  We theoretically analyze the quantum noise of signal-recycled laser
interferometric gravitational-wave detectors with additional input and output
optics, namely frequency-dependent squeezing of the vacuum entering the dark
port and frequency-dependent homodyne detection. We combine the work of
Buonanno and Chen on the quantum noise of signal-recycled interferometers with
ordinary input-output optics, and the work of Kimble el al. on
frequency-dependent input-output optics with conventional interferometers.
Analytical formulas for the optimal input and output frequency dependencies are
obtained. It is shown that injecting squeezed light with the optimal
frequency-dependent squeezing angle into the dark port yields an improvement on
the noise spectral density by a factor of exp(-2r) (in power) over the entire
squeezing bandwidth, where r is the squeezing parameter. It is further shown
that frequency-dependent (variational) homodyne read-out leads to an additional
increase in sensitivity which is significant in the wings of the doubly
resonant structure. The optimal variational input squeezing in case of an
ordinary output homodyne detection is shown to be realizable by applying two
optical filters on a frequency-independent squeezed vacuum. Throughout this
paper, we take as example the signal-recycled topology currently being
completed at the GEO600 site. However, theoretical results obtained here are
also applicable to the proposed topology of Advanced LIGO.
