Mixing of spin and orbital angular momenta via second-harmonic generation in plasmonic and dielectric chiral nanostructures

We present a theoretical study of the characteristics of the nonlinear spin-orbital angular momentum coupling induced by second-harmonic generation in plasmonic and dielectric nanostructures made of centrosymmetric materials. In particular, the connection between the phase singularities and polarization helicities in the longitudinal components of the fundamental and second-harmonic optical fields and the scatterer symmetry properties are discussed. By in-depth comparison between the interaction of structured optical beams with plasmonic and dielectric nanostructures, we have found that all-dielectric and plasmonic nanostructures that exhibit magnetic and electric resonances have comparable second-harmonic conversion efficiency. In addition, mechanisms for second-harmonic enhancement for single and chiral clusters of scatterers are unveiled and the relationships between the content of optical angular momentum of the incident optical beams and the enhancement of nonlinear light scattering is discussed. In particular, we formulate a general angular momenta conservation law for the nonlinear spin-orbital angular momentum interaction, which includes the quasi-angular-momentum of chiral structures with different-order rotational symmetry. As a key conclusion of our study relevant to nanophotonics, we argue that all-dielectric nanostructures provide a more suitable platform to investigate experimentally the nonlinear interaction between spin and orbital angular momenta, as compared to plasmonic ones, chiefly due to their narrower resonance peaks, lower intrinsic losses, and higher sustainable optical power.

Figure 1

Schematic of the nonlinear spin-orbital angular momentum (AM) coupling induced by second-harmonic generation (SHG) in a chiral cluster with $N$-fold rotation symmetry. Both the frequency and angular momentum have doubled during the SHG process. In addition, the quasi-angular-momentum, defined as the order of the rotational symmetry of the chiral cluster ($N$), can be transferred to the scattered SH wave to conserve the total angular momentum. $q$ is an integer and $q=0,\pm 1,\pm 2,\pm 3,\cdots$.

Figure 2

Spectra of the normalized fundamental (top panels) and SH (bottom panels) scattering cross section vs radius of the gold sphere illuminated by LCP ($\sigma =+1$) and RCP ($\sigma =-1$) LG beams with the OAM of $l=1$. The LG beam waist is $w_0=2{\lambda }_{\mathrm{FF}}$. Dashed lines correspond to the resonant responses in the bulk. Panels are plotted on a logarithmic scale.

Figure 3

Amplitudes and phases of the near-field longitudinal-field component ($E_z$) of the fundamental (top panels) and SH field (bottom panels) scattered from a gold sphere illuminated by the LCP ($\sigma =+1$) and RCP ($\sigma =-1$) LG beams with the OAM of $l=1$. The LG beam waist is $w_0=2{\lambda }_{\mathrm{FF}}$ and the wavelength of the beam is ${\lambda }_{\mathrm{FF}}=520\mathrm{nm}$. Radius of the gold sphere is $100\mathrm{nm}$. Panels (a), (b), (e), and (f) correspond to LCP beam ($\sigma =+1$), whereas panels (c), (d), (g), and (h) to RCP beam ($\sigma =-1$).

Figure 4

Spectra of the normalized fundamental (top panels) and SH (bottom panels) scattering cross section vs radius of the silicon sphere illuminated by LCP ($\sigma =+1$) and RCP ($\sigma =-1$) LG beams with OAM of $l=1$ and $l=4$. The waist of the LG beam is $w_0=2{\lambda }_{\mathrm{FF}}$. Figures are plotted on a logarithmic scale.

Figure 5

Normalized scattering cross section, $C_{\mathrm{sca}}$, spectra of a silicon nanosphere with radius of $500\mathrm{nm}$. The incoming excitation is a linearly polarized plane wave or circularly polarized LG beams. (a) Fundamental $C_{\mathrm{sca}}$ and its multipole expansion for linearly polarized plane wave excitation. (b) Fundamental $C_{\mathrm{sca}}$ for linearly polarized plane wave and circularly polarized LG beam excitations. (c) SH $C_{\mathrm{sca}}$ for linearly polarized plane wave and circularly polarized LG beam excitations.

Figure 6

Spectra of the normalized scattering cross section vs radius of the silicon sphere illuminated by linearly polarized plane waves. (a) Fundamental field and (c) SH field with incident wavelength ranging from 1300 nm to 4800 nm; (b) overlap between maps (a) and (c); (d) overlap between map (c) and the map at the FF (${\lambda }_{\mathrm{FF}}$ ranges from 650 nm to 2400 nm); (e) fundamental and SH scattering cross section of the silicon sphere with radius fixed to 500 nm surrounded by background material with different relative permittivity ${\epsilon }_{0r}$ (${\epsilon }_{0r}=1$ and ${\epsilon }_{0r}=1.2$). In (b) and (d), the left (${\lambda }_{\mathrm{FF}}$) and right (${\lambda }_{\mathrm{SH}}$) axes are used for the fundamental and SH scattering cross section, respectively. Panels (a)–(d) are plotted in a logarithmic scale. Panel (e) is plotted in a linear scale.

Figure 7

Spectra of the normalized scattering cross section, $C_{\mathrm{sca}}$, of dielectric chiral nanostructures illuminated by LCP ($\sigma =+1$) and RCP ($\sigma =-1$) LG beams with the OAM quantum number of $l=4$. The LG beam waist is $w_0=2{\lambda }_{\mathrm{FF}}$. (a) $C_{\mathrm{sca}}$ at the FF. (b) $C_{\mathrm{sca}}$ at SH. (c) Spatial distributions of the nonlinear polarization currents at the surface.

Figure 8

Comparisons between the fundamental and SH scattering spectra of the silicon chiral cluster with 3-fold rotational symmetry and the fundamental scattering spectra of a single silicon nanosphere. (a) ${\mathrm{LG}}_{04}$, $\sigma =+1$. (b) ${\mathrm{LG}}_{04}$, $\sigma =-1$.

Figure 9

Multipolar decompositions of the fundamental and SH scattered waves from the silicon C3-D chiral cluster in terms of angular momentum channels. MM and EM denote magnetic multipole and electric multipole, respectively. The silicon C3-D chiral cluster is illuminated by the RCP ${\mathrm{LG}}_{04}$ beam (left panels) and LCP ${\mathrm{LG}}_{04}$ beam (right panels). (a), (b), (e), and (f) correspond to scattered waves at FF, whereas (c), (d), (g), and (h) correspond to wave scattering at SH.

Figure 10

SH conversion efficiencies of C3-D chiral clusters illuminated by LCP and RCP ${\mathrm{LG}}_{04}$ beams. (a) Plasmonic (gold) chiral cluster. (b) Dielectric (silicon) chiral cluster.