KAGRA Logbook

In the context of the O3GK paper, Kazuhiro Yamamoto-san and I reviewed the calculation of the effect of coil driver noise and DAC noise on DARM. The reason for reviewing is that I don't think we did it correctly back then and more discussion was necessary.

The effect of coil driver noise and DAC noise in DARM can be calculated as follows:

Measurement of the individual transfer functions from each coil to DARM,
Propagation of coil driver noise and DAC noise for each coil using the transfer functions and
Addition of all the contributions in quadrature given that the noise in different coil driverchannels and DAC channels are not expected to be correlated.

During O3GK the individual transfer functions from each coil to DARM were not measured, therefore, it is not possible to calculate the noise as described above. Nevertheless, transfer functionsfrom the virtual degrees of freedom L, P and Y of mirrors were measured and they can be used to calculate upper limits. The measurement of transfer functions in L, P and Y use well defined phase relationships between the signals in the different coils, whereas in the case of the actuation noise the phases are random. This difference yields an overestimation of the propagated noise. For example, when using the transfer function along L, it is implicitly assumed that the phases are sucht hat the forces from the coils move the mirror in the same direction along L at any given moment. In reality, however, the phases are random and produce forces that move the mirror in +L and -L simultaneously, yielding a motion with a smaller amplitude. As the same applies to P and Y, the total upper limit becomes

$F_{i} = \left( T_{L} + T_{P} + T_{Y} \right) N_{i},$ (1)

where $i$ stands for either DAC or coil driver, $N_{i}$ and $F_{i}$ are the noise at the origin and the propagated one respectively and $T_{L}$ , $T_{P}$ and $T_{Y}$ are the transfer functions from actuation in L, P and Y to DARM respectively.

It is important to point out that in entry 14875 the formula

Sqrt(4) * (TF/4) * Noise = TF * Noise / 2

is wrong. As explained above, the different contributions of the individual coils should not be added in quadrature because the transfer function TF corresponds to either L, P or Y. It should be

4 * (TF/4) * Noise = TF * Noise,

which is where formula (1) comes from. Also note that the contributions from L, P and Y should not be added in quadrature either.

In the case of the BS, the calculation using formula (1) yields the attached plot.

I'll add the Jupyter notebook and the other files used in the calculation in the following directory:

/kagra/Dropbox/Subsystems/VIS/TypeBData/BS/DAC_coil_noise/