LIGO: How can laser interferometry (wavelength >$10^{-7}$m) detect length changes of arms <$10^{-18}$ m?
The LIGO interferometer uses a homodyne detection technique. Basically, the light travelling in each arm of the interferometer is derived from the same laser source and is combined in the output channel and falls onto a photodiode.
The interferometer is operated so that when there is no gravitational wave (GW) passing through the instrument, the beams combine to produce a dark fringe (i.e. they are set to destructively interfere). There is a small offset from this, but basically the phase difference between the combiniing beams is close to $\pi$.
The phase difference caused by a GW, due to the changing length of one arm with respect to the other, can be derived as
$$
 \Delta \phi \simeq 2\pi \left(\frac{2hL}{c}\right) \left(\frac{c}{\lambda}\right) = \frac{4\pi}{\lambda} hL \ ,
$$
where $L$ is the length of the arms, $\lambda$ is the laser wavelength and $h$ is the strain amplitude of the fravitational wave signal. Actually, it is a little bit more complex than this, since the arms act as Fabry-Perot resonators which means the light effectively travels backwards and forwards many times in the arms (about 300 for LIGO, i.e. $L$ is effectively 1200 km).
For a typical dimensionless GW strain of $h \sim 10^{-21}$, $\lambda = 1064$ nm, then $\Delta \phi \sim 10^{-8}$ and is modulated at the frequency of the GW (typically 20-2000 Hz).
The problem then reduces to combining
$$ E_{\rm tot} = E_0\sin (\omega_l t) + E_0 \sin (\omega_l t + \alpha + \Delta \phi)\ ,$$
where $E$ is the electric field in each arm, $\omega_l$ is the angular frequency of the laser and $\alpha$ is the offset phase between the arms (close to $\pi$).
Using the identity $\sin a + \sin b = 2 \cos[(a-b)/2] \sin[(a+b)/2$ and squaring the total E-field to get an intensity:
$$I = 4E^2 \cos^2[(\alpha + \Delta \phi)/2]\, \sin^2[\omega_l t +(\alpha + \Delta \phi)/2] $$
Since $\omega_l$ is much greater than the GW frequency and much higher than can be sampled by any photo-sensitive detector, then the second term in the product above can be replaced by its time-average of $1/2$. If we now identify the total power $P_{\rm in}=E^2$  as the average input power to each arm of the interferometer and note that $\cos^2 (a/2) = (\cos(a)+1)/2$ and $\Delta \phi \lll 1$
$$ I = P_{\rm in} \left[1 + \cos(\alpha + \Delta \phi ) \right] \simeq P_{\rm in} \left[1 + \cos(\alpha) -\Delta \phi \sin(\alpha)\right] = 2P_{\rm in} \left[ \cos^2 (\alpha/2) - \frac{\Delta \phi}{2}\sin \alpha \right]\ .$$
It is the second term inside the bracket that contains the signal of the GW. That signal is proportional to the power in the interferometer and the phase difference between the arms. Note that although the signal-to-(shot) noise is mathematically maximised when $\alpha=\pi$, this would mean the SNR was 0/0 ! In practice there is always some other noise present so $\alpha$ is shifted a little bit away from $\pi$ - Fricke et al. (2012) suggests that $\alpha \sim \pi+ 6\times 10^{-5}$ is used.
The power input into each arm is about 600 W (the 100s of kW Steve Linton mentions in a comment is after accounting for the Fabry-Perot resonator, which I did above by talking about an &quot;effective $L$&quot;).
In the absence of other forms of noise then photon counting (shot noise) becomes the limiting factor and is proportional to the square root of the power.
The output signal is the modulated GW signal discussed above which is recorded by detecting photons with the photodiodes. The response function that translates the photodiode signal into a strain is determined by acting on the test masses/mirrors with precisely calibrated lasers modulated at GW frequencies that can produce monochromatic phase shifts in the arm lengths.