Photonic Dirac cavities in all dielectric photonic crystals with spatially inhomogeneous mass terms

Yuting Yang; Sijie Yin; Bin Yang; Liwei Shi; Enyuan Wang; Zhi Hong Hang

doi:10.1117/1.APN.5.4.046004

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Abstract

In photonic systems, the synthetic pseudo-magnetic field can influence electrically neutral photons in a manner analogous to real magnetic fields on electrons, thereby offering an approach for light manipulation. Here, we construct two-dimensional photonic crystals composed of all-dielectric materials, where engineering rotation angles result in spatially varying gradients of opened Dirac cones corresponding to position-dependent mass terms. The induced in-plane pseudo-magnetic field gives rise to chiral zeroth-order Landau levels. We design and experimentally demonstrate photonic Dirac cavities based on the bulk state propagation of chiral Landau levels, exhibiting large-area Dirac mode distributions. The proposed photonic cavities demonstrate a significant capability to control confinement strength and spatial distribution of the trapped Dirac modes induced by the variation of the mass term. These findings represent an important step toward utilizing Dirac cavities for light confinement in integrated photonics and enhancing light–matter interactions, thereby contributing substantially to various device applications.

© The Authors. Published by SPIE and CLP under a Creative Commons Attribution 4.0 International License. Distribution or reproduction of this work in whole or in part requires full attribution of the original publication, including its DOI.

Keywords

photonic crystal pseudo-magnetic field chiral Landau level photonic cavity

1 Introduction

Artificial periodic structures exhibiting Dirac-like conical dispersion have given rise to intriguing physical phenomena.1^,2 The emulation of the Dirac physics relies on the construction of lattice symmetries within metacrystals, which is essential for realizing topological insulators.3^–7 Similar to the mechanical strain applied to the grapheme,8^–13 the inhomogeneous modulation in meta-crystals in recent studies can generate an effective gauge field of vector potential, with the resulting artificial pseudo-magnetic field perpendicular to the metacrystals, leading to the formation of discretized and flat Landau levels.14^–30

In contrast to the perpendicular pseudo-magnetic field generated by the linear shift of Dirac points in two-dimensional Dirac materials, an in-plane pseudo-magnetic field can be induced by a spatially varying mass term.31^–37 This results in a chiral zeroth-order Landau level38^–42 that displays linear dispersion and represents topologically protected one-way propagating bulk states, which are distinct from the topological edge states localized at the boundaries of the topological insulators. This engineered synthetic gauge field has been utilized to design photonic Dirac cavities and waveguides for the manipulation of photonic modes.

In this work, we investigate the construction of synthetic gauge fields within two-dimensional all-dielectric photonic crystals (PhCs). By introducing a rotation degree of freedom, we obtain a spatially inhomogeneous mass term, enabling the flexible construction of in-plane pseudo-magnetic fields in PhCs. This results in chiral zeroth-order Landau levels that support the directional light propagation, which belongs to the quantum valley-Hall topological system.43^–45 Concurrently, we utilize the Dirac modes to realize expanded and shrunken photonic ring cavities. The experiment is performed to demonstrate the propagation characteristics of Dirac cavities. These cavities are implemented in an all-dielectric platform at microwave frequencies and extended to the optical frequency range, facilitating practical applications.

