Spectral separation of optical spin based on antisymmetric Fano resonances

We propose a route to the spectral separation of optical spin angular momentum based on spin-dependent Fano resonances with antisymmetric spectral profiles. By developing a spin-form coupled mode theory for chiral materials, the origin of antisymmetric Fano spectra is clarified in terms of the opposite temporal phase shift for each spin, which is the result of counter-rotating spin eigenvectors. An analytical expression of a spin-density Fano parameter is derived to enable quantitative analysis of the Fano-induced spin separation in the spectral domain. As an application, we demonstrate optical spin switching utilizing the extreme spectral sensitivity of the spin-density reversal. Our result paves a path toward the conservative spectral separation of spins without any need of the magneto-optical effect or circular dichroism, achieving excellent purity in spin density superior to conventional approaches based on circular dichroism.


Supplementary Note 1. Real implementation using indefinite birefringent mirror
Supplementary Figure  For the practical implementation of Fano-resonant spin separation structures, we consider following requirements for highly-birefringent mirrors; (1) a 'flat' film-type mirror, for the serial deposition of a bottom mirror, a chiral medium, and a top mirror; (2) the indefinite (ε x < 0 and ε y > 0) non-resonant effective permittivity tensor of the mirror; (3) tunability of the effective material parameters for the control of the degree of spin separation.
Under these requirements, here we employ the non-resonant metallic grating embedded in a dielectric film ( Supplementary Fig. 1a).

Supplementary Figures 2 shows the fabrication process of the suggested indefinite
mirror. Firstly, the dielectric layer can be constructed for example, by using the spin coating method with soluble polymers 3 to achieve μm-scale mirror thickness (~λ 0 /100) for the THz operation. For the IR or visible regime applications, thin-film deposition methods for insoluble dielectrics 4 or soluble polymers 5 could be used to achieve tens of nm mirror thickness (~λ 0 /100) (e.g. based on the chemical vapor deposition (CVD) 4 or atomic layer deposition (ALD) 5 both guaranteeing sub-nm-scale precision). A gold layer can then be overlaid using electron-beam or thermal evaporation ( Supplementary Fig. 2b). After the lithography or imprinting process for the formation of grating pattern ( Supplementary Fig.   2c), the shielding top dielectric is overlaid (Supplementary Fig. 2d) using the same method as in Supplementary Fig. 2a, completing the flat birefringent film. We note that, in contrast to the spin coating method, thin-film deposition may result in the surface roughness (due to the gold grating pattern, ~ tens of nm in the IR regime), which however can be removed completely using the chemical-mechanical polishing (CMP) technique offering the sub-nm flatness 6 .

Supplementary Note 2. Material properties of indefinite birefringent mirrors in IR and THz regimes
In this Supplementary Note 2, we investigate the effective material parameters 1 of the mirror structure in Supplementary Fig. 1a and 2 Supplementary Fig. 4a). To compare, Re[ε y ] is relatively stable against the change of metal width (Re[ε y ] = 2.5 to 10 for 0.1 ≤ F t ≤ 0.7, Supplementary Fig. 4b) exhibiting the expected dielectric behavior. We also note that, because the loss part of permittivity does originate only from the induced current density in the metal region, Im[ε x ] is highly-stable (~ -10 dB, Supplementary Fig. 4c); meanwhile, Im[ε y ] varies significantly (-30 to -10 dB, Supplementary Fig. 4d) dependent on F t .
The effective permittivity can also be controlled with the variation of h poly or h Au ( Supplementary Fig. 5). Identical to the case of Supplementary Fig. 4, the absolute value of the real part of permittivity ( Supplementary Fig. 5a,5b,5e,5f) increases when the amount of metal increases. In contrast to the case of F t control, the increase of h Au results in the reduced loss. This effect originates from the decrease of the Ohmic resistance due to larger metal thickness.
Supplementary Figure 6 shows the result of frequency-dependent birefringent permittivity in the THz regime, which provides similar features with Supplementary Fig. 4, but with much larger magnitude of ε x and ε y , from the PEC-like property of metal in the THz regime. As can be seen, at F t = 0.1 (w = 400 nm and P = 4 μm), wide spectral range is guaranteed near 1.5 THz, also providing reasonable effective parameters (Re[ε x ] ~ −10 3 and Re[ε y ] ~ 15, with loss parameters of -10 dB and -15 dB, respectively).

Supplementary Note 3. Fano-resonant spin separation for arbitrary linear polarizations
Here we investigate the Fano-resonant spin separation for arbitrary linear polarizations, using scattering matrix calculations. Supplementary Figures 7a and 7b show the optical SAM density (σ) of the reflection beam as a function of the frequency and the state of linear polarizations. We can see that the chiral resonator system not only derives Fano spectral asymmetry for arbitrary polarization angle (Fig. 7b vs. 7a), but also achieve the non-zero SAM (σ unpol ) for the unpolarized incident beam (Fig. 7d vs. 7c). Such a difference originates from the chiral material, which realizes the Fano mixing of narrow-(x-axis) and broad-(yaxis) scattering paths from its spin-based eigenvectors with different effective indices.