# Michelson Morley Experiment- how are fringe shifts created?

Currently, I understand that under the aether model, the two perpendicular light rays (which were created when a monochromatic light was split by a half silvered mirror) will be out of phase when they reach the interferometer. I also understand that the light rays interact in some way at the interferometer to create a pattern of white and black bands which we call interference patterns.

Here's my question. All sources which I have encountered say that when you rotate the apparatus, it should be expected that there should be a fringe shift. I assume that they mean that the white bands literally move left and right? Or did I misunderstand this?

Assuming that I have interpreted it correctly, could someone either explain/ redirect me to a source which explains how exactly the interference patterns move left and right as we rotate the apparatus?

Thank you very much.

## 1 Answer

Short answer: We are looking for any fringe shift in the interferometer induced by this rotation which we cannot attribute to Earthly / lab causes such as weight induced variable strain. In the presence of the aether, the Earth's speed through space would account for an effective index difference of of the order of $30{\rm km\,s^{-1}} / 3\times 10^5{\rm m s^{-1}}\equiv 10^{-2}\%$ as the two arms swapped roles, i.e. it would give rise to hundreds of fringe change cycles with a meter arm length instrument for each quarter turn.

Details

The aim of the Michelson-Morley experiment is to test whether there is a direction dependence for the speed of light, i.e. whether the phase delay through an interferometer arm depends on whether the arm lies along or athwart Earth's motion relative to the putative luminiferous aether.

In theory, we could set up the interferometer with precisely equal mechanical arm lengths and then check whether this arm length begets different phase delays in the separate arms of the stationary (relative to Earth) interferometer. But in practice there is no way to set the arms to be equal accurately enough for this experiment, aside from with interferometry itself. You would need to have some way of ensuring that two orthogonal mechanical lengths were the same to within much less than a wavelength of light. At the same time, you want the arm lengths to be as long as possible so that the putative vacuum refractive index direction dependence has as much length to act on as one can arrange for.

Instead of striving for this impracticable mechanical goal, we simply repeatedly swap the rôles of the two arms, which we can do by rotating the whole interferometer. That way we do not have to control for unequal arm lengths. We are looking for any fringe shift in the interferometer induced by this rotation which we cannot attribute to Earthly / lab causes such as weight induced variable strain. One generally sets the interferometer up so that there are light-dark fringes rather than aligned so that we have uniform brightness across the whole field of view, which latter condition we achieve with perfect equality of arm lengths and zero relative beam tilt. See my answer here for an analysis of the fringe shapes. As long as we have tilt induced (linear) or defocus induced by slightly unequal arm lengths (round) fringes, we simply look for the bright and dark regions swapping roles. This makes it much easier to unambiguously detect phase shift: light and dark fringes swap places every $\pi$-radian phase change, as you can see from my other answer. A whole region's uniformly becoming bright and dark is harder to unambiguously detect, particularly if this is done by eye alone, so this is the reason we don't use a perfectly aligned interferometer. In Michelson's time, a perfect arm length equality was needed to see interference with incoherent light sources, but with a high coherence source this is not necessary.

On very good bearings, this rotation can indeed be done without disturbing the interferometer, not even by subwavelength amounts. One must ensure that the interferometer's rotation plane is indeed precisely normal to the gravitational field so that the instrument's weight does not induce a varying stress in the instrument's frame as it spin. Michelson of course floated the whole kit on a pool of quicksilver (I hope he and his experimenters wore gas masks to avoid ending up like the Hatter in Alice), thus achieving this goal by ingenious design. Modern replications of the experiment use air bearings.

I say more about modern versions of the MM experiment (which we are trying all the time with steadily improving accuracy) in my answer here and here.

• Hello, thank you for answer. But the first link you posted isn't quite what I am looking for, I can see that you're dealing with some mathematics but what I am looking for is a more first principle based, intuitive explanation of how the fringe patterns move as the apparatus is rotated slowly (I was kind of expecting an explanation using light ray diagrams etc.). Have you seen any such diagrams? – Curiouslearner Jan 21 '18 at 12:38
• @Curiouslearner There really isn't a ray explanation. Inteferometry is about phase comparison, so you really do need to write down the phases of the interfering fields to work out what the fringe pattern will be. – Selene Routley Jan 22 '18 at 0:16