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2 Results

We propose two-dimensional PhC heterostructures with three types of variation of the effective mass terms, denoted as structures I, II, and III, illustrated in Fig. 1(a). A unit cell of the PhC is constructed by etching three identical regular hexagonal airholes (side length $r = 1.7 mm$ ) into a hexagonal background composed of alumina materials ( $ε = 11.2$ ), in which the lattice constant of the triangular lattice is set as $a = 7 mm$ . The rotation angle of the air hole is represented by $θ$ . When $θ = 0 \deg$ , the degenerate Dirac cones exist at the $K / K^{'}$ valleys in the first Brillouin zone in the calculated TM ( $E_{z}$ along $z$ direction) bands (see Supplementary Material). According to the $k \cdot p$ theory,46^–48 the effective Hamiltonian around $K / K^{'}$ valleys is $H_{K / K^{'}} = ν_{D} (k_{x} σ_{x} \pm k_{y} σ_{y}) + m (\vec{r}) σ_{z},$ where $v_{D}$ is the group velocity at the Dirac point, $σ_{x, y, z}$ are the Pauli matrices, $m$ is the effective mass term, and $k$ is the displacement of the wave vector to the $K / K^{'}$ valleys. As the airhole in a unit cell is gradually rotated from 0 to $\pm 30 \deg$ , the broken mirror symmetry introduces an effective mass term, leading to the progressive opening of the topological bandgap, which eventually reaches its maximum separation at $θ = 30 \deg$ ( $m > 0$ ) and $θ = - 30 \deg$ ( $m < 0$ ) at $K$ points. The band inversion of these two PhCs actually corresponds to a sign reversal of the mass term.

Figure 1.(a) Three profiles of the position-dependent effective mass term induced by the variation of rotation angles. The PhC structures I, II, and III correspond to linear, nonlinear, and square-root rotation gradient, respectively, indicated by the color bar. The rotation angles vary from 30 deg at the bottom to $- 30 \deg$ at the top. (b) and (c) Electric-field amplitude distribution of the chiral zeroth-order Landau level for three PhC structures at the same frequency of 0.238 c/a and the wave vector $k_{x} = 0.7 π / a$ . (d) Dispersion curves of the chiral zeroth-order Landau levels for these three PhC structures. The number of layers in the PhCs is $L_{M} = 37$ . The colormap denotes the quality factor $Q$ of the bulk modes. Inset: The effective mass term is a function that depends on the angular gradient.

To realize the synthetic gauge field in the PhC, a linear variation in the rotation angles of the airholes is applied along the $y$ direction, introducing a position-dependent mass term, whereas the translational symmetry is preserved along the $x$ direction. When the total number of layers is set as $L_{M}$ in the PhC heterostructure, comprising upper and lower regions with opposite rotational gradients, the corresponding rotation angle $θ$ at each layer $L$ is $θ = - Δ θ \cdot L (- \frac{L_{M} - 1}{2} < L \leq \frac{L_{M} - 1}{2})$ , where the gradient of the rotation angle is $Δ θ$ . For the PhC structure I, $Δ θ = 5 \deg / 3$ and $L_{M} = 37$ . The size of the opened band gap ( $Δ ω = 2 υ_{D}^{2} | m |$ ) at $K / K^{'}$ points exhibits a linear relation with the coordinate $y$ . Consequently, the effective mass term that varies from negative to positive is linearly dependent on $y$ ( $m = q y$ ). One can obtain an effective Hamiltonian ${H = v}_{D} ({\tilde{k}}_{x} σ_{x} \pm {\tilde{k}}_{y} σ_{y} {) + m σ}_{z}$ , where ${\tilde{k}}_{x, y}$ denote the wave-vector operators, and ${\tilde{k}}_{y} = - i \partial_{y}$ ( ${\tilde{k}}_{x} {= k}_{x}$ ) in the direction with the broken (preserved) translational symmetry. The two-dimensional Dirac Hamiltonian can be regarded as a subsystem of a synthetic Weyl system $H_{w} = v (k_{x} σ_{x} \pm k_{y} σ_{y}) + {\tilde{k}}_{z} σ_{z}$ at ${\tilde{k}}_{z} = 0$ , where ${\tilde{k}}_{z}$ is a virtual wave vector used for constructing the Weyl Hamiltonian.33 As the effective mass term introduces a term in the Dirac Hamiltonian, it is equivalent to a vector potential $A$ that is introduced along the $z$ direction with $A_{z} = m (y)$ .34 Hence, we can obtain the in-plane uniform pseudo-magnetic field $B_{x} = \nabla \times A_{z}$ .

