Tutorial of Tolerancing Analysis

Using Commercial Optical Software

Ping Zhou

OPTI 521

December 3, 2006

Abstract

Tolerancing analysis is a very important step in optical system design. Since all the optical elements can not be perfectly manufactured, tolerances must be specified in such a way that the optical elements can be fabricated within the tolerances. Tolerances decrease the design merit functions, and also affect image quality. Therefore, a well-specified tolerance can maintain good system performance, and also make the optical elements easy to make. In this tutorial, the current tolerances techniques that most optical software packages use are introduced. It is also discussed that how to use the optical software to explore the tolerances analysis. A case study on tolerance for a null-corrector system is given using commercial software--Zemax.

1. Introduction

A critical step in the design of anoptical system destined to bemanufactured is to define a fabricationand assembly tolerance budget, and toaccurately predict the resulting as-builtperformance, including theeffects of compensation (e.g., refocus). Good tolerance can decrease the fabrication cost and still maintain good system performance. If smallvariations in the values of the lensparameters result in significant loss ofperformance, the cost to build the designcan be prohibitively high. Tominimize production costs, the idealoptical system design will maintainthe required performance withachievable component and assemblytolerances, using well-chosen post-assemblyadjustments. This complex processis often called, “tolerancing.”

ZEMAX, CODE V and OSLO arethree comprehensive softwarepackages for the design, analysis,tolerancing, and fabrication support ofoptical systems. All those software packages provideflexible and powerful tolerance development and sensitivity analysis capability. Thetolerances available for analysis include variations in construction parameters such as curvature, thickness,position, index of refraction, Abbe number, aspheric constants, and much more. They also support analysisof decentration of surfaces and lens groups, tilts of surfaces or lens groups about any arbitrary point, irregularityof surface shape, and variations in the values of any of the parameter. For a simple optical system, we can do the tolerances analysis manually with the help of those software packages, like what we did in OPTI 521 homework. However, for a complex optical system, which may have tens of optical elements, it is hard to tolerance the system manually. Most optical software packages provide the tools to do the tolerance automatically, which can decrease the calculation time dramatically. It is very tricky to use the tolerancing features that the software provides. However, it will be more accurate and save your time once you figure out how to correctly manipulate the software. In this tutorial, I will discuss how these software packages do the tolerance analysis.

In section 2, the three main tolerancing techniques that are available from the software packages are introduced. The methods of tolerance calculation that most software packages use are discussed and compared in section 3. Standard tolerances for optical elements, glass and assembly are summarized in section 4. The procedure of using optical software to do tolerancing and an exampled are given in section 5, 6.

2. Tolerancing techniques

Most optical software packagesprovide tolerances may be evaluated by several different criteria, including RMS spot radius, RMS wavefront error,MTF response, boresight error, user defined merit function, or a script which defines a complex alignment andevaluation procedure. Additionally, compensators may be defined to model allowable adjustments made to the lens after fabrication. Position of the image plane is one of the most commonly-used compensator. Radius of curvature, conic constant and other parameters can be defined as compensators.

Most software, like Code V, assumes the ray optical path differences due to tolerance perturbations vary linearly with tolerance change. This assumption is typically valid if the tolerance perturbation results in a small degradation of the nominal performance.

Basically, there are three ways the software provides us to explore the impact of manufacturing errors on our optical design.

  • Sensitivity analysis: For a given set of tolerances, the change in criteria is determined for each tolerance individually. It reports the effect that each parameter has on the error function and gives a number of “worst offenders”. This helps us to find out which parameter is the most sensitive to the system performance, and then we may specify a tight tolerance on that parameter.
  • Inverse Sensitivity: For a given permissible change in criteria, the limit for each tolerance is individuallycomputed. Inverse sensitivity may be computed by placing a limit on the change in the criteria from nominal,or by a limit on the criteria directly.
  • Monte Carlo Analysis: The sensitivity and inverse sensitivity analysis considers the effects on systemperformance for each tolerance individually. The aggregate performance is estimated by a root-sum-squarecalculation. As an alternative way of estimating aggregate effects of all tolerances, a Monte Carlo simulationis provided. This simulation generates a series of random lenses which meets the specified tolerances, thenevaluates the criteria. By considering all applicable tolerances simultaneously and exactly, highly accurate simulation of expectedperformance is possible. The Monte Carlo simulation performs iterations where all parameters are varied by an amount chosen at random, within the range of tolerance limits according to a specified distribution like normal,uniform, or user defined statistics.

