Abstract
The photonic spin Hall effect (PSHE), featured by a spindependent shift driven by its polarization handedness, is proposed to facilitate the applications in precision metrology and quantum information processing. Here, due to the magnetoelectric coupling of the chirality, the PSHE is accompanied with Goos–Hänchen and Imbert–Fedorov effects. Taking advantage of this superiority, the transverse shift (TS) and longitudinal shift (LS) can be applied simultaneously. Rearranging the PTsymmetric scattering matrix, the responsive PSHE near the exceptional points and their basic physical mechanisms are discussed in detail in the case of complex chirality κ. Re[κ] and Im[κ] regulated the rich (at multiangle), gaint (reach upper limit) and tunable (magnitude and direction) TS and LS, respectively. Based on the chiralitymodulated PSHE, the novel applications in binary code conversion and barcode encryption are proposed systematically. By incorporating the quantum weak measurement technology, our applications provide new mechanisms to realize optoelectronic communication.
1 Introduction
The spin–orbit interaction of photons, corresponding to the interplay between the spin degree of freedom of light and the extrinsic orbital angular momentum, leads to a spindependent shift of light, namely the photonic version of the spin Hall effect [1, 2]. The photonic spin Hall effect (PSHE) will split the shifts of parallel and perpendicular to the incident plane of light simultaneously in some specific materials, such as anisotropy and twodimensional materials [3–8]. The transverse shift (TS) splitting, implied by universal angular momentum conservation, is different from the Imbert–Fedorov (IF) shift, while the longitudinal shift (LS) occurs in the case of crosspolarization, which is also different from the Goos–Hänchen (GH) shift [9, 10]. Because the GH shift is polarizationdependent, which is described in terms of evanescentwave penetration, while the LS is spindependent, it takes place as a result of an effective spin–orbit interaction [11, 12]. The interesting PSHE phenomena show promising applications in various realms, including imaging, precise metrogy and optical sensing [13–17]. However, the PSHE is generally weak and is always subwavelength magnitude, due to the weak spin–orbit interaction. Therefore, to enable and further extend these applications, manipulating a giant and controllable PSHE is desirable [18, 19]. In order to achieve that featured PSHE, many studies have been proposed by using the special properties of metamaterials, such as hyperbolic metamaterials [20, 21], paritytime (PT)symmetric metamaterials [22, 23], epsilonnearzero metamaterials, and chiral metamaterials, etc. [24–27].
Chiral metamaterials are composed of particles that cannot be superimposed with their mirror images using translation and rotation [28], which has different responses to a left circularly polarized (LCP) and a right circularly polarized (RCP) waves (eigenwaves of the wave equation in such media), due to the strong magnetoelectric coupling characteristic. The interesting and intriguing effects in chiral metamaterials are induced by circular dichroism (i.e., different absorption of LCP and RCP waves) and optical activity (i.e., the rotation of the polarization of an initially linearly polarized wave). We know that chirality appears in natural materials, but metamaterials will have stronger magnetoelectric coupling and huge effects. In 2004, Pendry discussed the possibility of negative refraction in chiral metamaterials [29]. Since then, chiral metamaterials have attracted much interest because of their features, such as negative refractive indices, asymmetric transmission, and broadband circular polarizers [30–32].
On the other hand, Bender and Boettcher’s pioneering work showed that a wide class of nonHermitian Hamiltonians had an essentially real valued energy spectrum below the exceptional points (EPs) if they commute with the
When the beam interacts with matter, PSHE is a useful and intuitive metrological tool to characterize the physical properties of the structure. Although PSHE in PTsymmetric metamaterials and chiral metamaterials has attracted extensive research, respectively [22–25]. However, under the concept of PTsymmetry, an important class of metamaterials that has rarely been explored is the socalled chiral metamaterials [39, 40], let alone its PSHE. Compared with passive chirality, can circularly polarized waves preserve their handedness under the action of
In this paper, the chiralitymodulated PSHE in PTsymmetric system is studied and its new applications in photoelectric signal devices are explored. Firstly, using the newly arranged scattering matrix and the analytical solutions of PSHE shifts derived by angular spectrum method, we reveal the basic physical relationship between the variation law of the PTphase and the PSHE distribution. Differently, both the magnitude and direction dimensions of the multiple PSHE can be modulated by Re[κ] and Im[κ], respectively. Interestingly, the similarities and differences between TS and LS are found: both LS and TS are symmetrically distributed with Re[κ] and are asymmetrically distributed with Im[κ]. Finally, its potential applications in binary code conversion and barcode encryption are presented completely.