Under the application of the in-plane pseudo-magnetic field, the Dirac cones evolve into discrete energy level plateaus, exhibiting the chiral zeroth-order Landau levels with linear dispersions, as depicted in Fig. 1(d). The quantization of these energy levels is expressed as follows33^,40: $ω_{n} = {\begin{cases} χ sign (q) v_{D} k_{x}, & n = 0 \\ \pm \sqrt{v_{D}^{2} k_{x}^{2} + 2 n | q | v_{D}}, & n \geq 1 \end{cases},$ where $χ$ is the chirality corresponding to the $K$ and $K^{'}$ valleys. $n = 0$ corresponds to the zeroth-Landau level, and the dispersions of the higher-order Landau levels ( $n \geq 1$ ) have the square-root relation.

The PhC structures II and III are respectively constructed by modulating the rotation angle with nonlinear and square-root gradients, corresponding to spatially inhomogeneous mass terms. Nonlinear effective mass terms lead to nonuniform pseudo-magnetic fields in the PhCs. The rotation angles at each layer for the PhC II with the nonlinear mass terms are $θ = - Δ θ_{1} \cdot L (0 < L \leq \frac{2 (L_{M} - 1)}{3})$ and $θ = - Δ θ_{2} \cdot L (- \frac{L_{M} - 1}{3} \leq L \leq 0)$ , where the number of layers $L_{M} = 37$ and $Δ θ_{1} (Δ θ_{2})$ is 1.25 deg (2.5 deg) for the upper (lower) regions. For the PhC III with square-root gradient, the rotation angles are $θ = 30 \deg - \sqrt{Δ θ \cdot (9 + L)} (0 < L \leq \frac{3 (L_{M} - 1)}{4})$ and $θ = \sqrt{Δ θ \cdot | L |} (- \frac{L_{M} - 1}{4} \leq L \leq 0)$ , where $Δ θ$ is 100 deg for the PhC III with $L_{M} = 37$ .

Figure 1(b) depicts the electric field distributions of the zeroth-order Landau level at the wave vector $k_{x} = 0.7 π / a$ under three types of position-dependent mass terms. The chiral zeroth-order Landau level is the bulk state, and its Gaussian-like eigen-electric field is distributed in a large area in the domain wall of the PhC I with smooth linear variation of the mass term. It shows distinct transport properties with the step-like domain wall with sharp transition of the mass term in the photonic valley crystals, exhibiting strong localized topological interface states and exponential decay perpendicular to the domain wall. The distributed location of this chiral bulk state can be manipulated at the upper or lower regions by the nonsymmetric mass terms in the PhC II. This bulk state also tends to be confined around the domain wall with a sharp transition of the mass term in the PhC III with a square-root profile. Consequently, a photonic Dirac waveguide is designed. The light beam emitted from the output port of the straight waveguide exhibits a small divergence angle and adaptable beam shaping capability (see the Supplementary Material). By tuning the mass terms, the width and position of the emitted light beams can be precisely controlled. The ability to control the confined strength and distribution of the electric field plays a useful role in device applications.

The mass term distribution induced by distinct rotation methods impacts the properties of eigenmodes. In engineering, the quality factor $Q$ is employed to assess the energy storage capability of a device. The mathematical expression for the quality factor $Q$ is given by $Q = ω_{r} / 2 ω_{i}$ , where $ω_{r}$ and $ω_{i}$ represent the real and imaginary parts of the central resonant frequency, respectively. We observe that although the edge state dispersions remain unchanged, the $Q$ factors vary under different mass term profiles, denoted by the colormap in Fig. 1(d). The linear variation of the mass term results in more extended electric field distributions compared with a sharp transition. The smoother mass term enables the adiabatic preservation of symmetry across the domain wall, thereby suppressing valley spin mixing. This leads to reduced scattering losses and consequently enhances the quality factor, corresponding to a longer mode lifetime. This indicates that we can control the localized position and the linewidth of bulk states, which guide us in the subsequent design of multifunctional photonic cavities.

