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This paper presents a nanometer-sized metallic film periodically pierced by narrow slits with ellipse walls deposited on a substrate that demonstrates special optical properties of broadband extraordinary optical transmission (BEOT). Compared to slits with straight walls, the metal slits with nonlinearly tapered ellipse walls can collect more light on the upper surface, which is coupled into a gap plasmon polariton propagating along the ellipse walls, then delivers the light at the smaller exit slit opening. In the visible spectral region, BEOT of TM-polarized light is achieved with up to 80% transmission at resonance, which is resulted from the simultaneous enhancement of zero-order slit resonance and higher-order slit resonances excited due to the existence of the substrate. The spectral range of BEOT is limited by Wood-Rayleigh anomalies and surface plasmon polariton resonances (SPPs). The BEOT spectrum of oblique incidence with small incident angle that is divided into two separate bands are also presented and analyzed theoretically. This metallic grating overcomes the low optical transmission limit of the structures with wavelength-sized grating period in visible and near-IR regions. It can be used to design nanostructured BEOT polarizer, which is an important component in novel biomimetic-based optoelectronic systems especially those in skylight polarized environment.
©2013 Optical Society of America
1. Introduction
Extraordinary optical transmission (EOT) through a periodic array of subwavelength holes or slits in a metal film has become a subject of great interest since its discovery in 1998 [1, 2]. García-Vidal et al. gave a detailed review on the recent developments in the EOT of subwavelength apertures [3]. Compared to a two-dimensional array of holes, a one-dimensional (1D) periodic array of slits can be more easily fabricated and theoretically analyzed. Interestingly, even very narrow slits can support gap plasmon polariton waves and provide high resonant transmission. Many applications based on metal slit arrays have been demonstrated, including color filters [4–6], planar lenses [7] and a photon sorting element [8]. Currently, most research work in the area of EOT on slits array have been mainly focused on straight slits [9–16]. Recently, a periodic array of tapered slits has been concerned and demonstrated which shows further enhanced EOT phenomenon [17, 18]. Broadband and wide-angle transmission in the infrared can be achieved by applying a metallic grating device with linearly tapered slits [19]. However, in order to obtain broadband high transmission for TM-polarized light in visible and near-IR spectral regions, which is important for novel bio-inspired polarized-based systems for skylight-based environmental navigations [20], small period metallic gratings (grating period less than 200 nm) need be fabricated. Although metallic gratings with tapered slits can obtain higher transmission for TM-polarized light [21], it is difficult to achieve higher extinction ratio. We reported an EOT phenomenon of one-dimensional freestanding subwavelength metallic grating by using a combination of surface plasmon polariton (SPP) with a Fabry-Perot (FP)-like model [22].
In this paper, based on our previous work, we present a new metallic grating structure with nonlinearly tapered ellipse walls that can provide EOT phenomenon in visible region by overcoming some drawbacks of the existing metallic grating. This nanostructure-sized metallic grating consists of metallic film periodically pierced by narrow slits with ellipse walls deposited on a substrate. Theoretical analysis predicts that broadband extraordinary optical transmission (BEOT) in visible and near-IR regions with up to 80% transmission at resonance can be achieved for TM-polarized light (i.e., magnetic field parallel to the slits) and transmission is close to zero for TE-polarized light. The mechanism of this phenomenon can be explained as that, compared to the slits with straight walls, the slits with nonlinearly tapered ellipse walls collect more light on the incident surface, which greatly enhance the excitation of both zero-order and higher-order slit resonances due to the existence of the substrate. We also investigate the influence of grating period and obliquely-incident light on BEOT spectrum in this grating design. Maximum BEOT spectrum range is predicted, which is quantitatively explained by the Wood-Rayleigh (WR) anomalies and SPPs excitation on the metal/dielectric surface. Furthermore, we find that the BEOT spectrum is divided into two bands for obliquely-incident light with small incident angle (less than 17°), but not for large incident angle. SPPs excitations on the metal/dielectric surface contribute to this specific optical phenomenon reasonably.
