## Abstract

Electromagnetic resonance as the most important characteristic of metamaterials enables lots of exotic phenomena, such as invisible, negative refraction, man-made magnetism, etc. Conventional LC-resonance circuit model as the most authoritative and classic model is good at explaining and predicting the fundamental resonance wavelength of a metamaterial, while feels hard for high-order resonances, especially for resonance intensity (strength of resonance, determining on the performance and efficiency of metamaterial-based devices). In present work, via an easy-to-understand mass-spring model, we present a different and comprehensive insight for the resonance mechanism of metamaterials, through which both the resonance wavelengths (including the fundamental and high-order resonance wavelengths) and resonance intensities of metamaterials can be better understood. This developed theory has been well verified by different-material and different-structure resonators. This perspective will provide a broader space for exploring novel optical devices based on metamaterials (or metasurfaces).

© 2015 Optical Society of America

## 1. Introduction

In metamaterials, resonance can bend light, slow light, absorb light, and produce anomalous refraction or reflection [1, 2
], subwavelength-focusing [3], cloaking [4], wavefront-shaping [2, 5–9
], perfect absorber [10–15
]. In fact, resonance can also appear in conventional bulk materials, such as Ag, SiO_{2}, GaAs and CaCO_{3}, at ultraviolet wavelengths; however, which are too weak and hardly tunable, and not suitable for optical devices, especially for tunable optical devices. Metamaterials as manmade resonators, both the structure and material of which are designable, can exhibit a prominent resonance at any position of whole electromagnetic spectrum. As such, more and more high-performance and different-band optical devices and components, such as super lens [16–18
], super-compact phase shifter [19], ultra-thin absorber [20, 21
] and high-speed modulators [22, 23
], gradually appear in real life.

Here, we should bear in mind, the resonance wavelength and resonance intensity as the two most important characteristics of a metamaterial, radically determining the work band and performance of metamaterial-based devices, respectively. However, they are seldom systematically and comprehensively investigated by current reports. For example, LC-resonance circuit model was once thought to be the most effective model for understanding the working mechanism of metamaterials; moreover, so far it is still the most widely used model. Nevertheless, this model can only explain the fundamental resonance frequency, not predict high-order resonances often appearing in simulated and experimental results [14, 24–39 ], also can’t estimate the relative resonance intensities for different metamaterials. Similarly, in metamaterials, Mie resonance model [27] is often used to describe the high-order resonance behaviours; plasmonic resonance model [29] is good at explain the resonance frequencies, but not suit for predicting the resonance intensities of metamaterials; electrodynamics model [37] can perfectly interpret the multi-resonance effect of metamaterials; Babinet’s principle [38] is often used to predict resonance frequencies of complementary structure; and high order harmonic model [32, 35, 40 ] can only well explain the high-order resonance frequencies. In addition, to date, few reports clearly and specifically describe the occurrence regularity of resonances in a metamaterial; and more importantly, the resonance intensity of a metamaterial, determining the performances and efficiencies of metamaterial-based devices, the mechanism of which seems to be too complexity to attract researchers’ attention to explore so far.

Obviously, a simple, systematical and comprehensive resonance mechanism of metamaterials will be significant for the whole metamaterial research. In fact, the mass-spring resonance model as a well-known and classical physical model can explain most of resonance phenomena, such as the resonance motion of a simple pendulum as well as molecular vibration. And in metamaterials, for example, via this model, Kats *et al* [41] have accurately captured the near- and far-field spectral features of linear optical antennas; Genevet *et al* [42] have investigated the two orthogonal resonance modes of V-shaped antennas. However, as far as we know, no one used this model to systemically analysis the resonance frequencies and resonance intensities of metamaterials.

