Jose Sasian
`
`
`
`APPLE V. COREPHOTONICS
`IPR2020-00897
`Exhibit 2004
`Page 1
`
`

`

`Introduction to Lens Design
`
`Optical lenses have many important applications, from telescopes and spectacles, to
`microscopes and lasers. This concise, introductory book provides an overview of the
`subtle art of lens design. It covers the fundamental optical theory, and the practical
`methods and tools employed in lens design, in a succinct and accessible manner. Topics
`covered include first-order optics, optical aberrations, achromatic doublets, optical
`relays, lens tolerances, designing with off-the-shelf lenses, miniature lenses, and zoom
`lenses. Covering all the key concepts of lens design, and providing suggestions for
`further reading at the end of each chapter, this book is an essential resource for graduate
`students working in optics and photonics, as well as for engineers and technicians
`working in the optics and imaging industries.
`
`is Professor of Optical Design at the James C. Wyant College of Optical
`JOSE SASIAN
`Sciences at the University of Arizona in Tucson, AZ. He has taught a course on lens
`design for more than 20 years and has published extensively in the field. He has worked
`as a ccmsultant in lens design for the optics industry, and has been responsible for the
`design of a variety of successful and novel lens systems.
`
`APPLE V. COREPHOTONICS
`IPR2020-00897
`Exhibit 2004
`Page 2
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`

`

`Introduction to Lens Design
`
`JOSE SASIAN
`University of Arizona
`
`CAMBRIDGE
`UNIVERSITY PRESS
`
`APPLE V. COREPHOTONICS
`IPR2020-00897
`Exhibit 2004
`Page 3
`
`

`

`CAMBRIDGE
`UNIVERSITY PRESS
`
`University Printing House, Cambridge CB2 8BS, United Kingdom
`
`One Liberty Plaza, 20th Floor, New York, NY 10006, USA
`
`477 Williamstown Road, Port Melbourne, VIC 3207, Australia
`
`314-321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre,
`New Delhi - 110025, India
`
`79 Anson Road, #06-04/06, Singapore 079906
`
`Cambridge University Press is part of the University of Cambridge.
`
`It furthers the University's mission by disseminating knowledge in the pursuit of
`education, learning, and research at the highest international levels of excellence.
`
`www .cambridge.org
`Information on this title: www.cambridge.org/9781108494328
`DOI: 10.1017/9781108625388
`
`© Jose Sasian 2019
`
`This publication is in copyright. Subject to statutory exception
`and to the provisions of relevant collective licensing agreements,
`no reproduction of any part may take place without the written
`permission of Cambridge University Press.
`
`First published 2019
`
`Printed in the United Kingdom by TJ International Ltd, Padstow Cornwall
`
`A catalogue record for this publication is available from the British Library.
`
`Library of Congress Cataloging-in-Publication Data
`Names: Sasian, Jose M., author.
`Title: Introduction to lens design / Jose Sasian, University of Arizona.
`Description: Cambridge, United Kingdom; New York, NY, USA: University Printing
`House, 2019. I Includes bibliographical references and index.
`Identifiers: LCCN 20190194841 ISBN 9781108494328 (hardback)
`Subjects: LCSH: Lenses-Design and construction.
`Classification: LCC QC385.2.D47 S27 2019 I DDC 681/.423-dc23
`LC record available at https://lccn.loc.gov/2019019484
`
`ISBN 978-1-108-49432-8 Hardback
`
`Cambridge University Press has no responsibility for the persistence or accuracy
`of URLs for external or third-party internet websites referred to in this publication,
`and does not guarantee that any content on such websites is, or will remain,
`accurate or appropriate.
`
`APPLE V. COREPHOTONICS
`IPR2020-00897
`Exhibit 2004
`Page 4
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`

