Enhanced spin Hall effect due to the redshift gaps of photonic hypercrystals

Jiaoqiao Xia; Yu Chen; Yuanjiang Xiang

doi:10.1364/OE.420907

1. Introduction

The left and right-handed circularly polarized components split and produce transverse displacement in the direction perpendicular to the refractive index gradient when a Gaussian beam is reflected or transmitted at the interface. This phenomenon is called the photonic spin Hall effect (PSHE) [1–4]. It is an optical analogy of SHE in the electronic system, the spin of photons and refractive index gradient are equivalent to the spin of electrons and the electric potential, respectively [5]. The photonic SHE has attracted extensive interest in the past decade and has been widely applied to the various fields, including high-energy physics [6,7], optical physics [8,9], semiconductor physics [10–12], and so on. However, the SHE is very difficult to be observed by traditional experimental equipment. Thus, many effective and practical methods have been proposed to enhance the spin-dependent splitting.

Generally, the spin-dependent splitting is limited in scale of optical wavelength. It was first observed at the air-glass interface via weak measurement [13], but the measured displacement is very small. There are some methods were proposed to significantly enhance the displacement of the SHE. For example, Xiang et al. put forward the enhancement of the photonic SHE using the guided-wave surface plasmon resonance (GWSPR) [14], compared with the ordinary SPR, transverse displacement is extended to 11.2 µm. Slobozhanyuk et al. demonstrated theoretically and experimentally that the zigzag array with the odd number of particles can greatly improve the photonic SHE due to the exciting subwavelength edge states [15]. Luo et al. revealed that near the Brewster angle on reflection the enhanced SHE of light can be realized [16]. Kim et al. experimentally demonstrated that compared to the spin Hall effect on the horizontal hyperbolic metamaterial, a vertical hyperbolic metamaterial can enhance the optical spin Hall effect by several orders of magnitude [17]. Actually, there have already been various experimental and theoretical works of PSHE on HMMs [18–22]. In addition, there are also using other anisotropic materials and structures to enhanced PSHE, including polymer films [23], uniaxial crystals [24,25], polarizers [26], liquid crystals [27], etc.

In this paper, the physical model we built is a one-dimensional photonic hypercrystal (1DPHC). This is the first use of a hypercrystal to study the enhancement of photonic SHE. PHCs is proposed by E. E. Narimanov in 2014 [28]. It is an artificial media that combines features of metamaterials and photonic crystals, and also provides a platform for light-matter interaction [29–31]. By numerical simulation, we show that SHE of light on PHCs is much stronger than that on all-dielectric crystals. Our research has played a positive role for the practical application of SHE in optical devices.

2. Theoretical models and methods

The photonic SHE of reflected light is shown in Fig. 1. It is shown that a light beam cannot reflect immediately but splits into two components when light is incident on the xy plane. These components are left- and right-spin circular polarized light and they produce the displacements of equal value in the direction perpendicular to the incident surface, that is transverse displacement. In addition, there is also displacement in the vertical direction, which is called Goos-Hänchen (GH) effect.

Fig. 1. Schematic of the SHE of the reflected light, the incident plane is xy plane and ${\delta ^ \pm }$ are the transverse displacements of the light for the left- and right-spin components. Substructure is HMMs composed of silver and trititanium pentoxide. Figure 1. Schematic of the SHE of the reflected light, the incident plane is xy plane and ${\delta ^ \pm }$ are the transverse displacements of the light for the left- and right-spin components. Substructure is HMMs composed of silver and trititanium pentoxide.