In this paper, we firstly discuss the fundamental principle of resonance in metamaterials in detail, and propose a modified mass-spring resonance model for metamaterials, which can predict not only the fundamental resonance frequency and high-order resonance frequencies, but also the resonance intensity of metamaterials. To verify this modified model, we investigate the resonance characteristics of several classical different-structure and different-material resonators (or antennas). Results show that, resonance wavelengths (including fundamental and high-order resonance wavelengths) are approximately proportional to electrons’ effective path length (EPL); and resonance intensities are largely determined by the electric potential differences induced by incident waves and the density of electron participating in resonance motions in unit cells. In these validation procedures, we also find the resonance intensities of 2D metamaterials that made from doped semiconductors are usually lower than the resonance intensities of that made from metals, while semiconductor metamaterials can work as an intensity-tunable metamaterials distinctively different from common frequency-tunable metamaterials. We believe this easy-to-understanding resonance mechanism presented in this work may inspire a new insight in the design and optimization of metamaterial structures, and will be helpful for more-quickly exploring high-performance metamaterial-based devices.

## 2. Description of resonances in metamaterials

Here, in order give a more comprehensive understanding for resonance behaviors in metamaterials, we firstly consider a simple and common metamaterial structure, such as a rectangular metallic antenna, as shown Fig. 1(a)
. This type of antenna appears in lots of applications, such as focusing light from quantum cascade lasers [43], SEIRA spectroscopy [44], and infrared photo detectors [45]. When a normally incident wave with frequency *ω* impinges on the antenna with length *l*, assuming an electrical potential difference ∆*U* forms in the antenna, and is given by

*E*

_{c}is the coupling electric field proportional to external incident electric field

*E*

_{i}(that is

*E*

_{c}∝

*E*

_{i}) [41, 46 ],

*θ*is the polarization angle between polarization direction and antenna axis in x-y plane, as shown in Fig. 1(a). Due to the incident electric

*E*is a harmonic electric field, that is

*E*

_{i}=

*E*

_{0}

*e*

^{iωt}, according to Eq. (1), ∆

*U*will also be a harmonic potential difference. Consequently, under this variational potential difference ∆

*U*, electrons will make a reciprocating motion along the axis of the antenna. Actually, this reciprocating motion of electrons is analogy to the resonance motion of a spring vibrator [8, 41, 42, 47–49 ], as shown in Fig. 1(b).

Assuming a charge *q* with effective mass *m* locates at *x*(*ω*, *t*), we have

*k*is the spring constant;

*γ*is damp coefficient. Assuming harmonic motion

*x*(

*ω*,

*t*) =

*x*

_{0}e^{i}

*and solving Eq. (2), we get*

^{ωt}*x*

_{0}(

*ω*) contains the amplitude and phase of the oscillator and resonance frequency

*ω*

_{0}= (

*k*/

*m*)^(1/2).

As we know, the condition for resonance is that, the phase of an oscillator after a roundtrip should be equal to its initial phase, such that oscillator’s resonance motion will be reinforced. In a roundtrip, assuming an electron’s distance travelled is 2*s*, where *s* denotes the electron’s effective path length (or EPL for short), and its average velocity is *v*
_{F} assumed to be an approximate constant [50–52
], as illustrated in Fig. 1(c). Then, the resonance frequency *f*
_{0} of the oscillator can be expressed by

*λ*

_{0}denotes the resonance wavelength of the antenna and

*c*is velocity of light in vacuum. From Eq. (4), we can readily get that the resonance wavelength

*λ*

_{0}of the oscillators is directly proportional to the electron’s EPL, that is

Note that, for a metallic structure with the size larger than tens nanometers, electron’s maximum EPL (*s*) is usually proportional to its size *l* [50–52
], that is *s* ∝ *l*. Thus, according to Eq. (5), we have

In fact, this result (*λ*
_{0} ∝ *l*) is just the reason of ‘scale effect’ of metamaterials; metamaterials’ scale effect reveals that the resonance wavelength of an metamaterial is proportion to the size of the metamaterial unit structure, which has been verified by numerous simulations and experiments [53–57
]. Note that, when the length scale *l* of the continuous metal portion of the metamaterial unit structure is comparable to or smaller than *s*, that is *l* < *s*, the movement of free electrons is further limited by the physical boundary of the metal structure, and the electron’s EPL in Eq. (4) and Eq. (5) will be reduced, and be represented by size-limited free path *s*
_{i} according to Refs [52, 58
],

*a*is the prefactor on the order of one, and depends on the geometry specifics and some other factors. We want to emphasize that, this conclusion is available for not only the fundamental resonance, but also the second order resonance and even higher-order resonances, which will be verified in following section.

