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Lasing from complete set of topological states in two dimensional photonic crystal structure

Recently, topologically engineered photonic structures have garnered significant attention as their eigenstates may offer a new insight on photon manipulation and an unconventional route for nanophotonic devices with unprecedented functionalities and robustness. Herein, we present lasing actions at all hierarchical eigenstates that can exist in a topologically designed single two-dimensional (2D) photonic crystal (PhC) platform: 2D bulk, one-dimensional edge, and zero-dimensional corner states. In particular, multiple topological eigenstates are generated in a hierarchical manner with no bulk multipole moment. The unit cell of the topological PhC structure is a tetramer composed of four identical air holes perforated into an InGaAsP multiple-quantum-well epilayer slab. A square area of a topologically nontrivial PhC structure is surrounded by a topologically trivial counterpart, resulting in multidimensional eigenstates of one bulk, four side edges, and four corners within and at the boundaries. Spatially resolved optical excitation spontaneously results in lasing actions at all nine hierarchical topological states. Our experimental findings may provide insight into the development of sophisticated next-generation nanophotonic devices and robust integration platforms.

and was followed by experimental demonstrations in various systems, such as mechanical resonators 25 , electrical circuits 26,27 , and photonic microring arrays 28 . An alternative to form higher-order insulating TESs without bulk multipole polarisations has been suggested 24 , where a nontrivial topology induced by bulk dipole moments generates higher-order TESs in a hierarchical manner. Specifically, a 2D bulk with nonzero polarisations induces 1D edge states, which consequently generates a 0D corner state where two 1D edge states with nonzero polarisations encounter 29-34 . This type of higher-order TESs is useful when multipole polarisations are absent or difficult to realise.
In photonics, extreme light confinement using photonic crystals (PhCs) or surface plasmons is a fascinating research area that may result in strongly enhanced light-matter interactions in an ultimately small scale, which will significantly affect the development of high-density photonic integrated circuits. In this context, the recent experimental demonstration of topological 0D corner states in a 2D PhC slab structure 34 is noteworthy, although stimulated emission at the topological 0D state was not demonstratedeven at a cryogenic temperature.
In this study, we demonstrate lasing actions at all hierarchical TESs (bulk; edge; corner states) that exist in a topological 2D PhC structure. The coexistence of multiple hierarchical TESs in a topological PhC structure has been suggested theoretically 31 and proven experimentally in the microwave range 32,33 . We demonstrate lasing at multiple topological states in the optical frequency range by realising such a 2D PhC topological structure in an InGaAsP-based multiple-quantum-well epilayer using nanofabrication technology. The resultant lasing modes are not only small, but also multiplexed both spectrally and spatially, which are highly desirable and useful for applications such as high-density photonic integrated circuits or sophisticated photonic sensors.

Topological 2D PhC structure
In designing a 2D topological structure, it is natural to generalise the celebrated 1D SSH model 30,39 . We implemented our 2D SSH structure in a PhC slab platform, in which photonic atoms were tetramerised to form a unit cell, as shown in Fig. 1(a); it is noteworthy that our structure exhibits a hole-in-slab configuration, not a cylinder-in-air configuration 32,33 . Squareshaped holes (side length: s = 0.33a) were drilled into a dielectric slab (thickness t = 230 nm; refractive index n = 3.4) to form a unit cell (lattice constant: a = 500 nm) composed of four air holes arranged in a square symmetry. The four holes were spaced equally apart by distance d from the centre of the unit cell in the x-and y-directions; d is the key parameter that controls topological details of the structure. We assume that the modes are transverse-electric (TE) as the slab waveguide supports the fundamental TE-guided mode only.
Figure 1(b) shows the photonic band structures calculated for three unit cells with different d values: d1 = 0.25a -Δ, d2 = 0.25a, and d3 = 0.25a + Δ, where Δ was selected to be 0.07a. The calculations were performed using a full three-dimensional finite-difference time-domain (FDTD) method (see Methods). When d = d2, we obtained a gapless double-degenerate band structure, which was due to band folding as the lattice constant (a) was twice the actual periodicity (2d). For d = d1 or d3, the bandgap opens. In fact, the two band structures are identical because d1 and d3 are complementary (d1 + d3 = a/2); two different unit cells can be selected to produce the same PhC structuresee the insets in Fig. 1(b). Nonetheless, they possess different bulk polarisations. It is clear that the bulk polarisation is nonzero (topological) for d > d2, which results in a TES at the boundary of a finite structure; however, it becomes zero (trivial) for d < d2 (see Supplementary Information for details).

