## Abstract

Spectral control of the emissivity of surfaces is essential for efficient conversion of solar radiation into heat. We investigated surfaces consisting of sub-wavelength V-groove gratings coated with aperiodic metal-dielectric stacks. The spectral behavior of the coated gratings was modeled using rigorous coupled-wave analysis (RCWA). The proposed absorber coatings combine impedance matching using tapered metallic features with the excellent spectral selectivity of aperiodic metal-dielectric stacks. The aspect ratio of the V-groove can be tailored in order to obtain the desired spectral selectivity over a wide angular range. Coated V-groove gratings with optimal aspect ratio are predicted to have thermal emissivity below 6% at 720K while absorbing >94% of the incident light. These sub-wavelength gratings would have the potential to significantly increase the efficiency of concentrated solar thermal systems.

©2010 Optical Society of America

## 1. Introduction

The parabolic trough is one of the most common concentrated solar thermal systems. In this configuration sunlight is concentrated by an array of parabolic mirrors onto a heat collection element (HCE) which runs along its focal point. The HCE consists of a steel tube which is coated with a solar selective absorber coating. An evacuated glass envelope surrounds the steel tube in order to reduce thermal losses and to extend the lifetime of the absorber coating. Incoming solar radiation is absorbed by the HCE absorber coating and converted into heat. The collected heat is extracted using a heat transfer fluid (HTF) and can be converted into steam in order to drive a turbine and generate electricity. The Carnot efficiency, and thus the maximum temperature of the working fluid, sets a thermodynamical limit to the overall efficiency of the system. Parabolic troughs have concentration ratios of ~80, and operate at temperatures up to 660K [1,2]. The working temperature is currently limited by the thermal stability of the available absorber coatings and the cracking temperature of the synthetic oil used as HTF. Cracking oil produces hydrogen which can permeate the steel tube and lead to accelerated degradation of the absorber coating. Alternative working fluids such as molten salts are being pursued to allow for operation at higher temperatures. To achieve these higher temperatures, an increased spectral selectivity of the absorbing coating on the HCE is required to prevent re-emission of the absorbed energy as infrared (IR) radiation, while ensuring that most photons are absorbed.

Numerous selective coatings have been deployed including intrinsic absorbers, semiconductor-metal tandems and cermets [3–6]. Recent advances in nanofabrication have led to the exploration of spectral selectivity in one-dimensional (1D) periodic multilayer stacks [7–13], 1D gratings [14–18], two-dimensional (2D) gratings [19–22], photonic cavities [23] and three-dimensional (3D) photonic crystals [24–26]. In all these approaches the absorbing medium is nanostructured with feature sizes of the order of 100nm to 1μm in order to tune the response of the medium in the visible and near-infrared (NIR) spectral range. Structuring the surface of refractory metals as 1D and 2D gratings has been suggested to enhance the performance in solar thermal systems [14–16,20]. Here, we study the performance of sub-wavelength V-groove metal gratings coated with metal-dielectric stacks. We show that the combination of a nanostructured surface with an aperiodic metal-dielectric coating results in high performance solar thermal absorbers for operating at 720K.

## 2. Ideal absorber

Controlling this parasitic thermal emission from the absorber surface is crucial to increasing the overall conversion efficiency of solar thermal systems. An ideal absorber coating behaves as a perfect absorber (${\alpha}_{solar}$=1) for wavelengths shorter than a cutoff wavelength${\lambda}_{c}$to optimize light absorption, and suppresses thermal emission (${\epsilon}_{thermal}$=0) for wavelengths longer than${\lambda}_{c}$to minimize losses through infrared (IR) emission [27]. In accordance with Kirchhoff’s law we assume here that the directional spectral absorptivity$\alpha (\theta ,\lambda )$is equal to the directional spectral emissivity$\epsilon (\theta ,\lambda )$for a system in thermal equilibrium. In order to determine${\lambda}_{c}$and to evaluate the performance of solar selective coatings during optimization, a merit function needs to be defined. The following merit function, *F,* was previously suggested [27]:

