  Frequency dependent squeezing is the main upgrade for achieving broadband
quantum noise reduction in upcoming observation runs of gravitational wave
detectors. The proper frequency dependence of the squeezed quadrature is
obtained by reflecting squeezed vacuum from a Fabry-Perot filter cavity detuned
by half of its linewidth. However, since the squeezed vacuum contains no
classical amplitude, co-propagating auxiliary control beams are required to
achieve the filter cavity's length, alignment, and incident beam pointing
stability. In our frequency dependent squeezing experiment at the National
Astronomical Observatory of Japan, we used a control beam at a harmonic of
squeezed vacuum wavelength and found visible detuning variation related to the
suspended mirrors angular drift. These variations can degrade interferometer
quantum noise reduction. We investigated various mechanisms that can cause the
filter cavity detuning variation. The detuning drift is found to be mitigated
sufficiently by fixing the incident beam pointing and applying filter cavity
automatic alignment. It was also found that there is an optimal position of the
beam on the filter cavity mirror that helps to reduce the detuning
fluctuations. Here we report a stabilized filter cavity detuning variation of
less than 10$\,$Hz compared to the 113$\,$Hz cavity linewidth. Compared to
previously published results [Phys. Rev. Lett. 124, 171101 (2020)], such
detuning stability would be sufficient to make filter cavity detuning drift
induced gravitational wave detector detection range fluctuation reduce from
$11\%$ to within $2\%$.
