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Abstract
Bound states in the continuum (BICs) have been widely observed in many symmetric geometries in the optical system during the last decade. Here, we consider a scenario in which the structure is designed asymmetrically with anisotropic birefringent material embedded in one-dimensional photonic crystals. This kind of new shape opens the possibility of obtaining symmetry-protected BICs (SP-BICs) and Friedrich–Wintgen BICs (FW-BICs) form in tunable anisotropy axis tilt. Interestingly, these BICs can be observed as high-Q resonances by variation of the system’s parameters, such as the incident angle, which means the structure without being injected at Brewster's angle can also achieve BICs. Our findings might achieve active regulation and are easy to manufacture.
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1. Introduction
Bound states in the continuum (BICs) are a newly emerging area in the recent decade that has realized the process from theoretical model to experimental verification [1,2]. BICs can be described as radiationless localized states in an open system and unable to couple to radiation channels propagating outside the system, otherwise, they are radiation modes. Neumann and Wigner first originally predicted this phenomenon in quantum mechanics in 1929 [3]. Successively, Friedrich and Wintgen demonstrated BICs through the Feshbach projection of the whole Hilbert space onto the subspace of eigenmodes of a closed system that defines the effective non-Hermitian Hamiltonian. [4,5]. Later, BICs were generally found in the fields of mechanics, electronics, water waves, acoustics, and photonic systems [6–11]. Plotnik et al. discovered and analyzed symmetry protected BICs in 2011 [12]. The properties of BICs make them widely available in various photonic systems, such as gratings and photonic crystals [13–17]. Recently, there have been many methods to produce BICs with anisotropic birefringent media (ABM) [18–22].
There are two mutually orthogonal modes in the ABM consisting of ordinary and extraordinary waves. The coupling of the waves to the radiation channel can be feasibly controlled by rotating the axis of the ABM. Therefore, the bound states can be realized if the leakage is suppressed by appropriate mechanisms [19]. For example, using ridge on dielectric surface to realize BICs or using multilayered microcavities based on 1D photonic crystals can provide the continuum and localized state at the same frequency [23–25]. Recently, ABM has been proposed to combine with one-dimensional photonic crystals (PhC) and is incident at Brewster’s angle to support BICs [26–29].
BICs in asymmetric systems have been proven in layered waveguide structures [17–21]. However, in the system of photonic crystals, symmetry is usually a critical factor to achieve BICs. In addition, the existing fabrication technology is difficult to guarantee the precision of nanomaterials, even more demanding for symmetrical nanomaterials. The two sides of the structure are required to be completely symmetrical, otherwise, the desired accurate results cannot be achieved. A system injected at Brewster’s angle in the one-dimensional symmetric system has been reported previously [27]. In this paper, focusing on breaking spatial symmetry, we design a 1D-layered asymmetric system structure to prove the existence of BICs. Distinct from the PhC placed on two sides as a Fabry–Pérot cavity, the function of the PhC in our research is to filter different light polarization. As a result, by analyzing the transmittance of 1D PhC to different polarization, we propose to use the reflectance spectra vs anisotropy axis rotation angle to search symmetry-protected BICs (SP-BICs) and Friedrich–Wintgen BICs (FW-BICs), then calculate the Q factor to prove them. Besides, we continuously vary the incident angle and demonstrate that the SP-BICs and FW-BICs can exist. Distinct from the symmetric shape, our work provides convenience in the manufacturing process.
