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the motivation for this project is to design a novel optical system for quasi - real time alignment of tracker detector elements used in high energy physics ( hep ) experiments . fox - murphy _ et.al . _ from oxford university reported their design of a frequency scanned interferometer ( fsi ) for precise alignment of the atlas inner detector @xcite . given the demonstrated need for improvements in detector performance , we plan to design an enhanced fsi system to be used for the alignment of tracker elements in the next generation of electron positron linear collider ( ilc ) detectors . current plans for future detectors require a spatial resolution for signals from a tracker detector , such as a silicon microstrip or silicon drift detector , to be approximately 7 - 10 @xmath1@xcite . to achieve this required spatial resolution , the measurement precision of absolute distance changes of tracker elements in one dimension should be on the order of 1 @xmath1 . simultaneous measurements from hundreds of interferometers will be used to determine the 3-dimensional positions of the tracker elements . we describe here a demonstration fsi system built in the laboratory for initial feasibility studies . the main goal was to determine the potential accuracy of absolute distance measurements ( adm s ) that could be achieved under controlled conditions . secondary goals included estimating the effects of vibrations and studying error sources crucial to the absolute distance accuracy . a significant amount of research on adm s using wavelength scanning interferometers already exists @xcite . in one of the most comprehensive publications on this subject , stone _ _ describe in detail a wavelength scanning heterodyne interferometer consisting of a system built around both a reference and a measurement interferometer@xcite , the measurement precisions of absolute distance ranging from 0.3 to 5 meters are @xmath0 250 nm by averaging distance measurements from 80 independent scans . detectors for hep experiment must usually be operated remotely for safety reasons because of intensive radiation , high voltage or strong magnetic fields . in addition , precise tracking elements are typically surrounded by other detector components , making access difficult . for practical hep application of fsi , optical fibers for light delivery and return are therefore necessary . we constructed a fsi demonstration system by employing a pair of single - mode optical fibers of approximately 1 meter length each , one for transporting the laser beam to the beam splitter and retroreflector and another for receiving return beams . a key issue for the optical fiber fsi is that the intensity of the return beams received by the optical fiber is very weak ; the natural geometrical efficiency is @xmath2 for a measurement distance of 0.5 meter . in our design , we use a gradient index lens ( grin lens ) to collimate the output beam from the optical fiber . we believe our work represents a significant advancement in the field of fsi in that high - precision adm s and vibration measurements are performed ( without a _ priori _ knowledge of vibration strengths and frequencies ) , using a tunable laser , an isolator , an off - the - shelf f - p , a fiber coupler , two single - mode optical fibers , an interferometer and novel fringe analysis and vibration extraction techniques . two new multiple - distance - measurement analysis techniques are presented , to improve precision and to extract the amplitude and frequency of vibrations . expected dispersion effects when a corner cube prism or a beamsplitter substrate lies in the interferometer beam path are confirmed , and observed results agree well with results from numerical simulation . when present , the dispersion effect has a significant impact on the absolute distance measurement . the limitations of our current fsi system are also discussed in the paper , and major uncertainties are estimated . the intensity @xmath3 of any two - beam interferometer can be expressed as @xmath4 where @xmath5 and @xmath6 are the intensities of the two combined beams , @xmath7 and @xmath8 are the phases . assuming the optical path lengths of the two beams are @xmath9 and @xmath10 , the phase difference in eq . ( 1 ) is @xmath11 , where @xmath12 is the optical frequency of the laser beam , and c is the speed of light .    