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Electromagnetically induced transparency (EIT) and Autler-Townes splitting (ATS) are two phenomena that can affect the transmission of a probe field in the presence of a stronger field, both yielding transparency in the absorption profile. Being able to discriminate these two similar but distinct phenomena is of vital importance. Here we propose a scheme to describe the EIT and ATS phenomena in a plasmonic system. The proposed system consists of one radiative resonator and one subradiant resonator in metal-insulator-metal waveguide, and the transition is observed from the ATS model to the EIT model through three qualitative regions as the coupling strength decreases. In addition, we apply the method proposed by Anisimov to the induced transparency spectrum in our model, and numerically discerning EIT from ATS based on the Akaike’s information criterion in a clear way.
© 2015 Optical Society of America
1. introduction
Quantum coherence in atomic and radiation physics has led to many interesting and unexpected results. For example, an atomic system prepared in a coherent superposition state yields the cancellation of absorption by atomic interference, which could be further used in various applications, such as coherent population trapping [1], lasing without inversion [2] and electromagnetically induced transparency (EIT) [3, 4]. Since the emergence of EIT in 1990, the phenomenon has been associated with quantum destructive interference between two excitation pathways that could control the optical response of an atomic system. We can in this way eliminate the absorption and refraction susceptible of a probe field at the resonant frequency during the transition by using a control field. Moreover, the EIT effect with a large reduction in group velocity and giant enhancement of Kerr nonlinearity provides many promising applications in nonlinear optical processes, ultrafast switching, and optical storage owing to the strong dispersion in the transparency windows [5, 6]. However, the practical realization of the original EIT is quite challenging, due to the strict limitation. In the past two decades, EIT has been observed in a variety of systems, including quantum dots [7], nanoplasmonics [8], superconducting circuits [9], metamaterials [10, 11], and optomechanics [12]. Among these physical systems, plasmonic devices have emerged as a novel candidate in constructing smaller sizes owing to their inherent property of strong field confinement. By far, there has been numerous investigations in the realization of a novel phenomena analog to EIT transmission can occur in the coupled resonator systems due to the coherent interference of coupled resonators [13–18 ].
Surface plasmon polarizations (SPPs) are the electromagnetic waves trapped on metaldielectric interfaces and coupled to propagating free electron oscillations in the metals. Recently, it has shown the great potential to control light and strong nonlinearity, attributing to its significant enhancement of optical field intensity and the ability of light manipulation in a nanoscale domain [19–27 ]. Plasmonic waveguides such as the metal-insulator-metal (MIM) structure paved the way for the plasmonic analog of EIT, due to their nanoscale manipulation and deep-subwavelength confinement of light [28, 29]. Moreover, MIM waveguide resonator systems blaze new roads for the realization of optical functionality in metallic nanostructures [30–35 ]. As discussed in [36], the EIT phenomenon occurring in the plasmonic waveguide-resonator system can be considered as two alternative ways: radiative (coupled to a bus waveguide) and subradiant (not coupled to the waveguide) resonators mutually coupled by being closely placed, or two detuned resonators coupled to single bus waveguide.
In general, EIT effect is observed in three-level atomic systems in which there are two coherent routes for absorption that can interfere destructively, thus leading to the transparency of the transmission spectrum. However, there is a similar effect on the transmission spectrum with EIT called Autler-Townes splitting (ATS) which is the field-induced splitting of the optical response and exhibits no interference effects [37–39 ]. In the presence of a stronger control field, one of the atomic transitions will split into two Lorentzian lines (a doublet), which can be probed by a weak field. And ATS has been investigated in atomic systems [40], molecular systems [41], and quantum dots [42]. This similarity of the spectrum attracts much confusion and discussions on the distinction between EIT and ATS, thus distinguishing the difference between the phenomena analog to EIT and ATS has become crucial, because of their differences in slow light, optical storage, and quantum information processing. Motivated by this, Abi-Salloum [43] set the threshold of separation between EIT and ATS in a unified study of four different three-level atomic systems, using the method in [44]. Later in 2010, Anisimov et al. [45] proposed an objective method to distinguish these two phenomena from experimental data for three-level atomic systems. Later, an experimental investigation of the transition between EIT and ATS was carried out by Giner et al. [46], resorting to the method proposed by Anisimov et al. [45]. Subsequently, Zhu et al. [47] discovered that EIT may occur and a crossover from EIT to ATS existed in hot molecules. In 2014, Peng et al. experimentally realized the discerning of EIT and ATS in coupled whispering gallery mode microtoroid resonators [48]. They demonstrated the pathways to all-optical analogue of EIT, ATS and Fano resonance, and clarified the transition between them.
