Broadband Spin Hall Effect of Light in Single Nanoapertures
Supplementary Information for
Broadband spin Hall effect of light in single nanoapertures
Xiangang Luo*, Mingbo Pu, Xiong Li, Xiaoliang Ma
State Key Laboratory of Optical Technologies on Nano-Fabrication and Micro-Engineering, Institute of Optics and Electronics, Chinese Academy of Sciences, Chengdu 610209, China.
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Supplementary Methods
Supplementary Text
Figures S1-S2
Supplementary Methods
Sample fabrication
All the samples were fabricated on 1-mm-thick quartz substrates. A 3-nm-thick Cr film and 120-nm-thick Au film were subsequently deposited on the cleaned substrates by magnetron sputtering in a same sputter chamber. The apertures were then milled on the Au/Cr film using a Ga+ focused ion beam (FIB, FEI Helios Nanolab 650).
Sample measurement
All measurements were performed on a home-built microscope in a transmission mode (see supplementary Fig. S2e). Firstly, the incident collimated beam was converted into CPL via a cascaded polarizer and a quarter-wave plate. The transmitted intensity patterns were then imaged by using a 100× objective lens and a tube lens, and collected by a silicon-based CCD camera (1600×1200 pixels, WinCamD-UCD15, DataRay Inc). A quarter-waveplate and a polarizer were used to acquire the cross-polarized components.
Supplementary Text
Supplementary Note1:Anisotropic transmission induced geometric phase
The key of the geometric phase in the nano-aperture is the anisotropic and space-variant transmission. By matching the electromagnetic modes at the boundary, Martín-Moreno et al. gave an analytical formula to calculate the anisotropic transmission of a rectangular hole in a perfect conduction screen with zero thickness1:
(S1)
where ax and ay are the horizontal and vertical lengths of the aperture, λ is the operational wavelength, τ = ax/ayis the aspect ratio. When τ is within the range [1/3, 3], there is C(τ) = 0.0132 + 0.2127/τ + 0.2174/τ2. Clearly, the transmittance ratio of the two polarized stated is C(τ)/C(1/τ).
With the Jones matrix formalism, the transmission property for polarized light can be easily accessed. First, the Jones matrix for a subwavelength aperture with a width of w and main axes along the local u-v coordinates takes the form of:
(S2)
where tu and tv are the transmission along the two main axes.
Supposing that the u direction has a rotation angle ofζ with respect to the x-axis, we can obtain the general Jones matrix in x-y coordinates,
(S3)
Finally, the output fields for the circular polarized input can be written as:
(S4)
As can be seen from Equation (S1), there is tu= 0 andtv= 1 for very slim apertures, thus equation (S4) can be simplified:
(S5)
Obviously, the output electromagnetic fields are a superimposition of both the co- and cross-polarizations with nearly equal amplitudes,and there is an additional phase term of -2σζ(x) for the cross-polarized light.
In the visible regime, the light fields confined in the aperture (or slit) are gap plasmons. In principle, the propagation constants of the plasmons vary for different width, as determined by the characteristic equation:
(S6)
where k0 stands for the vacuum wavenumber, εm and εd are the permittivities for the metal and dielectric media, respectively, and w is the slab width. The dielectric is air and the permittivity of gold at λ = 632.8 nm is assumed as -11.6+1.27i.
As can be seen in Fig. S1, the plasmon-induced phase shift is relative small since the thickness is only 120 nm in our experiments. In the evaluation of the plasmon-induced phase, a straight slit with varying widthw is assumed (the unit is nanometre):
(S7)
where Λ is the same as that of the catenary (2 μm).
Figure S1 | Comparison of the geometric phase and plasmon-induced phase.The geometric phase is plotted for the catenary aperture under LCP illumination (σ = -1). The plasmon-induced phase is calculated for the straight aperture shown in the inset.
Supplementary Note2: Comparison the diffraction from the catenary aperture and a U-shaped aperture.
In order to see the importance of the aperture shape on the diffraction, we would like to compare the diffraction of a U-shaped aperture with the catenary aperture (Fig. S2). Once again, we use the PEC approximation in the numerical simulation. As depicted in the SEM image (Fig. S2d), the horizontal length of the U-shaped aperture is 2 μm.
For the co-polarized light which does not acquire geometric phase, the diffraction is mainly determined by the traditional Kirchhoff’s diffraction theory. Different from the catenary aperture (the left column of Fig. S2a), the U-shaped aperture can generate many diffraction orders for the co-polarization, as shown in both the numerical simulation (the left column of Fig. S2b) and experimental measurement (the left column of Fig. S2c).
For the cross-polarized light, the geometric phase dominates the diffraction behaviour. Once again, there is many unwanted diffraction orders for the U-shaped aperture (the right column of Fig. S2b,c)because the geometric phase is not linearly distributed, in contrast to the catenary aperture.
Figure S2 | Diffraction patternsof the catenary aperture and a U-shaped aperture under LCP illumination. (a)(b) Co- and cross-polarized components of the far-field diffraction of the catenary aperture (a) and U-shaped aperture (b) with Λ = 2 μm. (c) Experimental intensity of the U-shaped aperture for the co- and cross-polarized components in the xy-plane at z = 8 μm. (d) SEM image of the U-shaped aperture. Scale bar: 500 nm. (e) Schematic of the experimental set-up.
References：
1 Nikitin AY, Zueco D, García-Vidal FJ, Martín-Moreno L. Electromagnetic wavetransmission through a small hole in a perfect electric conductor of finite thickness.Phys Rev B 2008; 78: 165429.