`

`10
`Lens Tolerancing
`
`A lens manufacturer requires tolerances in the dimensions of a lens to be able
`to provide a cost estimate and be able to manufacture the lens. Further, for the
`lens to meet the lens specifications after it is built, it is necessary that the actual
`lens dimensions do not depart from the nominal design ones by some amounts
`known as fabrication and assembly tolerances. Thus, the task of the lens
`designer is not only to provide a lens design that meets image quality require(cid:173)
`ments, but to also provide tolerances, so that the as-built lens actually meets
`the specifications and satisfies the needs of the application. Critical goals of the
`lens tolerancing process are to provide tolerances to each of the constructional
`parameters of the lens, and to find out the statistics of the as-built lens so that
`the fabrication yield, and final cost, can be estimated. This chapter provides a
`primer into the lens tolerancing process. Commercial lens design software
`allows for the lens tolerancing analyses discussed below.
`
`10.1 Lens Dimensions and Tolerances
`
`A lens designer needs to develop an understanding of physical dimensions and
`their measurement so that realistic tolerances can be assigned. He or she needs
`to have insight into linear and angular dimensions, such as how big a microm(cid:173)
`eter is, or one-arc second is. In lens fabrication, both of these magnitudes often
`separate what is very difficult to make from what is reasonable to make. One
`must find out how a given lens dimension will be achieved and measured in the
`optics shop. If it cannot be measured, it probably cannot be made to
`specification.
`Twenty-five micrometers (25 µm) is a useful reference. The minimum
`measurement division of many instruments and machining tools is 0.001", or
`about 25 µm. Asking for an optical element to be made with a tolerance of
`
`110
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`10.1 Lens Dimensions and Tolerances
`
`111
`
`Table 10.1 Tolerance guidelines for glass lenses, 10 mm to 100 mm
`in diameter
`
`Lens parameter
`
`Diameter (mm)
`
`Central thickness (mm)
`Edge thickness
`difference (mm)
`Surface radius (rings)
`Wavefront error from
`surface figure
`
`Low
`precision
`
`+0.0
`-0.25
`±0.12
`0.12
`
`Precise
`
`+0.0
`-0.l
`±0.05
`0.012
`
`High
`prec1s10n
`
`Requires
`special process
`
`+0.0
`-0.025
`±0.012
`0.006
`
`+0.0
`-0.005
`±0.002
`0.003
`
`5% (10)
`1% (3-5)
`0.5A RMS 0.07ARMS
`
`0.1%(1)
`0.04ARMS
`
`0.01 % (0.25)
`0.02A RMS
`
`25 µm is considered doable. Asking for that element to be made to 50 µm or
`more is considered easy. However, asking for an optical part to be made to
`12.5 µm starts to become difficult, to 2.5 µm becomes very difficult, and to
`0.25 µm extremely difficult. Similarly, by dividing 25 µmover a lens diameter
`of 25 mm, we get an angular tolerance of about 3.3 arc-minutes, which is
`doable. One order of magnitude up or down makes the angular tolerance easy
`or difficult to achieve.
`Different optics shops can make a given lens dimension, such as lens
`diameter, lens thickness, surface radius, or wedge between the lens surfaces,
`with a tight tolerance for a given cost, or cannot achieve a given tolerance. The
`lens designer needs to have effective communication with the lens manufac(cid:173)
`turers, to find out how well they can achieve lens tolerances, and their
`associated cost. Lens manufacturers provide guidelines for the different lens
`tolerances they can achieve under some assumptions. Generally, the tighter the
`tolerances, the costlier the lens becomes. What a tight tolerance is also depends
`on the manufacturing process. For example, state-of-the-art, mass produced
`injection molded lenses for mobile phones routinely achieve micrometer level
`tolerances. Table 10.1 provides some guidelines for the level of tolerances for
`lenses with spherical surfaces in the order of 10-100 mm in diameter, made out
`of glass, and which are not mass produced.
`The lens diameter refers to the actual lens diameter, in comparison to the
`clear aperture of the lens that performs the optical function of refracting or
`reflecting light rays. A common surface polishing problem is to have the very
`edge of the surface turned down. To overcome this figuring problem, there is a
`tendency to specify a lens diameter larger, say 10-20% larger, than the clear
`aperture. However, usually packaging requirements and lens cost win and the
`
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`112
`
`Lens Tolerancing
`
`diameter of the lens is minimized to only allow for enough clearance to
`properly mount the lens. It is imperative that a bevel, or protective chamfer,
`is specified to avoid the lens edge easily chipping.
`The central lens thickness is measured from surface center to surface center,
`i.e. along the optical axis. Measuring central thickness requires finding the
`central portion of the lens, and this contributes to making a precise measure(cid:173)
`ment difficult.
`Edge thickness difference, or lens wedge, is measured by supporting the lens
`in a kinematical mount so that its position is well-defined, and rotating the lens
`while a micrometer measures the position of the lens edge as the lens rotates.
`This produces the micrometer reading to oscillate between a minimum and a
`maximum value, which is the edge thickness difference, called the total indicator
`runoff. This difference, divided by the lens diameter, gives the lens wedge.
`Measuring the radius of curvature of a surface requires an optical bench.
`Alternatively, optics shops have a collection of test plates with radii of
`curvature measured with accuracy in an optical bench. Then the lens designer,
`in a final lens optimization run, fits the radii of curvature of the surfaces of the
