Oscillations, Waves, and Interactions - GWDG

280 Schreiber
not moving along with the Earth. In order to investigate the possibility of a dragged
ether, George Sagnac set up a different experiment. He generated a coherent beam
of light, which he guided around a contour with a predetermined area of 0.086 m2 .
The entire apparatus was then rotated with a frequency of approximately 2 Hz [1].
With the help of a beam splitter and several mirrors he managed to generate two
counter-propagating beams passing around the same optical path. He observed a
shift in the interferogram of 0.07 ± 0.01 fringes and found that the measured shift
was directly proportional to the rate of rotation. This observation, known as the
‘Sagnac Effect’ today, however would require an ether at rest and was in contradiction
with Michelson’s findings. As a result of both experiments the ether theory was
concluded.
A full description of the Sagnac effect is based on General Relativity [2], however
in this case a classical interpretation yields the same result [3]. The observed phase
difference is
δφ = 8πA
n · Ω, (1)
λc
where A is the area circumscribed by the laser beams, λ the optical wavelength,
c the velocity of light, n the normal vector upon A and Ω the rate of rotation of
the interferometer. Equation (1) relates the obtained phase difference to the rate of
rotation of the entire apparatus and can be interpreted as the gyroscope equation [4].
Fibre-optic gyros (FOG) are modern representatives of this kind of optical gyroscopes.
Because glass fibres with a length of several hundred meters are used, the
scale factor can be made very large by winding the fibre to a coil and the sensitivity
for rotational excitations is therefore much larger than for G. Sagnac’s experiment.
While the rotation rate of the Earth would have generated a fringe shift of as little as
1
300
on his historic instrument, which was well outside the range of sensitivity, Earth
rotation can be observed to about an accuracy of 10% even on relatively modest
FOGs. Based on the experiment of G. Sagnac it was possible to estimate the required
size of an instrument capable of resolving an angular velocity of ≈ 50 µrad/s,
which corresponds to the amount of Earth rotation experienced at mid-latitude.
In this context the famous experiment 1 of Michelson and Gale [5] in 1925 must
be viewed. Figure 1 shows a design draft of this experiment. A beam path in an
evacuated rectangular arrangement of pipes with a length of about 603 m by 334 m
was used for that purpose. The incoming coherent light beam was split into two
counter-propagating beams with the help of the beamsplitter A and then guided
around the contour A, D, E and F by three more mirrors. The rotation rate of
the Earth at the location of Clearing (Illinois) generated a shift of 0.23 fringes,
measured with an uncertainty of no more than 0.005 fringes. This corresponds to a
measurement error of only 2%. From a historical point of view it is very interesting
to note, that this concept contained a substantial experimental challenge. Since the
Earth rotation rate is absolutely constant at this level of sensor resolution, Michelson
and Gale had to prove that the observed fringe shift was indeed a measurement
quantity and not an artifact generated from multiple reflections in the interferometer
1 Please note that the goal was to measure a very small, nearly constant, angular velocity.
The experiment was not intended to proof Earth rotation as such.