Y. Imai and K. Tanaka Vol. 16, No. 8 /August 1999 /J. Opt. Soc. Am. A 2007

Direct velocity sensing of flow distributionbased on low-coherence interferometry

Yoh Imai

Department of Computer Science and Electronics, Kyushu Institute of Technology, 680-4 Kawazu, Iizuka-shi,Fukuoka-ken 820-8502, Japan

Kazuhiro Tanaka

Department of Mechanical System Engineering, Kyushu Institute of Technology, 680-4 Kawazu, Iizuka-shi,Fukuoka-ken 820-8502, Japan

Received November 6, 1998; revised manuscript received April 7, 1999; accepted April 19, 1999

A new sensing method for measuring flow velocity distribution directly by using low-coherence interferencetechniques is proposed and demonstrated. In this method a temporally fluctuating signal, not the Dopplerfrequency shift, is detected. Theoretical analysis shows that a spectrum of light backscattered from a particletakes a Gaussian form whose width is simply proportional to the flow velocity. The measured velocity is ingood agreement with the actual flow velocity derived from the flow rate. The dynamic range of this sensingmethod is governed by the frequency range of the fast-Fourier-transform processor used and is estimated to be1.4 3 1024 –14 m/s. The depth position can be adjusted with an accuracy of approximately 30 mm, which isdetermined by the coherence length of the light source. © 1999 Optical Society of America[S0740-3232(99)02208-5]

OCIS codes: 060.2370, 280.2490, 280.7250.

1. INTRODUCTIONOptical velocimetry has been studied and developedextensively.1–9 Laser Doppler velocimetry (LDV) is oneof the well-developed methods for measuring flowvelocity.1–3 LDV is based on the Doppler frequency shiftof scattering waves caused by moving particles in theflow. Another of these methods is spatial filtering veloci-metry, which measures the modulation frequencies ofscattering waves generated by moving particles that crossan interference fringe or a transmission grating.4–6 Nei-ther type of velocimeter can be applied to a fluid with ahigh particle density, since the detectable scatteringpower and measuring accuracy of the instruments are re-duced by multiple scattering. In addition, it is difficult touse these approaches to measure a velocity distributionalong depth because additional traversing equipment isneeded.

Another method is speckle velocimetry, in which fluc-tuations of random interference among multiple compo-nents scattered in flow are detected.7–9 Relative velocityis obtained in this method. Moreover, the velocity distri-bution along depth cannot be measured in principle be-cause all scattering components along the depth contrib-ute to the formation of a speckle pattern.

Recently, interference techniques that use a low-coherence light source have been applied successfully tooptical velocimetry.10–12 In these applications, distrib-uted sensing along the depth has been achieved in LDV.12

Also, the influence of a fiber probe on flow has been elimi-nated in velocity measurements made by LDV.11 Thediffusion coefficient and the particle size of the Brownianmotion also have been measured by low-coherence

0740-3232/99/082007-06$15.00 ©

interferometry.13 A low-coherence source, such as a su-perluminescent diode (SLD) has a broad spectrum, sothat an interference output is formed only when the opti-cal path difference between a reference and a scatteringwave is negligible. Therefore a depth position at which alight component is scattered can be specified by adjustingthe path length of the reference arm in a low-coherenceinterferometer.

In this paper we propose and demonstrate new an-emometry that is based on optical low-coherence interfer-ence techniques. Unlike conventional velocimetry, theproposed method utilizes the temporal fluctuations of aninterference signal of the backscattered light,10 ratherthan a Doppler frequency shift. One can measure theflow velocity along the depth directly by analyzing thespectrum of an interference signal. A simple theory for aspectrum whose width is proportional to the flow velocityis in good agreement with experiments.

2. THEORYLet us consider a low-coherence interferometer of theMichelson type, which is shown in Fig. 1. A target flow isset in a probe arm. A reference arm is adjusted in lengthby changing the position of the mirror to choose a detect-ing point in the flow depth. The reference and back-scattering waves are expressed as (see Fig. 1)

Er~t ! 5 ArE a~n!exp~2i2pnt !dn, (1)

Es~t 2 t! 5 AsE a~n!exp@2i2pn~t 2 t!#dn, (2)

1999 Optical Society of America

2008 J. Opt. Soc. Am. A/Vol. 16, No. 8 /August 1999 Y. Imai and K. Tanaka

where a(n) is the spectrum of the low-coherence source,Ar is the amplitude of the reference wave, As is the scat-tering coefficient of the probe wave, which fluctuates tem-porally in correspondence to the motion of tracing scatter-ers contained in the flow, and t is the time difference thatresults from a path difference between the reference andbackscattered waves. Here the amplitudes of the refer-ence and probe waves are assumed to be identical. More-over, the fluctuation of As is much slower than t. Con-stant phase terms are neglected in Eqs. (1) and (2).

