HTML
-
BSWs are generated by a 1D PC consisting of a stack of six pairs of silicon dioxide and silicon nitride layers, with refractive indices of 1.45 and 1.79 (at λ = 1.55 µm), and thicknesses of 492 nm and 263 nm, respectively. The multilayer lies on a glass substrate (refractive index of 1.5) and is covered with a thin 80 nm-thick layer of silicon nitride. Figure 1a shows the dispersion diagram of the structure (calculated using the impedance approach49). This diagram shows a photonic bandgap, which contains the dispersion curve for a BSW. Given the 1D PC design, this surface mode is TE-polarized, as shown in Fig. 1b-e. We observe that Hy (Fig. 1d) is shifted by a quarter wavelength with respect to Hz (Fig. 1e) along the propagation direction y of the surface wave, thus revealing the helicity of the optical magnetic field along the transverse x direction, as shown in Fig. 1b. The transverse SAM for the surface wave is thus carried solely by its rotating magnetic field; the electric field shows no helicity (see Fig. 1c).
We numerically study the coupling of single ED and MD to a TE-polarized BSW. To this end, the dipoles are considered to be positioned 10 nm above the top surface of the 1D PC described above, which radiate light in a continuous wave regime at λ = 1.55 µm (details in Materials and Methods section). The ED is oriented along the x direction along the surface, i.e., parallel to the electric field of the TE-polarized BSW. The MD rotates in the (yz) plane perpendicular to the surface, i.e., in the helicity plane of the rotating magnetic field of the TE-polarized BSW. Figure 2 shows a snapshot of the resulting electric field amplitude along the (yz) plane, for the ED (Fig. 2a) and MD (Fig. 2b, c) excitations. The MD, whose dipole moment is $\vec m \propto \vec e_y \pm j \ast 0.53\vec e_z$ ($j = \sqrt { - 1}$), rotates either anti-clockwise (Fig. 2b) or clockwise (Fig. 2c). In these figures, the field distributions around the dipoles are saturated to provide a better view of the light distributions at the structure surface. The simulations are carried out using the 2D FDTD method. With the ED, the BSW is symmetrically excited on both sides of the point-like source. No directionality is observed in the optical coupling process. In contrast, the optical coupling process becomes unidirectional for the spinning MD. Figure 2d shows the ratio of the electric intensities for the left and right BSWs, for various MD polarizations. The MD polarization is tuned along the path shown in red in the Poincare sphere (see figure inset). In this case, only the ellipticity angle (2χ) for the MD moment varies. Our directionality factor (intensity ratio) becomes larger than 103 at 2χ = 55.8° and smaller than 10−3 at 2χ = 304.2°. Therefore, at these two particular angles, the portion of the incoupled power that propagates in one of the two possible directions becomes larger than 99.9% of the total incoupled power. These results reveal a tunable unidirectional optical coupling controlled by the magnetic field of light. They also confirm that the rotating magnetic field of a TE-polarized BSW carries SAM.
Fig. 2
TE-polarized Bloch surface waves excited with ED and MD sources. Simulation by FDTD of the coupling of a an ED source oriented along the x axis and b, c a spinning MD source, to a TE-polarized BSW. All three results show, in false colors, the absolute value of the real part of the electric field ( ). The MD rotates either b anti-clockwise or c clockwise. d Directionality factor (ratio of the electric intensities for the left and right BSWs) for various MD polarizations. The MD polarization ellipticity is changed along the path shown in red in the Poincare sphere (see inset). The MD moment is also expressed at specific values of the ellipticity angle 2χ related to the Poincare sphere$\left| {Re\left({E_x} \right)} \right|$ From an experimental point-of-view, it is possible to realize a tunable directional coupling for BSW with a spinning MD using a dielectric sphere showing magnetic resonances6, 50, 51 directly deposited on top of the 1D PC. This bead can then be illuminated with a circularly polarized beam at near-grazing incidence, following the electric spin-controlled excitation process of surface plasmons29. We will show here that magnetic directional coupling can be clearly demonstrated even with a standard subwavelength groove directly engraved on top of the 1D PC.
