There are some cases in which default thickness boundary constraints are not always sufficient when optimizing aspheric components. This article explains how the FTGT and FTLT optimization operands can be used to successfully constrain the thickness of a surface at intermediate aperture locations.

**Authored By Andrew Locke**

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## Introduction

Successful optimization usually involves the bounding of variables to ensure that optimized systems are physically realizable. Probably the most common variable type that requires bounding is thickness. Unbounded, thickness variables will generally produce either very thin, unmountable lenses or lenses that are unreasonably thick, heavy and expensive.

Thickness boundary constraints for glass and air are so common they are built directly into the Optimization Wizard. While these default constraints can be useful, there are cases where they will not sufficiently bound the thickness of a given surface. As we will see, polynomial aspheric surfaces generally cannot be successfully bounded using the default constraints.

## Default thickness boundary constraints

To examine the thickness boundary constraint capabilities of OpticStudio, please download the file attached to this article.

This Sequential file provides a model of a Schmidt camera. While Schmidt cameras are generally used for wide field of view applications, we will just work on-axis for the purposes of this article. This particular Schmidt camera is a standard configuration. It consists of an aspheric lens along with a spherical mirror:

We want to improve the imaging performance of our Schmidt camera so we will optimize for best RMS spot size. Variables have already been set for the front radius of curvature of the aspheric lens, along with the r^{4} coefficient of the asphere:

With variables set, we will now build a default Merit Function for RMS spot radius. In the Optimize tab, select **Optimization Wizard****.** Be sure to set the number of “Rings” to 4 given that we are optimizing the r^{4} aspheric coefficient. While we are not going to optimize the thickness of the lens, changes to the radius of curvature and 4th order aspheric coefficient will change the effective thickness of our lens. As such, we will incorporate default boundary constraints on thickness in our Merit Function. We will bound a minimum thickness of 1 mm at the center and edge along with a maximum thickness of 5 mm.

## Initial optimization results

With our Merit Function setup, we can now optimize our Schmidt camera using the standard (local DLS) optimizer. Click **Optimize...Optimize!** , select “Automatic” for the cycles, and then click “Start”. OpticStudio adjusts the radius of curvature and r^{4} coefficient of the front of our lens to improve image quality. Looking at the boundary constraint operands in the Merit Function Editor, it is clear that all of our boundary constraints are being met. The values of all three thickness boundary constraint operands have no contribution to the total Merit Function value:

Looking at the 3D Layout, though, there appears to be a problem with our lens’ thickness. While the thickness of the lens appears reasonable (i.e. neither too thick nor too thin) at the center and edges, the lens appears to be very thin at intermediate aperture locations:

By using the active cursor on the layout, you can establish that the thickness of the lens is less than 0.5 mm at these intermediate locations! Clearly this lens is so thin that it would likely be difficult to manufacture and/or mount.

The problem is that we did not establish any boundary constraints which restrain the thickness of the lens at these intermediate aperture locations. The default boundary constraints are designed for use with spherical and conically aspheric surfaces. Thus, they only constrain surface thicknesses at the center and edge. To address this limitation when using polynomial aspheric surfaces, we need to use a different type of boundary constraint operand, one that is more appropriate for polynomial aspheres.

## Constraining the "full" thickness

Remove the default boundary constraints by rebuilding the default Merit Function with the glass boundary constraints de-selected:

We will instead use the “full thickness” boundary constraint operands FTGT (Full Thickness Greater Than) and FTLT (Full Thickness Less Than). Unlike the default boundary constraint operands, the FTGT/FTLT operands measure the thickness of a surface at 200 points between the vertex and the edge along the +y radial direction.

For our FTLT and FTGT operands, we will use the same thickness constraints as we used for the default boundary constraints (minimum thickness of 1 mm and maximum thickness of 5 mm). Set the weight of both operands to 10. Here is the Merit Function with the FTLT and FTGT operands added:

The value and contribution of the FTGT operand is indicative of what we already know; the thickness of the lens is too thin at intermediate locations. The value of the FTGT operand indicates that the thinnest part of the lens is only 0.4079 mm thick.

We can now optimize the system with the improved boundary constraints in place. Click on the Optimize! button in the button bar and select “Automatic” optimization. The results are encouraging. The thickness of the lens has increased noticeably at the intermediate locations:

The active cursor in the layout as well as the value of the FTGT operand in the Merit Function both indicate that the intermediate thickness values are still less than 1 mm. The value of the FTGT operand indicates that the thinnest part of the lens is 0.8265 mm thick. Increasing the target and/or the weight of the FTGT operand and re-optimizing will increase the thickness further.

## Other options

While useful, the FTGT and FTLT operands do have their own limitations. These operands only check the thickness between the vertex and the +Y radial aperture. As such, FTGT and FTLT can only be used with rotationally symmetric surfaces such as radial aspheres. For polynomial aspheric surfaces that are not rotationally symmetric, the more general STHI operand can be used. The STHI operand can be used to constrain the thickness of a surface measured from any location on the surface. When using this operand, you specify a surface number as well as an X, Y location:

Other useful operands for constraining the thicknesses of non-rotationally symmetric aspheric surfaces include:

XNEA/XNEG/XNET (minimum edge thickness constraints)

XXEA/XXEG/XXET (maximum edge thickness constraints)

KA-01681

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