Artificial metal with effective plasma frequency in near-infrared region

Xingzhan Wei; Haofei Shi; Qiling Deng; Xiaochun Dong; Chunheng Liu; Yueguang Lu; Chunlei Du

doi:10.1364/OE.18.003370

1. Introduction

Surface plasmons (SPs), originating from the free electrons of the metal, are electromagnetic surface excitations localizing near the metal/dielectric interface [1]. Due to two key features, namely, local field enhancement and subwavelength resolution capability, SPs has attracted dramatic attentions recently and are known to play a great role in many domains, such as biochemical sensing [2], near-field microscopy and spectroscopy, superlens [3,4] and hyperlens [5] that enable optical imaging with unprecedented resolution, nanolithography [6] at deep subwavelength scale, and highly integrated optical devices and circuits [7,8]. However, SPs can only be excited and function at frequencies close to the intrinsic plasma frequency, which generally lies in the UV region for natural metals. At lower frequencies, especially in the far-infrared and microwave regions, the metals resemble the perfect conductor which has infinite conductivity and the local fields are only weakly confined in the dielectric, thus SPs are not supported. This restriction apparently limits the aforementioned SPs-based applications to a wider frequency regime. However, by texturing metal or even perfect-conductor surface with subwavelength holes, corrugations, or dimples, the surface wave modes were obtained which can mimic the properties of SPs [9,10].

Differently, we regard that it is more fascinating to construct a kind of artificial metal which can be used to engineer SPs at any frequency. The first relevant work was proposed by Pendry et al., in their work, the theory was specifically designed to deal with very thin wires with an effective plasma frequency shifted into the GHz range [11,12]. Since then a number of alternative simple analytic theories have been proposed [13–15], which are related with GHz or THz region. However, they all insufficiently consider the skin effect phenomenon, which is of considerable importance on the interaction of the electromagnetic wave with metal, and especially has considerable influence on the modulation of effective plasma frequency. In this paper, we demonstrate that the effective plasma frequency can be modulated into near-infrared region by a series of two-dimensional lattice structures, consisting of an array of circular metal rods whose diameters are not thin yet compared with skin depth. Thus, we must fully consider the skin effect and prove that only the effective active electrons near the rod surface (a little deeper than the skin depth) will work and play a great role in the modulation of effective plasma frequency. Accordingly, an analytical model is proposed by significatively introducing the parameter δ (namely skin depth), which is expected to be helpful both for clarifying the interior physical mechanisms and for providing more precise instructions in simulational and experimental investigation. This new artificial composite medium may open possibilities for many metal-based applications in near-infrared regime.

2. Artificial near-infrared metal

We study an artificial medium consisting of a rectangular array of circular metal rods embedded in a dielectric host, as shown in Fig. 1 . This structure was chosen because it is typical and assumed to be well understood. The rods are parallel with y direction, and are periodically arranged along x direction with a period p. The spatial extension of the layer in the z direction is assumed to be a. The waves illuminate the structure at normal incidence along z direction, with electric fields along the rods (TE mode). Then, the motion of electrons in the metal rods can take place, which creates a magnetic field circling the rod [11]. Vacuum is taken as the dielectric host first, and then, the influence of dielectric host on effective plasma frequency will be particularly considered in the latter discussion part. We carried out 3D Finite-Difference Fime-Domain (FDTD) calculations. A sufficiently large number of orders were retained in order to ensure convergence of all quantities derived from the calculation. As usual, the silver optical properties are described by the free-electron Drude model [16] with the parameters plasma frequency ω_pl = 1.37 × 10¹⁶s⁻¹ and collision frequency ω_col = 8.50 × 10¹³s⁻¹.

Fig. 1 Outline of the geometry under consideration. The basic geometrical parameters are rods period p, rod diameter d, and layer depth a.

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Calculated transmittance spectrums for TE mode as a function of the number of metal rod layers are shown in Fig. 2(a) , when d = 100nm, a = 300nm, and p = 600nm. The transmittance is nearly zero in the lower frequency region, and ascends rapidly at about 216THz, which is known as effective plasma frequency or cut-off frequency. When the number of functional layers N changes from 3 to 9, the effective plasma frequency will almost not shift. Thus, we can consider this medium as a high pass filter, that is to say, the electromagnetic wave whose frequency above cut-off frequency is allowed to propagate through the structure. Also, some small dips at higher frequency (above effective plasma frequency) can be observed, and we believe that this phenomenon is due to the F-P cavity effect, arising from multiple reflections in this medium. However, some minima in the transmittance spectrum occur at high frequency domain for TM polarization, as shown in Fig. 2(b). This phenomenon is due to electric and magnetic responses [17], resulting from the interaction of incident light with the periodical metal rods. Obviously, there is no effective plasma or cut-off frequency for this polarization.

