= Phase convention for complex amplitudes
Reference <wave phases> label <complex amplitudes> without changing measured <powers>. If $\mathbf b=S\mathbf a$, choose <diagonal matrices> $D_{\rm in},D_{\rm out}$ whose entries have <modulus> one, and set $\mathbf a^{\prime}=D_{\rm in}\mathbf a$, $\mathbf b^{\prime}=D_{\rm out}\mathbf b$. Then $S^{\prime}=D_{\rm out}SD_{\rm in}^{-1}$ 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.
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