1. Introduction

Surface plasmon polaritons (SPPs) are quasi-particles resulting from the coupling of electromagnetic waves with oscillations of conduction in a metal and propagate along the interface between a dielectric and a noble metal with amplitudes decaying into both layers. One of the most attractive aspects of SPPs is that they can be used to overcome the diffraction limit. This leads to miniaturized photonic devices with scales much smaller than those currently achieved. Previously, various kinds of optical devices, such as waveguides, reflectors, filters as well as beam splitters, have been proposed [1,2]. The analogy between electromagnetic wave propagation in multidimensionally periodic structures and electron wave propagation in real crystals has proven to be a valuable one. Similar to electrons in a crystal, band gaps occur when electromagnetic waves propagate in a periodic dielectric structure with a period comparable to the wavelength [3]. Construction of metallic surfaces by periodically modulating the thickness of metal films, known as relief modulation, have attracted the attention of researcher in both fundamental physics and applied sciences, mainly due to the appearance of photonic band gap as SPPs propagate on such relief surfaces[4]. Amplifying surface plasmons has opened up a new way for constructing a lasing device. The corrugation opens band gaps in the dispersion relation and provides coupling of the plasmon mode with free-space radiation. The wide band gap lowers the group velocity of the surface plasmons, resulting in a decrease of the lasing threshold [5]. Such band gaps have been reported before [6–8] and have been used for different applications such as surface enhanced Raman scattering [9], wave guiding, bending and splitting [10] as well as for biosensing applications [11]. Therefore, plasmonic structures with wide range of tunability for plasmon-polariton band gap may have useful applications for photonic devices. Controlling light propagation in a medium has great theoretical importance and potential applicability. Slow light accompanied with compression of energy density can be used to enhance nonlinear properties of materials with a small feature size [12]. In addition, the slow light usually leads to a large effective refractive index, which is a very promising characteristic in optical switching [13]. Measurements of such group velocities were reported in Refs. [14,15].

In this paper, we study the plasmon-polariton band gaps of T-shaped plasmonic gratings and we show how the localized surface plasmon polariton (LSPP) modes are constructed and exploit this phenomenon to control the plasmon band gaps of this T-shaped structure. In our previous work, we showed in the plasmon-polariton band structure there was a LSPP mode of the T-shaped structure [16]. Here, we modify the T-structure by introducing a gap (made of SiO₂) with thickness (G_t) in the post (made of metal). (See Fig. 1b) The reflectance of the structure is calculated by using the Rigorous Coupled Wave Analysis (RCWA) [17]. It is found that around 50 plane waves are needed to obtain convergent results due to the localized nature of the plasmonic modes. The input light is TM polarized in which the magnetic field is parallel to the grating grooves (i.e., parallel to the y-axis). The frequency-dependent complex dielectric constants of SiO₂ and Ag are taken from Ref. [18]. The material of the T-structure and substrate is Ag. The periodicity of the array is denoted by Λ_g. The width and thickness of the top slab are denoted W and t. The width of the post is W_T, and the thickness of SiO₂ layer is t _w. Silver was our choice due to its small absorption losses in the near infrared region.

2. Device descriptions

 

Fig. 1. Basic geometry of the investigated structures, (a) plasmonic multilayer structure, (b) Symmetric T-shaped array and (c) Asymmetric T-shaped array.

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We deal with three types of structures. First, when G_t = t _w the structure becomes a plasmonic multilayer structure which has a SiO₂ layer deposited on the metal layer (substrate) and followed by one corrugated metal layer (Fig. 1a). Second, when G_t < t _w and the post (broken into two pillars of equal length) is in the middle of structure (at x=0), it becomes a symmetric T-shaped grating (Fig. 1b). The last, when G_t < t _w and the post is not in the middle of structure (with a displacement d = 50 nm), it becomes an asymmetric T-shaped grating (Fig. 1c). The gap in the post plays an important role in controlling the effect of LSPP and SPP modes when its size and location are varied. The geometric parameters used in this study are Λ_g=1000 nm, W=500 nm, t=100 nm, W _T=70 nm, t _w=330 nm and (G_t) varying from 0 to t _w. To fabricate the T-shaped array, nano-fabrication techniques such as electron beam lithography, plasma-assisted dry-etching can be used during the fabrication. Silver and SiO₂ can be deposited by the electron gun evaporator and the RF sputter system followed by lifting off process.

