High-Yield Optimization improves the optical design workflow by accounting for tolerance sensitivities in the optimization stage. This produces designs that are easier to tolerance and more manufacturable. This article explains how to use this technique in OpticStudio to reduce costs and improve yields for your optical systems.

Note that this feature requires Opticstudio 19.8 or later. Also, it is exclusive to OpticStudio Professional or Premium edition subscription licenses. If your license is not eligible for this feature, this option will be greyed out. Contact your Zemax account representative to see how to become a subscriber.

**Authored By Chris Normanshire**

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## Introduction

Traditionally optimization of an optical design has focused improving the performance of a nominal system in the absence of manufacturing and assembly errors. Assessment of the as-built performance, including these errors, is left to a subsequent tolerancing step in the design process. This methodology can lead to designs that are more sensitive to errors which increase assembly time, reduce product yield, or require redesign, all of which add cost and delay time to market. High-Yield Optimization accounts for the as-built performance of a system directly in the optimizations stage, leading to designs that are less sensitive.

## Traditional design methodology

The as-built performance of an optical system is the sum of nominal performance (RMS spot sizes, wavefront error, MTF, etc.) plus the effect of aberrations induced by factors such as manufacturing defects and assembly errors (tolerance defects). It is the as-built performance that determines whether a single product will meet specification, or in the case of larger produciton, its likely yield.

Improving the as-built performance can be achieved both by improving the nominal system performance and reducing the effect of induced aberrations. Typically, a design is optimized to improve the nominal performance with the goal of eventually improving the as-built performance. Once the nominal design is complete, a set of predicted tolerance defects is applied and the resulting as-built performance of the imperfect system computed. See "How to perform a sequential tolerance analysis" for more details on this process. The better the nominal performance the greater the allowable ‘margin for error’ for performance degradation due to tolerances. If the predicted as-built performance does not meet specification or results in an unacceptable production yield, then is it possible to reduce the impact of tolerance defects by reducing the allowable magnitude of the defects. For example, this could mean tightening the allowable error in the radius of curvature of a lens from 0.2% to 0.1%, or tightening the surface irregularity tolerance from a maximum of 1 wave to 0.25 waves. While this approach will reduce the impact of induced aberrations on the as-built performance, it does come at a cost. Tighter tolerances can significantly increase the cost of optical components can have an unacceptable impact on the profitability of a product.

However, the amount of aberration a tolerance introduces depends upon the design form. Therefore, a better approach is often to reduce the impact of tolerance defects within the optimization stage of the design. This allows for looser tolerances, and subsequently reduced costs and improved performance. The key to reducing the impact of tolerance defects is to understand how they induce aberrations and why some designs are more sensitive than others.

## Where do aberrations come from?

Aberrations derive from the non-linearity of Snell’s Law:

where `θ` is the angle between the ray and the surface normal. The linear term is first-order optics and the higher order terms are aberrations. The higher the angle, the larger the aberrations

### Nominal means “net” performance

When optimizing a design, the focus is often on net aberrations at the image surface. For example, when optimizing for RMS spot radius at the image surface alone, the aberrations at other intermediate surfaces are ignored. This allows abberations introduced at those intermediate surfaces to be corrected by other surfaces later in the system.

One common downside of this approach, however, is that large angles of incidence at intermediate surfaces are not penalized. While this does not affect the nominal design, it will cause the induced abberations to increase for a fixed set of tolerances. That is, optimizing solely on nominal performance can produce designs that are overly sensitive to errors and more difficult to manufacture.

### High ray angle singlet

The following image shows an aspheric f/2 lens with EPD = 100 mm and RMS Spot Radius ≈ 0 at the image plane. As can be easily inferred from the steep ray-intercept angles in the diagram, this design adds huge aberration at the front lens surface, but it compensates for this at the back surface, resulting in a net near zero RMS spot radius.

### Low ray angle singlet

The following image shows another aspheric f/2 lens with EPD = 100 mm and RMS Spot Radius ≈ 0 at the image plane. Again, the net aberration is near zero, but this time the ray angles on the front lens surface are much lower.

While these designs have identical nominal performance, their as-built performance is drastically different. We can quantify this difference via a tolerance analysis by applying an identical set of default tolerances to both designs in OpticStudio. After doing this, the predicted as-built performance is computed by Root-Sum-Square (RSS) of all the individual induced aberration contributions.

The high angle design has predicted RMS Spot Radius of 185 microns.

