Measuring the Speed of Light

Transcription

1 Physics Teaching Laboratory Measuring the peed o Light Introduction: The goal o this experiment is to measure the speed o light, c. The experiment relies on the technique o heterodyning, a very useul tool or making high requency measurements. s the main goal o the experiment is to measure c as accurately as possible, it will be very important or you to keep track o errors and to calculate the inal uncertainty in your result. Conceptually the experiment is very simple: measure the time t it takes to send a pulse o light rom a source to a detector a distance d away. The speed o light is simply c = d / t. The obvious problem is that c is very large, about m/s, so the travel time is very small. You might manage to it a 3m long apparatus in the lab, but even this means the time will be about 0ns, which is just at the resolution limit o a good oscilloscope. I you want to determine c to % accuracy (a modest goal) you ll need timing resolution o 00ps, which is very ambitious. Clearly, a more subtle measurement technique is required. First consider a more mundane experimental problem. You ll recall rom the electronics standard experiment that electronic devices (and even cables) can introduce time delays o tens o nanoseconds. This delay will be a large instrumental oset to the timing measurement, since we are interested in nanosecond precision, but can be cancelled by making a dierential measurement. First, measure at a distance d so that the time delay is t = d / c + t0, where t 0 is the unknown delay due to the source and detector electronics. Then, without changing the electronics setup measure again at distance d so t = d / c + t0. Taking the dierence in the measurements cancels the unknown delay, t t = ( d d ). () c Measuring time delays at many distances d will be even better. plot o the delay vs. i d i d should give a straight line with slope /c. However, we still need to measure the time dierence with sub-nanosecond resolution, so we haven t solved the undamental problem. We use heterodyne detection to solve the timing problem. Imagine we send not a single pulse o light but a train o pulses at a signal requency. We compare the arrival time o the pulses with a reerence clock running at a slightly dierent requency, ( stand or local oscillator). Periodically, the signal pulses and the clock pulses will coincide; the requency at which this occurs is called the beat requency, B. The top part o igure illustrates this- the important point is that the beat requency is much lower than either or i they are similar. What happens when we delay the signal? The bottom hal o igure shows the result. In this example the signal is delayed by about hal o the signal period, this shits the beat pattern since we haven t changed anything about the reerence clock. Notice that the beat

2 pattern shits by about hal the period o the beat signal; a small shit in time o the pulse signal has translated into a large shit in time o the beat signal. It will be much easier to measure this time shit using our relatively slow electronic equipment. It is useul to think o this in terms o phase shits. In igure the phase o the pulse signal has been shited by 80º, which leads to a shit in the beat signal o 80º. By comparing the pulses to a reerence clock the requency is greatly reduced but the phase shit stays the same. The next section proves this mathematically. Figure. Compare the beat requency between the signal and the local oscillator in the top to the bottom, where the signal has been delayed by hal a period. Heterodyne Detection Rather than treat pulsed sources, we will consider signals which are sine waves. This is both easier mathematically and also corresponds more closely with how the apparatus t = sin π t + ϕ and a reerence clock actually works. Consider a source signal ( ) ( ) R( t) B sin( π t) = (we deine the phase o the reerence to be zero). For our apparatus = 60MHz and = 59.9MHz. We apply these signals to a device called a mixer, which multiplies them. The output (beat) signal is then B( t) = B sin ( π t + ϕ) sin( π t). Using standard trigonometric identities it is easy to show that this is equal to B B B( t) = cos[ π ( ) t + ϕ] cos[ π ( + ) t + ϕ]. Typically the beat signal is iltered to eliminate the sum requency term, as shown in igure. The output is then just B B( t) = cos[ π B t + ϕ], where in our case the beat requency B = is about 00kHz. * Note that Fig. ) Heterodyne detection. * In radio engineering lingo this is the IF or intermediate requency.

3 the phase ϕ o the original signal is now the phase shit o the beat requency. Phase is the integral o requency with respect to time. ince the signal requency is constant, we can simply multiply both sides o eq. by the angular requency π and interpret the time dierence t t as a phase dierence: π π = ( d d ) c = d c ϕ ϕ or ( d ). () ϕ is the phase shit with distance o signal requency, 60MHz, but o course the heterodyne detection translates this into the phase shit o the 00kHz beat requency. The pparatus schematic layout o the apparatus is shown in igure 3. The LED is modulated at 60MHz, light rom it travels a distance d to the detector. Note that both the transmitter and the detector signals are mixed with the 59.9MHz reerence oscillator. Both low requency outputs should be monitored with your scope. The output rom the transmitter mixer deines zero phase. You can apply an additional arbitrary phase shit using the phase knob. Fig. 3. The experimental apparatus. Experimental Procedure We suggest the ollowing procedure. We also give some hints or things you should note in your lab book.. Make sure the rails are horizontal and lined up. Make sure all lenses, LED s and detectors are at the same height. What is an eective way to do this?. witch the LED on and adjust it to give a parallel beam. How? Does it matter i it is parallel? 3. Focus the LED beam onto the diode inside the receiver box. The detector diode is said to be 3 mm inside the box. How accurately do you need to know this distance? Can you check it?