2 Model and theory
Chiral media belong to a wide range of biisotropic media, which is characterized by the constitutive relations: D = ɛɛ_{0}E + i(κ/c)H and B = μμ_{0}H − i(κ/c)E, where ɛ, μ and κ refer to the relative permittivity, permeability and the chirality parameter (which quantifies the magnetoelectric coupling), respectively. ɛ_{0}, μ_{0} are the vacuum permittivity and permeability, respectively, and c is the speed of light in vacuum. In PTsymmetrical system with chiral response, analogy to the Schrödinger equation in quantum mechanics, the eigenproblem of Maxwell’s equations ∇ × E = iωB and ∇ × H = −iωD is written in the form of
Then, we need to know the specific expressions of reflection and transmission coefficients of PTsymmetric chiral system before deriving the PSHE shifts. According to the continuity of the tangential components of E and H at the interfaces, by solving the 12 × 12 system of linear equations, we obtain the reflection and transmission coefficients of the transverse magnetic (TM) and transverse electric (TE) polarized incident waves, and the detailed processes refer to Ref. [39]. Due to the circular dichroism of the chiral metamaterials, the reflection and transmission coefficients of the circular polarizations can be obtained [42]:
Because of two possible circularly polarized waves on both sides, the system can be described by four input and four output ports, so it can be given by a 4 × 4 description of scattering matrix, S, consisting of eight reflection and eight transmission amplitudes
In Eq. (3),
Figure 1:
Now a horizontally (H)polarized Gaussian beam is illuminated on the PTsymmetric chiral structure, as shown in Figure 1b. Its angular spectrum is
Here, ζ = i/(z_{
R
} + iz_{
r
}), U = (r_{
sp
} − r_{
ps
}) cot θ, V = (r_{
pp
} + r_{
ss
}) cot θ, Rayleigh length
After some mathematical calculations, we obtain:
where
Both δ_{x±} and δ_{y±} contain two terms. The first term of δ_{x±} is the conventional GH shift, while the first term of δ_{y±} is the IF shift. They are spinindependent shifts, moving the RCP and LCP components of the reflected beam together. The second terms of δ_{x±} and δ_{y±} are spindependent, originating from the spin–orbit interaction. The cooperation effect of the spinindependent and spindependent terms of the centroid shifts will result in more abundant splitting. RCP and LCP components will shift toward opposite directions in these terms. Here, the nondiagonal reflection coefficients r_{sp} and r_{ps} play an important role in the spin–orbit interaction of light [24]. In other words, due to the magnetoelectric coupling caused by chirality, the conversion of intrinsic spin angular momentum and extrinsic orbital angular momentum on the surface of PTsymmetric chiral metamaterial can be modulated by the chirality. These characteristics will provide theoretical support for the following discussion of rich PSHE phenomena and the related applications.
3 Results and discussion
In PTsymmetry system, a large amount of work concerns scattering configurations rather than paraxial beam propagation systems to discuss the PTphase change and the singularity of the EPs. We make the dimensionless frequency ωd/c = 6.5 and assume the simplest case chirality κ = 0 first. Note that in this case the S matrice is
Figure 2:
The solutions of the scattering matrix S, i.e., σ_{1,2} and σ_{3,4}, in Figure 3a and Figure 3b, are distributed with incident angle θ and both Re[κ] and Im[κ] (κ = ±Re[κ] + Im[κ]i). In the full PTsymmetric phase, the energy eigenvalues are real and the circular polarizations are preserved without attenuation and amplification (
Figure 3:
Figures 3c and d show the distribution of δ_{
y
} and δ_{
x
} of LCP light with incident angle θ and both Re[κ] and Im[κ]. δ_{
y
} is not completely symmetrical distributed in the range of 6°, 45–62°, and 72° of Re[κ] and Im[κ] from −0.1 to 0.1. Combined with the variation law of PTphase distribution, it can be known that the large δ_{
y
} are distributed near EP_{1}, EP_{3}, and EP_{4} curves. It is worth noting that the sign of shifts near EP_{3} is switchable with the chirality κ, which is another PSHE regulation method in dimension of direction besides its magnitude. Here, the reason for the large δ_{
y
} is that
Whether the δ_{ y } or δ_{ x }, the underlying physics is associated with the nearzero absolute value and abrupt phase jump of the r_{pp} near the EPs. Through the analysis, the giant and switchable PSHE can be obtained by independently adjusting the chirality κ to change the position of EPs, other than ɛ and μ. This finding represents direct relevance of PSHE related to spontaneous PTsymmetry breaking in PTsymmetric chiral system.