`lens to the radii of curvature of the optics shop test plates. The optics shop tests
`for radii of curvature errors by observing the Newton rings formed by the test
`plate and a given lens surface. In this method, the surface radius of curvature is
`given a tolerance in Newton rings at a given wavelength of light. One Newton
`ring represents 1/2). of sag difference at the edge of the lens between the test
`plate and the lens surface.
`Surface figure, or irregularity, refers to the departure of a surface from the
`spherical shape, or from the nominal designed aspheric shape. There are many
`types of figure error, such as surface cylindrical deformation, which would
`introduce astigmatism aberration, an axially symmetrical error, which would
`introduce spherical aberration, such as turned down edge, periodic surface
`errors, which could diffract light and introduce image artifacts, asymmetric
`surface errors, and others. These figure errors depend on the lens manufactur(cid:173)
`ing method. For example, single point diamond turning produces periodic high
`spatial frequency figure errors.
`A change in the glass index of refraction of a lens element will change the
`first-order properties of a lens system and will introduce wavefront changes.
`A change in the glass v-number of a lens element will change the chromatic
`correction. To minimize errors, the index of refraction of the glass to be used in
`the lens manufacture is measured, and the lens is re-optimized to reflect the
`actual index of refraction. For optical systems with glass elements larger than
`about 80 mm in diameter, and that are diffraction limited, index of refraction
`homogeneity within the glass is also a concern.
`
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`10.3 Sensitivity Analysis
`
`113
`
`Figure 10.1 Parameter error distributions. From left to right, uniform, end limited,
`truncated normal, skewed, parabolic.
`
`Each of the constructional parameters of a lens can have a given error
`distribution. For example, the error in central lens thickness may be biased
`to the thicker side to allow room for regrinding a lens in case the surface
`becomes scratched. Some parameter error distributions are uniform, end(cid:173)
`limited, truncated normal, shifted-skewed, and parabolic, as shown in
`Figure 10.1.
`
`10.2 Worst Case
`
`It is perhaps tempting to try to determine the worst case performance of a lens
`that will be manufactured under a variety of fabrication errors. Determining the
`worst case estimate is not practical, because it would require us to compute the
`effects of all combinations of errors, and this can take an excessive amount of
`time, even for simple systems.
`Alternatively, if there are, say, n causes of errors, a worst case can be set by
`adding all the effects of the errors in the same direction. However, this
`approach is pessimistic.
`Therefore, the approach that is taken in practice for tolerancing is statistical
`in nature. Consequently, one goal in tolerancing a lens is to estimate the
`statistics of the as-built lens.
`
`10.3 Sensitivity Analysis
`
`For tolerancing a lens it is necessary to define a criterion of performance such
`as, for example, the error function used to optimize the lens. It is important to
`properly reflect relevant aspects of the lens in the tolerancing criterion. An
`insufficient criterion may lead to a faulty tolerancing analysis.
`A sensitivity analysis uses a list of all the constructional parameters that can
`have actual fabrication errors, such as lens thickness, lens spacing, surface
`curvature, and index of refraction. Then, tolerances are assigned and used to
`vary the constructional parameters of a lens, one at each time, and determine
`how much the tolerancing criterion has changed. This is done for each of the
`
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`114
`
`Lens Tolerancing
`
`Table 10.2 Sensitivity analysis
`
`Surface
`
`Item
`
`Nominal value
`
`Tolerance
`
`Criterion change
`
`1
`2
`3
`
`Radius
`Thickness
`Index
`
`50mm
`8mm
`1.51
`
`0.01 mm
`0.1 mm
`0.001
`
`0.3
`0.005
`0.001
`
`Table 10.3 Inverse sensitivity analysis
`
`Surface
`
`Item
`
`Nominal value
`
`Tolerance
`
`Criterion change
`
`1
`2
`3
`
`Radius
`Thickness
`Index
`
`50mm
`8mm
`1.51
`
`0.003 mm
`0.2mm
`0.01
`
`0.01
`0.01
`0.01
`
`constructional parameters, and the changes in the criterion are ranked to
`determine which parameters produce the larger changes in the criterion.
`Table 10.2 provides an example of the data produced in a sensitivity analysis.
`A sensitivity analysis produces two useful pieces of information: the lens
`parameters that worst offend the performance of the lens, and the criterion
`changes which can be used to estimate the statistics of the as-built lens.
`
`10.4 Inverse Sensitivity Analysis
`
`In an inverse sensitivity analysis, tolerances are determined that would produce
`a given change in the tolerancing criterion. Table 10.3 provide an example of
`the data produced in an inverse sensitivity analysis. Such analyses provide
`information about the levels of tolerance needed for a given performance, and
`indicate which parameters require loose or tight tolerances.
`
`10.5 Compensators
`
`In order to relax tolerances and reduce manufacturing cost, some compensators
`such as an air-space, or a lens decenter, can be used to improve a lens system
`after the lens elements have been made. For example, the back focal length is
`used to best focus the image, and an airspace can be used to restore the focal
`length or to correct for residual spherical aberration. However, for mass
`produced
`lenses,
`it is desirable
`to not specify compensators, as their
`
`APPLE V. COREPHOTONICS
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`