The interference signal in a photodetector is written byusing Eqs. (1) and (2) as

I~t;t! 5 ^uEr~t ! 1 Es~t 2 t!u2&,

5 I0 1 2Ar As~t !Re@ G~t!#, (3)

where I0 5 ^Ar2& 1 ^As

2&, ^...& means the time-averageprocessing, and G(t) is the temporal coherence function.In the derivation of Eq. (3) the averaging time T is rea-sonably assumed to be larger than the period of electricfield.

The coherence function is expressed by a Fourier trans-form of the power spectrum of the light source:

G~t! 5 E G~n!exp~2i2ptn!dn, (4)

where G(n) 5 ua(n)u2 is the power spectrum of the source.The coherence length is estimated by the relationship

lc ' l2/Dl, (5)

where l and Dl are the center wavelength and the spec-tral width, respectively, of the light source.

From a practical point of view, 2Ar As @ As2, since Ar

@ As . Then the temporally fluctuating signal in the in-terference output is affected primarily by the cross (sec-ond) term in Eq. (3). In Eq. (3) the second term, contain-ing G(t), appears only when the path lengths of thereference and the backscattered waves are almost identi-cal. Hence the depth position at which the flow velocityis measured can be chosen by adjusting the path lengthof the reference arm. Only the desired signal2ArAs(t)Re$G(t)% is fed to the Fourier processor with accoupling. The spectrum of the temporally fluctuatingscattering amplitude is obtained to evaluate the flow ve-locity from the spectrum width.

The Fourier spectrum at a specific depth position@G(t) ; 1# is then expressed as

Fig. 1. Schematic of a low-coherence interferometer for measur-ing distributed flow velocity: PD, photodetector; L, lens.

S~ f ! 5 ArE As~t !exp~2i2pft !dt, (6)

where As(t) is regarded as a random series with a Gauss-ian pulse. The scattering amplitude of a single particleis Gaussian, since the probe beam has a Gaussian inten-sity distribution across the cross section. Hence As(t) isexpressed as

As~t ! 5 (n

An expF2~t 2 tn!2

2T2 G , (7)

where An is the scattering amplitude of the nth particle,tn is the time when the nth particle crosses the probebeam, and T is the time period necessary for the singleparticle to cross the probe beam and is defined by a ratiow0 /u, where w0 is the beam waist of the probe beam andu is the flow (particle) velocity. By substituting Eq. (7)into Eq. (6) we obtain

S~ f ! 5 A2pArT exp~22p2T2f 2!(n

An exp~2i2pftn!.

(8)

Here the spectral profile of As(t) is governed primarily bythe Gaussian term exp(22p 2 T 2f 2) and is approximatedby a Gaussian form, because the summation term(An exp(22piftn) is thought to distribute randomly anduniformly in the frequency domain in correspondence torandom An and tn . Hence the spectral width is given ap-proximately by 1/(A2pT) 5 u/(A2pw0) and is propor-tional to the flow velocity. The flow velocity can be esti-mated directly from the spectrum width of S( f ).

To analyze the spectral profile of As(t) in detail, let usassume that An 5 A (constant) and that tn 5 nDt, whereDt is the mean period for the two successive particlescrossing the probe beam and is derived as

Dt 51

w0lcur, (9)

where lc is the coherence length and r is the particle den-sity, defined as the number of particles in a unit volume.Then Eq. (8) becomes

S~ f ! 5 A2pArTA exp~22p2T2f 2!

