Investigation of Fano resonance in planar metamaterial with perturbed periodicity

Mingbo Pu; Chenggang Hu; Cheng Huang; Changtao Wang; Zeyu Zhao; Yanqin Wang; Xiangang Luo

doi:10.1364/OE.21.000992

1. Introduction

Metamaterials, as an artificial engineered material based on the strong interaction between electromagnetic radiation and conduction electrons at metallic interfaces or in subwavelength metallic structures, have attracted much attention in the last decade due to their unique electromagnetic properties such as negative [1–3] and zero refractive index [4, 5]. One most important property of plasmonic metamaterial is that the transmission spectrum and working bandwidth can be controlled by the shape and size of the subwavelength structure. While broadband property have been intensively studied for applications such as invisibility cloak [6], perfect absorber [7–10], extremely narrow bandwidth and high Quality factor (Q-factor, defined as f/Δf) was pursued in recent work concerning biological sensing [11], slow light [12], etc. Unfortunately, ultrahigh Q-factor is difficult to realize for two dimensional metamaterial due to the lack of large volume confinement of electromagnetic fields and strong coupling to free space [13].

Quite recently, it is found that sharp trapped mode with ultrahigh Q-factor can be obtained in planar metamaterial by introducing symmetry breaking in the shape of its structural elements [13]. The Q-factor of such trapped mode is reversely related to the asymmetry factor (defined as the relative difference between structural elements). Later, pronounced Fano resonance and extraordinary optical transmission (EOT) are observed in asymmetric structure and attributed to the coexistence and interference of super-radiant and sub-radiant modes [14, 15]. The asymmetric metamaterial is termed coherent since the electromagnetic response is a collective phenomenon and strong interactions between metamolecules exist [16, 17]. The combination of sharp trapped mode and coherence paves the way for lasing spaser, a coherent source supported by gain [18, 19]. In addition, asymmetric Fano resonance has been utilized recently for ultrasensitive spectroscopy and identification of molecular monolayers [20]. Theoretically, the physical explanation of Fano resonance in metamaterial can be given by using classical analogy [21] or ab initio theory based on Feshbach formalism [22]. The classical analogy is a general technique but not rigorous enough. The Feshbach formalism, however, is much more complicated and the relation between spectrum and structure is not so clear.

In this paper, a planar metamaterial with perturbed periodicity and sharp Fano resonance is proposed. Generalized impedance theory and equivalent circuit theory are used to investigate the Fano-type resonance. Electric-magnetic coupling and periodicity perturbation induced magnetic-magnetic coupling are demonstrated unambiguously. Full-wave simulations agree well with the theoretical results. We expect that the approach proposed here can also be used to analyze other asymmetric metamaterial structures.

2. Structure and simulation

As illustrated in Fig. 1(a) , the structure proposed here is a periodic metallic wire pair configuration, which has been widely used to create artificial magnetic response and negative refractive index [23–25]. Here different widths of wire pairs (depicted by grey, orange, and red colors) are used to perturb the periodicity. In this case, three wire pairs actually act as a new unit cell. The center-to-center distances between adjacent wire pairs are kept as p = 5 mm and the thickness of the dielectric spacer is set as d = 0.75 mm. Metal is assumed as perfect conductor while the permittivity of dielectric material is chosen as 12 with loss tangent τ = 0 (the material loss will be considered later).

Fig. 1 (a) Schematic of the wire pairs with perturbed periodicity. The grey, orange, and red colors depict different widths. (b) Sketch map of the surface current distributions for the two subradiant modes.

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As is well known, metallic wire pairs have strong magnetic response which can be described by effective magnetic dipoles. Giant magnetic response can be obtained with anti-parallel surface currents when the electric field is perpendicular to the long axis of wire pairs. Different subradiant modes, which couples weakly with the external field, may exist at different resonance conditions (Fig. 1(b)). In numerical simulation, Comsol Multiphysics, a commercial finite element method (FEM) solver is used with periodic boundary condition in x direction and perfectly matched layer in ± z direction. The structure is assumed to be infinite long in y direction while the electric and magnetic fields are along x and y directions, respectively (i.e. normal incidence). The incident magnetic field is set as unit.

