1. Introduction

The manipulation and localization of electromagnetic waves through the optical resonators with high quality factors have been of fundamental importance for nano-optics and photonics [1,2]. Photonic crystal (PhC) slabs are utilized as promising Fabry-Pérot resonators that allow for engineering artificial dispersion and benefit from low propagating losses [3,4]. In such devices, two main quality factors related to the out-of-plane radiation losses should be taken into consideration [5]: one is the quality factors ${Q_\parallel }$ associated with the mirror-loss caused by imperfect reflection parallel to the waveguide direction, and the other is quality factors ${Q_ \bot }$ governed by waveguide-loss due to radiative leakage perpendicular to the slab surface. By acquiring small group-velocity mode or even zero group-velocity (ZGV) mode at the edge of Brillouin zone, the former has been greatly enhanced due to the significant reduction in mirror losses [6,7]. However, further elimination of waveguide losses requires that nonleaky guided modes are not only located at the Brillouin boundary but also below the light cone [8], which greatly limits the degree of freedom of their operating frequencies and momenta in practical applications.

It is however possible to overcome this limitation by utilizing the exotic nonleaky states above the light cone, i.e., bound states in the continuum (BICs), which are localized eigenmodes of Maxwell’s equations with the diverging quality factors embedded into the continuous spectrum of free space [9–14]. Recently, robust BICs supported by PhC slab have been extensively employed for manipulating the electromagnetic waves due to their fascinating and diverse optical properties, such as ultra-high Q factor [15,16], near-zero resonant width [17–20], topological natures [21,22], and enhanced nonlinear effects [23–26]. According to the formation mechanism, BICs discovered in slab-type photonic lattices so far can be divided into three types: (i) single-resonance parametric BICs [27–31], (ii) symmetry-protected BICs [32–34], and (iii) Friedrich-Wintgen BICs [9,35–41]. Among these types, Friedrich-Wintgen BICs are typically generated at the spectral vicinity of the avoided crossings of two nonorthogonal resonances [35], resulting from complete destructive interference of leaked radiation modes through the radiation channels. Furthermore, in the spectral vicinity of avoided crossings an anomalous dispersion relation that includes a point of ZGV at a generic k point can also be acquired [42]. Therefore, a question is naturally raised: whether or how to design perfect 2D nanoscale resonators by combining nonleaky BICs with lossless ZGV modes, i.e., the waveguide losses and mirror losses are completely suppressed. Thus far, however, BICs and avoided crossings have been widely studied in diverse 1D photonic structures [40,41], the systematic investigations based on 2D photonic platforms are relatively few.

The aim of the present article is to explore the fundamental properties of avoided crossings and Friedrich-Wintgen BICs in 2D high-index-contrast leaky-mode PhC slab. It should be noted that the high-index-contrast photonic lattices considered here provide much stronger field localization and lower propagation loss when comparing with their low-index-contrast counterpart [39]. In this work, based on the mode expansion method, we systematically investigate the generation and evolution of Friedrich-Wintgen BICs and avoided crossings by adjusting the geometrical parameters of PhC slab. The numerical results suggest that a pair of BIC and ZGV modes can coalesce each other at suitable values for the diameter of array holes and the thickness of dielectric slab, leading to a prefect optical resonator with completely suppressed radiation losses. The designed devices will become an important instrument in various applications relying on high-quality resonances.

2. Theory and methods

Figure 1(a) illustrates the 2D high-index-contrast PhC slab utilized for simultaneously generating BIC and ZGV modes at two orthogonal crystalline directions. Figure 1(b) and (c) show the conceptual illustration of group indices and quality factors corresponding to the guided resonances (GRs) at both $\Gamma K$ and $\Gamma M$ directions. The blue and red peaks represent the BICs (with diverging radiative Q factor) and ZGV modes (with diverging group index), respectively. In theory, when these two peaks coincide with each other at identical angular frequency and momentum, this simple photonic lattice supports a special mode that is characterized by the two exotic properties: non-radiative loss and ultra-low group velocity. In other words, the Bloch modes within the PhC slab are confined longitudinally by small group-velocity propagation and transversely by BICs. To accurately understand the physical mechanism of the coherent radiation from all Bloch waves in 2D PhC slab, we extend the formalism of BICs and GRs described by the mode expansion method (MPM) from 1D photonic lattice [43] into its 2D counterpart. The MPM algorithm provides detailed propagating components from different resonances, thereby disclosing quantitatively the underlying formation mechanisms of BICs at $\Gamma $ and off-$\Gamma $ point of the Brillouin zone. The 2D photonic-lattice slab in Fig. 1(a) is periodic in horizontal directions x and y and homogeneous in the lateral direction z. The permittivities ${\varepsilon _1}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _\parallel })$ and ${\varepsilon _2}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _\parallel })$ in region I and region II are uniform distribution in the direction orthogonal to the xy plane, where ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _\parallel }$ denotes the 2D position vector (x, y). Thus, the entire system is mirror-symmetric on the xy plane, and the eigenmodes of system can be divided into the even-in-z modes and odd-in-z modes. With an outgoing boundary condition in z, the electric field (or magnetic field) for even-in-z TM (or TE) mode with 2D wavevector ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over k} _{/{/}}}$ can be written as [44]:

