1. Introduction

Photonic crystal waveguides (PCWs) have attracted considerable attention due to their potential for various applications [1–3]. One of the key features of PCWs is their ability to trap light, which can be achieved through various mechanisms, such as topological edge states, waveguide modes, and localized modes [4–13]. Rainbow trapping, a specific type of light trapping, occurs when light of different wavelengths is trapped at different positions within a PCW. This can be achieved by designing a PCW with a specific dispersion relationship. Fano resonance is an important phenomenon that can often occur in PCWs, which arises from the interaction between a localized mode and a broadband background mode [14–16]. Fano resonances can have extremely high-quality $(Q )$ factors, making them ideal for on-chip nanophotonic devices [15,17,18].

Despite the significant progress made in Fano resonance devices [17,19–24] and rainbow trapping [4–7,25–27], several key challenges remain. First, realizing high-Q factor Fano resonances for cavities inside waveguides remains elusive. This is because the coupling strength between the cavity and waveguide is highly sensitive to their separation distance. Second, multi-mode rainbow trapping has yet to be demonstrated. This is a crucial requirement for on-chip multi-functional devices, as it enables the efficient and robust trapping and demultiplexing of multiple wavelengths of light. Meanwhile, the ability to trap and demultiplex multiple wavelengths of light compactly to disorders and efficiently is highly desirable to develop optical communication systems that can transmit and receive multiple wavelengths of light simultaneously. This could significantly increase the bandwidth of the systems. To date, the above problems have not yet to be solved.

In this study, we have realized robust multi-mode rainbow trapping with ultra-high-Q factor Fano resonances. We have proposed a structure that is formed by simply stacking two PCs on top of each other to form a new semi-bilayer PC. This creates a structure where the light will propagate through the two PCs in sequence. The new semi-bilayer PC is used to create PCW to support new photonic modes that are not possible in individual PCs. It allows PC devices to be more flexible, perform better, and resist defects and imperfections. By meticulously engineering a chirped PC within the PCW, multi-mode light trapping at distinct positions for different frequencies along the waveguide effectively creates a rainbow of light. This will allow for the efficient trapping and demultiplexing of multiple wavelengths of light. Fano resonance is observed at each localized mode due to the interference between the localized mode and a broadband background mode. We have emphasized that the formed rainbow trapping is still feasible and efficient after introducing small and large defects, which provides evidence that the rainbow trapping of this kind of new PC is robust to defects and different from traditional PCs. Finally, the proposed semi-bilayer PC structure addresses both challenges above simultaneously. By carefully designing a chirped PC within the PCW, we achieve ultra-high- high-Q factor Fano resonances with the cavity inside the waveguide. Moreover, we demonstrate multi-mode rainbow trapping for the first time. This breakthrough paves the way for a new generation of on-chip nanophotonic devices with unprecedented capabilities.

2. Results and discussions

Different unit cells are used in 2D PC with C-4 symmetry of the square lattice. The first unit cell, with four squares $Si$ rods of length $w\; = \; 0.2a$, where $a = 1\mu m$ is the lattice constant, is arranged in a square around the center of the unit cell. The relative permittivity of the $Si$ rods is 11.97, and the background material is air, a simple example of a C-4 symmetric unit cell. The second unit cell of the four rods is displaced near the corners of the unit cell, which is a more complex example, but it still maintains the C-4 symmetry, as shown in the inset of Fig. 1(a).

 

Fig. 1. (a) The schematic diagrams of the two different unit cells and band structures, (b) the schematic diagram of the combined unit cell and band structure, (c) the proposed supercell with the chirped PC of the four rods $({w_c})$ which is gradually changing along the propagation direction as shown by the red color, (d) the dispersion curves of the three guided modes, M1, M2, and M3 (e) the group velocity (${v_g}/c)\; $ of the three guided modes, M1, M2, and M3.

