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Abstract
This section contains the title page, table of contents, introduction, and glossary.

Library of Congress Cataloging-in-Publication Data

Ellis, Jonathan D. (Jonathan David)

Field guide to displacement measuring interferometry / Jonathan D. Ellis.

pages cm. – (The field guide series)

Includes bibliographical references and index.

ISBN 978-0-8194-9799-4 (print : alk. paper) – ISBN 978-0-8194-9800-7 (ebook : alk. paper) – ISBN (invalid) 978-0-8194-9801-4 (epub : alk. paper)

1. Interferometry. 2. Optical measurements. I. Title.

II. Title: Displacement measuring interferometry.

QC415.E45 2014

535'.470287–dc23

2013030249

Published by

SPIE

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The content of this book reflects the thought of the author(s). 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.

First printing

Introduction to the Series

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The Field Guide Series

Keep information at your fingertips with all of the titles in the Field Guide Series:
  • Adaptive Optics, Second Edition, Robert Tyson & Benjamin Frazier

  • Atmospheric Optics, Larry Andrews

  • Binoculars and Scopes, Paul Yoder, Jr. & Daniel Vukobratovich

  • Diffractive Optics, Yakov Soskind

  • Displacement Measuring Interferometry, Jonathan D. Ellis

  • Geometrical Optics, John Greivenkamp

  • Illumination, Angelo Arecchi, Tahar Messadi, & John Koshel

  • Image Processing, Khan M. Iftekharuddin & Abdul Awwal

  • Infrared Systems, Detectors, and FPAs, Second Edition, Arnold Daniels

  • Interferometric Optical Testing, Eric Goodwin & Jim Wyant

  • Laser Pulse Generation, Rüdiger Paschotta

  • Lasers, Rüdiger Paschotta

  • Lens Design, Julie Bentley & Craig Olson

  • Microscopy, Tomasz Tkaczyk

  • Nonlinear Optics, Peter Powers

  • Optical Fabrication, Ray Williamson

  • Optical Fiber Technology, Rüdiger Paschotta

  • Optical Lithography, Chris Mack

  • Optical Thin Films, Ronald Willey

  • Optomechanical Design and Analysis, Katie Schwertz & James Burge

  • Physical Optics, Daniel Smith

  • Polarization, Edward Collett

  • Probability, Random Processes, and Random Data Analysis, Larry Andrews

  • Radiometry, Barbara Grant

  • Special Functions for Engineers, Larry Andrews

  • Spectroscopy, David Ball

  • Terahertz Sources, Detectors, and Optics, Créidhe O’Sullivan & J. Anthony Murphy

  • Visual and Ophthalmic Optics, Jim Schwiegerling

Introduction

This Field Guide to Displacement Measuring Interferometry delves into a subfield of optical metrology that is prevalent in many precision systems. Precision systems that require accurate positioning knowledge use displacement measuring interferometry either through direct measurement or calibration of alternative metrology systems. Displacement measuring interferometry offers high-accuracy measurements with a wide bandwidth and direct traceability to international length standards.

The aim of this Field Guide is to provide a practical treatment of the fundamental theory of displacement interferometry along with examples of interferometry systems and uses, to outline alignment techniques for optical components, and to discuss measurement uncertainty with a practical example.

For practicing engineers, this will serve as a refresher manual for error sources and uncertainty budgets. For researchers, this will hopefully bring new insight to ways in which this technology can be useful in their field. For new engineers, researchers, and students, this Field Guide will serve as an introduction to basic alignment techniques for breadboard-based optical systems.

I would like to thank Vivek Badami for his helpful insight and for being a great mentor and friend. I am grateful for a thorough review of this manuscript by Steven Gillmer. I am indebted to many professors for training me in precision engineering and metrology, especially Stuart T. Smith, Robert J. Hocken, and the other faculty members of the Center for Precision Metrology at UNC Charlotte.

This Field Guide is dedicated to Kate Medicus for reducing my uncertainty budget in life.