The first factor${\alpha}_{solar}$is the fraction of the solar irradiance (AM1.5) absorbed by the stack at normal incidence. Solar absorptivity at normal incidence was chosen because for relatively low concentration factors, the incidence angle of the solar radiation on the absorber coating will be close to normal. This is in accordance to literature where absorptivity is typically measured and cited for normal incidence [3]. In the second factor, ${\epsilon}_{thermal}(T)$ is the integrated thermal emissivity at temperature *T*. This is given by the ratio of the emittance from the surface of the selective coating over the emittance from a perfect blackbody (BB) radiator at the same temperature *T*. This merit function *F* was chosen because it is independent of geometry and operation [27]. The geometry of the design determines the concentration factor, the uniformity and the angular distribution of the solar radiation on the absorber. The relative importance of thermal emissivity and solar absorptivity is also determined by system operation. For example, if the molten salts are pumped through the HCE at night time to avoid solidification, achieving a low HCE emissivity is a priority. Therefore the product in Eq. (1) results in a good performance evaluation when no assumptions are made about geometry and operation.

Based on the merit function *F*, an ideal cutoff wavelength can be determined for a specific operation temperature. Next generation parabolic solar troughs with molten salts are expected to operate around 720K [1,2]. Therefore, the spectral performance will always be evaluated at 720K in the following sections. According to Eq. (1) an ideal absorber for operation at 720K has a cutoff wavelength${\lambda}_{c}$=2.24µm.

## 3. Aperiodic metal-dielectric stacks

Recently, we proposed aperiodic one-dimensional metal-dielectric multilayer stacks as selective emitters in thermophotovoltaics [28] and selective absorbers for concentrated solar thermal (CST) applications [27]. The metals inside the stacks act as emitters/absorbers and the dielectrics as optical spacers, creating interference effects that enhance emission/absorption in a desired spectral range. We first summarize briefly the results obtained in [27] because in the next section, we will analyze whether the performance of the aperiodic multilayer stacks proposed in [27] can be further improved by deposition onto metallic V-groove gratings. The planar aperiodic stacks described in [27] were modeled using a standard transfer matrix method (TMM) [30] based on complex dielectric permittivity data at room temperature obtained from the literature [31,32]. Dielectric dispersion data at room temperature was used because of a lack of broadband data at elevated temperatures. It should be noted that changes in dielectric properties upon heating modify the optical path lengths as well as the amplitude and phase relationships at every interface. Generally the extinction coefficient of conductors increases with increasing temperature [33]. In contrast the emissivity of dielectrics can either increase or decrease upon heating depending on the material used. It is known from theory that small variations in extinction coefficient have only minor influence on spectral properties in spectral regions of high reflectance (IR) [34]. However we must point out that deviations in extinction coefficients might influence the position of the cutoff between the spectral region of high absorption and the region of high reflectance. In addition to dielectric properties, layer thicknesses will also vary under thermal expansion. Since the coating is typically less than a micron thick, thermal expansion will be largely dominated by the substrate. We have investigated the influence on the spectral performance when individual layer thicknesses are varied up to 10% and found that spectral performance or merit function *F* shows variations typically below 1%. It is however outside the scope of this manuscript to study and quantify the effect of heating on the dielectric properties and geometry of the structure and therefore these variations were not included in the optical modeling.

Optimization was performed using the needle optimization method [35,36] with Eq. (1) as performance merit. Because of their inherent spectral selectivity and stability at elevated temperatures, Molybdenum (Mo) and Tungsten (W) were used for the metal substrate and the thin metal layers in the stacks. For the dielectric spacer layers, Magnesium Fluoride (MgF_{2}) (*n*=1.37 at *λ=*1 µm) and Titanium Dioxide (TiO_{2} - Rutile) (*n*=2.75 at *λ=*1 µm) were selected in order to achieve high refractive index contrast ($\Delta n$=1.38 at *λ=*1 µm). In Fig. 1
the spectral absorptivity at normal incidence is shown for stacks composed of layers of Mo, MgF_{2} and TiO_{2} (a) and layers of W, MgF_{2} and TiO_{2} (b). The stacks have respectively 5, 7, 9 or 11 layers with layer thicknesses varying from 5 to 100 nm. When determining the number of layers in a stack, the substrate also counts as one layer. The dotted black line in Fig. 1 represents the spectral absorptivity of the ideal absorber at T=720K with a cutoff wavelength${\lambda}_{c}$=2.24µm. Increasing the number of layers leads to improved spectral absorptivity, more closely approximating the spectrum of the ideal absorber. The spectral behavior of the stacks represented in Fig. 1 will now be studied when used as coatings on top of sub-wavelength V-groove gratings.