In this paper, we consider the analogue of the radiative and subradiant resonators in MIM waveguides to the three-level atomic system in realizing the analogue of EIT in integrated plasmonics. The bus waveguide and two resonators perform as the counterparts of the ground state, the metastable state and the excited state in the atomic system. Evanescent coupling relies on the radiative and subradiant resonators. By changing the coupling strength between the two resonators, we numerically discern EIT from ATS based on our model by means of computational fitting technology [45].
2. The theoretical model of plasmonic waveguide-resonators coupled system
To illustrate the physical picture of the transparency resonance dip, we first consider the plasmonic waveguide-resonators coupled system consisting of two nanoresonators, which are separated by distance t as shown in Fig. 1. The first resonantor A with length L and width wr is evanescently coupled to the second resonator B with length d and width wc. The resonator B is side-coupled to the bus waveguide with a stronger coupling than that between the resonators, indicating it behaves as a radiative resonator. The width of the bus waveguide is w and the coupling distance between the resonator B and the bus waveguide is g. The resonator A can only couple to the bus waveguide through resonator B, working as a subradiant resonator. SPP waves are aroused on the metal interfaces and confined in the waveguide when a transverse-magnetic(TM) polarized plane wave is injected and coupled into the bus waveguide. As the SPP waves pass through the coupled bus waveguide, the energy can be coupled into the resonator B through the dielectric aperture. The metallic claddings are assumed as silver, whose frequency-dependent complex permittivity is determined by the Drude model [49],
where ε ∞ = 3.7, ωp = 9.1 eV,γ = 0.018 eV. Here, ε ∞ is the dielectric constant at the infinite frequency, γ and ωp denote the electron collision and bulk plasma frequencies, respectively. ω is the angular frequency of incident light in the vacuum.For the waveguide-coupled resonator, the localized resonance can be excited when the incident light approaches the intrinsic resonance frequency. By adjusting the separation between the two resonators, the transmission spectrum will exhibit a narrower or broader transparency dip at the resonance wavelength due to the destructive interference between the incident wave and escaped power from the resonator; this is the EIT-like transmission spectrum. The propagation and coupling losses are neglected and not involved in our simulations. The transmission properties of the waveguide system are numerically simulated by using the finite element method (FEM) with COMSOL Multiphysics.
In our case, the spectral features of waveguide-resonator systems can be investigated by the temporal coupled-mode theory [50]. The motion of the resonator B mode a with the central frequency ω 0 is
where κ 0 and κei represent the decay rate correlated to the internal loss of the resonator B and energy leaking into the waveguide and the resonator A. θi stands for the phase shift after propagation between the resonators and waveguide. The subscripts ± of S±ij(i, j = 1,2) denote the incoming and outgoing waves, respectively. The amplitudes of incoming and outgoing waves satisfy the following relationships:Before proceeding, it should be noted that the incoming waves to the resonator B should satisfy the relationships in a steady state:
where σ is the amplitude attenuation and ϕ is the phase shift term of the SPP mode, which could be expressed as ϕ (ω) = ωLRe(neff)/c +θ. Here, c is the speed of light in a vacuum and θ stands for the additional phase shift. neff denotes the effective refractive index of the fundamental SPP mode in the waveguide. It should be mentioned that the electromagnetic wave coupled into the resonant waveguide exhibits Fabry-Perot oscillations and satisfies the phase matching condition, thus we have ϕ (ωm) = mπ/2, where m is an integer and ωm is the resonant frequency of the standing-wave modes.In our case, we assume the light is injected from the left port of the bus waveguide (S +22 = 0). When θ 1 = θ 2, the transmission of the output port at the steady state can be derived as
Obviously, the transmission function depends on the decay rates κ 0, κe 1, κe 2 and the phase term ϕ. The presence of the intrinsic loss results in the nonzero transmission at the transparency dips, while has little impact on the width of transmission. The priority of our paper is to discern EIT from ATS, which relies on the coupling strength between two resonators, κe 1. For the sake of simplicity, we set κ 0 = 0 as an unchanged parameter with different κe 1 during the simulation. Fig. 2 illustrates the transmission spectra of the MIM plasmonic dual-resonator-coupled waveguide calculated by the analytical model with different coupling rates between the two resonators. When the decay rate κe 1 is zero, the effect between the two resonators can be neglected. A broad Lorentzian dip appears at the resonance wavelength of the side-coupled cavity. When we take κe 1 into consideration, a broad transmitted peak occurs. As κe 1 decreases, a narrower resonant transmitted peak appears between two transmitted dips and the linewidth of the spectrum relies on the smaller coupling rate.
