1. Introduction

Surface plasmons are electromagnetic excitations associated with charge density waves on the surface of a metal or a semi-conductor [1–15]. They have attracted tremendous interest to a wide spectrum of scientists, ranging from physicists, chemists to material scientists and have been explored for potential applications in nanophotonic devices [2, 3], quantum information [10] and solar cells [11]. One of the most attractive aspects of surface plamsons is the subwavelength character, which brings them unique capability to localize and manipulate electromagnetic field.

The simplest surface plasmonic structure consists of a plane interface between noble metal and dielectric. For such a structure, the transverse magnetic (TM)-polarized surface plasmonic dispersion relation can be obtained by solving Maxwell’s equations under the appropriate boundary conditions and have the form: $k_p=k_0\sqrt{{\epsilon }_d{\epsilon }_m/\left({\epsilon }_d+{\epsilon }_m\right)}$. Here k_p and k₀ are the plasmonic and vacuum wave numbers, respectively. The permittivities of the metal, ε_m, and the dielectric medium, ε_d, must have opposite signs if surface plasmon has to become possible at such interface. Since most of the traditional materials have positive permeability, transverse electric (TE)-polarized surface plasmons exist in nonmagnetic materials. However, several systems supporting TE-polarized plasmons were found recently such as in metamaterials [12, 16] and graphene layers with appropriate doping density [17]. However, these kinds of TE-polarized surface plasmons suffer from huge losses and strong retarding effects. The proper utilization of such TE-polarized surface plasmons is therefore very challenging.

In this article, we investigate the plasmonic properties at an interface constructed by media with strong chirality [18–25] on one side and dielectric or metal on the other side. The chirality results from the quantum coherence between atomic levels [26, 27] as discussed in [22–25]. In their model, coupling a magnetic dipole transition coherently with an electric transition, which is similar to electromagnetically induced transmission, leads to electromagnetically induced chirality. We find that under certain conditions when both the permittivity and permeability of the atomic medium are positive, the interface of such a coherent medium and a typical dielectric can still support a plasmonic mode. This is in contrast to the traditional chiral-dielectric (metal) structure which demands for opposite signs of the permittivities on the two sides of the interface. In our model, strong chirality is prerequisite. As a consequence, we call this phenomenon electromagnetically induced plasmon. Besides, due to the strong cross coupling between the electric and magnetic components of the plasmonic field, the surface plasmons are no longer purely TM-polarized. The surface plasmons also have a much larger TE-polarized component compared to the traditional chiral medium [28–34], such as organic molecule with comparatively weak chirality. Meanwhile, as the chiral strength and the effective refractive indices of the atomic medium are field intensity and phase dependent, the plasmonic properties can be highly tunable. The investigation can play an important role in realizing low loss, controllable, and all dielectric plasmonic devices.

The paper is organized as follows: In Sec. 2, we present the detail of our model and discuss electromagnetically induced chiral materials with analytical expressions. In Sec. 3, we obtain the dispersion relation for the surface plasmons at the interface between such chiral media and a metal or a dielectric. The numerical results of the study are presented in Sec. 4 while Sec. 5 gives the concluding remarks.

2. Chirality

Our model is schematically shown in Fig. 1, where two infinite half space media join at x = 0 in the xz plane. The medium below the interface, with relative permittivity ε₂, is considered to be a nonmagnetic noble metal or a dielectric, which is termed as medium 2. The medium above the interface, with relative permittivity ε₁ and magnetic permeability μ₁, is composed of atomic gas with strong chirality, which is termed as medium 1. In order to realize strong chirality, we choose the model investigated in [22–24] to be the atomic gas (medium 1) of the type as discussed in the following.

