## Abstract

We studied the surface plasmon resonance properties of transverse electric (TE) wave in a μ-negative (MNG) material/dielectric /μ-negative (MNG) material waveguide with a finite length which works as a subwavelength cavity. The wavelength of the surface plasmon becomes shorter when decrease the thickness of the dielectric core and decrease the plasma frequency of MNG material. The resonance in this cavity can be understood as a Fabry-Perot resonance caused by the reflection of the TE guided mode at the entrance and the exit surfaces. The electromagnetic fields and power flow are concentrated around the dielectric core at the resonant frequency, the magnetic field is maximized at the dielectric core entrance and exit. When a subwavelength magnetic resonator is put at the core entrance and the resonance frequency is tuned to the plasmon cavity mode, Rabi splitting and Rabi oscillation can appear because of the strong coupling between this resonator and the cavity mode.

©2010 Optical Society of America

## 1. Introduction

Rabi splittings are of strong interest for studies on interaction effects between matter and electromagnetic (EM) wave in recent years [1–11]. When an atomic-like two-level system such as quantum wells or quantum dots exhibiting a pronounced exciton resonance is put into a cavity with strong localized field, the resonant coupling between the transition of the two levels and the cavity mode will lead to Rabi splitting and Rabi oscillation [1–11]. One can see two transmission peaks and two reflection dips for the coupled atom-cavity system. Rabi oscillation can clearly demonstrate the discrete nature of the coherent exchange of energy between the atom and the cavity. Many experiments about Rabi splitting have been reported because it has promising applications in the atom detector and infrared photodetectors [9]. It is well known that the coupling efficiency is proportional to the strength of optical field of the cavity mode, which is inversely proportional to the cavity volume [6]. For studies of fundamental light-matter interaction, it is desirable to achieve a cavity with highly localized field and small volume. Recently, a subwavelength metal-dielectric-metal microcavity was used to observe Rabi splitting in terahertz frequency [2]. In the previous studies of Rabi splitting, most of the atom or the semiconductor quantum well are equivalent to the electric oscillating resonators. However, new kinds of artificial materials called metamatrials can be seen as electric and (or) magnetic oscillating resonator [8]. In addition, by using metamaterials, the subwavelength cavity with strong field intensity can also be realized by using the waveguide structure for TE and transverse magnetic (TM) wave in the frequency region of interest.

Metamaterials, including negative-index materials (ε<0 and μ<0) [12,13] and single negative (SNG) materials [14–20] have attracted intensive studies in the past few years, due to their unique EM properties and potential applications. There are two kinds of SNG materials: one is ε-negative media with ε<0 and μ>0 (ENG), and the other is μ-negative media with ε>0 and μ<0 (MNG). SNG materials can be realized by metallic wires [14], split ring resonator (SRR) structures [15] and special transmission lines [16–18]. A number of unique properties such as resonance, complete tunneling and transparency have been found in the structure containing SNG materials [16,19,20]. A single-mode cavity with sub-wavelength size has also been proposed based on the ENG-MNG pair and lead to unusual Rabi splitting [7]. Furthermore, the localized resonance is also realized based on the surface plasmon waveguide cavity containing ENG material [21–23]. A surface plasmon is associated with the EM waves that propagate along the interface and decay exponentially into both media [24,25], which can occur only when either permittivities (for TM wave) or permeabilities (for TE wave) of the two adjacent media have opposite signs. The excitation of surface plasmons causes a resonant transfer of the photon energy to a surface wave and is manifested in an attenuated total reflection configuration as a sharp drop in the reflectance [24]. Besides the attenuated total reflection structure, surface plasmons can also be excited from a free space without momentum matching simply by perpendicular incidence to the end face of a waveguide structure containing metamaterial [21,26]. The symmetric metal/dielectric/metal waveguide with a finite length has been reported as a TM plasmon resonance cavity with strongly localized field in optical region [21,22]. In this paper, the MNG material is used to make the finite length waveguide MNG/dielectric/MNG for TE wave in microwave frequency, Rabi splitting and Rabi oscillation induced by the localized resonance mode are theoretically studied when a magnetic resonator is put on the entrance of the plasmon cavity.

