Optimization

The function of the optimization part of the program is to take a starting design and modify its construction so that it meets a given set of specifications. The starting design may be the result of a previous design task, a lens from the library, or a new design based on general optical principles and the designer's intuition.

The performance of the design must be measured by a single number, often known in optics as the merit function, although the term error function is more descriptive and will be used here. The error function is the sum of squares of quantities called operands that characterize the desired attributes. Examples of typical operands include paraxial constants, aberration coefficients, and exact ray displacements. Sometimes, the operands are broken into two groups: those that must be satisfied exactly, which may be called constraints, and others that must be minimized. Examples of constraints might include paraxial conditions such as the focal length or numerical aperture.

The constructional parameters to be adjusted are called variables, which include lens curvatures, thicknesses, refractive indices, etc. Often the allowed values of the variables are restricted, either by requirements of physical reality (e.g. positive thickness) or the given specifications (e.g. lens diameters less than a prescribed value). These restrictions are called boundary conditions, and represent another form of constraint.

Usually, both the operands and constraints are non-linear functions of the variables, so optical design involves non-linear optimization with non-linear constraints, the most difficult type of problem, from a mathematical point of view. A great deal of work has been carried out to develop efficient, general methods to solve such problems. Detailed consideration of these methods is beyond the scope of this article, and the reader is referred to a paper by Hayford.

In a typical optical design task, there are more operands than variables. This means that there is, in general, no solution that makes all of the operands equal to their target values. However, there is a well-defined solution called the least squares solution, which is the state of the system for which the operands are collectively as close to their targets as is possible. This is the solution for which the error function is a minimum.

The damped-least-squares method

Most optical design programs utilize some form of the damped least squares (DLS) method, sometimes in combination with other techniques. DLS was introduced to optics in about 1960, so it has a history of more than 30 years of (usually) successful application. It is an example of what is known as a downhill optimizer, meaning that in a system with multiple minima, it is supposed to find the nearest local minimum. In practice, it sometimes suffers from stagnation, yielding slow convergence. On the other hand, many designers over the years have learned to manipulate the damping factor to overcome this deficiency, and even in some cases to find solutions beyond the local minimum.

We  consider first the case of unconstrained optimization. Let the system have M operands f_i and N variables x_i. The error function is given by

Define the following:

With these definitions, we have

If we assume that the changes in the operands are linearly proportional to the changes in the variables, we have

At the solution point, the gradient vector is zero, since the error function is at a minimum. The change vector is thus

These are called the least-squares normal equations, and are the basis for linear least-squares analysis. When nonlinear effects are involved, repeated use of these equations to iterate to a minimum often leads to a diverging solution. To prevent such divergence, it is common to add another term to the error function, which limits the magnitude of the change vector x. In the DLS method, this is accomplished by defining a new error function

A key property of DLS is that the minimum of is the same as the minimum of , since at the minimum, the change vector x is zero. By differentiating and setting the derivative equal to zero at the minimum, we arrive at the damped least squares equations

which look like the normal equations with terms added along the diagonal. These terms provide the damping, and the factor p is called the damping factor. This particular choice of damping is called additive damping, but more generally it is possible to add any terms to the diagonal and still maintain the same minimum. Some optical design programs multiply the diagonal elements of the  matrix by a damping factor, while others make them proportional to second derivative terms. Although theoretical arguments are sometimes advanced to support the choice of a particular method of damping, in practice the choice of damping factor is an ad hoc way to accelerate convergence to a solution by limiting the magnitude (and changing the direction) of the change vector found from the normal equations.

In practical optical design work, it has been found that no single method for choosing the damping factor works best in all cases. In a particular problem, one method may be dramatically better than another, but in a different problem, the situation may be completely reversed. Every optical design program has its unique way of choosing the optimum damping, which makes each program different from the others, and gives it a raison d'être.

Although the principal use of the damping factor is to accelerate convergence by limiting the magnitude of the change vector, the damping factor has also been used routinely to increase the magnitude of the change vector to escape a local minimum. During the course of a minimization task, if the solution stagnates, or does not converge to what the designer believes to be an acceptable configuration, it may be possible to force the solution into another region by running one or more iterations with reduced damping in which the error function increases.

Constraints and boundary conditions

There are two general methods used in optical design programs for handling constraints and boundary conditions. The first is to add a term (called a penalty function) to the error function that targets the constraint to its desired value. In the case of boundary violations, "one-sided" terms can be added, or special weighting functions can be constructed that increase in magnitude as a violation goes farther into a forbidden region. The other method augments the number of equations by the number of constraints, and solves the resulting equations using the Lagrange multiplier method. This produces a minimum that satisfies the constraints exactly, and minimizes the remaining error function.

The penalty function method is more flexible and faster (since there are fewer equations) than the Lagrange multiplier method. On the other hand, the Lagrange multiplier method gives more precise control over the constraints. Both are commonly used in optical design software.

Other methods

Although DLS is used in the vast majority of optical design applications, other methods are occasionally used[i], and two warrant mention. These are orthonormalization, which has been used to overcome stagnation in some DLS problems, and simulated annealing, which has been used for global optimization.

Orthonormalization

The technique of orthonormalization for the solution of optical design problems was introduced by Grey. Although it solves the same problem as DLS does, it proceeds in a very different fashion. Instead of forming the least-squares normal equations, Grey works directly with the operand equations

To understand Grey's method, it is best to forget about optics and consider the solution of these equations strictly from a mathematical point of view. The point of view that Grey uses is that f represents a vector in m dimensional space. The columns of A can be regarded as basis vectors in this m-dimensional space. Since there are only n columns, the basis vectors do not span the space. The change vector x represents a projection of f on the basis vectors defined by A. At the solution point, the residual part of f will be orthogonal to its projection on the basis vectors.