To fabricate the 1D PC described above, thin layers of silicon oxide and silicon nitride are deposited alternately by plasma-enhanced chemical vapor deposition onto a glass wafer. A cross-section of the multilayer realized by a focused ion beam (FIB) reveals the design detailed above (Fig. 3a). Then, the sample is covered by a 100 nm-thick chromium layer and a 600 nm wide and deep groove is milled by FIB over a length of 20 µm. Finally, the chromium is removed. The inset of Fig. 3d shows a scanning electron microscope (SEM) image for the resulting structure.
Fig. 3 A simple all-dielectric platform unveiling the magnetic spin-orbit interaction.
a SEM cross-section of the 1D photonic crystal (scale bar: 1 µm). b Schematic diagram of the elliptic polarization (electric optical field) generated by the polarizer (fixed) and the quarter-wave plate (rotating). c Schematic diagram of the magnetic field handedness in the helicity planes of the right and left BSWs for an incident right-handed polarization. The handedness is defined here by the parameter v: we have v = + 1 and v = − 1 for the clockwise and anticlockwise rotating magnetic fields, respectively. d Far-field optical image of the BSW obtained via excitation at λ = 1.55 µm of a 600 nm wide and deep groove. The groove is 20 μm long. The laser beam is incident from air onto the top surface of the 1D PC at almost the grazing angle (incidence angle: 80°, see Supplementary Fig. S2). This image originates from light scattering at the sample top surface. The incident light is linearly (TM) polarized here, to reveal the two symmetric BSW propagation directions provided by the phase-matching condition. Figure inset: SEM top view of the groove milled into the top surface of the 1D photonic crystalThe structure is characterized in the far-field by projecting a slightly focused beam of controlled polarization onto the sub-wavelength groove at an incidence angle of approximately 80°. The structure is imaged in reflection mode with an objective (× 20, numerical aperture = 0.4) coupled to an infrared camera (see details in the Supplementary Material and Supplementary Fig. S2). Due to light scattering at the free surface of the 1D PC, a direct real-time mapping is possible for the surface waves excited on both sides of the groove. Figure 3d shows the far-field images for the surface around the groove under illumination. The incident beam is TM linearly polarized with respect to the sample surface, leading to a symmetric scattering pattern. It is noteworthy that the definition of the TE/TM polarizations for the incident light and the BSWs are related to different local coordinate frames, and thus they should not be directly compared. In the context of our study, a TM-polarized incident wave can excite a TE-polarized BSW. The bright elongated spot along the y axis is the cross-section of the excitation beam along the surface. The two narrow rays on both sides of the excitation spot are traces of the BSWs excited by the subwavelength groove. Linear momentum conservation imposes a tilt angle for the BSW propagation direction with respect to the groove direction (y) that is predicted to be approximately 33.8°; it is measured to be approximately 36°.
