Physics R&A

Aims:

The aim of this project is to investigate the theory behind the Sagnac effect, which is used by Ring Laser Gyroscopes (RLG). There will also be comparisons made with traditional rotating mass gyroscopes, to determine why RLG have been preferred in military applications and are now more popular and beginning to be widely used in commercial aircraft. The future for gyroscopes and using interferometers as a means analysing the rate of rotation will also discussed.

Basic operations of RLGs

The ring in an RLG, is often a polygon with mirrors at the vertices, which turn the laser beam and make it travel in a closed circuit path. The laser source is in the path of the laser around the ring and emits a beam in both directions. The laser source lases spontaneously and emits in both directions.

When the assembly is stationary the beams travel the same distance around the ring and are recombined at the detector, the beam will be the same as when it left the transmitter. However, when the assembly is rotating with a component perpendicular to the plane of the laser path, there will be co- and counter- rotating beams. There are different ways of detecting the differences in the two beams. The current popular method used, is the detection of difference in frequency.

The difference occurs due to a Doppler shift. The detector is moving away from the co-rotating beam and thus it will shift towards the red side of the spectrum. For the counter-rotating beam, the detector is moving towards it and so the opposite happens with a blue shift. At the detector the beams are recombined and the interference pattern analysed. The difference in the frequencies of the two beams would give the rate of rotation. This effect is exactly the same as that used in determining whether celestial objects are moving away or towards the Earth by analysing the light from it, except for there is no beam combiner.

Another difference that can be detected is the difference in time of arrival. Since the co-rotating effectively has a longer path than the counter-rotating beam, it will take longer to arrive. However, the frequency difference can be measured more accurately than time differences, it makes frequency based ring lasers more sensitive and accurate. Therefore, nearly all commercial and research RLGs are frequency based.

The Sagnac Effect:

This effect is named after G, Sagnac who both predicted and observed it in the early 20th century. He realised that there would be a time difference between the arrival of the Co-rotating and the counter-rotating beams, and from this, the phase shift could be calculated. Using classical physics he arrived at the equations below. (For general circuits)

Where:

W=Angular frequency(rad s-1)

v =Undragged beam velocity (ms-1)

A= Area enclosed by ring (m2)

dt = Time difference (s)

Where:

W = Angular frequency (rad s-1)

v =Undragged beam velocity (ms-1)

A= Area enclosed by ring (m2)

dt = Time difference (s)

l=Wavelength of beam without rotation (m)

df = Phase shift

It can be seen from these equations that the smaller the wavelengths of the light beam, the greater the phase shift for any given rotation rate. This means that the smaller the wavelength of light used the greater the accuracy of the gyroscope

The Sagnac effect not only causes phase shift, the optical frequency of the beams is also changed. This is due to the phase shift being compensated by change in wavelength. The change in frequency is derived from the original Sagnac effect. The equation for this shift is:

Where:

df = frequency change (Hz)

A =area enclosed by ring (m2)

W= Angular frequency (rad s-1)

l =Wavelength of beam without rotation(m)

P = optical path length (m)

When this was first demonstrated in the 1960\'s, it was realised that a frequency based ring laser gyroscope would be much more accurate than a phase shift based system. The reason for this can be easily seen by comparing the two equations for the phase shift and the frequency change. The frequency change equation does away with the v term in the denominator, which is very large (v » the speed of light in a vacuum). Although this is not the only reason for its greater accuracy,

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