Observing 0D subwavelength-localized modes at ~100 THz protected by weak topology.

Topological photonic crystals (TPhCs) provide robust manipulation of light with built-in immunity to fabrication tolerances and disorder. Recently, it was shown that TPhCs based on weak topology with a dislocation inherit this robustness and further host topologically protected lower-dimensional localized modes. However, TPhCs with weak topology at optical frequencies have not been demonstrated so far. Here, we use scattering-type scanning near-field optical microscopy to verify mid-bandgap zero-dimensional light localization close to 100 THz in a TPhC with nontrivial Zak phase and an edge dislocation. We show that because of the weak topology, differently extended dislocation centers induce similarly strong light localization. The experimental results are supported by full-field simulations. Along with the underlying fundamental physics, our results lay a foundation for the application of TPhCs based on weak topology in active topological nanophotonics, and nonlinear and quantum optic integrated devices because of their strong and robust light localization.

INTRODUCTION

Topological photonic crystals (TPhCs) provide robust light control against structural disorders and could relieve the crucial fabrication requirements limiting common photonic crystals (–). The idea originates from the seminal works by Raghu and Haldane (, ), who generalized the topological band theory, discovered first in solid-state electron systems, to classical photonics based on the observation that band structure topology arising from waves in a periodic medium is regardless of the waves’ classical or quantum nature. Their discovery transferred the key feature of an electronic model—the quantum Hall effect—to the realm of photonics, and they theoretically proposed the possibility of equipping classical waves with topological protection (). With this extendedly robust protection of photons, lots of applications including robust light transmission (–), topological delay lines (), robust lasing (, ), and quantum interference (–) can be achieved, promising further development of optical devices based on the topological design theory in the future.

On the other hand, state-of-the-art nanophotonics based on optical zero-dimensional (0D) localized modes with strong light trapping is ubiquitously useful in applications such as nonlinear enhancement (), photonic device miniaturization (–), photonic or quantum chip integration (, ), and quantum electrodynamics (, ). However, the widely followed traditional strategy of using photonic crystal defect designs is notoriously fragile even to small structural changes, which lead to considerable detuning of resonance frequency and mode volume (). Hence, the implementation of robust topological protection of the 0D localized mode would boost the progress in these related fields. Recently, emerging seminal concepts with higher-order topological insulators (HOTIs) (–) and the Dirac vortex method () verified the promising routes for topological light localizations, e.g., showing several advantages in topological lasing. However, HOTIs need multiple domains of different unit cell designs to localize modes at the boundary between the domains. Another promising strategy to achieve topologically protected lower-dimensional localized modes lies in the design of weak topological insulators, which only exhibit these states for specific structural conditions (–). These weak topological insulators need only one topologically nontrivial unit cell design in addition to, e.g., an intentionally introduced dislocation, whereas at the dislocation centers, strongly localized modes (0D) can be generated (, ). Still, the design and the experimental characterization of a weak topology–protected localized mode has yet to be demonstrated at optical frequencies. First characterizations of TPhCs with a scattering-type scanning near-field optical microscope (s-SNOM) have recently been reported for corner states in HOTIs () and edge states in a valley photonic crystal (), while aperture-based near-field optical microscopy has been used to map edge modes (). However, a spectroscopic study by s-SNOM of the evolution of the eigenmodes within a TPhC over a wide spectral range including the bandgap and bulk modes has yet to be demonstrated.

