1. INTRODUCTION

The term “efficiency achromatization” describes the process of tailoring a diffractive structure to provide high diffraction efficiency in a selected diffraction order over a wide wavelength range by optimizing geometrical parameters in combination with choosing appropriate materials. This “efficiency achromatic” behavior is an essential requirement for using diffractive optical elements (DOEs) in high-quality broadband imaging systems. Only if high diffraction efficiency can be ensured over a wide spectral range can the advantages of imaging DOEs, such as their small negative Abbe number, be exploited to reduce aberrations, shrink volume, and decrease the number of employed elements in sophisticated optical systems. Efficiency-achromatized DOEs are already being used with great success in commercially available high-end photographic lenses fabricated by the Japanese companies Canon and Nikon [1,2]. Suitable diffractive structures combine dual- or multi-layers of different materials with interlaced or stacked sawtooth profiles (e.g., [3–7]). Different design rules and “dispersion engineering” strategies have been proposed to find an optimized design of multilayer structures, which includes minimizing their profile depths [8–10]. In addition to transmissive diffractive lenses for imaging applications, the approach of achromatizing efficiency was also applied to reflective dispersion gratings [11,12]. Besides investigations on the efficiency behavior of structures based on established materials, such as optical polymers and compression molding glasses [13], research has also been focused on enhancing optical materials. In particular, the ability to tailor the dispersion properties of nanocomposites offers very high potential for beneficial use in efficiency achromatization [14–16] and optimization of optical imaging systems [17]. In addition to the broadband applications already mentioned, DOEs offer further advantageous possibilities for imaging systems that are impossible or difficult to achieve with purely refractive systems. For example, for certain applications, the working principle of the imaging system is based on the wavelength selectivity of the applied DOE. The principle of wavelength selection does not simply imply a broadband achromatization of efficiency, but requires a tailored efficiency distribution in different specific diffraction orders for selected wavelengths or wavelength ranges. For example, a DOE can be integrated into an imaging system that provides high diffraction efficiency in a first order for a first wavelength, while simultaneously, high diffraction efficiency is achieved in a second order for a different wavelength. In particular, when two separate wavelength ranges address the zeroth and first orders, “wavelength selectivity” allows partial decoupling in the lens design and nearly independent correction for the two wavelengths. This approach has been applied to an optical data storage pick-up system including a DOE that provides high diffraction efficiency in the first diffraction order for 650 nm and simultaneously in the zeroth order for 405 nm [18]. Tailoring the element and therefore distributing the efficiency between the different diffraction orders allows to obtain different numerical apertures and different image distances for the two wavelengths. Other examples concern microscope lenses used for automated deep-ultraviolet (UV) inspection comprising two complementary DOEs [19,20]. Hereby, the first DOE is acting as a simple refractive element (zeroth order) for a near-infrared (IR) wavelength used for the autofocus system, and on the other hand, for the deep-UV wavelength, the first order is used for imaging. The second DOE shows the reversed behavior. So far, detailed approaches for the selection of suitable diffractive structures to achieve a defined wavelength selectivity are missing.

In this paper, we investigate concepts for a tailored contribution of diffraction efficiencies to selective diffraction orders and across different wavelength ranges. The basic structures used for this efficiency engineering are interlaced double-layer single-relief blazed gratings. An essential characteristic is that the two materials of the double layer show intersecting or converging dispersion curves. In the first part of the paper, fundamental considerations of chromatic diffraction efficiency of a double-layer single-relief grating composed of materials featuring intersecting dispersion curves are presented as a basic concept for flexible adjustment of chromatic diffraction efficiency. Subsequently, analytic examinations of chromatic diffraction efficiency of the double-layer single-relief blazed grating based on scalar approximations reveal a guideline to optimize diffraction efficiency for various diffraction orders and across different wavelength ranges by choosing specific material dispersions and adjusting the grating depth. Finally, these findings are transferred to scalar efficiency calculations of combinations of genuine materials including inorganic glasses, optical polymers, nanocomposites, and specific optical materials. The analytic investigations and scalar calculations revealed that, in fact, efficiency maxima can be generated in the zeroth, ±1st, and ±2nd diffraction orders at specific wavelengths or across different wavelength ranges by combining selected materials listed above with various spectral positions of the intersection of the dispersion curves and different dispersion properties and by adjusting the grating depth. These tailored efficiency contributions enable wavelength selection and partial decoupling of the optical design for different wavelengths in optical systems.