In order to investigate the influence of structure asymmetry, the transmission coefficients of both symmetric and asymmetric metamaterial at normal incidence are calculated and shown in Fig. 2 . For the symmetric structure (w₁ = w₂ = w₃ = 4 mm), the transmission peak at 9.78 GHz is induced by the magnetic resonance. Similar with Fano resonance, the transmission profile is asymmetric and a transmission dip occurs at 11.1 GHz.

Fig. 2 Transmission coefficients for the symmetric and asymmetric metamaterial slabs. Two regions are highlighted. Region I shows typical asymmetric line shape of Fano resonance. Region II demonstrates the ultra-sharp resonance peak induced by inter-element coupling.

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For the asymmetric structure, the widths of three wires can have three or two different values. Without loss of generality, we choose w₁ = 4 mm, w₂ = 3.6 mm and w₃ = 3.56 mm as an example. Compared with the symmetry case, the line shape around 9.8 GHz becomes more asymmetric. Moreover, ultra sharp transmission dip at the center of a transmission peak is observed at frequencies around 10.8 GHz with Q-factor as large as 100. Interestingly, this is in contrary to the metamaterial analogy of electromagnetic induced transparency (EIT) [26], where a transmission sharp peak arises in a reflective frequency region.

3. Analysis and discussion

3.1 Generalized impedance theory

The above exotic transmission properties can be understood by using generalized surface impedance theory, which has been used to retrieve effective material parameter for thin sheets such as graphene and single-layer metamaterials [27].

As illustrated in Fig. 3 , thin metamaterial slab can be treated as a combination of electric impedance and magnetic impedance with the following boundary condition:

\begin{array}{l} {\vec{E}}_{i} + {\vec{E}}_{r} = {\vec{E}}_{t} + \hat{n} \times {\vec{j}}_{m}, \\ {\vec{H}}_{i} + {\vec{H}}_{r} = {\vec{H}}_{t} + \hat{n} \times {\vec{j}}_{e} . \end{array}

By definition, there are E_r = rE_i, H_r = rH_i, E_t = tE_i and H_t = tH_i, where E_i = Z₀H_i is the incident electric field, Z₀ = 1/Y₀ = 377 Ω is the vacuum impedance, r and t are the reflection and transmission coefficients. The electric sheet current j_e and magnetic sheet current j_m are related with the average electric and magnetic field through the corresponding electric admittance Y_e and magnetic impedance Z_m:

\begin{array}{l} {\vec{j}}_{e} = \frac{1}{Z_{e}} {\vec{E}}_{a v e r a g e} = Y_{e} {\vec{E}}_{a v e r a g e} = Y_{e} (1 + r + t) {\vec{E}}_{i} / 2, \\ {\vec{j}}_{m} = Z_{m} {\vec{H}}_{a v e r a g e} = Z_{m} (1 - r + t) {\vec{H}}_{i} / 2. \end{array}

Combining Eqs. (1) and (2), one can obtain that:

\begin{array}{l} Y_{e} = 2 Y_{0} \frac{1 - r - t}{1 + r + t}, \\ Z_{m} = 2 Z_{0} \frac{1 + r - t}{1 - r + t}, \end{array}

\begin{array}{l} r = \frac{1}{2} (\frac{2 Y_{0} - Y_{e}}{2 Y_{0} + Y_{e}} + \frac{Z_{m} - 2 Z_{0}}{Z_{m} + 2 Z_{0}}), \\ t = \frac{1}{2} (\frac{2 Y_{0} - Y_{e}}{2 Y_{0} + Y_{e}} - \frac{Z_{m} - 2 Z_{0}}{Z_{m} + 2 Z_{0}}), \end{array}

and

Fig. 3 Schematic of sheet impedance description for thin metamaterial. The magnetic and electric responses are described by equivalent magnetic and electric sheet currents.

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\begin{array}{l} j_{e} = \frac{2 Y_{0} Y_{e}}{2 Y_{0} + Y_{e}} E_{i}, \\ j_{m} = \frac{2 Z_{0} Z_{m}}{Z_{m} + 2 Z_{0}} H_{i} . \end{array}

Equations (3)-(5) are the central results of the generalized sheet impedance theory. Using Eq. (3), the effective magnetic and electric impedances can be retrieved from simulated or measured S parameters (r and t). In the following, the generalized impedance theory is used to interpret the formation of Fano resonances in the planar metamaterial as shown in Fig. 2.

