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Abstract
Extraordinary transmission (EOT) through one- and two-dimensional deep metallic grating in the infrared range is explored by numerical electromagnetic (EM) field analysis, such as the finite difference time domain method. The transmittance at normal incidence is greater than the porosity that is defined as a proportion of the gap area to the total cross-section when the gap is much smaller than the wavelength of incident infrared light. The EOT mechanism is investigated using two approaches—the equivalent electrical circuit (EEC) model and the effective medium approximation (EMA). The transmittance and reflectance profiles calculated using the EEC model agree with those obtained by the numerical EM analysis. EMA is applied on the basis of the idea that the deep metallic grating can be regard as hyperbolic metamaterials. Then, the effective refractive index of the deep metallic grating is real with a negligible imaginary part, in infrared range. This means that the metallic grating behaves as a dielectric medium, resulting in the great transmission and existence of a Brewster angle.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Considerable attention has been directed toward metamaterials of metallic subwavelength structures. They express interesting optical properties, such as negative refractive index [1–4], optical super resolution [5,6], cloaking [7–9], efficient light absorbers [10,11] and emitters [12]. Such unique optical functionality can be obtained by designing the metamaterials. Among various optical properties of metamaterials, extra-ordinary transmission (EOT) has been investigated extensively [13–17]. After the discovery of EOT of a metallic film through ordered nanometer-sized holes, many researchers have investigated the mechanism [18–21], and have tried to apply this phenomenon to functional optical devices. Most EOT phenomena occur in a narrow wavelength range, suggesting that it stems from resonance phenomenon of surface plasmons. This narrow-band EOT is useful for the optical devices such as optical bandpass filters and optical-energy concentrator [22,23]. On the other hand, non-resonant EOT with broadband transmission has been also reported for metallic patches or rods assembled with subwavelength gaps, mainly in the infrared range [24–32], where metals have a negligible imaginary component of refractive index (RI). The non-resonant EOT is based on the transmission waveguide mode through the gap between two metallic structures at transverse magnetic (TM) polarization. This transmission mechanism has been used for wire-grid polarizers [33–35], which usually have wide gaps between the metallic lines, to support high transmittance.
An effective way to understand the non-resonant EOT is to represent the subwavelength structure by the equivalent electrical circuit [36]. In 1960s, the equivalent electrical circuits (EEC) were proposed to investigate the EOT of an infinitely thin metallic grid and showed the agreement of the calculated transmission spectra with the measured ones [37–39]. Alù et al. treated the thin metallic grating as a transmission line (TL) and explained the optical response [24,25]. They considered the grating as a waveguide to obtain the propagation constant and characteristic impedance. However, the metallic grating with a thickness have not been investigated using the modeling of EEC.
In this paper, we studied spectral transmission characteristics of a one- or two-dimensional (1D or 2D) deep grating made of metallic rods arranged with narrow gaps, as shown in Fig. 1(a) and 1(b), by various means: numerical EF analysis, TL analysis based on the EEC model and the effective medium approximation (EMA). The numerical EF analysis showed that the deep grating with the narrow gap much smaller than the wavelength has the transmittance higher than the porosity that is defined as a proportion of the gap area to the total cross-section. This is one of the EOT phenomena. When the gap is comparable to the wavelength, however, transmittance decreases with increasing the gap distance and is lower than the porosity. When the gap is larger than the wavelength, the transmittance approaches the porosity, as expected. To understand these properties, we proposed an EEC model, and calculated the transmittance and the reflectance. The calculated results based on this model almost agree with the numerical EF analysis. We also adopted the EMA because the metallic deep grating can be regarded as hyperbolic metamaterials (HMM) [40]. The EMA approach was proposed [41] for the grating of perfect conductor, we used the approach for the existing material such as gold. Then, the effective RI has a dominant real part with a negligible imaginary part, for the transverse magnetic (TM) polarization. Namely, the metallic grating can be regarded as a transparent dielectric slab, resulting in the non-resonant EOT.