We then study the distribution of the incoupled power between the two surface waves as a function of the incident polarization. The polarization is defined by the angle θ between the fast axis of the QWP and the polarizer. When = k 90° k=0, 1, 2, 3, the polarization is linear whereas a circular polarization is realized for θ = 45° + k90°, k = 0, 1, 2, 3. For intermediate angles, the polarization is elliptical. On the one hand, the incident polarization is defined by a rotating QWP and a fixed polarizer. In that case, the polarization ellipse for the emerging light (that is incident onto the 1D PC) rotates with the crystalline axes of the QWP (see Fig. 3b). Such a polarization property leads to an incident light field whose electric and magnetic amplitudes have, by projection, a 2θ dependence. The energy coupling to the right and left BSWs thus undergoes a 4θ dependence as it is homogeneous to the intensity (this point will be discussed in detail later). As our near-field coupler (the groove, i.e., dielectric scatterer) is mainly driven by an ED moment (per unit length), the energy coupling to the BSWs can be assumed to be mediated by the electric optical field. The energy coupling is, therefore, helicity independent since the electric field of a TE-polarized BSW is linearly polarized (Fig. 1b). On the other hand, the handedness of the light waves leaving the quarter wave plate shows a 180° periodicity with respect to the angle θ. Such a property originates from the universal 180° periodicity for the helicity dependent optical phenomena. As only the magnetic field of the TE-polarized BSW is rotating (Fig. 1b), any helicity dependent contribution to the excitation process for the surface wave will arise from a magnetic optical effect. Importantly, the incident magnetic field shows opposite handedness when projected onto the helicity planes of the right and left BSWs (see Fig. 3c). For instance, a right-handed incident polarization leads to a magnetic field rotating clockwise (v = + 1) and anticlockwise (v = − 1) in the helicity planes of the left and right BSWs, respectively (Fig. 3c). Such a configuration is favorable for spin-controlled unidirectional excitation of the BSWs, with the magnetic field of the BSW being described by the handedness vBSW = + 1. A right-handed (left-handed, respectively) incident polarization would thus direct TE-polarized BSWs to the left side (to the right side, respectively) of the groove. Such a spin-controlled contribution to the coupling process leads to a 2θ-dependent power distribution between the two BSWs excited on both sides of the groove.
We experimentally acquired images of the structure while varying the polarization of the incident beam. For each image recorded at a specific polarization state, we integrate the signal detected over two square areas located symmetrically with respect to the groove (shown in light red and blue colors in Fig. 3d). Finally, the resulting values Sr and Sl measured on the right and left BSWs, respectively, are plotted as a function of the angle θ (Fig. 4a). The experimental plots are represented by solid lines together with the simulation results obtained with the 3D FDTD method (see details in the Materials and Methods section).
Fig. 4 Magnetic spin-orbit interaction steers Bloch surface waves.
a Detected signals (circles) and simulated intensities (FDTD method, solid lines) on the right and left BSWs as a function of the angle θ between the quarter-wave plate and the polarizer. The curves related to the right and left BSWs are represented by red and blue colors, respectively. b, c Spectrum (amplitude) of the experimental blue and red curves of a, obtained by Fourier transformation. Coefficient u defines the harmonic orders for the Fourier series. d, e Representation in the real space of the non-null harmonics for the two Fourier series shown in b and c: d second harmonics and e fourth harmonics. Experimental and numerical curves are shown by the solid and dashed lines, respectivelyFigure 4a shows Sr and Sl as a function of θ Sr and Sl are described by sinusoidal functions, shifted by ~ 30° from each other, with the amplitudes modulated by a sinusoidal function. The experimental results and simulation predictions are in good agreement. As expected, the coupling process for light in the BSWs is asymmetric except for the linear (TM) incident polarization (θ=k 90° k=0, 1, 2, 3). In this case, the optical system is fully symmetric with respect to the groove and the two curves for Sr and Sl merge. As discussed above, the electric and magnetic contributions to the BSW excitation process show a 4θ and 2θ dependence, respectively. Therefore, a Fourier analysis for Sr and Sl may assist identification of these electric and magnetic optical effects. By Fourier transforming these two functions (cf. Figure 4b, c), we see that they can be simply expressed analytically as:
$$ S_i\left( \theta \right) = A_i^{(0)} + A_i^{\left( 2 \right)}\sin (2\theta + \phi _i^{\left( 2 \right)}) + A_i^{\left( 4 \right)}\sin (4\theta + \phi _i^{\left( 4 \right)}) $$ (1) where i=r, l, and the coefficients $A_i^{(u)}$ and $\phi _i^{(u)}$(u = 0, 2, 4) are constant. Coefficients $A_i^{(u)}$ and $\phi _i^{(u)}$ (u = 2, 4) are given by the Fourier transform of Sr and Sl.
