Optical absorbers based on metal nanostructures are widely used to realize surface-enhanced Raman scattering, high-sensitivity optical detection, and high-conversion solar cells because they can confine the incident light energy to a specified region.1^–5 Electromagnetic resonance is the condition for high-efficiency absorption in metal nanostructures, mainly attributed to metals’ high-loss properties and surface plasmon resonance (SPR).6 SPR are collective oscillations of free electrons on a metal surface caused by the interaction of free electrons in the metal with incident light when the light is incident on the surface. SPR was first discovered in 1902 by Wood on a metal diffraction grating,7 and the phenomenon was first theoretically explained by Fano.8 SPR has played a significant role in various fields, such as biological probes, optical imaging, and enhanced Raman scattering. Generally, it is almost impossible to observe SPR using planar metal surfaces because the surface plasmon excitations have propagation constants larger than those of the incident light, and this does not allow the condition of their phase matching. Therefore, prism couplings in the Otto and Kretschmann configurations are often used to avoid mismatches between the respective propagation constants. However, they have low spectral resolution and require bulky instrumentation.

As an alternative to the Otto and Kretschmann configurations, the grating coupling method can utilize the grating structure on the metal surface to change the horizontal momentum component of the incident electromagnetic wave.9 Thus, it can make up for the difference between the transverse wavevector of the surface plasma and the wavevector of the electromagnetic wave in the medium. Currently, the excitation of SPR using metal gratings has become the main means of obtaining high absorption.10^–14 Meng et al. achieved an ultra-narrow absorption with a full width at half maxima of only 0.4 nm in the near-infrared band using SPR excited by an Ag shallow slot grating.11 Zhou et al. proposed a two-dimensional Al grating that achieved polarization-independent high absorption under cavity resonance induced by coupled SPR.12 Wu et al. designed an Al shallow groove grating, which realized broadband absorption in the 220 to 800 nm band under the combined effect of SPR and Fabry-Pérot (FP) cavity resonance.13 Feng et al. proposed a metal-dielectric grating consisting of double Ag grating ridges.14 This grating can excite propagating surface plasma modes with low dissipation rates, resulting in perfect absorption near 1550 nm wavelength. Notably, most metal gratings currently used to achieve high absorption are at the subwavelength scale.10^–15 This phenomenon is because diffraction properties occur when the grating scale deviates from the subwavelength,16 resulting in lower absorption in metal gratings. However, as the study wavelengths continue to get smaller, gratings at subwavelength scales will require increasingly high-precision processes. To our knowledge, only two works have concerned the absorption properties of superwavelength gratings,17^,18 and no work has simultaneously investigated the absorption properties of metal gratings at different scales. Suppose high absorption can be realized in gratings at different scales. In that case, it will be possible to design grating scales that are more suitable for fabrication processes, thus reducing the difficulty of device fabrication. In addition, as a typical representative of superwavelength gratings, echelle gratings are often used to improve spectral resolution through high diffraction orders at large angles. Therefore, studying the absorption effect of superwavelength gratings at large incident angles will bring new perspectives to the design and fabrication of high-diffraction-efficiency echelle gratings.

In this paper, we find the strong absorption of TM-polarized light by Al triangle-groove gratings (TGGs) at different scales under grazing incidence. Based on the electric field distribution, the mechanism of the strong absorption at different grating scales is unified and is dominated by SPR on the grating surface. In addition, we find that the absorption of TGGs is symmetrical to the blaze angle. Taking this as a starting point, we generalize the equivalence rule of grating grooves to subwavelength scales.

Figure 1(a) shows the schematic structure of the blazed grating or triangle-groove grating, in which the apex angle, blaze angle, groove depth, and grating period in the $x$-direction, are denoted by $\phi$, $\alpha$, $h$, and $d$, respectively. According to the scale of the grating period, we categorize the gratings into three main types in Figs. 1(b)–1(d). The superwavelength grating, with deep or shallow groove depth, has multi-diffraction orders due to the long period to wavelength and requires nanometers, even sub-nanometers positioning accuracy; otherwise, the low precision of grating periods will lead to strong scatter light between diffraction orders. Though the grating period of the superwavelength grating is longer than the incident wavelength, the simulation for diffraction efficiency needs to be considered by rigorous vector diffraction theory based on electromagnetic theory if the groove depth is deep or the incident angle is apart from normal incidence due to the obvious polarization effect. The near-wavelength grating is normally called two diffraction orders grating with only 0th and −1st, and subwavelength grating has no other diffraction orders except for 0th order reflection.