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The propagation process of light in the whole structure must be deduced before calculate the transverse displacement. According to the angle spectrum theory, any complex amplitude distribution of light field can be regarded as the spectrum distribution obtained by Fourier transform. Therefore, the general propagation model of the incident gaussian beam can be written as:

(1)$${\tilde{E}_i} = \frac{{{\omega _0}}}{{\sqrt {2\pi } }}\exp \left[ { - \frac{{\omega_0^2({k_{ix}^2 + k_{iy}^2} )}}{4}} \right], $$

in this formulation, ω₀ = 0.5 mm, which is the beam waist. k_ix and k_iy are components of the wave vector in x and y axes, respectively. Besides, in spin basis set, the polarization components of the incident light are as follows,

(2)$$\tilde{E}_i^H = \frac{1}{{\sqrt 2 }}({\tilde{E}_{i + }} + {\tilde{E}_{i - }}), $$

(3)$$\tilde{E}_i^V = \frac{1}{{\sqrt 2 }}i({\tilde{E}_{i - }} - {\tilde{E}_{i + }}), $$

where $\tilde{E}_i^H$ and $\tilde{E}_i^V$ are the horizontal and vertical polarization of the incident spectrum, respectively. ${\tilde{E}_{i + }}$ and ${\tilde{E}_{i - }}$ denotes the left- and right-handed circularly polarized components. By a series of matrix calculate, the polarized components $\tilde{E}_r^H$ and $\tilde{E}_r^V$ of the reflect light are obtained. We know through the transmission matrix method that in the layered periodic structure, the reflection coefficient has the following expression,

(4)$${r_{p,s}} = \frac{{({{q_0}{M_{22}} - {q_s}{M_{11}}} )- ({{q_0}{q_s}{M_{12}} - {M_{21}}} )}}{{({{q_0}{M_{22}} + {q_s}{M_{11}}} )- ({{q_0}{q_s}{M_{12}} + {M_{21}}} )}}, $$

where ${q_0}$ and ${q_s}$ are the coefficients in vacuum and substrate, respectively. For TE wave, ${q_0} = \sqrt {{{{\varepsilon _0}} / {{\mu _0}}}} \cos {\theta _i}$ and ${q_s} = \sqrt {{{{\varepsilon _s}} / {{\mu _s}}}} \sqrt {1 - {{{{\sin }^2}{\theta _i}} / {{\varepsilon _s}{\mu _s}}}} $; for TM wave, ${q_0} = {{\sqrt {{\mu _0}} } / {\sqrt {{\varepsilon _0}} }}\cos {\theta _i}$ and ${q_s} = \sqrt {{{{\mu _s}} / {{\varepsilon _s}}}} \sqrt {1 - {{{{\sin }^2}{\theta _i}} / {{\varepsilon _s}{\mu _s}}}} $. When both the incident medium and the outgoing medium are vacuum, ${q_0} = {q_s} = \cos {\theta _i}$.

Now, the distribution of the electric field in real space can be calculated. Through applying a Taylor series expansion based on the arbitrary spectrum components, we can obtain the coefficient r_s and r_p,

(5)$${r_{p,s}}({{k_{ix}}} )= {r_{p,s}}({k_{ix}} = 0) + {k_{ix}}{\left[ {\frac{{\partial {r_{p,s}}({{k_{ix}}} )}}{{\partial {k_{ix}}}}} \right]_{{k_{ix}} = 0}}, $$

where ${k_{ix}} = {k_0}\sin {\theta _i}$, we have neglected the high-order infinitesimal of the above formula and keep the first-order terms.

Thus, according to Eqs. (2)–(5) we can obtain the two circular components of the reflected light by the inverse Fourier transform. Finally, according to the definition of the transverse displacements. We can calculate the shift of reflected light [14],

(6)$$\sigma _H^ \pm{=} \mp \frac{{{k_0}\omega _0^2\textrm{Re} ({1 + {{{r_s}} / {{r_p}}}} )\cot {\theta _i}}}{{k_0^2\omega _0^2 + {{|{{{\partial {ln} {r_p}} / {\partial {\theta_i}}}} |}^2} + {{|{({1 + {{{r_s}} / {{r_p}}}} )\cos {\theta_i}} |}^2}}}, $$

(7)$$\sigma _V^ \pm{=} \mp \frac{{{k_0}\omega _0^2\textrm{Re} ({1 + {{{r_p}} / {{r_s}}}} )\cot {\theta _i}}}{{k_0^2\omega _0^2 + {{|{{{\partial {ln} {r_s}} / {\partial {\theta_i}}}} |}^2} + {{|{({1 + {{{r_p}} / {{r_s}}}} )\cos {\theta_i}} |}^2}}}. $$

We can see that the magnitude of transverse displacements can be decided by the ratio of two reflection coefficients.