Here, we simulate and measure the topologically protected localized mode at optical frequencies in a TPhC design with weak topology as illustrated in Fig. 1. When the TPhC constructed by nontrivial unit cells has a Zak phase vector along the same direction as the Burgers vector describing the edge dislocation, mid-bandgap 0D light localization is achieved, with a mode area ~0.17 λ2 confined at the intentionally induced structural dislocation center. For differently extended dislocation centers, similar localization of light can be obtained, demonstrating the versatility and robustness of our design based on weak topology. For the experimental demonstration of the subwavelength localization, amorphous silicon (a-Si) nanopillars are placed on a chromium (Cr) film. The optical near fields on top of the nanopillar arrays are measured using an s-SNOM at multiple frequencies around 100 THz. Using a broadly tunable laser source (see Methods), we map the mode evolution within the TPhC over a range of about 35 THz from the bulk states to the topologically protected mode within the bandgap. The measured electric field distributions show excellent agreement with the simulated results that are extracted from an eigenmode solver. On the basis of the 0D light localization at optical frequencies, our work proves the possibility to access and analyze such subwavelength interaction. Insights of the work open up a promising path for the application of weak topology and s-SNOM characterization in TPhCs at optical frequencies, enabling broad prospects in nonlinear optics, quantum optics, and photonic device design and characterization.

Fig. 1.

Schematic illustration of the s-SNOM measurement on the localized topological state.

With the edge dislocation formed by nontrivial unit cells, a strong field localization around the dislocation center is visualized by s-SNOM measuring directly the optical near-fields. Inset: Schematic of the dislocation point with corresponding Burgers vector B.

RESULTS

To achieve 0D localization in the crystal dislocation design based on weak topology, we first calculate the band structure of the TPhC. In our design, we consider a square lattice with a unit cell (constant period P = 1334 nm) that consists of two rectangular pillars with a refractive index of 3.49 (corresponding to the value of a-Si determined from our measurement) placed on a perfect electric conductor as substrate (Fig. 2A). Using it as substrate reduces the requirements of fabricating dielectric nanostructures with a high aspect ratio.

Fig. 2.

Band structure analysis of the TPhC.

(A) Schematic illustration of the topologically trivial unit cell (d = 0 nm) and nontrivial unit cell (d = 870 nm) made of a-Si nanopillars on a Cr film. (B) Corresponding band structure of the TPhC with P = 1334 nm, W = 232 nm, and H = 673 nm for both trivial/nontrivial unit cells (the size parameters are same in the whole text). The light cone is shaded in gray. (C) Evolution of the full bandgap size [shadowed with orange in (B)] and Zak phase of the lower band for different d between the nanopillars. (D and E) Evolution of the two lowest bulk states at the X or Y point with different d. (F and G) Band structure of the supercells (with inset images showing the adjoining part) to verify the edge state, respectively. The red dotted line corresponds to the edge mode, and the light blue shaded area corresponds to the bulk modes. The edge state is present with initial nontrivial/trivial unit cells (F), while no edge state exists in the rotated nontrivial unit cell case (G). Inset images in (B), (D), and (E) show the z component of electric fields (E) for the first band at different k points.

The TPhC can be regarded as photonic realization of the 2D Su-Schrieffer-Heeger model, in which intracell and internal distances between two neighboring pillars correspond to the hopping amplitudes (, –). The topologically trivial (d = 0 nm) and nontrivial (d = 870 nm) unit cells here have the same band structure with a full bandgap (94.4 to 101.6 THz) as shown in Fig. 2B. By considering the first bulk band, their topology is distinguished by a 2D Zak phase (, )where = i⟨u_∣∂∣u_⟩ is the Berry connection, ∣u_⟩ is the periodic Bloch function, and it is integrated over the first Brillouin zone. The Zak phase is a Berry phase of a system acquired by the periodic Bloch function going a closed path in the 1D Brillouin zone, and the 2D case here is a generalization to both directions (k and k). The parity time symmetry of the unit cell requires the Zak phase to be only 0 or π (). Equivalently, the electric field profiles’ parity at the high symmetry points can indicate the Zak phase (, , ) by examining , in which is the mirror operator, and m is the mirror eigenvalue at mirror-symmetric Bloch moments. Since the fundamental bulk band at the Γ point is mirror symmetric in the static limit, the Zak phase for θΓ (equal to θ) is π only when m = −1 at the X point (mirror antisymmetric), and the same applies for θΓ (equal to θ). Hence, the differences in the electric field distribution at the X and Y points show that the nontrivial π Zak phase only emerges for θΓ (inset image of Fig. 2B). The result is identical to the numerical calculations by the Wilson loop approach in which complex electromagnetic fields obtained from simulations substitute the eigenstates in Eq. 1 (details in section S1). By dividing the closed loop in the Brillouin zone into a larger number of small segments, the Wilson loop approach provides a gauge invariant strategy to calculate topological invariants from numerical simulation, and the Zak phase is the summation of the Berry phase in all of the small segments. The full bandgap size together with the topological phase transition through changing d is further illustrated in Fig. 2 (C to E), with d = 435 nm as the topological transition point where the bandgap closes and reopens. Note that the parity inversion only occurs at the X point and is not observed at the Y point.