3.2 Fano resonance induced by the interference of magnetic and electric resonances

For the structure without asymmetry (w₁ = w₂ = w₃ = 4 mm), the line shape shown in Fig. 2 is asymmetric. In order to understand the physical meaning behind it, the electric and magnetic sheet impedances are retrieved using Eq. (3). As depicted in Figs. 4(a) and 4(b), the electric admittance is nearly flat in the whole frequency range and magnetic impedance manifests itself as a sharp resonance. We noted that this phenomenon is similar with Fano resonance, which stems from the interference of a discrete state with a continuum.

Fig. 4 (a) The electric admittance and (b) magnetic impedance of the symmetric structure (w₁ = w₂ = w₃ = 4 mm). (c) The transmission coefficients for the real structure and pure magnetic impedance. The fitted magnetic impedance using LC circuit model is also shown in (b).

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As given by Eq. (4), the transmission coefficient of any metasurface can be decomposed into two parts, one electric and the other magnetic. To analysis the influence of the electric admittance on the transmission spectrum, the electric admittance is set as ∞ (Z_e = 0) artificially and the corresponding transmission coefficient is plotted in Fig. 4(c). Obviously, the asymmetric line shape will transform into symmetric one when the electric resonance is not taken into account. Thus, one can conclude that it is the interference of the broadband electric resonance and narrowband magnetic resonance leads to the asymmetric Fano resonance.

In addition, to describe the magnetic resonance quantificationally, the magnetic impedance can be described by a simple equivalent LC circuit model in the form of:

Z_{m} = \frac{1}{1 / (i ω L) + i ω C} = \frac{i ω L}{1 - ω^{2} L C},

where ω is the angular frequency, L and C are the effective inductance and capacitance. The values of L and C can be evaluated by fitting the results retrieved from r and t. For the case of Fig. 4, L and C are 0.708 nH and 0.369 pF, which agrees well with the retrieved results.

3.3 Fano resonance induced by coupled magnetic resonances

For the structure with perturbed periodicity, the adjacent magnetic resonators will couple with each other. As shown in Fig. 5 , the electric and magnetic impedances as well as the corresponding sheet currents are calculated using Eqs. (3) and (5). Similar with the symmetric structure, the electric admittance is still rather flat. The magnetic impedance, however, shows complex resonant characteristics. Since there are three resonant peaks, the magnetic impedance shown in Fig. 5(c) can be written as a combination of individual magnetic resonances, as described by equivalent LC circuit model:

Z_{m} = \sum_{n = 1}^{3} Z_{m n} = \sum_{n = 1}^{3} \frac{i ω L_{n}}{1 - ω^{2} L_{n} C_{n}} .

Here, L₁, L₂, L₃, C₁, C₂ and C₃ are the corresponding inductances and capacitances. The resonant frequencies are

ω_{1} = {(L_{1} C_{1})}^{- 1 / 2}

,

ω_{2} = {(L_{2} C_{2})}^{- 1 / 2}

, and

ω_{2} = {(L_{3} C_{3})}^{- 1 / 2}

, respectively. By fitting Eq. (7) with the retrieved magnetic sheet impedance, the LC parameters can be obtained as L₁ = 0.295 nH, L₂ = 0.273 nH, L₃ = 0.27 nH, C₁ = 0.896 pF, C₂ = 0.8 pF and C₃ = 0.794 pF. The corresponding resonant frequencies are 9.79 GHz, 10.77 GHz and 10.87 GHz.

Fig. 5 (a) Electric admittance and (b) electric sheet current for the asymmetric structure. (c) The corresponding magnetic impedance and (d) magnetic sheet current.

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Similar with the magnetic sheet impedance, the magnetic sheet current is also a summation of all the individual sheet currents:

j_{m} = \sum_{n = 1}^{3} j_{m n},

where

j_{m n} = \frac{2 Z_{0} Z_{m n}}{Z_{m} + 2 Z_{0}} H_{i}, n = 1, 2, 3.

