We have studied electromagnetically induced transparency (EIT) in metamaterials for various schemes corresponding to those in an atomic medium. We numerically calculate a symmetric dolmen scheme of metamaterials corresponding to a tripod model of EIT-based optical switching and illustrate plasmonic double dark resonances. Our study provides a fundamental understanding and useful guidelines in using metamaterials for plasmonic-based all-optical information processing.
© 2010 OSA
Surface plasmon polaritons (SPPs) have been intensively studied from a simple transmission scheme of SPPs in a narrow metal strip interfacing with a dielectric material  to complicated SPP device applications . Localized surface plasmons have demonstrated interesting physics of resonance and fluorescence [3,4]. Emerging research in plasmonic mode control by means of light offers a step toward light-controlled nano optics in metamaterials. A recent observation of optical field-based SPP switching is a good example of light controlled plasmonics, where the physics lies in the saturation phenomenon . The saturation phenomenon, however, is limited in the strong field. Nano photonics inherently works at a low power limit.
Electromagnetically induced transparency (EIT) is a direct result of destructive quantum interference between two pathways induced by another light [6–8], where zero absorption probability at line center is obtained . Thus, EIT is free from the saturation phenomenon and is applied for ultra efficient nonlinear optics [10,11]. Group velocity control based on EIT has also been intensively studied for fundamental physics [12,13], as well as for various applications such as entanglement generation  and photon logic gates . Hence, EIT can be applied for surface plasmon-based nano photonics requiring low light power.
Recently, classical EIT based on the localized SPPs in metamaterials has been demonstrated in both theory and experiment [16,17]. Based on these results, we have studied EIT-like phenomena for two-, three- and four-level models using the finite-difference time-domain (FDTD) method. More interestingly, we illustrated plasmonic double dark resonances in our four-level tripod structure and this is explained as a result of perturbation by which the dark state destroys the transparency. It provides useful guidelines for plasmonic EIT in metamaterials and paves the way for its applications in delay lines, slow-light devices, selective storage, and optical switching.
2. Simulation method
For the study of EIT in metamaterials, we use the FDTD method  for numerical calculation, using Meep, a free software package [19,20]. The permittivity of silver was modeled using the Drude formula: , where the electric plasma frequency and the scattering frequency γ are rad/s and Hz, respectively [21,22]. The dispersion is implemented by the Drude-Lorentz model in Meep : , where is the electric conductivity, and are user-specified, and is a user-specified function of position giving the strength of the n-th resonance. To implement the Drude model, we take , , , very small , and the strength of resonance, , large to cancel in the numerator. For the unit cell, the perfectly matched layer absorbing boundary conditions are imposed in the propagation direction of light, and the Bloch-periodic boundary conditions are imposed on the other two directions perpendicular to the propagation axis. We discretized the unit cell including the metal strips and the surroundings with a high resolution, corresponding to 158.3 pixels/wavelength, in order to access to a fine meshing and a satisfactory convergence. The unit cells are arranged periodically, and the spatial separation of the unit cells (e.g., 200 nm in the case of the four-level structure) are such that the in-plane couplings are negligible and only zeroth-order transmission is investigated free from diffraction.
For the atom systems, simulations were carried out with a density-matrix approach. The time-dependent density matrix equations were derived from the interaction Hamiltonian.
