Open Access
Add to BibSonomy
Abstract
The basic properties in the extreme ultraviolet (EUV) of one-dimensional photonic crystals (Bragg reflectors) with incorporated superlattices are investigated by a numerical study using the multiple scattering method. The superlattice is realized in the “standard” Mo/Si system by periodically replacing certain Mo layers by Si layers. At 13.5 nm–the wavelength of interest for EUV lithography–the superlattice sharpens the reflection peak at normal incidence with only weak reduction of the peak value. Between normal incidence and total reflection at large angles, additional reflection peaks appear at certain angles where the reflection is zero for the “standard” Mo/Si system. By combining different superlattices and depth grading, the range of additional reflection peaks is extended towards all-angle reflection. The effect of interface imperfections is considered for the case of the interdiffusion of Mo and Si. The extension to other frequency ranges is addressed via band structure calculations.
© 2017 Optical Society of America
1. Introduction
Bragg reflectors are used in optical systems where simple refractive elements like lenses cannot be designed because of the real part of the refractive index being very close to unity. This is the case for the X-ray and extreme ultraviolet regime of the electromagnetic spectrum. In the language of photonics Bragg reflectors are one-dimensional photonic crystals whose periodicity is adjusted for the frequency range of interest. Generally, photonic crystals (PhC) are structures with a spatially periodic variation of the dielectric permittivity (the refractive index, resp.) resulting in special optical properties [1]. They can be designed in a very broad range of frequencies from microwaves (e.g [1–4].) to the visible light [5] and into the extreme ultraviolet (EUV) [6]. PhCs are also found in nature [7]. In the work presented here we combine for the EUV the concept of Bragg reflection with that one of superlattices well known in semiconductor physics. This means we modify the simple periodic variation, viz. the periodic arrangement of double layers of two different materials, by superimposing a photonic superlattice with a larger period than the simple one.
The extreme ultraviolet (EUV) is considered as crucial for next-generation lithography, i.e. transistor gate lengths of 7 nm or less [6]. EUV lithography uses a plasma source whose emission peaks at a wavelength λ of 13.5 nm. This wavelength is 14 times shorter than the wavelength of 193 nm used in present-day lithography systems promising greatly improved resolution. At 13.5 nm the refractive index n of all elements is governed by atomic-core level transitions and the value of its real part is close to unity. Therefore, simple refractive elements like lenses cannot be designed. Mirrors, however, can be made on the basis of Bragg’s law by employing the constructive interference of multi-layers with “large” index contrast of two consecutive layers. These Bragg reflectors are one-dimensional (1D) PhCs. The best Bragg reflector with reflectance R = 0.74 for normal incidence uses approx. 40 double layers of molybdenum and silicon. At 13.5 nm Mo has a relatively strong deviation (δ = −0.076) of the real part of n from 1(−0.15 for the real part of the permittivity ε). Si behaves almost like vacuum (δ = −0.001), yielding the necessary index contrast. The values of the imaginary parts β of n are 0.0064 and 0.0018 for Mo and Si, respectively. This causes weak but not negligible absorption and restricts the number of useful Mo/Si double layers to about 40 as mentioned above. Reducing the number below 40 increases the reflectance bandwidth and decreases the peak reflectance. Above 40 the peak reflectance increases only slightly and the bandwidth remains almost constant because of the increased absorption. Figure 1 shows a contour plot of the reflectance R of 1D PhCs in the complex permittivity plane for values representative for the EUV. R values are calculated for a wavelength of 13.5 nm by the multiple-scattering method as described in chapter 2. This is done for a 1D PhC with double layers consisting of an element with permittivity ε and silicon. A few1D PhCs (element/Si) are indicated. The total number of double layers is 40. The period of the PhC is 6.9 nm. This value corresponds to half of the free-space wavelength λ of 13.5 nm which is slightly increased in the material, λm = λ/n. The partial waves reflected at the Mo/Si and Si/Mo interfaces constructively interfere. For optimal interference the thickness of the silicon layer is 60% of the period. For elements other than Mo the peak value of R is lower and the peak wavelength varies slightly with the period unchanged at 6.9 nm, mainly because of a different value of the real part of the permittivity and the refractive index, respectively. Replacing Si by rubidium, Rb, increases the reflectance to 0.77. However, because of its properties Rb is not a practicable element. The “standard” PhC of present-day EUV lithography systems with a maximal reflectance R of 0.74 contains about 50 double layers. In our study we use 40 double layers since the number of layers is increased in the superlattice PhCs leading to an absorption of approximately the same strength as in the standard 50-layer system.