The absorption efficiency calculation part of this work was performed by PCGrate software based on the boundary integral equation method. At the same time, the calculation in terms of electric field distribution was done by COMSOL multiphysics based on the finite element method. Bloch boundary conditions are used in the $x$-direction, whereas perfectly matched layers are applied in the $z$-direction. The material used for the grating is Al, whose refractive index is taken from Palik’s data.19 It is defined as TE (TM) polarization when the electric (magnetic) field component of the incident light is parallel to the $y$-axis, and the incident angle is denoted by $\theta$. In the calculation process, the top angle of the grating and the thickness of the substrate are set to 90 deg and 200 nm, respectively. This thickness of the Al substrate completely prevents the transmission of the light wave, so the absorption of the grating can be expressed as $A=1-R$, where $R$ is the reflection.

First, we investigated the absorption properties of Al right-angle TGGs at different scales. Based on the results of Ref. 18, the absorption loss of Al material under TM polarization will be maximized at an incident angle of 83.4 deg and an incident wavelength ($\lambda$) of 806.4 nm (the maximum absorption is 48.4%). In addition, Ref. 17 has demonstrated the existence of strong absorption in triangular-groove super-wavelength metal gratings with large blaze angles. Therefore, we set the incident angle to 83.4 deg and the investigated wavelength to 806.4 nm. The blaze angle of the grating is set comparable to the incident angle, which is 84 deg. Figure 2(a) shows the absorption of the TGG with a period ranging from 20,000 to 200 nm. This range of periods corresponds to gratings spanning super-wavelength to subwavelength scales. As seen in Fig. 2(a), the Al TGGs at different scales exhibit low absorption for TE-polarized light. However, the gratings at different scales exhibit enhanced absorption for TM-polarized waves, with the absorption peak exceeding 74%. The absorptions at the absorption peaks are all substantially enhanced relative to the absorption of the Al material itself. In addition, the Al TGG at the subwavelength scale achieves perfect absorption of TM-polarized light, as seen in Fig. 2(b). To our knowledge, this is the first time that continuous absorption enhancement has been observed from super-wavelength gratings to subwavelength gratings. In particular, the absorption spectrum under TM polarization shows an apparent fluctuation phenomenon, the intensity gradually increasing from the super- to sub-wavelength direction. This phenomenon is associated with resonance anomalies in gratings. The resonance anomaly is created as the incident wave excites the surface wave propagating along the grating surface; this excitation is known as the coupling of the propagating wave to the surface wave, and it can result in the surface absorption of the grating and radiate outward.20^,21 The fluctuation of the absorption curve originates from the resonance anomaly of the grating diffraction efficiency under TM polarization. Moreover, the resonance and fluctuation anomalies are located at the same place, which can be obtained approximately by the following equation20: $$-\sqrt{\frac{\epsilon (\omega )}{1+\epsilon (\omega )}}=\sin \theta +(m-1)\frac{\lambda }{d},$$(1)where, $\epsilon (\omega )=n^2-k^2$, $n$ and $k$ are the real and imaginary parts of the metal’s complex refractive index, respectively, and $m$ is the diffraction order. It should be mentioned that Eq. (1) is only an approximate equation since it does not address the groove profile and groove depth that determine the diffraction efficiency curve. We have selected three mutation points with periods of 19,780 nm (superwavelength grating), 807 nm (near-wavelength grating), and 404 nm (subwavelength grating) as examples for our calculations. At the wavelength of 806.4 nm, the $n$ and $k$ of Al are 2.8089 and 8.4019, respectively. Because the incident angle is 83.4 deg, $m$ is $-48$, $-1$, and 0 when $d$ is 19,780, 807, and 404 nm, respectively. Substituting these data into Eq. (1) yields Geomorphon on the left side of the equal sign and $-1.0043$, $-1.0051$, and $-1.0027$ on the right side of the equal sign, respectively, which are approximately equal.

