2.1.

General Design Principle of High-Q Modes

Originally, Friedrich and Wintgen suggested that interference between two modes causes avoid-crossing and leads to the formation of BIC with an infinite $Q$-factor.²² The avoided crossing was used to realize BIC in quantum,²³ optics,^14–21,24 and acoustic systems.²⁵ When the system deviates from the ideal situation (i.e., destructive interference among different diffraction channels), it will convert the ideal BIC into QBIC with a finite $Q$-factor. The formation of BIC and QBIC can be well described by the two-level system (see Sec. 1 and Fig. S1 in the Supplemental Materials). Here, we demonstrated that from the leaky mode perspective, many QBIC can be found in a single dielectric nanocavity with a rectangular cross section by constructing avoid-crossing of pair modes. Without loss of generality, we consider the eigenmodes (named leaky modes) of a rectangular NW with refractive index $n=4$ under TE polarization with the electric field along the $z$ direction while the background medium is air. The cross section of the rectangle is in the $XOY$ plane while the NW is infinitely long along the $z$ axis, assuming the width and height of the NW are $a$ and $b$, respectively. The size ratio of the NW is defined as $R=b/a$. In previous work,^11,26,27 we have demonstrated that the leaky modes (also known as Mie resonance mode) supported by the single dielectric nanostructure play the dominant role in describing its optical properties (i.e., absorption/scattering). All the leaky modes can be rigorously calculated by the finite element method (FEM) with commercial software COMSOL-Multiphysics. The complex eigenvalues can describe them $N=n\omega b/c=N_{\text{real}}-i{N}_{\mathrm{imag}}$, where $\omega$ is the complex eigenfrequency of the leaky mode and $c$ is the speed of light. It allows expressing the $Q$-factor in the following form $Q=N_{\text{real}}/(2\times N_{\mathrm{imag}})$. Linear dependence between $N_{\text{real}}$ and the size ratio $R$ has been shown for modes $\mathrm{TE}(m,l)$,¹¹ where $m$ and $l$ correspond to the number of peaks of the electric field within the NW in the $x$ and $y$ dimensions. The expression can be written as $N_{\text{real}}\approx (m-1)\pi R+(n-1)\pi$. Due to the linear relationship, the avoided crossing of eigenvalues can be easily constructed for a pair of modes $\mathrm{TE}(m,l)$ and $\mathrm{TE}(m-2,l+2)$ or $\mathrm{TE}(m,l)$ and $\mathrm{TE}(m+2,l-2)$ while the size ratio is tuned. Consequently, high-$Q$ and low-$Q$ modes are realized at the critical size ratio, where avoided crossing occurs. Typically, high-$Q$ and low-$Q$ modes can be divided into four categories (see Table S1 in the Supplemental Materials): (1) type I: $l=m+2$; (2) type II: $m\le l<m+2$, (3) type III: $l>m+2$, and (4) type IV: $l<m$.

Figures 1(a) and 1(b) show $N_{\text{real}}$ and the $Q$-factor as a function of the size ratio $R$ for modes TE(3,5) and TE(5,3) which belong to type I. Interestingly, the $Q$-factor reaches its maximum value of 3300 at $R=1$ for TE(3,5) while the avoid-crossing occurs for $N_{\text{real}}$ in these two modes. Other high-$Q$ modes fall within the category of type I, such as TE(1,3) and TE(2,4), can be found at the same critical ratio $R=1$ (see Fig. S2 in the Supplemental Materials). Figures 1(c) and 1(d) show the $Q$-factor and $a/\lambda$ (or $ka/2\pi$) for mode $\mathrm{TE}(m,m+2)$ while the value of $m$ increases from 1 to 5. The $Q$-factor can be up to $2.3\times {10}^4$ for mode TE(5,7) while the resonant wavelength is still larger than the width of the square NW. Even higher $Q$-factors can be obtained for $\mathrm{TE}(m,m+2)$ with $m>5$. For instance, for $m=6$, the $Q$-factor can get up to $2.98\times {10}^5$. The resonant wavelength, however, will become smaller than the width of the square NW. Thus, there is a balance between the high-$Q$ and the dimensions of the structure. Another interesting point is that $a/\lambda$ shows a linear dependence on $m$. Such a linear relationship can be explained from the ray optics perspective (see Sec. S2 and Fig. S3 in the Supplemental Materials).^14,28 It can help to facilitate the process of finding modes even with a higher $Q$-factor for large $m$.

