= Beam splitter scattering matrix
{title2=$\mathbf b=S\mathbf a$}
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,
$$
\begin{pmatrix}b_1\\b_2\end{pmatrix}
=S\begin{pmatrix}a_1\\a_2\end{pmatrix},
\qquad S=\begin{pmatrix}t&r'\\r&t'\end{pmatrix}.
$$
Here $t,t'$ are transmission <complex amplitudes> per unit input and $r,r'$ 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>:
$$
S^\dagger S=I,\qquad
|t|^2+|r|^2=|r'|^2+|t'|^2=1,\qquad
t^*r'+r^*t'=0.
$$
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>: $\mathbf a^\dagger(S^\dagger S-I)\mathbf a=0$. 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 $S^\dagger S\leq I$ for a passive device.<Photon quantum mechanics and beam splitters by Charles H. Holbrow and others>[Photon quantum mechanics and beam splitters, §IV.A, printed p. 262, equations (1)–(5): unitary beam splitter matrix and phase constraints]{ref}
One consistent ideal symmetric 50:50 choice is
$$
S=\frac1{\sqrt2}\begin{pmatrix}1&i\\i&1\end{pmatrix},
\qquad b_1=\frac{a_1+ia_2}{\sqrt2},\quad b_2=\frac{ia_1+a_2}{\sqrt2}.
$$
Each 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 $D_{\rm out}SD_{\rm in}^{-1}$, 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.
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