= LIGO differential strain measurement
{c}
{title2=$h_D=(\delta L_x-\delta L_y)/L$}
Each <LIGO> instrument is a <Michelson interferometer> with approximately $4\,\mathrm{km}$ arms containing <Fabry–Pérot interferometers>. A stabilized $1064\,\mathrm{nm}$ <laser> illuminates suspended $40\,\mathrm{kg}$ fused-silica mirrors in <vacuum>. Four-stage suspensions and active isolation suppress ground motion; low-loss materials reduce <thermal noise>. Power recycling increases circulating <power>; signal recycling shapes the response to the <gravitational wave>.<Advanced LIGO by J. Aasi and others>[Sections 2–2.3, Table 1; 4.4 Test mass suspensions, Fig. 8; 4.5 Seismic isolation]{ref}
Why measure two perpendicular arms? In the long-<wavelength> limit, take a <gravitational wave> with <plus polarization> normally incident on ideal perpendicular arms. Its <transverse-traceless tensor> has $h_{xx}=h_+$ and $h_{yy}=-h_+$. Expanding each proper length to first order gives
$$
\frac{\delta L_x}{L}=\frac{h_+}{2},\qquad
\frac{\delta L_y}{L}=-\frac{h_+}{2},\qquad
\Delta L=\delta L_x-\delta L_y=Lh_+.
$$
Here $h_+$ is dimensionless, whereas $\Delta L$ is a <length>. A stipulated $h_+=10^{-21}$ and $L=4000\,\mathrm m$ imply $\Delta L=4\times10^{-18}\,\mathrm m$, with individual arm changes of opposite sign and magnitude $2\times10^{-18}\,\mathrm m$. These are equivalent differential arm-length changes, not a photograph of mirrors moving against a rigid external ruler. The observable compares returning light phases; <coordinate systems> can distribute the description between mirror coordinates and light propagation.
For unit arm directions $\mathbf u,\mathbf v$, define the detector <tensor> $D^{ij}=(u^iu^j-v^iv^j)/2$. If $e^+_{ij},e^\times_{ij}$ are the transverse polarization tensors, then
$$
h_D=D^{ij}h^{\rm TT}_{ij}=F_+h_++F_\times h_\times,\qquad
F_A=D^{ij}e^A_{ij}.
$$
The dimensionless factors $F_A$ encode sky direction and the polarization basis. Thus even identical sources can have different observed <amplitudes> and <wave phases>. Motion changing both arms equally cancels in this ideal differential observable, although real asymmetries can couple common disturbances into it.
Suspension supplies an inertial reference above its resonances. For example, a single ideal <harmonic oscillator> with support motion $y$ obeys $\ddot x+\omega_0^2(x-y)=0$. Its response at <angular frequency> $\omega$ is $x/y=\omega_0^2/(\omega_0^2-\omega^2)$, which decreases in magnitude as $(\omega_0/\omega)^2$ well above resonance. The mirror follows slow support drift but approximately free motion in the high-frequency limit. Real multistage suspensions also require damping and alignment control; resonant frequencies are not regions of good isolation.
<Rainer Weiss>'s 1972 report proposed suspended masses, multiple-pass optical delay lines, and a servo holding an interference fringe. Its noise analysis separately treats photon-counting fluctuations, <laser> frequency instability, mechanical <thermal noise>, <radiation pressure> and ground motion. Those delay lines should be distinguished from the resonant arm cavities of Advanced <LIGO>.<Electromagnetically coupled broadband gravitational antenna by Rainer Weiss>[Typeset transcription of 1972 report, sections 4–5.5, Fig. 1; original scan is linked from DCC record P720002]{ref}
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