Fig. 1

Properties of the high-$Q$ modes in the single rectangular NW. (a), (b) Real part and $Q$-factor of the eigenvalue of modes TE(3,5) and TE(5,3) (type I) as functions of size ratio $R$. (c), (d) $Q$-factor and $a/\lambda$ as functions of $m$ for high-$Q$ mode $\mathrm{TE}(m,m+2)$ at the critical ratio. (e), (f) Real part and $Q$-factor of the eigenvalue of modes TE(3,4) and TE(5,2) versus the size ratio $R$. (g), (h) $Q$-factor and $a/\lambda$, as functions of $m$ for high-$Q$ mode $\mathrm{TE}(m,m+1)$ at the critical ratio.

Following the same approach, many high-$Q$ modes belonging to types II and III can be found. As an example of type II, Figs. 1(e) and 1(f) show the $N_{\text{real}}$ and $Q$-factor as a function of size ratio $R$ for a pair of modes TE(5,2) and TE(3,4). Different from the case of type I, the anticrossing feature appears at $R=0.855$, and the $Q$ factor for TE(3,4) reaches the maximum value of 1309, while it is 33 for mode TE(5,2). More type II high-$Q$ modes are presented in Fig. S4 in the Supplemental Materials. Note that the critical ratio $R$ is always between 0 and 1 (see Fig. S5 in the Supplemental Materials). We also plot the $Q$-factor and $a/\lambda$ for a high-$Q$ mode TE$(m,m+1)$ as a function of $m$. Similarly, the $Q$-factor increases sharply with $m$ and can reach the value of $2.94\times {10}^4$ for the mode TE(6,7). We also found that $a/\lambda$ is directly proportional to $m$. Another example of the type II mode $\mathrm{TE}(m,m)$ is shown in Fig. S6 in the Supplemental Materials. The type III case is similar to type II and, therefore, the relevant results are put in Fig. S7 in the Supplemental Materials. For the type IV, note that the complex eigenvalue $N=n\omega b/c$ of TE$(m,l)$ for the size ratio $R=b/a$ is the same as the eigenvalue $N=n\omega a/c$ for $\mathrm{TE}(l,m)$ for the size ratio $R=a/b$. Therefore, if type II or type III pair modes TE$(l,m)$ and TE($l+2$, $m-2$) display avoid-crossing features at the critical ratio $R$ and the $Q$-factor of $\mathrm{TE}(l,m)$ has a maximum value, TE$(m,l)$ and $\mathrm{TE}(m-2,l+2)$ will show avoided crossing at $1/R$, and $\mathrm{TE}(m,l)$ will have the largest $Q$-factor.

We confirm the existence of such high-$Q$ modes at the critical size ratio by calculating the energy density mapping and scattering efficiency mapping versus both the size ratio and the normalized frequency $ka$. Here, the incident wave is TE polarization with the electric field along the $z$ axis. For some leaky modes such as TE(2,4) and TE(4,2), the eigenfield presents an antisymmetric distribution, and they cannot be excited by a normal incident wave. Therefore, the incident angle is set to 15 deg with respect to the $y$ axis to excite all of the eigenmodes. For a given size ratio, each peak in the energy density can be perfectly correlated to one of the leaky modes. From Fig. 2(a), an excellent agreement can be found between the resonant peaks in the mapping and the real part of the eigenvalue $N_{\text{real}}=ka$ for pair modes TE(3,5) and TE(5,3). Furthermore, the line width of the resonant peak indeed becomes the narrowest at $R=1$, which means that the $Q$-factor reaches the maximum value. The narrowing effect of linewidth can also be found in scattering spectrum mapping, as shown in Fig. 2(b). A similar phenomenon can also be applied to other pair modes TE(3,4) and TE(5,2) [Figs. 4(c) and 4(d)], TE(2,3) and TE(4,1) [Figs. S8(a) and S8(b) in the Supplemental Materials], TE(2,4) and TE(4,2) [Figs. S8(c) and S8(d) in the Supplemental Materials].

