The Sagnac Effect

The Sagnac Effect is based on a variation on the Michaelson-Morley experiment and involves light rays flying in opposite directions around a path that encloses a nonzero area in an apparatus that is rotating. Described theoretically by Max von Laue (1879 Oct 09 – 1960 Apr 24) in 1911 and observed experimentally by Franz Harress (no dates found) in 1911 and by Georges Sagnac (1869 Oct 14 – 1926 Feb 26) in 1913, the Sagnac effect denotes a phenomenon that some people believe locates an absolute frame of motion. In concept the experimental demonstration of the effect offers a beautiful example of simplicity in physics.

Imagine constructing a regular N-gon by connecting N flat mirrors edge to edge. At one of the joins mount a 50-50 beam splitter such that it lies perpendicular to the plane defined by the N-gon and on the straight line going from the join to the N-gon’s axis of symmetry, the axle on which it turns. A laser projects its beam of coherent light onto the beam splitter at such an angle that half of the light goes around the N-gon in the counterclockwise (positive) direction and the other half goes around the N-gon in the clockwise (negative) direction. Both half-beams come back to the beam splitter and recombine into two output beams that display an interference pattern, a series of light and dark bands, on a screen set up by the beam splitter as a detector.

Rotating the N-gon in the counterclockwise direction about its axis of symmetry and in the plane defined by the light beams will make the interference pattern shift its location on the detector screen. That shift happens because in the time it takes the half-beams to go around the N-gon the source/detector apparatus has moved; thus, one half-beam travels farther than it did when the apparatus stood at rest and the other half-beam travels less far than it did when the apparatus stood at rest. The difference between the lengths of those travels produces a phase shift between the two half-beams and that phase shift, in its turn, produces the displacement of the interference pattern on the detector.

Implicitly we have been viewing this apparatus through the perspective of an observer standing at rest relative to the N-gon’s axis of symmetry; in other words, an observer occupying an inertial frame of reference. Now consider how the system looks to an observer moving with the system’s source/detector.

That second observer occupies a non-inertial frame of reference, a fact that they can infer from the observation that they must exert a continuous centripetal force upon themselves in order to maintain their position relative to the axis of the N-gon. To translate their observations into those made by the first observer, the second observer will apply the principle of Relativity in the form that states that the dimensions of an entity may appear distorted between the two reference frames but that entity’s pattern will not. Thus, for example, if the second observer has made marks one millimeter apart on the detector screen and if one of the bright fringes of the interference pattern lies entirely between the 2-mm and 3-mm marks, those marks may not be one millimeter apart for the first observer, but the bright fringe will nonetheless lie entirely between the 2-mm and 3-mm marks.

So we know that the second observer detects the same shift in the interference pattern that the first observer does. The first observer has a ready explanation for that shift in terms of the distances the half-beams traveled around the N-gon. The second observer can’t use that explanation: in their view both beams traveled along paths of exactly the same length, whether the N-gon rotates or not. The second observer can only explain the fringe shift by asserting that the light in the beams traveled around the N-gon at different speeds, the clockwise beam traveling faster than the counterclockwise beam did. But although both observers have different explanations for the shift of the interference fringes, they will both use the same mathematical description of the phenomenon.

Let r represent the distance from the N-gon’s axis of symmetry (its center) to the center of any of the mirrors. If we draw straight lines from the center of the N-gon to the centers of two adjacent mirrors and then draw a straight line connecting the centers of those mirrors, we will have constructed an isoceles triangle whose vertex angle θ=360/N (in degrees)=2π/N (in radians) and whose base coincides with a segment of the path that light will follow around the N-gon. For the height of that triangle we have

eq’n i: h=rCos(θ/2)

and for the length of its base we have

eq’n ii: b=2rSin(θ/2),

so for the area enclosed within the triangle we have

eq’n iii: ½bh=r^{2}Sin(θ/2)Cos(θ/2).

For the length of the light path we have

eq’n iv: L=Nb=2NrSin(π/N)

and for the area enclosed within the light path we have

eq’n v: A=N(½bh)=Nr2Sin(π/N)Cos(π/N).

If we let N take on very large values, tending toward infinity, the N-gon will come increasingly to resemble a circle, so we have, to ever more accurate approximations, Cos(π/N)=1 and Sin(π/N)=π/N and the formulae above, in equation iv and v, become those associated with the perimeter of a circle and the area enclosed by that circle.

If the N-gon now rotates in the counterclockwise direction at an angular speed ω, the light path will shift under the light flying along it and thereby change the distance that the light travels in going from the source/detector and back to it. We assume that a given crest on the light wave traverses one segment of the light path in a time interval t. In that interval the N-gon has rotated through an angle

eq’n vi: ϕ=ωt

and the center of each mirror has moved a distance

eq’n vii: d=hωt

along a circular arc whose chord has the length

eq’n viii: d’=2hSin(ϕ/2).

