height by a variety of causes including evaporation, seepage, condensation, leakage, and thermal expansions or by a combination of these. In many cases, we need to measure the effect of one of these factors by itself. A solution to this problem can be found by taking advantage of periods when only one cause is present and the rest of them have little or no effect on the level change. A need arises in water manage-nient, for example, to differentiate seepage from evaporation, since these two may occur simultaneously. Sometimes it is very difficult to determine whether the evaporation is very high or a seepage or leakage problem is present. In such cases it is convenient to rely on a highly accurate method to measure evaporation rates. Interferometric methods are best suited for this purpose because of their high sensitivities.
There are times when it is not desirable for the measuring apparatus to be in physical contact with the liquid, such as floating devices that might interact with the sample either physically or chemically.
Some of the situations in which noncontact to surface type of measurements are called for include: (a) The liquid is chemically very active and may cause corrosion to the sensing element. (b) Foreign matter carried by the sensing element may contaminate the liquid, specially if it is highly pure. (c) Contact of the sensing element with the liquid may change its thermal conditions, for example, when the liquid is hot. (d)
Surface tension caused by the sensing element in mechanical methods changes the height of the liquid level.
Depending on the viscosity of the liquid, the free surface is more or less easily perturbed by the presence of environmental vibrations. However, it is expected that the free surface be more stable if placed in a low vibration environment such as that provided by the use of holographic tables or some adequate isolation against vibrations from the surroundings.
Sometimes the concern of the observer is focused on measuring level change rates over short periods of time rather than large total level changes over longer periods. This might be the situation in sampling the evaporation rate of liquids, for example.
Considering this, an inexpensive interferometer is proposed that combines a reference beam with the reflected beam of light from the surface in question to produce a pattern of fringes that are displaced when the liquid level changes.
The optical arrangement is shown in Fig. 1. The hypotenuse face of a right-angle prism is firmly positioned nearly parallel to the free surface of the liquid sample, the separation being a few millimeters.
A low power He-Ne laser is used to produce an expanded beam of collimated light at near normal incidence on the entrance face of the prism. The hypotenuse face acts as a beam splitter producing a reference reflected beam and a transmitted beam that reflects on the liquid surface and combines with the reference beam on entering the prism through the hypotenuse face. This we term probe beam. Both beams emerge from the prism producing a pattern of nonlocalized fringes that can be projected on a screen for observation.
The optical path difference between the interfering wave fronts is found to be
where h and α are defined in Fig. 1, and n is the index of refraction of the prism.
Bright fringes occur wherever OPD = mλ,m being an integer, and λ being the laser wavelength (0.6328 μm).
Differentiating Eq. (1) we get
This equation gives the number of displaced fringes dm produced by a change dh in the height of the level.
Note that the quantity dm/dh which is a measure of the sensitivity of the interferometer can be adjusted over a continuous range of values by changing the angle β (thus α). This
angle is restricted by the critical angle (~ 4l° for n = 1.52) and by considerations of contrast of the fringes.
To obtain high contrast fringes, the intensities of the interfering beams should be approximately equal. In this case, the probe beam turns out to be less intense than the reference beam since it undergoes two refractions before interfering. The refraction at the air-liquid interface is particularly weak due to the normally low indices of refraction of liquids as compared with solids.
We compensate this unbalance by increasing the angle β to a value slightly less than the critical angle. Thus angle α is relatively large, and consequently the light is strongly reflected at the air-liquid interface.
According to Fresnel equations, the relative intensities of the reflected and refracted beams depend on the state of polarization of the light. This means that we can also control the contrast of the fringes by changing the state of polarization of the laser light. As an example, we considered the case in which the liquid sample is water (n = 1.333) and assume that n = 1.5 and β = 40°. Direct calculations show that fringe visibilities of 0.83 and 0.99 are obtained for S and P polarizations, respectively, when the laser light is completely coherent.
If the laser is not linearly polarized, a piece of Polaroid sheet can be used as a linear polarizer that can be rotated for best contrast.
In our experiments we used this interferometer to observe variations of the level of water and glycerine due to evaporation. The prism (size 37 X 25 X 25 mm) was placed ~3 mm from the surface of the liquid, and the laser light was incident ~ 4O° from the normal to the hypotenuse.
Since the liquid surface is very sensitive to small vibrations, it is recommended that the sample be placed on a stand isolated from vibrations. Wavelets produce a rapid fringe motion that washes out the pattern.
