= LIGO displacement readout and control
{c}
An ideal single-round-trip <Michelson interferometer> converts differential <length> into <phase difference> $\delta\phi=4\pi\Delta L/\lambda$. If its dark-port <power> scale is $P_0$ and its static phase offset is $\phi_0$, then
$$
P_{\rm out}=P_0\sin^2\!\left(\frac{\phi_0+\delta\phi}{2}\right),\qquad
\delta P=\frac{P_0}{2}\sin\phi_0\,\delta\phi+O(\delta\phi^2).
$$
Exactly at a dark fringe, $\phi_0=0$, <power> is quadratic in small <displacement> and loses its sign. A deliberate offset supplies an optical reference field: interference with the small signal field produces a linear term. Equivalently, $|E_{\rm ref}+E_{\rm sig}|^2$ contains $2\operatorname{Re}(E_{\rm ref}^*E_{\rm sig})$. The <photodetector> measures <power>, not a signed <displacement> directly.
Advanced <LIGO> uses this DC homodyne principle with a small differential arm offset and an output mode cleaner. Its <feedback control> maintains optical resonance and alignment, using actuation distributed over the suspension stages.<Advanced LIGO by J. Aasi and others>[2.3 Gravitational wave readout; 4.4.2 Quadruple pendulum design; 4.8 Interferometer sensing and control]{ref} The simple formula above omits resonant storage: arm and recycling cavities give a frequency-dependent optical gain and delay, so no constant pass-count multiplier describes the entire observing band.
To see why a locked detector can still measure a disturbance, use a linear <feedback control> model. Let $x_{\rm ext}$ be the external differential <displacement>, $d_{\rm err}$ the residual readout, $d_{\rm ctrl}$ the recorded control command, $C(f)$ the sensing response and $A(f)$ the actuation response. With the convention $x_{\rm res}=x_{\rm ext}-A d_{\rm ctrl}$, reconstruction gives
$$
\widetilde x_{\rm ext}=C^{-1}\widetilde d_{\rm err}+A\widetilde d_{\rm ctrl},\qquad
\widetilde h_D=\widetilde x_{\rm ext}/L.
$$
Here tildes denote <Fourier transforms>; $C$ has readout units per metre and $A$ metres per command unit. Feedback suppresses residual motion but records the compensating command. Omitting that command would suppress the reconstructed low-frequency signal too. In the actual <GW150914> experiment, a modulated auxiliary <laser> exerted calibrated <radiation pressure> on a mirror to measure the response; periodic excitations monitored calibration.<Calibration of the Advanced LIGO detectors for the discovery of the binary black-hole merger GW150914 by B. P. Abbott and others>[II Control loop, equations (1)–(4); IV Photon calibrator; VII Time-dependent systematic errors]{ref}
\Image[/ligo-differential-strain-readout-control.png]
{title=From differential arm <length> to optical <power> and reconstructed <gravitational wave> strain}
{description=Original functional schematic, not a scale drawing. Upper panel: instrument components. Lower panel: ideal linear feedback reconstruction from residual error and applied control.}
Back to article page
