The motivation for this project is to design a noveloptical system for quasi-real-time alignment oftracker detector elements used in high-energy phys-ics experiments. Fox-Murphy et al. from Oxford Uni-versity reported their design of a frequency scannedinterferometer (FSI) for precise alignment of theATLAS Inner Detector.1 Given the demonstratedneed for improvements in detector performance, weplan to design an enhanced FSI system to be used forthe alignment of tracker elements in the next gener-ation of electron–positron linear collider detectors.Current plans for future detectors require a spatialresolution for signals from a tracker detector, such asa silicon microstrip or a silicon drift detector, to beapproximately 7–10 �m.2 To achieve this requiredspatial resolution, the measurement precision of ab-solute distance changes of tracker elements in onedimension should be of the order of 1 �m. Simulta-neous measurements from hundreds of interferom-eters will be used to determine the three-dimensionalpositions of the tracker elements.
We describe here a demonstration FSI system builtin the laboratory for initial feasibility studies. Themain goal was to determine the potential accuracy ofabsolute distance measurements (ADMs) that couldbe achieved under controlled conditions. Secondarygoals included estimating the effects of vibrationsand studying error sources crucial to the absolutedistance accuracy. A significant amount of researchon ADMs by use of wavelength-scanning interferom-eters already exists.3–8 In one of the most compre-hensive publications on this subject, Stone et al.describe in detail a wavelength-scanning heterodyneinterferometer consisting of a system built aroundboth a reference and a measurement interferometer3;the measurement precisions of absolute distanceranging from 0.3 to 5 m are �250 nm by the averag-ing of distance measurements from 80 independentscans.
Detectors for high-energy physics experiment mustusually be operated remotely for safety reasons be-cause of intensive radiation, high voltage, or strongmagnetic fields. In addition, precise tracking ele-ments are typically surrounded by other detectorcomponents, making access difficult. For practicalhigh-energy physics application of FSI, optical fibersfor light delivery and return are therefore necessary.
We constructed a FSI demonstration system byemploying a pair of single-mode optical fibers that areeach approximately 1 m in length, one for transport-ing the laser beam to the beam splitter and retrore-flector and another for receiving return beams. A key
The authors are with the Department of Physics, University ofMichigan, Ann Arbor, Michigan 48109-1120. H.-J. Yang’s e-mailaddress is [email protected].
Received 12 July 2004; revised manuscript received 24 Novem-ber 2004; accepted 25 November 2004.
issue for the optical fiber FSI is that the intensity ofthe return beams received by the optical fiber is veryweak; the natural geometric efficiency is 6.25� 10�10 for a measurement distance of 0.5 m. In ourdesign, we use a gradient-index (GRIN) lens to colli-mate the output beam from the optical fiber.
We believe our work represents a significant ad-vancement in the field of FSI in that high-precisionADMs and vibration measurements are performed(without a priori knowledge of vibration strengthsand frequencies) using a tunable laser, an isolator, anoff-the-shelf Fabry–Perot (F–P) interferometer, a fi-ber coupler, two single-mode optical fibers, an inter-ferometer, and novel fringe analysis and vibrationextraction techniques. Two new multiple-distance-measurement analysis techniques are used to im-prove precision and to extract the amplitude andfrequency of vibrations. Expected dispersion effectswhen a corner cube prism or a beam-splitter sub-strate lies in the interferometer beam path are con-firmed, and observed results agree well with resultsfrom numerical simulation. When present, the dis-persion effect has a significant effect on the ADM.The limitations of our current FSI system are alsodiscussed and major uncertainties are estimated.
2. Principles
The intensity I of any two-beam interferometer canbe expressed as
I � I1 � I2 � 2�I1I2 cos��1 � �2�, (1)
where I1 and I2 are the intensities of the two com-bined beams and �1 and �2 are the phases. Assumingthat the optical path lengths of the two beams are L1and L2, the phase difference in Eq. (1) is � � �1� �2 � 2�|L1 � L2|��c�, where � is the opticalfrequency of the laser beam and c is the speed of light.
For a fixed-path interferometer, as the frequency ofthe laser is continuously scanned, the optical beamswill constructively and destructively interfere, caus-ing fringes. The number of fringes N is
N � |L1 � L2|��c� � L�c, (2)
where L is the optical path difference (OPD) betweenthe two beams and is the scanned frequencyrange. The OPD (for absolute distance between thebeam splitter and the retroreflector) can be deter-mined by counting interference fringes while scan-ning the laser frequency.