3 exp@2i2p~N 2 1 !ftn#sin~pNDtf !

sin~pDtf !. (10)

The spectral profile is determined by a product of theGaussian term whose width is given by 1/(A2pT) and thelast term (sin NX/sin X). Since N is larger, a conditionthat is usually satisfied in measurements, the last termrepresents a pulse series with a period of 1/Dt. When1/(A2pT) . 1/Dt, the spectral width of S( f ) is deter-mined predominantly by 1/(A2pT), since the pulse seriesgiven by the last term is amplitude modulated by theGaussian envelope. When 1/(A2pT) , 1/Dt, the spec-tral width is affected primarily by the pulse width of thelast term (sin NX/sin X), which is derived as 1/(NDt).Then the spectral width of S( f ) is thought to be muchsmaller than 1/(A2pT) since N is larger. From a practi-cal point of view, however, An and tn in Eq. (8) are ran-dom variables. The spectral term of (An exp(22p if tn) is

Y. Imai and K. Tanaka Vol. 16, No. 8 /August 1999 /J. Opt. Soc. Am. A 2009

expected to take a continuous form with a random ampli-tude over a wide frequency range. Consequently thespectrum S( f ) is modulated again by the Gaussian enve-lope given by the term exp(22p 2 T 2f 2). Therefore thespectral width is also estimated to be 1/(A2pT), evenwhen 1/(A2pT) , 1/Dt. From these discussions, the re-sultant spectral width is proportional only to the flow ve-locity and is independent of the particle density.

3. EXPERIMENTSA SLD used as the low-coherence light source is charac-terized by an 841-nm center wavelength and a 20.8-nmlinewidth at 150-mA applied current. The visibility as afunction of the path difference of an interferometer usinga SLD is depicted in Fig. 2. The FWHM of the visibilitycurve is approximately 30 mm, which is in good agree-ment with the coherence length of 34 mm calculated fromrelation (5). This value determines the accuracy withwhich the depth position can be adjusted in the presentmethod.

Fig. 2. Visibility of a low-coherence interferometer as a functionof path difference.

Fig. 3. Configuration of the flow passage.

The velocity distribution is measured with a low-coherence interferometer as shown in Fig. 1, in which thelength of the reference arm is scanned by measurements.A single mode is excited in a 3-dB fiber coupler so thatmodal noise is eliminated. A low-coherence light beamemitted from the SLD is divided into the reference andprobe beams. The reference beam is reflected by a mirrorthat is mounted on a micropositioner. The microposi-tioner changes the length of the reference arm and ad-justs the depth position. Figure 3 shows a flow passage100 mm wide, 280 mm thick, and 600 mm long. Theprobe beam impinges normally on the passage so that theinfluence of the Doppler frequency shift is removed. Theprobe beam has a Gaussian intensity profile, and thebeam waist of the probe beam focused in flow is w05 40 mm. The probing position on the passage is centralin width and 100 mm away from the passage end. Tokeep the flow velocity constant, the height of the fluid inthe column is kept constant by a controlling valve. In thewater flow used, the Reynolds number is estimated to beRe ' 210 by using U 5 250 mm/s mean flow velocity and280 mm thickness of the flow passage. Hence the flow islaminar in every experiment, since Re is always smallerthan the critical number ;3000. A 0.2% water solutionof Intralipid, whose particle diameter is smaller than 1mm, is used as a target flow. Then the scattering ampli-tude of a single particle is regarded as having a Gaussianform whose width corresponds to the beam waist whenonly a small backscattering angle contributes to the de-tection. The interference signal, which fluctuates in cor-respondence to the motion of the scatterers, is fed to afast-Fourier-transform (FFT) processor with an ac cou-pling to produce the Fourier spectrum whose width isanalyzed to evaluate the flow velocity. The Fourier spec-trum is averaged over 64 samples to smooth the profile.It takes 40 ms to acquire a single spectrum when the fre-quency range is set at 10 kHz. This frequency rangeleads to a frequency resolution of 25 Hz. The averaging(integration) time to obtain the resultant spectrum is ;10s. Note that the acquisition and integration times de-crease as the frequency range, which is inversely propor-tional to the sampling time of the FFT, increases. Thefrequency range of the FFT processor was adjusted from 5to 20 kHz to correspond to the flow velocity.

4. RESULTSThe Fourier spectra of the interference signal at three dif-ferent depth positions, d 5 56, 75, 132 mm for a meanflow velocity of U 5 58 mm/s are shown in Fig. 4. Themean flow velocity is calculated by the ratio between theflow rate and the cross section of the passage. The loca-tion of the upper surface of the passage in which d5 0 mm is easily recognized because a Fresnel reflectionat the surface is much stronger than the backscatteredlight. In each mean velocity the spectrum broadens indepth up to the center (d 5 140 mm). In addition, thespectral width increases as the mean velocity increases.These results support the theoretical estimate that T be-comes smaller with an increase in velocity.