2. BASIC CONSIDERATIONS—DOUBLE-LAYER SINGLE-RELIEF BLAZED GRATINGS COMPOSED OF MATERIALS SHOWING INTERSECTING DISPERSION CURVES

For a transmissive diffractive structure of period $p$ illuminated by light of wavelength $\lambda$ at an angle of incidence ${\gamma _i}$, the grating Eq. (1) describes propagation angles ${\gamma _{t,m}}$ on the output side:

Here, ${n_{\rm{in}}}$ and ${n_{\rm{out}}}$ represent the refractive indices of the media in front of and behind the periodic structure. The set of propagation angles resulting from the different diffraction orders $m$ ($m$ being an integer number) characterizes the potential directions in which light can be detected. The grating equation does not provide any information about the distribution of light into the diffraction orders and therefore about efficiency. The efficiency mainly depends on the geometry of the diffractive profile and the optical material properties. Very high diffraction efficiencies of up to 100% for a selected diffraction order can be achieved for sawtooth-shaped (blazed) structures. As a rule of thumb for such blazed structures, high diffraction efficiency is achieved exactly when the local solution of the law of refraction to the sawtooth profile results in the same direction of deflection as the global application of the grating equation to the periodic structure.

This provides a simple and illustrative explanation for the effect of a double-layer single-relief blazed grating in which two materials with intersecting dispersion curves are used. Figure 1(a) schematically shows the dispersion curves of two materials used for the layout of such a double-layer single-relief blazed grating sketched in Fig. 1(b). The bottom-layer and top-layer materials are characterized by refractive indices ${n_1}(\lambda)$ and ${n_2}(\lambda)$, respectively. The grating is illuminated from below, and the corresponding diffraction orders propagate upwards. At the short wavelength ${\lambda _1}$ (blue), the refractive index of the first material is larger than that of the second material (${n_1}({{\lambda _1}}) \gt {n_2}({{\lambda _1}})$). This means that for a single period (single local prism), the refractive index difference ${n_1}({{\lambda _1}}) - {n_2}({{\lambda _1}})$ is positive, and therefore, the resulting phase change decreases from left to right. Therefore, light of wavelength ${\lambda _1}$ is diffracted efficiently to the upper left direction into positive orders. At a suitable profile depth, all light is deflected to the ${+}{1}$st order [indicated by the blue arrow in Fig. 1(b)]. In contrast, at the long wavelength ${\lambda _3}$ (red), the refractive index of the first material is smaller than that of the second material [${n_1}({{\lambda _3}}) \lt {n_2}({{\lambda _3}})$] and the refractive index difference ${n_1}({{\lambda _3}}) - {n_2}({{\lambda _3}})$ becomes negative. Hence, light of wavelength ${\lambda _3}$ is diffracted into the negative diffraction orders [red arrow in Fig. 1(b)]. Finally, at the intermediate wavelength ${\lambda _2}$ (green), both materials have the same refractive index ${n_1}({{\lambda _2}}) = {n_2}({{\lambda _2}})$ and the index difference ${n_1}({{\lambda _2}}) - {n_2}({{\lambda _2}})$ becomes zero. In this case, the outgoing light propagates efficiently in the direction of the zeroth order (green arrow).

 

Fig. 1. (a) Exemplary dispersion curves ${n_1}(\lambda)$ and ${n_2}(\lambda)$ for two materials and three wavelengths; (b) diffraction on a double-layer single-relief blazed grating according to the material characteristics from (a).

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The wavelength-dependent distribution of light into different diffraction orders can be particularly advantageous for hybrid optics in which refractive lenses and imaging DOEs are combined. For example, an imaging system can initially be designed to be completely classically refractive for the intermediate wavelength. To achieve a desired different imaging effect for a second wavelength, a double-layer single-relief blazed grating can be applied to a suitable surface of the already introduced refractive lenses. If the refractive index difference of the double-layer grating is zero at this intermediate wavelength, no diffraction effects need to be considered at that wavelength, and the imaging performance is preserved. On the other hand, the second wavelength is efficiently diffracted to a selected order ($m \ne 0$) by the grating structure. Here, the radial distribution of spatial frequencies of the diffractive structure across the selected lens area can be additionally used for beam shaping. Especially, the attribution of one wavelength to the zeroth order and a second one to a higher order (e.g., first order) allows, at least partially, to separate the optics design of the imaging system for both wavelengths. If instead of the zeroth diffraction order, a different order is selected and the wavelength distribution differentiates—between first and second orders, for example—then the spatial period distribution of the diffractive structure influences both wavelengths and their optical designs are linked.

3. GUIDELINE FOR DESIGNING MULTI-ORDER DOUBLE-LAYER SINGLE-RELIEF BLAZED GRATINGS

Applying scalar approximation, which is suitable for large grating periods ($p \gt 5\lambda$) [21] and small heights ($10h \lt p$) [9], the wavelength-dependent diffraction efficiency $\eta (\lambda)$ of a double-layer single-relief blazed grating in the far field for the $m$th diffraction order for normal incidence is given by [22]

Here, $h$ denotes the height of the sawtooth profile. In the following, the refractive index difference ${n_1}({\lambda _i}) - {n_2}({\lambda _i})$ at a specific wavelength ${\lambda _i}$ is symbolized by $\Delta n({\lambda _i})$. The first objective is to achieve maximum diffraction efficiency at a first wavelength ${\lambda _1}$ for a given diffraction order ${m_1}$. Since the sinc-function, and thus also the efficiency, is at its maximum where its argument is zero, the respective optimum grating depth $h$ can be calculated:

Simultaneously, the argument of the sinc-function must yield positive or negative integers to create minimum efficiency (corresponding to its roots) at a second design wavelength ${\lambda _2}$ for the same order ${m_1}$, which leads to Eq. (4):

From rearranging Eq. (4), it follows that minimum efficiency at design wavelength ${\lambda _2}$ in order ${m_1}$ corresponds to maximum efficiency in order ${m_2}$ at design wavelength ${\lambda _2}$ and the relation ${m_2} = {m_1} + z$.