3. Two-level system
To study EIT-like features in a metamaterial, we study a dipole-allowed transition in a single metal strip as a basic building block. Hecht et al. proposed that a single metal strip is likely to be resonantly excited by incident light . The electrons of the metal strip are driven by the time-varying electric field of light into a collective oscillation (SPP) that formed a standing wave with the incident light which works as an optical dipole antenna . In other words, this type of antenna adequately couples with the optical field, where a two-level atomic system is resonantly excited by the optical field whose energy is the same as the atom system. The Hamiltonian and the time varying density matrix operator are given by:
From the Liouville equation, we obtain the following equation for the density-matrix elements:Fig. 1 . The resonant wavelength of the rectangular antennas is shown in Fig. 1(b) as the function of the strips’ length B for. The plasmon resonance linearly depends on the length B. The dependence of the resonant wavelength on the width A is presented in Fig. 1(c), where the resonant wavelength is inversely proportional to A, resulting in a blue shift which agrees with the expectation that the resonant wavelength depends only on the aspect ratio in a quasistatic limit (i.e., the frequency of plasmon resonance only depends on the antenna shape via the aspect ratio, , if ) . As a general rule, the validity of quasistatic limit for the nanoantennas is also inspected, as shown in Fig. 1(d). Not all curves converge upon a common universal curve, which implies that the quasistatic limit does not hold perfectly here. However, we note that the divergence is relatively small at first and becomes larger with the increase of the aspect ratio. This discrepancy means that the approximation of the quasistatic limit is applicable only when the strips have a small aspect ratio.
For the studies of corresponding relations using a two-level atomic system as shown in Fig. 2(a) , we perform numerical simulations in Fig. 2. In Fig. 2(b), the absorption line becomes broadened as the decay rate γ increases, corresponding to that of atomic system in Fig. 1(c). Thus γ in the atomic system is analogous to A in rectangular nanoantennas. The blue shift corresponds to the increase of the detuning δ in Fig. 2(c). Thus, we can derive in the rectangular nanoantennas that a larger width A gives a bigger blue shift. As mentioned above, the decay of metallic nanoantennas results from both non-radiative and radiative damping , where the former is due to intrinsic metal loss, and the latter is due to strong dipole characteristics. The intrinsic metal loss cannot be altered by varying the shape of the nanoantennas. Therefore, the radiative damping shows the possibility of the broadening and attenuation, since the dipole characteristics are likely to change with increasing width A.
4. Three-level EIT system
The two-level system of Fig. 2(a) can be developed for a three-level scheme for EIT after adding a metastable state (see Fig. 3(b) ). The transitions and are dipole allowed, while is dipole forbidden. Consequently, two possible pathways to dressed states induced by the coupling field ΩC lenders the probe field ΩP transparent via destructive quantum interference, otherwise absorptive. Cho et al.  suggested that two magnetically excited parallel strips can work as the dark atom, since it cannot be excited directly by the external field but magnetically coupled with the bright state (the single strip). When the resonant wavelength of the two parallel strips is tuned to that of the single strip , they interact with each other, which is analogous to two-photon resonance process: the excited atom fall down from state to state , then pumped to state again. Here, and are the detunings of the probe and the pump beam , respectively .
For the three-level system, within the dipole approximation the atom-light interaction, , is often expressed in terms of the Rabi frequency , with being the amplitude of the electric field E, and μ the transition electronic dipole moment. After introducing the rotating-wave approximation, the Hamiltonian of the three-level atomic system interacting with a coupling laser whose Rabi-frequency is and with a probe laser whose Rabi-frequency iscan be expressed as:
Decreasing the length B(other parameters are fixed) in Fig. 3(a) resembles the enlarged energy difference between and (i.e., the elevated energy level) and the energy difference between and as well. As a result, both and are increased, which means an increase of both and. In contrast, the increased B gives rise to the decrease of both and, as shown in Figs. 4(a) and 4(b). The positions of the dip are nearly invariant, even though the feature of EIT becomes asymmetric. The position of the EIT feature is unambiguously determined by the frequency of the coupling field in the atomic system, other than and. Furthermore, when the width A increases, the resonant peak, like a two-level system, the shifts are towards the higher frequencies (i.e. blue shift), and the absorption peak becomes broader, which resembles the increased detuning and decay, respectively, as shown in Figs. 4(c) and 4(d). Here, since the geometric parameters of the two parallel strips, behaving as the coupling field in the metamaterial, are not varied, the peak position of EIT remains unchanged. Thus, the length of the two parallel strips determines the peak position of EIT and they are linearly dependent, as shown in Figs. 5(a) and 5(b). Following this picture, stretching the two parallel strips gives rise to the red shift of EIT, resembling the decreased frequency of the coupling field , which equivalently increased ; shrinking these strips causes the blue shift as the increased or the decreased. The EIT-like feature in the metamaterial [see Fig. 3(a)] arising from the interaction between the single strip and the two parallel strips implies the possibility for the coupling strength to be adjusted by controlling their spatial distance D. The larger the separation D, the weaker the interaction becomes. The interaction between the single strip and parallel strips resembles the Rabi frequency of the coupling field in an atomic system of Fig. 3(b) [see Fig. 5 (d)]. In contrast, the smaller distance D represents a better transparency window, as shown in Fig. 5(c). To have an entire and quick view, the parametric comparison is carried out in Table 1 .
