Observation of Polarization Singularities at the Nanoscale

With a phase-sensitive near-field microscope we measure independently the two in-plane electric field components of light propagating through a 2D photonic crystal waveguide and the phase difference between them. Consequently, we are able to reconstruct the electric vector field distribution with subwavelength resolution. In the complex field distribution we observe both time-dependent and time-independent polarization singularities and determine the topology of the surrounding electric field.

Figure 1

Schematic representation of the experimental setup. Light is coupled into a 2D photonic crystal waveguide. The near-field probe is scanned above the sample and collects light that is interferometrically mixed with a reference beam and detected using a heterodyne scheme. The polarization state in the reference branch is controlled using wave plates. Left inset: Scanning electron micrograph of the silicon membrane photonic crystal waveguide under investigation (lattice constant $a=450\mathrm{nm}$ and hole diameter $d=250\mathrm{nm}$). Right inset: Scanning electron micrograph of the aluminum-coated near-field probe with aperture of $\approx 200\mathrm{nm}$.

Figure 2

(a),(b) Detected amplitude pattern for two orthogonal polarizations (indicated by white arrows) in the reference branch. (c),(d) Amplitude pattern of the forward propagating mode, obtained after Fourier analysis of (a) and (b). (e),(f) Theoretical amplitude pattern of $E_x$ and $E_y$ 20 nm above the surface for the forward propagating mode. For all the images the depicted area is $4a\times 5a$ (the center of the waveguide is around $x=0$) and the amplitude is normalized. (g),(h) Crosscuts of the experimentally (blue) and theoretically (red) obtained real part of the complex field along the dashed lines of (c), (d), (e), and (f), respectively.

Figure 3

Experimentally and theoretically obtained instantaneous 2D vector plots of the electric field. The contour plots indicate the normalized magnitude of the electric field. The black arrows highlight a disclination. For both the figures the depicted area is $2a\times 1a$ (the center of the waveguide is around $x=0$).

Figure 4

(a) Representation of the polarization ellipse, where $v$ is the major semiaxis, $u$ is the minor semiaxis of the ellipse, $\epsilon =\mp u/v$ and $\alpha$ the orientation angle. (b) Experimentally and theoretically obtained $\epsilon$. Negative and positive values correspond to left- and right-handed polarization, respectively. Lines of linear polarization ($L$ lines) are shown in green. (c) Experimentally and theoretically obtained $\alpha$. The dotted ellipses indicate the orientation of the polarization and the dashed lines show the threefold symmetry of the system. The white dots indicate the position of $C$ points. The depicted area is $2a\times 1a$ (the waveguide is centered around $x=0$).