For detailed discussion, we further discuss the effect of only Im[κ] (Re[κ] = 0) on PSHE. As shown in Figure 4, with the change of weak chirality Im[κ] from −0.05 to 0.3, the distribution of TS and LS is still similar. In general, we know that there are two methods to enhance the spin splitting of light, according to:
Figure 4:
Similarly, the situation of δ_{ x } is like the above analysis, excepting that there is no large shift at the bottom of Figure 4b, because there are no U and V terms on the numerator of Eq. (6). According to the necessary conditions of PTsymmetric chiral system, layers B and C have the same chirality value in this case, which makes the PSHE varies asymmetrically with Im[κ] for both δ_{ y } and δ_{ x }. In general, by adjusting the Im[κ] in a PTsymmetric chiral system, two strategies can be achieved simultaneously to obtain a large shift.
When exploring the LS distribution in Figure 4b, in addition to the small angle, there are two large shifts near 62.3° and 71.2°, due to spontaneous PTsymmetry breaking, as shown in Figure 5. Obviously, the sign of LS corresponds inversely to the sign of the Im[κ] near 62.3°, while near 71.2°, it corresponds to the sign of the Im[κ]. Therefore, we can control the Im[κ] to obtain a positive or negative LS near 62.3° and 71.2°. This switchable PSHE phenomenon has practical applications. For example, in terms of materials, we control the positive or negative chirality of materials and use it as polarizing devices and optical switches, etc. Conversely, in optical field, the PSHE can be observed to analyze the properties and physical parameters of materials, including studying the exotic physics near the EPs in various photonic systems.
Figure 5:
Here, differently, we propose two novel functions of implementing twodigit binary code conversion and fourdigit barcode encryption by controlling the chirality parameter and observing its responsive PSHE (only LS is discussed here, and the TS discussion is placed in the Supplementary Material). First, we set our coding principle: the positive (negative) of Im[κ] or Re[κ] represents “1” (“0”), and the LCP (RCP) light represents “1” (“0”). These two constitute the binary parametric input code [chirality, polarization]. The positive (negative) of the second LS represents “1” (“0”), similarly, and the positive (negative) of the third LS represents “1” (“0”). These two phenomena form the observational output code [sign of the second LS, sign of the third LS]. Since the system is a four independent channel system, two binary code groups will combine into a “fourdigit barcode A”, based on the δ_{ x }: “chirality, polarization, sign of the second LS, sign of the third LS”. And “fourdigit barcode B” based on δ_{ y } is placed in Figure S1, using the same method.
We then illustrate the twodigit code conversion process: when analyze the LCP light and Im[κ] > 0, the input code is [1, 1], as shown in red line of Figure 5. It can be observed that the LS is negative at 62.3° and positive at 71.2°, so the output code is [0, 1]. By analyzing the LCP light with negative Im[κ] in blue line of Figure 5, the input code is [0, 1], then the output code is [1,0], according to the phenomenon of positive LS at 62.3° and negative value at 71.2°. Similarly, in the second line of Figure 5, the input code of green (black) line is [1, 0] ([0, 0]) and the output code is [1, 0] ([0, 1]). The encryption system of “fourdigit barcode A” has been marked in Figure 5. “fourdigit barcode B” based on the TS can be obtained “1 0 1 1” and “1 1 1 1” additionally. In this way, we obtained two sets of A and B barcodes.
