1. Introduction

Metamaterials have attracted great interest of researchers for the unique properties absent from naturally existing materials. The macroscopic properties of metamaterials are highly dependent on the geometry of unit cells, providing additional freedom for designing materials and controlling wave propagation [1,2]. Metamaterials have a wide range of applications, such as perfect lens [3,4], invisible cloak [5,6], transformation optics devices [7–9], polarization manipulation [10–12], electromagnetically induced transparency [13–15], etc. Based on the idea of metamaterials, Fano resonance, which is characterized by an asymmetric line shape, can be realized by asymmetric structures via near-field coupling [16–22]. It is significant that Fano resonance possesses sharp spectrum and generally brings on enhanced local field, allowing potential applications in bio sensing [23,24] and light modulating [25]. The introduction of asymmetry into microstructure enables the excitation of quadrupole mode, which weakly couples to the incident wave. Via interference between the quadrupole and dipole modes, a Fano feature is produced.

At higher frequencies, especially optical frequencies, the quality factor Q of plasmonic resonator decreases dramatically because of the dissipation in noble metal. Dielectric resonator based on Mie resonance is merely subject to small polarization loss and is a good alternative to plasmonic resonator [26–29].Tunable Fano resonance is of great significance in applications. We presented asymmetric dielectric wire pair and investigated the thermally tunable Fano resonance [30].

In this work, we introduce varactor diodes [31–33] into asymmetric wire pair, and investigate the tunable Fano-type resonance by both experiment and simulation. It is illustrated that Fano-type transmission spectrum with high quality factor Q appears as a result of interference between the dipole and quadrupole modes. Through full wave simulations, the absorption and far field radiation of electric dipole are also analyzed in detail. We demonstrate that, at the Fano-type resonance frequency, both the two metal wires exhibit the strongest electric resonance simultaneously, and the Fano-type resonance manifests a large group delay. It is shown that the Fano-type resonance frequency exhibits a prominent blueshift of 0.16 GHz, and the transmission experiences a modulation with a modulation depth of 97% as the bias voltage ranges from 0 V to 8 V.

2. Structure and dimensions of the asymmetric metal wire pair

Figures 1(a) and 1(b) depict the asymmetric structure composed of a pair of metal wires of different length. The asymmetric wire pair is fabricated with the standard PCB technology. To control the electric resonance of metal wire, a gap is engendered at the center of metal wire and a varactor diode (BBY52-02W, Infineon), which has a series resistance and inductance of 0.9 Ω and 0.6 nH, respectively, is soldered at the gap. By applying reverse bias voltage on varactor diode via bias lines, the capacitance can be altered and accordingly the electric or Fano-type resonance frequency can be manipulated. The scattering parameters are measured inside a standard rectangular waveguide of BJ32 and recorded using vector network analyzer (AV3629D). The TE₁₀ mode is normally incident on the asymmetric wire pair, which is located at the center of waveguide, with the electric field polarizing along metal wire.

 

Fig. 1 Schematic view (a) and photograph (b) of the asymmetric metal wire pair. The metal layer is copper with a conductivity of 5.8e7 S/m, and the depth is 0.018 mm. The substrate is Teflon with a relative permittivity of 2.45 and a loss tangent of 4e-4, and the dimension is 72.14*34.04*1 mm³. Two bias lines have a diameter of 0.1 mm. The dimensions of asymmetric wire pair are as follows: l₁ = 33 mm, l₂ = 31 mm, h = 5 mm, w = 3 mm, g = 1.3 mm and s = 5 mm.

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3. Fano-type resonance of the asymmetric metal wire pair

Figure 2(a) shows the transmission spectra of asymmetric wire pair, symmetric wire pair (SWP), and single wires. It is apparent that the left wire and right wire exhibit transmission dips at 2.89 GHz and 3.31 GHz, respectively, corresponding to the electric resonance of single wire. The quality factor Q, which is used to describe the sharpness of resonance response, is mere 11. By combining the left wire and right wire in parallel to form an asymmetric wire pair, a prominent asymmetric Fano spectrum emerges with a transmission peak approaching −1.29 dB at 3.03 GHz. The bandwidth is only 0.065 GHz as measured at 3 dB below the maximum and the quality factor Q is up to 47. Nevertheless, the transmission spectrum of symmetric wire pair, whose length l₁ and l₂ are both 33 mm, is analogous to that of left wire, but has a higher resonance frequency and broader resonance bandwidth.