We check the bulk-edge correspondence—which is used to determine the presence of edge states at an abrupt interface—as shown in Fig. 2 (F and G), by studying a single supercell with periodic boundary conditions only in the y direction. The supercell is formed by arranging seven (rotated) nontrivial unit cells followed by seven trivial unit cells along the x direction (the inset images show the adjoining section). Here, we find a unidirectional transport mode within the bandgap when the a-Si nanopillars in the trivial and not-rotated nontrivial unit cells are arranged as shown in the inset in Fig. 2F, corresponding to the nontrivial Zak phase θΓ = π. In contrast, no edge state is present if we rotate the nontrivial unit cell by 90°, as the Zak phase θΓ = 0 in such a design.

Next, we introduce a structural defect in a TPhC consisting of topologically nontrivial unit cells (with 26 ∙ 21 uniformly arranged unit cells) as shown in Fig. 3A: First, we partly remove the centerline of the unit cells (from the one shadowed in red, which sits at the 11th line in the y direction and the 10th unit cell in the x direction) in the uniform part of the TPhC. Then, all the units above and below the removed section are shifted linearly along the y direction toward the centerline (see red arrows) to compensate for the removal. This procedure creates an edge dislocation with a single pillar in the middle as the dislocation center. A closed loop, as shown by black arrows, results in a Burgers vector B = (0, P) (the light blue arrow) along the y direction in real space, which can be used to describe this dislocation. Last, the dislocation center is further extended with N more unit cells (here, N = 4, the number of unit cell from the one shadowed in red) as shown in the direction of the dashed arrow to reduce the mode scattering around the pillar.

Fig. 3.

Comparison of three kinds of TPhCs.

(A) Illustration of the defect formation with the nontrivial design as an example, with θ = (0, π/P) and B = (0, P) shown by the two solid arrows. The uniform area (corresponding to the uniformly arranged red shadowed unit cell in each design) is shadowed with green. The dark blue dashed arrow indicates the extension of the dislocation center with N = 4 more unit cells. (B) E field distribution obtained from the eigenmode calculation of the marked in-gap localized mode in the spectrum of the nontrivial design. The dotted purple line marks the area for the magnified view shown as inset. (C) Spectrum of the nontrivial design with the localized mode in the bandgap. (D) Layout at the dislocation center and spectrum of trivial 1 design with θ = (0, 0). (E) Layout at the dislocation center and spectrum of trivial 2 design with θ = (π/P, 0). The unit cell in the red box (A, D, and E) shows the starting position of the dislocation center extension for each design. The bandgap region obtained from the band structure of the unit cells is shadowed in orange.

Here, the weak topological invariant Q (which equals the number of localized modes) contains both parameters from the band structure topology and crystal dislocation ()where represents the Zak phase information of the band below the gap along the XM and YM line, respectively, in the Brillouin zone; B describes the crystal edge dislocation in real space. Q can only be 0 (no localized mode) or 1 (with a localized mode). According to this weak topological invariant Q, the case when the nonzero Zak phase vector θ and the Burgers vector B are parallel to each other can result in the topologically protected localized mode. This topological concept was termed “weak” to distinguish it from the well-known strong topological insulators (where “strong” is always omitted), which are solely characterized by a quantized scalar topological invariant (, ).