3. Demonstration System of the Frequency ScannedInterferometer
A schematic of the FSI system with a pair of opticalfibers is shown in Fig. 1. The light source is a NewFocus Velocity 6308 tunable laser �665.1 nm � �� 675.2 nm�. A high-finesse � 200� Thorlabs SA200F–P is used to measure the frequency range scannedby the laser. The free spectral range of two ad-jacent F–P peaks is 1.5 GHz, which corresponds to
0.002 nm. A Faraday isolator was used to reject lightreflected back into the lasing cavity. The laser beamwas coupled into a single-mode optical fiber with afiber coupler. Data acquisition is based on a NationalInstruments data-acquisition card capable of simul-taneously sampling four channels at a rate of 5 (MS�s)�channel (here MS means million samples) with aprecision of 12 bits. Omega thermistors with a toler-ance of 0.02 K and a precision of 0.01 mK are used tomonitor the temperature. The apparatus is supportedon a damped Newport optical table.
To reduce airflow and temperature fluctuations, atransparent plastic box was constructed on top of theoptical table. Polyvinylchloride (PVC) pipes were in-stalled to shield the volume of air surrounding thelaser beam. Inside the PVC pipes, the typical stan-dard deviation of 20 temperature measurements was�0.5 mK. Temperature fluctuations were suppressedby a factor of approximately 100 by employing theplastic box and PVC pipes.
The beam intensity coupled into the return opticalfiber is very weak, requiring ultrasensitive photode-tectors for detection. Considering the limited laserbeam intensity and the need to split into many beamsto serve a set of interferometers, it is vital to increasethe geometric efficiency. To this end, a collimator isbuilt by placing an optical fiber in a ferrule (1 mm indiameter) and gluing one end of the optical fiber to aGRIN lens. The GRIN lens is a 0.25 pitch lens with a0.46 numerical aperture, 1 mm in diameter, and2.58 mm in length that is optimized for a wavelengthof 630 nm. The density of the outgoing beam from theoptical fiber is increased by a factor of approximately1000 by use of a GRIN lens. The return beams arereceived by another optical fiber and amplified by asilicon femtowatt photoreceiver with a gain of2 � 1010 V�A.
4. Multiple-Distance-Measurement Techniques
For a FSI system, drifts and vibrations occurringalong the optical path during the scan will be mag-nified by a factor of � � �, where � is the averageoptical frequency of the laser beam and is thescanned frequency. For the full scan of our laser,� � 67. Small vibrations and drift errors that havenegligible effects for many optical applications mayhave a significant effect on a FSI system. A single-
Fig. 1. Schematic of an optical fiber FSI system. BS, beamsplitter.
frequency vibration may be expressed as xvib�t� �avib cos�2�fvibt � �vib�, where avib, fvib, and �vib are theamplitude, frequency, and phase of the vibration, re-spectively. If t0 is the start time of the scan, Eq. (2)can be rewritten as
N � L�c � 2�xvib�t��t� � xvib�t0��t0���c. (3)
If we approximate �t� � �t0� � , the measured OPDLmeas may be expressed as
where Ltrue is the true OPD without vibration effects.If the path-averaged refractive index of ambient airn� g is known, the measured distance is Rmeas �Lmeas��2n� g�.
If the measurement window size �t � t0� is fixed andthe window used to measure a set of Rmeas is sequen-tially shifted, the effects of the vibration will be evi-dent. We use a set of distance measurements in onescan by successively shifting the fixed-length mea-surement window one F–P peak forward each time.The arithmetic average of all measured Rmeas valuesin one scan is taken to be the measured distance ofthe scan (although more sophisticated fitting meth-ods can be used to extract the central value). For alarge number of distance measurements Nmeas, thevibration effects can be greatly suppressed. Of coursestatistical uncertainties from fringe and frequencydetermination, dominant in our current system, canalso be reduced with multiple scans. Averaging mul-tiple measurements in one scan, however, providessimilar precision improvement as averaging distancemeasurements from independent scans and is faster,more efficient, and less susceptible to systematic er-rors from drift. In this way, we can improve the dis-tance accuracy dramatically if there are nosignificant drift errors during one scan, caused, forexample, by temperature variation. This multiple-distance-measurement technique is called slip mea-surement window with fixed size, shown in Fig. 2.However, there is a trade-off in that the thermal drifterror is increased with the increase of Nmeas becauseof the larger magnification factor � for a smallermeasurement window size.
