Broadband and highly efficient enhancement of photonic spin Hall effect at a quasi-bound state in the continuum

Abstract

The efficiency of the photonic spin Hall effect (PSHE) has attracted increasing attention, as it plays a key role in the development of spin-selective devices. However, highly efficient enhancement of PSHE has always been achieved at a single wavelength. Here, we propose a method to achieve highly efficient enhancement of the photonic spin Hall effect over a broadband operating wavelength by utilizing a magnetic dipole quasi-bounded state in the continuum (q-MDBIC) with a high refractive index. Under both horizontal and vertical polarization incidence, when the wavelength is 50.9–51.9 μm and the incident angle is 0.5°–10°, the transverse shift of PSHE can reach 1.7λ–37λ, with high efficiency of more than 90%. Meanwhile, the quality (Q) factor can also be maintained around 55. Furthermore, different metasurface lattice periods can achieve different broadband PSHE, which provides favorable conditions for the control of broadband PSHE. Our work will have wide applications in devices with efficient spin selection.

Graphical abstract

Keywords

photonic spin Hall effect / bound states in the continuum / high efficiency

1 Introduction

As a result of the light-matter interaction, the photonic spin Hall effect (PSHE) is susceptible to the changes in physical parameters in different physical systems. In which, the transverse spin-related shift occurs naturally in real space when the light is reflected or refracted at an interface with a refractive index gradient [1−4], which arises from the splitting of the left and right circularly polarized components of linearly polarized light in the vertical incidence plane [2, 5]. To date, PSHE has been extensively studied in different physical applications, such as optical physics [6, 7], Metamaterials [8, 9], high energy physics [10, 11], semiconductor physics [12], and image-edge detection [13, 14].

Because of the naturally weak spin−orbit coupling, the transverse shift of PSHE is usually at the nanoscale, much smaller than the wavelength, which is often overlooked and hard to observe [2, 15−17]. Researchers have made many efforts to enhance the PSHE with different interfaces, such as surface plasmon [11], anisotropic material [3, 18], epsilon-near-zero material [19, 20], and birefringent material [21]. However, in the above-described method, the essential condition of enhancing PSHE is whether the Fresnel reflection coefficient of p wave or s wave is close to zero [20, 22, 23]. At this moment, the near-zero Fresnel reflection coefficient makes the reflected light used to observe the transverse shift very weak. Therefore, the efficiency of PSHE is always low, greatly choking the practical application of PSHE [24−26]. In order to solve the conflict between the enhancement condition of PSHE and the high efficiency, and to improve the efficiency of PSHE, researchers have focused on some new structures. Recently, metal wire arrays with anisotropic impedance mismatch have been used to demonstrate that the enhancement of PSHE has nearly uniform high efficiency in the microwave spectrum under horizontally polarized microwave incidence [27]. Then, based on the total internal reflection and external reflection around the critical angle, when a light incident from a dense medium to a sparse medium, the efficiency of PSHE could be largely improved under both horizontally and vertically polarized light incidence [28]. However, the previously proposed structures [27, 28] often need complex construction patterns (three-dimensional or stacked large structures are required for certain specific dielectric properties) and exhibit large optical losses (metal-dielectric). Therefore, an all-dielectric metasurface was proposed to regulate the Fresnel coefficient in a compact and versatile way. The efficiency of PSHE was improved to 71% and 79% at a single wavelength of 800 nm for horizontally and vertically polarized light, respectively [29]. Subsequently, a series of simple ways to enhance the efficiency have emerged, however, all at a single wavelength [27, 30, 31]. In the meantime, although broadband PSHE has also been studied and implemented, it does not exist in the case of high efficiency [32, 33]. Hence, the extension of the operating wavelength of the highly efficient PSHE has still been a burning issue. Recently, the proposal of bound states in the continuum (BIC) has brought new possibilities for achieving the highly efficient PSHE with broadband. When the BIC leakage is q-BIC, it can bring extremely high reflectivity, which makes it possible to further improve the efficiency of PSHE. In the previous work of using BICs for the shift of PSHE, due to the limitation of spin-dependent shift enhancement, previous work paid more attention to the high-quality factor characteristics of q-BIC, and did not use its extremely high reflection coefficient to achieve efficiency improvement [34]. Here, we use another enhancement mechanism to construct a special phase difference, so that the high reflectivity can be fully utilized. In addition, to further achieve broadband enhancement and high efficiency, the magnetic dipole bound state in the continuum (q-MDBIC) is used. Since the coupling (and scattering) of the magnetic dipole resonance in the in-plane electric field and magnetic field is completely allowed, the in-plane resonance will exhibit broadband characteristics, which expands the bandwidth [35]. In the meantime, q-MDBIC also can produce strong resonance (reflectance of 1) and high-quality factor (Q) [36−41], it can achieve a broadband polarization-independent total reflection (r_s = r_p = 1) effect.

