Polarization-sensitive and active controllable electromagnetically induced transparency in U-shaped terahertz metamaterials

Abstract

Electromagnetically induced transparency (EIT) phenomenon is observed in simple metamaterial which consists of concentric double U-shaped resonators (USRs). The numerical and theoretical analysis reveals that EIT arises from the bright-bright mode coupling. The transmission spectra at different polarization angle of incident light shows that EIT transparency window is polarization sensitive. More interestingly, Fano resonance appears in the transmission spectrum at certain polarization angles. The sharp and asymmetric Fano lineshape is high valuable for sensing. The performance of sensor is investigated and the sensitivity is high up to 327 GHz/RIU. Furthermore, active control of EIT window is realized by incorporating photosensitive silicon. The proposed USR structure is simple and compact, which may find significant applications in tunable integrated devices such as biosensor, filters, and THz modulators.

Key words： electromagnetically induced transparency (EIT); metamaterial; polarization-sensitive; active optical devices; sensor

1 Introduction

Electromagnetically induced transparency (EIT) was initially found in atomic system. EIT originates from quantum interference, which causes the decrease of absorption and produces transparent peak at the absorption frequency [1–3]. EIT has interesting applications in slow light technology [4], optical communication [5], optical switch [6,7], and signal transmission [8]. However, the strictly experimental conditions, such as ultra-low temperature, strong magnetic field and high-power laser, have greatly limited the applications of EIT.

Metamaterials are artificially electromagnetic materials which have unique properties, such as negative permeability [9], small antenna [10,11], perfect lens [12], invisibility cloak [13,14], and metamaterial absorber [15–17]. Those properties are impossible to acquire by use of naturally materials. Plasmonic analog of EIT phenomenon achieved in metamaterials is usually called plasmon-induced transparency (PIT) [18,19]. The bright-bright and bright-dark mode couplings are generally studied in metamaterials [20–22]. The bright mode with low quality factor can be directly coupled by the external field. On the contrary, the dark mode with high quality factor cannot be excited except through the coupling of bright mode [23,24]. The bright and dark modes resonate at the same frequency. The necessary condition to achieve PIT is their contrasting linewidths. For example, EIT-like effect was obtained by a cut wire and two oppositely oriented split ring resonators though bright-dark mode coupling [25]. PIT has been demonstrated by graphene ring and strip resonators though bright-bright mode coupling [26]. The frequency detuning is needed for PIT effect based on bright-bright coupling. Very recently, EIT-like effect based on a bright-dark-bright coupling was reported [27]. It is demonstrated that transparency window of EIT-like effect can be tuned by changing the structural parameters of metamaterials [28]. However, the reconstruction of structure is complicated. It is necessary to dynamically modulate EIT-like effect. Active control of the transparency window by applied voltage or by thermal actuation are proposed [29–31].

In this paper, the number of EIT peaks can be controlled by changing the polarization of incident light. The proposed planar metamaterial is composed of concentric double U-shaped resonators (USRs). Moreover, when silicon bar is added to the USR, the amplitude of EIT peaks can be modulated by changing optical power. The designed structure suggests applications in tunable THz modulator, biosensor.

2 Structure design and theoretical model

Figure 1 shows one period unit of the proposed metamaterial, which consists of concentric double USRs. The lengths of outer and inner USRs are a = 140 μm, b = 105 μm, respectively. The widths of outer and inner USRs are w₁ = w₂ = 10 μm. The distance between the two USRs is s = 7.5 μm. The periods along x- and y-axis are P_x = P_y = 160 μm. The yellow part stands for copper, whose thickness is 3 μm. The relative permittivity of copper is given by ε_m(ω) = ε_∞-ω_p²/(ω²+ iωγ_p), where ε_∞ = 1 and ε_∞ is the dielectric constant. The angle frequency of the incident light is ω and plasma frequency is ω_p = 6.03 × 10¹⁵ rad/s. The damping rate γ_p = 5.56 × 10¹³ rad/s. The relative permittivity of the substrate is 1.98 and the thickness of the substrate is h₁ = 50 μm. In this paper, the EIT-like effect of metamaterial is numerically studied by COMSOL Multiphysics software. Periodical boundary conditions in xOz and yOz planes are adopted. The extremely fine mesh is used to discretize the space and ensure the convergence of the calculated results. Floquet ports are added on the top and bottom faces of the simulation space.

