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Abstract
We demonstrate a conceptually simple polarization-independent mechanism for nearly perfect infrared light transmission through a subwavelength one-dimensional metal grating implemented in the grooves of a deep-subwavelength monolithic high-contrast grating (metalMHCG). We provide theoretical background explaining the transmission mechanism, which eliminates Fresnel reflection as well as significantly reduces metal absorption and the reflection of transverse electric and transverse magnetic light polarizations. Careful design of a metalMHCG implemented at the interface between the regions of high refractive index contrast enables the coincidence of high transmission conditions for both light polarizations, enabling up to 97% transmission of polarization-independent infrared radiation. Our analysis shows excellent electrical properties of the metalMHCG as evidenced by sheet resistance of 2 ΩSq−1 facilitating straightforward horizontal electron transport and vertical injection of the current into the semiconductor substrate on which the electrode is implemented.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Absorption and reflection are inevitably associated with the propagation of light through metal structures, which corrupt light transmission. In the case of transmission through a planar metal structure implemented at the flat interface between regions of high refractive index contrast, such as a semiconductor and air, a transmission is additionally deteriorated by Fresnel reflection. Reduction of parasitic metal absorption and reflection occurs in thin metal structures with periodic arrangements of one-dimensional slits or with two-dimensional holes, due to extraordinary optical transmission (EOT) [1–6]. This type of transmission occurs under illumination of the metal structure with the magnetic component of an electromagnetic (EM) field parallel to the walls of the slits (TM polarization) or to the walls of the holes in the metal layer. In this polarization, surface plasmons (SPs) can be resonantly excited [7–11] due to boundary conditions at the metal surface [12], supporting high transmission that at the same time deteriorates for orthogonal polarization (TE). These counteracting phenomena play an ambivalent role in the mechanism of transmission through two-dimensional metal structures [13]. However, in the case of one-dimensional metallic slotted gratings, several theoretical works have shown that transmission can be enhanced for TE polarization, mainly via the cavity resonance mechanism [14–18]. Unfortunately, these findings have been underutilized by the community. In [15], which is particularly important in the context of unpolarized transmission, it was shown that slits between metal stripes filled with high refractive index silicon implemented at the interface between silica and air enable up to 86% transmission of unpolarized light at a wavelength of 1550 nm. Although this concept shows the possibility of unpolarized light transmission through a one-dimensional metal grating, it does not eliminate either Fresnel or metal reflection. However, the great advantage of this design is the significant volume of metal present in the structure, which enables very low sheet resistance of a few ΩSq−1.
In this paper, we present a conceptually simple mechanism which is responsible for nearly perfect transmission of polarization-independent light through a one-dimensional metal grating integrated with our original concept of a monolithic high contrast grating (MHCG) [19]. Recently, our group proposed electrodes based on a semiconductor MHCG integrated with metal (metalMHCG), which enables low sheet resistance of less than 1 ΩSq−1 and transmittance above 95% or 99% for TE and TM polarized light, respectively [20]. We found that the semiconductor grating in the metalMHCG is mainly responsible for the transmission mechanism. Reduction of parasitic interaction of light with the metal, that is implemented between the semiconductor stripes in the metalMHCG, relies on funneling TE polarized light through the semiconductor stripes of the metalMHCG and funneling TM polarized light through the grooves between the semiconductor stripes. The great advantage of this approach, unlike others that have been reported, is that the metalMHCG eliminates Fresnel reflection. Here, we demonstrate by numerical analysis that careful tuning of the transmission mechanisms for TE and TM polarizations, [20] enables very high polarization-independent transmission through the metalMHCG. The design of the metalMHCG enables straightforward application as a transparent electrode with low sheet resistance, facilitating horizontal electron transport along the metal stripes and enabling current injection into the semiconductor below the electrode. To the best of our knowledge, there have been no previous reports of transparent electrodes implemented on high refractive index material enabling above 90% transmission of unpolarized infrared light and sheet resistance below 10 ΩSq−1. In Section 2 of this paper, we detail the metalMHCG design. In Section 3, we consider optical phenomena responsible for the high transmittance of the metalMHCG. In Section 4, we investigate the influence of the refractive index of the semiconductor and metal on the transmission of the grating. In Section 5, we evaluate the electrical properties of the metalMHCG.
