Hamiltonian treatment of the electromagnetic field in dispersive and absorptive structured media

We introduce a Hamiltonian formulation of electromagnetic fields in dispersive and absorptive structured media of arbitrary dimensionality; the Kramers-Kronig relations are satisfied by construction. Our method is based on an identification of the photonic component of the polariton modes of the system. Although the medium degrees of freedom are introduced in an oscillator model, only the susceptibility of the medium appears in the derived eigenvalue equation for the polaritons; the theory is applicable to both classical and quantum optics. A discrete polariton spectrum is obtained in the transparent regime below the absorption cutoff frequency, and the normalization condition contains the material dispersion in a simple way. In the absorptive regime, a continuous polariton spectrum is obtained. The expressions for the full electromagnetic field of the system can be written in terms of the modes of a limiting, nondispersive, nonabsorptive system, so the theory is well suited to studying the effect of dispersion or absorption on photonic dispersion relations and mode structure. Available codes for dispersive photonic modes can easily be leveraged to obtain polariton modes in both the transparent and absorptive regimes.

Figure 1

(Color online) Schematic diagram of the frequency domains of the various modes of the system before (left) and after (right) the canonical transformation. The nominal electromagnetic mode operators $a_m$ and medium modes ${\psi }_{\Omega }$ are plotted on the left, while the polariton mode operators $c_m$ and $c_{\Omega m}$ are plotted on the right.

Figure 2

(Color online) Schematic illustration of the scalar potential ${\varphi }^{\Omega m}$ for the longitudinal polarization modes. The hatched regions represent ${\mathcal{V}}_{\Omega }$, the region in which absorption is present at frequency $\Omega$. The scalar potential is constant within connected nonabsorbing regions.

Figure 3

(Color online) Schematic diagram of a 2D photonic crystal.

Figure 4

(Color online) In-plane polariton band structure for TM modes of 2D photonic crystal with lattice constant $a$ (black lines with symbols) and the photonic bands of a corresponding nominal, nondispersive crystal (pink lines). The crystal is composed of a triangular lattice of air cylinders of radius $r∕a=0.417$ in a dispersive background medium with dielectric response $\Gamma \left(\omega \right)$ shown inset (black curve), along with the nominal nondispersive response (pink), which corresponds to index $n=3.325$. In the frequency range $\nu a∕c∊[0,0.66]$ for which the bands are given, the medium exhibits normal dispersion and its index of refraction spans $n∊[3.1,3.59]$.

Figure 5

(Color online) For the dispersive photonic crystal with band structure given in Fig. 4, medium field amplitudes in a transparent-regime polariton with unitless frequency $\omega a∕2\pi c=0.8$. The medium contribution is given as $\mid {\mathbf{\alpha }}_{\omega }^m\mid$ (left scale) and $\mid {\mathbf{\beta }}_{\omega }^m\mid$ (right scale) in units of $\mid (a∕2\pi c)\sqrt{\mathrm{Im}\Gamma (\mathbf{r},\omega )∕\left(\hslash {\epsilon }_0\pi \right)}{\stackrel{\mathrm{̃}}{\mathbf{D}}}_m^{*}\left(\mathbf{r}\right)\mid$. The patterned area indicates the absence of medium oscillators, i.e., the transparent frequency regime, and the dashed line indicates the frequency of the polariton mode of interest. The electronic gap is at ${\omega }_{g}a∕2\pi c=0.85$.

Figure 6

(Color online) Schematic diagram of a GaAs waveguide structure.

Figure 7

(Color online) (a) Real (solid blue line) and imaginary (dotted red line) parts of the dielectric constant $\epsilon \left(\omega \right)$ of GaAs, as a function of polariton energy $\hslash \omega$. These data are due to ab initio calculations by F. Nastos [62]. (b) Real (solid blue line; left scale) and imaginary (dotted red line; right scale) parts of the dielectric response $\Gamma \left(\omega \right)$ of GaAs, as a function of energy.

Figure 8

(Color online) The lowest three effective broadened polariton bands for a GaAs waveguide with 1D confinement. Brightness indicates the degree of electromagnetic character of the polariton modes in the absorption regime $(\omega ⩾{\omega }_g)$. Dotted lines give the transparent regime bands below ${\omega }_g$; in the absorption regime above ${\omega }_g$ they give the three lowest bands of the transparent system obtained if absorption is switched off: $\Gamma \left(\omega \right)\to \mathrm{Re}\Gamma \left(\omega \right)$.

Figure 9

(Color online) Cross section through Fig. 8 at propagation constant $\overline{k}=25\mu {\mathrm{m}}^{-1}$, showing $V_{\overline{\omega u}\overline{\mathbf{k}}}\left(0\right)$ for the lowest three polariton modes as a function of mode energy $\hslash \overline{\omega }$.

Figure 10

(Color online) For the GaAs waveguide, $c{\max }_{\mathbf{r}}\mid {\mathbf{\alpha }}_{{\omega }^{\prime }}^{\omega m}\left(\mathbf{r}\right)\mid$ vs $\hslash {\omega }^{\prime }$ for $m=(u,k)=(2,25\mu {\mathrm{m}}^{-1})$, for $\hslash \omega =\{\hslash {\omega }_{\mathrm{disp}}-0.1\mathrm{eV},\hslash {\omega }_{\mathrm{disp}},\hslash {\omega }_{\mathrm{disp}}+0.1\mathrm{eV}\}$, where $\hslash {\omega }_{\mathrm{disp}}=1.82\mathrm{eV}$ is the energy of the corresponding discrete band in the dispersive but nonabsorptive limit, i.e., $\Gamma (\mathbf{r},\omega )\to \mathrm{Re}\Gamma (\mathbf{r},\omega )$.