Next, we further investigate the physical mechanism of the strong absorption of the TGG at different scales. Figure 3 shows the electric field distributions of the super-wavelength grating ($d=\mathrm{19,780}\mathrm{nm}$), the near wavelength grating ($d=807\mathrm{nm}$), and the subwavelength grating ($d=404\mathrm{nm}$) under TE and TM polarizations, respectively. As seen in Figs. 3(a)–3(c), the gratings at different scales have high reflective properties for TE-polarized light, and thus, most of the electric field energy is concentrated in the air above the gratings. However, in Figs. 3(d)–3(f), the surface of the Al grating at different scales under TM polarization can be seen to generate surface waves and undergo some transverse displacements, which is the phenomenon of SPR being excited.15^,22 Interestingly, the number of standing waves on the surface of the grating can be seen from the electric field diagrams to be approximately equal to the value of $d/\lambda$. Therefore, anomalously strong absorption of TM polarized waves by Al TGGs at different scales can be attributed to the presence of SPR on the grating surface to enhance the absorption by the Al material itself.

Further, we discuss the effect of blaze angle on the absorption properties of Al TGGs. The absorption of TM-polarized light by Al TGGs at different scales with the variation of blaze angle is calculated with other parameters constant, as shown in Fig. 4. At the subwavelength scale, the absorption of the Al TGG varies with the blaze angle from 1 deg to 89 deg is characterized by two absorption peaks and shows perfect symmetry. Among them, the TGG realizes perfect absorption at $\alpha$ of 5.9 deg and 84.1 deg, respectively. As the grating’s scale comes near wavelengths, the symmetry of the absorption curve is somewhat destroyed, but the two absorption peaks can still be seen (at $\alpha =4.2\mathrm{deg}$ and 84.6 deg). However, when the grating’s scale reaches the super-wavelength, the symmetry of the spectral lines is destroyed, with only one absorption peak at $\alpha =85.6\mathrm{deg}$. Overall, the change in the grating’s scale has a more significant effect on the absorption peaks at small blaze angles and a more minor effect on the absorption peaks at large blaze angles. In this case, the efficiency of the absorption peak at the large blaze angle is more than 80% for gratings of different scales. This shows that a large blaze angle is required for Al TGGs to achieve strong absorption at different scales. In addition, we speculate that the spectral symmetry breaking is related to the diffraction order of the gratings. In subwavelength gratings where only the 0th order exists, the spectra varying with the blaze angle tend to exhibit symmetry. As the scale of the grating increases, the number of diffraction orders increases, and thus, the symmetry of the spectrum is broken.

Furthermore, we note that the perfectly symmetric absorption of the subwavelength grating in Fig. 4 seems to coincide with the equivalence rule in grating theory. Therefore, we will demonstrate this by combining sinusoidal and rectangular gratings. First, we define the structures of sinusoidal and rectangular gratings in Fig. 5(a), where $d$ is the period, and $h_{\sin }$ and $h_{\mathrm{rcct}}$ are the sinusoidal and rectangular grating groove depths, respectively. In the equivalence rule, the face shape function $f(x)$ of the triangular, sinusoidal, and rectangular gratings can be expressed as23$$\mathit{f}(x)=\sum _{n=1}^{\infty }{b}_n\sin \frac{2\pi }{d}nx.$$(2)

Based on the equivalence rule, gratings with different face shapes will have similar efficiency with the same first-order Fourier coefficients.23 The first-order Fourier coefficient of the sinusoidal grating in Fig. 5(a) is $$b_1=\frac{h_{\sin }}{2}.$$(3)The first-order Fourier coefficient of the rectangular grating is $$b_1=\frac{2h_{\mathrm{rect}}}{\pi }.$$(4)The first-order Fourier coefficient of the triangular grating in Fig. 1(a) is expressed as $$b_1=\frac{d}{{\pi }^2}·[\tan \alpha -\tan (\phi +\alpha )]·\sin \left[\pi \frac{\sin (\phi +\alpha )\cos \alpha }{\sin \phi }\right].$$(5)Associating Eqs. (3)–(5), when the first-order Fourier coefficients of sinusoidal, rectangular, and triangular gratings are the same, the following relation will exist $$\frac{h_{\sin }}{d}=\frac{4h_{\mathrm{rect}}}{\pi d}=\frac{2}{{\pi }^2}·[\tan \alpha -\tan (\phi +\alpha )]·\sin \left[\pi \frac{\sin (\phi +\alpha )\cos \alpha }{\sin \phi }\right].$$(6)

