1. Introduction

In recent years, there are tremendous efforts seeking for the classical analogue of electromagnetically induced transparency (EIT) [1] in solid systems, including coupled resonators [2–4], electric circuits [2,4,5], optomechanical devices [6,7], whispering-gallery-mode microresonators [8], and various metamaterials [9–25]. Especially, the plasmonic analogue of EIT, called plasmon-induced transparency (PIT) [9–11], has become a very important platform for exploring EIT-like physical properties of plasmonic polaritons and for designing new types of metematerials.

Similar to EIT, PIT is resulted from a destructive interference between wideband bright and narrowband dark modes in artificial atoms (called meta-atoms). A typical character of PIT is the opening of a deep transparency window within broadband absorption spectrum, together with a steep dispersion and greatly reduced group velocity of plasmonic polaritons. PIT metamaterials can work in different frequency regions (including micro [10] and terahertz [11,14,17] waves, infrared and visible radiations [9,12,16]), and may be used to design novel, chip-scale plasmonic devices (including highly sensitive sensors [13,14], optical buffers [15,17], and ultrafast optical switches [17], etc.) in which the radiation damping can be significantly eliminated, very intriguing for practical applications.

However, the PIT in plasmonic metamaterials reported up to now [9–25] is only for the classical analogue of the simplest atomic EIT, i.e. the one occurring in a coherent three-level atom gas resonantly interacting with two laser fields. We know that atoms possess many (energy) levels, quantum interference effect may occur in atomic systems with level number larger than three and the number of laser fields larger than two [1]. In fact, in the past two decades the EIT has been extended into the atomic systems with various multi-level configurations, such as four-level systems of double Λ-type [26–36], tripod-type [37–39], Y-type [40–43], five-level systems of M-type [44–47], and six-level systems of double-tripod-type [48,49], etc. Thus it is natural to ask the question: Is it possible to find a classical analogue of the atomic EIT with the level number larger than three in a metamaterial?

In this work, we give a positive answer for the above question. The metamaterial we consider is an array of meta-atoms [see Fig. 2(a)], i.e. the unit cells consisting of two cut-wires (CWs) and a split-ring resonator (SRR) [see Fig. 2(b)], interacting with an electromagnetic (EM) field with two polarization components. We show that such plasmonic metamaterial system may be taken as a classical analogue of an EIT-based atomic medium with a double-Λ-type four-level configuration coupled with four (two probe and two control) laser fields [see Fig. 1(a) ], exhibits an effect of PIT and displays a similar behavior of atomic four-wave mixing (FWM).

 

Fig. 1 (a) Double-Λ-type four-level atomic system with the atomic states |j〉 (j = 1, 2, 3, 4), coupled with two probe fields (with Rabi frequency Ω_pn) and two strong control fields (with Rabi frequency Ω_cn) (n = 1, 2). Δ₃, Δ₂, and Δ₄ are respectively the one, two, and three-photon detunings. (b) $\text{Im}(K_a^{+})$ [imaginary part of $K_a^{+}$] as a function of ω for Ω_c1 = Ω_c2 = 20 MHz (red dashed line) and Ω_c1 = Ω_c2 = 60 MHz (green dashed-dot line). EIT transparency window is opened near the central frequency of the probe fields (i.e. at ω = 0). The blue solid cure is $\text{Im}(K_a^{-})$, which has always a large absorption peak at ω = 0 for arbitrary Ω_c1 and Ω_c2.

Download Full Size | PDF

 

Fig. 2 (a) Schematic of the plasmonic metamaterial, which is an array of meta-atoms. (b) The meta-atom consists of two CWs (indicated by “A” and “B”) and a SRR, where the parameters d_x, d_y, L_x, L_y, w_b, w_g, and w_s are given in the text. For generating nonlinear excitations, four hyperabrupt tuning varactors are mounted onto the slits of the SRR. (see Sec. 3). (c) The numerical result (blue dashed lines) of the normalized absorption spectrum of the EM wave as a function of frequency by taking ℰ_y0 = −ℰ_x0, d_x = d_y = 4.0 mm (first panel), and d_x = d_y = 3.4 mm (second panel). (d) The numerical result (blue dashed line) of normalized absorption spectrum for ℰ_y0 = ℰ_x0, d_x = d_y = 4.0 mm. Red solid lines in (c) and (d) are analytical results obtained from the formula Im(q₁₀) given by Eq. (26) in Appendix B.