Fig. 2 Transmittances for TE mode (a) and TM mode (b) versus different layer numbers (d = 100nm, a = 300nm, and p = 600nm). (c) Snapshot of electric field at three different frequencies, f = 200, 216, and 240THz, respectively. The wave vector is towards horizontal direction, and a period boundary condition along H direction was selected.

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In order to depict the response of our medium with electromagnetic wave vividly, we give the electric distribution patterns at three frequencies as shown in Fig. 2(c). When f = 200ΤΗz, the electric intensity falls off gradually with thickness, and becomes nearly zero in the right side. Therefore, no propagating mode can travel through this structure when the frequency is below effective plasma frequency, which is in well accordance with the transmittance curve depicted in Fig. 2(a). When f = 216ΤΗz, we can find that the effective wavelength in this material is especially large, accordingly there exists a near zero effective refractive index around effective plasma frequency as n_eff = λ_vaccum/λ_eff. When f = 240ΤΗz (above effective plasma frequency), we can find that the effective wavelength is about 2.7μm, and the relevant n_eff is around 0.46, which is in well accordance with the subsequent retrieved results in Fig. 3 (a) . Now, the electromagnetic waves are allowed to propagate through the structure. From above mentioned phenomenon, we can find that this structure can be regarded as an artificial metal, but the effective plasma frequency has been modulated into near-infrared region from original ultraviolet domain.

Fig. 3 Retrieved effective parameters of artificial structure with 3-layers rods. (a) effective refractive index, (b) effective impedance, (c) effective magnetic permeability, (d) effective electric permittivity. The real and imaginary parts of these complex parameters are shown as solid blue and dashed red curves, respectively. (d = 100nm, a = 300nm, and p = 600nm, layers number N = 3).

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We applied the well-known retrieval procedure [18] to calculate the effective refractive index n and the effective wave impedance z from the simulated reflection and transmission data. Then, the effective permittivity ε and permeability μ can be readily obtained in the light of ε = n/z and μ = nz. The retrieved results are depicted in Figs. 3(a)-(d). We can find that the real part of effective refractive index is zero when the frequency of incident light is smaller than effective plasma frequency. Thus, the very low transmittance T shown in Fig. 2(a) can be regarded mainly from the high reflection because of $R = (\frac{n - n_{v a c u u m}}{n + n_{v a c u u m}}) (\frac{n - n_{v a c u u m}}{n + n_{v a c u u m}}) * \approx 1$ . Also we can conclude that, when the frequency of incident light is above effective plasma frequency, optical responses are associated with a weak loss as Im(n) = 0, in addition the phase velocity exceeds the speeds of light in vacuum due to 0<Re(n)<1.

One noticeable feature in Fig. 2 (c) is that the electric field amplitudes of the rods center are always zeros at these three frequencies, namely the electrons in these locations cannot “see” the incident electromagnetic waves. Thus, the electromagnetic waves and these aforementioned electrons couldn’t interact, which is quite different from Pendry’s approximation [11]: all the electrons are uniformly distributed in the cross-section of the thin wire, and entirely attend the modulation of effective plasma frequency. It is no doubt that Pendry’s approximation is not suitable when the rod radius is larger than the skin depth. Therefore, it is very essential to introducing the skin effect. In order to better exhibit the influence of skin effect, we have performed corresponding calculations on the tubes array instead of above rods array as shown in the inset of Fig. 4 (a) . The outside diameter of tube is unchanged, and we gradually alter the tube thickness C, from below to greatly larger than the skin depth (about 22nm in near-infrared region) [19]. Figure 4(b) clearly shows that effective plasma frequency increases quickly when the thickness is below 20nm, whereas it nearly doesn’t vary when the thickness is bigger than 40nm. We confirm that this modulation of effective plasma frequency has great relationship with the skin effect. When the metal depth is bigger than the skin depth, only the effective electrons confined at a very thin surface (that is, a little deeper than the skin depth) take part in modulating the effective plasma frequency, and other electrons in the metal cannot “see” the electromagnetic waves and have little effect.

