## Abstract

Influence of a negative refractive index meta-material (NIM) capping layer on properties of Tamm plasmon-polariton at the interface of metal-Bragg reflector structure is investigated. Conditions for excitation of the plasmon-polariton is determined from reflectivity mapping calculation and analyzed with cavity mode theory. For specific thicknesses of capping layers, Tamm plasmon-polariton with negative group velocity is revealed in a wide region of frequency. Different from backward optical propagation induced by negative effective-group-refractive-index in dispersive media, negative group velocity of Tamm plasmon-polariton results from opposite signs of cross-section-integrated field energy and Poynting vector.

© 2014 Optical Society of America

## 1. Introduction

Photonic crystals represent a family of artificial structures with periodic dielectric coefficients that manipulate light in a similar way as electrons in crystals [1]. Distributed Bragg reflector (DBR), composed of repeated periods of high and low refractive index layer, is the simplest one-dimensional (1D) photonic crystal. In analogy with the electronic Tamm states formed at the boundaries of truncated periodic potentials, there exists an intriguing localized interface mode at surface of photonic crystals, known as Tamm plasmon-polariton or optical Tamm state. Tamm plasmon-polariton at surface of DBR has been predicted theoretically [2–6] and observed experimentally [7, 8] in the last ten years. Different from plasmon existed at metal-dielectric interface, confinement of light at interface of metal and DBR is due to reflection of metallic layer and multilayer reflector rather than total internal reflection. As a result, Tamm plasma-polariton at interface of metal-DBR may possess a smaller wave vector than the excited optics, and consequently can be generated in both TE and TM polarizations, even under the condition of zero in-plane wave vectors. It has spurred great attention for potential use such as resonant optical filters at multiple wavelength [6], optical switches [9], enhancement of Faraday rotation [10], optical absorption tuning [11–13], spin-optronics [14], and so on [15]. Dispersion of Tamm plasma-polariton is of great interest both physically and technically. It not only provides a promising tool for plasmon-polariton investigation but also play substantial role in the application of surface Tamm state as, for instance, a source of polariton lasers [16].

Recently, the influence of novel material such as negative refractive index meta-material (NIM) on the properties of optical Tamm state is extensively investigated. NIM is characterized by simultaneously negative effective dielectric permittivity and negative effective magnetic permeability, giving rise to abundant of unusual properties of electromagnetic wave (EM). By terminating conventional multilayer DBR with NIM, A. Namdar *et al.* demonstrated flexible control of the dispersion properties of linear or nonlinear Tamm plasmon-polaritons [17, 18]. At certain specified excitation condition, plasmon-polariton possesses negative group velocity and a vortex-like structure. Additionally, either positive or negative giant lateral Goos-Hänchen shift occurs depending on the energy flow of Tamm plasmon-polariton [19]. Optical propagation with negative group velocity is one of the most intriguing and counterintuitive phenomena. The peak of a laser pulse appears at the rear side of the sample before it enters the front side. The phenomena was firstly predicted and experimentally verified in anomalous dispersive media and recently in NIM [20]. The negative value of group refractive index is induced either by abnormal dispersion or negative phase refractive index. On the other hand, the mechanism of optical propagation with negative group velocity in the form of Tamm plasmon-polariton is unclear yet.

In the paper, we explore the properties of Tamm plasmon-polariton at interface of Bragg-reflector and metal film influenced by a conventional capping layer by positive refractive index materials (PIM) and an additional capping layer of NIM. With this structure, energy flow in the system is tuned for obliquely incident light by varying the thickness of NIM capping layer, enabling us to investigate the relationship between energy flow and dispersion properties of Tamm plasmon-polariton. We revealed that the excitation of Tamm plasmon-polariton is influenced by the relative thicknesses of NIM and PIM. At a certain condition, abnormal dispersion of Tamm plasmon-polariton is observed for a wide range of incident wavelength, which corresponds to a negative group velocity of plasmon-polariton. Different from backward propagation of optical pulse in anomalous dispersive media, opposite sign of cross-section-integrated field energy and Poynting vector is responsible for the negative group velocity of Tamm plasmon-polariton.

