Fano Resonances in Location-Dependent Terahertz Stub Waveguide

Abstract

The quest for simpler structures that do not require the use of nanofabrication techniques and exhibit high Q Fano resonances has attracted growing interest in the past decade. Here, we study an arrangement of coupled resonator waveguides that can excite Fano resonances. The results show that an odd mode, except for the usual even mode, is excited due to the symmetry breaking of the position stub intersection. The superposition of the even and odd modes generates a Fano-shaped spectrum with a very narrow linewidth. Coupled mode theory is used to analyze these waveguide-based Fano resonances. Experimental results obtained using VNA and VDI show good agreement with theory and simulations. Such waveguide-based Fano resonances can be tailored and are simple in structure and have potential applications in narrowband filtering, sensing, lasing, and nonlinearity enhancement.

1. Introduction

Fano resonance, a fundamental resonance mode, was originally discovered in quantum physics to describe asymmetrically shaped photoionization spectral lines of atoms and molecules [1,2]. Due to the fact that wave interference is specific not only to quantum mechanics, a great number of classical systems exhibiting Fano resonance have since been identified. Most recently, the concept of Fano resonances was introduced to the field of photonics by analogy with the original atomic system, such as photonic crystals [3,4], symmetry-broken metamaterials/metamolecules [5,6,7,8,9], metallic hole arrays [10,11,12], nanospheres and clusters [13,14,15], metal–dielectric multilayer structures [16,17,18], waveguide channels with side-coupled stubs [19,20,21,22,23,24], etc. Among these structures, Fano resonances based on waveguide channels with stubs have received more attention due to various applications in impedance matching, demultiplexing [20], biomedical sensing [22,23], and signal switching [24]. Qi et al. theoretically presented an asymmetric plasmonic structure composed of a metal–insulator–metal waveguide and a rectangular cavity that support Fano resonance induced by the asymmetry of the plasmonic structure [19]. Recently, extensive work on Fano resonance in waveguides has been reported [25,26,27,28]. However, the vast majority of work is theoretical. In spite of great efforts made on Fano resonances in a bus waveguide that are side-coupled to stubs, which have been widely reported theoretically, experimental verification has been insufficiently explored due to the difficult and time-consuming fabrication of the nanostructures, preventing their real applications. Therefore, it is highly demanded to experimentally exploit waveguide channels with a side-coupled stub structure exhibiting the Fano line shapes that can be fabricated with a low-cost, fast and easy method.

Here, by utilizing a two-stub structure in a parallel plate waveguide (PPWG), we theoretically and experimentally demonstrated the existence of Fano resonances in the side-coupled bus waveguide. The PPWG is an ideal bus waveguide that can enhance inside–stub interaction [29,30,31]. In asymmetric PPWG structures (the central points of the stubs overlap with small shift), both symmetric (even) and anti-symmetric (odd) waveguide modes can be supported. With coupled mode theory, a Fano-shaped spectrum is generated through the interference of a discrete mode (odd mode) and a continuous mode (even mode). Such a new micro-assembly of stubs provides much easier control of the resonance in terms of its shape and characteristics.

2. Theory and Simulations

The proposed structure and a photograph of the fabricated waveguide are shown (depicted) in Figure 1, which is composed of a PPWG structure (Al plate, ε_m) with two rectangular stubs separated by an air gap S. The permittivity of Al is described with the Drude model. The length and height of the rectangular cavity are w and h_y, respectively. The shift central points of the stubs overlap with small shift L. The length of the total waveguide is 5 mm. Terahertz radiation is polarized parallel to the plates to excite TE modes.

In order to investigate the transmission responses of the proposed structure, its transmission spectra were numerically calculated with COMSOL Multiphysics^® 6.0 using the Wave Optics module and frequency-domain solver based on the finite element method (FEM). In the simulations, the gap of the waveguide was set to S = 1000 μm and fixed throughout this study. We consider the case where the two stubs have the same geometry. The width and height of the cavity was set to w = 1100 μm and h_y = 700 μm, respectively. Perfect-electric-conductor (PEC) boundary conditions were applied to the upper and lower aluminum surfaces, while port boundary conditions with PML (perfectly matched layer) and scattering boundary conditions were used at the input and output facets of the waveguide to launch the incident wave and absorb the outgoing radiation without spurious reflections. The size of the mesh elements must be at most λ/3 for reliable results, though it is possible to use smaller elements, even as small as λ/50 for parallel-plate waveguide simulations. Because of the limitations of the computing power of our workstation, there is an upper limit to the number of elements that can be handled by the simulation, and so higher-frequency simulations are increasingly difficult. More details on the finite element method simulation techniques used in this thesis can be found in reference [32].

