Photon Pairs from Resonant Metasurfaces

All-dielectric optical metasurfaces are a workhorse in nano-optics, because of both their ability to manipulate light in different degrees of freedom and their excellent performance at light frequency conversion. Here, we demonstrate first-time generation of photon pairs via spontaneous parametric-down conversion in lithium niobate quantum optical metasurfaces with electric and magnetic Mie-like resonances at various wavelengths. By engineering the quantum optical metasurface, we tailor the photon-pair spectrum in a controlled way. Within a narrow bandwidth around the resonance, the rate of pair production is enhanced up to 2 orders of magnitude, compared to an unpatterned film of the same thickness and material. These results enable flat-optics sources of entangled photons—a new promising platform for quantum optics experiments.

: Sketch of one unit cell used in simulations.
To support our experimental results we performed numerical simulations using the finiteelement method software COMSOL Multiphysics. In our metasurfaces the nanoresonators were designed as nanocubes with side lengths/periods: QOM A 700/930 nm, QOM B 700/890 nm, QOM C 690/930 nm, QOM D 680/930 nm. But fabrication resulted in truncated pyramids on residual lithium niobate (LN) layer (see Fig. 1(b)). We modelled our metasurfaces as a periodic unit cell containing a LN pyramid and LN residual layer on the SiO 2 substrate as shown in Fig. S1. In Table 1 we provide the four model parameters used in simulations. Period is the period of the unit cell, Thickness is the thickness of the LN nanoresonator formed by LN pyramid and LN residual layer, Top length is the pyramid's top length and it's base length was equal to the period value, Residual layer thickness is the thickness of the LN residual layer under the pyramid. We measured the reflectance and transmittance spectra of our quantum optical metasurfaces (QOMs) using a custom-built white-light spectroscopy setup with collimated and linearly polarized excitation. Two Mie-type resonances appear in transmittance spectra (dotted lines) as dips and in reflectance spectra (solid lines) as peaks as can be seen in Fig. S2.
Their spectral positions are summarized for all QOMs in Table 2. Since the resonances at shorter wavelengths (λ ER ) are dominated by electric multipole components and the resonances at longer wavelengths by magnetic dipole contribution (as discussed below), we refer to them simply as electric resonances and magnetic resonances. In reflection, the electric resonances appear to be broader and flatter than in transmittance, which can be attributed to the scattering losses due to fabrication imperfections. Small fringes in the spectra are caused by Fabry-Pérot-like interferences in the setup.
As the linear spectra showed some constant background coming from scattering, in

Oblique incidence
In thin films photon pairs created via SPDC are generated within a broad angle. 1 In general, this also applies to our experiment but our optical system collects only pairs within the NA=0.14. However, for photons generated under an angle the resonances are effectively shifted. In Fig. S3 we plot experimental and simulated linear transmittance spectra for one sample metasurface QOM A where the polarization is kept along the LN optic axis. While at normal incidence in Fig. S3(a) we observe only two Mie-type resonances at 1591 nm and 1617 nm, with oblique incidence in Fig. S3(b,c) the situations drastically changes.
In case of excitation tilted in the xz-plane, Fig. S3(b), the electric resonance splits into two as clearly seen in Fig. S3(b)(ii) and becomes effectively wider. Additionally a new blue-shifted lattice resonance is appearing and moving towards shorter wavelengths with increasing angle of incidence. For the excitation tilted in the xy-plane, Fig. S3(c), the electric resonance moves to the shorter wavelengths by ≈10 nm for 2 • incidence angle. In case of QOM A where its position at normal incidence 1591 nm is red-shifted in relation to the degenerate photon-pair wavelength λ deg = 2 · λ p = 1576 nm, this means that the electric resonance moves closer to the degeneracy (at θ xy = 2 • it is at 1581 nm). This leads to more efficient sum-frequency generation (SFG) when signal and idler are incident under a small angle, as confirmed in the top panel of Fig. 4(b). Following the quantum-classical correspondence, SPDC for photon pairs emitted under a small angle should also be more efficient in QOM A. Figure S4: (a)-(c) Simulated transmittance spectra for QOMs B, C, D, respectively; for normal incidence (top row) and for the incidence angle 2 • in the xz-plane (θ xz , middle row) and in the xy-plane (θ xy , bottom row). Polarization along the LN optic axis, z-axis. The vertical dashed gray line indicates the degenerate photon-pair wavelength.
For other QOMs the tendency in the linear spectra is the same (see Fig. S4): under oblique incidence the electric resonance shifts to shorter wavelengths. But since for QOM B the electric resonance is very close to λ deg and for QOMs C and D the electric resonances are already blue-shifted with respect to λ deg , upon increasing the angle of incidence they will move even further away from the degenerate wavelength. Thus, in these metasurfaces SFG/SPDC will only decrease at larger angles of incidence/emission.
Measurements with oblique incidence were performed by rotating the sample around its y-axis with a manual rotation stage (Fig. S3(b)) and around its z-axis with a motorized rotation stage (Fig. S3(c)). The manual stage was less precise which led to the discrepancy between the experimental curves and simulations in Fig. S3(b). But the tendency is still clear. Additionally, we calculated the field enhancement 4 inside our QOMs as where V res is the volume of one LN nanoresonator, i = x, y, z and E i is the corresponding electric field component, and E 0 is the amplitude of the incident electric field. In

Correlation experiment in reflection
We pumped the nonlinear metasurfaces with a cw pig-tailed diode laser delivering up to ∼ 70 mW at 788 nm. The pump power was controlled using a half-wave plate (HWP) and a polarizing beam-splitter (PBS). We used another HWP to rotate the pump polarization.
The pump beam with a diameter of ∼ 2.5 mm was focused into the metasurfaces using a 90 The polarizer also served as an additional filter removing fluorescence.
A parabolic mirror identical to the one used for focusing and collection fed the radiation into the input facet of a 50:50 broadband single-mode fiber splitter (1550 nm ± 100 nm).
Due to the identity of the two parabolic mirrors, the NA of the fiber (0.14) also determined the collection angle of SPDC. The two outputs of the fiber splitter were sent to infrared superconducting nanowire single-photon detectors (SNSPD). We registered the arrival time differences between the two detectors using a Swabian Instruments time-tagger (not shown).

Simulations of sum-frequency generation
Simulations of sum-frequency generation were done in COMSOL Multiphysics using the undepleted pump approximation, and included three steps. The first two are linear simulations of electromagnetic field for a plane wave excitation at (1) the signal frequency ω s and (2) the corresponding idler frequency ω i = ω p − ω s , where ω p is the frequency of our pump laser.
Based on the electric fields from the first two steps, we calculated the nonlinear polarization (see Eq. 2) inside a LN nanoresonator (pyramid and residual layer, see Fig. S1) which in turn served as a source for the final SFG simulation. This algorithm was repeated by varying the signal frequency and setting the idler frequency accordingly. For the simulations of SFG with oblique incidence of signal and idler the excitation at the first step (signal study) was tilted by 2 • and at the second step (idler study) by −2 • , both either in xy or in xz planes (see sketches in Fig. S3). The components of the nonlinear polarization had the form 5 where E x,y,z (ω) were the components of the signal (ω s ) or idler (ω i ) electric field, and d 22 = 1.9 pm/V, d 31 = −3.2 pm/V, d 33 = 1 2 χ (2) zzz = −19.5 pm/V at 1313 nm. 6 The resulting SFG spectra were multiplied by the SPDC spectrum from the wafer (gray stars in Fig. 4(a)), to take into account the detectors sensitivity, and convolved with a Gaussian with FWHM 8.8 nm, to take into account the detectors timing jitter.