`10.6 Tolerancing Criterion Statistics
`
`115
`
`~~~ ~~~ ~~~ ~~~ ~~~ ~~~
`~~~ ~~~ ~~~ ~~~ ~~~ ~~~
`~~~ ~~~ ~~~ ~~~ ~~~ ~~~
`~~~ ~~~ ~~~ ~~~ ~~~ ~~~
`
`Figure 10.2 Twenty-four Cooke triplet lenses
`
`implementation requires testing and time to fix the problem. Best focusing of
`the lens by moving the lens assembly, or the image sensor, is most often
`specified as a compensator.
`
`10.6 Tolerancing Criterion Statistics
`
`Often lenses are manufactured in bulk, and the quality of each lens differs
`among the lenses because the manufacturing errors are not the same for all the
`lenses. Or, even, a single lens system where the lens is disassembled and
`reassembled, can result in a different lens because the lens element positions
`and air spaces vary. Figure 10.2 shows twenty-four Cooke triplet lenses. If the
`performance of these lenses were to be measured, one would find variation in
`the focal length and in the image quality.
`Theory shows that, when the manufacturing errors are very small, and for a
`given tolerancing criterion such as the RMS spot size, or RMS wavefront error,
`the histogram for a large number of lenses approaches a normal probability
`distribution, as shown in Figure 10.3 (left). However, in practice, as the errors
`are not very small, the histogram is skewed, as shown in Figure 10.3 (right).
`A reason for why, under very small errors, the histogram tends to be
`approached by a normal distribution is the central limit theorem. This theorem
`states that, for a set of independent and random variables having a mean and a
`variance,
`the probability density function of the sum of the variables
`approaches a normal distribution as the number of variables increases.
`A reason for why the histogram becomes skewed when the errors become
`larger is that, as the lens has been optimized, most combinations of changes
`
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`116
`
`Lens Tolerancing
`
`Figure 10.3 Left, histogram of RMS spot size for 1,000 Cooke triplets under very
`small fabrication errors. Right, histogram of the Cooke triplets under large
`fabrication errors. A best fit normal distribution has been overlaid with the
`histograms.
`
`will tend to degrade the performance, and very few, or none, will tend to
`improve it.
`For simplicity, a first estimate for the probability density function, p(S), of a
`tolerancing criterion, S, is a normal distribution defined by,
`
`(10.1)
`
`where (S) is the mean, and oj is the variance. The mean can be estimated by,
`(S) =So+ L (l:!.S;),
`
`(10.2)
`
`j
`
`i=I
`
`where S0 is the nominal value for the criterion, (l:!.S;) is the mean of the change
`of the criterion S, due to the error in the parameter i out of a number of j
`parameters. For small errors, the mean would approach the nominal perform(cid:173)
`ance, (S) = S0 • The variance can be estimated by,
`
`where a; is the variance of the change of criterion S, due to the error in the
`parameter i.
`
`(10.3)
`
`10.7 RSS Rule
`
`Out of the variance, a~ follows the Root Sum Square (RSS) rule. This
`estimates the standard deviation of the probability density function of the
`criterion. By using the square of the criterion change, !:!.Sf, due to the param(cid:173)
`eter, i, instead of the variance, a;, the RSS rule is written as,
`
`APPLE V. COREPHOTONICS
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`

