## An EUV beamsplitter based on conical grazing incidence diffraction |

Optics Express, Vol. 20, Issue 2, pp. 1825-1838 (2012)

http://dx.doi.org/10.1364/OE.20.001825

Acrobat PDF (3899 KB)

### Abstract

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 10^{2} 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 0^{th} order. For a sample made of SiO_{2} on silicon, measured data and simulated results from rigorous coupled wave analysis procedures are given.

© 2012 OSA

## 1. Introduction

1. J. Filevich, K. Kanizay, M. C. Marconi, J. L. A. Chilla, and J. J. Rocca, “Dense plasma diagnostics with an amplitude-division soft-x-ray laser interferometer based on diffraction gratings,” Opt. Lett. **25**, 356–358 (2000). [CrossRef]

2. J. Grava, M. A. Purvis, J. Filevich, M. C. Marconi, J. J. Rocca, J. Dunn, S. J. Moon, and V. N. Shlyaptsev, “Dynamics of a dense laboratory plasma jet investigated using soft x-ray laser interferometry,” Phys. Rev. E **78**, 016403 (2008). [CrossRef]

3. B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-ray interactions: photoabsorption, scattering, transmission, and reflection at E=50-30000 eV, Z=1-92,” At. Data Nucl. Data Tables **54**(2), 181–342 (1993). [CrossRef]

4. Y. Liu, X. Tan, Z. Liu, X. Xu, Y. Hong, and S. Fu, “Soft X-ray holographic grating beam splitter including a double frequency grating for interferometer pre-alignment,” Opt. Express **16**, 14761–14770 (2008). [CrossRef] [PubMed]

^{th}and (−1)

^{st}order as well [5

5. Y. Liu, H.-J. Fuchs, Z. Liu, H. Chen, S. He, S. Fu, E.-B. Kley, and A. Tünnermann, “Investigation on the properties of a laminar grating as a soft X-ray beam splitter,” Appl. Opt. **49**, 4450–4459 (2010). [CrossRef] [PubMed]

^{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 SiO

_{2}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

*k⃗*

_{inc}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*.

6. J. E. Harvey and C. L. Vernold, “Description of diffraction grating behavior in direction cosine space,” Appl. Opt. **37**, 8158–8160 (1998). [CrossRef]

6. J. E. Harvey and C. L. Vernold, “Description of diffraction grating behavior in direction cosine space,” Appl. Opt. **37**, 8158–8160 (1998). [CrossRef]

*r⃗*= (

*x*,

*y*,

*z*) in a cartesian system, Centered with its origin to the grating, we set the

*α*-axis roughly parallel to

*G⃗*and the

*β*-axis along the grooves, precisely

*ψ*≡ ∢(

*G⃗*,

*e⃗*). The

_{β}*γ*-axis is oriented as the surface normal. A schematic illustration is given on the left of 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 where the azimuthal and polar angles

*θ*

_{0}and

*ϕ*

_{0}are defined as in Fig. 1. From the grating equation – for a period

*d*and wavelength

*λ*– and its additional constraint to reflection, the intersection points of the diffracted rays into the

*m*

^{th}order with the unit sphere are found as (

*α*,

_{m}*β*) – where

_{m}*β*→

_{m}*β*

_{0}for

*ψ*= 90°. The spatial spectrum of apparent, “real” orders is limited by the inequality

*ϕ*

_{0}→ 90°, the projected grating period

*d*cos

*ϕ*

_{0}≪

*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

*θ*

_{0}= 0° and arrive at the configuration shown on the right. With

*α*= 0 but an implemented tilt

_{i}*ψ*= 90° ±

*δψ*with |

*δψ*| ≲ 1°, we obtain from Eq. (3) and using [6

6. J. E. Harvey and C. L. Vernold, “Description of diffraction grating behavior in direction cosine space,” Appl. Opt. **37**, 8158–8160 (1998). [CrossRef]