Fig. 2

Total energy density and scattering efficiency for the rectangular NW with different size ratios. (a), (b) Logarithm total energy density and scattering efficiency mapping versus $R$ and $ka$. Two modes TE(3,5) and TE(5,3) are labeled as black and red circles. (c), (d) Logarithm of total energy density and scattering efficiency mapping versus $R$ and $ka$. Two modes TE(3,4) and TE(5,2) are labeled as black and red circles.

2.2.

Physical Explanation of High-Q Mode by Multipole Decomposition

To get a better insight into radiative properties of the high-$Q$ modes, we employ the multipole expansion.^29–31 Here, we again consider the case of NW at oblique incidence ($\theta =15\mathrm{deg}$) with TE polarization. Figure 3(a) shows scattering efficiency contributed by multipoles for a square NW. Two resonant peaks can be observed at $ka=3.89$ and $ka=3.97$, which are related to the low-$Q$ mode TE(5,3) and high-$Q$ mode TE(3,5). The scattering efficiency around $ka=3.97$ is dominated by a single multipole, electric quadrupole ($m=2$), exhibiting a sharp Fano profile.^32,33 In contrast, there are two dominant multipoles in the scattering efficiency for the low-$Q$ mode around $ka=3.89$. This is also confirmed by the multipole analysis on the two eigenmodes TE(3,5) and TE(5,3). Indeed, from Fig. 3(b), there are two radiation channels ($m=0$ and $m=4$) for mode TE(5,3), but only one dominant radiation channel ($m=2$) exists for mode TE(3,5). Each multipole can be considered as an independent channel for the radiating decay. Thus, coupling to more leaky channels with a larger radiation intensity will, in general, reduce the $Q$-factor. That is why it is expected that high-$Q$ modes should couple to only one radiative channel with a small leakage, described by a single multipole.

Fig. 3

Multipole analysis of the high-$Q$ modes. (a) Multipolar contribution on the scattering cross section of the square NW under the excitation oblique incidence plane wave ($\theta =15\mathrm{deg}$). (b), (c) Multipole analysis and Fourier transformation on the eigenfields of two modes TE(3,5) and TE(5,3). (d), $E(k_0)$ obtained from a Fourier transformation of eigenfields for two modes. (e) Multipolar contribution on scattering cross section of the rectangular NW with $R=0.855$ excited by the obliquely incident plane wave ($\theta =15\mathrm{deg}$). (f), (g) Multipole analysis and Fourier transformation on the eigenfields of two modes TE(3,4) and TE(5,2). (h) $E(k_0)$ obtained from Fourier transformation of eigenfields for two modes. (i) Decomposition of TE(3,5) for the rectangular NW into ${\mathrm{TE}}_{61}$ and ${\mathrm{TE}}_{22}$ of eigenmodes for the circular NW. (j) Decomposition of TE(5,3) for the rectangular NW into ${\mathrm{TE}}_{42}$ and ${\mathrm{TE}}_{03}$ of eigenmodes for the circular NW.