3. Different methods to calculate tolerances

Different software packages use different methods to calculate the tolerances. The most common tolerancing methods are Finites Differences, Monte Carlo. Finite Differences can calculate the sensitivity and inverse sensitivity. Monte Carlo is the method to simulate many trials and get the statistical performance of the system. Code V has another method to calculate the tolerance, which is called Wavefront Differential.

The Finite Differences approachindividually varies each parameterwithin its tolerance range and predictsthe system performance degradationon a tolerance-by-tolerance basis. These individual results arestatistically combined to yield a totalsystem performance prediction. Thismethod accurately predictsperformance sensitivity to individualtolerances, which allowsdetermination of the parameters thatare “performance drivers.” However,since the Finite Differences methoddoes not consider how simultaneousparameter changes by multipletolerances will interact, its predictionof overall performance is typicallyoptimistic. The effects of toleranceinteractions on the systemperformance are known as “cross-terms.”

The Monte Carlo approach is to varyall of the parameters that have anassociated tolerance by randomamounts, but within each tolerancerange. The resulting systemperformance is analyzed. This processis repeated many times with differentrandom perturbations (each analysis isoften referred to as a “trial”). If manytrials are run (100 to 1000 is typical),an accurate statistical prediction of theprobability of achieving a particularperformance level can be generated. Since all the parameters are beingvaried at the same time, the MonteCarlo method accurately accounts forcross-terms. However, no informationcan be gleaned from the Monte Carloanalysis about individual tolerancesensitivities. Therefore, while youcan accurately predict system as-builtperformance, you cannot determinethe significant parameters that aredriving the performance, and thuscannot select the best set of tolerancesto minimize cost.

Both the Finite Differences and MonteCarlo tolerancing methods are verycomputationally intensive and can bevery slow. For Finite Differences, thesystem must be analyzed twice, oncefor each tolerance (the plus andminus perturbation). Thus, morecomplex systems will take longer to“tolerance” than simpler systems. Atriplet typically has over 50 tolerances,resulting in over 100 analysissimulations. For the Monte Carloapproach, the system must beanalyzed for every trial. Systemcomplexity becomes an issue when the system has many optical elements to tolerance.

Wavefront Differentialalgorithm introduced by Code V is another approach to calculate the tolerances. It provides information about bothindividual tolerance sensitivities (likethe Finite Differences method) and anaccurate performance prediction,including the effect of cross-terms(like the Monte Carlo method). Fortolerances that cause a small change tothe overall performance, the wavefrontdifferential method can also be moreaccurate than Finite Differences,which can suffer numerical precisionproblems when subtracting two largeperformance numbers to determine asmall difference.The WavefrontDifferential approach is fastcompared to either the FiniteDifferences or Monte Carlomethodologies, because the nominalsystem is ray traced once, and all therequired information for furtheranalysis is extracted from this ray trace of thenominal system.

4. Standard tolerances for lens, glass and assemble

The tolerances that a lens designer assigns not only affect the performance of the system, but also affect the cost of the system. The relationship between cost and performance is usually inversely related. Often a shop will publish different price schedules for base, precision, and high precision tolerances. The following tables from Dr. Burge’s class notes provide a general guideline for different levels of optical, mechanical tolerances.

Table 1 Optical assembly tolerance

Table 2 Lens tolerance

Table 3 Glass tolerances

Base: Typical, no cost impact for reducing tolerances beyond this.

Precision: Requires special attention, but easily achievable in most shops, may cost 25% more

High precision: requires special equipment or personnel, may cost 100% more

5. Procedure of tolerances analysis on optical software

The procedure of tolerancing usually consists of the following steps.

1) Define an appropriate set of tolerances for the lens. Usually, the default tolerance is a good place to start, or you can use the different tolerance levels given in table1-3.

2) Modify the default tolerances or add new ones to suit the system requirements.