## 2. Model and theory

The proposed waveguide structure is composed of a dielectric core with thickness of *d* sandwiched between two half-infinite SNG material claddings, it is similar with that in Ref. [22], in the reference, metal works as an ENG material in optical frequency and the surface plasmons exist just for TM wave. Here we consider the MNG/Dielectric/MNG system for TE mode in microwave frequency region as shown in the inset of Fig. 1(a)
. For the sake of simplicity, we call the waveguide structure as MDM hereafter. MNG material is described by Drude model: ${\epsilon}_{m}=1$, ${\mu}_{m}=1-{f}_{p}^{2}/({f}^{2}+i\chi f)$, where ${f}_{p}$ is the magnetic plasma frequency, *χ* denotes the damping. The dispersive characteristic can be realized by using L-C loaded transmission lines with low loss [16,18].

As is well known, surface plasmon can exist in a two-dimensional MDM and the dispersion relation of the MDM for TE wave can be written as [24]:

*β*is the propagation constant of the surface plasmon, ${k}_{0}=2\pi /\lambda $ is the wave number of light in the air,

*λ*is the wavelength of incident wave. ${\epsilon}_{d},{\mu}_{d}$, ${\epsilon}_{m}$ and ${\mu}_{m}$ are the permittivity and permeability of the dielectric core and MNG material, respectively. Effective refraction index of the MDM is defined as ${n}_{\mathit{eff}}=\beta /{k}_{0}$, which can be calculated by Eqs. (1) and (2), here the loss is not considered for simplicity, that’s $\chi =0$. When the two MNG-dielectric interfaces are brought closer to each other, the dispersion curve of a single interface will splits into high-(plasmon anti-symmetric) and low-energy (plasmon symmetric) modes. Surface plasmon can be excited from the left free space simply by perpendicular incidence to the end face of MDM. Since the perpendicularly incident plane wave can only excite the low-energy mode because of the matching of symmetry, we only discuss the symmetric low-energy mode. Figure 1(a) shows the real part of

*n*as a function of

_{eff}*d*and frequency

*f,*where air is chose as the dielectric core and

*f*

_{p}= 10GHz. As seen from the figure,

*n*decreases as the thickness

_{eff}*d*increases with the same frequency

*f*, and

*n*also varies when the value of

_{eff}*f*changes with

*d*fixed. From Eq. (1),

*n*is also affected by the permeability of the cladding which determined by the magnetic plasma frequency

_{eff}*f*of MNG material. Figure 1(b) shows the dispersion relations of the MDM with different magnetic plasma frequency,

_{p}*f*8GHz,10GHz and 12GHz respectively, where

_{p}=*d*= 1mm, the dielectric core is air. From Fig. 1(b) one can see that, with the same

*d*, the

*n*of the MDM with lower

_{eff}*f*is always greater than that with higher one. Taking one with another, the propagation constant of the surface plasmon mode in the MDM increases as the dielectric core is made narrower and as the permeability of MNG material has a reduced magnitude. For a MDM with a finite length, it can work as a Fabry-Perot resonant cavity [22], because the TE plasmon modes supported by MDM are reflected by the end faces, and the magnetic field will become the maximum at the ends. The resonance condition is $\beta L\approx m\pi $ or $2{n}_{\mathit{eff}}L\approx m\lambda $,

_{p}*m*is the resonance order, e.g.

*m*= 1, 2, 3 correspond to the first, second and third order resonance modes. Since the thickness of the dielectric core is very small, the cavity will produce high magnetic field enhancements in the core entrance.

## 3. Results and discussion

Since the propagating properties of the three-dimensional MDM do not vary so much from those of the two-dimensional structure even if the vertical height is reduced to subwavelength, we think the analysis mentioned above can elucidate the properties of the real MDM with a finite height without much loss of generality [27]. To study the propagation property of EM waves in our system, we perform a set of finite difference time domain calculations using a commercial software package CST (Computer Simulation Technology). In the calculation model shown in the inset of Fig. 1(a), we set the calculation region is 40mm in *y* direction, 10mm in *z* direction, 200mm + L in *x* direction, L is the waveguide length. The boundary condition is set as electric boundary (E_{t} = 0) at *z*
_{max} and *z*
_{min} plane and magnetic boundary(H_{t} = 0) at *y*
_{max} and *y*
_{min} plane. The selected boundary condition indicate that the incident plane wave is simple TEM wave. The incident plane wave coming from *x* = −100mm can excite the plasmon mode supported by the MDM and the plasmon mode will form standing wave between the two surfaces as predicted.

