1. Introduction

Guiding light is significant in the classical and modern optics. The concept of total internal reflection, as the basis of guided-wave optics, plays a key role in modern fiber optic communication networks [1,2]. Compared to the index guiding, the existence of photonic band gaps provides another mechanism to guide light [3–5]. Essentially, both of these two mechanisms are to design a mirror made of dielectrics to reflect and guide light, where outgoing waves are forbidden owing to the mismatch of momentum or band-gap effect [6]. Recently, another mechanism called coherent perfect reflection (CPR) is proposed. The CPR comes from the destructive interference of Bloch modes and can occur at the interface between the photonic crystal (PhC) and uniform dielectric medium [7].

If a periodic modulation is introduced in an optical waveguide or fiber, the original waveguide mode may turn into a guided resonance with a finite lifetime due to Bragg scattering. Guided resonances appear above the light line and can couple to external radiation modes [8], which have been studied by the multimode Fabry-Perot model in high-contrast gratings [9–11]. However, an exception is that some isolated state embedded in the continuum can be perfectly bounded with no radiation, known as bound state in the continuum (BIC) [12]. This concept was first proposed by von Neumann and Wigner for the electron in a hypothetical quantum system [13]. Since then, BICs have been found to be a widespread wave phenomenon and extended in various physical systems such as photonics [7,14–39], acoustics [40–42], and water waves [43,44]. The remarkable feature of BIC-inspired modes (e.g., quasi-BICs) with ultra-high quality factor renders many applications possible, including lasing [45–47], sensing [48,49], surface enhanced Raman spectroscopy [50] and nonlinear optics [51]. In the PhC slabs, BICs have been theoretically identified and experimentally observed on the frequency band of guided resonances [14–24]. The formation mechanism of the BICs in the PhC slab has been interpreted by the destructive interference of radiation from the bulk states of PhCs [7,19–21]. It is found that BICs can be determined by the polarization vortices of far-field radiation and therefore characterized by topological charges [18]. In addition to the topological properties of BICs in the far field, the near-field behaviors are also significant in the applications of BICs.

Here, the near-field properties of BICs in the PhC slab are analyzed by decomposing the BICs into bulk Bloch modes of PhC. By utilizing the coherent perfect reflection, the conventional waveguide condition (total internal reflection and resonance conditions) for the planar dielectric waveguide has been generalized to determine BICs [7]. Based on this generalized resonance condition, the indices, or the number of nodes, for each Bloch modes can be obtained. These mode indices depend on normal component of the wave vector, thickness of slab, and symmetry of BIC. Since the number of constituted Bloch modes is more than one, there are at least two mode indices for BICs. For the accidental BICs in the region with only two bulk Bloch modes and one diffraction order in the q-ω space, it is found that the difference between indices of Bloch modes is always fixed for the same structure. Furthermore, this analysis is also applicable to the guided resonances of PhC slab. For the guided resonances, the mode indices can undergo a sudden change on the same frequency band.

2. Theory and simulation

Firstly, we demonstrate the coherent perfect reflection for a semi-infinite one-dimensional (1D) PhC. This PhC is periodic in the y direction with a period a, uniform in the x direction, and truncated at the interface of z = 0, as schematically shown in Fig. 1(a). The dielectric at the space of z > 0 has a relative permittivity ε_b. The alternating dielectric layers of PhC have relative permittivities ε₁ and ε₂, and thicknesses a – d and d, respectively. Here, transverse electric (TE) waves are taken as an example, and similar results can be obtained for TM waves. For the PhC, the dispersion relation can relate the frequency ω, the normal component of the wave vector k_z, and the Bloch wave vector q as follows [1]:

 

Fig. 1. Coherent perfect reflection (CPR) for a semi-infinite photonic crystal (PhC). (a) Schematic of a semi-infinite 1D PhC truncated at the z = 0 plane. (b) Band structure of the 1D PhC for a fixed Bloch wave vector q = 0.2 (2π/a). In the region indicated in yellow, there are only two Bloch modes, while there exist three Bloch modes in the region marked in cyan. (c) Spatial field profiles of E_x at ω = 0.6 (2πc/a). The left two correspond to the single mode incidence with only the Bloch mode labelled respectively by point ① and ② in (b). The transmission of ① and ② cancel out each other perfectly and CPR is formed by the combination of the incidence of these two modes as shown in the right figure. (d) Spatial field profiles of E_x at ω=0.8 (2πc/a). The left three correspond to the single mode incidence with only the Bloch mode marked respectively by point ③, ④ and ⑤ in (b). The right one is the superposition of the former three in which the CPR occurs. Here, the parameters are chosen as ε₁ = ε_b = 1, ε₂ = 4.9, and d = 0.5a.