2. Numerical electromagnetic field analysis
Here we consider the 1D and 2D deep gold gratings as illustrated in Fig. 1(a) and 1(b), respectively. The 1D grating is made of gold slabs with width $w$ in $x$ direction, infinite width in $y$ direction and the depth of the grating length $l$ in $z$ direction. They are arranged with gaps $g$. The 2D grating is an assembly of rectangular parallelepiped rods with square cross-sections with sides $w$. The rods are arranged with gaps $g$, as shown in Fig. 1(b). In both cases, light is incident normal to the $x-y$ plane, unless mentioned, and transmittance $T$ and reflectance $R$ were calculated. The absorption $A$ was evaluated using the relation $A = 1 - T - R$, since $R$ and $T$ include the scattered and diffracted light intensity in the periodic boundary condition in $x$ and $y$ directions. For the numerical EF analysis, we adopted the following softwares, Full WAVE (Synopsys Inc.) and FDTD solutions (Lumerical Inc.) for FDTD, and COMSOL Multiphysics (COMSOL. Inc.) for finite element method (FEM) and GSOLVER (Grating Solver Development Company) for rigorous coupling wave analysis (RCWA). Optical constants of gold are installed from literature [42]. Each numerical EF analysis gives similar results, suggesting the validity of the EF analysis. Therefore the results of numerical EF analysis shown below were calculated using the Full WAVE software, unless mentioned.
We defined the 1D and 2D gold grating with $w = 0.03$ $\mu$m, $g = 0.01$ $\mu$m and $l =$ 0.2 $\mu$m as a primary structure. The periodicity $\Lambda$ is $w + g =$ 0.04 $\mu$m, and the porosity $p$ defined as a proportion of gap area to the cross-sectional area is calculated to be 0.33 for the 1D grating and 0.438 for the 2D grating. Figure 2 shows the transmittance $T$ spectra of the 1D and 2D primary structures for the polarization in the $x$-direction, which is TM polarization for the 1D structure. The 2D grating is independent of the polarization because of the symmetry. Both profiles are the same, indicating that the 2D grating behaves as 1D grating. This is because the light does not path through the gap aligning along the $x$ direction because the $x$-polarized light locally corresponds to the transverse electric polarization, which has negligible transmittance. Since the plane of the cross-section ($x$-$y$ plane) is isotropic, we mainly deals with the 2D grating in this paper, for simplicity.
Transmittance $T$, reflectivity $R$ and absorbance $A$ spectra are calculated for the 2D gold gratings at a constant width $w =$ 0.03 $\mu$m, length $l =$ 0.2 $\mu$m and different gap $g =$ 0.005, 0.01 and 0.03 $\mu$m, and the results are shown in Fig. 3(a)-3(c), respectively. The absorption is negligibly small in the infrared wavelengths $1-5$ $\mu$m, because of low imaginary permittivity of gold in the wavelength range longer than 1 $\mu$m. In each gap $g =$ 0.005, 0.01 and 0.03 $\mu$m, transmittance is greater than the the porosity $p =$ 0.265, 0.438 and 0.750, respectively, which are indicated with dotted line. The transmittance increases with increasing $g$, as expected. Figure 4(a)-(c) shows the $T, R$ and $A$ spectra for the gratings at a constant gap $g =$ 0.01 $\mu$m, length $l =$ 0.2 $\mu$m and different width $w =$ 0.01, 0.03 and 0.06 $\mu$m, respectively. The corresponding porosity are $p =$ 0.75, 0.438 and 0.265. Comparing Fig. 3 and Fig. 4, the spectra are identical at the same porosity, indicating that the transmittance is governed by the porosity $p$.
Fig. 3. Transmittance $T$, reflectivity $R$ and absorbance $A$ spectra of the 2D gold gratings at $w =$ 0.03 $\mu$m, length $l =$ 0.2 $\mu$m, and $g =$ 0.005, 0.01 and 0.03 $\mu$m. The dotted line corresponds to the porosity $p$.
Fig. 4. $T$, $R$ and $A$ spectra for the 2-D gold gratings at constant gap $g =$ 0.01 $\mu$m, length $l =$ 0.2 $\mu$m, and different width $w =$ 0.01, 0.03 and 0.06 $\mu$m. The dotted line corresponds to the porosity $p$.