Fig.1 Schematic diagram of a unit cell consisting of concentric double USRs. P_x = P_y = 160 μm, a = 140 μm, b = 105 μm, w₁ = w₂ = 10 μm, s = 7.5 μm, h₁ = 50 μm

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Theoretical analyzation is done based on coupled Lorentz oscillator (CLO) model [32,33]. The displacements vectors for particles 1 and 2 are x₁ = c₁exp(iω₁t) and x₂ = c₂exp(iω₂t). We consider two particles interacting with the incident electric field E = E₀exp(iωt), which can be described by the following equations:

(1){{\displaystyle \ddot{x}}}_{\mathrm{1}}(t)+{{\displaystyle \mathit{\gamma }}}_{\mathrm{1}}{{\displaystyle \dot{x}}}_{\mathrm{1}}(t)+{{\displaystyle \omega }}_{\mathrm{1}}^{\mathrm{2}}{{\displaystyle x}}_{\mathrm{1}}(t)+{\mathrm{\Omega }}^{\mathrm{2}}{{\displaystyle x}}_{\mathrm{2}}(t)={\displaystyle \frac{Q}{M}}E,

(2){{\displaystyle \ddot{x}}}_{\mathrm{2}}(t)+{{\displaystyle \gamma }}_{\mathrm{2}}{{\displaystyle \dot{x}}}_{\mathrm{2}}(t)+{{\displaystyle \omega }}_{\mathrm{2}}^{\mathrm{2}}{{\displaystyle x}}_{\mathrm{2}}(t)+{\mathrm{\Omega }}^{\mathrm{2}}{{\displaystyle x}}_{\mathrm{1}}(t)={\displaystyle \frac{q}{m}}E,

where (ω₁, ω₂) and (γ₁, γ₂) are resonance angular frequencies and loss factors of the bright and the quasi-dark modes, respectively. Ω stands for the coupling strength between the bright and quasi-dark particles. (Q,q) and (M,m) are effective charge and effective mass, respectively. External forces are denoted by f₁= Q/Mand f₂= q/m, which means free space coupling strengths of bright and quasi-dark modes with the incoming light. Dimensionless constants A and B are defined to describe the relative coupling of incident light with bright and quasi-dark modes A= Q/qand B= M/m.

Solving the above coupled equations, we obtain

(3){{\displaystyle x}}_{\mathrm{1}}={\displaystyle \frac{{\displaystyle \frac{B}{A}}{\mathrm{\Omega }}^{\mathrm{2}}+({\omega }^{\mathrm{2}}-{{\displaystyle \omega }}_{\mathrm{2}}^{\mathrm{2}}+\mathrm{i}\omega {{\displaystyle \gamma }}_{\mathrm{2}})}{{\mathrm{\Omega }}^{\mathrm{4}}-({\omega }^{\mathrm{2}}-{{\displaystyle \omega }}_{\mathrm{1}}^{\mathrm{2}}+\mathrm{i}\omega {{\displaystyle \gamma }}_{\mathrm{1}})({\omega }^{\mathrm{2}}-{{\displaystyle \omega }}_{\mathrm{2}}^{\mathrm{2}}+\mathrm{i}\omega {{\displaystyle \gamma }}_{\mathrm{2}})}}{\displaystyle \frac{Q}{M}}{{\displaystyle E}}_{\mathrm{0}},

(4){{\displaystyle x}}_{\mathrm{2}}={\displaystyle \frac{{\mathrm{\Omega }}^{\mathrm{2}}+{\displaystyle \frac{B}{A}}({\omega }^{\mathrm{2}}-{{\displaystyle \omega }}_{\mathrm{1}}^{\mathrm{2}}+\mathrm{i}\omega {{\displaystyle \gamma }}_1)}{{\mathrm{\Omega }}^{\mathrm{4}}-({\omega }^{\mathrm{2}}-{{\displaystyle \omega }}_{\mathrm{1}}^{\mathrm{2}}+\mathrm{i}\omega {{\displaystyle \gamma }}_{\mathrm{1}})({\omega }^{\mathrm{2}}-{{\displaystyle \omega }}_{\mathrm{2}}^{\mathrm{2}}+\mathrm{i}\omega {{\displaystyle \gamma }}_{\mathrm{2}})}}{\displaystyle \frac{Q}{M}}{{\displaystyle E}}_{\mathrm{0}}.

