Double Spectral Electromagnetically Induced Transparency Based on Double-Bar Dielectric Grating and Its Sensor Application

Abstract

The realization of the electromagnetically induced transparency (EIT) effect based on guided-mode resonance (GMR) has attracted a lot of attention. However, achieving the multispectral EIT effect in this way has not been studied. Here, we numerically realize a double EIT-ike effect with extremely high Q factors based on a GMR system with the double-bar dielectric grating structure, and the Q factors can reach 35,104 and 24,423, respectively. Moreover, the resonance wavelengths of the two EIT peaks can be flexibly controlled by changing the corresponding structural parameters. The figure of merit (FOM) of the dual-mode refractive index sensor based on this system can reach 571.88 and 587.42, respectively. Our work provides a novel method to achieve double EIT-like effects, which can be applied to the dual mode sensor, dual channel slow light and so on.

1. Introduction

In the past few decades, the electromagnetically induced transparency (EIT) effect caused by quantum destructive interference between two different excitation paths in a three-level atomic system has attracted extensive and in-depth research [1,2]. The EIT effect is that under the effect of strong resonance coupling light, the opaque medium becomes transparent for weak probe light and is accompanied by extraordinarily steep dispersion. In other words, the resonance probe light can propagate through the medium without being absorbed [3]. EIT has potential applications in many fields, such as sensing [4,5,6], slow light [5,7], nonlinear optics [8] and cavity quantum electrodynamics [9]. However, the realization of the EIT effect in atomic systems requires extremely strict experimental conditions, which limits its practical application. Therefore, researchers have turned to classical optical systems to achieve EIT-like effects. G. Shevts and J.S Wertele described the EIT analog in a classic plasma [10], and more of an EIT-like effect was subsequently observed in the plasmonic metamaterial [11,12]. The EIT-like effect has also been successfully achieved in other optical systems, such as waveguide cavity structures [13,14] and photonic crystals [15]. Sun-Goo Lee’s group realized the EIT-like effect based on the guided-mode resonance (GMR) effect for the first time in 2015 [16], the system consists of two planar dielectric waveguides and a subwavelength grating and a narrow transparent window appears in the transmission dip when high Q and low Q resonant waveguide modes are coupled.

In recent years, the realization of the EIT-like effect based on GMR has attracted more and more interest of researchers. Sun-Goo Lee’s group successfully realizes the polarization-independent EIT-like effect in two photonic systems, which are both composed of two planar dielectric waveguides and a two-dimensional photonic crystal [17]. Sun Y and Chen H et al. reported a planar metamaterial based on GMR achieve the EIT-like effect with a Q value exceeding 7000 [18]. Han Y and Yang et al. reported that a GMR system with two subwavelength silicon grating waveguide layers achieves the EIT-like effect with an ultra-high Q factor of 288,892 [19]. However, as is known to us, the realization of multispectral EIT effects by the subwavelength grating structure based on GMR has not been studied according to available works.

In this work, we numerically simulate the double spectral EIT-like effect in a coupled GMR system that consists of two silicon grating waveguide layers (GWLs) on a SiO₂ substrate. In addition, both GWLs contain double-bar dielectric gratings with unequal lengths in a cycle. When the distance between the two GWLs meets the phase matching condition, the GMR system works as a Fabry–Perot (F–P) cavity and GWLs as the reflection boundary of the cavity thus two EIT peaks appear in the near infrared band due to the top GMR mode and the bottom GMR mode are intercoupled. The effects of two grating widths and the separation between the two gratings are also studied. After the structural parameters optimization, two EIT peaks present ultra-narrow FWHMs of ${10}^{-2}$ nm and ultra-high Q factors of ${10}^4$. In terms of the sensor, this system has ordinary sensitivity and high figure of merit (FOM) with compact structure and a simple manufacturing process. This double spectral EIT-like effect may not only be applied to the sensor, but also to slow light and nonlinear optics.