In this metallic antenna, the polarization density *P* = *n*
_{e}
*qx*, where *n _{e}* is the density of electrons participating in resonance motion, is proportional to free-electron density

*n*

_{0}[41]; according to additional constitutive relation equation, we can immediately written down the emission pattern of a radiating electric dipole, that is

*D*=

*ε*

_{r}

*ε*

_{0}

*E*

_{e}=

*ε*

_{0}

*E*+

*P*, where

*E*

_{e}is the electric field emitted by the oscillator and

*ε*

_{r}is the effective relative permittivity of the antenna. Thereby, solving Eq. (3), we obtain,

From Eq. (8), we can deduce that resonance intensity is determined by not only the electric density *n*
_{e} but also the potential difference ∆*U* in metamaterials.

## 3. Results and discussion

#### 3.1 Resonance in simple-structure resonators

Assuming this rectangular antenna shown in Fig. 1(a) is made from gold with permittivity *ε*
_{Au}, here *ε*
_{Au} = 1 - *ω*
_{p}
^{2} / (*ω*
^{2} + *iωγ*
_{1}), and *ω*
_{p} = 2π × 2.175 × 10^{15} s^{−1}, *γ*
_{1} = 2π × 6.5 × 10^{12} s^{−1} [50], and its substrate is SiO_{2}, as shown in Fig. 2(a)
. Firstly, in the case of polarization angle *θ* = 0, we simulate the transmittance spectra for the antennas of different lengths *l* (400nm, 500nm, 700nm, 900nm and 1000nm), see Fig. 2(b).

As we expected, all these different-length antennas exhibit multi-resonance effects, such as the 1st-order, 2nd-order and 3rd-order resonances, the positions of which are labeled by three dash lines, respectively, as shown in Fig. 2(b). It is worthy to note that these multi-resonance phenomena of this shaped antennas also appear in other literatures’ experimental results [44, 45
]. And due to the electron’s EPL increasing with the antenna’s length *l*, antenna’s 1st-order, 2nd-order and 3rd-order resonances correspondingly shift to longer wavelengths.

In order to quantitatively analyze this shift phenomenon, we also evaluate these resonance wavelengths of the antennas with different lengths *l* in Fig. 2(c). Both the simulated and fitting results suggest that all the resonance wavelengths (i.e., 1st-order, 2nd-order and 3rd-order resonance wavelengths) are approximately direct proportional to the antenna’ length *l*. To investigate the underlying physical mechanism of multi-resonance effect, the current distributions and directions in these different-length antennas at 1st-, 2nd- and 3rd- order resonance wavelengths are simulated, respectively. We find the current distributions and directions in these different-length antennas for a given order resonance (such as 1st-, 2nd- or 3rd- order resonance) are similar, which in 400nm-length antenna are plotted in Fig. 2(d); moreover, the electrons’ EPLs in these antennas are proportional. For example all the electron’s EPLs (such as *s*
_{32}) in 400nm-length, 500nm-length, 700-length and 900nm-length antennas are actually proportional to their lengths *l*.

In order to further validate this model, we also simulate the transmittance spectra and current distributions of the gold antenna (geometry sizes of *l* = 375 nm, *w* = 25 nm and *t* = 20 nm) for different polarization angles *θ* (15°, 45°, 75°and 90°), as shown in Figs. 2(e) and 2(f), respectively. As expected, the resonance wavelength (*λ*
_{0} = 1.9 μm) indicated by the red dash line in Fig. 2(e) is independent of polarization angle *θ*; while the resonance intensity at resonance wavelength of 1.9 μm decreases with polarization angle *θ*, as is mainly result from the potential difference ∆*U* (∆*U* = *E*
_{c}
*l*cos(*θ*)) decreasing with the polarization angle *θ* according to Eq. (8).

