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I am trying to model the behavior of a Michelson-interferometer driven by a light source with a short coherence length (i.e. some centimeters at max.). By placing transmittive material in one of the arms, and adjusting the length of the other arm one then can measure the refractive index by using the fact that for an equal length of both arms the interference pattern has it's maximum value (and thus the reason for a source with a short coherence length.

Now, I would like to calculate some parameters for the interferometer, and therefore use a source defined as $$A(t)\exp(-i\omega_0 t)$$ Following the explanation given here (https://physics.stackexchange.com/a/12845/29546) I can define my amplitude $A(t)$ such that $A(t)A^*(t-\tau)\rightarrow0$ for large values of $\tau$. Assumed my amplitude is completely real and my amplitude $A(t)$ has a gaussian shape I then can write $$A(t) = A_0\exp\left(-\left(\frac{t}{t_0}\right)^2\right)$$ and $t_0$ can be defined based on the (estimated) interference length $l_0$ via $$t_0=\frac{l_0}{c}$$ and $c$ equal the speed of light. Is that approach correct?

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Yes, that should be correct, provided you have actually only one frequency $\omega_0$ involved here. Otherwise it would require that all the different frequencies are phase-locked with each other (in order for the Fourier theorem to be applicable) which in general is not the case (see my web page https://www.physicsmyths.org.uk/fourier.htm for more.) For natural light for instance, Michelson interferometry would indicate a much smaller value for the coherence time $t_0$ than actually physically present in the signal (see https://iopscience.iop.org/article/10.1088/0143-0807/24/4/363 )

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First: Is your $l_0$ representing the maximum difference in effective optical path length between the two legs (before coherence disappears)?

Next: is your approach to have such a short coherence length that you will see the interference max & min contrast go to zero when the optical path difference between reference and test leg is more than some distance? My concern is that will be at best a long fuzzy zone.

All said I think determining the index of refraction is much more easily found using the total internal reflection test. But if you want to do the interferometer test, don't depend on coherence length. Rotate your sample as described here and count the number of fringes.

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