We numerically study a dielectric coupled guided-mode resonant (GMR) system, which includes two silicon (Si) grating waveguide layers (GWLs) stacked on CaF2 substrates. It is confirmed that the coupling between the top and bottom GMR modes starts once a Fabry-Perot (F-P) resonator is introduced, and electromagnetically induced transparency (EIT)-like spectral responses occur in the coupled GMR systems. A very narrow transparency window with a high-quality (Q) factor EIT-like effect of up to 288,892 was demonstrated. Furthermore, EIT-like response wavelengths can be flexibly designed in wide wavelength range by modifying either the GMR resonance frequencies or the space between two GWLs. Therefore, this EIT-like response in coupled GMR systems would pave the way towards novel sensors with extremely high sensitivity.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
In atomic physics, the coherent coupling of wide resonances and narrow resonances leads to quantum interference, providing a formula for electromagnetically induced transparency (EIT). A sharp resonance of nearly perfect transmission can appear in the wide absorption spectrum. The EIT effect discussed above requires gas-phase three-level atoms, which greatly limits its applications. Recently, a lot of attention has been paid to the fact that EIT-like effects can occur in classical oscillator systems [1,2]. One fascinating development of EIT-like effect is the realization in coupled microresonators [3–6], a waveguide side-coupled to resonators [7–10], nanoscale plasmonic systems [11–13] and metamaterial structures [14–20].
In recent years, there has been much research interest in coupling and interference effects in plasmonic nanostructures [14–20]. However, the metallic structures inevitably suffer from Joule losses in the optical frequencies, which limits the achievable quality (Q) factors in conventional plasmonic EIT systems to values about 10, significantly hampering device performance. Because the intrinsic material losses in dielectric metamaterials is very low, the Q factor of resonance is mainly controlled by manufacturing, and may reach hundreds or even higher. Optical resonances and their coupling effects in dielectric nanostructures are attracting more and more attention [15,17].
With the rapid development of the micro/nano fabrication techniques, optical microcavities with ultrahigh Q factor are widely studied [21,22], which have many potential applications ranging from fundamental physics to applications, such as slow light , cavity quantum electrodynamics [23,24], nonlinear optics [2,25] and sensing applications [18,26,27]. Exploring new schemes to realize high Q factor and EIT-like effect is particularly significant.
Guided-mode resonant (GMR) effect in waveguide grating structures composed of planar dielectric waveguide and sub-wavelength grating has attracted great interest, due to its useful frequency selection function [28–30]. If the resonant condition is met when such structure is illuminated by a beam of light, a high reflection or transmission peak with narrow bandwidth can emerge in the corresponding wavelength band. Resonant frequency can be flexible modulated by varying the appropriate structural parameters. The realization of the EIT-like effect based on the GMR effect is an interesting research topic [31,32].
In this paper, we numerically simulate the EIT-like behavior in a coupled GMR system, which includes two silicon (Si) grating waveguide layers (GWLs) stacked on CaF2 substrates. The interesting phenomena arising from the coupled GMR systems is the EIT-like spectral response, where a narrow transparency window appears in the stop band due to destructive interferences. The space between two GWLs acts as a Fabry-Perot (F-P) cavity bounded by the partially reflecting GWLs. It is demonstrated that the coupling between the top and bottom GMR modes starts once an F-P resonator is introduced. The Q factor of the EIT-like spectrum can exceed 288892, which is remarkable compared to that of other plasmonic systems such as planar plasmonic metamaterials [14–20]. Furthermore, the position and lineshape of EIT-like spectrum are strongly dependent on the geometrical parameters. Most importantly, the EIT-like resonant modes can be designed in wide wavelength range.
2. Structure and materials
The optical responses in the systems are investigated with the 3D finite-difference time-domain (FDTD) method. In FDTD simulations, the mesh accuracy was set “λ/40”, and there is no changes in simulation results by increasing the number of grids. The mesh size in grating is Δx = Δy = 10 nm, and Δz = 2 nm, respectively. The “Boundary conditions” was set “Periodic” at x and y, and “PML” at z. Inset of Fig. 1(a) shows the real part of refractive index value for CaF2  and silicon  as a function of wavelength. CaF2 substrate in our model is treated as semi-infinite and lossless. All the devices are surrounded by a background medium of index ns = 1.
