1. Introduction

Extraordinary optical transmission (EOT) through a metallic film perforated with a nano-sized hole array has aroused great interest since it was first reported by Ebbesen et al. in 1998 [1]. Metallic films with various subwavelength hole arrays [2–4], slit and slit arrays [5–7] were studied to achieve an understanding of the wavelength-dependent transmission mechanism in this area. Attention of EOT is mostly focused on the periodic subwavelength structures composed of only one hole or slit in each primitive unit cell. It has been widely accepted that two kinds of transmission resonances, i.e. the coupled surface plasmons (SPs) resonance mode and the Fabry–Pérot(F-P)–like waveguided-mode resonance, have been involved in the explanation of EOT [8–10]. The former mechanism relies on the excitation of SPPs on either side of the metallic film, and the latter depends on the shape of single hole. In recent years, the compound periodic metallic structures composed of several slits or holes with in each cell have attracted much attention [11–14]. It is accepted that the phase resonances is responsible for the transmission dips in the spectrum. For a particular wavelength, the field distribution inside the different cavities/slits takes a particular form so that its phase in adjacent cavities can be opposite to each other and its amplitude maximizes the inner field.

For compound metallic gratings, previous works mostly focused on metallic films with slits, holes and grooves with straight channels. In the microwave range, Rance et al. discussed the electromagnetic response of metal transmission gratings comprised of identical but alternately orientated tapered slits [12,15]. Experimentally, in the millimeter wave regime, Skigin et al. provided evidence of phase resonances in metallic periodic structures [16] in which each period comprises several subwavelength slits of the same width. Many researchers demonstrated the transmission of light through subwavelength metallic square and rectangular holes and slit arrays with perpendicular bumps or cuts. However, transmission and plasmon properties of the compound periodic slit arrays with curving channels have been considered little up to now.

In this paper, we propose a metallic compound surface relief grating, in which each repeat period is comprised of two slits with different dielectrics or with different slit width, and explore the transmission behavior. We calculated the zero order transmission, reflection, and absorption spectra and show that the optical characteristics can tune by the dielectrics in the slits and the slit width. The underlying physics is also discussed by the simulated magnetic and electric field distributions.

2. Model and theory

A unit cell of a compound gold surface relief grating is presented in Fig. 1. In our two-dimensional finite-difference time-domain (FDTD) calculations [17,18], the computational space is truncated by using perfectly matched layer (PML) absorbing boundary conditions on the top and bottom boundaries along the y direction. And periodic boundary conditions are applied on left and right boundaries along the x direction due to the periodicity of the structure. In all cases, light is incident normal to the structure along the y direction with TM polarized [19]. We simulate the structure with an FDTD cube of size$L_x\times L_y=1200nm\times 2100nm$.The spatial and temporal steps are set at $\Delta x=\Delta y=1nm$ and $\Delta t=\Delta x/2c$(c is the velocity of light in vacuum). The structure is periodic in the x direction, and the periodic is p = 700nm. The widths of slit1 and slit2 with S-shaped channels are defined as d₁ and d₂, respectively. Geometric variables${\epsilon }_1$ and${\epsilon }_2$ represent the dielectric constants of the medium in the surface relief slit1 and slit2. The length of the slits and the radii of the humps and depressions in the S-shaped channels are h = 600nm and R = 75nm, which are fixed in the whole paper.

 

Fig. 1 Scheme of a unit cell of the compound gold surface relief grating defined for FDTD simulations. Parameters are defined in the text.

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The two-dimensional FDTD method is used to calculate numerically the transmission spectra and the time-dependent near fields of the compound gold surface relief slit arrays. The frequency-dependent permittivities of gold are approximated by the Drude model for the dielectric response of the metal which is mainly governed by its free electron plasmon [19]:

3. Results and discussion

For comparison, in Figs. 2(a) and 2(b), we first show the zero order transmission, reflection, and absorption spectra of gold grating consisting of bare slit without S-shaped channels and surface relief slit arrays, respectively, as a function of wavelength with same parameters (slits widths d₁ = d₂ = 300nm, the grating period p = 700nm and the grating thickness h = 600nm). The medium of the slit is assumed to be air (${\epsilon }_1$ = ${\epsilon }_2$ = 1). The inset in Fig. 2(a) display the grating consisting of bare slit without S-shaped channels structure. The absorption spectra were obtained via $A=1-T-R$, where A, T, and R express the absorbance, transmittance, and reflectance, respectively. Figure 2(a) shows two distinct transmission resonance peaks at wavelengths $0.8108\mu m$ and $1.565\mu m$ which correspond to the Fabry–Pérot–like waveguided-modes resonances and can labeled as $N=2$ (second-order) and $N=3$ (third-order), respectively, according to previously reported results [12, 13, 20]. There have strong absorption drops at the transmission wavelengths. The introduction of the slits with S-shaped channels have changed the wavelengths and intensity of the resonant peaks (as shown in Fig. 2(b)). Comparing Figs. 2(a) and 2(b), we can see the intensities of the resonant peaks labeled as N = 2 and N = 3 are decreased slightly and the splitting two peaks located at the short-wavelength region of $0.7-0.8\mu m$labeled as $N=4$. At the same time, the absorption of light increases with the introducing of S-shaped channels. This is obviously related to the absorption loss of light energy by more metal gold. It also accounts for the reduced intensities of transmission peaks. Further more, all the resonance peaks for the F-P cavity modes present an obvious red-shift. This phenomenon is differ from the results of the grating with semicircle bumps which we studied before [19] and that of the compound grating with perpendicular bars or cuts [12–14, 21].

 

Fig. 2 Transmission, reflection, and absorption spectra of the(a)bare slit without surface relief channels and (b) surface relief slit arrays.

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The localized waveguide resonance mode is characterized by constructive interference along the channel between all partial transmitted waves, in the way of the F–P cavity mode [22]. The F–P condition can be expressed as following equation [23]:

Figure 3 displays the calculated zero-transmission spectra of the compound gold surface relief slit arrays filled with different dielectric constant of (a) ${\epsilon }_2$ = 2, (b) ${\epsilon }_2$ = 3, (c) ${\epsilon }_2$ = 4 and (d) ${\epsilon }_2$ = 4.5, respectively. The medium in the surface relief slit1 is defined as air (${\epsilon }_1$ = 1). The widths of slit1 and slit2 with S-shaped channels are set to be d₁ = d₂ = 300nm. The figure shows an obviously red-shift of the transmission spectra and the intensity of the resonance peak located around $1.2\mu m$ increases gradually. For instance, the peak located at $1.063\mu m$in Fig. 3(a) shifts to $1.246\mu m$ in Fig. 3(d) and the peak intensity augment from 0.72 to 0.99 when the dielectric constant ${\epsilon }_2$ increases from 2 to 4.5. The finding that the presence of the medium induces a redshift of the plasmon resonances is not surprising. Quasi-statically, the plasmon resonances of this system can be thought of a homogenous electron gas oscillating over a fixed positive background, with induced surface charges providing the restoring force. When the surrounding medium is a dielectric, it polarizes in response to the resulting field, effectively reducing the strength of the surface charges and leading to a decreased restoring force and, consequently, lowering the plasmon energies [25]. Moreover, the larger the value of the dielectric constant is, the stronger the polarization effect of the dielectric is.

 

Fig. 3 The transmission spectra as a function of wavelength for compound gold surface relief slit arrays with different dielectric constant of (a) ${\epsilon }_2$ = 2, (b) ${\epsilon }_2$ = 3, (c) ${\epsilon }_2$ = 4 and (d) ${\epsilon }_2$ = 4.5. The medium in the surface relief slit1 is defined as air (ε₁ = 1). The widths of slit1 and slit2 are $d_1=d_2=300nm$.

