Experimental determination of the quadratic nonlinear magnetic susceptibility of a varactor-loaded split ring resonator metamaterial

Abstract

This letter presents a quantitative measurement of the second harmonic generated by a slab of varactor loaded split ring resonator metamaterial and the retrieval of the effective quadratic nonlinear magnetic susceptibility m ͑2͒ using an approach based on transfer matrices. The retrieved value of m ͑2͒ is in excellent agreement with that predicted by an analytical effective medium theory model. © 2010 American Institute of Physics. ͓doi:10.1063/1.3460919͔Metamaterials ͑MMs͒ are subwavelength, circuit-like elements whose electromagnetic response can be engineered by design. Near resonance, many MM elements, such as split ring resonators ͑SRRs͒, exhibit a significant concentration of the electric field within the capacitive regions, potentially enhancing the nonlinear properties of materials integrated into those regions. 1 Many nonlinear phenomena such as wave-mixing, 2 bistability, 3 and parametric generation 4 have been predicted in such MMs but implementations are still scarce. 5,6 At radio or microwave frequencies, the nonlinear material can be substituted by nonlinear electronic components. For instance, introducing a varactor into the gap of a SRR yields a MM element with nonlinear magnetizability, which, combined with like SRRs, produces a nonlinear magnetic MM. 7,8 To date, experimental studies of nonlinear MMs remain mostly qualitative. However, for applications, it is essential to characterize their nonlinearity precisely. Specifically, we want to characterize nonlinear MMs in terms of a series expansion of their nonlinear susceptibility, as a host of well known nonlinear phenomena connect to specific terms of this expansion and it allows direct comparison with conventional materials.Recently, Larouche and Smith have presented a retrieval method which uses the harmonics generated by a slab of a nonlinear medium to retrieve its effective nonlinear susceptibilities. 9 Here, we apply the method to retrieve the quadratic nonlinear magnetic susceptibility m ͑2͒ from quantitative measurements of the second harmonic generated by a varactor-loaded SRR ͑VLSRR͒ MM. The retrieved nonlinear susceptibility is then compared with the predictions of an analytical theoretical model proposed by Poutrina et al. 10 To study second harmonic generation ͑SHG͒, we reuse the VLSRR MM previously used to study the powerdependant resonance frequency shift, 11 which is related to the cubic nonlinear susceptibility. Each unit cell of this MM consists of a 17 m thick copper ring on a 0.2 mm thick FR4 PCB substrate. The ring has an internal radius of 4 mm, is 0.5 mm wide, and possesses a 1 mm gap. The gap is loaded with a Skyworks SMV1231 varactor, whose capacitance varies according towhere V D is the bias voltage, C 0 = 2.4 pF is the zero bias capacitance, V P = 1.5 V is the intrinsic potential, and M = 0.8 is the gradient coefficient. 12 The VLSRRs are arranged periodically to form a slab of 3 ϫ 15 cubic unit cells, 10 mm on a side. The slab is one unit cell in length in the direction of propagation. The unit cells are much smaller ...

Wave mixing in nonlinear magnetic metacrystal

“…The difference in the strengths of the two SHG signals results from the frequency-dispersive second order nonlinear susceptibilities which determine the conversion efficiency, described in detail in Ref. 8. Four-wave mixing signals 2f n − f m and 2f m − f n are also present in the spectrum.…”

“…Only 49% of the exciting power remains in the effective regime exciting the nonlinear metacrystal sample due to open-end design of the apparatus. 8 The transformation from power to the magnetic field intensity is taken from H = ͱ 2P / Z 0 S, where P is the power, Z 0 = 377 ⍀ is the free space impedance and S =18 cm 2 is the cross-sectional area of the transmission line apparatus.…”

Extreme stiffness tunability through the excitation of nonlinear defect modes

“…Although our study has focused on the effect of localized excitations on mechanical properties, we expect an analogous phenomenon to exist in electromagnetic systems, such as varactor loaded split ring resonator arrays (VLSRR). This is due to the fact that those systems present quadratic [33] and cubic [34] nonlinearities as well as a dependence of resonances on an external static bias [35].…”

Demonstration of nonlinear magnetoelectric coupling in metamaterials

“…These samples are placed, one at a time, into the impedance-matched transmission line appartus of Ref. 21, depicted in Fig. 3(a).…”