To extract the amplitude and frequency of the vi-bration, another multiple-distance-measurementtechnique called slip measurement window withfixed start point is shown in Fig. 2. In Eq. (3), if t0 isfixed, the measurement window size is one enlargedF–P peak for each shift; an oscillation of a set ofmeasured Rmeas values reflects the amplitude and fre-quency of vibration. This technique is not suitable fordistance measurement because there always existsan initial bias term including t0 that cannot be deter-mined accurately in our current system.
5. Absolute Distance Measurement
The typical measurement residual versus the dis-tance measurement number in one scan by use of theabove technique is shown in Fig. 3(a), where the scan-ning rate was 0.5 nm�s and the sampling rate was125 ksamples�s. Measured distances minus their av-erage value for ten sequential scans are plotted ver-sus number of measurements �Nmeas� per scan in Fig.3(b). The standard deviations (rms) of distance mea-surements for ten sequential scans are plotted versus
Fig. 2. Schematic of two multiple-distance-measurement tech-niques. The interference fringes from the femtowatt photoreceiverand the scanning frequency peaks from the F–P interferometer forthe optical fiber FSI system recorded simultaneously by a data-acquisition card are shown as waves and sharp peaks, respectively.The free spectral range of two adjacent F–P peaks �1.5 GHz� pro-vides a calibration of the scanned frequency range.
Fig. 3. Distance measurement residual spreads versus number ofdistance measurement Nmeas (a) for one typical scan, (b) for tensequential scans, (c) the standard deviation of distance measure-ments for ten sequential scans versus Nmeas.
number of measurements �Nmeas� per scan in Fig. 3(c).It can be seen that the distance errors decrease withan increase of Nmeas. The rms of measured distancesfor ten sequential scans is 1.6 �m if there is only onedistance measurement per scan �Nmeas � 1�. If Nmeas� 1200 and the average value of 1200 distance mea-surements in each scan is considered as the finalmeasured distance of the scan, the rms of the finalmeasured distances for ten scans is 41 nm for thedistance of 449828.965 �m; the relative distancemeasurement precision is 91 parts per billion(ppb).
Some typical measurement residuals are plottedversus the number of distance measurements in onescan �Nmeas� for open-box and closed-box data withscanning rates of 2 and 0.5 nm�s in Figs. 4(a)–4(d),respectively. The measured distance is approxi-mately 10.4 cm. It can be seen that the slow fluctu-ations of multiple distance measurements for open-box data are larger than that for closed-box data.
The standard deviation (rms) of measured dis-tances for ten sequential scans is approximately1.5 �m if there is only one distance measurement perscan for closed-box data. By use of the multiple-distance-measurement technique, the distance mea-surement precisions for various closed-box data withdistances ranging from 10 to 70 cm collected in thepast year are improved significantly. Precisions ofapproximately 63 nm are demonstrated under labo-ratory conditions, as shown in Table 1. All measuredprecisions listed in the Table 1 are the rms of mea-sured distances for ten sequential scans. Two FSIdemonstration systems, air FSI and optical fiber FSI,are constructed for extensive tests of the multiple-distance-measurement technique; air FSI means FSIwith the laser beam transported entirely in theambient atmosphere, and optical fiber FSI representsFSI with the laser beam delivered to the interferom-eter and received back by single-mode optical fibers.
On the basics of our studies, the slow fluctu-ations are reduced to a negligible level by using theplastic box and PVC pipes to suppress temperaturefluctuations. The dominant error comes from theuncertainties of the interference fringes’ numberdetermination; the fringes’ uncertainties are uncor-related for multiple distance measurements. In this
case, averaging multiple distance measurements inone scan provides a precision improvement similarto averaging distance measurements from multipleindependent scans, but is faster, more efficient, andless susceptible to systematic errors from drift.However, for open-box data, the slow fluctuationsare dominant, of the order of few micrometers in ourlaboratory. The measurement precisions for single-and multiple-distance open-box measurements arecomparable, which indicates that the slow fluctua-tions cannot be adequately suppressed by using themultiple-distance measurement technique. A dual-laser FSI system6,9 intended to cancel the drift error
Table 1. Distance Measurement Precisions for Various Setups with the Multiple-Distance-Measurement Technique
Fig. 4. Distance measurement residual spreads versus Nmeas inone scan: (a) for the open box with a scanning rate of 2 nm�s, (b) forthe closed box with a scanning rate of 2 nm�s, (c) for the open boxwith a scanning rate of 0.5 nm�s, (d) for the closed box with ascanning rate of 0.5 nm�s.
is currently under study in our laboratory (to bedescribed in a subsequent paper).
