A PURE and transparent liquid of unknown identity can often be identified by comparing the speed of light in the liquid with the speed of light in air. This ratio, which is termed the index of refraction, has been accurately measured and tabulated for thousands of substances by institutions such as the National Bureau of Standards. The indexes are listed in handbooks of chemistry and physics.

With this information and a modest collection of apparatus (an instrument for measuring the speed of light in a liquid, a thermometer and paper strips that have been treated to indicate acidity an alkalinity) the experimenter can identify many reagents and can also determine the concentration of some solutions. A homemade refractometer for measuring the speed of light in liquids has been built by Gene and Gary Frazier (2705 Gaither Street SE, Washington, D.C. 20031). The instrument includes as the source of light a helium-neon laser, a classroom version of which can now be bought for less than $100. A helium-neon laser can also be made at home, although the task is not an easy one [see The Amateur Scientist, SCIENTIFIC AMERICAN, September, 1964, and December, 1965]. The Frazier brothers have written as follows about the principles and the operation of their instrument.

"Our refractometer measures the speed of light indirectly. It is based on the fact that a ray of light changes direction when it passes at an oblique angle from one medium, such as air, into another, such as water, having a different optical density. That is why a straight stick that has been pushed into clear water at an angle appears to bend at the interface of the water and the air.

"The amount that a ray of light bends at the interface of adjoining mediums was first described quantitatively 354 years ago by the Dutch mathematician Willebrord Snell. Snell multiplied the index of refraction of each medium by the trigonometric sine of the angle made by the ray in each medium with respect to a line forming a right angle with the interface. He found that the two products are always equal. Expressed symbolically the relation is , in which and symbolize the indexes of refraction of the respective mediums and and symbolize the respective angles made by the ray with a line that is perpendicular to the interface. The relation, which is named Snell's law, is basic to the design of lenses and prisms. It also figures in the production of many familiar compounds, including sugar (in solution) and alcohol.

"A simple apparatus that demonstrates the principle of the refractometer can be set up with parts available at a novelty store. The demonstration requires only a circular protractor, a rectangular aquarium (or some other kind of rectangular tank made of glass or clear plastic) and a narrow beam of light that can be directed downward at an oblique angle into a liquid contained in the tank. Immerse the protractor to its midpoint in the liquid. The light should barely graze the upper part of the protractor at an angle of precisely 45 degrees with respect to the vertical. The table of trigonometric functions lists the sine of 45 degrees as .707107.

"The beam of light enters the liquid from the air. The index of refraction of air can be assumed to be 1.0000, because the speed of light in air differs from its speed in a vacuum by only three parts in 10,000. For this instrument the numerical values of the first medium in Snell's law are n = 1.0000 and =.707107. To demonstrate the operation of the instrument fill the tank with water to the middle of the circular protractor. Switch on the light beam and adjust it to strike the water at exactly 45 degrees. Read the angle made by the submerged beam with the vertical. If a laser is the light source, the experimenter should be able to read the angle to within one degree.

"In water the observed angle should be about 32 degrees. The table of trigonometric functions lists the sine of 32 degrees as .529919. According to Snell's law, the index of refraction of water, , is equal to or 1.0000 X .707107/.529919 = 1.33437. The speed of light in air is 186,228 miles per second. The speed of light in water is equal to the speed of light in air divided by the index of refraction of water: 186,228/1.33437 = 139,562.5 miles per second. This speed is lower than the best recorded value of the speed of light in water at 20 degrees Celsius by only 112 miles per second, or about .08 of 1 percent. The error stems from the difficulty of measuring angles closely with a protractor.

"Although an accuracy of .08 of 1 percent is adequate for rough measurements, the identification of unknown liquids and the study of the concentration of solutions require accuracies better than .004 of 1 percent. Finely wrought apparatus is needed to measure angles directly to this accuracy. Measurements of length to the same accuracy can be made with apparatus that is much easier to improvise at home.

"We cast about for a scheme that would enable us to measure angles indirectly to within at least a tenth of a degree. We finally hit on the idea of causing the specimen to displace a beam of light sideways and of employing a concave mirror to amplify the displacement. In this scheme the beam of light in air falls obliquely on one side of a glass cell with fiat, parallel walls. The cell contains the specimen liquid.

"The beam is bent in the horizontal plane at the interface of the cell and the air. It then propagates through the cell and emerges into the air through the opposite wall. At this interface the beam bends exactly the same amount as it did at the first one but in the opposite direction, because it now proceeds from a medium of relatively high refraction into a medium of lower refraction. The path of the beam after its encounter with the cell exactly parallels its path prior to the encounter. The displacement increases with the index of refraction of the specimen. The apparatus is aligned when the cell is empty so that the beam proceeds to the center of the concave mirror, which can be of spherical form.

