干涉测距仪测量不确定度评估

Osama Terra;Hatem Hussein;Mohamed Amer

【摘 要】This paper discussed the main parameters contributing to the measurement uncertainty of interferometric distance meter (IDM) .A simple and robust set-up is used to measure distance of about 12 cm with an expanded uncertainty (k=2) of ± 16 .4μm .The measurement uncertainty is found to be limited by the wavelength measurement accuracy .This set-up can be used to measure distances up to 56 m .It also enables easy determination of the point of equal path difference between the measuring and the reference arms .LabVIEW program is used for counting of the fringes and applying fast Fourier transfor-mation (FFT) to perform frequency selective filtration to the noise .Although the reported uncertainty does not represent the state-of-art uncertainty reached for similar distance , the measurement provides traceable measurement to the unit of length ,the meter .%本文对与干涉测距仪（IDM）测量不确定度有关的参数进行了讨论。采用一个简单、鲁棒装置测量了约12 cm距离，扩展不确定性为±16．4μm ，发现测量不确定度受波长测量精度制约。该设置可测量距离达56 m ，它也能够轻松确定测量臂和参考臂之间等光程差的点。LabVIEW程序用于条纹的计数，快速傅立叶变换（FFT）对噪声按频率进行选择性过滤。虽然本文对不确定性的测定不能代表同类测量距离的最高不确定度，但该测量方法提供了可溯源于长度单位为米的测量值。 【期刊名称】《测试科学与仪器》

【年(卷),期】2014(000)003 【总页数】6页(P73-78)

【关键词】距离测量;长度测量;光学干涉仪

【作 者】Osama Terra;Hatem Hussein;Mohamed Amer

【作者单位】Primary Length Standard and Laser Technology Laboratory, National Institute for Standard NIS, Giza 12211-136, Egypt;Primary Length Standard and Laser Technology Laboratory, National Institute for Standard NIS, Giza 12211-136, Egypt;Engineering and Surface Metrology Laboratory, National Institute for Standard NIS, Giza 12211-136, Egypt 【正文语种】中 文

【中图分类】TM9320.12+6

Distance measurement aims to measure the distance to a fixed object without moving it.It plays a crucial role in industry,surveying,building construction,military applications and metrology.

There are several techniques that can be used to measure distance absolutely,which can cover different distance ranges and different accuracies.These systems include differential global positioning systems (D-GPS),electronic distance meters (EDM) and laser range finder (LRF).In order to calibrate such systems,a more accurate and precise technique is required.In addition,some other metrologies applications require to

measure the length of an object accurately,for example,gauge blocks and fiber length measurements.Laser displacement interferometers are commonly used to accurately measure displacement of an object relative to its initial position.Resolution of around 1 nm or less can be obtained using commercially available interferometers.However,by using such interferometers it is not possible to measure distance to a fixed object.In order to enable such measurement,absolute distance interferometry using tunable lasers has been suggested previously to measure distance to a fixed target[1-4].Other setups are suggested to measure absolute distance using time of flight-of-flight and interferometry[5].Although these systems are accurate,they are complex and only possible to implement in few laboratories around the world.

In this work,the main parameters affecting the accuracy and the precision of interferometric distance meters will be discussed in details.A distance of about 12 cm will be measured using a simple and robust wavelength tuning interferometer.The distance measurement is traceable to the unit of length,the meter,through the calibrated wavemeter used for wavelength measurement.The measurement uncertainty will be calculated according to the guide to the expression of uncertainty in measurement (GUM)[6]. 1 Interferometric distance meter

Interferometric distance meter (IDM) mainly uses a Michelson-like interferometer and a mode-hope-free tunable laser.Distance in a distance interferometer can be described by[3]

(1)

where ΔΦ is the change in phase; N= ΔΦ/2π is the fringe number (integer+fractional); λ1 is the start wavelength; λ2 is the stop wavelength; Λ=λ1 λ2/(λ2-λ1) is called the synthetic wavelength and n is refractive index of air,which is calculated from the air parameters

(temperature,pressure,humidity) using the modified Edlen equation[7]. Therefore,by continuously sweeping the laser wavelength from λ1 to λ2 and counting the number of fringes (N) during the sweep,it is possible to calculate the distance to the measured target using Eq.(1).