3. Results and discussions

The proposed HMMs have been shown in Fig. 1. 1DPC is denoted by (AB)¹⁰, where A is SiO₂ and B is Ti₃O₅. The incident medium and substrate are air and BK7 glass (the refractive index is 1.52), respectively. The thicknesses of A and B are d_A and d_B, which can satisfy the 1/4 wave stack condition (i.e., n_Ad_A=n_Bd_B=λ₀/4, where the center wavelength of the first band gap λ₀=420 nm). The other parameters are chosen as follows: the refractive index of SiO₂ is 2.12 [32]; the dielectric constant of Ti₃O₅ is 5.83 [33].

As shown in Fig. 2(a), we calculated the reflection spectrum changing with the incident angle of both TM or TE waves. Whether TE or TM polarization, we can see that the gap edges are shifting toward shorter wavelengths with the increasing incident angle. This phenomenon with the blueshift bandgap always happened in the all-dielectric photonic crystal. Combining Eq. (6) and Eq. (7), the transverse displacement of the reflected light at three different incident wavelengths is shown in Fig. 2(b). The maximum displacement can be obtained at certain wavelength and incident angle. The transverse displacement can go up to a few microns. However, in order to obtain the larger and more obvious spin splitting, we have put forward a new model containing HMMs.

Fig. 2. (a) Schematic of the reflectance spectrum at (AB)^N-type 1DPCs, the green line separates the TM and TE waves. Whether TM or TE, the band gap moves to the short wavelength direction as the incident angle increases. (b) Spin splitting at three different incident wavelengths.

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The previous paper told us that all dielectric 1DPC cannot achieve a large polarization selection. As shown in the substructure of Fig. 1, for obtain the photonic hypercrystal (PHC), the layer A was replaced by a Ti₃O₅/Ag multilayer, the multilayer is denoted by (CD)^S. We choose the material of B is Ti₃O₅. The permittivity of silver in the multilayer is determined by Drude model [34], where the bulk permittivity at infinite frequency is 6, the plasma frequency and the collision frequency are 1.5×10¹⁶ rad/s and 7.73×10¹³ rad/s, respectively [35,36]. The whole structure is denoted by [(CD)^SB]^N, where S=4 and N=3. According to the effective medium theory [37], we know the effective permittivity tensors can be written as,

(8)$${\varepsilon _{xx}} = {\varepsilon _{yy}} = p{\varepsilon _m} + ({1 - p} ){\varepsilon _d}\,{\varepsilon _{zz}} = {\left( {\frac{p}{{{\varepsilon_m}}} + \frac{{1 - p}}{{{\varepsilon_d}}}} \right)^{ - 1}},$$

where $p = {{{t_m}} / {({t_m} + {t_d})}}$ is the volume filling ratio of the metal in a unit cell, t_m and t_d are the thicknesses of the metal layer and the dielectric layer, ε_m and ε_d are the permittivities of the metal and the dielectric, respectively. HMMs are an important class of anisotropic materials, which are created by subwavelength structuring and can be divided into two categories: Type Ι (ε_xx=ε_yy>0, ε_zz<0) and Type Π (ε_xx=ε_yy<0, ε_zz>0) [38–41]. It has been widely used in many fields [42–44]. We choose Type Ι HMM for constructing 1DPHC. Figure 3(a) represents the real parts of the dielectric tensors of HMM changing with the wavelength, where p=0.5. Real part of ε_Ax and ε_Az is shown by the red and blue solid lines. Therefore, according to the previous conclusions, we can get the wavelength range of HMM with type I, which is from 308 nm to 432 nm. Next, we determined the thickness of HMM and the dielectric.

Fig. 3. (a) Real parts of ε_Ax and ε_Az effective permittivity components for Ag/Ti3O5 multilayer with the filling ratio p=0.5; (b) The minimum thickness of Ag/Ti3O5 multilayer relations with wavelength.