Jonathan D. Ellis

Institute of Optics

University of Rochester

Table of Contents

Glossary of Terms and Acronyms xi

Fundamentals of Light and Interference 1

Basic Assumptions 1

Degrees of Freedom 2

The Meter 3

Electromagnetic Radiation 4

Electric Field 5

Polarization States 6

Complex Polarization 7

Superposition 8

Interference 9

Irradiance 10

Polarization Overlap 11

Fringe Contrast 12

Interferometer Components and Notation 13

More Interferometer Components 14

Polarization-Based Components 15

Waveplates 16

Ghosts, Absorption, and Scatter 17

Michelson’s Interferometer 18

Temporal Coherence 19

Displacement from Phase Change 20

Unwrapping and Folding 21

Basic Interferometry Systems 22

Interferometry Systems 22

Homodyne Interferometer 23

Retroreflector Homodyne Interferometer 24

Homodyne Optical Power Efficiency 25

Polarization-Sensitive Homodyne Interferometer 26

Directional Sensitivity 27

Direction-Sensitive Homodyne Interferometer 28

Homodyne Laser Encoder 29

Heterodyne Interferometry Systems 30

Basic Heterodyne Interferometer 31

Heterodyne Directional Sensitivity 32

Homodyne and Heterodyne Comparison 33

Interferometry System Characteristics 35

Unequal Plane Mirror Interferometer 35

Plane Mirror Interferometer (PMI) 36

PMI Variants 37

Beam Walkoff 38

Doppler Velocity 39

Dynamic Range and Acceleration Limitations 40

Laser Sources 41

Optical Power and Laser Modes 42

Zeeman-Stabilized Laser 43

Two-Mode Intensity-Balanced Laser 44

Heterodyne Frequency Generation 45

Phase Measurements 46

Interference Detection 47

Detection Bandwidth 48

Phase Quadrature Measurements 49

Time Interval Analysis 50

Lock-In Detection 51

Discrete Fourier Transform 52

Special Interferometer Configurations 53

Special Interferometer Configurations 53

Quad-Pass Interferometer 54

Differential Interferometer 55

Coaxial Differential Interferometer 56

Angle Interferometer 57

Straightness Interferometer 58

Refractometry 59

Wavelength Tracking 60

Refractive Index Tracker 61

Multiaxis Systems 62

Multi-DOF Interferometers 63

X-Y-Theta System 64

Tip-Tilt-Z System 65

Interferometer Alignment 66

Setup and Alignment Techniques 66

Commercial Interferometer Alignment 67

Vector Alignment and Breadboard Alignment 68

Beam Fly Height 69

Grid Alignment 70

Normal Mirror Alignment 71

45-deg Mirror Alignment 72

Mirror Steering 73

Beamsplitter Alignment 74

Polarizer Alignment 75

45-deg HWP Alignment 76

45-deg QWP Alignment 77

Polarization Flipping 78

In-line Beam Steering 79

Cosine Error 80

Cosine Mirror Alignment 81

Mixing and Periodic Error 82

Lissajous Figure 82

Source Mixing 83

Beam Leakage 84

Periodic Error 85

Assessing Periodic Error 86

Quantifying Periodic Error 87

Spatial Fourier Analysis 88

Measurement Errors and Uncertainty 89

Measurement Uncertainty 89

Probability Distributions 90

Combined Uncertainty 91

Uncertainty Sources 92

DMI Measurement Model 93

Source Vacuum Wavelength 94

Refractive Index Uncertainty 95

Cosine Error: Retroreflector Target 96

Cosine Error: Plane Mirror Target 97

Phase Change Uncertainty 98

Abbé Uncertainty 99

Measurement Axis Location 100

Interferometer Thermal Drift 101

Deadpath Uncertainty 102

Periodic Error Uncertainty 103

Surface Figure Error 104

Data Age Uncertainty 105

Error Corrections 106

Air Refractive Index Compensation 107

Error Budget 108

Measurement Uncertainty Example 109

Stage Measurement Uncertainty Example 109

Example Uncertainty Parameters 110

Example Uncertainty Propagation 111

Example Combined Uncertainty 115

Equation Summary 116

Bibliography 125

Index 129

Glossary of Terms and Acronyms

°C

degrees Celsius

%RH

percent relative humidity

A

amps

A1

first-order periodic error amplitude

A2

second-order periodic error amplitude

AD

photodetector area

ADC

analog-to-digital conversion/converter

AOM

acousto-optic modulator

BS

beamsplitter

c

speed of light

C

capacitance

CCD

charged-coupled device [camera]

