As a consequent work of the two-dimensional (2D) dendritic metamaterial which has been systematically studied in our previous work, a model of three-dimensional (3D) sphere-rod shaped structure is presented. Based on Drude model of the dielectric function of silver in the visible region, the parametric curves of electromagnetic response to the incident fields have been retrieved from detailed simulations. It is shown that the simultaneously negative values of permittivity and permeability in the optical range lead to a negative refractive index (NIM) through adjusting structural parameters, only the dimensions of the unit cells satisfy the effective medium theory. We therefore conclude that the proposed model offers a feasible route to fabricating 3D optical NIMs by ‘bottom-up’ approach.
©2010 Optical Society of America
Artificially engineered negative-index metamaterials (NIMs) of which compositions and morphology can be rationally designed and tailored play an increasingly important role in the physics, optics, and engineering communities, owing to the potential applications [1,2] resulting from their extraordinary properties [3–5]. Artificial magnetism and negative refractive index are two specific types of behavior in this man-made materials which have been shifted from gigahertz frequencies all the way to terahertz frequencies over the past decade [6–8], and now approach the visible part of the spectrum [9,10]. The real realization of negative refraction for optical frequencies would generate revolutionary impacts on present-day optical technologies. However, there exist some difficulties in the procedure of fabrication and it is still a challenge expected to be overcome at present.
Up to now, a relatively typical and extensively studied system is the so-called fishnet structure consisting of two perforated metal films separated by a thin layer of dielectric [7,8,11], pushing negative refraction up to the red-light wavelengths. This structure is often fabricated with the ‘top-down’ approach such as electron-beam lithography  or focused ion-beam technology . According to the effective medium theory, the interior structure of the medium must be on a sub-wavelength scale. This requires that the structural unit of artificial metamaterials should be constructed on the nanometer scale when operating at visible wavelengths. However, in current time this ‘top-down’ method could hardly meet this small precision, and moreover it often needs expensive equipments and involves complicated processes. Over the past several years, our group has proposed a ‘bottom-up’ approach of double template-assisted electrochemical deposition to fabricate NIMs which is convenient to operate and also can satisfy the nano-scale precision. In our previous work, the two-dimensional (2D) isotropic dendritic NIM has been successfully fabricated by using the ‘bottom-up’ approach and the negative refraction has been experimentally demonstrated at infrared frequencies. We have performed explicit study on 2D dendritic NIM. Negative permeability and subwavelength focusing of quasi-periodic dendritic cell metamaterial were studied in . The dendritic structure with simultaneously negative permeability and permittivity has been verified [15–17]. The infrared dendritic NIM was successfully fabricated by means of double template-assisted electrochemical deposition . NIM composed of random dendritic cells displays a multi-band resonance, negative refractive index and a high intensity of the pass-band at infrared frequencies . Recently we have experimentally demonstrated the trapped rainbow in tapered negative-index heterostructures at visible frequencies .
In addition to realizing negative refraction at optical frequencies, designing and fabricating three-dimensional (3D) NIMs is another challenging task. Until now, few 3D models have been reported, the 3D fishnet structure in  which can achieve negative refraction in the infrared part of the spectrum has attracted extensive attention of researchers worldwide. But it is not perfect for its absence of isotropy and the complexity in the procedure of fabrication. In this paper, we design a 3D isotropic NIM composed of sphere-rod shaped cells. We have performed numerical simulations with adjusting its geometrical parameters to study its electromagnetic response to the applied electromagnetic fields. It is found that this structure can yield a negative refractive index in the visible region. This model is so highly symmetric about itself that it possesses the property of isotropy, hence it is more propitious for actual applications. Because of the simplicity of the 3D structure and the convenient ‘bottom-up’ approach of fabricating 2D dendrtic NIM mentioned above, we are now attempting to fabricate this sphere-rod shaped NIM by means of chemical synthesization. So the purpose of this paper is to present a new 3D model for NIMs and also afford the theoretic support for after ongoing experiment.
2. Numerical simulation
Silver is chosen for the presented structure, because of its significantly low losses in the visible range, as well as easy to be fabricated with chemical synthesization. Numerical simulations are performed with a 3D electromagnetic simulative software based on the finite-difference time-domain method. At lower frequencies such as the gigahertz range, the silver behaves as nearly perfect conductor. Whereas in the terahertz range, its dielectric function should be characterized by Drude model with plasma frequency ωp = 1.37 × 1016 s−1 and collision frequency ωc = 8.5 × 1013 s−1 . The electromagnetic radiation is set as the transverse electric and magnetic (TEM) mode as shown in Fig. 1 . Because the unit cells are periodically repeated in the 3D space, we simulate one unit cell by applying the perfect electric and magnetic boundary conditions. After obtaining the reflection (S11) and transmission (S21) coefficients, we first extract refractive index n and impedance z from the two parameters, then the effective permittivity ε and the effective permeability μ can be directly calculated from μ = nz and ε = n/z .
The structural unit and its geometric dimensions are depicted in Fig. 1. It is a 3D sphere-rod shaped structure with the rods distributing uniformly around the nucleus. There are four geometrical parameters (L, r, t, a) on the order of nanometer, characterizing the length of the rod originated from the central nucleus, the radius of the nucleus which is assumed to be spherical, the diameter of the cross section at the end of the rod, and the lattice constant, respectively.