Fig. 1 Contour plot of the reflectance R at normal incidence in the complex permittivity plane for values representative for the EUV. R values are calculated for a PhC with double layers consisting of an element with permittivity ε and silicon. A few 1D PhCs (element/Si) are indicated. The total number of double layers is 40. The period of the PhC is 6.9 nm. The thickness of the silicon layer is 60% of the period. The “standard” PhC with Mo/Si layers has the highest reflectance (0.74).
A typical EUV lithography (EUVL) system consists of an EUV source, a collector, illumination optics, a reflective EUV mask, projection optics and finally a resist-coated wafer [6,9]. Several mirrors are used in this optical path, altogether ten or more. With the reflectance of 74% of each of the 10 mirrors only 4.9% of the original power reaches the wafer. Bragg reflectors are the established choice for most of the reflecting elements in the system. The goal for the bandwidth of the illuminating radiation is 2% at 13.5 nm, i.e. approx. 0.3 nm. A presently widely used EUV source– a CO2 laser excited tin plasma – has a bandwidth of about 1 nm. Narrow-band mirrors can help to meet the goal of 2%.
As a modification of the periodic multilayer structures non-periodic or periodic superstructures can be superimposed or the multilayer period can be varied (“depth grading”). Such modifications may result in richer spectral features and give rise to a number of interesting effects. Generally, periodic structures are used for achieving narrow bandwidths, depth-graded or aperiodic structures for broad-band response.
A non-periodic case can be realized by quasicrystals [8] which are made from building blocks that are arranged using well-defined patterns but lack translational symmetry. It was studied in the visible part of the spectrum. In the periodic case – investigated in the present work - the 1D PhC, described above, is modified by superimposing a superlattice. It means the superposition of a superstructure on the basic 1D periodic structure by replacing one element in certain layers periodically by another element, e.g. in every fifth double layer of a Mo/Si multilayer Mo is replaced by Si. This is a well-known concept in semiconductor physics [9]. From the anticipated analogy to modifications of electronic band structures interesting modifications of the photonic band structure can be expected. The main effect is that light of different wavelengths is reflected by the basic structure and by the superstructure. A “random” version of this concept – a so-called distributed Fabry-Perot etalon – was used for grazing incidence ReW/C mirrors in the X-ray regime [10]. Extra C periods were randomly distributed in sets of zero, one or two extra periods. Three-dimensional structures can be realized by etching trenches in a multilayer, that way combining vertical and lateral periodicities. Single-order operation of such lamellar structures was demonstrated in the soft x-ray range by van der Meer et al. [11].
In depth-graded multilayers light of different wavelengths is reflected at different depths in the stack. It was applied for extending the bandwidth in the EUV [12–15] and x-ray range [13,14,16,17]. For flat broad-band response numerical procedures involving the minimization of a merit function were used [16,18–20]. This resulted in specific gradings for the desired applications.
Huang et al. [21] reviews these multilayer designs in detail and discusses other concepts like multilayer gratings, multilayer zone plates, and diffraction pyramids. In the present work we apply the depth grading by gradually increasing the period with depth. Simultaneously, we combine two superlattices in order to modify the angle dependence of the reflection with the goal of achieving response between near-normal incidence and total reflection, i.e. approximately between 15° and 80° incident angle.
