We present an innovative grating design based on conical diffraction which acts as an almost perfect and low-loss beamsplitter for extreme ultraviolet radiation. The scheme is based on a binary profile operated in grazing incidence along the grating bars under total external reflection. It is shown that periods of a few 102 nm may permit an exclusive (±1)st order diffraction with efficiencies up to ∼ 35% in each of them, whereas higher evanescent orders vanish. In contrast, destructive interference eliminates the 0th order. For a sample made of SiO2 on silicon, measured data and simulated results from rigorous coupled wave analysis procedures are given.
© 2012 OSA
Precise and efficient beamsplitters for the extreme ultraviolet wavelength range are of particular interest for interferometric applications in plasma diagnostics [1,2]. In the absence of refractive optical components due to strong absorption in that spectral range , specialized diffractive elements have been developed . Used in grazing incidence, such lamellar structures provide reflected intensities of about 20% in the 0th and (−1)st order as well . However, their equalization, essential for a high fringe visibility, is extremely sensitive on the grating design parameters like groove depth and duty cycle. Moreover, the shadowing effect, enhanced by grazing incidence, causes significant diffraction losses.
In this paper, we suggest an alternative approach which circumvents those problems by means of conical diffraction. In our arrangement, the propagation direction of the incident field is oriented perpendicular to the grating vector. In this way, exactly symmetrized (±1)st order efficiencies near the theoretical limit are easily obtained. In Sect. 2, fundamentals of the conical mounting are reviewed. Design requirements for proper operation are discussed in Sect. 3 and applied to an optimized prototype made of SiO2 on silicon in Sect. 4. Simulation and measurement results for that sample are presented in Sect. 5 and 6, respectively. An outlook to further prospects of improvement is given in Sect. 7.
2. Basic features of conical diffraction
In most applications, one-dimensional transmission or reflection gratings are exclusively operated in the plane of incidence which is spanned by the wave vector k⃗inc of the incident field and the normal n⃗ to the grating surface. If this plane contains the grating vector G⃗ too, all diffracted orders are spread within this plane. This well-known situation changes under oblique incidence, namely arbitrary angles between the plane of incidence and G⃗.
We apply the elegant, purely geometrical concept of the “direction cosine space” (DCS)  to describe arrangements of that type. Since the conical incidence is often associated with wide-angle, i.e. non-paraxial diffraction, its linearization appears to be convenient. Following Harvey and Vernold , this can be achieved using the “direction cosines” of the actual spatial coordinates r⃗ = (x,y,z) in a cartesian system,Fig. 1 for the special case of reflection and ψ = 90°. The sum condition from Eq. (1) confines the problem to the unit sphere from which only the upper half is shown in Fig. 1. In that system, the incident ray is described by its coordinates Fig. 1. From the grating equation – for a period d and wavelength λ – and its additional constraint to reflection,
In grazing incidence with ϕ0 → 90°, the projected grating period d cosϕ0 ≪ d enables the construction of an almost perfect beamsplitter for short wavelengths: We specify the general case from the left of Fig. 1 to symmetric incidence with θ0 = 0° and arrive at the configuration shown on the right. With αi = 0 but an implemented tilt ψ = 90° ± δψ with |δψ| ≲ 1°, we obtain from Eq. (3) and using Fig. 1 differs from more conventional (near) normal incidence arrangements in diffraction. Whereas the 0th order undergoes an elementary reflection within the plane of incidence from (0,βi) to (0,β0), the (±1)st orders are mirrored at this point to (α±1,β±1) – both in direction and intensity, if there is no tilt (δψ = 0°).
3. Design constraints for EUV beamsplitters
An optimized conception based on this idea turns out to be straightforward now. In the EUV and soft X-ray band, all known materials are characterized by strong absorption and a weak refractive power, i.e. their refractive index is commonly written as n = 1 − δ − iβ, with real coefficients 0 < δ,β ≪ 1 . The peculiarity Re(n) = 1 − δ < 1 leads to total external reflection of an incident electromagnetic field with a critical incidence angle , more or less damped by the extinction β. Since numerous absorption edges preclude an efficient total external reflection in their vicinity throughout the soft X-ray and EUV spectrum, the grating composition and the wavelength of operation should be carefully selected. We choose silicon (Si) and its oxide SiO2 for the substrate and the grating bars, respectively, at a wavelength well beyond the SiL edge near 12.4 nm. In Fig. 2, the s-polarized reflectivity under grazing incidence, evaluated by means of the Fresnel formulae and the Henke data , is shown for both materials in the spectral range of interest. Within 15 nm ≲ λ ≲ 40 nm, the superior performance of pure Si is obvious, compared to SiO2. We discriminate between the substrate and the bars however, for reasons of technological simplicity (see Sect. 4) in the context of this “proof-of-concept” work. Furthermore, the limited precision in high-resolution nano-fabrication and constraints on the experimental setup for the measurements (see Sect. 6) suggest an accessible period of 400 nm, the spectral band from about 20 nm to 30 nm and a grazing angle of a few degrees; we set the initial test wavelength to λc = 25 nm and aim at an inclination near ϕ0 ≈ 85°. Zero-roughness reflectivities of 91% for Si and 76% in case of SiO2 are expected for those parameters. In average, an unsophisticated estimation predicts a summed efficiency in all propagating orders up to ∼ 80%, presumed a duty cycle of 50%.
