1. Introduction

The refractive index n is a key parameter describing the interaction between light and substance. For a natural medium, the Re(n) is positive. However, for a left-handed material (LHM), originally introduced by Veselogo [1] as a theoretical curiosity, the Re(n) is considered to be negative. Recently, the LHM has attracted the intense interest of researchers for a plenty of unique properties and valuable applications [2–7], especially after Smith et al [8,9] presented the first demonstration of the LHM at microwave frequencies.

Currently, increasing attention has been paid to the LHMs in the optical frequency band, and many researchers have proposed various the demonstrations. To date, there have been roughly three ways towards the realization of the LHMs in this band. The first one is to design one layer [10–13] or a few layers [14] of discrete resonator elements and incident waves are generally perpendicular to these layers. The negative indices of such LHMs were only limited in the infrared frequency band [10–12,14] and at the red end of the visible spectrum [13,15]. The second one is to utilize a metal-insulator-metal waveguide as a two-dimensional LHM [16]. In this manner, the negative refraction merely occurs in the blue-green region of the visible spectrum. The last one is a quantum method using either a single probe beam through multi-species media [17], or multiple beams through single species media [18]. This method is limited to theoretical research and has not been verified experimentally.

Below we present an approach to the realization of the LHM, a combination of the first two ways mentioned above. Namely, discrete rod resonators forming a sandwich structure lie in a metal waveguide. Our numerical simulations demonstrate that the negative-index band with high transmission ranges in the visible wavelengths. This approach opens new opportunities for designing negative-index materials in optics.

2. Structure model

As shown in Fig. 1(a), periodic discrete resonator elements, consisting of two metallic rods (24 nm thick Ag) separated by a dielectric layer (12 nm thick MgF₂ with the refractive index of 1.38), lie between the two same metallic plates (60 nm thick Ag). All the samples are located on a glass substrate (index n=1.5). This configuration can be fabricated by employing evaporation techniques and standard electron-beam lithography. Obviously, it can be viewed as a two-dimensional metal waveguide. Figure 2(b) exhibits one unit cell, and all dimensions are displayed in the caption under Fig. 1.

 

Fig. 1. (a) Schematic for a metal waveguide with array of H shape within it. (b) A unit cell with geometric dimensions L=600 nm, W=500 nm, a=120 nm, b=300 nm, g=40 nm, t=24 nm, and s=12 nm. The glass substrate is much thicker than the silver layer. The propagation of the polarized electromagnetic wave is along the z axis, and the electric field and the magnetic field are respectively in the x and y directions.

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3. Numerical calculations and discussion

An Electromagnetic Design System (Agilent Corp), based on a finite element method, a commercial electromagnetic mode solver, is employed in our simulations. Furthermore, the optical properties of silver are described by the Drude free electron model,

$\epsilon \left(\omega \right)=1-\frac{{\omega }_p^2}{{\omega }^2-i\omega \gamma },$

where ω_p is the plasma frequency and γ is the collision frequency. The ω_p and γ of silver are respectively 1.22×10¹⁶ and 9.0×10¹³ [19]. In our designed structure, the size of the unit cell ρ (i.e., 2t+s) is much smaller than an electromagnetic resonance wavelength λ, reaching ρ/λ<1/9. Thus, an effective medium model can be used in our research. Following the common method [20], to excite the negative magnetic response of resonators as well as the negative electric response of wires, the incident electromagnetic waves are polarized with the magnetic field parallel to the y axis (H‖y axis) and the electric field parallel to the x axis(E‖x axis). Consequently, the propagation direction (wave vector k) is along the z axis. The two wave ports are set in the planes perpendicular to the z axis. The symmetrical perfect magnetic boundaries (corresponding to the boundaries perpendicular to the y axes) is also applied.

