1. Introduction

High-Q-factor optical resonances are widely used for various applications such as high-sensitivity sensing [1–3], lasing [4–6], switching [7–9], filters [10,11], and nonlinear effects [12,13]. The inevitable losses in metals hindered the practical application of plasmonic-to-Mie resonances in dielectric nanoparticles. Therefore, higher Q factors are desirable for many all-dielectric nanophotonic applications [14]. The Q factor can be enhanced by increasing the size of the resonator [14]; for example, by confining waves in photonic crystal cavities [15] or by high-order F-P modes in elongated rods [16]. Another method involves arranging several resonators in space and exciting the collective modes [17]. An alternative method for improving the Q factor is the use of an anapole mode with spectrally overlapping electric and toroidal dipole modes [18]. However, the Q values of traditional photon structures are limited, even for large sizes.

In recent years, bound states in continua (BICs) have been employed as a new method for trapping and confining resonant optical modes from quantum mechanics [19], to photonics [19–22] in interfering systems [20,21]. The BIC represents a perfectly localized state with no leakage energy, even when it coexists with a continuous spectrum of radiating waves, which effectively manipulates the light–matter interaction and generates an ultrahigh Q-factor resonance [20,21]. In dark resonance mode, the optical BIC is invisible with zero linewidth and an infinite Q-factor in its optical spectra [23]. When it collapses to the quasi-BIC, it can be experimentally observed with a highly enhanced Q factor [23–26] and other interesting phenomena [20,21]. These unique features distinguish BICs from the traditional optical modes and significantly improve the performance of optical devices. However, BICS are locked in the passband of the continua in an open system, and these continua can be regarded as noise in practical applications, thus reducing the signal-to-noise ratio. Moreover, their transmission line shapes and background modes are difficult to control and filter, which limits their application.

Nevertheless, the manipulation of the line shape and Q factor of quasi-BICs has attracted considerable attention in a wide range of applications such as phase modulation [27], light control light [28], biosensing [29], enhancing optical force [30], and chiral response [31]. The control methods may include the variation of geometric parameters, such as dielectric metasurfaces [32], particle clusters [33], Dirac semimetal metasurfaces [34,35], and silicon film photonic crystal (PhC) slabs [36], and schemes using active tunable materials, such as liquid crystals [37,38] and graphene [39,40]. However, quasi-BIC line shapes and Q factors cannot be tuned separately, which limits the application of BICs. In particular, Ma [34] and Liao [23] demonstrated a method to realize both BIC-Fano and BIC-EIT (electromagnetically induced transparency) in a cell. However, the Fano and EIT modes cannot be tuned from one mode to another, and the tuning range of the asymmetry factor q is limited, which hinders its further application.

Therefore, the design of fully controlled SBS modes in the bandgap is of interest for various applications. These SBSs exhibit quality factors that approach infinity and are free from the background noise of the continuum. The characteristics of SBSs and BICs are the same; however, an SBS exists within a bandgap, whereas a BIC is in a continuum of waves. This is an important difference between the states and produces a considerable advantage for the SBSs over BICs; that is, the SBSs are more practical for real applications.

We employed a two-dimensional PhC cavity with a rectangular defect column to generate SBSs. The cavity had top-bottom symmetry and supported an inverse-phase field pattern or dipole field. In addition, fields or waves were confined to waveguides that can only propagate in one direction.

These SBSs require only a small volume, and their quality factors are theoretically as infinite as those of BICs. Moreover, there is no stray interference frequency component in the wide frequency range next to the SBS modes, which is conducive to obtaining a high signal-to-noise ratio; thus, they have the potential for practical applications. However, ideal SBSs are not visible, which again limits their application. The symmetry can be slightly broken to obtain quasi-SBSs that are visible at the output port and have limited albeit very high quality factors; thus, they can be used for applications. A quasi-SBS is a very-high-quality-factor EIT-like mode; therefore, we refer to it as the SBS-EIT mode to distinguish it from ordinary EIT modes. Furthermore, we used SBS to interfere with the bright mode to generate SBS electromagnetically induced reflection (SBS-EIR) and SBS Fano resonance. As an example of a possible application, an optical switch based on SBS-EIR was demonstrated.

