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 wellspecified 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 an optical system destined to be manufactured is to define
a fabrication and assembly tolerance budget, and to accurately predict the resulting asbuilt performance, including the effects of compensation (e.g., refocus). Good tolerance
can decrease the fabrication cost and still maintain good system performance. If small
variations in the values of the lens parameters result in significant loss of performance,
the cost to build the design can be prohibitively high. To minimize production costs, the
ideal optical system design will maintain the required performance with achievable
component and assembly tolerances, using well-chosen post-assembly adjustments. This
complex process is often called, “tolerancing.”
ZEMAX, CODE V and OSLO are three comprehensive software packages for the design,
analysis, tolerancing, and fabrication support of optical systems. All those software
packages provide flexible and powerful tolerance development and sensitivity analysis
capability. The tolerances 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 analysis of decentration of
surfaces and lens groups, tilts of surfaces or lens groups about any arbitrary point,
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irregularity of 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 packages provide 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 and evaluation
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 individually computed. 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 system performance for each tolerance individually. The aggregate
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performance is estimated by a root-sum-square calculation. As an alternative way
of estimating aggregate effects of all tolerances, a Monte Carlo simulation is
provided. This simulation generates a series of random lenses which meets the
specified tolerances, then evaluates the criteria. By considering all applicable
tolerances simultaneously and exactly, highly accurate simulation of expected
performance 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 approach individually varies each parameter within its tolerance
range and predicts the system performance degradation on a tolerance-by-tolerance basis.
These individual results are statistically combined to yield a total system performance
prediction. This method accurately predicts performance sensitivity to individual
tolerances, which allows determination of the parameters that are “performance drivers.”
However, since the Finite Differences method does not consider how simultaneous
parameter changes by multiple tolerances will interact, its prediction of overall
performance is typically optimistic. The effects of tolerance interactions on the system
performance are known as “cross-terms.”
The Monte Carlo approach is to vary all of the parameters that have an associated
tolerance by random amounts, but within each tolerance range. The resulting system
performance is analyzed. This process is repeated many times with different random
perturbations (each analysis is often referred to as a “trial”). If many trials are run (100 to
1000 is typical), an accurate statistical prediction of the probability of achieving a
particular performance level can be generated. Since all the parameters are being varied
at the same time, the Monte Carlo method accurately accounts for cross-terms. However,
no information can be gleaned from the Monte Carlo analysis about individual tolerance
sensitivities. Therefore, while you can accurately predict system as-built performance,
you cannot determine the significant parameters that are driving the performance, and
thus cannot select the best set of tolerances to minimize cost.
Both the Finite Differences and Monte Carlo tolerancing methods are very
computationally intensive and can be very slow. For Finite Differences, the system must
be analyzed twice, once for each tolerance (the plus and minus perturbation). Thus, more
complex systems will take longer to “tolerance” than simpler systems. A triplet typically
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has over 50 tolerances, resulting in over 100 analysis simulations. For the Monte Carlo
approach, the system must be analyzed for every trial. System complexity becomes an
issue when the system has many optical elements to tolerance.
Wavefront Differential algorithm introduced by Code V is another approach to calculate
the tolerances. It provides information about both individual tolerance sensitivities (like
the Finite Differences method) and an accurate performance prediction, including the
effect of cross-terms (like the Monte Carlo method). For tolerances that cause a small
change to the overall performance, the wavefront differential method can also be more
accurate than Finite Differences, which can suffer numerical precision problems when
subtracting two large performance numbers to determine a small difference. The
Wavefront Differential approach is fast compared to either the Finite Differences or
Monte Carlo methodologies, because the nominal system is ray traced once, and all the
required information for further analysis is extracted from this ray trace of the nominal
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
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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
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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.
Relay lens
Field lens
Diverger
Fig.1 Layout of the null corrector
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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
Units
Airspace
Relay Lens:
Curvature 1 (a flat)
Thickness
Radius 2
Index
Wedge 1
Wedge 2
Decenter
Tilt
Design
value
Uncertainty
mm
143.89
0.05
0.000400
/mm
mm
mm
0
19.763
51.68
1.51509
0
0
0
0
2.00E-06
0.025
0.025
0.0005
0.05
0.05
0.05
0.05
256.2
0
4.566
206.72
1.51509
0
0
0
0
mm
mm
mm
mm
Airspace
Field Lens:
Curvature 1 (a flat)
Thickness
Radius 2
Index
Wedge 1
Wedge 2
Decenter
Tilt
/mm
mm
mm
mm
mm
mm
mm
Change in conic Change in conic
constant(ZEMAX) constant (manual
calculation)
Rms WFE
(waves)
(ZEMAX)
Rms WFE
(waves)(manual
calculation)
0.000394
0.00705
0.00711
-0.000160
0.000130
0.000050
0.001580
0.000000
0.000000
0.000000
0.000000
-0.000158
0.000133
0.000047
0.001569
0.000053
0.000024
0.000018
-0.000068
0.00738
0.00737
0.00715
0.00704
0.00731
0.00731
0.00731
0.00732
0.00743
0.00727
0.00720
0.00708
0.00998
0.00849
0.00821
0.01144
0.05
-0.000680
-0.000690
0.00699
0.00703
2.00E-06
0.025
0.025
0.0005
0.05
0.05
0.05
0.05
0.000410
-0.000220
0.000120
-0.001000
0.000000
0.000000
0.000000
0.000000
0.000436
-0.000230
0.000126
-0.001073
0.000029
0.000140
0.000014
0.000014
0.01107
0.00715
0.00807
0.01455
0.00738
0.00737
0.00731
0.00755
0.01111
0.00721
0.00813
0.01446
0.00858
0.01820
0.00814
0.00814
0.00730
0.00725
0.03528
22.325nm
0.04148
26.246nm
Residual Wavefront wv
RSS
0.002096
0.002139
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.
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Table 5 Statistics of the compensators
Conic
Nominal
Minimum
Maximum
Mean
Standard deviation
-1.0827
-1.085964
-1.078858
-1.082708
0.001444
Position of the mirror
(mm)
0
-1.1563
0.9667
0.0256
0.4702
Decenter of the mirror
(mm)
0
-0.3712
0.4288
0.0073
0.1572
Tilt of the mirror
(degree)
0
-0.0772
0.0588
0.0018
0.0294
Cumulative probability (%)
Monte Carlo (100 trials)
100
90
80
70
60
50
40
30
20
10
0
0.005
0.007
0.009
0.011
0.013
0.015
0.017
0.019
RMS wavefront (in waves)
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
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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 lens is 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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