The linear susceptibility (c) is then written as

(5)\chi ={\displaystyle \frac{k}{A^{\mathrm{2}}B}}\left({\displaystyle \frac{A(B+\mathrm{1}){\mathrm{\Omega }}^{\mathrm{2}}+A^{\mathrm{2}}(({\omega }^{\mathrm{2}}-{\omega }_{\mathrm{2}}^{\mathrm{2}})+B({\omega }^{\mathrm{2}}-{{\displaystyle \omega }}_{\mathrm{1}}^{\mathrm{2}}))}{{\mathrm{\Omega }}^{\mathrm{4}}-({\omega }^{\mathrm{2}}-{\omega }_{\mathrm{1}}^{\mathrm{2}}+\mathrm{i}\omega {{\displaystyle \gamma }}_{\mathrm{1}})({\omega }^{\mathrm{2}}-{{\displaystyle \omega }}_{\mathrm{2}}^{\mathrm{2}}+\mathrm{i}\omega {{\displaystyle \gamma }}_{\mathrm{2}})}}+i\omega {\displaystyle \frac{A^{\mathrm{2}}{{\displaystyle \gamma }}_{\mathrm{2}}+B{{\displaystyle \gamma }}_{\mathrm{1}}}{{\mathrm{\Omega }}^{\mathrm{4}}-({\omega }^{\mathrm{2}}-{\omega }_{\mathrm{1}}^{\mathrm{2}}+\mathrm{i}\omega {{\displaystyle \gamma }}_{\mathrm{1}})({\omega }^{\mathrm{2}}-{{\displaystyle \omega }}_{\mathrm{2}}^{\mathrm{2}}+\mathrm{i}\omega {{\displaystyle \gamma }}_{\mathrm{2}})}}\right).

The transmission coefficient (T) is obtained from T = 1 - Im(c).

3 Results and discussion

The incident light propagates along z direction and enters the metal plane at normal incidence. The electric field E is oriented along x-axis. The simulated results are plotted in Fig. 2. When a unit cell consists of only one USR, there is a dip in the transmission spectrum, as shown in Figs. 2(a) and 2(b). The corresponding resonant frequency is 0.32 THz for outer USR and 0.45 THz for inner USR. Both modes can be excited directly by the incident light. Considering their small difference in linewidth, we regard them as bright modes. The resonance frequency has a blue shift with the decrease of USR dimension. This can be explained by resonance function. It is given by the following simple relation:

(6){{\displaystyle \omega }}_{\mathrm{0}}\mathrm{Re}({{\displaystyle n}}_{\mathrm{e}\mathrm{f}\mathrm{f}}){{\displaystyle L}}_{\mathrm{e}\mathrm{f}\mathrm{f}}/c+{{\displaystyle \phi }}_r=\mathrm{2}m\mathrm{\pi },

where ω₀ is the resonant frequency of the cavity. Re(n_eff) is the real part of effective index for surface plasmon polaritons in USR. L_eff denotes the effective cavity length for one trip in USR. j_r means the phase shift caused by the reflection on the USR facets during one trip. Positive integer m stands for the order of resonant mode. Equation (6) shows that the resonance frequency becomes bigger with the decrease of USR length. This analysis agrees with the above simulation.

Figure 2(c) shows the transmission spectrum of concentric double USRs. It is observed that a transparency window appears among two dips. EIT-like phenomenon is observed. Two dips are denoted by dip₁ and dip₂ whose frequencies are 0.32 and 0.45 THz, respectively. The peak frequency is 0.395 THz. The highest transmittance is up to 0.82. To further understand the intrinsic mechanism of EIT characteristic, the distribution of electric field is displayed in Figs. 2(d) and 2(f), in which red arrows show the direction of the surface current. The electric field distribution at dip₁ shows that strong field energy is located around the outer USR. The current density in outer USR is greater than that in inner USR. At the frequency of dip₂, field energy is mainly confined in the inner USR, and the current density in the inner USR is more intense. The directions of current flow at two dip frequencies are clockwise. At peak frequency, the energy is confined in the area between two USRs. The redistribution of energy implies that the field in the outer USR has an influence on the field in the inner USR by near field coupling. The direction of surface current in inner USR keeps clockwise while the surface current in the outer USR is counterclockwise. The coupling between the bright and bright resonators leads to the induced transparency window.

