Retrieval of all effective susceptibilities in nonlinear metamaterials

Electromagnetic metamaterials offer a great avenue to engineer and amplify the nonlinear response of materials. Their electric, magnetic, and magnetoelectric linear and nonlinear response are related to their structure, providing unprecedented liberty to control those properties. Both the linear and the nonlinear properties of metamaterials are typically anisotropic. While the methods to retrieve the effective linear properties are well established, existing nonlinear retrieval methods have serious limitations. In this work, we generalize a nonlinear transfer matrix approach to account for all nonlinear susceptibility terms and show how to use this approach to retrieve all effective nonlinear susceptibilities of metamaterial elements. The approach is demonstrated using sum frequency generation, but can be applied to other second-order or higher-order processes.

Figure 1

The system considered consists of a series of slabs of potentially bianisotropic materials in which we need to determine the modes and their propagation constants. The propagation constant of the applied wave (${\mathit{k}}_1$ here) sets the in-plane propagation constants $k_x$ and $k_y$ (jointly indicated by $k_{x,y}$), which are invariant throughout the whole system. Inside every slab, there are four modes with potentially different propagation constants. Since $k_x$ and $k_y$ are invariant, only $k_z$ varies for each mode and can be determined using the approach described in the text. In medium 2, for example, we want to determine the $z$ component of the propagation constant of the four modes $k_{2,1,z}$, $k_{2,2,z}$, $k_{2,3,z}$, and $k_{2,4,z}$. The determination must be performed independently for each medium.

Figure 2

The VLSRR considered in the example. The unit cell is cubic with a side of $1\text{cm}$. The substrate is FR4 with a thickness of $254\mu \text{m}$, covered with $17\mu \text{m}$ of copper; the external dimension of the ring is $9.2\text{mm}$ and its linewidth is $0.5\text{mm}$. Varactor diodes are inserted in the $1\text{mm}$ gaps (see text for details). The propagation direction is $z$ while the metamaterial is considered infinite in the $x$ and $y$ directions.

Figure 3

Linear properties of the VLSRR of Fig. 2. The real part (continuous lines) and imaginary part (dashed lines) of the permittivity (top) and permeability (bottom) are all diagonal.

Figure 4

Norm of the second-order nonlinear susceptibilities of the VLSRR. To put all values on the same scale, nonlinear susceptibilities involving magnetic terms are normalized using the impedance of vacuum $Z_0$. For each type of nonlinearity, the results are presented in three slices, corresponding to the orientation of the field at $f_3$. In each slice, columns and rows correspond to the orientation of the fields at $f_1$ and $f_2$, respectively. All subplots share the same axes.

Figure 5

Typical retrieval validation results for a material with the linear properties presented in Table 3 and randomly selected nonlinear susceptibility tensors. Circles represent the normalized nonlinear susceptibilities imposed in the simulation, while crosses represent the normalized nonlinear susceptibilities retrieved from the simulation results. Each pair of a circle and a cross represents one of the 216 nonlinear susceptibilities. The retrieval would be perfect if all pairs overlapped.