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The Speed of Light

History

Galileo attempted to measure the speed at which a light beam travelled from a lantern on one hilltop to another, but was never able to observe a delay from which he could calculate the speed. In 1676, a Danish astronomer visiting the Paris Observatory directed by Giovanni Domenico Cassini (1625-1712) was studying the orbits of the Galilean moons of Jupiter in hopes of solving the longitude problem. He noticed a curious pattern in the moons' eclipses. The time of the eclipses was early by a few minutes when the earth was at points in its orbit closest to Jupiter, and a few minutes late six months later. Ole Roemer (1644-1710) correctly inferred that the shifts were due to the finite speed of light and in 1676 he estimated that speed as 140,000 miles per second (about 25% low).

Foucault devised a way to make a terrestrial measurement of the speed of light using a rapidly rotating mirror. In the time it took a ray of light to travel from the surface of the rotating mirror to a distant fixed mirror, reflect back, and bounce again off the rotating mirror, that mirror turned a little bit and the reflected beam was deviated slightly from the path it would take when the mirror was not spinning. Knowing the rotation rate and the distance between mirrors, it was possible to deduce the speed of light.

Other methods relying on measuring a distance and a time taken by light to cover that distance were developed and refined, leading to increasingly accurate measurements of the speed of light. Ultimately, the precision of time measurements outstripped that of distance measurements, which were tied to the "standard meter" defined as the distance between scratches on a platinum-iridium bar kept near Paris. In view of the success of Einstein's special theory of relativity, which holds that the speed of light is independent of the inertial reference frame in which it is measured, it was agreed in 1983 to define the value of the speed of light as 299 792 458 m/s, so that the length of the meter would be based on the defined value of c and the fundamental unit of time, the second.

In view of the accuracy of the method you will explore in this experiment, we will not adopt the current definition of the meter based on an assumed value of the speed of light, but will instead measure the speed of light in terms of a measured length and time.

Theory

A conceptually simple way to measure the time take by light to travel a fixed distance would be to send out a pulse of light to a distant mirror and record the time elapsed between sending the pulse and receiving its reflection. One could use a beamsplitter to send a portion of the pulse directly to the detector and the remainder out to the distant mirror. When this portion of the pulse arrives at the detector it produces a second voltage or current spike. By monitoring the output of the photodetector with an oscilloscope, and timing the interval between the spikes, one can measure the speed of light.

A less expensive alternative is to modulate at reasonably high frequency the output power of a laser beam, again sending a portion directly onto the detector and the remainder out to and back from a distant mirror. Because a periodic wave shifted by any whole number of periods or wavelengths looks exactly the same as the unshifted wave, to know exactly how many periods have elapsed requires some careful thought.

Apparatus

You will use a semiconductor laser modulated by the output of a function generator, which is a device that puts out a periodic voltage with a variable frequency. The setup is shown in the figure below. The beam will pass through a pair of lenses (for reasons to be explained below), bounce off a flat mirror, and travel down the corridor to a corner cube on the far wall. A corner cube sends an incident beam back parallel to itself, so the beam retraces its path, bounces off the mirror, passes through the second lens, reflects off the beamsplitter and a portion gets focused onto a photodiode detector. By comparing the signal from the photodiode with that from the function generator driving the laser modulator using the visual representation on the oscilloscope screen, you can determine the time it takes the laser beam to make the round trip.

All the optics are mounted on a rail, which gets mounted on the wooden mounting bracket on the wall with two 1/4-20 screws. This will be done for you the first day, but you may be pressed into service on subsequent days. Please take great pains not to drop the rail, as this will destroy the apparatus. Tighten the socket-head 1/4-20 screws well to prevent the rail from shifting as you adjust the top mirror.

Electronics

The laser diode is normally powered by 3 V from a pair of batteries in series. To modulate the output of the laser, a modulator is placed in series with the battery. By varying the resistance of the modulator, the current through the laser diode varies, causing the brightness of the diode to change. The batteries will probably be replaced by a 3-V transformer. Note that the polarity of the circuit is important; the diode needs to be reversed biased.

The circuit diagram for the modulator is shown below.

The laser/modulator can function up to a maximum frequency of 4 MHz, although the depth of modulation may diminish at the highest frequencies. Furthermore, the detector has a maximum frequency it can handle, so be aware that the shape of the modulated signal may vary with frequency.