To verify that the localization only appears in the nontrivial design based on weak topology, we study the spectra and electric field profiles of three kinds of TPhCs. On the basis of the 2D Zak phase or E field distribution in Fig. 2B (with the field distribution at the M point the same as at the X point), the nontrivial unit cell gives rise to θ = π, while θ = 0 as for the parity inversion difference; and both θ and θ are equal to 0 for the trivial unit cell. This results in the three TPhCs: nontrivial (Fig. 3, A and B), constructed by the initial nontrivial unit cell with θ parallel to B, and θ = (0, π/P); trivial 1 (Fig. 3D), constructed by the trivial unit cell with θ = (0, 0); and trivial 2 (Fig. 3E), constructed by the 90° rotated nontrivial unit cell with θ = (π/P, 0) perpendicular to B. Note here that the design strategy for these two trivial TPhCs is similar to that of the nontrivial design, explained above regarding Fig. 3A. Coinciding with our previous assumptions, a localized mode (f = 96.7 THz) in the bandgap (94.4 to 101.6 THz) around the dislocation center is only found for the nontrivial design as shown in Fig. 3 (B and C). In contrast, for the two comparative trivial designs (trivial 1 and trivial 2), no localized state inside the bandgap is found in the spectra (Fig. 3, D and E), verifying our design strategy to realize the in-bandgap–localized mode with the nontrivial Zak phase and edge dislocation. Furthermore, other trivial designs with B = (P, P) are also consistent with our assumption, as shown in section S2. The additional requirement of a certain lattice arrangement in addition to the nontrivial Zak phase indicates that the localization is protected by weak topology.

The electric field profile of the designed localized mode in Fig. 3B suggests a mode area of ~0.25 λ2 (λ is the wavelength in vacuum) around the dislocation, which indicates a more strongly confined mode compared to other topological designs like HOTIs and the Dirac vortex method (, , ). Here, the mode area of the localization is calculated with the formulawhere ∣E(r)∣ is the z component of the electric field, and the integration is performed on top of the structure (in the plane at z = 680 nm).

To realize the designed TPhC above, we fabricated a-Si pillars on a glass substrate coated with a 100-nm-thick Cr film as shown in the scanning electron microscopy image in Fig. 4A. The target unit cell dimension corresponds to the simulated size (lattice constant, P = 1334 nm; height, H = 673 nm; and size of each single pillar, W ∙ 2W = 232 nm ∙ 464 nm). In Fig. 4B of the related spectrum, the four investigated frequencies are indicated by the red markers. E fields (in the plane at z = 680 nm) from the eigenmode solver corresponding to the near-field distribution at each frequency are shown below the experimental s-SNOM results in Fig. 4C (for explanation, see Methods). A strongly confined field localized around the single pillar at the dislocation center is measured for the frequency of the state in the bandgap (f = 96.7 THz), whereas the other frequencies show a weaker localization and broader field distributions. The simulations reproduce the measurement results well (further comparison of electric fields can be found in sections S3 and S4). We also observe the two eigenmodes at the band edge (f = 93.9 and 102.0 THz), where a slight localization around the center pillar can be seen. At 100 THz, where no eigenstate is expected within the bandgap, the measured signal on the pillars is comparable to the center pillar. The light is scattered nonresonantly by the array. That is, the pillars at the center appear slightly uniformly brighter, which can be explained by the spectral resolution of ~1.5 THz of the tunable laser source. Therefore, the eigenstates within this spectral range around 100 THz, especially at the band edge (f = 102.0 THz) and in the bandgap (f = 96.7 THz), can also be slightly excited. This overlap is reflected in the measured signals, like the middle pillar in Fig. 4C, giving rise to bright regions in the measurements even though the actual field values might be relatively small (see simulations).

Fig. 4.

Frequency-dependent near-field results.

(A) Scanning electron microscopy image of the nontrivial TPhC (inset: magnified view of the dislocation center). (B) Corresponding spectrum with the measured frequencies marked with red markers. (C) Normalized s-SNOM amplitude images to show the measured electric field distribution (top) and the simulated E fields (absolute value) obtained from the eigenmode solver (bottom) for each frequency. The subwavelength localization with strong fields around the single pillar at the dislocation center is observed in the bandgap at f = 96.7 THz, while the fields for other frequencies show weaker localization and broader distributions. Scale bars, 4 μm.