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Supposing that a number of Bloch modes in the PhC with a fixed frequency ω and wave vector q impinge on the interface at z = 0, the electric field inside the PhC [region I of Fig. 1(a)] can be expressed as superposition of these modes:

In simulations, the Bloch waves in the semi-infinite PhC can be excited by periodic point sources. The incident wave with only one propagating Bloch mode can be achieved by introducing several dipole sources with appropriately chosen complex amplitude [7]. The spatial field profiles of E_x at ω = 0.6 (2πc/a) are shown for the incidence of different Bloch modes in Fig. 1(c). There exist only two Bloch modes at this frequency, which are labelled by point ① and ② in Fig. 1(b), respectively. The spatial profiles of the left two figures in Fig. 1(c) correspond to the single mode incidence with respectively only the propagating mode ① and propagating mode ② impinging on the interface. As the ratio of these two incident waves, namely, a₁/a₂, satisfies the CPR condition, their corresponding transmitted waves have the same amplitudes but opposite phases. Therefore, if these two Bloch waves impinge on the interface simultaneously, CPR can be achieved as shown in the third figure of Fig. 1(c). As the frequency increases to ω = 0.8 (2πc/a), there are three Bloch modes in the PhC, which are indicated by point ③, ④ and ⑤ in Fig. 1(b), respectively. If these three modes impinge on the interface individually with their relative ratio set to match the CPR condition, the spatial field profiles of E_x are shown by the left three figures of Fig. 1(d). Similarly, if the incident wave is the superposition of these three Bloch modes, the transmitted waves cancel out exactly and CPR is formed as shown in the last figure of Fig. 1(d). These simulated results confirm our theoretical predictions.

Based on the CPR mechanism, light can be further guided in the PhC slab with a finite thickness h as sketched in Fig. 2(a). The origin of the z axis is now set at the center of this slab for convenience. Assuming that the incident and reflected wave vectors are respectively {q, $k_z^{(n)}$} and {q, $- k_z^{(n)}$}, the corresponding coefficients at the upper interface are respectively ${a_n}{e^{ik_z^{(n)}h/2}}$ and ${r_n}{e^{ - ik_z^{(n)}h/2}}$ due to the shift of the origin by h/2 compared to the above semi-infinite case. The CPR condition can be maintained at the two interfaces of the PhC slab if the relative ratio between the two modes keep the same after a bounce, that is, r_n/a_n = const.

 

Fig. 2. Near-field analysis for the BICs in PhC slabs. (a) Schematic of a 1D PhC slab with a thickness h. (b-e) Simulated reflection spectra for the PhC slabs with different thickness when they are illuminated by external plane waves. The accidental BICs are marked by red dots, while the symmetry-protected BICs are marked by black dots. The two BICs highlighted by rhombuses are labelled by $\textrm{TE}_{0 + 2}^{( - 1)}$ and $\textrm{TE}_0^{( - 1)}$ respectively. The mode indices for the other accidental BICs are shown in the red brackets. The region with only two bulk Bloch modes in the PhC and one diffraction order outside the PhC is enclosed by the cyan and black dashed lines. Here, other parameters are the same as those in Fig. 1(f) and (g): theoretical (red solid) and simulated (blue dots) electric field along the red dashed line of (a). For the two BICs marked by rhombuses in (b), the total electric fields are shown in the left panel, and the electric fields of the constituent Bloch modes are given in the boxes. The black dashed lines in (f) and (g) correspond to the upper and lower interfaces of the PhC slab.

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When light bounces back and forth between the upper and lower interfaces without any leakage, it forms a perfectly guided mode along the y direction. In addition to the CPR, a generalized resonance condition is also required. At the interface of PhC slab, for each time of reflection, a phase shift $\varphi _\textrm{r}^{(n)} = \textrm{arg(}{r_n}/{a_n}{e^{ - ik_z^{(n)}h}})$ takes place. The term of ${e^{ - ik_z^{(n)}h}}$ comes from the redefinition of origin compared to the previous semi-infinite PhC case. For the guided mode, the total phase change of light for a round trip should be an integer multiple of 2π for each Bloch mode. This resonance condition is similar to that in the conventional waveguide condition. The generalized resonance condition for the n-th Bloch mode can be written as:

Table 1. Examples of mode indices for the BICs with different symmetries

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Combined the CPR and above resonance conditions together, we can obtain ${{{r_n}} \mathord{\left/ {\vphantom {{{r_n}} {{a_n}}}} \right.} {{a_n}}} ={\pm} \textrm{1}$ for all the modes, where the positive and negative signs correspond to respectively the even and odd symmetries of BICs. It is worth noting that the evanescent modes with purely imaginary $k_z^{(n)}$ obtained from Eq. (1) should also be taken into account in addition to the ordinary Bloch modes with real $k_z^{(n)}$ when the PhC is finite in the z direction. These evanescent modes will diverge and is not physical in an infinite system, but play a significant role near the interface of finite system.

If N modes are considered in total, for the 0-th diffraction order l = 0, Eqs. (4) and (5) for the BICs can be rewritten as

As is known, BICs possess a divergent radiative quality factor and are decoupled from the radiation modes in the free space. If the PhC slab is illuminated by external waves, the resonant peak will disappear at the BIC points. Here, the simulated reflection spectra as a function of q and ω for the PhC slab with different thickness h are shown in Figs. 2(b-e). We focus on those BICs in the region with only two Bloch modes in the PhC and one diffraction order outside the PhC slab, which is enclosed by the cyan dashed and black dashed lines. There are two types of BICs, one is accidental BICs marked by the red dots and the other belongs to symmetry-protected BICs marked by the black dots.

To examine the near-field properties of BICs, two BICs highlighted by the rhombuses in Fig. 2(b) are taken as examples. For the BIC marked by red rhombus, it is consisted of two bulk Bloch modes with wave vectors $k_z^{(1)}$ and $k_z^{(\textrm{2})}$ which can be obtained from Eq. (1) and the corresponding mode indices can be identified as m⁽¹⁾ = 0 and m⁽²⁾ = 2 when h = 1.4a. Hence, we use $\textrm{TE}_{\textrm{0 + 2}}^{(\textrm{ - }1)}$ to denote this BIC, where the subscripts are the indices of constituent Bloch modes and the superscript −1 corresponds to the index of band folding in the reduced-zone scheme. It should be emphasized that for the conventional waveguide mode, only one index, namely, number of nodes, is enough to characterize the corresponding spatial field profile normal to the axis of waveguide. However, for BICs in the PhC slab, at least two indices are required since there are more than one constituent Bloch modes to form coherent reflection and construct BICs.

The incident coefficients of all modes for the BIC can be solved by Eqs. (7–9) and the corresponding field distribution can thus be obtained. The electric field E_x along the red dashed line of Fig. 2(a) is shown by the red solid curves in the left panel of Fig. 2(f) for the BIC marked by the red rhombus in Fig. 2(b). The corresponding simulated results are plotted by the blue dots, showing excellent agreement with the theoretical results. Furthermore, the contributions of the n-th constituent Bloch mode $\textrm{2}{a_n}\cos (k_z^{(n)}z)u_q^{(n)}(y ){e^{iqy}}$ are displayed in the box of Fig. 2(f). It is clearly seen that the two indices m⁽¹⁾ = 0 and m⁽²⁾ = 2 stand for the number of nodes of the constituent Bloch modes. The mode indices for the other accidental BICs are also shown in the red brackets of Fig. 2(b-e). It is interesting that the index difference Δm = m⁽²⁾− m⁽¹⁾ for the accidental BICs is a constant for a fixed thickness h whether the BICs possess even or odd symmetry. For example, when the thickness is 1.4a or 2a, the difference between the two indices of Bloch modes Δm is 2; for the thickness of 2.5a or 3a, the difference is 4.

As for the BIC marked by the black rhombus, in Fig. 2(b) it is a symmetry-protected BIC, and the constituent mode u⁽¹⁾ is antisymmetric along the direction of periodicity (y direction). The corresponding zero-th order Fourier component of u⁽¹⁾ is $\tilde{u}_{q,0}^{(\textrm{1)}} = (1/a)\int_0^a {u_q^{(1)}(y)dy} = 0$. Therefore, the mode u⁽¹⁾ is decoupled from the radiation continuum and a_n vanish for any n>1 for this BIC. We label it by $\textrm{TE}_\textrm{0}^{(\textrm{ - }1)}$. Similar to that in Fig. 2(f), the corresponding field profiles are shown in Fig. 2(g).