## 4. Sub-wavelength V-groove gratings

#### 4.1 Introduction

It was recently shown that tapered metallic sub-wavelength gratings (SWGs) can be optimized to achieve excellent spectral selectivity [21,29]. The tapering increases absorption because it provides an impedance matching between free space and the bulk metallic absorber for wavelengths comparable to the period of the grating. For electromagnetic waves with a wavelength much larger than the period of the grating, the impedance matching and thus the absorption enhancement, is less pronounced. Therefore, by tuning the period of the grating and the aspect ratio of the tapering the desired spectral selectivity can be obtained. Here, we analyze whether the performance of aperiodic multilayer stacks, introduced above, can be further improved by deposition onto V-groove gratings. Such structures are expected to combine the performance advantages of aperiodic metal-dielectric stacks and tapered metal gratings, and may be economical to fabricate using a vacuum deposition process on grooved substrates. The resulting structure is illustrated in Fig. 2
. The structure is characterized by its period *a*, the angle${\theta}_{GR}$of the V-groove, the material and the thickness of each layer of the aperiodic stack. All layer thicknesses are measured normal to the faces of the grooves.

The melting temperatures of nanostructured materials are known to be lower than for bulk materials [37]. 2D and 3D nanostructures have larger surface to volume ratios, making them more prone to surface diffusion, especially at sharp edges, such as in V-groove and pyramid gratings. This could lead to a change in shape at temperatures well below the bulk melting temperature, which in turn would lead to a degradation of the spectral properties. Schlemmer *et al.* [38] have observed this effect. However they have shown that depositing a coating on top of nanostructured metal gratings can enhance the thermal stability of the grating. This is an additional motivation to study the effect of coatings on top of nanostructured refractive metal surfaces.

#### 4.2 Rigorous coupled-wave analysis

The emissivity of aperiodic stacks deposited into these V-grooves was evaluated using rigorous coupled-wave analysis (RCWA) based on the original algorithm [39] and generalized for simulating arbitrary aperiodic layered 2D photonic crystals. The enhanced transmittance matrix method [40] was used to enforce boundary conditions at the interface of two grating layers. In order to model the structure, 3D space was discretized into boxes with a defined dielectric permittivity, as shown schematically in Fig. 3
. The complex dielectric permittivity data for the materials simulated were obtained from the literature [31,32]. The step size in the Y direction was set to 0.5 nm in all simulations. Because we are dealing with a 1D grating, no spatial dependence of the dielectric permittivity in the X direction was incorporated. In the Z direction the structure was sliced into a number of grating layers *L*. In order to obtain good convergence, the number of grating layers *L* was increased with the height of the structure in the following way:

A convergence analysis was performed to check the number of diffraction orders or modes that needed to be retained in the RCWA simulations. The result of the convergence analysis for an uncoated Mo V-groove with ${\theta}_{GR}$=45° is shown in Fig. 4
. The change in absorptivity |$\Delta \alpha $| when the number of retained diffraction orders *m* is increased stepwise with $\Delta m$=7 is shown for various wavelengths. This convergence analysis was performed for a period *a*=300nm and 260 grating layers. The change in absorptivity |$\Delta \alpha $| is <10^{-2} for all wavelengths once >40 modes are retained. Similar results were obtained for the coated structures and for different V-groove angles and periods. Based on the convergence analysis the number of diffraction orders in all simulations was set to 47.

