Directional emission of down-converted photons from a dielectric nanoresonator

Creation of entangled photon pairs is one of the key topics in contemporary quantum optics. In the present article, we theoretically describe the generation of photon pairs in the process of spontaneous parametric down-conversion of light by a resonant spherical nanoparticle made of a dielectric material with bulk ${\widehat{\chi }}^{\left(2\right)}$ nonlinearity. We consider the nanoparticle size that satisfies the condition of excitation of resonant eigenmodes described by Mie theory. We demonstrate that a highly directional photon pair generation can be observed utilizing the nonlinear Kerker-type effect, and that this regime provides useful polarization correlations between the emitted photons.

Figure 1

Schematic of SPDC process in the collinear decay geometry in spherical and Cartesian coordinates: signal and idler photons are being detected at the point in the far-field region ${\mathbf{r}}_i={\mathbf{r}}_s=\mathbf{r}$. Incident plane wave propagates along the $z$-axis and polarized along the $x$-axis. The frequency of the incident field and the pump field inside the nanoparticle is ${\omega }_p$. The process is degenerate; therefore the frequencies of the signal and idler photons are equal, ${\omega }_s={\omega }_i={\omega }_p/2$.

Figure 2

(a) Elastic scattering cross section, including different multipoles, including total (black solid), electric dipole (ED, dashed red), magnetic dipole [MD, solid blue (light gray)], electric quadrupole (EQ, dashed-dot yellow), and magnetic quadrupole (MQ, dotted purple), depending on pump wavelength in vacuum ${\lambda }_p$. (b) $D$-coefficients normalized to the maximum for all possible dipole decays at the wavelength ${\lambda }_p=720$ nm. (c) The geometry of the considered problem including a spherical particle of wurtzite GaAs with a radius $a=110$ nm, the incident plane wave propagating along the $z$-axis, and the electric field oscillating along the $x$-axis. The panel also shows the orientation of the crystal lattice relative to the incident plane wave. (d) Possible decay channels in the considered geometry. Channels needed to obtain directivity are shown with gray or yellow (light gray). Solid lines show decay to the crossed dipoles, dashed lines show decay to the aligned dipoles.

Figure 3

(a) Far-field patterns of the collinear two-photon generation for different wavelengths ${\lambda }_p=660$ nm, 720 nm, 786 nm, 1086 nm. (b) Forward (dashed-dot red) and backward (dashed blue) contributions $w_{unpol}$ and their ratio $w_{\theta =0}^{\text{unpol}}/w_{\theta =\pi }^{\text{unpol}}$ (solid black) depending on the pump wavelength ${\lambda }_p$. (c) Amplitudes of coefficients $|c_1|$ (solid blue) and $|d_1|$ (dashed red) in the decomposition of pump field inside the nanoparticle ${\mathbf{E}}_p$. (d) Phases of the coefficients ${\phi }_{c_1}$ [solid blue (light gray)], ${\phi }_{d_1}$ (dashed red), and the difference ${\phi }_{c_1}-{\phi }_{d_1}$ (solid black).

Figure 4

The structure of the nonlinear tensor providing the directivity of collinear two-photon emission in the specified direction. The columns illustrate the following: (1) far-field patterns of the two-photon collinear detection along one of the $x$-, $y$-, or $z$-axes; (2) decay channels into crossed dipoles; (3) decay channels into aligned dipoles; (4) nonzero components of the second-order nonlinear susceptibility tensor ${\widehat{\chi }}^{\left(2\right)}$; and (5) crystal symmetry groups required to observe the directional emission.

Figure 5

(a) The detection geometry in the area limited by the maximum angle $\theta ={30}^{\circ }$ relative to the backward direction, radius of the sphere, and all other parameters are the same as in Fig. 2. (b, c) Polarization correlations at ${\lambda }_p=720$ nm in the linear basis (b) and in the circular basis (c).

Figure 6

(a) Amplitudes of the magnetic dipole coefficients in the decomposition of scattered field $|b_1|$ (solid black), pump field inside the nanoparticle $|c_1|$ [solid blue (light gray)], and dyadic Green's function $|a_1^{\left(2\right)}|$ (dashed red). (b) Phases of the mentioned coefficients ${\phi }_{b_1}$ (solid black), ${\phi }_{c_1}$ [solid blue (light gray)], and ${\phi }_{a_1^{\left(2\right)}}$ (dashed red). (c) Amplitudes of the electric dipole coefficients in the decomposition of a scattered field $|a_1|$ (solid black), pump field inside the nanoparticle $|d_1|$ [solid blue (light gray)], and dyadic Green's function $|b_1^{\left(2\right)}|$ (dashed red). (d) Phases of the mentioned coefficients ${\phi }_{a_1}$ (solid black), ${\phi }_{d_1}$ [solid blue (light gray)], and ${\phi }_{b_1^{\left(2\right)}}$ (dashed red). The following geometry is considered: a spherical particle of wurtzite $\mathrm{GaAs}$ has the radius $a=110$ nm, the incident plane wave propagates along the $z$-axis, and the electric field of incident wave oscillates along the $x$-axis.

Figure 7

All possible scalar products $|{\mathbf{W}}_{\mathbf{J}}·{\mathbf{W}}_{{\mathbf{J}}^{\prime }}|$ of dipole vector spherical harmonics.

Figure 8

(a) Dependence of overlapping coefficients $D_{{\mathbf{J}}_p\to {\mathbf{J}}_i,{\mathbf{J}}_s}$ on the pump wavelength ${\lambda }_p$ in vacuum. Dashed blue (light gray), decay into two magnetic dipoles along the $x$-axis; solid blue (light gray), decay into two magnetic dipoles along the $y$-axis; dashed black, decay into two electric dipoles along the $x$-axis; solid black, decay into two electric dipoles along the $y$-axis; dotted red, decay into crossed electric dipole along $y$-axis and magnetic dipole along the $x$-axis; dashed-dot red, decay into crossed electric dipole along the $x$-axis and magnetic dipole along the $y$-axis. (b) Dependence $\alpha =D_{{\mathbf{M}}_y\to {\mathbf{M}}_x,{\mathbf{N}}_y}{D}_{{\mathbf{N}}_x\to {\mathbf{M}}_x,{\mathbf{M}}_x}-D_{{\mathbf{M}}_y\to {\mathbf{M}}_y,{\mathbf{N}}_x}{D}_{{\mathbf{N}}_x\to {\mathbf{M}}_y,{\mathbf{M}}_y}$ (solid black) and $\beta =D_{{\mathbf{M}}_y\to {\mathbf{M}}_y,{\mathbf{N}}_x}{D}_{{\mathbf{N}}_x\to {\mathbf{N}}_x,{\mathbf{N}}_x}-D_{{\mathbf{M}}_y\to {\mathbf{M}}_x,{\mathbf{N}}_y}{D}_{{\mathbf{N}}_x\to {\mathbf{N}}_y,{\mathbf{N}}_y}$ (dashed blue) coefficients on the pump wavelength ${\lambda }_p$ in the vacuum.