2. Model construction
Figure 1 illustrates the schematics of the designed metallic grating based on slits with nanostructured nonlinearly tapered ellipse walls (Figs. 1(a) and 1(b)) and a comparable grating structure based on tapered slits (Fig. 1(c)). As shown in Figs. 1(a) and 1(b), the boundary of the ellipse walls can be mathematically described by x2/δ2 + z2/h2 = 1, where δ, and h are the two semi-axes, b is the slit opening at the bottom, P is the grating period, and 2δ + b is the slit opening at the top. Each ellipse wall is a quarter of elliptic curve described by the above equation. This metallic grating is deposited on a substrate. This structure makes the slit opening decrease nonlinearly from top to bottom. In contrast, Fig. 1(c) shows a linearly tapered metallic grating that has identical structural parameters with the nonlinearly tapered ellipse wall grating. The only difference between them is the shape of metallic walls. More specifically, the slit opening in the tapered metallic grating decreases linearly from top to bottom. The finite difference time domain (FDTD) method [23] is employed for modeling in this research. To achieve high precision in the simulation, we use one period of the grating in the following calculation, the periodic boundary condition is set in x direction and perfectly matched layer is used in ± z direction under normal incidence. For oblique incident, Bloch boundary condition in x direction is used. The structure is assumed to be infinite long in y direction and all simulation results have been normalized to the incident light power.
Fig. 1 Schematic of metallic grating with nonlinearly tapered ellipse walls and a comparable grating with tapered walls. (a) Metallic grating with ellipse walls slit array deposited on substrate. (b) Cross-section of metallic grating with ellipse walls slit array. (c) Cross-section of tapered metallic grating.
The frequency-dependent permittivity εm of the metal silver is expressed by Drude–Lorentz model, which is defined as [24]
In Eq. (1), the first two items are given by the Drude model, where ω is the angle frequency, ωP0 is the plasma frequency, and γ0 is the damping coefficient. The third term is the Lorentzian term where Ω0 and Γ0, stand for, respectively, the oscillator strength and the spectral width of the Lorentz oscillators, and Δε0 can be interpreted as a weighting factor. But, the Drude model cannot give a detailed description about the permittivity of metal in a wide frequency range. In order to overcome the limitation of the Drude model and be able to consider the interband transitions, one or several Lorentzian terms should be added to make a better fitting effect than only Drude model. In this paper, we use silver as the deposited metal film material because of its low dissipation in visible and near-IR regions compared to other metals and its well-defined plasmonic properties. Therefore, in Eq. (1) εr = 4.6, ωP0 = 1.37 × 1016 rad/s, γ0 = 1.62 × 1014 rad/s, Δε0 = 1.10, Ω0 = 7.43 × 1015 rad/s, Γ0 = 1.82 × 1015 rad/s. The permittivity of SiO2 substrate material is 2.25.The SPP resonances (SPPs) are surface waves which propagate at a metal/dielectric interface and are evanescent in the direction normal to the substrate. SPPs excitation condition on both surfaces of the metallic grating is
where λ is the incident light wavelength, n is diffraction order, θ is the incident angle and ε is the relative permittivity of the medium on the top and bottom surfaces of metallic grating (for the top air medium, ε = 1; for the bottom SiO2 medium, ε = 2.25). When the above resonance condition is satisfied, the energy carried by the incident wave can be transferred to the SPPs, which propagate along the metal/dielectric interface.Wood-Rayleigh (WR) anomalies [25] are abrupt changes in transmission as a function of period when a diffraction order lies in the plane of the grating
It’s worth noting that WR anomalies are not resonant phenomena, and instead, they are caused by geometrical parameters.