The calculations and results for superlattice PhCs described below are obtained with Mo/Si PhCs whereby Mo layers are replaced by Si layers which has the most pronounced effects on the reflection characteristics. The results on the basic properties are described in chapter 3. They concern peak reflections and reflection band width at 13.5 nm, changes of the high-reflectance ranges near normal incidence as well as additional reflection at intermediate angles between normal incidence and total reflection. The discussion of the results in chapter 4includes the effect of interface imperfections as well as an extension to other frequency ranges.
2. Numerical procedure
A superstructure superimposed on the basic periodic structure of the “standard” 1D PhC can be achieved by replacing every mth (m = 3,4, ...) layer of Mo by a layer of another element (“superlattice-m”). The top layer is Mo. The replacement starts with the second Mo layer. The period of the superstructure is m times 6.9 nm - the lattice constant of the basic structure. Best results were obtained replacing Mo by Si. Therefore we restrict the results presented here to this case. Figure 2 shows an example for superlattice-4, where the superstructure has a period of 4x6.9nm = 27.6nm. In order to keep the number of Mo/Si interfaces constant the number of double layers was increased in the superlattices compared to the basic structure with 40 double layers. In other words this means that only “Si/Si” double layers are added while the number of Mo/Si double layers (the number of interfaces) is kept constant. This leaves the basic interference properties of the 40 Mo/Si layers of the standard PhC unchanged. For superlattice-2, −3, −4, and −5 this leads to the total number of double layers to be 80, 60, 53 and 50, respectively.
Fig. 2 Example of a 1D PhC with superlattice where every 4th layer of Mo is replaced by Si (“superlattice-4”). The period of the superlattice is four times the lattice constant of the basic structure, i.e. in the “standard” PhC 4x6.9 nm = 27.6 nm
The reflection of the structures is calculated by means of the multiple scattering method (MSM) using the MULTEM2 program [22]. The program uses a text file containing first general parameters (e.g. angle of incidence, range and number of wavelength for the calculation and then descriptions of each layer (e.g. thickness, complex permittivity and permeability). The program calculates the transmission scattering matrix for each individual element and determines the total scattering matrix as the product of the individual matrices. This matrix is used to output transmittance, reflectance, and absorbance.
The MULTEM2 program [22] was modified to take into account the wavelength dependence of the complex dielectric permittivity ε. For Mo and Si the permittivity was calculated from the atomic structure factors f1 and f2 available from [23,24]. For the materials considered the magnetic permeability was set to 1. The dielectric constant of the medium in front of the superlattice (from where the incident wave comes) and that of the medium behind the superlattice was set to 1. With these boundary conditions the results are independent of a special choice of a substrate material. A substrate, e.g. SiO2, can be included in the calculation.
For the “standard” Mo/Si PhC (without a superlattice) also an online simulator [24] can be used. There, however, an approximation neglecting multiple reflections is applied [25]. In our MSM calculation the multiple reflections are taken into account.
3. Results
Results of the reflectance for normal incidence around the wavelength of 13.5 nm are shown in Fig. 3. The total number of layers is increased in the superlattice PhCs as described in chapter 2. For comparison results of the “standard” 1D PhC are also shown in Fig. 3. There, the peak value and the width are slightly larger. The peak values and the widths are reduced with decreasing superlattice period (Table 1). The narrowing is accompanied by a shift of the peaks towards larger wavelengths and the curves of the superlattice PhCs overlap on the long-wavelength side.
Fig. 3 Reflectance at normal incidence of superlattice-2 (a), −3 (b), −4 (c), −5 (d) and of the PhC without superlattice (e). The number of double layers is 80, 60, 53, 50 and 40, respectively. This way the number of Mo/Si interfaces is constant (40).
Table 1. Peak values and full widths at half maximum (FWHM) of the reflectance peaks at normal incidence (2nd and 3rd column). Angles of strong reflectance R near 13.5 nm (4th column): bold angles indicate additional reflection peaks due to the superlattices (details in Figs. 4 and 5).