The remainder of this paper is organized as follows. In Section 2, the generation mechanism of the SBSs is presented. In Section 3, a schematic for generating the SBSs, SBS-EIT, SBS-EIR, and SBS-Fano modes and the results are shown. In Section 4, the design of an optical switch based on the SBS-EIR mode is presented. Finally, Section 5 presents the conclusions of this study.

2. Mechanism of the generation of SBSs

The operating mechanism is as follows. There may be many defect modes in the bandgap in a defect-PhC cavity, wherein certain modes are odd modes. These odd modes can be viewed as two sources with nearly zero distance and a phase difference of 180 °, that is, there is phase inversion, between them, which is referred to as the dipole source. Moreover, the far-field waves are confined to propagating in one direction. Therefore, the waves from the two phase-inverted sources will interfere with each other in the far region away from the sources, as shown in Fig. 1. Thus, these waves are localized in a small region near the two sources, forming an SBS. As is known, a dipole can exhibit radiation as well. However, if the dipole field is confined by a waveguide, that is, in one direction, the output field or far field can disappear. For simplicity, we assumed far fields (at the center of the output port, where the output power is primarily obtained) from the two sources in the dipole source, as follows:

 

Fig. 1. Near and far fields of a dipole source that are confined to propagate only in the horizontal direction. The thick black horizontal lines represent the confining waveguide. The far field becomes zero, that is, no energy travels outside; the fields are confined in the near field region only owing to the interference of the waves from the two phase-inverted sources of the dipole source. The localized field near the dipole source form the SBS mode.

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As the far-field waves are confined to propagate in the x direction, for the far field waves, we obtain

Therefore, the total field in the far-field region is expressed as

However, near the dipole source, the fields (i.e., near fields) of the two sources are

Therefore, the total field in the region near the dipole is expressed as

As shown in Fig. 1, for near fields, ${\overrightarrow k _1}$ and ${\overrightarrow k _2}$ are different in direction; however, they have the same magnitudes for waves of the same frequency, and the position vectors ${\overrightarrow r _1}$ and ${\overrightarrow r _2}$ are different in both direction and magnitude. Thus, based on Eq. (8), evidently, the total near field is not zero. Consequently, the fields are localized near the dipole source, forming an SBS.

These SBSs require only a small volume, and their quality factors are theoretically as infinite as those of BICs. Moreover, there is no stray interference frequency component in the wide frequency range next to the SBS modes, which is conducive to obtaining a high signal-to-noise ratio; thus, it can facilitate practical applications.

However, ideal SBSs are not visible or detectable, which limits their applications. The symmetry can be slightly broken to obtain quasi-SBSs that are visible at the output port and have limited albeit very high quality factors; thus, they can be used for applications. By shifting the defect column slightly upward ($\Delta y$), the two waves (at the output center, where the output power is primarily obtained) from the dipole source can be expressed as

Thus, the total field in the far region will be

Noting that ${k_1} = {k_2}$, $- {\beta _1} = {\beta _2}$, we have

For a very small shift $\Delta y$, we have

This indicates that the output wave field was not zero; thus, it became visible or detectable.

Even modes are guided modes that can exist in the structure of the bandgap of a photonic crystal. Therefore, we can utilize the even modes to interfere with the SBS mode to obtain the Fano resonance mode (SBS-Fano mode) and EIR resonance mode (SBS-EIR mode) under the condition that the odd and even modes degenerate at a certain desired frequency. Furthermore, the structure parameters can be tuned to obtain SBS, SBS-Fano, and SBS-EIR modes, separately; that is, the transmitted line shapes can be steered with more degrees of freedom, or the line-shape asymmetry parameter q can be tuned from -∞ (quasi-SBS resonance) to non-zero normal value (SBS-Fano resonance), 0 (SBS-EIR resonance), and +∞, which is not reported in literature. Moreover, the Q-factor can be controlled by adjusting the offset of the defect column along the y axis. In addition to the full control of a single quasi-SBS, the transmitted line shapes and Q-factors of multiple quasi-SBSs can be independently steered in a similar manner.