Depending upon which way the light is traveling, we calculate the actual distance the light travels by adding part of that length to b or subtracting part of it from b. The line d’ tilts away from the extension of line b by the angle ϕ, so we have the adjusted light path

eq’n ix: b’=b±d’Cosϕ.

We use the infinite series expansions of the sine and the cosine and with the mirrors moving at speeds very much smaller than the speed of light we dismiss all but the first term in each series. So the time it takes the counterclockwise beam to traverse one segment of the light path comes from the statement that

(Eq’n 1)

We thus have properly

(Eq’n 2)

For the clockwise beam we make the same calculation, but we subtract the projection of d’ from b. Thus we get

(Eq’n 3)

and

(Eq’n 4)

Those equations show us how the two observers interpret the cause of the
shift in the interference pattern. The first observer uses Equations 1 and 3 and
attributes the difference between t_{1} and t_{2} to the
difference between the path lengths that the beams follow. The second observer
uses Equations 2 and 4 and attributes the difference between t_{1} and t_{2}
to the difference between the speeds at which the beams appear to propagate
around the N-gon.

Equations 1 through 4 describe the transit times associated with one segment of the light path around the N-gon. To obtain descriptions of the total transit times the beams require to go around the N-gon we multiply Equations 1 and 3 by N and re-solve them for the time. We thus get

(Eq’n 5)

and

(Eq’n 6)

The shift in the interference fringes comes from a phase shift between the two beams: the crests and troughs of the waves in one beam are shifted relative to those in the other beam by the motion of the N-gon. Imagine the full two pi radians of the circumference of the stationary N-gon laid out as a straight line and imagine the phase shift Δθ added to or subtracted from one end of that line. Likewise imagine drawing a straight line segment representing one wavelength of the light propagating around the N-gon and, in proper proportion, add to it or subtract from it the difference in the lengths of the two light paths. A single wavelength λ, crest to crest or trough to trough, hits the beam splitter, fissions into two fadewaves, each half as intense as the original, and the two fadewaves propagate in opposite directions around the N-gon. As the fadewaves propagate around the N-gon the phase difference between them grows smoothly and uniformly from zero to its full value of Δθ and the difference between the lengths of the paths they follow increases smoothly and uniformly from zero to the full value of cΔT (in which T represents the time they would take to traverse the light paths if the N-gon were not rotating.

To relate the total phase shift to the total change in path length we note that we want the phase of the fadewave as it appears on the detector. At a given instant the first fadewave comes to the detector at a certain phase in its cycle. The second fadewave, traveling farther, comes to the detector at the same phase in its cycle an interval ΔT later. In that interval the first fadewave has advanced its phase on the detector by an amount

eq’n x: Δθ=2πνΔT,

in which ν represents the frequency of the wave and the factor two pi converts fractions of a cycle into radians. We know that λν=c, so we can express that result in terms of the wavelength as

(Eq’n 7)

Subtracting Equation 6 from Equation 5 gives us

(Eq’n 8)

Combining Equations 7 and 8 tells us that the phase shift stands in direct proportion to the area enclosed by the light path and the angular speed at which the interferometer rotates. That is the Sagnac effect.

As long as the light propagates in a vacuum we don’t have to account for the Fizeau effect (involving light propagating in a moving material medium) and the Sagnac effect makes no distinction between classical physics and relativistic physics in its predictions concerning the position of the interference fringes. The Sagnac effect thus marks a place where Newtonian physics and Relativity come together. It also marks a place where Special Relativity and General Relativity come together.

That latter statement is reflected in the observation that, while the experiment takes place in a perfectly flat Minkowskian spacetime, the second observer uses a metric equation that looks warped. The first observer expresses the differential line element in their reference frame through the standard equation of the Minkowski metric,

(Eq’n 9)

in Cartesian coordinates and in cylindrical polar coordinates. Given that the second observer stands on the axis of the interferometer and rotates with it, we make the following transformation between the two observers’ coordinates,

(Eq’ns 10)

in which r represents radial distance measured from the axis of the interferometer, θ represents the angular distance measured in the counterclockwise direction from the line along which the second observer is looking, ù represents the angular velocity at which the second observer and their frame rotate, and t represents the second observer’s proper time. In differential form we have those equations as

(Eq’ns 11)

Using Equations 9 and 11, we can generate the metric equation that the second observer would use to characterize their frame of reference, getting the square of the differential line element as

(Eq’n 12)

That equation gives us two facts immediately. First, the cross term involving theta and tee together tells us that the longitudinal coordinates and the temporal coordinates are not orthogonal to each other in this frame, which means, in practical terms, that the second observer will run into difficulties in synchronizing clocks mounted on the ring supporting the interferometer. And second, the equation tells us that the second observer occupies a non-inertial frame of reference, a frame that involves an inherent acceleration (centripetal acceleration in this case). We can infer more facts about the second observer’s frame, but that exercise properly requires a separate essay on rotating frames of reference.

Thus we have the Sagnac effect and what it means for physics.

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