5. Four-level tripod system
In metamaterials, not only can a three-level-like system be established, but also a four-level-like system with a tripod configuration . In a tripod system, the EIT-induced single dark resonance is swapped for another dark resonance when a third light is applied, known as coherence swapping [30,31] or simply double dark resonance [32,33]. As a result, at line center of EIT, the quantum interference-based absorption cancellation results in strong absorption [30–33]. Here we consider replacing the third light with another magnetically excited metallic pair located at the right wing.
Figure 6 shows the numerical simulations for a tripod model of metamaterial and a corresponding atomic system. Figure 6(c) presents the simulation results for the metamaterial of Fig. 6(a). Figure 6(d) presents the corresponding tripod system of Fig. 6(b) using the following equations:31,32]. For a special case of no detuning, the degenerate dark state (and ) at line center [see the first row of Fig. 6(d)] is obtained analytically as follows:
The metamaterial with equal length () of symmetric double parallel metal strips in the first row of Fig. 6(c) satisfies a degenerate tripod atomic system, which proves the degeneracy in Eqs. (9) and (10): see the red dash curves in the Fig. 0.6 (without the control field). As expected, the peak-to-peak separation determined by Ω is wider than the separation determined by EIT by (). The general solution of Eq. (10) with a symmetric detuning () can also be obtained from numerical simulations by introducing an asymptotic relationship with a detuning δ as follows:Eq. (10): see Figs. 6 and 7 .
We now discuss double dark resonance, which gives an absorption enhancement otherwise dark state of Eqs. (8) and (9) when detuning δ is introduced: the length of the double parallel metal strips and changes. Suppose a length change of the metallic pairs, and , in an opposite direction: and, respectively. The fourth row in Fig. 6(c), shows respectively a single-dark mode (red dash line) and a double-dark mode (black line) of the localized plasmon interactions induced by a monochromatic light vertically incident on the metamaterial, where the scheme corresponds to a detuned tripod optical system interacting with three lights, as shown in Fig. 6(b). The double-dark mode [see black solid line in the fourth row of Fig. 6(d)] results in absorption enhancement at line center. The single dark modes appear at the side bands of the double-dark mode (probe resonance transition), where each position (energy splitting) is determined by each detuning or .
In the three-level system, we have already illustrated that the reduced length of L results in a blue shift of the detuning. For the second row of Fig. 6(c), where and (), a weak absorption enhancement appears near line center. Due to the detuning effect of the control, the spectral position of the double dark resonance shifts slightly to the right. Here the dark states (EIT transparency) appear at both and . The eigenvalue of the double dark state (enhanced absorption) is determined by the ,, and . On the third row of Fig. 6(c), the length of the parallel double metal strips on the right wing is increased by (). Because the detuning is still within the Rabi frequency (; the corresponding Rabi frequency is analyzed through the simulations), the quantum coherence can still be sustained by inducing the double dark resonance resulting in absorption enhancement at line center. For the rest of Fig. 6(c), all the parameters remain the same unless otherwise indicated. Compared with the second row, the frequency shift of the probe on the third row is zero due to the balanced detuning. As the length of the double parallel metal strips continues to change, the system behaves noncoherently indicating a broadened absorption linewidth at line center (see the fifth row). All the corresponding figures in the right column for a tripod atomic model match well those in the left column.