Importantly, we can also find these radiation channels by carefully comparing its eigenfield distribution to the electric field distribution of the eigenmode for an infinite cylinder. For example, for mode TE(3,5), its eigenfield may be regarded as a superposition of eigenmodes ${\mathrm{TE}}_{22}$ and ${\mathrm{TE}}_{61}$ for the cylinder in Fig. 3(i), which serve as the orthogonal basis for the multipole expansion method. The radiation contribution of ${\mathrm{TE}}_{22}$ is much larger than that of ${\mathrm{TE}}_{61}$ from the field distribution. That also explains that $m=2$ dominates the radiation for TE(3,5). At the same time, the eigenfield of TE(5,3) may be viewed as the superposition of eigenmodes ${\mathrm{TE}}_{03}$ and ${\mathrm{TE}}_{42}$ for the cylinder, as shown in Fig. 3(j). Moreover, the radiation intensity for both channels in TE(3,5) is much lower than the counterparts in TE(5,3). From the perspective of the radiation channel, we may attribute the extreme confinement of TE(3,5) to the fact that these two leaky channels are more confined compared with leaky channels of TE(5,3). This explanation also works for the modes TE(3,4) and TE(5,2) belonging to the category of type II [see Figs. 3(e) and 3(f) and Fig. S9 in the Supplemental Materials]. Ideally, the eigenmode with the closest field profile to the eigenmode ${\mathrm{TE}}_{ml}$ ($m>1$ and $l=1$) in the cylinder will always have a high $Q$-factor because there is only one leaky channel with the minimum radiation intensity. The larger mode number $m$ is, the higher the $Q$-factor. This concept has been successfully applied to design the whisper gallery mode with an ultrahigh $Q$-factor.² We also find a similar phenomenon in a relatively low-order mode. For example, mode TE(2,4) for NW with $R=1$ and TE(2,3) for NW with $R=0.775$ reach maximum values of 885 and 145, respectively. Multipole analysis of the eigenfield indicates that only one radiation channel exists for these two modes (see Fig. S10 in the Supplemental Materials). The major leaky channels for TE(2,3) and TE(2,4) are eigenmodes ${\mathrm{TE}}_{31}$ and ${\mathrm{TE}}_{41}$ of the cylinder, respectively (see Figs. S11(a) and S11(b) and Figs. S11(e) and S11(f) in the Supplemental Materials]. Both have better field confinement than eigenmode ${\mathrm{TE}}_{12}$ and eigenmode ${\mathrm{TE}}_{22}$ that are main radiation channel of modes TE(4,1) and TE(4,2), respectively [see Figs. S11(c) and S11(d) and Figs. S11(g) and S11(h) in the Supplemental Materials]. In fact, the suppressed electric dipole that is realized by an in-phase magnetic dipole and an electrical quadrupole³⁴ is essential for building a magnetic mirror.

The radiative properties of an arbitrary source can also be analyzed in momentum space. It is known that only the nonzero current “on-the-shell” in $k$-space contributes to the far-field radiation.³⁵ Thus, it is instructive to analyze the electric field in momentum space of the high-$Q$ mode at the critical ratio to get a more profound physical insight. To do this, we perform the Fourier transform on the eigenfield of the high-$Q$ modes shown in Figs. 3(c) and 3(g). Here, it is worth pointing out that the electric field $E(k_0)$ contributes to the outward radiation only when $k_x^2+k_y^2=k_0^2$. Therefore, we extract $E(k_0)$ on the white circle boundary $k_x^2+k_y^2=k_0^2$ and plot them in Figs. 3(d) and 3(h). Indeed, the radiation field $E(k_0)$ of the high-$Q$ mode has a much lower amplitude, and the radiation channel is narrower. For the resonant mode with an extremely high $Q$-factor, $E(k_0)$ approaches zero at the circle boundary ($k_x^{2 }+k_y^{2 }=k_0^2$).

2.3.