`

`10.8 Monte Carlo Simulation
`
`117
`
`(10.4)
`
`The RSS rule provides the following insights. First, the statistical worst case
`estimate for n errors that produce the same criterion changes is ,jnt:,.S;; this is
`not as pessimistic as adding all the changes as nl::,.S;. Second, it is the large
`criterion changes that count much more as they enter as their squares. Thus, if
`we have ten parameters that produce changes of 1, and one parameter that
`produces a change of 10, the RSS rule indicates that the impact on the standard
`deviation of the former parameters is v'Io, while the impact of the latter
`parameter is v'Ioo.
`The RSS rule also helps to allocate tolerance budgets to different aspects of
`a lens system. For example, for a diffraction limited system, the total allowed
`wavefront error budget might be set to 0.0707A RMS. This budget is allocated
`according to the RSS rule as 0.03}.. RMS for the lens design, 0.04}.. RMS for the
`assembly, and 0.05A RMS for the fabrication (0.032 + 0.042 + 0.052 =
`0.07072
`).
`
`10.8 Monte Carlo Simulation
`
`In a Monte Carlo simulation the constructional parameters of a lens are chosen
`randomly from ranges defined by the nominal parameter values and their error
`probability distribution. The parameters in error are used to construct a lens trial,
`compensators are applied, and the system tolerancing criterion change is deter(cid:173)
`mined. Many Monte Carlo trials are done to determine the statistics of the
`tolerancing criterion change. The mean of the tolerancing criterion and its
`standard deviation are determined from the list of criterion changes. Depending
`on the application a Monte Carlo simulation may start with 100 trials to check
`for the appropriateness of the lens modeling, then 1,000 trials, or more. As the
`trials increase, it is expected that the mean and the standard deviation converge
`as the square root of the number of trials, ✓#trials. A rule of thumb is to
`execute a number of trials in the order of the square of the number of parameters
`under error. The modeling of a lens system for tolerancing can be an art and a
`science, as it can be quite elaborated to properly reflect the environment,
`materials, fabrication and assembly errors, and more. As the lens system must
`be optimized for each trial using the compensators as variables, Monte Carlo
`simulations may take long times to run. At the end, the goal is obtaining the
`statistics of the as-built lens and to assign tolerances for fabrication.
`
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`118
`
`Lens Tolerancing
`
`Table 10.4 Monte Carlo trials, nominal criterion 0.34}c
`RMS, mean 0.42H RMS, standard deviation 0.047A RMS
`
`Trial#
`
`Criterion
`
`Change
`
`1
`2
`3
`4
`5
`6
`7
`8
`9
`10
`
`0.441
`0.480
`0.369
`0.396
`0.445
`0.409
`0.390
`0.357
`0.516
`0.403
`
`0.101
`0.140
`0.029
`0.056
`0.104
`0.069
`0.050
`0.017
`0.175
`0.063
`
`Each parameter error may have its own probability density function, such as
`uniform, truncated normal, end-limited, and others. Once the lens trial is
`constructed with the parameters in error, the lens is optimized using the
`compensators. When lens decenters, or surface tilts, are lens errors, the lens
`loses its axial symmetry and, therefore, it is important to properly sample the
`field of view to determine correctly the tolerancing criterion such as RMS spot
`size, or RMS wavefront error.
`
`10.9 Monte Carlo Simulation Example
`Consider a Cooke triplet lens, as shown in Figure 10.2. The focal length isf' =
`50 mm, the field of view (FOV) is ±24 degrees, and the optical speed is FIS.
`The tolerances assigned are: thickness ±0.1 mm, radius ±2.5 fringes, index
`±0.0005,
`surface figure ±0.5 fringe, and surface tilt ± 1.5 arc-minutes.
`A truncated normal distribution for these errors is assumed. The field of view
`is sampled at the field center and four full field positions. The back focal length
`was used as a compensator. A lens decenter can be decomposed as two surface
`tilts and a thickness change. However, for small surface tilts the thickness
`change is negligible. Thus, for simplicity and clarity, here only surface tilts in
`two directions are allowed.
`Table 10.4 shows the results of ten Monte Carlo runs, which give a mean
`value of 0.421A RMS, and a standard deviation of 0.04711. RMS. The nominal
`wavefront error is 0.34t.. RMS. Depending on the performance requirements,
`on the parameters that most degrade the tolerancing criterion, the optics shop's
`ability to meet tolerances, and cost, the tolerances can be made tighter or looser
`
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`