Using the S-parameter retrieval methods [21–23], the complex refractive index n, wave impedance z, effective permittivity ε and effective permeability µ have been retrieved in Fig. 2. Our retrieved indices in Fig. 2(b) confirm the negative-index band ranging from 517 nm to 564 nm. As shown in Figs. 2(e), we find that, in this negative-index band, Re(ε)<0, while Re(µ)>0. Negative Re(n) is achieved throughout as the condition Γ=ε ₁ µ ₂+µ ₁ ε ₂<0 is satisfied [24,25,10], where ε=ε ₁+iε ₂ and µ=µ ₁+iµ ₂ (see Fig. 1(f)). Figures 3(d) and 3(e) show the strong electromagnetic resonances, occurring near the wavelengths of 545 nm. This resonance can be regarded as an optical LC circuit, with the silver rods providing the inductance L and the MgF₂ layer between the rods acting as capacitive elements C. Moreover, capacitance can also be generated between the silver rods and the waveguide wall. Maybe because of this capacitance the electromagnetic resonance occurs near the higher frequencies than those in Ref. [10,11,13,15].

 

Fig. 2. Fig. 2. (a) Simulated S parameters for the unit cell in Fig. 1(b); (b) retrieved refractive index; (c) retrieved impedance; (d) retrieved permittivity; (e) retrieved permeability. (f) Distributions of the condition Γ and the Re(n).

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Fig. 3. Distribution of FOM in the negative-index band in Fig. 2(b).

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It is to be noted that an LHM with high transmission is to be expected. In other words, its figure of merit (FOM=-Re(n)/Im(n)) should be as large as possible, i.e., the higher the FOM is, the higher the transmission is. From Fig. 3, we can find that the maximum of FOM arrives at 1.12 near the wavelength of 543.5 nm, which means the highest transmission occurs at this wavelength.

 

Fig. 4. (a) Simulated S parameters for the unit cell with geometric dimensions L=700 nm, W=640 nm, a=200 nm, b=400 nm, g=40nm, t=30 nm, and s=15 nm; (b) retrieved refractive index; (c) retrieved permittivity; (d) retrieved permeability.

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Fig. 5. istribution of FOM in the negative-index band in Fig. 4(b).

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In addition, we investigate the influence on the refractive index when the geometric dimensions of the unit cell are changed (the changed parameters are displayed beneath Fig. 4). From Fig. 4(b), we obtain that the negative-index band ranges between 660 nm and 770 nm. The strong electromagnetic resonance shifts toward the wavelengths of 725 nm. Meanwhile, figure 4(d) exhibits a negative permeability band at 717–728 nm, implying the smaller scattering loss than the above case [10]. This is verified in Fig. 5. Namely, the maximum of FOM equals 1.41 (higher than 1.12) near 721 nm. This FOM value is quite high. Importantly, it occurs in visible wavelengths and this wavelength is shortest thus far. It is well known that researchers have experimentally and theoretically confirmed the LHMs in the optical range.

However, their FOMs are low (often below 1) [10,11,13]. Although the LHMs were demonstrated at 1400 nm with an FOM of 3 [26], at 1800 nm with an FOM above 1[27], and at 813 nm with an FOM of 1.3 [15], they still range in the infrared wavelengths. Our additional large numbers of simulations testify that different negative-index bands with high FOMs in visible wavelengths can be obtained by tuning the geometric dimensions of the unit cell.

4. Conclusions

In conclusion, a type of metamaterial composed of a metal waveguide with discrete rod resonators, has been reported in our work. Simulating the S parameters of its unit cell, we have retrieved the complex refractive index n, wave impedance z, effective permittivity ε and effective permeability µ. Our retrieved results have shown that the maximum of FOM reaches 1.12 at the 543.5 nm. For another set of dimensions of the unit cell, the maximum of FOM arrives at 1.41 at the wavelength of 721 nm. This FOM value is quite high, indicating that this metamaterial possesses high transmission. Importantly, it occurs in visible wavelengths and this wavelength is shortest thus far. Our additional large numbers of simulations testify that different negative-index bands in visible wavelengths can be obtained by tuning the geometric dimensions of the unit cell. This approach is greatly effective to the realization of negative refraction at visible wavelengths.