Multiplying this equation by t we obtain the relation between the distances Z2, h and (x — y) that appear in Fig. 1:
(11)
Using eq. (1), we can express y by (12)
Again, from Fig. 1
or
(13)
We obtain then that the overall transit time through arm 2 is
(14)
The difference in transit times between the two arms is given by
(15)
Expanding and retaining only terms of second order in /? we have
(16)
If the frequency of light is v and its wave length in vacuum A, the phase difference of the two beams will be
(17)
A rotation of 90° will interchange arms 1 and 2, but will not affect the frequency v of the light source [a laser] due to our assumption that the MME gives a negative result in vacuum. We thus have for the total observed fringe shift for a 90° rotation of the system,
(IB)
3. - The experimental system.
The Michelson-interferometer arms consisted of perspex rods, and the light source was a He-Ne laser. For sensitive detection of fringe shifts, the fringes
were projected onto a pair of photoresistors that consisted of two arms of a Wheatstone bridge. Such a set-up is capable of measuring fringe shifts with a sensitivity up to 10~5 fringe (14). The whole system rested on a heavy turntable (about 3.5t), which floated on mercury.
The output of the fringe-sensing system was connected to the ?/-input of an xy recorder, the x input being a voltage proportional to the sine of the angle of rotation of the table.
A positive result would have appear ed on the xy recorder as shown in
Fig. 3. The Figure represents the trace going from East, North, West, South and back again to East, the total fringe shift being A. The effect is a second-order one (there should be no difference between East and West, nor between North and South), thus the line EW should be horizontal and the lines ENW and WSE should coincide.
Fig. 3. - Idealized recorder trace.
4. - Measurements, results and conclusions.
In our measurements, the line E\Y was not exactly horizontal, indicating a drift during the measurement. This shift was isolated as thermal. It was small and not bothersome since we measured directly with respect to the line E W. The more bothersome difficulty was the first-order effect of the lines ENW and WSE not exactly coinciding (S and N lay on different sides of the EW line). After much investigation this effect could only be attributed to strains induced in the table by the earth’s magnetic field. The effect decreased when a magnetic shield was used.
The magnetic field effect was the limiting factor in the accuracy attainable by the system. The fringe-sensing system operated at a sensitivity of 3000 mm/fringe. The final experimental result was a A (see Fig. 3) measured to be less than one mm.
Thus,
(19)
From this we can place an upper limit to the velocity of the earth through the ether. Using eq. (18), we obtain
(20)
C14) J. Shamir, K. Fox and S. G. Lipson: Appl. Opt.. 8. 103 (1969).