Download Full Size | PDF

Based on this interesting classical analogue, we further show that, if a nonlinear varactor is mounted onto each gap of the SRR, the system can support a new type of nonlinear plasmonic polaritons, i.e. vector plasmonic dromions, which are (2+1)-dimensional plasmonic solitons with two polarization components for electric field and have very low generation power and are robust during propagation. This gives a significant generalization of our work [24], where plasmonic dromions with only a single polarization component was reported. The results presented here not only gives a close metamaterial analogue of the EIT and FWM in multi-level atomic systems, useful to find novel wave interference phenomena and nonlinear property in solid systems, but also provides a way to obtain new type of plasmonic polaritons via suitable design of plasmonic metamaterials.

The article is arranged as follows. In Sec. 2, we give a simple introduction of EIT-based FWM in a four-level atomic system, describe our metamaterial model allowing the occurrence of PIT and FWM, and show the equivalence between the two models. The propagation of linear plasmonic polaritons in the metamaterial is discussed in detail. In Sec. 3, we derive the coupled nonlinear envelope equations and present the vector plasmonic dromion solutions when the nonlinear varactors are mounted onto the gaps of the SRRs. The last section (Sec. 4) gives a discussion and a summary of our work.

2. EIT-based atomic FWM and its metamaterial analogue

2.1. EIT-based FWM in a double-Λ-type four-level atomic system

For a detailed comparison with the metamaterial model presented in the next subsection, we first give a brief introduction on an atomic gas with a double-Λ-type four-level configuration, shown in Fig. 1(a). In this system, two weak probe laser fields with central angular frequencies ω_p1 and ω_p2 and wavevectors k_p1 and k_p2 drive respectively the transitions |1〉 ↔ |3〉 and |1〉 ↔ |4〉, and two strong control laser fields with central angular frequencies ω_c1 and ω_c2 and wavevectors k_c1 and k_c2 drive respectively the transitions |2〉 ↔ |3〉 and |2〉 ↔ |4〉. The total electric fields in this system is given by E = e_p1ℰ_p1 exp[i(k_p1z − ω_p1t)] + e_p2ℰ_p2 exp[i(k_p2z − ω_p2t)] + e_c1ℰ_c1 exp[i(k_c1z − ω_c1t)] + e_c2ℰ_c2 exp[i(k_c2z − ω_c2t)] + c.c., where e_jn and ℰ_jn (j = p, c; n = 1, 2) are respectively the unit vector denoting the polarization direction and the envelope of the corresponding laser field. Note that for simplicity all the laser fields are assumed to be injected in the same (i.e. z) direction (which is also useful to suppress Doppler effect). Under electric-dipole approximation and rotating-wave approximation (RWA), the Hamiltonian of the system in interaction picture reads

The dynamics of the atoms is governed by the optical Bloch equation iħ (∂/∂t + Γ) σ = [Ĥ_int, σ], where σ is a 4 × 4 density matrix, Γ is a 4 × 4 decoherence (relaxation) matrix describing spontaneous emission and dephasing. The explicit expression of the Bloch equation is given in Appendix A. We assume that initially the probe fields are absent, thus for substantially strong control fields the atoms are populated in the ground state |1〉. The solution of the Bloch equation reads σ₁₁ = 1 and all other σ_jl are zero.

When the two weak probe fields are applied, the ground state |1〉 is not depleted much. In this case, the Bloch equation reduces to

The dynamics of the probe fields is governed by the Maxwell equation ∇²E−(1/c²)∂²E/∂t² = 1/(ε₀c²)∂²P/∂t². Here the polarization intensity is given by P = N₀[σ₃₁e^{i(k_p1z−ω_p1t)} + σ₄₁e^{i(k_p2z−ω_p2t)} + c.c.], with N₀ the atomic density. Under a slowly-varying envelope approximation (SVEA), the Maxwell equation reduces to

It is easy to understand the basic feature of the propagation of the probe fields through solving the Maxwell-Bloch (MB) equations (3) and (4) with σ_l1 (l = 1, 2, 3) and Ω_pj (j = 1, 2) proportional to the form exp[i(K_az − ωt)] [52]. We obtain