The spatial localization of the mid-bandgap eigenmode can be visualized by simulating the near-field distribution of the TPhC with excitation by an obliquely incident plane wave. The calculations show a near-field enhancement of more than 30 times above the pillars (z = 680 nm), and the electric field distribution agrees well with the s-SNOM measurements (for more details, see section S3). On the basis of these results, the designed mid-bandgap localization and bulk states at optical frequencies could be demonstrated and verified by s-SNOM measurements, making it a promising tool to characterize TPhCs at optical frequencies (more details in section S4) ().

To further demonstrate the versatility of our weak topology design to achieve the 0D localized state, we consider several design layouts based on the nontrivial unit cell, but with a different number N of unit cells extending the dislocation center (N = 3, 5, and 6 for the three layouts here; see illustration of N in Fig. 3A) as shown in Fig. 5. The calculated spectra (left column) still show a localized mode inside the bandgap as marked by the red dot, and similar field confinement around the dislocation center can be observed (center column). The smallest mode area of ~0.17 λ2 (see Eq. 3) is achieved for the design with N = 6 because of the scattering suppression with a smaller vacant area in the array (largest mode area ~0.36 λ2 for N = 3). To verify the simulation and to show the variation of the mode localization in the different layouts, the arrays were fabricated as before and measured with s-SNOM at the marked frequencies of the localized eigenstate inside the bandgap. As shown in the normalized field distributions (right column in Fig. 5), a strong decrease in the relative field amplitude on the surrounding pillars with respect to the single pillar can be observed with increasing N. The spectra and localized field distributions together show the prominent advantages of the weak topology design for tuning of the 0D localization properties, i.e., frequency, relative localization position, and mode area.

Fig. 5.

Versatility of the weak topological design.

Demonstration of the versatility of the design to achieve the 0D localized mode with three different unit cell extensions at the dislocation center for (A) N = 3, (B) N = 5, and (C) N = 6, respectively. The left column shows the spectra, middle column shows the simulated field distribution from the eigenmode solver, and right column shows the normalized s-SNOM field distribution. Simulated and experimental field distributions show similar localization at the mid-bandgap (red dot in each spectrum) around the dislocation center. The localization becomes stronger with larger N. The dotted lines mark the area of the magnified inset in the simulation images, corresponding to the section shown in the measurements. Scale bars, 4 μm.

In stark contrast to the nontrivial designs, no topologically localized mode can be found in all the trivial TPhCs as shown in the Supplementary Materials (section S5). Only topological modes can appear and follow the topological phase transition. The spectra and field distribution comparison between the corresponding trivial and nontrivial TPhCs indicates that the modes that only exist in the nontrivial design are topological modes. The versatility of the weak topological design provides a flexible strategy to access and further adjust the localization of light fields. Together with the robust nature of topological protection (see robustness of the mode in section S6), it is highly promising for future photonic device design.

DISCUSSION

The weak topology applied in our work enables a mid-bandgap 0D localized mode, which is present when the vector describing the nontrivial Zak phase is parallel to the Burgers vector of the edge dislocation according to the defined weak topological invariant. Furthermore, the designed optical localization with a subwavelength confined area is experimentally verified by s-SNOM measurements of a TPhC made of a-Si nanopillars placed on a layer of Cr. On the basis of the versatility of the weak topology, similar in-bandgap 0D localizations with a confined field at the intentionally induced structural dislocation center are designed with different dislocation extensions. With a larger number of unit cell extensions at the dislocation center, a reduced mode area is observed in the measurements.

