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Periodic Error in Heterodyne Interferometry

Permanent Link: http://ufdc.ufl.edu/UFE0041110/00001

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Title: Periodic Error in Heterodyne Interferometry Measurement, Uncertainty, and Elimination
Physical Description: 1 online resource (174 p.)
Language: english
Creator: Kim, Hyo
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: aom, displacement, error, heterodyne, interferometer, periodic, polarization, uncertainty
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

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Abstract: PERIODIC ERROR IN HETERODYNE INTERFEROMETRY: MEASUREMENT, UNCERTAINTY, AND ELIMINATION The purpose of this study is to construct, test, and verify a new heterodyne displacement measuring interferometer design that eliminates the current accuracy limitation imposed by periodic error, which appears as a cyclical oscillation of the measured displacement about the true value during motion. In the new design, the two (heterodyne) frequencies are generated and spatially separated using acousto-optic modulators. By removing the potential for overlap and frequency mixing within the interferometer, periodic error is eliminated. The new concept replaces the traditional method of 'polarization coding', where the two beams (with different frequencies) are initially coincident with orthogonal polarization states and then separated using polarization dependent optics. Experimental results are presented for two arrangements of the new design at multiple target velocities. Spectral content (collected using an analog spectrum analyzer) is analyzed to verify zero periodic error. These results are compared to data collected using a traditional polarization coded heterodyne interferometer with variations in the optical alignment to demonstrate different levels of first and second order periodic error. Again, frequency spectrum data are provided.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Hyo Kim.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Schmitz, Tony L.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0041110:00001

Permanent Link: http://ufdc.ufl.edu/UFE0041110/00001

Material Information

Title: Periodic Error in Heterodyne Interferometry Measurement, Uncertainty, and Elimination
Physical Description: 1 online resource (174 p.)
Language: english
Creator: Kim, Hyo
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: aom, displacement, error, heterodyne, interferometer, periodic, polarization, uncertainty
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: PERIODIC ERROR IN HETERODYNE INTERFEROMETRY: MEASUREMENT, UNCERTAINTY, AND ELIMINATION The purpose of this study is to construct, test, and verify a new heterodyne displacement measuring interferometer design that eliminates the current accuracy limitation imposed by periodic error, which appears as a cyclical oscillation of the measured displacement about the true value during motion. In the new design, the two (heterodyne) frequencies are generated and spatially separated using acousto-optic modulators. By removing the potential for overlap and frequency mixing within the interferometer, periodic error is eliminated. The new concept replaces the traditional method of 'polarization coding', where the two beams (with different frequencies) are initially coincident with orthogonal polarization states and then separated using polarization dependent optics. Experimental results are presented for two arrangements of the new design at multiple target velocities. Spectral content (collected using an analog spectrum analyzer) is analyzed to verify zero periodic error. These results are compared to data collected using a traditional polarization coded heterodyne interferometer with variations in the optical alignment to demonstrate different levels of first and second order periodic error. Again, frequency spectrum data are provided.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Hyo Kim.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Schmitz, Tony L.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0041110:00001


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1 PERIODIC ERROR IN H ETERODYNE INTERFEROMETRY: MEASUREMENT, UNCERTAINTY, AND ELIMINATION By HYO SOO KIM A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2009

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2 2009 Hyo Soo Kim

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3 To my wife, daughter, and son

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4 ACKNOWLEDGMENTS I would like to thank m y advisor, Dr. Tony L. Schmitz, for his research guidance and moral support. It would have been impossible to complete this work without my family, so I would like to thank them from the bottom of my heart. Also, I gratefully acknowledge partial financial support from the National Science Fo undation (Grant No. C MMI-0555645), the Korea Science and Engineering Foundation (Grant No. D00010), and a University of Florida Alumni Fellowship.

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5 TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................4 LIST OF TABLES................................................................................................................. ..........7 LIST OF FIGURES.........................................................................................................................8 ABSTRACT...................................................................................................................................13 CHAP TER 1 INTRODUCTION..................................................................................................................14 2 BACKGROUND.................................................................................................................... 19 Introduction................................................................................................................... ..........19 Light Waves.....................................................................................................................19 Interference of Light........................................................................................................ 21 The Michelson Interferometer................................................................................................23 Two Displacement Measuring In terferometer Types...................................................... 24 Homodyne Displacement Meas u ring Interferometer...................................................... 27 Heterodyne Displacement Measuring Interferometer..................................................... 28 Phase Measurement......................................................................................................... 31 Error Sources in Heterodyne Interferometry.......................................................................... 34 Atmospheric Error...........................................................................................................34 Material Thermal Expansion Error.................................................................................. 35 Optics Thermal Drift.......................................................................................................35 Deadpath Error................................................................................................................35 Abbe Error.......................................................................................................................36 Cosine Error.....................................................................................................................36 Laser Wavelength Stability............................................................................................. 36 Electronics Error.............................................................................................................. 37 Periodic Error..................................................................................................................37 An Example of Application....................................................................................................38 3 PERIODIC ERROR REVIEW............................................................................................... 61 Periodic Error Literature Review............................................................................................ 61 Frequency-Path Model............................................................................................................62

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6 4 POLARIZATION CODED INTERFEROMETER................................................................75 Setup Description.............................................................................................................. ......75 Traditional Setup Experimental Results................................................................................. 76 Displacement Data........................................................................................................... 77 Spectrum Analyzer Data.................................................................................................. 77 Comparison between Measurements and Model............................................................. 78 5 PERIODIC ERROR CALCULATION FR OM SPECTRUM ANAL YZER DATA............. 89 Error Calculation.............................................................................................................. ......89 Experimental Results..............................................................................................................97 Discussion.............................................................................................................................100 6 PERIODIC ERROR UNCERTAINTY................................................................................ 123 Periodic Error Formulation................................................................................................... 124 Periodic Error Measurements...............................................................................................128 Displacement Combined Standard Uncertainty.................................................................... 130 Abbe Error.....................................................................................................................130 Cosine Error...................................................................................................................131 Deadpath Error..............................................................................................................131 Atmospheric Error.........................................................................................................132 Material Thermal Expansion Error................................................................................ 132 Other Phase Errors......................................................................................................... 132 Laser Wavelength Stability........................................................................................... 132 7 NEW HETERODYNE INTERFEROMETER DESIGN..................................................... 143 Acousto-Optic Modulator.....................................................................................................143 New Interferometer Configuration....................................................................................... 144 Periodic Error Elimination.................................................................................................... 146 Size Reduction......................................................................................................................147 Experimental Results............................................................................................................150 8 CONCLUSIONS.................................................................................................................. 163 Completed Work................................................................................................................. ..163 Future Work..........................................................................................................................164 LIST OF REFERENCES.............................................................................................................167 BIOGRAPHICAL SKETCH.......................................................................................................174

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7 LIST OF TABLES Table page 2-1 Heterodyne displacement measuring in terferom eter system accuracy for the integrated circuit wafer stage application.......................................................................... 41 3-1 Summary of periodic e rror literature review. ....................................................................62 3-2 Interference terms in fully leaking single pass heterodyne interferom eter........................ 67 6-1 Apparent individual unc erta inty contributors for lpe.....................................................128 6-2 Monte Carlo simulation i nput values for first and second order error uncertainty evaluation (all distributions were no rmal except for and which were uniform)........ 130 6-3 Monte Carlo simula tion input values for uc(l) evaluation................................................ 133 7-1 Optical components and their prices for both the traditional in terferom eter and AOM DMI..................................................................................................................................149

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8 LIST OF FIGURES Figure page 1-1 A simple polarization coded heterodyne interferom eter where the two beams are overlapped and imperfectly separated............................................................................... 18 1-2 New heterodyne interferometer design that uses an acousto-op tic m odulator (AOM) to eliminate frequency mixing........................................................................................... 18 2-1 The electric (E) and magnetic (B) wave s are orthogonal to each other and propagate in the sam e direction.......................................................................................................... 42 2-2 An example of constructive interference........................................................................... 43 2-3 An example of destructive interference............................................................................. 43 2-4 Schematic of the Michelson interferometer....................................................................... 44 2-5 The Michelson interferometer with modern technology components. .............................. 45 2-6 Schematic of a single pass interferometer......................................................................... 46 2-7 Interference according to the target di splacem ent in a single pass configuration............. 47 2-8 Schematic of a double pass interferometer........................................................................ 48 2-9 Interference according to the target di splacem ent in a double pass configuration............ 49 2-10 Schematic of a homodyne displ acement m easuring interferometer.................................. 50 2-11 Schematic of a single pass heterodyne displacement m easuring interferometer............... 50 2-12 Spectrum analyzer outputs from two interferometers........................................................ 51 2-13 Schematic of heterodyne interferometer where displacem ent is determined by comparing the measurement and reference signals........................................................... 52 2-14 Phase difference between the measurement and reference signal..................................... 53 2-15 Flow chart for phase measurement.................................................................................... 54 2-16 Flow chart for initial frequency estimation........................................................................ 55 2-17 Error sources degrading the measuremen t accuracy in hetero dyne interferometry........... 56 2-18 An example of deadpath error............................................................................................57 2-19 An example of Abbe error................................................................................................. 58

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9 2-20 An example of cosine error................................................................................................ 58 2-21 Visual description of periodic error................................................................................... 59 2-22 A heterodyne displacement measuring in terfero meter system configuration for position feedback in application of an integrated circuit wafer stepper............................ 60 3-1 Schematics for ideal and fully fr equency leaking interferom eter......................................68 3-2 Frequency-path models...................................................................................................... 69 3-3 Intended a c interference signal.......................................................................................... 70 3-4 Leakage induced ac in terference .......................................................................................71 3-5 Description of ac reference term s......................................................................................72 3-6 Description of dc interference term s..................................................................................73 3-7 Power spectrum for constant veloci ty m otion in fully leaking heterodyne interferometer................................................................................................................. ....74 4-1 Setup for a single pass he terodyne interferom eter............................................................. 80 4-2 Displacement data for m isaligned system.......................................................................... 81 4-3 Displacement data for w ell aligned system....................................................................... 82 4-5 Example of frequency content fo r fully leaking interferom eter........................................ 84 4-6 Spectral content for nominal angular alignm ent of the half wave plate and linear polarizer...................................................................................................................... .......84 4-7 Periodic error frequency content at three different velocities. ........................................... 85 4-8 Periodic errors for half wave plate/linear polarizer param eter study................................. 86 4-9 Periodic errors for half wave plate/linear polarizer param eter study................................. 87 4-10 Periodic errors for half wave plate/linear polarizer param eter study................................. 88 5-1 Phasor diagrams............................................................................................................ ...101 5-2 Periodic error in the presence of 0 and 1 only for various nominal phase angles (rad).......................................................................................................................... ........102 5-3 Periodic error in the presence of 0 and 2 only for various nominal phase angles (rad).......................................................................................................................... ........103

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10 5-4 Phasor diagram for general case wh ere the intended and leakage induced ac inter ference and ac reference signals are present with arbitrary initial phases...............104 5-5 Comparison between periodic error magnit udes obtained from Eq. 5-8 and Eqs. 5-5 and 5-6 for 150 dBm, 501 dBm, and variable 2 ..........................................105 5-6 Comparison between periodic error magnit udes obtained from Eq. 5-8 and Eqs. 5-5 and 5-6 for 150 dBm, 301 dBm, and variable2 ...........................................106 5-7 Comparison between periodic error magnit udes obtained from Eq. 5-8 and Eqs. 5-5 and 5-6 for 150 dBm, 251 dBm, and variable2 ...........................................107 5-8 Difference between periodic error magnit udes obtained from Eq. 5-8 and Eqs. 5-5 and 5-6 for 150 dBm, 251 dBm, and variable2 ...........................................108 5-9 Comparison between periodic error magnit udes obtained from Eq. 5-8 and Eqs. 5-5 and 5-6 for 150 dBm, 502 dBm, and variable1 ..........................................109 5-10 Comparison between periodic error magnit udes obtained from Eq. 5-8 and Eqs. 5-5 and 5-6 for 150 dBm, 352 dBm, and variable1 ..........................................110 5-11 Periodic error for 0 = 10 deg = 0.17 rad with 150 dBm, 301 dBm, 452 dBm, and 1 = 2 = 0.......................................................................................111 5-12 Periodic error magnitudes for 0 = 10 deg = 0.17 rad with 150 dBm, 301 dBm, 452 dBm, and 1 = 2 = 0............................................................................. 111 5-13 Periodic error for 0 = 170 deg = 2.97 rad with 150 dBm, 301 dBm, 452 dBm, and 1 = 2 = 0.......................................................................................112 5-14 Periodic error magnitudes for 0 = 170 deg = 2.97 rad with 150 dBm, 301 dBm, 452 dBm, and 1 = 2 = 0............................................................................. 112 5-15 Variation in periodic error m agnitudes for random 0 values ( range) with 150 dBm, 301 dBm, 452 dBm, and 1 = 2 = 0................................... 113 5-16 Variation of 1 and 2 with linear polarizer (LP) angle.......................................... 114 5-17 Periodic errors calculated by Eqs. 5-5, 5-6, and 5-8 are com pared to magnitudes computed using the discrete Fourier transform of position data...................................... 114 5-18 Differences between magnitudes from Eq s. 5-5, 5-6, and 5-8 and discrete Fourier transform of position data................................................................................................115

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11 5-19 Variation of 1 and 2 with linear polarizer (LP) angle...........................................116 5-20 Periodic errors calculated by Eqs. 5-5, 5-6, and 5-8 are com pared to magnitudes computed using the discrete Fourier transform of position data...................................... 116 5-21 Variation of 1 and 2 with linear polarizer (LP) angle...........................................117 5-22 Periodic error calculated by Eqs. 5-5, 5-6, a nd 5-8 agree with the position data results because 1 and 2 do not vary with linear polari zer angle for the well aligned system..............................................................................................................................117 5-23 Variation of 1 and 2 with half wave plate (HWP) angle....................................... 118 5-24 Periodic error calculated by Eqs. 5-5, 56, and 5-8 are com pared to position data error magnitudes (the linear polarizer misalignment angle was 17 deg from nominal). ..........................................................................................................................................118 5-25 Differences between magnitudes from Eq s. 5-5, 5-6, and 5-8 and discrete Fourier transform of position data................................................................................................119 5-26 Variation of 1 and 2 with half wave plate (HWP) angle....................................... 120 5-27 Periodic error calculated by Eqs. 5-5, 5-6, and 5-8 agree with the position data error m agnitudes (the linear polarizer angle wa s fixed at its nominal orientation).................. 120 5-28 The spectrum of position data (normalized to error order) contains first, second, and third order periodic error. .................................................................................................121 5-29 Periodic errors calculated by Eqs. 5-5, 5-6, and 5 -8 are compared to the position data error magnitudes.............................................................................................................. 122 6-1 An example of periodic error variation with conditions that dom inates first error order.......................................................................................................................... .......135 6-2 An example of periodic error variation with conditions that dom inates both first and second error orders...........................................................................................................136 6-3 Histogram of lpe values for normal distributions of d1, d2, and with u(d1) = u(d2) = 0.1 deg, u() = u() = u() = 2 deg and zero mean values; and uniform distributions of and with ranges of 0.05 and mean values of 0.95..........................137 6-4 Measurement/model comparison for 39 deg pol arizer m isalignment first order error dominates...................................................................................................................... ...138 6-5 Measurement/model comparison for 10 deg ha lf wave plate misalignment first and second order errors are presen t. Model parameters were: n = 1, vac = 633 nm, d1 = d2 = 0 deg, = = 20 deg, = 2 deg, and = = 1...................................................139

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12 6-6 Comparisons between measurements (cir cles), model (squares), and Monte Carlo sim ulation (dotted line) for vari able polarizer angle tests............................................... 140 6-7 Comparisons between measurements (cir cles), model (squares), and Monte Carlo sim ulation (dotted line) for variable half wave plate angle tests..................................... 141 6-8 Monte Carlo simulation results for Eq 6-10 using the data in T able 6-3........................ 142 7-1 Acousto-optic modulator schematic................................................................................ 151 7-2 Frequency of diffracted beam..........................................................................................151 7-3 AOM DMI setup.............................................................................................................. 152 7-4 Desired ac interference term and dc power peaks........................................................... 153 7-5 No undesired frequency conten t is present for the AOM DMI. ...................................... 154 7-6 Errors for polarization coded DMI.................................................................................. 155 7-7 Results for the new beat frequency, f of 5 MHz. ..........................................................156 7-8 Schematic of the current AOM-DMI setup..................................................................... 157 7-9 Size reduced AOM DMI setup........................................................................................158 7-10 Technical drawing for the right angle prism.................................................................... 159 7-11 An electroplated diamond core drill................................................................................ 159 7-12 Schematic of drilling operation for a right angle prism using a diamond core drill in the submerged (water) environment................................................................................ 160 7-13 The right angle prism wi th hole at the center.................................................................. 160 7-14 Frequency contents of the measurement signal during the constant velocity of 10,000 mm/ min (Doppler frequency, fd, is equal to 0.53 MHz) at the beat frequency, f of 3.64 MHz.........................................................................................................................161 7-15 Frequency contents of the measurement signal during the constant velocity of 10,000 mm/ min (Doppler frequency, fd, is equal to 0.53 MHz) at the beat frequency, f of 20 MHz............................................................................................................................161 7-16 Snapshots of power spectra during oscill atory target m otion at the beat frequency, f of 3.64 MHz.....................................................................................................................162 8-1 Setup size comparison...................................................................................................... 166

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13 Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy PERIODIC ERROR IN H ETERODYNE INTERFEROMETRY: MEASUREMENT, UNCERTAINTY, AND ELIMINATION By Hyo Soo Kim December 2009 Chair: Tony L. Schmitz Major: Mechanical Engineering The purpose of this study is to construct, te st, and verify a new he terodyne displacement measuring interferometer design that eliminat es the current accuracy limitation imposed by periodic error, which appears as a cyclical oscillati on of the measured displacement about the true value during motion. In the new design, the two (heterodyne) frequencies are generated and spatially separated using acousto-optic modulators. By removing the potential for overlap and frequency mixing within the interferometer, pe riodic error is eliminated. The new concept replaces the traditional method of polarization coding, where the two beams (with different frequencies) are initially coincident with orthog onal polarization states a nd then separated using polarization dependent optics. E xperimental results are presented for two arrangements of the new design at multiple target velocities. Spectral content (collected using an analog spectrum analyzer) is analyzed to verify zero periodic error. These results are compared to data collected using a traditional polarization coded heterodyne in terferometer with vari ations in the optical alignment to demonstrate different levels of first and second order periodic error. Again, frequency spectrum data are provided.

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14 CHAPTER 1 INTRODUCTION Since its introduction in the m id-1960s, displacem ent measuring interferometry has offered high accuracy, range, and resolution for non-contact displacement measurement applications. Important examples include: 1) positional fee dback of precision stages on photolithographic steppers for integrated circuit fabrication; 2) transducer calibration; and 3) positional feedback/calibration for machine tools, coordi nate measuring machines, and other metrology systems. A common configuration choice in these s ituations is the heterod yne (or two frequency) Michelson-type interferometer w ith single, double, or multiple passe s of the optical paths. These systems infer changes in displa cement of a selected optical path by monitoring the optically induced variation in the photodetect or current, which is generated from the optical interference signal. The phase measuring electro nics convert this photodetector current to displacement using an assumed relationship between corresponding cha nges in detector current and displacement, where this relationship is defined by idealized performance of the optical elements. Many error sources inherent to displacement me asuring interferometers can be corrected or compensated by setup or additional metrology. Some of these errors are briefly introduced here and are described in detail in Chapter 2. Abbe if the moving target is not located at the point of interest (e .g., the tool point in a machine tool), rotational error motions are converted to displacement errors by any offset; cosine an angular misalignment between th e interferometers optical axis and motion direction leads to a measured displacement value which is smaller than the true value; Reprinted with permission from Kim H, Schmitz T, Beckwith J. Periodic error in heterodyne interferometry: Examination and Elimination. In: Halsey D, Raynor W, ed. Handbook of Interferometers: Research, Technology and Applications. Hauppauge, NY: Nova Science; 2009.

