Very New Current  Topic # 08

Sagnac Effect Revisited

Proof of Galilean Transformations of Velocities c' = c + v

and the concept of the

Rectilinear Motion of All Electromagnetic Waves 

 

 

posted March 09, 2009

 

The Sagnac effect may be graphically depicted as illustrated in Figure 1.

Figure 1

Sagnac Effect illustrated using 3 (n=3) point source emitters/re-emitters

The three (n=3) optical elements or point sources a, b and c serve as mirrors, emitters and re-emitters via interference of rectilinearly propagating spherical waves. The 3 optical elements rotate about a common center on a rigidly attached platform with an angular velocity of ω radians per second. We assume that the platform does not stretch or remain free of mechanical distortions so that the 3 point sources a, b and c remain at fixed distances between them. Let us assume that the platform rotates in a clockwise direction as indicated in the figures. It is important to note that in this illustration, 3 optical elements are described for the derivation of the Sagnac effect. It can be illustrated using the very same Mathematical Physics technique for four (n=4) optical elements to arrive at the very same phase shift for the Sagnac effect. Also, we can let n=6, or any larger numbers for point-like optical elements, emitting and re-emitting the rectilinearly propagating waves along looped paths. For the details, see the published reference to this web-site: Discouses & Mathematical Illustrations pertaining to the Extinction Shift Principle under the Electrodynamics of Galilean Transformations, Optical Gyroscopes (for n=3, n=4, n=6, n=large, n=very large for fiber media) pages 7 thru 13, Sagnac Effect (for n=4) page 14, 15 and 16. http://www.extinctionshift.com/Books.htm

It would also be helpful to review Topic #01 on the Sagnac Effect {posted May 2001} and Topic #03, On Rectilinear Path of the Photon {posted September 2002}

As illustrated in Figure 1, the photons or particles of waves of electromagnetism propagate rectilinearly, along a straight line path, from the point source a to b', i.e., that position where the point source b will be positioned when the photons moving rectilinearly intercept the point source b at b'. The re-emitted photons of the reflected wave on the same frequency, as would be noted in the frame of reference of the point source re-emitter b, momentarily at the position b', will propagate from b' to the position c'', i.e., to that position where c, will have moved during which time the reflected wave travels along the clockwise loop from a to b', and from b' intercepting the rigidly attached point source c at the position c''. So, for the clockwise loop, if the wave starts at point source a, as depicted in Figure 2, the path would then be from a to b' and then by reflection from b' to c'' and then by reflection from c'' back to the re-emitting point source a again, intercepting it at the position a'''Since the rigid platform rotates in the clockwise direction, the path of the waves moving along a clockwise loop would require an increased transit time to propagate along a round-trip clockwise path, starting from a rigidly attached point and then returning to the very same point to complete the loop.

Figure 2

Sagnac Effect for a clockwise rotating platform

illustrating a clockwise loop a, b', c'', a'''

 

For the counter-clockwise loop, the opposite sequence occurs as indicated in Figure 3. If the wave starts again at a, the path would then be from a to c', from c' to b'', and then from b'' back to a at the position of a'''Since the rigid platform rotates in the clockwise direction, the path of the waves moving in opposite sense along a counter-clockwise loop would require a decreased in transit time to propagate along a round-trip counter-clockwise path, starting from a rigidly attached point and then returning to the very same point to complete the loop.

Figure 3

Sagnac Effect for a clockwise rotating platform

illustrating a counter-clockwise loop a, c', b'', a'''

 

It is also very important to note here that for all cases where sources and mirrors are rigidly attached to a rotating platform, there is absolutely NO relative motion between the point sources a, b and c, which serve as emitters, re-emitters, mirrors or secondary sources. While the platform is rotating, there is a tangential velocity of v ≠ 0 for the 3 moving point source elements. Referring to Figure 1, along the lines connecting the points a, b and c, there is a velocity component of motion of +vCos600. Under Galilean transformations of velocities, the velocity of the wave emitted from point source b' would simply be c' = c + vCos600. The point source emitter c is moving away from emitter b with a velocity component of  -vCos600. Thus, under Galilean transformations, a net velocity of motion of the wave arriving at point source c would b would be c' = c. Although there is no relative motion between the point source emitters, there is an ω dependent coordinate, continually modifying the effective optical path of the waves moving in a rectilinear fashion from optical element to optical element. There is an ω dependent effective optical path change for both the clockwise and counter-clockwise loops.

