Analogue of double-\Lambda-type atomic medium and vector dromions in a plasmonic metamaterial

Analogue of double--type atomic medium and vector dromions in a plasmonic metamaterial

Qi Zhang, Zhengyang Bai, and Guoxiang State Key Laboratory of Precision Spectroscopy, East China Normal University, Shanghai 200062, China
NYU-ECNU Joint Institute of Physics at NYU-Shanghai, Shanghai 200062, China
July 16, 2019

We consider an array of the meta-atom consisting of two cut-wires and a split-ring resonator interacting with an electromagnetic field with two polarization components. We prove that such metamaterial system can be taken as a classical analogue of an atomic medium with a double--type four-level configuration coupled with four laser fields, exhibits an effect of plasmon induced transparency (PIT), and displays a similar behavior of atomic four-wave mixing (FWM). We demonstrate that when nonlinear varactors are mounted onto the gaps of the split-ring resonators the system can acquire giant second- and third-order Kerr nonlinearities via the PIT and a longwave-shortwave interaction. We also demonstrate that the system supports high-dimensional vector plasmonic dromions [i.e. (2+1)-dimensional plasmonic solitons with two polarization components, each of which has a coupling between a longwave and a shortwave], which have very low generation power and are robust during propagation. Our work gives not only a plasmonic analogue of the FWM in coherent atomic systems but also provides the possibility for obtaining new type of nonlinear polaritons in plasmonic metamaterials.

42.65.Tg, 05.45.Yv, 42.50.Gy

I Introduction

Electromagnetically induced transparency (EIT), a very intriguing phenomenon occurring in atomic gases, has been intensively investigated due to its interesting physical properties and promising practical applications. The basic mechanism of EIT is the existence of a destructive quantum interference effect between two pathways of atomic transitions induced by a control laser field, through which the absorption of a probe laser field can be largely cancelled Fleischhauer ().

In recent years, there are tremendous efforts seeking for the classical analogue of EIT in solid systems, including coupled resonators Alzar (); Spring (); Harden (), electric circuits Alzar (); Harden (); Souza (), optomechanical devices Weis (); Kronwald (), whispering-gallery-mode microresonators Peng (), and various metamaterials Zhangs (); Papa (); Tassin (); Gu (); han (); Chen0 (); Dong (); NLiu (); Liu (); TNakanishi (); Sun (); Chen (); Mats (); NakanishiPRApp (); bai (); Bai2 (); Bai3 (). Especially, the plasmonic analogue of EIT, called plasmon-induced transparency (PIT) Zhangs (); Papa (); Tassin (), 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 effect 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 Papa () and terahertz Tassin (); Gu (); Dong () waves, infrared and visible radiations Zhangs (); NLiu (); han ()), and may be used to design novel, chip-scale plasmonic devices (including highly sensitive sensors Dong (); Chen0 (), optical buffers Gu (); Liu (), and ultrafast optical switches Gu (), 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 Zhangs (); Papa (); Tassin (); Gu (); han (); Chen0 (); Dong (); NLiu (); Liu (); TNakanishi (); Sun (); Chen (); Mats (); NakanishiPRApp (); bai (); Bai2 (); Bai3 () 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 Fleischhauer (). 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 -type Lukin (); Korsunsky1 (); Korsunsky2 (); Merriam (); DengPRA1 (); Kang (); Ying (); DengPRA3 (); YingSoliton (); Kang1 (); IteYu (), tripod-type Petrosyan (); Rebic (); Beck (), Y-type Joshi (); GaoJY (); Kha (); LiuYM (), five-level systems of M-type Ott (); Mat (); Rebic1 (); HangC (), and six-level systems of double-tripod-type Rus (); Lee (), etc. Thus it is natural to ask the question: Is it possible to get a classical analogue of the atomic EIT with the level number more than three in a metamaterial?

In this article, we give a positive answer for the above question. The metamaterial we consider is assumed to be 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 atomic medium with a double--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).

