Fig. 1. 1D PT-symmetric photonic crystal with spatial translation. (a) One-dimensional photonic crystal considered as a 2D system by introducing a synthetic translation dimension Δx. (b), (c) Real and imaginary parts of complex band structure for a photonic crystal with εL=2+0.5i,εG=2−0.5i,dL=0.5a, and dG=0.5a. The blue and black curves highlight the PT-exact phase and the PT-broken phase, respectively. A frequency band gap (gray box) exists between the second and third bands. (d) 2D band structure in synthetic momentum space (kx,Δx). (e) Composite Berry curvature in 2D periodic momentum space (kx,Δx). The gap Chern number is −2.

Fig. 2. Edge states between translated photonic crystal and perfect electric conductor. (a) Photonic crystal under the perfect electric conductor (PEC) boundary condition with different Δx. (b) Edge spectrum as a function of translation Δx. Two gapless edge bands with complex frequencies are plotted in yellow. The blue dashed lines are their projections onto the real frequency plane. Black dotted lines along ωi direction are projection lines. As expected by the gap Chern number of −2, the edge bands traverse the band gap twice. The gray areas represent the bulk bands. (c) Electric field distribution of edge state for Δx=0.9a, corresponding to the magenta star in (b).

Fig. 3. Chiral edge states between band 1 and band 2. (a) Edge spectrum for the lowest two bands. (b), (c) Side view and top view of the band structure in (a). Red line in (a) and red circle in (b) correspond to the EPs in bulk band structure. (d) Edge spectrum for a PC without PT symmetry. Here εL=2+0.45i,εG=2−0.5i. (e), (f) Side view and top view of the band structure in (f). In these figures, the translucent gray/blue plane denotes the real/complex bulk band in PT-exact/broken phase. Yellow solid line denotes the gapless edge band while yellow dashed line is its projection onto the complex frequency plane.

Fig. 4. 4π reflection phase winding of the translated photonic crystal. (a) Reflection phase of translated photonic crystals with different Δx. (b) Reflection phase ϕr in the second band gap. From Δx=0 to Δx=a, the phase winds up by 4π. (c) Reflection phase at lower band edge [ω=0.691(2πc/a)], mid-gap [ω=0.7(2πc/a)], upper band edge frequency [ω=0.713(2πc/a)].

Fig. 5. Interpretation of 4π phase winding via the surface admittance of translated photonic crystals. (a) Band structure near the second band gap. Magenta star marks the band edge mode at ωu=0.713(2πc/a). (b)–(d) Total reflection of translated PCs with Δx=0,0.15a,0.3a, when a plane wave with frequency of ωu impinges onto the surface. The solid and dashed lines plot the magnetic field (purely real) and the electric field (purely imaginary) of the bulk propagating mode [magenta star in (a)]. (e) Surface electric/magnetic field Es/Hs for different truncations. The three cases in (b)–(d) are highlighted by colored circles. (f) Surface admittance ηs=Hs/Es. This purely imaginary admittance changes from −∞ to +∞ two times, as Δx increases. (g) Corresponding reflection coefficients r=(1−ηs)/(1+ηs) on the complex plane. As Δx increases from zero to a, reflection phase winds up by 4π.

Fig. 6. Frequency band gap induced by Bragg reflection. (a) Band structure using the extended zone scheme. Red for homogeneous medium (ε=2), and blue lines for PC with ε=2±0.5i. Different background colors represent different Brillouin zones with different orders. Frequency gaps open at Brillouin zone boundaries at k=2Nπ/a (N is integer). The band gaps are highlighted with blue areas in the two subgraphs. (b) Corresponding folded band structure in reduced Brillouin zone. Due to the periodic modulation in εi, two plane waves are mixed together forming the lower and upper band edge modes (standing wave modes). (c), (d) Corresponding eigen electromagnetic field in (b).

Fig. 7. 8π reflection phase winding in the fourth band gap. (a) Surface electric field Es for different truncations at the lower band edge frequency ω=1.38(2πc/a). The solid and dashed lines represent the real and imaginary parts of the electric field, respectively. (b) Corresponding surface magnetic field Hs. (c) Surface admittance ηs. There are four inflection points as Δx increases. (d) Corresponding reflection coefficients r on the complex plane. (e) Reflection phase ϕr=arg(r). As Δx increases from zero to a, reflection phase winds up by 8π. (f) Reflection phase in the entire band gap. White dashed line is the result of (e).

Fig. 8. Composite Berry curvature obtained from different eigenstates.