Fig. 1(b) shows $\text{Im}(K_a^{+})$ [i.e. the imaginary part of $K_a^{+}$] as a function of ω for Ω_c1 = Ω_c2 = 20 MHz (red dashed line) and Ω_c1 = Ω_c2 = 60 MHz (green dashed-dot line). When plotting the figure, Δ_j (j = 1, 2, 3) are set to be zero, and realistic parameters from ⁸⁷Rb atoms are taken, given by Γ₁₃ = Γ₂₃ = Γ₁₄ = Γ₂₄ = 16 MHz, κ₁₃ = κ₁₄ = 1 × 10¹⁰ cm⁻³ [54]. We see that a transparency window is opened in the profile of $\text{Im}(K_a^{+})$ near ω = 0; the transparency window becomes larger when the control fields are increased. The opening of the transparency window (called EIT transparency window) is due to the EIT effect contributed by the control fields. The blue solid cure in the figure is $\text{Im}(K_a^{-})$ as a function of ω, which however has always a large absorption peak near ω = 0 irrespective of the value of the control fields. Below, for convenience we shall call the normal mode with the linear dispersion relation $K_a^{+}$ ($K_a^{-}$) as EIT-mode (non-EIT-mode).

The double-Λ-type four-level system can be used to describe a resonant FWM process in atomic systems [1, 29–32]. The first laser field (i.e. the control field tuned to the |2〉 ↔ |3〉 transition with the half Rabi frequency Ω_c1) and the second laser field (i.e. the probe field tuned to the |1〉 ↔ |3〉 transition with the half Rabi frequency Ω_p1) can adiabatically establish a large atomic coherence of the Raman transition, described by the off-diagonal density matrix element σ₂₁. The third laser field, i.e. the control field tuned to the |2〉 ↔ |4〉 transition with the half Rabi frequency Ω_c2, can mix with the coherence σ₂₁ to generate a fourth field with the half Rabi frequency Ω_p2 resonant with the |1〉 ↔ |4〉 transition. For details, see Refs. [1,29–32] and references therein.

2.2. Metamaterial analogue of the double-Λ-type four-level atomic system

We now seek for a possible classical analogue of the above four-level atomic model by using a metamaterial, which is assumed to be an array [Fig. 2(a)] of unit cells (i.e. meta-atoms) [Fig. 2(b)] consisting two CWs (indicated by “A” and “B”) and a SRR. The CW A and CW B are, respectively, positioned along x and y direction, while the SRR is formed by a square ring with a gap at the center of each side. 20-μm-thick copper forming the CWs and the SRR is etched on a substrate with a height of H = 1.5 mm. Geometrical parameters of the meta-atom are L_x = L_y = 8 mm, w_s = 1.2 mm, w_g = 0.6 mm, and w_b = 1.2 mm [53].

We assume that an incident gigahertz radiation E = e_xE_x+e_yE_y [with E_j = ℰ_j0e^−iω_pt+c.c. (j = x, y)] is collimated on the array of the meta-atoms, with polarization component E_x (E_y) parallel to the CW A (CW B). In order to understand the EM property of the system, a numerical simulation is carried out by using the commercial finite difference time domain software package (CST Microwave Studio) and probing the imaginary part of the radiative field amplitude at the center of the end facet of the CW A [red arrow in Fig. 2(b)] [9]. The blue dashed lines in Fig. 2(c) are normalized absorption spectrum as a function of the incident wave frequency by taking the excitation condition ℰ_y0 = −ℰ_x0 for d_x = d_y = 3.4 mm (first panel), and ℰ_y0 = −ℰ_x0 for d_x = d_y = 4.0 mm (second panel). Here and below, ℰ_x0 and ℰ_y0 are taken to be real for simplicity. Fig. 2(d) shows the normalized absorption spectrum under the excitation condition ℰ_y0 = ℰ_x0 for d_x = d_y = 4.0 mm. The red solid lines in the figure are analytical results obtained from Im(q₁₀) based on Eq. (26) in Appendix B. We see that the absorption spectrum profile depends on excitation condition, which is quite different from the PIT absorption spectrum considered before.

