In a classic homodyne balanced scheme, the difference current is read out that contains only a GW signal and quantum noise of the dark port:
Whatever quadrature the GW signal is in, by proper choice of the homodyne angle However, in real interferometers, the implementation of a homodyne readout appears to be fraught with
serious technical difficulties. In particular, the local oscillator frequency has to be kept extremely
stable, which means its optical path length and alignment need to be actively stabilized by a
low-noise control system [79]. This inflicts a significant increase in the cost of the detector, not to
mention the difficulties in taming the noise of stabilising control loops, as the experience of
the implementation of such stabilization in a Garching prototype interferometer has shown
[60, 75, 74
].
Let us discuss how it works in a bit more detail. The schematic view of a Michelson interferometer with
DC readout is drawn in the right panel of Figure 11. As already mentioned, the local oscillator light is
produced by a deliberately-introduced constant difference
of the lengths of the interferometer arms. It
is also worth noting that the component of this local oscillator created by the asymmetry in
the reflectivity of the arms that is always the case in a real interferometer and attributable
mostly to the difference in the absorption of the ‘northern’ and ‘eastern’ end mirrors as well as
asymmetry of the beamsplitter. All these factors can be taken into account if one writes the
carrier fields at the beamsplitter after reflection off the arms in the following symmetric form:
In the case of a small offset of the interferometer from the dark fringe condition, i.e., for
, the readout signal scales as local oscillator classical amplitude,
which is directly proportional to the offset itself:
. The laser noise associated
with the pumping carrier also leaks to the signal port in the same proportion, which might be
considered as the main disadvantage of the DC readout as it sets rather tough requirements on the
stability of the laser source, which is not necessary for the homodyne readout. However, this
problem, is partly solved in more sophisticated detectors by implementing power recycling and/or
Fabry–Pérot cavities in the arms. These additional elements turn the Michelson interferometer into a
resonant narrow-band cavity for a pumping carrier with effective bandwidth determined by
transmissivities of the power recycling mirror (PRM) and/or input test masses (ITMs) of the arm cavities
divided by the corresponding cavity length, which yields the values of bandwidths as low as
10 Hz. Since the target GW signal occupies higher frequencies, the laser noise of the local
oscillator around signal frequencies turns out to be passively filtered out by the interferometer
itself.
DC readout has already been successfully tested at the LIGO 40-meter interferometer in Caltech [164]
and implemented in GEO 600 [77, 79, 55] and in Enhanced LIGO [61, 5]. It has proven a rather promising
substitution for the previously ubiquitous heterodyne readout (to be considered below) and has become a
baseline readout technique for future GW detectors [79].
Up until recently, the only readout method used in terrestrial GW detectors has been the heterodyne readout. Yet with more and more stable lasers being available for the GW community, this technique gradually gives ground to a more promising DC readout method considered above. However, it is instructive to consider briefly how heterodyne readout works and learn some of the reasons, that it has finally given way to its homodyne adversary.
The idea behind the heterodyne readout principle is the generalization of the homodyne readout, i.e.,
again, the use of strong local oscillator light to be mixed up with the faint signal light leaking out the dark
port of the GW interferometer save the fact that local oscillator light frequency is shifted from the signal
light carrier frequency by several megahertz. In GW interferometers with heterodyne readout,
local oscillator light of different than
frequency is produced via phase-modulation of the pumping
carrier light by means of electro-optical modulator (EOM) before it enters the interferometer as drawn
in Figure 12
. The interferometer is tuned so that the readout port is dark for the pumping
carrier. At the same time, by introducing a macroscopic (several centimeters) offset
of the
two arms, which is known as Schnupp asymmetry [134
], the modulation sidebands at radio
frequency
appear to be optimally transferred from the pumping port to the readout
one. Therefore, the local oscillator at the readout port comprises two modulation sidebands,
, at frequencies
and
, respectively. These two are detected
together with the signal sidebands at the photodetector, and then the resulting photocurrent is
demodulated with the RF-frequency reference signal yielding an output current proportional to
GW-signal.
This method was proposed and studied in great detail in the following works [65, 134, 60, 75, 74, 104, 116]
where the heterodyne technique for GW interferometers tuned in resonance with pumping carrier field was
considered and, therefore, the focus was made on the detection of only phase quadrature of the outgoing
GW signal light. This analysis was further generalized to detuned interferometer configurations in [36, 138]
where the full analysis of quantum noise in GW dual-recycled interferometers with heterodyne readout was
done.
Let us see in a bit more detail how the heterodyne readout works as exemplified by a simple Michelson
interferometer drawn in Figure 12. The equation of motion at the input port of the interferometer
creates two phase-modulation sideband fields (
and
) at frequencies
:
Unlike the homodyne readout schemes, in the heterodyne ones, not only the quantum noise
components falling into the GW frequency band around the carrier frequency
has to be
accounted for but also those rallying around twice the RF modulation frequencies
:
![]() |
where
It is instructive to see what the above procedure yields in the simple case of the Michelson
interferometer tuned in resonance with RF-sidebands produced by pure phase modulation:
and
. The foregoing expressions simplify significantly to the
following:
For more realistic and thus more sophisticated optical configurations, including Fabry–Pérot cavities in the arms and additional recycling mirrors in the pumping and readout ports, the analysis of the interferometer sensitivity becomes rather complicated. Nevertheless, it is worthwhile to note that with proper optimization of the modulation sidebands and demodulation function shapes the same sensitivity as for frequency-independent homodyne readout schemes can be obtained [36]. However, inherent additional frequency-independent quantum shot noise brought by the heterodyning process into the detection band hampers the simultaneous use of advanced quantum non-demolition (QND) techniques and heterodyne readout schemes significantly.
http://www.livingreviews.org/lrr-2012-5 |
Living Rev. Relativity 15, (2012), 5
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