All-Optical Switching Using Cavity Modes in Photonic Crystals Embedded with Hyperbolic Metamaterials

Abstract

Hyperbolic metamaterials (HMMs) are highly anisotropic materials with the unique property of generating electromagnetic modes. Understanding how these materials can be applied to control the propagation of light waves remains a major focus in photonics. In this study, we inserted a finite-size HMM rod into the point defect of two-dimensional photonic crystals (PhCs) and investigated the unique cavity modes of this hybrid system. The HMM enhances the efficiency of the cavity system in controlling light transmission. Numerical results demonstrate that the cavity modes based on HMMs can be categorized into various types, showing high Q-factors and promising potential for resonant modulation. Furthermore, the switching performance of the cavity with an HMM rod was examined, revealing that the finite-size HMM modes are highly frequency-sensitive and suitable for nonlinear controlled all-optical switching. These switches, characterized by low power consumption and high extinction ratios, are highly suitable for integration into photonic systems. Our investigation on the new type of HMM cavity illustrates that anisotropic materials can be effectively applied in cavity systems to generate highly efficient modes for filtering and switching.

1. Introduction

Nonlinear optics plays a crucial role in spectrum control, ultrashort pulse generation, all-optical signal processing, optical computing, microfabrication, and biophotonics, making it a vital area of modern optics [1,2,3,4,5,6,7]. Achieving stronger and faster nonlinear effects has been a primary focus of nonlinear optics research. One effective method for enhancing and controlling nonlinear optical responses is through surface plasmon polaritons (SPPs) [8,9,10,11]. SPPs are surface electromagnetic waves that propagate along metal–dielectric interfaces, caused by coherent oscillations of conduction electrons near the surfaces of metal nanostructures [12,13]. This resonance is influenced by the size and shape of the metal particles [14,15]. SPPs can enhance nonlinear optical effects in several ways. First, the coupling of light with SPPs generates a strong local electromagnetic field, enhancing nonlinear processes within the metal and its surrounding dielectric [14]. Second, SPPs are highly sensitive to the refractive index of metals and adjacent media, allowing small changes in the refractive index near metal surfaces to significantly impact SPPs [16,17]. This sensitivity can be leveraged to control light in nonlinear optical systems. Lastly, the response time of SPPs can be as short as a few femtoseconds [9], enabling ultra-fast optical signal processing. Metamaterials, with their periodically arranged metal microstructures, can excite SPPs to effectively enhance and control optical nonlinear effects. Specific metamaterials can support special nonlinear processes, further advancing the field of nonlinear optics.

Hyperbolic metamaterials (HMMs) are a unique class of metamaterials characterized by strong anisotropic properties in electromagnetic fields. Within certain frequency ranges, the in-plane permittivity of HMMs is positive, while the out-of-plane permittivity is negative. This results in hyperbolic dispersion curves, from which HMMs derive their name [18,19,20,21]. These hyperbolic dispersion curves give rise to several intriguing properties, such as epsilon-near-zero (ENZ) [22], a large Purcell factor [23,24], and enhanced spontaneous emission [25,26]. HMMs are typically constructed from nanostructures using specific metals that support surface plasmon polaritons (SPPs). Recently, significant attention has been given to the nonlinear properties of HMMs, leading to numerous studies and reported phenomena [27,28,29,30,31].

HMMs, known for their high anisotropy, can be considered as effective uniaxial anisotropic crystals. The dielectric constants of uniaxial anisotropic materials are represented by tensors. Consequently, the dielectric tensor of HMMs is expressed by Equation (1):

$${\epsilon }_{\mathrm{eff}}=\left(\begin{array}{ccc}{\epsilon }_{\parallel }& 0& 0\\ 0& {\epsilon }_{\parallel }& 0\\ 0& 0& {\epsilon }_{\perp }\end{array}\right),$$

(1)

where ${\epsilon }_{\parallel }$ is the in-plane (transverse) component of the permittivity tensor and ${\epsilon }_{\perp }$ is the out-of-plane (longitudinal) component. The equifrequency surfaces of HMMs can be calculated using the following equation:

$${\displaystyle \frac{k_x^2+k_y^2}{{\epsilon }_{\parallel }}}+{\displaystyle \frac{k_z^2}{{\epsilon }_{\perp }}}={\left({\displaystyle \frac{\omega }{c}}\right)}^2,$$

