## Abstract

The majority of spectrometers use reflective dispersion gratings with a metal-coated blazed grating profile for spectral decomposition. They achieve high diffraction efficiency at the design wavelength, which decays considerably in the adjacent longer and shorter wavelength ranges. We introduce a structured metal double-blazed grating with a high diffraction efficiency for a broadband spectral range, consisting of a sawtooth-like structured metal surface filled with a first dielectric transparent material. The planarized upper surface is covered with a second blazed profile of a different transparent material. We present a systematical theoretical analysis of the diffraction efficiency in reflection geometry, based on a scalar approach involving fundamental dispersion parameters such as Abbe numbers and relative partial dispersions of the materials. We find material combinations reducing the profile heights down to 1–2 µm.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. INTRODUCTION

Reflective blazed diffraction gratings are the key elements in numerous spectroscopic instruments and monochromator systems. For a selected diffraction order, the sawtooth-like profile shape of the blazed gratings is optimized for a maximum grating efficiency of 100% in scalar approximation. But the maximum efficiency is only achievable for a selected single design wavelength and decays considerably in the adjacent longer and shorter wavelength ranges, where light is diffracted into the higher or into the zeroth diffraction order. The reduced diffraction efficiency in the spectral regions other than the design wavelength reduces the sensitivity of spectrometers or lowers the effectiveness of monochromators. Additionally, the contributions of unwanted diffraction orders are often associated with disturbing stray light, which reduces the quality of the optical instruments. Therefore, it is highly advantageous to enable a high diffraction efficiency over an extended spectral range, a quality of the grating, which is denoted as “efficiency achromatization.”

For some years, efficiency-achromatized approaches have been introduced for diffractive elements [1–3] including fabrication techniques for required layer coatings [4]. The focus was mainly on transmissive efficiency-achromatized blaze structures. One particular solution of these efficiency-achromatized elements are nearly index-matched structures, which consist of a deep sawtooth-like profile, separating two materials showing a small difference in their refractive indices [5]. A second solution of these efficiency-achromatized transmissive diffractive elements combines two single-layered sawtooth-like structures of different materials and profile heights, which face each other and which are separated by an air gap of a few micrometers [6–8]. Alternatively, the two sawtooth structures are in direct contact and the air gap is transferred to the outside of the structure [9]. Spectral and angular influence of two-layered microstructures has been investigated [10]. In recent years, it was shown that material designing such as nanocomposite-enabled diffractive optical elements, e.g., are powerful tools to manipulate dispersion characteristics and achieve broadband efficiencies [11,12].

In contrast to the transmission blazed gratings, solutions for reflective spectral efficiency achromatization have only been little investigated. Metal gratings with a single leveled dielectric coating achieved broadband efficiency in the near-infrared spectral region using rigorous numerical simulations [1]. A hybrid annular folded lens with a reflective-diffractive optical element exploited reflective structured surface [13]. A combination of a sawtooth-like dielectric double-layer structure was investigated on top of a plane metallic mirror [14]. This approach is linked to the requirement of profile heights in the range of up to several micrometers, which may lead to a conflict in spectroscopic gratings, where the typical periods are in the same dimensions.

In this contribution, we present an alternative of an efficiency-achromatized approach for reflective gratings using two design wavelengths including a third design wavelength in between. In Section 2, we present the known effect that a reduction of the profile height of a simple reflective blazed grating can be realized by filling the structure with a high-index material. We briefly outline the basics of the scalar theory used in the subsequent sections. Depending on the dispersion characteristics of typical available materials, including glasses, polymers, and inorganic materials, we investigate the possibility of reduction of profile heights for several material combinations in Section 3 and find first characteristics of materials’ dispersion properties that lead to small profile heights. In Section 4 we now derive theoretical conditions for a systematical selection of optimum material combinations based on scalar diffraction efficiencies and within a thin element approximation, which guarantees both broadband efficiency achromatization together with small profile heights for the grating structures. We build the systematical approach on fundamental dispersion parameters such as Abbe numbers and relative partial dispersions of the materials and derive conditions for sums and differences of the involved Abbe numbers and relative partial dispersions. In Section 5 we discuss resulting broadband diffraction efficiencies depending on selected material combinations in the scalar approach (Section 5.A) and with rigorous simulations in Sections 5.B and 5.C. In particular, the rigorous simulations are applied under normal incidence (Section 5.B) and for a varying angle of incidence (Section 5.C).

## 2. CLASSICAL BLAZED GRATING VERSUS HIGH-INDEX-FILLED BLAZED GRATING

Neglecting reflection losses due to the applied specific metallic coating, reflective blazed gratings offer maximum diffraction efficiency, which achieves 100% in a scalar approximation but, unfortunately, only for a specific design wavelength $ \lambda_{\!B}$. For a selected diffraction order $ m $ the wavelength dependency of the diffraction efficiency $\eta_{m}({\lambda})$ can be written as