In this model, we need to coherently couple the electric dipole and magnetic dipole for the same probe field. This is achieved via a five-level atomic medium of the type shown in Fig. 1. Such a medium, as discussed in [21–24], can simultaneously control the sign of the refractive index, suppress the absorption, and are appropriate for media of interest (atoms, molecules, excitons, etc.). Here the levels |3〉 and |4〉 are opposite parity states that couple the electric field and the levels |2〉 and |1〉 are the same parity states that couple the magnetic field. For an electromagnetic wave propagating in medium 1, the electric and magnetic fields of the probe pulse can therefore couple with |4〉 − |3〉 and |2〉 − |1〉 transitions with electric dipole d₃₄ and magnetic dipole μ₂₁, respectively [24]. The magnetic dipole μ₂₁ and the electric dipole d₃₄ are coupled by the control laser which is resonant with the |3〉 − |2〉 transition. The associated Rabi frequency of the control laser is ${\tilde{\mathrm{\Omega }}}_c={\mathrm{\Omega }}_c{e}^{i\phi }$ with Ω_c and φ being its magnitude and phase, respectively and can be tuned conveniently. Two external laser fields with Rabi frequencies Ω₁ and Ω₂ can generate quantum coherence between levels |1〉 and |4〉 via a Raman transition through level |5〉 and prepares the atom in a superposition of these two states. Under this condition, the system behaves as a modified three-level electromagnetically induced transparency (EIT) like system with ground state being the superposition of states |1〉 and |4〉 as discussed in [22, 24]. A number of parameters, such as the Rabi frequencies Ω₁ and Ω₂, the control field intensity Ω_c and phase φ are tunable which makes the given model highly controllable.

 

Fig. 1 Two infinite half media conjunct at x = 0 plane. A surface plasmon propagates along the z direction. The inset circle is the atomic energy structure of medium 1. Two strong external laser fields with Rabi frequencies Ω₁ and Ω₂ prepare a coherent superposition of states |1〉 and |4〉.

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By taking into account the fully coupling Rabi frequencies, the Hamiltonian of the system can be expressed as [23]

Here, ${\omega }_i^A$ ($i=1,2,3,4,5$) is the energy frequency of level |i〉, $d_{34}=〈3|e\mathbf{r}\cdot {\widehat{\mathbf{e}}}_{\mathbf{E}}|4〉$ and ${\tilde{\mu }}_{21}=c{\mu }_{21}=c〈2|\mathbf{\mu }\cdot {\widehat{\mathbf{e}}}_{\mathbf{B}}|1〉$ are the electric and magnetic dipole moments, and E and B are the amplitudes of electric and magnetic components of the weak probe field having a frequency ω, ${\widehat{e}}_E$ and ${\widehat{e}}_B$ are corresponding unit vectors of the fields. The field associated with Rabi frequencies Ω₁ and Ω₂ oscillate at frequency ω₁ and ω₂, respectively. In a real atomic system discussed in [23], ${\tilde{\mathrm{\Omega }}}_c$ for left- and right-circularly polarized waves are opposite due to the opposite signs of the corresponding dipole moments. The result is that the model shown in Fig. 1 does not correspond to a chiral medium. We can however use a two-photon excitation process to couple levels |3〉 and |2〉 with a strong coupling field (here ${\tilde{\mathrm{\Omega }}}_c$) [35]. In such situation, ${\tilde{\mathrm{\Omega }}}_c$ is the same for both left- and right-circularly polarized waves which is responsible for strong coherence resulting in strong chirality.

We set Ω_c to be much stronger than the Rabi frequency of the probe field. The atomic model has similarities to resonant nonlinear optics based on EIT. This allows us to neglect the populations of states |3〉 and |2〉, i.e., ρ₃₃ and ρ₂₂, and we can solve the three-level subsystem |1〉, |4〉, |5〉 undisturbed by the probe field. The solution for the Λ-type subsystem corresponds to the pure state [23, 24]

Here, the external lasers with Rabi frequencies Ω₁ and Ω₂ are strong coupling lasers and throughout the paper, we take ${\mathrm{\Omega }}_1={\mathrm{\Omega }}_2$. It is also assumed that ${\gamma }_1,{\gamma }_4\ll {\gamma }_5$, where ${\gamma }_i\left(i=1,2,3,4,5\right)$ are the decay rates for the corresponding energy levels. The state given by the above equation can be considered as the ground state of the system.

