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Abstract
This front matter contains an introduction, table of contents, and glossary of symbols.

Library of Congress Cataloging-in-Publication Data

Goodwin, Eric P.

Field guide to interferometric optical testing / Eric P. Goodwin & James C. Wyant.

p. cm. -- (The field guide series; 10)

Includes bibliographical references and index.

ISBN 0-8194-6510-0

1. Optical instruments--Testing. 2. Interferometry. I. Wyant, James C. II. Title.

TS514.G66 2004

535′.470287--dc22

2006024169

Published by

SPIE—The International Society for Optical Engineering

P.O. Box 10

Bellingham, Washington 98227-0010 USA

Phone: +1 360 676 3290

Fax: +1 360 647 1445

Email: spie@spie.org

Web: http://spie.org

The content of this book reflects the work and thought of the author. Every effort has been made to publish reliable and accurate information herein, but the publisher is not responsible for the validity of the information or for any outcomes resulting from reliance thereon.

Printed in the United States of America.

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Introduction to the Series

Welcome to the SPIE Field Guides—a series of publications written directly for the practicing engineer or scientist. Many textbooks and professional reference books cover optical principles and techniques in depth. The aim of the SPIE Field Guides is to distill this information, providing readers with a handy desk or briefcase reference that provides basic, essential information about optical principles, techniques, or phenomena, including definitions and descriptions, key equations, illustrations, application examples, design considerations, and additional resources. A significant effort will be made to provide a consistent notation and style between volumes in the series.

Each SPIE Field Guide addresses a major field of optical science and technology. The concept of these Field Guides is a format-intensive presentation based on figures and equations supplemented by concise explanations. In most cases, this modular approach places a single topic on a page, and provides full coverage of that topic on that page. Highlights, insights, and rules of thumb are displayed in sidebars to the main text. The appendices at the end of each Field Guide provide additional information such as related material outside the main scope of the volume, key mathematical relationships, and alternative methods. While complete in their coverage, the concise presentation may not be appropriate for those new to the field.

The SPIE Field Guides are intended to be living documents. The modular page-based presentation format allows them to be easily updated and expanded. We are interested in your suggestions for new Field Guide topics as well as what material should be added to an individual volume to make these Field Guides more useful to you. Please contact us at fieldguides@SPIE.org.

John E. Greivenkamp, Series Editor

Optical Sciences Center

The University of Arizona

The Field Guide Series

Keep information at your fingertips with all of the titles in the Field Guide Series:

Field Guide to Geometrical Optics, John E. Greivenkamp (FG01)

Field Guide to Atmospheric Optics, Larry C. Andrews (FG02)

Field Guide to Adaptive Optics, Robert K. Tyson and Benjamin W. Frazier (FG03)

Field Guide to Visual and Ophthalmic Optics, Jim Schwiegerling (FG04)

Field Guide to Polarization, Edward Collett (FG05)

Field Guide to Optical Lithography, Chris A. Mack (FG06)

Field Guide to Optical Thin Films, Ronald R. Willey (FG07)

Field Guide to Spectroscopy, David W. Ball (FG08)

Field Guide to Infrared Systems, Arnold Daniels (FG09)

Field Guide to Interferometric Optical Testing, Eric P. Goodwin and James C. Wyant (FG10)

Field Guide to Interferometric Optical Testing

The material covered in the Field Guide to Interferometric Optical Testing is derived from a course taught by Dr. Wyant at the College of Optical Sciences at the University of Arizona. The material has evolved over the years as the underlying technologies and techniques have changed. This text is meant as a reference of interferometric principles and methods for the practicing engineer.

Eric Goodwin dedicates this Field Guide to his wife, Sam, and their daughter, Ryan.

James Wyant dedicates this Field Guide to the memory of Louise.

