Topological photonic crystals (PCs) have emerged as a rapidly growing field of research in the past few years, holding great potential for revolutionizing the way we mold the flow of light [1–9]. A key characteristic of topological PCs is the presence of topologically protected boundary modes localized at the crystal’s boundary. According to the conventional bulk-edge correspondence, these topological boundary modes are generally one dimension lower than the topological systems themselves. For instance, one-dimensional topological systems host zero-dimensional topological modes [10–12], while two-dimensional topological systems host one-dimensional topological edge modes [13–15]. These topological boundary modes exhibit robustness against various defects and perturbations, rendering them highly resilient. However, certain perturbations targeting their topological protection mechanisms can compromise their robustness. Strong symmetry-breaking or large-scale structural deformation is examples of such perturbations that can disrupt the robustness of the topological modes. For instance, edge modes of valley photonic crystals are resilient against perturbations that do not mix the two valleys [16,17]. On the contrary, the sidewall roughness of valley photonic crystals can introduce backscattering and impact the robustness of topological edge modes [18].

Recent advancements have unveiled a novel class of topological phases known as higher-order topological phases, which deviate the traditional bulk-edge correspondence [19–23]. In contrast to conventional topological materials, the boundary modes of higher-order topological materials are more than one dimension lower than the bulk. For example, two-dimensional second-order topological photonic crystals (STPCs) can host zero-dimensional corner modes [24–36]. These corner modes are localized at the intersection of two lower-dimensional boundaries within the STPCs and hold significant promise for various applications, including on-chip cavities [37–39], advanced lasers [40,41], and nonlinear optics [42–44]. However, in realistic PC cavities, there are some perturbations causing the frequency shifts of the corner modes. For example, the translation of rods or holes in PCs is a common fabrication error during the fabrication and also induces frequency shifts of the corner modes [41,45]. Although the translation dependent frequency shift is hard to predict, a recent theoretical proposal has shown that frequency shift of corner modes spanning the entire band gap (i.e., gapless corner modes) can be achieved by considering the translation as an additional synthetic dimension [46]. Note that the introduction of synthetic dimensions [47–49] in photonic systems enriches the exploration of topological phase of light in higher-dimensional space beyond three-dimensional real-space [50–56]. The topology in synthetic translation space ensures the gapless modes, e.g., the gapless boundary modes from two-dimensional synthetic space [57], and gapless corner and dislocation modes from four-dimensional synthetic space [46]. These gapless modes facilitate the topological rainbow to separate different topological modes into distinct locations [58,59]. However, the gapless corner modes in Ref. [46] are based on ideal two-dimensional PCs without direct experimental evidence. Here, we observe the gapless frequency shift of corner modes in the PC slab. The experimental paradigm developed for the PC slab is adaptable to optical frequencies, offering promising avenues for frequency-tunable nanocavity. As a practical application, we propose a topological rainbow device with a PC slab, which can configure the corner modes of different frequencies at different locations.

In this work, the PC slab consists of periodic ceramic square-rods placed on a metallic substrate. By introducing the translation (Δx, Δy) into the PC slab, we can tune the frequency of the corner mode, thus demonstrating the existence of the gapless corner mode in synthetic translation dimensions. To validate our findings, we employ the near-field scanning measurements to experimentally image the localized corner modes. We systematically explore and image corner modes in PC slabs under different translational deformations, confirming the frequency shift of the corner mode. All experimental results are further supported by full-wave simulations, ensuring the reliability of our findings. Our findings demonstrate the potential for the flexible frequency modulation of corner modes in synthetic translation dimensions.

To illustrate the presence of corner modes and their gapless dispersion in synthetic translation dimensions, we first consider the corner formed between a two-dimensional (2D) STPC and an ordinary photonic crystal (OPC) [Fig. 1(a)]. The unit cell of the STPC consists of four square-rods with a side length of 3.5 mm. The distances between two neighboring rods in the in-plane directions are dx=dy=14.5mm. In contrast, the unit cell of the OPC consists of a square-rod with a side length of 7 mm. The lattice constant of both PCs is a=18mm, and the relative permittivity of the rod is 5.9. The topology of these two PCs is given by the 2D Zak phase Z=(Zx,Zy) [25,26]: Zj=∫dkxdkyTr[A^(k)],(1)where j=x or y and A^j(k)=i⟨u(k)|∂kj|u(k)⟩, with |u(k)⟩ being the periodic Bloch function. The 2D Zak phases of the STPC and OPC are (π, π) and (0, 0), respectively, enabling the localized corner modes when the STPC and OPC are placed together to form a corner. The eigenfrequencies of this corner configuration are calculated and presented in Fig. 1(b). The corner mode is denoted by the red point, while the edge and bulk modes are denoted by blue and gray points, respectively. The inset of Fig. 1(b) shows the |Ez|2 field of the corner mode, demonstrating strong field confinement at the corner.