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Both of these unit cells have the same C-4 symmetry, and they both have a photonic band gap (PBG) in the same frequency range, as shown in Fig. 1(a) for the transverse magnetic (TM) mode with nonzero (${E_z},\textrm{}{H_x},\textrm{}{H_y})$. However, there is no band gap for TE mode. By combining the two-unit cells to form a new unit cell (we call it semi-bilayer unit cell), as shown in the inset of Fig. 1(b), we can create a PC with a PBG at a different frequency range than the PBG of the two-unit cells individually because it has a different structure and material property than that of the two-unit cells. The new unit cell can also be designed to have a band gap wider than the band gap of the two-unit cells, as shown in Fig. 1(b). This means that the new unit cell can be used to create PCW to support new photonic modes that are not achievable in the individual unit cells. When the length of the four rods in the middle is gradually changed (${w_c}$) along the propagation direction, as shown by the red color in Fig. 1(c), we have obtained three guided modes, M1, M2, and M3, as shown from the dispersion curves of Fig. 1(d). For M1 and M2, ${w_c}$ varying from $0.11a$ to $0.14a$; above this range, the guided modes are folded. For M3, ${w_c}$ varying from $0.15a$ to $0.18a$. The three modes move to a lower frequency range with increasing ${w_c}$, due to the increase in the effective medium index of refraction of the PCW. As the rods become longer, the PCW becomes denser, which increases the effective medium index of refraction. This causes the bandgap of the PCW to move to a lower frequency range, as shown in Fig. 1(d). M1 and M2 have inflection points in the dispersion curve and touch each other nearly at the same frequency value, so the group velocity $({v_g}/c)\; $ has positive and negative values, as shown in Fig. 1(e).

M1 and M2 are called counter-propagating modes or rainbow-trapping modes. Rainbow trapping occurs because the group velocity of light at a specific frequency is decreased to zero. This is because the chirped unit cell breaks the symmetry of the PCW, allowing modes with zero group velocity at the inflection point. M3 has a nearby flat and very low group velocity called slow light mode. Slow light modes occur because the effective medium parameters of the PCW are designed to be very close to the band edge. We have three modes that can be used to form a rainbow, i.e. the two based on inflection point and one based on slow light mode at the band edge, which refers to the two mechanisms of rainbow trapping [12].

To design a rainbow-trapping structure, the chirped supercell can be coupled together to form gradually PCW. This will trap light at that frequency in the PCW at the frequency where the group velocity of the slow light mode is zero. This is because the gradient structure causes the refractive index of the waveguide to vary gradually. As a result, different frequencies of light will travel at different speeds in the waveguide. Figure 2 shows the intensity of the electric $|E |\; $ field distribution to realize the rainbow of the three guided modes M1, M2, and M3, due to the chirped PC of ${w_c}$ from $0.11a$ to $0.14a$ for M1, M2, and from $0.15a$ to $0.18a$ for M3. Subsequently, multi-modes of rainbow trapping of M1, M2, and M3 are realized with high-light localization. For M1, as shown in Fig. 2(a), the modes look like waveguide modes where the light is strongly localized between the chirped rods in the gradient structure and parallel to the propagation direction. For M2, as shown in Fig. 2(b), the modes look like localized modes where the light is more strongly localized on the chirped rods and perpendicular to the propagation direction. In Fig. 2(c), the slow light mode is different due to the chirped rods’ isolation, which acts as an isolated cavity that prevents the light from scattering out and protected by the new designed PCs. For this kind of mode, there is specific localization for each cavity in the gradient structure along the propagation direction. This is because multi-mode rainbow trapping is a phenomenon that occurs when light of different frequencies is trapped at different positions along a waveguide.

 

Fig. 2. The intensity of the electric $|E |\; $ field distribution to realize the multi-mode rainbow trapping of the three guided modes M1, M2, and M3, due to gradually increasing of ${w_c}$ from $0.11a$ to $0.14a$ (a) M1, (b) M2, (c) ${w_c}$ from $0.15a$ to $0.18a$ for M3.

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Fano resonance is a well-known phenomenon in photonics that arises from the interference between a narrowband resonance mode and a broadband continuum mode [17]. The COMSOL simulation of Fano resonance involves modeling the structure and defining the EM propagation. The breakdown of the setup defines the geometry and materials of the proposed design based on the above unit cell properties along 20-unit cells in the propagation direction and 5-unit cells above and down the chirped PC. A normal mesh is chosen, which determines an optimal mesh size that balances accuracy and computational efficiency with appropriate boundary conditions to absorb outgoing waves and prevent reflections. Ports at both ends of the waveguide are defined to measure the transmitted and reflected waves. To measure the transmission spectrum of Fano resonance as a function of the resonance wavelength, a sweep of the resonance wavelength is modeled by varying the resonance wavelength within the desired range. The EM propagation problem is solved using COMSOL's wavelength-domain solver for each resonance wavelength. The transmission coefficient at each resonance wavelength using the port fields is computed. Finally, the transmission spectrum of Fano resonance can be obtained by plotting the transmission coefficient versus the resonance wavelength. The convergence threshold defines the stopping criterion for the iterative solver. It specifies the maximum allowable change in the solution variables between iterations. A smaller convergence threshold ensures high precision but increases the number of iterations. Conversely, a larger threshold may lead to premature termination and inaccurate results. The transmission characteristics with the intensity of the $|E |\; $ field distribution to realize the rainbow trapping of the proposed structure is shown in Fig. 3 for M3. It's clear that sharp different types of Fano resonance are observed with ultra-high Q factor in the range of ${10^7}$ at each localized point during the chirped PC along the propagation direction. The enhanced $|E |\; $ field distribution is typically confined to the chirped PC. The ultra-high- Q factor of the Fano resonance suggests that the localized mode is very well-confined.