Therefore, it is possible to achieve high efficiencies for various wavelengths or ranges of wavelengths in different diffraction orders, instead of using just a single order for efficiency achromatization (as was done in previous studies). For example, high efficiencies should be enabled at wavelength ${\lambda _1}$ in order ${m_1}$ and simultaneously at wavelength ${\lambda _2}$ in order ${m_2}$. For this purpose, design guidelines are developed that take into account the different dispersion curves of genuine materials. In addition to the specific combinations of materials with intersecting dispersion curves, there are also solutions for selected diffraction orders where intersecting dispersion curves are not required. For the diffraction order of wavelength ${\lambda _1}$, we assume ${m_1} = 1$. Inserting Eq. (3) into Eq. (4) yields Eq. (5), which is analyzed in the following:

Both cases ${\lambda _1} \lt {\lambda _2}$ and ${\lambda _2} \lt {\lambda _1}$ are distinguished, and the options ${m_2} = 0$, ${m_2} \lt 0$, and ${m_2} \gt 1$ are analyzed successively. Wavelengths are positive ($\lambda \gt 0$) at all times. Figure 2 provides an overview of the cases discussed. The first column lists the diffraction orders, whereas the second column contains the efficiency conditions for the wavelengths and refractive index differences derived from Eq. (5). The right side of the figure schematically shows the required dispersion curves of the materials involved.

 

Fig. 2. Efficiency conditions and possible dispersion curves for ${m_1} = 1$ and different ${m_2}$.

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Of particular interest is the case ${m_2} = 0$ (first row in Fig. 2). Here, the distribution of the periods of the diffractive structure has no influence on the imaging properties of the system. The lens design can be at least partially decoupled for both diffraction orders so that the optimization procedures can be performed separately for the two wavelengths. According to Eq. (5), the refractive index difference has to be zero at the second wavelength [$\Delta n({{\lambda _2}}) = 0$; see also column 2 of Fig. 2]. With respect to real dispersion curves, the requirement $\Delta n({{\lambda _2}}) = 0$ can be achieved in two ways. First, the two dispersion curves can simply intersect at a specific crossing point at wavelength ${\lambda _2}$ (see 1A in Fig. 2). This approach is advantageous for a lens design of an imaging system specified exactly for two different wavelengths. For potential applications, it is particularly interesting if the refractive index difference is nearly zero over an extended wavelength range (1B). The corresponding imaging system can be optimized over this extended wavelength range using classical approaches; for wavelength ${\lambda _1}$, the diffractive structure can be used highly efficiently for independent beam steering in the first diffraction order. It has to be mentioned that the correction of chromatic aberrations requires ${m_2} \ne 0$.

Next, the case ${m_2} = - 1$ is considered (second row in Fig. 2). Here, a positive optical power of the diffractive structure for wavelength ${\lambda _1}$ corresponds to a negative optical power for ${\lambda _2}$ and vice versa. The possibilities of advantageously using this approach for lens design will not be analyzed in detail here. In short, potentially, this approach can be used to efficiently exclude an interfering wavelength from the system (${-}{1}$ st order may direct disturbing light into light traps) or to create negative power for imaging. In this case, it is necessary that the two refractive index differences $\Delta n({{\lambda _1}})$ and $\Delta n({{\lambda _2}})$ have opposite signs so that the efficiency condition [Eq. (5)] is fulfilled. This means that the two target wavelengths must be located on opposite sides of the intersection of the dispersion curves. For ${\lambda _1} \lt {\lambda _2}$ (2A) follows $\Delta n({{\lambda _1}}) \lt |\Delta n({{\lambda _2}})|$. Since the slope of the dispersion curves is steeper for short wavelengths than for longer ones, ${\lambda _1}$ must be closer to the intersection than ${\lambda _2}$. To ensure small structure heights $h$ and provide additional design freedom, large absolute values for the refractive index differences are advantageous. This requires a combination of a highly dispersive material (small Abbe number) and a second material with a flat dispersion curve (high Abbe number). The selection of highly dispersive optical materials is limited, which also restricts the availability of potential combinations of materials. The second case ${\lambda _2} \lt {\lambda _1}$ (2B) implies that the longer wavelength ${\lambda _1}$ is efficiently diffracted to the first order, and it follows that the second material has to exhibit a smaller Abbe number than the first. Additionally, the distance between ${\lambda _2}$ and the intersection wavelength has to be smaller than the respective distance for ${\lambda _1}$.