for a fixed path interferometer , as the frequency of the laser is continuously scanned , the optical beams will constructively and destructively interfere , causing `` fringes '' . the number of fringes @xmath13 is @xmath14 where @xmath15 is the optical path difference between the two beams , and @xmath16 is the scanned frequency range . the optical path difference ( opd for absolute distance between beamsplitter and retroreflector ) can be determined by counting interference fringes while scanning the laser frequency . a schematic of the fsi system with a pair of optical fibers is shown in fig.1 . the light source is a new focus velocity 6308 tunable laser ( 665.1 nm @xmath17 675.2 nm ) . a high - finesse ( @xmath18 ) thorlabs sa200 f - p is used to measure the frequency range scanned by the laser . the free spectral range ( fsr ) of two adjacent f - p peaks is 1.5 ghz , which corresponds to 0.002 nm . a faraday isolator was used to reject light reflected back into the lasing cavity . the laser beam was coupled into a single - mode optical fiber with a fiber coupler . data acquisition is based on a national instruments daq card capable of simultaneously sampling 4 channels at a rate of 5 ms / s / ch with a precision of 12-bits . omega thermistors with a tolerance of 0.02 k and a precision of 0.01 @xmath19 are used to monitor temperature . the apparatus is supported on a damped newport optical table .    in order to reduce air flow and temperature fluctuations , a transparent plastic box was constructed on top of the optical table . pvc pipes were installed to shield the volume of air surrounding the laser beam . inside the pvc pipes , the typical standard deviation of 20 temperature measurements was about @xmath20 . temperature fluctuations were suppressed by a factor of approximately 100 by employing the plastic box and pvc pipes . the beam intensity coupled into the return optical fiber is very weak , requiring ultra - sensitive photodetectors for detection . considering the limited laser beam intensity and the need to split into many beams to serve a set of interferometers , it is vital to increase the geometrical efficiency . to this end , a collimator is built by placing an optical fiber in a ferrule ( 1 mm diameter ) and gluing one end of the optical fiber to a grin lens . the grin lens is a 0.25 pitch lens with 0.46 numerical aperture , 1 mm diameter and 2.58 mm length which is optimized for a wavelength of 630 nm . the density of the outgoing beam from the optical fiber is increased by a factor of approximately 1000 by using a grin lens . the return beams are received by another optical fiber and amplified by a si femtowatt photoreceiver with a gain of @xmath21 . for a fsi system , drifts and vibrations occurring along the optical path during the scan will be magnified by a factor of @xmath22 , where @xmath12 is the average optical frequency of the laser beam and @xmath23 is the scanned frequency . for the full scan of our laser , @xmath24 . small vibrations and drift errors that have negligible effects for many optical applications may have a significant impact on a fsi system . a single - frequency vibration may be expressed as @xmath25 , where @xmath26 , @xmath27 and @xmath28 are the amplitude , frequency and phase of the vibration respectively . if @xmath29 is the start time of the scan , eq . ( 2 ) can be re - written as @xmath30/c \end{array } \eqno{(3)}\ ] ] if we approximate @xmath31 , the measured optical path difference @xmath32 may be expressed as @xmath33 \times \\ \sin[\pi f_{vib}(t+t_0)+\phi_{vib } ] \end{array } \eqno{(4)}\ ] ] where @xmath34 is the true optical path difference without vibration effects . if the path - averaged refractive index of ambient air @xmath35 is known , the measured distance is @xmath36 . if the measurement window size @xmath37 is fixed and the window used to measure a set of @xmath38 is sequentially shifted , the effects of the vibration will be evident . we use a set of distance measurements in one scan by successively shifting the fixed - length measurement window one f - p peak forward each time . the arithmetic average of all measured @xmath38 values in one scan is taken to be the measured distance of the scan ( although more sophisticated fitting methods can be used to extract the central value ) . for a large number of distance measurements @xmath39 , the vibration effects can be greatly suppressed . of course , statistical uncertainties from fringe and frequency determination , dominant in our current system , can also be reduced with multiple scans . averaging multiple measurements in one scan , however , provides similar precision improvement as averaging distance measurements from independent scans , and is faster , more efficient , and less susceptible to systematic errors from drift . in this way , we can improve the distance accuracy dramatically if there are no significant drift errors during one scan , caused , for example , by temperature variation . this multiple - distance - measurement technique is called slip measurement window with fixed size , shown in fig.2 . however , there is a trade off in that the thermal drift error is increased with the increase of @xmath39 because of the larger magnification factor @xmath40 for a smaller measurement window size .    