Fig. 4 (a) Transmittances as a function of the tube thickness C. The cross-section of tube is depicted in the inset figure, geometrical parameters used in this simulation are: invariable outside diameter of tube (200nm), layer depth a = 300nm, tube period p = 600nm, and layers number N = 3. (b) Effective plasma frequencies versus tube thickness C by retrieving the calculated transmission and reflection data.

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3. Analytical model

We have found that the modulation of effective plasma frequency is greatly influenced by the skin effect. Next, we will derive an approximate analytic formula of effective plasma frequency by considering the skin effect, and the FDTD simulation results are also provided as a criterion to verify this new model. As is well known, the plasma angular frequency ω_p is given in terms of the effective active electron density $n'_{eff}$ , the effective electron mass $m_{eff}$ and charge e.

ω_{p}^{2} = \frac{n'_{eff} e^{2}}{ε_{0} m_{eff}}

When the skin effect is considered, supposing

n'

is the actual conduction electron density in this metal, the amount of the active electrons per unit length of the wire can be immediately expressed to be

N' = n' \int_{0}^{R} \exp^{- (R - r) / δ} 2 π r d r = n' δ 2 π (R - δ + δ \cdot \exp^{(- R / δ)}) = n' δ 2 π r_{eff}^{} = n' δ l_{eff}

Here R is rods radius, δ is the skin depth,

r_{eff}^{}

and

l_{eff}

are effective radius and perimeter, respectively. Then, the effective active electron density can be written as

n'_{eff} = \frac{n' δ l_{eff}}{a p}

From Ampere’s circuital law, here the rectangle approximately is regarded as a circle with the same area (

a p = b^{2}

), and the vector magnetic potential A at a given point outside the wire is [12]:

A (R') = \frac{μ_{0} δ l_{eff} n' e v}{2 π} [\ln (\frac{R' \sqrt{π}}{b}) - \frac{π R'^{2}}{2 b^{2}} + \frac{1}{2}]

Where

R'

is the distance between wire center and above interesting point. Therefore, with the same principle as in Pendry’s work [11], the momentum per unit length of the rod is

P = δ l_{eff} n' e A (r_{eff}) = δ l_{eff} n' e \frac{μ_{0} δ l_{eff} n' e v}{2 π} [\ln (\frac{r_{eff} \sqrt{π}}{b}) - \frac{π r_{eff}^{2}}{2 b^{2}} + \frac{1}{2}] = m_{eff} δ l_{eff} n' v

then the effective electrons mass

m_{eff}

is

m_{eff} = \frac{μ_{0} l_{eff} δ n' e^{2}}{2 π} [\ln (\frac{l_{eff}}{2 \sqrt{π} b}) - \frac{l_{eff}^{2}}{8 π b^{2}} + \frac{1}{2}]

Bringing Eq. (2) and (5) into (1), and we can finally yield

f_{p}^{2} = {(\frac{ω_{p}}{2 π})}^{2} = \frac{c_{0}^{2}}{2 π b^{2} [\ln (\frac{l_{eff}}{2 \sqrt{π} b}) - \frac{l_{eff}^{2}}{8 π b^{2}} + \frac{1}{2}]}

Note that the skin depth, acts as a crucial factor in the modulation of effective plasma frequency, has been involved in the effective perimeter

l_{eff}

. In Fig. 5 , the effective plasma frequency values derived by the combination of FDTD simulation and standard S-parameter retrieval method, are provided and compared with the predicting results calculated by our and Pendry’s model [11]. Our model results are in well accordance with the simulation results (discrepancy is less than 3%), thus it certainly confirms that our formula is very trustworthy in predicting effective plasma frequency.

Fig. 5 Comparison of effective plasma frequencies yielded by FDTD simulation, our analytic model and Pendry’s model. Purple solid line represents the simulation results, cyan dash line shows our model results, and pink dot line represents Pendry’s model results. (a = 300nm, P = 600nm layers number N = 3).

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4. Discussion

As is well known, metamaterials exhibit their properties through sub-wavelength structures rather than chemical and physical composition. Here, we demonstrate two dominant geometrical parameters for modulating the effective plasma frequency, namely, the rod period p and rod diameter d. Figure 6 shows the transmittances as functions of different rod periods, and the inserted picture displays the corresponding effective plasma frequency variation. We can find that effective plasma frequency red shifts as p increases. This phenomenon can be well explained by the effects of reducing the active effective electrons density and increasing their effective mass [11].