3. Results and discussion
We use FDTD method to obtain and analyze the optical characteristics of the designed metallic grating with nonlinearly tapered ellipse walls. In order to investigate the influence of the ellipse walls on its transmission spectrum, we simulate the transmission spectrum of different polarized light normally incident on the metal grating based on ellipse of different parameters. The results are shown in Fig. 2 . Figure 2(a) is the transmission spectrum of a TM-polarized light for a grating with fixed period P = 500 nm, slit opening at the bottom b = 100 nm, semi-major axis (metal film thickness) h = 228 nm, and semi-minor axis δ in the range from 0 to 200 nm. Visible and near-IR regions are considered in this paper that covers a wavelength range from 450 nm to 1500 nm. We notice that the spectrum is divided into two bands at the wavelength of 784 nm, which corresponds to the wavelength of SPP resonant excitation at the metal/SiO2 surface (Eq. (2). The transmission of the shorter wavelength band is continuously enhanced, while the transmission spectrum is broadened as the semi-minor axis δ increases. However, the transmission of the longer wavelength band is weakened; the wavelength of the transmission peak has an evident blue shift as the semi-minor axis δ increases. Additionally, the changes of the transmission intensity in these two bands are completely different. We mainly focus on the extraordinary optical transmission (EOT) of the short wavelength band. The transmission intensity of metallic slits array with δ = 160 nm is increased by more than three times compared to the slits with straight walls (δ = 0 nm), which is a valuable result for polarizer design in the visible region. The TE-polarized light transmission of the above grating is presented in Fig. 2(b). As shown in Fig. 2(c), when the semi-minor axis δ increases, an enhanced transmission intensity of TM-polarized light is obtained, but the extinction ratio of structure (rext = TTM/TTE) is reduced, which is disadvantageous to design a polarizer for visible region application.
Fig. 2 Transmission spectra of metallic grating with nonlinearly tapered ellipse walls for a range of semi-minor axis δ in the case of fixed P = 500 nm and b = 100 nm. (a) TM-polarized light with semi-major axis h = 228 nm. (b) TE-polarized light with semi-major axis h = 228 nm. (c) Extinction ratio with semi-major axis h = 228 nm. (d) TM-polarized light with semi-major axis h = 400 nm. (e) TE-polarized light with semi-major axis h = 400 nm. (f) Extinction ratio with semi-major axis h = 400 nm. (g) TM-polarized light with semi-major axis h = 700 nm. (h) TE-polarized light with semi-major axis h = 700 nm. (i) Extinction ratio with semi-major axis h = 700 nm.
To overcome the above reduction of extinction ratio, we modify the parameters of the designed metallic grating to achieve better polarization characteristics. In Figs. 2(d) and 2(e), we present the simulated transmission spectra for TM-polarized light and TE-polarized light, respectively. The structure parameters are identical to those in Fig. 2(a) except for the semi-major axis h (metal film thickness) that is changed to 400 nm. It is shown that the transmission spectra are also divided into two bands at 784 nm for TM-polarized light. The band of the short wavelength that we focus on is enhanced and the higher transmission intensity of the whole range is achieved with up to an 80% transmission when semi-minor axis δ is 160 nm. The transmission spectral range is further broadened compared with the metal slits with the semi-major axis h as 228 nm. But for TE-polarized light, as shown in Fig. 2(e), transmission intensity of short wavelength range is reduced significantly compared to the metal slit with the semi-major axis h as 228 nm. We also present the case of a metallic film thickness h = 700 nm in Figs. 2(g) and 2(h). As shown in Figs. 2(f) and 2(i), it is remarkable that extinction ratio of metallic grating increases significantly as semi-major axis (metal film thickness) h increases from 400 to 700nm, which demonstrates this metallic grating design is suitable for developing a BEOT based polarizer for visible region. Figures 2(a), 2(d) and 2(g) show that when the semi-minor axis δ increases to about 160nm, the TM-polarized light transmission in the range of short wavelength band can achieve up to 80% transmission and transmission bandwidth is also close to maximum. Therefore we select a metallic grating with nonlinearly tapered ellipse walls with δ = 160 nm as an appropriate parameter in the following section.
