1. Introduction

Echelle grating, or echelle [1,2], with a coarse groove density but nanopositioning precise under control of laser interferometric control [3–5], is ruled first by Harrison for high spectral resolution and dispersion by applying high diffraction orders and large sizes of gratings. The theoretical resolution of the echelle can be over 10⁶, so it is a kind of high-resolution dispersion element second only to the Fabry–Pérot interferometer. So echelle gratings are mostly applied in astronomy spectrometry [6], inductively coupled plasma (ICP) spectrometers [7,8], thermospheric imaging [9], and laser wavelength tuning under Littrow conditions, especially for spectral metrology of excimer laser lithographic light sources at 193 nm [10], and optical waveguide for augmented-reality display [11]. Broadband blaze characteristic from UV to IR is another advantage because of using high diffraction orders for different wavelengths [12]. Diffraction efficiency is one of the most important parameters of the echelle, and to improve it, different groove shapes are investigated. Echelle, with the right apex angle and a specified blaze angle, can reach perfect blaze both for TE and TM polarizations when choosing a specified wavelength and incident angle [13,14]. It is found that an echelle can achieve high diffraction efficiency for wideband if it contains a series of reflective facets with a specific tilt angle that is located far from the nonworking facet of the grating and have a deep groove depth [15,16]. The polarization effect of the echelle is deeply studied, which shows that increasing the groove depth can suppress the polarization degree and improve the diffraction efficiency [17]. A dual-facet and a four-facet, or called multifacet, echelle is also presented for intensity broadening based on the single-facet echelle that avoids the weak spectral signal of single-facet echelle grating (SFEG) spectrometers [18]. High diffraction efficiency corresponds to low scatter light and absorption, and scatter light depends on the micro-nano machining precision. In terms of absorption, the Brewster effect of shallow metal gratings was first predicted and experimentally verified in the 1970s [19,20], i.e., under specific conditions, shallow metal gratings supporting only one particular diffraction order can completely absorb incident monochromatic light. Subsequently, Popov et al. theoretically explained the absorption anomalies at grazing incidence for metallic gratings which supports two diffraction orders [21], and discussed resonance as well as non-resonance types of absorption anomalies for metallic gratings in conjunction with energy flow distributions [22]. However, the above and most of the current studies on absorption anomalies in metal gratings are based on subwavelength gratings (usually only 0th order or only 0th and 1st order diffracted light) [19–23]. Almost none of the research has been studied on the absorption of echelle with a very long grating period over the incident wavelength as far as we know. Just one paper shows that the Au film on the echelle grating structure will lead to an attenuation of the TM polarization due to the coupling of the field into a surface wave on the un-blaze facet of the groove shape [24].

In this paper, we find a strong absorption enhancement of energy coming from the grating groove structures and materials for the echelle with a high incident angle and a large blaze angle in TM polarization. The resonance between the grating anomaly and the pseudo-Brewster effect results in the appearance of surface plasmon polariton (SPP) and strong absorption. These results will guide the future design of echelle gratings in case of high energy loss when the light incidents with a high angle for high resolution and expand the application areas of echelle gratings.

2. Grating structure and simulation method

Figure 1 shows the structure of the studied echelle grating with a right-angle triangular groove. As shown in Fig. 1, the structure consists of a grating layer and a substrate. Here, the grating layer is arranged periodically along the x-axis, while the groove direction is parallel to the y-axis. The structural parameters to be considered include the period d, the blaze angle β, the apex angle α, and the groove depth h. In the research process, the period d and the apex angle α are set to constant values of 12.5 µm (80-line/mm) and 90°, respectively. The material of the entire grating consists of aluminum with the refractive index taken from Palik [25], and the medium for both the grating groove and the incident layer is air. The z-axis is the normal direction of the echelle grating, and the incident light source is a monochromatic plane wave (the angle of incidence is denoted by θ). In addition, the electric field is parallel to the y-axis when the incident light is TE polarization, while the magnetic field is parallel to the y-axis when the incident light is TM polarization.

 

Fig. 1. Schematic diagram of the echelle grating with right-angle triangular groove.