#### 4.3 Uncoated sub-wavelength V-groove gratings

We first investigate the spectral performance of uncoated Mo and W sub-wavelength V-groove gratings, which are similar to the tapered pyramid gratings reported by Rephaeli *et al*. [29]. As illustrated in Fig. 2, the V-grooves are 1D-gratings with discrete translational symmetry along the Y-direction and continuous translational symmetry along the X-direction. In general, separation into distinct modes is only possible when there is a plane *M* for which there is mirror symmetry for both the wavevector **k** and position vector **r** [14]. For the V-groove grating, we can only define one mirror plane which is independent of origin. This is the plane perpendicular to the X-axis, here called *M _{x}*. The TE and TM mode can thus be defined with respect to this mirror plane

*M*. The TE mode has field components

_{x}*E*,

_{y}*E*and

_{z}*H*and the TM mode has field components

_{x}*H*,

_{y}*H*and

_{z}*E*. As a consequence, light polarized along the Y-direction (

_{x}*E*) can only couple into TE modes, and light polarized along the X-direction (

_{y}*E*) can only couple into TM modes. This is also illustrated in the insets of Fig. 5(a) and 5(b).

_{x}In Fig. 5(a), the spectral absorptivity of uncoated Mo V-groove gratings with a period *a*=300nm is shown for normally incident light polarized along the Y-direction (TE mode). The spectral absorptivity is modeled for gratings with increasing depth of the groove (keeping the period *a*=300nm constant), varying from planar geometry planar geometry (${\theta}_{GR}=0\xb0$) (dark blue) to deep V-groove gratings with high aspect ratio (${\theta}_{GR}=80\xb0$) (dark red). Incident light can couple into radiation modes of the sub-wavelength grating which conserve the wavevector in the Y-direction (${k}_{y}$) up to a reciprocal lattice vector, since there is discrete translational symmetry in this direction. This means that the normally incident light in this case (which has no wavevector component in the Y-direction) can couple into modes with ${k}_{y}=n\cdot 2\pi /a$, where *n* is an integer. This means that diffraction peaks are expected at wavelength$\lambda =a,\text{}a/2,\text{}a/3,\text{}\mathrm{...}\text{,}a/n$. Therefore a sharp absorption peak is observed at *λ*=*a*=300nm in Fig. 5(a).

For the TE mode, absorptivity is also enhanced for wavelengths larger than the grating period, especially for deeper grooves (dark red). Since the incoming electric field is polarized along the Y-direction, perpendicular to the groove, no continuous boundary matching is required. As a consequence, the EM wave can propagate deep into the V-groove, even for wavelengths larger than the grating period. The result is that the absorption edge red-shifts when the depth of the groove increases (higher aspect ratio). The gradual impedance matching strongly enhances the absorption of the TE mode up to wavelengths similar to the depth of the groove.

In Fig. 5(b), the spectral absorptivity is shown for normally incident EM waves polarized along the X-direction (TM mode). Again, the curves represent V-groove gratings with increasing depth of the groove, varying from planar geometry (${\theta}_{GR}=0\xb0$) (dark blue) to deep V-groove gratings with high aspect ratio (${\theta}_{GR}=80\xb0$) (dark red). Since the electric field is now polarized along the groove, the field must be continuous across the metal-air boundary. As a consequence, EM waves with a wavelength much larger than the grating period will not be able to penetrate deeply into the V-groove. More incoming radiation is reflected and less absorption enhancement is observed for the TM mode at longer wavelengths. No significant red-shift is observed for grooves with a higher aspect ratio (dark red).