2. Structure and model
The transparent electrode proposed in this paper is composed of an infinitely thick monolithic transparent semiconductor layer, with evenly spaced grooves etched on the surface that form a one-dimensional grating. Metal stripes are deposited on the bottom of the grooves (Fig. 1). The thickness of the air above the grating is assumed to be infinite. The parameters of the grating are as follows: L – period of the grating; H – height of the semiconductor stripes; Hm – height of the metal stripes; F – the fill factor, as the ratio of the width (a) to the period (L) of the semiconductor stripes. The height of the metal stripes Hm is set to 100 nm unless stated otherwise in Section 5. We assume exemplary refractive indices for the semiconductor grating ns = 3.56 and metal nm = 1.5 – 24i, which are close to the refractive indices of antimonide-based materials and gold, respectively, in the mid-infrared range. The refractive indices are varied in Section 4. The calculations are performed for a wavelength of λ = 3.5 µm, except in Section 4 where dispersion of the metal refractive index is considered.
Fig. 1. Configuration of the modelled structure composed of semiconductor stripes implemented on a substrate made of the same material and metal stripes implemented between the semiconductor stripes.
To determine the optical transmittance through the gratings, we use the plane-wave admittance method (PWAM) [21,22]. This method is a fully vectorial optical model that transforms Maxwell’s equations into a characteristic equation, in which we assume that in the direction of propagation the structure consists of parallel layers. The electromagnetic fields, as well as the electrical permittivities, are decomposed in a complete set of exponential functions. In the calculations, we consider a single period of the gratings with periodic boundary conditions, which elongates the grating to infinity in the lateral direction. The optical field distribution in a single period can be expressed in the form of Bloch waves:
where f(y) is a periodic function with a period that corresponds to the period of the metalMHCG (L) and ky is the lateral component of the wavevector ranging from − λ/L to λ/L. The accuracy of the model has been validated by comparison with other numerical models used to simulate light transmission through metallic gratings [20] and in a number of experiments, showing very good agreement with the reflection spectra of subwavelength periodic structures in the form of a stand-alone monolithic high-contrast grating (MHCG) [19,23]. The model was also used to design the first electrically-injected MHCG VCSEL [24] and revealed very good agreement with its experimental characteristics [22]. Here, we consider two orthogonal polarizations, TE and TM, where electric component of the electromagnetic field of TE polarization is parallel to the grating stripes. We calculate the transmission of unpolarized light as the mean value of transmitted intensities for both TE and TM polarizations since Maxwell’s equations are linear and the general solution is a superposition of both orthogonal polarizations [25].3. Transmission mechanism
Figure 2 presents transmission maps of the metalMHCG for two orthogonal TE (Fig. 2(a)) and TM (Fig. 2(b)) polarizations and for transmission of unpolarized light (Fig. 2(c)), in the domain of the grating period (L/λ) and the height of the semiconductor stripes (H/λ), both normalized with respect to the wavelength for F = 0.78, which is an optimal value in this case. Figure S1 in the Supplement 1 presents transmissions maps in the same domain and different values of F. The region where more than one mode exists in the grating (L/λ > 1/ns = 0.28) is distinguished in the maps of TE and TM polarization by a pattern of low and high transmission regions (checkboard pattern [26]). This indicates interference effects between the modes propagating in the grating.
