Emission of circularly polarized light by a linear dipole

Theoretical description of a linear dipole

The electric field E(r) radiated by an oscillating monochromatic dipole moment p located at the origin of the Cartesian coordinate system can be written in SI units as (22)E(r)=exp(ik0r)4πε0rk02{(e^r×p)×e^r+[3e^r(e^r⋅p)−p](1k02r2−ik0r)}(1)with the wave number in free-space k₀ = 2π/λ and wavelength λ. The unit vector e^r points away from the dipole in radial direction and ε₀ is the vacuum permittivity. Alternatively, E(r) can be represented as a superposition of all plane waves launched by the dipole. The local field is thereby given by a Fourier integral (23)E(r)∝∬E˜(k)ei[kxx+kyy+kz∣z∣] dkx dky(2)E˜(k) is called the vectorial angular spectrum (VAS) of the dipole. The VAS of a linear dipole—here we consider a dipole parallel to the x direction—can be written as (23, 24)E˜(k)∝1kz(k02−kx2k02−kxkyk02∓kxkzk02)(3)This is the k-space representation of the dipole field with respect to the x-y plane in real space (k_x-k_y plane in k space). As a practical notation, we define the transverse k vector as k_⊥ = (k_x, k_y,0) and the longitudinal component by kz=(k02−k⊥2)1/2 with imaginary part Im (k_z) ≥ 0. The ∓ sign in Eq. 3 refers to regions with z ≷ 0, respectively.

We are specifically interested in the circularly polarized emission or, equivalently, the spin of the emitted light. In complex three-dimensional field distributions, the spin density describes the local orientation and strength of the circular polarization. For real and k space, we can use similar definitions (18–20)s(r)=Im (E*×E), s˜(k)=Im (E˜*×E˜)(4)where we omit all prefactors for the sake of simplicity. Regarding the VAS, we distinguish two different regions in k space. First, the propagating part of the VAS defined by ∣k_⊥∣ ≤ k₀ and k_z ∈ ℝ. In this angular region, all components of the polarization vector in Eq. 3 are real, and s˜=0. Second, for ∣k_⊥∣ > k₀, we result in an imaginary k_z = i∣k_z∣ (from now on, we consider the half space with z > 0), indicating the evanescent part of the VAS. Concerning the polarization vector in Eq. 3, only the z component contains k_z, which implies that Ez˜ is phase-shifted with respect to Ex˜ and Ey˜ by ±π/2, giving rise to a certain degree of circular polarization. To investigate the orientation of the polarization ellipse, we decompose the spin density s into components parallel (s˜k) and transverse (s˜t) with respect to the propagation direction defined as the real part of the k vector, Re(k) = k_⊥ (details can be found in Materials and Methods). In our representation, s˜k describes the electric field vector spinning around the propagation direction, similar to the circular polarization of paraxial beams of light (18–20). In contrast, s˜t describes fields that spin around a transverse axis, similar to the spokes of a wheel spinning around an axis perpendicular to the propagation direction of the wheel (20).

We plot both components of the spin density of the VAS in Fig. 1 (A and B). As already mentioned, in the propagating region of the k space—within the dotted black circles—the spin density is zero in all its components. However, in the evanescent part of the VAS (∣k_⊥∣ > k₀), the transverse s˜t shows a two-lobe pattern, and the maximum amplitude occurs along the x axis. This transverse spin occurring in evanescent waves is well understood and has been investigated in many scenarios (18–20). Besides the transverse component, we additionally observe a nonzero longitudinal component s˜k, which is strongest along the ±45° directions, exhibiting a four-lobe pattern with alternating sign (a geometrical model describing the origin of the distribution of s˜k can be found in the Supplementary Materials). The occurrence of s˜k is astonishing because the real-space field distribution of a linear dipole does not show any spin being parallel to the propagation direction of the spherical wave. For demonstration, we depict the spin density distributions surrounding a linear dipole in real space for several planes of observation (Fig. 1C). As we can see, the spin density indicated by red- and blue-colored cones exhibits a purely transverse, azimuthal distribution around the axis of the linear dipole moment p, which is marked by the green vector. No longitudinal spin component (this is a spin component pointing away from the dipole in radial direction e^r) occurs and se^r=0 [further discussion on s(r) and its derivation is provided in the Supplementary Materials]. That is, knowing that the evanescent part of the VAS dominates the near field in real space (24), although their relation is not straightforward (23, 24), it is unexpected that we observe locally longitudinal spin in only one description of the dipole field but not the other.