`

`10.9 Monte Carlo Simulation Example
`
`119
`
`408
`
`309
`
`0.33kRMS
`
`0
`
`I
`
`o.m.RMS
`
`Figure 10.4 Histogram of 1,000 Monte Carlo runs for a Cooke triplet lens.
`
`to meet the requirements and cost. This is a simple example to illustrate how
`the tolerancing criterion statistics are obtained. However, for a lens to be
`manufactured the tolerancing process is often elaborated to properly model
`the as-built lens.
`Figure 10.4 shows the histogram of 1,000 Monte Carlo trials for the same
`Cooke triplet lens, the mean value is 0.4A RMS, and the standard deviation is
`0.055A RMS. Each histogram bin has trials with performance within about
`0.04A RMS. Thus, there are 408 lens trials with a tolerancing criterion between
`0.33A RMS and 0.37A RMS. Therefore, under the tolerances specified there is
`a percentage probability of about 40.8% that the lenses will perform within
`11.8% from the nominal performance. Also, there is a probability of 71.7%
`that the lenses will perform within 23.6% from the nominal performance. If
`uniform distributions are chosen for the parameters then the mean would be
`0.42A RMS and the standard deviation would be 0.077A RMS. Thus, properly
`modeling the parameters error distribution provides a more accurate level of
`tolerancing.
`Because the errors in the fabrication of a lens can substantially degrade the
`lens performance, it is important to minimize as much as possible the nominal
`tolerancing criterion during the lens optimization, so that there is more room to
`accommodate for such errors. However, different forms of optical systems that
`satisfy the requirements for an application may have more or less sensitivity to
`fabrication errors for the same level of nominal image quality.
`Table 10.5 provides the mean and standard deviation when 1,000 trials at a
`time were performed for a given category of error. The change in the mean of
`the tolerancing criterion for errors in thickness, as well as its standard devi(cid:173)
`ation, are large. Clearly, and by far, the worst offender is the category of
`thickness errors. Tightening the tolerance in thickness will be a choice. For
`example, by decreasing the thickness tolerance to ±0.05 mm, the criterion
`mean would be 0.34A RMS, and the standard deviation would be 0.016A RMS.
`This would make 81 % of the lenses perform within 10% of the nominal
`
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`120
`
`Lens Tolerancing
`
`Table 10.5 Cooke triplet lens. Mean and standard deviation for categories of
`errors, nominal mean 0.3282 RMS
`
`Parameter category
`
`Mean A RMS
`
`Standard deviation A RMS
`
`Radius
`Thickness
`Surface tilt
`Figure
`Index
`
`0.329
`0.378
`0.334
`0.328
`0.328
`
`0.0038
`0.0520
`0.0056
`0.0033
`0.0022
`
`Table 10.6 Constructional data of the Cooke triplet lens, f = 50 mm, FOV =
`±24°, F/5
`
`Surface
`
`1
`2
`3 (Stop)
`4
`5
`6
`
`Radius (mm)
`
`Thickness (mm)
`
`26.6335
`426.1623
`-25.9915
`25.0718
`169.8704
`-23.2263
`
`3.25
`6.0
`1.0
`4.75
`3.0
`42.3551
`
`Glass
`
`N-LAK33
`
`TIF6
`
`N-LAK33
`
`criterion. However, lens manufacturers put a cost premium on tight tolerances
`for thickness because of the risk of over-grinding the lens and the need to start
`over with a new blank lens. A next step would be to explore using the airspaces
`as compensators to avoid tightening the lens thickness tolerance. This might
