An "I/Q" detector, also called a coherent detector, is a four-quadrant detector. Its purpose is unambiguously to measure the phase of a sinusoid relative to another of the same frequency. It is to measure the unknown phase $\phi$ of a sinusoid $x(t)=A\cos(\omega t+\phi)$ relative to a known reference, say $r(t)=R\cos\omega t$.
The measurement involves splitting $x(t)$ into two equal amplitude signals, multiplication and delaying one of it by exactly $\pi/2$ resulting in an "I/Q" pair, called in-phase and quadrature, say $x_I(t)=B\cos(\omega t+\phi)$ and $x_Q(t)=B\sin(\omega t +\phi)$ for some $B=kA$. Here $k=k(A,B,\omega)$ is a known calibrated function of $A,B,\omega$, characteristic of the splitter, but is independent of $\phi$.
Next each "arm" is multiplied by the reference $r(t)$ followed by a low-pass filter to remove the double frequency term $2\omega$. For example, when the product $$x_I r(t)=B\cos(\omega t +\phi)R\cos(\omega t)= \frac{1}{2}B \cos\phi + \frac{1}{2}B \cos(2\omega t+\phi)$$ is low-pass filtered you get $$y_I=[x_Ir(t)]_{LP}=\left[\frac{B}{2}\cos(\omega t +\phi)R\cos(\omega t)\right]_{LP}= mB \cos\phi,$$
and similarly for the other arm
$$y_Q=[x_Qr(t)]_{LP}=\left[\frac{B}{2}\sin(\omega t +\phi)R\cos(\omega t)\right]_{LP}= mB \sin\phi,$$
where $m=m(R,B,\omega)$ is a factor characteristic of the multiplier and filter, depending on $R,B,\omega$, but is independent of $\phi.$
From $y_I$ and $y_Q$ one gets unambiguously $\phi =\beta \ell=\arctan (y_I,y_Q).$
The actual optical or RF "multiplication" can be mechanized several ways but one possible method is based on the identity $(a+b)^2+(a-b)^2=4ab$ and consists of a pair of "square-law" or intensity detectors.
Of course, instead of $x(t)$ one could just as well split and phase shift the "reference" signal $r(t).$
