Weak Measurement of Elliptical Dipole Moments by $C$-Point Splitting

We investigate points of circular polarization in the far field of elliptically polarized dipoles and establish a relation between the angular position and helicity of these $C$ points and the dipole moment. In the case of highly eccentric dipoles, the $C$ points of opposite handedness exhibit only a small angular separation and occur in the low intensity region of the emission pattern. In this regard, we introduce an optical weak measurement approach that utilizes the transverse electric (azimuthal) and transverse magnetic (radial) far-field polarization basis. Projecting the far field onto a spatially varying postselected polarization state reveals the angular separation and the helicity of the $C$ points. We demonstrate the applicability of this approach and determine the elliptical dipole moment of a particle sitting on an interface by measuring the $C$ points in its far field.

Figure 1

Free-space far-field radiation of dipoles spinning around the $x$ axis. The upper row corresponds to a circular polarized dipole, $|p_z|/|p_y|=1$, with (a) showing the total far-field intensity $I$ (black line), $I_{\mathrm{TM}}$ of $p_z$ (green line), and $I_{\mathrm{TE}}$ of $p_y$ (gray line). (b) Left-and right-handed circular polarization, $I_{-}$ and $I_{+}$. The far-field $C$ points ($C_{\pm }$) are highlighted by red and blue arrows. (c) and (d) The corresponding intensities for a strongly elliptical dipole moment, $|p_z|/|p_y|=20$. The inset in (c) represents the central part magnified by a factor of 100, and (d) is magnified by a factor of 2. (e) Represents the far field for the postselected polarization states $I_{u-}$ and $I_{u+}$ magnified by 100.

Figure 2

Far-field radiation of dipoles spinning around the $x$ axis above a dielectric air-glass interface. (a) and (b) Show $I$, $I_{-}$, and $I_{+}$ for a circular polarized dipole, with red and blue arrows highlighting $C_{\pm }$. (c) The corresponding postselected polarization states, $I_{u-}$ and $I_{u+}$, optimized to generate perfect directionality ($1:0$) on the critical angle. (d)–(f) Corresponding far-field intensity patterns for a strongly elliptical dipole moment, $|p_z|/|p_y|=20$.

Figure 3

Experimental setup and measurement scheme. (a) Sketch of the experimental setup. (b) Projection onto the polarization states ${\mathbf{u}}_{\pm }=[\cos (\varphi ),ı\sin (\varphi )]$, with $\pm$ indicating $\varphi \lessgtr 0$.

Figure 4

Polarization-resolved back focal plane (BFP) images. (a) and (b) $I_{-}$ and $I_{+}$ distributions (normalized to their common maximum, depicted angular range defined by $0.95\le k_{\perp }/k_0\le 1.3$) for a particle position of $x\approx 0$ and $y\approx 200\mathrm{nm}$, where $|p_z|/|p_y|\approx 1$. Theoretical distributions are shown as insets (red and blue crosses indicate the $C$ points). (c) and (d) $I_{-}$ and $I_{+}$ for $x\approx 0$ and $y\approx 30\mathrm{nm}$. (e) and (f) BFP images for the same position as (c) and (d), but for a postselected polarization state of ${\mathbf{u}}_{\pm }=[\cos (\varphi ),𝚤\sin (\varphi )]$ with $\varphi =\mp 5°$. Theoretical distributions including the reconstructed $C$ points are shown as insets. (g) The dashed black line indicates the reconstructed polarization ellipse, with error margins, spin, and dipole moments indicated as gray background, black vector, and green ($p_z$) and gray ($p_y$) vectors, respectively.