Rational design of metallic nanocavities for resonantly enhanced four-wave mixing

Optimizing the shape of nanostructures and nano-antennas for specific optical properties has evolved to be a very fruitful activity. With modern fabrication tools a large variety of possibilities is available for shaping both nanoparticles and nanocavities; in particular nanocavities in thin metal films have emerged as attractive candidates for new metamaterials and strong linear and nonlinear optical systems. Here we rationally design metallic nanocavities to boost their Four-Wave Mixing response by resonating the optical plasmonic resonances with the incoming and generated beams. The linear and nonlinear optical responses as well as the propagation of the electric fields inside the cavities are derived from the solution of Maxwell’s equations by using the 3D finite-differences time domain method. The observed conversion-efficiency of near-infrared to visible light equals or surpasses that of BBO of equivalent thickness. Implications to further optimization for efficient and broadband ultrathin nonlinear optical materials are discussed.

light source. The linear measurements, however, are based on white light illumination with a source that is neither spatially nor temporally coherent. Furthermore, the calculations are performed on a single cavity, with periodic boundary conditions, which implies a perfect sample where all cavities are identical, obviously unachievable in experimental reality. To test the applicability of these approximations, we repeated some of the calculations for spatially incoherent light, and for nonuniform samples. The light source is simulated as follows: The light source is divided into 12x12 plane wave sources each one spans 2.1 µm in both x and y directions, each centered on a different hole and having a random phase. Sample nonuniformity is simulated by performing the calculation on an array with 12x12 holes. The periodicity of the square array is maintained at 510 nm, but the dimensions of each hole are allowed to randomly vary (with Gaussian distribution) within 7 nm in x and y around the nominal size of 290x138 nm 2 (AR = 2.1).

a)
The experimental transmittance of an array of rectangles + Melles Griot 03SWP412). The signal is then coupled to a multimode optical fiber and sent to the spectrometer+CCD system.

Supplementary discussion 1: Nonlinear coupled mode theory simulations
The energy exchange between the longitudinal modes inside the cavities is discussed within the nonlinear coupled mode theory. The model provides a semi quantitative analysis and captures the essence of the underlying mechanism leading to the enhancement of the FWM response. The propagating waves are assumed to be stronger inside the cavity than on its surface so that most of the FWM signal is generated inside the nanoholes, as is seen by the numerical solutions of the wave equations. As the two longitudinal modes propagate inside the nanoholes, they interact with the metal and exchange energy with the generated FWM mode. The observed/measured forward FWM signal is taken as the integrated intensity of the FWM modes at the cavity exit.
The electric field of the FWM mode inside a cavity obeys the wave equation (1) where the nonlinear polarization is given by eq. (1). The fields propagating inside the cavity can be expressed in terms of the dominant transverse lowest-order mode , , , with (j=1, 2 , FWM), and are the amplitude of the fields at z = 0 and is a slowly varying function of z. The propagation constants , , , and can be obtained from the 3D-FDTD calculations. As the incoming fields are polarized along the y axis and hence the fields inside the cavities are mostly polarized in the same axis (according to the FDTD calculations), the tensor product in eq. 1 can be turned into scalar (the diagonal term of the susceptibility tensor is much larger than the off-diagonal terms). Therefore, eqs.
(2) and (1) can be plugged in eq. (1) in the main text, assuming that the is called the spatial overlap factor and it is a degree of transverse mode matching between the longitudinal modes. In eq. (4), the integrations are performed on the cavity walls, where the fields are much stronger than anywhere else inside the metal. The phase matching condition ∆ 2 accounts for the momentum conservation of the FWM process.
The intensity of FWM signal is given by the spatial integration of the intensity of the FWM mode at the cavity exit (z = l) With the analytical approximation, the FWM process in the nanocavities is now explicitly divided into the several factors that are individually easy to understand. These factors, calculated for the different ARs using the FDTD method, are shown in the figure below. The differences in the propagation constants of the three longitudinal modes (parts a,b) causes a phase mismatch (part c) as the modes propagate inside the cavities. Combined with the attenuation of the FWM mode, it decreases the FWM efficiency. There is no explicit phase-matching resonance for a particular AR. Equally, the transverse mode profile are similar for the different AR and the spatial overlap factor (part d) weakly grows with the AR.
The analytic calculations assumed, for simplicity, that on the average the fields propagate and decay exponentially inside the nanocavities, but this is not fully supported by the numerical calculations. At the FP resonance, for example, the field decays to a minimum value and then grows again. It seems, however, that due to the phase matching, the use of an average value for the field still represents the dependence of the FWM signal on the cavity parameters. , where T and R are the calculated transmittance and reflectance respectively, and l is the cavity length (film thickness).
In conclusion, it is seen that the shape resonance at 800 nm, due to the fourth power dependence on the intensity of this field, is the dominant factor for the FWM field enhancement at AR = 2.
Theoretical results of the FWM shape resonance based on coupled-mode theory. Real (a) and imaginary (b