`result in a tedious and costly lens assembly. Table 10.6 provides the construc(cid:173)
`tional data of the Cooke triplet. Thus, increasing the lens production yield is
`most often a trade-off with cost.
`
`10.10 Behavior of a Lens under Manufacturing Errors
`
`Under fabrication errors, that is under lens perturbation, a lens system suffers
`from a number of optical effects. These can be divided as relating to axial
`symmetry and not relating to axial symmetry. Errors in radii of curvature, lens
`thickness, and index of refraction maintain the axial symmetry of a lens. Errors
`in surface tilt break the axial symmetry.
`The first-order effects to take place are that the focal length changes, and
`that the image is displaced laterally. This lateral image displacement is known
`as bore-sight error, or line of sight error, and arises from the lenses becoming
`
`APPLE V. COREPHOTONICS
`IPR2020-00897
`Exhibit 2004
`Page 15
`
`

`

`10.10 Behavior of a Lens under Manufacturing Errors
`
`121
`
`Table 10.7 Changes that take place under perturbation according to symmetry
`and to aberration order
`
`Changes that relate to axial
`symmetry
`
`Changes that relate to lack of
`axial symmetry
`
`First-order
`
`Aberration
`
`First-order
`
`Aberration
`
`Focal length
`
`Image size
`
`Spherical
`aberration
`Linear coma
`aberration
`
`Image lateral displacement
`
`Uniform coma
`
`Anamorphic image
`distortion
`Chromatic change of line
`of sight
`
`Uniform
`astigmatism
`Linear
`astigmatism
`Field tilt
`
`wedged. In addition, for each wavelength, the image displacement might be
`different. The second effects that take place are changes in the aberrations, and
`that new aberrations appear. Table 10.7 provides a summary of these effects
`according to the symmetry, and whether they are of first-order, or relate to
`aberrations. In the same way that spherical aberration W040 is uniform over the
`field of view, uniform coma and uniform astigmatism can now be present over
`the field of view. Linear coma W131 grows linearly with the field of view; now
`linear astigmatism and linear focus, this is field tilt, can take place.
`The change in focal length of a thin lens is given by,
`
`(10.5)
`
`M = ~f-
`n - l
`A change in the index of refraction of 0.001 results in a change of focal length
`of approximately 0.2%. Index ofrefraction can be measured to 1 x 10-5
`, and is
`usually sufficiently well known. Thus, system changes due to errors in the
`index of refraction are expected to be very small. However, for diffraction
`limited systems it is important to check the index of refraction of the materials
`being used. The index of refraction homogeneity is also of concern, as a
`difference in index of 0.0001 in a 10 mm glass blank produces an optical path
`difference of 0.001 mm, or about two wavelengths in the visible spectrum.
`Having an understanding of the effects that take place when a lens is
`perturbed can allow a lens designer to control, or mitigate, them to avoid
`specifying tight tolerances. For example, the tilt of an image sensor can be
`used to match field tilt aberration, or some radial adjusting screws can be
`designed into a lens barrel to laterally displace a lens and correct for uniform
`
`APPLE V. COREPHOTONICS
`IPR2020-00897
`Exhibit 2004
`Page 16
`
`