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The numerical calculations for this work are carried out using PCGrate software based on the boundary integral equation method. PCGrate is a one-dimensional grating simulation software that enables the calculation of diffraction efficiency and associated near-field parameters for phase diffraction gratings or relief gratings. Due to the metal substrate being thick enough, no light penetrates the grating structure. The absorption (A) of the grating can therefore be expressed as A = 1 - R, where R is the reflectivity.

3. Simulation result and discussion

3.1 Absorption enhancement in echelle grating by the large incident angle

As the period and apex angle is constant, only the incident wavelength (λ), the incident angle, and the blaze angle need to be considered in the study of grating absorption properties. Of these three factors, incident wavelength and incident angle dominate the incident light field to some extent, often determining the scenario and field of use for the grating. First, we set the blaze angle to 85° and calculated the variation of the grating's absorption with the incident angle for different incident wavelengths, which is shown in Fig. 2. It can be observed that the grating exhibits low absorption for three different wavelengths of TE-polarized light at the incidence angles from 0° to 89°. And there is a small absorption peak near the incident angle of 75°, which moves towards the small incident angle as the incident wavelength increases. In contrast to TE polarization, the absorption curves of the grating under TM polarization are very different. As can be seen from the figure, the absorption curves for different wavelengths at TM polarization show the same trend: as the angle of incidence increases, the absorption of the grating gradually increases, reaching a maximum near the incident angle of 80° and starting to decrease. In particular, the grating has anomalous absorption peaks of varying degrees in the absorption curves at different wavelengths, the intensity of which becomes progressively more pronounced as the angle of incidence increases. In addition, the absorption performance of the grating increases at most incidence angles as the incident wavelength increases. Especially the angular bandwidth at high absorption will increase with increasing wavelength. For example, the angular bandwidth of the grating at the wavelengths of 400 nm, 600 nm, and 800 nm is 9.5°, 14.6°, and 17.7°, respectively, when absorption is more than 50%. To summarize the above analysis, the target grating has absorption enhancement at large incident angles under TM polarization, and its absorption curve exhibits many anomalous absorption peaks. These phenomena are present at different incident wavelengths, although the overall absorption performance of the grating may vary at different wavelengths.

 

Fig. 2. Absorption versus incident angle at incident wavelengths of 400 nm, 600 nm, and 800 nm for the echelle grating with β = 85°.

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3.2 Absorption enhancement in echelle grating with the large-blazed angles

Based on the above analysis, the incident wavelength may have little effect on absorption enhancement in echelle grating. To further determine whether the absorption enhancement occurs only at large incident angles and is affected by other conditions, we set the incident wavelength to 300 nm and studied the effect of different incident angles and blaze angles on the absorption properties of the grating. Figure 3(a) shows the absorption as a function of blaze angle at the incident angles of 0°, 40°, and 80°, respectively. As shown in Fig. 3(a), the grating exhibits low absorption for TE-polarized light at different incident and blaze angles. Under TM polarization, the absorption curves at different incident angles have a distinct absorption peak, and the peak position shifts towards the large blaze angle as the incident angle increases. Notably, the absorption curves of the grating at both polarizations are symmetrically distributed at the ends centered on the 45° blaze angle for an incident angle of 0°. Due to the apex angle of 90°, the blaze angle and the other angle of the grating structure are complementary angles to each other. In other words, when the blaze angle is changed from 30° to 60° it can be seen as a grating with a blaze angle of 30° rotating 180° around the z-axis while the structure remains unchanged. Therefore, when the light source is normally incident, the absorption curves of the structure under both polarizations are symmetrically distributed and centered on a 45° blaze angle. In addition, the absorption performance of the grating at TM polarization is significantly improved compared to TE polarization, where the maximum absorption increases with increasing incident angle. Significantly when the incident angle increases to 80°, the absorption enhancement appears around the blaze angle of 87°.

 

Fig. 3. (a) Absorption versus blaze angle at incident angles of 0°, 40° and 80° for the echelle grating with λ = 300 nm. (b) Absorption versus blaze angle at incident wavelengths of 400 nm, 600 nm, and 800 nm for the echelle grating with θ = 80°.

