This article describes a fast method to identify which surfaces in a design will best benefit from aspherization.

**Authored By Mark Nicholson**

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## Introduction

Making one or more surfaces in a design aspheric is a common way to improve optical performance without increasing the number of surfaces. However, it is not always apparent which surfaces will most benefit from aspherization. Further, adding an aspheric surface can increase manufacturing costs. Thus, it is usually best to use the lowest degree of asphericity needed to give the desired performance.

OpticStudio's Find Best Asphere tool is a useful way to identify which surface will benefit most from aspherization. The tool replaces spherical surfaces with aspherical ones of a user-specified degree and may be run multiple times to find the best performance. This article will demonstrate the use of Find Best Asphere tool.

## An example

The example file (which is provided as an attachment to the article) shows a derivative of the Cooke triplet, optimized for best RMS wavefront error. All radii and thicknesses are variable, except the last radius of curvature which is controlled by an F/# solve and maintains the lens as f/5 during optimization.

The merit function for this design was built using the default merit function tool, and consists of RMS wavefront error and lens center/edge thickness boundary constraints. Five rings were used by the Gaussian Quadrature pupil sampling algorithm. Since n rings allows aberrations of order r^{2n - 1} to be controlled exactly, this gives control of wavefront aberrations up to r^{9}. The highest order aberration in the design currently is r^{6}, higher order spherical aberration.

The current value of the merit function is 0.102. We then run **Tools...Optimization...Find Best Asphere**:

The tool allows us to choose start and stop surfaces, and the maximum order of the selected polynomial. Each surface within the range is evaluated to see if it is a candidate asphere. To be considered, the surface must be of type Standard, have no conic value, define a boundary between air and glass (cemented surfaces usually make poor aspheres), and have a curvature that is either variable or controlled by a marginal ray angle or F/# solve. Surfaces that do not meet this test are ignored.

When a candidate surface is identified, the surface is converted into an asphere of the user-selected type. The aspheric terms are set as variables for optimization. The local damped-least squares optimizer is then called to optimize the modified system. If the resulting system has a lower merit function, the system is retained. The procedure repeats until all surfaces have been tested. Finally, the tool reports which surface, when converted to an asphere, provided the lowest merit function. For example:

Pressing Keep and Exit will change surface 1 to an Even Asphere type with optimised parameter up to the 6th order aspheric coefficient. Alternatively, changing the desired order of asphere and pressing Start again yields these results:

- Initial Design: 0.102
- Conic asphere: 0.087
- 4th order: 0.088
- 6th order: 0.084
- 8th order: 0.084
- 10th order: 0.083
- 12th order 0.082

The user can then choose what degree of asphere provides the most effective improvement in performance.

## Considerations for use

The current merit function is used as the initial value for comparison in the tool, and all parameters that are variable are re-optimized during this process. The current merit function should be appropriate for an aspheric design, which may require higher sampling than a non-aspheric design for good optimization. In addition, thickness controls other than just center and edge thickness may be required. The “full thickness” boundary constraint operands FTGT (Full Thickness Greater Than) and FTLT (Full Thickness Less Than) are useful for bounding aspheres: see __this Knowledgebase article__ and the Optimization chapter of the Help Files for more details.

Note also that, like all local optimization results, there is no way to know if the solution found is the "global minimum" for that combination of merit function, variables, and design parameters. For this reason, once the best candidate asphere is determined, it is usually a good idea to run the Hammer Optimizer on the resulting design to see if any further gains are possible.

Finally, keep in mind that no attempt is made by OpticStudio to determine whether the resulting asphere is practical to fabricate or is more or less costly to manufacture as compared to making other surfaces aspheric.

KA-01683

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