Using Eqs. (7) and (9), the sheet currents for different resonators can be easily calculated. As shown in Fig. 6(a) , the current of the first resonator dominates at frequencies around 9.79 GHz. On the contrary, the second and third resonators dominate the frequency region around 10.8 GHz. From Fig. 6(b) one can find that the first resonator is out of phase with the other two resonators for 9.79 GHz < f <10.77GHz. Also, For 10.77 GHz < f <10.87 GHz, the third one is out of phase with the others.

Fig. 6 (a) Amplitudes and (b) phases of the effective magnetic sheet current calculated using Eq. (5). The fitted parameters of inductances and capacitors are used (see text). The three frequencies where the phase changes occur are highlighted. At theses frequencies, the corresponding amplitudes are zero.

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In fact, the π phase shift between these resonators is the key of resonant enhancement of sheet current. As an example, at frequency $ω_{0} = {(L_{2} + L_{3})}^{1 / 2} {(L_{2} L_{3} (C_{2} + C_{3}))}^{- 1 / 2} = 1 0. 82 GHz$ there are $| Z_{m 1} | ≪ | Z_{m 2} | \approx | Z_{m 3} |$ and $| Z_{m 1} | ≪ | Z_{m} |$ . Thus, one can obtain that:

\begin{array}{l} j_{m 1} ≪ Z_{0} H_{i} \\ j_{m 2} = \frac{i \sqrt{L_{2} L_{3} (L_{2} + L_{3}) (C_{2} + C_{3})}}{L_{3} C_{3} - L_{2} C_{2}} H_{i} . \\ j_{m 3} = - j_{m 2} = \frac{i \sqrt{L_{2} L_{3} (L_{2} + L_{3}) (C_{2} + C_{3})}}{L_{2} C_{2} - L_{3} C_{3}} H_{i} \end{array}

The second and third sheet currents in Eq. (10) are out of phase and reversely proportional with $ω_{3}^{2} - ω_{2}^{2} = L_{3} C_{3} - L_{2} C_{2}$ . As a result, higher enhancement factor and narrower bandwidth can be achieved by decreasing the difference between ω₂ and ω₃. Using previous fitted parameters, the sheet current at 10.82 GHz can be calculated as j_m₁ = 3.3iH_i, j_m₂ = 2000i H_i, and j_m₃ = −2000i H_i, which agree well with that shown in Fig. 6(a).

In general, the magnetic field inside the planar metamaterial is proportional to the sheet current. Recalling Eq. (1) and the Maxwell equation:

\begin{array}{l} {\vec{E}}_{i} + {\vec{E}}_{r} - {\vec{E}}_{t} = \hat{n} \times {\vec{j}}_{m} \\ \oint_{L} \vec{E} \cdot d \vec{l} = \frac{- d}{d t} \int_{s} \vec{B} \cdot d \vec{S} \end{array}

One can obtain that:

\hat{n} \times {\vec{j}}_{m} \cdot L = \frac{- d}{d t} (\vec{B} \cdot L \cdot Δ),

where L = p is the unit cell length and Δ is the effective thickness of the metamaterial layer. As a result, the magnetic field H_z can be calculated as:

| {\vec{H}}_{z} | = \frac{| {\vec{j}}_{m} |}{i ω μ_{0} Δ} .

It should be noted that Eq. (13) is only an approximation under the condition Δ<<λ, for which the electric field integration along the thickness direction can be neglected.

In order to further prove above discussion, the magnetic fields at the center of the three resonators for different frequencies are calculated using Comsol Multiphysics and illustrated in Fig. 7 , which are very similar with the magnetic sheet currents calculated from S parameters. At f₁ = 8 GHz, all the three wire pairs are out of resonance and the maximum magnetic field is only 3.6 H_i. At f₂ = 9.79 GHz, the first resonator is resonant and the maximum magnetic field increases to 69 H_i. At f₃ = 10.77 GHz, and f₅ = 10.87 GHz, the second and the third wire pairs are resonant, respectively. Since f₃ and f₅ are spectrally close, a new resonant mode takes place between them. At f₄ = 10.8 GHz, the second and the third wire pairs are on resonance with opposite phase. The maximum magnetic field becomes as high as 140 H_i.