In this work, we design a high refractive index all-dielectric metasurface with magnetic dipole resonance quasi-bound states in the continuum (q-MDBIC) to achieve flexible regulation of PSHE. A broadband magnetic dipole (MD) resonance band can be realized near the small incident angle condition, and a high polarization-independent reflection band can be obtained. We prove that this all-dielectric structure with appropriate geometric parameters supports broadband efficient and large polarization-independent PSHE. At the same time, it can be realized in a wide incident angle domain, which will be conducive to our flexible manipulation of PSHE. It is worth noting that at an incident angle of 0.5°, the photonic spin Hall shift can reach 37 times the wavelength, the efficiency is 100%, and Q is 55. In addition, we obtained a stable large PSHE and high-efficiency zone by adjusting the period of the structure. Our design method opens up a very effective way to realize the coexistence of large PSHE and high efficiency, and expands the new functions of spin photonic devices.

2 Theory analysis

When a linearly polarized Gaussian beam reflects at an interface with a certain refractive index gradient, it will undergo the transverse spatial shift on the vertical incident plane. According to the angular spectrum theory, when the incident beam waist is large enough, the shift can be expressed as [42]

(1){\delta }_{\pm }^{\mathrm{H}}=\mp {\displaystyle \frac{\lambda }{2\pi }}\left[1+{\displaystyle \frac{\left|r_{\mathrm{s}}\right|}{\left|r_{\mathrm{p}}\right|}}\cos \left({\phi }_{\mathrm{s}}-{\phi }_{\mathrm{p}}\right)\right]\cot {\theta }_{\mathrm{i}},

(2){\delta }_{\pm }^{\mathrm{V}}=\mp {\displaystyle \frac{\lambda }{2\pi }}\left[1+{\displaystyle \frac{\left|r_{\mathrm{p}}\right|}{\left|r_{\mathrm{s}}\right|}}\cos \left({\phi }_{\mathrm{p}}-{\phi }_{\mathrm{s}}\right)\right]\cot {\theta }_{\mathrm{i}}.

Here, the superscripts H and V represent the horizontal and vertical polarized light, while the subscripts + and − indicate the left and right circularly polarized light. The symbols r_s and r_p represent the Fresnel reflection coefficients of TE and TM polarized light, while {\phi }_{\mathrm{p}}-{\phi }_{\mathrm{s}} is the corresponding phase difference. λ is the wavelength of the incident beam, and θ_i is the incident angle. The transverse shift is primarily determined by the ratio of the Fresnel coefficient (|r_{\mathrm{p}(\mathrm{s})}|/|r_{\mathrm{s}(\mathrm{p})}|) based on the above formula. To enlarge the shift, the reflection coefficient in the denominator should be minimized or even close to zero. Many works have been done to enhance PSHE by utilizing the near-zero reflection around the Brewster angle θ_B [22, 23, 42, 43]. Fig.1(a) illustrates the basic physical mechanism. When polarized light incidents on a regular interface with Brewster angle, the p-wave vanishes at the reflection plane, r_p is equal to 0. Consequently, the large transverse shift with H-polarized light incident can be achieved by obtaining a large value of |r_{\mathrm{s}}|/|r_{\mathrm{p}}|. This makes the reflectivity extremely low under the enhancement mechanism of PSHE.