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15 thermal expansion/contraction changes in th e optical dimensions with temperature can cause apparent displacements if the optical path lengths between the moving and fixed targets are not balanced; also, deformations of the support struct ure can cause errors; atmospheric air refractive index, which re lates the optical path difference to the geometric motion, depends on temperature, pressure, humidity, and composition; and deadpath unequal path lengths at initial ization (of the phase measuring electronics) accompanied by uncompensated refractive in dex variation during the measurement. Others, such as laser wavele ngth stability and electronics error, are typically small (although non-negligible in some cases). Periodic error, however, remains an intrinsic error source that prevents traditional configurations from achieving s ub-nanometer level accuracy. The purpose of this research is to validate the abse nce of periodic error in a new interferometer design that does not rely on pol arization coding, where the two (h eterodyne) optical frequencies are carried on coincident, linearly polarized, mutually orthogonal laser beams and are separated/recombined using polarization depende nt optics. Rather, the two frequencies are carried on spatially separate beams in a polarizat ion independent optical c onfiguration that also enables the user to select the beat (or split) frequency. The new design is based on two acoustooptic modulators with different driving frequencie s that function as beam splitters/recombiners. By eliminating the potential for mixing between the two heterodyne frequencies, the periodic error source is removed. Figures 1-1 and 1-2 depict the simplest se tups for the traditional polarization coded interferometer and the new design of the heterodyne displacement inte rferometer, respectively. In polarization coding, the f2 frequency beam is (ideally) linea rly polarized in the horizontal plane, while the collinear f1 frequency beam is linearly polarized in the vertical plane. A polarization dependent beam splitter is then used to separate the two beams and direct them toward the moving and fixed targets, respec tively. Inherent imperfections in the beam

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16 polarization states and optics allow a portion of both frequencies to travel to the two targets. The resulting periodic error degrades the measurement system accuracy. In this research, the two fre quencies are kept spatially separa te to avoid leakage into the unintended paths and eliminate period ic error. In the Figure 1-2, the f2 (up-shifted frequency) beam generated by the acousto-optic modulator (AOM) travels to the moving target, while the f1 beam from the single frequency, He lium-Neon laser source travels separately to the fixed target. After reflection, the two beams in terfere during their return pa th through the AOM without the possibility for frequency leakage and correspondi ng periodic error. No other optical components are required. This is an important considerati on because the existence of first or second order periodic errors1, with amplitudes that vary cyclically w ith the target position, leads to nonlinear performance and limits the achievable accuracy. In addition to constructing the new heterodyn e displacement measuring interferometer, the calculation of first and s econd order periodic error from spectrum analyzer data is also analyzed and the displacement measurement uncertainty is evaluated. Identifying the magnitude of periodic error for a particular setup enables a lignment adjustments to be made to minimize the error magnitude. The calculation of periodic er ror magnitude, which build s on a prior analysis [2] that considered each error order individuall y, is completed for the general case. A single expression is developed for calculating both first and second order error magnitudes from spectrum analyzer data. The typical approach for displacement interferom etry uncertainty analysis is to evaluate the uncertainty due to the indivi dual error sources and then combin e them in a root sum squares manner to represent the total uncer tainty [3, 4]. In this work, a single expression for displacement 1 First and second order periodic erro rs exhibit spatial frequencies of one and two cycles per displacement fringe, respectively.

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17 including all significant error sources is presented and its uncert ainty is evaluated using Monte Carlo simulation. The uncertainty due to periodic error, described using the analytical model developed by Cosijns et al [5], is also included. The remaini ng chapters describe this work in detail.

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18 Figure 1-1. A simple polarizati on coded heterodyne interferometer wher e the two beams are overlapped and imperfectly separated. Figure 1-2. New heterodyne interferometer design that uses an acousto-optic modulator (AOM) to eliminate frequency mixing. f2 f1 f1 + (f2) f2 + (f1) fd Fixed retroreflector Moving retroreflector Polarizer for interference Polarizing beam splitter Beam to photodetector From laser source Frequency leakage Orthogonal polarizations Moving retroreflector f1 f2 fd Beam to photodetector Beam from laser source (single frequency) f1 Fixed retroreflector fs AOM fs AOM driving frequency f2 = f1 + fs

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19 CHAPTER 2 BACKGROUND Introduction Interferom etry is a technique for diagnosi ng the properties (i.e., magnitude, phase, and frequency/wavelength) of two or more light waves by analyzing the pa ttern of interference generated by their superposition. Th e instrument used to generate interference of the light waves is called an interferometer. Inte rferometry plays an important ro le in the fields of astronomy, engineering metrology, optical metrology, and many others. One common interferometer is the Michelson interferometer used to measure displa cement [6]. In this chapter, two types of the Michelson interferometer are introduced. Also, th e error sources for one of the two types, the heterodyne displacement measuring interferometer, are described. Since interferometry is based on the physic al phenomenon of the in terference of light, understanding the basic concepts of interference is essential. These concepts are introduced in the following sections. Light Waves Light can be described as a transverse el ectrom agnetic wave propagating through space. Since the electric and magnetic fields are orthogonal to each other and propagate together as shown in Figure 2-1, it is usually sufficient to consider only the el ectric field at any point [6]. The electric field can be treate d as a time-varying vector perp endicular to the direction of propagation of the wave. If the fiel d vector always lies in the same plane, the light wave is said to be linearly polarized in that plane. The el ectric field at any point due to a light wave propagating along the z direction is then described by Eq. 2-1, 0(,,,)cos2/ ExyztEftz (2-1)

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20 where E0 is the amplitude of the light wave, f is its frequency, and is its wavelength. The term within the square bracket, called the phase of the wave, varies with time as well as with distance along the z -axis from the origin. With varying time, a light wave specified by Eq. 2-1 moves along the z -axis with a speed of [6], vaccf (2-2) where c is the speed of light in a vacuum, approximately 3 x 108 m/s and vac is the vacuum wavelength. In a medium with a refractive index, n the speed of the light wave is expressed by Eq. 2-3. c v n (2-3) Since its frequency remains unchanged, its wavelength is determined by Eq. 2-4, vacn (2-4) Equation 2-1 can be rewritten in a compact form according to Eq. 2-5, 0,,,cos ExyztEtkz (2-5) where f is the circular frequency and k is the propagation constant. The representation of a light wave in terms of a cosine function in Eq. 2-5 is easy to visualize, but not well adapted to mathematical manipulation. It is of ten convenient to use a complex exponential representation as shown in Eq. 2-6 [7], 0,,,ReitkzExyztEe ReitAe (2-6) where0iAEe is known as the complex amplitude and / kzz

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21 The energy of the electroma gnetic radiation per unit area is proportional to the time average of the square of the electric field as shown in Eq. 2-7 [7], 221 lim 2T T TEE d t T 2 01 limcos 2T T T E tkzdt T Expand to obtain: 2 0 22 00 time average of cosine function is zero 2 01cos2 lim 22 limcos2 24 2T T T T T Ttkz E dt T EE tkzdt T E (2-7) where T is the averaging time. The intensity of electri c field can be expressed as the square of the complex amplitude; see Eq. 2-8, 2 01 2 I E 21 2 1 2 AA A (2-8) where *A is the complex conjugate of A. Interference of Light When two light waves are superimposed, the re sultant intensity at any point depends on whether they interfere constructiv ely (in phase) or destru ctively (out of phase) [6]; see Figures 22 and 2-3. Given that the two light waves are propag ating in the same direction, both are linearly polarized, and they have different amplit udes and the same frequency with a phase difference, they can be described according to Eq. 2-9,

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22 101 202cos cos EEtkz EEtkz (2-9) where E01 and E02 are the amplitudes of E1 and E2, respectively. Equation 2-10 shows the complex exponential representations of the two waves, 11 22Re Reit itEAe EAe (2-10) where101 i A Ee and 202 i A Ee are the complex amplitudes and kz and 2kz The resultant wave, E, after superposition is then given by the sum of the two waves, 1Eand2E. See Eq. 2-11. 12 0102 12 cos cos ReReit itEEE EtkzEtkz AeAe (2-11) The intensity of the resultant wave is then half of the square of the sum of the two complex amplitudes as shown in Eq. 2-12, 2 12 ** 1212 **** 11221221 22 010201020102 12121 2 1 2 1 2 1 2 2cosiiIAA AAAA AAAAAAAA EEEEeEEe IIII (2-12) where 1 I and 2 I are the intensities due to the two waves a nd the third term is the intensity due to interference between the two waves.

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23 The Michelson Interferometer The Michelson interferometer was first introdu ced by Albert Michelson in 1881 [6]. It has played a vital role in the development of modern physic s. A schematic of the Michelson interferometer is shown in Figure 2-4. From an extended light source, S, beam 1 is split by a semireflective coating on the surface of a beam splitter. The same beam splitter is used to recombine the beams reflected back from the two mirrors, M1 and M2. The recombined beam (beam 4) leaves the interferometer and the interference pattern is imaged on the screen. To align the beams, at least one of the mirrors is equipped with a tilting mechanis m that allows the surface of M1 to be made perpendicular to that of M2. Also, one of the mirrors is movable along the beam direction to generate an optical path difference between beams 2 and 3. When using a white light source [8] to obtain the interference fringes in the Michelson interferometer, the two optical path lengths must be equal for all wavelengths [6]. Therefore, both arms must contain the same thickness of glass having the same dispersion. However, beam 3 traverses the beam splitter three times while beam 2 traverses it only once. Accordingly, a compensating plate (identical to the beam splitte r, but without the semireflective coating) is inserted in the path of beam 2. Some important developments have extende d the scope and accuracy of displacement measurements made using the Michelson interferom eter. These include the invention of the laser, photodetector, optical fibers, and di gital electronics for signal pr ocessing. The schematic of the Michelson interferometer equipped with thes e improvements is shown in Figure 2-5. The extended light source is repl aced by the laser, which produces monochromatic (single wavelength) light, and the screen used to obser ve the interference fringes is replaced by a photodetector. Digital electronics are added after the photodetector to interpret the measurement signal, either intensity varia tion or phase change depending on the interferometer type

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24 (homodyne or heterodyne, which are described in next section) and convert the fringe information to displacement. Two Displacement Measuring Interferometer Types Two types of the Michelson interferomet er are commonly used for displacement measurements. A homodyne displacement measur ing interferometer uses a single frequency laser head and the intens ity variation induced by interference is converted into displacement. A heterodyne displacement measuring interferometer, on the other hand, uses a two-frequency laser as a light source and determines displacement from the phase change between the measurement and reference signal. The homodyne approach is brie fly described in the ne xt section, while the heterodyne approach used in this research is discussed in more de tail in the subsequent section. Since the Michelson-type interf erometer is implemented with a certain configuration (i.e., single, double or multiple passes of the optical pa th), the characteristics and implementations of the single pass and double pass conf igurations are introduced before proceeding to a description of the two interferometer types. In the single pass configuration shown in Figure 2-6, a beam having both the vertical, V, a nd horizontal, H, polarization co mponents splits at a polarizing beam splitter (PBS). Conventionally, the polarizing beam splitter transmits the horizontally polarized beam while the vertica lly polarized beam is reflected. The reflected beam, referred to as the reference beam, travels toward the fixed re troreflector and returns to the polarizing beam splitter. The transmitted beam, referred to as the measurement beam, is reflected from the moving retroreflector where phase change occu rs during the target motion. The measurement and reference beams are recombined at the pola rizing beam splitter and interfered after passing through a linear polarizer (LP) which makes the same polarization plane for the two beams by orienting the transmission axis (TA) at 45 deg with respect to the vertical axis.

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25 The path change for the measurement beam is twice the displacement of the target in the single pass interferometer. In other words, the phase change,single of the measurement beam due to the target motion is proportional to twice the target displacement; see Eq. 2-13, single2d (2-13) where d is the displacement of the target. Assume that the initial phases of the measurement and reference beams are the same at the initial position as shown in Figure 2-7A. That is, the two waves interfere constructively so that bright light is detected for the static case. When the target moves where is the wavelength of the beam, the phase of the measurement beam changes by rad as shown in Figure 2-7B which leads to destructive interf erence (i.e., no light at the det ector). Another movement of the target by in the same direction changes the measurement beam phase by 2 rad so that the constructive interference occurs; see Figure 2-7C. The bright-dark-bright pattern induced by the interference of the two beams during the target motion is defined as a fri nge, where one fringe is referred to as the optical resolution. The optical resolution of the single pass system is ha lf of the source wavelength as shown in Figure 2-7D. Therefore, the total displ acement of the target in the si ngle pass interferometer can be determined by multiplying the number of fringes by ; see Eq. 2-14. (# of fringes)d (2-14) Figure 2-8 shows a schematic of a double pass configuration, where th e retroreflectors are replaced by plane mirrors and quarter wave plates are added. The quart er wave plate (QWP) retards the phase of an incident beam by When a linearly polarized beam transmits through a QWP, the polarization state is converted to circular [6].

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26 The vertically polarized reference beam is reflected at the polari zing beam splitter and passing through the upper QWP, conve rting the state of polarizati on to circular. The reflected beam from the fixed plane mirror passes through the QWP again, which converts the circular polarization state to linear, but in the horizontal plane so that the polarizing beam splitter now transmits it. The offset beam reflected from th e retroreflector passes th rough the polarizing beam splitter and the QWP, converting fr om linear (horizontal) to circul ar polarization. The circularly polarized beam passes back through the QWP on the return path from the fixed mirror, is converted back to a verti cal polarization, and is di rected to the detector by reflecting at the beam splitter. The transmitted horizontally polarized measurement beam from the source follows a similar path to the moving plane mirror. This be havior may be modeled analytically using the Jones vector/matrix notation. Readers may fi nd more information on describing polarization states using Jones vectors and J ones matrices in reference [6]. Unlike the single pass configura tion, the measurement beam tr avels four times more than the target in the double pass system. The phase change,double due to the target motion is therefore proportional to four tim es the target displacement. double4d (2-15) Again, assume that the initial phases of the measurement and reference beams are the same at the initial position as shown in Figure 2-9A. Then, constructive in terference occurs (i.e., bright light is detected) prior to mo tion. When the target moves by the phase of the measurement beam changes by rad which leads to the 180 deg out of phase condition with respect to the reference beam phase. In that case, the two beams destructively interfere a nd no light is detected as shown in Figure 2-9B. If th e target moves by an additional in the same direction, constructive interference occurs; see Figure 2-9C.

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27 In the double pass configuration, one fr inge is obtained for motion equal to /4 as shown in Figure 2-9D. That is, the optical resolution of the double pa ss interferometer is twice that of the single pass interferometer. The total displacement of the target in the doub le pass interferometer can be determined by multiplying the number of fringes by /4; see Eq. 2-16. (# of fringes) 4d (2-16) Homodyne Displacement Measuring Interferometer In homodyne interferometry, a singl e frequency laser is used as a light source and intensity variation due to constructive/destructive interf erence is converted into displacement [9]. A homodyne interferometer system consists, at minimum, of a laser source, beam splitter, retroreflectors, photodet ector and measurement electronics as shown in Figure 2-10. A single frequency, linearly polarized beam oriented at 45 deg with respect to th e vertical axis from, typically, a Helium-Neon laser head is split into two beams at the polarizing beam splitter. One of the beams is reflected to the fixed retrorefle ctor while the other beam is transmitted to the moving retroreflector. They recombine at the polarizing beam splitter where the beam reflected from the fixed retroreflector remains at the sa me frequency while the frequency of the beam from the moving retroreflector is shifted by the Doppler freque ncy. The interference signal is passed through intensity m easuring electronics to count fringes equivalent to half the wavelength of the laser source ( = 633 nm for a He-Ne laser) in the single pass system. Since the frequencies of the two beams are the same and th e phase change (or frequency shift) is caused by the Doppler shift due to target motion, the intensity variation de tected on the photodetector in the homodyne interferometer is the same as in Eq. 2-12. Some limitations to the simple homodyne system include the inabi lity to detect the direction sense of the target when motion is stopped and sensitivity to changes in the laser power

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28 and ambient light intensity. To resolve these limita tions, the detector configuration in the system can be modified, but this adds system complexity. Heterodyne Displacement Measuring Interferometer A heterodyne interferometer sy stem uses two frequencies an d measures the phase change between the fixed and moving arms [10]. A schema tic of a single pass heterodyne interferometer is provided in Figure 2-11. Tw o polarized beams having sligh tly different frequencies are generated using either a Zeeman approach, wh ere two frequencies are obtained by placing a magnetic field around the laser tube or by combining a single fre quency laser with an acoustooptic modulator, which produces a second diffracted beam with a modulated frequency [11] .The two beams with different frequencies are made collinear with orthogonal polarizations, which allows a polarizing beam splitter to sepa rate them based on polarization state. The frequency1 f beam is reflected at the polarizing beam splitter and travels to the fixed retroreflector, while the frequency2 f beam transmits through the beam splitter and travels to the moving retroreflector. The2 f beam is reflected from the moving retroreflector and is shifted (by the Doppler frequencyd f ) due to target motion. The two beam s are then recombined at the beam splitter and directed to the detect or. Since the polarization states are orthogonal, a linear polarizer (LP) with a 45 deg transmission axis is inserted in the beam path to produce interference. The intensity variation of the two interfered beams is now investigated. The two light waves, E1 and E2, with amplitudes, E01 and E02, and angular frequencies, 1 (2 f1) and 2 (2 f2), propagate to the fixed (path1) and moving (pat h2) retroreflectors (as shown in Figure 2-11), respectively. See Eq. 2-17. 101111 2022222cos cosEEtkFFx EEtkFFx (2-17)

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29 In Eq. 2-17, the second terms in the parentheses, ki(FF)xi, where i = 1 and 2, are the values of the phase changes due to the motions of both the fixed (x1) and moving (x2) retroreflectors. As described in the earlier section, ki = 2/i is the propagation constant (i is the wavelength) and FF is the fold factor (equal to 2 for a singl e pass and 4 for a double pass configuration). The phase change (k1(FF)x1) of the wave E1 is zero because there is no motion of the fixed retroreflector (x1 = 0). On the other hand, the phase change (k2(FF)x2) of the wave E2 is produced due to the motion of the moving retroreflector. The initial phases, and 2 described in Eq. 217, represent the kz term in Eq. 2-5. Equation 2-17 can be rewritten as shown in Eq. 2-18 by letting and substituting zero for k1(FF)x1. 111 22Re Reit itEAe EAe (2-18) The complex amplitudes are 101AE and 22202 itkFFxAEe. Note that the initial phases are assumed to zero. When they interfere by passing through the linear polari zer, the resultant wave, E, in Eq. 2-19, is the sum of the two waves. 12 12 ReReit itEEE AeAe (2-19) The interference between the wave s produces a sinusoidal current va riation in the photodetector with a frequency difference,21 f ff which is referred to as the beat or split frequency; see Eq. 2-20. The terms in the parenthesis give the phase of interfered waves. The phase change, is induced by motion of the target. If th ere is no motion, the phase change is zero and the output frequency of the interference signal remains at the split frequency. Therefore,

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30 the operating point of a system is shifted from 0 Hz (DC) for a homodyne interferometry system to the split frequency, f for a heterodyne system. 22 222 12 ** 1212 22 010201020102 121222 phase change, 1 2 1 = 2 1 =E 2 =2cositkFFx itkFFxIAA AAAA EEEeEEe IIIItkFFx (2-20) Figure 2-12 shows the interf erence signal as displayed by a spectrum analyzer for both homodyne (Figure 2-12B) and heterodyne interf erometer system (Figure 2-12C). The phase change, during target motion is given by Eq. 2-21, 2vacn FFvt (2-21) wherevac is the vacuum wavelength, n is the refractive index of the medium in which the measurement occurs, v is the velocity of the target motion, and t is time. The intensity expression, Eq. 2-20, can be rewr itten to isolate the frequency shift caused by target motion. See Eq. 2-22. 1212() 2cos2vacFFn I IIIIfvt (2-22) The resulting Doppler frequency shift, fd, is proportional to the target velocity. The sign of the frequency shift depends on the direction of moti on so that heterodyne system enables direction sensing by monitoring the sign of the Doppler frequency. See Eq. 2-23. d vacFFn f v (2-23)

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31 The total phase difference is then the integral of the frequency shift, as shown in Eq. 2-24, and is directly related to the displacement, d ; see Eq. 2-25. 02 t vac vacFFnv dt n FFd (2-24) 1 2vacd FFn (2-25) Therefore, if the total phase difference is measured, the displacement can be determined. In order to measure the phase difference, the reference signal1 is needed to be compared to the measurement signal. There are two ways to ge nerate the reference signal as shown in Figure 2-13. An internal reference signal can be directly derived within th e laser head (beam 1) [12] or an external reference signal can be obtained using a non-polarizing b eam splitter immediately after the laser output (beam 2) The measurement and reference beams are launched into photodetectors which convert the optical signals to electrical signals. The phase measuring electronics measure the total phase change for bot h the measurement and reference signals, then subtract the measured phase of the reference si gnal from that of the measurement signal to determine their phase difference; see Figure 2-14. The following section describes a phase measuring technique that is us ed in commercially-available phase measuring electronics. Phase Measurement There are several approaches to measure the ph ase of a signal such as the traditional analog phase measuring technique and the phase digitizi ng technique [13]. In th is section, the phase digitizing technique used in this research is described. Figure 2-15 is a flow chart that shows how 1 The interference signal of the two heterodyne frequencies (prior to reaching the interferometer) is 12122cos.ref I IIIIt Naturally, this signal does not contain the Doppler phase shift.