Figure 5 shows the spectra at constant gap $g =$ 0.01, a width $w =$ 0.03 $\mu$m and different lengths $l =$ 0.1, 0.2, 0.3, 0.5 and 1 $\mu$m. Both $T$ and $R$ spectra oscillate and the number of peaks and dips increases with $l$. The oscillation stems from the interference of light propagating in the grating. The minimal reflectivity are at the condition that $\lambda = 2n_\textrm {eff}l/m (m = 1, 2\dots )$, where $n_\textrm {eff}$ is the effective RI of the grating. $n_\textrm {eff}\sim 2$ in the wavelength range of interest, according to the calculation shown below.
Fig. 5. $T$, $R$ and $A$ spectra for the 2D gold gratings at constant gap $g =$ 0.01 $\mu$m and width $w = 0.03$ $\mu$m, and different length $l =$ 0.1, 0.2, 0.3 and 0.5 and 1 $\mu$m. The dotted line corresponds to the porosity $p$.
Calculated spectra of $T$, $R$ and $A$ at constant porosity $p =$ 0.438 ($w/g =$ 3) and different size are shown in Fig. 6. The size is characterized by periodicity $\Lambda$. Figure 6(a) shows the spectra of the principal structure in which $\Lambda =$ 0.04 $\mu$m. The spectra at $\Lambda =$ 0.2 and 0.8 $\mu$m are shown in Figs. 6(b) and 6(c), respectively. Interestingly, the transmittance decreases with increasing the size in spite of the constant $p$. Figures 6(d), 6(e) and 6(f) are the transmission spectra at $\Lambda =$ 2, 4 and 10 $\mu$m, for 1D and 2D gratings, calculated by RCWA and FDTD, respectively. The transmittance $T$ is below the porosity, and $T =$ 0 at $\lambda = \Lambda$. Sharp dips in transmittance spectra are also observed at $\lambda = \Lambda /n$ ($n =$ 1, 2$\dots$). The transmittance approaches to the porosity with increasing the size, indicating that the infrared light simply passes through the gap without any interaction to the grating.
Fig. 6. Transmittance $T$, reflectivity $R$ and absorbance $A$ spectra calculated for the 2-D gold gratings at constant porosity $p =$ 0.438 ($w/g =$ 3) and different size (the periodicity is $\Lambda =$ (a) 0.04, (b) 0.2, (c) 0.8, (d) 2.0, (e) 4.0 and (f) 10 $\mu$m). The dotted line corresponds to the porosity $p$.
Spectra at different cross-sectional shapes (square, octagon, circle) at constant porosity $p =$ 0.438 are shown in Fig. 7. The spectra are similar, indicating that the spectra are governed by the porosity and size. We also calculated the spectra of the grating made of different metals (gold, silver, copper and aluminum) and found that the spectra are identical. This is because of the fact that the metals are almost ideal, at the infrared wavelengths.
Fig. 7. (a) Transmittance $T$, reflectivity $R$ and absorbance $A$ calculated for the 2D gold gratings at constant porosity $p =$ 0.438 ($w/g =$ 3) and different cross-sectional shape: square, octagon and circle. The dotted line corresponds to the porosity $p$.
Figure 8(a) and 8(b) shows spectra of the grating ($p =$ 0.438 ($w/g =$ 3) and $l =$ 0.2 $\mu$m) at $\Lambda =$ 0.04 and 3 $\mu$m in different ambient refractive indexes, respectively. In the spectra of $\Lambda =$ 0.04 $\mu$m (Fig. 8(a)), the transmittance peak at around 1 $\mu$m shifts to the longer wavelength, with increasing the ambient RI. This character can be used for RI sensing. The RI sensitivity that is the amount of peak shift for unit RI change is 0.76 eV/RI (0.79 $\mu$m/RI). In the spectra at $\Lambda =$ 3 $\mu$m (Fig. 8(b)), the spectral shift is also observed. The RI sensitivity of the transmittance dip at around $\lambda =$ 3 $\mu$m is evaluated to be 0.34 eV/RI (3.18 $\mu$m/RI). These sensitivity is greater than or comparable with the values previously reported [47–50], as summarized in Table 1, where the figure of merit (FOM) is also given in addition to the sensitivity $\Delta \lambda /n$ and $\Delta E/n$. FOM is evaluated by the sensitivity divided by the peak width (full width at half maximum: FWHM).
