Motion Measurement B y JOHN C. ANGUS DAVID L. MORROW JOHN W. DUNNING, JR. MICHAEL J. FRENCH
8
INDUSTRIAL A N D ENGINEERING CHEMISTRY
elocity measurements may be made by detection of
Vthe Doppler shift of laser light scattered from a mov.
ing object. The resolution of this new technique is so great that frequency changes of 10 Hz out of absolute light frequencies of 5 X l O I 4 Hz may be observed. This paper is divided into two parts. First, we will discuss laser Doppler velocity meters, with which the motion of fluids or macroscopic objects is measured. Second, we will review techniques for determining molecular motions. This subdivision is natural because the experimental techniques used in the two sections are different in important details, although similar in basic principles. The principle of operation of these new devices is quite straightforward. The frequency of laser light scattered from a moving object is slightly shifted in frequency by the well known Doppler effect. The Doppler shift is detected by optical mixing of the scattered radiation with a reference beam from the same laser. The resultant heterodyne or “beat” frequency is equal to the difference in frequencies of the reference and scattered beams. Knowledge of this frequency shift and the geometry permits one to compute the velocity directly. The measurement is accurate and almost instantaneous, and no mechanical coupling between the meter and system is required. Very high and very low velocities may be measured. The lower limit at present is about cm/sec. There is minimal perturbation of the system being studied. The spatial resolution is fixed ultimately by diffraction limitations, which means the lower limit of the size of the volume element being sensed is only about l o w 3cm in diameter. Important applications have been demonstrated and more are certain to be developed. Among the most important are flow measurement, particle size determination, transport property measurements, reaction kinetic measurements, and vibration analysis.
MEASUREMENTS OF MOVEMENTS OF MACROSCOPIC OBJECTS: T H E LASER DOPPLER VELOCITY METER Basic Principles
The optical heterodyne detection of the Doppler shift is basic to the operation of the velocity meter. The mixing of two signals of different frequencies in a nonlinear detector to generate a difference or beat frequency is a well known phenomenon. The first suggestion that the same effect could be achieved with signals of optical frequencies was made by Forrester in 1947 and 1948 (39, 46). In 1955 he achieved detection of the beat frequency between the Zeeman components of the Hg 5461 A line (38). The nonlinear detector was a photomultiplier tube. Consider a situation in which two perfectly monochromatic plane light beams are superimposed on the surface of a photocathode. Suppose their angular frequencies are 0 1 and w2. The electric fields will be E1 = El0 COS wit (11 E2 = E20 COS wzt (2 ) The output current of the photomultiplier tube, i, is proportional to the square of the electric field incident upon it. i = (const) ( E 1 E z ) ~ (3)
+
Since the photomultiplier will not follow frequencies greater than several hundred megahertz, terms in the u2 will expansion of Equation 3 involving u1, W Z , or w 1 give rise only to a dc current proportional to the time average of those terms. If, however, (u1 - w2)/2a is below about 100 MHz, there will be an ac signal of frequency w 1 - w2.
+
+ EioEzo cos(wi -
i = (const)
WZ)~]
(4)
The first term in the brackets is the dc component of the
Laser Doppler Techmiques Fluid flow, transport property, and other dynamic measurements are made b y detection of the Doppler shift of scattered laser light
VOL. 6 1
NO. 2 F E B R U A R Y 1 9 6 9
9
Photomuhiplier T i 4
Figure 2. Schcmotic of laser Dopplcr ~clocitymet#
signal. The second term is the component of current Equation 4 is true only if the modulated at WI - WI. two incident beams are perfectly monochromatic. The situation in which the incident light has a spectral distribution will be discussed later. With conventional light sources, the optical heterodyne process was marginal at best. The relatively large bandwidths (lo' Hz) and the low intensity per unit bandwidth gave extremely poor signal-to-noise ratios and ruled out detection of Doppler shifts smaller than lo' Hz. These difficultieswere removed with the development of lasers. Laser radiation has high power levels contained within a narrow bandwidth. The use of lasers in optical heterodyne processes has been discussed by Forrester (37)and Stone (702). Spatial and coherence requirements for efficient optical heterodyning have been considered by many workers (74,50,59,60,68, 76,77,
84,97). A simple schematic diagram of one version of a laser Doppler velocity meter using optical heterodyne detection is shown in Figure 2. Light from a continuous gas laser is split at a beam splitter. One portion of the beam impinges on a moving object, in this case a diffusely reflecting rotating disk. Some of the scattered light is collected with a lens and focused on the surface of the photomultiplier tube. The other portion of the original beam travels to a mirror, from which it is reflected to another beam splitter where it is combined with the scattered beam and led to the photocathode surface. The frequency of the beam scattered from the moving object has been changed by the Doppler effect; the other beam is unshifted in frequency. The output of the photomultiplier tube will be modulated at the difference frequency. This frequency difference can range from less than 100 Hz to the kilomegahertz range and can be detected with conventional spectrum analyzers. Instabilities in the output frequency of the laser do no harm since they appear in each beam and cancel out if the optical paths are of equal length. Under normal operating conditions, lasers do not operate at a single frequency, but instead produce several discrete lines spaced several hundred megahertz apart. These lines, called axial modes, have instantaneous widths of only a few hertz. Multiaxial mode operation usually causes 10
INDUSTRIAL A N D ENGINEERING CHEMISTRY
Figure 3. Geometry for computing Doppler shift
no problems because the beats between modes produce signals with frequencies equal to the mode spacing and consequently are usually much higher than the frequencies of interest. The laser can also resonate in transvem or off-axis modes. These modes are more closely spaced, and sometimes their frequency separation is rather unstable. They should be avoided by proper laser design. The velocity resolution is ultimately limited by the instantaneous line widths of the individual axial modes which, as mentioned above, are only a few hertz. Numerous surveys of laser technology contain discussions of mode structure, optical mixing, and related topics
(7,8, 44,69,74,9298). The geometrical relationship governing the magnitude of the observed Doppler shift is easily shown. Consider the situation shown in Figure 3. Light of frequency, An yo, is incident upon a moving scattering center. oscillating field will be induced in the scattering center by the electric field of the incident radiation. The frequency of this oscillation, v, is given by the Doppler relation (702)
-
vo[l
$1
(5)
where Go is a unit vector in the direction of the incident light and v is the velocity of the scattering center. The scattering center acts as an antenna, radiating light at a frequency, Y . T o an observer at the position of the photomultiplier tube this frequency of scattered radiation is shifted because of the Doppler effect. The frequency, vI, observed at the photomultiplier tube is v, =
where
.[ + $1 1
; , is a unit vector in the direction of the scattered
John C. Angus and MicheI J . French are, respectively, Associate Professor of Engineering and Research Associate in the C h i c a l Engineering Science Division, Case Western Reserue Uniuersiy, Cleueland, Ohio. John W . Dunning, Jr., is an Aerospace Engineer, NASA La-wis Research Lobwatories, CIeueIand, Ohio. Capt. Dauid L. Mmow is with the Royal Canadian Engineers, CFB ChiIliwack, Vedder Crossing, B. C., Canada. AUTHORS
radiation. Eliminating v from Equations 5 and 6 and neglecting the second degree term because /vi