3. Results and analysis
3.1 Parametric optimization
Recently, parametric study shows that the spectra of reflection and transmission depend strongly on the structural parameters of the periodic structures . Optimization of the shape and size of metamaterial is essential to get an appropriate structure with better electromagnetic performance. We have studied the dependence of the reflection and transmission spectra on the structural parameters. In this 3D model, among the four geometric parameters (L, r, t, a) is the radius of the nucleus r playing a primary role in affecting the electromagnetic response of the structure, this is because that r is directly associated with the overall morphology of the model. In simulation, we start with choosing appropriate parameters L, t, a, and then change r from zero to the value of the nucleus fully covering the rods. Figures 2(a) and 2(b) display the curves of the retrieved permeability of the structure composed of only the rods and only the nucleus, respectively. It can be seen that the real parts of the permeability in these two cases are positive over the entire spectral bands, it obviously cannot give rise to a negative refractive index. Next, we fix a proper value of r and respectively change other three parameters (L, t, a) in order to get a primitive model. At last synthetically considering the effects of the four parameters, and further adjusting them is to find out the expected structure exhibiting a negative index. This procedure is intricate and time-consuming, however, this paper is purely focused on displaying the negative refraction in the optical domain, so we only take a brief depiction on adjusting parameters in this paragraph.
3.2 Response of visible region
Based on the strategy of geometric optimization mentioned above, rationally adjusting the four parameters (L, r, t, a) can give rise to a negative refractive index from the red-light to the blue-light region. Owing to space limitations, the detailed and complicated adjustment of parameters are not displayed. Below, the spectra of electromagnetic response to the red-light and to the blue-light are displayed in Fig. 3 and Fig. 4 , respectively.
3.2.1 Negative refraction for red-light
In this case, the specific dimensions of the structure are as follows: L = 65 nm, r = 39 nm, t = 19 nm, a = 140 nm. In Fig. 3 (a), it can be seen that there is a transmission peak of S21 and a reflection trough of S11 at bout 436.3 THz. In the vicinity of this frequency, the real part of the refractive index is negative with simultaneously negative real parts of permeability and permittivity as shown in Figs. 3(b), 3(c), 3(d). The frequency band of the negative refractive index is within the red part of the spectrum (630-780nm). The ratio of the operation wavelength to the lattice constant is equal to 4.9, satisfying the effective medium theory.
3.2.2 Negative refraction for blue-light
The following numerical results for the structure with L = 50 nm, r = 39 nm, t = 18 nm, a = 110 nm. In Fig. 4(a), there is a transmission peak with a high value close to 0.6 and the corresponding reflection trough has a low value of less than 0.05 at about 608.6 THz. As the same of Fig. 3, in the neighboring area of this frequency all the real parts of permeability, permittivity and refractive index simultaneously have a negative value, as illustrated in Figs. 4(b), 4(c), 4(d). The ratio of the operation wavelength over the lattice constant is 4.5, so the system can be regarded as an effective medium. The negative refraction is achieved within the wavelength rang of blue-light (470-500nm).
As Fig. 3 and Fig. 4 have been shown, let us analyze the underlying physical mechanism. The electric response is attributed to the electric dipole oscillation. With regard to magnetic resonance, there are two arc currents exist between the two neighboring rods of either side of the structure, every arc current excites a magnetic fields (Fig. 5 ), which generates a magnetic field opposing the original field when its phase is reversed. This diamagnetic response yields a negative permeability, further combining the negative permittivity at the same frequencies leads to a negative refractive index. In the other hand, this sphere-rod shaped structure can be thought of as a 3D dendritic structure with a nucleus, its detailed theoretical analysis is similar to that of the 2D dendritic structure mentioned above.
Further observing Fig. 3 and Fig. 4, we note here that the spectral width of negative index is broader than that of the simultaneously negative permeability and permittivity in the lower band of the overlapped frequencies. This can be explained that designing an artificial negative-index metamaterial usually requires the real parts of the permeability (μ = μ' + iμ”) and the permittivity (ε = ε' + iε”) are simultaneously negative over a certain frequency band. But strictly speaking, it is not a necessary condition. There exists a more relaxed relationship P = μ'ε” + μ”ε' < 0 can also achieve a negative refractive index . So we calculated the value of P for the two cases as shown in Fig. 6 , it can be found that the wavelength range of the negative refractive index is perfectly corresponding to that of the negative P. The curves of the impedance of the two cases can also interpret this point, as depicted in Fig. 7 . The real part of impedance more than zero implies this area is a pass-band which agrees with the negative refraction pass-band, the other regions of zero value of impedance are opaque bands which correspond with the single negative value of the permeability or the permittivity.
In summary, we propose a new 3D optical NIM consisting of sphere-rod shaped cells. Under the condition that the ratio of lattice constant to operating wavelength meets the requirement of the effective medium theory, adjusting its geometrical parameters can make the structure give rise to a negative refractive index. By further scaling down and optimizing its geometric parameters, it is found that the response of negative refraction can be shifted from red-light to blue-light frequencies. The spectral band of the calculated negative refractive index is broader than that of simultaneously negative permeability and permittivity, this phenomenon can be explained by calculating the value of P. The model has the advantage of simplicity and isotropy, it is expected to be obtained from the ‘bottom-up’ approach of chemical synthesization, which also offers a feasible route to massive and low-cost fabrication of optical NIMs.
We acknowledge support from the National Natural Science Foundation of China under Grant No. 50632030, 50872113, 50936002.
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