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The above results again demonstrate an absorption enhancement of the grating at large incidence angles and that this phenomenon may need to be accompanied by a large blaze angle. To confirm this, we set the incidence angle to 80° and provide the absorption versus blaze angle for the grating at different wavelengths in Fig. 3(b). As shown in Fig. 3(b), the overall absorption characteristics of the grating at TE polarization are similar to the above calculations, showing low absorption characteristics for different incident wavelengths and blaze angles. For TM polarization, the absorption at different wavelengths differs only in intensity. The absorption intensity increases with increasing incident wavelength at large blaze angles. Notably, the absorption curves of the grating at different incident wavelengths have the same trend, which is consistent with the results in Fig. 2. Specifically, as the blaze angle increases from 1° to 89°, the absorption first decreases, then begins to increase after reaching a minimum near the 45° blaze angle. Finally, the absorption will start to go down after a phenomenon of high absorption around the 85° blaze angle. This result further confirms that absorption enhancement in echelle grating is required to be accompanied by a large blaze angle and that it is insensitive to wavelength. To better illustrate this conclusion, we set the incident wavelength and blaze angle to 800 nm and 85° respectively, and present the absorption of the grating for incident angles from −89° to 89° in Fig. 4. One should mention that the blaze angle of the grating can be considered to change from 85° to 5° when the incident angle changes from greater than 0 to less than 0 (as shown in the inset in Fig. 4). From Fig. 4, the grating at a small blaze angle shows no strong absorption of TM-polarized light. Furthermore, an interesting phenomenon can be observed in the figure, where the absorption curve of the grating is smooth at a small blaze angle. What is different is that the grating has multiple absorption peaks in the absorption curve at a large blaze angle. This phenomenon may be since echelle grating with large blaze angles can have grating anomalies as the incident angle changes.

 

Fig. 4. Absorption as a function of incident angle for the echelle grating with β = 85° and λ = 800 nm.

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3.3 Broadband absorption enhancement in echelle grating

The above analysis demonstrates that the observed absorption enhancement can occur in multiple wavelengths. However, it has yet to be discovered whether this phenomenon can occur in a continuous spectrum. Therefore, we set the incident angle and blaze angle to 80° and 85° respectively, and calculated the absorption in the spectrum from 300 nm to 900 nm, which is shown in Fig. 5. As can be seen in Fig. 5, the grating continues to exhibit low absorption for TE-polarized light throughout the spectrum but shows a broadband absorption for TM-polarized light. The grating absorbs over 60% of TM-polarized light at wavelengths from 369 nm to 900 nm, with a maximum absorption close to 90%. This result confirms that the found phenomenon of absorption enhancement exists within a continuous broad spectrum. Furthermore, a fluctuation in the absorption curve of the grating under TM polarization can be observed in this figure, and its intensity gradually increases in the direction of the long wavelength. This phenomenon is related to the resonance anomaly of the grating. A resonance anomaly is formed when the incident wave excites a surface wave that propagates along the surface of the grating; this excitation is called coupling of the propagating wave to the surface wave, which can lead to the surface absorption of the grating and radiate out [26,27]. A fluctuation in the absorption curve comes from the resonance anomalies of the diffraction efficiencies of the grating under TM polarization. And the positions of fluctuation and resonance anomalies are the same, which can be approximately concluded from the following equation [26]:

 

Fig. 5. Absorption as a function of incident wavelength for the echelle grating with β = 85° and θ = 80°.

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3.4 Reasons for anomalous absorption of TM-polarized light in echelle grating

To understand the effect of the increase in light absorption by this echelle grating with a large blaze angle, a comparison of the absorption of the Al grating and the Al mirror at different incident angles is given in Fig. 6. As shown in Fig. 6, the absorption of the Al mirror under TM polarization increases gradually with increasing incident angle and reaches a maximum (47.86%) at an incident angle of 83.5° before starting to decrease rapidly. This angle is also defined as the pseudo-Brewster angle θ_B, referring to the angle of incidence at which a metallic conductor will reflect the least amount of TM-polarized waves. Compared to the Al mirror, the absorption of Al grating is improved over almost the incident angle between 0° and 89°, especially at large incident angles. Among them, the Al grating's highest absorption (87.42%) is improved by almost 40% relative to the Al mirror. We also note an interesting phenomenon in the plot where the absorption curves of the Al grating and the Al mirror at TM polarization show a similar trend, and the maximum absorption of both occurs at large incident angles. It is, therefore, possible that the absorption enhancement of the Al grating may be related to the absorption properties of Al itself.