Fig.2 Transmission spectra for (a) only the outer USR, (b) only the inner USR, and (c) concentric double USRs. The electric field distribution of concentric double USR for (d) dip₁ at 0.32 THz, (e) peak at 0.395 THz and (f) dip₂ at 0.45 THz

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The influence of USR dimension on EIT transmission spectrum is investigated. Figure 3 shows the transmission spectrum under different length a and b of USRs. As the length a increases, dip₁ gradually moves to lower frequency and dip₂ remains unchanged. Meanwhile, the amplitude of peak increases. With the increase of b, the position of dip₁ does not change, but dip₂ has a redshift. The transmittance of peak gradually decreases. This phenomenon can be explained based on the energy distribution of field. Figures 2(d) and 2(f) indicate the position of dip₁ is more affected by the outer USR and dip₂ is more affected by the inner USR. Moreover, Eq. (1) shows that the resonance frequency is dependent on the length of cavity.

Fig.3 Simulated transmission spectrum with different lengths of USR. (a) Length a of outer USR; (b) length b of inner USR. The other geometric parameters remain unchanged

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The CLO model is adopted to analyze tunable transmission property in Fig. 3. Because of the increased length a (or b), the resonance frequency ω₁ (or ω₂) will decrease. We theoretically analyze the evolution of transmission with frequencies ω₁ and ω₂. When resonance frequencies ω₁ and ω₂ vary with USR dimension, the loss factors of resonators used in the fitting change. The fitted transmission spectrums are plotted in Fig. 4. The analytical results are in good agreement with the numerically simulated results. As ω₁ increases, dip₁ gradually moves to high frequency and dip₂ remains unchanged. With the increase of ω₂, the position of dip₁ does not change, but dip₂ moves to high frequency.

Fig.4 Fitted transmission spectrum with different (a) ω₁ and (b) ω₂

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The utilization of THz wave in biosensor attracts much attention because of its non-destructive detection of cells. Normally refractive index sensors rely on the change of optical properties caused by the variation in refractive index of detection objects. Narrow EIT-like lineshape in metamaterial has the advantage of improved sensitivity. To study the sensing performance, different analytes are placed on metamaterials. The obtained transmission spectra are presented in Fig. 5(a). The variations in refractive index produce the shift in position of EIT-like peak. The frequencies of dip₁/peak/dip₂ shift from 0.320 to 0.305 THz, 0.395 to 0.380 THz, and 0.450 to 0.435 THz, respectively. They show a trend of redshift when the refractive index increases from 1.00 to 1.06. Figure 5(b) presents the resonance frequency changes of refractive index of the analyte. The frequency shift of the dip₁/peak/dip₂ is approximately linearly related to refractive index. The sensitivity of refractive index sensor is defined by S = \mathrm{\Delta }f/\mathrm{\Delta }n [34], which denotes the change in frequency of the unit refractive index (RIU). For dip₁/peak/dip₂, the obtained sensitivity S are 250, 250, 250 GHz/RIU, respectively. The sensitivity is comparable to previously reported THz metamaterial sensors [35,36].

Table 1 presents the comparison of our sensor with those previously reported. Considering the working frequencies of sensors are different, we define relative sensitivity to evaluate the sensing performance: S* = S/f₀, where f₀ is the resonance frequency. The working frequency band, refractive index sensitivity S, and relative sensitivity S* are given. Our proposed sensor has good sensing performance.