Hierarchical topological states
Typically, the TESs of the 2D SSH model structures are calculated by the tight-binding method 28,39 . However, the latter is not suitable for PhC-based SSH structures because photonic interactions easily extend beyond the nearest-neighbour atoms; the only exception may be when an array of PhC cavities is employed 20 . Therefore, we conducted FDTD simulations to accurately simulate the TESs in our structure. As shown in Fig. 2(a), we first investigated 1D edge states that may manifest themselves at the interface where two topologically distinguished PhC structures (d1 and d3) are joined together. The region surrounded by the red solid lines in After confirming the existence of the 1D edge state, which is the lowest-order TES of the topological 2D PhC structure, we performed another numerical simulation to verify the coexistence of the higher-order 0D TES at the corner. The simulation structure was designed such that the topologically nontrivial PhC structure (d3) was surrounded by the topologically trivial structure (d1). Figure 2(d) schematically shows a 90° corner interface of the structure.
Despite the lack of quantised bulk quadrupole moment, which is apparent from the fact that only a single band exists below the bulk bandgap, our structure may still possess a corner charge Q induced by two topologically nontrivial quantised dipole moments at the side edges, px and py, in a hierarchical manner 29-34 . FDTD simulations reveal a distinct resonant mode outside the edge state band but still inside the bulk bandgapsee the dotted red line in Fig.   2(b). It is noteworthy that the 1D edge state band is gapped, which distinguishes our PhC structure from conventional topological insulators, such as Chern, spin Hall, and valley Hall insulators, where the first-order TESs are gapless. In our opinion, this gap in the first-order TES enables the simultaneous existence of higher-order TESs. The field distribution simulated for the state, which is shown in Fig. 2(e), clearly demonstrates that the mode is strongly localised at the corner, a direct indication that it is indeed a confined 0D corner state.

Lasing from three topological states
To experimentally identify these multidimensional topological photonic states and explore their application possibility as photonic devices, we fabricated the proposed structures using an InGaAsP multiple-quantum-well (MQW) epilayer (see Methods for detailed fabrication procedures). Although a large Δ value is preferred for a wide bandgap and robust TESs, extremely narrow (in width) ribbons are required between the square-shaped air-holes, rendering device fabrication formidable, if not impossible. As a quick solution for the structural dilemma, we investigated the extreme condition where d = s/2 (or Δ = 0.085a for a fixed s = 0.33a) such that the four square holes in the unit cell abutted to each other, as shown in Fig.   2(f). It is noteworthy that the topological nature and associated topological properties discussed thus far are not altered even in this extreme situation because Δ > 0, i.e. the selected Δ value does not cross the topological phase transition point. Figure 2 The fabricated device structure was optically excited with a 1064 nm laser diode in pulsed mode (at 500 kHz frequency for 10 ns). The excitation laser beam was focused down to ~10 μm in diameter using a 50× objective lens. The focused beam size is substantially smaller than an individual device (~20 µm × 20 μm), which allows us to selectively excite different topological statesamong the bulk, edge, and corner statesby selecting an appropriate spot location for excitation. We excited nine different positions sequentially: one centre, four side edges, and four corners of the square-shaped topological boundary, which are depicted in the insets of Fig. 3(a). Figure 3(a) shows the microphotoluminescence (µPL) spectra measured, while the excitation spot was located at the nine different positions. All the spectra were obtained at excitation power levels above the lasing thresholds, resulting in sharp stimulated emission peaks from the corresponding topological eigenstates. The bulk exhibits a singlemode lasing action despite the absence of active optical confinement, a characteristic of bandedge laser 40,41 . Meanwhile, the emission spectra from the edges exhibit multiple modes, which we attribute to Fabry-Pérot oscillations along the side edge terminated by two corners. The Furthermore, it may imply that our bulk state lasing originates from the band-edge of the lower bulk band at a/λ ≈ 0.32, which corresponds to the first M-point band edge, as identified from the band structure in Fig. 1(b).