We would like to build a true beamsplitter which implies the elimination of the 0th order. In total external reflection, the required groove depth for that purpose can be roughly pre-selected by the geometrical condition for destructive interference from two rays reflected from the rectangular grating bars with a thickness Δt and the substrate, respectively. Their optical path difference is given as Δsopt = 2Δt cosϕ0. This length should be equal to an odd multiple of λ/2. For the values from above, we find the lowest thickness Δtmin ∼ 70 nm in this approximation.
The actual field-matter interaction is nonetheless governed by the wave nature of light, as described by the Maxwell equations. In agreement with the specifications of the experimental setup at the synchrotron beamline as discussed in Sect. 6, we constrain to s-polarized incidence. Starting from that coarse estimation for Δtmin and a duty cycle f = 50%, numerical rigorous coupled wave analysis (RCWA) techniques yield an optimized target thickness of ≈ 100 nm. As described in Sect. 4 and confirmed by the EUV measurements (Sect. 6), that goal is matched with an accuracy of a few nm for the fabricated film. We thus present the results here from an alternating iteration with respect to Δt and f, in terms of a best-fit algorithm to the empirical data set, in particular the incidence angle ϕ0 = 84.77° for which the 0th order intensity passes through its minimum. As it is shown on the left of Fig. 3, only the “lowest-order” interference reflection with a groove depth of 101 nm provides the highest (±1)st order efficiency in each of them, accompanied by an almost vanishing 0th order contribution. Towards multiples of this best thickness Δtopt, near 290 nm for instance, the contrast between the (±1)st and the 0th order decreases, due to stronger coupling losses within the grooves. On the right of Fig. 3, the duty cycle is varied for Δtopt within 0 ≤ f ≤ 1. Again, the highest contrast for a duty cycle of 54.8% is obtained with an efficiency of 35% in the (±1)st order, whereas the 0th order drops down to 3.4 × 10−4 % or less, depending on the numerical RCWA sample rate. It should be noted that an alternative design for p-polarized waves yields quite similar efficiencies in all orders, albeit for slightly shifted optimal values in groove depth and duty cycle. Typically, variations of a few nm in Δt and ∼ 1% in f arise, respectively. Such an effect is evident from the different coupling strength between the incident and the diffracted waves for s- and p-polarization .
Since all higher diffraction orders with |m| ≥ 2 are found as being evanescent, the losses in our simulated beamsplitter are minimized and mainly caused by absorption on the grating surface. Moreover, we neglect here the surface roughness which would lead to diffusely scattered photons. As it will be discussed in Sect. 4, this assumption is justified with an excellent accuracy. In summary, Fig. 4 gives an overview on the geometrical parameters and the diffraction characteristics. The total sample dimensions of 4.0 mm × 15 mm account for the special features of the experimental setup at the synchrotron facility, where the device is measured afterwards (Sect. 6). In particular, its length enables the full exploitation of the available beam intensity, whereas potential stitching errors from the e-beam writer can be still neglected at this size.