The dependence on the excitation condition for the absorption spectrum can be briefly explained as follows. A sole CW in the meta-atoms is function as an optical dipole antenna and thus serves as a bright (or radiative) oscillator, which can be directly excited by the incident radiation. The surface current in an excited SRR can be clockwise or anticlockwise direction, indicating that there is no direct electric dipole coupling with the incident radiation and hence the SRR serves as a dark or trapped oscillator with long dephasing time [55]. For the excitation condition ℰ_y0 = −ℰ_x0 [Fig. 2(c)], the surface current is cooperatively induced through the near-field coupling between SRR and CWs, resulting in a maximum enhancement of the dark-oscillator resonance and thus the substantial suppression of the absorption of the incident radiation, acting like a typical PIT metamaterial. However, for the excitation condition ℰ_y0 = ℰ_x0 [Fig. 2(d)], the surface current is suppressed due to an opposite excitation direction, leading to a complete suppression of the dark-oscillator resonance. As a result, the radiation absorption is significant (acting like a sole CW) and hence no PIT behavior occurs. For convenience, in the following we called the excitation mode under the ℰ_y0 = −ℰ_x0 [Fig. 2(c)] as the PIT-mode, and the excitation mode under the ℰ_y0 = ℰ_x0 [Fig. 2(d)] as the non-PIT-mode (or absorption mode).

The dynamics of the bright oscillators (i.e. CW A and CW B) and dark oscillator (i.e. SRR) in the meta-atoms can be described by the coupled Lorentz equations [9,17,20,23–25]

Based on the solution given by Eq. (26), we deduce that, in the case of ω₃ = ω_p, γ₃ = 0 and g₁ = g₂, Eq. (6) allows a “dark state” (i.e. the state where both the bright oscillators are not excited, i.e. q₁₀ = q₂₀ = 0) exists, if

The equation of motion of the EM wave is governed by the Maxwell equation

Assuming the central frequency of the incident radiation ω_p is near the natural frequencies of the Lorentz oscillators described by Eq. (6) [55], a resonant interaction occurs between the incident radiation and these oscillators. To deal with the propagation problem of the plasmonic polaritons in the system analytically, we assume E_j(r, t) = ℰ_j(z, t)e^{i(k_pz−ω_pt}) + c.c. and q_l(r, t) = q̃_l(z, t) exp[i(k_pz − ω_lt − Δ_lt)] + c.c., where ℰ_j(z, t) and q̃_l(z, t) are slowly-varying envelopes and Δ_l = ω_p − ω_l is a small detuning. With this ansatz and under RWA, Eq. (6) is simplified into the reduced Lorentz equation

Under SVEA, the Maxwell equation in the metamaterial reads

The propagation feature of a plasmonic polariton in the metamaterial can be obtained by assuming all quantities in the ML equations (9) and (10) proportional to exp[i(K_mz − ωt)]. It is easy to get the linear dispersion relation

The character of the above two normal modes can be clearly illustrated by plotting $K_m^{+}$ and $K_m^{-}$ as functions of ω. Shown in Fig. 3(a) are $\text{Im}\left(K_m^{+}\right)$ (blue dashed line) and $\text{Re}\left(K_m^{+}\right)$ (red solid line) for κ₂ = −κ₁ = 50 GHz² (first panel; corresponding to d_x = d_y = 4.0 mm) and κ₂ = −κ₁ = 250 GHz² (second panel; corresponding to d_x = d_y = 3.4 mm), respectively. When plotting the figure, the system parameters are taken from Appendix B, and additional parameters are chosen by κ₀ = 10¹⁰ kg/(cm · s² · C) [56] and Δ_j = 0 (j = 1, 2, 3). We see that $\text{Im}\left(K_m^{+}\right)$ displays a transparency window (called PIT transparency window) near ω = 0, analogous to the EIT transparency window in $\text{Im}\left(K_a^{+}\right)$ of the four-level double-Λ-type atomic system [red dashed line and green dashed-dot line in Fig. 1(b)]. The steep slope of $\text{Re}\left(K_m^{+}\right)$ indicates a normal dispersion and a slow group velocity of the plasmonic polariton. As the coupling strength between the CWs and the SRR gets larger (i.e. the separations d_x and d_y is reduced), the PIT transparency window becomes wider and deeper, and the slope of $\text{Re}\left(K_m^{+}\right)$ gets flatter. The opening of the PIT transparency window is attributed to the destructive interference between the two bright oscillators and the dark oscillator through cooperative near-field coupling. Shown in Fig. 3(b) is the imaginary (red solid line) and the real (blue dashed line) of $K_m^{-}$ which is nearly independent on the coupling constant κ₁ (κ₂ = −κ₁). We see that $\text{Im}\left(K_m^{-}\right)$ has a single, large absorption peak and $\text{Re}\left(K_m^{-}\right)$ has an abnormal dispersion near ω = 0, analogous to $\text{Im}\left(K_a^{-}\right)$ of the double-Λ-type atomic system [blue solid line in Fig. 1(b)].

