Excitation of optical tamm state for photonic spin hall enhancement

This work presents a dielectric material-based optical Tamm state (OTS) excitation technique with modified dispersion characteristics for photonic spin hall effect (PSHE) enhancement. The dispersion analysis of the structure is carried out to validate OTS’s localization and corresponding PSHE generation for a given polarization at 632.8 nm incident wavelength. The exceptional points are optimized by considering thickness-dependent angular dispersion analysis. PSHE-based transverse displacement (PSHE-TD) is dependent on the defect layer thickness. The optimized structure provides 10.73 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times \lambda$$\end{document}×λ (or 6.78 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upmu$$\end{document}μm) PSHE-TD at an incidence angle of 41.86\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}^{\circ }$$\end{document}∘. The PSHE-TD of the optimized structure is sufficiently high due to the much narrower resonance than the plasmonic-based structures. Further, the structure’s potential to function as a PSHE-TD-based optical sensor is assessed. The optimized structure shows an analytical average sensitivity of about 43,789 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upmu$$\end{document}μm/RIU showing its capability to detect the analytes with refractive index variations in the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$10^{-4}$$\end{document}10-4 range. The structure demonstrates a three-time sensitivity improvement compared to similar resonance designs. Considering only dielectric materials in the proposed structure and considerably enhanced PSHE-TD, the development of highly efficient PSHE-TD-assisted commercial structures is anticipated.


Device structure and analytical model
The schematics of the proposed 1D-PhC device and corresponding induced PSHE-TD at the top dielectric-air interface of the proposed structure is represented in Fig. 1.The devices consists of [Substrate (Glass) | (A SiO 2 , B Si 3 N 4 ) N | defect (D) | Air] configuration.The structure is a periodically alternating layers of materials ' A' ,'B' , with low RI ( n L ) and higher RI ( n H ), respectively.The structural symmetry is deliberately broken by considering a defective low-index material top layer (D).The proposed device is uniform in x-direction and periodic in the normal z-direction, with periodicity ( ) of ( A SiO 2 , B Si 3 N 4 ) repeated 'N' times.
Hence, the refractive index profile in the normal z-direction n(z + �) = n(z) can be calculated by Eq. (1) as: Here, A t and B t are the ' A' and 'B' layer's physical thicknesses.The Helmholtz equation is utilized further to calculate the electric field confinement within the structure 35 .To establish the dispersion relation for the OTS and to solve the eigenvalue problem, the Floquet theorem is employed 36 : Here, M n represents the eigenvalue matrix elements, whereas the Bloch wave vector is represented by 'K' .The real values of Bloch wave vector attributes to the propagating OTS, and imaginary values give information on evanescent OTS.Furthermore, the transfer matrix is formulated to calculate the reflected and transmitted wave amplitude.
When a monochromatic Gaussian beam of wavelength having beam waist of w 0 is incident at the proposed device, the angular spectrum is represented by Eq. ( 3).The PSHE effect divides this incident Gaussian beam into two corresponding circularly polarized components.
(1)  www.nature.com/scientificreports/here, k iy and k ix are the wave-vector components in the y i and x i direction, and +/− represents the correspond- ing left and right circular polarization components.The spin-basis set of incident Gaussian beam is represented in Eq. ( 4), where V and H represents the vertical and horizontal polarization states.Further, the mathematical relations in incident and reflected angular spectra is given by Eq. ( 5) 16 , where Here, ' M R ' is the rotational matrix.Utilizing Eqs. ( 4)-( 5), the reflected angular spectrum is calculated and the Fresnel reflection coefficient is calculated by utilizing Taylor series expansion in transfer matrix method (TMM) 37 , The Eq. ( 6) is used to determine the PSHE-TD shift with regard to geometrical optic prediction 19 , Further, considering first-order Taylor-series expansion approximation to expand the Fresnel coefficients of Eq. ( 6) and using Eqs.( 4)- (7), δ V ± is obtained and is represented in (8) 20 : The PSHE based transverse displacement δ + possesses the similar magnitude of δ − .Therefore, only δ V − is considered here for further calculation and the same results can be obtained for δ V + .