Figure 2
shows the calculated transmittance of the resonant cavity realized by using a finite length MDM (L = 15mm) with different dielectric core thickness, *d* = 0.5mm, 1mm, 2mm and 4mm respectively, where the loss is not considered and *f*
_{p} = 10GHz. Combining with Fig. 1(a), one can see that the transmission peaks satisfy the resonance condition $f\approx mc/(2{n}_{\mathit{eff}}L)$,*c*is the vacuum speed of light. For *d* = 0.5mm, the wave vector of the surface plasmon is much larger than that of *d* = 4mm, so there are two resonant peaks at 1.77GHz and 2.78GHz respectively for *d* = 0.5mm, while there is one peak at 3.51GHz within 5GHz of interest for *d* = 4mm. In this paper, we are just interested in the first order resonance. Note that, at the first order resonance frequency, the wavelength is much longer than the length of MDM, for example *d* = 1mm, the first order resonance wavelength is about 128.8mm, about 8.6 times larger than the cavity length L = 15mm, so we call the cavity based on surface plasmon mode as subwavelength plasmon cavity. As known from Fig. 1(b), the wave vector of the surface plasmon mode become larger when reduce the magnetic plasma frequency of MNG material, so the first resonance peak will shift towards higher frequency with *f*
_{p}. Figure 3
shows the calculated resonant transmittance of the cavity with *d* = 1mm, L = 15mm and different *f _{p}*,

*f*= 8GHz, 10GHz and 12GHz respectively. It is clear that, the smaller

_{p}*f*

_{p}will lead to the smaller cavity length under the same resonance frequency. It is also worthwhile to mention that the quality factor of the resonant mode increase when decrease the dielectric core thickness and decrease

*f*which indicates that the plasmon cavity will possess strong localized field in the dielectric core.

_{p},At a typical first-order resonance for L = 15mm, *d* = 1mm and *f _{p} =* 10GHz, the simulated results of the power flow distribution in the plasmon cavity by CST is displayed in the upper side of Fig. 4
. Perpendicular and longitudinal energy confinement in the dielectric core is clearly visualized. Furthermore, the electric and magnetic field distribution (at

*y*= 0 plane) in the lower plots indicate that the standing wave picture inside a MDM is evident. It is worthy to note that the magnetic field is maximized near the entrance and the exit surfaces, the electric field is maximum at the center. but the electric field nodes is not exactly located on the waveguide ends due to that the exact reflection phase shifts at the ends are not 0 [23]. In the lower plots, the electric (magnetic) field magnitude is normalized with that of the incident wave, the maximum of electric (magnetic) field at

*x*= 7.5mm (

*x*= 0.5mm) is enhanced by about 95 (63) times, by and large, the maximized magnetic is near the MDM entrance. Reduction of the thickness

*d*and

*f*would also lead to a larger field enhancement by the energy confinement in a much smaller dielectric core volume. Consequently, it is easy to study the cavity electrodynamics properties such as Rabi splitting if put a subwavelength magnetic resonator approximately to the MDM entrance (

_{p}*x*= 0)or the maximum field location, i.e.

*x*= 0.5mm.

Different from the standing-wave-form field in a conventional cavity, the field of the plasmon mode is strongly localized as shown in Fig. 4. When a magnetic resonator is placed in the entrance of the MDM dielectric core, Rabi splitting can be observed because of the strong coupling between the magnetic resonator and the plasmon cavity, which is similar to that between electric field and an electric resonator [8]. Here the magnetic resonator is described by Lorenz model with $\epsilon =1,$
$\mu =1-{f}_{mp}^{2}/({f}^{2}-{f}_{m}^{2}+i\gamma f)$, where ${f}_{m}$ is the resonance frequency, ${f}_{mp}$ is magnetic plasma frequency and *γ* is the collision losses. The dispersion parameters can be realized conveniently by using SRR structure [15,28]. The volume of the magnetic resonator affect Rabi splitting remarkably when the strong coupling happens between the resonator and the cavity field because the splitting is proportional to square root of the resonator density [6]. In our calculation model, to reduce the effect of magnetic resonator on the cavity mode, the resonator thickness should be very small in *x* direction and the resonator volume is chose to (0.1mm × 1mm × 3mm) in *x-y-z* direction, where the cavity parameters are the same as that in Fig. 4. ${f}_{m}={f}_{mp},$
$\gamma ={10}^{-4}{f}_{m}$, Which are of the order of magnitude as the ones used in [29]. It is necessary to tune the resonance frequency ${f}_{m}$ of the magnetic resonator or the plasmon cavity resonant frequency *f*
_{0} in order to study Rabi splitting and the resulting anti-crossing behavior. Fortunately, *f _{m}* can be tuned by the structure parameters [28]. Figure 5(a)
presents the simulated transmission spectra of the plasmon cavity embedded with an magnetic resonator with different