The unit cell of the OPC can also be considered to consist of four square-rods with a side length of 3.5 mm and the in-plane distances of dx=dy=3.5mm. As a result, the unit cell of the STPC can be obtained through the unit cell of the OPC by changing the in-plane distances of dx and dy from 3.5 mm to 13.5 mm [Fig. 7(b) in Appendix A]. Alternatively, the STPC’s unit cell can be obtained by translating the dielectric rod within the OPC along two directions, i.e., Δx and Δy. Note that, when the dielectric rod traverses the unit cell during translation, the patch exiting one boundary re-enters through the corresponding periodic boundary, ensuring the integrity of the entire bulk crystal remains unchanged (Fig. 6 in Appendix A). As the translations Δx and Δy are cyclic and restricted to [−a/2, a/2], they can be considered as two additional Bloch momenta. Combining two original momenta of kx and ky, we can define a four-dimensional synthetic (kx, Δx, ky, Δy) space [46]. In contrast to the 2D Zak phase defined within the two-dimensional (kx, ky) space, the topological invariant within the four-dimensional (kx, Δx, ky, Δy) space is the second Chern number [60]: C(2)=132π2∫εlmnoBlm(kx,Δx,ky,Δy)⁢Bno(kx,Δx,ky,Δy)dkxdΔxdkydΔy,(2)where εlmno is the Levi-Civita symbol and each index l, m, n, o takes values of kx, Δx, ky, or Δy. The second Chern number is nonzero and quantized to be 1, implying the gapless corner modes in synthetic translation dimensions by the bulk-edge correspondence. To see this, we replace the STPC with a translated OPC whose square-rods are all translated away from the center of the unit cell by (Δx, Δy) [Fig. 1(c)]. For simplicity, we consider the case of Δx=Δy and calculate the eigenfrequencies of corner modes with different Δx values [denoted by red in Fig. 1(d)]. As Δx varies from −9 to 9 mm, the corner modes bifurcate from the bulk band edge and traverse the entire photonic band gap. It is noteworthy that the corner modes highlighted by two green circles at Δx=Δy=−9 and 9 mm are the same as the corner mode presented in Fig. 1(b). This indicates that the corner mode in the STPC is a lower-dimensional manifestation of gapless corner modes under translational deformation. Note that the term “gapless” used here is related to the synthetic translation dimensions, not the momentum dimension k. The slope of corner mode dispersion is negative, but it only represents the frequency dependence of the corner mode frequency and the translation (Δx, Δy). In fact, corner modes with tunable frequencies can also be obtained by changing dx and dy [Fig. 7(c) in Appendix A]. However, the dispersion under variable dx and dy is limited to a small frequency range. We notice that another type of translation, quantified by the parameter g, has been introduced to the STPC to enhance the quality factor of the corner mode [41,45]. Here, g represents the gap distance between the trivial and nontrivial parts of the PC slab, which can be tuned for creating high-quality topological nanocavities. However, such translation also fails to ensure the gapless frequency shift of corner mode without altering the shape of air holes (or dielectric rods) of STPC. The existence of gapless corner modes is robust and determined by the topological pumping in synthetic translation dimensions [61], which is independent of specific boundary conditions. To further illustrate this point, we replace the OPC with a trivial perfect electric conductor (PEC) [Fig. 1(e)] and calculate its corner mode dispersion as a function of translation [Fig. 1(f)]. Remarkably, gapless corner modes that traverse the entire photonic band gap are still supported when Δx=Δy (red curve), and the complete frequency diagram of the corner modes in (Δx, Δy) space is shown in Fig. 8 in Appendix A. When (Δx, Δy) departs from (0 mm, 0 mm), the corner mode’s frequency keeps increasing, and its eigen fields are first extended, then localized, and finally are extended again, showing a crucial characteristic of topological pumping. The frequency shift of corner mode is a boundary effect, but the gapless frequency shift is a topological effect. To clarify this point, we investigate the evolution of the gapless corner modes by adding air layers of different thicknesses weakening the boundary effect (Fig. 9 in Appendix A).