## 2. Modeling

Geometry of the metal-NIM-Bragg reflector structure is illustrated in Fig. 1. The 1D DBR is composited of repeated unit cells composing of high refractive index material A with relative permittivity *ε _{1}* = 4 and relative permeability

*μ*= 1 (Ta

_{1}_{2}O

_{5}) and low refractive index material B with relative permittivity

*ε*= 2 and relative permeability

_{2}*μ*= 1 (SiO

_{2}_{2}), as shown in the right part of Fig. 1. In each unit cell the thicknesses of A and B layers are 133 nm and 180 nm, respectively. The Bragg frequency (

*ω*) of the DBR locates at wavelength approximately 1000 nm at normal incidence. Dispersions and energy dissipation due to absorption of both materials A and B are eliminated. The DBR terminates at material A forming a conventional PIM capping layer with thickness

_{B}*dc*= 800 nm throughout this study. An additional capping layer of NIM with thickness

*dn*is covered over the PIM capping layer. The thickness of NIM capping layer varies. For simplicity the relative permittivity

*ε*and relative permeability

_{n}*μ*of NIM are assumed to be −4 and −1, respectively, with the absolute value equal to those of PIM capping layer. On top of the NIM layer is a 30 nm metallic film (Ag). The optical dispersion of Ag layer is described by

_{n}*ε*is the relative permittivity of Ag,

_{m}*ω*is the photon frequency of the incident light,

*ε*is the background dielectric constant,

_{b}*ω*and

_{p}*γ*are the plasma frequency and plasma oscillation rate of Ag, respectively. Photonic energy and plasma energy are given by

*ћω*and

*ћω*, where

_{p}*ћ*is the Plank constant. In the region of interest, we assumed

*ε*= 5,

_{b}*ћω*= 9.3 eV and

_{p}*ћγ*= 0.01 eV. The relative permeability

*μ*of Ag film is unity. The refractive index of metal (

_{m}*n*) is obtained from square root of Eq. (1). For frequency region much smaller than plasma frequency as in this study, the plasma oscillation of Ag (

_{m}*γ*) is small, and

*n*is approximately given by ${n}_{m}=i{\varpi}_{p}/\varpi $. The entrance and exit media are both air. The incident light is Transverse Magnetic (TM) polarized.

_{m}Monochromatic TM wave incidents on Ag layer with angle *θ* measured from the normal. The structure is of high reflectivity by the metal film for incident light with frequency much smaller than plasma frequency of Ag. However, in the spectral band gap of DBR the reflectivity is greatly reduced if Tamm plasmon-polariton is excited. As a result, Tamm plasmon-polariton is obtained from the dips in reflectivity maps. The reflectivity of the system is calculated with Transfer Matrix Method [21]. For calculation the Bragg reflector is represented by 20 periods of unit cells. Previous work [2] as well as our calculation shows that the properties of optical Tamm state, such as dispersion, calculated with Transfer Matrix Method resemble to those calculated with semi-infinite model.

## 3. Results and discussion

Figure 2 shows the dependence of the existing condition of Tamm plasmon-polariton on the thickness of NIM capping layer, where the thickness of PIM capping layer is assumed to be 800 nm, and the incident angle is 45°. The bright stripes in Fig. 2 and inset represent the reflectivity dips (with the minimum reflectivity value of approximately 50%) due to the excitation of optical Tamm state. Two regions are distinguished in the map separated by a blue line at approximately 1340 nm thickness (denoted by *d _{0}* hereafter). For thickness below 1340 nm the excitation energy increases monotonously with the thickness of NIM capping layer. On the other hand, the excitation condition of Tamm plasmon-polariton changes into an ‘S’ shape for NIM thickness above 1340 nm in the reflectivity map. As the thickness of NIM capping layer deviates away from 1340 nm in both directions, multiple optical Tamm states are excitable (not shown). With the increasing incident angle, the thickness of NIM capping layer increased slightly while reflectivity map coincides with Fig. 2. The reflectivity map is in sharp conflict to the situation without NIM capping layer [6].