Fig. 7 (a) Normalized magnetic fields (H_z) at the center of these resonators. The inset shows the unit cell and the points where the magnetic fields are extracted. (b)(c)(d)(e)(f) Magnetic field distributions at frequencies of f₁ = 8 GHz, f₂ = 9.79 GHz, f₃ = 10.77GHz, f₄ = 10.82 GHz and f₅ = 10.87 GHz.

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In order to further understand the influence of the periodicity perturbation on the Fano resonance, the effect of wire width is investigated. Here, two wires have the same width of 4 mm while the third wire width is changed from 3.8 mm, 3.96mm to 4mm. As illustrated in Fig. 8 , as the asymmetry decrease, two isolated resonances become very close and a third sharp resonance occurs. When the three wire pairs all have the same widths, the transmission spectrum will degenerate to a single magnetic resonance.

Fig. 8 Transmission spectrum of the metamaterial slab for different asymmetry. Here the geometrical parameters are w₁ = w₂ = 4 mm, while w₃ is variable.

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Finally, it should be noted that the Fano resonance proposed here is a coherent process thus practical performance is highly dependent on the unit cell numbers of finite arrays [18]. In addition, the fabrication error may also limit the achievable bandwidth, especially if the structure is scaled to higher frequencies, where the intrinsic loss in metal is another restriction of high Q-factor resonance.

3.4 Influence of loss

In the above discussion, the material loss is neglected since the dielectric material is lossless and metal is set as perfect electric conductor. In general, the loss of material has great influence on the performance. In order to take it into account, a typical loss tangent of 0.003 is added in the dielectric material and metal is used as copper with conductivity of 5.7e7 S/m. As shown in Fig. 9(a) , obvious absorption is observed at the resonant frequencies 9.8 GHz and 10.8 GHz. Nevertheless, the magnetic field enhancement factor only decreases a little, as shown in Fig. 9(b).

Fig. 9 (a) Reflection (r), transmission (t) and absorbance (1-r²-t²) of the lossy metamaterial slab. (b) Normalized magnetic fields extracted from Comsol Multiphysics.

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In fact, the loss mechanism in the asymmetric structure can be utilized to realize wideband wide-angle absorber, which has been intensively studied in recent years [8, 9]. Typical structure is shown in the inset of Fig. 10 , where a metallic ground plane is added in the center of wire pair structure. Here, the geometric parameters are optimized as p = 5 mm, d = 0.75 mm, w₁ = 4 mm, w₂ = 4.12 mm and w₃ = 3.9 mm. The permittivity is set as 12 with a higher loss tangent of τ = 0.015. The achieved bandwidth for 90% absorption is about 0.5 GHz, which is larger than that of metamaterial absorber without asymmetry (0.2 GHz) at the same thickness (p = 5 mm, w = 4 mm, τ = 0.025).

Fig. 10 Absorption of the wideband absorber for different angles of incidence. The absorption for the symmetric structure is also shown for comparison. Inset is schematic of the absorber which is composed of asymmetric wires and a metallic ground plane.

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4. Conclusion

In conclusion, we demonstrated analytically and numerically that Fano-type resonance can be excited in planar metamaterial with perturbed periodicity. Generalized impedance surface theory is given to interpret these exotic phenomena unambiguously, which agrees well with the numerical results calculated using finite element method. It is found that the electric-magnetic and magnetic-magnetic coupling effects in planar metamaterial cannot only generate pronounced Fano resonance but also sustain giant field enhancement which has potential applications in slow light, nonlinear optics and biological sensing etc.

Acknowledgment

This work was supported by 973 Program of China (No. 2011CB301800) and National Natural Science Funds for Distinguished Young Scholar (No. 60825405).

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Investigation of Fano resonance in planar metamaterial with perturbed periodicity

Abstract

1. Introduction

2. Structure and simulation

3. Analysis and discussion

3.1 Generalized impedance theory

3.2 Fano resonance induced by the interference of magnetic and electric resonances

3.3 Fano resonance induced by coupled magnetic resonances

3.4 Influence of loss

4. Conclusion

Acknowledgment

References and Links

Cited By

Figures (10)

Equations (13)

Optics Express