To observe gapless corner modes, we implement the above idea in the PC slab that utilizes both the in-plane periodicity and out-of-plane total reflection to confine light in three dimensions [Fig. 2(a)]. The unit cell of the PC slab consists of a dielectric square-rod with the relative permittivity of εr=9 placed on a metallic substrate. The in-plane lattice constant is a=18mm, and the side length and height of dielectric rods are b=7mm and h=13.5mm, respectively. When the relative permittivity is configured from 5.9 to 9, the bulk bands and eigen fields of this PC slab resemble those of the 2D PC presented in Fig. 1 (Fig. 10 in Appendix B), indicating the potential existence of gapless corner modes in the PC slab. To explore this possibility, we construct the “corner” by placing two metallic bars along the x and y directions next to the PC slab. The “corner” wrapped by the OPC should also has gapless corner modes and be more predictable, so we do not perform the observation. We first calculate the eigenfrequencies for the sample with (Δx,Δy)=(−2mm,−4mm) [Fig. 2(b)]. Along the frequency axis, the bulk, edge, and corner modes are denoted by gray, blue, and red points, respectively. The frequency range of the band gap is outlined by a cyan rectangle. The |Ez|2 fields at z=0mm for five representative bulk, edge, and corner modes are shown in Fig. 2(c). These fields show the frequency-dependent mode evolution. The bulk modes extend throughout the crystal, the edge modes propagate along x (or y) direction while being confined along the other direction, and the in-gap corner mode with a frequency of 6.66 GHz is localized in both directions. To achieve gapless corner modes, we translate the PC slab with respect to two static metallic bars. For simplicity, we focus on the representative case of Δx=Δy. Figures 2(d)–2(f) show the eigenfrequencies and |Ez|2 fields of corner modes for the samples with (Δx,Δy)=(−3mm,−3mm), (Δx,Δy)=(−4mm,−4mm), and (Δx,Δy)=(−5mm,−5mm). In these three samples, corner modes are found within the bulk band gap and localized around the corner. In addition, the frequencies of corner modes increase from 6.67 GHz to 6.88 GHz to 7.07 GHz as Δx decreases, showcasing the manifestation of gapless corner modes in PC slabs.

The PC slab is free of the metallic cover and in favor for the observation of eigen modes. To observe the corner modes, we implement the experiment with the microwave near-field scanning platform [Fig. 3(a)]. The setup includes a source antenna and a probe antenna connected to a vector network analyzer via a coaxial-cable. The probe antenna, mounted on an automatic stepper motor, collects signals at z=15mm, which are sent back to the vector network analyzer and a computer for imaging the eigen modes. The sample consists of periodic ceramic rods and two metallic bars, which are placed on the metallic substrate. The corner of the sample is focused, and the source antenna is positioned at the center of four rods closest to the corner. We first consider the sample with (Δx,Δy)=(−2mm,−4mm) to demonstrate the capability of imaging the eigen modes. The experimental |Ez|2 fields of four representative eigen modes are shown in Fig. 3(b). At the frequency of 6.32 GHz, the bulk mode exhibits an extended field within the bulk PC slab. At 6.51 GHz, the y edge mode shows a confined field localized around the y edge, i.e., the boundary perpendicular to the y-axis. At 6.40 GHz, the mixed x&y edge mode exhibits an extended field along both the x and y edges. This is induced by the field overlapping between the x edge mode and the y edge mode at the same frequency (see details in Fig. 11 in Appendix C). Importantly, at 6.71 GHz, a strongly confined field localized at the corner is observed, confirming the existence of corner mode. To validate these experimental results, we also perform the full three-dimensional numerical simulation under the same excitation settings [Fig. 3(c)]. The simulated |Ez|2 fields closely reproduce the experimental observations, providing further confirmation of the validity of the experimental results.