"When the cell contains a liquid, the displaced beam falls on the mirror at a point some distance from the optical axis. The beam is reflected by the mirror at an angle that increases with the displacement. The reflected beam falls on a horizontal scale that can be positioned at any convenient distance from the mirror.

"The displacement of the spot of light on the scale increases with the distance between the scale and the mirror and inversely with the focal length of the mirror. Mirrors of spherical figure deflect the beam at disproportionately larger angles as the beam departs from the optical axis. For this reason the excursion of the spot of light does not increase from its zero position on the scale in direct proportion to the displacement of the beam by the specimen, as would be the case if the mirror were a paraboloid. The distortion is taken into account when the instrument is calibrated.

"The sensitivity of the instrument is determined by the thickness of the cell (T) and by the magnification of the displacement by the mirror. Our cell is made of standard microscope slides 25 millimeters wide, 75 millimeters long and one millimeter thick. The thickness of the cell (T) is 75 millimeters.

"The displacement (d) of the beam can be calculated with the formula that is included in the accompanying schematic illustration of the instrument [below]. For example, in water, which has an index of refraction of 1.3333, the calculation for displacement is millimeters If the mirror had a paraboloidal figure of 100 millimeters focal length (c) and the scale were located 1,500 millimeters from the focal plane (a), the displacement of the spot of light on the scale (d) would amount to , or 1,500 X 19.86/100 = 297.9 millimeters.

"A beam of light that falls at an angle of 45 degrees on water at a temperature of 20 degrees C. is actually bent about 32.027 degrees at the interface of the air and the water. Our instrument measures the angle in terms of the excursion of the light spot on the scale at the rate of 9.25 millimeters per degree of displacement. This sensitivity enables us to easily measure the angle of refraction to within .1 degree and to determine the refractive index of liquids to within one part in 1,000.

"A helium-neon laser is our light source for aligning the instrument and calibrating it and for measuring indexes of refraction. The output window of the laser is fitted with a short cylindrical cardboard tube that supports at its outer end a white cardboard disk. The disk is pierced with a centered pinhole about half a millimeter in diameter. A distant reflector that picks up the laser beam can be manipulated to reflect the beam back on itself. When the reflected spot of light is centered on the pinhole, the plane of the reflector is perpendicular to the beam.

"To make the displacement cell cut from window glass a 3-1/4 inch square. Support the glass on a 3/4-inch wood base measuring five by eight inches. Insert a wood screw through one end of the base at the center to function as a leg. The leg should support the base about half an inch above the workbench.

"Insert somewhat longer wood screws through the corners of the opposite end of the board. These screws serve as adjustable legs for leveling the wood base. Attach the glass square lightly to the center of the base with a dab of siliconerubber cement.

"Direct the beam of the helium-neon laser across the center of the glass square at a height of about 12 millimeters. Apply a few thin dabs of epoxy cement to one long edge of a clean microscope slide. Center the tacky edge of the slide above the glass square about six millimeters from one edge of the square and lower it into contact. Turn the wood base as required to reflect the laser beam from the microscope slide to the cardboard disk of the laser.

"Tilt the slide backward or forward to reflect the beam to the height of the pinhole. Center the beam exactly on the pinhole by manipulating both the base and the slide. Brace the slide in this position by placing one end of a light wood slat on the bench and resting the other end on the upper edge of the glass. Turn off the laser.

"When the epoxy has hardened, add microscope slides to form the sides of the cell. They need not be accurately aligned, but do not disturb the position of the wood base or the aligned slide. Check the alignment of the first side. If the alignment has been disturbed, adjust the base to recenter the beam on the pinhole. Complete the cell by adding and aligning the fourth side.

"Manipulate the slide to center the beam on the pinhole without disturbing the position of the base and the first slide. When both the front and the back slides center the spot on the pinhole, both must be perpendicular to the laser beam and optically parallel. Make the completed cell watertight by coating all joints lightly inside and out with silicone-rubber cement.

"One of the parallel sides must now be positioned at an angle of exactly 45 degrees with respect to the laser beam. To make the adjustment fasten the long side of a 45-90-45-degree prism to one of the parallel sides of the cell with two short pieces of sticky tape. Adjust the beam of the laser to fall on the prism at a height some eight millimeters above the bottom of the cell. Adjust the base of the cell assermbly to the position at which one 45-degree facet of the prism reflects the beam back on the pinhole. The side of the cell now makes a 45degree angle with the beam [see illustration at left].