2. Results and discussion
Figure 1(a) shows our considered asymmetric multi-layers composed of 1D PhC arm (alternating isotropic materials A and B with their thicknesses da and db and refractive indices na and nb) combining with the ABM (thicknesses L and the ordinary refractive index no and the extraordinary refractive index ne) along the z-axis. On the left side of the structure is a cladding prism with refractive index of nin. The light cone is therefore defined as f = ck0/2πnin, where f and c are the frequency and the vacuum speed of light, respectively, k0 is the vacuum wavenumber. The period of the PhC is Λ=da + db. Figure 1(b) is the side view of Fig. 1(a), θa and θb are the propagation angles of light in the PhC. The optical axis of the ABM a lies in the x-y plane and has the following form:
ϕ is the angle of a with respect to the y-axis. And the relative permittivity tensors of the ABM have the form:
Fig. 1. Asymmetric multi-layers supporting BICs. (a) 1D PhC and ABM. The incident angle is θB = arcsin(nb/ninsin(θb)) ≈ 53.13°,where θa+θb = π/2, θa = arctan(nb/na), nin = 1.52, A and B are silicon dioxide (SiO2) and titanium dioxide (TiO2) respectively. Their refractive indices na = 1.47, nb = 2.16, and the thicknesses parameters da = 145 nm, db = 94 nm. Optical properties of the ABM are defined by no = 1.52 + 0.001i and ne = 1.72 + 0.001i with a little loss and its thickness is L = 2.75 µm. ϕ is the angle of the ABM optical axis with respect to the y-axis. (b) The side view of (a). The period of the PhC is Λ = da + db. m is the number of the unit cell.
TM polarized light ei(kr-ωt) whose wave vector is k = (kx,0,kz) is incident at θB. The propagation angles in the A and B layers are θa and θb respectively. ω is the angular frequency of light. In a nonmagnetic medium, the solution of Maxwell's equations for each layer is given by:
In layers A and B, the incident plane x-z determines TE- polarized waves (components Ey, Hx, Hz) and TM- polarized waves (Hy, Ex, Ez). Different from isotropic layers, the direction of the optical axis of ABM a is supported by two orthogonal solutions in isotropic media. The component of the wave vector on the x-axis is defined as follows: kx is a constant value (nink0sinθB) when electromagnetic waves propagate in every layer. Thus, we can calculate the TE- polarized field components in layers A (DA$\; \in$ [L + mΛ, L + mΛ+da]) by Eq. (3):
Similarly, TE-polarized field components in layers B (DB$\in$ [L + mΛ+da, L + (m + 1)Λ]) are:
Then combining Eqs. (4)–(8) by utilizing the continuity condition on the boundary between layers A and B, after some algebraic derivation we can obtain:
Let the determinant of Eq. (9) be zero, the TE dispersion equation in PhC satisfies [27]:
Equation (11) is used to describe the band structure of the PhC. Thus, we can obtain the photonic bandgap when $|{\cos (K\Lambda )} |> 1$.And $|{\cos (K\Lambda )} |< 1$ means the light can propagate in the PhC. One can get the boundary of all bands by using $|{\cos (K\Lambda )} |{ = }1$.
Similarly, we can obtain the dispersion equation of TM-polarization in PhC involving layers A and B as follow:
After the same operation as TE-polarization above and replacing Eq. (10) with Eq. (16), then the corresponding TM band can be obtained.
To look into the band diagrams of the PhC, we examine the transmission spectrum by changing the frequency of incident light and the projection of the wave vector on the x-axis kx. We set 20 pairs of PhC layers in Fig. 2(a) and 10 pairs of PhC layers in Fig. 2(b). The red line corresponds to the boundary of the bands in Fig. 2(a). It has a significant difference between both basic polarizations above the light cone. One can discover that TM-polarization can propagate at Brewster’s angle (the white dash line) in PhC but TE-polarization can not. Consequently, we choose this frequency range for our study in Fig. 2(b) where the PhC can be viewed as a perfect mirror for TE-polarization. Then, the TE-localized mode can survive in the TM continuum. Figures 3(c) and 3(d) are the band diagram obtained by letting $|{\cos (K\Lambda )} |< 1$ in Eq. (11) for both TE- and TM-polarization.