2. Structure and Simulation

The 3D finite-difference time-domain (FDTD) method was used to study the phenomenon of this optical system. During the simulation with FDTD, the mesh accuracy was set to λ/34 to ensure the convergence of the simulated results. The schematic of the GMR system with a double-bar dielectric grating structure is shown in Figure 1a. In terms of boundary conditions, X and Y were set to periodic, and Z was the perfectly matched layer (PML). The mesh accuracy in GWL₁ and GWL₂ were Δx = Δy = 5 nm and Δz = 10 nm, respectively. The inset of Figure 1b shows the real part of the refractive index value for SiO₂ [20] and Si [21] as a function of the wavelength. The background index of the system was n_s = 1.

Light polarized in the X direction (TM polarization) incidents vertically above the system in the near-infrared spectral band. The period P of grating was 500 nm. A SiO₂ layer with a thickness of 1885 nm was designed on the top of the GMR structure in order to reduce the reflection of light. The distance (d₂) between the two GWLs was designed to be 2137 nm. Gratings G₁₁ and G₁₂ had the same depths of D = 55 nm, and different widths of W₁ = 100 nm and W₂ = 230 nm, respectively. The separation between two Si gratings was g = 30 nm. The Si waveguide thickness was t = 435 nm. Meanwhile, structural parameters of GWL₁ and GWL₂ were consistent.

3. Results and Discussion

We first calculated the transmission spectrum of the structure with only single GWL₁ (GWL₂), as the green solid line (blue solid line) shown in Figure 1b. It is clear that the two different transmission dips appeared at the wavelengths of 1547.97 nm and 1553.08 nm. GWL₁ and GWL₂ had the same structure, but their optical properties were easily affected by the surrounding medium, thus the transmission spectra were slightly different. The SiO₂ cover layer had an influence on the resonance frequency. Adding a SiO₂ cover layer can reduce the impact of the surrounding medium on its optical characteristics to reach the same resonance frequency [18].

When GWL₁ and GWL₂ are both present, and their distance (d₂) met the phase matching conditions, the top GMR mode (in GWL₁) and the bottom GMR mode (in GWL₂) were intercoupled, Therefore, two different sharp EIT resonances appear in the two resonance dips respectively due to destructive interference [12,19,22]. The transmission spectrum is shown by the solid red line in Figure 1b. We refer to the two EIT peaks as EIT₁ and EIT₂. The transmission characteristics of their magnified views are shown in Figure 1c,d, respectively. The resonance wavelengths of EIT₁ (λ₁) and EIT₂ (λ₂) were 1548.12 nm and 1553.36 nm, respectively, and their corresponding full width at half maximum (FWHM) was 0.0441 nm (∆λ₁ in Figure 1c) and 0.0636 nm (∆λ₂ in Figure 1d). The Q factor was an important parameter of the resonant cavity, which is defined as follows:

$$\mathrm{Q}=\frac{\mathsf{\lambda }}{\mathrm{FWHM}}$$

(1)

There, their corresponding Q factors reached 35,104 and 24,423, respectively. Obviously, our work has two higher Q factors than another double EIT-like effects work, which has two Q factors of 950 and 216 [6].

The analogy between our system and the atomic EIT system can help us understand the physics of the double EIT-like effect [23,24]. A double-Λ five-level model of the double EIT-like effect in our system is illustrated in Figure 2a. In the system, |1〉 represents the ground state, the field of the bottom GMR mode corresponds to the probability amplitudes of atoms in the metastable states |2〉 and |3〉, the field of the top GMR mode corresponds to the probability amplitudes of atoms in the excited states |4〉 and |5〉 [19,23,25]. The control field refers to the coupling between the top GMR mode and the bottom GMR mode, and the probe field refers to the input of the top GMR mode. We observed that two sharp EIT-like windows appeared in the probe area due to the introduction of the control field. The energy level of |5〉 — |1〉 was higher than |4〉 — |1〉, hence EIT₁ corresponded to the destructive interference between two different transition pathways |1〉 → |5〉 and |1〉 → |5〉 → |3〉 → |5〉, EIT₂ corresponded to the destructive interference between two different transition pathways |1〉 → |4〉 and |1〉 → |4〉 → |2〉 → |4〉.