3. Schematic and results

Figures 2(a) and 2(b) show the schematics and parameter symbols, respectively, of the proposed structure. Circular cylinders or columns of 5 × 15 were arranged in air to form a two-dimensional PhC and the central circular column was replaced with a rectangular column (defect column). The height of the columns in the z direction was much larger than the lattice constant of the PhC. The structure exhibited a mirror symmetry about the y axis at its center. Such a structure can only allow a wave input in the x direction to tunnel through the cavity in the x direction in the bandgap of the PhC. Therefore, it satisfies the condition for light confinement along a single axis, which is an important condition for generating SBSs using the aforementioned mechanism. For a rectangular defect column set up at the center in the y direction, that is, when the structure exhibits y-mirror symmetry about the x axis in the structure center, the photonic crystal cavity generates ideal SBS modes. When the rectangular column is shifted slightly along the y direction, quasi-SBS modes can be generated. There is no apparent waveguide in the structure. However, because the wave is input from the left side along the x direction, the proposed structure can only allow the transmission of the wave in the x direction based on the optical tunneling effect in the x direction for the PhC cavity. Therefore, this structure has an implied waveguide along the x direction for a specified input wave.

 

Fig. 2. (a) Schematic of the PhC cavity structure. (b) Cross-sectional view of the cavity structure. (c) TE band map of the simplified 2D perfect PhC along the Γ-X direction. (d) TE band map of the defect cavity mode TE₃₁ when h = 0.4a, w = 1.01a in the bandgap along the Γ-X direction.

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For simplicity, the simulation model was set to two dimensions (2D) as the height in the z direction was infinite. A TE plane wave with electric field in the z direction propagated from the port “in” to the port “out” along the x direction with wave vector k_x. The cavity was settled between two extremely large metal plates at the top and bottom in the x-y plane to sustain the propagating boundary condition of the TE wave, and perfectly matched layers were used to cover the cavity to absorb electromagnetic waves that scattered to free space. The lattice constant of the square lattice PhC was a and dielectric silicon columns had permittivity ε = 12.25. Further, the width (w) and height (h) of the rectangular defect pillar were the most important parameters for obtaining different SBS modes. The offset or shift of the rectangular defect column in the y direction, $\varDelta$y, is the parameter considered to control the Q factor of the quasi-SBS. Two ports marked as the blue solid line indicate the input and output ports of wave for the cavity. In Fig. 2(c), the band map of the simplified 2D perfect PhC was obtained using the plane-wave-expansion method. The ideal silicon PhC exhibits a large photonic bandgap in the range 0.274–0.429 in units of fa/c (c is the light speed in vacuum). In the cavity, there are many defect modes in the photonic bandgap, for example, an ordinary leaky mode TE₃₁ was shown in Fig. 2 (d) with h = 0.4a, w = 1.01a. The band map of this cavity was calculated by treating the entire structure as a supercell in plane-wave-expansion method.

Figure 3 shows the evolution of the transmission spectrum and Q factor of the odd SBS-mode TE₁₂. In Fig. 3 (a), the resonance frequency of the SBS mode TE₁₂ varies with h and w values in the bandgap. The white part in Fig. 3 (a) represents the resonant frequency of the TE₁₂ mode in the passband, which conforms to the definition of ordinary BICs; however, this is beyond the scope of this study. At the red point in Fig. 3 (a) (shown by the red dot in the figure, h = 0.4a, w = 1.01a), the evolution of transmission spectral profiles of the cavity structure with $\varDelta$y are shown in Figs. 3 (b) and (c). When $\varDelta$y = 0, it yields an SBS in the bandgap without continuum, which totally decouples with the structure with the Q factors tending to infinity. When $\varDelta$y ≠ 0, the symmetry of the entire structure is broken, the odd mode no longer possesses a cut off frequency, the bound state turns into a guided mode or partly transmitted mode, and the transmission spectrum exhibits a Lorentz line-shape.