Fig. 8. Change in the transmittance $T$ and reflectivity $R$ spectra of the 2D grating in different ambient RI, $n =$ 1, 1.3 and 1.5. The grating with (a) small periodicity ($\Lambda =$ 0.04 $\mu$m) and (b) large periodicity ($\Lambda =$ 3 $\mu$m). Porosity of both gratings are $p =$ 0.438 ($w/g =$ 3). The dotted line corresponds to the porosity $p$.
Table 1. Comparison of the sensitivity and FOM at the peak wavelength.
3. Equivalent electrical circuit
An effective way to understand the optical response is to represent the gratings by a lumped constant electrical circuit. We propose an EEC model of the repeating unit of the gold grating shown in Fig. 9(a). Because of the symmetry, the circuit can be reduced as shown in Fig. 9(b), which consists of two capacitors and the inductors of the rods L. The capacitor C$_\textrm {i}$ forms with the end surfaces of the rods and the capacitor C$_\textrm {p}$ does the side of the rods. The inductor is divided into two components: kinetic inductance L$_\textrm {k}$ and mutual inductance L$_\textrm {m}$, where $L=L_\textrm {k}+L_\textrm {m}$.
The capacitor $C_\textrm {i}$ at an angular frequency $\omega$ is described using a parameter $\alpha =g/\Lambda$ and vacuum impedance $Z_0$ [36],
According to the equivalent circuit shown in Fig. 9, the characteristic impedance of the grating $Z_\textrm {s}$ is
Reflectivity $R$ and transmittance $T$ calculated on the basis of the equivalent circuit are [26]
Fig. 11. The transmittance $T$ (a) and reflectivity $R$ (b) spectra calculated with the equivalent circuit. The calculated results of $T$ and $R$ by the FDTD method are also shown for comparison.
In order to clarify the reason why the transmittance $T$ decreases with the periodicity $\Lambda$ even at a constant $w/g$, as shown in Fig. 6, the impedance ${Z_\textrm {i}}$ and ${Z_\textrm {L}}$ are calculated as a function of the periodicity $\Lambda$. The results are shown in Fig. 12. The decrease in ${Z_\textrm {i}}$ brings about the increase in $T$ because it is a shunt capacitor, whereas the decrease in ${Z_\textrm {L}}$ brings about the decrease of $T$. Although they are a competitive relation, the decrease in ${Z_\textrm {i}}$ is larger than that of ${Z_\textrm {L}}$, with increasing the periodicity $\Lambda$. As a result, the transmittance $T$ decreases with the size.
Fig. 12. The impedance ${Z_\textrm {i}}$ and ${Z_\textrm {L}}$ as a function of $\Lambda$ at $\lambda = 3\ \mu$m.
4. Effective medium approximation
When the periodicity $\Lambda$ is much smaller than the wavelength, the gratings can be regarded as HMM [40]. The 1D grating illustrated in Fig. 1(a) is a uniaxial medium and has two effective RI components: $n_\textrm {eff, o}$ for ordinary ray and $n_{\textrm {eff, e}}$ for extraordinary ray. The optic axis is along x-direction. They are written as
where $n_\textrm {m}$ and $n_0$ are RIs of the metal and air, respectively. The 2D grating illustrated in Fig. 1(b) is also a uniaxial medium of which optic axis is along z-direction. The effective RIs areFig. 13. Transmittance $T$ and reflectivity $R$ calculated under the EMA, for the 2-D gold gratings at constant gap $g =$ 0.01 $\mu$m and width $w =$ 0.03 $\mu$m, and different length (a) $l =$ 0.2 $\mu$m and (b) 1 $\mu$m. Those calculated by FDTD is also plotted with dotted line for comparison.