4 4. The alignment is good when you can change the LED - detector distance without losing the signal. It is worth spending time to get this right as it makes the rest o the experiment much easier. What will the eect o a misalignment be on the measured value o c? 5. Display the reerence and detector signals on your scope. What scope settings give you the maximum phase inormation? Measuring the phase dierence Here are two methods you can use to measure the phase shit as a unction o distance between the source and detector: a) Using the crossover point Put the LED close to the detector lens, then adjust the phase shiter on the ront o the receiver box until the horizontal positions o zero volts o the signals coincide. It is critical that you don t have a DC oset as this can generate an enormous systematic error. Use the ND button to make sure your 0V position is known. Check your setup with your demonstrator. Measure d. From where to where? What is the uncertainty? Move the LED to a larger value d. Measure the phase shit and the new distance. Calculate d d and the uncertainty in the dierence. Move back to d and check that the phase shit goes back to zero. Is there a more rigorous check? d, ϕ at dierent distances. Record a ew points ( ) Calculate a value or c. Is it close to m s -? i i b) Using Lissajous igures witch the scope to Y mode. This plots V LED vs. V det rather than the voltage vs. time. You should see an ellipse, or the extreme orms o an ellipse, a straight line or circle. Change the LED - source distance and observe what happens to the ellipse. Watch out or the dual button on the scope. Move the LED close to the lens and set the phase shiter on the receiver so that you get a straight diagonal line. The phase between the two signals is zero. Expressed mathematically, x t V sin π t y t = V sin π t. Data analysis ( ) = ( ), ( ) ( ) LED Increase d until the trace has rotated 90º to produce another straight line. This corresponds to a phase shit o 80º. Formally, the scope trace is now given by x( t) = VLED sin( π t), y( t) = Vdet sin( π t + π ) = Vdet sin( π t). Measure the change in d. You can then calculate a value or c. Beore you go on you should calculate the uncertainty in each phase measurement method. Is one signiicantly better than the other? Does the uncertainty in the phase or the uncertainty in d d dominate the calculated value o c in eq.? Is it better to measure at longer distances? det

5 You should repeat the measurement o the phase shit or at least our or ive dierent distances. There are two reasons or this. The irst is that, assuming your errors are random, the standard error or a measurement repeated N times goes down roughly as N. veraging our measurements reduces the error by about a actor o. Multiple measurements are also an important way to check or systematic errors. Do your individual measurements agree within their uncertainties? How ar (measured in standard deviations) does your central value o c dier rom the accepted value? Does a plot o c versus d d or dierent distances reveal a systematic trend? The appendix to this script discusses the dierence between systematic and statistical (random) errors. The mystery substance The inal part o this lab is the identiication o the mysterious luid in the long tubes by measuring its reractive index. By comparing the value you get with a table or measured substances, you can identiy it. dmittedly, reractive index is not a unique identiier, but it can be part o a diagnostic strategy. You should use the more accurate phase measurement technique. Optical path length is the distance light travels as measured by its wavelength, i.e. d λ. In vacuum c = λ, but i the light propagates in a material with reractive index n then * its wavelength is shorter, λ = λ n, and so its velocity is v = c n. By inserting into the light path a tube o length d + d, where d is the length o substance and d the length o the glass ends, you impose an extra optical path length d n n + d n n = d n + d n, (3) ( ) ( ) ( ) ( ) where n, n, n are the indices o reraction o the mystery substance, the glass ( n =. 5 ) and air (we take n = ). You ll have to work out what this means or the phase shit given by eq.. The index o reraction o air is actually about +0-4 and depends on temperature, pressure and wavelength. Is your measurement accuracy high enough to worry about this?

6 ppendix: Random and systematic errors Physics advances by comparing mathematical models based on some underlying theory to experimental measurements. (I any o these links are missing then it is not physics, though it might be mathematics or engineering.) For the speed o light experiment, the underlying theory is Maxwell s electrodynamics, the model is the discussion leading up to eq. and the experiment is what you are working on. O course, sometimes the process works in reverse, an experimental discovery is modeled which then leads to a new theory. Random errors, noise, are due to luctuations in some part o the experimental equipment. These might be related to some undamental process, like the statistics o radioactive decay, or they might be the result o some compromise in the construction o the measurement apparatus. For example, at some level all measurements are limited by thermal luctuations, but it is rarely worth the bother and expense o cooling the experiment to 4K. In this experiment there is always a certain amount o uzz on the oscilloscope trace due to electrical pickup and ampliier noise. more expensive scope might eliminate some o this, but the trace would still have a inite width and so it will always be impossible to determine the phase shit with ininite accuracy. Repeated measurements should on average cluster around the true value but their distribution will have some width. This width is a measure o the random error. The ormulas you learn or combining errors are only valid or these random errors. ystematic errors result rom a ailure to model the experiment completely. For example, you might always read a scale at a small angle and so always suer rom parallax error. It would be possible to correct or this by recording the position o your eye with respect to the scale and doing a trigonometric calculation, but usually this would not be part o your model, or data. Repeated measurements won t reduce systematic errors- you either need to understand what you have let out and then elaborate your model to include the eect or to change your measurement technique. For this experiment you can at least conirm the presence o a systematic error by comparing your measured value o c to the true value. I it diers by many standard deviations then you ve let something out o your model. ystematic errors are generally much harder to deal with than random errors because they oten involve non-ideal behavior o equipment. It is diicult to understand in detail how everything works. Uncalibrated instruments (a badly printed meter rule, a bad timebase on your scope) are trivial examples o systematics. You can usually check or these by swapping equipment or ensuring that it has been properly calibrated. More insidious problems with this experiment might involve poor alignment o the source and detector, or a slow warm-up o the electronics leading to a phase drit. It takes a great deal o imagination to think o possible systematic errors and a great deal o technical knowledge to check i they are really inluencing your results. B. E. auer, 006 One could say that both random and systematic errors are eects which are ignored in the model. The dierence is that systematic errors are correlated with experimental control parameters while random errors are not. O course, you can t say anything until you ve computed an experimental uncertainty.