# Measuring the speed of light at Wiener Forschungsfest 2022

12 Sep 2022 - tsp
Last update 18 Jul 2024
14 mins

Imagine you could measure the speed of light with a bicycle! At the Wiener Forschungsfest 2022, we did just that.

This experiment was conducted at an exhibition held from September 9 to 11, 2022, in the city hall of Vienna. The primary audience was children, so this was not a precision measurement. However, it worked incredibly accurately given the conditions and methods used. The experiment aimed to illustrate how one can measure the speed of light and highlight that light travels incredibly fast but not infinitely fast. To reiterate: this was a demonstration experiment, not a serious measurement.

## Some history

Back in 1848 Hippolyte Fizeau determined the speed of light with an accuracy of about 5% (he estimated the speed of light to be too fast) which is pretty impressive for those times. For a detailed description see his original paper - or the way simpler readable version that one can find online. He measured the speed of light by using a gas lamp (not a laser back then - it’s even more impressive when one sees how hard it’s to use a laser even over a shorter distance and tries to imagine how hard it’s to see the reflection of a gas lamp on those distances), passing the focused light of this lamp through a gearwheel, passing it over a distance of roughly 8.5km, reflecting it on a mirror - i.e. a total distance a little bit over 19km - and passing it through the same gearwheel again. The whole apparatus had been driven by hand over a mechanical gear with a hand driven crankshaft (again impressive when one thinks about vibrations and other problems).

When one now turns the wheel the beam gets periodically chopped. At slow speeds the returned beam is still a pulsed version of the light one has sent out. But at some point the time it takes for the light to travel forth and back the distance to the mirror:

[ \delta t_{travel} = \frac{s_{travel}}{c} ]

is equal to the time it takes to turn the gearwheel one tooth forward. One can then calculate the time it takes to fully swap a tooth with a hole - note that this of course is only a coarse estimate to illustrate how it works, the transition will be (depending on focus of the beam) smooth. When the wheel turns at a frequency of $f_{gear}$ the teeth and holes swap with a frequency of $f_{swap} = n * f_{gear}$ ($n$ is the number of teeth)

This can be transformed into a period:

[ \begin{aligned} \delta t_{swap} &= \frac{1}{f_{swap}} \\ &= \frac{1}{n * f_{gear}} \end{aligned} ]

When one does not see any reflected light any more one can assume that it takes at least the swapping time for the light to travel forth and back - or a multiple of it:

[ \begin{aligned} \delta t_{swap} &= \delta t_{travel} \\ \frac{1}{n * f_{gear}} &= \frac{s_{travel}}{c} \\ \to c = s_{travel} * n * f_{gear} \end{aligned} ]

A variation of this method would be focusing the returning beam onto the gearwheel in a way that one blocks the returning beam as long as the outgoing beam passes between two teeth. When one turns the wheel fast enough the light that was allowed to travel outwards will take the same time for it’s travel as the wheel takes to turn far enough to allow it to pass through the space between too teeth.

For a full and correct interpretation one will have to take into account the shape of the gearwheel as well as the size of the laser beams dot or the size of the light cone reflected.

## Our experiment

Due to the nature of the exhibition we used a somewhat different assembly that still hinted to the usage of a gearwheel back in 1848 and was somewhat more attractive to the target audience. The idea was pretty simple:

• Use a laser beam in visible range ($630nm$)
• Chop it using a bicycle wheel (so the visitors had an interactive experiment) - this is the first major difference to a serious experiment.
• Split the chopped beam into a reference signal and a beam that is transferred through the exhibition hall and back to our assembly
• Detect the outgoing and incoming light pulses using separate photodiodes (i.e. not passing through the chopper again but cheating and sampling using a modern oscilloscope with a high resolution clock)
• Calculate the speed of light out of the correlation between the two signals
• Visualize everything on a fancy screen that attracts peoples attention

The interesting black and white images that you see also on this photograph had also been presented in conjunction with a video sequence. These show the propagation of a femto second long spherical wave of light (an ultra short laser pulse) imaged via a CCD camera that has been gated using a GEM foil (utilizing conversion from photons to electrons, then gating and then using an phosphor to convert the electrons back to photons). This has been done by SEEC photography.

### The laser source and the chopper

Even though the speed of light is always constant and it’s not measured like in the historical experiment by double passing through the wheel (due to the short distance we had available the rotation speeds even for a gearwheel would have been insane - not to speak about the rotational speed for a bicycle tire) the visitors had the ability to influence the measurement using the bicycle - the faster the tire turns the cleaner the oscilloscope measured the sloped due to the ADCs resolution and the better one can of course detect the time difference between the incoming and outgoing pulse (i.e. one gets a point measurement that should be near the real speed of light and an error margin that gets smaller the faster the wheel turns).

To realize this we first took a 630 nm laser that was fixed together with an exercise stand for the bike (the bike has been provided by Intersport Winninger for the duration of the exhibition) as well as two mirrors and an fiber port to align and focus the beam onto a fiber optics cable. The laser we used was the most expensive part of the experiment and has been borrowed from an experiment at the University of Vienna. This has been a Thorlabs laser that already coupled into a single mode fiber and provided optical output powers between 0 and 8 mW. This type of laser and controller is of course way overkill for such an project, a simple diode and collimator would do. In the beginning we tried to use it at low powers but saw there major noise caused by the laser driver - so in the end we used it at 8 mW which turned out to be just sufficient for the whole experiment. Coupling into the single mode fiber provides the advantage that one really only transmits a single mode into the experiment which makes focusing much more simple - the only thing required was a simple telescope built out of two $125mm$ lenses. The focal point of the two lenses was put exactly at our chopper.