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Furthermore, along with the dielectric constant ${\epsilon }_2$ increasing, there appear three more and more obvious dips in the wavelength regions around$0.9\mu m$, $1.5\mu m$ and $2.07\mu m$(as shown in Fig. 3(d)). In order to understand the physical origin of the transmission peaks and dips, in Fig. 4, we draw magnitude of magnetic field $\left|H\right|$, the electric field component $E_x$ and $E_y$distributions at three different wavelengths (as shown by arrows in Fig. 3). The resonant wavelengths are located at (a-c) $\lambda =1.317\mu m$ in Fig. 3(b), (d-f) $\lambda =2.104\mu m$ in Fig. 3(b) and (g-i) $\lambda =2.067\mu m$ in Fig. 3(d). The electric field component E_y distributions are obtained at a time T /4 (T is the time period) before the magnetic amplitude $\left|H\right|$. Firstly, we study the magnitude of magnetic field $\left|H\right|$ in Figs. 4(a), 4(d) and 4(g). We find that the FP-like phenomena have been found both inside the slit1 and slit2, but the orders of the FP-like modes inside the two slits are different. For instance, when the resonant peak located at $\lambda =1.317\mu m$(two slits are filled with dielectrics ${\epsilon }_1=1$ and${\epsilon }_2=3$), one is the second-order FP-like mode and the other is the fourth-order FP-like mode [as shown in Fig. 4(a)]. However, for the dip located at $\lambda =2.067\mu m$ (two slits are filled with dielectrics ${\epsilon }_1=1$ and${\epsilon }_2=4.5$), one is the second-order FP-like mode and the other is the third-order FP-like mode [as shown in Fig. 4(g)]. It is well known that the phase retardation of the coupled-SPPs transmitted through each metallic slit is determined mainly by $k_0\mathrm{Re}(n_{eff})l$ in Eq. (2). When the order of the FP-like mode inside one slit one bigger than inside the other, the phases at the exits of the two slits are opposite to each other, and then π resonances can be excited. So we obtain a dip in the transmission spectra. While, as shown in Figs. 4(a) and 4(d), when the order of the FP-like mode inside one slit is two bigger than inside the other, the electromagnetic waves at the exits of the two slits are still in-phase, resulting in the enhanced transmission. The phases at the exits and entrances of the two slits can understand by the distributions of electric field component $E_x$. The areas colored red denote positive field amplitude (labeled as “+”), while areas colored blue denote negative field amplitude (labeled as “-”) in the field figures. The positive and negative fields indicate the vibration directions of them are opposite. In Figs. 4(b) and 4(e), the electromagnetic waves at the entrances and the exits of the two slits are in-phase because of the identical vibration direction of the electric field, resulting in the enhanced transmission. However, the electromagnetic waves at the exits of the two slits are out-phase because of the opposite vibration direction of the electric field. The results showed by the $E_x$ distributions are according with the above conclusion.

 

Fig. 4 The magnitude of magnetic field $\left|H\right|$and the electric field component $E_x$, $E_y$ distributions at three resonant wavelengths shown by arrows in Fig. 3. The resonant wavelengths are (a-c) $\lambda =1.317\mu m$ in Fig. 3(b), (d-f) $\lambda =2.104\mu m$ in Fig. 3 (b) and (g-i) $\lambda =2.067\mu m$ in Fig. 3(d), respectively.

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Then, the spatial images of the instantaneous electric field component $E_y$at steady state for three resonant wavelengths in Fig. 3 are also shown in Figs. 4(c), 4(f) and 4(i). The positive and negative charge densities as measured by the electric field $E_y$is mainly accumulate at the bumps of the surface relief slit, but the electric field $E_y$is relatively weak at the depressions of the surface relief slit, which is a typical edge effect. There also exhibits an alternation of positive and negative charge densities in the two slits. Corresponding to the distribution of magnitude of magnetic field$\left|H\right|$ in Fig. 4(d), the electric field component $E_y$ is found mostly localized inside the slit1 in Fig. 4(f).

The dispersion equation for TM mode [20] in the straight and single waveguide is given by

