Dual-Band Topological Valley Cavity in Mid-Infrared Range

Abstract

Topological edge states, emerging at boundaries between regions with distinct topological properties, enable unidirectional transmission with robustness against defects and disorder. However, achieving dual-band operation with high performance remains challenging. Here, we integrate dual-band topological edge states into a valley photonic crystal cavity operating in the mid-infrared region, leveraging triangular scatterers. A key contribution of this work is the simultaneous realization of ultra-high Q-factors (up to 6.1593 × 10⁹) and uniform mode distribution (inverse participation ratio < 2) across both bands. Moreover, the dual-band cavity exhibits exceptional defect tolerance. These findings provide a promising platform for mid-infrared photonic integration, paving the way for high-performance optical cavities in multifunctional photonic systems.

1. Introduction

The field of valley topological photonics represents a novel and rapidly evolving area of research in the realm of topological photonics, with a notable focus on its capacity to manipulate light [1,2,3,4,5]. This distinctive capability is chiefly attributable to the existence of valley degrees of freedom, which can be utilized to engender novel topological phenomena in photonic systems [6,7,8,9]. Among these phenomena, topological edge states (TESs) are particularly salient.

TESs emerge at the interface between spatial domains with distinct topological properties in their band structures [10]. These states exhibit unidirectional and robust propagation, effectively suppressing backscattering in waveguides while demonstrating resilience to defects and disorder [10,11,12]. Considerable efforts have been made in photonics to realize such states, driven by their potential applications in robust optical delay lines [13], amplifiers [14], and other devices [15,16]. TESs can also coexist with other topological states, such as topological corner states [17,18,19] and bound states in the continuum (BIC) [20], providing a rich physical basis for multi-state coexistence and further expanding their application potential. Recent studies have also achieved multi-band [21,22,23,24,25,26,27,28] and tunable [29,30,31] topological edge states, significantly enhancing their adaptability and flexibility across different spectral ranges. Furthermore, the chiral excitation of the edge states has the potential to be combined with multifunctional metasurfaces [32], thereby enabling the realization of more complex functionalities. While these topological features have shown great potential, dual-band photonic devices have been primarily explored in waveguide transmission [33,34,35,36,37], especially since most topological photonic systems operating in the mid-infrared range are confined to a single frequency band [38,39], which limits the multifunctionality and operational bandwidth of mid-infrared photonic devices. Compact mid-IR sources have a variety of applications in industrial control, breath analysis, optical communications, IR scene projection, and threat detection [40,41,42]. The utilization of efficient mid-IR photonic devices is paramount to the efficacy of these applications. Valley photonic crystals (VPCs) offer a method to achieve dual-band topological edge states, thereby overcoming the limitations imposed by single-band solutions. Studies have demonstrated that modifying the structural characteristics of the photonic crystal can optimize topological edge states, thereby improving their adaptability and functionality in practical applications [43,44,45].

In this work, we propose a novel dual-band valley photonic crystal cavity design, leveraging topological edge states to achieve operation around 3.1 μm and 7.5 μm. The structure is based on a honeycomb lattice of scatterers resembling triangles. The design demonstrates high Q-factors in both bands, along with exceptional defect tolerance and mode stability. In comparison with previous cavity studies [46], the proposed design exhibits significantly higher Q-factors, with one frequency band exceeding 10⁹. This dual-band approach highlights significant potential for mid-infrared light source applications, offering enhanced performance and reliability for advanced sensing, communication, and imaging technologies. The ability to manipulate light across diverse wavelengths within the same structure heralds new possibilities for developing multifunctional photonic chips, advancing the frontiers of integrated photonics, nonlinear optics, and quantum technologies. The dual-band nature of this structure expands the potential of mid-infrared lasers for precision applications such as gas detection, environmental monitoring, and imaging, offering new technical support for nonlinear optics, precision measurement, and quantum optics in the mid-infrared range.

