= Solution
A <gravitational wave> produces a time-dependent tidal distortion. For a normally incident plus-polarized wave in the long-wavelength limit, two perpendicular freely suspended arms change by $\delta L_x/L=h_+/2$ and $\delta L_y/L=-h_+/2$. Their differential length is therefore
$$
\boxed{\Delta L=Lh_+},\qquad
\delta\Phi=\frac{4\pi\Delta L}{\lambda}
$$
for a simple round-trip <Michelson interferometer>. <LIGO> uses kilometre-scale orthogonal arms with <optical cavities> that enhance the phase response, a stabilized <laser> and suspended mirrors in <vacuum>. As an illustrative scale, $h=10^{-21}$ across a 4 km arm corresponds to differential length $4\times10^{-18}\,\mathrm m$. The measured quantity is returning light phase, not mirror displacement against a rigid external ruler. Optical storage and control loops give a frequency-dependent calibrated response.
Seismic and terrestrial disturbances dominate low <frequencies>, thermal motion of suspensions and mirror coatings contributes at intermediate <frequencies>, and <photon> <shot noise> limits high-frequency readout. Isolation and suspension systems reduce ground coupling; low-loss materials reduce <thermal noise>; <vacuum> suppresses gas scattering and <forces>. Greater optical <power> reduces <shot noise> but increases radiation-pressure fluctuations and thermal demands. Stable optical alignment and feedback are essential, yet their own sensor and actuator noise must be controlled. These are engineering requirements because the desired <strain> is extraordinarily small.<LIGO: The Laser Interferometer Gravitational-Wave Observatory by B. P. Abbott and others>[Instrument configuration and detector-noise discussion; primary initial-LIGO account]{ref}
A candidate should appear coherently at separated observatories with consistent propagation delay and antenna response. Auxiliary environmental monitors, <calibration> injections and time-shifted coincidence backgrounds test instrumental explanations. Matched filtering is powerful for modelled compact-binary chirps; coherent burst searches allow less predictable signals. Binary-pulsar orbital <energy> loss already provided indirect evidence for <gravitational radiation> before 2013, but it is distinct from directly measuring passing waves.<Binary Pulsars and Relativistic Gravity by Joseph Hooton Taylor Jr.>[Binary-pulsar orbital-decay test of relativistic gravity]{ref} At the examination date, direct detection remained a search objective. \b[Reliable detection requires both sufficient <strain> sensitivity and a demonstrated astrophysical signal rather than correlated or accidental noise.]
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