From Fig. 4(d) we observe periodic oscillation ofthe distance measurement residuals in one scan,where the fitted frequency is 3.22 � 0.01 Hz for thescan. The frequency depends on the scanning ratef � �scanning rate in nm�s� � 60��675.1 nm� 665.1 nm�. From Eq. (4) it is clear that the am-plitude of the vibration or oscillation pattern formultiple distance measurements depends on4avib� sin��fvib�t � t0��. If avib and fvib are constantvalues, it depends on the size of the distance measure-ment window. Subsequent investigation with a CCDcamera trained on the laser output revealed that theapparent �3 Hz vibration during the 0.5 nm�s scanarose from the beam’s centroid motion. Because thecentroid motion is highly reproducible, we believethat the effect comes from motion of the internalhinged mirror in the laser used to scan its frequency.
The measurable distance range is limited in ourcurrent optical fiber FSI demonstration system forseveral reasons. For a given scanning rate of0.25 nm�s, the produced interference fringes, esti-mated by N � �2 � L � ��c, are approximately26,400 in a 40 s scan for a measured distance �L� of60 cm, that is, �660 fringes�s, where is thescanned frequency and c is the speed of light. Thecurrently used femtowatt photoreceiver has a 3 dBfrequency bandwidth ranging from 30 to 750 Hz; thetransimpedance gain decreases quickly beyond750 Hz. There are two ways to extend the measurabledistance range. One straightforward way is to extendthe effective frequency bandwidth of the femtowattphotoreceiver; the other way is to decrease the inter-ference fringe rate by decreasing the laser scanningrate. There are two major drawbacks for the latter;one is that larger slow fluctuations occur duringlonger scanning times; the other is that the laserscanning is not stable enough to produce reliable in-terference fringes if the scanning rate is lower than0.25 nm�s for our present tunable laser. In addition,another limitation to distance range is that the in-tensity of the return beam from the retroreflectordecreases inverse quadratically with range.
6. Vibration Measurement
To test the vibration measurement technique, a pi-ezoelectric transducer was employed to produce vi-brations of the retroreflector. For example, thefrequency of the controlled vibration source was set to1.01 � 0.01 Hz with an amplitude of 0.14� 0.02 �m. For Nmeas � 2000 distance measurementsin one scan, the magnification factor for each distancemeasurement depends on the scanned frequency ofthe measurement window, ��i� � ��i�, where � isthe average frequency of the laser beam in the mea-surement window, and scanned frequency �i�� �4402 � Nmeas � i� � 1.5 GHz, where i runs from 1to Nmeas, shown in Fig. 5(a). The distance measure-ment residuals for 2000 distance measurements inthe scan are shown in Fig. 5(b); the oscillation of themeasurement residuals reflect the vibration of the
retroreflector. Since the vibration is magnified by afactor of ��i� for each distance measurement, thecorrected measurement residuals are measurementresiduals divided by the corresponding magnificationfactors, shown in Fig. 5(c). The extracted vibrationfrequencies and amplitudes that are used with thistechnique are fvib � 1.007 � 0.0001 Hz and Avib� 0.138 � 0.0003 �m, respectively, which is in goodagreement with expectations.
Another demonstration was made for the same vi-bration frequency, but with an amplitude of only9.5 � 1.5 nm. The magnification factors, distancemeasurement residuals, and corrected measurementresiduals for 2000 measurements in one scan areshown in Figs. 6(a)–6(c), respectively. The extractedvibration frequencies and amplitudes used with thistechnique are fvib � 1.025 � 0.002 Hz and Avib� 9.3 � 0.3 nm.