where $\eta_{{F},{B}}({\lambda})$ is the factor of Fresnel or Beer reflection, respectively, which all depend on the wavelength $ \lambda $. The factor $ \alpha $ is denoting the optical phase retardation, which in the case of a simple blazed (sB) profile with a profile height $ h $ yields The blaze efficiency is maximum if $\alpha = m$, which results in the optimum profile height: For a typical design wavelength in the visible wavelength range, e.g., for 587.6 nm (d-line of helium) and for the first diffraction order ($m = 1$), the optimal profile height measures nearly 300 nm. Although these height dimensions seem not too demanding, different aspects necessitate substantially smaller structure heights. Here, an important argument concerns the limitations of appropriate manufacturing technologies. For the manufacturing of reflective spectroscopic gratings interference lithography is a well-established technology offering advantages such as the possibility to address large areas in a single exposure step and the reduction of grating “ghosts.” Unfortunately, due to the geometrical limitations in the illumination setup, the accessible profile heights are restricted to values in the range of ${\sim} 150\;{\rm nm} $ [15]. Additionally, small profile heights may also be beneficial for applications in which large diffraction angles are used. Here, small profile heights allow the reduction of efficiency losses due to shadowing effects generated by the passive blaze facet.A very simple approach to reduce profile height and simultaneously retain maximum efficiency for the design wavelength is achievable by filling the blazed profile with a transparent high-index material [5] or an optical resin [1]. This approach reduces the profile height linearly with the refractive index of the transparent filling material ${{n}_0}({\lambda_{\!B}})$ at the design wavelength of the filled blaze ($ B $) in reflection geometry ($ R $) to

and the wavelength-dependent optical phase retardation becomes The design heights of a blazed grating profile for different high-index filling materials at the reference wavelength of 587.6 nm are displayed in Table 1. For compactness we have used the abbreviation “$C$” for diamond as material throughout the entire paper.Due to the dispersion characteristics of the filling material, the diffraction efficiency also depends on the wavelength. Figure 1 shows the efficiency curves for different high-index materials and for vacuum as a reference. It follows that only materials with exceptional small Abbe numbers such as ${\rm TiO}_2$ experience a noticeable reduction of the diffraction efficiency aside the design wavelength. For higher Abbe numbers (${\gt} 30$) the deviation of the efficiency curve from the reference curve for vacuum is in the low percentage range.

From this it follows that the filling of a blazed profile with a high-index material of medium or low dispersion allows significantly to reduce the profile height and, simultaneously, to maintain the wavelength dependency of the diffraction efficiency. It should be noted that potential refraction losses due to the interface from air to the high-index filling material can be reduced by anti-reflective layers at the interface or, e.g., by the introduction of “moth-eye” structures offering a gradient index transition between high-index material and surrounding air [21].

## 3. EFFICIENCY-ACHROMATIZED REFLECTIVE DISPERSION GRATING BY A DOUBLE-BLAZED CONFIGURATION

In comparison to alternative groove profiles such as sinusoidal or rectangular shaped profiles, a conventional blazed grating offers the advantage of a high diffraction efficiency for a pre-selected diffraction order but only for one specific design wavelength. In the adjacent long- and short-wavelength ranges, the diffraction efficiency decays significantly. To overcome this limitation and to allow “efficiency achromatization,” which means a high diffraction efficiency for an extended wavelength range, a specific grating structure is suggested that is useable for applications in reflection geometry. A schematic representation of the efficiency-achromatized grating profile is shown in Fig. 2. The basic structure combines a sawtooth-like structured metal surface as a first blazed profile, which is filled up with a first dielectric transparent material (material 1). On top of the planarized upper surface of material 1, a second blazed profile of a different transparent material (material 2) is deposited, which has the same spatial frequency and lateral phase position with respect to the lower grating structure. It is important that the optical properties of both transparent dielectrics, especially the wavelength-dependent refractive index ${{n}_{j}}({\lambda})$, differ from each other and so do the profile heights ${{ d }_{j}}$ of both grating structures with ${j} = 1$, 2.

In scalar approximation, which is reasonable for grating periods larger than a few times the working wavelength [22], the diffraction efficiency can be calculated by Eq. (1), but with an adapted optical phase retardation $\alpha= {\alpha_{Z}}$:

where $\alpha _1^R$ is the retardation phase of the lower substructure in reflection and $\alpha _2^T$ of the upper substructure in transmission, respectively. For pure transmission geometry, only the $\alpha _2^T$ is used for efficiency achromatization, and often dispersion is modeled by a Cauchy-series approximation [23].The first term represents the effect of the dielectric material, which is in contact to the metallized underlying blazed profile, and the second term describes the phase effect of the upper grating with contact to the surrounding medium of refractive index $ n_0 $, for example, the ambient air. Inserted in Eq. (6) this yields

In the following specific material combinations are discussed exemplarily. As design wavelengths for which the diffraction efficiency achieves the 100% value, the $\lambda_1 = 486.1\;{\rm nm} $ (hydrogen $ F $-line) and the $\lambda_2 = 656.3\;{\rm nm} $ (hydrogen $ C $-line) lines were selected. Based on the theoretical considerations (see Section 4) it was found that the combined dielectric materials should be, on one hand, high-index materials and, on the other hand, highly disparate in their dispersion characteristics to provide high diffraction efficiency for a broad wavelength range and also to minimize the profile heights. To describe quantitatively the dispersion properties of the different materials both the Abbe number $ \nu $ and the relative partial dispersion $ P $ have to be used:

Hereby, the subscripts of the refractive indices indicate the reference wavelengths: $ F $-line ${\lambda_{F}} = 486.1\;{\rm nm} $, $ C $-line ${\lambda_{C}} = 656.3\;{\rm nm} $, and $ d $-line ${\lambda_{d}} = 587.6\;{\rm nm} $. For most of the inorganic glasses used for optical instruments, there is a linear relation between $ \nu $ and $P$: The constants possess the values $A = - 6.146 \cdot {10^{- 4}}$ and $B = 0.7306$, especially with reference to the standard inorganic glasses BK7 [24] and F2 [25]. In the following, the deviation of the dispersion characteristics of a specific material (cf. Refs. [26,27]), given by the two parameters $\nu $ and $P$, from the normal linear relation, known as the “normal line of glasses” [cf. Fig. 3(b)], will be used to discuss the influence on the profile heights of both superimposed gratings.As a first material pair the combination of the two standard inorganic glasses BK7 and F2 was analyzed. These two glasses are typically used to reduce chromatic aberrations in imaging systems, especially for achromatization, which guarantees the same focus position for two distinguished wavelengths. With respect to possible manufacturing technologies, this combination of inorganic glasses is rather a theoretical calculation than an implementation possibility and serves here as a reference system.