Schematic of the coupled GMR systems and geometrical parameters is shown in Fig. 1(b). The proposed structure is designed to be polarized along the x direction (TM polarization) at the normal incidence light. Two Si GWLs (p = 1.6 μm) with a space D = 3.535 μm are set on a CaF2 substrate. We fixed the filling factor (f = 0.5), G1 (G2) grating depth (h = 40 nm) and waveguide thickness (H = 0.5 μm). Due to GMR properties are affected by surrounding media, a CaF2 cladding layer was designed by covering on the top GWL to make almost the same resonant wavelength in GWL1 and GWL2. Its thickness (1.8 μm) is optimized to give maximum transmission across the designed mid-wave infrared portion of the electromagnetic spectrum (3 ~4 μm).
3. Simulations and theory
We firstly proceed to calculate the transmission spectra of one single Si GWL1 (GWL2) by removing G2 (G1) grating from the structure (the planar waveguide layer still kept) as depicted in Fig. 1(b), and it can work as a GMR resonator. In Fig. 1(a), the transmission spectra for GWL1 (GWL2) (dark and blue lines) with broad resonances and high reflection display a typical GMR characteristic. Generally, GMR occurs under phase-matching of incident light to a leaky waveguide mode , and it can be highly reflected to the outside. Although GWL1 and GWL2 have identical structures, their spectral properties are affected by surrounding media and still slightly different. Especially, CaF2 cladding layer can affect the resonance frequency of the top GWL (GWL1). The resonance frequency of the top GWL will make a distinct shift when the CaF2 cladding layer was removed, and the coupling between GWL1 and GWL2 become invalid due to resonance detuning between them.
When there is a coupling between the top GMR mode (in GWL1) and bottom GMR mode (in GWL2), a sharp resonance of the EIT-like features (red line) is evident in the stop band of the GMR responses. Magnified view of the transmission features around resonant wavelengths for the coupled GMR systems are shown in Fig. 1(c). At the EIT resonant wavelength of λ = 3.410830 μm, full width at half maximum (Δλ) is 0.057 nm. The narrow linewidth indicates an ultrahigh Q factor in the coupled GMR system.
The interesting phenomena arising from the coupling of GMR modes is the EIT-like spectral response, where a narrow transparency window appears in the stop band due to destructive interferences [14,19]. The physics of the EIT-like effect can be better understood if we examine the analogy between our system and atomic EIT systems [2,20]. A prototype three-level model of EIT-like effect in our system is illustrated in Fig. 1(d). In the coupled GMR system, the field of the top (bottom) GMR mode corresponds to the probability amplitude of the atom at the excited (metastable) state [1,2]. The coupling between the top and bottom GMR modes corresponds to the control field, and the input of the top GMR mode corresponds to the probe field. We find that the sharp EIT-like window appear at the position of the control mode within the resonant region of the probe mode.
To better understand the origin of EIT-like behavior, we plot the electric field (a) - (c) and magnetic field (d) - (f) distributions for the coupled GMR system (p = 1.6 μm) near the resonant wavelengths [as indicated by blue dash lines in Fig. 1(c)] in Fig. 2. Near the resonant wavelengths, field distributions at the off EIT wavelengths of 3.410370 μm [in Figs. 2(a) and 2(d)] and 3.411120 μm [in Figs. 2(c) and 2(f)] are obviously observed. The main oscillations are excited inside the GWL2 and GWL1 respectively, and display a typical GMR mode characteristic. These spectra have low transmissions, because GMR mode can couple to the free-space electromagnetic wave easily.
At the resonant wavelength of λ = 3.410830 μm, the corresponding field distributions are shown in Figs. 2(b) and 2(e), and strong oscillations are simultaneously excited in the two GWLs. When the wavelength of incident light is at the resonance of EIT, the optical energy can be reflected back and forth between the coupled GMR resonators with high reflectivity, and electromagnetic energy is coupled into the two symmetric parallel GWLs and then strong oscillations are excited through coupling. Thus, the coupled GMR resonators shows a narrow transmission peak at 3.410830 μm in the transmittance spectra. Here, we note that the optical distance between GWLs is exactly 3/2 times as long as the EIT resonance wavelength (λ = 3.410830 μm). This may serve as direct evidence that the resonant enhancement occurs in an F-P resonator as plasmonic systems [13,15], and it is also demonstrated in subsequent discussions on transmittance spectra with varying optical distance between GWLs.