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32 the phase is determined from the incoming analog signal for both the reference signal (channel 1) and the measurement signal (channel 2). Since th e signal processing proce dure for both signals is the same, only the phase determination for the measurement signal is described. An optical interference signal, Eq. 2-26, is conv erted into an electrical signal, Eq. 2-27, by a photodetector, composed of photovoltaic cells whic h converts light (i.e., photo) into electricity (i.e., voltaic). 0 meascos2 EEft (2-26) 0 meascos2 VVft (2-27) In Eqs. 2-26 and 2-27, fmeas is the frequency of the incoming measurement signal (d f f ), is the initial phase (or phase error in [13]), and E0 and V0 are the amplitudes of the electric field and current, respectively. The analog electrical signal is digitized at a samp ling rate of 80 MHz by a 12 bit analog-to-digital converter (ADC). The dig itized data sampled at 80 MHz is then divided by 256 (with a modula-256 counter) which slows the phase to 312.5 kHz (80 MHz divided by 256). Within the 3.2 s time frame (the period of 312.5 kHz digitized data), the frequency, fmeas is assumed to be constant. The initial frequency estimation is illustrated in Figure 2-16. The first step is to capture N samples (2048 for the electroni cs used in this study) of the digitized signal sampled at 80 MHz. The discrete Fourier Transf orm (DFT) is then performed on the captured data set. Next, the cell number (o r index) of the frequency axis, k, with the largest transform modulus (i.e., the square root of the sum of the real and imaginary parts of the cell) is identified. Finally, the initial frequency is estimated by k [sampling rate (80 MHz)]/N. This initial frequency is used to begin the linear regres sion to determine the amplitude of the signal ( V0) and the initial phase ( ) in Eq. 2-27.

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33 Equation 2-27 can be expanded to form Eq. 228. There are 256 values on the left hand side of Eq. 2-28 0 meas 0 meas 0 meascos2 cos2cos()sin2sin VftVftVft (2-28) On the right hand side, meascos2 f t and meassin f t are provided by cosine and sine look-up tables for 256 values of time, t Note that the frequency, fmeas, in the cosine and sine functions is the initial frequency, which is estimated as descri bed in the previous paragraph and is used only once for the first segment of th e data analysis. This forms a system of 256 equations and two unknowns, Xc and Xs, where 0coscXV and 0sinsXV Linear regression is applied to determine the best-fit values of Xc and Xs. Accordingly, V0 and are obtained after a polar-torectangular transformation as shown in Eq. 2-29. 22 0 1tancs s cVXX X X (2-29) The output phase of the segment, seg is estimated by integrating the latest frequency estimation (a phase accumulator performs the integration) and updated by adding the initial phase, obtained by linear regression. See Eq. 2-30: segupdate b a f dt (2-30) where a = 0 and b = 3.2 s in this research and fupdate is the frequency updated using the latest phase of the segment. As noted, the initial freque ncy is used for the phase determination of the first segment of data. The phase of the first se gment is stored in a phase accumulator and the phases for the subsequent segments are estimated and accumulated to determine the total phase of the measurement signal, meas Since frequency is the derivativ e of phase, it can be estimated

PAGE 34

34 by taking the finite difference between the late st two successive phase outputs divided by the time increment (3.2 s). This updated frequency, fupdate, is used in the linear regression for the next segment. The total phase for the reference signal, ref can be determined using the same process. The phase difference, measref between the two phases is finally calculated and converted into displacement in the electronics using Eq. 2-25. Error Sources in Heterodyne Interferometry In this section, the errors in a heterodyne system are reviewed. Each error source is described briefly and compensations or correctio ns for the errors are discussed. Errors in heterodyne interferometry can be divided into three primary categories: environmental errors, geometry errors, and system errors, as shown in Figure 2-17 [14]. Atmospheric Error The refractive index of a medium is a functi on of its density, which depends, for air, on temperature, pressure, and composition. Since most displacement measuring interferometers operate in air, it is necessary to identify the local refractive index. A ny changes in the air's refractive index due to changes in the atmosphe ric conditions during a measurement introduce an apparent displacement which degrades the m easurement accuracy. Atmospheric error can be expressed as shown in Eq. 2-31, where n is the refractive index variation and PD is the physical displacement of the moving target. The error cause d by the refractive index variation is often the largest component in the e rror budget [14]. Therefore, it must be compensated. Atmospheric Error = nPD (2-31) The refractive index can be estimated by a dire ct measurement of index using a wavelength tracker, which measures changes in the air's refrac tive index, or by using an empirical expression.

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35 For the latter, the air pressure, temperature and relative humidity, are measured and these values are substituted into an appropria te equation [15] to approximately determine the refractive index during the measurement. Material Thermal Expansion Error Since the mechanical parts dimensions in th e interferometer are a function of temperature (due to thermal expansion or cont raction), it is required to correct for this dimensional variation. This requires that the temperature of the part and its coefficient of linear thermal expansion are known. Optics Thermal Drift Temperature changes of the optical compone nts during the measurement can also cause measurement error. When a change in temperature occurs, the physical size of the optical elements will vary which causes an apparent di splacement. Optical thermal drift can be reduced by either controlling the temperature of the m easurement environment or by using an optical configuration that is insensitive to temperature changes. Deadpath Error Deadpath error is caused by an uncompensated change in refractive index combined with a difference in length between the fixed and moving paths at initialization. A conventional linear interferometer with unequal path lengths is shown in Figure 2-18A. The measurement beam,2f, has a longer path length than the reference beam,1f, by the deadpath length, DP Even in the absence of moving target motion, an apparent displacement is caused by any uncompensated refractive inde x change over the deadpath, DP Deadpath error can be expressed as shown in Eq. 2-32. This leads to a shift in the initialized position and, therefore, causes

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36 measurement error. In most applications, dea dpath errors can be minimized by reducing the deadpath length, as shown in Figure 2-18B. The inte rferometer is located at its initial position. Deadpath Error = nDP (2-32) Abbe Error Abbe error occurs when the measurement axis of interest is offset from the actual measurement axis and angular errors exist in the positioning system. Abbe error can be expressed as shown in Eq. 2-33, Abbe error = offsettand (2-33) where doffset is the offset distance and is the angular change during motion. An example of Abbe error is shown in Figure 2-19. The measurement axis is offset by doffset from the displacement axis and angular motion of the targ et generates Abbe erro r. To eliminate Abbe error, the axis of measurement must pass through the point of interest. Cosine Error Angular misalignment between the measurement ax is (laser beam axis) and axis of motion results in an error. It is called cosine error because its magnitude is proport ional to the cosine of the angle of misalignment [16]. Figure 2-20 illustrates cosine error with an angle between direction of motion and the beam axis. The measured displacement, lm, is always less than the actual displacement, l See Eq. 2-34. Cosine error can be minimized by aligning the laser beam parallel to the axis of mo tion as closely as possible. Cosine error 1cosmlll (2-34) Laser Wavelength Stability The laser source of any interferometer system has some type of frequency stabilization to maintain its wavelength accuracy. As discussed previously, interferometer systems generate

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37 fringes when displacement occurs. Each fringe is equivalent to a fraction of a wavelength of the laser. If the wavelength changes, the result is an apparent displacement. This apparent movement appears as a measurement error. Electronics Error In a heterodyne displacement measuring interferometer, the phase change between measurement and reference signals is measured. The optical measurement resolution, one fringe ( for the single pass configuration) can be electrically or optic ally extended. The electronics measurement resolution is base d on how many points represent one complete period of phase. For example, if 2radians of phase are subdivided by 1024 parts, the resolution can be extended up to20482(1024) for the single pass configuration. The electronics error can be taken to be equal to the electronics measurement resolution. However, amplifier nonlinearity may also to be considered [12]. Periodic Error Imperfect separation of the two light frequenc ies into the moving and fixed paths has been shown to produce first and second order periodic error, or errors of one and two cycles per wavelength of optical path change, respectively. In other words, during the target motion, the measured displacement oscillates cyclically about the true displacement, typically with amplitude of several nanometers [1]. Sources of frequency leakage between paths include non-orthogonality between the linear beam polarizations, elliptical polarization of the individual beams, imperfect optical components/coatings, parasitic reflections from individual surfaces, and/or mechanical misalignment between the interferometer (laser, pol arization dependent optic s, and targets) [10]. Figure 2-21A shows a measured displacement (s olid line) where periodic nonlinearity is

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38 superimposed on the true value of displacement (dashed line). Figure 2-21B isolates periodic error by removing the least squares fit line from the constant velocity displacement data. Periodic error is described in more detail in Chapter 3. An Example of Application As noted, heterodyne displacement measuring in terferometry has been widely used in applications that require high precision displacement. Those applications include position feedback for integrated circuit (I.C.) wafer steppers in the se miconductor fabrication industry, precision cutting machines, coordinate measuring machines, and calibration of transducers. In this section, the interferometer system applied fo r the position feedback of an I.C. wafer stepper stage, one of the most accuracy demanding appl ication, is described using a commercially available interferometer system. The two ma jor suppliers for a displacement measuring interferometer system are Agilent Technologies and Zygo. In this example, an Agilent 5527A laser system is described and measurement errors are provided [14]. Figure 2-22 shows the typical conf iguration for the I.C. wafer stage application. A two axis (X and Y), double pass heterodyne displacement measur ing interferometer is used to measure the position of the wafer stage and provide position f eedback to a controller. A wavelength tracker measures changes in the air's refractive index which is used to compensate atmospheric and deadpath errors (which occur due to environmental changes). The interferometer system accuracy is determ ined using specifications provided in [14]. The list of parameters is summarized below. Maximum displacement measured (L): 0.2 m Environment: o Temperature: 20C 0.1C(temperature-controlled environment) o Pressure: 760 mmHg 25 mmHg (no pressure control) o Humidity: 50 % 10 % (humid ity-controlled environment)

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39 Deadpath length: 0.1 m Abbe error: none (assume zero offset) Cosine error: 0.05 ppm (worst case) Laser wavelength stability: o Short-term (< an hour): 0.002 ppm o Long-term (> an hour): 0.02 ppm Electronics error (measuremen t resolution): 0.15 nm Periodic error: 2.2 nm Note that the error components are divided into proportional and fixed terms. Proportional error terms are generally specified in parts-per-million (ppm) and these errors are a function of the target motion. Fixed error terms are noncumulativ e and do not depend on th e target displacement. Units of fixed terms are given in length, such as meters and nanometers. Each error component is calculated individually and then the errors are combined to determine the system accuracy. Atmospheric error depends on the air's refractive index variation due to changes temperature, pressure, humidity, and chemical composition. The wavelength tracker measures the air's refractive index varia tion and gives compensation information. The performance of the wavelength tracker is given in Eq. 235 for the compensation accuracy [14]. Compensation accuracy = [0.067 ppm + (0.06 ppm/C T) + (0.002 ppm/mm Hg P)] (2-35) In Eq. 2-35, T is the temperature variation about 20C and P is the pressure variation about 760 mm Hg. Using Eq. 2-35 and the environmenta l conditions, the compensation accuracy is 0.14 ppm. At the maximum displacement, the atmo spheric error due to the wavelength tracker compensation is calculated as 28 nm ( 0.14 10-6 0.2 m) using Eq. 2-31 where the compensated air's refractive i ndex variation is used for n. The material thermal expansion error

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40 is assumed to be zero and the optics ther mal drift is given as 4 nm (40 nm/C (.1C)) for the plane mirror used in this application. Geometric errors (deadpath, Abbe, and cosine error) are calculated next. The deadpath error is a function of deadpath length, method of compensation, and environmental conditions. Using the wavelength tracker compensation info rmation and Eq. 2-32, the corrected deadpath error can be calculated as 14 nm (( 0.14 10-6) 0.1 m). In the I.C. wafer stage application, it is usually possible to have the beam axis in li ne (collinear) with the axis of the stage motion. Therefore, the Abbe offset is zer o and there exists no A bbe error. The cosine error is assumed to be 0.05 ppm which is -10 nm ((-0.05 10-6) 0.2 m) at the maximum displacement in the worst case. The negative sign indicates that the measur ed displacement is always smaller than true displacement due to cosine error. Laser wavelength stability errors for short-te rm and long-term measurements are given as 0.4 nm ( 0.002 10-6 0.2 m) and 4 nm ( 0.02 10-6 0.2 m), respectively, at maximum displacement. For the wafer ste pper application, the process ti me for the wafer exposures is typically short (< 2 min) [14]. Therefore, the short-term error is used in this example. The electronics error is the measurem ent resolution for the system (w ithout considering the amplifier nonlinearity). The measurement resolution in th e phase measuring electronics used in this application is 0.15 nm. Finally, period ic error is measured to be 2.2 nm. Table 2-1 lists the error contributors associated with the I.C. wafer stepper application including and excluding atmospheric compensation. The error levels when the measurement occurs in a vacuum chamber are also included. Although measurement in a vacuum environment is costly, it is added to show how much the system accuracy can be improved and what errors still limit the system performance. In vacuum, th e errors related to environmental changes are

PAGE 41

41 eliminated and only cosine e rror and system errors remain. Moreover, if the measured displacement is small (i.e. nanometer or micromet er displacement which can be realized using piezo stage positioners), cosine error and laser wavelength stability error become negligible compared to electronics error (measurement resolution). For example, if the maximum displacement of a target is 100 m, the cosine error (0.05 ppm) a nd the laser wavelength stability error ( 0.002 ppm) become 0.005 nm a nd 0.0002 nm at the maximum displacement. However, periodic error is unchanged because it is a fixed error term. Therefore, periodic error limits to achieve the maximum system accuracy in th is situation. This provides the motivation to eliminate periodic error in heterodyne di splacement measuring interferometers. Table 2-1. Heterodyne displacement measuring interferometer system accuracy for the integrated circuit wafer stage application. Error With atmospheric compensation Without atmospheric compensation In vacuum (nm) (nm) (nm) Atmospheric error 28 1800 0 Material thermal expansion error 0 0 0 Optics thermal drift error 4 4 0 Deadpath error 14 900 0 Abbe error 0 0 0 Cosine error -10* -10 -10** Laser wavelength stability error 0.4 0.4 0.4** Electronics error 0.15 0.15 0.15 Periodic error 2.2 2.2 2.2 The measured displacement is always smaller than true displacement for this single-sided error. ** At the maximum displacement of 100 m, cosine error (0.05 ppm) becomes 0.005 nm and the laser wavelength st ability error ( 0.002 ppm) becomes 0.0002 nm.

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42 Figure 2-1. The electric (E) a nd magnetic (B) waves are orthogonal to each other and propagate in the same direction. A linea rly polarized beam is shown. E0B0z E B

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43 Figure 2-2. An example of constructive interference. A) Two waves with the same magnitude and phase (in phase). B) Constructive inte rference leads to a wave with double the magnitude. Figure 2-3. An example of destructive interference. A) Two wave s with the same magnitude but 180 deg phase difference (out of phase). B) Destructive interference leads to a wave with zero magnitude. -2 -1 0 1 2 -2 -1 0 1 2 -2 -1 0 1 2 E1 E2 E A Bt t -2 -1 0 1 2 -2 -1 0 1 2 -2 -1 0 1 2 E1 E2 E A Bt t

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44 Figure 2-4. Schematic of th e Michelson interferometer. 1 3 2 4 Extended source M2 M1 Beam splitter Compensator S Screen

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45 Figure 2-5. The Michelson interferomet er with modern technology components. Displacement Photodetector Laser head Fixed mirror Moving mirror Digital electronics

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46 Figure 2-6. Schematic of a single pass interferometer. Photodetector Fixed retroreflector Moving retroreflector PBS H V H+ V H LP d Vertically polarized beam Horizontally polarized beam Laser head

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47 Figure 2-7. Interference according to the target displacement in a single pass configuration. A) Constructive interference at the no target motion (d = 0). Bright light is detected. B) Destructive interference at the target displacement of d=/4 where =. No light is detected C) Constructive interferen ce at the target displacement of d=/2 where =2. Bright light is detected. D) A bright-dark-bright transition, one fringe or the optical resolution, corresponds to a half wave length of the light source in the single pass configuration. -2 -1 0 1 2 -2 -1 0 1 2 -2 -1 0 1 2 A Eref Emeas at d = 0 Eresulan t t + -2 -1 0 1 2 -2 -1 0 1 2 -2 -1 0 1 2 t + -2 -1 0 1 2 -2 -1 0 1 2 -2 -1 0 1 2 at d = /4 = at d = /2 = 2 t ErefEmeasEresulan t ErefEmeas Eresulan t B C Detecto r Fringe bright dark bright d = /2 D + ===

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48 Figure 2-8. Schematic of a double pass interferometer. Fixed p lane mirro r Movin g p lane mirro r PBS H V V + (H+ ) LP d Retroreflecto r Q WP Laser hea d

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49 Figure 2-9. Interference according to the target displacement in a double pass configuration. A) Constructive interference at the no target motion (d = 0). Bright light is detected. B) Destructive interference at the target displacement of d=/8 where =. No light is detected. C) Constructive interferen ce at the target displacement of d=/4 where =2. Bright light is detected. D) A bright-dark-bright transition, one fringe or the optical resolution, corresponds to a quarte r wavelength of the light source in the double pass configuration. -2 -1 0 1 2 -2 -1 0 1 2 -2 -1 0 1 2 A Eref Emeas at d = 0 Eresulan t t + -2 -1 0 1 2 -2 -1 0 1 2 -2 -1 0 1 2 t + -2 -1 0 1 2 -2 -1 0 1 2 -2 -1 0 1 2 at d = /8 = at d = /4 = 2 t ErefEmeasEresulan t ErefEmeas Eresulan t B C Detecto r bright dark bright + Fringe d= /4 D = = =

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50 Figure 2-10. Schematic of a homodyne di splacement measuri ng interferometer. Figure 2-11. Schematic of a single pass heterody ne displacement meas uring interferometer. Photodetector Fixed retroreflector Moving retroreflector PBS f1 f2 f2 (E2) f1 (E1) f2 fd (f2 f1) fd Two-frequency laser head Vertically polarized beam Horizontally polarized beam path 2 path 1 LP Photodetector Moving retroreflector Beam splitter Single-frequency laser head Fixed retroreflector

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51 Figure 2-12. Spectrum analyzer outputs from two interferometers. A) A simple setup for homodyne and heterodyne systems sh owing the measurement signal output. B) Homodyne interferometer spectrum. C) Heterodyne interferometer spectrum. Fiber optic pick-up Fixed retroreflector Moving retroreflector PBS Spectrum Analyzer Homodyne Heterodyne 0 fd Hz f f+fd Hz f-fd A B C

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52 Figure 2-13. Schematic of heterodyne in terferometer where displacement is determined by comparing the measurement and reference signals. Fiber optic pick-up Fixed retroreflector Moving retroreflector PBS He-Ne laser Phase measuring electronics Displacement (1) (2)

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53 Figure 2-14. Phase difference between the measurement and reference signal. Measurement signal ( f fd) Reference signal ( f ) 0 0

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54 Figure 2-15. Flow chart for phase measurement. 1 2vacd FFn measseg measref = Phase measurement for reference signal ADC ( 12 bit 80 MHz ) Modula 256 counte r Photodetecto r Linear regression Phase accumulato r Frequency estimation Channel 1 Optical interference signal (analog) Electrical signal (analog) Digitized signal sampled at 80 MHz Phase slows to 312.5 kHz Output and V0 Phase (seg ) estimation and update by Update frequency (fupdate) fupdate ref seg Channel 2 Initial frequency estimation ( f meas )

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55 Figure 2-16. Flow chart for in itial frequency estimation. Capture digital signal N samples Perform DFT on captured samples Identify cell, k with largest transform modulus Initial frequency f si g nal= k (sampling rate)/N

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56 Figure 2-17. Error sources degrading the measurement accuracy in heterodyne interferometry. Error sources Environmental errors Geometry errors System errors Atmospheric error Material thermal expansion error Optics thermal drift Deadpath error Abbe error Cosine error Laser wavelength stability Electronics error Periodic error

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57 Figure 2-18. An example of deadpath error. A) Unequal path length between the fixed and moving arms gives the deadpa th length. B) Deadpath error can be minimized by locating the interferometer at the initial position of the measurement. DP=0 f1 f2 L Photodetector Fixed retroreflector Moving retroreflector PBS Deadpath length D P f1 f2 Initial position Laser A B

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58 Figure 2-19. An example of Abbe error. Figure 2-20. An example of cosine error. l lm Motion axis doffset Motionaxis l doffset Abbe error Measurement axis Beam axis

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59 Figure 2-21. Visual description of periodic error. A) The constant ve locity displacement, x, is not linear. B) First and second or der periodic error is reveal ed after removing the least squares best fit line, fit. A B tx (displacement) 0 tx-fit (periodic error)

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60 Figure 2-22. A heterodyne displacement measuring interferomet er system configuration for position feedback in application of an integrated circuit wafer stepper. RR: retroreflector. QWP: quarter wave plate. PBS: polarizing beam splitter. Plane mirror X-Y stage Wafer Laser head Beam splitter Receivers Wavelength tracker QWP PBS RR Plane mirror X Y

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61 CHAPTER 3 PERIODIC ERROR REVIEW One funda mental accuracy limitation for the commonly selected heterodyne (or two frequency) Michelson-type interf erometer is periodic error. This non-cumulative error is inherent to the typical polarization codi ng approach, where the two (heter odyne) optical frequencies are carried on coincident/collinear, linearly polar ized, mutually orthogonal laser beams and separated/recombined using polarization de pendent optics. It is caused by frequency mixing/leakage between the fixed and moving paths and has been extensively explored in the literature. Periodic Error Literature Review Many studies of periodic error have been re ported. Eighty-one journal publications with direct relevance to periodic error in displacem ent measuring interferometry are identified here1. They are subdivided into the following eight cate gories (see Table 3-1): 1) early work [18-21] these papers describe early investigations of the ex istence of periodic error; 2) discussion of error sources [11, 16, 17, 22-25] publications that pr ovide descriptions of common error sources in displacement measuring interferomet ry; 3) index of refraction [15, 26-31] papers that show the calculation/measurement of the refractive index of air, which is critical for measurements not completed in vacuum or another gaseous environm ent, such as helium due to its reduced index sensitivity to environmental cond itions; 4) periodic error description and modeling [5, 12, 32-48] these articles are focused on describing and mode ling the physical sources of periodic error for various interferometer configura tions; 5) error measurement methods [2, 49-55] papers that Reprinted with permission from Kim H, Schmitz T, Beckwith J. Periodic error in heterodyne interferometry: Examination and Elimination. In: Halsey D, Raynor W, ed. Handbook of Interferometers: Research, Technology and Applications. Hauppauge, NY: Nova Science; 2009. 1 One conference proceedings [58] was included because it is the first reference that describes frequency domain evaluation of periodic error, which is applied in this research.

