1. Introduction

Metamaterials are artificially composite materials with electromagnetic properties significantly different from their individual inclusions. The properties of metamaterials are determined by the composite structure, more than the components. In past decades, metamaterials have exhibited a series of exotic properties, e.g. optical magnetic response [1] and negative refractive index [2–5], and have inspired impressive potential applications including optical invisibility and superlens et al [1,6,7]. Recently, the possibility of localizing electromagnetic field in extremely small volume to obtain a huge local field enhancement in metamaterial has attracted considerable research attention. Consequently, a number of inherently weak optical effects can be dramatically enhanced for practical applications. One of the prominent examples is the second-harmonic generation (SHG) in nonlinear optical materials. As the SHG is governed by photon-photon interactions, it is superlinearly dependent on the electromagnetic field and thus can be strengthened in environment that provides mechanisms for field enhancement. Similar ideas have been successfully demonstrated in plasmonic nanostructures [8–10]. The conversion efficiency of SHG has been improved in metal apertures filling with LiNbO₃ or GaAs nonlinear materials [11–15], and nano-antennas arrays with nonlinear materials in the gap with high electromagnetic field [16–18]. Till now, the researches on the enhancement of optical nonlinearities are mostly focusing on the electrical resonances. The influence of optical magnetic resonance on SHG has not been well explored and utilized except a few works on Kerr effect [19]

Here we demonstrate the impact of magnetic resonance on the SHG by inserting a second-order nonlinear material (SNM) into a novel magnetism structure. The magnetic resonance constructs strong local field environment inside the SNM, and thus enhances the generation of the second-harmonic (SH) and its re-emission to far-field. Our results show that the magnetic resonance can be an additional way to improve the SHG in nanostructural metamaterials.

2. Results and discussions

While many structures have shown magnetic resonance, the metamaterial (also known as meta-magnetics) consisting of arrays of paired thin silver strips has intrinsic advantages to achieve large magnetic resonance, especially in the optical wavelength region [19]. In this research, we used a similar structure to obtain an initial design with a significant magnetic resonance. Fig. 1 shows the schematic diagram of the metamaterial. Each sandwich unit consists of a pair of thin silver strips separated by an alumina layer. The thicknesses of the silver strips and alumina layer are t and d. And the width of the sandwich stack and the period are w and p, respectively. Two thin alumina layers of 10 nm are added respectively on the top and bottom of sandwich stacks to increase the adhesion of the bottom silver layer and decrease the degradation in real experiment.

 

Fig. 1 Structure of the coupled silver strips.

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2.1. Numerical calculations

The numerical analysis is performed using a free FDTD software package, known as MEEP [20]. The cross-section of metamaterial and the definition of polarization in the simulation are presented in Fig. 2(a). As the lengths of the silver strips far exceed the dimensions of the cross-section, the magnetic structure could be calculated as a two-dimensional object. The paired thin silver strips are positioned in the center of the rectangular simulation domain and periodic boundary conditions have been applied in x-direction. As the sandwiched nanostrip structure only exhibits magnetic and electric resonances under TM illumination (H in z direction) [19], below we focus on the TM polarized light with incident wave vector normal to the surface of strips. The optical property of silver is described with experimental data reported by Johnson and Christy [21]. And the refractive indices of alumina and substrate are set as 1.62 and 1.52, respectively.

 

Fig. 2 (a) Diagram of the model; (b) Simulated transmission (T) and reflection (R) and absorption (A) spectra of the structure for TM polarization; (c), (d) Magnetic field and the electric displacement at the electric resonance and the magnetic resonance.

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The transmission, reflection and absorption spectra of the structure are plotted in Fig. 2(b). Here we set t = d = 30 nm, w = 170 nm, and p = 400 nm. Two resonant peaks at 615 nm and 1040 nm can be observed. The color maps in Figs. 2(c) and 2(d) show the magnetic field normalized magnitude at two resonances, while the arrows represent the electric displacement. We can see that the electric displacement at λ_e = 615 nm is aligned nearly along one direction, whereas the current flow forms a closed loop at λ_m = 1040 nm. We thus know that resonances at 615 nm and 1040 nm correspond to the electrical resonance and magnetic resonance, respectively.