Also for ${m_2} = - 2$ (third row in Fig. 2), opposite signs for the refractive index differences $\Delta n({{\lambda _1}})$ and $\Delta n({{\lambda _2}})$, and thus intersecting dispersion curves, are required. The case ${\lambda _1} \lt {\lambda _2}$ (3A) is basically comparable to the corresponding preceding consideration for ${m_2} = - 1$. Here, the adapted relation $2\Delta n({{\lambda _1}}) \,\lt\def\LDeqbreak{} |\Delta n({{\lambda _2}})|$ even increases the requirement of a small distance between ${\lambda _1}$ and the intersection wavelength. The reverse situation ${\lambda _2} \lt {\lambda _1}$ requires further distinction. For ${\lambda _2} \lt {\lambda _1} \lt 2{\lambda _2}$ (3B), it follows that $2\Delta n({{\lambda _1}}) \gt |\Delta n({{\lambda _2}})| \gt \Delta n({{\lambda _1}})$, and the distance of wavelength ${\lambda _2}$ from the intersection wavelength can be increased. The situation in 3C describes the special case for ${\lambda _1} = 2{\lambda _2}$. Here, the refractive index differences for both reference wavelengths ${\lambda _1}$ and ${\lambda _2}$ are equal. If ${\lambda _1} \gt 2{\lambda _2}$ (3D), then $|\Delta n({{\lambda _2}})|$ is smaller than $\Delta n({{\lambda _1}})$ and the distance from ${\lambda _2}$ to the intersection wavelength becomes small again.

Finally, the case ${m_2} = 2$ is considered (last row in Fig. 2). Since ${\lambda _1} \gt 0$ and ${\lambda _2} \gt 0$, both refractive index differences must have the same sign, and the dispersion curves do not intersect. If ${\lambda _1} \lt {\lambda _2}$, the following relation results from the efficiency condition $2\Delta n({{\lambda _1}}) \lt \Delta n({{\lambda _2}})$ (4A). Again, materials with small and large Abbe numbers have to be combined, which corresponds to divergent dispersion curves in the regarded wavelength range. More precisely, both the refractive index and the Abbe number must be larger for material 1 than for material 2. Material combinations with these characteristics are not easily accessible. For the reverse situation ${\lambda _2} \lt {\lambda _1}$, again further distinction is necessary. If ${\lambda _2} \lt {\lambda _1} \lt 2{\lambda _2}$ (4B), the dispersion curves of both materials converge for increasing wavelength. Material 1 shows a steeper dispersion curve and larger values for the refractive index than material 2. Cell 4C shows the special case for ${\lambda _1} = 2{\lambda _2}$. Here, the refractive index differences for the two reference wavelengths ${\lambda _1}$ and ${\lambda _2}$ are equal. Taking also into account the grating Eq. (1), this particular case yields that both wavelengths are diffracted highly efficiently into exactly the same direction. The last option for ${m_2} = 2$, ${\lambda _1} \gt 2{\lambda _2}$ (4D) also requires the challenging combination of material 1 with a large refractive index and large Abbe number and highly dispersive material 2 with a low refractive index. Efficiency, contribution to the different diffraction orders, and propagation direction for other wavelengths depend on the specific shape of the two dispersion curves. Possibly of particular interest are combinations where both materials have similar Abbe numbers and one material is characterized by normal and the second by abnormal relative partial dispersion. In this case, the contributions of the reference wavelengths ${\lambda _1}$ and ${\lambda _2}$ can be separated from the other wavelengths.

4. MULTI-ORDER CHROMATIC DIFFRACTION EFFICIENCIES CALCULATED FOR COMBINATIONS OF GENUINE MATERIALS

Using the guidelines from the last section and considering actual available materials, the wavelength-dependent diffraction efficiencies for some of these materials in different diffraction orders are examined next. Even though it is not very simple to find materials with intersecting dispersion curves, some suitable pairs of materials are available. Relevant combinations can be characterized by very different Abbe numbers but comparable refractive indices. For our considerations, we included inorganic glasses, layer materials, crystals, polymers, nanocomposites, and specific high-index optical liquids. Aspects concerning the possible fabrication of double-layer single-relief blazed gratings from these material combinations are of only secondary interest.

Figure 3 shows an Abbe diagram in which optical materials are positioned as a function of their refractive indices ${n_d}$ and Abbe numbers ${\nu _d}$. The materials specifically considered in this contribution are represented by enlarged symbols. For orientation, a selection of typical inorganic SCHOTT glasses is also included (small spots). In addition to the individual values, two special curves are shown in the diagram. The red dotted curve displays the refractive index and the Abbe number for nanocomposites of a polystyrene (PS) matrix and indium tin oxide (ITO) nanoparticles for varying mixing ratios of up to 35%. The values for this curve are taken from [14] and are calculated theoretically. The green curve displays values from a series of commercially available refractive index liquids (series B of [23]). For our purposes, both curves are interesting: first, they are in the important range of small Abbe numbers, and second, their continuity allows the precise choice of a specific value. From both curves, we have selected individual suitable examples. It should be mentioned that both curves are exemplary. For nanocomposites, it is possible to combine different polymer matrices with different nanoparticles, and refractive index liquids can be selected from several series.