In Fig. 6
, the normal spectral absorptivity of uncoated Mo V-groove gratings is shown as a function of the angle ${\theta}_{GR}$ of the V-groove. The angle is varied from planar geometry (${\theta}_{GR}=0\xb0$) to a very deep groove (${\theta}_{GR}=80\xb0$), and for each structure the spectral absorptivity is shown at normal incidence. For these calculations, the grating period *a* was kept constant at 300 nm (top), 500 nm (middle) and 700 nm (bottom), respectively. Here, the absorptivity is averaged over the TE and TM polarization. The absorption for *λ* < 2.24μm is clearly enhanced with increasing V-groove angle, because a more gradual impedance matching is obtained, in agreement with Rephaeli *et al.* [29]. This would result in a more optimal absorber for operation at 720K. The periodicity of the grating influences the range over which enhanced absorption is achieved. Larger periods lead to a broader range of enhanced absorption, because EM waves with longer wavelengths can more easily penetrate into grooves with a larger period. In order to achieve optimal coatings, the selectivity of the enhancement is essential. Increased absorption for wavelengths above the cutoff wavelength${\lambda}_{c}$= 2.24 µm of the ideal absorber at 720K will lead to thermal losses by IR emission. The obtained spectral selectivity and the merit evaluations for these uncoated metal V-groove gratings are discussed in detail below.

For all uncoated Mo gratings in Fig. 6, distinct diffraction peaks can be observed, once the groove becomes sufficiently deep. As mentioned above, diffraction peaks are expected at wavelength $\lambda =a,\text{}a/2,\text{}a/3,\text{}\mathrm{...}\text{,}a/n$. For the top, middle and bottom plot of Fig. 6 the first diffraction peak is observed at 300nm, 500nm and 700nm respectively. For *a* = 700 nm (bottom), the second order diffraction peak can also be observed at *λ* = 350 nm.

#### 4.4 Coated sub-wavelength V-groove gratings

We have simulated the spectral properties of coated V-groove gratings for various depths and periods using RCWA. The coatings that were applied to the V-groove are those illustrated in Fig. 1 and their design for planar substrates was previously described in [27]. First, we will investigate the spectral selectivity for the 5-layer stack composed of layers of Mo, MgF_{2} and TiO_{2}, illustrated in Fig. 1(a) (cyan). In Fig. 7
, the normal spectral absorptivity is shown as a function of the angle of the V-groove for this 5 layer coating on top of a Mo V-groove grating. Again, the angle of the groove is varied from planar geometry (${\theta}_{GR}=0\xb0$) to a very deep groove (${\theta}_{GR}=80\xb0$), and for each structure the spectral absorptivity is shown at normal incidence. Similar to the case of uncoated Mo, the wavelength range of enhanced absorption increases with the depth of the groove. However, the enhancement is more subtle, because the 5-layer aperiodic metal-dielectric coating was already optimized for planar geometry.

To compare the spectral performance of the uncoated and coated structures, the merit function *F*, described in section 2, was evaluated. In Fig. 8
, the merit evaluation is shown for the uncoated Mo V-groove gratings (blue) with a grating period *a*=300nm. The angle of the groove is varied from planar geometry (${\theta}_{GR}=0\xb0$) to a very deep groove (${\theta}_{GR}=80\xb0$), and for each structure the merit *F* was evaluated at 720K, since we are optimizing the absorber for solar thermal applications at this temperature. A significant increase in spectral performance can be observed with increasing groove depth. This is due to the enhanced impedance matching for deeper grooves causing a selective increase in absorption, in agreement with Rephaeli *et al.* [29]. The merit *F* for the 5-layer stack (cyan) is also shown in Fig. 8. The 5-layer metal-dielectric coating was previously optimized for planar geometry (${\theta}_{GR}=0\xb0$) and thus the initial merit function is relatively high (*F*=0.85). However the spectral performance further increases with${\theta}_{GR}$or groove depth. The increase in spectral performance can be explained by the gradual increase in absorption for wavelengths shorter than the ideal cutoff wavelength${\lambda}_{c}$=2.24 µm, illustrated in Fig. 7. However, the absorptivity for wavelengths larger than${\lambda}_{c}$also gradually increases with increasing depth of the groove. This results in a loss in spectral selectivity for very deep grooves, as observed in Fig. 8. These two effects lead to a trade-off and therefore an optimal angle${\theta}_{GR,\mathrm{max}}$of the groove can be determined. Values for${\theta}_{GR,\mathrm{max}}$and the corresponding merit evaluation for all coatings are given in Table 1
for grating period *a*=300nm.