Tamm plasmon-polariton is considered as eigenmodes of EM bound by a semi-infinite metallic layer and a DBR [2, 4]. The eigenmodes of the system under consideration are determined in terms of the amplitude of right and left propagating wave:

*C*is a constant,

*φ*is the phase change of the monochromatic wave along the capping layer,

*r*and

_{L}*r*represent the reflection coefficients of a left propagating wave upon metal-NIM interface and a right propagating wave upon DBR, respectively. Eliminating

_{R}*A*from Eq. (2) givesFor normal incidence reflection coefficient at metal-NIM interface is given by [22]:

*n*= −2 is the refractive indices of NIM capping layer. Similar to Fresnel formula at metal-PIM interface [2], reflection coefficient at metal-NIM interface is approximately given by

_{n}*l*is an integer,

*n*,

_{c}*n*, and

_{1}*n*are the refractive indices of the conventional capping layer, layer A and layer B in the unit cell of DBR, respectively. For

_{2}*l*satisfying $2\pi (l-1)+\pi {n}_{1}/\left|{n}_{1}-{n}_{2}\right|<0$, the energy of Tamm plasmon-polariton increases with the thickness of NIM layer, and vice verse. The theoretical results accord well with the simulation results at normal incidence. For oblique incidence, simulation shows that the phase changes in

*r*,

_{L}*r*and in capping layer increase as the incident angle increases. As a result, the thickness of NIM layer is up-shifted as compared to normal incidence for creating Tamm plasmon-polariton corresponding to the same

_{R}*l*.

In-plane dispersion properties of the Tamm plasmon-polariton for different thicknesses of NIM is investigated by keeping the thickness of conventional capping layer at 800nm. The dispersion is strongly influenced by thickness of NIM capping layer. With increase of NIM thickness from zero, four different dispersions are revealed from reflection maps, as shown in Fig. 3. The thickness of NIM is smaller than *d _{0}* in Fig. 3(a), equal to

*d*in Fig. 3(b), slightly larger than

_{0}*d*in Fig. 3(c), and much larger than

_{0}*d*in Fig. 3(d). For thickness of NIM capping layer smaller than

_{0}*d*, the dispersion of Tamm plasmon-polariton is similar to the case without NIM capping layer, with a parabolic shape and always a positive slope. Meanwhile for thickness of NIM capping layer much larger than thickness of the conventional capping layer ($2\pi (l-1)+\pi {n}_{1}/\left|{n}_{1}-{n}_{2}\right|>>0$), the slope of the dispersive curve is still positive, but the shape showed in Fig. 3(d) is drastically different from Fig. 3(a). Interesting phenomena occurs for the thickness of NIM equals to

_{0}*d*. The Tamm plasmon-polariton turns from the lower edge to the upper edge of the band gap with increasing incident optical frequency. The dispersion of Tamm plasmon-polariton exhibits a negative slope in a wide range of frequency, indicating a negative group velocity in Fig. 3(b). With NIM thickness close to

_{0}*d*, $2\pi (l-1)+\pi {n}_{1}/\left|{n}_{1}-{n}_{2}\right|\approx 0$. As

_{0}*l*is always an integer, the condition is obtained by optimized the refractive indices of layer A and layer B of the unit cell of the Bragg reflector, such that ${n}_{1}/\left|{n}_{1}-{n}_{2}\right|$ close to an even integer just like in our case. For NIM thickness slightly larger than

*d*, a negative group velocity is present only near the edge of the bandgap of DBR as shown in Fig. 3(c).

