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Table of Contents
                            Table of Contents
Introduction
Chapter 1: Introduction to Mechanical Analysis Using Finite Elements
Chapter 2: Introduction to Optics for Mechanical Engineers
Chapter 3: 
Zernike and Other Useful Polynomials
Chapter 4: 
Optical Surface Errors
Chapter 5: 
Optomechanical Displacement Analysis Methods
Chapter 6:
 Modeling of Optical Mounts
Chapter 7: 
Structural Dynamics and Optics
Chapter 8: 
Mechanical Stress and Optics
Chapter 9: 
Optothermal Analysis Methods
Chapter 10: Analysis of Adaptive Optics
Chapter 11: Optimization of Optomechanical Systems
Chapter 12: Superelements in Optics
Chapter 13:

Integrated Optomechanical Analysis of a Telescope
Chapter 14: 
Integrated Optomechanical Analyses of a Lens Assembly
Index
About the Authors
                        
Document Text Contents
Page 1

P.O. Box 10
Bellingham, WA 98227-0010

ISBN: 9780819492487
SPIE Vol. No.: PM223

SPIE PRESS

The development of integrated optomechanical analysis tools has increased
significantly over the past decade to address the ever-increasing challenges in
optical system design, leveraging advances in computational capability.
Integrated Optomechanical Analysis, Second Edition presents not only finite
element modeling techniques specific to optical systems, but also methods to
integrate the thermal and structural response quantities into the optical model for
detailed performance predictions. This edition updates and expands the content
in the original SPIE Tutorial Text to include new illustrations and examples, as
well as chapters about structural dynamics, mechanical stress, superelements,
and the integrated optomechanical analysis of a telescope and a lens assembly.

Page 202

MODELING OF OPTICAL MODELS 181


such as that shown in Fig. 6.36(b):


2 2
2 1 ,A R R [6.19(a)]

3 3
2 1

2 2
2 1

sin2
,

3

R R
C

R R
[6.19(b)]

4 4
2 1

1 1
sin 2 ,

4 2XX
I R R [6.19(c)]

23 32
2 14 4

2 1 2 2
2 1

sin1 1 4
sin 2 ,

4 2 9YY
R R

I R R
R R

[6.19(d)]

3
1 2 1 2 ,6

J R R R R [6.19(e)]6


where A is the cross-sectional area, , R1, and R2 are as defined in Fig. 6.36(b), C
is the distance from the center of curvature of the flexure to the centroid of the
flexure cross-section, IXX and IYY are the moments of inertia at the centroid about
the x and y axes, respectively, and J is the torsional constant. It should be noted
that the expression for the torsional constant J is approximate based on an
assumption that the thickness of the flexure is much smaller than the nominal
radius of curvature. In addition, an expression for the location of the shear center
relative to the centroid is not given.

6.3 Modeling of Test Supports
The purpose of performing a test-support deformation analysis is often to assess
the surface-error contribution due to fabricating an optic to a desired prescription
while in a test support that does not adequately represent the optic’s in-use
support. This error contribution, however, is as much a function of the optic in its
in-use configuration as it is a function of the optic in its test support. Fig. 6.37
illustrates an optic that is tested on an air bag during the figuring process and
subsequently supported in operation using an inclined configuration on its
mounts. The error contribution of interest is the difference between the deformed
optical surfaces of these two states. Since a linear finite element analysis assumes
that the model begins in a stress-free and strain-free state, the deformation
analysis of the optic in each state is the deformation change relative to a perfectly
figured optic floating in a zero-gravity environment. To obtain the change in
surface figure between two deformed states, a node-by-node difference in the
finite element displacement results must be generated before deformed surface
characterization is performed. Most finite element codes allow users to
accomplish such a difference operation within the finite element analysis.
However, a simple program or spreadsheet application can be used to difference
the results of two analysis cases.

Page 203

182 CHAPTER 6


Air Bag

In-Use
Configuration

Test Configuration
Figure 6.37 The optic is figured to the environment in which it is tested, and it can
display different figures in its operational environments.