By adjusting the rotation angles of airholes in the PhCs, we can design a photonic Dirac cavity with spatially variable mass terms, as illustrated in Fig. 2(a). Here, the hexagonal PhC structure satisfies the $C_{6}$ rotational symmetry. The rotation angles of the airholes are a function of layers, featuring a square-root rotation angle $θ = \sqrt{Δ θ \cdot L}, (Δ θ = 90 \deg, 0 \leq L \leq L_{M} - 1)$ , in which the mass term of the Dirac cavity varies from zero at the center of the cavity to a negative maximum at the outermost layer. There are two degenerated bulk modes, indicated by red dots in Fig. 2(c), in the calculated eigenfrequency spectrum of the PhCs with $L_{M} = 11$ and $Δ θ = 90 \deg$ . Owing to the finite-size effect, the continuous frequency spectra in an infinite structure become a discrete localized state.49 The eigen-electric field distributes around the center, as shown in Fig. 2(b). At these two frequencies, the $Q$ factor of the cavity reaches $10^{3}$ , as shown in Fig. 2(d). This indicates that the resonance characteristics of our designed resonant cavity are strong, enabling more effective transmission of signals at specific frequencies while suppressing signals at other frequencies. In addition, we calculate the eigenstate spectrum of the photonic cavity with 31 layers and the gradient $Δ θ = 30 \deg$ , as shown in Fig. 2(e). Figure 2(f) shows that the increase in the cavity size leads to an increase in the $Q$ factor, which approaches $10^{5}$ . The energy loss in the photonic cavity arises from alumina material absorption and scattering boundary condition losses, primarily due to the finite-size effect. More fundamentally, this effect influences the $Q$ factor of the cavity, with larger photonic Dirac cavities exhibiting higher $Q$ factors.

Figure 2.(a) Schematic of the photonic Dirac cavity featuring a spatially varying mass term with the square-root profile. (b) Three-dimensional graph of the electric field distribution for the bulk state within an 11-layer PhC. The bulk state is confined to the central region of the cavity. The height represents the electric field intensity. (c) and (e) Calculated eigenstate spectra of the Dirac cavity, corresponding to 11- and 31-layer PhCs, respectively. (d) and (f) Quality factors of the cavity with 11 and 31 layers.

Next, we implement experiments to validate our proposed photonic Dirac cavity with bulk states. The experimental sample is shown in Fig. 3(a). The detailed parameters of the PhC sample are described in Fig. 1. The hexagonal sample has a side length of 39 mm, and the height of the structure is finite along the $z$ -direction. The sample is placed within a waveguide composed of two parallel metallic plates and is thus regarded as a two-dimensional system. The excitation source is placed at the center of the cavity, indicated by a red star. Absorber materials are placed around the sample to simulate scattering boundary conditions. The experimentally measured results are shown in Fig. 3(b), where the stronger electric fields distribute around the center, which agrees with the simulated results. There is a pronounced peak in the transmission spectra, demonstrating the existence of the localized bulk state around the center of the cavity, as shown in Figs. 3(c) and 3(d). The $Q$ factor can be calculated as the ratio of the resonance center frequency to the full width at half maximum extracted from the transmission spectrum. Experimentally measured $Q$ values are approximately 1 order of magnitude lower than their simulated counterparts. This discrepancy arises primarily from two experimental imperfections, including the three-dimensional printing errors of the fabricated alumina sample and absorbing material losses, which contribute to the reduced peak amplitude and spectral linewidth broadening of the measured transmission spectra. A narrow air gap between the photonic crystal sample and the waveguide plate in the experimental setup results in the resonance frequency shift between the experiment and simulations.