For ${\theta}_{GR}$≤ 70°, significant higher spectral selectivity is obtained for Mo V-groove gratings coated with the 5-layer stack composed of Mo, MgF_{2} and TiO_{2}. The interference effects created inside the aperiodic stack dictate the spectral properties and lead to an enhanced spectral selectivity compared to the bare grating. Since the aperiodic stacks have wide angular absorptivity [27], the interference effects dominate the spectral performance even for deep grooves. However, for ${\theta}_{GR}$> 70°, the angular spectral properties suffer in accordance to [27] and the spectral performance of the coated gratings becomes worse than that of the uncoated gratings (see Fig. 8). Similar trends were observed for the V-groove gratings with larger grating period and the best results were obtained for the 300nm grating period. The highest merit for the 5-layer stack was obtained at ${\theta}_{GR,\mathrm{max}}$≈ 40° (merit *F* = 0.87).

The angular dependence of the absorptivity of the V-groove gratings was also investigated. For the optimal coated Mo grating with period a=300nm and${\theta}_{GR}=40\xb0$the angular absorptivity $\alpha (\theta ,\varphi )$ is shown in Fig. 9(a)
and 9(b) at λ=0.8μm for the P and S polarization, respectively. Here the incident direction of radiation is defined by the polar angle *θ* and the azimuthal angle *ϕ*. By convention, P (S) polarization is defined as the electric field polarized parallel (perpendicular) to the plane of incidence. At λ=0.8μm, the coated grooves behaves as a wide-angular absorber, with little azimuthal dependence. Only for very large polar angles the absorptivity decreases notably. The angular absorptivity at λ=3μm is also shown in Fig. 9(c) and 9(d) for the P and S polarization, respectively. At this wavelength, the coated groove behaves as a low-emissivity surface for all angles.

We also studied the angular dependence of the absorptivity close to the spectral edge (1μm<λ<3μm) for the 5-layer stack on the optimal Mo V-groove with period a=300nm and${\theta}_{GR}=40\xb0$. At λ=1.4μm, we see a significant angular dependence in the absorptivity of the coated V-groove grating for the P and S polarization [Fig. 10(c) and 10(d)]. To understand this angular dependence, we need to first consider the spectral absorptivity at normal incidence. As explained above, light at normal incidence polarized along the Y-direction (perpendicular to the groove) will couple into the TE mode. In contrast, light polarized along the X-direction (parallel to the groove) will couple into the TM mode. The spectral edge for the TE mode is red-shifted compared to the TM mode because the TE waves penetrate more deeply into the groove. This is also illustrated in Fig. 10(a) and 10(b) where the spectral absorptivity at normal incidence is plotted for the TE and TM mode respectively. Let us now consider incoming light propagating in the ZY plane, for$\theta \approx 45\xb0$and$\varphi =90\xb0$. In this direction, the absorptivity is significantly different for the P and S polarization [Fig. 10(c) and 10(d)]. The P polarization will have an electric field component in the Y-direction and can thus effectively couple into the TE mode, for which the absorptivity is still >90%. As a consequence the absorptivity for the P-polarization will also by high. In contrast, the S polarization will have a field component along the X-direction and effectively couple into the TM mode. At λ=1.4μm, the TM mode has a lower absorptivity (≈50%), and therefore the S polarization has a lower absorptivity at this angle of incidence. When averaging over both S and P polarization for randomized incoming solar radiation, the angular dependence is smoothened.

It is clear from the results above that the performance of uncoated metallic V-groove gratings can be improved by coating the gratings with an optimized metal-dielectric multilayer stack. An optimal V-groove angle ${\theta}_{GR}$ can be determined for each selected coating. We modeled the spectral performance of all the aperiodic stacks that were introduced in section 3, (see Fig. 1). In Fig. 11(a)
, the merit evaluation at 720K is shown for the aperiodic stacks composed of Mo, TiO_{2} and MgF_{2} layers on top of Mo V-groove gratings with a period *a*=300nm. The stacks have 5, 7, 9 and 11 layers and were previously optimized for the planar geometry [27]. At${\theta}_{GR}=0\xb0$(planar geometry), the 11-layer stack has the highest merit function (*F*=0.88). Trends similar to those observed for the 5-layer coating above are observed for coatings with more layers. However the increase in merit is less pronounced, since the coatings had already excellent spectral performance in the planar geometry.