Additionally, metallic grating with nonlinearly tapered ellipse walls can provide higher extinction ratio than other metallic grating with different structures, such as linearly tapered metallic grating. Using a linearly tapered metallic grating design in Fig. 1(c), we simulate its transmission spectra using similar parameters as those used for the above non-linearly tapered ellipse wall based grating. The results are shown in Fig. 3 . The related transmission spectra of the tapered metallic grating are illustrated separately from Fig. 3(a) to Fig. 3(f) to compare with the performance of the ellipse-wall based grating (Fig. 2(a) to Fig. 2(f)) respectively.
Fig. 3 Transmission spectra of tapered metallic grating for a range of semi-minor axis δ in the case of fixed P = 500 nm and b = 100 nm. (a) TM-polarized light with film thickness h = 228 nm. (b) TE-polarized light with film thickness h = 228 nm. (c) Extinction ratio with film thickness h = 228 nm. (d) TM-polarized light with film thickness h = 400 nm. (e) TE-polarized light with film thickness h = 400 nm. (f) Extinction ratio with film thickness h = 400 nm. (g) TM-polarized light with film thickness h = 700 nm. (h) TE-polarized light with film thickness h = 700 nm. (i) Extinction ratio with film thickness h = 700 nm.
Although metallic grating with larger tapered slits also can obtain higher TM-polarized light transmission, it is difficult for linearly tapered slits to obtain higher extinction ratio. Both high TM-polarized light transmission and high extinction ratio are very important in many applications of polarizer devices. Therefore, we conclude that compared with other reported metal gratings, this nonlinearly tapered ellipse-wall based metallic structure can obtain higher BEOT for TM-polarized light and achieve higher extinction ratio in the visible spectral range.
The slit opening at the bottom can affect the transmission bandwidth of the designed grating. We investigate the influence of the silt opening at the bottom b on the transmission spectrum of the designed metallic grating with ellipse walls in Fig. 4 . We use the following grating parameters: P = 500 nm, h = 400 nm, silt opening at the top 340 nm, and b in the range from 340 nm to 40 nm. Note that decreasing b from 340 nm to 40 nm for a fixed silt opening at the top results in a change of δ from 0 nm to 150 nm because the slit opening at the top is 2δ + b. Figures 4(a) and 4(b) show the TM- and TE-polarized light transmissions respectively. The increase of the slit opening at the bottom b can enhance the transmission of the short wavelength band continuously and broadens the transmission bandwidth; but the transmission intensity and bandwidth of short wavelength band of metallic grating with ellipse walls have small changes, which always keep a high TM transmission and a wide bandwidth of short wavelength band. Extinction ratio of the structure is shown in Fig. 4(c). It illustrates that as b increases, extinction ratio decreases rapidly. But the extinction ratio in whole short wavelength band is over 1000 (corresponding to a logarithmic scale 3 in Fig. 4(c)) when slit opening b = 100 nm. In order to satisfy simultaneously higher TM-polarized light transmission and higher extinction ratio, we select a slit opening at the bottom b = 100 nm in the following investigation.
Fig. 4 Transmission spectra of metallic grating with nonlinearly tapered ellipse walls slits array change with its slit opening at the bottom (grating period P = 500 nm and height of the slits h = 400 nm). (a) Transmission of TM-polarized light. (b) Transmission of TE-polarized light. (c) Extinction ratio.