We see that in Fig. 4d the second harmonic components relative to the left and right BSWs are almost in opposition, i.e., shifted by 180°. The fourth harmonics (cf. Figure 4e) undergo a shift of approximately 30° initially evidenced in Fig. 4a. Importantly, the local maxima and minima of the second harmonic component closely coincide with the right and left-handed circular polarization states. Moreover, changing incident polarization handedness inverts the distribution for the incoupled light in the right and left BSWs. The second harmonic contribution to the BSW coupling is, therefore, helicity dependent. In contrast, the fourth harmonics of the Fourier series stay unchanged when the input polarization handedness is reversed (See Fig. 4e). Therefore, the fourth harmonic contribution to the BSW coupling is independent of the helicity of the light.
As subwavelength scatterers are optically governed by an ED moment, one may consider that a subwavelength groove on top of a 1D PC interacts with the electric field of an incoming wave to transfer energy to the BSWs. Following this pure electric model, and assuming an incident plane wave, the previously defined coefficients Sr and Sl become proportional to the coupling rates:
$$ R_i = \alpha \left| {\vec e_i \cdot \vec E_{inc}(\vec r_0)} \right|^2 $$ (2) where i = r, l denotes the right and left sides of the groove and α is a constant. $\vec E_{inc}(\vec r_0)$ is the incident electric field at a single point for the coordinate $\vec r_0$ along the subwavelength groove. $\vec e_i$ is the unit vector in the direction of the electric field of the emerging right and left BSWs. When plotted as a function of the angle θ, Rr, and Rl are described by two sinusoids showing a 4θ dependence and shifted by 8° (see Supplementary Material and Supplementary Fig. S4). Moreover, changing the polarization handedness does not interchange the values for the two coefficients, which indicates that pure electric coupling of the incident light to the TE-polarized surface waves is helicity independent. By comparing Supplementary Fig. S4 and Fig. 4e, we see that Rr and Rl closely match the fourth harmonic function of Eq. 1. The unbalanced electric coupling is due to the asymmetric projections for the electric field onto $\vec e_r$ and $\vec e_l$. The larger angular shift observed between the couple of experimental curves (30° vs. 10° with our model) may be because the scattering properties of our 600 nm large (i.e., λ/2.5) groove-like coupler slightly deviates from the dipole emission.
The modulation of the electric coupling by a helicity dependent optical process (Fig. 4) is not predicted by our analytical model. As noted above, only the magnetic field of the TE-polarized BSW is rotating, with the electric field showing zero helicity. A helicity dependent process for such waves, thus, solely involves the magnetic field of the light. We plotted, as a function of θ, the ellipticity factor for the magnetic field (plane-wave illumination) projected onto the helicity planes of the right and left BSWs (i.e., the planes perpendicular to the transverse spin momentum of the surface waves). Details of the calculation are given in the Supplementary Material. The ellipticity curves show a periodicity of 2θ and opposite values when the polarization handedness is changed (see the Supplementary Fig. S5). These curves closely resemble the second harmonic functions of Eq. 1 (Fig. 4d). Therefore, the second harmonic contribution to the optical coupling is solely controlled by the magnetic field of the light.
Remarkably, the magnetic effect is clearly visible using a dielectric scatterer described by an ED moment (per unit length). Figure 4b, c show that its contribution is larger than 45% of the electric contribution to the coupling. Despite the extremely low response of the scatterer to the magnetic field of the impinging wave, the rotating magnetic field incident at an electric scatterer provides the initial conditions to direct a large portion of the incoupled energy to the right or to the left BSW depending on the polarization handedness. The second harmonic curves shown in Fig. 4d thus describe a magnetic spin-directional coupling, as shown in Fig. 2. In the experimental case, however, the phase matching between the incident light and the BSW is mediated by the electric optical field given the ED nature of the scatterer. The rotating incident magnetic field incident at the groove, which is less affected by the scatterer, controls the directionality of the launched surface waves. This explains the 4θ dependence of the experimental coupling process (Fig. 4) that is comparatively not observed for MD excitation: Fig. 2c shows a directionality curve with only a 180° periodicity. The BSW excitation undergoes light-to-BSW electric field projection rules that accompany the phase matching. A subwavelength resonant particle or antenna, whose resonance is described by a MD moment, would cancel this electric component of the coupling. Such a configuration is, however, beyond the scope of this paper.