Based on Maxwell equation, we know that the closed current loop gives rise to an artificial magnetic moment. Thus the strong resonant behavior at the magnetic resonance wavelength will naturally result in strong and inhomogeneous electromagnetic field inside the structure. Remarkably, most of the electromagnetic energy will be confined inside the gap, which induces a giant local field energy enhancement and magnifies the interaction between matter and electromagnetic wave.

To verify the influence of magnetic resonance, we have studied the dependence of local field enhancement factor on the wavelength. Here we define the field enhancement factor as equation (1) [22]:

 

Fig. 3 The dependence of local field enhancement factor of electric field and components on the wavelength.

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Since the optical nonlinearities are greatly dependent on the electromagnetic enviroment, it is expected to be strengthened by the local field enhancement in metamaterial. In our study, we consider the SHG process in nonlinear metamagnetic structure with a nonlinear material embeded in the gap between metal strips. The purpose of this study is not to optimize the SHG conversion efficiency but to demonstrate that the structure under study can be used to enhance optical nonlinearities in adjacent materials. For simplicity, in the simulation, the real part of refractive index of the second order nonlinear material is set as n = 1.62 and the imaginary part has been neglected. The nonlinearity was included by introducing nonlinear polarization P⁽²⁾ = ε₀χ⁽²⁾E² into the governing time-dependent equation. The second-order nonlinear susceptibility to be χ⁽²⁾ = 3×10⁻¹⁰ m/V, consistent with the values available, for example organic nonlinear optical crystal DAST [23,24].

In SHG simulation, the amplitude of the incident fundamental harmonic source is set as 10 MV/m. The center wavelength of the fundamental waves is set at the peak position of magnetic resonance and the pulse duration is 1000 femtoseconds. Figure 4 illustrates the normalized electric and magnetic field distributions of the SH signals. While the thickness and nonlinearity of the nonlinear material layer are extremely small, impressive SH signal can still be observed in the gap area, where highly enhanced electromagnetic fields are localized. Moreover, the spatial distribution of magnetic field in the gap changes from fundamental resonance (see Fig. 2(b)) to higher order resonance (see Fig. 4(b)). Thus we know that the enhanced local field inside the metamagnetic structure plays a key role in the SHG process.

 

Fig. 4 Simulated SH (520nm) electric (a) and magnetic (b) field distribution with arrow representing the electric displacement.

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To quantitatively verify the impact of magnetic resonance, we have evaluated the efficiency of SHG signal radiated to the transmitted far field as a function of the wavelength of fundamental wave. We calculate the second-harmonic efficiency by far-field second-harmonic radiated power divided by fundamental input power. The far-field second-harmonic radiated power is taken from the transmitted wave, and defined as the integral of the SH Poyinting vector normal to the exit. And the fundamental input power is defined as the integral of the FH input Poyinting vector at the incident. All the results are shown in Fig. 5. There is no any SHG from metamaterial structure without nonlinear material inside, and the signal is treated as noise background. The efficiency of SHG from a single nonlinear material layer shows less dependent on the wavelength and is almost four orders of magnitude higher than the noise. By placing the nonlinear layer on top of a silver layer with thickness of 30 nm, the conversion efficiency of SHG reduces significantly due to the intrinsic loss of metal at optical region. Once the nonlinear material layer is inserted inside the metamaterial, and the structure is excited under TM-polarization fundamental wave, the conversion efficiency of SH signal is further increased by almost four orders of magnitude. And the peak position matches very well with the magnetic resonance at 1040 nm. For comparison, the SH signal excited under TE polarization fundamental wave (Ag-NL-Ag_TE) has also been studied. We can see that the signal is much weaker than the nonlinear material layer itself. This coincides with the linear properties of the metamagnetic structure, which can be viewed as dilute metal under TE-polarization. As a conclusion, placing a nonlinear material within metamatierals with magnetic resonance can dramatically increase its nonlinear response.

 

Fig. 5 The SH conversion efficiency as the function of the wavelength of fundamental wave under different settings.