The profile heights ${d}_1$ (lower grating part) and ${d}_2$ (upper grating part) that are associated with the specific choice of reference wavelengths $\lambda_1$ and $\lambda_2$ yield

For the material combination of BK7 and F2 the profile heights of ${d}_1 = 2.771\; {\unicode{x00B5}{\rm m}}$ and ${d}_2 = 6.289\; {\unicode{x00B5}{\rm m}}$ were found for each substructure. These profile heights are very large, especially when compared to the periods of diffractive gratings used for spectroscopic applications. Typically, these periods are in the range between 2 µm to 10 µm when the grating works in the first diffraction order. That means, the exemplary calculated profile heights will result in a high aspect ratio, which is challenging to manufacture and will cause shadowing effects for non-perpendicular incident light.

To reduce the profile heights a combination of inorganic glasses was selected in a next step, which offer very different dispersion properties. Specifically, one inorganic glass was the fluorite crown FK51 [28] with a very high Abbe number of $\nu_{{\rm FK}51} = 83$ and as the second, upper material, the highly dispersive dense flint SF67 [29] ($\nu_{{\rm SF}67} = 21.4$) was chosen. This material combination results in profile heights of ${d}_1 = 1.559\; {\unicode{x00B5}{\rm m}}$ and ${d}_2 = 2.220\; {\unicode{x00B5}{\rm m}}$. Although these inorganic glasses in their kind show extraordinary dispersion quantities, the resulting profile heights are still large.

For the above-mentioned inorganic glass combinations, and also for material pairs as will follow in the discussion below, Table 2 lists all dispersion characteristics of the involved materials and the resulting heights of the blazed structure parts. For optical material data, see [7,24–33].

Polymers are of enormous significance for modern optics, especially in the case of high-volume applications. Therefore, we also investigated different polymer combinations for the potential use as an efficiency-achromatized superimposed grating structure. As a reference, the combination of polystyrene (PS) and polymethylmethacrylate (PMMA) was chosen, where PMMA shows a moderate Abbe number ($\nu_{{\rm PMMA}} = 58.4$) and PS exhibits a small value ($\nu_{{\rm PS}} = 29.4$). The corresponding profile heights were found to be ${{ d }_{1\_{\rm PMMA}}} = 2.258\; {\unicode{x00B5}{\rm m}}$ and ${{ d }_{2\_{\rm PS}}} = 5.178\; {\unicode{x00B5}{\rm m}}$, which is significantly larger than the values for the inorganic material combinations.

In order to reduce the profile heights also for polymers, we chose materials that are characterized by extreme dispersion quantities. In particular, the first material was an amorphous fluoropolymer with the trade name “CYTOP” [30] with a high Abbe number $\nu_{{\rm CYTOP}} = 90$. The second material, a polyethersulfon (PESU) with the trade name “Ultrason E 2010,” exhibits a very low Abbe number $\nu_{{\rm PESU}} = 20.4$ [31]. In this case we found profile heights of ${{ d }_{1\_{\rm CYTOP}}} = 1.609\; {\unicode{x00B5}{\rm m}}$ and ${{ d }_{2\_{\rm PESU}}} = 2.833\; {\unicode{x00B5}{\rm m}}$. In the last case, the Abbe number of the specific PESU was derived from the wavelength-dependent curve of the refractive index [32].

For a further reduction of the profile heights, the material selection was extended to inorganic layer- and special-materials, which offer even more extreme dispersion characteristics (Table 2). In particular, the combination of the alkali metal lithium (Li) or the alkaline earth metals magnesium (Mg) and calcium (Ca) with halogens such as fluorine (F), chlorine (Cl), or iodine (I) are characterized by very high Abbe numbers. ${\rm MgF}_2$ is a frequently used low-index layer material with high Abbe number; the Li-combinations show even higher Abbe numbers but are difficult to process and are particularly vulnerable with regard to humidity. Moreover, ${\rm Nb}_2{\rm O}_5$ and ZnO have very small and titan oxide (${\rm TiO}_2$) or indium-tin-oxide (ITO) extremely low Abbe numbers.

From the analysis of the different material pairs it follows that the profile heights reduce significantly with the reduction of the Abbe number of the second material. For example, in the material pair of LiI and ${\rm TiO}_2$ the profile height of the LiI is calculated to be 0.705 µm and 0.760 µm for ${\rm TiO}_2$, respectively. Here, the overall change in the profile height is more than a factor of 3 smaller with respect to the previous case of the inorganic glass combination.

An alternative material combination, which also offers low profile heights, can be achieved by replacing the high Abbe number material by diamond. In particular, the combination of diamond ($ C $) as the first material and ${\rm TiO}_2$ as the second one results in corresponding profile heights of 0.623 µm and 0.850 µm, respectively. Although diamond ($ C $) is characterized only by a moderate Abbe number, an essential detail concerns the large distance of the specific $\nu{-}P$-value of diamond from the normal line of glasses regarding the relative partial dispersion.

For illustration, an Abbe diagram [Fig. 3(a)] and the Abbe numbers against values of relative partial dispersion are depicted in Fig. 3(b) together with the linear curve of normal relative partial dispersion. From Fig. 3(b) it seems obvious that small profile heights can be achieved when the first material has a low Abbe number and the second shows a strong deviation from the normal line of glasses concerning the values of the relative partial dispersion along the linear curve.