Procedure

Most laser beams can cause permanent eye damage. Never look directly into a laser beam. Even though our laser beam is fairly weak, it is important to take care not to expose yourself, your lab partner, or anyone else to a dangerous beam.

Never touch the surfaces of optics. The oils on your fingers will significantly degrade the transmission of lenses and the reflection from mirrors. Handle optics carefully by the edges, or by their holders or mounts.

Measuring the Speed of Light

The modulator has a maximum frequency of 4 MHz, but the amplitude of modulation decreases with frequency, so you will probably have better luck with frequencies closer to 1 MHz. Why do you need to modulate the output of the laser? Why do you need to vary the frequency of the sine wave you use to modulate it?
Connect the output of the detector to Channel 1 of the oscilloscope using a BNC cable. Use a tee to send the output of the function generator to both the modulator and Channel 2 of the oscilloscope. Press the trigger button on the scope and select Channel 2 for the trigger source. Adjust the gain setting on Channel 2 and the trigger level until you see a sine wave on the oscilloscope. See the Oscilloscopes page for more information on how to use an oscilloscope.
When you can see a sine wave from the detector on Channel 1, you can begin making your measurement. You will probably need to use a setting of about 5 mV/division for Channel 1 and 1 V/division for Channel 2. Start with a frequency of 100 kHz on the function generator, and adjust the time base of the oscilloscope and the trigger level to see two or three periods of oscillation.
Troubleshooting If you don't see a signal from the photodetector, see the troubleshooting section below.
Check whether the time base of the oscilloscope and the function generator agree. Using the Measure button you can set up a measurement of the frequency of the wave on either channel, or both channels. Make sure you have at least two cycles on the screen to use this feature.
Do the signals from the photodetector and the function generator have the same phase at 100 kHz? Do they basically look the same? If they aren't very close, see the troubleshooting section for tips on adjusting the apparatus.
Life would be simple if the phase of the function generator and the phase of the modulator agreed. Life is not simple. To convince yourself of this, place a reflector above the second lens to send the beam immediately back through the lens, off the beamsplitter, and onto the detector. You should have a very large signal. Vary the frequency from 100 kHz to 3 MHz to see how the phase of the light signal compared to the drive signal varies with frequency. To correct for this behavior, you should use the reflector at each frequency you investigate to correct for the electronic phase delay.
Make a rough measurement to see if you are close to the expected value. Just pace off the distance and use the shift in the wave you observe on the screen. Try more than one modulation frequency.
To refine your measurement, measure the round trip distance more carefully.

Troubleshooting

Wiring	Make sure that the output of the Video port of the detector is connected to Channel 1 of the scope and that the detector is switched on (the green LED is lit when the detector is on).
Alignment	Make sure that you can see a small, bright red dot of light on the gray square area of the photodiode directly under the Video output.
Oscilloscope	Check that the scope is set to look at Channel 1 by examining the display area for a "1" at the margin. If it is not shown, press the Channel 1 button until it is activated. Then make sure the gain setting on the channel is about 5 mV and that the beam is centered. To check, you can press the Channel 1 button until it is grounded, then look for the trace in the middle of the screen. Rotate the offset knob until the trace is centered.
Laser	The laser wants about 2.7 V. If you use the DC power supply to power the laser, set the voltage to 0 before turning on the power supply. Turn up the current knob a bit, then gradually turn the voltage knob until the laser begins emitting red light, at about 2.5 V. Put a white paper or card in the beam path and gradually raise the voltage until the beam becomes fairly bright. In no case raise the voltage above 3.0 V. You want the voltage to be set just at the onset of the bright regime. If you get up to 3.0 V, gradually lower the voltage until the beam begins to dim, then raise the voltage a touch to enter the bright regime.
Modulator	If the photodiode signal on the oscilloscope is not modulated at the same frequency as the function generator, the modulator potentiometer ("pot") may be adjusted incorrectly. Set the function generator frequency to 1.0 MHz. If that fixes the problem, then you probably had the frequency too high. If not, use a small flat-blade screwdriver to turn the pot counterclockwise until the laser beam dims noticeably. While watching the scope, gradually turn the screw clockwise until you see a signal modulated at the frequency of the function generator.
Noise	If the signal is very noisy, adjust the mirror to maximize the intensity of the reflected light on the photodiode. If the signal appears to have the right frequency, turn on averaging using the Measure button. If it doesn't have the right frequency, see the directions above for adjusting the laser voltage and the modulator.

Updated 10/18/00 by Peter N. Saeta .