In addition, vibration frequencies at 0.1, 0.5, 1.0, 5,10, 20, 50, and 100 Hz with controlled vibration am-plitudes ranging from 9.5 to 400 nm were studiedextensively using our current FSI system. The mea-sured vibrations and expected vibrations all agreewell within the 10%–15% level for amplitudes, and1%–2% for frequencies, where we are limited by un-certainties in the expectations. Vibration frequenciesfar below 0.1 Hz can be regarded as slow fluctuations,which cannot be suppressed by the above analysistechniques.
For comparison, nanometer vibration measure-ment by a self-aligned optical feedback vibrometry
Fig. 5. Frequency and amplitude of the controlled vibrationsource are 1 Hz and 140 nm. (a) Magnification factor versus num-ber of distance measurements, (b) distance measurement residualversus number of distance measurements, (c) corrected measure-ment residual versus number of distance measurements.
technique has been reported.10 The vibrometry tech-nique is able to measure vibration frequencies rang-ing from 20 Hz to 20 kHz with a minimal measurablevibration amplitude of 1 nm. Our second multiple-distance-measurement technique demonstratedabove has the capability to measure vibration fre-quencies ranging from 0.1 to 100 Hz with minimalamplitude on the level of several nanometers, with-out a priori knowledge of the vibration strengths orfrequencies.
7. Effect of Dispersion Effects on DistanceMeasurement
Dispersive elements such as a beam splitter or acorner cube prism in the interferometer can create anapparent offset in measured distance for a FSI sys-tem since the optical path length of the dispersiveelement changes during the scan. The small OPDchange caused by dispersion is magnified by a factorof � and has a significant effect on the absolute dis-tance measurement for the FSI system. The mea-sured OPD difference Lmeas can be expressed as
Lmeas � |L�t����t� � L�t0����t0�|c�,
L�t� � 2D1nair � D2n���t��corner cube, (5)
where L�t� and L�t0� refer to the OPD at times tand t0, respectively; ��t� and ��t0� are the wave-length of the laser beam at times t and t0; c is thespeed of light; D1 and D2 are true geometric dis-
tances in the air and in the corner cube prism; and nairand n���t��corner cube are the refractive index of ambientatmosphere and the refractive index of the cornercube prism for ��t�, respectively. The measured dis-tance Rmeas � Lmeas��2n� g�, where n� g is the averagerefractive index around the optical path.
The Sellmeier formula for dispersion in crown glass(BK-7)11 can be written as
n2(�) � 1 �B1�
2
�2 � C1�
B2�2
�2 � C2�
B3�2
�2 � C3, (6)
where the beam wavelength � is in units of micro-meters, B1 � 1.03961212, B2 � 0.231792344, B3 �1.01046945, C1 � 0.00600069867, C2 �0.0200179144, and C3 � 103.560653.
If we use the first multiple-distance-measurementtechnique described above to make 2000 distancemeasurements for one typical scan, where the cornercube prism is used as retroreflector, we observe ahighly reproducible drift in measured distance, asshown in Fig. 7 where the fitted distance drift is6.14 � 0.08 �m for one typical scan with a straight-line fit. However, there is no apparent drift if wereplace the corner cube prism by the hollow retrore-flector.
Numerical simulations have been carried out usingEqs. (5) and (6) to understand the above phenomena.For example, consider the case D1 � 20.97 cm andD2 � 1.86 cm (the uncertainty of D2 is 0.06 cm),where the first and last measured distances among2000 sequential distance measurements are denotedR1 and R2000, respectively. Using the Sellmeier equa-tion [Eq. (6)] to model the corner cube prism material(BK-7) dispersion, we expect R1 � �D1 � D2� �373.876 �m and R2000 � �D1 � D2� � 367.707 �m.The difference between R1 and R2000 is 6.2� 0.2 �m, which agrees well with our observed6.14 � 0.08 �m drift over 2000 measured distances.The measured distance shift and drift strongly de-pend on D2, but are insensitive to D1. A change of1 cm in D1 leads to a 3 nm distance shift, but thesame change in D2 leads to a 200 �m distance shift.If a beam splitter is oriented with its reflecting side
Fig. 6. Frequency and amplitude of the controlled vibrationsource are 1 Hz and 9.5 nm. (a) Magnification factor versus num-ber of distance measurements, (b) distance measurement residualversus number of distance measurements, (c) corrected measure-ment residual versus number of distance measurements.