3)Add compensators and set allowable ranges for the compensators. The default compensator is the back focal distance, which controls the position of the image plane. Other compensators, such as image surface tilt and decenter, may be defined.

4) Select appropriate criteria, such as RMS spot radius, wavefront error, MTF or boresight error. More complex criteria may be defined using a user defined merit function.

5) Select the desired mode, either sensitivity or inverse sensitivity. For inverse sensitivity, choose criteria limits or increments, and whether to use averages or computer each field individually.

6) Perform an analysis of the tolerances.

7) Review the data generated by the tolerance analysis, and consider the budgeting of tolerances. If required, modify the tolerances and repeat the analysis.

6. Case study

A two-element null corrector design with an F/2.54 is used to study the tolerances on Zemax. This null corrector is used to test a 4 m diameter mirror. The system layout is shown in Fig.1. The relay lens and field lens are the two lenses I am going to tolerance. In sum, there are 18 tolerances are analyzed. For the analysis, all the parameters are assumed to have an equal probability of having any value within the plus and minus tolerance limits.

Fig.1 Layout of the null corrector

Sensitivity analysis:

I tried to do the tolerancing manually and also use tolerance feature of Zemax to give us the information about individual tolerance sensitivities. The RMS wavefront is the criteria for both cases. Conic constant, the position of the test mirror and its tip-tilt are the compensators. The sensitivity mode is applied to find out the conic constant change and RMS wavefront error. As shown in Table 4, the changes in conic constant using two methods are very close. However, the RMS wavefront errors are slightly different. I am guessing the criteria to calculate the RMS wavefront is slightly different. In general, the calculation using Zemax tolerancing feature is consistent with those calculated manually.

Table 4 Tolerance comparison

The top-three “worst offenders” are index of refraction, the radii of the two surfaces for the field lens. When we tolerance this two-element null lenses, we should give tight tolerances on these worst offenders.

Monte Carlo analysis:

The 100-trial Monte Carlo simulation was run to analyze the overall performance of the system with the tolerance shown in Table 4. The simulation time took about 6 minutes. The statistics for the conic constant, position and orientation of the mirror is given in Table 5. The cumulative probability of the Monte Carlo analysis is given in Fig.2.

Table 5 Statistics of the compensators

Conic / Position of the mirror (mm) / Decenter of the mirror (mm) / Tilt of the mirror (degree)
Nominal / -1.0827 / 0 / 0 / 0
Minimum / -1.085964 / -1.1563 / -0.3712 / -0.0772
Maximum / -1.078858 / 0.9667 / 0.4288 / 0.0588
Mean / -1.082708 / 0.0256 / 0.0073 / 0.0018
Standard deviation / 0.001444 / 0.4702 / 0.1572 / 0.0294

Fig.2 Monte Carlo analysis

The nominal RMS wavefront is 0.0073 waves. The BEST and WORST trials give the RMS wavefront 0.0070 and 0.0245 waves, respectively. The MEAN RMS wavefront is 0.01220 waves with the standard deviation of 0.00468 waves. Among the 100 trials, 90% of the trials have the RMS wavefront less than 0.0193 waves. As we can see that the RMS wavefront from the Monte Carlo simulation is smaller than the Root-Sum-Square method.

Monte Carlo simulation is considered to be a more practical analysis, since it takes the cross-terms into consideration. To get a more accurate result, we can also increase the number of trials.

7. Conclusion

The tutorial introduced the tolerance features in the optical software packages. Three methods (Finite Different, Monte Carlo and Wavefront Differential) of calculating the tolerance are discussed and compared. The Wavefront Differential is the fastest method in calculating the tolerance, and it can provide both the sensitivity and Monte Carlo analysis. The software provides outstanding calculation speed, accuracy and flexibility in tolerancing, if we can use it appropriately. Good system tolerances can maintain optical system performance while reducing the cost during the process of manufacture and assembly.

Standard tolerances for the optics, glass and assembles are given in this tutorial, which can be a guideline when we tolerance the system.

An example of a two-element null lensis used to calculate the tolerance sensitivities and do the Monte Carlo analysis. The simulation results are consistent with the manual calculation on the tolerance sensitivities.

Reference:

  1. Zemax manual
  2. Code V tolerancing release
  3. OPTI 521 class notes

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