The tight thickness tolerance close to 100 nm from above suggests an atomic layer deposition (ALD) of the SiO2 film. We use an optical grade, polished 4” silicon wafer in 〈100〉 orientation. Heated to 400 K, the substrate is subsequently coated in an O2 plasma enhanced ALD procedure with a growth rate of 0.86 Å per cycle. Ellipsometric measurements show a final central layer thickness of 92 nm, whereat a thickness homogeneity of 0.9% is assured. Afterwards the 50 nm chromium metal hard mask is deposited by magnetron sputtering. In order to define the grating structure with a period of 400 nm and an implemented variation in the duty cycle 0.49 ≲ f ≲ 0.59 in steps of ≈ 2.6 × 10−2, we use conventional electron beam lithography (Vistec SB350 OS) with the chemical amplified resist FEP 171, as illustrated in Fig. 5. After exposure and developing, reactive ion etching (RIE) is performed on the chromium in a parallel plate reactor (SI 591, Sentech Instruments Berlin) by a chlorine-oxygen plasma. Because of the etch chemistry and – in consequence – the isotropic part of the etching process, there is a significant loss in the grating bar width which has been taken into account in the initial design. The chromium serves as a mask for the following etch step in SiO2 by reactive ion beam etching (RIBE) and CF4. This is done in a self-made tool with a 150 mm ion source (“Kaufmann-type”). The SiO2 etching process stops quite well on the silicon substrate, an important condition for an efficient grating performance. In Fig. 6, a sequence of SEM pictures illustrates each fabrication stage. Finally, the chromium mask is removed. This is done by wet etching to prevent damage on the silicon substrate, since the silicon etch rate would be comparable to that of chromium in the chlorine-oxygen dry etching tool.
Though not initially intended, the natural formation of an extremely thin “native” SiO2 layer with a typical thickness of about 1 nm on a blank Si surface in air is a well-known phenomenon  and inevitable in practice, for typical exposure times of several hours. An intermediate layer of that thickness is thus assumed to coat the bottom of the grooves and incorporated in all RCWA simulations presented in this work. Analogous numerical evaluations performed on bare Si indicate no significant effect on the 0th order and an acceptable diffraction loss of up to ≈ 2% in the (±1)st ones. That moderate diminishment can be understood from the relatively large penetration depth Δzp = 5 nm of the electromagnetic field in SiO2 . From the duty cycle series, an optimized specimen with f = 0.547 ± 0.017 is identified, which is close the modeled value of 0.548 within an error range of ±1% as described in Sect. 3. Its intrinsic uncertainty of almost ±7 nm has its origin in the line edge roughness caused by non-perfect etching. Compared to an ideal structure, both the 0th and the (±1)st order efficiencies are thus expected to degrade as it will be shown in Sect. 6. Moreover, scattering losses would arise in case of significant surface imperfections, i.e. an inevitable roughness of the SiO2 grating bars on the nm scale. However, scanned by an atomic force microscope (AFM), their thickness is determined to 98 ± 0.6 nm, as visualized in Fig. 7 – a few nm more than in the central wafer region, but in good agreement with the value Δtopt = 101 nm indirectly deduced from 0th order diffraction (Sect. 6). The estimated RMS level should nearly maintain the optical performance, as calculated from the Fresnel formulae. Indeed, the reflectivity of SiO2 degrades from 76.0% to 75.8%, whereas that of pure Si would only be reduced from 91.5% to 91.4% .
The two-dimensional intensity pattern obtained in conical diffraction impedes the direct experimental characterization of special sample features using synchrotron light. We thus start with an investigation of those properties by means of RCWA calculations, again. Slightly different coupling strengths of the E⃗- and H⃗-field are well known from the literature  for planar gratings of that type. The polarization is accordingly expected to affect the diffraction efficiencies at least in minor magnitude. We quantify this amplitude as follows: In agreement with Fig. 4, the orientation of the incident E⃗-field is defined by the angle ϑ with respect to the s-polarized direction (E⊥). As it is confirmed afterwards by the numerical evaluation, the total diffracted power into the mth order can be written asEq. (5), the “excess amplitude” describes the positive difference between the actual efficiency for a certain polarization and the global minimum within 0 ≤ ϑ < 2π. Towards the symmetry angle ϑ̃m, that excess goes down to zero, wherefore the maximum is found for ϑ̃m ± π/2. The results of the simulation for the propagating orders, combined with the model just described, are shown in Fig. 8. For simplicity, the RCWA data are calculated only in the half plane A⊥ ≥ 0 here and mirrored for A⊥ < 0. Given the binary profile, the symmetry of the (±1)st orders with respect to the plane of incidence is obvious, where the symmetry angle is found as ϑ̃±1 = ±58.5°. Since the excess amplitude turns out to be quite small, , the beamsplitter may be used in practice for arbitrary polarizations with an almost constant efficiency. In contrast, the 0th order shows the oppositional behavior; the amplitude P̃0 exceeds the global minimum for ϑ̃0 = 0° by two magnitudes.