Fig. 6 Transmittances for different rod periods, when d = 100nm, a = 300nm and layers number N = 3. Inset graph shows effective plasma frequencies versus rod period p.

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The diameter of silver rod is an equally important parameter to modulate effective plasma frequency. Figure 7 shows the transmittances of this artificial medium versus different rod diameters. When the diameter becomes bigger, a blue-shift of effective plasma frequency occurs as expected, which can be attributed to the increase of effective electrons density and the decrease of effective electrons mass.

Fig. 7 Transmittances for different rods diameters, when a = 300nm, p = 600nm and layers number N = 3. In the inset, we show effective plasma frequencies versus rods diameter.

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In our simulation and analytic model, the circular rods were chosen for simplicity. Actually, many other shapes, such as rectangle and ellipse, even arbitrary shape are all suitable for constructing this artificial metal. Besides, we just focus on the near-infrared region in this paper, actually our model can be applicable in other spectrum range, for instance, GHz region [20], and here we will not give more discussions. Considering the realistic fabrication, vacuum cannot be employed as the dielectric host, instead, other materials such as SiO₂ and MgF₂ can well act as the filling layer. Accordingly, a modulation factor ε_r called the relative permittivity is added. The ultimate expression of effective plasma frequency can be expressed as

f_{p}^{2} = \frac{c_{0}^{2}}{2 π b^{2} ε_{r} [\ln (\frac{l_{eff}}{2 \sqrt{π} b}) - \frac{l_{eff}^{2}}{8 π b^{2}} + \frac{1}{2}]}

In Fig. 8 , the simulated results are provided as the criterion and compared with the results calculated with Eq. (8). The relative permittivity of SiO₂ and MgF₂ is 2.31 and 1.9, respectively. We can find that the calculated results by our model are in well accordance with simulation. As for realistic manufacture, we confirm it is not difficult to meet the requirement of critical dimension and fabrication uniformity. These artificial structures can be realized by many different fabrication techniques, for example, standard electron-beam lithography, laser beam interference lithography and our localized surface plasmon nanolithography [21], which have to be combined with lift-off and other assistant procedures. Relevant experimental work is currently underway in an attempt to verify our theoretical predictions for the detailed artificial structures.

Fig. 8 The effective plasma frequencies as a function of rod diameters as obtained in simulation and model with realistic dielectric host. The solid and dash lines represent the simulation and model results, respectively. (a = 300nm, p = 600nm and layers number N = 3).

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In our above simulation, two-dimensional rods array was proposed and demonstrated by FDTD simulation to modulate effective plasma frequency into near-infrared spectrum region. Note that only the modes whose electric fields are parallel to the rods will play a great role in the modulation of effective plasma frequency. When electric field is perpendicular to the axis of the rods, there is no effective plasma frequency. Actually, we have only considered the electric field is parallel to the metal rods, and this artificial medium is anisotropic. However, this medium can be made to represent an approximately isotropic response by considering a lattice of rods oriented along the three orthogonal directions.

5.Conclusions

In conclusion, a kind of artificial metal with effective plasma frequency in near-infrared region has been presented by a series of structures consisting of an array of circular metal rods in a two-dimensional lattice. The influence of relevant geometrical parameters on the modulation of effective plasma frequency has been shown. It is important to underline that the skin effect is of considerable importance in the design of this metamaterials, but ignored in previous work. Thus, the theoretic model has been constructed by considering the skin effect, which is helpful both for clarifying the interior physical mechanisms and for providing more precise instructions in simulation and experimental investigation. The fascinating electrodynamics effects anticipated for such metamaterial are expected to be investigated further.

Acknowledgment

This work was supported by the National Basic Research Program (2006CB302900) and High Tech. Program of China (2007AA03Z332). The authors thank Dr. Shaoyun Yin and Ms. Lifang Shi for their kind contributions to the work.

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Artificial metal with effective plasma frequency in near-infrared region

Abstract

1. Introduction

2. Artificial near-infrared metal

3. Analytical model

4. Discussion

5.Conclusions

Acknowledgment

References and links

Cited By

Figures (8)

Equations (8)

Optics Express