Results and discussion
The 2, k A = 0, and k B = -0.0002.Initially, the impact of top layer thickness on OTS excitation for TE polarization and corresponding PSHE generation is investigated.Thus, the defect layer thickness-dependent, angular dispersion analysis of the 1D-PhC is represented in Fig. 2. The device demonstrates a cutoff defect layer thickness of about 150 nm beneath that, no OTS is excited.This is because the dispersion curve for OTS with 150 nm defect layer thickness approaches the air light line for operating wavelength (632.8 nm ).However, a strong OTS is excited beyond this point, as shown in Fig. 2. The OTS excitation characteristics at the top dielectric-air interface is investigated at three defect layer thicknesses, shown as 'X' , 'Y' , and 'Z' in Fig. 2. With an incident wavelength of 632.8 nm , point 'X' has D t = 155 nm , and θ i ≈ 41.86 • , point 'Y' is D t = 169.8nm , and θ i ≈ 42.55 • and point 'Z' is D t = 184.5 nm , and θ i ≈ 43.68 • , respectively.The structural reflectance response at considered three points is shown in Fig. 3. Thus, for D t ∈ (150 nm , 200 nm ), θ i ≥ θ b (Critical angle ≈ 41 • ) OTS propagation is sustained.From Eqs. ( 7)-( 8 |r s | for the selected points.Furthermore, the OTS confinement and electric field distribution analysis for these 'X' , 'Y' , and 'Z' points is presented in Fig. 3b.This is calculated using the finite element method (FEM) of COMSOL Multiphysics.The device exhibits a strong OTS localization for higher defect layer thickness (having normalized electric field intensity of around 5.5× 10 5 V/m).Decreasing defect layer thickness exhibits higher evanescent mode coupling in the air region (having normalized electric field intensity of around 1.52 × 10 5 V/m).Thus, by choosing proper incident wavelength and top layer thickness, the proposed device can also be used for both sensing (low thickness) and wave-guiding (high thickness) applications.The structure can generate enhanced PSHE-TD at all the defect layer thicknesses beyond the cut-off value (at different θ i ).
It is noteworthy to mention that the proposed structure is optimized to excite the OTS at a 632.8 nm incident wavelength.However, the structure can also be optimized to excite the OTS at other wavelengths.Figure 4 represents the wavelength dispersion analysis of the 1D-PhC for a given defect layer thickness.This demonstrates the structure's capability to excite an OTS at any user-defined wavelength by just changing the defect layer thickness.After OTS excitation, the structure capability to enhance PSHE-TD is evaluated.Thus, with = 632.8nm and D t = 155 nm , θ i ≈ 41.86 • , PSHE-TD analysis is further analyzed.The structure is showing a extremely small (4) full-width-half-maximum (FWHM) at |r TE | |r TM | ≈ 0.5 of around 0.003 • .This shows a very narrow region for produc- ing higher |r TM | |r TE | based on θ i .The higher reflection sensitivity towards narrower angle dependency exhibits its potential applications in both sensing and precision metrology.Figure 3a, shows that for small ∂ θ i at 41.86 • , the term ∂lnr s ∂θ i 2 ≈ 0. Thus, the zeroth-order Taylor series expansion of Eq. ( 6) can be used to obtain δ V ± .This leads to a simplified expression for PSHE calculation with sufficient accuracy 11,38,39 :  Further, for an incidence angle of 41.86 • and = 632.8nm , the structure exhibits a very high reflectance ( | r TM | ) for p-polarized and almost negligible reflection ( | r TE | ) for s-polarized light, which is shown in Fig. 3. Thus, a significantly higher value of |r TM | |r TE | of around 4340 is obtained around the resonance angle ( θ r ).This is represented in Fig. 5a.Since δ V ± also depends on both Fresnel coefficient phases ( φ s ,φ p ) and cos(φ s -φ p ), therefore θ i dependent ( φ s ,φ p ) and cos(φ s -φ p ) values are calculated and are illustrated in Fig. 5b.An sharp change in the cos(φ s -φ p ) is obtained at resonance angle, which is generally the observed case for enhanced PSHE-TD generation approach.
Finally, PSHE-TD is calculated for the proposed optimized structure.The calculated PSHE-TD for V polarized light normalized to wavelength is shown in Fig. 6a.The optimized parameters exhibit a maximum PSHE-TD of 10.73× at θ r = 41.86 • .This leads to a total PSHE-TD of around 6.78 µm for the proposed structure.The obtained �δ V − also demonstrates a narrower FWHM of around 0.003 • .Further, PSHE-based wavelength interrogation is utilized to demonstrate the PSHE-assisted sensing capability of the proposed structure.For sensing, an analyte with different dielectric constants is infiltrated at the top interface, which changes the top interface's effective RI.Thus, change in both PSHE-TD and operating wavelength at a given incidence angle is observed, which is used to calculate the devices' sensitivity.
The PSHE-TD based sensitivity ( S TD ) values for constant wavelength ( ) and incidence angle is analytically obtained using the PSHE-TD ( �δ V − ) shift and is given by the relation: Here n d represents the difference in the infiltrated analyte's RI.The proposed 1D-PhC is highly sensitive, and can detect a very small perturbation at the top interface.Therefore, the structure's sensitivity is measured by considering a 0.0001 variation in the RI (1.000-1.0001)at the top interface.This exhibits PSHE-TD shift ( �δ V − ) of around 6.92 (6.565 at 1.000 and − 0.356 at 1.0001) for respective RI variation ( n d ) of 10 −4 .This provides an average S V TD of about 43,789 µm/RIU, illustrated in Fig. 6b.The obtained sensitivity is much higher than the previously reported 1D-PhCR and SPR-based PSHE sensors [39][40][41][42] .Finally, the PSHE-based sensitivity of the 1D-PhC optimized structure is compared with the recently reported similar SPR and LMR-based structure and is reported in Table 1.In comparison to previously reported PSHE-TD sensors, the optimized 1D-PhC exhibits significantly improved PSHE-TD value, resulting in substantially better sensitivity.Additionally, the structure demonstrates its capability to detect a minute 10 −4 fluctuation in the higher RI range (1.0-1.5).Furthermore, the design can be fabricated by considering deposition or dip/spin coating techniques 43 .The proposed structure has several advantages due to its simple dielectric materials-based structure, easier fabrication and characterization with low optical losses compared with various 2D and 3D devices based on metasurfaces,and meta lenses.