3. Behavior of plasmon-polariton band structures

In Fig. 2, we show the TM-mode dispersion relations of the plasmon-polariton modes for an asymmetric T-shaped grating by calculating the reflectance as a function of energy E of a p-polarized incident plane wave and its in-plane wave vector k_x. Dark region means low reflectance (high absorbance) since the transmission is negligible for the metal substrate. In these plots, two grating-related photonic bands are revealed: one associated with the T-shaped cap and the other with the post. In the case of plasmonic multilayer (i.e. G_t= t _w) the SPP modes can be excited as shown in Fig. 2a and these SPP dispersion curves are crossing at k_x=0. These two bands are caused by the zone-folding associated with reciprocal lattice vectors -K_g and K_g. The SPP modes associated with plasmonic multilayer structure reveal the resonances due to the SPP excitation at either the SiO₂/Ag or Air/Ag interfaces of the metal film as described previously [19]. However, in such structure we can not find any band gap in the dispersion relation, because the grating structures are strongly coupled along the x direction. (The spacing of two adjacent grating structures is 500 nm) When the spacing (Λ_g-W) is increased to 1000 nm (as in Ref. [19]), the gap will open up. However, we found that simply changing the spacing of two adjacent grating structures did not lead to a gradual variation of the group velocity of the bands because part of the bands became undetectable. So a better solution to this problem is to introduce another modulation in the structure. When the second grating is added to the grating profile or in other word when the gap thickness (G_t) is decreased, the energy band gaps open up in the SPP dispersion curves at the zone center where the two branches cross [6]. The gap opening is due to the localization of plasmonic modes in the x-direction (along the post under the cap) and LSPP modes are formed. Consequently, the group velocities in x-direction, ∂ω/∂k _x (V_g) decrease. It is well known when the optical wave vector is equal to half of the Bragg vector corresponding to the corrugation periodicity, Bragg scattering results in both forward and backward traveling waves that interfere constructively to set up a standing wave. This standing wave within a periodic structure can have two configurations, one when the standing wave has optical field concentrate in the high index region, the other when it concentrate in the low index region [20]. The different refractive indices of the two regions mean that the two modes have different energies (and therefore frequencies) associated with them—a band gap opens up. In our case, the Bragg vector K=2π/Λ_g=2π/(1000 nm) ≈ 0.0063 nm^-1 so that when k_x ≈ 3 μm^-1, which is the Brillouin zone boundary, a standing wave is formed and the group velocity becomes zero as shown in Fig. 2c. An apparent band gap occurs for G_t below 160 nm.

 

Fig. 2. Various stages of conversion of SPP mode to LSPP mode by varying G_t for (a) G_t=t _w=330 nm, (b) G_t=250 nm, (c) G_t=160 nm, (d) G_t=80 nm, (e) G_t=50 nm and (f) G_t=0 nm.

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4. Comparison between symmetric and asymmetric T-shaped gratings

In this section we compare results between symmetric and asymmetric T-structures. The dispersion relation for the symmetric T-shaped grating is shown in Fig. 3a when G_t =160 nm. The major drawback of this structure is that its dispersion relation has a momentum band gap at small values of k_x (i.e. plasmon decouples with light) which makes the band gap at near normal incidence unobservable. This momentum gap exists for all values of G_t < f_w-10 nm. Momentum gap occurred after introducing the second grating material (i.e. T-shaped post). The appearance of the momentum gap can be explained by an analytic model for a corrugated metallic surface with a profile described by the function s(x)=d₁sin(Kx+φ₁)+d ₂sin(2Kx+φ₂) [20]. When φ ₂ = -90° the SSP band structure (Fig. 11 of [20]) behaves very similar to that shown in Fig. 3a. The surface profile of the T-shaped grating (upper part) can be described by the sum of two functions: s(x) = s_t(x) + s_p(x), where s_t(x) = t for |x-x ₀| < W/2, and 0 otherwise, while s_p(x) = t_p for |x-d-x ₀| < W _T/2 [t_p=(t _w-G_t)/2)], and 0 otherwise. Here, x ₀ is a shift in x to make φ₁ =0. We perform Fourier expansion of s(x) and require its projection into cosine basis functions to be zero up to second order, so it will fit the form adopted in [20]. This leads to

For the symmetric T-structure (d=0), we obtain Kx ₀ = π/2 and φ₂= -90°. Thus, the presence of the momentum gap is explained. For the asymmetric T-structure considered, we obtain Kx ₀ = 1.52 and φ₂ ≈ -120°, which leads to a complete band gap without momentum gap as seen in Fig. 3b. The insets of Fig. 3(a,b) show the reflectivity for symmetric and asymmetric T-shaped structures at normal incidence (θ=0) with the same geometrical parameters. There is a large dip in the case of the asymmetric case at 0.76 eV, whereas in the symmetric case there is no pronounce dip at the same energy, and the band gap cannot be observed at normal incidence. We find that φ₂ decreases with increasing d, and φ ₂ will pass through -180° (equivalent to 0° with a sign change in coefficient d ₂) at d ≈ 130 nm, and reach -270° (equivalent to 90°) when d≈ 280 nm, where we get a momentum gap in E⁺ band instead of E^-.

 

Fig.3 (a) Symmetric T-shaped dispersion relation at G_t=160 nm. Inset: reflectivity versus photon energy of symmetric T-shaped at normal incidence. (b) Asymmetric T-shaped dispersion relation at G_t=160 nm. Inset: reflectivity versus photon energy of asymmetric T-shaped at normal incidence.