(2)

where k_x, k_y, and k_z are the wave vectors in the x, y, and z directions, respectively. The symbol $\omega$ represents the frequency of the incident light. Equation (2) indicates that when ${\epsilon }_{\parallel }>0$, ${\epsilon }_{\perp }<0$, or ${\epsilon }_{\parallel }<0$, ${\epsilon }_{\perp }>0$, the equifrequency surfaces of HMMs exhibit two distinct shapes. This classification categorizes HMMs into type-I and type-II, depending on whether ${\epsilon }_{\parallel }>0$, ${\epsilon }_{\perp }<0$, or ${\epsilon }_{\parallel }<0$, ${\epsilon }_{\perp }>0$.

Krishnamoorthi et al. proposed that the effective permittivity of HMMs can transition from positive to negative values and reach zero at certain frequencies [31]. Electromagnetic waves propagating near these frequencies exhibit large wave vectors, leading to a strong resonance within the material. These transmission modes, known as high wave vector modes, are highly sensitive to parameters such as the effective refractive index and incident light wavelength. They find application in the design of optical devices like sensors and waveguides. Sreekanth et al. employed a structure combining optical gratings and HMMs to develop a highly sensitive biosensor [32]. Optical gratings couple incident light to the surface of HMMs, exciting high wave vector modes. The biosensor operates by detecting changes in the permittivity of a bio-liquid covering the optical grating, which alters the coupled light’s wave vector. Experimental results demonstrate the sensor’s sensitivity to variations in the liquid’s refractive index. The strong anisotropy of HMMs gives them a unique response to incident light with different polarizations. This property enables applications such as beam splitting, as demonstrated by Hu et al., using varied HMM arrangements to achieve controllable image changes based on polarization and wavelength [33]. Moreover, the nonlinear response of HMMs is notable due to their unique properties. Yu et al. showed that equifrequency curve shapes can be modified by incorporating graphene into certain layers of HMMs [34]. This capability can be harnessed in applications such as beam steering and far-field imaging.

Researchers have identified numerous unique properties of HMMs based on their permittivity tensors. However, much of the research has focused on one-dimensional HMMs composed of multi-layers of metal and dielectric materials, typically treated as infinite planes [35,36]. Two-dimensional arrays of metal nanorods can also constitute HMMs, but they are similarly analyzed as infinite planes. The properties of finite-size HMMs have not yet been extensively explored, despite their potential advantages in integrated systems. Unlike infinite HMM planes, finite-size HMM rods may exhibit distinct effects on light propagation. Consequently, the optical modes of cavities containing finite-size HMMs remain poorly understood. Given HMMs’ effective nonlinear control over light propagation, there is considerable anticipation regarding their application in optical switching to enhance the switch performance. Investigating the optical properties of HMM cavities could thus yield significant benefits.

This article introduces a novel resonance cavity based on photonic crystals (PhCs), employing finite-size HMMs to enhance efficiency and extinction ratios. The cavity features a rectangular finite-size HMM rod, demonstrating unique capabilities in controlling light transmission. We investigate the transverse and longitudinal components of the effective permittivity of HMMs with varying filling factors, identifying distinct cavity modes supported by the PhC. HMM modes exhibit several distinct properties compared to conventional modes. Firstly, HMM modes are surface plasmonic modes, which enhance nonlinear effects and generally result in higher Q-factors. This increased nonlinearity is a significant difference from conventional modes, which often do not exhibit such pronounced effects. Additionally, HMM modes are characterized by anisotropy, meaning their properties vary depending on the direction of propagation. This anisotropic behavior contrasts with the typically isotropic nature of conventional modes. The anisotropy of HMM modes also leads to unique field distributions that are not observed in conventional photonic modes. Additionally, we analyze the transmission spectrum of the HMM cavity across different filling factors. Based on these findings, we propose an all-optical switch utilizing HMM modes. This switch leverages the unique properties of HMM modes and the high Q-factor of the PhC system, demonstrating an excellent performance in terms of power consumption and extinction ratio. These HMM modes hold promise for integration into photonic systems. We have investigated the new type of HMM cavity, which, to our knowledge, has not been explored before. This cavity exhibits unique anisotropic properties, leading to the emergence of novel resonance modes. Additionally, the HMM cavity demonstrates the ability to modify its properties by adjusting the filling factor, offering a new level of adjustability. Since HMMs can support surface plasmonic modes, they enhance Kerr nonlinear effects. Compared to conventional cavities, these cavities are more efficient in modulation applications. Our proposed cavity system illustrates that anisotropic materials can be effectively applied in cavity systems to generate highly efficient modes for filtering and switching.

