Commercial Opical Design Software

The exact process involved in design optimization using each commercial optical design software differs from program to program. [43] [44] However, the fundamental principle is very similar across all design software.

Defining Error Function

First, a set of error functions (some referred to as the merit function) are defined. These functions dictates the quantities to be minimized during optimization, and does not nec-essarily have to be related to specific optical system properties in general. For example, if for some odd reason that an occasion should arise that for a system consisting of two lenses, the lens diameter of the second lens (D₂) is required to be twice as large as the diameter of the first lens (D₁), then that particular error function can simply be expressed as the quantity

ε = D2− 2D₁ (60)

to serve as the criterion of the optimization.

Commercial design software usually have a set of default merit functions dedicated to improve the image quality by reducing the aberration of the optical system. This allows optical designers to dedicate their efforts in defining the set of error functions to target specific goals that they might want to achieve in the design, while the formulation of the general image aberration error functions can be handled by the software. The total error

is then a weighted square sum of each of the individually defined errors

ε_total =

∑

n

ε_n² (61)

Defining Variable Parameters

After the error functions are defined, before optimization can commence the variable parameters must be defined. Parameters that defines an optical system are parameters such as

• Surface Profile Parameters

The surface profile parameters of the optical elements. Surface radius of curvature, conic constant, and aspheric polynomial coefficients.

• Element Thickness and Spacings

The thickness of the optical elements, and the spacing between each of them.

• Material Properties

Refraction indices and dispersion coefficients of the material of the optical ele-ments.

• Optical System Properties Effective focal length, field of view, NA, F-Number, entrance pupil and exit pupil location and diameter.

Optical Design Optimization

The act of optical design optimization is an attempt to find the optimal combination of optical system parameters such that the total error defined is minimized. Mathematically, if the number of variable parameters are few, this can be as simple as taking a derivative of the error function and finding where it goes to zero

d dρ

∑

n

ε_n(ρ)² = 0. (62)

However this is also apparent that when either (or both) the number of variable parameters or the number of defined error functions is large, this process becomes extremely tedious, or practically impossible to perform.

How the optimization algorithm tackles this task is by simply seeking for a solution that is better than the current solution. By tuning the variable parameters, the optimiza-tion algorithm checks if the total error is decreased each time when the parameters are varied. If so, the result is accepted, saved, and the optimizer moves on to find the next better solution; or else if the error function did not decrease, the result is discarded and again the optimizer moves on to find the next better solution. This way, if one continues to find solutions that are better than the current solution, then hopefully, given enough time, the optimal solution can be reached eventually. Up to this day, many optimization methods exist. A typical method of optimization employed by optical design software is the damped least square method. [45]

The Problem

By defining a set of error functions as the optimization criterion and allowing some pa-rameters of the optical system to be variable, the optical design software can optimize the design to the error functions given, by varying the available parameters. However, one glaring issue is that not all error functions are equal in importance or priority. For example, in the case of a lithographic projector, the property of telecentricity against a desired mask-wafer magnification. To the optimizer, some degree of loss in the telecen-tricity will have the same measure of “badness” as some degree of change in the system magnification, depending on the relative weightings given to the error functions associ-ated. However in terms of lithographic projectors, some level of difference between the desired magnification and the actual magnification is not a problem as long as the actual magnification can be known and measured, the loss of telecentricity on the other hand is a very serious offense leading to a multitude of undesired imaging quality deterioration.

As such, optics designers faces these questions constantly:

• How much weighting should be assigned to each error function?

• How large does the weighting has to be to ensure that the particular error function maintained unvioleted throughout the optimization?

• Is the assigned weighting too large (or too small) as to cause the design to distort beyond intended?

• What should the weighting balance be between the error functions associated with the desired optical properties, and the error functions associated with the image aberration in general?

These question have no simple straightford answers, and is likely to haunt the designers throughout the entire design process.

The optical design software Code V attemps to address this issue of unequalness of the importance of the error functions by separating them into two “tiers” of error func-tions. The lower tier error functions are weighted, and the higher tier error functions are hard solved by sacrificing one available variable as the compensator to guarantee its maintainance. In the case that the option to hard solve the solution is not possible, either due to the error function being overly complex or convoluted to calculate, or the number of defined error functions exceed the number of available variables, they will be given a considerably larger weighting to be optimized along with the lower tier error function.

In practice, except for extremly limited and simple cases, the number of defined error functions will always exceed the number available variable parameters. Also, the notion of setting one variable parameter aside purely as a compensator parameter to fulfill one particular error function, aside from being a luxurious choice that most designs could not afford, is also quite dangerous. First, there is the risk of the calculated value of the com-pensator parameter being unrealistic or impossible is also present, such as having negative propagation distances (e.g. point of convergence for a diverging optical system) and a lens with radial thickness larger than the surface radius of curvature of the lens. Second, if dur-ing the optimizdur-ing process that error function was somehow violated, the large weightdur-ing assigned to it would cause the optimization algorithm to focus all optimizing efforts on it until it was remedied, most of the time in complete disregard to all other error functions

and likely rending the design irrecoverable beyond recognition.

One possible roundabout to deal with this is to first deal with the problematic error functions first in the initial stage of the optimizations, before adding in the other error functions that are less likely to cause trouble. After a few rounds of optimization, the design should be at a point where the problematic error function are mostly satisfied, and only then start adding in other error functions while keeping the weighting on the prob-lematic error functions large and hope that the large weighting would deter the optimiza-tion algorithm from steering the soluoptimiza-tions away from them. Of course, from this other problem arises. For example, there is no way of knowing which defined error functions in the set are more problematic or sensitive prior to performing the optimization. And even then, since the addition of other error functions changes the solution space, problematic error functions in the previous case might not be in the new solution space, and vice versa error functions that did not cause problems may become problematic in the new solution space.

It is in this situation that GGC can be proven useful.