For incident angles different from 0° and for a larger wavelength range the results are presented in a contour plot for s and p polarization (Fig. 4). R values in steps of 1° and in 900 steps between 6 nm and 40 nm, resp., are plotted. The strongest reflectance in the “standard” PhC originates from the 1st order Bragg reflection of the basic structure (near 13.5 nm and incident angles up to about 15°) and from total reflection (in the full wavelength range at angles larger than about 80°). In between, superlattice-m causes m-1 additional reflection peaks (see also Table 1). At normal incidence the mth order peak of superlattice-m is at the position of the 1st order peak of the basic structure (13.5 nm). The positions of the low-order peaks are determined by the wavelength dependence of the real part ε’ of the permittivity (of constant ε’ the 1st order peak of superlattice-m would be at a wavelength m times 13.5 nm. The positions of the superlattice reflection peaks are well reproduced by using the Bragg formula, Eq. (1), valid for the case of zero imaginary part of ε and n, resp., i.e. no absorption (dashed curves in Fig. 4).
Fig. 4 Contour plot of the reflectance vs. wavelength and angle of incidence. From top to bottom: superlattice-3, −4 and −5. Positive (negative) angle: s (p) polarization. Dashed curves: positions of Bragg reflections according to Eq. (1). Solid curve: critical angle of total reflection according to Eq. (2). Details see text.
In this approximation, n is the real part and averaged over m consecutive double layers for superlattice-m. Λsl is the period, ksl the order of the Bragg reflections of the superlattice. α is the angle of incidence (measured against the surface normal). Also the onset of total reflection is well described by the formula for the critical angle of total reflection (Eq. (2), solid curves in Fig. 4):
Due to the dominance of the Mo value (the deviation of n from 1 is approximately twice that of Si) also the averaged value of n is smaller than 1 leading to total reflection in the whole wavelength range shown.
Figure 5 shows a plot of the R values (in 1° steps) for superlattice-4 and 5 at 13.5 nm. As in Fig. 4 the signature of the superlattice is clearly seen: Compared to the “standard” Mo/Si PhC additional reflection peaks occur increasing in number from superlattice-2 (one additional peak, not shown in Fig. 5) to superlattice-5 (four additional peaks); the peak reflectance at 0° is slightly reduced. The positions of these peaks for superlattice-2 and 3 are given in Table 1.
Fig. 5 Reflectance as a function of angle of incidence for superlattice-4 (middle) and 5 (bottom) at 13.5 nm. For comparison the reflectance for the “standard” PhC is shown (top). Positive (negative) angle: s (p) polarization. The dotted-blue and yellow curves for superlattice-5 are for an interdiffusion layer with σ = 0.07 and 0.35 nm, respectively. Details in the text.
In real structures interface imperfections are present to some extent. In order to estimate the sensitivity of our results to imperfections we performed calculations which consider one type, viz. interdiffusion at the interface between Mo and Si. The calculation simulates the transitions between Mo and Si by inserting intermediate layers as in the discrete roughness model described, e.g., in [27]. The roughness parameter of the interdiffusion layer is varied between σ = 0.07 and 0.35 nm which corresponds to the experimentally determined root-mean-square roughness [28]. Examples for the effect of interdiffusion (σ = 0.007 and 0.35 nm) are included in Fig. 5 for superlattice-5. For σ = 0.35 nm the near-normal incidence peak is reduced in height from 0.7 to 0.67 and is narrower than the peak without interdifussion. This peak is due to the basic period of 6.9 nm and therefore more sensitive to the imperfection than the superlattice-related peaks from 35° to 73° (s polarization). The larger-angle peaks are broadened and slightly higher with the interdiffusion. We attribute the higher reflectance at the peaks to the enhanced background caused be the imperfection of the interfaces. The lower bound of σ (0.07 nm) has minimal effect on the R curves, only weakly observable at the minima and maxima at large angles.
