Blog Ekkehard Friebe

By A. A. Faraj

The Sagnac effect and related topics are discussed in detail and expounded within the framework of the Emission Theory of light.

The Sagnac Experiment:
Three plane mirrors, M1, M2, and M3, and a beam splitter P, are mounted on a turntable in a circular configuration of 25 cm radius. A coherent beam of light, from a source S, is made parallel by a lens L, and sent to P.The initial beam is split by P into two beams, A and B, which traverse the same polygon path in opposite directions.

The beam A travels counterclockwise, in the direction of the rotating apparatus, and meets the clockwise beam B, at P. The two beams are focused on a photographic plate O, forming interference fringe bands there.Both the source of light S and the photographic plate O rotate with the same angular velocity as the apparatus.For a radius of 25 cm and angular velocity of 2.2p rads-1, a fringe shift of 0.034 is observed.The observed fringe shift varies linearly with the angular velocity, w, and the area enclosed by the light path, A, [Ref. #5]. 

1. General Considerations
Before applying the Emission Theory of light to the above experiment, the following points have to be clarified and made explicit:

1. The light path, as marked by the three mirrors and the beam splitter, forms a square whose side, by the Pythagoras‘ theorem, is equal to (21/2) of the radius of the rotating disc.

2. As dictated by the law of reflection, in order for the two beams, A and B, to loop through the Sagnac apparatus, the normal lines of the mirrors, M1 and M3, must be at right angles to the normal of the mirror M2, and at the same time made parallel to the normal of the beam splitter P. For illustration, see [Ref. #1]. Moreover, both the direction of the initial beam, from the source S, and the direction of the detector O, must make 45o and 135o with the normal of the beam splitter P, respectively.

3. For the Sagnac Experiment to be carried out properly, the rotation of the apparatus must be made as close to uniform circular motion as possible. From this basic requirement, it can be deduced that the tangential velocity of each mirror is perpendicular to its normal, and that of the beam splitter is parallel to its normal.

4. The Sagnac apparatus, in most trials, takes less than one second to make one rotation, in the laboratory. By contrast, the earth takes 86,164.09 seconds to make one rotation relative to the stars. Accordingly, the angular velocity of the Sagnac apparatus is overwhelmingly greater than the angular velocity of the earth. Furthermore, the flight time for both beams is vanishingly small as compared to the earth period of rotation. Therefore, it must be concluded that the effect of the earth rotation on the Sagnac Experiment is well below the apparatus sensitivity. Practically, for any Sagnac loop with radius less than 10 m, the earth tangential velocity is uniform and linear, and as such it cannot be detected through the use of the Sagnac apparatus.

5. In principle, more and more plane mirrors can be added to the Sagnac optical apparatus without altering the experimental results, [Ref. #6]. Geometrically, adding more plane mirrors adds more sides to the polygon path of light. As a result, the total length of the light path gets closer to the limit 2pr, where r is the radius of the Sagnac disc. In fact, this increasing approach to the above limit is the optical analogue to the Archimedes way of doing calculus. However, light loops of 2pr length is easily achieved in modern Sagnac interferometers through the use of circular foils of fiber optics. Based on total internal reflection, a fiber optic is essentially equivalent to a configuration of plane mirrors with infinitely small length.

6. The rotation of the Sagnac apparatus causes the normal lines of the mirrors to rotate with respect to the incident beams. For the counterclockwise beam A, during the travel time from P to M1, M1 to M2, M2 to M3, and from M3 to P once again, the normal lines rotate in the direction of the incident beam A. As a result, the angles of incidence, in the four cases, are reduced, and the total length of the light path is increased. The reverse is true for the clockwise beam B, where the normal lines are shifted away by rotation from the direction of the incident beam, and hence the total length of the light path is decreased. The amount of length change in both cases is directly proportional to the angular velocity of the apparatus w, and the area, enclosed by the light loop, A. It should be pointed out that within the value range of w used in the Sagnac Experiment, the mirrors would not get out of alignment enough to break the light loop. Furthermore, the rotation of the normal lines, relative to the incident beams, is the geometrical equivalence to the algebraic representation, which will be discussed later in this exposition.

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