*ϕ*

_{0}close to 90° restricts the set of possible orders to −1 ≤

*m*≤ +1 even for “macroscopic” periods, compared to the wavelength. In this way, the symmetric conical mounting on the right of Fig. 1 differs from more conventional (near) normal incidence arrangements in diffraction. Whereas the 0

^{th}order undergoes an elementary reflection within the plane of incidence from (0,

*β*) to (0,

_{i}*β*

_{0}), 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

*n*= 1 −

*δ*−

*iβ*, with real coefficients 0 <

*δ*,

*β*≪ 1 [3

3. B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-ray interactions: photoabsorption, scattering, transmission, and reflection at E=50-30000 eV, Z=1-92,” At. Data Nucl. Data Tables **54**(2), 181–342 (1993). [CrossRef]

*n*) = 1 −

*δ*< 1 leads to total external reflection of an incident electromagnetic field with a critical incidence angle

*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 SiO

_{2}for the substrate and the grating bars, respectively, at a wavelength well beyond the Si

_{L}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 [3

3. B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-ray interactions: photoabsorption, scattering, transmission, and reflection at E=50-30000 eV, Z=1-92,” At. Data Nucl. Data Tables **54**(2), 181–342 (1993). [CrossRef]

*λ*≲ 40 nm, the superior performance of pure Si is obvious, compared to SiO

_{2}. 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

*λ*= 25 nm and aim at an inclination near

_{c}*ϕ*

_{0}≈ 85°. Zero-roughness reflectivities of 91% for Si and 76% in case of SiO

_{2}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%.

^{th}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 Δ

*t*and the substrate, respectively. Their optical path difference is given as Δ

*s*

_{opt}= 2Δ

*t*cos

*ϕ*

_{0}. This length should be equal to an odd multiple of

*λ*/2. For the values from above, we find the lowest thickness Δ

*t*

_{min}∼ 70 nm in this approximation.

*s*-polarized incidence. Starting from that coarse estimation for Δ

*t*

_{min}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 Δ

*t*and

*f*, in terms of a best-fit algorithm to the empirical data set, in particular the incidence angle

*ϕ*

_{0}= 84.77° for which the 0

^{th}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 0

^{th}order contribution. Towards multiples of this best thickness Δ

*t*

_{opt}, near 290 nm for instance, the contrast between the (±1)

^{st}and the 0

^{th}order decreases, due to stronger coupling losses within the grooves. On the right of Fig. 3, the duty cycle is varied for Δ

*t*

_{opt}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 0

^{th}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 Δ

*t*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 [8

8. M. G. Moharam and T. K. Gaylord, “Three-dimensional vector coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. **73**, 1105–1112 (1983). [CrossRef]

*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.

## 4. Fabrication

_{2}film. We use an optical grade, polished 4” silicon wafer in 〈100〉 orientation. Heated to 400 K, the substrate is subsequently coated in an O

_{2}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 SiO

_{2}by reactive ion beam etching (RIBE) and CF

_{4}. This is done in a self-made tool with a 150 mm ion source (“Kaufmann-type”). The SiO

_{2}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.

_{2}layer with a typical thickness of about 1 nm on a blank Si surface in air is a well-known phenomenon [9

9. M. Morita, T. Ohmi, E Hasegawa, M. Kawakami, and M. Ohwada, “Growth of native oxide on a silicon surface,” J. Appl. Phys. **68**(3), 1272–1281 (1990). [CrossRef]

^{th}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 Δ

*z*= 5 nm of the electromagnetic field in SiO

_{p}_{2}[10

10. L. G. Parratt, “Surface studies of solids by total reflection of X-rays,” Phys. Rev. **95**, 359–369 (1954). [CrossRef]

*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 0

^{th}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 SiO

_{2}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 Δ

*t*

_{opt}= 101 nm indirectly deduced from 0

^{th}order diffraction (Sect. 6). The estimated RMS level should nearly maintain the optical performance, as calculated from the Fresnel formulae. Indeed, the reflectivity of SiO

_{2}degrades from 76.0% to 75.8%, whereas that of pure Si would only be reduced from 91.5% to 91.4% [11

11. Lawrence Berkeley National Laboratory’s Center for X-ray optics, Mail Stop 2R0400, 1 Cyclotron Road Berkeley, CA 94720 USA, http://henke.lbl.gov (2011).