High-Q Modes in Rectangular NW for TM Case

The above phenomenon can also be generalized to the rectangular NW with TM polarization. However, it is interesting that avoid-crossing is not always the prerequisite of realizing a high-$Q$ mode in TM cases. For example, as shown in Figs. 4(a) and 4(b), mode TM(2,3) reaches the maximum $Q$-factor while crossing happens between pair modes TM(2,3) and TM(4,1). More crossing cases can be found for pair modes TM(3,3) and TM(5,1) in Figs. S12(a) and S12(b) in the Supplemental Materials, TM(4,4) and TM(6,2) in Figs. S12(c) and S12(d) in the Supplemental Materials. Figures 4(c) and 4(d) show the case of pair modes TM(3,4) and TM(5,2), where the high-$Q$ mode happens at the avoided crossing. More avoided crossing cases can be found between modes $\mathrm{TM}(m,m+2)$ and $\mathrm{TM}(m+2,m)$, such as TM(2,4) and TM(4,2) in Figs. 4(e) and 4(f), and TM(3,5) and TM(5,3) in Figs. S13(a) and S13(b) in the Supplemental Materials. It is entirely different from the cases of TE polarization, in which avoid-crossing is the necessary condition to realize the high-$Q$ mode. Such a difference is also reflected in the mode evolutions (see Fig. S14 in the Supplemental Materials). For TE(2,3) and TE(4,1), the modes interchange with each other while the structure crosses the critical size ratio may suggest a relatively strong coupling between these two modes.^36,37 Nevertheless, modes TM(2,3) and TM(4,1) remain similar field profiles, which may indicate a weak coupling between them (see Sec. S3, Table S2, and Fig. S15 in the Supplemental Materials). In addition, we also perform the multipole analysis on the eigenmode at different size ratios for both TE and TM cases. It can be found that the channel $m=1$ for both cases is reduced to a minimum at a critical size ratio (see Fig. S16 in the Supplemental Materials). Thus, the high-$Q$ mode, on the other hand, can be regarded as an anapole state^38,39 for which the radiation can be significantly quenched due to the total elimination of one leaky channel. In addition, $Q$-factor and $a/\lambda$ versus $m$ for the high-$Q$ mode $\mathrm{TM}(m,m+2)$, $\mathrm{TM}(m,m+1)$, and $\mathrm{TM}(m,m)$ are calculated and shown in Fig. S17 in the Supplemental Materials. The $Q$-factor can be up to $2.15\times {10}^4$ for mode TM(5,5) with $a=0.975\lambda$ (subwavelength scale). Moreover, similar to the TE case, almost linear dependence can be found between $m$ and $a/\lambda$. Also, we plot the critical size ratio versus $m$ for high-$Q$ mode $\mathrm{TM}(m,m+2)$, $\mathrm{TM}(m,m+1)$, and $\mathrm{TM}(m,m)$, which can be found in Fig. S18 in the Supplemental Materials. The high-$Q$ mode for the TM case is also well explained by multipole analysis on the eigenmode (see Fig. S19 in the Supplemental Materials). Importantly, these high-$Q$ modes are not limited to the dielectric structure with the high refractive indices, such as for 4. Figure S20 in the Supplemental Materials shows the different high-$Q$ modes of a rectangular NW as the refractive index varies from 2 to 8. One interesting thing is that the $Q$-factor does not always monotonically increase with the increasing refractive index. For example, the $Q$-factor for the high-$Q$ mode TE(2,4) increases much faster than that of the high-$Q$ mode TE(3,5), which becomes saturated for $n\ge 4$. Therefore, the refractive index-dependent $Q$-factor can help us to immediately find which mode should be chosen when a high-$Q$ factor is desired for semiconductors with a different refractive index.

Fig. 4

High-$Q$ mode for the TM case. (a), (b) Real part and $Q$-factor of the eigenvalue for modes TM(2,3) and TM(4,1) as functions of the size ratio $R$. (c), (d) Real part and $Q$-factor of the eigenvalue for modes TM(3,4) and TM(5,2) versus the size ratio $R$.

2.4.