`

`122
`
`Lens Tolerancing
`
`Table 10.8 Lens configuration setting for desensitizing a Cooke triplet lens for
`lens element and tilt errors
`
`Configuration
`
`2
`
`3
`
`4
`
`5
`
`6
`
`7
`
`Focal length, mm
`Lens #1 decenter, mm
`Lens #2 decenter, mm
`Lens #3 decenter, mm
`Lens #1 tilt
`Lens #2 tilt
`Lens #3 tilt
`
`100
`
`0.025
`
`0.025
`
`0.025
`
`0.05°
`
`0.05°
`
`0.05°
`
`coma. This has been done in adjusting microscope objectives. Alternatively, a
`lens airspace can be adjusted to correct for residual spherical aberration.
`Uniform astigmatism depends on the square of the surface tilt. Since the lens
`tilts under consideration are small, uniform astigmatism is negligible. Thus, if
`this aberration is detected in a nominally axially symmetric lens system, it is
`likely due to surface figure error or to a lens being deformed due to improper
`mounting. Table 1 in Appendix 4 summarizes the primary aberrations that can
`take place in a plane symmetric system.
`
`10.11 Desensitizing a Lens from Element Decenter,
`Tilt, or Wedge
`
`Lens design programs allow us to set multi-configurations for a lens system.
`Each configuration may differ, for example, in constructional parameters, in
`field of view, in relative aperture, and in wavelength choice. An opto(cid:173)
`mechanical engineer is concerned about lens decenter and tilt tolerances. To
`desensitize a lens, say a Cooke triplet lens, seven configurations are defined.
`One configuration is the nominal configuration; three configurations are for
`lens element decenter, one for each lens; and three configurations are for lens
`element tilt, one for each lens; this setting is shown in Table 10.8. The error
`function for the nominal configuration has the first-order lens constraints and
`may include image quality performance. The error function for the remaining
`six configurations has only image quality performance.
`Lens decenters and lens tilts are set only in one direction, so as to reduce the
`lens system symmetry to plane symmetry. The field of view needs to be
`properly sampled, as there is no longer axial symmetry for six configurations.
`However, because the system becomes plane symmetric and the main effects
`
`APPLE V. COREPHOTONICS
`IPR2020-00897
`Exhibit 2004
`Page 17
`
`

`

`10.11 Desensitizing a Lens
`
`123
`
`Table 10.9 Lens configuration setting for desensitizing a Cooke triplet lens for
`lens wedge
`
`Configuration
`
`Focal length, mm
`Surface #1 tilt
`Surface #2 tilt
`Surface #3 tilt
`Surface #4 tilt
`Surface #5 tilt
`Surface #6 tilt
`
`2
`
`3
`
`4
`
`5
`
`6
`
`7
`
`100
`
`0.1 °
`
`0.1 °
`
`0.1°
`
`0.1°
`
`0.1°
`
`0.1 °
`
`Figure 10.5 Desensitized Cooke triplet lens. Left, standard lens solution; Right,
`desensitized lens solution. The front positive lens takes a meniscus form, and the
`rear positive lens takes a double convex form. Glasses are N-LAK33, T1F6, and
`N-LAK33. FOV = ±24° at FIS.
`
`of surface tilt are uniform coma and linear astigmatism, sampling three or five
`fields in the plane of sym