 

Fig. 3. Abbe diagram showing refractive indices and values for the Abbe number for a variety of materials used in this contribution (enlarged symbols). The green line represents refractive index liquids series B from [23], and red dots different nanocomposite mixtures of PS and ITO (see text). For orientation, a selection of typical inorganic SCHOTT glasses is displayed as small blue dots.

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The diffraction efficiencies of double-layer single-relief blazed gratings with parameters of genuine materials were calculated using Eq. (2). The wavelength ranges were selected according to the transparency of the material composition. The zeroth, $\pm 1$st, and $\pm 2$nd orders were considered and are displayed in false-color plots. The bottom of the figure shows a superposition diagram of all considered orders for a selected grating depth $h$ and specific design wavelength ${\lambda _0}$. Unless otherwise mentioned, the specific values of the dispersion curves for the individual materials were taken from [24] and are shown on top of each of the following figures. To calculate the efficiency for any wavelength, polynomial functions were derived from tabulated values for all materials. The developed and used software tool was based on the programming language Python.

The first plot in Fig. 4 shows the results for three combinations of materials with intersecting or overlapping dispersion curves with each column of the figure being assigned to one combination. First, the combination of the two inorganic glasses N-LaSF44 and N-SF6 was investigated (left column of Fig. 4). Both glasses have similar refractive indices at the d-line (587.56 nm) but different Abbe numbers (N-LaSF44: ${n_d} = 1.8042$, ${\nu _d} = 46.50$; N-SF6: ${n_d} = 1.8052$, ${\nu _d} = 25.36$). The dispersion curves intersect at $\lambda = 630\;{\rm nm}$, and their layout can be assigned to case 1A in Fig. 2. Grating depths deeper than 10 µm are necessary to obtain contributions for the first diffraction order. As the design wavelength for the first order approaches the intersecting wavelength, the grating depth increases sharply. The zeroth order naturally has a maximum at the intersection wavelength, but also achieves high efficiencies in the wavelength range above. The small change in zeroth order efficiency for long wavelengths can be attributed to the nearly parallel dispersion curves with small refractive index differences in this region. To get a contribution for the second order, grating depths deeper than 20 µm are necessary. In summary, the diagram below shows the efficiency for all orders for a grating depth of 15 µm, exhibiting a sharp efficiency peak for the first order at a short wavelength and high efficiencies over a wide range at longer wavelengths in the zeroth order. In a lens design for an optical system, this behavior can potentially be used to separate a continuous long wavelength region from a very specific short wavelength region by the diffraction orders.

 

Fig. 4. Dispersion curves (top row) and diffraction efficiency for different orders of two combinations of inorganic glasses (N-SF6/N-LaSF44 and N-SF6/N-LaF32, left and right columns, respectively), and a combination of polystyrene and an inorganic glass (PS/N-SK5, center column). The last row shows the multi-order chromatic diffraction efficiency for an exemplary design wavelength ${\lambda _0}$ and grating depth $h$ for each combination.

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Methods for fabricating such a double-layer grating from two inorganic glasses have not been established. It seems much easier to combine an inorganic glass and a polymer. Therefore, the center column of Fig. 4 shows the results for gratings based on the inorganic glass N-SK5 and PS. PS was chosen because of its small Abbe number. N-SK5 has a similar refractive index and has the largest difference in Abbe number from PS of any inorganic glass (N-SK5: ${n_d} = 1.5891$, ${\nu _d} = 61.27$; PS: ${n_d} = 1.5916$, ${\nu _d} = 29.53$). The dispersion curves intersect at 640 nm. Due to the small wavelength difference, a profile depth of about 20 µm is already required at the short design wavelength of 410 nm to obtain a first order contribution. With the exception of the intersection wavelength and the required profile depth, the efficiencies for the different orders show behavior similar to the material combination discussed before and can also be assigned to case 1A.

The third column of Fig. 4 again shows the results of a combination of two inorganic glasses. The two glasses (N-LaF32: ${n_d} = 1.7946$, ${\nu _d} = 45.53$; N-SF6: ${n_d} = 1.8052$, ${\nu _d} = 25.43$) have the peculiarity that their refractive indices differ significantly in the short-wave range, but overlap in the long-wavelength range (compare cases 1B and 4B of Fig. 2). Due to the great refractive index difference in the short-wavelength range, a contribution to the first diffraction order is already achieved with relatively moderate profile depths ($\approx 8\;{\unicode{x00B5}{\rm m}}$ depth at 400 nm). For larger profile depths, even a second order contribution can be observed. Due to the vanishing refractive index difference, a zeroth order efficiency of nearly 100% is achieved for all wavelengths above 770 nm. This approach could be optimally used for the lens design example mentioned above.