Based on this classical analogue, we further show that, if nonlinear varactors are mounted onto the gaps of the SRRs, the system can acquire giant second- and third-order Kerr nonlinearities many orders of magnitude larger than conventional nonlinear optical media. Using a method of multiple scales, we derive coupled envelope equations, which include dispersion, diffraction, and the Kerr nonlinearities and govern the evolution of the two polarization components of the EM field. We demonstrate that the system supports a new type of nonlinear plasmonic polaritons, i.e. high-dimensional vector plasmonic dromions, which are (2+1)-dimensional plasmonic solitons with two polarization components. Each polarization component has a coupling between a longwave and a shortwave, which have very low generation power and are robust during propagation. The results presented here not only gives a close metamaterial analogue of the EIT and FWM in multi-level atomic systems, useful to illustrate and find novel interference and nonlinear properties in solid systems, but also provides a way to obtain new type of plasmonic polaritons via suitable design of plasmonic metamaterials.

The main body of the article is arranged as follows. In Sec II, we give a simple introduction of the four-level atomic model allowing EIT and FWM, describe the metamaterial model, and show the similarity between the two models. The propagation of linear plasmonic polaritons in the metamaterial is discussed in detail. In Sec. III, 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. Lastly, in Sec IV we give a discussion and a summary of our work. Details of some calculating results are given in five appendixes.

Ii EIT-based atomic FWM and its metamaterial analogue

ii.1 EIT-based FWM in a double--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 a lifetime-broadened atomic gas with a double--type four-level configuration, shown in Fig. 1(a).

Figure 1: (color online) (a) Double--type four-level atomic system with the atomic states (), coupled with two probe fields (with Rabi frequency ) and two strong control fields (with Rabi frequency . , , and are respectively the one, two, and three-photon detunings. (b) [imaginary part of ] as a function of for (red dashed line) and (green dashed-dot line). EIT transparency window is opened near the central frequency of the probe fields (i.e. at ). The blue solid cure is , which has always a large absorption peak at for arbitrary and .

In this system, two weak probe laser fields with central angular frequencies and and wavevectors and drive respectively the transitions and , and two strong control laser fields with central angular frequencies and and wavevectors and drive respectively the transitions and . The total electric fields in this system is given by , where and () 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. ) 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


where ; , , and are respectively one-, two-, and three-photon detunings, with the eigenenergy of the atomic state (); , , , and are respectively the half Rabi frequencies of the probe and the control laser fields, with the electric dipole moment related to the transition . The Hamiltonian (1) allows three bright states and one dark state note00 (). The dark state reads , which is a superposition of only the two lower states and and has a zero eigenvalue. The condition yielding the dark state is Korsunsky2 ()


The dynamics of the atoms is governed by the optical Bloch equation , where is a density matrix, is a 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 . The solution of the Bloch equation reads and all other are zero.

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


with with (). Equations (3a)-(3c) describe the dynamics of three coupled harmonic oscillators note01 (), where and are bright oscillators due to their direct coupling to the probe fields and , but is a dark oscillator because it has no direct coupling to any of the two probe fields.

The dynamics of the probe fields is governed by the Maxwell equation . Here the polarization intensity is given by , with the atomic density. Under a slowly-varying envelope approximation (SVEA), the Maxwell equation reduces to


with and . For simplicity, we assume the two control fields are strong enough and thus have no depletion during the evolution of the probe fields; additionally, the diffraction effect is negligible, which is valid for the probe fields having large transverse size.

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 () and () proportional to the form  note02 (). We obtain


with and . We see that the MB equations allow two normal modes, with the linear dispersion relations given by and , respectively.

Fig. 1(b) shows [i.e. the imaginary part of ] as a function of for (red dashed line) and (green dashed-dot line). When plotting the figure, () are set to be zero, and realistic parameters from atoms are taken, given by ,  Rb87 (). We see that a transparency window is opened in the profile of near ; 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 as a function of , which however has always a large absorption peak near irrespective of the value of the control fields. Below, for convenience we shall call the normal mode with the linear dispersion relation () as EIT-mode (non-EIT-mode).

The double--type four-level system can be used to describe a resonant FWM process in atomic systems Fleischhauer (); Merriam (); DengPRA1 (); Kang (); Ying (). The first laser field (i.e. the control field tuned to the transition with the half Rabi frequency ) and the second laser field (i.e. the probe field tuned to the transition with the half Rabi frequency ) can adiabatically establish a large atomic coherence of the Raman transition, described by the off-diagonal density matrix element . The third laser field, i.e. the control field tuned to the transition with the half Rabi frequency , can mix with the coherence to generate a fourth field with the half Rabi frequency resonant with the transition. For details, see Refs.Fleischhauer (); Merriam (); DengPRA1 (); Kang (); Ying () and references therein.

ii.2 Metamaterial analogue of the double--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.

