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

P.O. Box 10

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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

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

*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.*

**SPIE Field Guides**Each * SPIE Field Guide* addresses a major field of optical science and technology. The concept of these

*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 Guides***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.*

**Field Guide**The * SPIE Field Guides* are intended to be living documents. The modular page-based presentation format allows them to be updated and expanded. We are interested in your suggestions for new

*topics as well as what material should be added to an individual volume to make these*

**Field Guide***more useful to you. Please contact us at fieldguides@SPIE.org.*

**Field Guides**John E. Greivenkamp, **Series Editor**

College of Optical Sciences

The University of Arizona

## The Field Guide Series

*Series:*

**Field Guide**, Second Edition, Robert Tyson & Benjamin Frazier**Adaptive Optics**, Larry Andrews**Atmospheric Optics**, Paul Yoder, Jr. & Daniel Vukobratovich**Binoculars and Scopes**, Yakov Soskind**Diffractive Optics**, Jonathan D. Ellis**Displacement Measuring Interferometry**, John Greivenkamp**Geometrical Optics**, Angelo Arecchi, Tahar Messadi, & John Koshel**Illumination**, Khan M. Iftekharuddin & Abdul Awwal**Image Processing**, Second Edition, Arnold Daniels**Infrared Systems, Detectors, and FPAs**, Eric Goodwin & Jim Wyant**Interferometric Optical Testing**, Rüdiger Paschotta**Laser Pulse Generation**, Rüdiger Paschotta**Lasers**, Julie Bentley & Craig Olson**Lens Design**, Tomasz Tkaczyk**Microscopy**, Peter Powers**Nonlinear Optics**, Ray Williamson**Optical Fabrication**, Rüdiger Paschotta**Optical Fiber Technology**, Chris Mack**Optical Lithography**, Ronald Willey**Optical Thin Films**, Katie Schwertz & James Burge**Optomechanical Design and Analysis**, Daniel Smith**Physical Optics**, Edward Collett**Polarization**, Larry Andrews**Probability, Random Processes, and Random Data Analysis**, Barbara Grant**Radiometry**, Larry Andrews**Special Functions for Engineers**, David Ball**Spectroscopy**, Créidhe O’Sullivan & J. Anthony Murphy**Terahertz Sources, Detectors, and Optics**, Jim Schwiegerling**Visual and Ophthalmic Optics**

## 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

**A**_{1}

first-order periodic error amplitude

**A**_{2}

second-order periodic error amplitude

**A**_{D}

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

CO_{2}

carbon dioxide

CSY

a coordinate system

**C**_{T}

thermal drift coefficient

CTE

coefficient of thermal expansion

**d**

displacement interferometer output estimate

**d**_{A}

Abbé offset

**d**_{A,x}

Abbé offset along * X* axis

**d**_{A,y}

Abbé offset along * Y* axis

dB

decibels

DC

direct current

deg

degree (angle)

DFT

discrete Fourier transform

**d**_{i}

displacement error contribution

DMI

displacement measuring interferometry/interferometer

DOF

degree of freedom

**d**_{SF}

surface figure error

DSP

digital signal processor

**d**_{TD}

thermal drift error

**d**_{ψ}

cosine error

**E**

electric field vector

**E**_{0}

electric field amplitude

**E**_{1}

electric field of beam 1

**E**_{2}

electric field of beam 2

**e**_{A}

Abbé offset error

**E**_{net}

net electric field of two-beam interference

**f**

optical frequency

F

farads

**f**_{0}

optical frequency of iodine-stabilized laser

**f**_{1}

first optical frequency

**f**_{2}

second optical frequency

**f**_{clk}

DFT clock frequency

**f**_{D}

Doppler frequency shift

FPGA

field-programmable gate array

**f**_{s}

heterodyne (split) frequency (or frequency difference)

FSR

free spectral range

**G**

transimpedance amplifier gain

GHz

gigahertz (10^{9} Hz)

**H**

humidity

HeNe

helium-neon laser

HPF

high-pass filter

HWP

half waveplate

Hz

hertz

**i**

complex number (= $\sqrt{-1}$ )