The data reported here were obtained in a relatively quiet laboratory in a second-story level. The interferometer was placed on top of an iron table supported by inflated tire tubes. The container was placed on top or layers of rubber foam alternated with aluminum plates to absorb further the vibrations of the table.
The hypotenuse face of the prism is adjusted to form a small angle with the free surface so that a convenient number of fringes appear in the field of view.
An automatic recording of the fringe motion can be obtained by monitoring the intensity at a fixed point while the fringes move. A flat piece of cardboard is used as a screen, and a small pinhole is made to let the light from a narrow
Fig. 2. Trace of the recorder output as a function of time due to evaporation of (α) water and (6) glycerine. Arrows show the presence
of marginal vibrations of the liquid surface.
portion of the fringe reach the detector which is placed just behind said pinhole. Thus variations of intensity resulting from the displacement of fringes across the pinhole are converted into voltage variations that are plotted with a strip chart recorder.
The up or down motion of the level corresponds to either of two directions in which fringes may be displaced.
When changes in the direction of the level motion are foreseeable, an electronic fringe counter with direction sensing has to be supplied to keep automatic and continuous track of directionality. We did not use such a device in our experiments. However, when monotonic level changes are guaranteed, there is no need for direction sensing, and a fringe count is sufficient. This is the case when evaporation or seepage through the walls of the container, for example, is the only possible cause of a level change.
Figure 2 shows an actual trace of the recorder output over a 11.7-min time interval when the liquid was (a) water or (b) glycerine. The atmosphere conditions were approximately equal.
Note that in (a), a total of 45 fringes are counted, whereas in (b) only four are. Using Eq. (2), this gives an average rate of change of water level of 0.69 and 0.062 μm/min for glycerine. Also note the fairly constant evaporation rate of water as compared with glycerine.
The maximum separation hm between the prism and liquid depends on the size of the prism. This is due to the displacement suffered by the probe beam on traversing the air gap that separates the prism from the liquid. Direct calculation shows that separation greater than hm = B/2 tana results in a reflected beam that misses the prism, and no interference is produced. Here B is the length of the hypotenuse.
The minimum detectable level change depends on how accurately fringe motions can be measured.
If we, conservatively, assume that at least half of a fringe can be detected (∆m = ½), from Eq. (2) we obtain
(∆h)m i n = 0.09 μm
for β ≅ 40°. This gives a resolving power of the order of 105
for a prism with B = 25 mm. However, it is expected that by using an appropriate fringe
detection scheme1 that this figure can still be increased. Methods for measuring liquid level change that use the
interference of light2 and the moire technique3 have been reported. In the first case, ⅛ -wavelength accuracy can be obtained, but the method involves the use of a floating reflector, and direct contact with the sample is inevitable. Although this approach is less sensitive to vibrations, it is expected to be considerably more expensive than the method we suggest.
The moire technique involves no physical contact and is inexpensive, but it is considerably less sensitive than this method.
This interferometer may be useful in the study of many physical processes. For example, it can be used to obtain information about the crystal growth process from variations of the level of the saturated solution from which the crystal is grown.
It is possible to monitor simultaneously level changes of two separate liquids by dividing the container in two compartments, one for each liquid in question. In this fashion, the two free surfaces give rise to corresponding fringe patterns that can be recorded independently for later correlation.
An improved version of this interferometer may include temperature sensing so that thermal variations may be compensated if needed.
The quality of the prism need not be very high, and ordinary prisms may be used. The flatness of the prism faces is not an important factor for it is fringe motion that matters regardless of fringe shape.
In summary, we have demonstrated the use of a right-angle prism as an inexpensive liquid level interferometer. The use of this device is suggested when changes in the level of a liquid reservoir need to be measured with interferometric sensitivity, and no physical contact with the sample is desired.
I gratefully acknowledge the support provided by Secretaría de Educación Pública of México. I would like to thank L. Yeomans and F. J. Espinoza for general support and K. Van Horn for revision of the English manuscript.
References 1. J. H. Bruning, in Optical Shop Testing, D. Malacara, Ed. (Wiley,
New York, 1978), p. 409. 2. S. F. Jacobs and J. G. Small, "Liquid Level Interfereometer," Appl.
Opt. 20, 3508 (1981). 3. A. Livnat and O. Kafri, "Moire Technique for Measuring Liquid