Eric P. Goodwin and James C. Wyant

College of Optical Sciences

University of Arizona

Table of Contents

Glossary x

Fundamentals of Interferometry 1

Two-Beam Interference Equation 1

Basic Concepts and Definitions 2

Conditions for Obtaining Fringes 3

Visibility 4

Spatial Coherence 5

Polarization 6

Beamsplitters 7

Plate and Pellicle Beamsplitters 8

Interferometers 9

The Interferometer 9

Classic Fizeau Interferograms 10

Newton’s Rings 11

Twyman-Green Interferometer 12

Compensating Plate 13

PBS-based Twyman-Green Interferometer 14

Laser-based Fizeau 15

Mach-Zehnder Interferometer 17

Beam Testing 18

Lateral Shear Interferometry 19

Rotating Grating LSI 20

Radial Shear Interferometer 21

Interferograms 22

Interferograms 22

Wavefront Aberration Coefficients 23

Zernike Polynomials 24

RMS Wavefront Error 26

Spherical Aberration Interferograms 27

Astigmatism Interferograms 28

Interferograms—Other Aberrations 29

Moiré 30

Moiré and Interferograms 31

Phase-Shifting Interferometry 32

Direct Phase Measurement 32

Methods for Phase Shifting 33

Continuous Phase Shifting 34

Liquid Crystal Retarder 35

Phase Shifting Algorithms 36

Basic Phase Unwrapping 37

Phase-Stepping vs. Phase-Ramping 38

Errors in PSI 39

Quantization Errors 40

Incorrect Phase Shift 41

Avoiding Vibrations 42

Spatial Synchronous and Fourier Methods 43

Spatial Carrier Interferometry 44

Ground Glass 45

Surface Microstructure 46

Surface Microstructure 46

Nomarski Interference Microscope 47

Fringes of Equal Chromatic Order (FECO) 48

Phase-Shifting Interference Microscope 49

Multiple-Wavelength Interferometer 50

Vertical Scanning Techniques 51

Flat Surface Testing 52

Flat Surface Testing 52

Mirrors—Continued 53

Windows—Continued 54

Prisms 55

Corner Cubes 56

Curved Surface Testing 57

Testing Curved Surfaces—Test Plate 57

Curved Surfaces—Twyman-Green 58

Curved Surfaces—Laser-based Fizeau 59

Testing Lenses or Lens Systems 60

Shack Cube Interferometer 61

Scatterplate Interferometer 62

Phase-Shifting Scatterplate Interferometer 63

Long-Wavelength Interferometry 64

Smartt Point Diffraction Interferometer 65

Phase Shifting a PDI 66

Sommargren Diffraction Interferometer 67

Curved Surfaces, VSWLI 68

Absolute Measurements 69

Absolute Measurements: Flats 69

Absolute Measurements: Spheres 70

Asphere Testing 71

Aspheric Surfaces 71

Aspheric Testing 72

Hyperboloid Null Tests 73

Offner Null 74

Holographic Null Optics 75

CGH Basics 76

CGH Design Guidelines 77

Non-Null Tests 78

Reverse Raytracing 79

Sub-Nyquist Interferometry 80

Long-Wavelength Interferometry 81

Appendices 82

Non-Interferometric Testing 82

Foucault (Knife-Edge) Test 83

Ronchi Test 85

Equation Summary 86

Bibliography 93

Index 97

Glossary

Frequently used variables and symbols:

a

Average phase shift between frames

A

Amplitude

An

Aspheric surface coefficients

b

Number of bits for quantization error

B

Obscuration ratio

c

Speed of light

C

Moiré fringe spacing

C

Curvature

d

Distance, displacement

D

Diameter

DHS

Diameter of Hindle Sphere

f

Focal length

f

Spatial frequency

f/#

F-number

F

Focal point

F

Coefficient of finesse

g[θ′]

Zernike angular component

G

G-factor

h

Height

H

Normalized field height

i

Step number, frame number

I

Irradiance

Lc

Coherence length

m

Diffraction order or fringe order

m

Fresnel zone plate zone number

m

Transverse or lateral magnification

n

Index of refraction

ne

Extraordinary index, uniaxial crystal

no

Ordinary index, uniaxial crystal

N

Number of algorithm steps

N

Integer number of 2p

NA

Numerical aperture

OPD

Optical path difference

OPL

Optical path length

p

p-polarization state

r

Non-normalized radial coordinate

rm

Radius of mth bright fringe

rp

Pupil radius

R

Radius of curvature

R/T

Reflection/transmission ratio

Rs

Radial shear coefficient

s

s-polarization state

s(r)

Sag as function of part radius

S

Fringe spacing

SNR

Signal to noise ratio

t

Thickness

tc

Coherence time

T

Lateral translation

υ

Speed of light in medium or velocity

V

Visibility

Vsc

Visibility factor due to spatial coherence

Wijk

Wavefront aberration coefficients

W(x, y)

Wavefront as function of spatial position

x

Spatial coordinate

xp

Pupil coordinate

xs

Pixel spacing

xw

Pixel width

y

Spatial coordinate

yp

Pupil coordinate

z

Object distance, axial position

z′

Image distance (lens)

Z

Zernike polynomial coefficients

α

Angle between two polarization states

α

Moiré angle, wedge angle

β

Tilt

δ β

Tilt difference

Γ

Fringe contrast

δ(x, y)

Grating errors, function of position

Δ

Fringe displacement

Δ

Integrated phase change

ε

Linear phase shift error

ε

Angle error for 90-degree prism

εz

Axial distance from paraxial focus

η

Diffraction efficiency

θ

Angle, shear angle, tilt orientation

θ

Angle, Zernike polynomial set

θd

Diffraction angle

θi

Incident angle

κ

Conic constant

λ

Wavelength

λc

Center wavelength

λeq

Equivalent wavelength

Λ

Diffraction grating or moiré grating period

υ

Frequency

Δυ

Frequency difference

ξc

Cutoff frequency

ξc,sa

Cutoff frequency for a sparse array detector

ξNy

Nyquist frequency

ρ

Reflectance (ratio of reflected irradiance)

ρ

Normalized pupil radius (0 < ρ < 1)

σ

RMS wavefront error

σ2

Wavefront variance

σϕ,i

Standard deviation, irradiance fluctuations

σϕ,q

Standard deviation, quantization phase error

ϕ

Phase

ϕ (t)

Phase shift as a function of time

Ω

Rotation rate

Ω

Solid angle

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