The results presented in Fig. 5 show a completely different wavelength-dependent efficiency behavior for the different diffraction orders of the double-layer grating. In each of the three examples shown, ITO is used as an essential component of the material combinations. ITO is widely used for opto-electronic applications and is one of the few materials that are both transparent and conductive. For our purposes, the extremely high dispersion of ITO is important, characterized by an Abbe number ${\lt}{10}$. ITO is used as a coating material with optical properties depending on the specific process conditions. For our investigations, we refer to the values from [24] (ITO: ${n_d} = 1.8270$, ${n_F} = 1.9282$, ${n_C} = 1.7590$, ${\nu _d} = 4.9$). The transmittance of ITO covers the wavelength range from the UV (${\approx} 350\;{\rm nm}$) up to the near-IR (${\approx}1 \;{\unicode{x00B5}{\rm m}}$).

 

Fig. 5. Dispersion curves (top row) and diffraction efficiency for different orders of two combinations of inorganic glasses and ITO (ITO/N-LaSF44 and ITO/N-SK5, left and center columns, respectively), and a combination of a nanocomposite and an inorganic glass (PS/ITO (65%/35%)/N-LaK7, right column). The last row represents the multi-order chromatic diffraction efficiency for an exemplary design wavelength ${\lambda _0}$ and grating depth $h$ for each combination.

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We investigated the combination of ITO with inorganic glasses: (a) ITO and N-LaSF44; (b) ITO and N-SK5 (N-LaSF44: ${n_d} = 1.8042$, ${\nu _d} = 46.50$; N-SK5: ${n_d} = 1.5891$, ${\nu _d} = 61.27$). In the upper left of Fig. 5, the dispersion curves for the three materials are shown, which can be compared to the cases 1A, 2A, and 3A in Fig. 2, as well as 4B as long as only the region left of the intersection point is considered. In the left and center columns of Fig. 5 the efficiency behavior of ITO/N-LaSF44 and ITO/N-SK5 grating combinations are shown, respectively. In addition to the zeroth, ${+}{1}$st, and ${+}{2}$nd orders, contributions are also visible in the ${-}{1}$st and ${-}{2}$nd orders. For the different orders, both combinations of materials show very similar behavior with sharply defined regions for the efficiency maxima. There is no diffraction order that provides high diffraction efficiency over an extended wavelength range for a selected grating depth. This behavior can be attributed to the strong divergence of the dispersion curves of the material combination near the intersection wavelength. The different intersection wavelengths of the dispersion curves of the inorganic glasses with ITO also shift the position of the zeroth order efficiency maximum (intersection wavelength and spectral position of the zeroth order maximum: ITO/N-LaSF44 at 0.61 µm; ITO/N-SK5 at 0.81 µm). Determining the intersection wavelength between ITO and inorganic glass cannot be done by choosing the specific glass type alone. An alternative is to use ITO as a component of a nanocomposite. The dispersion curve for the nanocomposite can be tailored by the choice of material for the polymer matrix and by the mixing ratio between polymer and nanoparticles [14]. As a theoretical example, we chose a nanocomposite of ITO and PS with a mixing ratio of 35/65. For the design of the double-layer grating, this nanocomposite was combined with the inorganic glass N-LaK7 (N-LaK7: ${n_d} = 1.6516$, ${\nu _d} = 58.52$). The dispersion curves of these materials are shown in Fig. 5, top right. For comparison, the original curve of ITO is additionally plotted. The dispersion curves of the nanocomposite and N-LaK7 intersect at about 650 nm. For the corresponding double-layer gratings, the maximum efficiency for the zeroth order can also be observed at this wavelength (see right column of Fig. 5). Similar to the ITO/glass combinations discussed before, the sharply defined regions for the efficiency maxima occur at the different diffraction orders. For all discussed combinations of ITO, including ITO nanocomposites, with inorganic glasses, moderate grating depths of a few micrometers are already sufficient to obtain contributions for higher diffraction orders. This behavior is due to the extremely steep dispersion curve of ITO. For imaging diffractive structures, it is often crucial that certain aspect ratios of grating depth and period are maintained to avoid shadowing effects. Therefore, a small grating depth is advantageous, and also usually easier to manufacture. The efficiency behavior of the different diffraction orders described here can be used for diffractive imaging systems where different imaging properties must be achieved for very specific wavelengths.