The experimental data of the spectrum must be fittedinto the Gaussian envelope, and the spectral width at 1/e

2010 J. Opt. Soc. Am. A/Vol. 16, No. 8 /August 1999 Y. Imai and K. Tanaka

is used to evaluate the flow velocity quantitatively. Themeasured flow velocity along the depth for four mean flowvelocity levels, U 5 29, 58, 120, and 246 mm/s, is shownin Fig. 5. The solid curves are the theoretical curves ofthe velocity profile in a laminar flow sandwiched betweentwo parallel surfaces. The theoretical profile is ex-pressed as a parabola:

u 5 2ax~x 2 b !, (11)

where x is the depth coordinate, a is a constant that de-pends on the flow velocity, and b is the thickness of thepassage. This expression is adequate for the flow used,because the thickness of the laminar boundary layer atthe probing point is estimated to be 25 mm, which is muchless than the thickness of the passage (280 mm). Thelaminar boundary layer is well developed at the probingpoint. The mean velocity U and the maximum velocityumax are derived as U 5 ab2/6 and umax 5 ab2/4 (53U/2)in laminar flow. The experimental data are in goodagreement with the theoretical curve in each mean veloc-ity case.

5. DISCUSSIONSA. Influence of Particle DensityThe density independence of the Fourier spectrum isshown in Fig. 6, for which the mean flow velocity and thedepth position are set at U 5 60 mm/s and d 5 150 mm,

Fig. 4. Typical Fourier spectra of interference output at threedepth positions, d 5 56, 75, and 132 mm, at mean flow velocityU 5 58 mm/s.

Fig. 5. Flow velocity distribution measured from spectral width.The mean flow velocity is set at the values of U shown. Solidcurves are theoretical curves calculated from Eq. (8) with themean velocity.

respectively. The spectral width does not change whenthe Intralipid density is increased from 0.1% to 0.4%.The measured velocity is independent of the particle den-sity, as we verified experimentally for the density rangefrom 0.05% to 10%. The thin limit is determined bywhether one particle passes through the measuring vol-ume in the flow; the thick limit is given by the minimumdetectable light power of the detection system. The back-scattered light becomes too weak to be detected as the In-tralipid solution thickens. The minimum detectablepower in this photodetector system is ;50 nW and can beimproved by use of the optical heterodyne detectionscheme.

Even if the power of the probe beam is enough that thescattering light is detected in the high-density flow, thetemporal fluctuation of the interference output becomesrelatively small and smooth and cannot be recognized, be-cause too many scatterers contribute to forming the out-put. In comparison with the conventional LDV and spa-tial filtering method, it is apparent in principle that thepresent method can be applied to a higher-density flow,because only the scatterers in the measurement volumerestricted by the coherence length contribute to the inter-ference output, whereas in the conventional methods allthe particles in the probe beam affect the output.

In addition, the high-density flow reduces the accuracyfor setting the depth position and leads to an incorrectspectral width. The path length of the multiply scatteredlight is long, even at a shallow scattering point. Whenthe setting depth d is smaller than half of the passagethickness b/2, pulses with wider widths are contained inIout(t). Consequently the spectrum of S( f ) consists ofnarrower components, in addition to components withtrue width, which yields a narrower spectral width, i.e., inthe underestimation of the flow velocity.

B. Dynamic Range and Accuracy of the VelocityMeasurementIn the present system the dynamic range of the flow ve-locity is determined by the time response (frequencyrange) of the FFT processor, because the time response ofthe photodetector is much faster than that of the FFT.The frequency range of the FFT processor used in the ex-periments is 1 Hz–100 kHz, which leads to minimum andmaximum measurable flow velocities of 1.4 3 1024 –14m/s with use of the relationship Df 5 u/A2pw0 . The

Fig. 6. Fourier spectra for three Intralipid density levels, r5 0.1%, 0.2%, and 0.4%, for depth position d 5 150 mm andmean velocity U 5 60 mm/s.

Y. Imai and K. Tanaka Vol. 16, No. 8 /August 1999 /J. Opt. Soc. Am. A 2011

highest measurable flow velocity can be improved by useof a faster photodetector and a high-speed signal proces-sor such as an analog spectrum analyzer.

In the velocity measurement, a total error and uncer-tainty of ;8% is included in each measured value in Fig.5, as has been confirmed experimentally. The error origi-nates in the instabilities in the light source and the flowrate, turbulence and vibration in the system, and varia-tion in the frequency resolution of the FFT processor. Inthe FFT processor used, the frequency resolution is de-fined as (frequency range)/400. Over a 5-kHz frequencyrange, for example, the frequency resolution becomesDfmin 5 12.5 Hz, followed by a velocity resolution of 2.2mm/s, which leads to a maximum 2–3% error, as shownin the data at U 5 29 mm/s in Fig. 5. Here the frequencyrange was set at 5 kHz for the flow with U 5 29 mm/s, at10 kHz for 58- and 120-mm/s flows, and at 20 kHz for a246-mm/s flow.

