Local symmetries and perfect transmission in aperiodic photonic multilayers

We develop a classification of perfectly transmitting resonances occurring in effectively one-dimensional optical media which are decomposable into locally reflection symmetric parts. The local symmetries of the medium are shown to yield piecewise translation-invariant quantities, which are used to distinguish resonances with arbitrary field profile from resonances following the medium symmetries. Focusing on light scattering in aperiodic multilayer structures, we demonstrate this classification for representative setups, providing insight into the origin of perfect transmission. We further show how local symmetries can be utilized for the design of optical devices with perfect transmission at prescribed energies. Providing a link between resonant scattering and local symmetries of the underlying medium, the proposed approach may contribute to the understanding of optical response in complex systems.

Figure 1

(a) Schematic of an aperiodic multilayer comprised of 16 planar slabs of materials $A$ (light gray) and $B$ (dark gray), having equal optical thickness $n_A{d}_A=n_B{d}_B={\lambda }_0/4$. The scattered monochromatic plane light wave of stationary electric-field amplitude $E$ propagates along the $z$ axis, perpendicularly to the $xy$ plane of the slabs. (b) 1D cross section of the multilayer in real space, showing its local symmetries. The arcs depict locally symmetric domains ${\mathcal{D}}_m$ of the device, with lengths $L_m$ and symmetry plane positions ${\alpha }_m$. For simplicity and figure clarity, only maximal local symmetries are shown, which are the ones of largest $L_m$ at a given ${\alpha }_m$ (i.e., any smaller arc, concentric to the ones shown, is also a local symmetry). The selected local symmetry decomposition into $N=3$ domains (dashed arcs) will constitute one of the examples in Sec. 3b.

Figure 2

Geometric representation of scattering in locally symmetric media. The scaled invariants ${\mathcal{V}}_m$ characterizing each symmetric subdomain ${\mathcal{D}}_m$ are represented by vectors (thin black lines) and added “head to tail” for increasing $m$, with sign ${(-1)}^{m-1}$ (see text), yielding $\mathcal{L}$ (thick colored lines). (a) For any non-PTR, the trajectory is open with arbitrary end $\ne 2ik$. (b) and (c) For any PTR, the trajectory formed by the vectors ends at 0 ($2ik$) for an even $N$ (odd $N$) LP decomposition of the scattering device. For (b) asymmetric PTRs, the trajectory explores the complex plane, while for (c) symmetric PTRs, it oscillates between 0 and $2ik$.

Figure 3

(a) Transmission coefficient $T$ as function of the scaled frequency $\omega /{\omega }_0$ for the photonic multilayer shown in Fig. 1 comprised of slabs $A$ and $B$ with refraction indices $n_A=2.12$ and $n_B=1.45$. (b) Field magnitude $E_0\left(z\right)$ across the multilayer at the frequencies marked in (a), corresponding to a non-PTR (${\omega }_{\square }=0.607{\omega }_0$), two $s$-PTRs (${\omega }_{▵}={\omega }_0$ and ${\omega }_{▿}=1.276{\omega }_0$), and an $a$-PTR (${\omega }_{◊}$), with scale on the left. The background shows the slabs $A$ (light gray) and $B$ (dark gray) along the device, and the vertical lines depict the considered decomposition into LP symmetric subdomains of the device. The $|Q_m|$ for each subdomain $D_m$ are plotted as thick solid lines (scale on the right). (c) The alternating sum $\mathcal{L}$ (thick colored arrows) of the ${\mathcal{V}}_m$ for each considered decomposition in (b), represented as a trajectory (thin black arrows) in the complex plane, like in Fig. 2. For the $a$-PTR ($◊$), an odd ($N=3$) and an even ($N=4$) decomposition is considered [solid and dashed lines in (b) and (c), respectively].

Figure 4

(a) Transmission spectrum around a central frequency ${\omega }_0=2\pi c/{\lambda }_0$, with ${\lambda }_0=700\mathrm{nm}$, of a photonic multilayer consisting of two different kinds of slabs $A$ ($n_A=2.15$, $d_A=0.08426{\lambda }_0$) and $B$ ($n_B=1.43$, $d_B=1.01348{\lambda }_0$), with two intervening gaps $C_1$ and $C_2$ of the ambient medium ($n_{C_1}=n_{C_2}=1$), of widths $d_{C_1}=0.1146{\lambda }_0$ and $d_{C_2}=0.04236{\lambda }_0$. (b) Field amplitudes at the $s$-PTR frequencies ${\omega }_{▵}=0.67{\omega }_0$ and ${\omega }_{▿}=1.38{\omega }_0$ marked in (a). The background shows the media $A$ (light gray), $B$ (dark gray), and $C$ (white) along the device, and vertical lines distinguish the considered LP decompositions into resonators at each $s$-PTR. Note that for ${\omega }_{▵}$, the gaps $C$ are parts of the resonators, while for ${\omega }_{▿}$, they are not (see text).