in order to extract the amplitude and frequency of the vibration , another multiple - distance - measurement technique called slip measurement window with fixed start point is shown in fig.2 . in eq . ( 3 ) , if @xmath29 is fixed , the measurement window size is enlarged one f - p peak for each shift , an oscillation of a set of measured @xmath38 values reflects the amplitude and frequency of vibration . this technique is not suitable for distance measurement because there always exists an initial bias term including @xmath29 which can not be determined accurately in our current system . the typical measurement residual versus the distance measurement number in one scan using the above technique is shown in fig.3(a ) , where the scanning rate was 0.5 nm / s and the sampling rate was 125 ks / s . measured distances minus their average value for 10 sequential scans are plotted versus number of measurements ( @xmath39 ) per scan in fig.3(b ) . the standard deviations ( rms ) of distance measurements for 10 sequential scans are plotted versus number of measurements ( @xmath39 ) per scan in fig.3(c ) . it can be seen that the distance errors decrease with an increase of @xmath39 . the rms of measured distances for 10 sequential scans is 1.6 @xmath1 if there is only one distance measurement per scan ( @xmath41 ) . if @xmath42 and the average value of 1200 distance measurements in each scan is considered as the final measured distance of the scan , the rms of the final measured distances for 10 scans is 41 nm for the distance of 449828.965 @xmath1 , the relative distance measurement precision is 91 ppb . some typical measurement residuals are plotted versus the number of distance measurements in one scan(@xmath39 ) for open box and closed box data with scanning rates of 2 nm / s and 0.5 nm / s in fig.4(a , b , c , d ) , respectively . the measured distance is approximately 10.4 cm . it can be seen that the slow fluctuations of multiple distance measurements for open box data are larger than that for closed box data . the standard deviation ( rms ) of measured distances for 10 sequential scans is approximately 1.5 @xmath1 if there is only one distance measurement per scan for closed box data . by using multiple - distance - measurement technique , the distance measurement precisions for various closed box data with distances ranging from 10 cm to 70 cm collected in the past year are improved significantly , precisions of approximately 50 nanometers are demonstrated under laboratory conditions , as shown in table 1 . all measured precisions listed in the table 1 . are the rms of measured distances for 10 sequential scans . two fsi demonstration systems , air fsi and optical fiber fsi , are constructed for extensive tests of multiple - distance - measurement technique , air fsi means fsi with the laser beam transported entirely in the ambient atmosphere , optical fiber fsi represents fsi with the laser beam delivered to the interferometer and received back by single - mode optical fibers . .distance measurement precisions for various setups using the multiple - distance - measurement technique . [ cols="^,^,^,^,^ " , ]     based on our studies , the slow fluctuations are reduced to a negligible level by using the plastic box and pvc pipes to suppress temperature fluctuations . the dominant error comes from the uncertainties of the interference fringes number determination ; the fringes uncertainties are uncorrelated for multiple distance measurements . in this case , averaging multiple distance measurements in one scan provides a similar precision improvement to averaging distance measurements from multiple independent scans , but is faster , more efficient and less susceptible to systematic errors from drift . but , for open box data , the slow fluctuations are dominant , on the order of few microns in our laboratory . the measurement precisions for single and multiple distance open - box measurements are comparable , which indicates that the slow fluctuations can not be adequately suppressed by using the multiple - distance - measurement technique . a dual - laser fsi system@xcite intended to cancel the drift error is currently under study in our laboratory ( to be described in a subsequent article ) .    from fig.4(d ) , we observe periodic oscillation of the distance measurement residuals in one scan , the fitted frequency is @xmath43 hz for the scan . the frequency depends on the scanning rate , @xmath44 . from eq.