In a next step, we investigated the efficiency behavior of a diffractive double-layer structure consisting of a substrate grating of an inorganic glass filled with a high-index liquid. As an example, we chose the liquid named “B 1.67” from the manufacturer Cargille [23]. The high-index liquid B 1.67 is composed of 1-bromonaphthalene and 1-iodonaphthalene. The wavelength-dependent refractive index of this liquid was taken from the manufacturer’s data. The number in the product name indicates the refractive index for the d-line (refractive index liquid B 1.67: ${n_d} = 1.670$, ${\nu _d} = 19.8$). First, the combination of B 1.67 with the inorganic glass N-LaK7 is analyzed. For small wavelengths, the refractive indices of both materials deviate significantly from each other; in the range larger than 1 µm, both dispersion curves show nearly identical behavior (see Fig. 6, top left; compare Fig. 2 1B, 4B). The efficiency characteristics for the different diffraction orders are similar to those reported for the combination N-SF6/N-${\rm LaF32}$ (compare right column in Fig. 4). Due to large refractive index difference in the short wavelength range, smaller grating depths are sufficient for the liquid–glass combination. A sharp efficiency contribution in the first order is already observable for grating depths larger than 4 µm in the short wavelength range. For a grating depth of more than 10 µm, a second order contribution also occurs. The indistinguishable dispersion curves in the range larger than 1 µm are associated with an efficiency of 100% for the zeroth order.

 

Fig. 6. Dispersion curves (top row) and diffraction efficiency for different orders of three combinations of a high-index liquid and inorganic glasses (B 1.67/N-LaK7, B 1.67/N-LaK34, and B 1.67/N-LaK12, left, center, and right columns, respectively). The last row represents the multi-order chromatic diffraction efficiency for an exemplary design wavelength ${\lambda _0}$ and grating depth $h$ for each combination.

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The combination of B 1.67 with the inorganic glass N-LaK34 (N-LaK34: ${n_d} = 1.7292$, ${\nu _d} = 54.50$) shows different characteristics (see center column of Fig. 6). Their dispersion curves do not intersect in the visible spectral range (VIS), but diverge at wavelengths above 400 nm (cases 4A and 4D in Fig. 2). Due to the missing intersection in the dispersion curves, no efficiency contribution occurs at the zeroth order. The efficiency plots for the first and second orders show a “concave” form with a steep slope for short wavelengths and a flat slope for long wavelengths. Due to the flat minimum, high efficiency is achieved in the first order in the range between 480 and 760 nm for a grating depth of $\approx 10\;{\unicode{x00B5}{\rm m}}$. For a grating depth of 20 µm, a multi-faceted distribution of efficiencies between diffraction orders is obtained. The first order shows a first peak efficiency at 410 nm and a second broad maximum in the range of 1.5 µm. For the second order, a continuous high efficiency is observed between 500 and 750 nm, which moderately drops for longer wavelengths. As a final example of the liquid–inorganic glass system, the combination of B 1.67 with N-LaK12 is studied (N-LaK12: ${n_d} = 1.6779$, ${\nu _d} = 55.20$) [see right column of Fig. 6; compare cases 1A, 2A, and 4B (left from the intersection point)]. The dispersion curves of both materials intersect at 550 nm, which coincides with a zeroth order efficiency maximum. Due to the large differences in the refractive indices of both materials in the short-wavelength range and due to the different slope of the dispersion curves, the ${+}{1}$st order shows a sharp maximum already at a moderate depth. In contrast to the other efficiency diagrams presented in this contribution, grating depths of up to 50 µm are considered for this special case, which allows to observe also a contribution to the ${-}{1}$st order. In particular, at a grating depth of 40 µm, a broad efficiency maximum is observable in the ${-}{1}$st order ranging from 700 nm to 1 µm (see bottom diagram in the right column of Fig. 6). Finally, the second order provides a contribution for diffraction efficiency in the small wavelength range.

 

Fig. 7. Dispersion curves (top row) and diffraction efficiency for different orders for the combination of ${{\rm CaF}_2}$ and fused silica (left column) and the combination of a SCHOTT chalcogenide glass and an optical crystal (IRG27/diamond, right column). The last row represents the multi-order chromatic diffraction efficiency for an exemplary design wavelength ${\lambda _0}$ and grating depth $h$ for each combination.

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Figure 7 shows the results for two double-layer gratings designed of materials that exhibit high transparency across a very broad wavelength range. Particularly, both combinations are characterized by dispersion curves intersecting in the IR and can be attributed to cases 1A and 4B (left from intersection) from Fig. 2. The grating combination of fused silica and ${{\rm CaF}_2}$ (fused silica: ${n_d} = 1.4585$, ${\nu _d} = 67.82$; ${{\rm CaF}_2}$: ${n_d} = 1.4338$, ${\nu _d} = 94.99$), with a transmission range extending from the deep-UV (${\lt}{200}\;{\rm nm}$) to wavelengths ${\gt}\;{4}\;{\unicode{x00B5}{\rm m}}$, shows the intersection of the two dispersion curves at 3.1 µm. The second grating in this series combines diamond with chalcogenide glass IRG27 (diamond: ${n_d} = 2.4168$; ${\nu _d} = 63.69$; IRG27: ${n_{(1.523{\unicode{x00B5}{\rm m}})}} = 2.4441$; not transparent in VIS).