Figure 2: (color online) (a) Schematic of the plasmonic metamaterial, which is an array of meta-atoms. (b) The meta-atom consists of two cut-wires (CWs) (indicated by “A” and “B”) and a split-ring resonator (SRR), where the parameters , , , , , , and are given in the text. For generating nonlinear excitations, four hyperabrupt tuning varactors are mounted onto the slits of the SRR. (Sec. III). (c) The numerical result (blue dashed lines) of the normalized absorption spectrum of the EM wave as a function of frequency by taking , (first panel), and (second panel). (d) The numerical result (blue dashed line) of normalized absorption spectrum for , . Red solid lines in (c) and (d) are analytical results obtained from the formula given by Eq. (32) in Appendix B.

The CW A and CW B are, respectively, positioned along and direction, while the SRR is formed by a square ring with a gap at the center of each side. 20--thick copper forming the CWs and the SRR is etched on a substrate with a height of . Geometrical parameters of the meta-atom are , , , and  note1000 ().

We assume that an incident gigahertz radiation [with ()] is collimated on the array of the meta-atoms, with polarization component () 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 [red arrow in Fig. 2(b)] Zhangs (). 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 for (first panel), and for (second panel). Here and below, and are taken to be real for simplicity. Fig. 2(d) shows the normalized absorption spectrum under the excitation condition for . The red solid lines in the figure are analytical results obtained from based on Eq. (32) 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 note03 (). For the excitation condition [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 [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 [Fig. 2(c)] as the PIT-mode, and the excitation mode under the [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 Zhangs (); Gu (); Chen (); bai (); Bai2 (); Bai3 ()


where are displacements from the equilibrium position of the bright oscillators () and the dark oscillator (), with and  note03 () respectively the damping rate and the natural frequency of th oscillator; () is the parameter describing the coupling between the CW A (CW B) and the -polarization (-polarization) component of the EM wave, and ( is the parameter describing the coupling between CW A (CW B) and SRR. The numerical values of these coefficients can be obtained from the numerical result presented in Fig. 2, by using the method described in Appendix B.

Based on the solution given by Eq. (32), we deduce that, in the case of , and , Eq. (6) allows a “dark state” (i.e. the state where both the bright oscillators are not excited, i.e. ) exists, if


This “dark state” condition is equivalent to the one obtained in the four-level double--type atomic system, given by Eq. (2). Obviously, the PIT-mode shown in Fig. 2(c) corresponds to the case , where the minus symbol can be understood as a -phase difference resulting in a cooperative coupling effect, which is assumed in all the numerical calculations carried out below.

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


where , with the unit-cell density, the unit charge, and the optical susceptibility of the hosting material. We assume the distance between the meta-atoms is large so that the interaction between them can be neglected.

Assuming the central frequency of the incident radiation is near the natural frequencies of the Lorentz oscillators described by Eq. (6note03 (), 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 and , where and are slowly-varying envelopes and is a small detuning. With this ansatz and under RWA, Eq. (6) is simplified into the reduced Lorentz equation


with . We see that the reduced Lorentz equation (9) describing the unit cell has the same form as the optical Bloch equation (3) describing the four-level double- atom. Consequently, each unit cell in the metamaterial is analogous to a four-level double--type atom in the atomic gas presented in the last subsection. That is to say, the unit cell is indeed a meta-atom, where the bright-oscillator excitation in the CW A (CW B) driven by () is equivalent to the dipole-allowed transition () driven by the probe field (), and the dark-oscillator excitation in the SRR is equivalent to the dipole-forbidden transition in the four-level double--type atom. We also see that the coupling between the CW A (CW B) and the SRR, described by (), is equivalent to the control field () driven the atomic transition ().

Under SVEA, the Maxwell equation in the metamaterial reads


with , . Obviously, Eq. (10) has similar structure as Eq. (4). Thus, a complete correspondence between the MB equations (3) and (4) described the four-level double--type atomic gas and the Maxwell-Lorentz (ML) equations (9) and (10) described the plasmonic metamaterial is established.

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 . It is easy to get the linear dispersion relation


where , with and (). As expected, the metamaterial system allows two normal modes with the linear dispersion relation respectively given by and . In fact, () is a PIT-mode (non-PIT-mode) of the system, as explained below.