 

Fig. 3. The transmission spectrum of Fano resonance as a function of the resonance wavelength due to chirped PC in the PCW with the intensity of the $|E |\; $ field distribution to realize the rainbow trapping.

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To illustrate the relation between the robust ultra-high-Q factor of our proposed structure and sharp Fano resonance, which can be attributed to the following factors. A chirped PC inside the PCW, often used to create slow-light devices. This causes the light to propagate at different speeds at different points along the waveguide, which traps the light more effectively, as shown in the second panel of Fig. 3. This reduction in the speed of light allows for strong interactions between light and matter. This leads to a longer lifetime of the photons in the PCW and, hence, a higher Q-factor. The strong coupling between the cavity and the waveguide and the protection from the surrounding PCs allows the light to be efficiently transferred back and forth between the two, further enhancing the resonance's localized nature. Besides, using two PCs stacked on each other can provide additional confinement for the light, which can further reduce scattering losses and improve the Q-factor. Finally, the cavity inside the waveguide makes the device more compact, easier to fabricate and allows for better control over the coupling between the cavity and continuum modes. This means the cavities are part of the waveguide structure, compacting the proposed structure. The proposed structure also has the advantage of generating different Fano resonances along the propagation direction. The waveguide's chirping profile can vary along the propagation direction. This allows the different modes of the waveguide to be coupled in different ways, resulting in different types of Fano resonances as shown from as shown in the first panel of Fig. 3. The proposed way opens up new possibilities for the design of on-chip nanophotonic devices based on Fano resonances.

The fabrication of the proposed chirped PC in PCW is compatible with advanced CMOS technology in the electronics industry, making it a promising candidate for integrating electronic devices [28–30]. The rod-type design offers advantages for fabricating semiconductor active devices [31]. Electrochemical etching of silicon in hydrofluoric acid (HF) solutions is a well-established technique for fabricating 2D-PhCs [32,33]. The rods are fabricated on the silicon dioxide cladding using electron beam lithography (EBL) [34–36]. A thin ultra-small electron beam resist layer is coated onto the silicon dioxide. An ultra-thin conductive layer of aluminum is then evaporated onto the resist, followed by EBL exposure to define the desired PC pattern [37]. The resist is developed, and a thick layer of dust is deposited onto the patterned resist. Finally, the resist is lifted off, leaving behind the fabricated rods on the substrate [38]. Careful control of fabrication parameters is crucial to minimize errors and ensure the desired rod size and geometry. High-precision techniques, such as high-quality pattern and high-resolution lithography, are essential for achieving accurate fabrication.

3. Verification of robustness

The robustness of the forming rainbow refers to how well the rainbow maintains its shape and quality in the presence of defects. The proposed structure with a robust forming rainbow will be able to form rainbows even in the presence of these defects. Introducing defects into a PC can be useful for studying the robustness of rainbow trapping and developing new PC devices with improved performance. Once the defects have been introduced, the performance of the rainbow trap can be reasonable by measuring the field profile of the light. If the rainbow trap is still visible and efficient after the defects have been introduced, then this provides evidence that the rainbow trap is robust to defects. This is important because rainbow traps can be used in real-world applications where defects are inevitable. The proposed structure is not topological in the strict sense, as it comprises only one type of PC. However, it does exhibit some topological properties, such as robustness to defects and immunity to backscattering. This is due to the unique design of the semi-bilayer PC structure, which supports new photonic modes that are not possible in conventional PCs. Thus, our work also reflects that robustness does not always require topology.

To verify the robustness of the rainbow to small defects in the waveguide, three cases of defects are presented after each $5a$ along the propagation direction on the first row up and down the waveguide, as shown in Fig. 4(a). First, two rods are reduced by $0.05a$ as shown in inset (I) of Fig. 4(a); after $5a$ the second defect is introduced, two rods are shifted in ${\pm} y$ by ${\pm} 0.05a$ as indicated in inset (II). Finally, the third defect is two rods are shifted in ${\pm} {\boldsymbol x}$ by ${\pm} 0.05a$. These kinds of disordered mean that the PC is not perfectly periodic and that the irregularities are randomly distributed throughout the PC. Figures 4(b-d) show the intensity of the E field profile of the M1, M2, and M3 in the presence of defects. The frequencies of the three modes are unchanged compared with Figs. 2(a-c) unless the last frequency value of M2 is slightly affected to be at 226.1 THz instead of 225.9 THz. The efficiency of rainbow trapping is still high in the disordered PC, demonstrating that rainbow trapping is robust to disorder. To verify the robustness of the rainbow to larger defects, such as missing rods, dislocation, in the chirped waveguide and first row up and down of the waveguide as shown by the black arrows in the zoom-in of Fig. 4(e). We have shown only M3; it's seen that away from the dislocation, the rainbow is highly localized and protected by the surrounding PCs at the values 218.1 and 239.6 THz; it's the same values of localized modes in Fig. 2(c). While near the dislocation, as shown in panel 2 of Fig. 4(e), the rainbow is distorted and incomplete, significantly impacting the formation of rainbows and cannot even prevent the rainbow from forming. Rainbow trapping is a promising new area of research, but its potential applications can only be realized if their robustness and protection are confirmed.