## 5. Simulations

*E⃗*- and

*H⃗*-field are well known from the literature [8

8. M. G. Moharam and T. K. Gaylord, “Three-dimensional vector coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. **73**, 1105–1112 (1983). [CrossRef]

*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

*m*

^{th}order can be written as In Eq. (5), the “excess amplitude”

*ϑ*< 2

*π*. Towards the symmetry angle

*ϑ*̃

*, that excess goes down to zero, wherefore the maximum is found for*

_{m}*ϑ*̃

*±*

_{m}*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

*ϑ*̃

_{±1}= ±58.5°. Since the excess amplitude turns out to be quite small,

^{th}order shows the oppositional behavior; the amplitude

*P̃*

_{0}exceeds the global minimum

*ϑ*̃

_{0}= 0° by two magnitudes.

*y*-axis, i.e. for

*s*-polarization again, the experimental setup permits as well the variation of the incidence angle

*ϕ*

_{0}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 0

^{th}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 0

^{th}order within about ±0.1°. Depending on the concrete setup, enlarged non-diffracted – as for misadjusted incidence angles

*ϕ*

_{0}– or unbalanced orders for operation – as for tilted gratings around

*e⃗*– can reduce the visibility of an EUV interferometer [5

_{z}5. Y. Liu, H.-J. Fuchs, Z. Liu, H. Chen, S. He, S. Fu, E.-B. Kley, and A. Tünnermann, “Investigation on the properties of a laminar grating as a soft X-ray beam splitter,” Appl. Opt. **49**, 4450–4459 (2010). [CrossRef] [PubMed]

12. P. Lemaire, “Ultraviolet conical diffraction: a near-stigmatic tandem grating mounting spectrometer,” Appl. Opt. **30**, 1294–1302 (1991). [CrossRef] [PubMed]

13. Y.-Y. Yang, F. Süßmann, S. Zherebtsov, I. Pupeza, J. Kaster, D. Lehr, H.-J. Fuchs, E.-B. Kley, E. Fill, X.-M. Duan, Z.-S. Zhao, F. Krausz, S. L. Stebbings, and M. F. Kling, “Optimization and characterization of a highly-efficient diffraction nanograting for MHz XUV pulses,” Opt. Express **19**, 1954–1962 (2011). [CrossRef] [PubMed]

14. R. K. Heilmann, M. Ahn, E. M. Gullikson, and M. L. Schattenburg, “Blazed high-efficiency x-ray diffraction via transmission through arrays of nanometer-scale mirrors,” Opt. Express **16**, 8658–8669 (2008). [CrossRef] [PubMed]

^{st}order without any background photons from higher orders and a strongly suppressed 0

^{th}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*cos

*ϕ*

_{0}≪

*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 Δ

*x*along the grating vector should measure at least 4 × 10

^{−3}m for our sample to receive a resolving power

*λ*/Δ

*λ*∼ 10

^{4}as desired for future plasma diagnostics in astronomy, for instance. The longitudinal dimension is determined by the beam diameter, Δ

*y*= ∅

_{beam}/ cos

*ϕ*

_{0}. 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 Δ

*y*= 1.5 × 10

^{−2}m, as it was already shown in Fig. 4.