Based on Eq. (6), the relationship between $h_{\sin }/d$, $h_{\mathrm{rect}}/d$, and $\alpha$ is given in Fig. 5(b). The curve in the figure is calculated for $\phi =90\mathrm{deg}$. As shown in Fig. 5(b), the curves are perfectly symmetrically distributed with $\alpha =45\mathrm{deg}$ as the symmetry axis. That is, two triangular gratings with blaze angles $\alpha$ and 90 deg–$\alpha$, respectively, have the same first-order Fourier coefficients and thus should have the same efficiency. This result is consistent with the curves of the subwavelength gratings in Fig. 4 and the reciprocity theorem and energy conservation. Moreover, we give the variation of absorption with groove depth for sinusoidal and rectangular gratings under TM polarization in Figs. 5(c) and 5(d), respectively. As seen in Figs. 5(c) and 5(d), the absorption curves of the two gratings first show a perfect absorption peak. Then, with the increase of the groove depth, the absorption peaks of the sinusoidal and rectangular gratings appear with quasi-periodic and standard periodicity rules, respectively, and the efficiency of the absorption peaks gradually decreases. Because the equivalence rule is more applicable when $h/d$ is relatively small, we only use the equivalence rule to analyze the first peaks in Figs. 5(c) and 5(d). In Fig. 4, the subwavelength triangular grating achieves perfect absorption at $\alpha$ of 5.9 deg and 84.1 deg, respectively. In Fig. 5(b), $h_{\sin }/d=0.065$ and $h_{\mathrm{rect}}/d=0.051$ are obtained when $\alpha$ is 5.9 deg (84.1 deg). In Figs. 5(c) and 5(d), the first perfect absorption peaks of the sinusoidal and rectangular gratings correspond to $h_{\sin }/d$ and $h_{\mathrm{rect}}/d$ of 0.058 and 0.046, respectively. This result is in agreement with the analytical results of the equivalence rule. The equivalence rule of grating grooves has previously been applied only to the gratings with only two diffraction orders, i.e., near-wavelength gratings. Here, we extend the equivalence rule of grating grooves to subwavelength gratings for the first time.

To understand the mechanism of the absorption peaks in Figs. 5(c) and 5(d), we calculated the electric field distribution at the top four peaks, as shown in Fig. 6. In Figs. 6(a) and 6(e), it can be seen that the sinusoidal and rectangular gratings with shallow grooves share similar electric field distribution characteristics with the triangular grating [Fig. 3(c)], with SPR present on the grating surfaces. As the groove depth increases, the SPR’s intensity on the surfaces of both the sinusoidal and rectangular gratings becomes weaker. In addition, increasing groove depth induces FP cavity resonance in the grating groove,24^,25 as shown in Figs. 6(b)–6(d) and 6(f)–6(h). According to the FP model, when the groove depth satisfies the phase-matching condition, the absorption will take to reach the maximum value, which is26$$2k_0{\int }_0^{-h}\mathrm{Re}[n\mathrm{eff}(z)]\mathrm{d}z+\arg (r_a)+\arg (r_m)=(2q+1)\pi ,$$(7)where $\arg (r_a)$ and $\arg (r_m)$ denote the phase change of the fundamental mode in the groove when it is reflected from the air interface at the top of the groove and the metal interface at the bottom of the groove, respectively, $k_0$ is the number of waves in the vacuum, $n_{\mathrm{eff}}$ is the equivalent refractive index of the fundamental mode, and $q$ denotes an arbitrary integer that ensures that the depth of the groove is a positive value. The increase of the groove depth will make the real and imaginary parts of $n_{\mathrm{eff}}$ of sinusoidal grating increase at the same time, so the groove depth to satisfy the resonance effect will show a quasi-periodic variation (the peak spacing becomes smaller with the increase of $h_{\sin }$). In contrast, the $n_{\mathrm{eff}}$ of the rectangular groove is independent of the depth, so the groove depth to satisfy the FP resonance condition shows a standard periodic variation.

In summary, the absorption properties of metal gratings at different scales under grazing incidence are studied in this paper. It is shown that the Al triangular grating under grazing incidence has perfect absorption of TM polarized waves at the subwavelength scale. At the same time, there is a continuous strong absorption of more than 74% in the near wavelength to the superwavelength scale range. Based on the electric field distribution results, the strong absorption phenomenon is caused by the SPR excited by the surfaces of gratings at different scales. Furthermore, we find the novel phenomenon that the absorption of the subwavelength Al TGG shows perfect symmetry to the blaze angle. This symmetry is gradually broken as the grating period scale shifts from subwavelength to superwavelength. Moreover, combining the absorption properties of sinusoidal and rectangular gratings, we extend the equivalence rule to subwavelength gratings and explain the symmetric absorption properties. These findings provide a new perspective on the design of metal grating absorbers and will help improve the designability of the devices and reduce the difficulty of processing the devices.

Biographies of the authors are not available.