 

Fig. 1. Schematic diagram of a 2D PhC slab with high-index-contrast for exciting both BIC and ZGV modes at two orthogonal crystalline directions, (b) $\Gamma K$ and (c) $\Gamma M$ directions. The blue and red peaks in (b) and (c) correspond to the BICs and ZGV states (associated with the diverging radiative Q factor and diverging group index, correspondingly). The symmetric photonic-lattice system in (a) is consisted of the triangular-lattice array holes embedded in the high-index dielectric slab (region I) and the homogeneous low-index background medium (region II). The permittivities in regions I and II are indicated by ${\varepsilon _1}$ and ${\varepsilon _2}$, respectively. With periodic dielectric constant modulation, the guided resonances are described by the complex frequency $\omega = {\omega _r} - i{\omega _i}$, where ${\omega _i}$ denotes the decay rate of the radiation mode.

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On the other hand, the expansion coefficients ${T_{mn}}$ and ${S_{mn}}$ in Eqs. (1,2) can be related via the continuity of ${E_z}$ (or ${H_z}$) and $\partial {E_z}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _\parallel })/\partial z$ (or $\varepsilon _s^{ - 1}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over G} _\parallel } - \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over G} {^{\prime}_\parallel })\partial {H_z}({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _\parallel })/\partial z$) at $|z |= 0.5d$:

If we stack the expansion coefficients $\{ {T_{mn}}\}$ and $\{ {S_{mn}}\}$ into the single column vector T and S, and defining the transfer matrices J and L as the sets of eigenmodes $\{ {j_{mn}}\}$ and $\{ {l_{mn}}\}$ in Fourier space. Then the systems of Eqs. (12–14) can be reduced to the following matrix equations:

Once all of the solutions of the determinant are given, the corresponding guided modes, radiating modes and BICs supported by the periodic photonic system can be achieved simultaneously. Furthermore, inserting the solutions of Eq. (19) and Eq. (20) correspondingly into $({i\eta {L^{ - 1}}J + {L^{ - 1}}J\sigma } )T = 0$ and $({i\eta {L^{ - 1}}J + {L^{ - 1}}{\varepsilon_1}\varepsilon_2^{ - 1}J\sigma } )T = 0$ yields the corresponding vector T, then substituting T into Eq. (16) can determine the S. Finally, the field patterns as given in Eqs. (1,2) can be obtained owing to the associated vectors T and S are known.

3. Results and discussions

3.1 Formation of avoided crossings and BICs at different crystalline directions

The 2D triangular-lattice photonic system we consider is composed of high-index silicon slab embedded in an arbitrary homogeneous low-index cladding (liquid, silica or air). The radii of hole R and slab thickness d of dielectric slab are assumed as 0.26a and 1.4a, respectively, where a is the lattice constant. The effective refraction indices of dielectric slab and homogeneous cladding are taken as 3.4 and 1.0, respectively. Figure 2(a) shows the photonic bands of TM-like modes calculated by using the mode expansion algorithm, including avoided crossings and BICs at both $\Gamma K$ and $\Gamma M$ directions. The yellow shaded region in Fig. 2(a) denotes the continuum with an only leaky channel in the surrounding medium. i.e., only one Fourier order m and n for real $\sigma _{mn}^2$. The red and blue lines represent the GRs at $\Gamma K$ and $\Gamma M$ directions, respectively. As mentioned previously, the symmetric photonic lattices support two types of GRs with different transverse parities (mirror-symmetric with respect to the xy plane): symmetric (even) modes and asymmetric (odd) modes [45]. The avoided crossings marked by the black solid circles in Fig. 2(a) originate from the mutual repulsion between two even modes, whereas these dispersion curves can also cross each other, as shown by the black dashed circle in Fig. 2(a). This is because the odd mode and even mode are perfectly orthogonal in symmetric photonic waveguide. Figure 2(b) exhibits the magnified view of the avoided crossing at $\Gamma K$ direction within the spectral range marked by the black solid circle in Fig. 2(a). The green dotted lines indicate the uncoupled resonances of two photonic modes. The obvious mode separation in Fig. 2(a) can be further explained as follows: the strong interference between the two resonances push them apart, which leads to an avoided crossing of the resonance position and also affects the resonance width. As a result, the resonance modes can acquire an anomalous dispersion relation that includes a region of backward-wave propagation, as well as a point of zero group velocity (i.e., ZGV mode with zero slope).