 

Fig. 6. Absorption versus incident angle for Al mirror and Al grating with β = 85° at an incident wavelength of 800 nm. The corresponding angles of incidence at θ₁, θ₂, θ₃ and θ₄ are 83.5°, 54.2°, 71.4° and 78.3°, respectively.

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To further investigate the physical mechanism of the absorption enhancement by the Al grating, the electric field distributions of the Al mirror at θ₁ and Al grating at θ_2-4 under TE and TM polarizations, respectively, are given in Fig. 7. Here, θ₁ is the pseudo-Bruster angle of the Al mirror (θ_B=θ₁, specific location of θ_1-4 is shown in Fig. 6). In Fig. 7(a), the Al mirror reflects almost all the TE-polarized waves into the air at θ₁; thus it can be seen that the air above the Al mirror surface gathers most of the electric field energy. In contrast, it can be seen in Fig. 7(b) that the electric field energy of the Al mirror under TM polarization is mainly confined to the metal surface. This is because the TM-polarized light at the pseudo-Bruster incidence angle can interact better with the Al mirror interface compared to other incidence angles [28]. As seen in Figs. 7(c, e, g), Al grating has an excellent reflection of TE-polarized light at θ_2-4 angles of incidence, making most of the electric field energy distributed mainly in the air. Compared with TE polarization, the electrical field distribution of Al grating under TM polarization is very different. As shown in Figs. 7(d, f, h), when the TM-polarized light is incident at angles of θ_2-4, respectively, surface waves are generated on the surface of the Al grating and undergo a certain transverse shift, which is the phenomenon that the SPP is excited [29]. In which, the intensity of the surface wave on the surface of the Al grating is gradually increased from Fig. 7(d) to Fig. 7(f) and Fig. 7(h) in turn. This indicates that the intensity of the SPP increases from the incident angle θ₂ to the incident angle θ₄. This phenomenon can be attributed to the fact that as the incident angle gets closer to the pseudo-Bruster angle, more TM-polarized light can interact with the metal interface, resulting in further strengthening of the SPP on the grating surface. The following conclusions can be obtained by combining the previous analysis and these electric field distribution phenomena. Under TM polarization, the surface of the echelle grating with a large blaze angle will exist SPP, thus achieving the absorption enhancement effect on the incident light. And this absorption enhancement is progressively more potent as the incident angle increases and reaches its strongest near the pseudo-Brewster angle. In addition, it can be seen from Fig. (6) that the absorption peaks caused by the grating anomaly appear in the absorption curve with the change of the incident angle. In particular, when these grating anomalies occur near the pseudo-Brewster angle, they will be coupled to the strong SPP of the grating structure, which further enhances the light absorption effect of the grating.

 

Fig. 7. Electric field distribution of the Al mirror under (a) TE and (b) TM polarizations when the incident angle is θ₁. Electric field distribution of the Al grating under (c, e, g) TE and (d, f, h) TM polarizations when the incident angle is θ₂, θ₃, and θ₄, respectively. The observed ranges in the x and z directions are −12500 nm to 12500 nm and 0 nm to 3000 nm, respectively.

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Hence, the absorption enhancement of the echelle grating under large blaze and high incident angles can be attributed to the combined coupling of the SPP and grating anomaly. This explains why in Fig. 6 the Al grating shows absorption enhancement only in the absorption curve under TM polarization but not under TE polarization.