 

Fig. 2 (a) Transmission for the asymmetric wire pair, symmetric wire pair (SWP), and single wires. (b) Induced surface current, which is indicated by arrows, on the asymmetric wire pair and x component of local magnetic field H_x at the transmission peak frequency of the asymmetric wire pair. Whole absorption (c) and partial absorption (d) in the asymmetric wire pair. Magnitude (e) and phase (f) of the normalized scattering wave of electric dipole or the normalized electric dipole in the asymmetric wire pair. All these results are acquired by simulation under the case that the capacitance of varactor diode is 2.63 pF, which is corresponding to the bias voltage of 0 V.

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To explore the underlying physics of Fano-type resonance of asymmetric wire pair, the surface current on metal wires and x component of local magnetic field H_x at the transmission peak frequency are calculated and shown in Fig. 2(b). It is evident that the induced surface currents on the left wire and right wire oscillate approximately out of phase, forming an electric quadrupole. Through interference between the electric dipole and quadrupole modes of the asymmetric wire pair, a Fano-type resonance with asymmetric line shape and high quality factor Q is obtained. Furthermore, the Fano-type resonance results in remarkably enhanced local field, which is one of the most significant characteristics of Fano-type resonance. The enhanced local field arising from Fano-type resonance is of great significance in chemical and biological sensing applications as well as optical nonlinearity in plasmonic nanostructures.

The absorption of asymmetric wire pair is also examined. By integrating the surface loss density R_s|J_s|²/2 or volume loss density ωε”|E|²/2 over the surface or volume, the absorption of each component is calculated. Herein, R_s represents the surface resistance of varactor diodes or metal wires, J_s represents the surface current density on varactor diodes or metal wires, ω is angular frequency, ε” is the imaginary part of permittivity of dielectric substrate, and E is electric field. Figures 2(c) and 2(d) display the whole absorption of asymmetric wire pair and the partial absorption of each component, respectively. It is apparent that the whole absorption is the largest at the transmission peak frequency and the absorption of varactor diode is an order of magnitude larger than that of metal wire and substrate. To enhance the quality factor Q of Fano-type resonance, therefore, it is necessary to lower or offset the series resistance in varactor diode.

In addition, the electric dipole scattering of asymmetric wire pair is investigated in detail, and the magnitude and phase of scattering field normalized to incident field E₀ or electric dipole moment normalized to the effective electric dipole source p₀ of incident field are demonstrated in Figs. 2(e) and 2(f), respectively. It is distinct that the electric dipole resonance occurs in each metal wire simultaneously at the transmission peak frequency and consequently results in the enhanced local field and maximum absorption presented in Figs. 2(b) and 2(c), respectively. The electric dipole moments of the left wire and right wire are equal in magnitude and have a phase difference slightly deviating from 180°, resulting in a minimum but non-negligible net electric dipole moment. The net electric dipole moment has a phase difference of ± 90° with that of left wire or right wire; the scattering field of net electric dipole has a phase difference of 180° with the incident field and interferes with it destructively, resulting in a transmission less than 0 dB. The high transmission of asymmetric wire pair is attributed to the low radiative loss of net electric dipole for the transmission mainly rests with the radiative loss rather than absorption. Moreover, both the transmission dips at 2.89 GHz and 3.40 GHz are attributed to the electric resonance, whereas they have an obvious difference in the configuration of electric dipole, i.e. the electric dipoles on the left wire and right wire oscillate out of phase at 2.89 GHz, but in phase at 3.40 GHz.

Figures 3(a) and 3(b) show the transmission phase and group delay of asymmetric wire pair and single wires, respectively. At the electric resonance frequencies, the scattering field E_s and incident field E₀ are out of phase, and thus the transmission field E_t, which is the sum of E_s and E₀, and incident field E₀ are in phase, suggesting a transmission phase of zero; similarly, the transmission phase of asymmetric wire pair is also zero at the Fano-type resonance frequency. Furthermore, at the electric resonance frequencies, the transmission phase varies rapidly with frequency, resulting in a large and negative group delay; at the Fano-type resonance frequency of asymmetric wire pair, however, the rapid variation of transmission phase with frequency results in a large but positive group delay. The phase-frequency characteristic of transmission field can be interpreted by resorting to the inset in Fig. 3(a). Herein, the blue arrows represent phasors, i.e. complex amplitudes, not vectors; the phasor of E₀ is assumed to be unit, and the end point of phasor of E_s moves along the direction indicated by black arrows as the frequency increases according to the red curves in Figs. 2(e) and 2(f). Thus, the phasor of E_t also represents S₂₁. It is significant that as the electromagnetic wave travels through the 0.018-mm-thick asymmetric wire pair, it undergoes a maximum positive group delay of 5.01 ns at the Fano-type resonance frequency, corresponding to a 1.09-m-long transmission line. Nevertheless, the single wires do not possess such group delay.

 

Fig. 3 Transmission phase (a) and group delay (b) of the asymmetric wire pair and single wires. The inset in (a) illustrates how E_s and E₀ compose E_t for asymmetric wire pair. These results remain acquired by simulation under the case that the capacitance of varactor diode is 2.63 pF.