_{0}A. Namdar *et al.* termed Tamm plasmon-polariton with negative group velocity as Backward Tamm state according to the direction of total energy flow [17]. However, our results show that the group velocity of Tamm plasmon-polariton is not directly correlated to the energy flow direction. Figure 4 presents the cross-section integrated Poynting vector and field energy density along the vertical line in Fig. 2 for different NIM thicknesses. Here the sign of Poynting vector is positive if its direction is the same as direction of Poynting vector in air, and vise versa. For NIM thickness much larger than *d _{0}*, the Poynting vectors are negative while the slope of dispersion is positive. On the contrary, for NIM thickness equals to

*d*, the Poynting vector is positive while the slope of dispersion is negative. Further calculation reveals that the negative group velocity occurs if total energy flow is close to zero. For example, at position D in Fig. 3(b) the total energy flow is small because the negative energy flow in NIM layer is compensated by energy flow in PIM capping layer and in DBR. Meanwhile, the total energy flow is approximately zero at point E in Fig. 3(c) for much deeper penetrating depth of electromagnetic wave into the DBR in the vicinity of edge of bandgap, which compensates the negative energy flow in NIM layers.

_{0}Negative group velocity of Tamm plasmon-polariton in the system can be explained by the vibration theory, which is used to describe group velocity in photonic crystals [1]. For this Maxwell’s equations is written as

*H(r)*denotes a realizable EM mode, and

*ε(r)*is the permittivity at position

*r*,

*ω*and c are angular frequency and phase velocity of the light, respectively. The operator $\nabla \times \left(1/\epsilon (r)\right)\nabla \times $ at left hand of Eq. (8) is a Hermitian operator, which is denoted by

*Ξ*in the following. By differentiating Eq. (8) with respect to wavevector

*k*after transforming

*H(r)*into wavevector space, and then taken the inner product with

*H*on both side, we obtain

*S*is the average Poynting vector,

*u*and

_{E}*u*are the cross-section integrated energy density of electric field and magnetic field, respectively. A negative group velocity is achieved in such system only if the total energy flow has an opposite sign to the integrated energy density. From Fig. 4 we see that the cross-section-integrated energy flux and energy density have similar trend with increasing NIM capping layer, and only at the condition that the energy density is nearly zero the negative group velocity is produced. The theory can also explain the negative group velocity in Refs [17]. and [18].

_{H}NIM is artificially engineered with sub-wavelength microstructures. For optical frequency, NIM generally composes of fishnet structures fabricated by lithographic methods in optically thin metal-insulator layers [23–25]. Detailed structure of the NIM can be designed with Finite-Difference Time-Domain (FDTD) simulation method. Recently advance has been made in design of novel three-dimensional (3D) NIM at optical frequency. For example, S. P. Burgos *et al.* has proposed an approach to fabricate thick NIM composing of a single layer of coupled plasma coaxial waveguides with low optical loss. With the NIM negative refractive index (approximately −2) insensitive to both polarization and incident angle up to 50° can be realized. The relative permittivity *ε _{n}* and relative permeability

*μ*of the NIM are close to −4 and −1 in the blue spectral region [26]. The NIM can be used to construct the proposed metal-NIM-Bragg reflector structure for demonstration of the negative propagation of Tamm plasmon-polariton. If the absolute value of the refractive index of NIM differs from that of PIM, more complicate dispersion of Tamm plasmon-ploration is theoretically revealed. Still negative group velocity is realizable at conditions according to our theory.

_{n}## 4. Conclusion

In conclusion, we have developed a metal-NIM-Bragg reflector model for studying the influence of NIM capping on the properties of Tamm plasmon-polariton. With this model we further investigated the excitation condition and dispersion of Tamm plasmon-polariton with respect to the thickness of NIM capping layer with cavity theory and reflection mappings calculation. It was revealed that Tamm plasmon-polariton with negative group velocity could be created with a certain NIM thickness. The negative group velocity presents much larger absolute values and exists in a wider spectral region than previous reports. From calculation of the cross-integrated Poynting vectors and energy density, we found that the negative group velocity is not directly correlated to the direction of cross-integrated Poynting vectors but originated from reverse of the sign of cross-integrated Poynting vectors and energy density. We have also proposed a possible routine for construction of the metal-NIM-Bragg reflector structure for demonstration the negative group velocity of Tamm plasmon-polariton. Detailed investigation on negative propagation of Tamm plasmon-polariton could be further carried out by FDTD simulation based on our model.

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