In order to fill the reflection gap between normal incidence and total reflection further, different superlattices can be combined and/or the superlattice period can be depth graded. An example is shown in Fig. 6. There, superlattice-4 and 5 are combined and the period is linearly depth graded (bottom graph). The structure alternates between the subunits of SL-5 and SL-4 starting with SL-5 and consists of 8 (SL-5 SL-4) groups. For the top double layer the width is set to 6.6 nm. The thickness is gradually (linearly) increased, for the final double layer by a factor of 1.4 (final thickness 9.24 nm). The change of the period from 6.9 nm of the single superlattice PhC to 6.6 nm is necessary to position to maximum of the reflection at 13.5 nm. The effect of the combination is to increase the number of additional peaks, the effect of the depth grading is to broaden the peaks and cover an increased range of angles. With the grading the near-normal incidence peak is broadened by roughly a factor of two and there is no zero reflection in the whole range of s polarization. However, at a few sharp minima the reflectance is near one percent only. This wide-angle to all-angle reflection is accompanied by a reduction of the reflectance of the near-normal incidence peak.
Fig. 6 Reflectance as a function of angle of incidence for combined superlattices-4 and 5 without (top) and with depth grading (bottom) at 13.5 nm. Positive (negative) angle: s (p) polarization. Details of the grading see text.
4. Summary and discussion
4.1 The EUV range
Results for 1D PhCs with superlattices having 2, 3, 4 and 5 times the basic period are presented. In certain double layers Mo is replaced by Si periodically. In order to keep the number of Mo/Si interfaces constant the total thickness is increased by adding Si/Si double layers periodically to the basic structure with 40 Mo/Si double layers. The superlattice has pronounced influence on the reflection behaviour of 1D PhCs. At normal incidence reflection peaks at wavelengths larger than 13.5 nm occur according to the Bragg condition for the periodicity of the superstructure. In the angular dependence additional reflection peaks appear in the gap at angles between the 1st order peak of the basic structure and the onset of total reflection. For superlattice-m their number is m-1. Increasing the period above 5 fills the gap further. However, the strength of the higher-order peaks decreases with increasing order. The higher-order peaks are also present in the constant-period multilayers. In [26], Fernández-Perea et al. used them to achieve triple-wavelength performance in the range 25 – 80 nm with multilayers of the proper period. In the present work these higher-order peaks are the additional peaks in the angular dependence mentioned above.
The effect of the superlattices on the normal-incidence spectra is a) a reduced peak width (in superlattice-5 it is 78% of the value of the “standard” PhC, in superlattice-2 it is 48%), b) a small reduction of the peak value of the reflectance (4% for superlattice-5 and 14% for superlattice-2), c) a small peak shift towards longer wavelengths (≤ 0.1nm), increasing with decreasing superlattice period (decreasing superlattice number m), and d) an overlap of the superlattice curves on the long-wavelength side of the reflectance peak.
The changes of the peak values and of the width in the superlattice PhCs (Table 1) are explained as follows. The Bragg condition for constructive interference is fulfilled at 13.5 nm in the superlattice PhCs as it is in the standard PhC. The reduction of the peak values from the standard PhC to SL-5, −4, −3 and −2 is attributed to the increasing total thickness, i.e. more absorption by additional Si layers. As regards the peak width, with increasing deviation of the wavelength from 13.5 nm the waves reflected from the Mo/Si interfaces come increasingly out of phase and the constructive interference is weakened. In the superlattice PhCs with increasing distance between the first and last interface this effect of dephasing is enforced closer to the peak wavelength leading to a peak width progressively reduced towards SL-2.
A narrowing of the peaks could also be achieved by reducing the fraction of Mo and correspondingly increasing that one of Si in the standard constant-period PhC. Results are shown in Table 2 for “standard” PhCs with certain Mo fractions and the superlattice PhCs. At the same peak width the peak reflectance is higher in the superlattice PhCs, particularly those with small number m. Another concept of reducing the peak width is to use higher-order reflection of constant-period PhCs with increased period [29]. E.g. for the second order the period is doubled with the Mo thickness optimized. With the number of double layers kept constant it leads to similar results as for the superlattice PhCs in Table 2 [29]. Obviously, this increases the necessary total thickness of the PhC by a factor of two for the second order and more for the higher orders.
Table 2. Comparison of peak reflectance and peak width of superlattice PhCs and standard PhCs for normal incidence at 13.5 nm. Details see text.