Here h is the width of waveguide, ${\beta }_{spp}$ and $n_{eff}$ are propagation constant and effective index for SPPs, k₀ is the free-space wave vector, ${\epsilon }_m$ and ${\epsilon }_d$are relative dielectric constants of the metal and dielectric material, respectively. In Fig. 5, we present the variation of the real and imaginary parts of $n_{eff}$ calculated using Eqs. (3)–(5) for straight metal–insulator–metal (MIM) waveguide with width h = 200 nm (i.e. d = 200nm in the paper). We can roughly estimate the values of the resonance or anti-resonance wavelengths by Eq. (2). For example, the central slit length $l=2\times 2\pi R$in an S-shaped slit. When the wavelength is $1.317\mu m$, $\mathrm{Re}(n_{eff})$ equal to 1.107 as ${\epsilon }_d=1$ and equal to 1.925 as ${\epsilon }_d=3$. The above numerical values is put in Eq. (1), we get the n value is 1.6 and 2.75, respectively. When the wavelength is $2.104\mu m$, $\mathrm{Re}(n_{eff})$ equal to 1.105 as ${\epsilon }_d=1$ and equal to 1.917 as ${\epsilon }_d=3$. The above numerical values is put in Eq. (2), we get the n value is 1 and 1.7, respectively. We can find that the numerical values less than the above graphic result. There are two reasons accounting for this phenomenon. One is $\arg (\rho )$ resolved by taking the approximate value ($\arg (\rho )=0$). The other is Eq. (2) is suitable for straight and single MIM. Considered the structural differences in straight and curved slits, the effective widths of the S-shaped slits should be less than 200nm. $\mathrm{Re}(n_{eff})$will increase with the widths of the slits decrease. This lead to the numerical values increase and will accord with our graphical results. From Fig. 5 we can find that the imaginary part of $n_{eff}$ grows with increasing ${\epsilon }_d$, which implies the energy loss increases.

 

Fig. 5 Variation of the real and imaginary parts of $n_{eff}$ with wavelength for coupled-SPPs inside the slit filled with different dielectric ${\epsilon }_d=1,2,3,4,4.5$, respectively.

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To demonstrate the influence of different slit width on the plasmon resonance coupling, the slit1 width $d_1=300nm$ and the dielectric constant in two slits ${\epsilon }_1={\epsilon }_2=1$in all the structures are fixed. We depict the transmission spectra of the compound gold surface relief slit arrays as a function of wavelength for different slit2 width in Fig. 6(a) $d_2=200nm$, (b) $d_2=240nm$, (c) $d_2=300nm$and (d) $d_2=340nm$. Firstly, the increasing slit2 width results slightly in a blue-shift of the transmission spectra. This phenomenon still explained by Eq. (2). The coupling between the charge densities of the two surface relief walls becomes weaker with the wider slits, and the effective refractive index decreases. Smaller $n_{eff}$ needs larger$k_0$or smaller wavelength to meet the F–P condition, so the transmission spectra blue-shifts when the two slits become broader. Secondly, the resonance peak located at $0.8\mu m$ splits into two peaks when the slit2 width becomes the narrowest (see Fig. 6(a), $d_2=200nm$). It caused by the stronger surface plasmon couplings between the two walls in slit2 with the narrower slit width. Finally, there appears a new resonant peak at $0.54\mu m$ when the slit2 width increases to 300nm. Furthermore, the intensity of the new resonant peak augments with the slit2 width further increasing (see Fig. 6(d), $d_2=340nm$).

 

Fig. 6 The transmission spectra as a function of wavelength for compound gold surface relief slit arrays with different widths of slit2 of (a) $d_2=200nm$, (b) $d_2=240nm$, (c) $d_2=300nm$and (d) $d_2=340nm$. The slit1 width and the dielectric constant in two slits are set to be $d_1=300nm$ and ${\epsilon }_1={\epsilon }_2=1$.

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4. Conclusions

In summary, we propose a metallic compound surface relief grating, in which each repeat period is comprised of two slits with different dielectrics or with different slit width. We numerically and theoretically explorer the optical transmission in the compound gold surface relief slit arrays by using FDTD method. Compare to the bare slit without S-shaped channels structure, the intensities of transmission peaks reduce and the resonance peaks for the F-P cavity modes present an obvious red-shift as we introduce the surface relief channels. When the two slits fill different dielectrics, the presence of the medium induces a red-shift of the plasmon resonances. Along with the dielectric constant ${\epsilon }_2$ increasing, there appear obvious dips in the transmission spectra. Based on the magnetic and electric field distributions, we show that when the order of the FP-like mode inside the one slit is one bigger than inside the other, the intensity of the transmission will be significantly weakened. We attribute this phenomenon to the phase resonance. On the contrary, when the order of the FP-like mode inside the one slit is two bigger than inside the other, the intensity of the transmission will be enhanced because the light waves at the exits of the two slits are in-phase. Finally, the influence of different slit width on the plasmon resonance coupling of the surface relief slit arrays is also discussed. Our results may find applications in designing channel-selecting devices, new types of actively-controlled nano-optic devices, and enhanced SPP sensors, and so on.