To provide a sharp cut the tire was covered with tape - and at the only gap that has been left opened a small knife blade has been fixed. It doesn’t really matter on which side of the gap one uses the knife edge - it’s only important to use only that edge for triggering and calculating. If one doesn’t do this the signal will get smeared out the same way as an out of focus cut would do. The main influence users had was the speed of the tire and thus the steepness of the signal edge. When one does not take into account discretization at the ADCs this would not have much influence on the accuracy of the result - but when one takes a look at the discretized curves one can immediately see the accuracy of the calculation increases the steeper the edge was.

After passing through the wheel the light just got again coupled into a single mode fiber (in our case standard telecommunication G652.D low water profile fiber - this accounted for a major part of losses in our setup since this fiber is not optimal for visible wavelengths - but it’s cheap and was sitting in a drawer - and was long enough for our application while still having already spliced FC/APC connectors at both ends that where compatible with out fiber ports). Due to the used fiber only less than a $mW$ of power arrived at our telescope.

The housing that you see above has been build purely for safety reasons. Since we targeted younger people and then had a very fast spinning wheel (even with a blade attached) we wanted to prevent anyone reaching into or catching in the tire. During the course of this event this has turned out to be a very good idea - very often shoes hit the enclosing, children had been very curious and also reached near the wheel and since we shot through the chopper with an 8 mW laser beam some kind of housing was required out of laser safety reasons anyways so no one was able to look into the beam of reach in with reflective stuff. All of the assembly was mounted on a very heavy optical breadboard to dampen vibrations by mass.

### Our beamsplitter, photodiodes, telescope and beamline

The main part of our experiment resided on a small breadboard that unfortunately in this variant was not accessible to the users since access to the gallery of the city hall has been restricted. It has been built on a small optical breadboard and consisted of a transmitter and receiver part.

The transmitter part consisted of an outcoupler, a beamsplitter that directed about half of the chopped beam on our first photo diode that also provided the trigger pulse and the second half of the chopped beam through two mirrors (for alignment as usual for optics setups) through a beam widening telescope (consisting of a pair of lenses - 50mm and a 150mm lens. This also provides a beam expansion ratio of 1:3).

As one can see from the photographs on the final setup we’ve added a micrometer stage on the outgoing path to be able to adjust the collimation of our outgoing beam. We also added some anodized aluminum foil around the photo diodes for ambient light shielding. The beam for the outgoing photodiode was somewhat misaligned since the power difference between both arms was very huge - one could have used a phase plate and polarizing beamsplitter to adjust power in both arms but that would have complicated this setup and we only had such phase plates and polarizing beamsplitters available for the near infrared range due to the experiments our labs usually perform.

The beam then one was directed onto a single 50mm silver mirror (surface coated, broadband) that has been mounted on a kinetic mount for easier alignment. The beamline was 72 meters away so the total distance traveled was 144m (on some days this has been adjusted).

The reflected light was then again passed through a telescope consisting of a 200m and another 50mm lens. Afterwards it was again directed via two mirrors onto a 20mm focusing lens and then onto our photodiode.

### Our backup beamline

In case one wants to use such a setup where one is not able to run a free-space beamline (for example when not having spacing in height to people - always keep laser safety in mind - or simply not having space available) we also built a variant using 1 km of telecom fiber. The setup works about the same, the main difference is that the telescopes for in and outgoing beam are replaced by fiber couplers that couple the collimated beam into and out of a fiber. One also has to adjust the refraction coefficient in the software when using a fiber (from 1 to about 1.47 for our fiber)

To access the fiber we just attached to FC/APC connectors to both ends of the fiber, polished and used it in wrapped up state. This allows for a very compact setup.

Our photo diodes had been attached via two long coax cables to two input channels of an Rigol MSO3000 oscilloscope. This device has been controlled from our presentation computer via Ethernet where a small Python program ran. This program is available on GitHub. The software awaits new data after a hardware trigger from the first (outgoing) photo diode. The two traces are queried. Then there are two different ways to extract the time delay and speed of light:

• The first is the more classical variant. The program can calculate the difference of the outgoing and reflected signal. This yields a Gaussian shape. By fitting and locating the maximum as well as the full width at half maximum one can calculate the delay between the two signals. This is less robust but mimics the optical readout via plain eye more close than the second method.
• The more robust version normalizes both signals and then calculates the (auto)correlation of the signal with its reflected part. This yields a maximum at the time delay. This method is mathematically more challenging but allows way better measurements. In fact it does not only work when using a chopper - one can also just introduce noise into the transmitted signal (keeping the chopper open and just applying vibrations to the fiber coupler on the chopping setup). The noise is then correlated with itself with gives high statistics average of the time delay between both signals. This is about the idea behind radar when one neglects for frequency shifts that get important there for moving objects.

The software was designed to be more eye-candy than usual scientific software though as one can see from the screenshots:

Note that the above screenshot was actually a simulation. During first tests the performance was way better and achieved an accuracy down to 0.2% from the correct value of the speed of light:

## Conclusion

As one can see building such an experiment that can be used to measure the speed of light is not really challenging today - and even though it has been done many times as of today with incredible high precision one can still bring in some interactivity into such an demonstration experiment while keeping results accurate. Though providing no real scientific gain this experiment hopefully provided some insight into the finite speed of light and some fun for the visitors. On the other hand it’s of course always fun to build experiments even when they only resemble historical ideas.