## 6. Measurement technique and results

15. J. Tümmler, G. Brandt, J. Eden, H. Scherr, F. Scholze, and G. Ulm, “Characterization of the PTB EUV reflectometry facility for large EUVL optical components,” Proc. SPIE **5037**, 265–273 (2003). [CrossRef]

16. F. Scholze, B. Beckhoff, G. Brandt, R. Fliegauf, R. Klein, B. Meyer, D. Rost, D. Schmitz, M. Veldkamp, J. Weser, G. Ulm, E. Louis, A.E. Yakshin, S. Oestreich, and F. Bijkerk, “The new PTB-beamlines for high-accuracy EUV reflectometry at BESSY II,” Proc. SPIE **4146**, 72—82 (2000). [CrossRef]

17. B. Beckhoff, A. Gottwald, R. Klein, M. Krumrey, R. Müller, M. Richter, F. Scholze, R. Thornagel, and G. Ulm, “A quarter-century of metrology using synchrotron radiation by PTB in Berlin,” Phys. Status Solidi B **246**, 1415–1434 (2009). [CrossRef]

*λ*/Δ

*λ*∼ 10

^{3}) 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)mm

^{2}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

*ϕ*

_{0}= 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.

^{th}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) mm

^{2}, 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% ≲

*f*

_{exp}≲ 59%. For each sample, the angle of incidence

*ϕ*

_{0}is varied from 83.8° to 86.2° in steps of 0.04°. Subsequent to those scans, the incidence angle for minimized 0

^{th}order reflectance

*P*

_{0}at each duty cycle is used to fit the measured results with an RCWA model. The simulated thickness Δ

*t*

_{sim}, continuously increasing with

*f*

_{exp}, indicates an almost constant aspect ratio 𝒜 = 0.47 ± 0.01 of the grating bars for this set of samples. The 0

^{th}order yields a distinct global minimum of 0.12% for the specimen with the almost optimal duty cycle

*f*

_{exp}= 54.7%. Analogous RCWA calculations yield an even lower value. The experimentally obtained result for

*P*

_{0}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.

*f*

_{exp}= 54.7% from above is adjusted to the corresponding angle of incidence for minimized 0

^{th}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 [6

**37**, 8158–8160 (1998). [CrossRef]

*ψ*≈ 90°, i.e. for small variations

*δψ*≈ 0°, the set of equations from (6) may be – in 1

^{st}order of the Taylor expansion – written as Whereas the azimuthal angle

*θ*and thus the horizontal position of the diffracted spot along the

_{m}*x*-axis is almost not affected under the assumption |

*δψ*| ≲ 1°, both the vertical positions and the diffraction efficiencies

*P*

_{±1}follow approximately linear functions of the tilt

*δψ*. This fact permits a simplified and precise alignment of the grating in an interferometric configuration. are measured indeed, 2.2% less than expected for the optimized design as shown in Fig. 4. In an arrangement as it was described before [5

5. Y. Liu, H.-J. Fuchs, Z. Liu, H. Chen, S. He, S. Fu, E.-B. Kley, and A. Tünnermann, “Investigation on the properties of a laminar grating as a soft X-ray beam splitter,” Appl. Opt. **49**, 4450–4459 (2010). [CrossRef] [PubMed]

*V*using the conical beamsplitter would thus differ from the maximum

*V*= 1 by only 4.6 × 10

^{−8}.

*P*

_{±1}do not coincide exactly with the minima for

*P*

_{0}, with respect to the groove depth Δ

*t*and duty cycle

*f*as well. That feature is confirmed indeed by RCWA calculations, based on the duty cycle series data for

*ϕ*

_{0}and Δ

*t*

_{sim}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

*f*

_{exp}= 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*,

*ϕ*

_{0}) is located actually in the vicinity of an incidence angle

*ϕ*

_{0}≈ 84.92°, where the variation of

*P*

_{±1}with

*f*and

*ϕ*

_{0}vanishes, i.e. ∇⃗

*P*

_{±1}(

*f*,

*ϕ*

_{0}) = 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

*f*

_{exp}= 54.7% towards 48.9% is probably negligible in practice, and an appropriate beamsplitter design should maximize the power ratio

*P*

_{±1}/

*P*

_{0}instead. As it is shown in Tab. 2 too, an ultimate ratio far beyond 10

^{4}can be obtained, depending on the 0

^{th}order suppression. For the measured sample in comparison, we obtain an experimental power ratio of 2.75 × 10

^{2}at

*f*

_{exp}= 54.7% especially.

