Fig. 1. Corner mode and its gapless dispersion in synthetic translation dimensions. (a) Schematic illustration of the corner between STPC and OPC. Parameters: lattice constant a=18  mm; the side length of square-rod of OPC (blue region) is 7 mm, while that of STPC (orange region) is 3.5 mm; in-plane distances between two neighboring rods in STPC are dx=dy=14.5  mm; relative permittivity of rods εr=5.9. (b) Calculated eigenfrequencies of configuration in (a). The corner, edge, and bulk modes are denoted by red, blue, and gray points, respectively. Inset: simulated |Ez|2 field of the corner mode (outlined by a green circle). (c) Schematic illustration of the corner between the translated OPC and the untranslated OPC. All square-rods of the translated OPC are translated away from the center of the unit cell by (Δx, Δy). (d) The gapless dispersion of corner modes of configuration in (c) when Δx=Δy. Corner modes outlined by two green circles at Δx=Δy=−9 and 9 mm are the same as the corner mode in (b). (e) Schematic illustration of the corner between the translated OPC and PEC to confirm the robustness of the existence of gapless corner modes. (f) The gapless dispersion of corner modes of configuration in (e) when Δx=Δy.

Fig. 2. Realization of gapless corner modes based on PC slab. (a) Schematic illustration of the corner between the PC slab and two metallic bars. Both the PC slab and metallic bars are placed on a metallic substrate. Parameters: lattice constant a=18  mm; in-plane side length of dielectric rods b=7  mm; height of dielectric rods h=13.5  mm; and relative permittivity of dielectric rods εr=9. (b) Calculated eigenfrequencies for the sample with (Δx,Δy)=(−2  mm,−4  mm). The cyan region represents the bulk band gap of PC slab. The eigen modes whose |Ez|2 fields will be shown in (c) are outlined by circles. (c) |Ez|2 fields at z=0  mm for bulk, edge, and corner modes in (b). (d)–(f) Calculated eigenfrequencies and |Ez|2 fields of the corner modes for the sample with (Δx,Δy)=(−3  mm,−3  mm), (−4  mm,−4  mm), and (−5  mm,−5  mm).

Fig. 3. Observation of eigen modes of the corner between the PC slab and two metallic bars. (a) Left panel: photograph of the fabricated experimental sample. Right panel: photograph of the corner of the sample. (b) Experimental |Ez|2 fields at z=15  mm of four eigen modes for the sample with (Δx,Δy)=(−2  mm,−4  mm), representing the bulk mode, the edge mode, and the corner mode. (c) Simulated |Ez|2 fields of four representative eigen modes, which are consistent with the experimental results.

Fig. 4. Observation of gapless corner modes within the translation dimensions. (a) Response spectrum for the sample with (Δx,Δy),=(−3  mm,−3  mm). The red line is the simulated result, and the blue is the experimental result. The cyan region represents band gap. The response intensity is defined as |Ez|corner2/|Ez|source2, whose peak indicates the excitation of corner mode. For the sample with (Δx,Δy)=(−3  mm,−3  mm), the central frequencies of corner modes are 6.70 GHz (simulation) and 6.71 GHz (experiment). Inset: illustration of two positions at which electric field is used to determine the response intensity. (b), (c) Measured and simulated |Ez|2 fields of the excited corner mode for the sample with (Δx,Δy)=(−3  mm,−3  mm). (d)–(f) and (g)–(i) are similar to (a)–(c), while (d)–(f) correspond to the sample with (Δx,Δy)=(−4  mm,−4  mm) and (g)–(i) correspond to the sample with (Δx,Δy)=(−5  mm,−5  mm).

Fig. 5. Realization of photonic topological rainbow. (a) Schematic illustration of topological rainbow. (b) Calculated |Ez|2 fields at z=0  mm for different frequencies.

Fig. 6. Schematic illustration of the translation of dielectric rods.

Fig. 7. Comparison of corner mode dispersions obtained by varying (dx, dy) or (Δx, Δy). (a) Schematic illustration of the corner between the STPC and OPC. Here, dx and dy are varied as parameters, no longer fixed at dx=dy=14.5  mm. (b) Schematic illustration of the unit cells of PC as (dx, dy) changes from (3.5 mm, 3.5 mm) to (14.5 mm, 14.5 mm). (c) Non-gapless dispersion of corner modes of configuration in (a) when dx=dy. (d) Schematic illustration of the corner between the translated OPC and the untranslated OPC [same as Fig. 1(c) in the main text]. (e) Schematic illustration of the unit cells of translated OPC as (Δx, Δy) changes from (−9  mm, −9  mm) to (9 mm, 9 mm). (f) Gapless dispersion of corner modes of configuration in (d) when Δx=Δy [same as Fig. 1(d) in the main text].

Fig. 8. Frequency diagrams of the corner modes in (Δx, Δy) space. (a) Top panel, schematic illustration of the corner between the translated OPC and the untranslated OPC [same as Fig. 1(c)]. Bottom panel, frequency diagram of corner modes in (Δx, Δy) space under the OPC boundary condition. (b) Top panel, schematic illustration of the corner between the translated OPC and PEC [same as Fig. 1(e) in the main text]. Bottom panel, frequency diagram of corner modes in (Δx, Δy) space under the PEC boundary condition.

Fig. 9. Corner mode dispersion with inserted air layers of different thicknesses. (a) Schematic illustration of the corner with an inserted air layer between the translated OPC and PEC. The thickness of the air layer is denoted by dair. (b) The gapless dispersion of corner modes with dair=0  mm. (c) The gapless dispersion of corner modes with dair=9  mm. (d) The gapless dispersion of corner modes with dair=18  mm.

Fig. 10. Comparison of band structures and eigen fields of the 2D PC and 2D PC slab. (a) The band structure and eigen fields (the first band) of 2D PC, which is discussed in Fig. 1 in the main text. (b) The band structure and eigen fields (the first band) of 2D PC slab, which is applied in Figs. 2–5 in the main text.

Fig. 11. Dispersion of x edge mode and y edge mode for the PC slab with (Δx,Δy)=(−2  mm,−4  mm). (a) 2D schematic illustrations of the edge between a metallic bar and the PC slab perpendicular to x and y axes, respectively. (b) The x edge mode and y edge mode dispersion.

Fig. 12. Evolution of corner mode when (Δx, Δy) changes.