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The occurrence of longitudinal spin in the evanescent part of the VAS has direct consequences for a variety of experimental scenarios. In particular, whenever linear dipole emitters like atoms, quantum dots, molecules, and nanoantennas are coupled to waveguides or an optically denser medium, they might emit circularly polarized photons into defined angular regions, depending on the orientation of the dipole moment.

Experimental verification of the longitudinal spin

For an experimental demonstration, we use a dipole-like spherical plasmonic antenna positioned on a glass substrate. The interface between the air and glass half-spaces represents the x-y plane, with the surface normal e^z pointing into the glass substrate (Fig. 2A). The antenna is excited by a focused x-polarized beam of light, which induces a dipole moment parallel to the x axis. Because the glass substrate has a higher refractive index than the surrounding medium, a part of the evanescent VAS of the dipole can couple to propagating waves in the optically denser glass (25, 26). An oil immersion microscope objective index-matched to the glass substrate collects the emitted VAS of the dipole up to ∣k_⊥∣/k₀ = 1.3. Because the VAS of the excitation beam is limited by the numerical aperture (NA) of the focusing objective (∣k_⊥∣/k₀ ≤ 0.9; see gray area in Fig. 2A), we can be sure that we measure only the scattered light within the angular range defined by 0.9 < ∣k_⊥∣/k₀ < 1.3. The experimental results are presented in Fig. 2B, where we show, from left to right, the distributions of the total intensity I, the left- and right-handed circularly polarized intensity components I₋ and I₊, and the longitudinal spin density (pointing in radial direction), here defined by s_r ∝ I₊ − I₋. All four experimental distributions show a very good overlap with their theoretical counterparts (Fig. 2C). A derivation of the theoretical far-field distributions similar to (25–28) can be found in the Supplementary Materials. In particular, the spin density shows the expected four-lobe pattern similar to the evanescent VAS in Fig. 1B. We conclude that linearly polarized dipoles emit circularly polarized light in the evanescent part of the VAS. This notion can be crucial, when the dipole emitter is coupled to a waveguide by evanescent waves.

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Coupling of circular polarization to waveguides

As an example, we consider the crossing of two high–refractive index waveguides (n = 4) with quadratic cross sections and edge lengths of 100 nm. On top of the crossing—20 nm above the waveguides—we place a linear dipole emitter (λ = 600 nm), whereby the orientation of the dipole along the x axis represents the angle bisector between the waveguides. The geometry of the system is depicted in the top-right inset of Fig. 3, where we furthermore plot the numerically calculated radial component of the spin density, sr=se^r, and the electric field intensity w = ∣E∣² (bottom-right inset) in the x-y cross section.

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Once more, we observe the coupling of elliptically polarized evanescent waves, in this case, to the waveguide modes propagating along the ±45° directions, which verifies the occurrence of longitudinal spin in the VAS of a linear dipole. The degree of circular polarization can be calculated by normalizing the spin density with the field intensity (29). In the center of the waveguides and far away from the dipole moment, we obtain ∣s_r∣/w ≈ 55%. Our results therefore indicate the possibility to control the circular polarization of the guided mode with the orientation of the dipole. For example, switching from a dipole moment along the x direction to a dipole in the y direction also changes the sign of the spin in the coupled modes. Further examples can be found in the Supplementary Materials, where we also optimize the dipole moment orientation to achieve ∣s_r∣/w ≈ 85%.