#### 3.2 Resonance in complex-structure resonators

As a matter of fact, this model is also valid for more complicated metamaterial structures. To confirm this point, we consider another four types of antenna, as shown with **A**~**D** in Fig. 3(a)
. Note that, all the lengths, widths and thicknesses of these four C-shaped antennas are the same with that of the linear antenna **E** in Fig. 3(a), are *l* = 420 nm, *w* = 25 nm and *t* = 20nm, respectively, and the split angles (*α*) of which are 45°, 90°, 120° and 180°, respectively. We simulate the transmittance spectra and current distributions of the five antennas, as shown in Fig. 3(b) and Fig. 3(c), respectively.

From Fig. 3(b), due to electron’s EPLs (*s* ∝ *l*) in these five antennas are same, according to Eq. (5), we can easily conclude that their resonance wavelengths (i.e., the 1st-order resonance wavelength of 2.1 μm) are also almost identical. However, their resonance intensities are different each other. As we discussed above in Eq. (8), the resonance intensity of a metamaterial is largely determined by the electric potential difference ∆*U* (∆*U* = *E*
_{c}
*l* cos(*θ*) shown in Eq. (1). We assume that the electric fields *E*
_{c} in these five antennas are equal because their lattice constants (450 nm) and contact areas (*l* × *w* = 10500 nm^{2}) between light and an antenna are the same, respectively; in addition, to simplify the analysis of the part of *l*cos(*θ*) in Eq. (1), we can use the effective length *l*
_{eff} to replace the *l* cos(*θ*) in Eq. (1), and get ∆*U* = *E*
_{c}
*l*
_{eff}, and from Fig. 3(b), we can see the effective length *l*
_{eff} of the antenna (from **A** to **E**) becomes longer and longer. Consequently, we can readily understand why the resonance intensity gradually increases.

Of particular interest is that, LC-circuit model thinks the resonance wavelength is determined by the effective inductance *L _{eff}* and capacitance

*C*of resonators, which mainly rely on the line length and split width of a metamaterial unit cell [59, 60 ], respectively. As we can see from Fig. 3(a), the so-called effective capacitances

_{eff}*C*of these five antennas are quite different, while their resonance wavelengths are the same with each other. Obviously, this result is not as LC-circuit model expected.

_{eff}#### 3.3 Resonance in different-material resonators

Additionally, from Eq. (8), we also find that, the resonance intensity of a metamaterial is also related to the free-carrier density of metamaterial unit cell. Before to confirm this point, we need note that, the resonance current in metamaterials is displacement current rather than the conductive current [61], and metals (such as gold, silver and copper) have usually been selected for metamaterials largely ascribed to that they have high free-electron concentrations [2]. In fact, common non-metal plasmonic materials, such as doped semiconductors and metal oxide materials, can replace the metal in metamaterials. For example, Han *et al* [62] construct a manganic-tunable semiconductor split-ring resonators; Gregory *et al* [63] fabricated U-shaped SRRs that are made from an indium-tin oxide (ITO) material. Our further simulation results suggest that subwavelength structures, such as the C-shaped antenna in Fig. 4(a)
, made from doped semiconductors (GaAs) can be also used as building blocks for metamaterials to excite electromagnetic-resonance behaviors, as shown in Fig. 4(b). As one can see, the resonance intensity increases with the doped density *n*
_{e}. Additionally, because carrier density *n*
_{e} (doped density or electron’s density) almost can’t alter electron’s EPL, the resonance wavelength (*λ*
_{0} = 24 μm, the 1st-order resonance wavelengths) thus keeps unchanged.