 

Fig. 3 (a) Linear dispersion relation of the $K_m^{+}$-mode (PIT-mode). $\text{Im}(K_m^{+})$ (blue dashed line) and $\text{Re}(K_m^{+})$ (red solid line) are plotted as functions of ω for κ₂ = −κ₁ = 50 GHz² (first panel) and κ₂ = −κ₁ = 250 GHz² (second panel). (b) Linear dispersion relation of the $K_m^{-}$-mode (non-PIT-mode) for arbitrary κ₁ (κ₂ = −κ₁).

Download Full Size | PDF

2.3. Propagation of linear plasmonic polaritons via an analogous FWM process of atomic system

As indicated above, the meta-atoms in the present metamaterial system are analogous to the four-level atoms with the double-Λ-type configuration, and hence an analogous resonant FWM phenomenon for the plasmonic polaritons is possible. That is to say, if initially only one polarization-component of the EM wave (e.g. x-component) is injected into the metamaterail, a new polarization-component (e.g. y-component) will be generated through two equivalent control fields (i.e. the couplings between the SRR and CWs, described by κ₁ and κ₂). To illustrate this, we present the solution of the ML equations (9) and (10)

The conversion efficiency of the FWM is given by $\eta (L)\equiv {\int }_{-\infty }^{+\infty }dt{\left|{ℰ}_y(L,t)/{ℰ}_x(0,t)\right|}^2$, where L is the medium length. For the case κ₂ = −κ₁, one has $\text{Im}\left(K_0^{-}\right)\gg \text{Im}\left(K_0^{+}\right)$, which means that the $K_m^{-}$ mode decays away rapidly during propagation and hence can be safely neglected.

Then Eq. (14) is simplified as

 

Fig. 4 FWM conversion efficiency η as a function of the dimensionless optical depth (κ₀g_f1/γ₁)L for Δ₁ = Δ₂ = 0 (blue dashed line), and Δ₁ = Δ₂ = 5γ₁ (red solid line). Inset: FWM conversion efficiency η for optical depth up to 300 for Δ₁ = Δ₂ = 5γ₁.

Download Full Size | PDF

3. Vector plasmonic dromions in the PIT metamaterial

Note that when deriving Eq. (10), the diffraction effect has been neglected, which is invalid for the plasmonic polaritons with small transverse size or long propagation distance; furthermore, because of the highly resonant (and hence dispersive) character inherent in the PIT metamaterial, the linear plasmonic polaritons obtained above inevitably undergo significant distortion during propagation. Hence it is necessary to seek the possibility to obtain a robust propagation of the plasmonic polaritons in the PIT metamaterial. One way to solve this problem is to make the PIT system work in a nonlinear propagation regime.

In recent years, nonlinear metamaterials have attracted much attention due to their potential applications (see Ref. [58] and references therein). One suitable way to design a nonlinear PIT metamaterial in microwave and lower THz ranges is to use nonlinear insertions onto the meta-atoms [59,60]. Here, as suggested in Refs. [24,59,60], we assume the nonlinear insertion in the PIT metamaterial are varactor diodes, which are mounted onto the gaps of the SRRs [60] [see Fig. 2(b)].