Fig. 7. Residuals of 2000 distance measurements for one typicalscan; the corner cube prism is used as retroreflector.
facing the laser beam, then there is an additionaldispersive distance shift. We verified this effect with1 and a 5 mm beam splitters. When we insert anadditional beam splitter with 1 mm thickness be-tween the retroreflector and the original beam split-ter in the optical fiber FSI system, we observe a500 �m shift on measured distance if ng is fixed con-sistent with the numerical simulation result. For the5 mm beam splitter (the measured thickness of thebeam splitter is 4.6 � 0.05 mm), the first 20 scanswere performed with the beam splitter’s antireflect-ing surface facing the optical fibers and the second 20scans with the reflecting surface facing the opticalfibers. The expected drifts �R2000 � R1� for the firstand the second 20 scans from the dispersion effect are0 and �1.53 � 0.05 �m, respectively. The measureddrifts obtained by averaging the measurements from20 sequential scans are �0.003 � 0.12 �m and�1.35 � 0.17 �m, respectively. The measured valuesagree well with expectations. In addition, the disper-sion effect from air12,13 is also estimated by usingnumerical simulation. The expected drift �R2000� R1� from air dispersion is approximately �0.07 �mfor an optical path of 50cm in air; this effect cannot bedetected for our current FSI system. However, itcould be verified by using a FSI with a vacuum tubesurrounding the laser beam; the measured distancewith air in the tube would be approximately 4 �mlarger than for an evacuated tube.
In summary, dispersion effects can have a signifi-cant effect on absolute distance measurements, butcan be minimized with care for elements placed in theinterferometer or corrected for once any necessarydispersive elements in the interferometer are under-stood.
8. Error Estimations
Some major error sources are estimated in the fol-lowing:
(1) Error from uncertainties of fringe and scannedfrequency determination. The measurement preci-sion of the distance R (the error due to the air’srefractive-index uncertainty is considered separatelybelow) is given by ��R�R�2 � ��N�N�2 � ����2,where R, N, , �R, �N, and � are measurementdistance, fringe numbers, scanned frequency, andtheir corresponding errors. For a typical scanningrate of 0.5 nm�s with a 10 nm scan range, the fullscan time is 20 s. The total number of samples for onescan is 2.5 MS at a sampling rate of 125 kS�s. Thereis an approximate 4–5 sample ambiguity in fringepeak and valley position due to a vanishing slope andthe limitation of the 12 bit sampling precision. How-ever, there is a much smaller uncertainty for the F–Ppeaks because of their sharpness. Thus the estimateduncertainty is �R�R � 1.9 parts per million (ppm) forone full scan for a magnification factor � � 67. If thenumber of distance measurements Nmeas � 1200, thedistance measurement window is smaller and thecorresponding magnification factor is �* � �,where � is the average frequency of the laser beam,
� �4402 � Nmeas� � 1.5 GHz. One obtains �*� 94 and �R�R � 1.9 ppm � �*����Nmeas� 77 ppb.
(2) Error from vibrations. The detected amplitudeand frequency for vibration (without a controlled vi-bration source) are �0.3 �m and 3.2 Hz. The corre-sponding time for Nmeas � 1200 sequential distancemeasurements is 5.3 s. A rough estimation of theresulting error gives �R�R � 0.3 �m��5.3 s� 3.2 Hz � 4��R � 10 ppb for a given measureddistance of R � 0.45 m.
(3) Error from thermal drift. The refractive indexof air depends on air temperature, humidity, andpressure (fluctuations of humidity and pressure havenegligible effects on distance measurements for the20 s scan). Temperature fluctuations are well con-trolled down to �0.5 mK (rms) in our laboratory bythe plastic box on the optical table and the pipeshielding the volume of air near the laser beam. Fora room temperature of 21 °C, an air temperaturechange of 1 K will result in a 0.9 ppm change of airrefractive index. For a temperature variation of0.5 mK in the pipe, Nmeas � 1200 distance measure-ments, the estimated error will be �R�R� 0.9 ppm�K � 0.5 mK � �* � 42 ppb, where themagnification factor �* � 94.
The total error from the above sources, when addedin quadrature, is �89 ppb, with the major errorsources arising from the uncertainty of fringe deter-mination and the thermal drift. The estimated rela-tive error agrees well with measured relative spreadsof 91 ppb in real data for a measured distance of�0.45 m.