The shift of the peaks towards longer wavelengths is due to the mean refractive index increasing with decreasing superlattice number m (more Si with n larger than that of Mo). This interpretation is supported by results on a hypothetical “Mo”/Si PhC with ε’>1 (n>1) for “Mo”, i.e. an “inverted” contrast. In this case the peak shift is towards shorter wavelengths since the mean refractive index decreases with decreasing m (more Si with smaller n than “Mo”), and the curves for the superlattice PhCs overlap on the short-wavelength side.
Depth-grading, in the present work increasing the period linearly with increasing depth, has pronounced effects on the near-normal incidence peak and on the filling of the reflection gap at medium angles of incidence. The near-normal incidence peak is broadened with the reflectance being larger than 0.1 up to 30°. Unavoidably this is accompanied by a reduction of the peak at 0°, from 0.71 to 0.46 for superlattice-5. Depth-grading of constant-period multilayers can have a similar effect on the near-normal incidence peak. In [30], by a thickness optimization of each layer by a stochastic method a reflectance of more than 0.3 was obtained from 0 - 20°. With such a method or the minimization of a merit function results for the superlattice PhCs could be improved. The filling of the reflection gap, however, is not possible with the constant-period multilayers, and a unique feature of the superlattice PhCs.
4.2 Other frequency ranges
The consequences of the EUV results for other frequency ranges are discussed here in two ways where both are approximations for zero absorption. The first one considers Eq. (1). Despite neglecting absorption this is a very good estimate of the positions of reflection peaks as a function of wavelength and incident angle with the wavelength dependence of the refractive index n included. The second way is a band structure calculation using the plane-wave-expansion (PWE) method which is performed for an infinite PhC. Here, the absorption and the wavelength dependence of n are neglected. The advantage is to obtain results for the full Brillouin zone of the PhC and for a reduced frequency so that the band structure is valid for all frequencies (for the index contrast chosen). A section of the band structure for the basic structure and superlattice-4 is shown in Fig. 7.
Fig. 7 Band structure (reduced frequency a/λ vs. wavevector kz) of the basic structure (dashed) and superlattice-4 (solid) for kx = 0. The lowest gap of the basic structure is at kz = π/a, the lowest mini-gap of superlattice-4 at kz = π/4a. The mini-gaps are indicated by red arrows.
Propagation is in the z direction. Figure 7 is calculated for kx = 0 which corresponds to normal incidence. Due to the superlattice “mini-gaps” appear within the bands of the basic structure. To make them clearly visible the deviation δ of n from 1 of the Mo layer is enlarged from −0.076 to −0.3. In the basic structure the average absolute value of δ is smaller than 0.3 (four Mo layers replace three Mo layers and one Si layer of the superlattice). To compensate for this effect the deviation is reduced to 0.225 = 3/4 * 0.3.This aligns the lowest gap of the basic structure (centred at the reduced frequency a/λ = 0.547) to the corresponding one of superlattice-4. a/λ = 0.547 corresponds to the wavelength λ = 12.6 nm with the lattice constant a = 6.9 nm. For superlattice-4 three mini-gaps appear within the first band of the basic structure. From the higher bands which also contain mini-gaps only the lower part of the second band is shown. Comparing with the MSM results, gaps in the band structure correspond to strong reflection.
The PWE calculation also yields results for kx≠0, i.e. the dependence on the angle of incidence. Calculations of the band structure with n > 1 and similar contrast give essentially the same results. Thus, the PWE method is a valuable tool to get a general overview of the positions of gaps and mini-gaps in the full frequency regime. The case n<1 concerns part of the x-ray and EUV regime, the case n>1 for non-metals most of the regime from deep UV to microwaves.