2. Materials and Methods

The schematic of the PhC cavity containing HMMs is shown in Figure 1a. The PhC consists of square dielectric rods in a silicon (Si) background medium, with a refractive index of n_Si = 4.1020 and an extinction coefficient of k_Si = 0.043853. The cubes in Figure 1a represent air holes with a refractive index of n₀ = 1. The lattice constant of the PhC is a = 0.2 μm, and the side length of the air holes is r = 0.8a. A line-defect waveguide is formed by filling one row of air holes. In the middle of the waveguide, two air holes and one HMM rod create the resonant cavity. Calculations show that the PhC has a bandgap between 370 THz and 540 THz, which is the operating band for the waveguide with the HMM cavity. The HMM rod consists of metal and dielectric layers, with Ag as the metal and Si as the dielectric. The thicknesses of the metal and dielectric layers are denoted as d_m and d_d, respectively. The length of the HMM unit cell, d_m + d_d, is 0.075 mm, representing the combined thickness of the dielectric and metal layers. Each HMM rod contains 4 layers of metal and 4 layers of dielectric, giving the HMM rod a side length of r₁ = 0.3a. The effects of varying the number of layers have been discussed in our previous work [37]. We conducted full-wave simulations to investigate HMM rods with 4, 5, and 6 layers, comparing these results with those obtained using the effective medium approximation. The findings show that the eigenmodes and frequencies calculated by the effective medium method closely match those obtained from full-wave simulations. This confirms the effectiveness of the effective medium approximation in describing HMMs. Furthermore, as the number of layers increases, the results from the two methods converge more closely. Based on our previous work, 4 layers are sufficient for the effective medium approximation to accurately capture the properties of the HMM rod. As shown in Figure 1b,c, the layers of HMM can be oriented in the y–z plane or x–y plane. The structure shown in Figure 1c, with layers in the x–y plane, is easier to fabricate. In this configuration, layers of metal and dielectric are arranged in the x–y plane, and once multiple layers of the HMM are made, the rod arrays can be constructed using photoetching technology. This structure is more practical and shares similar properties with the one shown in Figure 1b. The following context will use the structure shown in Figure 1c, with layers in the x–y plane, while the HMM is described by Equation (1).

The geometry of the proposed structure is designed to meet the specific requirements of the cavity system. The structure consists of a waveguide and a cavity containing an HMM rod, which is composed of a multi-layered structure of dielectric and metal materials. This multi-layered design of the HMM rod induces anisotropic properties in scattering waves, leading to the generation of anisotropic modes within the cavity. The waveguide serves as a platform for the resonant cavity. The proposed system can be fabricated using a method that has been successfully employed by researchers to create photonic hypercrystals [38]. These techniques involve growing HMM substrates using a Kurt Lesker PVD 75 electron beam evaporation system and using focused ion-beam milling to etch holes into the HMM structure. In our proposal, the photonic crystal rods can be replaced with HMM rods to meet the fabrication requirements while still achieving similar results. The key is to ensure that the HMM rods exhibit the same properties as dielectric rods, which can be accomplished by adjusting their size and ensuring their compatibility with the waveguide system. Since our focus is on the cavity resonance of the HMM rod, the photonic crystal structure could also be substituted with a bulk HMM coupled with external fiber waveguides. This alternative approach may be easier to realize and would yield similar results, as we are primarily concerned with the resonance properties. In essence, any cavity-waveguide system can serve as a platform for the HMM rod to generate cavity modes, yielding comparable outcomes. We chose the photonic crystal platform to minimize background noise and better observe the resonance performance of the HMM rod.