Fig. 2. Maps of metalMHCG transmittance (T) under normal incidence of TE polarization a), TM polarization b) and polarization-independent c) in the domain of the grating period and height of the semiconductor stripes for ns = 3.56, nm = 1.5 – 24i, F = 0.78, Hm = 0.029λ, and λ = 3.5 µm. The black cross indicates the design with maximal polarization-independent transmittance.
In the deep subwavelength regime of the metalMHCG, L/λ < 1/ns, only one grating mode exists and the pattern of transmission bands resembles the Fabry-Perot (FP) mechanism, with a low-quality factor etalon created between the two interfaces of the metalMHCG, one at the interface of the grating with cladding and the other at the interface with air. High transmission and hence elimination of Fresnel reflection relates to coupling of the incident plane wave to low-quality factor cavity resonances (CRs), which are located in the high refractive index stripes in the case of TE polarization and in the air slits between the stripes in the case of TM polarization [5], as shown in Fig. 3. Figure S2 in the Supplement 1 presents analogous distributions in MHCG.
Fig. 3. a) Distributions of ExEx* and b) EyEy* under normal incidence represented by colors within a single period of the metalMHCG. The parameters of the metalMHCG are as follows: L = 0.18λ, F = 0.78, H = 0.37λ, Hm = 0.029λ, λ = 3.5 µm.
A condition for maximal transmission relates the wavelength (λg) in the grating layer with the thicknesses of the grating layer (Hp), by the relation 4Hp/λg = 2p – 1, where p is a positive integer. The difference between successive heights of the stripes admitting high transmittance (ΔH = Hi – Hi−1) differs for each polarization, due to their different propagation paths through the grating. This results in larger ΔH in the case of TM with respect to TE polarization. Tuning H and F enables the coincidence of high transmission conditions for both polarizations, enabling high transmission of polarization-independent light, indicated by dark red regions in Fig. 2(c). The optimal design enabling the highest transmission is indicated by black cross in Figs. 2(a-c) and corresponding distributions of TE and TM polarisations in that configuration are presented in Fig. 3. In the optimal configuration metal stripes are positioned in the node of the standing wave pattern reducing interaction of light with the metal, which minimizes absorption and eliminates plasmonic effects. Analogous transmission maps of the MHCG are presented in Fig. S3 in the Supplement 1. Comparison of the metalMHCG and MHCG maps suggests that the mechanism for light transmission of both polarizations based on CR remains similar in both configurations. This mechanism is responsible for the elimination of Fresnel reflection. Nevertheless, metal introduces optical absorption and the reflection of impeding light that may potentially reduce the transmittance of the metalMHCG compared to the MHCG. If the relative permittivity of the metal stripes (ε = εr + iεi) meets the condition |εr| < |εi|, as is the case with Ag, Au, Al, and Cu in the infrared region, the metal stripes contribute to light confinement outside the metal stripes. This effect is responsible for the stronger confinement of TE polarization in the semiconductor stripes and of TM polarization in the air slits between the stripes compared to an MHCG that is without metal. If the parameters of the grating are chosen carefully, absorption and metal reflection can be almost totally eliminated for both polarizations, thanks to the significant reduction of light–metal interaction in the metalMHCG. This mechanism has been discussed in greater detail in [20].
The influence of the geometry of metal stripes on maximal transmission is visualized in Fig. 4, which collects maximal values for TE and TM transmittances as a function of F in the deep subwavelength region of a metalMHCG and of an MHCG for comparison. For each point in the figure, L and H are used as optimization parameters, whereas L and H are chosen from the ranges (0.015, λ/ns) and (0, 0.45λ), respectively. In the case of the MHCG, maximal transmittance of TE and TM polarizations occurs for significantly different F, which is in line with polarization-dependent transmission in the case of the metamaterial limit (L/λ→ 0). Perfect transmission is achieved for F ranging from 0.15 to 0.2 in the case of TE polarization and from 0.5 to 0.8 in the case of TM polarization. This discrepancy between the grating parameters limits total transmission of unpolarized light through the MHCG to 90%.