In such a system, the electromagnetic constitutive relations relating the polarization P or magnetization M of the medium with the electromagnetic fields E and $H$ are [25, 27] (see Appendix A)

The physical scenario is that the polarization, given by the coherence of the transition |3〉 − |4〉, is not only induced by the electric field component but also by the magnetic field component of the probe field. Similarly, the magnetization, induced by the coherence of the transition |2〉 − |1〉, also gets a contribution from the electric field component. The coefficients in the above equations can be written as

with the denominator

In Appendix A, we show that the coefficients ${\alpha }_{EB}$ and ${\alpha }_{BE}$ are proportional to the density matrix element ρ₄₁, which refers to the quantum coherence between levels |1〉 and |4〉. If ${\rho }_{41}=0$, $i.e.$, if there is no quantum coherence, the chirality will be zero and no surface plasmons exist on the interface between this kind of medium and the dielectric if both the permittivity and permeability are positive. However, as discussed above, we can achieve maximum coherence (${\rho }_{11}={\rho }_{44}=-{\rho }_{14}=-{\rho }_{41}=1/2$) when ${\mathrm{\Omega }}_1={\mathrm{\Omega }}_2$ [see Eq. (2)], thus leading to maximum chirality.

Next we derive the expressions for the refractive indices. A consequence of chirality is that the refractive indices for two states of polarization are different. First we derive a propagating equation for the field by using the Maxwell’s equations

and the constitutive relations, i.e., $D={\epsilon }_{0}E+P$ and $H=B/{\mu }_0-M$, along with Eqs. (3) and (4). We obtain

The resulting equation,

The solution of the above equation are two circularly polarized waves with propagation wave numbers (see Appendix B)

3. Dispersion relation of the surface plasmons

To derive the dispersion relation for the surface plasmon propagating in the z direction along the interface x = 0 between the two media [see Fig. (1)], we assume the field profiles of the form $E_1=\left(E_1^x{\widehat{e}}_x+E_1^y{\widehat{e}}_y+E_1^z{\widehat{e}}_z\right){\mathrm{e}}^{\text{ikz}}{\mathrm{e}}^{\mathrm{i}{\beta }_1\mathrm{x}}{\mathrm{e}}^{-\mathrm{i}\omega \mathrm{t}}$ in medium 1 and $E_2=\left(E_2^x{\widehat{e}}_x+E_2^y{\widehat{e}}_y+E_2^z{\widehat{e}}_z\right){\mathrm{e}}^{\text{ikz}}{\mathrm{e}}^{\mathrm{i}{\beta }_2\mathrm{x}}{\mathrm{e}}^{-\mathrm{i}\omega \mathrm{t}}$ in medium 2. Here, ${\beta }_{1\left(2\right)}=\sqrt{{\epsilon }_{1\left(2\right)}{k}_0^2-k_{z1\left(z2\right)}^2}$ is the normal component of the field $E_{1\left(2\right)}$ having wavenumber $k_{1\left(2\right)}$ inside the medium 1(2) and accounts for the the transverse decay of the field. Note that the z components of k₁ and k₂, i.e, $k_{z1}$ and $k_{z2}$, are the only field components which contribute to the eigen plasmonic wavevector such that $k_z=k_{z1}=k_{z2}\equiv k$. We assume that the electric field component along y direction in medium 1 is of the form

where a₁ and b₁ denote the left- and right-circularly polarized components, respectively. By using Eq. (12), we obtain the magnetic field component as

where $m_{\pm }$ is given by

The electric and the magnetic fields in the z direction are obtained to be

and

Here ${\beta }_{\pm }=\sqrt{n_{\pm }^2{k}_0^2-k^2}$ are the transverse components of the corresponding left- and right-circularly polarized beams inside the medium 1. Similarly, we assume the electric fields in y and z directions in medium 2 to be in the form

The corresponding magnetic fields can be obtained by Maxwell’s equations as,

Making use of the boundary conditions for the field components in the two media along the interface, the continuity of the electric and magnetic field components requires

By solving the above set of equations, we obtain

This equation for the plasmonic wavenumber k (via the expressions for ${\beta }_{\pm }$ and β₂) represents the dispersion relation of surface plasmons propagating along the interface shown in Fig. 1. A possible experimental setup for the realization of these electromagnetically induced plasmonic modes is to use Otto-type configuration. It is important to mention here that even though the present study develops an approach to study the properties of SPPs using electromagnetically induced chiral systems, the formalism can be equally applied to media with high structural chirality as well as metamaterials [19, 20, 36].