3.1. Nonlinear envelope equations

Since the introduction of the nonlinear element onto the SRRs, Eq. (6c) should be replaced by [60]

Due to the quadratic and cubic nonlinearities in Eq. (17), the input EM field (with only a fundamental wave) will generate longwave (rectification), and second harmonic components, i.e. E_l(r, t) = ℰ_dl(r, t) + [ℰ_fl(r, t)e^{i(k_pz−ω_pt)} + c.c.] + [ℰ_sl(r, t)e^iθ_p + c.c.] (l = x, y), with θ_p = (2k_p +Δk)z −2ω_pt and Δk a detuning in wavenumber. The oscillations of the Lorentz oscillators in the meta-atoms have the form q_j(r, t) = q_dj(r, t)+[q_fj(r, t)e^iθ_j +c.c.]+[q_sj(r, t)e^2iθ_j +c.c.] (j = 1, 2, 3), with θ_j = k_jz − ω_jt − Δ_jt. Substituting these expressions into Eqs. (6a), (6b), (8), and (17), and adopting RWA and SVEA, we obtain a series of equations for the motion of q_μj and ℰ_μl, listed in Appendix C.

We solve the equations for q_μj and E_μl by using the standard method of multiple scales [61]. Take the asymptotic expansion $q_{fj}=∊q_{fj}^{(1)}+{∊}^2{q}_{fj}^{(2)}+\cdots$, $q_{dj}={∊}^2{q}_{dj}^{(2)}+\cdots$, $q_{sj}={∊}^2{q}_{sj}^{(2)}+\cdots$, ${ℰ}_{fl}=∊{ℰ}_{fl}^{(1)}+{∊}^2{ℰ}_{fl}^{(2)}+\cdots$, and ${ℰ}_{dl}={∊}^2{∊}_{dl}^{(2)}+\cdots$ (here ∊ is a dimensionless small parameter characterizing the amplitude of the incident EM field), and assume all quantities on the right sides of the asymptotic expansion as functions of the multiscale variables [61] z_l = ∊^lz (l = 0, 1, 2) and t_l = ∊^lt (l = 0, 1). Substituting the expansion into the equations for q_μj and ℰ_μj and comparing powers of ∊, we obtain a chain of linear but inhomogeneous equations which can be solved order by order.

The leading order [i.e. O(∊)] solution reads ${ℰ}_{fx}^{(1)}=F_{+}{e}^{i{\theta }_{+}}$ and ${ℰ}_{fy}^{(1)}=G^{+}{F}_{+}{e}^{i{\theta }_{+}}$, where ${\theta }_{+}=K_m^{+}{z}_0-\omega t_0$ and F₊ is a slowly-varying envelope function to be determined in higher-order approximations. The expression of G⁺ is given in Sec. 2.3. Here we consider only the PIT (i.e. $K_m^{+}$) mode because the non-PIT (i.e. $K_m^{-}$) mode decays rapidly during propagation, as indicated in the last section. The solution for $q_{\mu j}^{(1)}$ is presented in Appendix D.

At the second order [i.e. O(∊²)], a solvability condition yields $i\left[\partial /\partial z_1+(1/V_g^{+})\partial /\partial t_1\right]F_{+}=0,$ where $V_g^{+}$ is the group-velocity of the fundamental wave. Solution for the longwave is ${ℰ}_{dx}^{(2)}=Q_{+}$ and ${ℰ}_{dy}^{(2)}=G^{+}{Q}_{+}$, with Q₊ the slowly-varying envelope to be determined yet. Explicit expressions of the solutions for other quantities at this order are listed in Appendix D.

At the third order [i.e. O(∊³)], a solvability condition results in the equation

At the fourth order [i.e. O(∊⁴)], a solvability condition results in the equation for the longwave Q₊

Under the PIT condition [i.e. (κ_j/2ω_p)² ≫ γ_jγ₃/4, j = 1, 2], the real parts of the nonlinearity susceptibilities ${\chi }_{+}^{(2)}$ and ${\chi }_{+}^{(3)}$ are greatly enhanced and their imaginary parts are much smaller than their real parts. Interestingly, ${\chi }_{+}^{(3)}$ can be further enhanced via a strong coupling between the longwave and the fundamental wave. This can be seen from the expression of the effective third-order nonlinearity susceptibility ${\chi }_{\text{eff}}^{(3)}={\chi }_{+}^{(3)}+R_0{\left[{\chi }_{+}^{(2)}\right]}^2/\left\{c^2\left[\left(\frac{1}{V_p^{+}}\right)-{\left(\frac{1}{V_g^{+}}\right)}^2\right]\right\}$, obtained by neglecting the diffraction term in Eq. (20) and then plugging the derived expression for Q₊ into Eq. (18). One sees that when longwave-shortwave resonance occurs (i.e. $V_g^{+}\approx V_p^{+}$), ${\chi }_{\text{eff}}^{(3)}$ can be enhanced largely, which can be realized by adjusting the coupling parameters κ₁ and κ₂. It is easy to show that the coupling strength of the longwave-shortwave interaction in the present FWM-based system is $\sqrt{2}$ time larger than that in the metamaterial used Ref. [24], and hence ${\chi }_{\text{eff}}^{(3)}$ here is larger than that obtained in Ref. [24].