The low angle design predicted RMS Spot Radius of 50 microns.

The high angle design is much more sensitive to tolerance defects than the low angle design, demonstrating the key point that identical nominal performance and tolerances may yield radically different as-built performance!

## Optimizing for as-built performance

The High-Yield optimization feature in OpticStudio includes the effects of induced aberrations in the merit function and allows optimization of as-built rather than nominal performance. The method applies a penalty term to the merit function discourages high angles of incidence or exitance at optical surfaces.

Where:

- n and n’ are the indices of refraction before and after surface
- N is the surface normal vector
- R is the ray vector
- If n’ > n, this is the incident ray vector, else the refracted ray vector
*R**.**N*is the cosine of the ray angle- Cosine expands as
- τ goes as the ray angle squared, weighted by surface refractivity

The penalty term acts as a surrogate for the induced aberrations in a design. Its value can be computed at every surface and can be optimized toward zero at every surface.

The penalty term value is integrated into OpticStudio as the HYLD operand and easily added to the merit function using a simple "Improve Manufacturing Yield" option in the Optimization Wizard.

If the "Improve Manufacturing Yield" option is grayed out and cannot be selected, your license is not eligible for this feature. High Yield Optimization is a capability exclusive to OpticStudio Professional or Premium subscription licenses. If you are a perpetual license user, contact your Zemax account representative to see how to become a subscriber.

This approach has numerous benefits:

- High-Yield will work with any image quality criterion (Wavefront, Contrast, Spot Size, etc.).
- The HYLD operands are computed with the same rays as are used for the image quality criterion, meaning there is no speed penalty.
- The HYLD operands utilize the same pupil integration techniques (Gaussian Quadrature, or Rectangular Array) as the other criteria and so benefit from the same computational efficiency and accuracy.
- HYLD operands thus optimize the penalty term at every surface, and over the entire field of view and pupil.

## Optimization example

High-Yield optimization enables the discovery of robust systems that would not have been found using only traditional optimization methods. The following example demonstrates the process and how significantly improved as-built performance can be achieved.

### Specification

The task was to design a lens with the following specifications:

- 9 spherical air spaced elements
- Stop after 5th element
- f/3.0, 100 mm EFL
- 28 degrees full FOV
- F, d, C wavelengths
- Thickness constraints
- 2 mm minimum air/glass edge/center; 100 mm max

- Max 1% distortion, no vignetting
- Glasses: subset of 60 Schott preferred glasses
- Design for smallest RMS Spot size. Gaussian Quadrature sampling: 4 rings, 6 arms

The starting point is a series of plane parallel plate which was optimized using the Global Search tool to find novel design forms. The attached sample files contain a starting point for this design.

First this starting point was optimized with a traditional merit function targeting nominal performance with the following result.

Then the starting point was optimized with the addition of HYLD operands to the merit function. A weight of 10 was used for the Improve Manufacturing Yield setting in the merit function to increase the impact and the following design form design form resulted.

In both cases Global Search was run for 3 hours on a 6 core machine. The final designs are included on the attached sample files.

### Results and tolerancing

After optimization a tolerance analysis was performed on both designs to assess the as-built performance. A default set of tolerances was applied using the Tolerance Wizard and 181 individual defects considered.

Using RMS Spot Radius as the tolerance criterion, the design optimized with traditional method achieved a nominal performance of 1.65 microns and an as-built performance (based on RSS method) of 81.9 microns. The results for the design optimized with High-Yield were nominal 3.8 microns and as-built 22.9 microns. As the following chart illustrates, the as-built performance of the High-Yield design is significantly better even though the nominal spot size is slightly larger.

This indicates that the High-Yield design is much less sensitive to tolerance defects. An important step to understanding if this reduced sensitivity translates into a higher yield is to perform a Monte Carlo analysis. 500 Monte Carlo systems were simulated for the Traditional and High-Yield designs, with the following results.

- Traditional system: Mean RMS spot size 38.1 microns; 90% of systems have RMS spot size below 55.8 microns
- High-Yield system: Mean RMS spot size 16.0 microns; 90% of systems have RMS spot size below 23.1 microns

The High-Yield method clearly has a higher yield of better performing systems.

While this example used Spot Size as an optimization and tolerancing criterion, because the High-Yield approach is not based on HYLD operands will work just as well with any other imaging criterion or merit function (Wavefront, Contrast, MTF, etc.).

KA-01837

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