The TS and LS are asymmetrically distributed with the Im[κ] as discussed ahead. On the contrary, in terms of Re[κ] change (Im[κ] = 0), the TS is axisymmetrically and the LS is centrosymmetrically distributed with Re[κ], as shown in Figure 6. With the increase of incident angle θ, the number of shift peaks does not change for both δ_{
y
} and δ_{
x
}. The reason for the symmetry of the PSHE spectrum is that the Re[κ] of layers B and C have the same value but opposite signs. Unlike the previous discussion, the maximum of TS and LS occurs at
Figure 6:
As can be seen from Figure 7, when the Re[κ] is positive (negative), the LS is positive (negative) at 67° and negative (positive) at 71.3°, which is opposite to the change of LS with Im[κ]. The process of Re[κ] modulated PSHE to realize code conversion and barcode encryption is discussed below. Obviously, when the Re[κ] is positive (negative), the input code of the LCP light is [1, 1] ([0, 1]), showing the output code [1, 0] ([0, 1]) that LS is positive (negative) at 67° and negative (positive) at 71.3°, such as the red (blue) line in Figure 7. In addition, it is easy to find that the input and output codes of green (black) line are [1, 0] ([0, 0]) and [0, 1] ([1, 0] [0, 1]), for RCP. Their combination “fourdigit barcode A” has been marked in Figure 7. In addition, “1 1 0 0” and “1 0 0 0” can be obtained based on δ_{ y }, in Figure S2.
Figure 7:
Binary coding is easy to implement in technology and simple in operation rules, which is widely used in the field of computer technology and communication. In terms of binary code conversion function, only two output codes [1, 0] [0, 1] are modulated by adjusting the Re[κ] and Im[κ] based on δ_{ x }. However, all four input and output codes [0, 0], [0, 1], [1, 0] [0,1], and [1, 1] are perfectly completed based on δ_{ y }. In the barcode encryption function, two sets of A and B barcodes form a very comprehensive barcode encryption system. Based on the above analysis, in the PTsymmetric chiral system, by adjusting the Re[κ] and Im[κ] and observing δ_{ x } and δ_{ y }, two functions of binary fourdigit barcode encryption and twodigit code conversion are realized simultaneously, which provides positive significance for photoelectric communication encryption and photoelectric integration system.
Finally, it is concluded that chirality modulates the rich, giant LS and TS. The observation of PSHE shifts by quantum weak measurement technology has been maturely realized in experiments. In order to demonstrate the feasibility of code conversion and barcode encryption in the experiment, we propose the possible experimental scheme of PSHE in PTsymmetric chiral system, as shown in Figure 8. The Gaussian beam generated by the laser enters the quantum weak measurement system in Figure 8e. The centroid shifts of the LCP reflected beam can be observed through the intensity distribution recorded by CCD, and the corresponding encrypted barcode can be read, as shown in Figure 8a–d. Combine Figures 8a and b, “fourdigit barcode A”: “1 1 1 0”, “fourdigit barcode B”: “1 1 0 0”. Figures 8c and d, “fourdigit barcode A”: “1 1 0 1” and “fourdigit barcode B”: “1 1 1 1”.
Figure 8:
4 Conclusions
In this paper, the novel chiralitymodulated PSHE and its applications in code conversion and barcode encryption have been studied in the PTsymmetric chiral system. After deriving the analytical PSHE shifts and discussing the PTphase change distribution law, we revealed that the huge PSHE shifts occur near the EPs. In detail, the large and tunable TS and LS can be obtained not only by incident angle, but also by Re[κ] and Im[κ]. The similarities and the differences between TS and LS are revealed. LS and TS are symmetrically distributed with Re[κ] and are asymmetrically distributed with Im[κ]. Based on these PSHE phenomena, the intact code conversion and barcode encryption functions of the system are obtained. Finally, by incorporating the quantum weak measurement technology, the applications in photoelectric signal processing are put forward, which opens up the prospect for novel and adjustable photoelectric devices.
Funding source: Key Program for Guangdong NSF of China
Award Identifier / Grant number: 2017B030311003
Funding source: Natural Science Foundation of Guangdong Province
Award Identifier / Grant number: 2018A030313480
Award Identifier / Grant number: 2022A1515012377
Funding source: Science and Technology Program of Guangzhou City
Award Identifier / Grant number: 201707010403
Acknowledgment
This work was supported by the Natural Science Foundation of Guangdong Province, China (2022A1515012377, 2018A030313480), Key Program for Guangdong NSF of China (2017B030311003), and Science and Technology Program of Guangzhou City, China (201707010403).

