Total internal refraction of laser beam through sugar solution with refractive index gradient; representative of seismic waves through Earth's mantle.
What it shows:
As a simulation of atmospheric refraction, this demonstration shows the gradual and continuous bending of light due to a gradient in the optical density of the medium. In this case the variable refracting medium is a tank of sugar water with a vertical gradient in the concentration of sugar and a HeNe laser provides the light beam. It can be used as a model of mirage formation (except that the direction of increasing refractive index is in the opposite direction) or even as a representation of the refraction of seismic waves through the Earth's mantle. Total internal reflection, total internal refraction, as well as periodic beam defocusing and refocusing are also shown.
How it works:
A long, narrow, plastic tank 1 is filled two-thirds full of warm water adulterated with a light-scattering substance. Sugar cubes are added to the tank and allowed to dissolve undisturbed. This establishes a strong gradient in the index of refraction ranging from n=1.50 (84% sugar at the bottom of the tank) to n=1.33 (pure water, a few centimeters off the bottom). A laser beam, directed horizontally down the length of the tank, will be refracted downward in an arc and "bounce" by total internal reflection at the plastic-air interface at the outside surface of the bottom of the tank. Following the bounce, the beam will arc upward and back down by "total internal refraction" and so on. Depending on the gradient that has been established and the angle and height at which the beam enters the tank, the beam may bounce anywhere from once to several times in the length of the tank.
A closer observation reveals another interesting effect: a defocusing (divergence) of the beam as it arcs downward toward the higher refractive sugar water and a refocusing as it arcs upward. This effect is only in the vertical direction (direction of the gradient) and not in the horizontal, and is periodic with every bounce. Thus this demo can also be used as a model of long-distance laser beam transmission (periodic focusing and defocusing with gas lenses) and possibly as an analogy to accelerator strong focusing.
Setting it up:
The tank of sugar water must be prepared at least 24 hrs before intended use to allow enough time for all the sugar to completely dissolve. Fill the tank to about two-thirds full of warm water and stir in a few drops of milk to make the laser beam visible by scattering. 2 A good rule-of-thumb is, if the water looks cloudy (milky) you've added way too much milk and the beam will become attenuated (scattered out of the path) before it reaches the end of the tank (that's bad). If you don't have a feel for what looks too cloudy, look down the length of the tank...if you can't see the other end of the tank, the laser beam won't make it that far either (for the same reason). Having satisfied yourself on the proper concentration of milk, drop sugar cubes in so that 50% of the bottom area of the tank is evenly covered with cubes. 3 Of course they'll start dissolving immediately but it takes a long time to dissolve completely and you don't want any undissolved sugar remaining on the bottom as this will spoil the "bounce" of the beam.
Once the gradient has been established it will be quite stable and the tank can be moved into the lecture hall, albeit very carefully so as not slosh out the water. Best results are obtained 24 to 30 hrs after preparation, but the demo will still work even after two to three days. However, expect fewer bounces with increasing time as the sugar diffuses upwards and slowly degrades the gradient. Also, the sugar water will begin to reek after a few days as the milk begins to spoil.
In an emergency (short time notice), you can use Karo™ light corn syrup or prepare a concentrated sugar solution in a beaker of water 4 and add it to the tank by gently pouring it down a stirring rod to minimize mixing with the water. However, this results in too strong a gradient and it's best to still wait a few hours for diffusion to take its course and "soften" the interface between sugar solution and pure water.
Set the tank on the lecture bench supported by a block of wood on each end. Setting directly on the lecture bench may result in frustrated total internal reflection, especially if some spilled water allows for intimate optical contact between tank and bench-top. As for which laser to use, the more power the better: 2 mW is fine for a classroom but lecture halls require 10 mW and upward for good visibility. The 35 mW HeNe and the 125 mW Argon work brilliantly. The Argon laser is a bit unwieldy and noisy, however and requires warmup/cooldown time.
Originally conceived by William M. Strouse, 5 we have modified it only in scale. If you happen to use corn syrup and the Argon laser combination, you will be in for a surprise. It turns out that corn syrup fluoresces yellow when excited by the blue/green laser light. As the Argon beam arcs downward, it appears to turn yellow in color, bounces off the bottom and becomes blue/green again as it arcs upward. And this happens with every bounce! This would be an interesting puzzler to present to the class.
1 It measures 36" × 5" × 1.5" (91.4cm × 12.7cm × 3.8cm). The side walls are made from 1/4" thick acrylic, the end walls from 3/8", and the bottom from 1/8". Size is, of course, not important to the demonstration except that the thickness of the bottom of the tank should be kept to a minimum; no more that 1/8". This is because one gets a partial internal reflection off the inside surface as well as the total internal reflection off the outside surface; if the distance between these reflections becomes much larger than 1/8", the double reflection becomes noticeable and degrades the effect.
2 Rather than milk, you can of course use other scattering substances. For example, Edmund Scientific Co. sells a "Scatter Liquid Concentrate" (Cat. No. 38,938) which is a concentrate of polystyrene spheres. Mix 5 drops per 2 liters of water. We have also used DuPont "Ludox", which is a colloidal silica; mix 10 ml per 2 liters of water.
3 The tank size and water volume is irrelevant. 50% coverage (as measured by eye) is what counts...if you overdue it you'll find that the excess sugar won't dissolve.
4 Use about 800 g of sugar to 500 ml of water. Continuously stir and heat to 40-50°C; let cool before adding to tank.
5 W. M. Strouse, Am J Phys 40, 913 (1972) "Bouncing Light Beam"