Fig.5 (a) Spectral transmission at different refractive indices; (b) resonance frequency of dip₁/peak/dip₂ versus the refractive index

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Tab.1 Comparison of performance of reported EIT-based sensor

	our work	Ref. [27]	Ref. [36]	Ref. [37]	Ref. [38]
working frequency band/THz	0.1−0.7	1−3	0.4−0.7	20−36	15−45
sensitivity/(THz·RIU−1)	0.25	0.59	0.112	26.6	6.4
sensitivity (S* = S/f0)	0.633	0.32	0.164	0.85	0.285

Fig.6 (a) Transmission spectrum under different polarizations; (b) transmission spectra at different refractive indices. The polarization angle q = 60°

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Figure 6(a) shows the transmission spectrum at different polarization angle. It is obvious that EIT-like effect is polarization-sensitive. For y-polarized THz field (corresponding to q = 90°), EIT transparency window can still be observed but it has a blueshift about 0.365 THz compared with that of x-polarization (q = 0°). At other polarization angles, multiple transparency windows and dips are observed. This provides a simple method to obtain multi-channel filtering since it does not need the reconfiguration of geometry. Furthermore, asymmetric lineshape appears around 0.81 THz as the polarization angle changes from 30° to 75°. This kind of lineshape is the typical characteristic of Fano resonance [39]. We attribute the generation of Fano resonance to the combined effect of x-polarized and y-polarized incident light. Take the example of q = 45°. In this case, electric field will decompose into two components along x and y axis. Both x-polarized and y-polarized light have a dip around 0.91 THz in transmission spectra. The combined impact leads to the change in line shape. The sharp and asymmetric Fano lineshape is high valuable for improving sensitivity [40]. The transmission spectra with different refraction index are shown in Fig. 6(b) when polarization angle is 60°. Refractive index increases from 1.00 to 1.15 with spacing of 0.05. The obtained sensitivity is high up to 327 GHz/RIU, which is twice as much as the above result. The sensing performance is highly improved.

Silicon bars are added to the U resonators to form closed squares, as shown by the red part in Fig. 7(a). The constant permittivity of Si is 11.7. Silicon is photosensitive whose conductivity is pump-power-dependent. A simple conductivity model of silicon is adopted. The photoconductivity d_Si varies from 1 to 4000 S/m. Figure 7(b) shows evolution of transmission spectra with the conductivity of the silicon. When d _Si is gradually increased, the amplitude of transparency window gradually is reduced and EIT window almost disappears at 4000 S/m. The transmittance at 0.386 THz decreases from 0.816 to 0.313. Dynamical modulation of the EIT window is realized. Define modulation depth DT/T₀ = (T₀−T_p)/T₀, where T₀ is the transmission of the sample without external light pump and T_p is the transmission with applied optical power. The modulation depth for peak amplitude is 61.8% when d_Si = 4000 S/m. This attractive modulation may provide prospective applications in THz imaging, communications, and so on.

Strong dispersion is a remarkable characteristic of EIT response. Slow light effect is physically related to large group index which can be produced by extremely steep normal phase dispersion. Figure 7(c) depicts the simulated phase spectra at different silicon conductivity. Strong phase dispersion around the transmission window is observed. With the increase of d_Si, the phase shift changes. The group delay (t_g) can be retrieved from t_g = - dj(w)/dw, w = 2pf, where j(w) is the transmission phase shift. Figures 7(d) shows the group delay as a function of frequency at different silicon conductivity. It is observed that at the resonance frequencies of w₁ and w₂, group delay is negative. When d _Si = 1 S/m, large positive group delays are obtained around the transparency peak, indicating potential applications in slow-light devices. With increasing the conductivity, group delay is reduced. Thus flexible control on group delay and slow effect are achieved.

Fig.7 (a) Geometric diagram of structure. The red region is photosensitive silicon; (b) transmission spectrum, (c) phase shift, and (d) group delay of the EIT metamaterial with increasing the conductivity d_Si

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4 Conclusions

In conclusion, we have demonstrated EIT-like effect by two concentric USRs based on the interaction between bright-bright modes. The transparency window and dips have a redshift with the increase of USR length. Multiple resonance dips are obtained with the change of polarization angle of incident light, which can be used for multi-channel filter. Fano lineshape is observed at certain polarization angles. The polarization-sensitive property may find its applications in flexible tuning. The sensing performance based on EIT-like and Fano resonance is discussed. The sensitivity is about 327 GHz/RIU. In addition, active control on EIT-like effect is demonstrated by adding photosensitive silicon bars into unit cell. The amplitude and phase of EIT window are adjusted by changing the pump power. Thus dynamically modulation is achieved under photoexcitation. This property may open up a new way in designing active devices such as sensors and modulators.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 11104200) and the Natural Science Foundation of Tianjin (No. 18JCYBJC17000).

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