As shown in Fig. 5 , to further understand this BEOT behavior for TM-polarized light, we calculate the spatial distribution of the magnetic and electric intensities normalized to the incident field intensities at the wavelength of 670 nm, where transmission maximum is achieved in Fig. 2(a). Figure 5 shows two unit cells of the metallic grating, in which we compare the spatial field distribution of the magnetic and electric intensities between a metallic grating based on nonlinerly tapered ellipse walls (semi-minor axis δ = 160 nm) and a metallic grating based on straight walls (δ = 0 nm). For spatial distribution of the magnetic intensity, we can see that the magnetic field is localized on the top surface of the metallic grating with straight walls (Fig. 5(b)) and little energy can be transmitted into the SiO2 substrate. However, in nonlinearly tapered ellipse-wall based metallic grating with δ = 160 nm (Fig. 5(a)), there is no localized magnetic field on the top surface. Strong magnetic field is localized on the surface just below the grating which leads to significant magetic field distribution in the substrate SiO2. For the spatial distribution of the electric intensity, there is a distinct difference between the two structures. In the ellipse-wall based metallic grating with δ = 160 nm (Fig. 5(c)), electric field is localized inside the metal slits and is mainly distributed at the metal ellipse walls/air boundaries. It shows that EM energy mainly propagates along the metal ellipse walls/air surfaces and then transmits into the substrate, which is an evident difference from the case of straight slits (Fig. 5(d)). In Fig. 5(d), most of the electric field is distributed above the metallic grating and in the top part of the metal slit and electric field intensity is relatively low at the surface between the grating and the substrate. Figure 5 illustrates that sharp corners only exist around the bottom surface of the ellipse-wall slits; on the contrary, sharp corners appear on the upper and lower surface of the straight slits. Because the optical energy is always strongly localized around the sharp corners, most energy can flow toward the metal/substrate interface in the nonlinearly tapered metallic grating compared to the metal grating with straight slits. These phenomena might be explained as follows: the nonlinearly tapered ellipse-wall based metallic grating (δ = 160 nm) has a larger slit opening which leads to smaller reflection on the top surface of grating compared to the straight-wall based metallic grating (δ = 0 nm). The small reflection might allow most of energy flows inside ellipse slits and is subsequently coupled into the SPP waves on ellipse walls/air surfaces. Both SPP waves on slit walls react with each other to form gap plasmon polariton and as the distance between metal surfaces decreases, the interaction of both SPP waves on slit walls become stronger. As a result, then electromagnetic field is inclined to be distributed toward the bottom part of the slits. But for the case of straight-wall slits, most of the optical energy is reflected before it penetrates inside straight walls, which eventually leads to simultaneous enhancement of zero-order and higher-order transmission. We may conclude that the transmission in the case of straight-wall slits is relatively lower than that of nonlinearly tapered ellipse-wall slits. More investigation needs be conducted to further understand the physical origin of this unique phenomenon for the optimization of the designed metal grating with ellipse walls slits.
Fig. 5 Spatial field distribution at the wavelength 670 nm with fixed P = 500 nm, b = 100 nm and semi-major axis h = 228 nm. (a) Magnetic intensity with δ = 160 nm. (b) Magnetic intensity with δ = 0 nm. (c) Electric intensity with δ = 160 nm. (d) Electric intensity with δ = 0 nm.
We make further investigation of above BEOT behavior for TM-polarized light as shown in Fig. 6 . Figure 6(a) illustrates the total transmission versus wavelength of normally –incident TM-polarized light and film thickness h with fixed grating parameters of P = 500 nm, b = 100 nm, and δ = 160 nm. According to Eq. (3), higher-order diffraction of light will appear in transmission for wavelength shorter than ndP = 1.5 × 500 = 750 nm, where nd = 1.5 is the refractive index of the SiO2 substrate. For wavelength larger than 750 nm, there is only zero-order diffraction in transmission. The red dashed line in Fig. 6(a) indicates a minimum transmission of metallic grating appears at the wavelength of 784 nm, corresponding to the wavelength of SPPs resonant excitation at the metal/SiO2 surface (Eq. (2). Another transmission minimum is at 500 nm (grating period), which is the WR anomalies and corresponds to ε = 1 in Eq. (3). We notice that BEOT can be achieved between 500 to 784 nm. The two wavelengths corresponding to the two transmission minima are regarded as the boundary of BEOT for the designed grating. From 500 to 784 nm, BEOT always appears as film thickness h increases and the width of the BEOT spectrum have little periodic changes.
Fig. 6 Transmission versus wavelength of normally incident light and film thickness h with fixed P = 500 nm, b = 100 nm. (a) Total transmission with δ = 160 nm. (b) Total transmission with δ = 0 nm. (c) Zero-order transmission with δ = 160 nm. (d) Zero-order transmission with δ = 0 nm. (e) High-order (one-order) transmission with δ = 160 nm. (f) High-order (one-order) transmission with δ = 0 nm.
