Topology versus element shape and geometry-faithful band-structure control in 3D capacitively coupled metallo-dielectric photonic crystals

Sievenpiper, Yablonovitch, Winn, Fan, Villeneuve, and Joannopoulos (Phys. Rev. Lett. 80, 2829 (1998)) demonstrated that a three-dimensional metallo-dielectric photonic crystal built from stacked printed-circuit boards, in which tetrahedral metallic islands are linked through strong metal/insulator/metal capacitors on a diamond-like lattice, exhibits a wide omnidirectional microwave band gap (measured 12-30 GHz) with a lattice constant far smaller than the free-space wavelength. The paper left three linked issues open: the origin of quantitative discrepancies between its finite-difference time-domain band-structure computations and the measured higher-frequency bands; the extent to which the band structure is governed by the topology (connectivity) of the capacitive network rather than by the detailed shape of the metallic elements, which the authors stated as a conjecture without a systematic study; and the performance limits (losses, scaling to lower radio frequencies, other material systems) of such compact capacitive lattices.

Background

A photonic crystal is a periodic electromagnetic structure in which the periodicity produces a photonic band gap, a frequency interval in which propagating electromagnetic modes are forbidden, analogous to the electronic band gap of a semiconductor. In ordinary dielectric photonic crystals the lattice constant must be comparable to half the wavelength at the band edge, so structures useful at radio or low microwave frequencies become bulky. Metallic and metallo-dielectric (metal-in-dielectric) photonic crystals relax this constraint: a continuous metallic wire mesh excludes fields down to zero frequency with a cutoff set by the lattice constant, while isolated metallic islands interact through their mutual capacitance. In Phys. Rev. Lett. 80, 2829 (1998), Sievenpiper, Yablonovitch, Winn, Fan, Villeneuve, and Joannopoulos combined these ingredients deliberately: they built a three-dimensional lattice of metallic tetrahedral islands, each connected to its four neighbors (diamond connectivity) through metal/polyimide/metal parallel-plate capacitors, fabricated as 13 stacked printed-circuit boards (Rogers Duroid, $\varepsilon = 2.2$ ) with 25 $\mu$ m polyimide dielectric layers between overlapping metal patches. The measured inter-island capacitance was $C = 0.54$ pF and the unit-cube length $a = 3.73$ mm. Transmission measurements with plane waves and with monopole (short wire) probes found a complete band gap from 12 to 30 GHz with up to 35 dB rejection; because the valence (lower-edge) wavelength is about 25 mm, the periodicity satisfies $a/\lambda \approx 0.15$ , far below the half-wavelength rule of dielectric crystals. The band structure was computed with the finite-difference time-domain (FDTD) method, which integrates Maxwell's equations on a space-time grid (here $68\times68\times104$ cells with Bloch-periodic boundary conditions $E(r+R)=e^{ik\cdot R}E(r)$ ); for computational economy most of $k$ -space was computed with the capacitor plate spacing enlarged threefold, and only the $\Gamma$ -Z direction was recomputed with the true 25 $\mu$ m spacing, giving a computed gap of 13-28 GHz in good agreement with experiment. The physical picture is that the valence band edge behaves as a lumped $LC$ resonance, $\omega_{\mathrm{valence}}\approx1/\sqrt{LC}$ , with the electric field concentrated in the narrow capacitive gaps, while the conduction (upper) band edge is a half-wavelength standing-wave mode set by the lattice periodicity, $\omega_{\mathrm{conduction}}\approx\pi c/(na)$ with refractive index $n$ of the dielectric background; the two edges can therefore be tuned independently. The paper acknowledged that the computed and measured higher-frequency bands disagreed quantitatively (candidate causes being field-discretization error, the enlarged capacitor spacing used in most computations, and experimental spacing and fabrication uncertainties); that the authors believed the band-structure behavior to be governed mainly by the topology of the capacitive network rather than by the detailed shape of the metallic elements, but supplied no systematic shape study; and that the practical performance limits of such structures at lower radio frequencies, in lossy media, and in other material systems remained to be explored. The 1999 companion work of the same group reduced the concept to two dimensions: the mushroom-like high-impedance electromagnetic surface (a lattice of metal patches shorted to a ground plane by vias), described by a parallel resonant $LC$ circuit with sheet impedance $Z = j\omega L/(1-\omega^2 LC)$ and bandwidth $\Delta\omega/\omega_0 = Z_0/\eta$ , whose surface-wave band gap and in-phase reflection enabled low-profile antennas at gigahertz frequencies. That two-dimensional line, and three-dimensional diamond-lattice metallodielectric crystals realized by other fabrication routes, form the adjacent later literature.

Problem Statement

For three-dimensional metallo-dielectric photonic crystals whose periodic metallic islands are linked by strong inter-island capacitors, as realized by the 1998 stacked-printed-circuit-board diamond lattice, establish the quantitative, geometry-faithful design rules that the source paper left open, namely: (1) compute the full-wave band structure of the actual fabricated geometry (correct 25 $\mu$ m capacitor spacing, adequately fine discretization) over the entire Brillouin zone (the full set of Bloch wave vectors $\mathbf{k}$ in reciprocal space) - not only the $\Gamma$ -Z line, the high-symmetry line from the zone center to the zone boundary along the stacking direction - and determine, by comparing the converged computed band structure with the measured spectra reported in the source paper, which of the computational candidate causes (field discretization or the enlarged capacitor spacing used in the original computations) dominates the discrepancies between computed and measured higher-frequency bands, with the experimental spacing and fabrication uncertainties fixed and known rather than assumed negligible; (2) determine systematically the extent to which the band structure and gap behavior are fixed by the topology (connectivity) of the capacitive network alone, as the authors conjectured, versus depending quantitatively on the detailed shape of the metallic capacitive elements; the conjecture holds if, at fixed lattice connectivity and fixed effective inter-island capacitance $C$ and inductance $L$ , the computed band structure (gap edges and omnidirectionality) agrees across element shapes - so that the large-gap, sub-wavelength-period behavior generalizes to other capacitive-element geometries on the same lattice connectivity - and is refuted to the extent those features differ; and (3) characterize the performance envelope of such three-dimensional capacitively coupled lattices as compact photonic-band-gap structures, including ohmic and dielectric losses, scaling to lower radio frequencies, and realization in other material systems and fabrication routes. Later literature has already realized part of (3) through two-dimensional high-impedance surfaces and other-material-system three-dimensional diamond lattices. The parts serve one objective: turning the 1998 demonstration into validated, geometry-faithful design rules that specify which geometric parameters control the valence edge $\omega_{\mathrm{valence}}\approx1/\sqrt{LC}$ , the conduction edge $\omega_{\mathrm{conduction}}\approx\pi c/(na)$ , the gap width and omnidirectionality, and the practical performance limits.

References

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