Author contributions: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

Research funding: None declared.

Conflict of interest statement: The authors declare no conflicts of interest regarding this article.
References
[1] K. Bliokh and A. Aiello, “Goos–Hänchen and Imbert–Fedorov beam shifts: An overview,” J. Opt., vol. 15, p. 014001, 2013. https://doi.org/10.1088/20408978/15/1/014001.Search in Google Scholar
[2] X. Ling, X. Zhou, K. Huang et al.., “Recent advances in the spin Hall effect of light,” Rep. Prog. Phys., vol. 80, p. 066401, 2017. https://doi.org/10.1088/13616633/aa5397.Search in Google Scholar PubMed
[3] W. Zhang, W. Wu, S. Chen et al.., “Photonic spin Hall effect on the surface of anisotropic twodimensional atomic crystals,” Photon. Res., vol. 6, pp. 511–516, 2018. https://doi.org/10.1364/prj.6.000511.Search in Google Scholar
[4] W. Zhang, Y. Wang, S. Chen, S. Wen, and H. Luo, “Photonic spin Hall effect in twisted fewlayer anisotropic twodimensional atomic crystals,” Phys. Rev. A, vol. 105, p. 043507, 2022. https://doi.org/10.1103/physreva.105.043507.Search in Google Scholar
[5] M. Liu, L. Cai, S. Chen, Y. Liu, H. Luo, and S. Wen, “Strong spinorbit interaction of light on the surface of atomically thin crystals,” Phys. Rev. A, vol. 95, p. 063827, 2017. https://doi.org/10.1103/physreva.95.063827.Search in Google Scholar
[6] L. Cai, M. Liu, S. Chen et al.., “Quantized photonic spin Hall effect in graphene,” Phys. Rev. A, vol. 95, p. 013809, 2017. https://doi.org/10.1103/physreva.95.013809.Search in Google Scholar
[7] T. Tang, J. Li, L. Luo, P. Sun, and J. Yao, “Magnetooptical modulation of photonic spin Hall effect of graphene in terahertz region,” Adv. Opt. Mater., vol. 6, p. 1701212, 2018. https://doi.org/10.1002/adom.201701212.Search in Google Scholar
[8] H. Lin, B. Chen, S. Yang et al.., “Photonic spin Hall effect of monolayer black phosphorus in the Terahertz region,” Nanophotonics, vol. 7, pp. 1929–1937, 2018. https://doi.org/10.1515/nanoph20180101.Search in Google Scholar
[9] X. Zhou, S. Liu, Y. Ding, L. Min, and Z. Luo, “Precise control of positive and negative Goos–Hänchen shifts in graphene,” Carbon, vol. 149, pp. 604–608, 2019. https://doi.org/10.1016/j.carbon.2019.04.064.Search in Google Scholar
[10] W. Zhen and D. Deng, “Goos–Hänchen shifts for Airy beams impinging on graphenesubstrate surfaces,” Opt. Express, vol. 28, pp. 24104–24114, 2020. https://doi.org/10.1364/oe.400939.Search in Google Scholar PubMed
[11] Y. Qin, Y. Li, X. Feng, Y. Xiao, H. Yang, and Q. Gong, “Observation of the inplane spin separation of light,” Opt. Express, vol. 19, pp. 9636–9645, 2011. https://doi.org/10.1364/oe.19.009636.Search in Google Scholar
[12] K. Bliokh, F. RodríguezFortuño, F. Nori, and A. Zayats, “Spin–orbit interactions of light,” Nat. Photonics, vol. 9, pp. 796–806, 2015. https://doi.org/10.1038/nphoton.2015.201.Search in Google Scholar
[13] T. Zhu, Y. Lou, Y. Zhou et al.., “Generalized spatial differentiation from the spin Hall effect of light and its application in image processing of edge detection,” Phys. Rev. Applied, vol. 11, p. 034043, 2019. https://doi.org/10.1103/physrevapplied.11.034043.Search in Google Scholar
[14] X. Zhou, Z. Xiao, H. Luo, and S. Wen, “Experimental observation of the spin Hall effect of light on a nanometal film via weak measurements,” Phys. Rev. A, vol. 85, p. 043809, 2012. https://doi.org/10.1103/physreva.85.043809.Search in Google Scholar