However, about the efficiency of PSHE, it can be represented as [27]

(3){\eta }_{\mathrm{H}}={\left|r_{\mathrm{p}}\right|}^2,

(4){\eta }_{\mathrm{v}}={\left|r_{\mathrm{s}}\right|}^2.

Obviously, the methods of enhancing PSHE by minimizing the reflection coefficients [11, 44−46] inevitably decrease the efficiency. To maximize the efficiency, the enhancement mechanism that is not affected by the reflection coefficient must be used, and the both reflection coefficient under the TM and TE polarization must be equal to 1 (r_s = r_p = 1) shown in Fig.1(b). This also means that the phase difference of the reflection coefficient must be guaranteed to be 0. Then, the transverse shift can be simplified as

(5){\delta }_{\pm }^{\mathrm{H}}=\mp \frac{\lambda }{\pi }\cot {\theta }_{\mathrm{i}},

(6){\delta }_{\pm }^{\mathrm{V}}=\mp \frac{\lambda }{\pi }\cot {\theta }_{\mathrm{i}}.

Different from relying on near-zero reflection coefficient to realize the enhancement of PSHE. Currently, the enhancement of PSHE only depends on the incident angle θ_i. To achieve high efficiency besides with a large transverse shift, we only need to reduce the incident angle.

3 Results and discussion

Based on the high reflectivity and high Q factor brought by q-BIC, and the wide wavelength range of high refractive index materials, we design an almost lossless slab with a high refractive index. After parameter optimization, a periodic structure composed of rectangular blocks with a dielectric constant of ε = 78 + i0.05 (in the GHz waveband) is designed. The slab is placed on a substrate with a thickness of h and a dielectric constant of 1, as shown in Fig.1(c). The structure parameters of the cuboid block are set as L = 6 mm, W = 6 mm, and H = 4 mm. The lattice period P = 12 mm, and the substrate thickness h = 1 mm, as shown in the lower right corner (the single atom schematic).

Initially, we analyze the behavior of a linearly polarized beam incident on the designed metasurface. Fig.1(d) illustrates the relationship between the reflectance and the operating wavelength at different incident angles. When the incident angle is 0° (normal incidence), the spectrum shows a sharp q-BIC resonance in the wavelength range of 50−52.5 mm. As the incident angle increases, the resonance peak of q-BIC gradually narrows and finally disappears into BIC. In this process, the Q factor gradually increases. When the incident angle is 90°, the BIC is formed at 51.5 mm. The corresponding electric field distribution is shown in Fig.1(e), and the black arrow indicates the displacement current density. In the BIC mode, the radiation channel is completely closed, and the electric field energy is completely limited around the structure. We use the characteristic frequency analysis of COMSOL to calculate the corresponding Q factor, as shown in Fig.1(f). During the process of approaching the BIC (k from 0.8 to 1, black dotted line), the Q factor gradually increases with k and eventually diverges to infinity at BIC. Simultaneously, under small angle conditions (k from 0 to 0.1, blue dotted line), the q-BIC resonance remarkably maintains a high Q factor exceeding 50. High Q factor can enhance the coupling between light and matter, and significantly improve the performance of nanophotonic devices.