Figure 6(b) shows the total transmission versus wavelength of normally TM-polarized incident light and film thickness h for the case of straight wall (δ = 0 nm). Other grating parameters are the same as used in Fig. 6(a). Two red dashed lines in Fig. 6(b) correspond to the SPPs resonant excitations at the upper (metal/air) and bottom (metal/SiO2) surfaces of the metallic grating. It is obvious that the transmission minimum at the wavelength 530 nm is not present in Fig. 6(a). This is because SPPs resonant excitation at the upper surface is not sustained as the slit opening at the top increases. The wavelength of the transmission minimum is moved from 530 nm to 500 nm (grating period). The transmission spectrum broadens in the visible wavelength range. In Fig. 6(b), the transmission from 530 to 784 nm is overall lower and shows small changes as film thickness h increases. There is only an abrupt change in the transmission spectrum near the cutoff of higher-order diffraction order at 750 nm (indicated by white straight dashed line). For wavelength longer than 750 nm, there is only a single diffraction order in the transmission. Using the proposed model based on a combination of surface Plasmon polariton (SPP) resonance and Fabry–Perot (FP)-like resonance [22], the maximal transmission can be predicated. SPP waves on both slit walls react with each other to form gap plasmon polariton in the slit. The constant of gap plasmon polariton propagation depends on the distance between the metal surfaces. Gap plasmon polariton propagating from top to bottom in the slit satisfies the excitation of FP-like under certain conditions. The condition for slit resonances in Fig. 6(b) is shown as follows
where βGP is the gap plasmon polariton propagation constant and ΦR is the total reflection phase at both slit terminations. For the case of straight wall in Fig. 6(b), βGP is a constant. The predicted wavelengths of the transmission maxima are indicated in Fig. 6(b) by the white dashed lines, which are calculated according to Eq. (4). Results from above model agree with that of FDTD simulation.However, for the case of Fig. 6(a), βGP is very sensitive to the width of slit. It decreases as the slit width increases, and increases as the wavelength decreases. In Fig. 6(a) the transmission is relatively lower and the transmission maxima shift toward longer wavelength for wavelength longer than 750 nm. This is because the larger opening at the top of the slit leads to a smaller reflection of the gap plasmon polariton on the top surface of the grating, which weakens the resonance, and the decreasing of βGP in Eq. (4) related to the case of straight walls leads to a shift toward longer wavelength. For wavelength longer than 750 nm, metallic grating with nonlinearly tapered ellipse walls also can be explained by the model combining the SPP and the FP-like resonance. Transmission for wavelength larger than 750 nm is out of the scope of this paper. In addition, this is only valid for a single diffraction order. For wavelength shorter than 750 nm, the transmissions of the structures have more transmission orders, so the model is invalid. Figures 6(c) and 6(e) present the zero-order transmission and the higher-order transmission of the structure, respectively, which have the same parameters as used in Fig. 6(a). In Figs. 6(d) and 6(f), we also illustrate the zero-order transmission and the higher-order transmission of the structures with the same parameters as used in Fig. 6(b). Figures 6(e) and 6(f) only show + 1 order transmission which is identical to the −1 order transmission for normally incident light. Comparing Fig. 6(c) with Fig. 6(d) and Fig. 6(e) with Fig. 6(f), BEOT can be generated because higher-order slit resonances and zero-order transmission are enhanced simultaneously. Higher-order transmission appears because of the presence of the SiO2 substrate. EM energy mainly propagates along the metal ellipse walls-air surface and then transmits into substrate (Fig. 5(c)), which enhances the higher-order transmission greatly. Moreover, simultaneous enhancement of zero-order and higher-order transmission is also originated from the decreased reflection of the incident light on the top surface of the metallic grating with ellipse walls due to a larger slit opening. It is interesting that high transmission seems to favor the presence of higher-order diffraction.
