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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 Bellingham, Washington 98227-0010 USA Phone: +360.676.3290 Fax: +360.647.1445 Email: books@spie.org 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 SeriesWelcome 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 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 College of Optical Sciences The University of Arizona The Field Guide SeriesKeep information at your fingertips with all of the titles in the Field Guide Series:
IntroductionThis 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 ContentsGlossary 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 (= ) 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 u(Δn/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 u(λnom) uncertainty in nominal wavelength u(λstab) uncertainty in wavelength stability u(τDA) uncertainty in data age u(φx) uncertainty in target angle error about X axis u(φy) 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 |
CITATIONS
Interferometers
Interferometry
Error analysis
Optical alignment
Beam guidance systems
Homodyne detection
Mirrors