 

Fig. 2. Photonic bands of TM-like modes computed with MEM, including avoided crossings and Friedrich-Wintgen BICs at both $\Gamma K$ and $\Gamma M$ directions. (b) The enlargement of avoided bands (i.e., anomalous dispersion with ZGV mode) at $\Gamma K$ direction in (a), the blue dots denote the TM-like GRs calculated by FDTD simulation. (c) and (d) Radiative quality factors and group indices for the GRs of the upper avoided band marked by black solid circles in (a) at $\Gamma K$ direction and $\Gamma M$ direction, respectively. Inset shows the absolute value of dominated radiation-coefficient $Abs({S_{0,0}})$ for the corresponding Friedrich-Wintgen BICs. (e-l) The spatial profiles of electric field at both XZ and YZ planes, corresponding to the BIC, ZGV and GR states indicated by the black arrows in (c) and (d), respectively. The diagonal and dashed frames represent the Si slab and air hole, respectively.

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In the spectral vicinity of avoided crossing, the BIC mode supported by the presented photonic system can also be observed due to coherent interference between two nonorthogonal leaky modes. Such BICs are commonly referred to as the typical Friedrich-Wintgen BICs [9]. The numerical results are in accordance with the simulation results performed by finite-difference time-domain (FDTD) method, as shown by the blue dots in Fig. 2(b) and (c). Here, we select the spatial resolution of 32 grid point per area of the elementary cell to ensure the convergence of the FDTD simulations. However, the FDTD algorithm takes greatly longer than the rigorous mode expansion analysis. It should be noted that although the presented MPM theory is difficult to deal with the complex and aperiodic photonic systems due to the requirement for a large number of plane waves, it is highly efficient for the periodic photonic system and can provide sufficient theoretical analysis. In addition, our periodic system is assumed to be a Hermitian system without loss or gain, otherwise it is difficult to obtain the real eigenvalues in Eqs. (19) and (20).

Figure 2(c) and (d) demonstrate that the existence of both BIC and ZGV modes can be verified by calculating the corresponding quality factors and group indices associated with the GRs of upper bands marked by black solid circles in Fig. 2(a). The radiative quality factors are obtained from the accurate numerical calculation $Q\textrm{ } ={-} {\omega _r}/(2{\omega _i})$ (${\omega _r}$ and ${\omega _i}$ are the real part and imaginary part of the complex-frequency $\omega$) [45], and the group indices are deduced by numerical differentiation of bands ${n_g}\textrm{ = c}dk/d\omega$ ($c$ is the speed of light in vacuum) [46]. From Fig. 2(c) and (d), it is seen that the BICs correspond to the isolated resonance peaks with extremely high Q factors at both $\Gamma K$ and $\Gamma M$ directions, denoted by the red and blue lines, respectively. By calculating the corresponding radiation-coefficient terms in matrix representation of Eq. (16), ${S_{m^{\prime},n^{\prime}}} = \sum\nolimits_{m,n}^M {{K_{m^{\prime}n^{\prime},mn}}{T_{m,n}}}$, where $K = {L^{ - 1}}J$ is coupling coefficient between eigenmodes $\{ {j_{mn}}\}$ and $\{ {l_{m^{\prime}n^{\prime}}}\}$ inside and outside slab, we can confirm that the BICs stem from the destructive interference from the radiation modes, which are formed by the in-slab modes leaking out through all radiation channels. As shown in inset, the absolute value of dominated radiation-coefficient $Abs({S_{0,0}})$ at ${k_x} = 0.18034(2\pi /a)$ along $\Gamma K$ direction and ${k_y} = 0.3627(2\pi /a)$ along $\Gamma M$ direction that correspond exactly to the resonance positions of the Friedrich-Wintgen BICs are both equal to zero. Also, compared with $\Gamma M$ direction, we can clearly observe one atypical group-index peak corresponding to the ZGV mode at nonzero wavevector ${k_x} = 0.173835(2\pi /a)$ along $\Gamma K$ direction, as shown by the black solid lines in Fig. 2(c) and (d). This anomalous mode arises from a standing-wave resonance, i.e., the forward and backward travelling components agree in phase and amplitude, resulting in a stationary-wave pattern whose radiative field is destructively interfered. It is noteworthy that the standing-wave mode away from Brillouin boundary does not transmit energy but does have moving phase front. Therefore, the presented ZGV mode with nonzero longitudinal wavevectors are capable of fulfilling the critical requirement of Fabry-Pérot resonance [47].