 

Fig. 2 The permittivities, permeabilities, refractive indices and eigen surface plasmons versus the light frequency. Here, $\phi =-\pi /2$ and in (a) ${\mathbf{\Omega }}_c=60{\gamma }_p$ whereas in (b) ${\mathbf{\Omega }}_c=6{\gamma }_p$. γ_p represents the non-radiative dephasing rate and is taken to be ${10}^3/$s.

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Fig. 3 The permittivities, permeabilities, refractive indices and eigen surface plasmons versus the control field Rabi frequency. (a) $\phi =-\pi /4$, ${\mathbf{\Delta }}_B={\mathbf{\Delta }}_E$ (b) $\phi =-\pi /2$ and (c) $\phi =-\pi /4$ but ${\mathbf{\Delta }}_B=2{\mathbf{\Delta }}_E$. The other parameters are the same as in Fig. 2.

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4. Numerical results and discussions

In Fig. 2, we plot the real parts of the permittivity and permeability of medium 1 with two different values of ε₂. We assume that $-{\mathrm{\Delta }}_E=-{\mathrm{\Delta }}_B=\mathrm{\Delta }$ [21–27] and the atomic density is taken to be $5\times {10}^{16}$cm⁻³, which is two orders of magnitude lower than those of the similar works [37, 38]. In the absence of any coupling field, the system effectively remains a three-level scheme with states |2〉, |3〉 and the dark state, i.e, the linear superposition of states |1〉 and |4〉 which allows us to assume the spontaneous decay rates of the two states ${\gamma }_1={\gamma }_4=0$. Similarly, the magnetic response of a level transition is suppressed by four orders as compared to the electric response, therefore, the decay rates are taken to be ${\gamma }_3={\gamma }_5={137}^2{\gamma }_2$, where ${\gamma }_2={10}^3/$s, which accounts for realistic approaches [37, 38]. The plasmonic frequency is $\omega =3.14\times {10}^{15}$ rad/s. Subsequently, the effective refractive indices for left- and right-circularly polarized lights can be obtained from Eq. (13), which are shown in Fig. 2(a2) and 2(b2). In Fig. 2(a), we assume that the medium 2 is a metal with permittivity ${\epsilon }_2=-2+0.1i$ and suppose that ε₂ varies slowly in the frequency region that we are interested in. We can see that the real parts of ε₁ and μ₁ are always positive over a great range of probe field detunings and, therefore, the left- and right-circularly polarized fields are both right handed having positive refractive indices. In contrast to [22, 23], no negative refraction is achieved for any of the two beams inside the medium at these specific detunings. By using the dispersion relation given by Eq. (22), we can obtain the surface plasmon eigen modes, which are shown in Fig. 2(a4). The structure here behaves like a traditional chiral-metal interface and a single surface plasmon is supported when the detuning is small. Even though the refractive indices of the two circularly polarized fields have huge losses, the plasmonic wave number has a small imaginary part and can propagate a large distance.

In Fig. 2(b), the medium 2 is replaced by a dielectric with permittivity ${\epsilon }_2=1$. We notice that as the detuning decreases, ε₁ changes from negative (metal) to positive (dielectric) values while μ₁ remains positive. This is a direct consequence of the field profiles inside medium 2 which can alter the properties of the above medium (medium 1). In contrast to the previous case where medium 2 was supposed to be a metal, the left-circularly polarized field inside medium 1 shows left-handed property when the medium 2 is a dielectric. The later geometry can be of particular interest for plasmonic studies as: when $\text{Re}\left({\epsilon }_1\right)<0$, the structure is similar to traditional metal-dielectric interface and supports a surface plasmon. However, a remarkable property is noted for certain detunings when $\text{Re}\left({\epsilon }_1\right)>0$, we can still excite a surface plasmon on the interface. This result is contrary to the traditional surface plasmonic condition that at least one of the permittivities or permeabilities on the two sides of the interface should be negative. This phenomenon is analogous to the electromagnetically induced chirality in [22] and we call the results electromagnetically induced plasmon. This exciting property means that we can realize an all dielectric plasmonic structure using our model.

 

Fig. 4 The corresponding wave numbers perpendicular to the interface of the surface plasmons as shown in Figs. 3(a4)–(c4).