The PhC cavity system with an HMM rod can be analyzed using the plane-wave expansion method and the finite-element method for electromagnetic waves. This work considers only TM waves with H_z polarization, where light propagates in the x–y plane. For the HMM rod shown in Figure 1c, the transverse and longitudinal components of the effective permittivity of the HMM have the following relationship: ${\epsilon }_{\parallel }={\epsilon }_{xx}={\epsilon }_{yy}$, ${\epsilon }_{\perp }={\epsilon }_{zz}$, corresponding to Equation (1). Note that if the structure shown in Figure 1b is applied, with the HMM layers in the y–z plane, the polarization and wave vector should be adjusted accordingly. The effective permittivity of the HMM, as shown in Equation (1), is given by the following equations:

$${\epsilon }_{xx}={\epsilon }_{yy}={\displaystyle \frac{{\epsilon }_{\mathrm{m}}{\epsilon }_{\mathrm{d}}}{(1-f){\epsilon }_{\mathrm{m}}+f{\epsilon }_{\mathrm{d}}}},$$

(3)

$${\epsilon }_{zz}=f{\epsilon }_{\mathrm{m}}+(1-f){\epsilon }_{\mathrm{d}},$$

(4)

where ${\epsilon }_{\mathrm{m}}$ and ${\epsilon }_{\mathrm{d}}$ are the permittivities of the metal (Ag) and the dielectric (Si), respectively. The permittivity of Ag is determined from experimental data and can also be described by the Drude model. The coefficient f in Equations (3) and (4) is the ratio of the thickness of the metal layer $d_{\mathrm{m}}$ to the length of the HMM unit cell, defined as the filling factor:

$$f=d_{\mathrm{m}}/(d_{\mathrm{d}}+d_{\mathrm{m}})$$

(5)

Note that the silver layers in HMM exhibit nonlinearity under strong fields, which has been accounted for and utilized to achieve tunability of the system.

According to Equations (3) and (4), the transverse and longitudinal components of the effective permittivity of HMMs can be calculated as functions of frequency. At different filling factors, a transition point in the effective permittivity may exist. For instance, with a filling factor of 0.3, the transition point occurs at 331 THz, where the longitudinal component of the effective permittivity ${\epsilon }_{\perp }$ changes from a negative to a positive value. This transition marks a shift in the HMM’s properties: before the transition, it behaves as an effective type-I HMM; after the transition, it acts as an effective dielectric. This property is crucial for modifying wave propagation in HMMs. Not only does the transition point of ${\epsilon }_{\perp }$ change, but the transition point of ${\epsilon }_{\perp }$ also shifts with varying filling factors, altering the HMM type and its properties. Near the transition point of ${\epsilon }_{\perp }$, its value changes dramatically, indicating that ${\epsilon }_{\perp }$ significantly affects the HMM’s properties. This sensitivity near the transition point can be observed in Figure 2d–i. For example, in Figure 2f, with a filling factor of f = 0.6, the transition point is at 409 THz, where ${\epsilon }_{\perp }=0$ and the curve of ${\epsilon }_{\perp }$ has a steep slope. The HMM transitions from a type-I HMM to an effective metal. Typically, an HMM showing properties of an effective metal does not support propagation modes. Near the transition point, the permittivity changes dramatically. For example, the transverse component of the permittivity near 4.10 THz, as shown in Figure 2f, undergoes a significant shift. A small frequency change from 4.00 THz to 4.20 THz results in a permittivity change from 275 to −275, which substantially alters the properties of the HMM and modifies wave propagation. Operating near the transition point allows for easier control of wave propagation, enabling the HMM to function effectively as a switch.