**i**

incident beam direction

**I**_{amp}

amplitude of the interference signal

**I**_{D}

detected irradiance

**I**_{FC}

interference fringe contrast

**I**_{i}

input irradiance

**I**_{m}

measurement irradiance

**I**_{max}

maximum interference signal

**I**_{mean}

average interference signal

**I**_{min}

minimum interference signal

**I**_{o}

output irradiance

i-V

current-to-voltage amplifier

**k**

uncertainty coverage factor

K

Kelvin

**K**_{H}

air refractive index sensitivity from humidity

km

kilometer (10^{3} m)

**K**_{P}

air refractive index sensitivity from pressure

**K**_{T}

air refractive index sensitivity from temperature

**l**_{c}

laser cavity length

**L**_{c}

long coherence length

**L**_{D}

distance between interferometers

LHC

left-hand circular (polarization)

**L**_{offset}

offset length for cosine error

LPF

low-pass filter

**L**_{range}

target displacement range

**L**_{RR}

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

**n**_{air}

air refractive index

**n**_{f}

final refractive index (during a measurement)

**n**_{i}

initial refractive index (during a measurement)

**n**_{i}

refractive index of medium **i**

NIST

National Institute of Standards and Technology

nm

nanometers (10^{–9} m)

**n**_{o}

refractive index of medium **o**

nrad

nanoradians (10^{–9} rad)

**n**_{RR}

retroreflector refractive index

ns

nanoseconds (10^{–9} s)

nW

nanowatts (10^{–9} W)

OPD

optical path difference

OPL

optical path length

**O**_{pmi}

PMI axis offset

**P**

optical power

**P**

pressure

Pa

Pascals

PBS

polarizing beamsplitter

PD

photodiode

PD_{m}

measurement photodiode

PD_{r}

reference photodiode

PLL

phase-locked loop

pm

picometers (10^{–12} m)

PMI

plane mirror interferometer

PSD

position-sensitive detector

QWP

quarter waveplate

**R**

resistance

**r**_{1}

amplitude of beam 1

**r**_{2}

amplitude of beam 2

rad

radians

RH

relative humidity

RHC

right-hand circular (polarization)

**r**_{net}

amplitude of two-beam interference

RR

retroreflector

**RR**_{h}

retroreflector height

s

seconds

**t**

time

**T**

temperature

THz

terahertz (10^{12} Hz)

TIR

total internal reflection

**U**

expanded uncertainty

* u*(

**A**_{1})

uncertainty in first-order periodic error

* u*(

**A**_{2})

uncertainty in second-second periodic error

* u*(

**C**_{T})

uncertainty in thermal drift

* u*(

**d**_{A})

uncertainty in Abbé offset

* u*(

**d**_{A,x})

uncertainty in Abbé offset along * X* axis

* u*(

**d**_{A,y})

uncertainty in Abbé offset along * Y* axis

* u*(

**d**_{SF})

uncertainty in surface figure

* u*(

*)*

**H**uncertainty in relative humidity

* u*(

*)*

**n**uncertainty in refractive index

* u*(

**n**_{air})

uncertainty in air refractive index

* u*(

*)*

**P**uncertainty in pressure

* u*(

*)*

**T**uncertainty in temperature

* u*(

**x**_{i})

uncertainty in input estimates

* u*(

**z**_{DP})

uncertainty in deadpath distance

* u*(

**α**_{A})

uncertainty in Abbé angle

* u*(

**α**_{cosine})

uncertainty in cosine angle

* u*(

**α**_{N})

uncertainty in beam normality angle

* u*(Δ

*/*

**n**

**n**_{f})

uncertainty in fractional refractive index change

* u*(Δθ)

uncertainty in phase change

* u*(Δλ/λ

_{i})

uncertainty in fractional wavelength change

* u*(λ)

uncertainty in wavelength

* u*(λ

_{nom})

uncertainty in nominal wavelength

* u*(λ

_{stab})

uncertainty in wavelength stability

* u*(τ

_{DA})

uncertainty in data age

* u*(φ

_{x})

uncertainty in target angle error about * X* axis

* u*(φ

_{y})