The transmission spectra calculated by varying L are displayed in Figure 2a. At the symmetric position (L = 0), there is one prominent resonance (even mode), whose response is Lorentzian in the transmission spectrum with a frequency of 230 GHz. Upon increasing L, the power transmission shows two distinct resonances with significantly different linewidths. The THz wave transmitted different top and bottom cavities, which makes the symmetry of the structure broken, resulting in the generation of Fano resonances. In order to understand the underlying physics of the Fano resonances, the corresponding field distribution of |E_z|² at the transmission dips with L = 50 μm is simulated and displayed in the inset of Figure 2b. The unit of the field intensity is V/m, which represents a surface electric field z component. At both resonant frequencies, energy is confined almost equally in both grooves. However, the electric field has the opposite polarity in both grooves for the lower-frequency resonance (odd mode) with an asymmetric line profile and narrow resonance linewidths, while the field has the same polarity in the two grooves for the higher-frequency resonance (even mode). Figure 2a also lists transmission spectra with L = 100, 150, and 200 μm. With the increase in L, the linewidth is generally increased and the Q is decreased dramatically.

In the asymmetric PPWG structures (the central points of the stubs overlap with small shift), both symmetric (even) and anti-symmetric (odd) waveguide modes can be supported. When L = 0, only symmetric mode (even mode) is excited, which can be considered an ideal BIC. An ideal BIC is strictly a mathematical concept that results in an infinite quality (Q) factor, and it only exists in an ideal structure with a highly confined non-radiative mode that does not couple with the free space. Thus, ideal BIC states are symmetry-protected. When L > 0, both symmetric mode (even mode) and anti-symmetric (odd) mode are excited. Using coupled mode theory, a Fano-shaped spectrum is generated through the interference of a discrete mode (odd mode) and a continuous mode (even mode). With BIC theory, the structural symmetry of two rectangular stubs is broken, which can create a leakage channel and transform the ideal BIC into a quasi-BIC (super-cavity) radiating in the far field [33,34,35].

In order to provide an adequate theoretical explanation for the Fano phenomenon, the coupled oscillator model is employed to qualitatively understand the interference of the hybrid spectra. In this theory, the low-Q even mode provides a continuous spectrum, and the waveguide couples with a low-quality stub; the high-Q odd mode provides a discrete mode, and the two stubs act as a single inline coupled cavity. The interference of a discrete mode (odd mode) and a continuous mode (even mode) in this stub-coupled system is expressed as [36]

$${\displaystyle \frac{{\partial }^2{A}_e\left(t\right)}{{\partial t}^2}}+{\gamma }_e{\displaystyle \frac{\partial A_e\left(t\right)}{\partial t}}+{\omega }_e^2{A}_e\left(t\right)+{\kappa }^2{A}_o\left(t\right)=f_e{e}^{i\omega t}$$

(1)

$${\displaystyle \frac{{\partial }^2{A}_o\left(t\right)}{{\partial t}^2}}+{\gamma }_o{\displaystyle \frac{\partial A_o\left(t\right)}{\partial t}}+{\omega }_o^2{A}_o\left(t\right)+{\kappa }^2{A}_e\left(t\right)=f_o{e}^{i\omega t}$$

(2)

where (A_e, A_o), (ω_e, ω_o), and (γ_e, γ_o) are the scattering amplitudes, angular resonance frequencies, and dissipation factors of the even and odd modes, respectively. κ defines the coupling coefficient between the even and odd modes. The external force f_e excites only the even mode, while the narrow-resonance mode (odd mode) is excited only through the coupling interaction (f_o = 0). The coupled equations provide insights into the interactions that lead to Fano-type destructive interference in amplitudes A_e and A_o for the contrasting damping rates γ_b and γ_d of the even and the odd modes, respectively. After solving for the scattering amplitudes A_e and A_o in coupled Equation (1), the total transmission spectrum can be expressed in the following form:

$$t=1-K·Im\left({\displaystyle \frac{{\omega }^2-{\omega }_o^2+i\omega {\gamma }_o}{{\kappa }^4-\left({\omega }^2-{\omega }_e^2+i\omega {\gamma }_e\right)\left({\omega }^2-{\omega }_o^2+i\omega {\gamma }_o\right)}}\right)$$

(3)

where K is the normalizing constant. Fano resonance depends on the coupling coefficient κ, which in turn relates to parameter L in the side-coupled waveguide structures. Figure 2b depicts the amplitude transmission predicted by Equation (2), where the intrinsic loss terms for the even mode and the odd mode at L = 50 μm are γ_e= 1.24 × 10¹⁰ rad/s⁻¹ and γ_o = 5 × 10⁸ rad/s⁻¹, respectively. The theoretical results show good agreement with simulated ones, indicating that the origin of the Fano resonance is due to the interference of even and odd modes.