3.2. Vector plasmonic dromions

Equations (18) and (20) can be converted into the dimensionless form

In favor of the formation of plasmonic dromions, we take the following two assumptions. First, we shall assume R_x ≪ R_y and thus g_d0 ≪ 1 so that the original (3+1)-dimensional nonlinear problem can be reduced into a (2+1)-dimensional one. Second, we assume the contribution of the dispersion, diffraction and the nonlinearity effects are of the same level, which can be achieved by taking L_diff = L_disp = L_nln and thus we obtain ${\tau }_0=R_x\sqrt{-{\omega }_p{n}_{\text{D}}\text{Re}\left(K_2^{+}\right)/c}$ and $U_0=\left[c/\left({\omega }_p{R}_x\right)\right]\sqrt{2/\text{Re}\left[{\chi }_{+}^{(3)}\right]}$. In fact, these two assumptions can be realized by taking the realistic set of parameters, i.e. Δ₁ = Δ₂ = −5γ₁, κ₂ = −κ₁ = 4260 GHz², R_x = 1.8 cm, R_y = 10.2 cm, U₀ = 8.25 V/cm, V₀ = 1.3 V/cm, τ₀ = 4.1 × 10⁻¹¹ s, and hence we have $K_2^{+}=\left(-1.30+i0.10\right)\times {10}^{-19}{\text{cm}}^{-2}\text{s}$, ${\chi }_{+}^{(3)}=1.20\times {10}^{-3}+i2.26\times {10}^{-7}{\text{cm}}^2/{\text{V}}^2$, L_diff = L_disp = L_nln = 13.4 cm, g₁ = g₂ = g_d1 = g_d2 = 1, g₃ = 4, and g_d0 ≪ 1. For such case, Eq. (22) can be simplified into standard Davey-Stewartson-I (DS-I) equation $i\partial u/\partial s+{\partial }^{2}u/\partial {\xi }_1^2+{\partial }^{2}u/\partial {\tau }_1^2+v_{1}u=0$, $\left({\partial }^2/\partial {\xi }_1^2+{\partial }^2/\partial {\tau }_1^2\right){\left|u\right|}^2={\partial }^2{v}_1/\partial {\xi }_1\partial {\tau }_1$, with v₁ = v + 2|u|², where for convenience ${\xi }_1=(\xi +\tau )/\sqrt{2}$ and ${\tau }_1=(\xi -\tau )/\sqrt{2}$. The DS-I equation can be exactly solved via the Hirota’s bilinear method [62]. A single-dromion solution reads [63] u = G/F, v₁ = V₁₁ + V₁₂, with $V_{11}=2{\partial }^2\text{ln}(F)/\partial {\xi }_1^2$, $V_{12}=2{\partial }^2\text{ln}(F)/\partial {\tau }_1^2$, and $F=1+\text{exp}\left({\eta }_1+{\eta }_1^{*}\right)+\text{exp}\left({\eta }_2+{\eta }_2^{*}\right)+\gamma \text{exp}\left({\eta }_1+{\eta }_1^{*}+{\eta }_2+{\eta }_2^{*}\right)$, G = ρ exp(η₁ + η₂). Here η_j = (k_jr + ik_ji)(r_j − r_j0) + (Ω_jr + iΩ_ji)s (j = 1, 2), with (r₁₍₀₎, r₂₍₀₎) = (ξ₁₍₀₎, τ₁₍₀₎), Ω_jr = −2k_jrk_ji, ${\mathrm{\Omega }}_{1i}+{\mathrm{\Omega }}_{2i}=k_{1r}^2+k_{2r}^2-k_{1i}^2-k_{2i}^2$, and $\rho =2\sqrt{2k_{1r}{k}_{2r}(\gamma -1)}$. Here k_jr, k_ji, r_j0 and γ are free real parameters.

