Modal expansions in periodic photonic systems with material loss and dispersion

We study band-structure properties of periodic optical systems composed of lossy and intrinsically dispersive materials. To this end, we develop an analytical framework based on adjoint modes of a lossy periodic electromagnetic system and show how the problem of linearly dependent eigenmodes in the presence of material dispersion can be overcome. We then formulate expressions for the band-structure derivative $\left(\partial \omega \right)/\left(\partial \mathbf{k}\right)$ (complex group velocity) and the local and total density of transverse optical states. Our exact expressions hold for 3D periodic arrays of materials with arbitrary dispersion properties and in general need to be evaluated numerically. They can be generalized to systems with two, one, or no directions of periodicity provided the fields are localized along nonperiodic directions. Possible applications are photonic crystals, metamaterials, metasurfaces composed of highly dispersive materials such as metals or lossless photonic crystals, and metamaterials or metasurfaces strongly coupled to resonant perturbations such as quantum dots or excitons in 2D materials. For illustration purposes, we analytically evaluate our expressions for some simple systems consisting of lossless dielectrics with one sharp Lorentzian material resonance added. By combining several Lorentz poles, this provides an avenue to perturbatively treat quite general material loss bands in photonic crystals.

Figure 1

Example for the difficulty of deriving the DOS from common band-structure plots. The system is a gas of strong Lorentzian oscillators in vacuum. The frequency band-structure is really a complex function of complex wave number. (a) shows two typical (and equally correct) projections: purely real frequency vs real part of the (complex) wave number and real part of the (complex) frequency associated with purely real wave numbers. (b) shows the correct DOS (derived from the Green tensor) and incorrect results found by naively computing a “DOS” from the apparent group velocities of either curve in (a). The example is taken from the discussion in Sec. 5a.

Figure 2

Sketch of the process of transforming the $\mathbf{k}$-space integration for the case of a one-dimensional (1D) system stacked along the $\widehat{z}$ axis with an arbitrarily small $k_x\ne 0$. The top left box shows the conventional integration procedure, where the different bands are distinguished by colors. (a) Conventional band structure for a lossless system. (b) Outline of the reduced zone scheme integration path in the complex $k_z$ plane; all bands are integrated over the first BZ. Each stop band corresponds to a pair of branch points (indicated as “x” symbols) at the BZ boundary connected by a branch cut (thin dashed line). The resonance pole of the Green function is indicated by a “+” symbol and lies away from the $k_z$-real axis for lossy systems. The top right box shows the transformation to two integrals (one solid, one dotted line) in an extended zone scheme, i.e., integration over the $i\mathrm{th}$ band is shifted to the $i\mathrm{th}$ BZ; the individual integral segments remain disconnected. (c) Conventional dispersion diagram for a loss-less system. Each integral covers eigenvalues with opposite sign for positive and negative $k_z$ to match the symmetry of adjoint modes [see Eq. (23)]. (d) Extended zone scheme integration path in the complex plane. The bottom left box (e) shows the integration contour after connecting the individual integral segments around the stop band branch points except the one at $k_z=0$. In the limit ${\mathbf{r}}^{\prime }\to \mathbf{r}$, the integrand is symmetric under $(\mathbf{k},\lambda )\to (-\mathbf{k},\lambda )$ and the hairpins are equivalent to a set of closed integral loops, each of which vanishes. The complex plane above this integration contour includes exactly one branch point at $\Re \left\{k_z\right\}=0$, i.e., closing the contour around the top brings us from the positive $\lambda$ bands to negative bands and vice versa. Integrating twice along the contour $\mathcal{C}$ shown in (f) provides a closed integral to which the residue theorem can be applied. See the main text and Appendix pp3 for further details.

Figure 3

Complex band structure of an isotropic homogeneous medium with Lorentz resonance $\left(\mathrm{\Omega }=1.0,\gamma =0.03\right)$. The line colors represent different oscillator strengths $A$: $-0.4\times {10}^{-3}$ (black), $-1.0\times {10}^{-3}$ (red), $-1.8\times {10}^{-3}$ (blue), and $-4.0\times {10}^{-3}$ (green). The left column shows views of the real-$\omega$ curve, i.e., the part of the path in complex $k$-space that is associated with real frequencies. In contrast, the right column shows views of the real-$k$ band structure, i.e., the complex frequencies associated with real wave numbers. Crosses and thin solid lines in panel (b) indicate the resonance-related branch points and branch cuts. Solid and dashed lines indicate corresponding solutions in panels (c) and (d). See text for further discussion.

Figure 4

Comparison of the proper tDOS [Eq. (48), black solid line] and incorrect expressions based on the group slowness derived from “sanitized” real-$\omega$ (red dashed) and the real-$k$ (blue dotted) contours for an isotropic bulk medium with a Lorentz resonance $\left(\mathrm{\Omega }=1.0,\gamma =0.03\right)$ for four different oscillator strengths (annotated in the graphs). See text for further discussion.

Figure 5

Comparison of the loss spectrum (top left panel), tDOS (bottom left), and two projections of the complex dispersion relation (right column) of hybrid system composed of a two-level system strongly coupled to an isotropic band edge. The parameter varied is the oscillator strength density $\alpha$: $-2\times {10}^{-7}$ (black), $-1\times {10}^{-6}$ (red), $-3\times {10}^{-6}$ (blue), and $-1\times {10}^{-5}$ (green). The tDOS of the unperturbed band edge is shown as a shaded area. See text for further discussion.