When the filling factor is 0.1, the HMM shows positive ${\epsilon }_{\parallel }$ and ${\epsilon }_{\perp }$ at frequencies between 171 THz and 600 THz, behaving as an effective dielectric. As the filling factor increases, the effective permittivity curves change. For a filling factor of 0.2, ${\epsilon }_{\parallel }$ remains positive, but the frequency at which ${\epsilon }_{\perp }$ is zero shifts from 171 THz to 254 THz. Now, the HMM exhibits different properties at different frequencies. From Figure 2, we can see that the HMM can assume four different types: (a) effective metal, such as the HMM with a filling factor of 0.8 at a frequency of 400 THz; (b) effective dielectric, such as the HMM with a filling factor of 0.2 at a frequency of 400 THz; (c) type-I HMM, such as the HMM with a filling factor of 0.5 at a frequency of 400 THz; and (d) type-II HMM, such as the HMM with a filling factor of 0.3 at a frequency of 400 THz. Each type of HMM supports different modes and has distinct properties. Based on the properties of HMMs shown in Figure 2, we can investigate the characteristics of a cavity containing a finite-size HMM rod according to the different types of HMMs. Given the rich properties of HMMs, the HMM cavity is expected to support various modes. Moreover, the dramatic change in the ${\epsilon }_{\perp }$ curve indicates enhanced dispersion and nonlinearity in controlling wave propagation. Switching based on the HMM cavity will exhibit high-frequency sensitivity, resulting in high efficiency.

3. Results and Discussion

Through mode finding processes, we identified multiple modes supported by the cavity system, as depicted in Figure 3. As expected, these modes fall into four categories where the HMM acts as an effective dielectric, effective metal, type-I HMM, and type-II HMM, respectively. These modes span different orders and exhibit distinct field profiles. It is important to note that some modes exist within the cavity system but cannot be excited by the waveguide due to symmetry requirements. The system exhibits even symmetry, allowing only even modes to be generated, as shown in the figure. Odd modes could be excited using a different type of source and configuration that possesses odd symmetry. When the HMM acts as an effective dielectric, as shown in Figure 3a–c, the HMM rod shapes the flow of light similarly to a conventional dielectric rod. These modes exhibit minimal absorption by the HMM rod, minimizing power loss and influencing the reflection and transmission of waves. Conversely, when the HMM acts as an effective metal, the HMM rod blocks wave propagation. Modes of this type typically exhibit field distributions that avoid contact with the HMM rod to minimize losses, as exemplified in Figure 3d where most of the mode’s energy resides on the sides rather than the center where the HMM rod is located. In type-I or type-II HMM configurations, as shown in Figure 3e, Figure 3d, and Figure 3g, Figure 3h, respectively, the modes exhibit oriented field distributions along one axis. In a type-I HMM cavity, the field distribution primarily aligns along the x-axis since modes along the y-axis would be absorbed by the HMM rod. Conversely, a type-II HMM cavity exhibits the opposite behavior with field distributions predominantly along the y-axis.

Different types of modes exhibit distinct properties. Figure 3a–c depict dielectric modes of various orders characterized by high transmission within the cavity system. In contrast, Figure 3c,d show HMM modes, which typically exhibit lower transmission. However, due to the HMM’s high sensitivity to frequency, these modes offer advantages in achieving switching with a large extinction ratio. The HMM cavity system can capitalize on these modes, making them unique in controlling the propagation of light. Metal modes within the HMM cavity system are rarely found because effective metal HMMs typically absorb light heavily, reducing power output within the cavity. However, the mode shown in Figure 3d is an exception. As described in the previous paragraph, this mode has a unique field distribution that avoids energy absorption by the HMM rod. It is important to note that the modes depicted in Figure 3 can be generated under various conditions, which are not limited to the parameters provided. For instance, the mode shown in Figure 3d can also be generated at filling factors of 0.4, 0.5, and 0.7, maintaining a frequency of 466 THz.