Next, we investigate the relationship between the absorption enhancement of the Al grating and the absorption properties of the metal itself. Figure 8(a) presents the absorption of the TM polarization wave by the Al mirror as a function of the incident angle at the incident wavelengths of 400 nm, 600 nm, and 800 nm. As shown in Fig. 8(a), the Al mirror's pseudo-Brewster angle is near 80° at all three wavelengths. And the absorption efficiency of the Al mirror gradually increases as the wavelength increases, with the most pronounced increase, particularly around the pseudo-Brewster angle. Of these, the enhancement of the absorption by the Al mirror is not significant as the wavelength increases from 400 nm to 600 nm within an incidence angle of 0-50°. However, as the wavelength is further increased to 800 nm, there is a significant enhancement in the absorption of the mirror. Combining Fig. 2 and Fig. 8(a), it can be observed that the absorption curves of the Al grating and the Al mirror under TM polarization show the same trend for the same incident wavelengths. Furthermore, the maximum absorption of Al grating for different incident wavelengths occurs around the pseudo-Brewster angle. As the incident wavelength increases from 400 to 800 nm, the grating and mirror absorption show the same trend for small incident angle ranges. At large incident angles, the angular bandwidth of the grating's absorption is also consistent with that of the mirror, which increases with increasing incident wavelength. Thus, the absorption of the Al grating with a large blaze angle at different incident angles is to some extent determined by the absorption properties of the Al itself (absorption of the Al mirror at different incident angles). In other words, the maximum absorption of the Al grating is influenced to a certain degree by the maximum absorption of Al itself (the absorption at the pseudo-Brewster angle). Furthermore, the maximum absorption of Al itself does not get larger and larger as the wavelength increases. Figure 8(b) shows the pseudo-Brewster angle of the Al mirror in the wavelength of 700-900 nm and the absorption at the pseudo-Brewster angle. As shown in Fig. 8(b), the pseudo-Brewster angle of the Al mirror remains at about 83° for the wavelengths from 700 nm to 900 nm. The absorption of the Al mirror at the pseudo-Brewster angle first increases with increasing wavelength and then decreases after reaching a maximum of around 800 nm. This phenomenon coincides with the calculations in Fig. 5. The peak in the absorption curve increases and then decreases as the wavelength increases from 700 nm to 900 nm, with the inflection point occurring around the 800 nm wavelength. These results further confirm the above inference. In addition, we can screen the target wavelengths based on the above conclusions before the design and optimization of the grating.

 

Fig. 8. (a) Absorption of Al mirror at TM polarization as a function of incident angle for incident wavelengths of 400 nm, 600 nm, and 800 nm, respectively. (b) The pseudo-Brewster angle θ_B of the Al mirror at incident wavelengths from 700 nm to 900 nm and the absorption of the Al mirror at θ_B.

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Finally, to investigate whether the above results can be extended to other metals, the absorption versus incident angle for the grating with different metallic materials is presented in Fig. 9. The refractive indices of Cu, Au, and Ag at 800 nm are 0.2543 + i4.9255, 0.1803 + i5.1287, and 0.1421 + i5.2888, respectively, while the refractive indices at other wavelengths can be obtained from the Ref. [25]. As shown in Fig. 9, the absorption enhancement found for the echelle grating with a large blaze angle holds for Al, Cu, Au, and Ag, and the highest absorption occurs near the pseudo-Brewster angle. The location of the anomalous absorption peaks in the absorption curves is essentially the same for Al, Cu, Au, and Ag. Moreover, the absorption behaviors of the four metal gratings at different incident angles are consistent with the metals’ absorption, which agrees with the above conclusions.

 

Fig. 9. Absorption of Al, Cu, Au, and Ag mirrors and Al, Cu, Au, and Ag gratings with β = 85° at an incident wavelength of 800 nm as a function of incident angle for TM polarization

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4. Conclusion

In summary, the absorption characteristics of the Al echelle grating with different structural parameters and incidence conditions have been investigated. The results show that the Al grating with a large blaze angle will show an absorption enhancement effect for TM-polarized light, and the intensity will be enhanced with the increase of the incident angle. At an incident wavelength of 800 nm, for example, the Al grating can achieve the highest absorption of 87.42%, nearly 40% higher than that of an Al mirror without the micro-nano structure. More interestingly, this absorption enhancement will exist within a continuous broad spectrum, achieving over 60% absorption of TM-polarized light in the spectrum from 369 nm to 900 nm. In addition, the absorption enhancement was found to be caused by the SPP through analysis of the electric field distribution of the Al grating and Al mirror. In addition, we observe that the grating anomaly appears on the absorption curve of the grating, and the absorption enhancement effect of the grating is further enhanced when the grating anomaly is coupled with this SPP. On this basis, by analyzing the absorption curves of the Al grating and the Al mirror at different wavelengths, the absorption enhancement effect of the Al grating is related to the absorption of its metal itself. Therefore, the coupling between the grating anomaly and the grating's SPP is most potent when both are present near the pseudo-Brewster angle, resulting in high grating absorption. Moreover, we have found this absorption enhancement effect in Cu, Au, and Ag metallic materials. These results will guide the future design of echelle gratings in case of high energy loss when the light incidents with a high angle for high resolution and expand the application areas of the echelle.