The advantage of the wavelength tuning technique is that the zero-point of the interferometer (where measurement arm=reference arm,see Fig.1),could be easily determined.This is done by tuning the wavelength of the laser while searching the position of M2 at which no fringe shift appears at the photodetector.At this position N=0 and hence LIDM=0. Fig.1 Zero-point of the interferometer (M: mirror,BS: beam splitter) 2 Experiment

2.1 System configuration

Fig.2 shows the system used to measure the distance to a fixed mirror M2 (without moving the mirror like common displacement interferometers). Fig.2 IDM system (BPD: balanced photodetector,M: mirror,S: beam splitter,FC: fiber coupler,L: lens)

The system shows a typical set-up for a simple Michelson interferometer with mirror M1 at the reference arm and mirror M2 at the measurement arm.The wavelength of a tunable laser (New Focus,Velocity 6308) is swept

from 665 nm to 675 nm.The laser has a linewidth less than 300 kHz.The laser operates over the whole range without mode hoping.The fringes are counted using a balanced photodetector (BPD) (New Focus nirvana 2007).By using a reference beam from the laser,the laser power fluctuations can be eliminated using the BPD.Part of the laser beam is directed to a fiber coupler and then to a wavemeter (Burleigh 1500) to measure tunable laser wavelength with accuracy of about 0.2 pm.A travelling stage is used to measure the distance to the mirror M2 at different positions.The fringes are acquired using an oscilloscope (Yokogawa DLM2022).The fringes are then counted with a developed LabVIEW program.The signal is first transformed to the frequency domain using Fourier transformation,as shown in Fig.3.

Fig.3 Fringes generated by tunable laser interferometer and corresponding FFT

The frequency domain enables frequency selective filtration of our signal from the combined noise which is generated by the interferometer and electronics.The highest peak in the frequency domain corresponds to the fringe frequency of our signal.The program then applies an algorithm to accumulate the phase of the fringes and transform it to fringe number to be used in Eq.(1).A peak counting algorithm is used also for comparison. 2.2 Distance measurement limit

Although the oscilloscope has a frequency bandwidth of 200 MHz

(sampling rate=1 GSample/s),it will be able only to acquire fringes with a maximum rate of 250 kHz.The reason for this limitation is explained as follows: the maximum oscilloscope frame acquisition rate is only 20 kHz.Therefore,in order to preserve the continuity in fringe counting,the fringes must be acquired in a single frame.Since the laser takes below 10 s to finish the 10 nm wavelength scan,the oscilloscope will be adjusted to a measurement time of 10 s.The memory of the oscilloscope can hold only 1.25×107 samples (record length=1.25×107 points) in 10 s.According to the Nyquist sampling theorem,the sampling rate must be at least twice the highest frequency being counted to avoid aliasing.The limit is taken to be at least 5 times the signal frequency (sampling limit=5) due to practical considerations.Therefore,the maximum number of fringes in a single frame of 10 s can be described by (2)

From the above equation,for a record length of 1.25×107 (in 10 s),the maximum number of fringes in one measurement frame is 2.5×106 fringes.Consequently,the maximum fringe rate will be 250 kHz.From Eq.(1),a maximum fringe number of 2.5×106 is equivalent to a maximum measurement distance of 56 m.This range can be extended by using an oscilloscope with the same bandwidth but with larger record length. 3 Results

At the beginning,the zero-point of the interferometer is determined by tuning laser wavelength and searching for the position of M2 at which no

fringes appear during tuning.Afterwards,the mirror M2 is moved to five different positions and the distance is measured at each position.The traveling stage motion range is limited to around 12 cm.The five positions are chosen to be distributed along this range but concentrated more near the end.The five positions are at 11.78,10.43,7.84,5.39 and 0.37 cm from the zero-point.A set of four measurements are made at each position and the standard deviation is calculated.Fig.4 shows the calculated standard deviation of the four measurements.The standard deviation is calculated at each position to be 2.9,3.27,6.15,7.8 and 7.85 μm.According to GUM,these values represent the statistical standard uncertainty (type-A standard uncertainty).As will be discussed in the next section,the main contribution to the uncertainty comes from the uncertainty in wavelength measurement.This is mainly caused by the wavemeter measurement accuracy.

