Riemann-Encircling Exceptional Points for Efficient Asymmetric Polarization-Locked Devices

Dynamically encircling exceptional points (EPs) have unveiled intriguing chiral dynamics in photonics. However, the traditional approach based on an open manifold of Hamiltonian parameter space fails to explore trajectories that pass through an infinite boundary. Here, by mapping the full parameter space onto a closed manifold of the Riemann sphere, we introduce a framework to describe encircling-EP loops. We demonstrate that an encircling trajectory crossing the north vertex can realize near-unity asymmetric transmission. An efficient gain-free, broadband asymmetric polarization-locked device is realized by mapping the encircling path onto $L$-shaped silicon waveguides.

Figure 1

(a) Hamiltonian parameter space described by $\zeta =(\beta /\kappa ,\gamma /\kappa )$. The yellow line shows an evolution path along the parameter space boundary (blue dashed line). (b) RSP ($x_s^2+y_s^2+z_s^2=2^2$, $x_s=2\sin {\phi }_1\cos {\phi }_2$, $y_s=2\sin {\phi }_1\sin {\phi }_2$, and $z_s=2\cos {\phi }_1$) describing the Hamiltonian parameter space. (c)–(f) The evolution loops on RSP (black curves) and in the 2D parameter space (yellow curves) when encircling (c) no EP, (d) two EPs, and (e), (f) one EP. The loop in (f) locates in the center of the two EPs.

Figure 2

(a) APLD made of silicon waveguides. The full height of the $L$-shaped waveguide is $h_a=340\mathrm{nm}$, and the height of the lower waveguide is $h_b=220\mathrm{nm}$. The width of the upper waveguide and the full waveguide is $w_{\mathrm{La}}$ ($w_{\mathrm{Ra}}$) and $w_{\mathrm{Lb}}$ ($w_{\mathrm{Rb}}$) for the left (right) waveguide, respectively. The gap separation between the two waveguides is $d$. (b) The polarization directions for $R1$ and $R2$ modes. (c) The cross sections of the right waveguide at $A$, $B/C$, and $D$, and their associated electric field intensity distributions for the R1 and R2 eigenmodes.

Figure 3

(a) and (b) CW loops on RSU formed by the real part of the eigenvalues of $H$ when the input state is (a) ${[1,1]}^T$ and (b) ${[1,-1]}^T$. The RSU is extracted by casting the hemisphere with $y_s\ge 0$ (pure lossy case) of the RSP onto the $x_s\text{−}{z}_s$ plane. The color of the surface indicates the imaginary part of the eigenvalues of $H$, in which a smaller imaginary part indicates a larger loss. (c) and (d) ACW loops when the input state is (c) ${[1,1]}^T$ and (d) ${[1,-1]}^T$.

Figure 4

(a) The cross-sectional electric field intensity distributions at $A$, $B$, $C$, and $D$. (b) Optical microscope images of the fabricated APLD, where the $L$-shaped waveguides are partially depicted by the SEM image on the right panel. (c),(d) Measured transmittance spectra over 1535–1575 nm wavelength range as (c) TE and (d) TM modes are injected, respectively.