To further demonstrate the presence of gapless corner modes in synthetic translation dimensions, we perform measurements on samples with Δx=Δy. Since the source antenna is placed near the corner, there are field enhancements at frequencies where no eigen modes are expected. In order to accurately identify the resonantly excited corner modes, we define the response intensity R=|Ez|corner2/|Ez|source2, representing the ratio of the field intensity at the corner to that at the source. The positions of the corner and source are illustrated in the inset of Fig. 4(a). The response intensity reaches its maximum value when the corner mode is excited, and the peak of the response spectrum confirms the existence of the corner mode. Figure 4(a) shows the response intensity spectrum for the sample with (Δx,Δy)=(−3mm,−3mm), where the red line shows the simulated result and the blue shows the experimental result. The experimental and simulated frequencies of corner modes are 6.71 GHz and 6.70 GHz, respectively, demonstrating consistency between these two results. The measured and simulated |Ez|2 fields of the excited corner modes for this sample are shown in Figs. 4(b) and 4(c), respectively, revealing a strongly localized field around the corner. Similarly, the response spectra, as well as the measured and simulated |Ez|2 fields, are presented for the sample with (Δx,Δy)=(−4mm,−4mm) in Figs. 4(d)–4(f), and for the sample with (Δx,Δy)=(−5mm,−5mm) in Figs. 4(g)–4(i). The frequencies of these corner modes are almost the same (with an error less than 0.8%), and the |Ez|2 fields of these corner modes exhibit negligible differences, confirming the high consistency between the experimental and simulated results. By summarizing the results in Figs. 4(a), 4(d), and 4(g), we observe that the frequencies of corner modes increase from 6.71 GHz to 6.89 GHz and 7.17 GHz as (Δx, Δy) changes from (−3mm, −3mm) to (−4mm, −4mm) and (−5mm, −5mm). When (Δx,Δy)=(0mm,0mm) and (−5.5mm, −5.5mm), we observe the disappearance of the corner mode in the band gap (Fig. 12 in Appendix D). This provides clear evidence of the observation of gapless corner modes in synthetic translation dimensions.

Gapless corner modes can be employed to spatially separate topological corner modes. A recently published research work reported an acoustical topological rainbow by employing the same idea of translational deformation [59]. Similarly, we propose a photonic topological rainbow with gradient translation, taking advantage of the strong localization and gapless nature of corner modes in the PC slab. As shown in Fig. 5(a), we present a schematic illustration of the topological rainbow, which is divided into six gradient regions (I, II, III, IV, V, VI) and one non-gradient region (VII). Region VII serves as a barrier region formed by OPC to provide a band gap preventing the leakage of light. In region I, there are a total of 5×5 dielectric rods. The dielectric rods in the first to fifth columns are translated by (Δx1, Δy1), (3Δx1/4, 3Δy1/4), (Δx1/2, Δy1/2), (Δx1/4, Δy1/4), and (0, 0), respectively. This gradient architecture offers two advantages: first, the bottom-left corner of region I can be approximated to form a corner with translation (Δx1, Δy1); second, the right side of region I can be considered as OPC to assist in forming the corner of the next region. Regions II, III, IV, V, VI are designed based on the similar principle. The (Δxn, Δyn) (n=1, 2, 3, 4, 5, 6) for the six gradient regions are (−6.5mm, −6.5mm), (7.5 mm, −7.5mm), (−8.3mm, −8.3mm), (−8.9mm, −8.9mm), (−9.3mm, −9.3mm), and (−9.5mm, −9.5mm), respectively. They respectively support six corner modes of different frequencies. Consequently, when light is incident from the bottom, the frequency component around the frequency of the corner mode will be trapped in the corresponding corner, as shown in Fig. 5(b). Note that the variation of (Δxn, Δyn) is not linear. Owing to the high quality factor of corner modes, we can effectively extract and separate two frequency components (7.28 and 7.34 GHz) with a frequency interval of 0.06 GHz into distinct spatial locations.

In contrast to previous works predominantly focused on the “corner mode in a single structure,” this work delves into the “corner modes in multiple structures.” Starting from the STPC, we find that topological corner modes exhibit robust gapless dispersion in synthetic translation dimensions and directly observe the gapless corner modes in PC slabs system. In dielectric photonic crystal, the breaking of chiral symmetry destroys the zero-energy character of the corner mode. Although it allows the corner mode to be embedded in bulk spectrum (manifested as BICs), the underlying mechanisms are different from the gapless corner modes. Utilizing the microwave near-field scanning platform, we map the mode evolution of the sample over a range of about 6–7 GHz from the bulk mode to the topological corner mode within the band gap. We verify the gapless characteristic of corner modes by translating rods and measuring response intensities. All measured results are in good agreement with the simulated results. The demonstrated gapless corner modes under synthetic translational deformations are universal and can be applied to other lattice structures. Our study demonstrates a feasible strategy for accessing and tuning topological corner modes at microwave frequencies. Although our implementation involved a metal-containing microwave system, it can be extended to nanophotonic systems working at optical frequencies. For example, the in-plane excitation of a topological corner mode at a telecommunications wavelength has been implemented by a cross-coupled PC cavity based on STPC [38]. Applying the same translation method to this nanophotonic structure, we can also observe the gapless corner modes at optical frequencies and further explore its applications in nonlinear optics and quantum optics.