"Remove the prism without disturbing the position of the cell. A substantial amount of subsequent labor can be saved by making a metal plate containing three detents that fit the three adjustment screws of the cell assembly. The plate can be aligned, with the adjustment screws in the detents, and screwed to the bench. Thereafter the cell can be removed (for cleaning) and automatically replaced in alignment by returning the tapered ends of the alignment screws to the detents.

"As we have mentioned, the horizontal displacement of the laser beam by the specimen cell is magnified at the position of the scale by reflection from a concave mirror. Ideally the mirror should be a paraboloid. Parallel rays that fall on all parts of the paraboloid pass through the focal point. For maximum convenience the mirror should have a small ratio of focal length to aperture, say f/l, because for a given magnification the distance between the focal plane of the mirror and the scale varies inversely with the f ratio. The smaller the f ratio, the more compact the instrument.

"On the other hand, the difficulty of making parabolic mirrors varies inversely with the focal ratio. It is all but impossible for an amateur to make a good parabolic mirror. Mirrors of spherical figure not only are easier to make but also are available commercially at modest cost from suppliers such as the Edmund Scientific Co. Rays that fall on the edge of such mirrors, however, are reflected at angles corresponding to significantly shorter focal lengths than rays reftected from zones closer to the center. We accepted this spherical aberration and compensated for it by calibrating the scale of the instrument with a series of test liquids of known refractive index.

"The mirror is mounted on a wood L bracket fitted with adjustment screws that resembles the cell bracket in principle. We attached the back of the f/1 mirror to a wood disk with silicone-rubber cement. The disk fits tightly into a wood rod 1-1/4 inches in diameter. The rod is supported by an eyepiece holder of the rack-and-pinion type used in reflecting telescopes.

"The scale, which can be a 450-millimeter length cut from a meterstick, is also supported by a wood L bracket, but no screws are needed for making fine adjustments. The scale is fastened to a small sub-base of 1/4-inch plywood that is attached to the main bracket with wood screws. The screws pass through slots in the sub-base. The slots make it possible to adjust the height of the scale through a range of about an inch. The laser beam passes under the 'zero' end of the scale on the way to the mirror. The mirror deflects the beam laterally and also upward at an angle of about one degree.

"To set up the instrument turn on the laser and mark the path of its beam above the bench with two widely separated pushpins. Draw a straight line on the workbench between the pins. With a carpenter's square draw a perpendicular to this line a few inches from the laser. Place the zero end of the scale above the line and adjust the position of the scale to parallel the perpendicular. If the focal length of the mirror is about 100 millimeters, place the mirror about 1,600 millimeters from the scale.

"With the cell empty switch on the laser. Shift the mirror to the position where its center intercepts the beam. Adjust the angle of the mirror to center the reflected beam on the pinhole. Tilt the beam upward so that it falls on the scale. Adjust the scale sideways to center the beam on the zero graduation. The beam is not displaced significantly by the empty cell.

"Switch off the laser. Measure the interior thickness of the cell to determine T as accurately as possible, at least to within .04 millimeter. Fill the cell with clear water at room temperature, preferably at 20 degrees C.

"Switch on the laser. In our instrument water causes the beam to appear upscale about 300 millimeters from the zero mark. The beam should remain fixed. Any movement of the beam may indicate temperature differences in the specimen liquid that give rise to corresponding differences in the index of refraction and even convection currents. The effect can be minimized by working with specimen liquids that are at room temperature.

"The instrument can now be calibrated. We begin with a graduated scale such as a meterstick. To make the calibration we measure a series of increasingly refractive known solutions. An inexpensive series can be made up with cane-sugar syrup in increasing concentration.

"Tabulate the known index of refraction of each test solution and the corresponding excursion of the light beam on the scale. The spot of light that indicates the position of the beam varies in diameter with its distance from the laser. Center the spot exactly on the zero graduation of the scale and thereafter read all upscale excursions with respect to the center of the spot.

"In general, indexes of refraction range from 1.0010 for acetone at zero degrees C. through 1.8033 for a sugar solution of 85 percent at 30 degrees C. Within this range we established 15 calibration points at approximately equal intervals on the scale. On a strip of paper we plotted the calibration points to make a replacement scale divided into increments of refractive index. We cemented the strip to the meterstick to make the instrument read directly. Alternatively, the tabulated data could have been plotted as a calibration graph."

Bibliography

FUNDAMENTALS OF OPTICS. Francis A. Jenkins and Harvey E. White. McGraw-Hill Book Company, 1950.

APPLIED OPTICS AND OPTICAL DESIGN. A. E. Conrady. Dover Publications, Inc., 1960.

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