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In 1DPCs, the total propagating phase of the photonic crystal cell is $\Phi = {k_{Az}}{d_A} + {k_{Bz}}{d_B}$, where ${k_{A,Bz}} = {{2\pi } / \lambda }\sqrt {{n^2} - {{\sin }^2}\theta } $. Owing to the Bragg reflection condition, the total phase at the band gap must be an integral multiple of π. In an all-dielectric 1DPCs, because of the iso-frequency curves of medium A and B, i.e., ${{\partial {k_{A,Bz}}} / {\partial {k_x} < 0}}$, so that the propagating phase will decrease with the increasing incident angle and the gap will shift toward short wavelengths for ensuring the Bragg condition. This is the blueshift bandgap what mentioned above.

On the contrary, in a 1DPCs containing HMMs (i.e., 1DPHCs), the iso-frequency curves of HMM is a hyperbola for TM polarization, i.e., ${{\partial {k_{Az}}} / {\partial {k_x}}} > 0$. Therefore, the total phase of TM wave will increase with the increasing incident angle if $|{{{\partial {k_{Az}}} / {\partial {k_x}}}} |> |{{{\partial {k_{Bz}}} / {\partial {k_x}}}} |$. Similarly, to maintain the Bragg condition, the band gap shifts to longer wavelength. This phenomenon is called redshift and occurs only at TM wave but still blueshift at TE wave. That is why the larger radio of r_s/r_p can be obtained when the incident light propagated through a 1DPCs. Under the condition of redshift bandgap, ${{\partial \Phi } / {\partial {k_x}}} = {{\partial {k_{Az}}} / {\partial {k_x}}}{d_A} + {{\partial {k_{Bz}}} / {\partial {k_x}}}{d_B} > 0$, we can obtain the thickness of the A and B layers by combining the Bragg conditions [45],

(9)$${d_A} > \frac{{{\lambda _{Bragg}}}}{2}\frac{1}{{\sqrt {{\varepsilon _{Ax}}} [{1 - {{{\varepsilon_B}} / {{\varepsilon_{Az}}}}} ]}}\,{d_B} = \frac{{{{{\lambda _{Bragg}}} / 2} - \sqrt {{sprt}({\varepsilon _{Ax}}){d_A}} }}{{\sqrt {{\varepsilon _B}} }},$$

where λ_Bragg is the Bragg wavelength. From Eq. (9), the thickness of dielectric B is related to λ_Bragg and thickness of A. But the thickness of A must be greater than a certain value. As shown in Fig. 3(b), we know that how the minimum thickness of A changes with the wavelength. The results are the same under both lossy and lossless conditions [46].

Now, we can calculate the thicknesses of A and B layers by the permittivity tensor x and y components of HMM. Here, we choose λ_Bragg = 360 nm to obtain the corresponding dielectric components in Fig. 3(b). Besides, the minimum value of the A layer with 73 nm can be obtained by Eq. (9). d_A must be greater than 73 nm so that the band gap can be redshift in 1DPCs. But if d_A is too large, the effective medium theory will not be applicable. In order to get the maximum r_s/r_p, after a series of calculations. We choose the optimal thickness of A layer is 96 nm and it is further calculated that the thickness of B layer is 21 nm. Since the period number of HMMs is four, the thickness of the metal layer and the dielectric layer are both 12 nm. Finally, we obtained the reflection spectrum with the varying incident angle for TM and TE waves (Fig. 4(a)).

Fig. 4. (a) Schematic of the reflectance spectrum at 1DPHCs; (b) The reflection coefficient r_s of TE changes with wavelength when incident angle is ${15^\circ}$, ${30^\circ}$, ${45^\circ}$ and ${60^\circ}$. (c) the reflection coefficient r_p of TM changes with wavelength when incident angle is ${15^\circ}$, ${30^\circ}$, ${45^\circ}$ and ${60^\circ}$. (d) Spin splitting at four different incident angles.