Consequently, there basically coexist three types of slab modes (i.e., BIC, ZGV and GR modes) within the continuum above the light line indicated by the yellow shaded region in Fig. 2(a). Panels (e-l) plot the spatial electric field profiles corresponding to the above three slab modes at both xz and yz planes, where panels (e, f) and (g, h) describe the electric field patterns for the BICs at ${k_x} = 0.18034(2\pi /a)$, ${\omega _r} = \textrm{0}\textrm{.41952}(2\pi c/a)$ along $\Gamma K$ direction and ${k_y} = 0.3627(2\pi /a)$, ${\omega _r} = \textrm{0}\textrm{.37123}(2\pi c/a)$ along $\Gamma M$ direction, respectively. It is clearly seen that the radiation waves outside the slab disappear completely, and the propagation modes are perfectly localized inside the waveguide. Correspondingly, panels (i, j) and (k, l) give the spatial electric field distributions for the ZGV state at ${k_x} = 0.173835(2\pi /a)$, ${\omega _r} = \textrm{0}\textrm{.41913}(2\pi c/a)$ along $\Gamma K$ direction and the GR state at ${k_y} = 0.355(2\pi /a)$, ${\omega _r} = \textrm{0}\textrm{.37202}(2\pi c/a)$ along $\Gamma M$ direction. Compared with the BICs, both ZGV and GR modes are radiative outside the photonic lattice due to the intrinsic symmetry properties, while exhibiting the significant leaky patterns at both xz and yz planes.

3.2 Modulation of BICs and ZGV modes at both $\Gamma K$ and $\Gamma M$ directions

Now we investigate the influences of geometric parameters of the presented periodic structure on the avoiding crossings and Friedrich-Wintgen BICs at both $\Gamma K$ and $\Gamma M$ directions. Figure 3(a) and (a’) show the evolution of GRs and BICs near the avoiding crossings as the hole radius R is continuously modified, while keeping the slab thickness d unchanged. By elaborately selecting the tunable range of geometric parameters, it is possible to guarantee the simultaneous presence of the BIC and ZGV states at two crystallographic directions. As mentioned previously, the yellow shaded regions represent the continuum above the light line. The solid dots denote the BICs generating from the various modified PhC slabs. Several intriguing phenomena can be noted in Fig. 3(a) and (a’). Firstly, the resonance positions of avoiding bands and BICs shift simultaneously to the lower angular frequencies and higher wavevectors as the value of R decreases, which can be explained that the effective index of the entire system is enhanced to fulfill the dispersion relation $k/\omega = {n_{eff}}/c$. Moreover, the BIC point gradually moves close to the bottom of the upper avoiding band with the continuously decreasing R, which implies that the ultra-high Q factors in the photonic system are accompanied by substantially increased ${n_g}$ due to the extremely flattened dispersion relations. The calculated results are agree well with the numerical simulations associated with radiative Q factors and group indices ${n_g}$, calculated by the GRs of upper avoided bands in Fig. 3(a) and (a’), as shown in Fig. 3(c, d) and (c’, d’). By inspecting the series of sharp peaks, one can clearly find that there exists a one-to-one correspondence between the BIC and ZGV modes, and the deviations between their resonance positions gradually decline with the decreasing R. Especially when R is reduced to 0.26a in Fig. 3(c, d) and 0.34a in Fig. 3(c’, d’), the two green and two orange peaks almost coincide with each other due to the extremely tiny differences between a pair of wavevectors {${k_x} = 0.126765(2\pi /a)$, ${k_x} = 0.126782(2\pi /a)$} or {${k_y} = 0.23567(2\pi /a)$, ${k_y} = 0.23681(2\pi /a)$}. We also note that in Fig. 3(a’), as R is further reduced to 0.30a and 0.26a, the BIC point moves from one side of the avoiding band through the lowest point to the other side, and the ZGV modes disappear completely with the vanishing flatten band due to the attenuation of near-field coupling between the guided modes, as shown in Fig. 3(d’). As a result, it can be concluded that the Friedrich-Wintgen BICs appearing in presented photonic structure not only effectively possess ultra-high Q factors but also the varying slow-down factors, i.e. the tunable ${n_g}$ ranging from (22∼1794) along $\Gamma K$ direction and (19∼84) along $\Gamma M$ direction.