In Fig. 7, we discuss whether the enhanced absorption results from the sum of two single classical oscillators. In the inset of Fig. 7 we assume that the tripod model (M) can be decomposed into two independently detuned EIT models (M1and M2). Figures 7(a) and 7(b) are for metamaterial and atomic systems, respectively. Then the average sum of both detuned EIT spectra is compared with that of the tripod model. As a result, each probe spectrum for the detuned model shows a mirror image of the other across line center [see the red dotted and blue dash-dot curves in Fig. 7(a) and 7(b)]. Compared with the third row of Fig. 6 (c), no absorption enhancement appears in the sum spectrum at line center in Fig. 7(a) and 7(b). Clearly, the enhancement cannot be attributed to the simple superposition of two single classical oscillators.
In order to provide a qualitative explanation of the enhanced absorption, the classical oscillator picture is applied to the four-level system. The model consists of three damped harmonic oscillators follows : One of the oscillators of mass (the particle 1) is driven by a harmonic force and attached to a wall by a spring with a force constant ; the other two of and (the particles 2 and 3) are attached to it by springs with force constants and , respectively; and they are fixed from the other side to a wall by springs with force constants and, respectively. Alzar et al. indicated that the power, absorbed by the driven oscillator (i.e., the particle 1) as a function of the frequency ω, presents a EIT-like profile considering the particles 1 and 2, which resembles a three-level system . Compared with their model, the particle 3 is involved here to take into account another dark mode when a third light is applied. Setting in the driven force and for simplicity without any loss of generality, the equations of motion are written as follows,Eq. (12), finally, the absorbed power during one period of the driven force is expressed as
Equation (13) shows that the real part of has the characteristic of Autler-Townes effect without any couplings. In contrast, the profile is dramatically modified when two couplings, and , take effect. As a result, a double dip-shape evolves into a triple dip-shape as shown in Fig. 6, arising from the double dark resonance. According to these numerical calculations, it is reasonable to assume and . The enhanced absorption becomes evident when the detunings of the coupling and the pumping field have the opposite signs (i.e., ). This can be understood as a result of destructive interaction of two dark modes according to Eq. (13), since the sum of the last term in the denominator is close to zero considering the finite detunings (e.g., within the Rabi frequency), as shown in Fig. 8 . Therefore, the effects of the coupling and the pumping fields on the bright atom can be negligible and the absorbed power at the resonant frequency has a similar value to that of Autler-Townes peak, as shown in Fig. 9 . As a result, the transparency was destroyed by the perturbation.
The plasmonic tripod system discussed in Fig. 6 might be attractive for selective storage of excited energy into either the left dark cell or the right cell. Specifically, the field can be concentrated into the left cell if the frequency of incident light is equal to the resonant frequency of the left cell; if the frequency of the incidence light is tuned to the same frequency as that of the right cell, the field can be localized into the right cell. This outcome may have application in plasmon-based photonic switching.
We numerically analyzed single dark (EIT) and double dark (EIT-based photonic switching) states in metamaterials compared with corresponding atomic models. From numerical calculations, geometrical parameters of the dolmen shaped metal strips can be used for absorption control of incident light for plasmonic switching. Based on the comparison, a plasmon-induced tripod model is presented for a double dark resonance. This model may be applied for the selective storage of excited energy and plasmon-based all-optical information processing.
We acknowledge that this work was supported by the Center for Photon Information Processing of the Korean Ministry of Education, Science and Technology via KOSEF. We thank S. Zhang and X. Zhang of UC Berkeley for helpful discussions.
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