**Tolerancing in Zemax**

## Introduction

Being able to design a good optical system is important as an optical engineer, but equally as important is being able to tolerance it. Without this skill it is near impossible to bring any theoretical system that you have designed to reality. For simple systems consisting of perhaps one or two lenses, this is a relatively easy task because there a limited number of degrees of freedom. Designs can get complicated very quickly though, making tolerancing by hand an extremely tedious and time consuming process. Computer programs such as Zemax can be very helpful in these situations. They save time and prevent calculation mistakes that are inherent when tabulating everything by hand. In this paper, I will go over using the tolerancing feature of Zemax.

## Sources of Error

Once your optical system is completely designed, the first step in tolerancing is determining all your degrees of freedom. That is, where are all of the sources of error. They can come from misalignment of the optics (tilt,lateral decenter, or axial decenter), manufacturing errors of the actual elements (curvature, thickness, wedge, etc.), defects of the element material (eg, index and dispersion), surface finish, the list goes on and on. Zemax is fully capable to combining all these variables into one large tolerancing computation, however, it is often easier to break them up into groups (alignment, lenses, operational changes, etc) and allocate each group a specific error budget. You can then tolerance each group separately, making it easier to keep track of what is going on, and make sure the program is doing what you want it to. As you will see, this is what was done in the example I will show latter.

## The Tolerance Data Editor

Tolerance information in Zemax is inserted into the Tolerance Data Editor (TDE), which is very similar to the Merit Function Editor (MFE). Each line of the TDE spreadsheet is one operand that represents one degree of freedom. As with the MFE, there is often extra data to insert to specify each of these operands, such as surface number. Then, you must enter the minimum and maximum departure from nominal for this operand. For example, suppose you were specifying the tolerance on the separation of two lenses. Nominally, this distance is say 5 mm, but you are going to allow it to shift +/- 0.2 mm. In the TDE, you would have a thickness operand for the surface spanning between the two lenses, with the minimum value set at -0.2 and the maximum value set at +0.2. A list of the different operands is provided in the Zemax manual, or the Help function in Zemax. A good starting point for a list of degrees of freedom are the default tolerances in Zemax. Especially for a beginner, it is a good way to, perhaps not catch all of the possible variables, but it will certainly list most of them. It is similar to the default merit function, in that is opens a window that has different fields for you to fill in. In the case of the default tolerances, it lists the alignment variables for each element, as well as for each surface. The latte is used to specify the lens itself, in that one surface can be off center or tilted wrong with respect to the other. It is the users job to simply enter minimum and maximum values for each of these, and what range of surfaces to apply them to. Then, once in the TDE, you can refine them for each specific surface.

## Compensators

Often when you build a system, there is a way to compensate for errors. Probably the most common of these is focus, meaning the the position of the image plane can be adjusted to make the best image when the system is put together. In the TDE, you can add your own compensators with the operand COMP. For example, if designing a telescope, the spacing between the primary and secondary mirror can be adjusted to make the best image. In that case, you would add a compensator to that surface. Zemax then uses the compensators in the following way. When your system is perturbed from it’s nominal state by the amount you indicate in the TDE, the image quality drops. The entire purpose of tolerancing is to determine how much things are allowed to be wrong without the image degrading too much. If you were to measure the image quality strictly from the perturbations, your analysis will not be correct. You have to take into account that fact that you can “fix” or compensate for at least a little bit of the error. Thus, after perturbing the system, Zemax then looks at all the compensators you have defined (keep in mind you do not have to define any, your system might not have any.) and optimizes them to make the best image.

## Tolerancing

Once your TDE is filled, the next step is to open the tolerancing window (Ctrl + T, or Tools – Tolerancing – Tolerancing…) In the tolerance window, there are 4 different tabs, I will discuss each one here

**Set-Up**

Zemax supports a few different modes of tolerancing. I will discuss the most common, Sensitivity. In this mode, Zemax perturbs each of the specified degrees of freedom in the minimum and maximum direction and measures the change in the criterion you set (discussed below). This difference can be calculated as either a Root-SumSquare (RSS) change or a Linear Change.