Finally, the material combination, for which we find the minimum profile height, combines diamond ($ C $) or LiI with gallium phosphide (GaP). GaP has an extreme low Abbe number of $\nu_{{\rm GaP}} = 6.6$. It has to be mentioned, however, that GaP possesses a high absorption below 490 nm, so that this material is opaque in the blue wavelength range. We find the absolute minimum for the combination of diamond and GaP with the structure heights of ${d}_1 = 0.384\; {\unicode{x00B5}{\rm m}}$ and ${d}_2 = 0.261\; {\unicode{x00B5}{\rm m}}$.

## 4. THEORETICAL CONDITIONS FOR SYSTEMATICAL SELECTION OF MATERIAL COMBINATIONS

The discussions in the previous sections have revealed that material combinations with high refractive indices together with certain Abbe numbers and certain relative partial dispersions seem to be more appropriate for the efficiency achromatization by a double-blazed grating than other material combinations. Particularly when small structure heights are required for use in spectral applications. In this section we will derive theoretical conditions for an optimal material selection to obtain small structure heights together with a broadband efficiency achromatization in the wavelength range in between the two wavelengths ${\lambda_1}$ and ${\lambda_2}$.

However, from the complicated dependence of the structure heights on the refractive indices as in Eqs. (11) and (12), it can hardly be seen what materials are optimal to obtain small structure heights. Moreover, it is not clear what optical properties in terms of Abbe numbers and partial dispersions these materials should have for a successful efficiency achromatization. We therefore rewrite these two equations using the optical key parameters: the Abbe number $\nu$ and the relative partial dispersion $ P $ of both materials, explicitly, which then yields for the lower and upper grating heights, respectively,

The substructural heights ${h_1} = {\lambda_2}/({2{{n}_{12}}})$ and ${h_2} = {\lambda_2}/({2({{{n}_{22}} - 1})})$ turn out to be the heights of the corresponding (non-achromatized) index-filled single-blazed gratings [cf. Eq. (7)] at the wavelength $\lambda_2$. From the heights ${h_1}$ and ${h_2}$ it is clear that choosing high refractive indices ${n_{12}}$ and ${n_{22}}$ is a first design step that will immediately reduce the grating profiles ${d_1}$ and ${d_2}$. The factors $ f $ and ($1 + f\,$), however, scale these upper and lower substructural heights, respectively, and will increase the profile heights ${d_1}$ and ${d_2}$ again. The function $f({a,b}) = ({1 + a})b/({a - b})$ is related to the coupled dispersion of the two materials. It involves the parameters $ a $ and $ b $ containing modified Abbe numbers $\nu _1^C$ and $\nu _2^C$ of each material, respectively, relative to the grating’s modified Abbe number $\nu _G^C$:

Due to the refractive-index dependence of the expressions in Eqs. (11)–(12) for the double-blazed grating on the two boundary wavelengths $\lambda_1$ and $\lambda_2$ only, the classical Abbe numbers ${\nu_j}$ are not advantageous for the evaluation of the dispersion properties of the optical materials, because they would, on one hand, generate extremely complicated expressions and, on the other hand, Eqs. (13)–(14) are now applicable for any wavelength range desired. Thus, it is more appropriate to shift $\lambda_0$ to $\lambda_2$ in the numerator of the Abbe number and introduce modified Abbe numbers $\nu_j^C$ defined as and the relative partial dispersion as where the first index represents the material ${j} = 1$, 2 and the second index represents the wavelength $\lambda_{i}$ with the subscript ${i} = 0$, 1, or 2. The modified Abbe numbers $\nu_j^c$ for each material $ j $ are related to the classical Abbe number ${\nu_j}$ through a shift by the corresponding relative partial dispersion ${P_j}$: For the specific spectroscopic wavelengths used in Section 3, i.e., $\lambda_0 = {\lambda_{d}} = 587.6\;{\rm nm} $, $\lambda_1 = {\lambda_{F}} = 486.1\;{\rm nm} $, and $\lambda_2 = {\lambda_{C}} = 656.3\;{\rm nm} $, one immediately obtains the classical Abbe number $ \nu_j = (\nu_d)_j$ as defined in Eq. (8) and the relative partial dispersion $ P_j = (P_{d,C})_j = 1 - (P_{F,d})_j$ as defined through Eq. (9), for each material $j$.One can also define an Abbe number for a diffraction grating (cf. [34]):

The grating’s Abbe number is negative due to the inverse spectral dispersion property of a diffraction grating, since $\lambda_1$ is the blue wavelength and thus smaller than the red wavelength $\lambda_2$. Analogously, we define the relative partial dispersion and a modified Abbe number of a grating as follows: The relation between classical and modified Abbe numbers [Eq. (19)] equivalently holds for the grating when one puts the subscript $j = G$.In finding the optimal material combination, the ratios of the material Abbe numbers to the grating Abbe number are decisive for a plane metal grating [14]. However, for the structured double-blazed grating as in Fig. 2 the refractive index ${n_{12}}$ appears in the lower grating part and thus the parameter $ a $ as in Eq. (15). It turns out that the modified Abbe number $\nu _{G1}^C$ of the lower grating structure filled with material 1 determines the achromatized property rather than Abbe number $\nu _1^C$ of the filling material itself, and $\nu _{G1}^C$ is a “natural” variable of the filled structured grating, since it uses the wavelengths within the filling material 1:

So the parameter $ a $ as defined in Eq. (15) can completely be written with modified grating Abbe numbers and without using the refractive index ${n_{12}}$, explicitly: Equations (13) and (14) determine the height of each of the two substructures of the double-blazed grating and explicitly depend on the choice of the two wavelengths ${\lambda_1}$ and ${\lambda_2}$. This does not necessarily ensure that the efficiency remains high for any intermediate wavelengths ${\lambda_0}$. Only if the phase retardation $ \alpha_0 = \alpha (\lambda_0) $, as given in the following Eq. (25), remains close to 1, the efficiency achromatization is optimal:Before we will now discuss specific material selection properties, two general aspects can be immediately seen from these results and the properties of the function $f({a,b})$, in particular, the denominator $a - b$ in $ f $ must be large to obtain small grating heights. As a condition for material selection, if the blazed diffraction order $m \gt 0$, then requires

The first condition of material selection states that $\nu_2^C$ has to be as close to $| {\nu_G^C} |$ as possible. So material 1 should possess a large Abbe number and material 2 should possess a low Abbe number. This means for most optical materials, in particular inorganic glasses, that material 1 is a low-refractive-index material and material 2 is a high-refractive-index material. Unfortunately, this is contrary to the general goal to choose high-index materials. Thus, only very specific materials such as diamond, e.g., can fulfill this condition [Fig. 3(a)].The second condition requires that the difference of the modified Abbe numbers $\Delta {\nu^C}$ should be considerably different in order to obtain small grating heights ${d_1}$ and ${d_2}$. The latter resembles the condition for choosing crown and flint glasses when building a refractive achromat or apochromat, respectively.

The third and important condition for material selection concerns the efficiency for the intermediate wavelength. For achieving 100% diffraction efficiency at $\lambda_0$, it is required that $\alpha_0 = 1$ and, as a consequence, the numerator in Eq. (26) has to vanish. This leads to a third condition:

An overview of the resulting substructural profile heights ${d_1}$ and ${d}_2$ of each blazed substructural grating part together with the total height ${d_1} + {d_2}$ of the entire double-blazed grating depending on the cases of the different material combinations that we have discussed in this work and listed in Tables 2 and 3 are displayed in Fig. 4. The high refractive index of the filling materials of both grating parts reduces the grating heights ${h_1}$ and ${h_2}$ of the underlying single-blazed grating substructures, so that the total grating height ${d_1} + {d_2}$ significantly decreases with the parameter $ b $ approaching 1 and the factor $ f $ becoming small (Table 3). This requires that the high-index material of the upper grating substructure has an Abbe number close to the modulus of the gratings Abbe number according to the first selection condition. The second selection condition with $ \Delta \nu^C$ large reduces the profile heights down to 2 µm. However, for a further reduction of the heights below 1 µm, the first selection condition turns out to be more decisive in favor of materials like ${\rm TiO}_2$, ITO, and GaP.

The broadband efficiency for intermediate wavelength with ${\alpha_0}$ close to 1 is fulfilled for four material combinations with a negative difference in partial dispersions that we have considered here (Table 3). They involve ITO as the material 2 with a small Abbe number. ITO is located significantly below the line of normal glasses in Fig. 3(b), so that in combination with a material 1 leads to the $ \Delta P \lt 0$.

Optical polymers may deviate from the linear rule of normal glasses depending on an appropriate polymer design. So we expect that it should be possible to chemically design new polymers in a way that the achromatization is achieved for a very broadband wavelength regime fulfilling the condition given in Eq. (32).

## 5. RESULTING PROFILE HEIGHTS OF THE STRUCTURED METAL DOUBLE-BLAZED GRATING AND DIFFRACTION EFFICIENCIES FOR SELECTED MATERIAL COMBINATIONS

From the determined profile heights, the efficiency curve as a function of the wavelength was obtained for each material combination. Effects due to Beer reflectivity and reflection losses, due to the change of the refractive index at the interfaces, were neglected in the scalar approach. In real implementations, these losses can be reduced by the introduction of anti-reflective layers between the dielectric materials and on the top surface.

#### A. Efficiency of Metal Double-Blazed Grating from Scalar Diffraction Theory

Figure 5 shows the efficiency curves for the selected inorganic glass combinations and the polymers. The efficiency curve of the conventional blazed profile was added for comparison. The overall view [Fig. 5(a)] shows that the different material combinations are very similar in their efficiency dependency. Magnified views [Fig. 5(b)] show some minor differences between the material combinations with a potential influence in the shorter wavelength range.

A very similar behavior was found for the efficiency curves of the inorganic material combinations. Figure 6 presents the efficiency curves for all remaining material combinations from Table 2. The overall view [Fig. 6(a)] shows again very similar characteristics for all curves. The magnified views depicted in Fig. 6(b) (different material compositions split into two diagrams), show only some smaller differences in details. The curves for the material combinations, which are not mentioned in the magnified views, coincide with the presented curves in the same figure. In particular, the efficiency dependency of the combinations ${\rm LiI}/{\rm TiO}_2$ and ${\rm CaF}_2/{\rm TiO}_2$ differs by less than 0.5% over the plotted spectral range from the depicted ${\rm LiCl}/{\rm TiO}_2$ curve, and the LiI/ITO dependency is nearly identical to the ${\rm MgF}_2/{\rm ITO}$ curve. The material combinations in which GaP was used show a very similar efficiency dependency compared to the corresponding curves which include ${\rm TiO}_2$.

A remarkable effect was found when ITO was used as the material with the low Abbe number (instead of ${\rm Nb}_2{\rm O}_5$, ZnO, ${\rm TiO}_2$, or GaP). In this case, the efficiency decay in the longer wavelength range is very small and the efficiency curves show nearly 100% from the blue reference wavelength (486.1 nm) up to the near-infrared regime, according to the material selection conditions [Eqs. (31)–(32)].