In order to help us understand the origin of the double EIT-like effect in this GMR system, electric field distribution diagrams of the GMR system near two EIT peaks (such as the transmission spectra of Figure 1c,d are given in Figure 2b. Near the EIT₁, electric field distribution at the off—resonant wavelength of EIT₁ 1547.98 nm (the blue dash I in Figure 1c) and 1548.24 nm (the blue dash III in Figure 1c) was very easily observed and excitation mainly occurred in GWL₁ (I in Figure 2b) or GWL2 (III in Figure 2b), which was consistent with the characteristics of GMR. The corresponding transmittance was very low since the GMR mode was easily coupled to electromagnetic waves in free space. Near the EIT₂, the electric field distribution at the off—resonant wavelength of EIT₂ 1553.12 nm (the blue dash IV in Figure 1d) and 1553.54 nm (the blue dash VI in Figure 1d) corresponded to IV and VI in Figure 2b. The principles were the same as near EIT₁.

At the EIT₁—resonant wavelength of λ₁ = 1548.12 nm (the blue dash II in Figure 1b) and the EIT₂—resonant wavelength of λ₂ = 1553.36 nm (the blue dash V in Figure 1d), the electric field distribution diagrams were II and V Figure 2b, respectively. Obviously, very strong oscillations occurred simultaneously in GWL₁ and GWL₂. When the incident light was at two EIT wavelengths, the light was reflected back and forth between GWL₁ and GWL₂, and electromagnetic energy was coupled into the two GWLs, so strong oscillations were excited through coupling. Once the top GMR mode and the bottom GMR mode are intercoupled, and this system works like as an F–P cavity [19,26,27], therefore, two very narrow transmission peaks could be obtained at the wavelengths of 1548.12 nm and 1553.36 nm in the transmission spectrum, where transmissions reached 96.23% and 94.48%, respectively.

The transmission spectra of the coupled GMR system with different d₂ are shown in Figure 3, and other parameters are the same as Figure 1. In the range of d₂ from 1970 to 2220 nm, with the increase of d₂, the two resonance wavelengths (λ₁ and λ₂) were both red-shifted. When d₂ was increased from 2020 (in Figure 3b) to 2120 nm (in Figure 3d), λ₁ was increased from 1547.67 to 1548.06 nm and λ₂ was increased from 1552.51 to 1553.22 nm. This verified that the GMR system worked like an F–P cavity and GWLs as the reflective boundary of the cavity when d₂ met the phase matching condition. Obviously, for the bottom GMR mode in GWL₂, a displacement of about one hundred nanometers could achieve a favorable coupling with the top GMR mode in GWL₁, two EIT peaks appeared in the two transmission dips when the F–P cavity was introduced. Generally, a phase matching condition corresponds to a resonance wavelength when the distance and refractive index are determined [19]. Here, it is worth mentioning that when the determined d₂ satisfies the phase matching condition, two EIT peaks with different resonant wavelengths can be generated simultaneously because the refractive index of Si is different at different wavelengths.

The transmission spectra and resonance wavelengths under different W₁ and W₂ are shown in Figure 4, and other parameters are the same as Figure 1. Distinctly, with the increase of W₁ from 85 to 145 nm or the increases of W₂ from 180 to 240 nm, both λ₁ and λ₂ appeared red shifted. For a clearer observation, we show two resonance wavelengths under different W₁ and W₂ in Figure 4c,d. When W₁ increased from 85 to 145 nm, λ₁ moved from 1547.33 to 1549.93 nm and λ₂ moved from 1552.16 to 1558.13 nm. When W₂ increased from 180 to 240 nm, λ₁ moved from 1545.44 to 1549.2 nm and λ₂ moved from 1549.41 to 1554.54 nm. Thus we could easily control the two resonance wavelengths by adjusting the parameters of W₁ or W₂. The reason for the red shift of the two resonance wavelengths is that as W₁ or W₂ increase, the fill factor increases, which changes the average refractive index of the grating layer [28].