2. Design of Valley Photonic Crystals

We present a topological cavity design for dual-band valley edge modes. The valley photonic structure is based on a honeycomb lattice composed of scatterers resembling triangles. Conventional mid-IR interband cascade lasers (ICLs) are typically built on an InAs/GaSb/AlSb substrate with a refractive index of 3.3. The scatterers are modeled as perfect electric conductors, achieved by applying thin metal coatings on the air hole boundaries. The lattice period is $a$ = 985 nm, with each scatterer consisting of three legs, each of length $l$ = $y·a$ and width $w$ = $x·a$, and an angle of $\phi =120°$ between adjacent legs, as shown in Figure 1a. We used COMSOL (Multiphysics 6.2) to simulate the band structure of the valley photonic structure for transverse electric (TE) polarization. The band diagram for the primitive unit cell is shown in Figure 1b, revealing two Dirac cones near the K (K′) points, protected by mirror inversion symmetry. The parameters x and y are chosen to control the position of the Dirac points, which directly affect the bandgaps. Figure 1c,d show the positions of the accidental frequency degeneracy Dirac cones at the K and K′ points for different values of y and x. When $y=0.46$ and $x=0.07$, by rotating the central scatterer around the origin by an angle $\theta$, the mirror symmetry is broken, leading to the splitting of the degenerate bulk Dirac modes at the K point, as illustrated in Figure 1f, leading to the splitting of the degenerate bulk Dirac modes at the K point. Figure 1g shows the frequency spectra of the first, second, fourth, and fifth bulk modes at the K and K′ points as a function of $\theta$. To achieve the largest possible bandgap while limiting the working frequency range to the mid-infrared band, $\theta =\pm 5°$ is chosen, and the VPC1 and VPC2 are formed. A lower band gap from 37.186 THz to 42.632 THz (7.037 μm to 8.068 μm) and an upper band gap from 94.714 THz to 101.39 THz (2.958 μm to 3.167 μm) are observed, which are indicated by gray areas in Figure 1f. Their topologies are determined by the valley Chern numbers of the bulk bands below the band gaps. Using the $k·p$ perturbation method (where $k$ is the wave vector and $p$ is the momentum operator) [47,48]. the Berry curvature of the energy bands below Gap I and Gap II can be expressed as

$$\Omega \left(\delta k\right)=m_i{\nu }_{Di}/\left[2{\left(\delta k^2+m_i^2+{\nu }_{Di}^2\right)}^{3/2}\right]$$

(1)

Here, ${\nu }_{Di}$ is the Dirac cone dispersion velocity at the rotation angle $\theta =0°$, $\delta k$ is the momentum offset at the point K in the Brillouin zone, and $m_i$ denotes the effective mass, $m_i=\left({\omega }_{q+}^i-{\omega }_{q-}^i\right)/\left(2v_{Di}^2\right)$, which is proportional to the difference in frequency between the two eigenstates ${\varphi }_{q+}^i$ and ${\varphi }_{q+}^i$. Here ${\omega }_{q+}^i$(or ${\omega }_{q-}^i$) is the frequency of the edge state in the counterclockwise (or clockwise) energy-flow state. The valley Chen number can be obtained by numerically integrating the Berry curvature near the valley:

$$C_K^i=\left(1/2\pi \right)\iint \Omega \left(k\right)d^{2}k=\left(1/2\right)\mathit{sgn}\left(m_i\right)$$

(2)

When $\theta =5°$ and $\theta =-5°$, their effective masses in the lower gap and upper gap are of opposite signs, and the Berry curvatures of VPC1 and VPC2 near K and K′ in each band gap are opposite as shown in Figure 1h,i. Therefore, the edge states can be formed in the range of these two band gaps from these two photonic crystals with opposite valley Hall phases.

