1. Introduction

The quantum phenomenon of electromagnetically induced transparency (EIT) has been a subject of intensive investigations in recent years due to the EIT-associated features of strong dispersion and slow-light propagation within the transparency (spectral) window [1], promising a variety of potential applications, e.g. in nonlinear mixing and optical storage. However, there are several specific and strict restrictions on the realization of the original EIT (based on the quantum interference of atomic resonances), making the experimental realization of it rather challenging. Recently, a number of classical configurations have been suggested for the realization of EIT-like transmission under less demanding experimental conditions, including coupled dielectric resonators [2,3], metamaterial-induced transparency [4,5] and phase-coupled plasmon-induced transparency [6]. Among these approaches, on-chip photonic analogues of EIT in integrated photonic circuits are of special interest because potential EIT applications can be realized with guided optical devices, introducing novel functionalities into optical integrated circuits. Experimental waveguide-based realization of EIT-like transmission was reported using coupled Si ring resonators, featuring however quite large footprints of hundreds of square microns [3]. Plasmonic devices are more promising in constructing structures with smaller sizes because of their inherent property of strong field confinement. Two metal stripes (different in length) on top of a Si waveguide were suggested to be used as phase-coupled surface plasmon resonators that would ensure EIT-like transmission, resulting in considerably smaller footprints of ~μm² [6].

It should be noted that the phenomenon of EIT can be considered using two alternative ways: as resulting from the destructive interference between two pathways involving the bare, dipole-allowed and metastable, states or, equivalently, the doublet of dressed states (created by the strong pump radiation) representing two closely spaced resonances decaying to the same continuum [1,7]. While these two physical pictures are equivalent when dealing with the EIT in atomic systems, their realization with classical systems, whose responses are determined by their configurations and not electromagnetically induced as in the EIT, depends on the EIT mechanism that is imitated [8]. The first (bare-state) picture suggests employing radiative (coupled to a bus waveguide) and subradiant (not coupled to the waveguide) resonators that are mutually coupled by being closely placed [2]. Alternatively viewed, the EIT is achieved due to the cancelation of opposite contributions from two detuned resonances, which are equally spaced but with opposite signs of detuning from the probe frequency (with the detuning being close to the resonance linewidth), due to the Fano-like interference of the decay channels [7]. Consequently, the second (dressed-state) picture suggests using detuned resonators that are both coupled to a bus waveguide [3, 6]. In the latter realizations, however, the two detuned resonators were separated by an integer number of mode wavelengths because of the design constraints [3, 6], but such a separation is, in principle, not required for realization of EIT-like behavior.

2. Aperture-coupled Fabry-Perot resonators in MIM waveguides

In this paper, we demonstrate that on-chip EIT-like transmission can be realized with detuned aperture-side-coupled Fabry-Perot resonators (FPRs) in MIM waveguides. MIM waveguides utilize gap surface plasmons allowing extremely tight mode confinement with moderate propagation loss [9], and are thereby especially suitable for deep subwavelength photonic integration. Various plasmonic waveguides of the MIM type have been experimentally investigated, including V-groove channel plasmon-polariton waveguides [10] and plasmonic slot waveguides [11]. When MIM waveguides are terminated within a metal background, the reflectivity at the end is quite high (≈1) due to the small skin depth of metals in optical regime, which makes MIM waveguides ideal candidates for FPRs. FPRs with ultra-compact lateral dimension and ultra-short lengths can then be constructed, which lead to FPRs with ultra-small footprints. The length of the FPRs can be estimated with the following equation:

Note that, due to the tight mode confinement in MIM waveguides, resonator excitation via evanescent coupling between adjacent MIM waveguides becomes inefficient, when the gap between the bus and resonator waveguides is large enough for practical experimental processing [13]. Aperture coupling was proposed to realize efficient coupling without increasing the waveguide-resonator interaction length and, thereby, ring resonators with high extinction ratios and low modal volumes [14] can be achieved. Here, the aperture-coupling scheme is applied to the FPR excitation.

We start with the consideration of individual MIM-based FPRs utilizing the lowest cavity mode supported in the FPR, i.e. m = 1 in Eq [1]. The upper inset in Fig. 1(a) shows the schematic figure for the aperture-coupled FPR, where the background material in yellow is silver, whose permittivity is described by the Drude model${\epsilon }_r={\epsilon }_{\infty }-{\omega }_p^2/({\omega }^2+j\gamma \omega )$, with ε_∞ = 3.7, ω_p = 9.1eV and γ = 0.018eV (parameters obtained by fitting the experimental data [15] at the infrared frequencies). The insulator index is assumed to be 1.45 and the width w for both the bus waveguide and the FPR is 100nm. The gap g between the waveguide and the resonator is 50nm, which allows one to easily make the structure with state-of-the art nanofabrication facilities, and this value is assumed to be constant throughout this paper. Note that the magnetic field in the lowest mode is anti-symmetric along the central plane of the FPR, while the magnetic field into the FPR due to the aperture is symmetric. In order to excite this mode, one has to shift the aperture (over a distance s) away from the FPR center. The finite-difference time-domain (FDTD) method with fast Fourier transform is used to obtain the spectral response of the FPR. The transmission and reflection spectra, when the FPR length L is 300nm and the aperture (width d = 60nm) is 50nm away from the center of the FPR, are shown in Fig. 1(a). One can see that a strong resonance appears at the wavelength around 1.089μm with a high transmission extinction ratio (≈20dB). The quality factor at the resonant wavelength is not quite large due to the intrinsic loss in the FPR, as well as the fast decaying rate of power from the FPR to the bus waveguide. The latter can be simply adjusted by changing the aperture width d. Note that the transmission spectrum suffers from a much lower extinction ratio in this spectral range when there is no aperture (d = 0) and the gap g is 50nm because of the weak evanescent coupling. The magnetic field in the structure at the resonance is shown as the lower inset in Fig. 1(a), from which we can see that a strong cavity mode with one magnetic field node is formed in the FPR.