For each coating, an optimal V-groove angle${\theta}_{GR,\mathrm{max}}$and corresponding merit *F* can be determined. The best result is obtained with the 9-layer stack (orange) on top of a Mo V-groove with ${\theta}_{GR,\mathrm{max}}=36\xb0$. This structure is predicted to have a solar absorptivity of >94% while having a thermal emissivity as low as 6% of that of a blackbody at 720K (*F*=0.89). The results are summarized in Table 1. In Fig. 11(b), the merit evaluation is shown for stacks composed of W, TiO_{2} and MgF_{2} layers on top of W V-groove gratings with a period *a*=300nm. Again, similar trends are observed and an optimal V-groove depth can be determined. Here, the results obtained for the 7, 9 and 11 layer coatings are similar, with an optimal merit *F* around 0.89.

In Fig. 12 , the spectral absorptivity at normal incidence is shown for the coated V-groove gratings with optimal aspect ratios (${\theta}_{GR}={\theta}_{GR,\mathrm{max}}$) for each stack under investigation. Here, the absorptivity is averaged over both polarizations. If we compare these spectral curves to the planar geometry (Fig. 1), we observe an increased absorption for short wavelengths and a slight red-shift of the absorption edge. This red-shift is related to the TE mode as discussed above.

The merit evaluations of the coated V-grooved structures with period *a* = 300nm are summarized in Table 1. In general, stacks with more layers perform better. These coated sub-wavelength gratings are predicted to have performance in pair with commercially available cermet coatings (PTR70) [2] and complex multilayer coatings previously suggested by Kennedy *et al.* [4] (NREL 6A). For comparison, their spectral data and merit is also given in Table 1.

For the W-based stacks, the increase in spectral selectivity obtained by depositing the stacks onto V-groove gratings might be insufficient to validate the more complex fabrication. However, in this work we have limited our variable space. We believe that potential improvements lie in optimizing multilayer coatings specific for each V-groove. One potential way of fabricating sub wavelength grooves is to use selective etching processes which results in grooves where the angle of the groove is set by the preferential crystal planes. In this case the V-groove angle would be fixed, and the multilayer stack could be further optimized to achieve optimal spectral performance. Other improvements lie in optimizing the periodicity of the grating, as well as investigating 2D grating structures.

## 5. Conclusion

We previously reported on planar aperiodic metal-dielectric stacks for use in solar thermal systems. Here, the optical behavior of metallic V-groove gratings coated with these aperiodic stacks was modeled using RCWA. The spectral selectivity of these coated gratings was investigated and compared to the planar geometry. We conclude that the performance of uncoated metallic V-groove gratings can be significantly improved by coating them with a metal-dielectric multilayer stack. The grating period *a* and groove angle ${\theta}_{GR}$ have a significant impact on the spectral absorptivity. For coated V-grooves, the interference effects of the coating still dominate the spectral behavior. However, optimized metal-dielectric coatings combined with the gradual impedance matching provided by the V-groove, exhibit improved performance. Even though the improvement in adding a V-groove texture to the multilayer stack is subtle, it is important to note that fewer layers in the coating are required to achieve high performance. For a specific operational temperature, an optimal grating period *a* and V-groove angle ${\theta}_{GR}$can always be determined depending on the selected coating. It is important to note that alternative figures of merit that take geometry and temperature into consideration will result in different optimal values for grating period and V-groove angle than the ones described in this manuscript. We must point out that the operation lifetime of absorber coatings used in heat collection elements is critical and thus the thermal stability of the proposed coated gratings will need to be verified. If they prove to be thermally stable, these coated structures are good candidates for use in solar thermal applications because of their potential to achieve excellent spectral performance.

## Acknowledgements

The authors gratefully acknowledge the financial support from GCEP at Stanford University. N.P.S. acknowledges financial support as a Francqui Foundation Fellow from the Belgian American Educational Foundation.

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