In addition, the velocity gradient within the coherencelength in the flow causes the measurement accuracy todeteriorate. The velocity gradient broadens the spec-trum because the faster scatterers contribute more to thespectrum, which is remarkable in the steeply gradient ve-locity flow. This effect is greater than predicted theoreti-cally in the measured velocity points at depth positionsfar from the center, as shown in Fig. 5. Other factors,such as the random amplitude in the scattering pulsesand the distortion of the pulse from the Gaussian profile,also cause the spectrum to deviate from a Gaussian pro-file and make the spectrum bumpy, as shown in Fig. 4.However, the bumpy profile can be reduced by averagingof the spectrum.

The accuracy of velocity measurement is further de-creased as the depth position approaches the surface ofthe flow. When the depth position is set within half ofthe coherence length from the surface, a strong dc compo-nent caused by Fresnel reflection at the surface appearsin the spectrum. This phenomenon makes the spectrumnarrower, followed by a decreased flow velocity. We cansolve this problem, however, by processing the spectrumdata so as to remove the influence of the Fresnel reflec-tion, by setting the probe beam to be oblique to the sur-face of the flow, or both. It becomes difficult to calibratethe depth position exactly when the probe beam is ob-liquely incident. When the scattering particles are de-posited on the surface, which is likely to occur for envi-ronmental measurements, the reduction in themeasurement accuracy occurs even in the case of the ob-liquely incident probe beam. However, the sediment di-rectly affects the power of the scattered light rather thanthe accuracy of its measurement.

C. Optical Heterodyne Detection SchemeIn the sensing method described here it is possible to uti-lize the optical heterodyne technique to detect weak back-scattering light from flow. When the reference beam ismodulated in intensity by an optical chopper at frequencync , the interference signal becomes

I~t; t! 5 I0 112 Ar

2 cos~2pnct ! 118 Ar

2 cos~4pnct !

1 Ar As~t !@1 1 cos~2pnct !#Re@ G~t!#. (12)

Then this interference signal is fed to a lock-in amplifierwith a reference signal, and the amplitude at frequency ncis determined as

Ie~t; t! 512 Ar

2 1 Ar As~t !Re@ G~t!#. (13)

In this case, only the cross term that contains the back-scattering amplitude can be FFT analyzed by an ac cou-pling in the FFT processor. The optical heterodyne de-tection improves the sensitivity but degrades the dynamicrange. In the optical heterodyne scheme the maximummeasurable flow velocity is limited in particular by thetime constant of the lock-in amplifier because the flow ve-locity is faster when the scattering pulse width is nar-rower. The minimum time constant of a typical lock-inamplifier is 1 3 1023 s. Hence the measurable flow ve-locity is limited, so w0 /u . 1 3 1023 s. In this case themaximum measurable velocity is estimated as ;40 mm/sunder the condition that w0 5 40 mm. The low limit ofthe measurable velocity is again determined by the mini-mum frequency range, 1 Hz, of the FFT processor.

6. CONCLUSIONSA new anemometer that uses low-coherence interferencetechniques has been proposed and demonstrated. Thisanemometer can measure the velocity profile along depthin flow and is capable of measuring flow velocity indepen-dent of particle density.

It was verified theoretically and experimentally thatthe flow velocity can be obtained directly from the spec-tral width of the interference signal. The dynamic rangeof the flow velocity in this experimental system was esti-mated as 1.4 3 1024 –14 m/s. The accuracy of the mea-sured velocity was 8%, as confirmed experimentally. Theaccuracy for adjusting the depth position was 30 mm forthe SLD used. This high accuracy enables us to applythe present method to measurements for velocity distri-butions of a boundary layer in liquid flow, which is impor-tant in the field of fluid mechanics and is difficult toachieve by conventional methods. The present methodcan also be applied to gas flow.

We can also use an autocorrelation function of the in-terference signal to evaluate flow velocity. In this casethe correlation length becomes shorter as the flow velocityincreases. Improvements for measuring the two-dimensional velocity distribution and velocity vector dis-tribution are under consideration.

Y. Imai’s e-mail address is imai@leo20.cse.kyutech.ac.jp.

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