 

Fig. 4. (a) Schematic diagram of the proposed three cases of defects in the first row up and down the waveguide of (I) two rods reduced by $0.05a$, (II) two rods are shifted in ${\pm} y$ by ${\pm} 0.05a$, (III) two rods are shifted in ${\pm} x$ by ${\pm} 0.05a$, the intensity of the E field profile of the (b) M1, (c) M2, and (d) M3, (e) M3 when two rods are missing in the chirped waveguide and in first row up and down of the waveguide.

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Despite introducing enlarged three types of defects in the range of ${\pm} 0.1a$ instead of ${\pm} 0.05a$ to study the robustness of the rainbow to larger disordered. The rainbow trapping phenomenon persists with minimal disruption to the trapped mode frequencies compared to the defect-free case, demonstrating their resilience to disorder. To quantify the impact of defects on rainbow trapping, we measured the frequencies of the trapped modes as shown in Fig. 5. The frequencies of M1, M2, and M3 remain virtually unchanged compared to the defect-free case, except for the last frequency value for M1 and M2. This slight deviation is attributed to the reduced length of cavities in the upper and lower rows, which resonates with the chirped PC and causes a localized disturbance in the rainbow pattern, as shown in the inset of Fig. 5(b). Nevertheless, the overall rainbow trapping behavior remains intact, demonstrating its exceptional robustness to defects. In addition to maintaining the trapped mode frequencies, rainbow trapping exhibits remarkable efficiency in the presence of defects. The intensity of the E field profile of M3 remains highly localized within the chirped PC, even when defects are introduced, as shown in the inset of Fig. 5(c) due to the isolation of the chirped rods, which acts as an isolated cavity which prevents the light to scattering out. This light confinement ensures efficient trapping and minimizes scattering losses, further demonstrating the robustness of our approach.

 

Fig. 5. The intensity of the E field profile with three cases of defects in the first row up and down the waveguide, two rods reduced by $0.1a,$ two rods are shifted in ${\pm} y$ by ${\pm} 0.1a$, and two rods are shifted in ${\pm} x$ by ${\pm} 0.1a\; $ of the (a) M1, (b) M2, and (c) M3.

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Multi-mode rainbow trapping with ultra-high Q factor holds immense potential for revolutionizing various fields, including laser technology, optical sensing, and optical communication. Multi-mode rainbow trapping can pave the way for developing next-generation lasers with unprecedented capabilities by harnessing the ability to trap and manipulate multiple wavelengths of light simultaneously, which can perform multiple tasks, such as identifying different chemicals in a sample or precisely sculpting materials with intricate patterns. Secondly, it can transform optical sensing by developing sensors capable of simultaneously detecting and identifying multiple substances. By utilizing the distinct frequencies of trapped light modes to interact with different materials, these sensors could revolutionize various industries, including environmental monitoring, medical diagnostics, and food safety inspections. Finally, it can empower the development of optical communication systems that transmit data at unprecedented rates. By exploiting the ability to trap and transmit multiple data channels simultaneously, these systems could significantly boost the capacity and speed of data transmission, enabling advancements in telecommunications.

4. Conclusion

In conclusion, the proposed structure is a promising new way to realize multi-mode rainbow trapping in PC. It is compact, efficient, versatile, and has many potential applications. A significant breakthrough is demonstrating multi-mode rainbow trapping at each point along the graded structure in the waveguide and Fano resonance with ultra-high-quality factor. By carefully designing the chirped PC, it is possible to create structures that can trap different frequencies of light at different positions, forming a “rainbow” of light. This can be used to create a variety of novel devices, such as optical filters, optical demultiplexers, and optical delay lines. The proposed structure is the first to demonstrate multi-mode rainbow trapping and ultra-high-quality factor Fano resonance at each localized mode. This is a significant advance in PC research, as it opens up new possibilities for developing high-performance optical devices.