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Since most of nonlinear crystals are highly anisotropic, it is interesting to explore the alignment of the nonlinear crystal axis to the local electric field in order to achieve optimal design for SHG. For simplicity, we neglect the birefringence in the simulations and two settings are used. In one setting, the axis of symmetry of the crystal is parallel to x and nonlinear susceptibility in x axis is set as ${\chi }_{xx}^{(2)}=3\times {10}^{-10}\hspace{0.17em}\hspace{0.17em}\text{m}/\text{V}$, with zero in any other direction. In another setting, the nonlinear susceptibility in y axis is set as ${\chi }_{yy}^{(2)}=3\times {10}^{-10}\hspace{0.17em}\hspace{0.17em}\text{m}/\text{V}$ with zero in any other direction. The calculated results are shown in Fig. 6. We can see that both conditions exhibit apparent enhancements of SHG around the magnetic resonance. More interesting is that the enhancement factor of SHG signal from Ag-χ_yy-Ag with axis of symmetry of the nonlinear crystal vertical to the surface is around four, which is in the same order as the one with a isotropic nonlinear material in the gap demonstrated in Fig. 5. And the SHG signal for Ag-χ_xx-Ag with axis of symmetry of the crystal parallel to the surface is comparable to that for a single nonlinear material layer, much lower compared to Ag-χ_yy-Ag setting. This indicates that there is no enhancement effect under Ag-χ_xx-Ag setting.

 

Fig. 6 The SH conversion efficiency as the function of the wavelength of fundamental wave for Ag-χ_xx-Ag and Ag-χ_yy-Ag.

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The normalized electric field distributions for above simulations are presented in Fig. 7, where the arrows represent the electric field polarizations. Similar to the studies in Fig. 2, most of the fundamental wave at λ = 1040 nm is concentrated in the gap between silver strips and the electric polarization mainly aligns along y direction. Significant difference can be observed for the SH signals in these two settings. For the condition of Ag-χ_xx-Ag, the direction of the SH polarization in the gap is mostly aligned along x axis, but much smaller compared to that for Ag-χ_yy-Ag condition, which is paralleled to the y axis. As a result, the peak magnitude of the electric field of the SH signal in Ag-χ_xx-Ag setting is two orders smaller than that of Ag-χ_yy-Ag setting, consistent with the quantitative studies shown in Fig. 6.

 

Fig. 7 Electric field distribution and polarization of Ag-NL-Ag (a), Ag-χ_xx-Ag (b) and Ag-χ_yy-Ag (c).

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2.2. Discussions

We know that the second-order nonlinear polarization in the nonlinear material leads to the generation of SH. The second order nonlinear polarization depends quadratically on the electric field of fundamental wave and can be described as equation (2) [25]:

In our simulation, only χ_xx and χ_yy components are non-zero for the nonlinear materials doped in the gap under Ag-χ_xx-Ag, and Ag-χ_yy-Ag settings, respectively. As a result, the second-order nonlinear polarizations are reduced to be as equation (3).

The second-order nonlinear polarization has a component with twice the input frequency, and can be approximated as a source oscillating at doubled frequency. The frequency-doubled radiation wave can be described as equation (4).

As discussed above, the paired silver strips will obtain a strong local electric field along y axis under the excitation of normal incident fundamental harmonic wave with TM-polarization. Then it is obvious that enhanced local field with electric field along y direction will bring up strong second order polarization for the nonlinear material with non-zero nonlinear tensor component along y axis, the same with Ag-χ_yy-Ag setting. As a result, the SH signal in Ag-χ_yy-Ag setting is highly superiority to that in Ag-χ_xx-Ag setting, which accords with what is shown in Fig. 7.

3. Conclusions

In summary, we have investigated the wavelength dependent electromagnetic field enhancement in the gap between paired silver strips and studied its influences on the SHG conversion efficiency by embedding a nonlinear material in the highly local field area. Due to strong oscillatory currents induced in the paired silver strips at resonance, electric field enhancement factors is as high as 11, which has dramatically enhanced the SHG conversion efficiency by almost four orders of magnitude. In additional to the conventional studies on the electric resonance, the impacts of magnetic resonance has provided a further way to improve the generation of weak nonlinear optical effect such as SHG.