 

Fig. 3. (a) Evolution of resonance frequency of odd mode TE₁₂ with different h and w values. (d) Band map and Electrical field pattern in z-direction of SBS mode TE₁₂ when h = 0.4a, w = 1.01a. Evolution of (e) Q factor and (f) modulation depth of the SBS mode TE₁₂ with different $\varDelta$y values and structure sizes when h = 0.4a, w = 1.01a. For h = 0.4a, w = 1.01a, evolution of transmission spectral profile of the SBS mode TE₁₂ with different $\varDelta$y values when the structure size is (g) 5 × 13, (b, c) 5 × 15, (h) 5 × 17, (i) 3 × 15.

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Figure 3 (d) shows the Electric field distribution pattern in z direction and the band map of the SBS mode in the cavity. According to Huygens’ principle, the two phase-inverted field regions can be viewed as two phase-inverted sources, and the output wave can be regarded as a superposition of the waves from the two phase-inverted sources. Subsequently, the generation mechanism of the SBS and quasi-SBS modes in the structure can be understood based on the theoretical description presented in Section 2. From Fig. 3 (c), evidently, when $\varDelta$y = 0, the waves from the two phase-inverted sources interfere to result in disappearance as predicted by Eq. (5). Consequently, a completely localized bound state is formed. Conversely, for $\varDelta$y ≠ 0, the two secondary waves cannot cancel each other as predicted by Eq. (13); thus, a quasi SBS emerges in the transmission with an extremely high Q factor value. From Fig. 3 (c), the quasi SBSs exhibit the same property as that of EIT modes. However, these quasi SBSs (SBS-EITs) exhibit considerably higher quality factors than that of conventional EIT modes.

We not only studied the transmission spectrum of the 5 × 15 structure, but also studied the influence of different column numbers in the x- and y-directions on the transmission spectrum, as shown in Fig. 3 (e-i). In momentum space, Q is shown to decay quadratically ($Q \propto \,1/{k^2}$) with respect to the distance k from a single isolated BIC [41]. Here, there is a similar phenomenon: when $\varDelta$y tends to 0, Q increases quadratically concerning (a/$\varDelta$y) ($Q \propto {\left(\,a/ {\varDelta y} \right)^2}$), as shown in Fig. 3 (e) where B(a/$\varDelta$y)² is used to fit the Q-value curve with the fitting constant B. In Fig. 3 (e), the structures of 5 × 13, 5 × 15 and 5 × 17 have the same number of columns in the x-direction, and their Q-value curves have a similar trend, with corresponding B-values around 90. From Fig. 3 (c) and Fig. 3 (g-h), it can be seen that when the number of columns in the x-direction is fixed, the number of columns in the y-direction changes, and the transmission spectrum only have slight frequency shift, while the transmission spectrum line remains unchanged. After the number of columns in the x-direction changes, as shown in Fig. 3 (e), the Q-value curve trend of the 3 × 15 structure changes (gray line), and its corresponding B-value decreases to 4 (orange dashed line). The frequency of the defect mode also changes significantly, and the transmission spectrum becomes a Fano line shape as shown in Fig. 3 (i). For Fig. 3 (f), as $\varDelta$y tends to 0, the maximum transmission in the spectrum (A) of the quasi-SBS of mode TE₁₂ decreases rapidly, which is appropriate for the amplitude modulator. We will use the 5 × 15 structure in subsequent calculations.

At different parameters of h, w, and $\varDelta$y, the SBS mode and an even mode can both exist in a small spectrum, such that the SBS mode (acting as dark mode) can interfere with the even mode (acting as bright mode) to generate SBS-EIR and SBS-Fano modes, as shown in Fig. 4. In Figs. 4 (a)–(c), three typical contour maps of transmission coefficient exhibit sharp resonances with the Q value tending to infinity when $\varDelta$y = 0. A significant difference among the maps is the line shape, which corresponds to asymmetric Fano (right peak), EIR, and asymmetric Fano (left peak) line shapes, respectively, which is realized by fixing h = 0.4a, and considering w = 0.9945a, 0.9952a, and 0.996a, respectively. Furthermore, Figs. 4 (d)–(f) coupled with Fig. 3(c) show that the SBS-EIT, SBS-EIR, and SBS-Fano modes can be tuned by fixing h = 0.4a, and considering w = 1.01a, 0.9945a (or 0.996a), and 0.9952a, respectively. Moreover, the mode field patterns in the insets in Figs. 4 (d), 4 (e), and 4 (f) clearly show that these modes are the results of interference of the odd SBS mode (dark and TE₁₂ modes) and even bright mode (TE₃₁ mode).