In Grey's orthonormalization method, the solution of the equations is found by a technique similar to Gram-Schmidt orthogonalization, but during the solution process, the actual error function is evaluated several times in an effort to use the best variables to maximum advantage. Because of this, the method is computationally intensive; compared to DLS. However, the extra computation is justified by a more accurate solution. The common wisdom is that orthogonalization is superior to DLS near a solution point, and inferior to DLS when the solution is far removed from the starting point.

Simulated annealing

Simulated annealing has been applied to optical design optimization, chiefly in problems where the task is to find a global minimum. The method varies drastically from other techniques. It makes no use of derivative information, and takes random steps to form trial solutions. If a trial solution has a lower error function than the current system, the new system replaces the old. If a trial solution has a higher error function than the current system, it may be accepted, depending on how much worse it is. The probability of acceptance is taken to be , where T is an experimentally determined rate. In general simulated annealing, T is provided by the user. In adaptive simulated annealing, T is reduced automatically according to algorithms that hold the system near statistical equilibrium.

Error functions

Obviously, the choice of an error function has a major impact on the success of an optical design task. There are a number of requirements that an error function should meet. Most importantly, the error function should accurately characterize the desired properties of the system under design. There is little chance of success if the program is optimizing the wrong thing. Yet this is an area of great difficulty in computer-aided optical design, because it is at odds with efficiency. In order to obtain more accuracy, more extensive computations should be carried out, but this takes time.

There are two schools of thought concerning the implementation of error functions in optical design programs. The first holds that the designer should have complete control over the items included in the error function, while the second holds that the program itself should set up the basic error function, allowing the designer some degree of control through weighting functions. Neither school has demonstrated superiority, but the approach to error function construction taken by various optical design programs accounts for user allegiances that are sometimes remarkably strong.

The different ways that optical design programs handle error functions makes it difficult to discuss the topic here in anything other than broad detail. At one extreme are programs that provide practically no capability for the user to insert operands, displaying only the value of the overall error function, while at the other extreme are programs that make the user enter every operand individually. Regardless of the user interface, however, there are some general concepts that are universally relevant.

Error functions can be based on either aberration coefficients or exact-ray data (or both). In the early stages of design, aberration coefficients are sometimes favored because they provide insight into the nature of the design, and do not suffer ray failures. However, the accuracy of aberration coefficients for evaluating complex systems is not very good, and exact-ray data are used in virtually all final optimization work.

So far as exact-ray error functions are concerned, there is the question of whether to use ray displacements or optical path difference (or both). This is a matter of user (or programmer) preference. The use of ray displacements leads to minimizing geometrical spot sizes, while the use of optical path difference leads to minimizing the wavefront variance.

For exact-ray error functions, a suitable pattern of rays must be set up. This is often called a ray set. There are three common methods for setting up a ray set. The first is to allow the designer to specify the coordinates (object, pupil, wavelength, etc.) for a desired set of rays. This gives great flexibility, but demands considerable skill from the user to ensure that the resulting error function accurately characterizes performance.

The other two methods for setting up ray sets are more automatic. The first is to allow the user to specify object points, and have the program define a rectangular grid of rays in the aperture for each point. The second uses a gaussian integration scheme proposed by Forbes to compute the rms spot size, averaged over field, aperture, and wavelength.[ii] The Forbes method, which is restricted to systems having plane symmetry, leads to dividing the aperture into rings and spokes. For systems having circular pupils, the Forbes method has both superior accuracy and efficiency, but for vignetted pupils, there is little difference between the two.

Multiconfiguration optimization

Multiconfiguration optimization refers to a process in which several systems having some common elements are optimized jointly, so that none of the individual systems are optimized, but the ensemble of all of the systems is optimized. The archetype of multiconfiguration systems is the zoom system in which changing the separation between certain elements changes the focal length. The system is optimized simultaneously at high, medium, and low magnifications to produce the best overall performance.

Most of the larger optical design programs have the capability to carry out multiconfiguration optimization, and this capability is probably used more for non-zoom systems than for zoom systems. A common use of this feature is to optimize a focal system for through-focus performance, to minimize sensitivity to image plane shifts. In fact, multiconfiguration optimization is used routinely to control tolerances.

Tolerancing

Beyond the task of desensitizing a given design, considerations of manufacturing tolerances become increasingly important as the complexity of optical designs increases. It is quite easy to design optical systems that cannot be built because the fabrication tolerances are beyond the capability of optical manufacturing technology. In any case, specifying tolerances is an integral part of optical design, and a design project cannot be considered finished until appropriate tolerances are established.

Tolerancing is closely related to optimization. The basic tolerance computation is to calculate how much the error function changes for a small change in a construction parameter, which is the same type of computation carried out when computing a derivative matrix. Even more relevant, however, is the use of compensators, which requires reoptimization. A compensator is a construction parameter that can be adjusted to compensate for an error introduced by another construction parameter. For example, a typical compensator would be the image distance, which could be adjusted to compensate for power changes introduced by curvature errors.

There is considerable variation in how different optical design programs handle tolerancing. Some use the reoptimization method described here, while others use Monte Carlo techniques. Some stress interaction with the designer, while others use defaults for more automatic operation.

Global optimization

After several years during which there was little interest in optimization methodology, the tremendous increase in the speed of new computers has spawned a renewal of efforts to find global, rather than local, solutions to optical design problems. Global optimization is a much more difficult problem than local optimization. In the absence of an analytic solution, one never knows whether a global optimum has been achieved. All solution criteria must specify a region of interest and a time limit, and the method cannot depend on the starting point. The simulated annealing method described above is one area of continuing interest. Several methods for what might be called pseudo-global optimization have been used in commercial optical design programs, combining DLS with algorithms that allow the solution to move away from the current local minimum.