Figure 3.(a) Experimental sample of the photonic Dirac cavity composed of an 11-layer PhC with the square-root variable mass terms. The cavity is fabricated using 3D printing of alumina dielectric material with a relative permittivity $ε = 11.2$ . A red star marks the position of the excitation source. (b) Experimental measurement of $E_{z}$ electric field amplitude at 10 GHz. (c) and (d) Simulated and experimental results of the normalized transmittance within the Dirac cavity.

By manipulating the rotation angles of airholes in PhCs, we can construct two anti-gradient PhCs, exhibiting a nonuniform pseudo-magnetic field. Based on these, we design a photonic Dirac ring cavity with nonlinear position-dependent mass terms. We can selectively control the confined location of the chiral bulk states of the zeroth-order Landau level at the domain wall of two PhCs, which allows for the flexible design and tunable manipulation of the ring cavity. Figure 4(a) shows the designed expanded ring cavity. The hexagonal PhC structure consists of 31 layers, corresponding to $Δ θ_{1} = 1.5 \deg$ for the inner region and $Δ θ_{2} = 3 \deg$ for the outer region beside the domain wall, as depicted in the inset. In the calculated eigen field, indicated by red and blue dots in the eigen-frequency spectrum in Fig. 4(c), the electric field of the zeroth-order Landau levels is localized around the domain wall, forming an expanded ring cavity. The widespread use of the chiral bulk states effectively improves the energy capacity of the photonic ring cavities. This makes the guided Dirac modes especially pronounced compared with other states, such as the more tightly localized topological edge state and delocalized bulk states. We also design a shrunken ring cavity with the reversed structure, that is, $Δ θ_{1} = 3 \deg$ and $Δ θ_{2} = 1.5 \deg$ , as shown in Fig. 4(b). Due to the time-reversal symmetry, two eigenmodes within the gap are doubly degenerated, corresponding to two counter-propagating edge states, which characterize the quantum valley spin Hall states. We observe that over a broad frequency range of 0.23 to 0.245 c/a, the electric fields are localized at the domain wall, indicating that the cavity possesses a wide operational bandwidth. These bulk states correspond to high $Q$ factors, as shown in Fig. 4(d). This ensures a high working lifetime and transport efficiency for our ring cavity. For comparison, we calculate the eigenstate spectrum and $Q$ factors of the expanded ring cavity with 16 layers ( $Δ θ_{1} = 3 \deg$ and $Δ θ_{2} = 6 \deg$ ), displayed in Figs. 4(e) and 4(f). It is found that an increase in the size of the cavity results in a greater number of degenerated modes and higher $Q$ factors.

Figure 4.(a) Expanded photonic ring Dirac cavity with nonlinear position-dependent mass terms. (b) Shrunken ring cavity. The electric field distributions of the bulk states correspond to the chiral zeroth-order Landau level. The left insets indicate the rotation angle of the airholes in the PhCs with 31 layers. (c) and (d) Eigenfrequency spectrum and $Q$ factor of the discrete bulk states in the expanded ring cavity with 31 layers. (e) and (f) Calculated eigenstate spectrum and $Q$ factor of the expanded ring cavity with 16 layers.

We conduct experiments to further validate the large-area chiral modes in the aforementioned photonic Dirac ring cavities. Due to size constraints of the experimental platform, we downsize the expanded ring cavity to 16 layers, with rotational gradients $Δ θ_{1} = 3 \deg$ and $Δ θ_{2} = 6 \deg$ . Due to the symmetric distribution of bulk modes, our sample is processed as a semi-hexagonal cavity, as shown in Fig. 5(b). The position of the excitation sources is marked by two blue stars. The simulated and experimentally measured electric fields in the ring cavity are shown in Figs. 5(a) and 5(c). The chiral bulk states, corresponding to the zeroth-order Landau level, are not confined at the interface and are distributed among large regions near the domain wall. The simulated and measured normalized electric field amplitude along the $x$ -direction at $x = 140 mm$ and along the $y$ -direction at $y = 24 mm$ are shown in Figs. 5(d) and 5(e), respectively, which demonstrate that the electric field is uniformly concentrated and distributed around the domain wall.