Shown in Fig. 5(a) and Fig. 5(b) are, respectively, intensity distributions of the shortwave profile |u|² and the longwave profile |v₁|² as functions of ξ₁ and τ₁ at s = 0, when taking γ = 9, ξ₁₀ = τ₁₀ = 0, k_1r = k_2r = 1, and k_1i = k_2i = 0. Obviously, the shortwave profile |u|² denotes a localized envelope function, which decays exponentially in all spatial directions [Fig. 5(a)]; the longwave profile |v₁|² denotes two interacting plane solitons (kinks), which decay in their respective traveling directions [Fig. 5(b)].

 

Fig. 5 Plasmonic dromions and their interaction. (a) [(b)] is the intensity profile of the shortwave |u|² (longwave |v₁|²) as functions of ξ₁ and τ₁ at s = 0. (c1), (c2), (c3), (c4) [(d1), (d2), (d3), (d4)] are intensity profiles of the shortwave |u|² (longwave |v₁|²) during the interaction between two dromions, respectively at s ≡ z/(2L_diff) = 0, 1, 2, 3. System parameters are given in the text.

Download Full Size | PDF

We proceed with the investigation on the interaction between two plasmonic dromions by using a numerical simulation. Shown in Fig. 5(c1)–Fig. 5(c4) [Fig. 5(d1)–Fig. 5(d4)] are results for the evolution of the shortwave |u|² (longwave |v₁|²) during the collision between two dromions at s ≡ z/(2L_diff) = 0, 1, 2, 3, respectively. When doing the simulation we have taken a superposition of two dromion solutions as an initial input [i.e. Fig. 5(c1) and Fig. 5(d1)], and initial speeds and positions of the two dromions are setting to be k_1i = −k_2i = 1.8 and r₁₀ = −r₂₀ = 3.2. From the figure we see that two initial dromions become four dromions after the collision, which are, respectively, located around the four intersections of the longwave v₁, and these four dromions gradually separate and propagate almost stably, indicating that the collision between dromions is inelastic. The reason is that the four intersections of v₁ are the most attractive points in the entire region, which attract the EM wave intensities of the main peaks of the shortwaves u during the collision, resulting in the appearance of four pulses for the shortwave located around the four intersections after the collision [63].

Within the forth-order approximation, the explicit expression for the EM field in the metamaterial takes the form

The threshold of the power density of the vector plasmonic dromion given above can be estimated by using Poynting vector. Based on the above system parameters, the average power of the vector plasmonic dromion is given by P̄ = 6.1 mW. We see that due to the resonant character of the PIT effect in the system, extremely low generation power is required for generating the vector plasmonic dromion.

4. Discussion and summary

It should be mentioned that in writing the dark-state condition (7), the damping coefficient γ₃ of the dark oscillator in the meta-atoms is assumed to be small. However, due to the Ohmic loss inherent in the metal, by our numerical calculation the value of γ₃ is about 0.18 GHz, which, although smaller comparing with the damping coefficients of the CWs (γ₁ = γ₂ = 2.1 GHz), is still large and has inevitably detrimental impact on the PIT quality. In order to improve the performance of the PIT, one can suppress γ₃ by introducing a gain element into the SRR of the meta-atoms. One possible way is the use of tunneling diodes that have negative resistance and hence may provide gain to the PIT-based metamaterial [66,67]. Such method has been recognized to be useful for suppressing and even cancelling γ₃, particularly in microwave and THz regimes.

In conclusion, we have considered a plasmonic metamaterial interacting with a radiation field with two polarization components. We have shown that such metamaterial can be taken as a classical analogue of an EIT-based atomic gas with a double-Λ-type four-level configuration, displays an PIT effect and allows an equivalent process of atomic FWM. We have also shown that, when the nonlinear varactors are mounted onto the gaps of the SRRs, the system may support vector plasmonic dromions, which have very low generation power and are robust during propagation. Our work not only contributes a plasmonic analogue of atomic EIT and FWM but also provides a way for generating novel plasmonic polaritons, and hence opens a new avenue on the exploration of PIT effect in metamaterials.