In order to quantitatively analyze the influence of the doped density *n*
_{e} on the resonance intensity of metamaterials, we retrieve their effective permittivity (real part of *ε*
_{eff}) spectra from above transmission-reflection data via an extraction procedure [64–66
], as depicted in Fig. 4(c); then use the effective permittivity variation Δ*ε* (Δ*ε* = *ε*
_{max} - *ε*
_{min}, where *ε*
_{max} and *ε*
_{min} denote the maximum and minimum values of the real part of *ε*
_{eff} around the resonance wavelength of 24 μm, respectively) to represent the strength of resonance in metamaterials, and the evaluated results (Δ*ε*) for this GaAs metamaterials are depicted with the red solid curve in Fig. 5
. With the same method, we also simulate and evaluate the effective permittivity variations Δ*ε* for some metallic metamaterials made from Ag, Al, Au, Co, Cu, Fe, Ni, Pd, Pt, Ti and W, respectively, as denoted with different-color circles in Fig. 5, respectively. Note that, the geometries of the unit cell shown with the inset (b) in Fig. 5 in these metallic metamaterials are same with that of the unit cell shown with inset (a) in Fig. 5 in GaAs metamaterials, and the parameters (such as plasmonic frequencies and collision frequencies) of these bulk metals in the Ref [67] are used in these simulations.

The results in Fig. 5 indicate that the resonance intensity (permittivity variation Δ*ε*) of the GaAs metamaterial almost linearly increases with doped density log(*n*
_{e}); additionally, the resonance intensity of GaAs metamaterials is usually weaker than that of metallic metamaterials due to the carrier densities *n*
_{e} in metals are usually above 10^{22} cm^{−3} higher than that in doped semiconductors (not greater than 10^{19} cm^{−3}). Moreover, it also implies that doped-semiconductor metamaterials can act as alternative materials to achieve the tunable optical devices; however, their working principles are distinctively different from that of currently tunable metamaterial-based devices.

As we know, currently tunable metamaterials are usually made from metals, and due to the properties (such as permittivity or permeability) of which are mostly unchangeable by external stimuli, so changing the properties (such as the refractive and conductivity) of the substrate of metamaterials to tune its resonance wavelength via stimuli (such as light, electric field, temperature etc.) naturally becomes the major method, and such tunable metamaterials are actually frequency-tunable metamaterials. However, doped-semiconductor metamaterials presented in this work, the resonance properties (i.e., resonance intensity) of which can be tuned by controlling the free-carrier density of semiconductors (not the substrate) through external stimuli (such as injection current, pump light and temperature), are actually considered to be intensity-tunable metamaterials. This find may greatly broad current research ranges of tunable metamaterials, and will bring lots of new types of tunable optical device.

Additionally, it is worth noting that, strictly speaking, detail physical processes, such as scattering effect, wave coupling and boundary condition etc., in different-shaped and different-material resonators, especially for more complex-structure resonators (such as square SRRs and Ω-shaped resonators), will be different, which will surely influence electrons’ resonance motion in unit cells, and change the resonance behavior (such as resonance frequency and resonance intensity) of metamaterials in some ways. Nevertheless, in order to get some more accurate results, according to the specific situation (such as the shape of unit cell and incident angle), one should make a proper modification for the parameters in this model by further analyzing these physical processes. For example, as we know, electrons in different shaped resonators usually don’t interact equally with the driving and the scattering fields, and according to the skin depth effect, we can approximately calculate the densities (*n _{e}*) of electrons participating the resonance motion for these resonators [41], and can finally modify some values of corresponding parameters (such as effective mass

*m*) in this model.

## 4. Conclusions

From the investigation of resonance mechanisms in this work, in essence, arbitrary-shaped subwavelength structures made from arbitrary plasmonic materials can be utilized to construct metamaterials. And the resonance wavelengths (such as the fundamental resonance wavelength and high-order resonance wavelengths) of metamaterials are radically governed by the electrons’ free path lengths (generally proportion to the size of the metamaterial unit cell), while resonance intensities are largely determined by the electrical potential differences induced by incident waves and the densities of electrons participating in resonance motion in metamaterial unit cells. This concept will subvert traditional designs of metamaterials, and open a broader designable space for metamaterial structures; furthermore, it will also provide more opportunities to achieve higher-performance and more-novel optical metamaterial-based devices or components.

## Acknowledgments

This work was supported by the National High Technology Research and Development Program of China (No. 2013A014401), Specialized Research Fund for the Doctoral Program of Higher Education (SRFDP) (No. 20120142110064) and Natural Science Foundation of Hubei Province (No. 2012FFB02209). National Natural Science Foundation of China (Nos. 11104093 and 11474116).

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