 

Fig. 3. The evolution of the avoided crossings and Friedrich-Wintgen BICs at both $\Gamma K$ and $\Gamma M$ directions. (a, b) and (a’, b’) The MEM simulated the dispersion relations near avoided crossings vs a serial of varying geometrical parameters R and d. (c, d) and (c’, d’) Radiative quality factors and group indices corresponding to the GRs of upper avoiding bands for different R in (a) and (a’). (e, f) and (e’, f’) the same as in (c, d, c’, d’) but for the upper avoiding bands under different d in (b) and (b’).

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Similarly, Fig. 3(b) and (b’) show the evolution scenario of BIC and ZGV modes under the slab thickness d gradually changes but with the same R. We can clearly observe that the evolution trajectories are red shifted as the d value increases, and the BIC and ZGV points simultaneously migrate to lower wavevectors until they intersect with the axis of vanishing wavevector ($\Gamma $ point). It is apparent that the band gap still opens at resonance position where two unperturbed dispersion curves cross each other, but its size dramatically decreases as d increases, indicating that a relatively small frequency separation is favorable for a strong mode interaction. By contrast, when the modes are sufficiently close together in the vicinity of the $\Gamma $ point, this repulsion will lead to unusual features such as a clear Dirac cone dispersion and a relatively flat band, for which an effectively zero-refractive-index response can be expected [48]. As in the previous cases, Fig. 3(e, f) and (e’, f’) exhibit that the resonant positions between the corresponding BIC and ZGV modes also have a one-to-one correspondence at almost the same wavevectors. Further details regarding the impact of subtly differences in their resonant wavevectors will be provided later. More interestingly, Fig. 3(e) and (e’) demonstrate that the interconversions between the symmetry protected BICs at $\Gamma $ and topologically protected BICs at off-$\Gamma $ can be revealed via the parameter tuning. Therefore, in this article we have successfully engineered the exotic ZGV modes, since they are also symmetry-protected or topologically protected BICs.

3.3 Propagation loss for designed Fabry-Pérot resonator

Furthermore, to demonstrate the significance of the dispersion engineering for potential applications in Fabry-Pérot resonators, we compute the theoretical loss from out-of-plane radiation per unit of propagation distance. As described previously, the waveguide-loss ${L_ \bot }$ is qualified by the quality factors ${Q_ \bot }$ at the lateral direction z. It is reasonable to expect that the loss stemmed from radiative leakage can be significantly suppressed by the Friedrich-Wintgen BICs. We estimate ${L_ \bot }$ in unites of decibel per millimeter as ${L_ \bot }(dB/mm) = 10({\omega _r}/c{Q_ \bot })/\ln 10 \times 1mm$. On the other hand, the mirror-loss ${L_\parallel }$ is governed by the quality factors ${Q_\parallel }$ along the propagation direction x or y. This loss caused by imperfect reflection is proportional to the group index ${n_g}$, becoming infinitesimal at a ZGV point. Similarly, one can calculated numerically ${L_\parallel }$ in per unit of propagation distance as ${L_\parallel }(dB/mm) = 10(2\pi /\lambda {Q_\parallel })/\ln 10 \times 1mm$ with ${Q_\parallel } ={-} 2\pi {n_g}l/(\lambda \log {R_0})$, where the reflection coefficient ${R_0}$ is assumed as 0.3 [5], and the length of waveguide l is required to obey the Fabry-Pérot resonance condition that the round-trip accumulated phase $2{k_{x,y}}l$ is a multiple of $2\pi$. To tune the ZGV response into the telecommunication band around wavelength 1550 nm, we set the appropriate lattice parameters. As shown in Table 1(below), the tunable structural arguments, operating frequencies and wavevectors, as well as round-trip phase shifts are summarized, corresponding to the ZGV states in Fig. 4(a-d), correspondingly.