The properties of the HMM cavity were further investigated by calculating the transmission spectrum. Based on the four types of modes identified, four distinct types of resonant peaks were observed in the transmission spectra. The first type consists of resonant peaks corresponding to dielectric modes, which occur when the HMM acts as an effective dielectric. These resonant peaks are observed at the following values: 455 THz and 492 THz for f = 0.1 (Figure 4a); 452 THz, 466 THz, and 488 THz for f = 0.2 (Figure 4b); 438 THz, 447 THz, 482 THz, and 539 THz for f = 0.3 (Figure 4c); and 417 THz, 440 THz, 466 THz, 472 THz, and 509 THz for f = 0.4 (Figure 4d). These resonant peaks exhibit either a large full width at half maximum (FWHM), indicating small Q-factors, or a small FWHM, indicating large Q-factors. Modes with larger Q-factors are particularly suitable for optical switching applications due to their high extinction ratios. The second type consists of resonant peaks corresponding to metal modes, which correspond to the unique modes in Figure 3g–i observed when the HMM acts as an effective metal. These resonant peaks are found at the following values: 464 THz and 481 THz for f = 0.6 (Figure 4f); 466 THz, 474 THz, and 522 THz for f = 0.7 (Figure 4g); 465 THz, 469 THz, and 515 THz for f = 0.8 (Figure 4h); and 461 THz, 466 THz, 504 THz, and 539 THz for f = 0.9 (Figure 4i). Resonant peaks of metal modes typically exhibit large FWHMs. Comparing mode profiles reveals that metal modes experience higher material losses than dielectric modes, indicating smaller Q-factors for the cavity. The third type consists of resonant peaks corresponding to type-I HMM modes, observed when the HMM acts as a type-I HMM. These resonant peaks are observed at 427 THz and 466 THz for f = 0.5 (Figure 4e). The fourth type consists of resonant peaks corresponding to type-II HMM modes, observed when the HMM acts as a type-II HMM. These resonant peaks are observed at 496 THz for f = 0.5 (Figure 4e). Resonant peaks of type-I HMM and type-II HMM modes typically exhibit small FWHMs and high sensitivity to frequency, showing great potential for application in optical switching. There are several transmission peaks in Figure 4 with very narrow widths. For instance, the resonance peak at 466 THz, as shown in Figure 4e, corresponds to a mode with a Q-factor of 932, indicating a strong potential for use in filtering and switching applications. The high Q-factors of the modes observed in the transmission spectra indicate their strong potential for use in highly sensitive optical switching. When nonlinearity is introduced, the resonant peaks will shift, and the high Q-factor or narrow width of these peaks suggests that even a slight shift will result in a dramatic change in wave amplitude. For example, in Figure 4e, if a shift occurs, the resonant peak at 466 THz may disappear, causing the transmission at 466 THz to drop to nearly zero.

Hybrid modes are also observed in the HMM cavity system. As shown in Figure 4a, the peak at 455 THz exhibits two modes. Hybridization can occur between modes of the same type or between modes of different types. This phenomenon arises because the cavity system itself is hybrid, incorporating both dielectric rods and an HMM rod. When the cavity system supports different modes at the same frequency, hybrid modes can form. Typically, this results in a mode with a low Q-factor and another with a high Q-factor, leading to the occurrence of Fano resonance.

The effective permittivity of the HMM can be altered by the filling factor, as depicted in Figure 5. The figure illustrates how the effective permittivity of the HMM rod inside the cavity varies with the filling factor. At a specific operating frequency, the HMM can exhibit different types of behavior depending on its effective permittivity value. For example, as shown in Figure 5a, when f is less than 0.38 at 400 THz, the HMM behaves as an effective dielectric; when f is between 0.40 and 0.60, it behaves as a type-II HMM; and when f exceeds 0.62, it behaves as an effective metal. Thus, the properties of HMMs can undergo significant changes. This characteristic of HMMs can be leveraged to design components tailored to specific operating frequencies. The transverse component of the permittivity changes dramatically with modifications to the filling factor. For example, near f = 0.55 in Figure 5b, a small filling factor change of 0.01 causes a significant permittivity shift from approximately 100 to −100, demonstrating the high adjustability of the HMM.