Fig. 2. Band structure of 1D PhC. (a) Reflectance spectra for TE (left) and TM (right) polarization wave calculated by transfer-matrix method. The green dashed line corresponds to the light cone and the white dashed line corresponds to Brewster’s angle. (b) The operating frequency range of the PhC we selected. Red pluses are the BICs. (c) - (d) The color regions are the result of $|{\cos (K\Lambda )} |< 1$. The yellow one corresponds to the TE band, and the red one is the TM band.
Here, we use finite element method to check the BICs. The Bloch boundaries are set in the x and y direction. Figure 3(a) illustrates the existence of BICs in the asymmetric structure with the incident wavelength λ and the rotation of ABM ϕ. Unlike bilateral PhC structures, PhC in the asymmetric system simply acts as a filter for TE and TM waves instead of the Fabry- Pérot cavity in symmetric system. At some specific rotation angles of the ABM optical axis, the TM-polarized incident light reaches the interface between the ABM layer and the air, forming a total reflection. At the same time, the TE localized mode cannot radiate out. Hence, we can observe some Fano resonance collapses in the reflectance picture. Furthermore, we verify the high-Q factor when the incident wavelength is 550 nm as shown in Fig. 3(b), which can also verify the existence of the BICs.
Fig. 3. Illustration of BICs. (a) Reflectance spectra of the incident wavelength vs the rotation of the AMB axis when illuminated by TM-polarized light (b) Quality factor at the wavelength of 550 nm. (c) The electric field distribution of the eigenmodes when SP-BICs occur.
When ϕ = 0°, the ABM optical axis is along the y-axis and the TE-polarization can only inspire the e-wave but cannot contribute to the TM continuum. At the same time, TM-polarization excites the o-wave only. Then, the TE mode is decoupled to the TM continuum which is the existence of the SP-BICs. Moreover, rotating the ABM axis from 30° to 60° can result in the interference of o and e-wave with the aid of the coupling with radiation channel. Under specific conditions, the TM wave inside the PhC could be decoupled from the localized mode in the ABM. The electric fields in the y-direction composition have a non-zero value, but in the x-direction, the sum of both waves is zero. For the eigenstates of the system, both the energy of SP-BICs and FW-BICs are confined in the ABM. In other words, their field distributions are quite similar. The field patterns agree with what we have discussed.
Then, we demonstrate that when the incident light deviates from Brewster’s angle, BICs can also exist. To see the behaviors of this conclusion, we calculate the reflectance spectra for the whole system (PhC plus ABM). In Fig. 3(a), we can see four BICs at ϕ = 0°. We plot these four BICs in the full kx-f diagram in Fig. 4(a). Above the light line, BICs all lie on the Brewster’s angle line. The red dotted line represents the band boundary of the 1D PhC for the TM-polarization and the blue dashed line corresponds to the TE-polarization. It should be noticed that the green lines represent the angle of total internal reflection between the ABM and the air. All BICs are located between the green lines and the black lines, which means they will not leak into the air from the bottom side as shown in Fig. 1(a). In Fig. 4(a), we keep the optical axis of ABM to 0°. Then we slightly change the optical axis of ABM to 1° and depict the detailed reflection spectra in Fig. 4(b). It is clearly shown that several reddish lines emerge in the area enclosed by the blue and red dashed lines. All of these lines are located in the stopband of TE-polarization and at the same time in the passband of TM polarization. This is the evidence of the existence of BICs line in such a system. We choose the points (magenta crosses in Fig. 4(b)) which have the same kx with the two known BICs (a and d) to demonstrate the analysis.
Fig. 4. Illustration of SP-BICs with non-Brewster’s angle incidence. (a) BICs’ locations (red plus) when ϕ = 0° in the reflectance spectrum with kx vs incident wave frequency. Red dashed line corresponds to TM-polarized incident light and blue dashed line represent the TE case. Green line is the total internal reflection of ABM. Black lines correspond to the light lines. (b) BICs’ locations by non-Brewster’s angle incidence (magenta crosses) at ϕ = 1° (c) – (d) Reflectance spectra of kax = 1.119 × 107 m-1 and kdx = 1.5327 × 107 m-1 to illustrate the BICs. The part inside the solid blue line is the photonic band gap. (e) – (f) The evidence of high Q-factor for BICs correspond to (c) - (d).
