Phase convention for complex amplitudes

Reference wave phases label complex amplitudes without changing measured powers. If $\mathbf{b}=S\mathbf{a}$, choose diagonal matrices $D_{\mathrm{in}},D_{\mathrm{out}}$ whose entries have modulus one, and set ${\mathbf{a}}^{\prime }=D_{\mathrm{in}}\mathbf{a}$, ${\mathbf{b}}^{\prime }=D_{\mathrm{out}}\mathbf{b}$. Then $S^{\prime }=D_{\mathrm{out}}S{D}_{\mathrm{in}}^{-1}$ and each component has unchanged squared modulus. For a beam splitter scattering matrix, transforming $S$ without transforming input complex amplitudes changes the stated physical input.