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62 detail the measurement of periodic error and related parameters; 6) periodic error compensation and correction [9-10, 56-82] text s that describe variations in optical/electrical configurations and/or data processing algorithms that yield reduced peri odic error levels; 7) uncertainty evaluation [83-84] descriptions of uncertainty analyses for he terodyne interferometry; and 8) measurement applications [85-89] examples of displacement measuring interferometry applied in practice with considerat ion of periodic error. Table 3-1. Summary of period ic error literature review. Topic References Early work 18-21 Discussion of error sources 11, 16, 17, 22-25 Index of refraction 15, 26-31 Periodic error description and modeling 5, 12, 32-48 Error measurement methods 2, 49-55 Periodic error compensation and correction 9, 10, 56-82 Uncertainty evaluation 83-84 Measurement applications 85-89 Frequency-Path Model As briefly described in Chapter 2, in heter odyne Michelson-type inte rferometers that rely on polarization coding, imperfect separation of the two light frequencies into the moving and fixed paths has been shown to produce first and second order periodic errors. Unwanted frequency leakage may occur due to a numbe r of influences, including non-orthogonality between the ideally linear beam polarizations, e lliptical polarization of the individual beams, imperfect optical components, parasitic reflect ions from individual optical surfaces, and/or mechanical misalignment between the interferometer elements (laser, polarizing optics, and targets). In a perfect system, a single frequency travels to a fixed target while a second, single frequency travels to a moving target. Interference of the combined signals yields a perfectly sinusoidal trace with phase that va ries, relative to a reference phase signal, in response to motion of the moving target. However, the inherent frequency leakage in actual implementations

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63 produces an interference signal which is not purel y sinusoidal (i.e., contai ns unintended spectral content) and leads to periodic error in the measur ed displacement. The ideal case is depicted in Figure 3-1A, where f1 and f2 represent the two frequencies and fd indicates the Doppler frequency shift due to motion of the moving target (the right retroreflector in the schematics). Note that/dv a cfFFnv, Eq. 2-22, where FF is the fold factor (equal to 2 for the single pass heterodyne system in Figure 3-1), vac is the vacuum wavelength, and v is the measurement target velocity. In Figure 3-1B, the frequency l eakage is indicated by the dashed lines (leakage) superimposed on the solid (intended) paths. Schmitz and Beckwith [12] presented the Frequency-Path model that predicts the number of interference terms that may be expected at the photodetector output. The F-P model for a single pa ss heterodyne interferometer is described in the following paragraphs for both the ideal ( no frequency leakage) a nd non-ideal (frequency leakage) cases. In the ideal case, there are two source frequencies, f1 and f2, and only one possible path for each frequency from the source to the detector. This yields two FP elements; one FP element represents the propa gation of the first frequency along its in tended path to the fixed target, while the other FP element represents the propagation of the second frequency along its intended path to the moving target. In the model notation, th e electric field amplitude of each element is designated by the letter E appended with two subscripts, Eij. The first subscript denotes the source frequency, while the second gives the path from the source to detector. For example, E11, displayed as a line in Figure 3-2A, gives the electric field amplitude of frequency 1 that reaches the detector via path 1. For the ideal representa tion shown in Figure 3-2A, the two FP elements give a total of (m)(m + 1)/2 = (2)(2 + 1)/2 = 3 interference terms: two optical power terms (due to the squaring action of the photodetector ), which occur at zero frequency (dc), and the desired

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64 ac interference term, which appears at the beat frequency (f = f2 f1) with no motion and is Doppler shifted up or down in frequency duri ng motion of the moving target (depending on the direction). In the non-ideal case displayed in Figure 32B, there are two sour ce frequencies and two possible paths for each frequency from the source to detector (assuming multiple extraneous reflections are neglected) due to imperfect fre quency separation at the polarizing beam splitter. In other words, both frequencies can travel to eith er the fixed or moving ta rget in a fully leaking system. This gives four F-P elements and (4 )(4+1)/2 = 10 distinct interference terms. The following paragraphs describe the origin s of the 10 interference terms in a fully leaking interferometer. The fixed path, whic h ideally contains light of only frequency f1 (expressed in Hz), propagates two si gnals due to frequency leakage, 1111111cosEtkFFx (3-1) 2122121cos EtkFFx where ij are the initial phases of the signals Eij, x1 represents motion of the reference target, k1 and k2 are the propagation c onstants equal to 2/1 and 2/2, respectively, and 1,2 = 2 f1,2 (rad/s). The two wavelengths 1 and 2 correspond to the two heter odyne frequencies. Recall that in the Eij notation, the first subscript denotes frequency, while the s econd indicates the path (1 for fixed and 2 for moving) [12]. Similarly, the moving path ideally composed of only f2 light, also contains two signals, 2222 222cosEtkFFx (3-2) 1211 212cosEtkFFx

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65 where the parameter definitions are analogous. Th e photodetector current is obtained by squaring the sum of the four Eij terms (the two intended signals ha ve equal subscripts while the two leakage induced signals have une qual subscripts). See Eq. 3-3. 2 11111112122121 2 22222221211212cos cos cos cos IEEtkFFxEtkFFx EtkFFxEtkFFx (3-3) 2 11 11111 2 21 22121 2 22 22222 2 12 11212 =cos2221 2 cos2221 2 cos2221 2 cos2221 2 E tkFFx E tkFFx E tkFFx E tkFFx 2122112211 2211 22112 21 1 2112212112 2112 12212 11 2 2 2212cos cos cos cos cos ttkFFxkFFx EE tkFFxkFFx ttkFFxkFFx EE tkFFxkFFx t EE 122122212 2 21 22212 2121112111 2111 21112111 1121111 1112cos cos cos cos2 tkFFxkFFx tkFFxkFFx ttkFFxkFFx EE tkFFxkFFx tkFFxkFFx EE 12 12111112 222212122 2122 22212122cos cos2 coskFFxkFFx tkFFxkFFx EE kFFxkFFx Several simplifications may be applied to Eq. 3-3. First, x1 is ideally zero because there is no motion of the fixed target. Second, for a re latively small split frequency between the two heterodyne signals, 12 (on the order of MHz for commercial systems), the

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66 propagation constants, k1 and k2, are nearly equal and a single value, k, may be substituted for each. Third, due to detector bandwidth limitations terms that oscillate at twice the optical frequency (i.e., 21t, 22t, or 1t+2t) may be neglected. See Eq. 3-4. 22222 12 2 22 2 21 2 11 2EEEE EI optical power (3-4) 2211 22211cos EEtkFFx ac interference (intended) 2112 22112cos EEtkFFx ac interference (leakage induced) 1222 1222cos tEE ac reference (due to measurement path) 1121 1121cos tEE ac reference (due to reference path) 1112 21112cos EEkFFx dc interference (frequency 1) 2122 22122cos EEkFFx dc interference (frequency 2) Equation 3-4 lists all 10 interference terms in a fully leaking two fre quency interferometer. These 10 terms are summarized in Table 3-2. The intended ac interference term is defined by the interference of E22 and E11. It represents the signal of choice in heterodyne systems. Under constant velocity target motion, this term appears at a frequency2 of f fd in the spectrum analyzer display. A second ac interference term is obtained due to interference between the leakage terms E21 and E12. This term represents second orde r periodic error and includes a Doppler phase shift, k(FF)x2, of equal value but opposite si gn relative to the intended ac interference term. Therefore, at constant velocity this term is seen at a frequency of f + fd. The ac reference terms represent first order periodic error and occur due to interference between the intended and leakage terms of differ ent frequencies that exist in ei ther the fixed or moving paths. They appear at the split frequency f. 2 The intended ac interference signal may also be up-shifted depending on the target motion direction. In this case, the leakage induced ac interference will be do wnshifted.

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67 Two dc interference terms also exist because f1 and f2 appear in both the moving and fixed paths. For a single frequency, or homodyne system, the corresponding dc interference term is the selected measurement signal (e.g., E11E12 for frequency f1). These terms exhibit a positive Doppler phase shift of equal value to the ac interference terms regardless of target motion direction and exist at zero fre quency when the measurement target is at rest. Finally, the optical power terms contribute a zero frequency offset to the photodetector current regardless of optical path changes. Due to their frequency dependence, a spectru m analyzer provides a powerful tool for visualizing the contributions of the various te rms [2, 55]. Figures 3-3 through 3-6 depict each interference term and their appear ance in a spectrum analyzer di splay. A representation of a typical frequency spectrum for the fully leaki ng interferometer is provided in Figure 3-7. Table 3-2. Interference terms in fully leak ing single pass heter odyne interferometer. Type Active F-P elements optical power E11E11 optical power E21E21 optical power E22E22 optical power E12E12 ac interference (intended) E22E11 ac interference (leakage induced) E21E12 ac reference (path 2) E22E12 ac reference (path 1) E21E11 dc interference (frequency 1) E11E12 dc interference (frequency 2) E21E22

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68 Figure 3-1. Schematics for ideal and fully freque ncy leaking interferometer. A) Ideal heterodyne interferometer behavior. B) The leakage terms are shown in parentheses. The two frequencies, f1 and f2, are ideally linearly polarized and orthogonal. This enables the polarizing beam splitter to separate them based on their polarization states. A f1 f2 fd f2 f1, f2 Polarizing beam splitter Fixed retroreflector Moving retroreflector Path 1 Path 2 f2 (+f1) f1 (+f2) f2 fd f1 B f2 (+f1) fd f2(+f1) fd, f1 (+f2)

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69 Figure 3-2. Frequency-path models. A) For ideal behavior. B) For fully leaking case. Source Paths Detector f1 f2 1 2 E22 E11 Source Paths Detector f1 f2 1 2 E22 E11 E12 E21 A B

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70 Figure 3-3. Intended ac interference signal. A) Interference between ideally separated beams (no frequency leakage). B) Power spectrum for the ac interference term. f1, f2 f2f1 E11+E22 Path 1 Path 2 0 f fd f Log power ac interference intended A B

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71 Figure 3-4. Leakage induced ac interference. A) Interference between leaking frequencies. B) Power spectrum for the leakage induced ac interference term. f1, f2 f1f2 E12+E21 Path 1 Path 2 0 f + fd f Log power ac interference leakage induced (second order periodic error) A B

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72 Figure 3-5. Description of ac reference terms. A) Interference be tween intended and leaking frequencies on the same path. B) Power spectrum for ac reference terms. f1, f2 f1 f2 E11+E21 Path 1 Path 2 f1, f2 E22+E12 f2 f1 0 Log power ac reference (first order periodic error) A B f

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73 Figure 3-6. Description of dc interference terms. A) Interference between the same frequency in both paths. B) Power spectrum for the dc interference terms. f1, f2 f1f1 E11+E12 Path 1 Path 2 f1, f2 f2f2 E22+E21 0 fd f Log power dc interferenceA B

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74 Figure 3-7. Power spectrum for constant ve locity motion in fully leaking heterodyne interferometer. 0 Log poweroptical power dc interference ac interference leakage induced ac interference intended ac reference f fd f fdf + fd

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75 CHAPTER 4 POLARIZATION CODED INTERFEROMETER As described in the previous chapter, a polarization coded heterodyne interferom eter system is a common choice for measuring displaceme nt. In this chapter, a setup for a traditional single pass heterodyne interferometer and periodic error measurement results are presented. The setup is specifically designed to isolate peri odic error by minimizing th e other error sources described in Chapter 2. It can also vary the magn itude of periodic error by using a phase retarder (half wave plate) which rotates the orientation of linear polarized light. Test results from both phase measurement (for displacem ent calculation) and spectrum analyzer data are provided. Time domain periodic error is obtained using th e measured displacement. It is then converted into the frequency domain (using the Fourier transform) to identify the first and second error orders. Periodic error magnitudes from the sp ectrum analyzer, the displacement data, and a model developed by Cosijns et al. [5] are compared for various setup conditions. Setup Description A photograph and schematic of a traditio nal heterodyne displacement measuring interferometer setup are provided in Figure 4-1. The orthogonal, li nearly polarized beams with a frequency difference of approximately 3.65 MH z (Helium-Neon laser source with a Zeeman split) first pass through a ha lf wave plate. Rotation of the half wave plate enables variation in the apparent angular alignment (about the beam axis) between the polarization axes and polarizing beam splitter; deviations in this alignment lead to frequency mixing in the interferometer. The light is then incident on a non-pol arizing beam splitter (80% transmission) that directs a portion of the beam to a fiber optic pickup after passing through a fixed angle sheet polarizer (oriented at Reprinted with permission from Kim H, Schmitz T, Beckwith J. Periodic error in heterodyne interferometry: Examination and Elimination. In: Halsey D, Raynor W, ed. Handbook of Interferometers: Research, Technology and Applications. Hauppauge, NY: Nova Science; 2009.

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76 45 deg to the nominal laser orthogonal polarizati ons). The pickup is mounted on a two rotational degree-of-freedom flexure which enables efficient coupling of the light into the multi-mode fiber optic. This signal is used as the phase reference in the measurement electronics. The remainder of the light continues to the polarizing beam splitter where it is (ideally) separated into its two frequenc y components that travel separately to the moving and fixed retroreflectors. Motion of the moving retroreflect or is achieved using an air bearing stage. After the beams are recombined in the polarizing beam splitter, they are di rected by a 90 deg prism through a polarizer (or analyzer) with a variable rotation angle. Finall y, the light is launched into a fiber optic pickup. This serves as the measurem ent signal in the measur ement electronics (0.3 nm resolution for the single pass configuration). The intent of this setup was to minimize other well-known error contributors and set various first and second order peri odic error magnitudes. To isolat e periodic error, the setup was constructed with zero dead path difference (i .e., the distance between the polarization beam splitter and the moving retroreflector was equal to the distance between the polarization beam splitter and the fixed retroreflector at initia lization) and small Abbe offset (25 mm). The measurement time (~100 ms) and motion excurs ions were kept small to minimize the contribution of air refractive index variations due to the environmental changes. Additionally, careful alignment of the ai r bearing stage axis with the optical axis resulted in small beam shear. Traditional Setup Experimental Results In this section, results obtained from the traditional polarization coded design are presented. These results, which exhibit various pe riodic error levels, were collected using both a spectrum analyzer and typical phase measuring electronics. The experimental data is also compared to predictions from the model developed by Cosijns et al. [5].

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77 Displacement Data First and second order periodic error magnitudes can be extracted from the displacement data obtained from the interference signal. Disp lacement was determined using standard phase measuring electronics (312.5 kHz sampling frequency) during constant velocity motion of the moving target retroreflector. Figure 4-2A shows the displace ment data for a misaligned system (5 deg difference from the nominal orientation fo r both the half wave plate and linear polarizer). In Figure 4-2B the least squares best fit line is re moved to reveal the periodic error. To identify the first and second order error magnitudes, the discrete Fourie r transform of the error was computed and the spatial frequency axis was norma lized to the periodic error order; see Figure 42C. For the selected misalignments, both first an d second order periodic e rror is present. Figure 4-3 shows the displacement obtained for nominal a ngular alignment of both the linear polarizer and half wave plate. The second order error domin ates in the well aligned system, which agrees with the spectrum analyzer data shown later in Figure 4-6. Spectrum Analyzer Data Figure 4-4 shows a schematic diagram of an analog spectrum analyzer setup to collect measurement signals from the interferometer. The measurement signal was carried via the multimode optical fiber to a battery operated photodet ector (to minimize noise), which converted the optical signal to an electrical signal. The electrical signal was carried to the spectrum analyzer (Model: HP1850A) by a (shielded) BNC cable and the corresponding spectrum was recorded. The spectrum analyzer consists of three modules: 1) display section to control the signal display, 2) RF section to control freque ncy, and 3) IF section to cont rol magnitude. Figure 4-5 shows the frequency content from the fully leaking interferom eter pictured in Figure 4-1. The spectrum in Figure 4-5 was measured with a 5 deg misalig nment of the half wave plate and 10 deg misalignment of the linear polarizer from their nominal orientations. The velocity, v, of the

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78 moving target retroreflector was 5,000 mm/min. The constant velocity motion of the target produces a Doppler frequency shift wh ich is proportional to the velocity, 2d vacnv f for the single pass setup where n 1 is the air refractive index and vac = 633 nm is the laser source vacuum wavelength. In this case, the Doppler freque ncy is positive when the stage moves in the + x direction (as identified Fi gure 4-1A) and its magnitude is 0.26 MHz. The intended ac interference signal is up-shifted by the Doppler frequency, f + fd. The ac reference (first order periodic error) and leakage induced ac interference (second order periodic error) signals appear at the beat frequency, f = 3.65 MHz, and the beat frequency downshifted by the Doppler frequency, f fd, respectively. Figure 4-6 demonstrates the spec tral content for a well aligned system. In this case, the second order periodic error dominates, which ag rees with the result in [2]. The Doppler frequency shift is now 0.53 MHz ( v = 10,000 mm/min). Figure 4-7 shows that the periodic error magnitudes are independent of target velocity. Measurements were performed at {5,000, 10,000, and 15,000} mm/min. The interferometer configur ation was not changed: the misalignments were 10 deg (from nominal) for both the half wa ve plate and linear polar izer. Although the peak magnitudes are constant, the Doppler frequency shift changes with velocity, as expected. Comparison between Measurements and Model Several tests were completed to compare the periodic error magnitudes obtained from the spectrum analyzer and phase measuring electronic s (position data) for various frequency mixing levels. Periodic error calculation from the spectrum analyzer is described in Chapter 5. Data were collected for variation of the ha lf wave plate orientation of 10 deg from nominal (fast axis vertical in Figure 4-1) and linear polarizer variation of 17 deg from the nominal angle (45 deg

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79 from vertical). Note that a 10 deg rotation of th e half wave plate yields a 20 deg rotation of the polarization vector. The results ar e also compared with the mean periodic error values from a Monte Carlo evaluation [84] of the Cosijns et al model [5]. The parameters used to calculate the periodic error in this model is described in detail in Chapter 6 when analyzing the uncertainty of periodic error. Figures 4-8 through 4-10 show the error magnitudes of first and second order periodic error resulting from multiple measurements. The variables and are the orientation of the linear polarization vectors with respect to the polarizing beam splitter axes and the angular deviation of the linear polarizer transmission axis from 45 deg, respectively. They can be varied by rotating the half wave plate ( variation) and linear polarizer ( variation). The discrete Fourier transform magnitudes of periodic error for the position data (phase measuring electronics) is shown in Figure 4-8, the spectrum analyzer data is provided in Figure 4-9, and the mean values from the Monte Carlo evaluation of the model for each combination is plotted in Figure 4-10. Good agreement is seen for both period ic error orders in all cases. The preceding analyses were completed to: 1) demonstrate the spectral content of displacement measuring interferometer position data that includes periodic error; and 2) show that spectrum analyzer evaluation of periodic error is comparable to both discrete Fourier transform calculations of error magnitudes from position data and an established model. The spectrum analyzer evaluation approach is used in the experimental validation of the new acoustooptic modulator-based displacement measuring interferometer (AOM DMI) design. The configuration and test re sults for the AOM DMI are presented in Chapter 7.

