Christian M. Lange

Macroscopic quantum electrodynamics (QED) describes light-matter interactions in dispersive, absorptive media. The Green's function propagates the field and noise operators arising from material absorption account for quantum fluctuations. Standard macroscopic QED represents the field through bulk noise operators alone and for a finite object in a lossless exterior it is extended by a limiting-absorption construction. The modified Langevin noise approach instead retains an independent incoming field explicitly but expands it in free-space plane waves in the far field. Nanophotonic input-output systems, however, naturally have dielectrics extending to the boundary: Waveguide ports, not free-space plane waves in the far field, define the channels through which quantum fields enter and leave. Here we extend macroscopic QED to finite systems in which the field reaches a boundary that may itself contain dielectrics. We express the interior quantum field in terms of two independent noise sources: bulk Langevin operators from material absorption and field operators on the boundary. We show that their commutators close into the exact relation $[\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathcal{E}},{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathcal{E}}}^{\ifmmode\dagger\else\textdagger\fi{}}]\ensuremath{\propto}\mathrm{Im}\phantom{\rule{0.16em}{0ex}}g$, so that the theory satisfies the fluctuation-dissipation theorem with the total fluctuations partitioned between bulk absorption and boundary flux. Carrying the boundary fields requires retaining both $\mathbf{E}$ and $\mathbf{H}$, so we recast Maxwell's equations as a first-order operator equation for the dual field $\mathcal{E}={[\mathbf{E},{Z}_{0}\mathbf{H}]}^{T}$, propagated between surfaces by a first-order Green operator $g$. Treating the problem at the operator level makes the otherwise involved algebra tractable: Symmetries of the Maxwell operator under energy and reciprocal inner products yield the interior-field representation, Lorentz reciprocity, and a generalized optical theorem with minimal vector calculus. This formalism provides the rigorous field expression underlying a future modal input-output theory for nanophotonic systems.