Vectorial vortex generation and phase singularities upon Brewster reflection

We experimentally demonstrate the emergence of an azimuthally polarized vectorial vortex beam with a phase singularity upon Brewster reflection of focused circularly polarized light from a dielectric substrate. The effect originates from the polarizing properties of the Fresnel reflection coefficients described in Brewster's law. An astonishing consequence of this effect is that the reflected field's Cartesian components acquire local phase singularities at Brewster's angle. Our observations are crucial for polarization microscopy and open avenues for the generation of exotic states of light based on spin-to-orbit coupling, without the need for sophisticated optical elements.

Figure 1

(a) Focusing scheme of an aplanatic lens for incident circular polarization (CP). The paraxial input field ${\mathbf{E}}_{\mathrm{in}}$ in the back focal plane (BFP) is projected onto a reference sphere of radius $f$, where the local wave vector is tilted towards the geometrical focus under an angle $\theta$ and the CP is preserved for each wave vector. (b) The beam reflected from the substrate is collected by the same objective and analyzed polarization resolved in the BFP. The transverse electric and transverse magnetic (TE and TM) polarized fields $E^{\mathrm{TE}}$ and $E^{\mathrm{TM}}$ are aligned with the azimuthal and radial cylindrical unit vectors, respectively. For Brewster's angle ${\theta }_{\mathrm{B}}$, the TM component $E^{\mathrm{TM}}$ vanishes. (c) Snapshot in time of CP electric-field components in the reflection BFP around the intersection point of the $y$ axis (origin centered in BFP) and Brewster ring. In contrast to the azimuthal (TE) field, the radial (TM) component exhibits a phase jump of $\pi$ at ${\theta }_{\mathrm{B}}$, due to the properties of the Fresnel reflection coefficient $r^{\mathrm{TM}}$. (d) Projection of panel (c) onto the $y$ axis. The projected TE polarized field components on opposite sides of the intersection point possess opposite phase and are inherently $\pm \frac{\pi }{2}$ out of phase with respect to TM in CP light. Consequently, a vortex with topological charge of $\ell =\pm 1$ emerges, with the sign of the phase topological charge depending on the handedness of the incoming polarization state. (e) Time evolution of the polarization distribution in the BFP at Brewster's angle for the reflected vortex beam referred to throughout this paper. Note that for reflection the incident CP field on a ring is multiplied by the corresponding Fresnel reflection coefficient, thus there is no TM polarized field $E^{\mathrm{TM}}$ at the ring corresponding to ${\theta }_{\mathrm{B}}$.

Figure 2

Simplified sketch of the experimental setup utilized for measuring the BFP in reflection. The BFP is imaged on a variable spiral plate (VSP) in order to convert radial and azimuthal (TM and TE) field components to the $x$ and $y$ laboratory axes. The final linear polarizer before the CCD camera allows for selecting the intensity distribution of the desired polarization state to be recorded. In addition, the phase profile may be reconstructed via interference with a reference beam. A variation of the setup without VSP is used to directly project the polarization distribution in the BFP on Cartesian $x$ and $y$ axes.

Figure 3

(a, b) Experimental and (c, d) theoretical intensity distributions in the BFP of the $\mathrm{NA}=0.9$ microscope objective, for reflection of a focused CP Gaussian beam from a BK7 glass substrate. In contrast to the rather homogeneous intensity distribution for TE polarized light (a, c), the projection on TM (b, d) shows a null intensity ring with radius corresponding to Brewster's angle ${\theta }_{\mathrm{B}}$. The insets in panels (b) and (d) show cropped areas of the BFP surrounding ${\theta }_{\mathrm{B}}$ with enhanced contrast. (e, f) Cross-sectional view of the calculated TE and TM projections for the incident and reflected beam's intensity distributions $I_{\mathrm{in}}$ and $I_{\mathrm{r}}$ (normalized to their respective maximum), alongside the absolute value of the corresponding Fresnel reflection coefficient $|r^{\mathrm{TE}/\mathrm{TM}}|$.

Figure 4

(a, b) Interference patterns of TE polarized reflected light with a planar-wavefront reference beam for incident left- and right-hand CP (LCP and RCP) in the BFP of the $\mathrm{NA}=0.9$ microscope objective, displaying fork holograms of opposite orientations. (c, d) Corresponding experimental and (e, f) theoretical phase fronts, verifying the presence of a central singularity surrounded by a helical phase distribution.

Figure 5

(a, b) Experimental and (c, d) theoretical intensity distributions in the BFP of the $\mathrm{NA}=1.4$ oil immersion microscope objective, for reflection of a focused CP Gaussian beam from the glass-air interface at the bottom of a BK7 substrate. As for the case with the dry $\mathrm{NA}=0.9$ objective in Fig. 3, a null intensity ring at Brewster's angle ${\theta }_{\mathrm{B}}$ shows up for TM polarized light (b, d), contrary to the projection on TE polarization (a, c). Furthermore, the sharp transition at the critical angle for total internal reflection ${\theta }_c$ and the high intensity above it are prominent in all images. (e, f) Cross-sectional view of the calculated TE and TM projections for the incident and reflected beam's intensity distributions $I_{\mathrm{in}}$ and $I_{\mathrm{r}}$ (normalized to their respective maximum), alongside the absolute value of the corresponding Fresnel reflection coefficient $|r^{\mathrm{TE}/\mathrm{TM}}|$.

Figure 6

(a, b) Interference patterns of $x$ and $y$ polarized projections of the reflected light with a planar-wavefront reference beam for incident RCP in the BFP of the $\mathrm{NA}=1.4$ microscope objective. Two horizontally ($x$) or vertically ($y$) aligned forks emerge at the intersection points of the Brewster ring and respective Cartesian axis. (c, d) Phase reconstruction around these points affirms the presence of vortices with topological charge of $\ell =\pm 1$.