We keep the value of kx at 1.119 × 107 m-1 and 1.5327 × 107 m-1, respectively, and rotate ϕ from -1° to 1°. The reflection spectra are depicted in Figs. 4(c) and 4(d). The resonance width diminishes to zero at ϕ = 0° for all these magenta crosses points. To verify whether these are BICs and whether they can exist in the system, we calculate the Q-factor and depict the results in Figs. 4(e)–4(f). The ultra-high Q-factors of these points give the evidence of the existence of the BICs.
For a’ and d’ as shown in Figs. 4(c)–4(d), the incident angles are 44.41° and 61.22° which have deviated from Brewster’s angle. In order to prove BICs can exist at such parameters, we fix the incident angle at 44.41° and 61.22° and calculate the reflectance spectra with variables λ and ϕ. These zero resonant widths in Figs. 5(a)–5(b) show that there are multiple BICs except for a’ and d’. The BICs at 0° or 90° are SP-BICs and FW-BICs when ϕ is between 30° and 60°. We demonstrate that the BICs exist when the light incidents at non-Brewster’s angle.
Fig. 5. Reflectance at non-Brewster’s angle incidence. (a) The reflectance spectra when the incident angle is fixed at 44.41°. The part inside the white dashed line is the photonic band gap. (b) The incident angle is fixed at 61.22°.
Finally, we turn to study the FW-BICs. As an example, we choose the point e in Fig. 3(a) (ϕ = 38.34°, λ = 545.8 nm). We keep ϕ = 38.34° unchanged and calculate the whole band diagram kx-f as shown in Fig. 6(a). The inset is the band diagram in a bigger range. Different from the SP-BICs lines in Fig. 5(b), the gray lines are broken at specific points. The resonance width diminishes to zero, which means the FW-BICs occur. Such FW-BICs appear as points in the kx-f diagram. Except for the red plus e, we can also observe other BICs (magenta crosses). These magenta crosses also deviate from Brewster’s angle. Then, we rotate ϕ from 30.84° to 46.84°, and the reflection spectra are shown in Fig. 6(b). Outside the photonic band gap, there are multiple magenta crosses where Fano collapses, which demonstrate the existence of the BICs.
Fig. 6. The evidence of FW-BIC. (a) FW-BIC location in the band structure of the whole system red plus e is the FW-BIC when the incident wavelength is λ = 545.8 nm and ϕ is 38.34° The blue and red dashed lines are the stopband of TE polarization and the passband of TM polarization respectively (b) The reflectance spectra with frequency f and ϕ while kx is fixed at 1.3993 × 107 m-1. Magenta crosses correspond to the FW-BICs at non-Brewster’s angle incident. The blue dashed line area is the photonic band gap.
3. Conclusion
In conclusion, we demonstrate that the SP-BICs and FW-BICs can exist in an asymmetric structure composed of 1D PhC and anisotropic birefringent material. We give an explanation about how they are generated and show the difference from those in the symmetric structure. Moreover, both BICs can also exist whether the angle of the incident is greater or less than Brewster's angle. Actually, BICs are also available in the symmetric system at non-Brewster’s angle. The PhC could even be designed to use as a mirror for TM-polarization. The asymmetric 1D layers system may open new methods for the study of BICs generation which may provide convenience for the fabrications of BIC devices.
Funding
Fundamental Research Funds for the Central Universities (3072022CFJ2506); Foundation of Heilongjiang Province in China (LH2022A009); National Natural Science Foundation of China (11974428, 12074087).
Disclosures
The authors declare no conflicts of interest.
Data availability
The datasets presented in this paper are available from the corresponding authors upon reasonable Request.
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