Fig. 2 Transmission spectra of the MIM plasmonic dual-resonator-coupled waveguide calculated by using the theoretical modeling with different κe 1. The decay rates are taken to be κe 1 = κe 2 (the red line), 28κe 1 = 9κe 2 (the blue line), 28κe 1 = 3κe 2 (the green line) and 28κe 1 = κe 2 (the yellow line). Other parameters are kept as ϕ = 0.08ω, σ = 0.97 and κ 0 = 0.
3. Discerning EIT from ATS in the plasmonic waveguide-resonators coupled system
In order to give a detailed picture about the transition from EIT to ATS in the plasmonic waveguide-resonators coupled system, we numerically simulated the transmission characteristics on the coupling strength between the two resonators using the finite element method(FEM). One simple way of describing coupling strength is using the coupling gap between the resonators. The other parameters of the structure are chosen as L = 400 nm, d = 150 nm, w = wr = wc = 50 nm and g = 5 nm, which remain unchanged throughout the simulation. As illustrated in Fig. 3, we obtain the evolution of the transmission spectrum as the coupling distance t between the resonators increases from 26 nm to 50 nm with a step of 8 nm. The transparency peak reaches 0.7 when t = 26 nm and exhibits a decrease to 0.6, 0.46 and 0.32 as the coupling distance becomes larger. Meanwhile, it’s easily observable that narrower transmission dips appear, which is consistent with the theoretical result. These results clearly demonstrate that the larger t will result in a weaker and narrower transmission peak at the same wavelength, which implies a smaller resonance splitting due to weaker coupling between the two resonators.
To get more insight into the physics of the transparency spectrum observed, we further investigated the field distributions of magnetic field at the transmitted peak wavelength of 631.6 nm and transparency-dip wavelength of 610 nm and 659.2 nm with t = 26 nm, respectively. Fig. 4(a)–4(c) indicates the distribution of the magnetic field for the wavelengths of the two transmission dips and the transmission peak. The middle subplot in Fig. 4(b) corresponds to the transparency peak at 631.6 nm while Fig. 4(a) and 4(c) correspond to the dips at 610 nm and 659.2 nm, respectively. It is quite distinct that at the two transmission dips both the resonators are strongly excited. Nevertheless, the magnetic fields in the two resonators are in-phase at 610 nm, while out-of-phase at 659.2 nm. Notably, at the transmission peak only the resonator A is excited while the other resonator is not excited. The explanation describing this phenomenon can be given as the destructive interference experienced by different optical pathways from the bus waveguide to the two supermode resonances, which cancels the original resonance peak at the transmission spectrum.
Fig. 4 (a) The distribution of the magnetic field of the transmission dip for the wavelength λ1 = 610 nm. (b) The amplitude of the magnetic field of the transmission peak at λ2 = 631.6 nm. (c) The strength of the magnetic field of the other transmission dip at λ3 = 659.2 nm).