In Fig. 7 , we present the simulated result of transmission versus the wavelength of normally incident light and the period P of the grating, which has a fixed b = 100 nm and film thickness h = 350 nm. Figures 7(a) and 7(c) illustrate the transmission versus wavelength and period for δ = 160 nm and δ = 0 nm, respectively. To clearly indicate the transmission minimum, Figs. 7(b) and 7(d) are logarithmic graphs of Figs. 7(a) and 7(c). There are five branches of transmission minima in Fig. 7(a). When light is normally incident on the metallic grating with δ = 160 nm, the resonance condition for the excitation of SPP wave at the metal/SiO2 surface is satisfied. Due to the large slit opening on the top of metallic grating, SPPs wave cannot be excited at the metal/air surface. The transmission minimum will be transferred to the WR anomaly, which is demonstrated in Fig. 2(a). As the semi-minor axis δ increases from 0 nm to 160 nm, transmission minimum moves from the excitation of SPPs on the top surface (530 nm) to the WR anomaly (500 nm) gradually.
Fig. 7 Transmission versus wavelength of normally incident light and period P with fixed b = 100 nm, film thickness h = 350 nm. (a) Transmission with δ = 160 nm. (b) Logarithmic result of transmission with δ = 160 nm. (c) Transmission with δ = 0 nm. (d) Logarithmic result of transmission with δ = 0 nm.
In Fig. 7(a), there are three white solid lines (blue solid lines in Fig. 7(b)) from right to left that correspond to the excitation of SPPs wave at the metal/SiO2 surface for n = 1, 2, and 3 in Eq. (2). The two white dashed lines (blue dashed lines in Fig. 7(b)) from right to left are P = λ and P = 1.94λ (WR anomalies), respectively, which are only dependent on the period of the structure. Both above theoretical model and FDTD simulation results explain the presence of the transmission minima. As the period of the structure P increase, BEOT can also be obtained. The two wavelengths corresponding to the transmission minima are regarded as the boundary of BEOT explained in Fig. 6(a). In contrast, we present the transmission for δ = 0 nm in Fig. 7(c). There are two main differences between Fig. 7(a) and Fig. 7(c): on one hand, the two white dashed lines in Fig. 7(c) (blue dashed lines in Fig. 7(d) from right to left correspond to the excitation of SPPs wave at the metal/air surface for n = 1, 2 in Eq. (2). The corresponding locations of WR anomaly are transmission peaks, which are close to the white dashed lines shown in Fig. 7(c); on the other hand, the transmission maxima appear in longer wavelength range in Fig. 7(c). It means that the change of δ can shift the transmission maxima from longer wavelength to shorter wavelength. This can be explained as follows: larger δ means larger slit opening at the top. A larger opening at the top of the slit leads to a smaller reflection of gap plasmon polariton on the top surface of grating, which breaks the FP-like resonance in the slit. Weak resonance results in lower transmission in longer wavelength. In shorter wavelength range, the larger opening at the top of the slit leads to higher-order slit resonances and zero-order transmission is enhanced simultaneously, which provides higher transmission.