 

Fig. 1 Transmission and reflection spectra for an aperture-coupled FPR with length L being 300nm (a) and 600nm (b); In both figures, upper Inset is the schematic figure of the structure and lower inset shows magnetic field at resonance.

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For the resonance wavelength of 1.089μm, with Eq [2]. we can calculate the FPR length and find it to be 311nm, which is quite close to the value of 300nm that we use in the FDTD simulation. The small discrepancy can be attributed to the difference between the approximation of $k_{MIM}$and its exact value for the gap width of 100nm, the additional phase introduced by the aperture which is not taken into account in Eq [1], as well as some numerical errors in the FDTD simulations. However, we note that Eq [1]. can still provide a good estimation of the FPR length and thus can be used to roughly design the FPR. We can also use the 2nd order FPR mode, whose magnetic field is symmetric along the central plane of FPR and then the aperture can be placed at the center of the resonator. The transmission and reflection spectrums in this case where FPR length is 600nm, and aperture width d = 50nm are shown in Fig. 0.1(b). One can see that comparable results to Fig. 1(a) are achieved and the lower inset of Fig. 1(b) clearly shows the characteristic field distribution of second order mode.

3. Plasmon-induced-transparency with aperture-coupled FPRs

We further found that, when two aperture-coupled FPRs with slightly different resonant frequencies are placed on the opposite side of the bus waveguide, the EIT-like behavior shows up in transmission spectra (Fig. 2 ). Detuned FPRs (DFPRs) can be realized by choosing different geometrical parameters. Here, for simplicity, we consider DFPRs with different FPR lengths while keeping their width the same. We put the two apertures away with the same distance s from the center of the DFPR [see inset in Fig. 2(a)]. Numerical results, for the DFPR transmission spectra with one FPR length L ₁ = 300nm while the length L ₂ of the other FPR is 310nm/320nm, are shown in Fig. 2(a). The reflection spectra are omitted here for the sake of clarity. One can see that pronounced EIT-like transmission occurs for both cases, with a high-transmission peak appearing in a wide low-transmission band. The peak transmission is higher than 0.75 when the length deviation between the two FPRs is just 10nm, and increases as the detuning of resonant wavelengths becomes larger. The quality factor for the transmission peak corresponding to L ₂ = 310nm is about 60, twice of that for a single aperture-coupled FPR. Although the tradeoff between the quality factor and the peak transmission still exists, both of them can be adjusted by changing the aperture width, which in turn changes the coupling coefficient. Detailed investigations over this topic show that the optimal condition for the EIT-like spectrum is the case when the detuning δ of the two transmission-dip wavelengths from the central wavelength λ₀ (λ₁ = λ₀−δ,λ₂ = λ₀ + δ) equals to 0.5Γ, where Γ is the width of the transmission dip when there is only one FPR as shown in Fig. 1 with the resonance wavelength around λ₀. This criterion can be used as a guideline for the designing of these kinds of devices. It should be noted, that a similar condition of the detuning to be close to the broadening of two resonances is also found in the dressed-state picture of the EIT for atomic media [1,7] as well as in that realized with plasmonic metamaterials based on detuned electrical dipoles [8].

 

Fig. 2 (a) EIT-like transmission spectra for two FPRs with L₂ equaling to 310nm (solid line), 320nm (dashed line) and L₁ being kept as 300nm. Inset is the schematic figure of the structure. (b) Snapshots of the magnetic field when the light at three different wavelengths is incident into the structure in which L₁ = 300nm and L₂ = 310nm. The black dashed line indicates the incident plane.