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80 Figure 4-1. Setup for a single pass heterodyne interf erometer. A) Photograph of single pass heterodyne interferometer experimental set up. B) Schematic of setup where the beam paths are shown. Air bearing stage Polarizing beam splitter Rotating polarizer Rotating half wave plate Laser head 90 deg prism Reference signal Non-polarizing Beam Splitter Measurement signal Rotating half wave plate Rotating polarizer 90 deg prism Moving retroreflector Polarizing beam splitter Non-polarizing b eam s p litte r Fixed retroreflector Fixed polarizer Laser head Measurement signal A B x (+) (-) Fixed retroreflector Moving retroreflector Reference signal

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81 Figure 4-2. Displacement data for misaligned sy stem. A) Measured displacement from typical phase measuring electronics. B) Periodic error revealed by su btracting the least squares best fit line from the measured displacement data. C) Magnitudes of the first and second order periodic erro rs in the frequency domain where the frequency axis is normalized by error order. 0 0.05 0.1 0.15 0 0.1 0.2 x (mm)t (sec) 0 1 2 3 4 5 x 10-3 -5 0 5 x-fit (nm)t (sec) 0 1 2 3 4 0 1 2 3 error ordererror (nm)A B C

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82 Figure 4-3. Displacement data for well aligned sy stem. A) Measured displacement from typical phase measuring electronics. B) Periodic er ror is revealed by s ubtracting the least squares best fit line from the measured di splacement data. C) Magnitudes of first and second order periodic error in the frequency domain wher e the frequency axis is normalized by error order. A B C 0 0.1 0.2 0.3 0.4 0.5 0 0.5 1 x (mm)t (sec) 0 1 2 3 4 5 x 10-3 -5 0 5 x-fit (nm)t (sec) 0 1 2 3 4 0 0.5 1 1.5 error ordererror (nm)

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83 Figure 4-4. Schematic of an analog spectrum analyzer setup to collect measurement signals from the interferometer. Measurement signal Rotating half wave plate Rotating polarizer Moving retroreflecto r Polarizing b eam s p litte r Fixed retroreflecto r Fixed polarizer Laser head Reference signal Optical fiber BNC cable Analog spectrum analyzer Battery operated photodetector RF Section IF Section Display Section

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84 Figure 4-5. Example of frequency content for fully leaking interferometer. The intended ac interference f + fd, ac reference (first order periodic error), f and leakage induced ac interference (second order periodic error), f fd, signals are observed during + x direction motion (see Fig. 4-1A) at 5,000 mm/min. Figure 4-6. Spectral content for nominal angular alignment of the half wave plate and linear polarizer. The target velo city is 10,000 mm/min. 2 2.5 3 3.65 4 4.5 5 -80 -60 -40 -20 0 frequency (MHz)power (dBm) f fd f f + fd 2 2.5 3 3.65 4 4.5 5 -80 -60 -40 -20 0 frequency (MHz)power (dBm) f fd f f + fd

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85 Figure 4-7. Periodic error frequency content at three different velocities. The magnitude is independent of target velocity. The data was collected with 10 deg half wave plate and linear polarizer angular misalignments. 2.5 3 3.5 4 4.5 5 -80 -60 -40 power (dBm) 2.5 3 3.5 4 4.5 5 -80 -60 -40 power (dBm) 2.5 3 3.5 4 4.5 5 -80 -60 -40 frequency (MHz)power (dBm)v = 5,000 mm/min v = 10,000 mm/min v = 15,000 mm/min

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86 Figure 4-8. Periodic errors for ha lf wave plate/linear polarizer parameter study. The errors were obtained from the discrete Fourier tran sform of position data. The variables and are the angles of the half wave plate and linear polarizer from their nominal orientations. A) First or der periodic error. B) Se cond order periodic error. -20 0 20 -20 0 20 0 5 10 15 20 (deg) (deg) error (nm) -20 0 20 -20 0 20 0 10 20 30 (deg) (deg) error (nm) A B

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87 Figure 4-9. Periodic errors for ha lf wave plate/linear polarizer parameter study. The errors were obtained from the spectrum analyzer data. A) First order periodic error. B) Second order periodic error. -20 0 20 -20 0 20 0 10 20 30 (deg) (deg) error (nm) -20 0 20 -20 0 20 0 5 10 15 20 (deg) (deg) error (nm) A B

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88 Figure 4-10. Periodic errors for ha lf wave plate/linear polarizer parameter study. The errors were obtained from mean values of the Cosijns et al. model Monte Carlo evaluation. A) First order periodic error. B) Second order periodic error. -20 0 20 -20 0 20 0 10 20 30 (deg) (deg) error (nm) -20 0 20 -20 0 20 0 5 10 15 20 (deg) (deg) error (nm) A B

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89 CHAPTER 5 PERIODIC ERROR CALCULATION FR OM S PECTRUM ANALYZER DATA This chapter describes the Monte Carlo evaluatio n of a single equation that can be used to determine periodic error magnitude s from spectrum analyzer data. In this approach, the optical interference signal is recorded during constant ve locity target motion using a spectrum analyzer and the magnitudes of spectral peaks are used to calculate periodic error magnitudes. This approach builds on prior work [2] by treating th e general case where both first and second order error components exist and arbitr ary initial phase values are considered. Significant experimental results are presented which verify the new approach. Error Calculation Phasor diagrams present a revealing graphical approach to analyzing periodic error. As described in the Chapter 3 (Frequency-Path mode l), the photodetector curr ent contains not only the desired ac interference term, 2211 22211cos EEtkFFx but also the leakage induced ac interference term, 2112 22112cos EEtkFFx and two ac reference terms, 1222 1222cos tEE and 1121 1121cos tEE. Because the frequency offset is the same (or nearly so) for the two ac reference terms, they cannot generally be individually distinguished in the spectrum analyzer display. Therefore, they are considered as a single term with identical frequency and phase in this analysis. These three terms (intended and leakage induced ac interference and ac reference signals), depicted in the Figure 3-7 spectrum in Chapter 3, may be described using three separate phasors in the complex plane. Reprinted with permission from Kim, H.S. and Schmitz, T., 2009, Periodic Error Calculation from Spectrum Analyzer Data, Precision Engineering, doi:10.1016/j.precisioneng.2009.06.001.

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90 First, consider the intended ac interference term. It can be described as the phasor 22211 000 0itkFFx iee where 0 is the magnitude (photode tector current units of Amperes), 2tkFFx (rad) is the nominal phase change due to the measurement target motion and 11220 (rad) is the (arbitrary) initial phase. This phasor rotates at f in the complex plane with no motion, due to the split frequency difference, and f fd depending on the direction while the moving target is in moti on. Alternately, the e xponential notation may be replaced by the rectangula r coordinate representation shown in Eq. 5-1, 0000000cos siniejk (5-1) which specifically identifies the real ( j axis) and imaginary (k axis) components. The moving target position is ideally determined from the instantaneous phase of 0 Under constant velocity conditions, for example, the instantaneous phase grows linearly with time, as does the target position. In practice, the phase m easuring electronics frequency shift this term back to zero for no motion, or near zero during targ et motion, by subtracting the reference (split) frequency. For this analysis, it is convenient to consider this frequency shifted condition so that the 0 phasor is rotating at fd for constant velocity motion; a counter-c lockwise rotation for the selected target direction is assumed here. Note that after the frequency translation, 2kFFx. See Figure. 5-1A. Similarly, the ac reference term can be expressed in rect angular coordinates as shown in Eq. 5-2. k j 11 111sin cos (5-2)

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91 The orientation of this phasor (see Figure 5-1B) does not vary with time (recall that the phasor after the frequency translat ion step is considered); its di rection is fixed by the arbitrary initial phase 1 which, in general, differs from 0. Finally, the leakage induced ac interference term can be expre ssed in rectangular coordinates as shown in Eq. 5-3. k j 2 2 2 22sin cos (5-3) This phasor (see Figure 5-1C) rotates in the clockwise direction (f or counter-clockwise 0 rotation) due to the opposite sign of the Doppler shift. (Again, the frequency translated version of the signals is considered for convenience of explanation.) Its ar bitrary initial phase 2 differs, in general, from both 0 and 1. Prior to determining the periodic error in the general case, consider the presence of only 0 and 1 and then only 0 and 2 individually. It is assumed that the initial phases are zero for now to enable direct comparison to reference [2]. Figure 5-2 depicts the vector sum of the intended ac interference and ac reference phasors (0 and 1 respectively) at progressing times during constant velocity motion. In Figure 5-2A, an arbitrary time is selected where the nominal phase (from the intended ac interference term) is zero. For zero initial phases, 0 and 1, both phasors are directed along the positive real axis. At a later time in Figure 5-2B, the nominal phase is 2 rad, but the actual phase, is less than the nominal due to the vector addition of 0 and 1 Recall that the orientation of 1 does not change for the frequency translated condition. The phase error, is therefore positive and depends on the magnitude of 1 Similar to Figure 5-2A, the phase error in Fi gure 5-2C is again zero In Figure 5-2D, the

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92 error is negative, but equal in magnitude to th e situation depicted in Figure 5-2B. Figure 5-2E demonstrates the corresponding single cycle of phase error variation per 2 rad of nominal phase change for first order periodic error. By vector addition, the phase error is, 1 0 0 1cos sin tan (rad) (5-4) and the corresponding first order periodic error is, 1 0 0 1 1cos sin tan 2 1 FF (nm). (5-5) If the maximum first order periodic error, max,1, is assumed to occur when 2 (see Figure 52B), then it can be expressed as, 1 0 1 1max,tan 22 1 FF (nm) (5-5A) which is equivalent to Eq. 5 in reference [2] for the small angle approximation. Equation 5 in reference [2] identifies the maximum phase error magnitude as 0 1 Figure 5-3 shows the situation when only 0 (intended ac interference signal) and 2 (leakage induced ac interference signal) are considered. Again zero initial phases are assumed and an arbitrary time is selected when both phasors are directed along the positive real axis; see Figure 5-3A. Because the vectors are counter-rota ting, the geometries shown in Figures 5-3B through 5-3H are obtained for nominal phase values of { 4 2 4 3 4 5 2 3 and 4 7 } rad. The characteristic two cycl e phase error variation per 2 rad of nominal phase change (second order periodic error) is depicted in Fi gure 5-3I. The phase error is calculated according

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93 to: cos sin tan20 20 1 (rad) and the correspondi ng second order periodic error is, cos sin tan 2 120 20 1 2FF (nm). (5-6) If the corresponding maximum periodic erro r is assumed to be obtained when = 4 rad (see Figure 5-3B), so that 2 2 cossin then the maximum second order periodic error, max,2, is, 20 20 1 2max,tan 42 1 FF (nm) (5-6A) which agrees with Eq. 6 from reference [2] for the small angle approximation. Equation 6 in reference [2] identifies the maximum phase error magnitude as 0 2 In general, however, all three phasors, 0 1 and 2 are present and the initial phases, 0, 1,and 2, are nonzero and unequal. In this case, Eqs. 5-5 and 5-6 may not accurately describe the first and second order period ic error magnitudes in the m easured phase/position for all combinations of input parameters. To treat the general case, an expression for the phase error must first be determined. Figures 5-1A, 5-1B and 5-1C show the individual (frequency translated) phasors with arbitrary phases. They are superimposed in Figure 5-4. Based on this geometry, the phase error can be calculated as, 2 2110 0 2 2110 0 1 0 0cos cos cos sin sin sin tan (rad) (5-7) and the corresponding periodic error is,

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94 2 1 FF (nm). (5-8) Note that the error is dependent on the nominal phase (of the intended ac interference signal), the three phasor magnitudes, and the initial phases of the three phasors. To evaluate Eq. 5-8, and identify the periodic error order magnit udes, Monte Carlo simulation is applied. This enables the unknown (uniformly dist ributed), uncorrelated initial phases to be incorporated. The required steps are: 1. define the values for FF, 0 1 and 2 ; 2. select random, uniformly distributed values of 0, 1,and 2 from the range i, where i = 0, 1, 2; 3. compute from Eq. 5-7; 4. compute from Eq. 5-8; 5. calculate the discrete Fourier transform of the result from step 4 and normalize the frequency axis to error order (multiply by /FF ) to identify the individual periodic error contributions from each order; and return to step 2; and 6. after many iterations, the periodic error magnit ude for each order is determined from the resulting distributions. As a first comparison between Eqs. 5-5 and 5-6 (equivalent to Eqs. 5 and 6 from reference [2]) and Eq. 5-8, zero initial pha se values are select ed. This removes the requirement for Monte Carlo simulation; the periodic error magnitude s are computed directly from the relevant equations, which depend on the phasor magnitude s. Additionally, because spectrum analyzers typically display power data usi ng a logarithmic (dBm) scale, si gnal amplitudes in this format are applied. To convert from magnitudes, i, in dBm to the (linear) Ampere units for i included in the previous descriptions, the conver sion shown in Eq. 5-9 is applied, where i = 0, 1, 2. 2010ii (5-9)

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95 Figure 5-5 displays first and s econd order error magnitudes for 150 dBm, 501 dBm, and values of 2 from {-50 to -20} dBm. In a quali tative sense, the 35 dBm difference between 0 and 1 corresponds to the attenuation for a well aligned system. While 1 and 2 do not necessarily vary independently wi th changes in the optical setup, this approach provides an initial numerical comparison of the different equations. Numerical resu lts for a conventional setup are provided in Figures 5-5 through 5-10. Figure 5-5 shows picometer (pm) level agreement between Eq. 5-8 and Eqs. 5-5 and 5-6 for the case where the 1 magnitude is negligible. The expected st rong variation in second or der error with changes in 2 is also observed. In Figure 5-6, the attenuation between 0 and 1 is reduced to 15 dBm, 150 dBm and 301 dBm. This is characteristic of a misa ligned system; note that the first order periodic error is now 10 tim es larger. The values of 2 are varied over the same range. In this case, there is an approximately constant offset in the second order period ic error of 0.8 nm. The residual differences (beyond the second or der offset) are at the pm level. The results for a significantly misaligned system, 150 dBm and 251 dBm, are provided in Figure 5-7. It is seen that the second order e rror for Eq. 5-8 does not increase monotonically with decreas ed attenuation between 0 and 2. Rather, a local minimum is seen at 412 dBm. Figure 5-8 is included to show the di fferences between Eq. 5-8 and Eqs. 5-5 and 5-6 (the Eq. 5-5 and 5-6 results are subtracted from the Eq. 5-8 results). Next, the 0 and 2 magnitudes are fixed, and 1 is varied. Figure 5-9 displays the results for 150 dBm and 502 dBm (well aligned setup), while 1 is varied from {-50 to -20} dBm. As expected the first order periodic error grows with1 ; the agreement between Eq. 5-8

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96 and Eq. 5-5 is at the pm level. For Eq. 5-8, however, the second order error is strongly influenced by the presence of significant 1 spectral content. The difference between the Eq. 5-8 and Eq. 56 results exceeds 6 nm for the largest misalignment (201 dBm). Also, the second order error behavior is again non-monotonic a nd reaches a local minimum at 301 dBm when Eq. 5-8 is applied. Interesting second order error beha vior is also observed for 150 dBm and 352 dBm (misaligned setup) a nd the same variation in1 Figure 5-10 shows that the second order error for Eq. 5-8 is again influenced by1 However, the error now decreases with increasing 1 magnitude until a local minimum at 22 dBm is r eached, when the error begins increasing again. Before comparing the equations with experiment al results, the effect of arbitrary initial phase on the periodic error calculations is demonstrated. First, a nonzero initial phase of 0 = 10 deg = 0.17 rad is considered with 150 dBm, 301 dBm, 452 dBm, and 1 = 2 = 0. The resulting periodic error over 2 of nominal phase change is shown in Figure 5-11. The corresponding discrete Fourier tr ansform magnitude (with frequenc y normalized to error order) is provided in Figure 5-12. The first order periodic error dominates with a magnitude of 8.96 nm. The second order magnitude is 0.82 nm. Small thir d order content (0.19 nm ) is also observed, although this is not typically consider ed in most analyses. However, if 0 is changed to 170 deg = 2.97 rad, for example, the associated error wave form differs; see Figure 5-13. The second order error magnitude now increases to 2.38 nm as shown in Figure 5-14. The third order magnitude also increases to 0.38 nm. Using the previously described Monte Carlo simulation, the variation in periodic error magnitudes with 0 variation between (uniformly distributed) is determined. See Figure 5-15, where 1000 iterations were completed. The first order error changes very little,

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97 while the second order error varies between 0.80 nm and 2.39 nm. Analogous results are obtained if the other initial pha se values are varied, but are not included here for brevity. Experimental Results In this section, spectrum analyzer data (0 1 and 2 spectral peaks measured in dBm) are used to calculate first and s econd order periodic error magnitude s via Eqs. 5-5, 5-6, and 5-8. These results are compared with periodic error ma gnitude values determined from the discrete Fourier transform of position data obtained fr om traditional phase measuring electronics. Data were collected for different levels of frequency mixing by varying the linear polarizer and half wave plate angles from their nominal orientations. Figure 4-1A in Chapter 4 shows the test setup that was used to collect data. As noted Eqs. 5-5, 5-6, and 5-8 were applied to compute the corresponding periodic error. Note that Monte Carlo simulation was used to evaluate Eq. 5-8, which enabled the uniformly distributed, uncorrelated initial phases to be randomly selected over many iterations. In the following analyses, the maxi mum values from the simulation distributions are presented. Figure 5-16 displays the case wh ere the linear polarizer angle was varied about its nominal orientation (indicated as zero), while the half wave plate angle was fixed at 10 deg from its nominal angle. A st rong variation for the ac reference term, is observed while 0 and the intended ac interference and leakage induced ac interference terms, respectively, are nearly constant. The first order errors calculated by Eqs. 5-5 and 5-8 increase with larger misalignment angles and agrees with the magnitudes calcula ted from the position data using the discrete Fourier transform. See Figure 5-17A. However, the second order er rors computed using Eqs. 5-6 and 5-8 do not agree. As shown in the Figure 5-17B, the Eq. 5-8 results more closely follows the second order error calculated from the position data.

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98 Figure 5-18 shows the difference between th e Eq. 5-5, 5-6 and 5-8 calculations and position data (discrete Fourier transform) magnit udes for first and second error order errors; the data from Figure 5-17 was analyzed. It is seen that the Eq. 5-5, 5-6 and 5-8 results agree with the position data for small linear polarizer angul ar misalignments. For large misalignments, however, Eqs. 5-5 and 5-6 provide less accurate estimates (2.6 nm differences for first order error and 9.5 nm differences for second order erro r at the largest misalignm ent). Equation 5-8, on the other hand, agrees to within 1.2 nm for first and 3.0 nm for second order error. These results show that Eq. 5-8, which consid ers all three spectral peaks, pr ovides a more accurate estimate of the first and second order periodic errors than Eqs. 5-5 and 5-6, respectively, which consider only two periodic error components either0 and (first order, Eq. 5-5) or 0 and (second order, Eq. 5-6) especially for si gnificant misalignments from nominal. Results for a medium misalignment case (5 de g half wave plate angular misalignment) are provided in Figures 5-19 and 5-20. Trends in0 ,1 and 2 variation similar to those identified in Figure 5-16 are observed. This yields the same first and second order periodic error behavior shown in Figures 5-17 and 5-18. Again, Eq. 5-8 more closely agrees with the position data periodic error magnitudes. When the half wave plate is oriented at its nominal angle, the axes of the two polarized light frequencies emitted from the laser head are well aligned with the axes of the polarization dependent optics. This naturally leads to signif icantly reduced frequency leakage. Figure 5-21 shows the power level of the three interference term s as a function of the linear polarizer angle. Very little change in the individual power leve ls is observed and the attenuation between the intended ac interference signal,0 and ac reference, and leakage induced ac interference, signals is on the order of 33 dB m. Because the signal power levels are constant with the linear

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99 polarizer angle, the first and sec ond order errors calculated by Eqs. 5-5, 5-6, 5-8 are also constant and agree with the periodic error magnitudes dete rmined from the position data; see Figure 5-22. Next, variation of the half wave plate angle is considered. Figure 5-23 displays results for a 17 deg angular misalignment of the linear polarizer. This figure shows that, while both the ac reference, and leakage induced ac interference, terms vary, exhibits higher sensitivity to the half wave plate angle. Figure 5-24 show s the corresponding first and second order periodic error comparisons. As seen previously, the Eq. 5-8 results agree more closely with the position data error magnitudes, particularly for sec ond order error under sign ificant misalignments. Figure 5-25 shows the associated error magnitude differences. In Figure 5-26, the linear polarizer was set at its nominal orientation and the half wave plate angle was varied. It is s een that only the leakage induced ac interference term, varies. The first order error is therefore constant and th e Eq. 5-5 and 5-8 results agree with the position data first order error magnitudes. See the top pa nel of Figure 5-27. Similarly, both the Eq. 5-6 and 5-8 results agree with the pos ition data second order error ma gnitudes as seen in the bottom panel of Figure 5-27. The Eq. 5-6 agreement occurs because does not change with the half wave plate angle. As a final example, Figure 5-28 shows the di screte Fourier transform of position data collected using a significantly misaligned system (half wave plate angle is 5 deg from nominal and the linear polarizer is 10 de g from nominal). The existence of third order error is observed. The magnitudes are: 6.3 nm (first order), 2.1 nm (second order), and 0.3 nm (third order). The power levels of the three interference terms usi ng the same configuration were also measured using the spectrum analyzer; the values were: 5 .450 dBm, 8.621 dBm, and

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100 5.742 dBm. The corresponding maximum error magnitudes calculated us ing a Monte Carlo evaluation of Eq. 5-8 are: 6.1 nm (first order), 2.0 nm (second order), and 0.2 nm (third order). Discussion The data shown in Figure 5-29 were collected with a 10 deg linear polarizer misalignment, while the half wave plate angle was varied. Maxi mum, minimum, and mean values of the first and second order error magnitudes determined fr om a Monte Carlo evaluation of Eq. 5-8 are presented, together with the Eq. 5-5 and 5-6 result s and position data magnit udes. It is seen that the spread in first order error values obtained from Eq. 5-8 (Figure 5-29A) is small. This suggests that the calculation is not particularly sensitiv e to the initial phases of the three interference terms. For the second order error calculations (F igure 5-29B), however, the spread is significant. As reported in the previous figures, the maxi mum errors (the upper bound of the band) from simulation agree well with the position data e rror magnitudes. The mean values, however, track more closely with the Eq. 5-6 results. This outcome suggests that the a ssumption of uncorrelated arbitrary phases may be incorrect. However, in a practical sense, Eq. 5-8 still provides accurate estimates for both first and second periodic er ror magnitudes (under arbitrary misalignments) provided the maximum value from the Monte Carlo evaluation is applied.

