= Coherent beam splitter recombination
{title2=$P_{1,2}=P\mp P\sin\delta$}
Suppose the two input beams share <frequency>, <electromagnetic polarization> and overlapping transverse profiles at each output, with stable relative <wave phase> during the <photodetector>'s averaging interval. For the ideal <beam splitter scattering matrix> above, expanding the squared <complex amplitudes> gives
$$
P_1=\frac{|a_1|^2+|a_2|^2}{2}-\operatorname{Im}(a_1^*a_2),
\qquad P_2=\frac{|a_1|^2+|a_2|^2}{2}+\operatorname{Im}(a_1^*a_2).
$$
The <imaginary part> terms describe <wave interference>. Applying the single-input power split to each input separately and adding the results would discard these terms. Their opposite signs ensure $P_1+P_2=|a_1|^2+|a_2|^2$: <conservation of energy> survives even when one output is dark.
For an original example, let $a_1=\sqrt P$, $a_2=\sqrt P e^{i\delta}$ and $P=1\,\mathrm{mW}$. Then $P_1=P(1-\sin\delta)$ and $P_2=P(1+\sin\delta)$:
|| Relative input <wave phase> $\delta$
|| Output <power> $P_1$
|| Output <power> $P_2$
| $0$
| $1\,\mathrm{mW}$
| $1\,\mathrm{mW}$
| $+\pi/2$
| $0$
| $2\,\mathrm{mW}$
| $-\pi/2$
| $2\,\mathrm{mW}$
| $0$
At $+\pi/2$, $a_2=i\sqrt P$ makes $b_1=(\sqrt P-\sqrt P)/\sqrt2=0$ and $b_2=i\sqrt{2P}$; at $-\pi/2$ the dark output swaps. Thus the two inputs can feed all their combined <power> into one output without gain. At zero relative input <wave phase>, the splitter's own relative <wave phase> leaves the contributions in quadrature, so neither output is dark.
\Image[/beam-splitter-coherent-input-phase-powers.png]
{title=Relative input <wave phase> redirects the combined <power> of two coherent beams}
{description=Original ideal-model curves for $a_1=\sqrt P$, $a_2=\sqrt P e^{i\delta}$ and $S=2^{-1/2}\begin{pmatrix}1&i\\i&1\end{pmatrix}$. Output <powers> are normalized by one input's <power> $P$; their sum is always $2P$. Markers show the three worked cases. These are calculated curves, not measured data.}
For fluctuating fields, average the cross term as $\operatorname{Im}\langle a_1^*a_2\rangle$. It vanishes for mutually uncorrelated inputs, making the independent power-splitting rule valid for the averaged output. Partial <first-order optical coherence> reduces it. Orthogonal <linear polarizations> eliminate the cross term in a polarization-insensitive power readout, and spatial mismatch reduces the overlap. Detector averaging can also erase rapid relative <wave phase> changes. A fixed relation between two inputs can be prepared by splitting one <laser> beam and recombining the two arms.
An actual teaching protocol in Michelson Interferometer & Precision Wavelength Measurement — Week 1, University of Colorado Boulder, §§4–6, starts with an ideal-model prediction, then builds a <Michelson interferometer> using a <helium-neon laser>, a <nonpolarizing beam splitter>, adjustable mirrors, a micrometer translation stage and a <photodetector>. Steering mirrors align beam overlap at near and far planes; far-plane <interference fringes> refine alignment. Moving the arm mirror changes the returning relative <wave phase>; detector voltage is watched with a live acquisition display or <oscilloscope>. The guide compares measured arm <powers> with 50:50 splitting and asks for a revised field model. It tests <electromagnetic polarization> dependence, optionally changing one arm with a <quarter-wave plate>, and estimates noise from repeated readings at fixed mirror position. These controls test whether fringe changes reflect coherent recombination rather than splitter imbalance or misalignment. The guide supplies a protocol, not a numerical dataset. Its detector noise, imperfect overlap and <electromagnetic polarization> mismatch limit displacement inference.<Michelson Interferometer & Precision Wavelength Measurement — Week 1 (University of Colorado Boulder laboratory guide)>[Michelson Interferometer & Precision Wavelength Measurement — Week 1, §§4.1, 5.1–5.3, 6.1–6.4; figure 1 and live voltage monitor]{ref}
Additional <background subtraction> controls follow from the field model: block both arms to record the detector offset and ambient <light>, then unblock each arm separately to establish its detected <power> at fixed gain. With both open, scan the mirror and compare extrema after subtracting that background. Keep the detector below saturation and repeat the scan to check drift. A changing signal alone cannot distinguish source <optical coherence> from motion or changing overlap. For a <vacuum> round trip, a mirror displacement $x$ changes relative <wave phase> by $4\pi x/\lambda$; the useful signal is its modulation after detection, not an optical-frequency oscillation that the detector must resolve.
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