Figure 5.(a) Simulated electric field of the chiral zeroth-order Landau level within the expanded photonic ring Dirac cavity. The number of layers is $L_{M} = 16$ and the rotation gradients $Δ θ_{1} = 3 \deg$ and $Δ θ_{2} = 6 \deg$ . (b) Photograph of a semi-hexagonal cavity sample fabricated using the alumina material. Two blue stars mark the positions of the excitation sources. (c) Experimentally measured $| E_{z} |$ electric field, corresponding to the red rectangular section in panel (a). (d) and (e) Simulated and experimental normalized electric field intensity at $x = 94$ and $y = 24 mm$ indicated by green solid and dashed lines in panel (a).

Our proposed photonic Dirac cavity and waveguide can be extended to the optical frequency range by utilizing the micro-nano fabrication. As shown in Fig. 6(a), we consider a PhC consisting of air holes embedded in a silicon background material, which are identical to those in Fig. 1. The lattice constant is $a = 400 nm$ , the side length of the air hole is $r = 97 nm$ , and the relative permittivity is $ε = 11.68$ . The straight Dirac waveguide comprises nonlinear gradient PhCs with $L_{M} = 10$ layers with rotational gradients $Δ θ_{1} = 10 \deg$ and $Δ θ_{2} = 5 \deg$ for the upper and lower regions. The triangular expanded Dirac cavity consists of 19-layer rotational air holes with the nonlinear rotation angle of $Δ θ_{1} = 2.5 \deg$ and $Δ θ_{2} = 5 \deg$ , in which the dispersion curves are shown in Fig. 6(b). The operational bandwidths of the Dirac cavity and waveguide overlap precisely. The large-area propagating bulk states in the waveguide satisfy chirality-matching with the trapped modes in the cavity. Consequently, the bulk states of the chiral Landau level in the triangular expanded and shrunken cavities can be efficiently excited via evanescent coupling with the straight Dirac waveguide, as displayed in Figs. 6(c) and 6(d).

Figure 6.(a) Schematic diagram of a silicon-based triangular expanded photonic Dirac cavity coupled with a straight Dirac waveguide, where the colormap represents the nonlinear rotation gradients. (b) Dispersion curves of the 19-layer PhC with nonlinear mass terms. (c) and (d) Electric field distributions at 177 and 177.6 THz within the expanded and shrunken triangular cavities and waveguides.

3 Conclusion

In this work, we experimentally realize a two-dimensional PhC structure with a spatially variable mass term by manipulating the gradient of the rotation angle, which generates an in-plane synthetic pseudo-magnetic field, leading to the quantization of Landau levels. The zeroth-order Landau level is a chiral bulk state with a large-area distribution, which is used to realize photonic Dirac waveguides, hexagonal Dirac cavities in microwave, and triangular Dirac cavities in optical waves. The linear and nonlinear mass terms lead to uniform and nonuniform pseudo-magnetic fields, respectively, exhibiting distinct field distributions of the Dirac modes. Our work introduces a useful approach to obtaining artificial pseudo-magnetic fields and provides an ideal platform for the manipulation of light. The distributions of mass term profiles can be controlled according to specific design requirements,50 which can manipulate the light and bring out abundant application devices. The large distribution of Dirac modes in a photonic cavity allows us to optimize the excitation efficiency by overlapping the spot size of the laser beam with the mode profile in the optical device in a nanostructure. In addition, the trapped Dirac modes can be used for control of light–matter interactions on a photonic chip, which paves the way for the practical design of advanced photonic devices for future applications.

Yuting Yang is an associate professor in the School of Materials and Physics at China University of Mining and Technology. She received her PhD from Soochow University in 2019. During 2018–2019, she worked as a visiting PhD student at Nanyang Technological University. Her research interest includes metamaterials, photonic crystals, and topological insulators.

Biographies of the other authors are not available.

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