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101 Figure 5-1. Phasor di agrams. A) Intended ac interference signal. B) The ac reference signal. C) Leakage induced ac interference signal. 0 Im 0 fd A 1 1B 2 2fdC Re

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102 Figure 5-2. Periodic error in the presence of 0 and 1 only for various nominal phase angles (rad). A) 0. B) 2 C) D) 2 3 E) The single cycle of phase error variation per 2 rad of nominal phase change. 0 fd A 1B 0 1 0C 1 D 0 1 0 E /2 3 /2 2 Im Re

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103 Figure 5-3. Periodic error in the presence of 0 and 2 only for various nominal phase angles (rad). A) 0. B) 4 C) 2 D) 4 3 E) F) 4 5 G) 2 3 H) 4 7 I) The two cycles of phase error variation per 2 rad of nominal phase change are shown. 0 fd A 2 B 0 2 0 I C 2 0 D 2 0 0E 2 0F 2 0G 2 H 0 2 fd /2 3 /2 2 Re Im

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104 Figure 5-4. Phasor diagram for general cas e where the intended and leakage induced ac interference and ac reference signals are present with arbitrary initial phases. 1 1 2 2 0 0 fd fd 'Re Im

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105 Figure 5-5. Comparison between periodic error magnitudes obtaine d from Eq. 5-8 and Eqs. 5-5 and 5-6 for 150 dBm, 501 dBm, and variable 2 The agreement is at the picometer level. A) The magnitude of firs t order periodic erro r variation. B) The magnitude of second order periodic error variation. -50 -45 -40 -35 -30 -25 -20 0.894 0.896 0.898 1st order error (nm) Eq. 5-8 Eq. 5-5 -50 -45 -40 -35 -30 -25 -20 0 10 20 30 2 (dBm)2nd order error (nm) Eq. 5-8 Eq. 5-6 A B

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106 Figure 5-6. Comparison between periodic error magnitudes obtaine d from Eq. 5-8 and Eqs. 5-5 and 5-6 for 150 dBm, 301 dBm, and variable2 There is approximately a 0.8 nm offset in the second order periodic error. A) The magnit ude of first order periodic error variation. B) The magnitude of second order peri odic error variation. -50 -45 -40 -35 -30 -25 -20 8.956 8.958 8.96 1st order error (nm) Eq. 5-8 Eq. 5-5 -50 -45 -40 -35 -30 -25 -20 0 10 20 30 2 (dBm)2nd order error (nm) Eq. 5-8 Eq. 5-6 A B

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107 Figure 5-7. Comparison between periodic error magnitudes obtaine d from Eq. 5-8 and Eqs. 5-5 and 5-6 for 150 dBm, 251 dBm, and variable2 The second order error no longer increases monotonica lly for the Eq. 5-8 calculati ons. A) The magnitude of first order periodic error variation. B) The magnitude of seco nd order periodic error variation. A B -50 -45 -40 -35 -30 -25 -20 15.928 15.93 15.932 1st order error (nm) Eq. 5-8 Eq. 5-5 -50 -45 -40 -35 -30 -25 -20 0 10 20 30 2 (dBm)2nd order error (nm) Eq. 5-8 Eq. 5-6

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108 Figure 5-8. Difference between periodic error ma gnitudes obtained from Eq. 5-8 and Eqs. 5-5 and 5-6 for 150 dBm, 251 dBm, and variable2 A) The first order periodic error variation with respect to various2 B) The second order periodic error variation with respect to various2 -50 -45 -40 -35 -30 -25 -20 0 1 2 x 10-3 1st order diff. (nm) -50 -45 -40 -35 -30 -25 -20 -4 -2 0 2 2 (dBm)2nd order diff. (nm)A B

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109 Figure 5-9. Comparison between periodic error magnitudes obtaine d from Eq. 5-8 and Eqs. 5-5 and 5-6 for 150 dBm, 502 dBm, and variable1 The second order error is influenced by the variable 1 magnitude when using Eq. 5-8. A) The magnitude of first order periodic error variation. B) The magnitude of seco nd order periodic error variation. A B -50 -45 -40 -35 -30 -25 -20 0 10 20 30 1st order error (nm) Eq. 5-8 Eq. 5-5 -50 -45 -40 -35 -30 -25 -20 0 5 10 1 (dBm)2nd order error (nm) Eq. 5-8 Eq. 5-6

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110 Figure 5-10. Comparison between periodic error magnitudes obtained from Eq. 5-8 and Eqs. 5-5 and 5-6 for 150 dBm, 352 dBm, and variable1 The second order error from Eq. 5-8 is again influenced by the variable 1 magnitude; it reaches a local minimum at dBm. A) The magnitude of fi rst order periodic er ror variation. B) The magnitude of second order periodic error variation. -50 -45 -40 -35 -30 -25 -20 0 10 20 30 1st order error (nm) Eq. 5-8 Eq. 5-5 -50 -45 -40 -35 -30 -25 -20 0 2 4 6 1 (dBm)2nd order error (nm) Eq. 5-8 Eq. 5-6 B

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111 Figure 5-11. Periodic error for 0 = 10 deg = 0.17 rad with 150 dBm, 301 dBm, 452 dBm, and 1 = 2 = 0. Figure 5-12. Periodic error magnitudes for 0 = 10 deg = 0.17 rad with 150 dBm, 301 dBm, 452 dBm, and 1 = 2 = 0. 0 0.5 1 1.5 2 2.5 3 3.5 4 0 1 2 3 4 5 6 7 8 9 Error orderError (nm) 0 0.5 1 1.5 2 -10 -5 0 5 10 /Error (nm)

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112 Figure 5-13. Periodic error for 0 = 170 deg = 2.97 rad with 150 dBm, 301 dBm, 452 dBm, and 1 = 2 = 0. Figure 5-14. Periodic error magnitudes for 0 = 170 deg = 2.97 rad with 150 dBm, 301 dBm, 452 dBm, and 1 = 2 = 0. 0 0.5 1 1.5 2 2.5 3 3.5 4 0 1 2 3 4 5 6 7 8 9 Error orderError (nm) 0 0.5 1 1.5 2 -10 -5 0 5 10 /Error (nm)

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113 Figure 5-15. Variation in period ic error magnitudes for random 0 values ( range) with 150 dBm, 301 dBm, 452 dBm, and 1 = 2 = 0. -1 -0.5 0 0.5 1 8.957 8.958 8.959 0/1st order error (nm) -1 -0.5 0 0.5 1 0 1 2 3 0/2nd order error (nm)

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114 Figure 5-16. Variation of 1 and 2 with linear polarizer (LP) angle. The half wave plate (HWP) was fixed at 10 deg from its nomi nal orientation (a large misalignment configuration). Strong variation of 1 is observed. Figure 5-17. Periodic errors calculated by Eqs. 55, 5-6, and 5-8 are compared to magnitudes computed using the discrete Fourier tran sform of position data. The agreement is good for first order error, but only Eq. 5-8 re produces the second order error. A) The magnitude of first order periodic error. B) The magnitude of second order periodic error. -20 -15 -10 -5 0 5 10 15 20 -80 -75 -70 -65 -60 -55 -50 -45 -40 power (dBm)LP angle (deg) 0 1 2 -20 -15 -10 -5 0 5 10 15 20 0 10 20 30 1st order error (nm) Eq. 5-8 Eq. 5-5 Pos. data -20 -15 -10 -5 0 5 10 15 20 5 10 15 20 LP angle (deg)2nd order error (nm) Eq. 5-8 Eq. 5-6 Pos. data A B

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115 Figure 5-18. Differences between magnitudes from Eqs. 5-5, 5-6, and 58 and discrete Fourier transform of position data. A) The first orde r error. B) The sec ond order error. The differences were calculated from the errors displayed in Figure 5-17. -20 -15 -10 -5 0 5 10 15 20 0 1 2 3 1st order diff. (nm) Pos. data Eq. 5-8 Pos. data Eq. 5-5 -20 -15 -10 -5 0 5 10 15 20 0 5 10 HWP angle (deg)2nd order diff. (nm) Pos. data Eq. 5-8 Pos. data Eq. 5-6 A B

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116 Figure 5-19. Variation of 1 and 2 with linear polarizer (LP) angle. The half wave plate was fixed at 5 deg from its nominal orientation (a medium misalignment configuration). Strong variation of 1 is again observed. Figure 5-20. Periodic errors calculated by Eqs. 55, 5-6, and 5-8 are compared to magnitudes computed using the discrete Fourier tran sform of position data. The agreement is good for first order error, but only Eq. 5-8 re produces the second order error. A) The magnitude of first order periodic error. B) The magnitude of second order periodic error. -20 -15 -10 -5 0 5 10 15 20 -80 -75 -70 -65 -60 -55 -50 -45 -40 power (dBm)LP angle (deg) 0 1 2 -20 -15 -10 -5 0 5 10 15 20 0 5 10 15 1st order error (nm) Eq. 5-8 Eq. 5-5 Pos. data -20 -15 -10 -5 0 5 10 15 20 1 2 3 4 LP angle (deg)2nd order error (nm) Eq. 5-8 Eq. 5-6 Pos. data A B

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117 Figure 5-21. Variation of 1 and 2 with linear polarizer (LP) angle. The half wave plate was fixed at its nominal orie ntation (a well aligned confi guration). All three signals are nearly constant. Figure 5-22. Periodic error calculated by Eqs. 5-5, 5-6, and 5-8 agree with the position data results because 1 and 2 do not vary with linear polarizer angle for the well aligned system. A) The magnitude of first order pe riodic error. B) The magnitude of second order periodic error. -20 -15 -10 -5 0 5 10 15 20 -80 -75 -70 -65 -60 -55 -50 -45 -40 power (dBm)LP angle (deg) 0 1 2 -20 -10 0 10 20 0 1 2 1st order error (nm) Eq. 5-8 Eq. 5-5 Pos. data -20 -10 0 10 20 0 1 2 LP angle (deg)2nd order error (nm) Eq. 5-8 Eq. 5-6 Pos. data A B

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118 Figure 5-23. Variation of 1 and 2 with half wave plate (HWP) angle. The linear polarizer was misaligned by 17 deg from nominal. Both 1 and 2 vary. Figure 5-24. Periodic error calculated by Eqs. 5-5, 5-6, and 5-8 are compared to position data error magnitudes (the linear polarizer misalignment angle was 17 deg from nominal). Improved agreement is observed for the Eq 5-8 results, particul arly in the second order error case at large misalignments. A) The magnitude of first order periodic error. B) The magnitude of second order periodic error. -10 -5 0 5 10 -80 -75 -70 -65 -60 -55 -50 -45 -40 power (dBm)HWP angle (deg) 0 1 2 -10 -5 0 5 10 0 10 20 30 1st order error (nm) Eq. 5-8 Eq. 5-5 Pos. data -10 -5 0 5 10 0 10 20 HWP angle (deg)2nd order error (nm) Eq. 5-8 Eq. 5-6 Pos. data A B

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119 Figure 5-25. Differences between magnitudes from Eqs. 5-5, 5-6, and 58 and discrete Fourier transform of position data. A) The first orde r error. B) The sec ond order error. The differences were calculated from the errors displayed in Figure 5-24. -10 -5 0 5 10 0 1 2 3 1st order diff. (nm) Pos. data Eq. 5-8 Pos. data Eq. 5-5 -10 -5 0 5 10 0 5 10 HWP angle (deg)2nd order diff. (nm) Pos. data Eq. 5-8 Pos. data Eq. 5-6 A B

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120 Figure 5-26. Variation of 1 and 2 with half wave plate (HWP) angle. The linear polarizer was fixed at its nominal orientation. Only 2 varies. Figure 5-27. Periodic error calculated by Eqs. 5-5, 5-6, and 5-8 agree with the position data error magnitudes (the linear polarizer angle was fixed at its nomin al orientation). Note that only 2 varies, while 1 remains constant. A) The magn itude of first order periodic error. B) The magnitude of second order periodic error. -10 -5 0 5 10 -80 -75 -70 -65 -60 -55 -50 -45 -40 power (dBm)HWP angle (deg) 0 1 2 -10 -5 0 5 10 0 1 2 1st order error (nm) Eq. 5-8 Eq. 5-5 Pos. data -10 -5 0 5 10 0 5 10 HWP angle (deg)2nd order error (nm) Eq. 5-8 Eq. 5-6 Pos. data A B

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121 Figure 5-28. The spectrum of position data (normalized to error order) contains first, second, and third order periodic error. The data was obt ained for 5 deg half wave plate and 10 deg linear polarizer misalignments from their nominal angles. 0 0.5 1 1.5 2 2.5 3 3.5 4 0 1 2 3 4 5 6 7 Error orderError (nm)

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122 Figure 5-29. Periodic errors calculated by Eqs. 55, 5-6, and 5-8 are compared to the position data error magnitudes. The Eq. 5-8 Monte Carlo simulation results include the full distribution of values (indi cated by the gray band). A) The magnitude of first order periodic error. B) The magnitude of second order periodic error. -10 -5 0 5 10 0 5 10 15 1st order error (nm) Mean Eq. 5-8 Eq. 5-5 Pos. data -10 -5 0 5 10 0 2 4 6 8 10 12 2nd order error (nm)HWP angle (deg) Mean Eq. 5-8 Eq. 5-6 Pos. data A B

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123 CHAPTER 6 PERIODIC ERROR UNCERTAINTY Evaluation of the overall uncertainty in displacement m easuring interfero metry has traditionally followed the error budget technique where the individual error contributors are individually determined (either through statisti cal or other analyses) a nd then combined, often using a root sum squares, or RSS, approach [3,4]. These contributors, which may include Abbe error, cosine error, deadpath error, environmen tal error, air (or other medium) turbulence, beam shear, thermal effects, electronics linearity, lase r wavelength stability, an d periodic error, are tabulated so that primary offenders may be identif ied and compensated or corrected [83]. This is an effective and time-proven method. However, a single analytical expression that describes displacement, l, in terms of the multiple inputs that determine its value has not been presented. This precludes the use of a Taylor series expa nsion of the measurand [90, 91] and/or Monte Carlo simulation to evaluate the combined standard uncertainty, uc(l), for the measurement result. In this chapter, the analytical peri odic error expression presented by Cosijns et al. [5] with terms that describe the other error sources listed previously (Chapter 2) is modified to arrive at a single expression for displacement. Uncertainty contributors are then propagated through this equation to determine the comb ined standard uncertainty, uc, in the measurement result. The chapter is organized as follows: first, the Cosijns et al. expression is presented and example error distributions are shown for vari ous periodic error conditions; s econd, a single pass, heterodyne interferometer setup, as describe d in Chapter 4 (the polarization coded interferometer), is used here again and measurement results are provided; and, third, the additional uncertainty Reprinted with permission from Schmitz T, Kim HS. Monte Carlo evaluation of periodic error uncertainty. Precision Engineering 2007; 31(3):25159.

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124 contributors are appended to the displacement e quation and the uncertainty is evaluated using Monte Carlo simulation. Periodic Error Formulation As noted, this chapter focuses on traditional heterodyne Michelson-type interferometers with a two frequency laser source. As previous ly described, this system produces first and second order periodic error by imperfect separati on of the two light frequencies into the moving and fixed paths. The two frequencies are typically carried on collinear, mutually orthogonal, linear polarizations, i.e., polarizat ion coded. As described in Chap ter 3, unwanted leakage of the reference frequency into the moving path, a nd vice versa, may occur due to a number of influences, including non-orthogonality between the id eally linear beam polarizations, elliptical polarization of the individual b eams, non-ideal performance of th e optical components, and/or mechanical misalignment between the interferometer elements (laser, polarizing optics, and targets). The inherent frequency leakage in ac tual implementations produces an interference signal which leads to periodic erro r in the measured displacement. The Cosijns et al. [5] analysis propagates: ellipticity of the two (nominally linear) polarizations; non-orthogonality be tween the two polarizations; rota tion of the polarization axes relative to the polarizing beam splitter (which idea lly separates the collinear frequencies into the measurement and reference paths); transmission co efficient variations for the polarizing beam splitter; and rotation of the measurement polarizer, which causes interference of the measurement and reference beams, relative to its nominal 45 deg orienta tion (for vertical and horizontal source polarizations), th rough the interference eq uations to arrive at an expression for the periodic phase error, pe. See Eq. 6-1, where is the deviation of the polarizer angle from 45 deg and the variables A-F are defined in Eqs. 6-2 through 6-7.

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125 2cos2sin 2cos2sin tan1F ED C BApe (6-1) A = (-(2sin()2+2cos()2)cos(d1/2)sin(d2/2)-(2cos()2+2sin()2)sin(d1/2) (6-2) cos(d2/2))cos()+(2cos()sin()+2sin()cos())cos(d1/2+ d2/2)sin() B = ((2sin()2-2cos()2)cos(d1/2)sin(d2/2)+( 2cos()2-2sin()2)sin(d1/2) (6-3) cos(d2/2))cos()+(-2cos()sin()+2sin()cos())cos(d1/2+ d2/2)sin() C = (cos()sin()cos(d1/2)sin(d2/2)(1-cos(2))+sin()sin()cos(d1/2) (6-4) cos(d2/2)sin(2)-cos()cos()sin(d1/2)sin(d2/2)sin(2)-sin()cos() sin(d1/2)cos(d2/2)(1+cos(2))) D = ((2sin()2+2cos()2)cos(d1/2)sin(d2/2)+( 2cos()2+2sin()2)sin(d1/2) (6-5) cos(d2/2))sin()+(2cos()sin()+2sin()cos())cos(d1/2+ d2/2)cos() E = ((-2sin()2+2cos()2)cos(d1/2)sin(d2/2)+(2cos()2+2sin()2)sin(d1/2) (6-6) cos(d2/2))sin()+(-2cos()sin()+2sin()cos())cos(d1/2+ d2/2)cos() F = (cos()sin()cos(d1/2)sin(d2/2)sin(2)+cos()cos()(cos(d1/2) (6-7) cos(d2/2)-sin(d1/2)sin(d2/2)cos(2))+sin()sin()(-sin(d1/2)sin(d2/2)+ cos(d1/2)cos(d2/2)cos(2))+sin()cos()sin(d1/2)cos(d2/2)sin(2)) In these equations, d1 and d2 are the ellipticities of the tw o collinear beams (ideally zero), and are the orientation of the two polarizations relative to the polarizing beam splitter axes (together the two ideally zero angles dete rmine both non-orthogonali ty between the two polarizations and rotation of the polarization axes relative to the polarizing beam splitter), and are the transmission coefficients for the polari zing beam splitter (ideal ly equal to one), and 4vacnl is the phase change introdu ced by a given displacement, l (vac is the source vacuum wavelength and n is the refractive index for the pr opagating medium) for a single pass configuration of the interferometer. The di splacement error, lpe, due to the periodic phase error is given in Eq. 6-8.