The character of the above two normal modes can be clearly illustrated by plotting and as functions of . Shown in Fig. 3(a)

Figure 3: (color online) (a) Linear dispersion relation of the -mode (PIT-mode). Im() (blue dashed line) and Re() (red solid line) are plotted as functions of for (first panel) and (second panel). (b) Linear dispersion relation of the -mode (non-PIT-mode) for arbitrary ().

are Im() (blue dashed line) and Re() (red solid line) for (first panel; corresponding to ) and (second panel; corresponding to ), respectively. When plotting the figure, the system parameters are taken from Appendix B, and additional parameters are chosen by  Mikhail () and . We see that displays a transparency window (called PIT transparency window) near , analogous to the EIT transparency window in of the four-level double--type atomic system [red dashed line and green dashed-dot line in Fig. 1(b)]. The steep slope of 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 and is reduced), the PIT transparency window becomes wider and deeper, and the slope of 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 , which is nearly independent on the coupling constant (). We see that has a single, large absorption peak and has an abnormal dispersion near , analogous to of the double--type atomic system [blue solid line in Fig. 1(b)].

ii.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--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. -component) is injected into the metamaterail, a new polarization-component (e.g. -component) will be generated through two equivalent control fields (i.e. the couplings between the SRR and CWs, described by and ). To illustrate this, we present the solution of the ML equations (9) and (10)


which can be obtained by using Fourier transform Ying (); LiHJ (). Here with , and is the initial amplitude of the normal mode determined by given excitation condition. We assume initially only the -component of the EM field in input to the system, i.e. the initial condition for the EM field is given by , . By Eq. (12) we have


where . For simplicity, we consider the adiabatic regime where the power series of and on converge rapidly. By taking and , we readily obtain


where , with being the group-velocity of the normal mode .

The conversion efficiency of the FWM is given by , where is the medium length. For the case , one has , which means that the mode decays away rapidly during propagation and hence can be safely neglected. Then Eq. (14) is simplified as


We see that the -and -polarization components of the EM wave have matched group velocity . The expression of the FWM conversion efficiency reduces into


Shown in Fig. 4

Figure 4: (color online) FWM conversion efficiency as a function of the dimensionless optical depth for (blue dashed line), and (red solid line). Inset: FWM conversion efficiency for optical depth up to 300 for .

is the FWM conversion efficiency as a function of the dimensionless optical depth for (blue dashed line) and for (red solid line). When plotting this figure, we have set and in order for a better analogue to the atomic system. The influence of can be effectively reduced by introducing a gain element into the gaps of the SRRs (see the discussion in Sec. IV). From the figure, we see that for the case of exact resonance (i.e. ), the FWM efficiency increases and rapidly saturates to when the dimensionless optical depth (i.e. ), indicating a unidirectional energy transmission from to . For the case of far-off resonance (i.e. ), the FWM efficiency displays a damped oscillation in the interval , indicating a back-and-forth energy exchange between and ; eventually the efficiency reach to the steady-state value when (see the inset). Interestingly, the value of the FWM conversion efficiency may reach to at (i.e. ).

Iii 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. Lapine () 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 Shadrivov (); Wang (). Here, as suggested in Refs. Shadrivov (); Wang (); Bai2 (), we assume the nonlinear insertion in the PIT metamaterial are varactor diodes, which are mounted onto the gaps of the SRRs Wang () [see Fig. 2(b)].

iii.1 Nonlinear envelope equations

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


where and are nonlinearity coefficients, described in Appendix E.

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. (), with and a detuning in wavenumber. The oscillations of the Lorentz oscillators in the meta-atoms have the form (), with . 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 and , listed in Appendix C.

We solve the equations for and by using the standard method of multiple scales Jeffery (). Take the asymptotic expansion , , , , and (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 Jeffery () and . Substituting the expansion into the equations for and 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. ] solution reads and , where and is a slowly-varying envelope function to be determined in higher-order approximations. The expression of is given in Sec. II.3. Here we consider only the PIT (i.e. ) mode because the non-PIT (i.e. ) mode decays rapidly during propagation, as indicated in the last section. The solution for is presented in Appendix D.

At the second order [i.e. ], a solvability condition yields , where is the group-velocity of the fundamental wave. Solution for the longwave is and , with 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. ], a solvability condition results in the equation


where describes the group-velocity dispersion of the fundamental wave; is a coefficient characterizing the coupling between the fundamental and long waves, with the expression given in Appendix D; and are, respectively, the second-order and third-order nonlinear susceptibilities of the mode, with the form