Those polarization characteristics might be of interest in sophisticated interferometric applications – in most cases however, the two-dimensional alignment of the grating to the beam around the both remaining axes affects its performance even more. If fixed with respect to the y-axis, i.e. for s-polarization again, the experimental setup permits as well the variation of the incidence angle ϕ0 and its orthogonal counterpart ψ. Fig. 9 visualizes the results of RCWA simulations. The modest misalignment around the x-axis is apparently not critical for the (±1)st order, tolerances up to ±0.5° would degrade their brightness by a few percent at most. A nearly singular behavior is observed instead for the central 0th order. As it is shown in the mid of Fig. 9, this sharp minimum runs through two magnitudes within the narrow band of ±0.1°. On the other hand, accidental tilts around the z-axis evoke an asymmetric distribution between the (−1)st and the (+1)st order, as depicted on the left and right of Fig. 9. Now the wide-range minimum would suppress the 0th order within about ±0.1°. Depending on the concrete setup, enlarged non-diffracted – as for misadjusted incidence angles ϕ0 – or unbalanced orders for operation – as for tilted gratings around e⃗z – can reduce the visibility of an EUV interferometer . Its design should thus match the technical feasible error budget in angular coordinates.
Besides its interferometric application, the conical grazing incidence device also works as an ordinary spectroscopic instrument within a certain range of wavelengths , wherever the somewhat complicated off-plane detection is acceptable. In comparison to state-of-the-art blazed reflection  or transmission gratings  used in classical orientation with an incidence along the grating vector, it may provide a similar or even higher net efficiency in the (±1)st order without any background photons from higher orders and a strongly suppressed 0th order contribution which can both degrade the achievable signal-to-noise ratio. Fig. 10 shows the simulated efficiencies which propagate within 21 nm ≤ λ ≤ 29 nm. The great advantage of the grazing incidence condition relies on the projected or “effective” period d cosϕ0 ≪ d, enabling the simple construction of high-throughput gratings with eliminated higher orders by means of standard e-beam lithography techniques. Nevertheless, the price that has to be paid for this comfort is an enlarged grating area. In particular, the lateral dimension Δx along the grating vector should measure at least 4 × 10−3 m for our sample to receive a resolving power λ/Δλ ∼ 104 as desired for future plasma diagnostics in astronomy, for instance. The longitudinal dimension is determined by the beam diameter, Δy = ∅beam/ cosϕ0. Based on the technical constraints at the EUV synchrotron beamline, i.e. for sufficient brightness of the diffracted light, we set this minimum length to Δy = 1.5 × 10−2 m, as it was already shown in Fig. 4.
6. Measurement technique and results
The measurements are performed using synchrotron radiation in the reflectometer  at the EUV radiometry beamline  in the laboratory of PTB at the storage ring BESSY II . Highly monochromatic (λ/Δλ ∼ 103) radiation of the constant wavelength λ = 25 nm, linearly polarized in s-orientation with a degree of polarization of 99.3%, is collimated to better than 0.5 mrad in both directions and has a cross section of (1.6 × 1.1)mm2 in x- and z-direction, respectively. Fig. 11 shows the respective edge scans of the intensity profile in both directions. For an incidence angle of ϕ0 = 85°, the beam is spread to about 12 mm in vertical direction in projection on the grating surface plane. The diffraction pattern is recorded using the photodiode detector of the EUV reflectometer which moves at a circle with a radius of 550 mm around the x-axis (see Fig. 4) with an accuracy of ±0.01°. An additional drive moves the detector parallel to the x-axis, perpendicular to the plane of reflection. This allows to move the detector to the out-of-plane diffraction orders in the conical diffraction geometry. Besides the rotation around the x-axis, the sample can also be rotated around its normal, the z-axis. Fig. 12 illustrates the effect of small angular variations δψ of this type in the direction cosine space.
We start with an investigation of the 0th order efficiency as a function of the duty cycle. For this purpose, a series of different duty cycles was fabricated (see Sect. 4). The measurement is a relative measurement by comparing the radiant power in the incoming beam and the diffracted or reflected beam. Both measurements are done using the same photodiode detector with a sensitive area of at least (5.5 × 5.5) mm2, significantly larger than the photon beam size as given above. Therefore, the measured ratio of the photodiode signals directly equals the ratio of the incident and diffracted radiant power, i.e. the diffraction efficiency, because the detector sensitivity cancels in this ratio. Table 1 gives an overview for empirical duty cycles 49% ≲ fexp ≲ 59%. For each sample, the angle of incidence ϕ0 is varied from 83.8° to 86.2° in steps of 0.04°. Subsequent to those scans, the incidence angle for minimized 0th order reflectance P0 at each duty cycle is used to fit the measured results with an RCWA model. The simulated thickness Δtsim, continuously increasing with fexp, indicates an almost constant aspect ratio 𝒜 = 0.47 ± 0.01 of the grating bars for this set of samples. The 0th order yields a distinct global minimum of 0.12% for the specimen with the almost optimal duty cycle fexp = 54.7%. Analogous RCWA calculations yield an even lower value. The experimentally obtained result for P0 is therefore not yet perfect albeit likely deep enough for real applications even if that non-diffracted light contaminates the signal from the (±1)st order.