3. Valley Edge States Characteristics

Based on VPC1 and VPC2, topologically protected edge states at both the lower and higher band gaps (approximately 7.5 µm and 3.1 µm, respectively) can be constructed by splicing the two structures. In this approach, the upper part is VPC1 and the lower part is VPC2 to form supercell I (SC I), while the upper part is VPC2 and the lower part is VPC1 to form supercell II (SC II). Periodic boundary conditions are applied to the left and right sides during the calculation of the supercell energy bands. Additionally, spurious modes arising from the upper and lower scattering boundary conditions are excluded from the results. As shown in the previous section, the two VPCs are topologically distinct, and edge modes are expected to appear at the domain wall between them. Figure 2a,d show the dispersion relations for the two edge modes of the supercells, respectively. The gray area represents the bulk state, while the blue (red) curve indicates the valley edge mode at the K (K′) valley. From Figure 2a, it is evident that for SC I, near 7.5 µm, the group velocity of the edge state near the K (K′) valley is negative (positive), while at approximately 3.1 µm, the group velocity near the K (K′) valley is positive (negative). In contrast, from Figure 2d, it can be observed that for SC II, the group velocity of the edge state near the K (K′) valley at both the lower and upper band gaps is opposite to that of SC I. Figure 2b,c,e,f illustrate that the energy flow directions at points K and K′ are opposite. The field plots confirm that the edge states are strongly localized at the domain wall, i.e., the interface between two regions with opposite valley Chern numbers.

Next, we examine the transmission characteristics of the edge states. Valley photonic crystals, which enable selective excitation of phase vortices, generate unidirectional coupling at the interface. The vortex field in the upper domain interacts with the vortex field in the lower domain, resulting in the formation of a chiral-flow edge state. Therefore, chiral sources can be employed to achieve unidirectional coupling of valley-dependent edge states. The two-dimensional planar structures of the valley-dependent waveguide, constructed using the valley edge states described in the previous section, are shown in Figure 3a. To demonstrate the topologically robust transport of valley-dependent edge states, topological edge states are excited using both left circularly polarized (LCP) chiral sources, with the position of the chiral source indicated by an star. Figure 3b presents the electric field intensity distribution transmitted through different valley photonic crystal waveguides at various frequency bands under LCP and excitation. It is evident from the figure that the energy is primarily concentrated at the spatial domain boundaries and is transmitted in different directions across the waveguide structures and frequency bands. This phenomenon also occurs in Ω-shaped waveguide structures (with four sharp bends in the interface). As shown in Figure 3d, robust transport is maintained even when the edge states encounter sharp bends. Furthermore, the transmission of various waveguides within the operating frequency range was calculated to analyze their transport characteristics. The electromagnetic energy that passes through the planes 1 and 2 can be calculated by $U=1/2{\int }_l\mathit{Re}\left(E\times H^{*}\right)·\mathrm{d}l$. The transmission is then defined as the ratio of electromagnetic energy passing through plane 2 ($U_2$) to that passing through plane 1 ($U_2$). The results shown in Figure 3e,f demonstrate highly efficient energy transmission within the operating frequency range.

4. Valley Photonic Crystals Cavity Based on Valley Edge States

Based on this framework, we can develop photonic triangular cavities with topological protection. As depicted in Figure 4a,b, the cavity consists of a closed loop that encircles the domain wall. When triangular VPC1 is embedded within rectangular VPC2, a cavity leveraging SC1 is created. Similarly, embedding triangular VPC2 within rectangular VPC1 results in a cavity utilizing SC2. Analysis of the Q-factor distributions reveals that both the lower and upper frequency bands of the SC1 and SC2 cavities exhibit high-Q eigenmodes with regular spacing, characteristic of modes circulating in a triangular configuration akin to whispering gallery modes observed in disk or ring cavities. For these cavities, the Q-factor in the lower frequency band is approximately 10⁴, while in the upper frequency band, it surpasses 10⁹, reaching values of 6.1593 × 10⁹ and 3.545 × 10⁹, respectively.