At short wavelengths, the transmission range of IRG27 is limited to $\approx 1\;{\unicode{x00B5}{\rm m}}$. Both components have a very large refractive index, and their dispersion curves intersect at a wavelength of about 11 µm. The top row of Fig. 7 shows the similar shapes of the dispersion curves for both material combinations, the main difference being the wavelength range covered in each case. In both cases, large differences in the refractive indices of the grating components appear at the short-wavelength end of the respective spectral range. Therefore, high diffraction efficiencies in the first order can be observed already at grating depths below 3 µm. Efficiency contributions for the second order are already apparent at a grating depth of $\approx 5\;{\unicode{x00B5}{\rm m}}$. Due to the strong decline of the refractive index difference with increasing wavelength, the first order efficiency contribution is distinct only over a narrow wavelength range. The zeroth order, on the other hand, shows high efficiency over a wide spectral range. Since diamond is also a suitable material for creating layers, a potential grating combination of IRG27 and diamond may be technically feasible. A combination of fused silica and ${{\rm CaF}_2}$, though, appears very challenging.

Finally, the resulting efficiency behavior is presented for two additional material combinations that are also suitable for the IR range. In contrast to the previous combinations, here the dispersion curves of the materials involved intersect at the extreme shortwave limit or not at all, corresponding to 1A, 4A, and 4D in Fig. 2. In particular, double-layer gratings of diamond and zinc sulfide (ZnS) (left column of Fig. 8) and of KRS 6 and ${{\rm ZrO}_2}$ (right column of Fig. 8) are investigated. In the upper part of Fig. 8, the dispersion curves for both material combinations are plotted. For the left diagram, the length of the axis attributed to the wavelength corresponds to the transparency range of ZnS. ZnS is highly dispersive in the VIS and near-IR regions (ZnS: ${n_d} = 2.3677$, ${\nu _d} = 15.43$). For the second combination, the transmission window of KRS 6 defines the lower limit of the considered wavelength range; the upper limit is set by the transmission properties of ${{\rm ZrO}_2}$ [KRS 6: ${n_d}$ (600 nm) $= \;{2.1968}$; ${{\rm ZrO}_2}$: ${n_d} = 2.1588$, ${\nu _d} = 33.54$]. In general, the thallium halogenide crystal KRS 6 is mainly used for near- and mid-IR applications, and ${{\rm ZrO}_2}$ is a high-index material suitable for optical coatings. The differences in the refractive indices of the materials involved, which increase for longer wavelengths, correlate with a small efficiency contribution to the zeroth order (second row of Fig. 8). The zeroth order efficiency contribution occurs only for small grating depths. As the depth of the diamond/ZnS double-layer grating increases, an efficiency contribution to the first order occurs. With further increasing the grating depth, the wavelength-dependent efficiency is distributed between the first and second orders. In principle, a comparable behavior can also be observed for the KRS ${{\rm 6/ZrO}_2}$ combination. However, the wavelength- and depth-dependent efficiency shows a specific appearance for the first order. For larger structure depths ($\approx 20\;{\unicode{x00B5}{\rm m}}$), a sharp maximum occurs at short wavelengths and a second broad maximum is observable in the long-wavelength range. In the wavelength range in between, a significant contribution to the second order efficiency is found.

 

Fig. 8. Dispersion curves (top row) and diffraction efficiency for different orders for two combinations including optical crystals (diamond/ZnS and KRS $6/{{\rm ZrO}_2}$, left and right columns, respectively). The last row represents the multi-order chromatic diffraction efficiency for an exemplary design wavelength ${\lambda _0}$ and grating depth $h$ for each combination.

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5. CONCLUSION

Diffractive structures offer optical properties that are significantly distinct from those of refractive or reflective optics and can be used for a wide variety of optical applications. For example, diffractive lenses have become established in imaging systems where their very small and negative Abbe numbers are used for color correction. In this contribution, we have focused on wavelength selectivity of diffractive structures, meaning the tailored efficiency distribution of light into different specific diffraction orders for selected wavelengths or wavelength ranges. We have shown that double-layer single-relief blazed gratings of materials with intersecting dispersion curves are particularly suitable for achieving a certain wavelength selectivity for diffraction efficiency. By selecting suitable material combinations and adjusting the grating depth, different wavelengths or wavelength ranges can be attributed to the zeroth, first, and second or even the ${-}{1}$st and ${-}{2}$nd orders. It is possible to assign very small specific wavelength ranges to individual diffraction orders, but a distribution over very different sized wavelength ranges can also be chosen. The selection of material combinations with intersecting dispersion curves is limited. If the manufacturability of the double-layer single-relief blazed grating is also taken into account, the challenge of finding a suitable material combination increases. Solutions for double-layer single-relief blazed gratings based on bulk inorganic glasses are not obvious. On the other hand, it seems possible to find manufacturing processes in which a saw-tooth structured inorganic glass is filled with layered materials, polymers, or nanocomposites. In particular, our investigations have shown that high-refractive-index liquids offer great potential for controlling wavelength selectivity. As mentioned earlier, “engineering wavelength selectivity” in optical imaging systems allows partial decoupling in lens design for different wavelengths. However, there are also numberless new opportunities for applications in broadband lighting systems, and even new security features for banknotes or personal documents could be deduced.