[15] X. Zhou, X. Ling, H. Luo, and S. Wen, “Identifying graphene layers via spin Hall effect of light,” Appl. Phys. Lett., vol. 101, p. 251602, 2012. https://doi.org/10.1063/1.4772502.Search in Google Scholar
[16] W. Zhu, H. Xu, J. Pan et al.., “Black phosphorus terahertz sensing based on photonic spin Hall effect,” Opt. Express, vol. 28, pp. 25869–25878, 2020. https://doi.org/10.1364/oe.399071.Search in Google Scholar PubMed
[17] C. Liang, G. Wang, D. Deng, and T. Zhang, “Controllable refractive index sensing and multifunctional detecting based on the spin Hall effect of light,” Opt. Express, vol. 29, pp. 29481–29491, 2021. https://doi.org/10.1364/oe.435775.Search in Google Scholar PubMed
[18] M. Kim, D. Lee, B. Ko, and J. Rho, “Diffractioninduced enhancement of optical spi Hall effect in a dielectric grating,” APL Photon., vol. 5, p. 066106, 2020. https://doi.org/10.1063/5.0009616.Search in Google Scholar
[19] M. Kim, D. Lee, Y. Kim, and J. Rho, “Generalized analytic formula for spin Hall effect of light: Shift enhancement and interface independence,” Nanophotonics, vol. 11, pp. 2803–2809, 2022. https://doi.org/10.1515/nanoph20210794.Search in Google Scholar
[20] P. Kapitanova, P. Ginzburg, F. RodríguezFortuño et al.., “Photonic spin Hall effect in hyperbolic metamaterials for polarizationcontrolled routing of subwavelength modes,” Nat. Commun., vol. 5, p. 3226, 2014. https://doi.org/10.1038/ncomms4226.Search in Google Scholar PubMed
[21] M. Kim, D. Lee, T. Kim, Y. Yang, H. Park, and J. Rho, “Observation of enhanced optical spin Hall effect in a vertical hyperbolic metamaterial,” ACS Photonics, vol. 6, pp. 2530–2536, 2019. https://doi.org/10.1021/acsphotonics.9b00904.Search in Google Scholar
[22] Y. Fu, Y. Fei, D. Dong, and Y. Liu, “Photonic spin Hall effect in PT symmetric metamaterials,” Front. Physiol., vol. 14, p. 62601, 2019. https://doi.org/10.1007/s1146701909388.Search in Google Scholar
[23] X. Zhou, X. Lin, Z. Xiao et al.., “Controlling photonic spin Hall effect via exceptional points,” Phys. Rev. B, vol. 100, p. 115429, 2019. https://doi.org/10.1103/physrevb.100.115429.Search in Google Scholar
[24] H. Wang and X. Zhang, “Unusual spin Hall effect of a light beam in chiral metamaterials,” Phys. Rev. A, vol. 83, p. 053820, 2011. https://doi.org/10.1103/physreva.83.053820.Search in Google Scholar
[25] G. Xu, T. Zang, H. Mao, and T. Pan, “Transverse shifts of a reflected light beam from the airchiral interface,” Phys. Rev. A, vol. 83, p. 053828, 2011. https://doi.org/10.1103/physreva.83.053828.Search in Google Scholar
[26] H. Chen, D. Guan, W. Zhu et al.., “Highperformance photonic spin Hall effect in anisotropic epsilonnearzero metamaterials,” Opt. Lett., vol. 46, pp. 4092–4095, 2021. https://doi.org/10.1364/ol.433332.Search in Google Scholar PubMed
[27] H. Luo, S. Wen, W. Shu, Z. Tang, Y. Zou, and D. Fan, “Spin Hall effect of a light beam in lefthanded materials,” Phys. Rev. A, vol. 80, p. 043810, 2009. https://doi.org/10.1103/physreva.80.043810.Search in Google Scholar
[28] I. Lindell, A. Sihvola, S. Tretyakov, and A. Viitanen, Electromagnetic Waves in Chiral and Biisotropic Media, MA, Boston, USA, Artech House Publishers, 1994.Search in Google Scholar