*f*at the center of

_{m}*x*= 0.5mm plane where the magnetic field is maximal, the data of successive simulations are offset by −10dB. The plasmon cavity resonance mode is spitted in the presence of magnetic resonator: the transmission spectra of the coupled-system are clearly double-peaked, and the frequency splitting depends on the detuning between

*f*and

_{m}*f*

_{0}. When decrease

*f*

_{m}from 2.8GHz to about 2.4GHz, the upper peak

*f*

_{H}shifts to low frequency more quickly than that of the lower peak

*f*

_{L}, while the

*f*

_{L}shifts faster than

*f*

_{H}as

*f*

_{m}decreases further from 2.4 GHz. The two modes repel each other in the vicinity of 2.4GHz. The splitting is always symmetric when the magnetic resonator is in resonance with the cavity mode, while the splitting and spectra shape become asymmetric at off resonance. The simulated splitting mode peaks of the coupled resonator-cavity system are plotted as a function of

*f*in Fig. 5(b). The clear anticrossing behavior is the characteristic of Rabi splitting, where the Rabi splitting (

_{m}*Ω*) is about 0.154GHz.

Rabi splitting may be seen to follow from the energy exchange between the resonator and the plasmon cavity. In the transient regime, this exchange is manifest as a temporal oscillation on the EM wave transmitted through the coupled cavity system [1,5]. The frequency domain response of the coupled cavity system to an incident pulse is connected with the time domain response by Fourier transform. See from Fig. 5, the transmittance of the coupled cavity system has symmetric double peaks when the magnetic resonator frequency is tuned to about 2.4GHz, then if the input pulse is sufficiently short, the output pulse will contain an oscillation with a period of $1/\Omega $ [1]. The transient response of our resonator-cavity system is investigated by injecting a short pulse in the left end of the MDM and detecting the temporal evolution of the EM wave emitted out through the other end. The temporal evolution of the input (normalized output) pulse intensity is shown in Fig. 6(a) and 6(b), which shows that the energy exchanges between the cavity and magnetic resonator is evident. There are obvious oscillations in the transmitted pulse with time, meanwhile its amplitude exponentially decays with time due to the loss and radiation and the decay time is about 9ns. The oscillation periodic is about 6.5ns in the simulation, which closes to the expected oscillation period ($1/\Omega $) for Rabi splitting of 0.154GHz. Moreover, it is reasonable that the special cavity and the EM properties of the resonator-cavity system can be studied in any frequency region of interest if the metamaterials is realized in corresponding frequency.

## 4. Conclusion

The resonance EM properties of the plasmon cavity based on MDM are theoretically studied. It is shown that the wavelength of the TE-polarized surface plasmon can be adjusted by changing the thickness of the dielectric core or the permeability of MNG material. At the resonant frequency, the magnetic field is strongly localized at the dielectric core entrance and exit. Whereas in a conventional TM-polarized plasmon cavity, the electric field is strongly localized at the dielectric core entrance and exit. When put a magnetic resonator at the dielectric core entrance and tune the resonance frequency to the cavity mode, the strong magnetic resonance coupling happens. In addition, the Rabi oscillation with oscillation periodic of 6.5ns also appears for the transmitted pulse.

## Acknowledgements

This research was supported by National Basic Research program (973) of China (No. 2011CB922001), by the National Natural Science Foundation of China (Nos. 10904032 and 10874129), by Natural Science Foundation of Henan Educational Committee (No. 2010B140005) and by Science and Technology Program of Henan province (No. 092300410215).

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