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Fig. 5 The permittivities, permeabilities, eigen surface plasmons, versus the quantum coherence ρ₄₁. Here, Ω_c=6.145γ_p and ${\rho }_{41}=-\left|{\rho }_{41}\right|$.

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In Fig. 3, we show that the properties of medium 1 and, consequently, the plasmonic wave number k as well as the propagation distance, defined by $L=1/2\pi$ Imk, can be controlled by either controlling the coupling laser intensity or phase. In Figs. 3(a) and 3(b), we set the frequency detuning to be $\mathrm{\Delta }=-0.0215{\gamma }_p$. Figures 3(a) correspond to the case where $\phi =-\pi /4$ while 3(b) correspond to the case where $\phi =-\pi /2$. When Ω_c is small, the medium shows metallic characteristics with Re$\left({\epsilon }_1\right)<0$. However, when Ω_c is large, the medium becomes pure dielectric with Re$\left({\epsilon }_1\right)>0$. Under both initial phases, the left-circularly polarized field displays left-handed property by having negative refractive index. A surface plasmon mode can be found at certain Ω_c region. In Figs. 3(c), we set ${\mathrm{\Delta }}_B=2{\mathrm{\Delta }}_E=-0.043{\mathrm{\Gamma }}_p$ and $\phi =-\pi /4$. We can see that compared with Fig. 3(a), where we assume ${\mathrm{\Delta }}_B={\mathrm{\Delta }}_E$, a slight detuning shift of transition frequencies from |3〉 to |4〉 and |2〉 to |1〉 has a small influence on the dispersion relation of the surface plasmonic mode. However, large detuning difference means that the cross coupling effect diminishes, resulting in reducing the chirality. As a consequence, the detuning cannot be too large in our model. All the numerical results show that the properties of the chiral material and consequently the surface plasmons are highly dependent on Ω_c as well as φ, hence the plasmonic properties of the structure can be controlled efficiently.

 

Fig. 6 The electric field distributions as a function of the distance to the interface. The electric fields decay as $e^{-Im\left(k\right)z}$ along z direction. (a) is the field intensity. (b) and (c) are the real parts of the electric field at time 0 and $\pi /2\omega$. Here $\left|{\mathbf{\Omega }}_c\right|=5.93{\gamma }_p$ and $\phi =\pi /4$. Time dependent variations of the plasmonic field can be seen more clearly in the supplementary Visualization 1 and Visualization 2 .

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In Fig. 4, we plot the imaginary parts of the transverse $\left(x\right)$ component of the plasmon wave number corresponding to different phases. It is shown that for the case when $\phi =-\pi /4$, both ${\beta }_{-}$ and ${\beta }_{+}$ have nearly the same values even at different detunings of the probe field [Figs. 4(a), 4(c)]. This again reveals the fact that the medium’s responses remains almost unchanged by slightly varying the probe field detuning from ${\mathrm{\Delta }}_B={\mathrm{\Delta }}_E$ [Fig. 4(a)] to ${\mathrm{\Delta }}_B=2{\mathrm{\Delta }}_E$ [Fig. 4(c)]. This means that the left- and right-circularly polarized fields are almost symmetric and contribute only by a small fraction to the TE-polarized component of the electric field intensity. On the other hand, when the phase of the controlling field is taken to be $\phi =-\pi /2$, ${\beta }_{-}$ and ${\beta }_{+}$ have much difference and will contribute more [Fig. 4(b)]. This property defines the nature of the plasmonic modes and will be shown in Fig. 7.

 

Fig. 7 The ratio between the energy densities of TE- and TM-polarized components of the surface plasmons. The parameters are the same as in Fig. 3.

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As discussed in [22, 23], the quantum coherence ρ₄₁ plays a key role in the strong chirality of the medium. In order to investigate the role of ρ₄₁ on the surface plasmons, we plot the surface plasmonic mode dispersion as a function of ρ₄₁ in Fig. 5(a). When ρ₄₁ is small, even the medium shows metallic characteristics, no surface plasmons are supported. However, when ρ₄₁ reaches a certain specific value, surface plasmons can be found beyond that region. Note that as ρ₄₁ approaches its maximum value, i.e., $0.5$ [see Eq. (2)], the medium displays dielectric characteristic but surface plasmons are still supported. As a consequence, this special surface plasmons coming from the strong chirality is highly related to quantum coherence. We have previously mentioned that the coherence term ρ₄₁ is directly dependent on the Rabi frequency of the driving field Ω_c, therefore making a comparison between Fig. 5(a) and Fig. 3(b2), we see that both the results are in good agreement. Figure 5(b) shows the longitudinal and transverse components of the left- and right-circularly polarized beams in the chiral medium. The two components decay faster when the coherence is maximum. Similarly, both the real and the imaginary parts of the plasmonic wavevector are higher when the coherence is high.