The occurrence of the transmission dip is a trade-off between EIT which corresponds to the destructive interference between the electromagnetic fields into the resonators and ATS which attributes to the strong coupling between two resonators modulating the phase and splitting the degenerated eigenmodes into two modes, both of which can be mediated by the coupling distance. Having the transparency spectrum at hand, we would like to put more emphasis upon the issue how to discriminate EIT from ATS on the behalf of the absorption profile of the plasmonic radiative and subradiant resonators in MIM waveguide, resorting to the computational fitting technology [51]. The absorption profiles for the EIT model attributing to the Fano interference can thus be written as
Similarly, the corresponding absorption for the ATS model with a splitting of the exited states can be expressed as
where C and C± stand for the amplitudes of the Lorentzian curves, γ and γ± represent the linewidth, respectively. δ and δ± denote the detunings from resonant frequency.Here exploiting the functions AEIT and AATS, we fit the absorption curves of the waveguide-resonators coupled system in our simulation, altering all the aforementioned parameters. The evolution of absorption, as well as how well these two generic models fit the absorption profile is depicted in Fig. 5. In the strong coupling regime as t = 26 nm, the resonator B is operating in a overcoupled regime. When the TM polarized wave is scanned across the resonance of the resonator, a broad and well-separated absorption dip appears. This peak is attributed to ATS effect (the blue solid line). As t further increases to reach t = 50 nm, the resonator B goes into the undercoupled condition, and the resonance peak becomes shallower and transforms into two within-reach resonance dips. This window arises from the destructive interference between the two optical pathways. One path passes through the resonator A, the other path bypasses the resonator B. The above results manifest that the absorption spectrum of the output port experiences a transition from ATS to EIT as the gap between the resonators becoming wider, i.e., with weaker coupling coefficient. According to the increment of the coupling distance, one can clearly see that the EIT model fits the absorption better in the weaker coupling regime, otherwise, the ATS model conforms much better.
Fig. 5 Absorption profiles (red dot curve) and fitting lines of the two models for two different values of the coupling distances. (a) t = 26 nm with a good fit to AATS (blue dashed curve) model with {C +, C −, γ +, γ −} = {52.5, 34.8, 27.9,18.7}. (b) t = 50 nm with a good fit to AEIT (green solid curve) with {C, γ, δ +, δ −} = {9.3, 12.5, 14.4, 12.1}.
To quantitatively discern the models, we then employ the Akaike’s information criterion (AIC) to fit our data. This criterion, quantifies the information lost when model Ai with Ki fitting parameters Ii= 2log(σ i 2)+2Ki (i = EIT or ATS), where Ki denotes the number of unknown parameters and σi is the variance of the likelihood function obtained from the considered model [51]. In our case, we have four unkown parameters in both models as can be seen from equation (6) and (7), thus having Ki=4 as a constant. The relative likelihood of model Ai out of two models is measured by the relative weights , as presented in Fig. 6. With the relative weights, we can obtain the Akaike weights of ATS and EIT models corresponding to Fig. 5(a) and 5(b) at this stage. When the coupling distance is t = 26 nm, the weights of ATS and EIT models is 0.9687 and 0.0313, respectively. As for the coupling distance 50 nm, the weight corresponding to the EIT model weighs over the ATS model, being 0.5755 and 0.4245, respectively. It better verifies the conclusion we’ve just got in the last paragraph. Readily visualized by the curve representation, the Akaike weights in our simulations for a plasmonic waveguide-resonators coupled system are in good qualitative consistency with the speculations given in [45].
Fig. 6 Akaike weights ωi as a function of coupling distance for ATS model (blue stars) and for EIT model (red circles).
Essentially, one can obtain three different regions of the absorption spectrum according to the value of coupling distances. The first is the ATS region (t < 29 nm), where the well-separated double dips are contributed only to the two different channels corresponding to the two Lorentzians. Then the ATS model decreases and a crossing is observed where ATS and EIT coexist (29 nm< t < 31 nm). The third region can be interpreted as the EIT region (t > 31 nm) where the quantum destructive interference dominates, as expected. The weights corresponding to ATS and EIT models saturate approximately at 0.46 and 0.54.
4. Conclusion
Here exploiting MIM waveguide and coupled resonators system, we numerically simulated the EIT and ATS effects. Our proposed system consists of one radiative resonator and one subradiant resonator in MIM waveguide, and we have analyzed and numerically observed the smooth transition from the ATS model to the EIT model through three qualitative regions as the coupling strength decreases in a plasmonic waveguide-resonators coupled system. The mutual coupling of the two resonators and the waveguide shows the splitting of the original resonance into two supermode resonances. And both the two distinct effects result in a transparency peak in the absorption window. Based on the Akaike information criterion, we can objectively clarify EIT and ATS in a clear way. This proposed new scheme demonstrates a great complement to the differentiation between EIT and ATS in integrated plasmonics using detuned resonators, and we expect that such electro-optical plasmonic circuits could be a future key component in chip-scale photonic circuits.
Acknowledgments
This work is supported by China National Natural Science Foundation Grant Nos. 11404031, 61205117, and 61471050, Beijing Higher Education Young Elite Teacher Project No. YETP0456, and the Fundamental Research Funds for the Central Universities No. 2014RC0903. The project was supported by Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications), China.
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