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From the results of Fig. 4(a), we can see that with increases of the incident angle, band gap of the TE wave is almost same as the theoretical estimation. The edge is shifted toward the short wavelength with the increased incident angle. This is easier to see from Fig. 4(b). The solid points in the gray area are edges of the band gap. However, Different from the traditional 1DPCs, band gap edge of the TM wave is redshift at 1DPHCs. The explanation of the redshift phenomenon is more obvious in Fig. 4(c). With the increase of the angle, the edge of the band gap moves to the long wavelength. Then according to Fig. 4(d). It’s obvious that the ratio of reflection coefficient r_s/r_p is large at the maximum value point of transverse displacement, which is same as the conventional 1DPCs. For TM polarization in Fig. 4(a), the lower gap edge shifts from 323.5 nm to 351 nm when the incident angle increases from 0 to 90°.

The redshift bandgap of 1DPCs is obtained, which can be used to enhance spin-split and get a greatly transverse displacement. In the figure above, we know that the lower edges of band gap with TM and TE polarizations are very different. This means that the reflection coefficients r_p and r_s of TM and TE are quite different. Actually, in the wavelength range of the lower edge of band gap with TM polarization, r_s is much larger than r_p. Hence, we can achieve a huge shift of horizontal polarization state in this band.

Finally, in Fig. 5(a), the reflection coefficient ratio r_s/r_p of the two polarization states is plotted as a function of angle and incident wavelength. We compared the ratios under the four angles and find the ratios of the coefficients are all reach 10 to the 3rd power, as shown in Fig. 5(b). But at 79.71° and 344.95 nm, we can clearly see that there is a peak in the figure. The peaks’ value is much larger than other points because the reflection coefficient r_s is much larger than r_p. Therefore, the transverse displacement value must also be large at the same coordinate position. By data processing, we get the maximum value of r_s/r_p is 31380. Then, we simulated the values of the transverse displacement of the horizontal polarization state at various angles and incident wavelengths (Fig. 6(a)). It is found that there is indeed a maximum displacement at the maximum point of r_s/r_p, the maximum displacement can reach to 14.95 µm. This value is larger than other points (Fig. 6(b)) and is 1.8 times higher than the displacement value of the previously proposed all-dielectric 1DPC model. It is at least three orders of magnitude higher than that of traditional devices.

Fig. 5. Reflection coefficient r_s and r_p: (a). Numerical distribution of reflection coefficient ratio of TE and TM waves at different incident angles and wavelengths; (b). The ratio of r_s/r_p under 79.69°, 79.705°, 79.720°, 79.735°.

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Fig. 6. (a) The change of transverse displacement with wavelength and incident angle. (b) Comparison of the maximum displacement and the displacement at three selected wavelengths.

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Using HMMs to replace dielectric materials in photonic crystals, we can not only achieve the red shift bandgap, we can also achieve the zero-shift bandgap when the appropriate thickness of HMMs is applicated. The so-called zero-shift means that the edge of the photonic band gap is almost horizontal, that is, it will not move to the longer or shorter wavelength with the increase of the incident angle. This band gap has no dispersion characteristic. It has important applications in all-angle reflectors, high-Q filters excited with finite-sized sources, and nonlinear wave mixing processes [46]. According to the three shifts mentioned in this paper, we can also propose a new method to modulate SHE.

4. Conclusion

In this article, we have proposed a new structure that contains type-I hyperbolic metamaterials to greatly enhanced spin-dependent split of the reflection light. When the thickness of layer A satisfies condition, the value of transverse displacement can reach up to 14.95 µm. This result is preferable to that reported in the past, and it is achieved by using the anisotropy of hyperbolic metamaterials to increase the ratio of r_s to r_p. We have achieved enhancement or suppression of split of the reflected light by replacing the material of layer A. Besides, our work provides a new train of thought for the preparation of nano photonic devices based on spin optics and the control of photons.

Funding

National Natural Science Foundation of China (11874269, 61875133); Science and Technology Planning Project of Shenzhen Municipality (JCYJ20180508152903208, JCYJ20190808143801672, JCYJ20190808150803580).

Disclosures

The authors declare no conflicts of interest.

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Enhanced spin Hall effect due to the redshift gaps of photonic hypercrystals

Abstract

1. Introduction

2. Theoretical models and methods

3. Results and discussions

4. Conclusion

Funding

Disclosures

References

Cited By

Figures (6)

Equations (9)

Optics Express