CLK

clock

CO2

carbon dioxide

CSY

a coordinate system

CT

thermal drift coefficient

CTE

coefficient of thermal expansion

d

displacement interferometer output estimate

dA

Abbé offset

dA,x

Abbé offset along X axis

dA,y

Abbé offset along Y axis

dB

decibels

DC

direct current

deg

degree (angle)

DFT

discrete Fourier transform

di

displacement error contribution

DMI

displacement measuring interferometry/interferometer

DOF

degree of freedom

dSF

surface figure error

DSP

digital signal processor

dTD

thermal drift error

dψ

cosine error

E

electric field vector

E0

electric field amplitude

E1

electric field of beam 1

E2

electric field of beam 2

eA

Abbé offset error

Enet

net electric field of two-beam interference

f

optical frequency

F

farads

f0

optical frequency of iodine-stabilized laser

f1

first optical frequency

f2

second optical frequency

fclk

DFT clock frequency

fD

Doppler frequency shift

FPGA

field-programmable gate array

fs

heterodyne (split) frequency (or frequency difference)

FSR

free spectral range

G

transimpedance amplifier gain

GHz

gigahertz (109 Hz)

H

humidity

HeNe

helium-neon laser

HPF

high-pass filter

HWP

half waveplate

Hz

hertz

i

complex number (= 1 )

i

incident beam direction

Iamp

amplitude of the interference signal

ID

detected irradiance

IFC

interference fringe contrast

Ii

input irradiance

Im

measurement irradiance

Imax

maximum interference signal

Imean

average interference signal

Imin

minimum interference signal

Io

output irradiance

i-V

current-to-voltage amplifier

k

uncertainty coverage factor

K

Kelvin

KH

air refractive index sensitivity from humidity

km

kilometer (103 m)

KP

air refractive index sensitivity from pressure

KT

air refractive index sensitivity from temperature

lc

laser cavity length

Lc

long coherence length

LD

distance between interferometers

LHC

left-hand circular (polarization)

Loffset

offset length for cosine error

LPF

low-pass filter

Lrange

target displacement range

LRR

length between retroreflectors for angle optics

LSB

least significant bit

m

meters

m

number of observations

M

1D or 2D cosine uncertainty parameter

MHz

megahertz

mm

millimeters (10–3 m)

mrad

milliradians (10–3 rad)

n

refractive index

N

interferometer fold constant

N

mirror normal

nair

air refractive index

nf

final refractive index (during a measurement)

ni

initial refractive index (during a measurement)

ni

refractive index of medium i

NIST

National Institute of Standards and Technology

nm

nanometers (10–9 m)

no

refractive index of medium o

nrad

nanoradians (10–9 rad)

nRR

retroreflector refractive index

ns

nanoseconds (10–9 s)

nW

nanowatts (10–9 W)

OPD

optical path difference

OPL

optical path length

Opmi

PMI axis offset

P

optical power

P

pressure

Pa

Pascals

PBS

polarizing beamsplitter

PD

photodiode

PDm

measurement photodiode

PDr

reference photodiode

PLL

phase-locked loop

pm

picometers (10–12 m)

PMI

plane mirror interferometer

PSD

position-sensitive detector

QWP

quarter waveplate

R

resistance

r1

amplitude of beam 1

r2

amplitude of beam 2

rad

radians

RH

relative humidity

RHC

right-hand circular (polarization)

rnet

amplitude of two-beam interference

RR

retroreflector

RRh

retroreflector height

s

seconds

t

time

T

temperature

THz

terahertz (1012 Hz)