**Criterion **

This is where you specify what Zemax looks at when it does the tolerancing. Meaning, what the criterion is. The criterion can be many things, including the spot radius, wavefront error, MTF, or even your own merit function. Sampling is how many rays are traced in the tolerancing. Comp is the compensator setting, there are 3 options: Optimize All, which is just like optimizing in Zemax. Zemax uses the compensators you have set as variables and minimizes the criterion you selected. There are two different optimization choices: DLS which is Damped Least Squares, and OD, Orthogonal Decent. The third compensator setting is paraxial focus. If this is selected, then only the focus position is adjusted as a compensator.

**Monte Carlo **

The Monte Carlo Simulations are what take the most time in the Zemax tolerancing. After each sensitivity is computed and the total criterion change found, Zemax will run a user defined number of simulations. In each of the simulations, each of the degrees of freedom is perturbed a random amount between the minimum and maximum set values. The the compensators are adjusted and the criterion evaluated. You can choose what kind of statistics the Monte Carlo analysis uses. The options are Normal (Gaussian), Parabolic, or uniform distributions. If selected, each of the trials are saved in it’s own file. That way, you can go back and look at them to make sure Zemax is doing what you are expecting, and investigate in more detail any unusual or curious results.

**Display**

This tab lets you choose how many of the tolerance operands are displayed, whether descriptions of each operand are printed, and whether or not the compensator data is shown for each trial of the Monte Carlo Analysis.

## Tolerance Results

Once all four of these tabs are filled with the appropriate information, you can press the “OK” button and let Zemax loose. When it completes the analysis, it will pop up a Text Viewer Screen with the results. The sensitivity analysis is shown first, with each tolerance operand listed with the change in criterion for its maximum and minimum values. These are then ordered in a list called “Worst Offenders,” which lists the operands from most to least effect on the criterion. A statistical analysis is then performed on the data, estimating the change in criterion using a Root Sum Square calculation. 2 After the Sensitivity analysis comes the Monte Carlo Analysis for for ever many trials were selected. For each trial, the criterion and change in criterion is listed. And then a distribution of the data is computed depending on the statistical method selected previously. Finally, the compensator statistics are are shown. This in information regarding the nominal value of the compensator variable, and its maximum, minimum, and average change.**Example**** **

Recall that this assignment was only concerned with the alignment tolerances of the lens. First I set up the optical system, below shows the LDE and layout.

Next, each degree of freedom was entered into the TDE as a different operand. Below shows what that looks like:

My original hand calculated tolerances are attached in the appendix, and if you have a look at that, you’ll notice a decentration tolerance of 0.2mm on each lens. Above however, you don’t see that. What I realized, is that decentration has no x- or y- orientation, it is just decentering. In Zemax however, you have to specify whether you are decentering in x or y and by how much. Because I wanted to be able to make a comparison between my hand calculations and the Zemax output, I had to figure out how to represent the same thing I had done by hand. What I realized is that the total decenter is just the x and y decenters summed in quadrature. Thus I set the x and y decenters equal and computed what their decenters would have to be to sum to the decenter I had originally specified. As one would expect, the result is just the total decenter divided by root two. One mistake I almost made was to assume that because my system is axially symmetric, I could just use the x-or just the y- decenters in the TDE. But, because a tilt and decneter can compensate each other, not including tilt and decenter in both x- and y- does not represent the system properly, and the correct results were not obtained. each of these.

Then the tolerancing was run. The out put of the sensitivity analysis and Monte Carlo Analysis for 20 trials is shown below. As you can just barely make out in the text below, the estimated RMS Wavefront after errors is 0.0394 waves, very close to the 0.0398 waves I calculated by hand. This is nice reassurance that I successfully modeled in Zemax what I had previously done by hand.

The results of the Monte Carlo Analysis are below.

## Conclusions

Probably the most important thing to remember when tolerancing is to set it up like you would assemble the actual system. And this goes for the entire set-up and running of the tolerances. As you define your operands in the TDE, think about how you are assembling the system, and the possible ways you could be wrong in alignment. Imagine making the optic (unless you are making it yourself), and think about all the ways you could make it wrong. Then, after assembly, how do you look at the image. Is there a focuser on the eyepiece, and adjustment of an element position? Or anything else to produce the best image possible? This is the best way to model your system as accurately as possible.

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