To explore the specific origin of the q-BIC strong resonance, we adopt the multi-machine decomposition method [47, 48] and analyze its electromagnetic field distribution. We calculate the moments of electric dipole (ED), magnetic dipole (MD), toroidal dipole (TD), electric quadrupole (EQ), and magnetic quadrupole (MQ) under TM and TE polarizations to illustrate the composition of the excitation resonance, as shown in Fig.2(a) and (b), respectively. Fig.2(a) shows that under TM polarization, MD has a strong resonance, similar to the reflectance curve, and the maximum value also appears at 51.5 mm. The radiation powers of other dipoles are almost kept around 0 in the whole wavelength range. We can conclude that the contribution of MD to the far-field scattering power is dominant in the generation of the strong resonance. The corresponding electric and magnetic field distributions are shown in Fig.2(c) and (d). The white arrow represents the displacement current density. Due to the leakage mode, the electric field energy is concentrated around the cuboid block, while the magnetic field is mainly concentrated inside the cuboid block. The displacement current and conduction current create a closed-loop circulation in the x−z plane within the center of the cuboid block as can be seen from the electric field’s polarization direction. The corresponding magnetic field distribution shows the MD mode. The head of the displacement current is directly connected to the tail, and each cell has an MD moment in the z-direction, which is a characteristic of q-MDBIC [49]. This q-MDBIC provides a moderately uniform electromagnetic field distribution on the metasurface. While for TE polarization, it also exhibits similar behavior shown in Fig.2(b), (e), and (f), further proving the polarization-independent characteristics.

From Eqs. (5) and (6), we have known that under the set reflectivity and phase conditions, the increase of the shift value of PSHE is no longer related to the ratio of the Fresnel coefficient, but only depends on the incident angle. To demonstrate the broadband and highly efficient PSHE, the relationships among the reflectance, wavelength, and incident angle are shown in Fig.3(a) and (b) under TM and TE polarization, respectively. Since the spin-orbit coupling of PSHE will compete and confuse with that of the vortex light when the incident angle is near 0° (the normal incidence) [49], we set the incident angle range as 0.5°−60°. For TM polarization in Fig.3(a), the high reflectance of R_p ≈ 1 appears at the wavelength range of 50.9−51.9 mm and 55.5−57.5 mm under a wide incident angle range, forming a broadband q-MDBIC. At the former band, the wavelength range of R_p ≈ 1 decreases as the incident angle increases from 0.5° to 60°. In the meantime, R_s ≈ 1 also appears during the incident angle of 0.5°−60°, as shown in Fig.3(b). The broadband operating wavelength is derived from the previously analyzed MD resonance in Fig.2(a) and (b), which shows that the strong resonance possesses a broadband property. Obviously, during this wavelength range, it is beneficial to obtain broadband and highly efficient PSHE.

On the contrary, around 56.5 mm, the wavelength range of R_p ≈ 1 is wider, as shown in Fig.3(a). With the decrease of the incident angle, it gets narrower and gradually disappears below 8.2°, and a BIC state appears. Unfortunately, during this wavelength range, R_s always approach zero. Therefore, the highly efficient PSHE cannot be obtained within this area. The corresponding cosine value of the phase difference is shown in Fig.3(c) and (d). Due to the polarization-independent nature of the metasurface, cos(φ_s−φ_p) and cos(φ_p−φ_s) are roughly the same over the entire wavelength and incident angle range. Most importantly, in the wavelength range of 50.9−51.9 mm, the phase difference between r_s and r_p remains almost 1 within the incident angle range of 0.5°−10°. So far, the conditions for achieving broadband and highly efficient PSHE are all satisfied according to Eqs. (5) and (6).

After substituting the reflection coefficient and phase difference obtained in Fig.3 into Eqs. (1) and (2), the transverse shifts of PSHE under TM and TE polarization incidence can be calculated, as shown in Fig.4(a) and (b), respectively. From Eqs. (5) and (6), the shift value of PSHE is proportional to \mathrm{c}\mathrm{o}\mathrm{t}{\theta }_{\mathrm{i}} and is no longer related to the reflection coefficient. Since the above mentioned PSHE is inversely proportional to the incident angle, we only calculate the transverse shift within 0.5°−10°. For TM polarization incidence [Fig.4(a)], at a small angle, the PSHE can always be enhanced in the entire 45.5−60 mm waveband. The smaller the incident angle is, the larger the transverse shift become. When the incident angle is 0.5°, the maximum can reach 37λ.