Figure 5 underscores the versatility of metamaterials in optical component designs and suggests the potential for modifying HMM properties by adjusting their structure. As the filling factor (f) changes, the properties of the HMM transition from behaving like a dielectric to a type-II HMM, and finally to a metal. They have different properties listed in this section. (a) Effective dielectric: When the HMM behaves as an effective dielectric, its properties resemble those of conventional dielectric materials. In this regime, the HMM allows waves to propagate through it, and surface plasmons are not generated. The material supports typical dielectric modes without significant absorption or anisotropy. (b) Type-II HMM: As the filling factor increases, the HMM transitions to a type-II HMM, which exhibits anisotropic properties. In this regime, the HMM supports a limited number of modes due to its directional dependence. Some modes may not satisfy the required profile or may experience heavy absorption, leading to a reduced number of supported modes compared to the dielectric regime. (c) Effective metal: At higher filling factors, the HMM behaves like an effective metal. In this regime, wave propagation inside the material is not supported, but surface plasmons are generated. This characteristic allows the HMM to enhance nonlinear effects, making it useful for applications such as nonlinear optical switching.

Furthermore, we introduce nonlinearity into the cavity system to investigate the nonlinear response of the HMM cavity. In the previous section, we used Ag as the material for the metal layers of the HMM rod. Ag itself exhibits nonlinear behavior under strong fields, characterized by a Kerr coefficient of 2.49 × 10⁻⁸ + 7.16 × 10⁻⁹i. Assuming a homogeneous pump, the effective permittivity of the HMM can be calculated using Equations (3) and (4), which vary with different pump powers. Figure 6a–c depict the effective permittivity of HMMs as a function of frequency. Note that infrared light is used as the pumping source, with the amplitude represented by the square of the electric field of the infrared light. Therefore, the electric field unit (V/m) is used to describe the amplitude of the pump. It is evident from the figures that when the pumping power reaches 1 × 10⁸ V/m, the effective permittivity of HMMs changes significantly. Consequently, the transmission characteristics of the cavity system also undergo dramatic changes, as illustrated in Figure 6d,e. This phenomenon can be leveraged for tunable light filtering applications, as evidenced by the modification of the center frequency and bandwidth of the transmission curves in Figure 6d,e.

The results shown in Figure 6 suggest that high-performance switches can be realized using the HMM cavity for light signal transmission. For example, at a frequency of 438 THz, with a low pump power, the transmission is 5.085 × 10⁻⁶ and shows minimal power at the output port. Increasing the pump power to 7 × 10⁷ V/m boosts the transmission power to 5.502 × 10⁻⁵, which is ten times the transmitted power without pumping—the HMM cavity switches from an “off” to an “on” state. Further details are illustrated in Figure 7. At an operating frequency of 440 THz, the transmission of the HMM cavity varies with the pumping power, achieving an extinction ratio of 16.6 dB, demonstrating its promising application in all-optical integrated systems.

4. Conclusions

In summary, we applied a finite-size HMM rod in the point defect of two-dimensional PhCs and investigated the unique cavity modes of the HMM. The HMM enhances the efficiency of the cavity system in controlling the transmission of light. Numerical results demonstrate that the cavity modes based on the HMM can be categorized into four types: dielectric modes, metal modes, type-I HMM modes, and type-II HMM modes. Some of these modes exhibit high Q-factors, indicating promising potential for resonant modulation. Furthermore, the switching performance of the cavity with an HMM rod has been examined, and the results show that the finite-size HMM modes are highly sensitive to frequency changes. The proposed system can be applied in nonlinear controlled all-optical switching, achieving an extinction ratio of 16.6 dB. The achievement of an extinction ratio of 16.6 dB indicates a significant contrast between the ON and OFF states of the HMM cavity. This high extinction ratio ensures that the cavity effectively suppresses unwanted signals while clearly distinguishing between different states, which is crucial for reliable switching operations. In addition to its compact design, the high extinction ratio makes the HMM cavity highly suitable for integrated photonic systems, where space and efficiency are paramount. This capability allows the HMM cavity to function effectively as a switch within compact photonic circuits, improving the overall performance and integration in photonic devices. Future research and development in this field will focus on two key aspects: Firstly, integrating such high-performance switches into integrated systems using the structures proposed here. By adjusting the filling factor f, any desired operating frequency can be achieved; Secondly, further investigations into complex HMM cavity systems need to be carried out. For example, exploring multiple coupled cavities could lead to interactions between different modes, potentially giving rise to new phenomena.