User:Physchim62/Speed of light experiment

How to measure the speed of light in SI units

Principle

To measure the speed of light, c, based on:

c = fλ

where f is the frequency of light from a given source and λ is its wavelength.

For any measurement we need a standard. To make a measurement in SI units, we need practical realizations of the corresponding SI units, here the metre and the second. The practical standards will be:

• for the metre, the wavelength of light from a source of precisely known frequency, specifically a laser stabilized to the unperturbed transition observed with a trapped and cooled ⁸⁸Sr⁺ ion of the type used at the Optical Frequecy Standards Laboratory of the National Research Council Canada [1] and at the UK National Physical Laboratory.
• for the second, a caesium masar, more commonly known as an "atomic clock"

Details

The accepted value for the frequency of the strontium-stabilized laser is [2]

f_s = 444779044.0954846(30) MHz

which gives a wavelength of

λ_s = 674.025590863136(5) nm

In effect, my laboratory realization of the metre is 1483623.194068090(11) wavelengths of the light corresponding to the 5²S_1/2—4²D_5/2 transition of a a trapped and cooled ⁸⁸Sr⁺ ion.

For the test source, we can take a methane-stabilized helium–neon laser operating in the infra-red region of the spectrum. I measure the wavelength of the light from my test laser against the standard wavelength, and find

λ_t/λ_s = 5.032793149(18)

corresponding to

λ_t = 3.392231376(12) μm

I measure the frequency of the light from my test laser against the standard frequency from the atomic clock, and find

f_t = 88376181.627(50) MHz

By multiplication, and standard propagation of uncertainties, we have

c = 299792.4562(11) km/s

Discussion

The value of the result, and its uncertainty, might look familiar to some readers. That's because I took my experimental results directly from the paper of Evenson et al. (1972), the work that lead to the current value of the speed of light in SI units. If I'd have used more up-to-date values for the frequency of the methane-stabilized helium–neon laser [3], I would have

f_t = 88376181.60018(27) MHz

λ_t = 3.392231397327(10) μm

c = 299792.4560000(13) km/s

The string of zeros going right to the is indeed an indication that something funny is going on, even if I still have a measurement uncertainty. In the second example, the wavelength is not a measured figure but is simply calculated from the (measured) frequency! So let's get one thing clear:

To measure the speed of light by this method, you must separately measure the frequency and the wavelength from the same source.

So long as you do that, this method will give you a measured value of the speed of light in SI units, complete with a measurement uncertainty, that you can compare to the value given in the definition of the metre.

Secondly, and although it might seem counter-intuitive, I would have got exactly the same result had I switched the two light sources around, that is used my methane-stabilized laser as the standard light source and the (more accurate) strontium-stabilized laser as the test source. The precision of the final result depends on the precision of the standard as well as on the characteristics of the test source.

That's important, because the experiment I describe here is exactly equivalent to the experiment of Evenson et al., except that the NIST team didn't use a strontium-stabilized laser as their standard – such things didn't exist in 1972 – but instead used the krypton light source that, at that time, was the basis for the definition of the metre. Their work showed that the krypton light source was less precise than the methane-stabilized laser – in effect, the "meterstick" was a bit fuzzy at the ends – and it would be impossible to have a more precise figure for the speed of light in metres per second while using the krypton wavelength as the definition for the metre.

Finally, is there any point? Well not really, people don't measure the speed of light in SI units, and haven't done since 1972. It all depends what you're trying to prove. For example, this method only works because the speed of light in vacuum is independent of its frequency. You could try to use this method to look for frequency dependence of the speed of light, but there's plenty of other evidence that any such dependence (if it exists) is too small to be detected with our current techniques and equipment. There's also plenty of theory, some of it dating back more than 150 years, to say that there's no such dependence. You might have some other theory that you want to test, but then you'd have to ask yourself if this experiment is the best way to test it. However, if in the end, you decided you wanted to do this measurement, and you had access to the necessary equipment, you could do it. Pointless is not the same as impossible.