= Beam splitter
{wiki}
A <beam splitter> is an <optical instrument> that routes an incident beam of <light> into transmitted and reflected beams. It also combines incident <complex amplitudes>: two inputs can contribute to the same output. A common geometry has four <optical ports>, with two chosen as inputs and two as outputs. A <beam splitter> changes the field; a <photodetector> supplies a measurement.<Photon quantum mechanics and beam splitters by Charles H. Holbrow and others>[Photon quantum mechanics and beam splitters, §IV, printed p. 262, figure 4: two input and two output ports]{ref}
For one illuminated input of a passive <beam splitter>, at specified <wavelength>, <angle of incidence> and <electromagnetic polarization>, define power <transmittance> $T=P_{\rm transmitted}/P_{\rm in}$, <reflectance> $R=P_{\rm reflected}/P_{\rm in}$ and <radiometric absorptance> $A=P_{\rm absorbed}/P_{\rm in}$. In steady operation, with no gain or other source, no net stored <energy> change, and all escaping <light> included in the transmitted or reflected collection,
$$
T+R+A=1.
$$
This is <conservation of energy> divided by incident <power>. If $T$ and $R$ count only the intended output beams, include a separate scattered or otherwise uncollected fraction $L$: then $T+R+A+L=1$. Missing output <power> need not be <wave absorption>. For an original specified example, $P_{\rm in}=10\,\mathrm{mW}$, $T=0.45$, $R=0.50$ and $A=0.05$ give $4.5\,\mathrm{mW}$ transmitted, $5.0\,\mathrm{mW}$ reflected and $0.5\,\mathrm{mW}$ absorbed; these are assumed values, not measurements.
A 50:50 <beam splitter> specifies equal output <powers> for one input, not equal <complex amplitudes> without a relative <wave phase>. Recombination needs the <beam splitter scattering matrix>, including its <wave phases>. A <nonpolarizing beam splitter> approximately divides both orthogonal <linear polarizations> in the same power ratio over its specified operating range; a <polarizing beam splitter> deliberately routes the two differently.
\Image[/beam-splitter-four-port-interferometer-geometry.png]
{title=Four <optical ports> of a <beam splitter> and a <Michelson interferometer>}
{description=Original schematic. Left: the transmitted paths are $a_1\to b_1$ and $a_2\to b_2$; the reflected paths exchange direction. This ordering defines the <beam splitter scattering matrix> below. Right: one source is split into two mirror arms and the returning fields meet the same <beam splitter> again; the <photodetector> supplies the readout. Separate arrows show outward and returning paths, not separate physical beams. The schematic omits alignment steering mirrors and is not to scale.}
The useful surprise is that the same component divides <power> with one input and steers <power> by <wave interference> with two inputs. In a <Michelson interferometer> or <Mach-Zehnder interferometer>, mirror motion changes relative <wave phase>, turning a small change in <optical path length> into a measurable change in output <power>.
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