Due to the asymmetry of the left- and right-circularly polarized fields, the plasmons at a chiral-metal interface are no longer purely TE- or TM-polarized. In Fig. 6, we plot the electric field distributions as a function of the distance to the interface. Electric field along y direction can be found. At different intervals, the behaviors of the electric field components in three directions are different, which satisfies the signature of circularly polarized fields. A time dependent visualization of these variations can be seen in the supplementary Visualization 1 and Visualization 2. Visualization 1 shows the real parts of the electric fields along x axis due to the translational symmetry along y direction; while Visualization 2 shows the real part of the electric field E_y

in the x − z plane. These figures clearly display the circularly-polarized signatures of the plasmonic fields.

In Fig. 7, we calculate the ratio of the two components of the plasmons at the interface, defined by the quantity

which represents the ratio between the electric energy densities of TE- and TM-polarized components. In this figure, we see that for $\phi =-\pi /4$, the ratio is much smaller than that for $\phi =-\pi /2$. The reason is that the relative difference between the x components of the wave vectors for the left- and right-circularly polarized fields was much larger at $\phi =-\pi /2$ [see also Fig. 4]. At this specific phase, the asymmetry of the two fields is large and, consequently, a large y component of the field. Having similar variations as in Fig 4(a), when $\phi =-\pi /4$ and ${\mathrm{\Delta }}_B=2{\mathrm{\Delta }}_E$, we see that the relative difference of the x components of the wave vectors decreases first and then increases by increasing the control field. As a consequence, in order to increase the TE-polarized component, the asymmetry of the left- and right-circularly polarized fields can be effectively controlled by varying the control laser intensity and phase. Therefore, the results presented in Fig. 7 are in good agreement with that of Fig. 4.

A wide range of atomic, molecular, and condensed matter systems, such as dysprosium [22], have similar energy structures as used in our model. Here the probe field at 710nm is coupled with the electric transition between $4f^{10}5d6s$, J = 8 and $4f^{9}5d^{2}6s$, J = 7 and magnetic transition between $4f^{10}6s6p$, J = 7 and $4f^{10}6s6p$, J = 6. The Rabi frequencies are dependent upon different external laser fields, which can be varied independently. On the other hand, even the relation between Δ_B and Δ_E depends upon the atomic level structure, however, as discussed and shown above, a small detuning difference between them has a slight (almost negligible) influence on our results. In contrast, large detuning means that the cross coupling effect diminishes, resulting in reducing the chirality. Therefore, the detuning cannot be too large in our model. In a real experiment, a dielectric slab can be used to trap the atomic gas medium. To observe the surface plasmon between the atomic gas and the dielectric, we can use the Otto configuration with prism located in the dielectric. The incident field from the prism to the the atomic medium can be coupled tothe surface plasmons at the interface and, consequently, the reflection can be modified efficiently.

5. Conclusion

In this article, we investigated the electromagnetic properties of surface plasmons at the interface between a multilevel medium and a metal or a dielectric. The cross coupling resulting from the quantum coherence leads to strong chirality in the atomic system. In contrast to the traditional chiral material, the surface plasmonic mode can be found at the interface between the atomic medium and a dielectric even the permittivities and permeabilities of both media are positive. As the chiral strength and effective refraction depend on the control laser intensity and initial phase, we can control the plasmonic properties conveniently. In principle, these special surface plasmonic modes can also be found using structurally chiral media and metamaterials having strong chirality [19, 20, 36]. To implement this work experimentally, a possible scheme is to use the Otto configuration to observe the influence of the surface plasmons on the reflection of an incident beam. These results may have important applications on the controllable and an all dielectric plasmonic devices.