(4 ) , it is clear that the amplitude of the vibration or oscillation pattern for multiple distance measurements depends on @xmath45 $ ] . if @xmath26 , @xmath27 are constant values , it depends on the size of the distance measurement window . subsequent investigation with a ccd camera trained on the laser output revealed that the apparent @xmath03 hz vibration during the 0.5 nm / s scan arose from the beam s centroid motion . because the centroid motion is highly reproducible , we believe that the effect comes from motion of the internal hinged mirror in the laser used to scan its frequency . the measurable distance range is limited in our current optical fiber fsi demonstration system for several reasons . for a given scanning rate of 0.25 nm / s , the produced interference fringes , estimated by @xmath46 , are approximately 26400 in a 40-second scan for a measured distance ( @xmath47 ) of 60 cm , that is @xmath48 fringes / s , where @xmath23 is the scanned frequency and @xmath49 is the speed of light . the currently used femtowatt photoreceiver has 3 db frequency bandwidth ranging from 30 - 750 hz , the transimpedance gain decreases quickly beyond 750 hz . there are two ways to extend the measurable distance range . one straightforward way is to extend the effective frequency bandwidth of the femtowatt photoreceiver ; the other way is to decrease the interference fringe rate by decreasing the laser scanning rate . there are two major drawbacks for the second way ; one is that larger slow fluctuations occur during longer scanning times ; the other is that the laser scanning is not stable enough to produce reliable interference fringes if the scanning rate is lower than 0.25 nm / s for our present tunable laser . in addition , another limitation to distance range is that the intensity of the return beam from the retroreflector decreases inverse - quadratically with range . in order to test the vibration measurement technique , a piezoelectric transducer ( pzt ) was employed to produce vibrations of the retroreflector . for instance , the frequency of the controlled vibration source was set to @xmath50 hz with amplitude of @xmath51 . for @xmath52 distance measurements in one scan , the magnification factor for each distance measurement depends on the scanned frequency of the measurement window , @xmath53 , where , @xmath12 is the average frequency of the laser beam in the measurement window , scanned frequency @xmath54 ghz , where i runs from 1 to @xmath39 , shown in fig.5(a ) . the distance measurement residuals for 2000 distance measurements in the scan are shown in fig.5(b ) , the oscillation of the measurement residuals reflect the vibration of the retroreflector . since the vibration is magnified by a factor of @xmath55 for each distance measurement , the corrected measurement residuals are measurement residuals divided by the corresponding magnification factors , shown in fig.5(c ) . the extracted vibration frequencies and amplitudes using this technique are @xmath56  hz , @xmath57 , respectively , in good agreement with expectations . another demonstration was made for the same vibration frequency , but with an amplitude of only @xmath58 nanometers . the magnification factors , distance measurement residuals and corrected measurement residuals for 2000 measurements in one scan are shown in fig.6(a ) , fig.6(b ) and fig.6(c ) , respectively . the extracted vibration frequencies and amplitudes using this technique are @xmath59  hz , @xmath60 nanometers . in addition , vibration frequencies at 0.1 , 0.5 , 1.0 , 5 , 10 , 20 , 50 , 100 hz with controlled vibration amplitudes ranging from 9.5 nanometers to 400 nanometers were studied extensively using our current fsi system . the measured vibrations and expected vibrations all agree well within the 10 - 15% level for amplitudes , 1 - 2% for frequencies , where we are limited by uncertainties in the expectations . vibration frequencies far below 0.1 hz can be regarded as slow fluctuations , which can not be suppressed by the above analysis techniques .    for comparison , nanometer vibration measurement by a self - aligned optical feedback vibrometry technique has been reported@xcite . the vibrometry technique is able to measure vibration frequencies ranging from 20 hz to 20 khz with minimal measurable vibration amplitude of 1 nm . our second multiple - distance - measurement technique demonstrated above has capability to measure vibration frequencies ranging from 0.1 hz to 100 hz with minimal amplitude on the level of several nanometers , without a _ priori _ knowledge of the vibration strengths or frequencies . dispersive elements such as a beamsplitter , corner cube prism , etc . in the interferometer can create an apparent offset in measured distance for an fsi system , since the optical path length of the dispersive element changes during the scan . the small opd change caused by dispersion is magnified by a factor of @xmath40 and has a significant effect on the absolute distance measurement for the fsi system . the measured optical path difference @xmath32 may be expressed as @xmath61 where @xmath62 and @xmath63 refer to the opd at times t and t0 , respectively , @xmath64 and @xmath65 are the wavelength of the laser beam at times t and t0 , c is the speed of light , d1 and d2 are true geometrical distances in the air and in the corner cube prism , @xmath66 and @xmath67 are the refractive index of ambient atmosphere and the refractive index of the corner cube prism for @xmath64 , respectively . the measured distance @xmath36 , where @xmath35 is the average refractive index around the optical path . the sellmeier formula for dispersion in crown glass ( bk7)@xcite can be written as , @xmath68 where , the beam wavelength @xmath69 is in unit of microns , @xmath70 , @xmath71 , @xmath72 , @xmath73 , @xmath74 , @xmath75 . if we use the first multiple - distance - measurement technique described above to make 2000 distance measurements for one typical scan , where the corner cube prism is used as retroreflector , we observe a highly reproducible drift in measured distance , as shown in fig.7 , where the fitted distance drift is @xmath76 microns for one typical scan using a straight line fit . however , there is no apparent drift if we replace the corner cube prism by the hollow retroreflector . numerical simulations have been carried out using eq.(5 ) and eq.(6 ) to understand the above phenomena . for instance , consider the case d1 = 20.97 cm and d2 = 1.86 cm ( the uncertainty of d2 is 0.06 cm ) , where the first and the last measured distances among 2000 sequential distance measurements are denoted @xmath77 and @xmath78 , respectively . using the sellmeier equation ( eq.(6 ) ) for modeling the corner cube prism material ( bk7 ) dispersion , we expect @xmath79 = 373.876 microns and @xmath80 = 367.707 microns . the difference between @xmath77 and @xmath78 is @xmath81 microns which agrees well with our observed @xmath76 microns drift over 2000 measured distances . the measured distance shift and drift strongly depend on d2 , but are insensitive to d1 . a change of 1 cm in d1 leads to a 3-nanometer distance shift , but the same change in d2 leads to a 200-micron distance shift . if a beamsplitter is oriented with its reflecting side facing the laser beam , then there is an additional dispersive distance shift . we have verified this effect with 1-mm and a 5-mm beam splitters . when we insert an additional beamsplitter with 1 mm thickness between the retroreflector and the original beamsplitter in the optical fiber fsi system , we observe a 500 microns shift on measured distance if @xmath35 is fixed consistent with the numerical simulation result . for the 5-mm beam splitter ( the measured thickness of the beam splitter is @xmath82 mm ) , the first 20 scans were performed with the beamsplitter s anti - reflecting surface facing the optical fibers and the second 20 scans with the reflecting surface facing the optical fibers . the expected drifts ( @xmath83 ) for the first and the second 20 scans from the dispersion effect are 0 and @xmath84 microns , respectively . the measured drifts by averaging measurements from 20 sequential scans are @xmath85 microns and @xmath86 microns , respectively . the measured values agree well with expectations . in addition , the dispersion effect from air@xcite is also estimated by using numerical simulation . the expected drift ( @xmath83 ) from air dispersion is approximately -0.07 microns for an optical path of 50 cm in air , this effect can not be detected for our current fsi system . however , it could be verified by using a fsi with a vacuum tube surrounding the laser beam ; the measured distance with air in the tube would be approximately 4 microns larger than for an evacuated tube .    in summary , dispersion effects can have a significant impact on absolute distance measurements , but can be minimized with care for elements placed in the interferometer or corrected for , once any necessary dispersive elements in the interferometer are understood . some major error sources are estimated in the following ;    \1 ) error from uncertainties of fringe and scanned frequency determination . the measurement precision of the distance @xmath87 ( the error due to the air s refractive index uncertainty is considered separately below ) is given by @xmath88 . where @xmath87 , @xmath13 , @xmath23 , @xmath89 , @xmath90 , @xmath91 are measurement distance , fringe numbers , scanned frequency and their corresponding errors . for a typical scanning rate of 0.5 nm / s with a 10 nm scan range , the full scan time is 20 seconds . the total number of samples for one scan is 2.5 ms at a sampling rate of 125 ks / s . there is about a 4@xmath05 sample ambiguity in fringe peak and valley position due to a vanishing slope and the limitation of the 12-bit sampling precision . however , there is a much smaller uncertainty for the f - p peaks because of their sharpness . thus , the estimated uncertainty is @xmath92 for one full scan for a magnification factor @xmath93 . if the number of distance measurements @xmath94 , the distance measurement window is smaller , the corresponding magnification factor is @xmath95 , where , @xmath12 is the average frequency of the laser beam , @xmath96 ghz . one obtains @xmath97 , @xmath98 . \2 ) error from vibrations . the detected amplitude and frequency for vibration ( without controlled