Fig. 1. Schematics used to derive the generalized Snell's law of refraction^[14]

Fig. 2. Real (imaginary) part of refractive index is an even (odd) function of the x axis^[18]

Fig. 3. Schematic representation of a generic two-level system composed of two coupled entities^[21]

Fig. 4. Optical wave propagation when the system is excited at either channel 1 or channel 2 in a PT-symmetric system light propagates in a non-reciprocal manner both below and above threshold^[22]

Fig. 5. Eigenvalues of coupled dual waveguide systems with the same gain/loss and the eigenvalue varies with the gain/loss coefficient. Real parts (“Re”, solid lines) and imaginary parts (“Im”, dashed lines) of the two normalized eigenvalues, and the position of the exceptional points^[29]

Fig. 6. A parity-time-symmetric ternary micro-ring system with equidistantly spaced cavities. The side resonators experience balanced gain and loss whereas the middle one is neutral^[24]

Fig. 7. Real parts (left) and the imaginary parts (right) of the eigenfrequencies of the ternary parity-time-symmetric system as a function of the normalized gain/loss contrast g/κand the detuning ε/κ

Fig. 8. Photograph of PT symmetric metasurface composed of 300 nm thick silver (yellow or light gray) and lead (turquoise or dark gray) SRRs on silicon substrate ^[32]

Fig. 9. Schematic of PT symmetric metasurface with less lossy dipoles (blue) and more lossy dipoles (red) that is symmetric about y equals minus x

Fig. 10. Eigenpolarization states of transmission through an ideal PT symmetric metasurface when G_xy changes

Fig. 11. Schematic of the metasurface design. Each unit cell contains two strip antennas with thickness t=30 nm in the out-of-plane direction, lengths a₁=401.75 nm anda₂=435 nm, and widths w₁=50 nm and w₂=100 nm. Both antennas are free-standing, with a lattice periodd=600 nm

Fig. 12. Schematic of the metasurface unit cell geometry. The dimensions of each unit are set to L₁=L₃=30 μm, L₂=55 μm, L₄=80 μm, d₁=10 μm, d₂=50 μm, t_Si=525 μm

Fig. 13. Schematics of (a) the PT-symmetric system based on a pair of amplifying and attenuating metasurfaces, and (b) its enabled reciprocal and unidirectional reflectionless transparency. PT symmetry is satisfied with constrains: G₂=−G₁=−γY₀ and B₂=B₁=χY₀,where G_i and B_i are the surface conductance and susceptance of the ith metasurface; if the gain-loss parameter γ>0, metasurface 1 and 2 provide loss and gain, respectively, and vice versa. In the THz regime, the optically pumped graphene metasurface represents an active metasurface (G₂<0), while the metallic filament represents a resistive sheet (G₁>0)

Fig. 14. Schematic illustration of a PT-symmetric metasurface. The images on the right show the top and side views of the metasurface. Two colors represent, respectively, two kinds of metal (purple: titanium, yellow: gold) to obtain a high loss-contrast of conductivity. The outer diameter and the gap size of the SRR is denoted by d_m and g_m (m=1, 2), respectively, and the width of the arc is denoted by w. s represents the distance between the two outer arcs as one SRR is projected in the plane where the other SRR is

Fig. 15. Schematic of the non-Hermitian metasurface showing extremely asymmetrical reflections at the EP

Fig. 16. Schematic of unit cell of the non-ideal PT metasurface structure. The parameters are h=20 nm, r₁=61 nm, R₁=122 nm, r₂=63 nm, R₂=170 nm and t=640 nm, respectively. The distance s is variable. The incident wave is in x − z plane and has an angle θ with +zaxis

Fig. 17. Schematic diagram of the unit cell, highlighted by black dotted line, on a glass substrate. Thicknesses of gold, PMMA, and ITO layers are 45 nm, 180 nm, and 65 nm, respectively. s denotes the separation between the two orthogonal slots/bars with lengthsl₁=140 nm, l₂=170 nm and widths w₁=98 nm, w₂=110 nm

Fig. 18. Schematic of graphene metasurface. Each unit cell contains two orthogonal graphene stripes

Fig. 19. Re(κ) and Im(κ) as a function of δx at λ=111.5 μm