Fig.4 Standard deviation of the five measurements and a linear interpolation

A linear interpolation is performed on the measured data with an intercept of 2.06 μm and slope of 4.9×10-4 (0.49 μm/cm).Therefore,a standard deviation of 51 μm is expected when 1 m distance is measured.This value is higher than the reported in literature,however,this measurement provides traceable measurement to the unit of length through the calibration of the used wavemeter,as going to be discussed in the next section.

4 Uncertainty analysis

Different parameters should be considered during distance measurement which can degrade the accuracy and the precision of the measurement.As described in Eq.(1),these parameters are the wavelength measurement,the fringe number counting and the refractive index determination.To determine the sensitivity coefficients of the measured distance to each parameter,Eq.(1) is partially differentiated.Since the uncertainties could be added in quadrature as[3] uB(LIDM)=δLIDM= (3)

where δN is the uncertainty in the counted fringes,δΛ is the uncertainty in the measured wavelength (synthetic wavelength),δn is the uncertainty in the refractive index and uB(LIDM) is the systematic standard uncertainty (type-B uncertainty).

By partially differentiating Λ=λ1 λ2/Δλ and performing some modifications to the above equation,one can write it as (4)

where δλ is the uncertainty in λ1 and λ2 since they are measured with the same wavemeter.

The fringe count consists of an integer part and fractional part.As long as the counting algorithm is justified to avoid miscounting errors,the uncertainty in fringe count is determined by the fractional part

(phase<2π).Since the wavelength tuning range is 10 nm,this parameter has

an influence of less than 22 μm on the measured distance.This value is totally independent on the measured distance and does not increase by increasing the distance,as demonstrated by Eq.(4).Practically,with the setup discussed in the previous section,it is possible to detect phase with about 0.1 radian resolution using a digital oscilloscope.This will contribute uncertainty in distance by about 0.4 μm.

The next parameter is the uncertainty in determining the refractive index.It is possible using Edlen equation to determine the air refractive index from the air parameters (temperature,pressure and humidity) with uncertainty of 10-8[7].However,the uncertainty in determining the refractive index will be limited by uncertainty in temperature,pressure and humidity

measurements.The uncertainty contribution to the distance measurement is demonstrated in Table 1 based on the uncertainty in the measured air parameters.

Table 1 Air parameters ane its effect on distance measurementsAir parameterTypicalvalueMeasurement accuracyEffect on distanceDaily variationsEffect on distanceTemperature20 ℃±0.2 ℃±1.85×10-7LIDM±1℃±9.25×10-7LIDMPressure1 013.25 hPa± 2 hPa±5.37×10-7LIDM±20 hPa±5.36×10-6LIDMRelative humidity50%±5%±4.3×10-8LIDM±15% ±1.2×10-7LIDM

From Table 1,it is clear that,by insuring accurate measurements of air parameters,it is possible to make the effect of the air-index change negligible (<0.07 μm) for the measured distance of around 12 cm.Even for the possible daily variation inside a laboratory,the uncertainty contribution

will be less than 0.7 μm.

The main contribution to the uncertainty in distance measurement comes from the uncertainty in the wavelength measurement.

The wavelength measurement uncertainty is limited by the accuracy and the measurement speed of the wavemeter,which are 0.2 pm/s and 0.5 nm/s,respectively.

From Eq.(4),wavelength accuracy will contribute to the total measurement uncertainty by 3.4 μm.This value scales with distance.The second limiting factor is the wavemeter measurement speed.A wavemeter with

measurement rate of 1 Hz (1 measurement/s) will be too slow to measure scanning laser with speed of 0.5 nm/s.Even if the measurement is done at the end of the scan,laser drift will lead to inaccurate determination of the measured wavelength.Although the use of wavemeter limits the measurement uncertainty,it offers traceable measurement to the unit of length,the meter.This originates from the fact that the wavelength measurement in the wavemeter is online verified through an internal laser and is also previously calibrated using an iodine stabilized laser,which is the Egyptian primary length standard[8].