 

Fig. 4. Evolution of transmission spectral profile of the mode TE₁₂ with different $\varDelta$y values when h = 0.4a, (a, d) w = 0.9945a, (b, e) w = 0.9952a, and (c, f) w = 0.996a. The evolution of the Q factor and A (modulation depth) of the mode TE₁₂ with different $\varDelta$y values when h = 0.4a, (g) w = 0.9945a, (h) w = 0.9952a, and (i) w = 0.996a.

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From Figs. 4 (d)–(f), it can be concluded that the structure becomes symmetric and there is only a bright mode for $\varDelta$y = 0, such that no interference can occur and the EIR and Fano modes disappear at $\varDelta$y = 0. Simultaneously, Fig. 3 (c) indicates that the SBS and quasi SBS modes are not reliant on the interference of the two waves. This reveals an important fact that the Fano and EIR modes operate according to the same mechanism, which is different from the operating mechanism for SBS modes. Moreover, the structure exhibits even-symmetry along the x-axis in the center, whereas the TE₁₂ mode exhibits an odd symmetry along the x-axis. Consequently, the TE₁₂ mode cannot be coupled outside the structure, that is, it is completely localized in the structure. Furthermore, Figs. 4 (d)–(f) show that the dark modes for the Fano (Figs. 4 (d) and 4 (f)) and EIR (Fig. 4 (e)) resonances have a common characteristic in that they are all SBS modes with an odd mode pattern. Therefore, the Fano modes and the EIR mode can exhibit very high Q factors because the odd mode can almost not be coupled to the outside of the cavity and thus can considerably enhance the localization of waves in the cavity. As such, the Fano and EIR modes are considerably different from conventional ones, and are referred to as SBS-Fano modes and SBS-EIR modes, respectively. In Figs. 4 (g)–(i), regardless of the value of w, the Q value increases quadratically with respect to (a/$\varDelta$y) and the modulation depth of the transmission spectra (A) decreases rapidly with decrease in the absolute value of $\varDelta$y.

As can be seen that, it is difficult to find a suitable Full Width at Half Maximum (FWHM) for the Fano curve in the Fig. 6 with h = 0.63a, w = 0.943a, and $\varDelta$y = -0.006a, because the flat side of Fano resonance is more affected by background light and is very large. Moreover, the steepness of the steep side of the Fano resonance mainly depends on the dark mode, and for any line-shape of the Fano resonance, we can get a suitable single-side width at half maximum. Furthermore, it can be noted that the steep side gives the key feature for applications as in sensing, switching, and filtering, which indicates that the single-side quality factor is more helpful for designs of devices. Therefore, we adopt the definition of single-side Q-factor [42] for Fano mode:

For the Fano- and EIR-like transmission line shapes, the spectral profile can be fitted by a Fano resonance modulated by the Lorentz resonance using [43,44]:

According to Eqs. (16) and (17), the q factor expresses the line shape more intuitively than the plotted curve. The simulation results were fitted well using Eqs. (16) and 17 (red dashed lines), as shown in Figs. 5 (a)–(e). Here, q varies from a negative to a positive finite value with the line shape transforming from Lorentz, left-side asymmetric Fano, EIR, and right-side asymmetric Fano to a Lorentz line shape. Moreover, the sign of q factor indicates the position on the left side of the bright mode (for negative q, Fig. 5 (b)) or the right side (for positive q, Fig. 5 (d)). Moreover, q = 0 corresponds to the case wherein the resonance frequency of the SBS mode (dark mode) is equal to that of the guided mode (bright mode), exhibiting a symmetric EIR line shape with an ultra-sharp peak (dip) over a broad peak in the reflection (transmission) spectrum, as shown in Fig. 5 (c). In Fig. 5 (e), the absolute value of q is considerably larger than that in Figs. 5 (b)–(d). This corresponds to the case wherein the resonance frequency of the SBS mode is far from that of the guided mode, exhibiting a Lorenz-like line shape, which is actually an EIT mode. In addition, when q is approaching negative infinite in Fig. 5 (a), it exhibits an EIT line shape, that is, the line shapes for q → -∞ and q → ∞ are of the same type, EIT line shape.