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To get more insight into the physics of the EIT-like transmission observed, the FPR lengths were kept to be 300nm and 310nm, respectively, while one FPR was shifted along the other in the propagation direction. Simulations for the shift being 0nm, 50nm and 100nm showed that the EIT-like transmission deteriorated with the shift increase, indicating that one cannot gain by introducing the phase-coupling scheme [6], at least not for relatively small FPR displacements. Magnetic field distributions shown in Fig. 2(b) calculated at the two transmission dips (λ₁ = 1.089μm and λ₂ = 1.152μm) and the transmission peak (λ₃ = 1.119μm) for DFPRs with lengths L ₁ = 300nm and L ₂ = 310nm indicate that the main reason for the occurrence of the transmission peak is related to circumstance that the two FPRs resonate out of phase with each other at the intermediate frequency λ₃. It is the destructive interference between the electromagnetic fields in the two FPRs, which allows the incident light to propagate through the structure uninhibited. Judging from Babinet’s principle, we also believe that these FPRs with lengths smaller to the resonant wavelengths can be regarded as slot dipole antennas. Thus the DFPR configuration is similar to the case of detuned electrical dipoles [8,16], and the EIT-like transmission can be attributed to the interference between the two detuned slot dipoles, similar to what happens in the metamaterial-induced transparency. At the same time, the coupling is mediated by the bus waveguide between the two FPRs and enhanced with the two apertures. Also note in Fig. 2(b) that for λ₁ = 1.089μm and λ₂ = 1.152μm large magnetic fields exist on the left side of the incident plane which indicates large reflections while this is not true for λ₃ = 1.119μm.We further investigated two coupled DFPRs resonating at the 2nd order FPR modes, whose configuration is schematically shown as the inset in Fig. 3 . Here, as in Fig. 1(b), the apertures with width d = 50nm should be located at the center of the DFPRs. The transmission spectra, when L ₁ is kept at 600nm and L ₂ changes from 610 to 630nm, exhibit pronounced EIT-like transmission that is rather similar to that found in the previous case [cf. Figure 2(a) and Fig. 3]. Note that, in both cases, the whole structure is ultracompact with the footprint being smaller than 0.15μm² (0.25μm²) for the DFPRs resonating at the lowest (2nd order) modes, values that can be advantageously compared with those previously reported [3, 6].Having on-chip EIT-like transmission at hand, the next important issue to be considered is related to the expected phenomenon of slow-light propagation, i.e. to the level of group indexes that can be achieved with these structures. Using the continuous wave excitation in the FDTD simulations, a phase distribution for the magnetic field at a specific wavelength can be obtained throughout the whole structure. The effective phase index $n_p$ at this wavelength can then be calculated by considering the phase difference between the fields at bus waveguide positions before and after the DFPRs. Repeating this procedure, the wavelength-dependent effective phase index in the spectral region of our interest can be determined. The results calculated for both DFPR configurations under consideration are shown in Fig. 4 along with the effective phase index for a straight MIM waveguide obtained (with the same method) and plotted for comparison. One can see that, in both cases, $n_p$ varies rapidly near the transmission peak wavelength, deviating considerably from the MIM effective index. In the domain of normal dispersion, the group index $n_g$ can be calculated according to $n_g(\omega )=n_p(\omega )+\omega d{n}_p(\omega )/d\omega$ and is found to be up to 26 for the lowest FPR mode and 25 for the 2nd order mode at the wavelengths corresponding to the transmission maxima. The high group index achieved here makes the structure very promising for on-chip applications, e.g. compact delay lines and slow-light components. The bandwidth-delay product estimated with $\mathrm{\Delta }f\tau \simeq L\mathrm{\Delta }\lambda (n_g-n_{g0})/{\lambda }_0^2$, where L is the distance between two planes where the phases are used to calculate $n_p$ (and can also be considered as the unit cell length), ${\lambda }_0$is the wavelength where maximum $n_g$ is achieved, $\mathrm{\Delta }\lambda$ is the FWHM of the transparency window and $n_g{}_0$ is the group index at ${\lambda }_0$ for the MIM waveguide, is found to be 0.31 and 0.30 at the transmission-peak wavelength, for the two cases respectively. Note that these values are also favorably comparable to the bandwidth-delay product calculated for the waveguide configuration with spatially separated plasmonic resonators [6].

 

Fig. 3 EIT-like transmission spectra for two FPRs with L₂ equaling to 610nm (solid line), 620nm (dotted line) and 630nm (dashed line) and L₁ being kept as 600nm. Inset is the schematic figure of the structure and the two apertures are located at the center of the two FPRs.

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Fig. 4 Wavelength-dependant phase index obtained with the structure in which the solid line is for L₁ = 300nm and L₂ = 310nm while the dotted line is for L₁ = 600nm and L₂ = 610nm; the dashed line is the effective index for the MIM waveguide in this spectral range.

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4. Conclusion

In summary, we have demonstrated the on-chip plasmonic EIT analogue with well-pronounced EIT-like transmission in integrated plasmonic devices using aperture- coupled DFPRs realized with MIM waveguides. Group indexes of up to 26(25) can be achieved at near-infrared wavelengths with the footprints being less than 0.15μm² (0.25μm²) for the lowest (2nd order) FPR modes. It is relatively straightforward to employ the same design principle for constructing more sophisticated configurations known from (dielectric-based) photonics, e.g. balanced side-coupled sequences of resonators [17]. We believe that the plasmonic DFPR-based components will find broad applications in nanophotonics and introduce novel functionalities into guided plasmonic circuits, advancing further the subwavelength photonic integration with surface plasmons.

Note added in proof: After submission of our manuscript, we have learned about simulations demonstrating the EIT-like effect of optical transparency achieved in a similar system of side-coupled detuned waveguide-ring resonators [18].