[29] J. B. Pendry, “A chiral route to negative refraction,” Science, vol. 306, pp. 1353–1355, 2004. https://doi.org/10.1126/science.1104467.Search in Google Scholar PubMed
[30] E. Plum, J. Zhou, J. Dong et al.., “Metamaterial with negative index due to chirality,” Phys. Rev. B, vol. 79, p. 035407, 2009. https://doi.org/10.1103/physrevb.79.035407.Search in Google Scholar
[31] C. Menzel, C. Helgert, C. Rockstuhl et al.., “Asymmetric transmission of linearly polarized light at optical metamaterials,” Phys. Rev. Lett., vol. 104, p. 253902, 2010. https://doi.org/10.1103/physrevlett.104.253902.Search in Google Scholar
[32] J. Gansel, M. Thiel, M. Rill et al.., “Gold helix photonic metamaterial as broadband circular polarizer,” Science, vol. 325, pp. 1513–1515, 2009. https://doi.org/10.1126/science.1177031.Search in Google Scholar PubMed
[33] C. Bender and S. Boettcher, “Real spectra in nonhermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett., vol. 80, pp. 5243–5246, 1998. https://doi.org/10.1103/physrevlett.80.5243.Search in Google Scholar
[34] M. Miri and A. Alù, “Exceptional points in optics and photonics,” Science, vol. 363, p. eaar7709, 2019. https://doi.org/10.1126/science.aar7709.Search in Google Scholar PubMed
[35] Y. Chong, L. Ge, and A. Stone, “PTsymmetry breaking and laserabsorber modes in optical scattering systems,” Phys. Rev. Lett., vol. 106, p. 093902, 2011. https://doi.org/10.1103/physrevlett.106.093902.Search in Google Scholar
[36] L. Ge, Y. Chong, and A. Stone, “Conservation relations and anisotropic transmission resonances in onedimensional PTsymmetric photonic heterostructures,” Phys. Rev. A, vol. 85, p. 023802, 2012. https://doi.org/10.1103/physreva.85.023802.Search in Google Scholar
[37] Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. Christodoulides, “Unidirectional invisibility induced by PTsymmetric periodic structures,” Phys. Rev. Lett., vol. 106, p. 213901, 2011. https://doi.org/10.1103/physrevlett.106.213901.Search in Google Scholar PubMed
[38] Y. Sun, W. Tan, H. Li, J. Li, and H. Chen, “Experimental demonstration of a coherent perfect absorber with PT phase transition,” Phys. Rev. Lett., vol. 112, p. 143903, 2014. https://doi.org/10.1103/physrevlett.112.143903.Search in Google Scholar
[39] I. Katsantonis, S. Droulias, C. Soukoulis, E. Economou, and M. Kafesaki, “PTsymmetric chiral metamaterials: asymmetric effects and PTphase control,” Phys. Rev. B, vol. 101, p. 214109, 2020. https://doi.org/10.1103/physrevb.101.214109.Search in Google Scholar
[40] S. Droulias, I. Katsantonis, M. Kafesaki, C. Soukoulis, and E. Economou, “Chiral metamaterials with PT symmetry and beyond,” Phys. Rev. Lett., vol. 122, p. 213201, 2019. https://doi.org/10.1103/physrevlett.122.213201.Search in Google Scholar PubMed
[41] G. Castaldi, S. Savoia, V. Galdi, A. Alù, and N. Engheta, “PT metamaterials via complexcoordinate transformation optics,” Phys. Rev. Lett., vol. 110, p. 173901, 2013. https://doi.org/10.1103/physrevlett.110.173901.Search in Google Scholar
[42] J. Lekner, “Optical properties of isotropic chiral media,” Pure Appl. Opt., vol. 5, pp. 417–443, 1996. https://doi.org/10.1088/09639659/5/4/008.Search in Google Scholar
[43] M. Pan, Y. Li, J. Ren et al.., “Impact of inplane spread of wave vectors on spin Hall effect of light around Brewster’s angle,” Appl. Phys. Lett., vol. 103, p. 071106, 2013. https://doi.org/10.1063/1.4818816.Search in Google Scholar