 

Fig. 5. Transmission (T, gray solid line) and reflection (R, black dotted line) spectra of the structure and the fitted transmission line (fitted T, red dashed line) with parameters h = 0.4a, $\varDelta$y = 0.004a and (a) w = 0.991a, (b) w = 0.9945a, (c) w = 0.9952a, (d) w = 0.996a, and (e) w = 1.01a, respectively.

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As indicated earlier, the height of the defective column is fixed at h = 0.4a. In the following, we show that results with similar properties can be obtained for other values of h. Figure 6 shows the SBS mode obtained at h = 0.63a and h = a. In Figs. 6 (a) and (d), all modes exhibit antisymmetric field patterns about the x axis at the center of the structure (red dashed line), which are different from the cavity structure in terms of symmetry. Therefore, they are all SBS modes and their coupling is forbidden provided the symmetry of the structure is preserved [21,45]. A quasi-SBS can be realized by destroying the symmetry of the structure; for example, by moving the defective rectangular pillar in the y direction. The offset parameter $\varDelta$y is a facile tuning tunnel to research the SBS states.

To view the evolution of SBS to quasi SBS modes, the transmission spectral and the Q factor with different $\varDelta$y for the two modes are shown in Fig. 6. The odd modes TE₂₂ and TE₃₂ in Fig. 6 (b), (c), (e), and (f) exhibit different symmetry with the structure when $\varDelta$y = 0. These odd modes decouple with the structure as superbound states. When $\varDelta$y $\ne $ 0, the symmetry of the entire structure is broken, and scattering odd modes can be observed in transmission with sharp peaks. With decrease in the absolute value of $\varDelta$y, the amplitude decreases and Q factors increase rapidly. This demonstrates that the odd modes in the bandgap are SBS modes for $\varDelta$y = 0 and quasi SBS modes for $\varDelta$y $\ne $ 0 with the quasi SBS modes being very sensitive to the parameter $\varDelta$y. When $\varDelta$y decreases in Figs. 6 (c) and (f), with a sharp increase in the Q value (black line), the amplitude (red line) decreases rapidly, which is appropriate for an amplitude modulator.

 

Fig. 6. (a) SBS mode TE₂₂ at h = 0.63a and w = 0.98a, (d) SBS mode TE₃₂ at h = a and w = 1.03a. The evolution of transmission spectral profile and Q factor with different $\varDelta$y values of the SBS mode (b, c) TE₂₂ when h = 0.63a, w = 0.98a, (e, f) TE₃₂ when h = a, w = 1.03a.

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To obtain the Fano (SBS Fano) and EIR-like (SBS EIR) modes, we determine the overlapping points of the odd and even modes in the structure through the evolution of the resonance frequencies of these modes with the structure parameters h and w, as shown in Fig. 7. The red dotted lines in Fig. 7 indicate the overlap points of the odd and even modes, where the two modes can interfere with the generation of Fano- or EIR-like modes.

 

Fig. 7. Evolution of resonance frequency of odd modes (a) TE₂₂, (b) TE₃₂ with different h and w. The red dotted lines represent the evolution of frequency overlap points of (a) TE₂₂ and TE₃₁, (b) TE₂₃ and TE₃₂ with different h and w.

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Figure 8 is derived from the typical value ranges of h and w at the black elliptical points in Fig. 7. The two pairs of degeneration mode groups are named I (TE₂₂ and TE₃₁) and II (TE₂₃ and TE₃₂), corresponding to the intersection bands in Figs. 8 (a)–(d). The odd band can encounter an even band, as in Fig. 8 (a) and (c), which creates Fano resonances with the SBS modes (odd modes) as dark modes. Through minute adjustments to the geometric parameter w, the two bands of a pair no longer intersect, as in Figs. 8 (b) and (d). Therefore, the transmission line shape of the quasi-SBS can be controlled, as shown in Figs. 8 (e) and (g). The red dashed lines show the quasi-SBSs caused by the odd modes. Moreover, changing $\varDelta$y can facilitate the control of the Q factor and the modulation depth of the transmission spectra (T) of the quasi-SBSs, as in Figs. 8 (e)–(h). The Q values of quasi-SBSs from modes TE₁₂, TE₃₂, and TE₂₂ can modulate within ranges of 10⁴–10⁶, 10⁶–10⁷, and 10⁷–10⁹, respectively. For the same mode, the modulation range of Q value is similar under different line-shapes.

