Electromagnetic vacuum of complex media: Dipole emission versus light propagation, vacuum energy, and local field factors

We offer a unified approach to several phenomena related to the electromagnetic vacuum of a complex medium made of point electric dipoles. To this aim, we apply the linear response theory to the computation of the polarization field propagator and study the spectrum of vacuum fluctuations. The physical distinction among the local density of states which enter the spectra of light propagation, total dipole emission, coherent emission, total vacuum energy, and Schwinger-bulk energy is made clear. Analytical expressions for the spectrum of dipole emission and for the vacuum energy are derived. Their respective relations with the spectrum of external light and with the Schwinger-bulk energy are found. The light spectrum and the Schwinger-bulk energy are determined by the Dyson propagator. The emission spectrum and the total vacuum energy are determined by the polarization propagator. An exact relationship of proportionality between both propagators is found in terms of local field factors. A study of the nature of stimulated emission from a single dipole is carried out. Regarding coherent emission, it contains two components. A direct one which is transferred radiatively and directly from the emitter into the medium and whose spectrum is that of external light. And an indirect one which is radiated by induced dipoles. The induction is mediated by one (and only one) local field factor. Regarding the vacuum energy, we find that in addition to the Schwinger-bulk energy the vacuum energy of an effective medium contains local field contributions proportional to the resonant frequency and to the spectral line width.

Figure 1

Diagrammatic representation of the Dyson propagator, $\mathrm{Ḡ}$.

Figure 2

(a) Feynman rules. Only two-point irreducible correlation functions, $h(r)$, have been used for the sake of simplicity. (b) Diagrammatic representation of the equivalence between multiple-scattering processes amounting to $\overline{\mathcal{G}}(\mathrm{r⃗},\mathrm{r⃗})$. Reciprocity applies. $\mu$ represents the virtual presence of the emitter. (c) and (d) Diagrammatic representations of the integration in $q$ of Eqs. (22) and (23), respectively.

Figure 3

(a) Five-scattering closed loop with undefined reference frame. (b) Fixing the reference frame at the position of the uncorrelated (removed) scatterer labeled as $1$ yields a polariton in $\mathrm{Ḡ}(\mathrm{r⃗},\mathrm{r⃗})$. It can be excited either by external light or by dipole emission. (c) Fixing the reference frame at the position of the correlated (removed) scatterer labeled as $3$ yields a polariton in ${\overline{\mathcal{G}}}^{\mathrm{VC}}(\mathrm{r⃗},\mathrm{r⃗})$ which is not in $\mathrm{Ḡ}(\mathrm{r⃗},\mathrm{r⃗})$. It can be excited by dipole emission only.

Figure 4

(a) Feynman’s rules for the classical regularization scheme of Appendix . (b) Diagrammatic representation of Eq. (A3). (c) Diagrammatic representation of the dressing up of ${\chi }_e$ leading to ${\alpha }_0$. Approximation symbols denote that the field within the emitter is taken uniform. (d) Diagrammatic representation of one of the self-polarization cycles which enter the series in (b).