5. Conclusions
Superlattice PhCs with varying period of the superstructure are simple modifications of constant-period double-layer PhCs. Compared to the latter ones only certain layers of one element have to be replaced by layers of the other element without changing the basic double-layer thicknesses and period. This concept is not restricted to the EUV wavelengths around 13.5 nm. Three most important effects compared to the standard Mo/Si PhC are observed at 13.5 nm: a) Narrowing of the bandwidth of the normal-incidence peak with only slight reduction of the peak reflection. Maintaining a high reflectance is an advantage over other concepts for reducing the bandwidth. b) Filling the reflection gap between near-normal incidence and total reflection with reflection peaks at certain angles. Their number is increased when combining different superlattices (in this work superlattice-4 and 5). On the wavelength scale these peaks appear on the long-wavelength side of the peak of the standard PhC. For superlattice-5 this means up to about five times the wavelength of the standard-PhC peak at normal incidence. c) Depth grading of the periods of the superlattices even leads to reflection at all the angles where it is zero in the standard Mo/Si PhC. It is accompanied by a reduced but broader reflectance of the near-normal incidence peak and minima with about one percent reflectance at a few angles. This indicates that the total strength of the near-incidence peak is distributed over a wide range of angles in the depth-graded superlattices.
From a) optical systems can benefit where a narrow-band mirror response is needed. This could be in EUV lithography or in monochromators. Results b) and c) allow multi-angle and all-angle, equivalently multi- or broadband-wavelength applications where maximum reflectivity is not required, e.g.in EUV spectroscopy or metrology for EUV sources, and higher integral reflectivity in the combination with a broad-band EUV plasma source [21,31]. The depth grading can extend the usefulness of near-normal incidence mirrors to larger angles. Also, when high reflectance is not an issue the combination of superlattices and depth grading can be the basis for all-angle mirrors.
Funding
Austrian Science Fund (FWF) (TRP 284-N30).
References and links
1. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58(20), 2059–2062 (1987). [PubMed]
2. F. Kuchar, R. Meisels, P. Oberhumer, and R. Gajic, “Microwave Studies of Photonic Crystals,” Adv. Eng. Mater. 8, 1156–1161 (2006).
3. R. Meisels, O. Glushko, and F. Kuchar, “Tuning the flow of light in semiconductor based photonic crystals by magnetic fields,” Photonics and Nanostructures – Fundamentals and Applications 10, 60– 68 (2012).
4. R. Meisels, R. Gajic, F. Kuchar, and K. Hingerl, “Negative refraction and flat-lens focusing in a 2D square-lattice photonic crystal at microwave and millimeter wave frequencies,” Opt. Express 14(15), 6766–6777 (2006). [PubMed]
5. M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature 404(6773), 53–56 (2000). [PubMed]
6. V. Bakshi, ed., EUV Lithography (Wiley, 2009).
7. B. Winter, B. Butz, C. Dieker, G. E. Schröder-Turk, K. Mecke, and E. Spiecker, “Coexistence of both gyroid chiralities in individual butterfly wing scales of Callophrys rubi,” Proc. Natl. Acad. Sci. U.S.A. 112(42), 12911–12916 (2015). [PubMed]
8. Z. V. Vardeny, “Optics of Photonic Quasicystals,” Nat. Photonics 7, 177–187 (2013).
9. H. T. Grahn, ed., Semiconductor Superlattices (World Scientific, 1995).
10. M. P. Bruijn, J. Verhoeven, and W. J. Bartels, “Improved resolution of multilayer x-ray coatings: a distributed Fabry-Perot etalon,” Opt. Eng. 26, 679–684 (1987).
11. R. van der Meer, I. Kozhevnikov, B. Krishnan, J. Huskens, P. Hegeman, C. Brons, and B. Vratzov, “Single order operation of lamellar multilayer gratings in the soft x-ray spectral range,” AIP Adv. 3, 012103 (2013).
12. D. L. Windt, “EUV multilayer coatings for solar imaging and spectroscopy,” Proc. SPIE , 9604, 96041 (2015).
13. Z. Wang and A. G. Michette, “Optimization of depth-graded multilayer designs for EUV and x-ray optics,” Proc. SPIE 4145, 243 (2001).