Fig. 4. Maximal transmittance (T) under normal incidence as a function of fill factor F of TE (red lines) and TM (blue lines) polarizations for an MHCG (dashed lines) and a metalMHCG (solid lines).
In the case of the metalMHCG, the maximal transmission of TE polarization shifts to larger F due to strong lateral confinement of light induced by the metal stripes. This effect has been studied in detail in [20]. The maximal transmission of TM polarization is significantly deteriorated by the presence of metal in the range F < 0.7. In this range the light does not enter the semiconductor stripes. This results in significant reflection from the metal as Fig. S4 in the Supplement 1 illustrates. However, although light is concentrated in the air slits in the range F > 0.7 it also penetrates the semiconductor stripes enabling metal bypassing and transmission. In the range F > 0.7 the functions of the TM transmission for the MHCG and metalMHCG coincide, suggesting that the influence of the metal on transmission is reduced in the metalMHCG. Consequently, the fill factors corresponding to maximal transmission of TE and TM polarizations approach the same value, enabling high transmission of polarization-independent transmission in the metalMHCG.
Careful tuning of L, F, and H in the case of the metalMHCG with exemplary refractive indices (ns = 3.56 and metal nm = 1.5 – 24i) enables the coincidence of high transmission for TE and TM polarizations with the following parameters: L = 0.18λ, F = 0.78, and H = 0.37λ. This enables transmittance of unpolarized light as high as 0.97 and 0.11λ spectral width of transmittance above 0.8 (Fig. 5). The presence of absorption at the level of 0.03 is caused by penetration of the metal stripes by TE polarization. Such high transmission is observed only in the deep subwavelength regime. In the subwavelength regime, polarization-independent transmittance not larger than 0.8 can be observed.
Fig. 5. Spectra of transmittance (T, red), reflection (R, green) and absorption (A, blue) under normal incidence of the metalMHCG for L = 0.18λ, F = 0.78, H = 0.37λ, Hm = 0.029λ, and λ = 3.5 µm.
Figure 6 combines maps of the transmission, reflection, and absorption of both polarizations and of polarization-independent (PI) light in the planes (λ, kx) and (λ, ky) for the metalMHCG. See Fig. S5 in the Supplement 1 to compare analogous transmission maps for an MHCG of the same L, F, and H parameters. Transmission maps of the metalMHCG in the deep-subwavelength range (L/λ < 0.3) at kx = ky = 0 reveal several pronounced transmission maxima, corresponding to the FP phenomenon. In this region, the presence of the metal introduces no additional features in the maps of transmission, which show close correspondence to the MHCG maps. For L = 0.18λ, corresponding to the optimal design of the metalMHCG, reflection reaches nearly 0, confirming the elimination of Fresnel and metal reflection. Nearly 0 absorption is observed for TM polarization and absorption of a few percent is present in the case of TE polarization, related to weak penetration of the metal by light. The absorption of TE light is responsible for non-total transmission of the metalMHCG. For kx ≠ 0, ky ≠ 0 the metalMHCG shows broad angles, ranging 60°, enabling transmission above 80% (also see Fig. S6 in Supplement 1 which illustrates the angular and spectral dependence of the transmission). The transmission is limited mainly by the increased reflection of TE polarization when the incidence is tilted in the y direction and by increased reflection of TM polarization when the incidence is tilted in the x direction. In the range L/λ < 0.18 transmission maxima are smaller with respect to the optimal one at L/λ = 0.18 due to significant absorption of TE polarization. In this spectral region of long wavelengths, TE polarization penetrates the metal which contributes to larger absorption. In the range of L/λ > 1/ns = 0.28, sharp features in the transmission maps of the metalMHCG appear, which are accompanied by significant absorption related to the creation of plasmons at the surface of the metal.