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126 4 p evac pel n (6-8) An example of the variation in periodic erro r with nominal displacement is provided in Figure 6-1A. The conditions are: d1 = d2 = 0, = = 2 deg, = = 1, = 20 deg, vac = 633 nm, and n = 1. It is seen that first order error dom inates. This is highlighted by computing the spatial discrete Fourier transfor m and normalizing the frequency ax is to error order. See Figure 6-1B, where the first and second order error amplitudes are 2.95 nm and 0.15 nm, respectively, for the given conditions. The fr equency distribution in error values (Figure 6-1C) was determined by Monte Carlo simulation. Because it is equally likely that displacement is recorded at any location along the moving path, l is described using a uniform distribution and, in each iteration of the Monte Carlo simulation (100,000 to tal), is randomly sampled to determine the nominal phase This value is then used to compute the periodic error using Eqs. 6-1 through 6-8. The strongly non-normal distribution seen in Figure 6-1C is obtained due to the profile slopes around zero values in the sinusoidal error. The standard deviation is 2.09 nm. A second example is provided in Figure 6-2. In this case, both firs t and second order error are significant for the conditions: d1 = d2 = 0, = = 20 deg, = = 1, = 2 deg, vac = 633 nm, and n = 1. Their amplitudes are 2.56 nm and 6.74 nm, respectively. The distribution is again non-normal, but now contains four peaks rather than two due to the second order error contribution. The standard deviation is 5.12 nm Many other distributions are possible depending on the frequency mixing conditions. To determine the periodic error uncertainty, M onte Carlo simulation is used. This approach is applied, rather than the Taylor series e xpansion method described in references [17, 91], because the periodic error phase in Eq. 6-1 is identically zero for ideal values of the input variables. Selection of ideal mean values pres umes that all misalignments/imperfections have

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127 been corrected to within the uncertainty limits (i.e., all known biases have been removed to within the applicable limits). For demonstration purposes, standard uncertainties of u(d1) = 0.1 deg, u(d2) = 0.1 deg, u() = 2 deg, and u() = 2 deg were selected. Normal dist ributions were assumed in all cases. The corresponding mean values were d1 = d2 = 0 deg, = = 0 deg, and = 0 deg. The transmission coefficients, and, are bounded by a maximum value of 1. Therefore, a uniform distribution with a range of 0.05 was selected about a mean value of 0.95 for each1. Uncertainties in and n were not considered at this stage; these are treated in the section that evaluates the combined standard uncertainty in displacement where the remaining uncertainty contributors are added. Finally, vac = 633 nm and n = 1 were used and a uniform distribution for l with a range from zero to was again applied. Results are provided in Figure 6-3, which shows the distribution of lpe values for the selected input uncertainties. The mean value is zero and the standard deviation is 1.77 nm (100, 000 iterations). The reader may note that the distribution is non-normal, with a higher like lihood of obtaining zero error than a normal distribution would suggest. As an exercise, the products of the sensitivities, x lpe and standard uncertainties, xu, for each input x were calculated; see Table 6-1. The individual contributors were isolated by setting all uncertainties except the term in questi on equal to zero. Presumably, this would enable the individual contributors to be compared, similar to the e rror budget and analytical Taylor series approaches. However, as seen in the ta ble, the apparent indivi dual contributions for, and are zero. This is clearly not the case. Rath er, these terms are only zero for ideal mean 1 The standard deviation for this case is 0.05/3 [91].

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128 values and no variation in all other inputs. For any other case, their contributions are non-zero. This emphasizes the utility of using the Monte Ca rlo technique to simultaneously consider all uncertainties for this evaluation. Table 6-1. Apparent individua l uncertainty contributors for lpe x x u(x) xu x lpe (nm) d1 0 0.1 deg 0.03 d2 0 0.1 deg 0.03 0 2 deg 1.25 0 2 deg 1.25 0.952 3 050. 0 0.953 3 050. 0 0 2 deg 0 Periodic Error Measurements Experimental data were collected using the se tup shown in Figure 4-1 in Chapter 4 to show the various periodic error leve ls depending upon the angular misa lignment of the linear polarizer and half wave plate. Example comparisons between measurement results and the Cosijns et al. model are provided in Figures 6-4 and 6-5. For Fi gure 6-4, the polarizer was rotated 39 deg from its nominal 45 deg orientation in or der to provide a scenario with si gnificant first order error. The other model parameters were: n = 1, vac = 633 nm, d1 = d2 = 0 deg, = = 1.5 deg, and = = 1. In Figure 6-5, the half wave plate was adjusted 10 deg from its nominal orientation (fast axis vertical). In this case, both first and second order errors were present. The model parameters were: n = 1, vac = 633 nm, d1 = d2 = 0 deg, = = 20 deg (a 1 deg rotation of the half wave plate gives a 2 deg change in the linear polarization angle), = 2 deg, and = = 1. Good 2 The mean value was set equal to 1 when evaluating other inputs.

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129 agreement is seen. In both cases, the phase meas uring electronics used a sampling frequency of 312.5 kHz. Also, the displacement measuring resolution was 0.3 nm for the single pass interferometer configur ation implemented here. To evaluate the Monte Carlo approach to peri odic error uncertainty ev aluation, two sets of measurements were performed. First, the polariz er angle was varied from -41 deg to +37 deg about the nominal value and the first and sec ond order errors identifi ed using the Fourier transform approach described previously. Second, the half wave plate angle was varied from -16 deg to +14 deg about its nominal value and the periodic error determined. These measurement results were compared to model predictions us ing nominal input values: 1) Figure 6-6 shows results for the polarizer tests with n = 1, vac = 633 nm, d1 = d2 = 0 deg, = = 1.5 deg, = 41 deg to +37 deg, and = = 1; and 2) Figure 6-7 displays the half wave plate results with n = 1, vac = 633 nm, d1 = d2 = 0 deg, = = -32 deg to +28 deg, = 2 deg, and = = 1. These figures also show mean values from Monte Carlo si mulations, where the Monte Carlo results include one standard deviation (1 ) error bars. In Figure 6-6A, good agreement is seen. Additionally, for polarizer angles near the nominal value (axis value of zero) it is observed that the Monte Carlo means are larger than the mode l values. This trend matches the experimental results. In Figure 6-6B, reasonable agreement is obs erved, but the measured errors lie outside the 1 error bars for small angles. These error levels approach the resolution limit (0.3 nm), which was not considered as an uncertainty contributor at this stage. In Figur e 6-7A, the larger Monte Carlo mean errors near the nominal half wave plate angle again more closely agree with experiment than the model values determined from mean inputs. In Figure 6-7B, although the general trends agree, the experimental errors generally fall outside the Monte Carlo error bars. This could be the result of incorrect estimates of the mean input values.

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130 The Monte Carlo simulations for first and seco nd order uncertainty evaluation required the following steps: 1) sample the input values for d1, d2, , and from the predefined distributions (see Table 6-2); 2) calculate the pe riodic error profile using the Cosijns et al. model; 3) identify the first and second order periodic error amplitudes using the Fourier transform approach; and 4) record the results for each simu lation iteration. The mean values and standard deviations from all iterations (10, 000) were then taken to represent the best estimates of the first and second order error expect ation values and standard uncertainties, respectively. Table 6-2. Monte Carlo simulati on input values for first and second order error uncertainty evaluation (all distributions were normal except for and which were uniform). Polarizer tests Half wave plate tests x x u(x) x u(x) d1 0 0.1 deg 0 0.1 deg d2 0 0.1 deg 0 0.1 deg 1.5 deg 1 deg -32 to +28 deg 1 deg -1.5 deg 1 deg +32 to -28 deg 1 deg 0.95 3 050. 0.95 3 050. 0.95 3 050. 0.95 3 050. -41 to +37 deg 1 deg 2 deg 1 deg Displacement Combined Standard Uncertainty In this section, the additional displacement uncertainty contributors described in Chapter 2 are reviewed and the analytical displacement equation is provided. Abbe Error The potential for Abbe error exists whenever the measurement beam is not collinear with the motion axis (assume they are parallel here). The relationship between the true, l, and measured, lm, displacement is offsettanmlld where doffset is the Abbe offset between the

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131 measurement beam and motion axis and is uncompensated rotation about a line normal to the plane containing both the measurement beam and motion axis. Cosine Error Cosine error is inherent to displacement measurement interfer ometry because the beam and motion axis cannot be perfectly aligned (i.e ., some uncertainty always remains). The corresponding true/measured displacement relationship is secmll where is the positive angular misalignment. The reader may note that, for any value of the measured displacement is smaller than the true displacement (i.e., a bias is introduced). The reported value, lr, can be corrected for the bias using 21ullmr where ml is the expectation value of lm and 2u is the variance of [17]. Deadpath Error Deadpath error occurs when the path lengths from the polarizing beam splitter to the fixed and moving targets are unequal at initialization and there is an uncompensated change in the refractive index, n, of the propagating medium (air is c onsidered here) during the measurement. The true/measured displ acement relationship is DPnllm where DP is the deadpath, or difference between the path lengths. The value of the refractive index for air may be expressed as a function of absolute temperature, T (K), pressure, P (Pa), percent relative humidity, RH, and carbon dioxide content, CO2 (ppm) as shown in Eq. 6-9 [18]. An evaluation of this equation at conditions of standard temp erature and pressure (20 de g C and 101323.2 Pa) with 50% RH and an assumed CO2 level of 355 ppm yields an air index value of 1.0002713. RH CO T P n8 6 2 6101 101 300 54.01 15.293 101325 108.2711 (6-9)

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132 The reader may note that Eq. 6-9 does not implicitly consider air tu rbulence, which affects index through localized time-depe ndent fluctuations in temperature and pressure [15, 23]. This could be incorporated, however, by adding noi se (in addition to the variations caused by uncertainties in the temperature, pressure, and relative humidity transducers) to the index values within the Monte Carlo simulation. Atmospheric Error This error occurs when there is an uncompen sated change in index during a measurement. The error relationship is PDnllm where PD is the physical displacement of the moving retroreflector after init ialization. Resolution limits in the displacement measurement system can be conveniently included as perturbations in PD. Material Thermal Expansion Error Changes in temperature, and the associated thermal deformations, can lead to errors associated with the interferometer optics. The corresponding relationship is th mCTll where T is the change in temperature and Cth is a constant typi cally supplied by the interferometer manufacturer. Other Phase Errors In addition to periodic error, nonlinearities can also be introduced by the phase measuring electronics, elect [12, 24]. Also, a change in overlap between the reference and measurement beams, or beam shear, during a measurement can l ead to errors due to the imperfect, non-planar wavefronts, shear. Laser Wavelength Stability Variation in the source wavelength during a measurement naturally leads to errors. The level of variation is typically small and provided by the la ser manufacturer.

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133 The final displacement relationship is provide d in Eq. 6-10. This equation considers periodic error as well as the terms identified in the previous sections. To demonstrate the application of Eq. 6-10, a Monte Carlo simulation is completed to evaluate uc(l) for a range of displacements. The inputs are provided in Table 6-3 and the simulation results are shown in Figure 6-8. offsetsectan 4peelectshearvacld n D P n P D T C n (6-10) Figure 6-8A shows both the difference between the Eq. 6-10 mean value and nominal displacement, l0, and the 1 error bars (100,000 iterations). As expected, the uncertainty increases substantially over the 1 mm to 1000 mm interval. It is also seen that the mean displacement from Eq. 6-10 is consistently smaller than the nominal value. This is caused by the single-sided cosine error distribution (note that th e bias is present even though the mean value of is zero). Figure 6-8B highlights this bias (s quares) and also shows the bias removal using 2501u.llmr (circles). Table 6-3. Monte Carlo si mulation input values for uc(l) evaluation. x x u(x) Distribution type d1 0 0.1 deg Normal d2 0 0.1 deg Normal 1 deg 1 deg Normal -2 deg 1 deg Normal 0.95 3 050. Uniform 0.95 3 050. Uniform 2 deg 1 deg Normal elect 0 0 shear 0 0 vac 633 nm 6x10-6 nm Normal

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134 Table 6-3. Continued P 101323.2 Pa 3 50 Pa Uniform T 20 deg C 3 20. deg C Uniform RH 50% 3 2 % Uniform CO2 355 ppm 0 ppm AO 0 mm 1 mm Normal DP 50 mm 1 mm Normal PD 1 to 1000 mm 3 30. nm* Uniform Cth 25 nm/deg C 1 nm/deg C Normal 0 0.03 deg Normal 100 rad 10 rad Normal *This uncertainty represents the resolution of the interferometer, which depends on the phase measuring electronics and optical configuration.

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135 Figure 6-1. An example of periodic error varia tion with conditions that dominates first error order. A) Periodic error as a f unction of nominal displacement for d1 = d2 = 0, = = 2 deg, = = 1, = 20 deg, vac = 633 nm, and n = 1. B) Spatial Fourier transform of periodic error with the frequency axis normalized to error order (i.e., 1 represents first order error). C) Distribution of periodic error. A B C

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136 Figure 6-2. An example of periodi c error variatio n with conditions that dominates both first and second error orders. A) Periodic error for d1 = d2 = 0, = = 20 deg, = = 1, = 2 deg, vac = 633 nm, and n = 1. B) Spatial Fourier tran sform of periodic error. C) Distribution of periodic error. A B C

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137 Figure 6-3. Histogram of lpe values for normal distributions of d1, d2, and with u(d1) = u(d2) = 0.1 deg, u() = u() = u() = 2 deg and zero mean values; and uniform distributions of and with ranges of 0.05 and mean values of 0.95.

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138 Figure 6-4. Measurement/model comparison for 39 de g polarizer misalignment first order error dominates. Other model parameters were: n = 1, vac = 633 nm, d1 = d2 = 0 deg, = = 1.5 deg, and = = 1.

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139 Figure 6-5. Measurement/model comparison for 10 de g half wave plate misalignment first and second order errors are presen t. Model parameters were: n = 1, vac = 633 nm, d1 = d2 = 0 deg, = = 20 deg, = 2 deg, and = = 1.

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140 Figure 6-6. Comparisons between measurements (circles), model (squares), and Monte Carlo simulation (dotted line) for variable polarizer angle tests. A) First order errors. B) Second order errors. A B

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141 Figure 6-7. Comparisons between measurements (circles), model (squares), and Monte Carlo simulation (dotted line) for variable half wave plate angle tests. A) First order errors. B) Second order errors. A B

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142 Figure 6-8. Monte Carlo simula tion results for Eq. 6-10 using the data in Table 6-3. A) Difference between mean Eq. 6-10 valu es and nominal displacement with 1 error bars. B) Uncorrected (squares) a nd corrected (circles) difference. A B

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143 CHAPTER 7 NEW HETERODYNE INTERFEROMETER DESIGN The presence of periodic error in a traditiona l heterodyne interferom eter system that uses polarization dependent optics to separate orthogonal and linearly polarized beams was shown in Chapter 4. Although numerous studies have been completed to correct/com pensate for periodic error, it remains an inherent er ror source due to the potential for frequency mixing. In this chapter, a new heterodyne interferometer design is described that eliminates periodic error. The setup and results for the new acousto-optic modulator-based displacement measuring interferometer (AOM DMI) design are detailed. The absence of periodic error is demonstrated in the frequency domain. Acousto-Optic Modulator In the AOM DMI design, a pair of AOMs is used to generate, and keep spatially separate, the heterodyne frequencies. The basic elements of an AOM are a glass body with a piezoelectric transducer (PZT) attached at one end and a freq uency source (stable quartz oscillator) to drive the PZT [92]. Actuation of the PZT at the oscillator driving frequency generates acoustic waves within the glass which leads to periodic spatial va riations in the glass re fractive index due to the changes in density (i.e., a moving diffraction grating is produced). When laser light is incident on the moving diffraction grating, it is diffracted in to multiple separate beams (or orders). The diffraction angle, is given by Eq. 7-1, where m = 0, 1, 2, is the order number and is the acoustic wavelength. Under high driving frequencie s and by proper design, a portion of the light is diffracted into the 1st order (m = 1) beam while the rest (0th order beam) pa sses through the glass; see Figure 7-1. The amplitude of the ac oustic waves within the glass determines the Reprinted with permission from Kim H, Schmitz T, Beckwith J. Periodic error in heterodyne interferometry: Examination and Elimination. In: Halsey D, Raynor W, ed. Handbook of Interferometers: Research, Technology and Applications. Hauppauge, NY: Nova Science; 2009.

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144 amount of light diffracted into the 1st order beam. Adjusting the pow er level of the PZT driver can therefore be used to balance th e intensity between the two beams. sin 2 m (7-1) The diffracted wave can be considered to originate from the moving diffraction grating. Since the motion of a source cause s a Doppler shift, the frequenc y of the diffracted wave is shifted from the incident light frequency. Th e Doppler shifted frequency due to the moving diffraction grating,,dsoundf, is the ratio of the velocity, v, and the wavelength, of the sound wave,,/dsound f v so that the Doppler frequency is e quivalent to the os cillator driving frequency. When the incident light encounters an approaching acoustic wave, the frequency of the diffracted beam,dif f is up-shifted as shown in Fi gure 7-2A, and when it encounters a receding wave, the frequency of the diffracted be am is down-shifted as shown in Figure 7-2B. The frequency of the diffracted wave for each or der can be identified by Eqs. 7-2 (approaching) and 7-3 (receding). 0,dif dsoundffmf (7-2) 0,dif dsoundffmf (7-3) New Interferometer Configuration A photograph and schematic diagram of the new AOM DMI setup are provided in Figure 7-3. There were three fundamental requirement s for the AOM DMI design: 1) frequency mixing was to be eliminated by keeping the two frequenc ies spatially separate; 2) both measurement and reference signals needed to be ge nerated; and 3) optical feedback into the laser cavity had to be avoided. Additionally, it was preferred that the beat frequency be adjustable. The solution to these requirements is describe d in the following paragraphs.

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145 With reference to Figure 7-3B, a single freque ncy laser is used as the light source rather than the two frequency source ap plied in polarization dependent configurations. The beam passes through a half wave plate and is incident on th e first acousto-optic modulator (AOM1). Setting the angle of AOM1 (f1 driving frequency) to the incident beam as specified in Eq. 7-1 produces the 0th order and 1st order beams. The frequency of the 0th order beam is simply the optical frequency, f0, while the 1st order beam frequency is f0 + f1. The 0th order beam continues toward the second acousto-optic modulator (AOM2) which is driven at the frequency f2, while the 1st order beam is directed toward the fixed retror eflector (RR) for the measurement signal and one of the two retroreflectors fo r the reference signal. The 0th order beam from AOM1 passes through AOM2. The 1st order (diffracted) beam from AOM2 with a frequency of f0 + f2 is directed toward the moving retroreflector and th e second retroreflector for the reference signal. The undiffracted beam is not used. Both the 1st order beams from AOM1 and AOM2 are retroreflected and recombined at AOM1 (i.e., it functions essentially as a beamsplitter). The reference interference signal (Ref) is produced after passing through a linear polarizer oriented at 45 deg with respect to the vertical axis. It is the solid line in Figur e 7-3B and has a beat frequency equal to two times the difference between the two AOM (user selected) driving frequencies as shown in Eq. 7-4. The measurem ent interference signal (Meas) is produced using the same linear polarizer. It is the dashed line in Figure 7-3B and is Doppler shifted by, fd (due to the target motion) from the beat frequency as show n in Eq. 7-5. In both cases, the beat frequency is determined as the difference between the freque ncies of the two interfered beams; the signals are collected using two photodete ctors with fiber-optic pickups. The reader may note that the angle of the 1st order diffracted beam in Figure 7-3B is exaggerated. Sufficient distance is needed

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146 for the beams from the AOMs to be separated an d directed onto each retroreflector due to the small diffraction angle from the AOMs. 01 021 212 Ref2 ff fff ff (7-4) 01 021 212 Meas2d dff ffff f ff (7-5) Periodic Error Elimination In this section, the frequenc y content of interference signals from the AOM DMI (collected using a spectrum analyzer) is presented. The und esired interference terms, which cause periodic error, are not observed in the spectrum. Data wa s collected while the ta rget displaced in the +x direction (see Figure 7-3A ). The beat frequency, 212 f ff, for the first alignment was 3.64 MHz; it was obtained by setting f1 to 40 MHz and f2 to 41.82 MHz. Figures 7-4A and 7-4B show the frequency content for velociti es of 5,000 mm/min and 10,000 mm/min. The corresponding Doppler shifts (upshifted) are 0.26 MHz and 0.53 MH z, respectively. It is seen that only the desired ac interference signal, 3.64 + 0.26/0.53 MHz, and dc power peaks are present with no content at the ac reference signal (first order period ic error) frequency of 3.64 MHz or the leakage induced ac interference signal (second order periodi c error) frequency of 3.64 0.26/0.53 MHz. In Figure 7-5, spectra for a single target ve locity (10,000 mm/min) for various: A) linear polarizer angles; and B) half wave plate angles are shown. Ag ain, unwanted interference signals

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147 are not present for any angular orientation. This result indicates that the new interferometer does not depend on polarization of the laser source. The test results of the polarization coded displacement measuring interferometer (see Fi gure 4-1A) for the same test conditions are provided in Figure 7-6 for comp arison purposes. It is seen th at both first and second order periodic error are present in both cases. One of the advantages of the new interferomet er design is the ability to tune the beat frequency by adjusting the driving frequencies of the two acousto-optic modulators. This makes the new design compatible with existing phase m easuring hardware/software independent of the system specific beat frequency (common comm ercial options are 3.65 MHz and 20 MHz). To demonstrate the variable beat frequency cap ability, the driving frequency for AOM2 was modified to f2 = 42.5 MHz, while f1 was maintained at 40 MHz. This gave a new beat frequency of f = 5 MHz. Figure 7-7 shows the spectra for this new configuration at target velocities of A) 5,000 mm/min; and B) 10,000 mm/min. No periodic error content is observed. Size Reduction Although the elimination of frequency leakag e-induced periodic error was successfully demonstrated for the AOM DMI, the size of the ne w design is too large for practical application. Reducing the setup foot print is necessary. The size of the AOM-DMI is enlarged by the small beam separation from the first AOM (AOM1) in the setup. As discu ssed, the diffracted beam angle is proportional to the wavelength of the incident beam (633 nm for He-Ne laser) and inversely proportional to the wavelength of acoustic wave propagating in the glass body for an AOM. The diffraction angle of the first order beam in the setup is 7 mrad (0.4 deg). Theref ore, a significant distance (1500 mm) is needed

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148 between the first AOM and second AOM (AOM2) to direct the first diffr acted and undiffracted beams into the reference and measurement retroreflectors, respectively. See Figure 7-8. A photograph and schematic diagram of the reduced size AOM DMI setu p are provided in Figure 7-9. In Figure 7-9B, a singl e frequency stabilized He-Ne laser is again used as the light source. The output beam of the He-Ne laser is collimated and passes th rough the first acoustooptic modulator (AOM1), oriented at the Bra gg angle, which produces the 0th order and 1st order (diffracted) beams. The right angle prism with hole is inserted at th e shortest distance from AOM1 where the 0th and 1st order b eams are visually separated. The 1st order beam is directed toward the fixed retroreflector for the m easurement signal and one of the two (fixed) retroreflectors for the reference signal (Ref. RR1 ) by reflecting at the prism surface, while the 0th order beam continues toward th e second acousto-optic modulator (AOM2) by passing beside the prism. The 1st order (diffracted) beam from AOM2 is directed toward the moving target retroreflector and second retroreflector for th e reference signal (Ref. RR2). The retroreflected beam from Ref. RR2 returns to AOM2 and pass es through the hole of the right angle prism to recombine and interfere at AOM1 with the reflected beam from th e Ref. RR1. The frequency of the reference signal is two times the freque ncy difference between the two AOMs driving frequencies. The retroreflected beams from both fixed and moving retroreflectors are also recombined at AOM1 to serve as the measurem ent signal. The frequency of the measurement signal is the same as the reference signal for no target motion, but up or down-shifted by the Doppler frequency during target motion (dependi ng on direction). Finally, both the measurement and reference signals are passed to the photodete ctors using multi-mode fiber optics. With the new setup (1400 mm (L) x 620 mm (W)), approximate ly a 65% area reduction from the original setup (2240 mm (L) x 1100 mm (W)) is achieved.