TIR

total internal reflection

U

expanded uncertainty

u(A1)

uncertainty in first-order periodic error

u(A2)

uncertainty in second-second periodic error

u(CT)

uncertainty in thermal drift

u(dA)

uncertainty in Abbé offset

u(dA,x)

uncertainty in Abbé offset along X axis

u(dA,y)

uncertainty in Abbé offset along Y axis

u(dSF)

uncertainty in surface figure

u(H)

uncertainty in relative humidity

u(n)

uncertainty in refractive index

u(nair)

uncertainty in air refractive index

u(P)

uncertainty in pressure

u(T)

uncertainty in temperature

u(xi)

uncertainty in input estimates

u(zDP)

uncertainty in deadpath distance

u(αA)

uncertainty in Abbé angle

u(αcosine)

uncertainty in cosine angle

u(αN)

uncertainty in beam normality angle

un/nf)

uncertainty in fractional refractive index change

u(Δθ)

uncertainty in phase change

u(Δλ/λi)

uncertainty in fractional wavelength change

u(λ)

uncertainty in wavelength

unom)

uncertainty in nominal wavelength

ustab)

uncertainty in wavelength stability

uDA)

uncertainty in data age

ux)

uncertainty in target angle error about X axis

uy)

uncertainty in target angle error about Y axis

u(ψ)

uncertainty in cosine error

uA(d)

displacement uncertainty from Abbé errors

uc(d)

combined displacement uncertainty

uDA(d)

displacement uncertainty from data age

uDP(d)

displacement uncertainty from deadpath errors

uEdlen

uncertainty in the Edlén equation

ui

standard uncertainty

un(d)

displacement uncertainty from refractive index

uPE(d)

displacement uncertainty from periodic error

uSF(d)

displacement uncertainty from surface figure

uTD(d)

displacement uncertainty from thermal drift

uΔθ(d)

displacement uncertainty from phase change

uλ(d)

displacement uncertainty from wavelength

uψ(d)

displacement uncertainty from cosine error

V

volts

Vi

interference signal converted to volts

W

watts

y

output estimate

Y

measurand

z

actual displacement

z

direction of target motion

ZCD

zero-crossing detector

zDP

deadpath distance

zi,m

initial measurement arm length

zi,r

initial reference arm length

zm

measured displacement

zo

optical path length

zp

physical path length

αA

Abbé angle

αcosine

angle between target and interferometer axes

αN

target normal beam angle

αp

polarizer angle

αs

Wollaston prism angle

ασ

probability distribution half-width

ασ,H

humidity probability distribution half-width

ασ,P

pressure probability distribution half-width

ασ,T

temperature probability distribution half-width

β

Bragg angle

γ1

first-order mixing amplitude

γ2

second-order mixing amplitude

Γ1

interference signal Fourier magnitude

Γ2

first-order mixing Fourier magnitude

Γ3

second-order mixing Fourier magnitude

ΔH

change in humidity

ΔIf

change in optical power between laser modes

Δlc

change in laser cavity length

Δn

change in refractive index

ΔP

change in pressure

ΔRφ

walkoff between beams

ΔT

change in temperature

Δx

straightness error in X direction

Δy

straightness error in Y direction

Δθ

change in phase

Δλ

change in wavelength

Δλstab

change in wavelength from stability

Δφ

change in target angle

Δφx

change in target angle about X axis pitch (along path)

Δφy

change in target angle about Y axis yaw (along path)

Δφz

roll (along path)

ε0

vacuum permittivity

εmedium

permittivity of propagation medium

εr

relative permittivity

η633

silicon responsivity at 633 nm

θ1

phase of beam 1

θ2

phase of beam 2

θm

measured phase

θm

phase of the reference beam

θnet

phase of two-beam interference

λ

wavelength

λ0

wavelength of iodine laser

λf

final wavelength (during a measurement)

λi

initial wavelength (during a measurement)

λnom

nominal wavelength

λstab

wavelength stability

μ0

vacuum permeability

μm

micrometers (10–6 m)

μmedium

permeability of propagation medium

μr

relative permeability

μrad

microradians (10–6 rad)

μs

microseconds (1 6 s)

μW

microwatts (10–6 W)

ν

velocity of light

ν

target velocity

νD

Doppler velocity

νo

voltage output

τDA

measurement data age

φ

target angle

φx

target angle error about X axis

φy

target angle error about Y axis

φz

target angle error about Z axis

ψ

displacement scale error

ψi

angle of incidence from medium i

ψo

angle of refraction into medium o

ω

optical frequency in angular units

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