Fig. 6. Dispersion diagrams of transmission (T) in first column, reflection (R) in second column, and absorption (A) in third column for a metalMHCG in the cases of TE polarization in first row, TM polarization in second row and polarization-independent (PI) light in third row for air-side light incidence. The parameters of the metalMHCG are as follows: L = 0.18λ, F = 0.78, H = 0.37λ, Hm = 0.029λ, and λ = 3.5 µm.
4. Influence of the refractive index of the grating on light transmission
Figure 7 presents the dependence of the light transmission of the metalMHCG on the refractive index ns of the grating and on the height of the semiconductor stripes H/λ. For each point of the map, the grating period L and fill factor F have been optimized with respect to the maximal transmission for a given H. In the calculations, we assumed that L ranges from 0.05λ to λ/ns and F ranges from 0.1 to 0.9. As can be seen, the transmission of the metalMHCG is significantly dependent on the refractive index of the semiconductor stripes, with higher values facilitating lower optimal H. There is an island of high transmission (IHT) corresponding to transmission above 0.93 that ranges in ns from 2.7 to over 4 (indicated by 1 in the figure). This range of refractive indices is particularly important for infrared applications, in which high refractive index materials such as silicon, arsenide-, phosphide-, and antimonide-based are widely used. Structures that belong to this IHT require H in the range of between 0.33λ to 0.48λ, L from 0.15λ to 0.26λ and F from 0.76 to 0.83. Two other IHTs (indicated by 2 and 3 in the figure) relate to lower refractive index materials (ns < 3). These require high values for H. However, the maximum transmission reaches 0.985 and 0.989 in the case of IHTs 2 and 3, respectively. Reducing the refractive index ns of the metalMHCG to 1.5 results in transmission comparable to that through a plane interface between air and the substrate, since Fresnel reflection is lower than 0.04 in this configuration. Such high transmission can be achieved with a purely metal grating, although its period tends to 0 whereas to sustain transmittance above 0.95 the period of the metalMHCG may be as large as 0.2λ. In Supplementary information S1 we present influence of refractive index dispersion of gold on transmission of the metalMHCG and Fig. S7 illustrates the transmission spectra of the metalMHCG realized for exemplary chosen materials.
Fig. 7. Polarization-independent transmission (T) under normal incidence of the metalMHCG in the domain of the refractive index ns of the semiconductor and of the stripe height H. The black crosses with numbers indicate the designs with locally maximal polarization-independent transmittance within islands of high transmission (IHT).
5. Sheet resistance
The electrical properties of the electrode can be characterized by the sheet resistance Rs, which is defined according to the formula:
where ρ is the electrical resistivity of the metal stripes. To consider metal stripes of nanoscale cross-sectional dimensions, it is necessary to take into account conductivity-size effects which deteriorate the electrical resistivity of bulk gold (2.2·10−8 Ωm), as demonstrated in [26] for nanowires with a circular cross-section. We assumed that the resistivity of the metal stripe of height Hm and width Wm is equal to the resistivity of a nanowire of diameter min(Hm,Wm). Such simple approach provides close agreement with the experimental results for various gold nanostructures (see Table S2 in the Supplement 1). The dependence of resistivity versus the cross-sectional dimensions of the metal stripes based on [27] are presented in Fig. S8 in the Supplement 1. Figure 8 depicts the relation between transmission and sheet resistance in the case of the metalMHCG. For each point in the figure, L, F, H, and Hm are used as optimization parameters. The parameters L and H are freely chosen from the ranges (0.015λ, λ/ns) and (0, mλ), respectively. Blue lines indicate m = 0.4, corresponding to less technologically demanding designs, and m = 1.2 is indicated by red lines. The F and Hm parameters are related, due to the assumed value of sheet resistance. The figure reveals maximal optical transmission of 97% for sheet resistance of 2 ΩSq−1. The maximum is present due to the ambiguous influence of the metal, which enables better light confinement in the semiconductor stripes or air slits, but when the volume of metal becomes too large