As shown in Fig. 8 , we also investigate the influence of oblique incident light on transmission spectra in the designed grating. The structure parameters include P = 500 nm, b = 100 nm, h = 228 nm and δ = 160 nm, incident angles range from 0° to 40°. When the incident angle increases, BEOT always appears in the transmission spectra and the transmission intensity is further enhanced with up to 90% transmission at resonance. At the same time, transmission spectra are further broadened but there are two transmission minima whose wavelengths are regarded as the boundary of BEOT. When the incident angle is relatively smaller (less than 17°), the transmission spectra is divided into two bands by the transmission minimum. As shown in Fig. 8(a), there are four branches of transmission minima, indicated by the three white solid lines and one white dashed line. The first white solid line in the right refers to the excitation of SPPs wave at the metal/SiO2 surface for n = 1, obtained in our calculation by Eq. (2), using period P = 496 nm (less than 500 nm). The middle white solid line also refers to the excitation of SPPs wave at the metal/SiO2 surface for n = 1 in Eq. (2); but wavelength vector in the x direction is opposite to the first white solid line and the value of period P is 486 nm (which is also less than 500 nm). These results illustrate that the period condition of the excitation of SPPs is less than period of grating for oblique incidence. To illustrate clearly the physical origin of the excitation of two SPPs at the metal/substrate interface, Figs. 8(c) and 8(d) show the distribution of the real part of the electric field (Ey) and the magnetic field (Hz) with incident angle 10° for wavelength 692 nm and 856 nm, respectively. Apparently, electromagnetic field is localized strongly on the metal/substrate interface. We observe the exponential dependence of electromagnetic field in the z direction. The spatial distribution of electric field (Ey) for wavelength 692 nm is the same as that of 856 nm; on the contrary, the spatial distribution of magnetic field (Hz) is just opposite. According to the basic nature of the electromagnetic plane wave, kx, Ey and Hz satisfy right-hand screw rule, where kx refers to wave vector of SPPs at the metal/substrate interface in the x direction. We conclude that SPP waves at the metal/substrate interface for wavelength 692 nm and 856 nm propagate along two opposite directions, which confirms that the excitation of two SPPs under two conditions have opposite wavelength vectors in the x direction. The third line in the left refers to the excitation of SPPs wave at the metal/SiO2 surface for n = 2 in Eq. (2). The white dashed line is calculated according to Eq. (3), where ε is the relative permittivity of air. We also present all these results in a logarithmic system as shown in Fig. 8(b), which further demonstrates that the theoretical explanations we presented quantitatively agree with FDTD simulation results.
Fig. 8 Transmission versus wavelength and incident angle and Spatial field distribution with fixed P = 500 nm, b = 100 nm, h = 228 nm and δ = 160 nm. (a) Transmission in a linear map. (b) Transmission in a logarithmic map. (c) Real part distribution of electric field (Ey) and magnetic field (Hz) with incident angle 10° and wavelength 692 nm. (d) Real part distribution of electric field (Ey) and magnetic field (Hz) with incident angle 10° and wavelength 856 nm.
Based on all above results, we conclude that SPPs have a negative effect on broadband transmission spectrum of light with small oblique incident angle. To avoid transmission spectrum divided into two bands, we should consider larger incident angles (at least larger than the angle corresponding to the crossing point between blue dashed line and blue solid line in Fig. 8(b)) in the design of visible and near-IR nanostructured polarizer.
4. Conclusions
In summary, we propose a nanostructured metallic grating with nonlinearly tapered ellipse walls possessing the attractive features of non-resonant broadband extraordinary optical transmission in visible and near-IR regions. Compared to other subwavelength metallic gratings such as those with straight walls or tapered slits, the grating structure presented here possesses BEOT spectrum for TM-polarized light with up to 80% transmission in visible spectral range, which is the result of the simultaneous enhancement of zero-order and higher-order diffraction transmission. Maximum range of transmission spectrum is analyzed theoretically by means of physical model of Wood-Rayleigh anomalies and SPPs excitation on the metal/dielectric surface. Because of the SPPs excitation on the metal-dielectric surface, the transmission spectrum of the designed grating is divided into two separated bands for oblique incident light with small incident angle. Such structure will find application in nanostructured optical polarizer design for visible and near-IR regions. Further investigation will be focused on the optimization and development of this metallic grating with nonlinearly tapered ellipse walls by using nanofabrication technologies such as micro-electro-mechanical system and laser 3-dimension microstereo lithography. The results presented here is valuable for developing novel polarizer with broadband extraordinary optical transmission, which is a kind of general photonic components that has extensive applications in biomimetic navigation systems especially those used in skylight polarized environments.
Acknowledgments
The authors would like to thank financial supports from the National Nature Science Foundation of China (Grant Nos. 61137005 and 60977055) and the Ministry of Education of China (Grant Nos. NCET-09-0255, DUT11Z102, and SRFDP 20120041110040).
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