#### B. Efficiency of Metal Double-Blazed Grating from Rigorous Numerical Simulations

In the previous sections we applied scalar diffraction theory to the metal double-blazed grating to find the appropriate preconditions for high diffraction efficiency over a broad spectral range and simultaneously to minimize profile heights of the involved sawtooth structures. On the other hand, the used scalar approach is limited with respect to accuracy and validity. In particular, disturbing edge effects at the step facet and reflections at the transition of the material interfaces are neglected, which may reduce the efficiency in the desired diffraction order in a real practical application.

For comparison and to assess the significance of the previous findings, in the following a rigorous treatment was applied to selected examples of double-blazed gratings. For all rigorous analyzed model structures an initial blazed aluminum grating was assumed, which is completely filled up with ${\rm MgF}_2$ and finally planarized. The material of the second blazed grating was either ${\rm TiO}_2$ or ITO. For each of the examples, the facet heights of both involved blazed gratings were optimized with respect to the material composition and design wavelengths. Although this contribution focuses on theoretical considerations, for the rigorous simulations we chose widely used thin film materials which could potentially be used in practical realizations.

In contrast to the scalar approximation, the rigorous approach includes also reflection effects at the interfaces. To minimize these disturbing reflection effects, suitable anti-reflective coatings have to be introduced at both the interface between the blazed gratings and at the transition from the top grating to air. For a practical realization these anti-reflective coatings would most probably be sophisticated and well-adapted dielectric multilayer stacks.

Because in this contribution the main purpose is to demonstrate the broadband high efficiency of the metal double-blazed grating, we waive the development of perfectly adapted multilayer systems but used a simplified approach of more artificial anti-reflection coatings. In particular, we introduced a graded-index transition consisting of 20 artificial layers with a single layer thickness of 50 nm between both dielectric sawtooth profiles, which changes the refractive index linearly from one material to the other. Analogously a final anti-reflective layer stack was introduced to the top blazed grating. In particular, the final top layer was composed of 20 artificial layers, each with a thickness of 50 nm, and assuming a graded index transition with a linearly changing refractive index between ${\rm TiO}_2$ or ITO, respectively, and ambient air.

Figure 7 shows schematically the cross section of the unit cell of the entire combined profile structure. The grating period was varied between 10 µm and 100 µm. The numerical calculations were carried out with the rigorous coupled-wave analysis (RCWA)-based grating solver UNIGIT [35]. Aluminum was assumed for the substrate and air for the superstrate, respectively. The entire structure was vertically divided into 200 slices and laterally into a grid with 100 elements. For all calculations, perpendicular incident light was assumed at angle 0° and the entire wavelength range was captured in steps of 20 nm. Preliminary convergence tests showed that the limit of 300 Rayleigh orders is sufficient.

Figure 8 shows the calculated efficiency of the first diffraction order for the selected metal double-blazed gratings and for a classical Al-coated blazed grating as a reference. The various metal double-blazed gratings differ in terms of periodicity, design wavelengths, and grating period. The displayed efficiency is calculated as the mean value of the contributions from the reflected TE and TM polarizations. The classical blazed grating with a grating period of 10 µm shows maximum efficiency at the design wavelength of 650 nm and strong decays in the shorter and longer wavelength ranges. The maximum efficiency is approximately 90% and therefore considerable below the 100% value, which is mainly attributed to the material characteristics of the Al coating.

The efficiency curve for ITO as the material for the upper blazed profile (orange curve) shows a somewhat reduced maximum efficiency of ${\sim} 80 \%$ but the efficiency remains nearly constant across the entire wavelength range. Especially in the wavelength ranges below 540 nm and above 840 nm, the efficiency of the ${\rm MgF}_2{-}{\rm ITO}$ double-blazed grating exceeds the reference value of the blaze grating significantly. Here, the grating period was 10 µm and the facet heights of the ${\rm MgF}_2$ and ITO structures were set to 738 nm and 799 nm, respectively, calculated for the design wavelengths of 486 nm and 656 nm. The curve ends at a wavelength of 980 nm since ITO absorbs at larger wavelength.

To get also access to a higher wavelength range we changed the materials of the upper blazed grating from ITO to ${\rm TiO}_2$. Keeping the design wavelength and the grating period constant, we get adapted facet heights for both blazed gratings of 1002 nm (${\rm MgF}_2$) and 762 nm (${\rm TiO}_2$), respectively. The corresponding efficiency curve (curve in red) shows a similar behavior across the visible wavelength range as before with ITO as the upper material. In the wavelength range larger than approximately 900 nm the efficiency curve begins to modulate, whereby the efficiency is always higher compared to the reference curve. The occurring modulation in the efficiency curve may be attributed to the specific design of the chosen anti-reflective layers, which offers potential for optimization and higher overall efficiency. The potential for increased efficiency becomes evident by changing the upper design wavelength to 900 nm. This modification leads to adapted facet heights for both blazed gratings of 1539 nm (${\rm MgF}_2$) and 1255 nm (${\rm TiO}_2$), respectively. Compared to the previous case, the efficiency curve (blue curve) is slightly lower in the visible range but significantly higher in the infrared region and also no or reduced modulations appear.

Finally, when changing additionally the grating period to 100 µm (green line) a strong increase in the efficiency is observable in the near-infrared region (approximately at around 1 µm). In particular, for this double-blazed grating the integrated efficiency across the whole wavelength range exceeds the corresponding value of the reference simple blazed grating by far. Consequently, the metal double-blazed grating approach allows to achromatize the diffraction efficiency in the reflection geometry over a large wavelength range. The deviation from the expected maximum value may be attributed to the insufficient approach chosen for the anti-reflective coatings. Here it is assumable that optimized anti-reflective coatings offer potential for efficiency improvements.