To examine the suitability of conical grazing incidence gratings as beamsplitters, the sample with fexp = 54.7% from above is adjusted to the corresponding angle of incidence for minimized 0th order reflection again. The position and intensity of the (±1)st order is now measured via angular and lateral movements of the photodiode around and along the x-axis, respectively. For each measurement, the EUV spot is carefully centered to the detector. For various tilt angles −0.2° ≤ δψ ≤ + 0.2°, the results are shown in Fig. 13. In general, the diffraction from a tilted grating  is described by Eq. (4), i.e.Fig. 12. In the vicinity of ψ ≈ 90°, i.e. for small variations δψ ≈ 0°, the set of equations from (6) may be – in 1st order of the Taylor expansion – written as Fig. 4. In an arrangement as it was described before , the fringe visibility V using the conical beamsplitter would thus differ from the maximum V = 1 by only 4.6 × 10−8.
From Fig. 3, the maximal values for P±1 do not coincide exactly with the minima for P0, with respect to the groove depth Δt and duty cycle f as well. That feature is confirmed indeed by RCWA calculations, based on the duty cycle series data for ϕ0 and Δtsim from Tab. 1 again. The corresponding (±1)st order efficiencies are listed in Tab. 2 with their “ ± ” uncertainty as it follows from the standard deviation in f. The extremum in P±1 is no longer reached for fexp = 54.7% but rather near the lowest duty cycle of 48.9%. Presumed the constant aspect ratio from above, the global, differentiable maximum in P±1 (f,ϕ0) is located actually in the vicinity of an incidence angle ϕ0 ≈ 84.92°, where the variation of P±1 with f and ϕ0 vanishes, i.e. ∇⃗P±1 (f,ϕ0) = 0. In Fig. 14, that region – from which only about one half is plotted – is found in the upper left corner. However, the gain of less than 2% in P±1 from fexp = 54.7% towards 48.9% is probably negligible in practice, and an appropriate beamsplitter design should maximize the power ratio P±1/P0 instead. As it is shown in Tab. 2 too, an ultimate ratio far beyond 104 can be obtained, depending on the 0th order suppression. For the measured sample in comparison, we obtain an experimental power ratio of 2.75 × 102 at fexp = 54.7% especially.
Due to the field-grating interaction length of 12 mm, the lowest 0th order level should strongly correlate with profile imperfections in fact, in particular the statistical standard error of the duty cycle, i.e. the line edge roughness. However, its effect on absorption and stray light losses is not yet known in detail and demands for further analysis. Fig. 15 illustrates the measured results for the values from above (Tab. 1), together with numerical predictions for the calculated thickness data Δtsim. In each case, an interpolation up to the 3rd order ensures the straightened appearance of the contour plots and the minimum, drawn in red, roughly follows a linear dependence across the (f − ϕ0) plane. Despite of deviations especially for f ≲ 0.50 and f ≳ 0.58 and a less pronounced minimum around f ≈ 0.54 than expected from the RCWA model, an acceptable agreement is observed.
We introduce an alternative design for EUV beamsplitters, based on conical diffraction. Despite their technological simplicity, the optical performance exceeds that of well-known “classical” lamellar devices with an incidence along the grating vector. As predicted by RCWA simulations and confirmed by measurements on a fabricated sample, equalized efficiencies beyond 30% are obtained in the (±1)st order, providing an extraordinary high interference contrast in corresponding setups. Moreover, any diffraction losses from higher orders or background contributions from the 0th order are absent. Potential improvements might target the useable (±1)st order output and an even better suppression of the non-diffracted light. Further research steps should thus include the look for alternative materials aside from Si and SiO2 with higher reflectivity and advanced fabrication procedures for high-precision etching of binary profiles.
This work was part of the projects “Zentrum für Innovationskompetenz ultra optics” and “Photonische Nanomaterialien”, funded by the German Federal Ministry of Education and Research (BMBF) with the fund numbers 03Z1HN32 and 03IS2101A, respectively. We thank the team of the EUV beamline, Annett Barboutis, Martin Biel, Christian Buchholz, Jana Puls, and Christian Stadelhoff, for performing the measurements. The authors would also like to thank Frank Fuchs, Hans-Jörg Fuchs, Robert Hähle, Manuela Holz, Maria Oliva, Werner Rockstroh, Adriana Szeghalmi and Uwe Zeitner for their support and fruitful discussions.
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