In contradiction,the reference Fabry Perot cavities,which are used in several papers[1,3],require operation in vacuum,cavity stabilization and free spectral range measurements to be traceable to the meter,which is hard and not practical to achieve.

Table 2 summarizes the sources of uncertainty and its contribution to the measured length of 12 cm.

The combined uncertainty in distance measurement is obtained by summing in quadrature the uncertainty contributions in Table 2 for a length of 0.12 m as (5)

Therefore,the expanded uncertainty will perform the same calculation for the rest of the four measured positions,the expanded uncertainty is calculated to be 6.4,7.3,12.8 and 16.2 μm).Fig.5 shows the expanded uncertainty values and the corresponding distance values. Table 2 Uncertainty budget for distance of 12 cmSource of

uncertaintyValue Probability distributiondivisorUncertaintySensitivity coefficientStandarduncertaintyWavelength±0.2 pmRectangular±0.12 pm±1.4×108 L±1.68×10-5

LTemperature±0.2 ℃Rectangular±0.12 ℃±2.6×10-6 L ±1.1×10-7 LPressure±2 hPaRectangular±1.2 hPa±2.6×10-9 L±3.1×10-7 LRelative humidity±5%Rectangular±2.9%±1.5×10-8 L±4.3×10-8 LFringe

counting±0.1 radRectangular±0.06 rad±2.24×10-5±1.34×10-6Repeated measurements (0.12 m)±7.8 μmNormal1±7.8 μm1±7.8×10-6Combineduncertainty±8.2 μmExpanded uncertainty (k=2)±16.4 μm Fig.5 Expanded uncertainty (k=2) of five positions 5 Conclusion

In this paper,a tunable laser is used to measure a distance

interferometrically.This setup can be used to measure distances up to 56 m.It also enables easy determination of the point of equal path difference

between the measuring and the reference arms.LabVIEW program is used for counting of the fringes and to apply FFT to perform frequency selective filtration to the noise.The uncertainty contributions from different parameters to the distance measurement are discussed in the paper.The parameters considered here are the tuning wavelength,the fringe counting and the refractive index.Wavelength measurement is found to be the dominant contribution to the uncertainty in distance measurement.A set of four measurements are made at each position of the five positions (0.37,5.39,7.84,10.43 and 11.78 cm) from the point of equal path.

The expanded uncertainty (k=2) of this interferometer is calculated at each position to be 6.4,7.3,12.8,16.2 and 16.4 μm,respectively.Although the reported uncertainty does not represent the state-of-art uncertainty reached for similar distance,the measurement provides traceable measurement to the unit of length,the meter. References

[1] YANG Hai-jun,Deibel J,Nyberg S,et al.High-precision absolute distance and vibration measurement with frequency scanned interferometre.Applied Optics,2005,44(19): 3937-3944.

[2] Abou-Zeid A,Badr T,Balling P,et al.Towards new absolute long distance measurement systems in air.In: Proceedings of NCSL International Workshop and Symposium,Orlando,Floride,2008.

[3] Cabral A,Abreu M,Rebordo J M.Absolute distance metrology for long distances with dual frequency sweeping interferometry.In: Proceedings of XIX IMEKO World Congress Fundamental and Applied

Metrology,Lisbon,Portugal,2009: 1942-1947.

[4] Alzahrani K,Burton D,Lilley F,et al.Absolute distance measurement with micrometer accuracy using a Michelson interferometer and the iterative synthetic wavelength principle.Optics Express,2012,20 (5): 5658-5682. [5] Coddington I,Swann W C,Nenadovic L,et al.Rapid and precise absolute distance measurements at long range.Nature Photonics,2009,3: 351-356. [6] BIPM,IEC,ISO,et al.Guide to the expression of uncertainty in measurement.ISO,1995.

[7] Bønsch G,Potulski E.Measurement of the refractive index of air and comparison with modified Edlens formulae.Metrologia,1998,35: 133-139. [8] Hussein H,Sobee M A,Amerand M.Calibration of a Michelson-type laser wavemeter and evaluation of its accuracy.Optics and Lasers in Engineering,2010,48(3) : 393-397.