A displacement of a few hundreds of nanometers for GWL2 is sufficient to realize good coupling efficiency, and a sharp transparency window appears at the frequency of the high Q resonance. In Fig. 3, the transmittance spectra of the coupled GMR systems for p = 1.4 μm (a) - (e) with space D = 3.135, 3.235, 3.335, 3.435 and 3.535 μm, and for p = 1.6 μm (f) - (j) with space D = 3.335, 3.535, 3.585, 3.635 and 3.735 μm are presented respectively. In each spectra, we can see a vivid transparency window arising in the stop band. When the value of space D changes from 3.135 μm (3.335 μm) to 3.535 μm (3.735 μm) for p = 1.4 μm (p = 1.6 μm), a transmittance peak experiences a redshift near the center of the broad GMR response.
The EIT-like spectral response of the coupled GMR system can be altered in two ways shown in Fig. 3. First, it can be tailored by modifying the GMR resonance frequencies through a choice of their geometrical parameters (e.g., grating periods). Second, the peak transmittance can be modified by changing the value of space between two GWLs.
It is well known that even little losses can become important for a high value of Q factor. However, the losses of Si are extremely low in the studied spectral range . Thus, we have ignored losses of Si in the above simulations. As visible in the case of D = 3.335 μm (3.585 μm) for p = 1.4 μm (p = 1.6 μm), the transparency window at 3.177825 μm (3.411419 μm) displays a narrow linewidth, indicating an ultrahigh Q factor for the EIT peak. By further enlarging the influence of space between GWLs, The full width at half maximum (Δλ) for the EIT resonances are discussed. The Δλ for the EIT resonances shown in Figs. 3(b) - 3(d) and 3(g) - 3(i) are 0.346 nm, 0.011 nm, 1.007 nm, 0.057 nm, 0.012 nm and 0.152 nm, and corresponding Q factors are about 9181, 288892, 3157, 59839, 284285 and 22447 (Q = λ0/Δλ, where λ0 is the wavelength of maximum transmission at the EIT window).
The coupling between the top and bottom GMR modes starts once an F-P resonator is introduced, and the coupling strength have a great influence on the F-P cavity length. As a result, the transparency window around 3.177825 μm (3.411419 μm) grows in strength and becomes more and more prominent. Furthermore, the top and bottom GMRs are uncoupled, giving rise to a single transmittance dip in the spectrum (not shown).
It is demonstrated that EIT-like window appears in the stop band of the GMR responses, and the GMR resonance frequency can be modulated in a wide range through a choice of their geometrical parameters, such as grating period. In Fig. 4(a), GMR responses of the structure with one single GWL1 for different periods (p = 1.4, 1.6, 1.8 and 2.0 μm) are discussed. These spectra have reflection maximums more than 99% at resonant wavelength. The transmittance dips in each spectra corresponding to the resonant wavelength show a broad resonance with a relatively large Δλ, and make an obvious move from 3.177 μm to 3.833 μm with varying periods from 1.4 μm to 2.0 μm. Resonant wavelength with respect to grating period are shown in Fig. 4(b), and a linear fit is expressed by the red line . It means EIT-like response wavelengths can be flexibly designed in wide wavelength range by varying grating periods. Furthermore, the same EIT-like effect still occur when G1 and G2 are on top and bottom of the waveguide layers respectively, or both are on top (bottom) of the waveguide layers.
4. Summary and conclusion
In summary, we numerically simulate the EIT-like behavior in a coupled GMR system. The interesting phenomena arising from the coupled GMR systems is the EIT-like spectral response, where a narrow transparency window appears in the stop band due to destructive interferences. It is demonstrated that the coupling between the top and bottom GMR modes starts once an F-P resonator is introduced. The Q factor of the EIT-like spectrum can exceed 288892. Most importantly, EIT-like response wavelengths can be flexibly designed in wide wavelength range by modifying the GMR resonance frequencies or the space between two GWLs. An access to the coupling medium exposed between two coupled GMR resonators can facilitate new types of passive sensing structures and a dynamic control of their EIT-like response.
National Natural Science Foundation of China(NSFC) (60907003); Foundation of NUDT (JC13-02-13); Natural Science Foundation of Hunan Province (13JJ3001); Program for New Century Excellent Talents in University(NCET) (NCET-12-0142).
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