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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.^[ref]

For one illuminated input of a passive beam splitter, at specified wavelength, angle of incidence and electromagnetic polarization, define power transmittance $T=P_{\mathrm{transmitted}}/P_{\mathrm{in}}$, reflectance $R=P_{\mathrm{reflected}}/P_{\mathrm{in}}$ and radiometric absorptance $A=P_{\mathrm{absorbed}}/P_{\mathrm{in}}$. 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,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, $P_{\mathrm{in}}=10\mathrm{mW}$, $T=0.45$, $R=0.50$ 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.

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.

For a linear, fixed-frequency beam splitter, choose power-normalized complex amplitudes $a_1,a_2$ at its input optical ports and $b_1,b_2$ at its output optical ports: $|a_j{|}^2$ and $|b_j{|}^2$ are powers. The scalar description treats one specified electromagnetic polarization and matching transverse profiles; other fields require additional components. With the geometrical ordering in the diagram,Here $t,t^{\prime }$ are transmission complex amplitudes per unit input and $r,r^{\prime }$ are reflection complex amplitudes per unit input. In this power normalization, a single input gives $T=|t{|}^2$ and $R=|r{|}^2$. Raw electric field amplitudes in different media generally need additional power-normalization factors.

If these two output optical ports account for all outgoing power and the device has no loss or gain, $S$ is a unitary matrix:The dagger is the Hermitian transpose and a star is complex conjugation. To prove the condition, substitute $\mathbf{b}=S\mathbf{a}$ into conservation of total power for every input vector: ${\mathbf{a}}^{†}(S^{†}S-I)\mathbf{a}=0$. Inputs $(1,0)$ and $(0,1)$ fix the diagonal entries; $(1,1)$ and $(1,i)$ fix the real and imaginary parts of the off-diagonal entry. A lossy two-output description instead has $S^{†}S\le I$ for a passive device.^[ref]

One consistent ideal symmetric 50:50 choice isEach reflection carries $+\pi /2$ relative wave phase in this convention. The columns have unit vector norm and zero inner product, directly verifying that $S$ is a unitary matrix. Choosing a different phase convention for complex amplitudes at the optical ports replaces $S$ by $D_{\mathrm{out}}S{D}_{\mathrm{in}}^{-1}$, where the $D$ are diagonal matrices whose entries have modulus one; the input complex amplitudes must be transformed consistently. Observable powers are unchanged. The displayed $i$ is a convention for this ideal model, not a universal reflection rule for every coating or incidence geometry.

A polarizing beam splitter sends two orthogonal linear polarizations to different output paths. Acting coherently on $\alpha |H⟩+\beta |V⟩$, it correlates path with photon polarization instead of measuring photon polarization unless a path is actually detected or discarded.

For an ideal polarizing beam splitter with one input, choose orthogonal unit electromagnetic polarization vectors ${\mathbf{e}}_H,{\mathbf{e}}_V$ and an incident power-normalized field $a_H{\mathbf{e}}_H+a_V{\mathbf{e}}_V$. If $H$ is transmitted and $V$ reflected, the output powers are $P_{\mathrm{transmitted}}=|a_H{|}^2$ and $P_{\mathrm{reflected}}=|a_V{|}^2$. The associated complex amplitudes can acquire different wave phases; the routing is not an equal-power split for every incident electromagnetic polarization. For linear polarization at angle $\theta$ to $H$, input power $P$ gives $P{\cos }^2\theta$ and $P{\sin }^2\theta$, respectively. At $\theta =30^{\circ }$, a stipulated $4\mathrm{mW}$ input gives $3\mathrm{mW}$ transmitted and $1\mathrm{mW}$ reflected. A $45^{\circ }$ input gives equal powers even though an $H$ input goes entirely into the transmitted output.^[ref]

Real polarizing beam splitters have leakage into the unwanted output and losses. Each linear polarization has its own field transformation and power coefficients. To make an ordinary power wave interference term between the two output contributions, first bring them into the same path with overlapping transverse profiles and a common electromagnetic polarization, for example using a polarizer. Merely placing two orthogonal linear polarizations on a polarization-insensitive photodetector adds their powers. Coherent routing can be reversed by suitable optics before detection; discarding a path loses access to that part of the field but supplies no recorded measurement outcome by itself.

// Target: optics.bigb