The distances between the rigidly attached optical elements are constant, assuming that there are no mechanical deformations or stretching occurring in the rotating platform due to centrifugal forces.  Let the unchanging spatial distances be represented as given in Table 1.

distance from a to b

1)

distance from b to c

2)

distance from c to a

3)

Table 1

What actually does change is the path of the waves due to the rectilinear nature by which all electromagnetic waves move in an inertial space. From inspection, it is clearly seen that geometry of the looped waves being reflected by the rigidly attached moving optical elements a, b and c varies as a function of the angular velocity ω. We can see that the path lengths between the positions in space a, a', a'', ... , b, b', b'', ... , and c, c', c'', ... , result from the rectilinear motion of all electromagnetic waves and vary as a function the angular velocity ω.

 

Figure 4

Graphical Illustration of the Sagnac Effect due to the Rectilinear Motion of Wave Fronts

and Rotation Dependent Optical Path

 

 

The Sagnac Effect is graphically illustrated in Figure 4 to be a direct consequence of the rectilinear motion of the wave front, moving from optical element to optical element. As the spherical wave front moves from optical element a to b, the rigidly attached element a moves to a', b moves to b' and c moves to c'. During the transit time of the wave from optical element a to b, a slight modification in the geometry of the optical path occurs.  The transit times and effective optical path change due to the rotational motion of the mechanical platform are given in Table 2.  

 

transit time of wave from segment a to b

4a)

transit time of wave for loop path abca

4b)

effective optical path change due to rotational motion

as function of ab' and ab

5)

effective optical path change due to rotational motion

as function of bb'

6)

Table 2

 

The effective optical path change can be obtained from the geometry of the Sagnac effect apparatus shown in Figure 4 and the basic equations of rotational motion. Using the tangential velocity v of motion of the optical elements from Figure 1, the length of the segment bb' is obtained. 

using 4a), the length of segment bb'

7)

from the effective path change 6), and the length of the segment bb' 7), the phase shift is gotten for path  a to b', a portion of the optical loop

8)

the clock-wise loop abca is obtained from 8) by applying a factor of 3

9)

Table 3

 

CW loop abca plus CCW loop acba

a net phase shift from 9) using factor of 2

10)

Simplifying with area A of the equilateral triangle abc. note: The area A has nothing to do with the Physics of the problem: it just a simplification. 

11)

a net phase shift from 10) and 11)

12)

Table 4

 

It is seen that the Sagnac effect is clearly a direct consequence of the rectilinear nature of the interfering electromagnetic radiation that moves from optical element to optical element; an important principle of optics that is virtually forgotten by too many researchers in this discipline. The fact is: the rectilinear motion of the propagating electromagnetic radiation, emitted and re-emitted by the optical elements is not affected at all by the rotational motion of the mechanical platform. The path of the wave does not curve or rotate with the rotating mechanical apparatus that houses the rigidly attached optics. We recall an illustration from Topic#03, On Rectilinear Path of the Photon.

The Nature of the Rectilinear Motion of All Waves

and their constituent Photons

Note that the wave fronts and the photons, the constituent parts of the wave fronts, move along straight-line paths. The observed phase shifts are clearly due to changes in the geometry of the effective optical paths of the waves, as the waves are being emitted and re-emitted by the rigidly attached optics. The photons of emitted wave fronts move rectilinearly, along straight line paths, connecting the optical elements rigidly attached to the rotating platform. The photons belonging to the wave front that  fly by and miss the point-like optical elements do not contribute to the Sagnac effect. These photons have nothing at all to due with the process of displaying the effect. To an observer that is attached to the rotating platform, the photons belonging to the wave fronts will appear to move along curved paths. A clever and intelligent observer that is attached to the rotating platform will correctly apply the Galilean transformations and realize that the observations of the apparent curves paths are just illusions due to rotational motion and that all waves actually move along rectilinear "straight-line" paths in an inertial space.