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4. Tunability of Fano-type resonance of the asymmetric metal wire pair

Finally, we investigate the dynamic tunability of Fano-type resonance of asymmetric wire pair by applying bias voltage on varactor diodes. The single wire can be modeled as a LC circuit, and the electric resonance frequency is determined by ω = (LC)^-1/2, where 1/C = 1/(C_d + C_g) + 1/(C_m), C_d is the capacitance of varactor diode, C_g is the gap capacitance, C_m is the coupling capacitance, and L is the inductance of single wire. It is evident that, therefore, the electric resonance frequency of single wires and accordingly the Fano-type resonance frequency of asymmetric wire pair can be manipulated by applying bias voltage to alter the capacitance of varactor diodes.

Figure 4 depicts the tunability of electric resonance of single wires. When the bias voltage varies from 0 V to 8 V, and accordingly the capacitance of varactor diode ranges from 2.63 pF to 0.76 pF, the experimental results demonstrate that the electric resonance frequency of left wire shifts from 2.94 GHz to 3.21 GHz, exhibiting a blueshift of 0.27 GHz; similarly, the electric resonance frequency of right wire shifts from 3.34 GHz to 3.62 GHz, exhibiting a blueshift of 0.28 GHz. The simulated results agree well with experimental results, verifying the effective manipulation of electric resonance frequency by varactor diode.

 

Fig. 4 Tunable electric resonance of single wires by altering the bias voltage on varactor diode and accordingly altering the capacitance. Transmission spectra of left wire (a, b) and right wire (c, d) acquired with both experiment (a, c) and simulation (b, d) under the bias voltage ranging from 0 V to 8 V and accordingly the capacitance ranging from 2.63 pF to 0.76 pF.

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The tunability of Fano-type resonance of asymmetric wire pair is represented in Fig. 5. When the bias voltage varies from 0 V to 8 V, the experimental results illustrate that the Fano-type resonance frequency of asymmetric wire pair shifts from 3.11 GHz to 3.27 GH, displaying a blueshift of 0.16 GHz; meanwhile, the lower and upper electric resonance frequency display blueshifts of 0.19 GHz and 0.23 GHz, respectively. The simulated results are in good agreement with experimental results, confirming the effective control of Fano-type resonance frequency by varactor diode. Moreover, it is significant that the high transmission and sharp spectrum are preserved as the Fano-type resonance frequency is tuned.

 

Fig. 5 Tunable Fano-type resonance of asymmetric wire pair by altering the bias voltage on varactor diodes and accordingly altering the capacitance. Transmission spectra of asymmetric wire pair acquired with both experiment (a) and simulation (b) under the bias voltage ranging from 0 V to 8 V and accordingly the capacitance ranging from 2.63 pF to 0.76 pF.

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Owing to the sharp transmission spectrum arising from Fano-type resonance, the transmission of asymmetric wire pair is subject to considerable variation around the Fano-type resonance frequency as the bias voltage varies. As shown in Fig. 6(a), at the frequency of 3.11 GHz, for example, the transmission of asymmetric wire pair varies from the maximum −2.4 dB to −19.4 dB via the minimum −35.1 dB as the bias voltage varies from 0 V to 8 V, demonstrating a transmission modulation with a modulation depth of 97%. By comparison, Fig. 6(b) demonstrates that the transmission of single wire, for instance left wire at 3.21 GHz, varies from the maximum −10.3 dB to the minimum −39 dB as the bias voltage varies from 0 V to 8 V. It is obvious that the maximum transmission of single wire is much lower than that of asymmetric wire pair. From this point of view, the asymmetric wire pair can act better as a switch for microwave controlled by bias voltage.

 

Fig. 6 Relation between the transmission and the bias voltage on varactor diode for asymmetric wire pair at 3.11 GHz (a) and left wire at 3.21 GHz (b) measured by experiment.

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5. Conclusion

In summary, we present asymmetric wire pair loaded with varactor diodes, and investigate the tunable Fano-type resonance by both experiment and simulation. Fano-type transmission spectrum with high quality factor Q occurs as a result of the interference between the dipole and quadrupole modes. The absorption mainly results from the ohmic loss of series resistance in varactor diode. At the Fano-type resonance frequency, both the two metal wires exhibit the strongest electric resonance simultaneously. In virtue of the rapid variation of transmission phase with frequency, the Fano-type resonance manifests a large group delay. As the bias voltage ranges from 0 V to 8 V, the Fano-type resonance frequency exhibits a conspicuous blueshift of 0.16 GHz, and the transmission experiences a modulation with a modulation depth of 97%. The results in this work may be useful in application areas such as microwave modulating.