The primary feature of a topological laser cavity is its ability to support a traveling wave mode akin to a whispering gallery mode, even in the presence of three sharp corners. In contrast, a conventional cavity cannot maintain such a mode due to significant back-reflection at the corners, which results in the localization of the electromagnetic field across various sections of the cavity. This phenomenon can be quantitatively analyzed by calculating the inverse participation ratio ($IPR$) along the one-dimensional curve representing the triangular loop. The $IPR$, a widely recognized metric for assessing mode localization, is defined as follows [49]:

$$IPR\left(\omega \right)={\displaystyle \frac{{\int }_L{\left|Hz\left(\omega ,\xi \right)\right|}^4\mathrm{d}\xi }{{\left[{\int }_L{\left|H_z\left(\omega ,\xi \right)\right|}^2\mathrm{d}\xi \right]}^2}}L.$$

(3)

where the formula $\xi$ is the coordinate parametrizing the 1D curve of length L. The denominator in Equation (3) ensures normalization. For modes confined to a length L0, the $IPR$ is $L/L0$, while for completely delocalized modes $L0\approx L,$ the $IPR\approx 1$, as the degree of localization increases, $L0$ decreases, resulting in an increase in $IPR$.

To demonstrate that the inherent mode of the topological cavity typically exhibits a lower $IPR$ and distributes more uniformly along the boundary, we designed a trivial cavity (Figure 5a) for comparison. In the trivial cavity, the same photonic crystal type is employed at both the inner and outer boundaries of the triangular ring. Scatterers at the boundary are eliminated, and defects are introduced to generate a localized mode, facilitating strong light resonance within the defect region and forming a waveguide. Numerical results for the $IPR$ of the triangular loop cavity are displayed in Figure 5b. As anticipated, the topological modes exhibit substantially lower $IPR$ compared to non-topological modes. It is observed that the eigenmodes of the topological cavity consistently demonstrate lower $IPR$, indicating more uniform extension along the loop. Moreover, the $IPR$ of the SC2 cavity is lower than that of the SC1 cavity at higher frequency bands, making it more suitable for photonic crystal lasers. We analyzed several high Q-factor electric field distributions from trivial cavities at characteristic frequencies (Figure 5d). In comparison with topological cavities (Figure 5e,f), the electric field distribution in trivial cavities is noticeably uneven. This uneven distribution of energy leads to increased losses within the cavity, thereby diminishing the output power and efficiency of the laser. Topological cavities significantly mitigate these issues. Furthermore, defects were applied to topological cavities to evaluate their topological protection capabilities. The outcomes, illustrated in Figure 5e,f, reveal that topological cavities provide exceptional protection against both edge and corner defects, with energy distributions remaining stable despite the presence of applied defects. This underscores the robust topological protection offered by energy valley edge states.

5. Conclusions

In summary, a VPC structure incorporating triangular scatters was developed, tailored for the mid-infrared spectrum at wavelengths of 3.1 µm and 7.5 µm. This structure can function as a unidirectional optical propagation device. Within the topological Z-/Ω-shaped VPC interfaces, robust transport with significant suppression of backscattering has been demonstrated. Moreover, the topologically protected triangular cavity exhibits exceptionally high Q-factor across different orders of magnitude. Triangular cavity modes manifest in both wavelength bands, maintaining a uniform energy distribution even in the presence of defects, thereby showcasing high robustness.

By further tuning the structural parameters, the operating wavelengths of the device can be adjusted within a certain range, demonstrating the design’s remarkable flexibility and adaptability. The dual-band operation enhances its potential for mid-infrared applications such as gas detection, environmental monitoring, and imaging, where simultaneous multi-wavelength operation can improve selectivity and sensitivity. In addition, future studies could explore further modulation through temperature and applied electric field adjustments, potentially expanding the device’s dynamic tunability. This tunability enhances its practical application potential in dynamic environments and extends the usability of the device across diverse scenarios. Our research provides a critical theoretical and technical foundation for enhancing the performance and broadening the application scope of mid-infrared lasers, thereby fostering the evolution and innovation of mid-infrared laser technology.