14. A. E. Yakshin, I. V. Kozhevnikov, E. Zoethout, E. Louis, and F. Bijkerk, “Properties of broadband depth-graded multilayer mirrors for EUV optical systems,” Opt. Express 18(7), 6957–6971 (2010). [PubMed]
15. J. I. Larruquert, A. G. Minette, Ch. Morawe, Ch. Borel, and B. V. Vidal, in A. Erko, M. Idir, T. Krist, and A. G. Michette, eds., Modern Developments in X-Ray and Neutron Optics, Springer Series in Optical Sciences (Springer, 2008), Ch. 25.
16. I. V. Kozhevnikov and C. Montcalm, “Design of X-ray multilayer mirrors with maximal integral efficiency,” Nucl. Instrum. Methods Phys. Res. A 624, 192–202 (2010).
17. F. E. Christensen, W. W. Craig, D. L. Windt, M. A. Jimenez-Garate, F. A. Harrison, P. H. Mao, J. M. Ziegler, and V. Honkimaki, “Measured reflectance of graded multilayer mirrors designed for astronomical hard X-ray telescopes,” Nuclear Instrum. Methods Phys. Res. Section A 451, 572–581 (2000).
18. Z. Wang and A. G. Michette, “Broadband multilayer mirrors for optimum use of soft x-ray source output,” J. Opt. A, Pure Appl. Opt. 2, 452 (2000).
19. A. E. Yakshin, I. V. Kozhevnikov, E. Zoethout, E. Louis, and F. Bijkerk, “Properties of broadband depth-graded multilayer mirrors for EUV optical systems,” Opt. Express 18(7), 6957–6971 (2010). [PubMed]
20. S. A. Yulin, T. Kuhlmann, T. Feigl, and N. Kaiser, “Spectral reflectance tuning of EUV mirrors for metrology applications,” Proc. SPIE 5037, 286 (2003).
21. Q. Huang, V. Medvedev, R. van de Kruijs, A. Yakshin, E. Louis, and F. Bijkerk, “Spectral tailoring of nanoscale EUV and soft x-ray multilayer optics,” Appl. Phys. Rev. 4, 011104 (2017).
22. N. Stefanou, V. Yannopapas, and A. Modinos, “Heterostructures of photonic crystals: frequency bands and transmission coefficients,” Comput. Phys. Commun. 113, 49–77 (1998).
23. http://henke.lbl.gov/optical_constants/asf.html
24. http://henke.lbl.gov/optical_constants/multi2.html
25. V. G. Kohn, “On the Theory of Reflectivity by an X-Ray Multilayer Mirror,” Phys. Status Solidi 187, 61–70 (1995).
26. M. Fernández-Perea, R. Soufli, J. C. Robinson, L. Rodríguez-De Marcos, J. A. Méndez, J. I. Larruquert, and E. M. Gullikson, “Triple-wavelength, narrowband Mg/SiC multilayers with corrosion barriers and high peak reflectance in the 25-80 nm wavelength region,” Opt. Express 20(21), 24018–24029 (2012). [PubMed]
27. https://www.classe.cornell.edu/~dms79/refl/XR-Roughness.html
28. A. Haase, V. Soltwisch, S. Braun, C. Laubis, and F. Scholze, “Interface morphology of Mo/Si multilayer systems with varying Mo layer thickness studied by EUV diffuse scattering,” Opt. Express 25(13), 15441–15455 (2017). [PubMed]
29. T. Feigl, S. Yulin, N. Benoit, and N. Kaiser, “EUV multilayer optics,” Microelectron. Eng. 83, 703 (2006).
30. T. Kuhlmann, S. A. Yulin, T. Feigl, N. Kaiser, H. Bernitzki, and H. Laut, “Ion beam sputter deposition of low-defect EUV mask blanks on 6-in. LTEM substrates in a real production environment,” Proc. SPIE 4688, 509 (2002).
31. S. Yulin, T. Kuhlmann, T. Feigl, and N. Kaiser, “Spectral reflectance tuning of EUV mirrors for metrology applications,” Proc. SPIE 5037, 286 (2003).
