Oscillations, Waves, and Interactions - GWDG

296 Schreiber
and the last term accounts for nonlinear contributions such as backscatter-related
coupling, which are neglected in the following discussion.
Usually, the gain factor G is considered constant with respect to time in ring
laser theory. However, for the reasons outlined above, one has to account for the
progressive compensation of gas impurity related losses by setting for example
G = G0e αt . (13)
This choice of G is arbitrary and motivated from the behaviour of the loss as shown
in Ref. [13]. The beam output power of Eq. (12) in the required form for large ring
lasers with an 1:1 isotope mixture of 20 Ne and 22 Ne becomes
Po = 2IsAbT
�
G
µ · κ1Zi(ξ1)
�
+ κ2Zi(ξ2)
− 1
Zi(0)
, (14)
with Is the saturation intensity, Ab the beam cross section and T the transmission
of the laser mirrors. Zi is the imaginary part of the plasma dispersion function with
lasing at a frequency detuning of ξn with respect to the corresponding line centers of
the two Neon isotopes, each having a partial pressure of κ1 and κ2, respectively. The
most important part in this equation is the factor G/µ, which represents the gain–
loss ratio. This factor is approximately constant over the time of the measurements
because of the feedback loop operation. Since there is no drift in the optical frequency
involved, the contribution of the plasma dispersion function also remains constant.
Therefore, the denominator in Eq. (12) can be approximated by 1 + xPo ∼ = 1 so that
this equation reduces to
∆KA
=
KA
� �
∆K
− aG0e
K N
αt . (15)
Applied to the dataset of Fig. 16 one obtains a corrected dataset as shown in Fig. 17
after a nonlinear fitting procedure is performed. The result from Fig. 17 outlines the
best mid-term performance obtained from UG2 so far.
6.6 Ring laser orientation
The Sagnac frequency in Eq. (2) contains contributions from three distinctly different
mechanisms. Most obviously it relates the experienced rotation rate with the
frequency difference observed between the two senses of propagation in the ring laser.
Variations of the scale factor modify the proportionality between rotation rate and
observed Sagnac frequency. The inner product between the normal vector on the
ring laser plane and the vector of rotation finally determines how much of the rotation
rate is projected onto the ring laser. For a large instrument rigidly attached to
the ground and monitoring the Earth rotation rate this means, that any changes in
orientation between the ring laser and the instantaneous Earth rotation vector show
up in the Sagnac frequency.
Solid Earth tides and diurnal polar motion cause such orientation changes. A
detailed description is given in Refs. [21, 22]. The direction of the Earth’s rotation axis
varies with respect to both Earth- and space-fixed reference systems. The principal