The construction of BICs by bulk Bloch modes is also applicable to the guided resonances in the PhC slab. Normally, the guided resonances have finite lifetimes because of the leakage to the free space. The generalized waveguide condition can also work for guided resonances if the condition r_n/a_n=const. is relaxed from all the modes to only the propagating modes. In other words, r_n/a_n is no longer a constant for the evanescent modes with purely imaginary $k_z^{(n)}$. This will in fact break the rigorous CPR condition and lead to the leakage to the free space. Remarkably, guided resonances can be obtained from this relaxed condition. By solving Eqs. (4–5) together with r_n/a_n = ±1 for only the propagating modes, the band structure of guided resonances of a PhC slab with h = 1.4a are achieved and shown as solid curves in Fig. 3(a). For comparison, the dispersion determined by the loci of the reflection maxima in Fig. 2(b) are shown as dots, which coincide with the theoretical results. The region marked in yellow or cyan in the q-ω space corresponds to the existence of two or three bulk Bloch modes in the PhC. For the guided resonance with q = 0.3 (2π/a) and ω = 0.468 (2πc/a) indicated by the green rhombus, it has even symmetry in z, possesses two constituent Bloch modes with the indices m⁽¹⁾ = 0 and m⁽²⁾ = 2, and is labelled by $\textrm{TE}_{\textrm{0 + 2}}^{(\textrm{ - }1)}$. As ω increases to the point marked by a green circle, m⁽²⁾ varies from 2 to 4 and this mode changes to $\textrm{TE}_{\textrm{0 + 4}}^{(\textrm{ - }1)}$. For the band with odd symmetry shown by the black curve, all the guided resonances can be characterized by the indices m⁽¹⁾ = 1 and m⁽²⁾ = 3, and thus labelled by $\textrm{TE}_{1 + 3}^{(\textrm{ - }1)}$. It is worth emphasizing that the third band plotted in purple stretches across the regions with two and with three bulk Bloch modes. In the cyan region, one more Bloch mode with the smallest k_z is involved in comparison to that in the yellow region. The indices of the guided resonance vary from $\textrm{TE}_{2 + 4}^{(\textrm{ - }1)}$ at the point of purple rhombus to $\textrm{TE}_{\textrm{0 + 2 + 4}}^{(\textrm{ - }1)}$ at the point of purple circle. Similar to Fig. 2(f), the electric fields for the five marked points in Fig. 3(a) are shown as red solid curves at the left panel of Fig. 3(b). The corresponding simulated results are given by the blue dots, which agree well with the theoretical ones. The field of constituent Bloch modes, ${a_n}({e^{ik_z^{(n)}z}} \pm {e^{ - ik_z^{(n)}z}})u_q^{(n)}(y ){e^{iqy}}$, are also displayed in the boxes in Fig. 3(b), manifesting clearly the corresponding mode indices.

 

Fig. 3. Mode indices for guided resonances in a PhC slab. (a) Theoretically calculated (solid curves) and numerically simulated (dots) band structure. The region with only one diffraction order and two (or three) propagating Bloch modes in the PhC is highlighted in yellow (or cyan), while the gray shaded area in the lower right part denotes the region below the light line. The five guided resonance modes marked by symbols are labelled by $\textrm{TE}_{0 + 2}^{(\textrm{ - }1)}$, $\textrm{TE}_{0 + 4}^{(\textrm{ - }1)}$, $\textrm{TE}_{1 + 3}^{(\textrm{ - }1)}$, $\textrm{TE}_{2 + 4}^{(\textrm{ - }1)}$, $\textrm{TE}_{0 + 2 + 4}^{(\textrm{ - }1)}$, respectively. (b) Calculated (red solid) and simulated (blue dots) electric field for the five guided resonances. The total fields are shown in the left panel, and the field for the constituent Bloch modes are plotted in the boxes. Here, the parameters are the same as those in Fig. 2(b).

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3. Conclusion

In conclusion, we have analyzed the near-field properties of BICs in PhC slabs based on the constituent Bloch modes. The BICs can be characterized by the mode indices of these Bloch modes. An interesting feature for the accidental BICs in the region with only two Bloch modes is discovered, that is, the index difference of the two Bloch modes is a fixed value for the same PhC slab. This analysis is also applicable to the guided resonances of the PhC slab. The guided resonances can also be characterized by mode indices. It has been found that the guided resonances on the same band can have different mode indices. Our results can have potential applications in guided-wave optics, enhanced light-matter interaction, and high-Q cavity sensing.