Simulations on topological states
We conducted FDTD simulations to theoretically confirm the eigenstates supported by the structure. Several randomly oriented dipole sources were deployed at appropriate locations, while monitors were set above the slab to record signals emitted from the corresponding area.
Simulation results are summarised in Fig. 3(b). It is noteworthy that the spectra from the four edges as well as those from the four corners are identical, which is due to the 4-fold rotational symmetry of the simulation model structure. The bulk and edge states support multiple modes whereas the corner state is singular. The multiple modal nature of the bulk state can be explained by the finiteness in sample size, which effectively lifts the degeneracy of an otherwise single band-edge mode in the ideal infinite PhC structure; the modal profiles that correspond to the three bulk modes in Fig. 3

Laser characteristics
To assess the application possibility of the hierarchical topological states, especially of the higher-order corner state, as quality optical resonators, we examined their lasing characteristics.

Hierarchical corner states
Although FDTD simulations and µPL measurements unambiguously proved the simultaneous presence of 0D and 1D TESs as well as 2D bulk states in our structure, the origin of their stability and robustness as optical modes still remains unclear. Hence, we performed analytic model calculations. SSH model structures are often oversimplified in tight-binding calculations such that the interaction strength is nonvanishing only between the nearest neighbours. The correspondingthus unrealisticsituation for our tetramerised PhC structure is depicted in Fig. 5(a), where there are only two kinds of the nearest neighbours (or interaction strengths). Such oversimplified model calculations preserve the chiral symmetry and yields a symmetric eigenvalue spectrum with respect to zero energy 28,39 . Figure 5

Numerical simulation
The The long and short distances between the nearest neighbours were arbitrarily selected to be 3.5 and 1.5, respectively, while all on-site energies were set to zero.

Device fabrication
An epi-wafer for device fabrication was procured commercially, in which an InP buffer,     where m and n represent the indices of all the bands below the band-gap of interest. We numerically calculate this value of our system by using a finite-difference time-domain (FDTD) software. Note that numerical evaluation of derivatives and integrations with respect to discrete k points relates this calculation to Wilson-loop approach, which has been introduced in previous studies [2,3]. It is also wellknown that a mirror symmetry in ( ) direction quantizes ( ) to 0 or 0.5 [2,3]. We first obtain eigenvectors | ( , )⟩ by calculating the Hz field distribution in the unit cell at each discrete ( , ) point, and then numerically compute the bulk polarizations. Figure S1

Performance characteristics of the topological state lasers
To evaluate the qualities of the topological state lasers, light-in versus light-out characteristics were measured. Typical L-L curves for the nine different topological states, all measured from a single device, are shown in Fig. S2(a). Although the topological states are amply different, their lasing thresholds are rather similar, ~7 kW/cm 2 . This experimental observation results are qualitatively consistent with the simulation results, which are shown in Fig. S2(b): the simulated cavity Q-factors are in the order of 10 3 , regardless of the topological states. The Q-factors are calculated for the modes shown in Fig. 3(c) in the main manuscript, which belong to the realistic model structure with disorder.

Identification of the bulk states
An ideal band-edge laser structure should be infinitely periodic. In reality, however, a band-edge laser is finite. When the excitation area is large enough, the device boundaries affect the formation of normal modes and lift the modal degeneracy of the ideal band-edge laser, as demonstrated by FDTD simulationssee Fig. 3(b) in the main manuscript, which is copied here as Fig. S3(a). In order to prove that those three eigenmodes are indeed originated from a band-edge mode, we calculate their modal profiles, which are shown in Fig. S3

Modal profiles of the edge states
In order to clarify the physical characteristics of the edge state lasers, we perform FDTD simulations. Figure S4(a) is the replot of a part of Fig. 3(b) in the main manuscript, the simulated modal spectrum of the edge states in our topological PhC structure. Shown in Fig. S4(b) are the mode profiles of the three shortest-wavelength modes, which are identified by the arrows in Fig. S4(a). One can see that the modal periodicity in each figure is commensurate with the underlying PhC lattice, which is modulated by an envelope function with a much longer period. We believe that the envelope function is a result of mode beating between multiple cavity lengths of L, 2L, 3L, and 4L, where L is the side length of the square boundary.

Modal robustness of the corner state
Modal robustness of the corner state has been investigated in a few previous studies [4,5]. In the present work, we examine an extreme situation where an edge defect, a missing atom from a tetramer, is very close to a corner. Despite the existence of such a strong defect near the corner, the corresponding corner state is still well defined with the modal profile rarely disturbed, as can be seen in Fig. S5. The calculated Q-factors are ~3,100 and ~2,500 without and with the defect, respectively.