Nonlinear Interference and Unidirectional Wave Mixing in Metamaterials

When both electric and magnetic mechanisms contribute to a particular nonlinear optical process, there exists the possibility for nonlinear interference, often characterized by constructive or destructive interference in the radiation pattern of harmonics and mix waves. However, observation of a significant effect from nonlinear interference requires careful balancing of the various contributions. For this purpose, we propose an artificial metamaterial, using the formalism of nonlinear magnetoelectric coupling to simultaneously engineer the nonlinear polarization and magnetization. We confirm our predictions of nonlinear interference with both simulations and experiment, demonstrating unidirectional wave mixing in two microwave metamaterials. Our results point toward an ever wider range of nonlinear properties, in which nonlinear interference is just one of many potential applications.

Figure 1

Graphical illustration of the nonlinear parameter space. The insets depict nonlinear generation (arrows indicate electric fields) from a thin nonlinear slab for four limiting cases, corresponding to the areas under the colored cones. When the second-order polarization (magnetization) is dominant, as in the vertical (horizontal) cones, the nonlinear generation resembles an electric (magnetic) dipole. When a second-order polarization and magnetization are present, however, the interference can favor nonlinear generation in a particular direction, illustrated by the insets next to the diagonal cones. This is an oversimplification, however, as both the polarization and magnetization are, in general, complex valued.

Figure 2

(a) The nonlinear SRR proposed in Ref. 19 for observation of nonlinear interference. Both electric (left) and magnetic (right) incident fields interact with the nonlinear dielectric, resulting in nonlinear magnetoelectric coupling. (b) Photograph of the analogous VLSRRs from Ref. 27.

Figure 3

Second-order susceptibilities calculated for symmetric and antisymmetric nonlinear SRRs in Ref. 19 as a function of ${\omega }_i/{\omega }_p=1-{\omega }_s/{\omega }_p$, for ${\omega }_p$ fixed at the SRR resonance. The resonant susceptibilities (involving the magnetic component of the pump field) are dominant over most of the spectrum in each VLSRR, so that DFG is dominated by their interference, as in Eqs. (6, 8). The insets graphically illustrate the resulting nonlinear interference in each sample, with the optimum occurring in the neighborhood of ${\omega }_s\approx {\omega }_i\approx {\omega }_p/2$.

Figure 4

BPF (a) Schematic of the experimental setup for measuring DFG. (b) Plots of the forward and backward experimental DFG spectra generated by the symmetric VLSRRs (left) and the antisymmetric VLSRRs (right). The unidirectionality of the spectra confirm the presence of strong nonlinear interference, as illustrated in the insets. The solid lines show the results of simulations for comparison.