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To understand the conversion of SPP mode to LSPP mode and its angular independence we analyze the magnetic field distributions (|H_y|²) for the E⁺ and E^- modes, which are the two resonance dips at normal incidence and compare them with the field distribution in the multilayer structure. As shown in the dispersion relation of the multilayer structure there is no band gap. The wave can freely propagate in the multilayer structure. Figure 4a shows the free-propagation like field distribution in x-direction at normal incidence. Figure 4b, 4c illustrate the formation of LSPP for E⁺ and E^- modes for the asymmetric T-shaped structure with G_t =160 nm. When the gap thickness is decreased the wave propagation is perturbed by the post of the structure and start to localize in two different locations according to the SPP mode excitation: one between the SiO₂ and Ag in the base of structure and the other between the SiO₂ and Ag in the post of the structure and these LSPP modes can be identified as the n=1 and n=2 Fabry-Perot cavity modes. From Fig. 4c we can see most of field concentrate on the lower pillar. So, if we simplify the structure by removing the upper pillar and increase the length of the lower pillar (to keep G_t=160 nm) we will face two consequences. First, the E^-band becomes flattened due to increasing the length of lower pillar. Second, removing the upper pillar causes part of the higher energy bands (e.g. n=3) to be missing, because the upper pillar is responsible for localizing the higher-energy plasmonic modes.

 

Fig. 4 |H_y|² distribution within two periods for (a) multilayer structure at the crossing point of SPP mode, and (b) E⁺ and (c) E^- for asymmetric T-shaped array with G_t = 160 nm at band edges.

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5. Tunability of band gap and group velocity

To determine the parameters that control the plasmonic band gap we vary three geometrical parameters t _w, W _T and G_t individually. The E^- and E⁺ are resonance energies of the lower and upper plasmon-polariton modes (or n=1 and 2 cavity modes) whose corresponding wave lengths are λ^- and λ ⁺, respectively. As shown in Fig. 5a, varying W _T from 25 nm to 200 nm results in a change of E^- (λ^-) from 0.7706 eV (1.61 μm) to the 0.7523 eV (1.65 μm) and E⁺ (λ⁺) from 0.8709 eV (1.42 μm) to 0.9 eV (1.37 μm). The plasmonic band gap (band gap=E⁺-E^-) is raised from 0.1 eV to 0.148 eV as shown in Fig. 5a. On the other hand, changing t _w will affect E⁺ and E^- equally because changing the thickness of SiO₂, where both E^- and E⁺ modes are present, has the same effect on them so that the plasmonic band gap only changes from 0.183 eV to 0.19 eV, which is not as pronounced as shown in Fig. 5b. We expect that when any change occurs in the post length of the T-shaped structure, E^- (λ^-) will be affected more than E⁺ (λ⁺) due to the location of the Fabry-Perot resonance. This is illustrated in Fig. 5c, where it is seen that E^- mode is much more sensitive to the variation of G_t than the E⁺ mode, because the field is concentrated near the corner of the post and the substrate for the E^- (n=1) mode whereas for the E⁺ (n=2) mode the field is concentrated in the region away from the post. It is also noted that the effect of G_t is more pronounced than that of other parameters. By changing the gap thickness from 0 to 250 nm E^- (λ^-) is shift from 0.4754 eV (2.61 μm) to 0.8387 eV (1.48 μm) whereas E⁺ (λ⁺) is shifted slightly from 0.8801 eV (1.41 μm) to 0.871 eV (1.42 μm). The plasmonic band gap shrinks from 0.4047 eV to 0.0323 eV. The plasmonic band gap obtained here has much larger range of variation than that obtained in previous works [7, 8] which allows more freedom to locate the cavity or waveguide mode inside the band gap.

 

Fig. 5 (a) Photon energy of band gap edge [i.e. E⁺ (solid line) and E^- (dashed line)] versus post width, WT, at G_t=160 nm and t _w=330 nm. (b) Photon energy of band gap edge versus SiO₂ thickness, t _w, at G_t=160 nm and t _w=330 nm. (c) Photon energy of band gap edge versus gap thickness, G_t, at WT=70 nm and t _w=330 nm.

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Figure 6 shows group velocities along x-direction as functions of k _x for the E⁺ and E^-bands of the asymmetric T-shaped structure with G_t =160 nm [referring to the Fig. 2c for band structures]. The group velocity of the E⁺ band varies from 0 at k _x=0 μm^-1 to 0.1 c at k _x =1.2 μm^-1 and 0.0006 c at k _x =3.1 μm^-1, where c is the speed of light in vacuum. We got much smaller group velocities comparing with previous works, where group velocities around 0.5 c [14] and 0.44 c [15] were found.

 

Fig. 6. Group velocity versus k _x for asymmetric T-structure with G_t =160 nm for (a) E⁺ and (b) E^- modes.

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6. Conclusion

In summary, the plasmon-polariton band structures of symmetric and asymmetric T-shaped structures have been investigated. Comparing with previous works [7,8] we obtained wider band gaps with better tunability of the group velocity and we showed this band gap has strong dependence on the geometric gap in the post of the T-shaped structure. Plasmon-polariton band gap in these structures can vary from 0.4 eV to 0.0323 eV by changing the gap length inside the structure from 0 to 250 nm. Wide photonic band gap is an important consideration for photonic devices such as filters, waveguides, splitters and lasers. We have illustrated that by introducing a gap in the post of an asymmetric T-shaped structure, optical devices with a tunable and small group velocity can be achieved.