A. B. C.
imaging resource guide imaging lenses filters microscopy cameras illumination targets
Tolerancing Methods and Assumptions
Statistical modeling has become a more widely used tool in the optics
industry over the last decade. Monte Carlo simulation is the most
common analytical method for modeling the performance probability
of an optical system. Therefore, it is important to understand the
statistical models and probability distributions that may be used to
control speci cation tolerances.
Like the diameter of a lens, the center thickness is usually kept intentionally
thick during early stages of fabrication. This provides the manufacturer
with a larger tolerance window within which to work when
re ning other dimensions or speci cations, such as surface accuracy or
surface quality, which naturally reduce the center thickness dimension.
For this reason, the distribution for the center thickness of a batch of
lenses will likely be skewed to the higher side of the tolerance (Figure 3).
During a Monte Carlo analysis it is often falsely assumed that the distribution
for center thickness follows a symmetric normal distribution, without
skew or kurtosis. The truth is that a number of factors actually in uence
the shape of the distribution, including the number of lenses in the batch,
the fabrication process (e.g. diamond turning vs batch process), and a
particular technician’s style. A sensitivity analysis can help reveal which
factors have a greater impact on the design and can help designers prioritize
Modeling Surface Irregularity
Surface irregularity (SI) modeling can be as simple or complex as needed.
Two commonly used simpli ed SI models include tting a surface to
either a 50/50 combination of spherical aberration and astigmatism, or
100% astigmatism. Disregarding coma, trefoil, and other higher order
e ects is not recommended for lens assemblies with a large number of
elements or for optically sensitive systems, since simpli ed SI models
often do not su ciently reproduce the wavefront error or irregularity in
the system. Fitting real surface irregularity maps to Zernike coe cients
is a modeling method with much higher accuracy, and most design
software include this tolerancing functionality (Figure 4).
This modeling method does require surface measurements of the
lenses, resulting in additional time and e ort, and therefore expenses.
It is also important to understand exactly which Zernike scheme a
lens design software uses. There are many Zernike schemes, all inconsistent
with each other, and with slightly di erent expressions
and coe cients. The Standard (or Noll) Zernike and Fringe Zernike
schemes are just two Zernike schemes often used in lens design.
Stack-ups of Assembled Systems
Manufacturers assemble lenses into assemblies and must be able to
ensure that groups of lenses still perform within speci cation, even
with tilt and decenter e ects that may be present. Optical assemblies
require additional attention to individual element wedge and tilt as
well as system-level stack-ups as elements and spacers push against
each other but are subjected to limitations of the inner diameter of
the barrel. Stack-up models should attempt to accumulate tilt and decenter
e ects, while keeping elements anchored to the optical axis,
for additional accuracy. To model a system, ensure each Monte Carlo
iteration is con gured with the correct stack-up of element tilts according
to the element arrangement in the assembly (Figure 5).
Roll and decenter of an element can a ect subsequent elements in
the barrel. Connected elements will be “coupled” to a single rolling
element and will move together. Only elements with convex rear surfaces
contacting spacers have a coupled decenter. Elements with annuli
or at surfaces resting against a spacer can move independently
(not coupled) from an initially decentered element (Figure 6).
Depending on the application for which the assembly is designed, roll
and decenter may largely– or not at all– a ect performance. Tolerancing
the components, however, can provide valuable insight into the
possible assembly process and may aid in alternative methods, such
as grouping di erent elements together into sub-cell assemblies or
installing elements in a di erent order, should alternatives be needed.
Over simplifying tolerance models and designs can overlook possible
manufacturing issues. Doing so, however, increases the chances that
designs will need revisions or additional iterations with increased levels
of complexity. Moreover, assemblies may be manufacturable, yet
yield poor, non-robust products that fail to meet design speci cation.
Such instances inevitably extend the life of a project and increase its
cost. Increasing system model accuracy and using high- delity tolerancing
methods early in design will require additional e ort up front,
but will reduce expensive mistakes and save time in the end.
1. H.H. Karow (2004). Fabrication Methods for Precision Optics. J. Wiley & Sons, Inc.
2. R. Bean (April 28, 2017). How Companies Say They’re Using Big Data. Harvard
Business Review, https://hbr.org/2017/04/how-companies-say-theyre-using-big-data.
3. M.I. Kaufman et al. (September 19, 2014). Statistical distributions from lens manufacturing
data, Proc SPIE 9195, Optical System Alignment, Tolerancing, and Veri cation
VIII, 919507; https://doi.org/10.1117/12.2064582.
4. Zemax LLC (2018) Zemax OpticStudio 18.4 User Manual. Kirkland, Wash.
5. Synopsys (2018) CODE V Tolerancing Reference Manual. Mountain View, Calif.
Figure 3: A lens will often be oversized to allow for downstream corrections,
skewing the distribution of the value of dimensions like center
thickness or diameter within a given batch.
Figure 4: A. Four surface irregularity patterns based on the 5th to
11th Zernike coe cients. B. The approximate models t to Zernike
coe cients simulated in Zemax OpticStudio.
Figure 5: Three approaches to lens element tilt in a drop-together
assembly. All elements are tilted by 2° in the same direction to illustrate
the di erences. A. Tilts are modeled independently. B. Tilts and
decentration are accumulated in the order of assembly. C. Tilts are
accumulated in the order of assembly, with no additional decentration;
this motion is called shearing.
Figure 6: A. Roll motion of a lens element. B. Coupled roll motion.
C. Decenter motion of a lens element. D. Coupled decenter motion.