3. Experiments

To confirm the Fano resonance experimentally, we fabricated a waveguide with the geometry based on this two-stub geometry, with two 1100 × 700 μm grooves 1 mm apart (as shown in Figure 1b in detail and discussed above). A diagram of the design and a photograph of the fabricated waveguide are shown in Figure 1b. This structure is simple to machine, requiring only a machine shop mill and a saw blade of appropriate width. The waveguide geometry is on a more manageable scale, millimeters instead of 10 s of microns (with a tolerance of ±2 μm, exerting no impact on the performance of the devices). The only issue relating to the experimental result is the waveguide gap, which can be controlled with a micrometer. This waveguide was placed between the transmitter and receiver modules. The experimental setup is based on the end-fire coupling method. We used the Agilent N5227A PNA network analyzer (Keysight, Colorado Springs, CO, USA) with IF and RF cables and a set of WR-5.1 VNA extenders(Keysight, Colorado Springs, CO, USA), attached to an electric-controlled translation stage (the detailed parameters of the device can be referred to in [37]), to characterize the device transmission performances, as shown in Figure 3. After the calibration, a THz wave was coupled with the device under test from the left waveguide port. The input polarization was optimized for the TE mode. The output wave power was coupled from the right waveguide port and finally detected by the network analyzer.

Figure 4a depicts the measured power transmission spectrum of a side-coupled waveguide with L = 0, 50, 100, 150, and 200 μm in the frequency band from 160 GHz to 200 GHz. The bottom plot provides a closer look at the low-frequency dips, as shown in Figure 4b. We can obtain the following results from the experimental data: (1) For L = 0, the Fano resonance cannot be excited due to the symmetry of the top and bottom stubs. (2) The Fano-like resonance evolves once the symmetry of the structure is broken along the x axis. (3) A further increase in the displacement of the two stubs from the center results in the reduction in the Q factor, as shown in Figure 4c. This is due to the increased asymmetry/shift between the top and bottom stubs. The experimental results fit the simulation results well. To illustrate the good agreement between experiment and simulation, we overlay the results of the simulation (Sim.) and experiment (Exp.) for L = 50 μm in Figure 4d for direct comparison. The figure shows that the experimental result agrees well with the simulation. The minor discrepancies are mainly attributed to fabrication and measurement tolerances. There are also some slight deviations, which are mainly caused by fabrication and measurement tolerances. The Q factors in Figure 4c,d are calculated by Q = f₀/Δf, where f₀ is the central frequency of the Fano resonance and Δf is its full width at half maxima. We use the Fano formula (Equation (4)) [38] to fit the Fano resonance peak and extracted Δf and f₀ from the fitting parameters for the calculation of the Q value.

$$T={\left|t_B\right|}^2{\displaystyle \frac{{\left(\frac{f-f_0}{∆f/2}+q\right)}^2}{{\left(\frac{f-f_0}{∆f/2}\right)}^2+1}}$$

(4)

Figure 4d shows that the fitting curve agrees well with simulation, and the fitting parameters of the simulated Fano resonant of L = 50 μm are |t_B|² = 0.9, f₀ = 0.1798 THz, Δf = 0.00034 THz, and q = −0.1554.

4. Summary

We proposed a symmetry-broken side-coupled waveguide to achieve Fano resonance excitation. This system is composed of waveguide-based stubs, and the stubs are separated by an air gap. Simulation results show that by shifting bottom stubs, both of the even and odd modes can be excited in the stubs. Due to the interaction between the even and odd modes, the transmission spectra possess a sharp asymmetrical Fano-type profile. The Fano resonance exhibits different dependences on the overlapped length and can be easily tuned, which is also experimentally proved using the PNA network analyzer platform. In addition, the proposed waveguide generates an ultra-sharp Fano resonance, making it highly promising for sensing applications. For example, filling the bottom stub with liquids of different species or concentrations would modify the local refractive index. An increase in the refractive index induces a pronounced red shift in the Fano resonance, enabling the device to function as a micro-fluidic sensor [39]. The mechanism based on stub shift paves a new route toward exciting Fano resonance, and the utilization of the odd mode in the waveguide provides a new possibility for inducing Fano resonances.