

∑j µ j (x j − x jL ) ⎬ 2

(4)

⎭

Here, xjL is the coordinate of the local minimum, µj is the weight imposed on parameter xj , H is the height of the fE2 (contribution of the escape function to the merit function), and W corresponds to the width of fE2 as shown in Fig. 1. This function enables the design to get out of the basin of the merit function to find a different local minimum. This algorithm for finding plural solutions is named “Global Explorer”.1,2

Fig.1

Contribution of an escape function to get out of a local minimum

The above optimization method is purely of mathematical nature; it can be applied not only to optical design but also to many general problems, provided the functions treated are one-valued, continuous and differentiable. However, our merit function has some features peculiar to optical design. Our empirical knowledge of optical topography will be utilized to establish a new strategy of finding practical design solutions.

2.2 Topological feature of merit function in parameter space

In actual lens design, good local minima of the performance merit function (Eq. 1) are located within a certain winding string that stretches in the multidimensional parameter space as in Fig. 2 (a). This has been empirically known for these 40 years.4,5 Fig.2 (a) is the image of the winding string and (b) shows contour lines of a merit function for a two dimensional parameter space. (a)

(b)

Fig.2 Winding string and local minima

As stated above, good local mimima lie densely along the bottom of the twisted valley, the both sides of which are generally steep walls. In most cases, those local minima of good designs have nearly the same values of merit function. The designer’s next job is to find out the most favorable solution among many acceptable local minima. To select the one that has the smallest merit function is practically not always recommendable because very small differences in the merit function are meaningless. The merit function is an indispensable tool to optimize the system, but it is no more than an averaged error functions or aberrations; it is not good to pursue minimizing the value too persistently. There should be some other criterion for choosing the practical solution at the final stage of design such as cost, size, or robustness of the system against manufacturing errors. Our investigation lies on this line; the detail of which will be described in following section. 2.3 Sensitivity reduction

When tolerances of a system is given by ∆x1, ∆x2,…,∆xn, the sensitivity S is defined as

⎛ ∂φ S = ∑ ⎜⎜ ∆x j j ⎝ ∂x j

2

⎞ ⎟ , ⎟ ⎠

(5)

where φ is the merit function of the system.6 In designing lenses, what is important is not only to reduce the merit function φ but also to make S as small as possible. However, it is not practical to try to reduce those two factors simultaneously with DLS or GE (Global Explorer), because it takes too much time to calculate S and its change table with respect to design parameters. Instead of using this complicated function S, we employed a root mean square value of incident and refracted angles for some sample rays denoted by θ as

k

θ=

(i s ∑ s

2

=1

+ rs2

) ,

2k

(6)

where is and rs are respectively incident angle and refracted angle of a sample ray at a surface s. These sample rays may typically consist of the marginal ray to the image center, as well as the upper and lower rim rays to the image corner. Angles at cemented surfaces are excluded. Previous studies have suggested that the value θ has good correlation with S, and we can use this value tentatively to obtain a robust solution. It is important to remember that this correlation is only empirically known, therefore some irregularities, exceptionally, can creep into this relation; we have to check it at the final stage. The proposed method (GE2) is to reduce the merit function φ under the condition θ < θL , (7) where θL is the upper limit. Several solutions are obtained by successively reducing this limiting value. Firstly, GE2 is applied to the starting design without the restriction of Eq. (7), or under a limiting value θL =90 (degrees). The solution (No.0) has the angle of θ0 and the merit function φ0 . GE2 is applied to the solution (No.0) with a limiting value θL1 which is a little smaller than θ0. The resulting solution (No.1) has θ1 and φ1. The process is repeated until the value φ becomes too big to be accepted. The results are shown in Table 1. For example, if θ0=23.4 (degree) for the solution (No.0), the first θL1 can be set to 23.0. Then, θL 2=22.5, θL 3=22.0, θL 4=21.5,… and so on. In most cases θk≌ θLk except the solution No. 0. This means that topography of the θ in the parameter space is rather simple compared with that of the merit function. Table 1 List of solutions

Solution No. 0 No. 1 No. 2 No. 3 …. No. k

Limiting angle(θL) 90.0

θL1 θL2 θL3 …

θLk

Angle(θ)

θ0 θ1 θ2 θ3

Merit function(φ)

…

θk

φ0 φ1 φ2 φ3 …

φk

Sensitivity(S) S0 S1 S2 S3 … Sk

After reaching an acceptable solution with the escape function, it often seems impossible to reduce the sensitivity in tolerance without degrading the image quality already acquired. Generally speaking, performance and robustness are in a trade-off relation, i.e. reduced sensitivity can be obtained only by increasing the merit function of the system. In actual experiments, however, this general tendency is not always kept. Sometimes the value of merit function is a little smaller than φ0 for the first few solutions (e.g. No.1, No.2), and after that φ increases with the increase of the solution numbers. Due to this ‘fortunate’ abnormality, the designer can find a solution in which both performance and stability can be improved. The reason for this irregularity could be explained as follows. The solution No. 0, which is reached by GE2 without angle limitation, is not the real optimum solution having the smallest merit function. It is only the best solution among the those obtained by GE within a limited computer time. In the next run, a better solution (No. 1) may be reached by the same algorithm within the same limited time, because the 