[44] J. Ren, B. Wang, Y. Xiao, Q. Gong, and Y. Li, “Direct observation of a resolvable spin separation in the spin Hall effect of light at an airglass interface,” Appl. Phys. Lett., vol. 107, p. 111105, 2015. https://doi.org/10.1063/1.4931093.Search in Google Scholar
[45] J. Ren, B. Wang, M. Pan, Y. Xiao, Q. Gong, and Y. Li, “Spin separations in the spin Hall effect of light,” Phys. Rev. A, vol. 92, p. 013839, 2015. https://doi.org/10.1103/physreva.92.013839.Search in Google Scholar
[46] L. Feng, Y. Xu, W. Fegadolli et al.., “Experimental demonstration of a unidirectional reflectionless paritytime metamaterial at optical frequencies,” Nat. Mater., vol. 12, pp. 108–113, 2013. https://doi.org/10.1038/nmat3495.Search in Google Scholar PubMed
[47] M. Kim, D. Lee, and J. Rho, “Spin Hall effect under arbitrarily polarized or unpolarized light,” Laser Photon. Rev., vol. 15, p. 2100138, 2021. https://doi.org/10.1002/lpor.202170037.Search in Google Scholar
[48] M. Kim, D. Lee, T. Nguyen, H. Lee, G. Byun, and J. Rho, “Total reflectioninduced efficiency enhancement of the spin Hall effect of light,” ACS Photonics, vol. 8, pp. 2705–2712, 2021. https://doi.org/10.1021/acsphotonics.1c00727.Search in Google Scholar
[49] M. Kim, D. Lee, H. Cho, B. Min, and J. Rho, “Spin Hall effect of light with nearunity efficiency in the microwave,” Laser Photon. Rev., vol. 15, p. 2000393, 2021. https://doi.org/10.1002/lpor.202000393.Search in Google Scholar
[50] X. Ling, W. Xiao, S. Chen, X. Zhou, H. Luo, and L. Zhou, “Revisiting the anomalous spinHall effect of light near the Brewster angle,” Phys. Rev. A, vol. 103, p. 033515, 2021. https://doi.org/10.1103/physreva.103.033515.Search in Google Scholar
[51] M. Kim, D. Lee, and J. Rho, “Incidentpolarizationindependent spin Hall effect of light reaching half beam waist,” Laser Photon. Rev., vol. 16, p. 2100510, 2022. https://doi.org/10.1002/lpor.202100510.Search in Google Scholar
[52] X. Ling, F. Guan, X. Cai et al.., “Topologyinduced phase transitions in spinorbit photonics,” Laser Photon. Rev., vol. 15, p. 2000492, 2021. https://doi.org/10.1002/lpor.202000492.Search in Google Scholar
[53] M. Kim, D. Lee, Y. Yang, Y. Kim, and J. Rho, “Reaching the highest efficiency of spin Hall effect of light in the nearinfrared using alldielectric metasurfaces,” Nat. Commun., vol. 13, p. 2036, 2022. https://doi.org/10.1038/s4146702229771x.Search in Google Scholar PubMed PubMed Central
[54] X. Ling, F. Guan, Z. Zhang, H. Xu, S. Xiao, and H. Luo, “Vortex mode decomposition of the topologyinduced phase transitions in spinorbit optics,” Phys. Rev. A, vol. 104, p. 053504, 2021. https://doi.org/10.1103/physreva.104.053504.Search in Google Scholar
[55] X. Ling, H. Luo, F. Guan, X. Zhou, H. Luo, and L. Zhou, “Vortex generation in the spinorbit interaction of a light beam propagating inside a uniaxial medium: origin and efficiency,” Opt. Express, vol. 28, pp. 27258–27267, 2020. https://doi.org/10.1364/oe.403650.Search in Google Scholar
Supplementary Material
The online version of this article offers supplementary material (https://doi.org/10.1515/nanoph20220229).
© 2022 the author(s), published by De Gruyter, Berlin/Boston
This work is licensed under the Creative Commons Attribution 4.0 International License.