Conclusion
The work presents an optical Tamm state (OTS)-assisted PSHE generation and corresponding PSHE-TD enhancement.The work focused on optimizing top layer's optical thickness parameters to generate and localize OTS and corresponding PSHE generation.The impact of working wavelength, incidence angle and optical thickness of top interface layer is thoroughly investigated to enhance the PSHE-TD.The analytical results demonstrate the confinement of the OTS for 632.8 nm wavelength providing a maximum PSHE-TD of around 6.78 µm (10.73 × ).Finally, the structure's capability to detect a minute change in the surrounding analyte refractive index is demonstrated.The analytical results exhibit a 43,789 µm/RIU average sensitivity with a 0.0001 change in the analyte refractive index unit.This demonstrate its potential to detect the analyte having refractive index variations in the 10 −4 range.This envisages the development of high-resolution, greater PSHE-TD and low-cost dielectric material-based PSHE devices for commercial applications.

Figure 1 .
Figure 1.The schematics of induced PSHE-TD because of OTS localization at the top layer of the 1D-PhC structure [Substrate (Glass) | (A SiO 2 , B Si 3 N 4 ) N | D | Air].
), it can be shown that the PSHE-TD shift is a function of |r TE | |r TM | (or |r TM | |r TE | ) depending on the considered TE (or TM) polarization.The reflectance response at the three considered points 'X' , 'Y' , and 'Z' is illustrated in Fig. 3a.The 1D-PhC structure shows a significantly higher reflection for TM polarization and very low value for TE-polarization.This gives a very high |r TM | |r TE | or |r p |

Figure 4 .
Figure 4.The defect layer thickness-dependent angular dispersion for TE polarisation at = 632.8nm.

Figure 5 .
Figure 5. (a) Incident angle-dependent Fresnel reflectance coefficients ratio variations, and (b) cosine of phase difference with variation in the incident angle for 632.8 nm wavelength.

Figure 6 .
Figure 6.(a) The PSHE-TD at 41.86 • incident angle for the optimized structure at 632.8 nm wavelength, and (b) the PSHE-TD-based sensitivity performance analysis.

Table 1 .
PSHE-TD and sensing performance comparison of proposed structure with recently reported literature.