In the meanwhile, we also give out the corresponding efficiency in Fig.4(c) and (d). For TM polarization incidence [Fig.4(c)] during the whole waveband of 50.9−51.9 mm at an incident angle of 0.5°−10°, the efficiency of PSHE enhancement all reaches more than 90%. Most importantly, in the wavelength range of 51.1−51.7 mm, the efficiency can reach more than 98%. When the incident wavelength is 51.5 mm, r_s = r_p = 1, so the efficiency can reach 100%. While for TE polarization [Fig.4(d)], it exhibits the same behavior as that for TM polarization, which further confirms the polarization-independent characteristics of our q-MDBIC metasurface.

Then, we select the wavelengths of 56.5 mm and 51.5 mm for comparison, shown in Fig.4(e) and (f). In Fig.4(e) for 56.5 mm, although the PSHE can be largely enhanced at the incident angle of 0.5°−10°, the reflectance is very small, resulting in a very low efficiency (close to 0). However, for 51.5 mm [Fig.4(f)], within an incident angle range of 0.5°−10°, the PSHE can be largely enhanced, also, the efficiency can always approach 100%. When the incident angle is 0.5°, the maximum transverse shift value can reach as high as 37λ, simultaneously, the Q factor also remains at a high level of 55. When the incident angle is 10°, the transverse shift of PSHE is reduced to about 1.7λ. Therefore, at the incident angle of 0.5°−10° and the wavelength of 50.9−51.9 mm, we can obtain high-efficiency PSHE and control it.

To flexibly enhance the PSHE, we further explore the influence of the lattice period P on the PSHE. The incident angle is fixed at 5°, P is reduced from 12 mm to 6 mm, and the reflectance curves under TM and TE polarization are shown in Fig.5(a) and (b), respectively. According to Eqs. (3) and (4), the efficiency of PSHE is the square of the reflection coefficient, so Fig.5(a) and (b) can also be used to represent the efficiency of PSHE. As the lattice period decreases, the cuboid columns become closer to each other. When P changes from 12 mm to 8 mm, the resonance wavelength of q-MDBIC keeps nearly unchanged around 51.5 mm (the dotted line). Moreover, R_p and R_s all approach 1 in the wavelength range of 50.9−51.9 mm. When P is reduced below 8 mm, the resonance position begins to diverge.

Then, we calculate the transverse shifts of PSHE under TM and TE polarization incidence versus the lattice period P, as shown in Fig.5(c) and (d). For TM polarization [Fig.5(c)], due to the polarization-independent characteristics, when the incident angle is 5° around the wavelength of 51.5 mm (50.9−51.9 mm), most of the transverse shifts are around 3.61λ in the entire 6−12 mm period range, besides with high efficiency. In addition, within the wavelength of 55.5−60 mm, the transverse shifts can be enhanced larger with a maximum of 5.2λ, however, the efficiency is always low. In the range of 8−12 mm, when the wavelength is 50.9−51.9 mm, the efficiency (>90%) can be obtained besides with a large transverse shift (3.61λ). Furthermore, when the wavelength is located at 51.1−51.7 mm, the efficiency can even reach more than 98%. Similar behavior can be found for the TE polarization, shown in Fig.5(d). Therefore, we can also obtain different broadband and efficient PSHE enhancement by changing the period of the metasurface lattice. This brings favorable conditions for the flexible control of broadband PSHE.

4 Conclusions

In conclusion, we theoretically examine the conditions necessary to achieve efficient and large coexistence of PSHE. A method for designing a q-MDBIC metasurface using the magnetic dipole resonance and lossless properties of a high refractive index rectangular blocks is proposed to simultaneously achieve a wide band efficient and large polarization-independent PSHE. At an incident angle of 0.5°−10° and a wavelength range of 50.9−51.9 mm, we obtain H- and V-polarization PSHE shifts with efficiencies above 90%, and the maximum values can reach 37λ and 36.8λ with high Q (55), respectively. When the wavelength is 51.1−51.7 mm, the efficiency can even reach more than 98%. In addition, by changing the lattice period, we can still obtain such broadband PSHE. The great enhancement of PSHE and the high efficiency in broadband will greatly improve the performance of spin-related photonic devices.

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