 

Fig. 4. The propagation loss of the engineered Fabry-Pérot resonator as a function of wavevector at two crystallographic directions, (a, b) $\Gamma K$ direction and (c, d) $\Gamma M$ direction. The red and blue lines denote the leaky waveguide-loss ${L_ \bot }$ (along the transverse direction z) and imperfectly reflecting mirror-loss ${L_\parallel }$ (along the longitude direction x or y), respectively. The arguments involved are displayed in the following Table 1. (i, j) The spatial profiles of electric field for the lossless resonant modes at the operating wavelength 1550 nm within a segment of simulated photonic system at both $\Gamma K$ and $\Gamma M$ directions (corresponding to the zero-loss points in (a) and (c), respectively).

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Table 1. The geometrical parameters for ZGV responses at the operating wavelength 1550 nm and round-trip accumulated phase shift of Fabry-Pérot resonators

View Table

The numerical results are shown in Fig. 4 for both propagations along $\Gamma K$ and $\Gamma M$ directions. One can clearly find from Fig. 4(a) and (d) that the mirror losses ${L_\parallel }$ are approximately comparable to the waveguide losses ${L_ \bot }$, as indicated by the blue and red lines, respectively. Moreover, Fig. 4(a) and (c) demonstrate that both ${L_\parallel }$ and ${L_ \bot }$ go to zero at two orthogonal crystallographic directions due to the coalescence of a pair of BIC and ZGV modes. By contrast, when these two modes depart from one another, a significant radiation loss is inevitably produced either at transverse direction z or longitudinal directions x and y, as shown in Fig. 4(b) and (d). As expected, panels (i) and (j) explain very well the field evolution behavior at both $\Gamma K$ and $\Gamma M$ directions, including the lossless interfering patterns in a segment of the finite-sized Fabry-Pérot cavities, where the relevant variables are shown in (a) and (c) of Table 1. It can be clearly seen such waveguide modes are localized longitudinally by ZGV modes and transversely by BICs. Noted that the group velocity in the Fabry-Pérot resonator can be expressed as a constant gain per unit time and reduction of loss of a resonator. Therefore, the small group velocity in this article leads to the high Q primarily due to the increasing round-trip travel time and the decreasing power lost at each reflection (even though no conventional high-quality mirrors in the propagation direction). The numerical results suggest that our dispersion engineering is not only expected to have outstanding properties above the light cone, but it could also be relevant for efficient laser applications. Furthermore, this structure with both BIC and ZGV modes has ultra-high Q (implying ultra-low loss and large field enhancement), making it potentially capable for suppression of radiation in the biological sensor with 2D periodicity. It has also wavevector and wavelength selectivity, making it potentially useful for filters, modulators and waveguides. It is noteworthy that these optical and photonic devices possess unique advantage of both non-radiative loss and ultra-low group velocity in the fields of nonlinear and quantum optics, as well as planar integrated photonics.

Next, we will consider the influence of material dispersion from more practical experimental system on the study of BICs, especially parity-time (PT) symmetric system with the complementary gain and loss, in which the gain medium is considered as the semiconductor material pumped by the light source, such as InGaAsP with stimulated radiation wavelength of 1.55um at room temperature [49], and the loss material can be described by Drude or Lorenz model. After the theoretical model is improved, the BIC in PT symmetric systems can be verified experimentally. In addition, previous study has developed BIC lasers by employing photonic-crystal processing technology at the infrared wavelength of 1550 nm [49]. Therefore, using similar techniques we believe that the BIC-ZGV laser modes is expected to be further discovered in PT symmetric structures through experiments

4. Conclusion

In conclusion, we have shown a way for engineering the perfect nanoscale optical resonators that can completely eliminate the out-of-plane radiation from 2D high-index-contrast PhC slab. Using the mode expansion method, we have demonstrated that the designed devices are governed by BICs, which are also ZGV modes at nonzero Bloch wavevectors. The radiation losses are completely suppressed in the transverse direction by BICs and longitudinal direction by zero-velocity modes. By adjusting geometric parameters, we have achieved the essential characteristics including the formation and modulation of the ZGV and BIC modes, whose radiative fields are destructively interfered in both parallel and vertical to the waveguide plane, resulting in the ultra-high Q factors. The presented nanoscale optical resonators are thus ideal for applications in linear and nonlinear optics and radiation control at near-infrared frequencies.