 

Fig. 8. Band maps of the (a) intersection and (b) separation of the modes TE₂₂ and TE₃₁ of pair I. Band maps of the (c) intersection and (d) separation of the modes TE₂₃ and TE₃₂ of pair II. Evolution of spectral profile and Q factors of the intersecting modes (e, f) for pair I and (g, h) for pair II.

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Thus, it can be concluded that the odd modes are superbound states that function as dark modes because of topological protection. Conversely, the even modes are continua that function as bright modes. Further, SBS-Fano and SBS-EIR are generated by the intersection points of the odd and even modes, and quasi-SBS is produced by the scattering of lone odd modes. We designed an operating mode with suitable geometric parameters for the required transmission line shape and Q-value tuning range of the quasi-SBS. Because of the high Q-value and other transmission characteristics, it can be used for filters, cavities, and lasers with ultra-narrow linewidth, and sensors, switchers, and amplitude modulators with high sensitivity, and so on.

4. Design of optical switch based on the SBS-EIR mode

As an example of the application of the studied structure, we designed an optical switch scheme in Figs. 9 (a) and (b) using the SBS-EIR mode and the leaky mode in Figs. 9 (c) and (d). The lattice constant of the PhC was a = 1 mm and the model operated at a frequency of 0.11 THz. The positions corresponding to the circular silicon column on the pedestal were all circular holes; however, the center was a hollow square hole, as shown in Fig. 9 (b). The rectangular column in the center of the cavity was longer than that in the other columns. It was pinned to a piezoelectric ceramic actuator (PZA) through the square hole of the pedestal. The distance from the center was $\varDelta$y. When the PZA drives the rectangular column 4 µm ($\varDelta$y = 4 µm) along the y axis under suitable voltage, the optical switch changes from ON to OFF, as shown in Fig. 9 (d). The extinction ratio (ER) of the optical switch can be calculated according to the formula:

 

Fig. 9. (a, b) Schematic of the optical switch. (c) Band maps of the SBS-EIR mode TE₂₂ and the leaky mode TE₃₁. (d) Optical switch property.

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For T_on = 0.9993 and T_off = 0.0117, ER = 19.3 dB is obtained.

5. Conclusions

This study placed a rectangular column in a top–bottom symmetric PhC cavity, constructed a symmetric resonator, and obtained multiple defect modes in the bandgap. Consequently, an SBS mode generation method was proposed based on the interference of two phase-opposite (or with a phase difference of 180 °) sources (dipole sources) with a confined propagation along a single axis. SBS modes were found in such a structure, whereas quasi-SBS modes (which can be regarded as SBS-EIT modes) were found in structures with broken symmetry. The SBS-Fano and SBS EIR-like modes were obtained when the odd-mode SBSs and even modes in the structure contained overlapping points. The interference of the odd mode SBSs, which were dark modes, and the even modes, which were continua as bright modes, generated the SBS-Fano or SBS EIR-like modes. The four mode types (SBS, SBS-EIT, SBS-EIR, and SBS-Fano) and the value of the Q factors of the modes were controllable via the structure parameter $\varDelta$y as required. The Q factor values of quasi-SBSs for modes TE₁₂, TE₃₂, and TE₂₂ can be modulated in the ranges of 10⁴–10⁶, 10⁶–10⁷, and 10⁷–10⁹, respectively. For the same mode, the modulation range of Q value was similar under different line shapes. Finally, as an application of the study, an optical switch scheme using a PZA provided the potential application operating at terahertz frequencies. Moreover, this structure was not limited to a rectangular dielectric column defect cavity, and can be of non-centrosymmetric shapes such as ellipse, diamond, cross, etc., using the same principle.