Confining light at optical frequencies to a point (0D) is promising for applications in nonlinear optics, quantum optics, and miniature active photonic device integration. Our study demonstrates a feasible strategy for accessing and tuning a topologically protected 0D localized state at infrared frequencies. We also show that s-SNOM is a valuable tool for characterizing TPhCs with nanoscale resolution at optical frequencies. Going further, programmable phase-change materials (–) with tunable refractive index could be included into TPhCs to realize robust and active topological nanophotonics. Furthermore, the rapid progress of nanotechnology makes it possible to equip the defect center with quantum emitters, nanocrystals, or molecules, which is highly promising for hybrid nanophotonic devices for single photon generation.

METHODS

TPhC fabrication

First, a thin Cr film with a thickness of 100 nm was deposited on a cleaned glass substrate by electron beam evaporation. An a-Si layer of thickness 673 nm was deposited by plasma-enhanced chemical vapor deposition, immediately on top of the Cr film. The sample with deposited a-Si on top of the Cr film was then used to perform the subsequent processes to transfer the desired patterns onto a-Si. A polymethyl methacrylate (PMMA) resist layer was spin coated onto the a-Si film and baked on a hot plate at 170°C for 2 min to remove the solvent. The desired structures were patterned by using a standard electron beam lithography and developed in 1:3 methyl isobutyl ketone:isopropyl alcohol solution. Next, another 20-nm-thick Cr mask was deposited by electron beam evaporation. After a liftoff process in hot acetone, the patterns were transferred from PMMA to Cr. Last, the structures were transferred onto a-Si using an inductively coupled plasma reactive ion etching and the following removal of the Cr mask by a commercially purchased Cr etch solution.

Simulation

The band structure and edge state were calculated by using a commercially available COMSOL Multiphysics software with the radiofrequency module using the eigenmode solver. The perfect electric conductor boundary condition was used for the bottom, and the periodic boundary condition was used for each pair of the unit cells or supercells. The spectra and corresponding near-field distributions were simulated using the eigenmode solver in the same way, but all with scattering boundary conditions outside the structure. For the refractive index of the a-Si, we use 3.49 obtained from ellipsometric measured data in the range of 600 to 1500 nm fitted with a Lorentz model.

Near-field measurement

s-SNOM records the scattering amplitude and phase of the optical near-fields at the atomic force microscopy (AFM) tip at certain laser frequency by lock-in detection (). The signal is then demodulated at higher harmonics of the tip’s oscillation frequency.

Here, we deployed an s-SNOM setup by Neaspec GmbH in pseudoheterodyne detection mode in combination with an LN2 cooled InSb detector optimized for the wavelength range up to 5.4 μm (Infrared Associates) to record amplitude and phase data simultaneously. The s-SNOM AFM was operated in tapping mode at oscillation frequencies between 220 and 270 kHz, with commercially available metal-coated AFM probes. The tapping amplitude is 40 to 60 nm. The laser source is a commercially available Optical Parametric Oscillator/Amplifier (OPO/OPA) pulsed laser (Alpha Module by SI Instruments) tunable between 65 and 217 THz. The laser system has a repetition rate of 42 MHz and a pulse duration of up to 1 ps. This provides a spectral resolution of 1.5 THz and allows for recording images in s-SNOM in pseudoheterodyne detection mode around 100 THz (~3 μm) and above as recently shown (). We extracted the signal from the second demodulation order amplitude. The spectral resolution of the laser leads to a smearing of the states compared to the simulations that work at a single frequency. Thus, not every eigenstate is perfectly localized as expected from simulations.

Evaluation of s-SNOM measurements

Between the pillars, the AFM probe picks up the strong background signal of the Cr substrate; thus, the substrate area is not used for evaluation and set to zero. Instead, the second-order normalized amplitude images in Figs. 4 and 5 were obtained by evaluating the amplitude signal on top of the pillars. The top pillar position was determined by setting a threshold value in AFM images and overlayed with the corresponding optical images, setting the surrounding to zero. Afterward, the signal was averaged over two neighboring pixels with a Gaussian filter to smooth the pixels and normalized to the signal on top of the middle pillar (dislocation center with a single pillar) to highlight the field distribution. The step-by-step data analysis is shown in section S7.