transmission reduces due to the increasing absorption of the metal. Sheet resistance can be reduced to about 0.4 ΩSq−1 while sustaining transmission at 95%. When the volume of gold is very low (F = 0.49, Hm = 6·10−5λ), corresponding to sheet resistance of 2·103 ΩSq−1, the minimum transmittance falls below the level of MHCG transmittance, showing that if volume of the metal is too small it does not contribute to light confinement outside the metal and introduces absorption instead. For the purposes of comparison, we determined the corresponding transmission function for a one-dimensional grating consisting of gold stripes deposited on a semiconductor substrate of the same refractive index as in the metalMHCG. The L, F, and Hm for a given sheet resistance were optimized with respect to maximal transmission. We found that the maximal unpolarized transmission occurs for a fill factor F = 0.9. In this case, TM transmission is not as high as would be expected due to EOT, since the structure is optimized with respect to unpolarized light transmission. The maximal transmission reaches 70%, which corresponds to transmission limited by Fresnel reflection. The horizontal magenta line illustrates light transmission through the air/sapphire interface, showing that the metalMHCG deposited on a high refractive index substrate (ns = 3.6) reaches a transmission larger than any of the possible transparent electrodes deposited on a low refractive index material.Fig. 8. Polarization-independent transmittance (T) under normal incidence versus sheet resistance of a metalMHCG assuming H < 0.4λ, indicated by blue lines, corresponding to designs with moderate technological demands and H < 1.2λ (red lines). The transmittance of the one-dimensional metal grating is indicated by green lines. TE and TM polarizations and polarization-independent transmissions (PI) are indicated by dotted, dashed, and solid lines, respectively. The maximal polarization-independent transmittance of the MHCG is indicated by a black solid line and transmission through the air/sapphire interface is indicated by a solid magenta line.
6. Conclusions
This paper has proposed a design for a new transparent electrode for unpolarized light, in which a one-dimensional metal grating is integrated with a deep-subwavelength monolithic high-contrast grating (metalMHCG). By conducting a numerical analysis for unpolarized infrared radiation, we have shown that the metalMHCG enables 97% transmission, with a transmission band of over 80% for a spectral width of 0.11λ of the incident wavelength, and admits above 80% transmission for angles of incidence reaching 60°.
High transmittance is attributed to light funneling through the semiconductor stripes, in the case of TE polarization, and through the air slits between semiconductor stripes, for TM polarization. The design of the semiconductor grating enables both polarisations to bypass the metal stripes. The metal stripes additionally contribute to light confinement in the grating (TE polarisation) or in air slits (TM polarisation). Both mechanisms reduce light absorption by the metalMHCG. Elimination of Fresnel reflection is governed by coupling of the incident plane wave with low quality factor cavity resonances (CR) occurring within the metalMHCG. The conditions of high transmittance for each polarization are different, due to their different channels of transmission through the metalMHCG. However, tuning the geometrical parameters of the grating enables the coincidence of high transmission conditions for both polarizations.
The periods and heights of the optimal gratings are 0.2λ and 0.3λ, respectively, which can be facilitated by contemporary electrolithography or nanoimprint technologies. Unlike existing transparent electrodes which are implemented on low refractive index materials, the metalMHCG can be implemented on the surface of high refractive index semiconductors, facilitating straightforward horizontal electron transport and the injection of current into the semiconductor. The metalMHCG offers the prospect of > 95% transmittance of unpolarized infrared radiation for a sheet resistance of 0.4 ΩSq−1 and 97% transmittance for a sheet resistance of 2 ΩSq−1.
Funding
Narodowe Centrum Nauki (OPUS 018/29/B/ST7/01927); Narodowe Centrum Badań i Rozwoju (HybNanoSens no. DZP/POL-SINIV/283/2017).
Disclosures
The authors declare no conflicts of interest.
See Supplement 1 for supporting content.
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