Layered photonic systems typically acquire one topological transition of the isofrequency contour (IFC) per external parameter tuned. Nonlocal effective-medium studies show that a fixed cell can change its IFC topology more than once but treat that topology as a single open-versus-closed condition, which names neither the invariant that labels a change nor the parameter that moves it. Here, combining a four-layer transfer matrix with finite-element dipole simulations and the normal-incidence spectrum of a finite stack, we show that the IFC of a fixed silica/silver/silica/silicon cell is one cycle on the Brillouin cylinder $S_{k_x}^1\times {\mathbb{R}}_{k_y}$ carrying two independent indices: the winding number $w$ about the $k_x$ circle, and the intersection number ${\nu }_{\parallel }$ with the mirror-invariant line $k_y=0$. Two saddle-type transitions follow. At ${\lambda }_{\mathrm{I}}\approx 621\mathrm{nm}$, $(w,{\nu }_{\parallel })$ changes from (1,0) to (0,1); at ${\lambda }_{\mathrm{II}}\approx 987\mathrm{nm}$, ${\nu }_{\parallel }$ changes from 0 to 1 at fixed $w=1$, inverting the forbidden direction by ${90}^{\circ }$. Because ${\nu }_{\parallel }$ counts the propagating Bloch channels at normal incidence, a four-regime reflectance-transmittance fingerprint measures it directly, with the simulated momentum spectra falling on the analytic contours with no fitted parameter. The same transfer matrix also locates the multiple Dirac degeneracies of this superlattice, so the IFC saddles and the Dirac points are the constant-$\omega$ and constant-$\mathit{k}$ critical points of one band manifold. Two deposition thicknesses thereby add a passive, deposition-defined structural route alongside twist, ferroelectric, and phase-change tuning for wavelength-multiplexed routing.

Close

PrevNext

View figure in article

• Plasmonics
• Topological effects in photonic systems
• Waveguide arrays
• Transfer matrix method