#### C. Efficiency of Metal Double-Blazed Grating from Rigorous Numerical Simulations for Varying Angle of Incidence

In Section 5.B we proofed the validity of different scalar designs by rigorous simulations. In this section we compare the angular behavior of two metal double-blazed gratings with considerably different facet heights by rigorous simulations. The grating period was set to 10 µm for both analyzed models and the design wavelengths are 486 nm and 656 nm. The first analyzed model is the known ${\rm MgF}_2 {\text -} {\rm TiO}_2$ grating with a full height of 1.764 µm. As the second model, we chose the combination of FK51 and SF67 with the heights of 1.559 µm and 2.220 µm, respectively. The total height of 3.779 µm is more than 2 times higher than the height of the ${\rm MgF}_2/{\rm TiO}_2$ combination. For these calculations we sampled the full wavelength range in steps of 50 nm. The angle of incidence (AOI) was varied from 30° to ${-}30^\circ$ in steps of 15°. It is presumed that larger facet heights are connected to disturbing edge effects which results in a lower diffraction efficiency, especially with increasing AOI.

Figures 9 and 10 show the calculated efficiencies of the first diffraction order for the ${\rm MgF}_2/{\rm TiO}_2$ and FK51/SF67 combination, respectively. Both gratings work well for an AOI of 0° in the spectral range of the design wavelengths with maximum efficiencies of approximately 80% (${\rm MgF}_2/{\rm TiO}_2$) and 72% (FK51/SF67). Besides the design wavelengths, the diffraction efficiency decreases, especially in the infrared range. As expected, the maximum efficiency decreases with increasing AOI. Furthermore, the maximum efficiency peak shifts to longer wavelengths for the considerably large AOIs of 30° and ${-}30^\circ$.

To analyze the simulated diffraction efficiencies in detail, we focus on the visible range of 400 nm to 700 nm for which the designs are optimized. A comparison of both gratings reveals that the stronger efficiency decay can be observed for the larger double-blazed grating made of FK51 and SF67. In particular, the maximum efficiency of the FK51/SF67 grating reduces to 61% and 54% for the AOIs of ${-}15^\circ$ and 15°, respectively. A further efficiency drop down to 7% and 18% is observed for the AOIs of 30° and ${-}30^\circ$, respectively. In contrast, the maximum efficiency of the ${\rm MgF}_2/{\rm TiO}_2$ combination is reduced to 73% and 69% for the AOIs of ${-}15^\circ$ and 15°, respectively, and decreases to 29% and 41% for the large AOIs of 30° and ${-}30^\circ$, respectively. This is approximately 2 times higher than for the FK51/SF67 combination. These findings confirm the expectation that metal double-blazed gratings with lower facet heights show a better performance in terms of overall diffraction efficiency and angular behavior.

## 6. CONCLUSIONS

We proposed a metal double-blazed grating combination offering a high and largely constant diffraction efficiency across a large spectral range for the use in reflection geometry, which may be advantageous for applications in spectrometers. The presented analyses of the diffraction efficiencies show that a suitable choice of the specific involved materials allows us to minimize the profile heights of both sawtooth subgratings. Specifically, for minimum profile heights the material of the first blazed substructure should be characterized by a low Abbe number as close as possible to the grating’s Abbe number, and for the second blazed substructure the material has to show a strong deviation from the normal line of glasses concerning the values of the relative partial dispersion. Systematic conditions of material selection are presented and discussed for specific material combinations of the filled grating. The tailored choice of specific dispersion characteristics allows the fundamental design of double-grating structure with a high and uniform diffraction efficiency across a broad wavelength range.

Disturbing effects, such as reflection losses at the interfaces of the involved materials or at the ambient air, have been neglected hereby. In order to take these aspects into account and simultaneously to attenuate their efficiency reduction effect, it is possible to introduce additional intermediate and top layers. Graded-index structures or tailored multilayer stacks may serve as such intermediate and top layers. Although this aspect is not in the center of the present contribution, we discussed an exemplary implementation by rigorous numerical calculations. The non-optimized layer stack added to the metal double-blazed grating shows in the maximum a slightly reduced efficiency compared to classical Al-blazed structure, but it is also characterized by a high efficiency across a very broad wavelength range. Besides the maximum efficiency, a high efficiency over a broad spectral range may be also advantageous for the application in spectroscopy. The important next steps will involve an optimization of the intermediate and top layer structure as well as the development of a microstructuring process for the realization of the metal double-blazed grating. The presented theory allows us to choose specific materials with respect to the targeted application and serves as a good starting point for advanced optimization procedures.

## Disclosures

The authors declare no conflicts of interest.

## REFERENCES

**1. **B. Kleemann and R. Güther, “Metal gratings with dielectric coating of variable thickness within a period,” J. Mod. Opt. **38**, 897–910 (1991). [CrossRef]

**2. **C. Londono and P. P. Clark, “Modeling diffraction efficiency effects when designing hybrid diffractive lens systems,” Appl. Opt. **31**, 2248–2252 (1992). [CrossRef]

**3. **D. A. Buralli and G. M. Morris, “Effects of diffraction efficiency on the modulation transfer function of diffractive lenses,” Appl. Opt. **31**, 4389–4396 (1992). [CrossRef]

**4. **R. Steiner, “Die Entwicklung der holografischen Gitter bei Carl Zeiss in Jena. Teil 1: Grundlagen und Technologie der Plangitter,” in *Jenaer Jahrbuch zur Technik- und Industriegeschichte*, M. Steinbach, ed. (Vopelius Jena, 2014), Vol. 17, pp. 55–124.