High-Q Modes in a Single Nanoparticle

So far, we only discuss how to find the QBIC-induced high-$Q$ mode in a single rectangular NW. The above approach can also be applied to a 3-D nonspherical structure, including the cuboid and cylinder with finite thickness. Here, we use a single cuboid as an example to demonstrate how to find a high-$Q$ mode. For the sake of convenience and without loss of generality, $a=b$ and $R=c/a$ are assumed. Also, the mode number along $x$ is chosen as 1 for simplicity. We first consider magnetic eigenmodes $M$(1,2,3) and $M$(1,4,1) in the $YOZ$ plane. Remarkably, from Figs. 5(a) and 5(b), a similar anticrossing feature can be observed for these two modes, and the $Q$-factor reaches a maximum of 325 at $R=0.795$. The results of multipole analysis on the modes $M$(1,4,1) and $M$(1,2,3) indicate that the main radiation channels are $l=1$ and $m=2$, $l=3$ and $m=1$, respectively [see Fig. 6(a)]. Careful examination on the eigenfield distribution tells us that these two channels can be linked to eigenmodes $M_{12}$ and $M_{31}$ for sphere nanoparticles. $E(k_0)$ obtained from a Fourier transform on the eigenfield has a much lower amplitude for $M$(1,2,3) than that of mode $M$(1,4,1) [see Fig. 6(b)]. It again confirms that the radiation is suppressed to a minimum. Another example of high-$Q$ and low-$Q$ modes, $M$(1,4,2) and $M$(1,2,4), are shown in Figs. 5(c) and 5(d). The $Q$-factor is up to $>1000$. Different from the 2D NW case, a single cuboid can support both electric and magnetic eigenmodes. The conclusion drawn from magnetic modes can also be applied to electric modes. Following a similar approach, we find another two electric high-$Q$ modes, $E$(1,2,3) [see Figs. 5(e) and 5(f)] and $E$(1,2,4) [see Figs. 5(g) and 5(h)]. However, these two are a little bit different. The mode features for $E$(1,2,3) and $E$(1,4,1) are crossing while anticrossing occurs for pair electric modes $E$(1,4,2) and $E$(1,2,4). Other high-$Q$ modes in a single cuboid can be found in Fig. S21 in the Supplemental Materials. Besides, we want to point out that such a strategy also works for a cuboid with $a\ne b$. Those high-$Q$ modes can be constructed using a similar way. In fact, we can treat the single cuboid as a truncated rectangular NW. When the length of the $z$ axis becomes finite for the NW, we will expect the increase of $Q$-factor because there is confinement in the third dimension for the 3D cuboid compared to the NW, which is demonstrated in Fig. S22 in the Supplemental Materials. Last, another typical example of a single particle is a cylinder with finite thickness. We present two pairs of magnetic high-$Q$ modes of a single disk in Fig. S23 in the Supplemental Materials. One is magnetic mode $M$(1,2,3) and $M$(1,4,1) while the other is magnetic mode $M$(1,2,4) and $M$(1,4,2). Similarly, electric high-$Q$ modes, such as $E$(1,2,3) and $E$(1,2,4), can be found and shown in Fig. S24 in the Supplemental Materials. The design principles are the same, and we shall not repeat them here. Note that in Refs. 17 and 36, the modes $M$(1,2,3) and $M$(1,4,1) are also described as Mie mode and Fabry–Perot mode. However, not all pair high-$Q$ and low-$Q$ modes can be regarded as Mie mode and Fabry–Perot mode. For example, both modes $M$(1,4,2) and $M$(1,2,4) belong to Mie mode. A similar phenomenon can also be found in a single 2D rectangular NW. For example, pair modes TE(2,4) and TE(4,2), TE(3,4) and TE(5,2), TE(3,5) and TE(5,3), TM(2,4) and TM(4,2), TM(3,4) and TM(5,2), cannot be categorized into Mie mode and Fabry–Perot mode. Therefore, our strategy represents the perfect solution of finding all high-$Q$ modes in a single dielectric nanocavity with a rectangular cross section.

Fig. 5

High-$Q$ mode for the single 3D nanoparticle. (a), (b) Real part and $Q$-factor of the eigenvalue for the magnetic eigenmode $M$(1,2,3) and $M$(1,4,1) in the single cuboid as functions of the size ratio $R=c/a$ while $a=b$. (c), (d) Real part and $Q$-factor of the eigenvalue for the magnetic eigenmode $M$(1,2,4) and $M$(1,4,2) as functions of the size ratio $R=c/a$ of a cuboid. (e), (f) Real part and $Q$-factor of the eigenvalue for the electric eigenmode $E$(1,2,3) and $E$(1,4,1) as functions of the size ratio $R=c/a$ of the cuboid. (g), (h) Real part and $Q$-factor of the eigenvalue for the electric eigenmode $E$(1,2,4) and $E$(1,4,2) as functions of the size ratio $R=c/a$ of a cuboid.

Fig. 6

(a) Multipole analysis on the eigenfields for two modes of the cuboid, $M$(1,4,1) and $M$(1,2,3). (b) $E(k_0)$ distribution for two modes.