In addition to the above error sources, othersources can contribute to systematic bias in the ab-solute differential distance measurement. The majorsystematic bias comes from the uncertainty in thefree spectral range of the F–P used to determine thescanned frequency range. The relative error would be�R�R � 50 ppb if the FSR were calibrated by awavemeter with a precision of 50 ppb. A wavemeterof this precision was not available for the measure-ments described here. The systematic bias from themultiple-distance-measurement technique was alsoestimated by changing the starting point of the mea-surement window, the window size, and the numberof measurements; the uncertainties typically rangefrom 10 to 50 nm. Systematic bias from uncertaintiesin temperature, air humidity, and barometric pres-sure scales are estimated to be negligible.
9. Conclusion
An optical fiber FSI system was constructed to makehigh-precision absolute distance and vibration mea-surements. A design of the optical fiber with a GRINlens was presented that improves the geometric effi-ciency significantly. Two new multiple-distance-measurement analysis techniques were presented toimprove distance precision and to extract the ampli-tude and frequency of vibrations. ADM precisions ofapproximately 50 nm for distances ranging from 10
to 70 cm under laboratory conditions were achievedusing the first analysis technique. The second anal-ysis technique measures vibration frequencies rang-ing from 0.1 to 100 Hz with a minimal amplitude of afew nanometers. We verified an expected dispersioneffect and confirmed its importance when we placeddispersive elements in the interferometer. Major er-ror sources were estimated, and the observed errorswere found to be in good agreement with expectation.
This research is supported by the National ScienceFoundation and the U.S. Department of Energy.
References1. A. F. Fox-Murphy, D. F. Howell, R. B. Nickerson, and A. R.
Weidberg, “Frequency scanned interferometry (FSI): the basisof a survey system for ATLAS using fast automated remoteinterferometry,” Nucl. Instrum. Methods A 383, 229–237(1996).
2. American Linear Collider Working Group, “Linear ColliderPhysics, Resource Book for Snowmass 2001,” hep-ex�0106058,SLAC-R-570 299-423 (Stanford Linear Collider Center, Stan-ford University, 2001), pp. 299–423.
3. J. A. Stone, A. Stejskal, and L. Howard, “Absolute interferom-etry with a 670�nm external cavity diode laser,” Appl. Opt. 38,5981–5994 (1999).
4. D. Xiaoli and S. Katuo, “High-accuracy absolute distance mea-
surement by means of wavelength scanning heterodyne inter-ferometry,” Meas. Sci. Technol. 9, 1031–1035 (1998).
5. G. P. Barwood, P. Gill, and W. R. C. Rowley, “High-accuracylength metrology using multiple-stage swept-frequency inter-ferometry with laser diodes,” Meas. Sci. Technol. 9, 1036–1041(1998).
6. K. H. Bechstein and W. Fuchs, “Absolute interferometric dis-tance measurements applying a variable synthetic wave-length,” J. Opt. 29, 179–182 (1998).
7. J. Thiel, T. Pfeifer, and M. Haetmann, “Interferometric mea-surement of absolute distances of up to 40 m,” Measurement16, 1–6 (1995).
8. H. Kikuta and R. Nagata, “Distance measurement by wave-length shift of laser diode light,” Appl. Opt. 25, 976–980 (1986).
9. P. A. Coe, “An investigation of frequency scanning interferom-etry for the alignment of the ATLAS semiconductor tracker,”Ph.D. dissertation (St. Peter’s College, University of Oxford,Oxford, UK, 2001).
10. K. Otsuka, K. Abe, J. Y. Ko, and T. S. Lim, “Real-timenanometer-vibration measurement with a self-mixing micro-chip solid-state laser,” Opt. Lett. 27, 1339–1341 (2002).
11. SCHOTT’96 for Windows Catalog Optical Glass (Schott Gla-swerke Mainz, Germany, 1996), http://us.schott.com/sgt/english/products/catalogs.html.
12. E. R. Peck and K. Reeder, “Dispersion of air,” J. Opt. Soc. Am.62, 958–962 (1972).
13. P. E. Ciddor, “Refractive index of air: new equations for thevisible and near infrared,” Appl. Opt. 35, 1566–1573 (1996).