**5. **S. M. Ebstein, “Nearly index-matched optics for aspherical, diffractive, and achromatic-phase diffractive elements,” Opt. Lett. **21**, 1454–1456 (1996). [CrossRef]

**6. **Y. Arieli, S. Noach, S. Ozeri, and N. Eisenberg, “Design of diffractive optical elements for multiple wavelengths,” Appl. Opt. **37**, 6174–6177 (1998). [CrossRef]

**7. **Y. Arieli, S. Ozeri, T. Eisenberg, and S. Noach, “Design of diffractive optical elements for wide spectral bandwidth,” Opt. Lett. **23**, 823–824 (1998). [CrossRef]

**8. **T. Nakai and H. Ogawa, “Research on multi-layer diffractive optical elements and their application to camera lenses,” in *Diffractive Optics and Micro-Optics* (Optical Society of America, 2002), paper DMA2.

**9. **H. P. Herzig and A. Schilling, “Optical systems—design using microoptics,” in *Encyclopedia of Optical Engineering*, R. G. Driggers, ed. (Marcel Dekker, 2003), Vol. 2, pp. 1830–1842.

**10. **G. I. Greisukh, E. G. Ezhov, S. A. Stepanov, V. A. Danilov, and B. A. Usievich, “Spectral and angular dependences of the efficiency of diffraction lenses with a dual-relief and two-layer microstructure,”J. Opt. Technol. **82**, 308–311 (2015). [CrossRef]

**11. **D. Werdehausen, S. Burger, I. Staude, T. Pertsch, and M. Decker, “Dispersion-engineered nanocomposites enable achromatic diffractive optical elements,” Optica **6**, 1031–1038 (2019). [CrossRef]

**12. **D. Werdehausen, S. Burger, I. Staude, T. Pertsch, and M. Decker, “General design formalism for highly efficient flat optics for broadband applications,” Opt. Express **28**, 6452–6468 (2020). [CrossRef]

**13. **B. Zhang, M. Piao, and Q. Cui, “Achromatic annular folded lens with reflective-diffractive optics,” Opt. Express **27**, 32337–32348 (2019). [CrossRef]

**14. **O. Sandfuchs and R. Brunner, “Efficiency-achromatized reflective dispersion grating by a double-blaze configuration: theoretical conditions for optimal material selection,” Asian J. Phys. **25**, 897–906 (2016).

**15. **T. Glaser, “High-end spectroscopic diffraction gratings: design and manufacturing,” Adv. Opt. Technol. **4**, 25–46 (2015). [CrossRef]

**16. **J. R. DeVore, “Refractive indices of rutile and sphalerite,” J. Opt. Soc. Am. **41**, 416–419 (1951). [CrossRef]

**17. **M. Zhang, G. Lin, C. Dong, and L. Wen, “Amorphous TiO_{2} films with high refractive index deposited by pulsed bias arc ion plating,”Surf. Coat. Technol. **201**, 7252–7258 (2007). [CrossRef]

**18. **H. R. Phillip and E. A. Taft, “Kramers-Kronig analysis of reflectance data for diamond,” Phys. Rev. **136**, A1445 (1964). [CrossRef]

**19. **D. L. Wood and K. Nassau, “Refractive index of cubic zirconia stabilized with yttria,” Appl. Opt. **21**, 2978–2981 (1982). [CrossRef]

**20. **I. H. Malitson and M. J. Dodge, “Refractive index and birefringence of synthetic sapphire,” J. Opt. Soc. Am. **62**, 1405 (1972).

**21. **R. Brunner, O. Sandfuchs, C. Pacholski, C. Morhard, and J. Spatz, “Lessons from nature: biomimetic subwavelength structures for high-performance optics,” Laser Photonics Rev. **6**, 641–659 (2012). [CrossRef]

**22. **O. Sandfuchs, A. Pesch, and R. Brunner, “Rigorous modelling of dielectric and metallic blaze gratings in the intermediate structure regime,” Proc. SPIE **6675**, 667501 (2007). [CrossRef]

**23. **B. H. Kleemann, M. Seesselberg, and J. Ruoff, “Design concepts for broadband high-efficiency DOEs,” J. Eur. Opt. Soc. **3**, 08015 (2008). [CrossRef]

**24. **http://www.schott.com/advanced_optics/english/abbe_datasheets/schott-datasheet-n-bk7.pdf, 2020.

**25. **http://www.schott.com/advanced_optics/english/abbe_datasheets/schott-datasheet-f2.pdf, 2020.

**26. **P. Hartmann, R. Jedamzik, S. Reichel, and B. Schreder, “Optical glass and glass ceramic historical aspects and recent developments: a Schott view,” Appl. Opt. **49**, D157–D176 (2010). [CrossRef]

**27. **P. Hartmann, “Optical glass: deviation of relative partial dispersion from the normal line—need for a common definition,” Opt. Eng. **54**, 105112 (2015). [CrossRef]

**28. **http://www.schott.com/advanced_optics/english/abbe_datasheets/schott-datasheet-p-sf67.pdf, 2020.

**29. **http://www.schott.com/advanced_optics/english/abbe_datasheets/schott-datasheet-n-fk51a.pdf, 2020.

**30. **“Cytop Technical Information," AGC Chemicals Europe, Ltd., 2020, http://www.agcce.com/cytop-technical-information/.

**31. **C. Maletzko, *Ultrason Optical Properties* (BASF Aktiengesellschaft, 2020).

**33. **https://refractiveindex.info/, 2020.

**34. **D. C. O’Shea, T. J. Suleski, A. D. Kathman, and D. W. Prather, *Diffractive Optics—Design, Fabrication, and Test* (SPIE, 2004), Chap. 4.4.1.

**35. **Osires Software, “UNIGIT—a rigorous grating solver,” 2020, https://www.unigit.net/.