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149 As noted, in the new setup design, a right angl e prism with a hole is used to reduce the size relative to the initial setup. Figure 7-10 shows the prism with dimensions of 15 mm 15 mm 21.2 mm (A B C). The prism material is BK7 glass with an enhanced aluminum coating on the hypotenuse. To fabricate th e required hole through the pr ism center, an electroplated diamond core drill was used. Figures 7-11A a nd 7-11B show the diam ond core drill and its surface, respectively. The diameter of the beam passing through the prism is approximately 2 mm so that a 4.8 mm (3/16) hole provided suffici ent clearance. Figure 7-12 shows the drilling operation setup, where the right angle prism was submerged in the water to reduce heating. The drilling was operated at a feedrate of 0.23 mm/min. The prism afte r drilling is shown in Figure 713. As comparison, Table 7-1 lists the optical components and approximate costs for both the traditional interferometer and the AOM DMI. The main components that differentiate the two interferometers are the la ser, polarizing beam splitter, and aco usto-optic modulator. In traditional interferometers, overlapping beams that are gene rated within a two-fre quency He-Ne laser head are separated by a polarizing beam splitter, whil e two acousto-optic modu lators are used to separate beams from a single frequency stabilized He-Ne laser in the AOM DMI. The total costs are similar, but the AOM DMI offers improve d accuracy by eliminating periodic error. Table 7-1. Optical components and their prices for both the traditional interferometer and AOM DMI. Traditional interferometers AOM DMI Components Quantity Price/ea Components Quantity Price/ea Two-frequency He-Ne laser 1 $ 8,242.00 Single frequency He-Ne laser 1 $ 3,861.00 Polarizing beam splitter 1 $ 180.00 Acousto-optic modulator 2 $ 697.00 Non-polarizing beam splitter 1 $ 180. 00 Modulator driv er 2 $ 877.00 Retroreflector 2 $ 275.00 Non-polarizing beam splitter 2 $ 180.00 Right angle prism 1 $ 52.50 Retroreflector 4 $ 275.00 Right angle prism 8 $ 52.50

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150 Experimental Results In this section, frequency content of the meas urement signal collected using an analog spectrum analyzer for the new setup is presented. Sin ce the beat frequency, or frequency difference between two beams, has been shown to be tuna ble in the AOM DMI, frequency content of the measurement signal for the reduced-size setup is again reported for two different beat frequencies that are used in commercial ly-available phase measuring electronics. Figures 7-14 and 7-15 show the frequency cont ent for constant velocity target motion of 10,000 mm/min for beat frequencies of 3.64 MHz and 20 MHz, respectively. The Doppler frequency for the selected velocity is 0.53 MHz. Only the intended ac interference signal at the Doppler up-shifted frequency for the selected moti on direction (+x in Figure 7-9A) is seen. No extra peaks (i.e., leakage induced interference terms) are observed. The frequency content during oscillatory targ et motion, where the Doppler frequency can be both up-shifted and down-shifted, is presented. Figures 7-16A and 7-16B show snapshots of power spectra while the target m oves in the x (down-shifted) and +x directions (up-shifted) in Figure 7-9A, respectively. It can be seen that th e magnitude of the Doppler frequency is varied with respect to the target velo city. Again, there exists no frequency leakage induced interference terms.

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151 Figure 7-1. Acousto-optic modulator schematic. Th e incident beam is diffracted into the 0th and 1st order beams. Figure 7-2. Frequency of diffracted beam. A) Up-shi fted. B) Down-shifted. Acoustic waves 633 nm laser light PZT 0t h order beam 1s t order beam 2 f0 fdi f =f0 +fd,soundf0 f0f0 B Acoustic wave p ro p a g ation direction A PZT PZT fdif =f0 fd,sound

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152 Figure 7-3. AOM DMI setup. A) Experimental setup of heterodyne interferometer using acoustooptic modulators to spatially separate th e beams. B) Schematic of acousto-optic modulator based interferometer showing the beam separation and combination at each AOM. (RR is used to abbreviate retroreflector.) Single frequency laser Rotating half wave plate AOM1 AOM2 Fixed RR for meas. RRs for ref. Moving RR for meas. (air-bearing stage) Fiber optic pickup for ref. Fiber optic pickup for meas. Rotating polarizer = 2( f2 f1 ) = 2( f2 f1) fd f0 + f1 f0 + f1 f0 + 2 f2 f1 f0 + 2 f2 f1 fdf0 f0f0 + f1f0 + f1f0 + f2f0 + f2 f0 + 2 f2f0 + 2 f2 fdf1f2 Ref. Meas. Single frequency lase r AOM1 AOM2 RRs for meas. RRs for ref. x( )A B ( + ) LP HWP x

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153 Figure 7-4. Desired ac interference term and dc power peaks for. A) 5,000 mm/min ( fd = 0.26 MHz). B) 10,000 mm/min ( fd = 0.53 MHz). No other content is present. 0 1 2 3 3.64 5 -80 -60 -40 -20 0 power (dBm)frequency (MHz) 0 1 2 3 3.64 5 -80 -60 -40 -20 0 frequency (MHz)power (dBm) A B f f+fdf-fdf f+fdf-fd

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154 Figure 7-5. No undesired frequenc y content is present for the AO M DMI. A) Linear polarizer (LP) angle variation. B) Half wa ve plate (HWP) angle variation. 2 3 4 5 6 0 5 10 -80 -60 -40 -20 0 frequency (MHz) LP angle (deg)power (dBm) 2 3 4 5 6 0 5 10 -80 -60 -40 -20 0 frequency (MHz) HWP angle (deg)power (dBm)A B

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155 Figure 7-6. Errors for polarization coded DM I. The leakage induced interference terms (ac reference and ac interference ) accompany the desired ac interference signal. The data was collected at the same angular orientati ons as in Figure 7-5. A) Linear polarizer (LP) angle variation. B) Half wa ve plate (HWP) angle variation. 2 3 4 5 6 0 5 10 -80 -60 -40 -20 0 frequency (MHz) LP angle (deg)power (dBm) 2 3 4 5 6 0 5 10 -80 -60 -40 -20 0 frequency (MHz) HWP angle (deg)power (dBm)

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156 Figure 7-7. Results for th e new beat frequency, f of 5 MHz. No periodic error frequency content is observed for two target velo cities. A) 5,000 mm/min. B) 10,000 mm/min. 0 1 2 3 4 5 6 7 -80 -60 -40 -20 0 frequency (MHz)power (dBm)f f+fdf-fdf f+fd f-fd A B 0 1 2 3 4 5 6 7 -80 -60 -40 -20 0 power (dBm)frequency (MHz)

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157 Figure 7-8. Schematic of the current AOM-DMI setup. (RR is used to abbreviate retroreflector.) From lase r AOM 1 AOM 2 Meas. RR 1 Meas. RR 2 Ref. RR 2 Ref. RR 1 1500 mm Meas. si g nal Ref. si g nal BS BS

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158 Figure 7-9. Size reduced AOM DMI setup. A) The size reduced setup using the right angle prism with the hole. B) Schematic of the size reduced setup showing the beam separation and combination at each AOM. (RR is us ed to abbreviate retroreflector.) From laser AOM 1 AOM 2 Fixed RR for meas. Moving RR for meas. Ref. RR 2 Ref. RR 1 BS BS Meas. signal Ref. signal 400 mm Right angle mirror with hole f0 f0+ f1 f0 f0+ f1 f0+ f2 f1 f2 f0+ f2 f0+ f2 fd f0 + 2 f2 f1 fd f0 + f1 2(f2-f1) fd f0+2 f2 fd f0+2 f2 f0+2 f2-f1 f0+ f1 2(f2-f1) f0+ f1 B Single frequency laser Rotating half wave plate AOM1 AOM2 Fixed RR for meas. Ref. RR 1 Moving RR for meas. (air-bearing stage) Fiber optic pickup for ref. Fiber optic pickup for meas. Rotating polarizer ( ) x (+) Right angle prism with hole A Ref. RR 2

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159 Figure 7-10. Technical drawing for the right angle prism. Figure 7-11. An electroplated diamond core drill. A) An electropl ated diamond core drill with an outer diameter of 4.8 mm. B) Diamonds coated on the surface of the drill. 4.8 mm A B B A Enhanced aluminum coating C

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160 Figure 7-12. Schematic of drilling operation for a right angle pris m using a diamond core drill in the submerged (water) environment. Figure 7-13. The right angle pr ism with hole at the center. The hole was produced using a diamond core drill. Water container Right angle prism Prism fixture Electroplate diamond core drill

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161 Figure 7-14. Frequency contents of the measur ement signal during the c onstant velocity of 10,000 mm/min (Doppler frequency, fd, is equal to 0.53 MHz) at the beat frequency, f of 3.64 MHz. Only intended ac interferen ce and optical power terms are present. Figure 7-15. Frequency contents of the measur ement signal during the c onstant velocity of 10,000 mm/min (Doppler frequency, fd, is equal to 0.53 MHz) at the beat frequency, f of 20 MHz. Only intended ac interferen ce and optical power terms are present. 0 5 10 15 20 25 -80 -60 -40 -20 0 frequency (MHz)Power (dBm) f f-fd f+fd 0 1 2 3 3.64 5 -80 -60 -40 -20 0 frequency (MHz)Power (dBm) f f-fd f+fd

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162 Figure 7-16. Snapshots of power spectra during oscillatory target motion at the beat frequency, f of 3.64 MHz. A) The target moves toward -x direction (Doppler down-shifted). B) The target moves toward +x direction (D oppler up-shifted) in Fig. 9A. Only the intended ac interference signals are observed. 0 1 2 3 4 5 -80 -60 -40 -20 0 frequency (MHz)Power(dBm) 0 1 2 3 4 5 -80 -60 -40 -20 0 frequency (MHz)Power(dBm)A B

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163 CHAPTER 8 CONCLUSIONS Completed Work In this research, periodic e rror in heterodyne in terferometry was studied. Periodic error calculation, uncertainty evaluation, and its elimin ation were investigated. First, a generalized approach was described to calculate periodic error magnitudes by Monte Carlo evaluation. This improved upon prior work [2] where the first and second order error magnitudes were separately calculated. In the new approach the general case was treated where both the first and second order error components were considered simu ltaneously. Experiments showed good agreement between the new approach and periodic error magnitudes determined from the discrete Fourier transform of position signals collected usi ng traditional phase measuring electronics. Second, a single analytical expression for th e displacement recorded using a traditional heterodyne interferometer in terms of the various uncertainty contributor s was presented. These included: 1) periodic error; 2) Abbe error; 3) cosine error; 4) deadpath error; 5) atmospheric error; 6) material thermal expansion error; 7) laser wavelength stab ility; and 8) phase nonlinearities from the phase measuring elec tronics and beam shear. The displacement uncertainty was then evaluated using Mont e Carlo simulation. A numerical example demonstrated the well-known cosine error bias, as well as the correction of this bias using the variance in the misalignment angle. In this analys is, the analytical expres sion reported by Cosijns et al [5] was used to describe the periodic error. Comparisons between th e periodic error model and experiments for a variety of frequency mixing conditions were provided Third, a new displacement measuring interf erometer design with no periodic error was demonstrated. In the new design, the two (heter odyne) frequencies were ge nerated and spatially separated using acousto-optic m odulators to avoid the potential for frequency mixing. This

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164 contrasted with the traditional setup, where th e two optical frequencies are initially coincident with orthogonal polarization states and then separated using polarization dependent optics (polarization coded). Experimental results at multiple target velocities were presented for two arrangements of the new design, which also show ed the capability to ar bitrarily set the beat frequency. Spectral content was collected using a spectrum analyzer to verify zero periodic error in the new design. These results were compared to data obtained from a traditional polarization coded heterodyne interferometer. It was shown that variations in the optical alignment caused different levels of first and sec ond order periodic error in the tradi tional setup, but not in the new design. However, the small diffraction angle (7 mrad) between the two beams exiting the acousto-optic modulator caused the large setup size of the initial design. Therefore, the footprint of the new heterodyne displacement measuring in terferometer was reduced using a right angle prism with a hole through its center. It is anticipated that the new, periodic error-free interferometer design will enable improved measurement accuracy for various applications, including position feedback for lithographic stepper stages, precision cutting machines, and coordinate measuring machines, as well as transducer calibration, for example. To summarize, the contributions to periodic error research in the field of displacement measuring interferometry are: calculation of first and second order periodic error magnitude s for the general case using spectrum analyzer data; evaluation of uncertainty for he terodyne displacement measur ing interferometry via Monte Carlo simulation; and elimination of periodic error by a new interferometer design. Future Work An acousto-optic modulator-based heterodyne displacement measuring interferometer (AOM DMI) realized successful periodic error elimin ation. Due to the large setup size, the initial

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165 design was modified to reduce the footprint using a right angle pr ism with a hole at the center. Although the modified design redu ced the setup size significantly compared to the original design (65% area reduction), it is s till large compared to traditi onal interferometers. Figures 81A and B show setup sizes for both the typical tr aditional interferometer and the modified AOM DMI, respectively, for single axis measurement. The traditional interferometer setup area (800 mm 350 mm) is approximately three times sma ller than that of the AOM DMI (1400mm 620 mm). Acousto-optic devices have been used in lase rs for Q-switching [92], laser scanning, and in spectroscopy for frequency control [93] because they provide diffraction, frequency modulation, and intensity modulation of light. For these applications, only the diffracted beam is used and the undiffracted beam is blocked. Therefore, the di ffraction angle has not been an issue in these cases. However, the acousto-optic modulator in the new heterodyne displacement measuring interferometer uses both the diffracted and undiffracted beams to gene rate the interference signal. The size of the new interferometer se tup heavily depends on the diffraction angle (as described in Chapter 7). Conse quently, research on new acousto-optic modulator designs to increase the beam separation would enab le miniaturization of the AOM DMI.

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166 Figure 8-1. Setup size co mparison. A) Traditiona l interferometer (800 mm 350 mm). B) AOM DMI (1400 mm 620 mm). AOM 1 AOM 2 Fixed RR for meas. Moving RR for meas. Ref. RR 1 BS BS Fiber optic pick-up for meas. Fiber optic pick-up for ref. He-Ne lase r Fiber optic pick-up for meas. Reference retroreflecto r Measurement retroreflector PBS He-Ne lase r Fiber optic pick-up for ref. Fixed polarizer Rotating polarizer Air-bearing stage BS 800 mm A B 1400 mm 350 mm 620 mm Ref. RR 2

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167 LIST OF REFERENCES [1] Kim H, Schmitz T, Beckwith J. Periodic error in heterodyne in terferometry: Examination and Elimination. In: Halsey D, Raynor W, ed Handbook of Interferometers: Research, Technology and Applications. Haupp auge, NY: Nova Science; 2009. [2] Badami V, Patterson S. A frequency domain me thod for the measurement of nonlinearity in heterodyne interferometry. Preci sion Engineering 2000; 24(1):41. [3] Donaldson R. Error budgets. In: Hocken R, editor. Technology of Machine Tools. Machine Tool Accuracy, vol. 5. Machine Tool Task Force (UCRL); 1980. [4] Slocum A. Precision Machine Design. Englewood Cliffs, NJ: Prentice Hall; 1992. [5] Cosijns S, Haitjema H, Schellekens P. Mode ling and verifying non-lin earities in heterodyne displacement interferometry. Prec ision Engineering 2002; 26:448. [6] Pedrotti L, Pedrotti M, Pedrot ti S. Introduction to Optics, 3rd Ed. New Jersey: Pearson Education; 2007. [7] Hariharan P. Basics of Interferometry, 2nd Ed. Boston: Academic Press; 2007. [8] Hlubina P. White-light spectral interferom etry with the uncompensated Michelson interferometer and the group refractive i ndex dispersion in fused silica. Optics Communications 2001; 193:1. [9] Eom T, Kim J, Jeong K. The dynamic compensation of nonlinearity in a homodyne laser interferometer. Measurement Sc ience and Technology 2001; 12:1734. [10] Schmitz T, Houck III L, Chu D, Kalem L. Bench-t op setup for validation of real time, digital periodic error correction. Pr ecision Engineering 2006; 30:306. [11] Peggs G, Yacoot A. A review of recent wo rk in sub-nanometer displacement measurement using optical and X-ray interferometry. Philosoph ical Transactions of the Royal Society of London Series A-Mathematical Physical and Engineering Sciences 2002; 360(1794):953 968. [12] Schmitz T, Beckwith J. An investigation of two unexplored periodic error sources in differential-path interf erometry. Precision Engi neering 2002; 27(3):311. [13] Chu D. Alta P. Phase digitizer. United States Patent No. US 6 480 126 B1, 2002 [14] Agilent Technologies. Achieving maximum accur acy and repeatability. Santa Clara, CA 2001. [15] Edlen B. The refractive index of air. Metrologia 1966; 2:71. [16] Bobroff N. Residual errors in laser interferometry from air turbulence and nonlinearity. Applied Optics 1987; 26(13):2676

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174 BIOGRAPHICAL SKETCH The author was born in Changnyeong, Kye ongnam Korea in 1976. After finishing his m iddle school education at his hometown, he we nt to Changwon where he completed his high school education. He remained there to pur sue his undergraduate degree in mechanical engineering at Changwon National University (CNU). At CNU, he received a full scholarship, including tuition waiver and stipend, thr oughout his undergraduate st udies. As a mandatory requirement, he served in the military (Republic of Korea Navy) for 28 months. While studying at CNU, he was sponsored by th e project named Brain Korea 21st to go to University of Nebraska for one year as a full scholar exchange student. After graduation, the author worked for three years before joining the Ph.D. program at University of Florida (UF), where he was awarded a UF Alumni Fellowship. During hi s Ph.D. studies, he received several honors including the NAMRI/SME Outstanding Paper Fi nalist Award, Samsung Electromechanical 4th 1nside Edge International Paper Competition Silver Prize, Stud ent Scholarship for the Annual American Society for Precision Engineering Conference, and a UF Outstanding Academic Achievement Award. The author plans to seek a university faculty posit ion after graduation.