The gap between the gratings G₁₁ and G₁₂ (gratings G₂₁ and G₂₂) is also an important parameter, being defined as “g”. The transmission spectrums under different g are shown in Figure 5 and other parameters are the same as Figure 1. Obviously, as g varied from 10 to 50 nm in steps of 10 nm, the resonance wavelength of EIT₁ (λ₁) appeared blue shifted, while the resonance wavelength of EIT₂ (λ₂) appeared red shifted. Two resonance wavelengths and two resonance wavelengths difference (λ₂-λ₁) when g changed from 10 to 50 nm are shown in Figure 5f. As g increased, λ₁ moved from 1549.93 to 1547.19 nm and λ₂ moved from 1552.55 to 1553.93 nm (as shown on the left vertical axis in Figure 5f), and “λ₂-λ₁ ” increased from 2.62 to 6.74 nm (as shown in the right vertical axis in Figure 5f). The resonance wavelength of the two EIT peaks could also be flexibly adjusted by adjusting the parameter g.

Sensing Performance

Considering the resonance wavelength difference between the two EIT peaks and the transmittance of the two EIT peaks (not shown here), we chose a structure with W₁ = 100 nm, W₂ = 230 nm and g = 30 nm to study the corresponding sensing performance. At the same time, the SiO₂ between GWL₁ and GWL₂ changed to the dielectric sample. In order to make the refractive index change of the dielectric more sensitive, d₂ was set to 3740 nm. The performance of the sensor was evaluated by two factors, sensitivity (S) and FOM [29]:

$$\mathrm{S}=\frac{\mathsf{\Delta }\mathsf{\lambda }}{\mathsf{\Delta }\mathrm{n}}$$

(2)

$$\mathrm{FOM}=\frac{\mathrm{S}}{\mathrm{FWHM}}$$

(3)

Here, S refers to the resonance wavelength shift caused by the change in the refractive index unit of the dielectric, and FOM represents the optical resolution of the sensor.

Figure 6 shows the transmission spectrum with a different refractive index of the dielectric, the insert shows the resonance wavelengths of two EIT peaks at a different refractive index of the dielectric. It is obvious that the resonance wavelengths of the two modes had a linear relationship with the refractive index of the dielectric. The changes in EIT₁ and EIT₂ with the refractive index of the dielectric are referred to as mode 1 and mode 2, respectively. Therefore, the average FWHM of the mode 1 and mode 2 from the refractive index of the dielectric from 1.440 to 1.452 was 0.051 nm and 0.061 nm. The S of mode 1 and mode 2 were 29.166 nm/RIU and 35.833 nm/RIU, respectively, so FOM of mode 1 was 571.88, and FOM of mode 2 was 587.42. Compared with the other sensors, this sensor had ordinary sensitivity due to the narrow line width of two EIT peaks [30]. However, it had higher FOM than other previous sensors [5,6,31,32], as shown in

Table 1. Figure of merit (FOM) compared with previous sensors.

Reference	FOM1	FOM2
[5]	42.00
[6]	330.00	281.00
[31]	39.00
[32]	20.00
this work	571.88	587.42

. So, this sensor had a super high optical resolution.

4. Conclusions

In summary, we proposed a GMR system with a double-bar dielectric grating structure to achieve a double spectral EIT-like effect. The F–P cavity was introduced due to the distance (d₂) between the two GWLs satisfying the phase matching condition, and two EIT peaks with ultra-narrow FWHM and ultra-high Q factors could be obtained. The influences of the width and separation of the double-bar gratings on two EIT peaks were also investigated. Meanwhile, the performance of the ultra-high FOM sensors based on this system was also studied. This work provided a way to achieve double spectral EIT-like effect, which has potential applications in the dual mode sensor, dual channel slow light and so on.