uncertainty in target angle error about * Y* axis

* u*(ψ)

uncertainty in cosine error

**u**_{A}(* d*)

displacement uncertainty from Abbé errors

**u**_{c}(* d*)

combined displacement uncertainty

**u**_{DA}(* d*)

displacement uncertainty from data age

**u**_{DP}(* d*)

displacement uncertainty from deadpath errors

**u**_{Edlen}

uncertainty in the Edlén equation

**u**_{i}

standard uncertainty

**u**_{n}(* d*)

displacement uncertainty from refractive index

**u**_{PE}(* d*)

displacement uncertainty from periodic error

**u**_{SF}(* d*)

displacement uncertainty from surface figure

**u**_{TD}(* d*)

displacement uncertainty from thermal drift

**u**_{Δθ}(* d*)

displacement uncertainty from phase change

**u**_{λ}(* d*)

displacement uncertainty from wavelength

**u**_{ψ}(* d*)

displacement uncertainty from cosine error

V

volts

**V**_{i}

interference signal converted to volts

W

watts

**y**

output estimate

**Y**

measurand

**z**

actual displacement

**z**

direction of target motion

ZCD

zero-crossing detector

**z**_{DP}

deadpath distance

**z**_{i,m}

initial measurement arm length

**z**_{i,r}

initial reference arm length

**z**_{m}

measured displacement

**z**_{o}

optical path length

**z**_{p}

physical path length

α_{A}

Abbé angle

α_{cosine}

angle between target and interferometer axes

α_{N}

target normal beam angle

α_{p}

polarizer angle

α_{s}

Wollaston prism angle

α_{σ}

probability distribution half-width

α_{σ,H}

humidity probability distribution half-width

α_{σ,P}

pressure probability distribution half-width

α_{σ,T}

temperature probability distribution half-width

β

Bragg angle

γ_{1}

first-order mixing amplitude

γ_{2}

second-order mixing amplitude

Γ_{1}

interference signal Fourier magnitude

Γ_{2}

first-order mixing Fourier magnitude

Γ_{3}

second-order mixing Fourier magnitude

Δ**H**

change in humidity

Δ**I**_{f}

change in optical power between laser modes

Δ**l**_{c}

change in laser cavity length

Δ**n**

change in refractive index

Δ**P**

change in pressure

Δ**R**_{φ}

walkoff between beams

Δ**T**

change in temperature

Δ**x**

straightness error in * X* direction

Δ**y**

straightness error in * Y* direction

Δθ

change in phase

Δλ

change in wavelength

Δλ_{stab}

change in wavelength from stability

Δφ

change in target angle

Δφ_{x}

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

Δφ_{y}

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

Δφ_{z}

roll (along path)

ε_{0}

vacuum permittivity

ε_{medium}

permittivity of propagation medium

ε_{r}

relative permittivity

η_{633}

silicon responsivity at 633 nm

θ_{1}

phase of beam 1

θ_{2}

phase of beam 2

θ_{m}

measured phase

θ_{m}

phase of the reference beam

θ_{net}

phase of two-beam interference

λ

wavelength

λ_{0}

wavelength of iodine laser

λ_{f}

final wavelength (during a measurement)

λ_{i}

initial wavelength (during a measurement)

λ_{nom}

nominal wavelength

λ_{stab}

wavelength stability

μ_{0}

vacuum permeability

μm

micrometers (10^{–6} m)

μ_{medium}

permeability of propagation medium

μ_{r}

relative permeability

μrad

microradians (10^{–6} rad)

μs

microseconds (1^{–}
^{6} s)

μW

microwatts (10^{–6} W)

**ν**

velocity of light

**ν**

target velocity

**ν**_{D}

Doppler velocity

**ν**_{o}

voltage output

τ_{DA}

measurement data age

φ

target angle** **

φ_{x}

target angle error about * X* axis

φ_{y}

target angle error about * Y* axis

φ_{z}

target angle error about * Z* axis

ψ

displacement scale error

ψ_{i}

angle of incidence from medium **i**

ψ_{o}

angle of refraction into medium **o**

ω

optical frequency in angular units