A test-support deformation analysis may be important if surface figuring

methods such as ion figuring or small tool polishing are employed. The inverse
of the shift in surface figure from the test-support configuration to the in-use
configuration can be fabricated into the surface of the optic, thereby lessening the
effects caused by testing the optic in an environment different from the in-use
environment. This process is accomplished by generating an analytically
computed prediction representing the deformation change caused by the test
support relative to the operational configuration and adding this array to the
interferogram results of each test measurement performed during fabrication of
the optic. As each figuring pass is performed, the optical figure will converge to
the desired prescription minus the anticipated deformation change. The surface
figure error contribution associated with going from the test state to the in-use
state would then be the error with which the analytical prediction was made.

In addition, various optical testing procedures require limits on the deviation
of the optical surface from its intended shape. Such requirements may impose
restrictions on how the test support should be designed to adequately support the
optic or optical system so that accurate test results can be obtained. Therefore,
analysis prediction of how optical systems deform in their test supports can be
very important.


6.3.1 Modeling of air bags

Air bags are commonly used to simulate a 0-g environment during an optical test.
Methods of modeling air bags stem from the fact that the pressure inside the air
bag is either assumed constant or is a function of the hydraulic head h, as
illustrated in Fig. 6.38. Therefore, for an axisymmetric optic supported by an air
bag, the air bag can be represented by a uniform pressure applied normal to the
supported face of the optic. This method, however, assumes that test engineers
have inflated the air bag such that tangency is achieved at all points around the

Page 404

Keith Doyle has over 25 years of experience in the field of
optomechanical engineering, working on a diverse range of
high-performance optical instruments specializing in the
multidisciplinary analysis and integrated modeling of
optical systems. He is currently a Group Leader in the
Engineering Division at MIT Lincoln Laboratory. He
previously served in a variety of roles including Vice
President of Sigmadyne, Inc., Senior Systems Engineer at
Optical Research Associates, and a Structures Engineer at
Itek Optical Systems. He received his Ph.D. from the

University of Arizona in Engineering Mechanics with a minor in the Optical
Sciences in 1993, and he holds a BS degree from Swarthmore College received
in 1988. Dr. Doyle is an active participant in SPIE symposia, teaches short
courses on optomechanics and integrated modeling, and has authored and co-
authored over 30 technical papers in optomechanical engineering.


Dr. Victor Genberg PE has over 45 years of experience in
the application of finite element methods to high-
performance optical structures, and is a recognized expert
in optomechanics. He is currently President of Sigmadyne,
Inc. Prior to starting Sigmadyne, Dr. Genberg worked at
Eastman Kodak for 28 years, serving as a technical
specialist for commercial and military optical instruments.
He is an author of SigFit, a commercially available

software product for optomechanical analysis. Dr. Genberg is also a full
professor (adjunct) of Mechanical Engineering at the University of Rochester,
where he teaches a variety of courses in finite elements, design, optimization, and
optomechanics. He has over 50 publications. He received his Ph.D. from Case
Western Reserve University in 1973.


Gregory Michels PE has worked for twenty years in
optomechanical design and analysis, and is currently Vice
President of Sigmadyne, Inc. He received his MS degree in
Mechanical Engineering from the University of Rochester
in 1994. He specializes in finite element analysis and
design optimization of high-performance optical systems.
Mr. Michels is also a software developer and technical
support engineer for Sigmadyne’s optomechanical analysis

software product, SigFit. Prior to co-founding Sigmadyne, he worked at Eastman
Kodak for five years as a structural analyst on the Chandra X-Ray Observatory.
Mr. Michels has authored or co-authored over 25 papers in the field of integrated
optomechanical analysis and teaches short courses on finite element analysis and
integrated modeling.

Page 405

P.O. Box 10
Bellingham, WA 98227-0010

ISBN: 9780819492487
SPIE Vol. No.: PM223

SPIE PRESS

The development of integrated optomechanical analysis tools has increased
significantly over the past decade to address the ever-increasing challenges in
optical system design, leveraging advances in computational capability.
Integrated Optomechanical Analysis, Second Edition presents not only finite
element modeling techniques specific to optical systems, but also methods to
integrate the thermal and structural response quantities into the optical model for
detailed performance predictions. This edition updates and expands the content
in the original SPIE Tutorial Text to include new illustrations and examples, as
well as chapters about structural dynamics, mechanical stress, superelements,
and the integrated optomechanical analysis of a telescope and a lens assembly.

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