Transmittance $T$ and reflectivity $R$ as a function of the angle of incidence, calculated based on the EMA, for the 2-D gold gratings are shown in Fig. 14 for (a) p-polarization and (b) s-polarization. The gap, width and length are g = 0.01 $g = 0.01\: \mu\text{m}$, $w = 0.03\: \mu\text{m}$, and $l = 0.2\: \mu\text{m}$, respectively. The grating is assumed as a slab of uniaxial medium with $n_\textrm {eff, o}$ and $n_\textrm {eff, e}$, and $T$ and $R$ are calculated following the manner reported by Kidwai et al. [51]. With increasing the angle of incidence, $\theta _\textrm {in}$, the transmittance $T$ for p-polarization increases and is unity at around $\theta _\textrm {in} = 60^\circ$, which corresponds to the Brewster angle, as shown in the previous reports [25–27,41], in which different calculation method was adopted. The reflectance $R$ decreases with $\theta _\textrm {in}$ and is zero at the Brewster angle. It increases to unity at $\theta _\textrm {in} = 90^\circ$. For s-polarization, the transmittance decreases with $\theta _\textrm {in}$ monotonically. The $T$ and $R$ profiles are similar to those of the dielectric slab. Figures 15(a) and 15(b) show $n_\textrm {eff, o}$ and $n_\textrm {eff, e}$ spectra of the 2D grating, respectively. In Fig. 15(a), $n_\textrm {eff, o} \sim 2$, similar to the results calculated by the EEC model shown in Fig. 10. Although the 2D grating consists of gold, it behaves as a transparent dielectric with negligible loss for ordinary ray.
Fig. 14. Transmittance $T$ and reflectivity $R$ as a function of the angle of incidence, calculated based on the EMA, for the 2-D gold gratings at (a) p-polarization and (b) s-polarization. The gap, width and length are $g =$ 0.01 $\mu$m, $w = 0.03$ $\mu$m, and $l =$ 0.2 $\mu$m, respectively.
Fig. 15. Real and imaginary part of the effective RI of the grating, in (a) $x$-direction and (b) $y$-direction. The gap, width and length are $g =$ 0.01 $\mu$m, $w = 0.03$ $\mu$m, and $l =$ 0.2$\mu$m, respectively.
The EMA is also applicable to the EOT of a grating with a cross-sectional shape of octagon. The spectrum is shown in Fig. 7. Consider regular octagons with side length $a$, which are separated with a gap $g'$, as a cross-section of the grating. Since the porosity of the octagon is the same as that of the square grating, $w = a \sqrt {2 s^+}$, where $s^\pm = \sqrt {2} \pm 1$. Then the effective refractive index of the octagon grating for o-ray, $n_\textrm {eff,o}$, is
The effective refractive index of the octagon grating for e-ray, $n^2_\textrm {eff,e}$, is given by Eq. (14), which is rewritten with $a$ and $g'$ as,
5. Conclusion
Non-resonant broadband EOT through one-dimensional and two dimensional deep metallic grating was investigated in infrared range, by the EEC model, the EMA and numerical the EF analysis. The numerically-calculated transmittance and reflectance spectra can be explained by the EEC model proposed in this paper. The EEC model consists of the input/output capacitors and kinetic/mutual inductors. Since the deep metallic grating can be also regard as HMM, the EOT mechanism can be also explained by the calculation under the EMA. Although the 2D grating consists of metals, the metallic grating behaves as a transparent dielectric slab with negligible loss at normal incidence, resulting in the great transmission and existence of a Brewster angle.
Appendix
A characteristic impedance ${Z_\textrm {s}}$ for a four terminal circuit, as depicted in Fig. 16, is written with the components of F-matrix of the circuit as
The F-matrix is defined as where $V_1$ and $I_1$ are the input voltage and current, and $V_2$ and $I_2$ are the output voltage and current, respectively.The F-matrix of the EEC in Fig. 9(b) is
Acknowledgments
We thank Dr. Takayuki Okamoto of RIKEN for helpful discussion on the FDTD calculations.
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