vibration source ) are about 0.3 @xmath1 and 3.2 hz . the corresponding time for @xmath94 sequential distance measurements is 5.3 seconds . a rough estimation of the resulting error gives @xmath99 for a given measured distance @xmath100 meters . \3 ) error from thermal drift . the refractive index of air depends on air temperature , humidity and pressure ( fluctuations of humidity and pressure have negligible effects on distance measurements for the 20-second scan ) . temperature fluctuations are well controlled down to about @xmath20(rms ) in our laboratory by the plastic box on the optical table and the pipe shielding the volume of air near the laser beam . for a room temperature of 21 @xmath101 , an air temperature change of @xmath102 will result in a 0.9 ppm change of air refractive index . for a temperature variation of @xmath20 in the pipe , @xmath94 distance measurements , the estimated error will be @xmath103 , where the magnification factor @xmath104 . the total error from the above sources , when added in quadrature , is @xmath105 , with the major error sources arising from the uncertainty of fringe determination and the thermal drift . the estimated relative error agrees well with measured relative spreads of @xmath106 in real data for measured distance of about 0.45 meters .    besides the above error sources , other sources can contribute to systematic bias in the absolute differential distance measurement . the major systematic bias comes from the uncertainty in the fsr of the f - p used to determine the scanned frequency range . the relative error would be @xmath107 if the fsr were calibrated by a wavemeter with a precision of @xmath108 . a wavemeter of this precision was not available for the measurements described here . the systematic bias from the multiple - distance - measurement technique was also estimated by changing the starting point of the measurement window , the window size and the number of measurements , the uncertainties typically range from 10 to 50 nanometers . systematic bias from uncertainties in temperature , air humidity and barometric pressure scales are estimated to be negligible . an optical fiber fsi system was constructed to make high - precision absolute distance and vibration measurements . a design of the optical fiber with grin lens was presented which improves the geometrical efficiency significantly . two new multiple - distance - measurement analysis techniques were presented to improve distance precision and to extract the amplitude and frequency of vibrations . absolute distance measurement precisions of approximately 50 nm for distances ranging from 10 cm to 70 cm under laboratory conditions were achieved using the first analysis technique . the second analysis technique measures vibration frequencies ranging from 0.1 hz to 100 hz with minimal amplitude of a few nanometers . we verified an expected dispersion effect and confirmed its importance when dispersive elements are placed in the interferometer . major error sources were estimated , and the observed errors were found to be in good agreement with expectation . fox - murphy , d.f . howell , r.b . nickerson , a.r . weidberg , `` frequency scanned interferometry(fsi ) : the basis of a survey system for atlas using fast automated remote interferometry '' , nucl . inst . meth . a383 , 229 - 237(1996 )    american linear collider working group(161 authors ) , `` linear collider physics , resource book for snowmass 2001 '' , prepared for the department of energy under contract number de - ac03 - 76sf00515 by stanford linear collider center , stanford university , stanford , california . hep - ex/0106058 , slac - r-570 299 - 423(2001 )                p. a. coe , `` an investigation of frequency scanning interferometry for the alignment of the atlas semiconductor tracker '' , doctoral thesis , st . peter s college , university of oxford , keble road , oxford , united kingdom , 1 - 238(2001 )
in this paper , we report high - precision absolute distance and vibration measurements performed with frequency scanned interferometry using a pair of single - mode optical fibers . 
 absolute distance was determined by counting the interference fringes produced while scanning the laser frequency . 
 a high - finesse fabry - perot interferometer(f - p ) was used to determine frequency changes during scanning . 
 two multiple - distance - measurement analysis techniques were developed to improve distance precision and to extract the amplitude and frequency of vibrations . under laboratory conditions , measurement precision of @xmath0 50 nm 
 was achieved for absolute distances ranging from 0.1 meters to 0.7 meters by using the first multiple - distance - measurement technique . 
 the second analysis technique has the capability to measure vibration frequencies ranging from 0.1 hz to 100 hz with amplitude as small as a few nanometers , without a _ priori _ knowledge .