^{th}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 Δ

*t*

_{sim}. In each case, an interpolation up to the 3

^{rd}order ensures the straightened appearance of the contour plots and the minimum, drawn in red, roughly follows a linear dependence

*f*−

*ϕ*

_{0}) 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.

## 7. Conclusion

^{st}order, providing an extraordinary high interference contrast in corresponding setups. Moreover, any diffraction losses from higher orders or background contributions from the 0

^{th}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 SiO

_{2}with higher reflectivity and advanced fabrication procedures for high-precision etching of binary profiles.

## Acknowledgments

## References and links

1. | J. Filevich, K. Kanizay, M. C. Marconi, J. L. A. Chilla, and J. J. Rocca, “Dense plasma diagnostics with an amplitude-division soft-x-ray laser interferometer based on diffraction gratings,” Opt. Lett. |

2. | J. Grava, M. A. Purvis, J. Filevich, M. C. Marconi, J. J. Rocca, J. Dunn, S. J. Moon, and V. N. Shlyaptsev, “Dynamics of a dense laboratory plasma jet investigated using soft x-ray laser interferometry,” Phys. Rev. E |

3. | B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-ray interactions: photoabsorption, scattering, transmission, and reflection at E=50-30000 eV, Z=1-92,” At. Data Nucl. Data Tables |

4. | Y. Liu, X. Tan, Z. Liu, X. Xu, Y. Hong, and S. Fu, “Soft X-ray holographic grating beam splitter including a double frequency grating for interferometer pre-alignment,” Opt. Express |

5. | Y. Liu, H.-J. Fuchs, Z. Liu, H. Chen, S. He, S. Fu, E.-B. Kley, and A. Tünnermann, “Investigation on the properties of a laminar grating as a soft X-ray beam splitter,” Appl. Opt. |

6. | J. E. Harvey and C. L. Vernold, “Description of diffraction grating behavior in direction cosine space,” Appl. Opt. |

7. | D. Attwood, |

8. | M. G. Moharam and T. K. Gaylord, “Three-dimensional vector coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. |

9. | M. Morita, T. Ohmi, E Hasegawa, M. Kawakami, and M. Ohwada, “Growth of native oxide on a silicon surface,” J. Appl. Phys. |

10. | L. G. Parratt, “Surface studies of solids by total reflection of X-rays,” Phys. Rev. |

11. | Lawrence Berkeley National Laboratory’s Center for X-ray optics, Mail Stop 2R0400, 1 Cyclotron Road Berkeley, CA 94720 USA, http://henke.lbl.gov (2011). |

12. | P. Lemaire, “Ultraviolet conical diffraction: a near-stigmatic tandem grating mounting spectrometer,” Appl. Opt. |

13. | Y.-Y. Yang, F. Süßmann, S. Zherebtsov, I. Pupeza, J. Kaster, D. Lehr, H.-J. Fuchs, E.-B. Kley, E. Fill, X.-M. Duan, Z.-S. Zhao, F. Krausz, S. L. Stebbings, and M. F. Kling, “Optimization and characterization of a highly-efficient diffraction nanograting for MHz XUV pulses,” Opt. Express |

14. | R. K. Heilmann, M. Ahn, E. M. Gullikson, and M. L. Schattenburg, “Blazed high-efficiency x-ray diffraction via transmission through arrays of nanometer-scale mirrors,” Opt. Express |

15. | J. Tümmler, G. Brandt, J. Eden, H. Scherr, F. Scholze, and G. Ulm, “Characterization of the PTB EUV reflectometry facility for large EUVL optical components,” Proc. SPIE |

16. | F. Scholze, B. Beckhoff, G. Brandt, R. Fliegauf, R. Klein, B. Meyer, D. Rost, D. Schmitz, M. Veldkamp, J. Weser, G. Ulm, E. Louis, A.E. Yakshin, S. Oestreich, and F. Bijkerk, “The new PTB-beamlines for high-accuracy EUV reflectometry at BESSY II,” Proc. SPIE |

17. | B. Beckhoff, A. Gottwald, R. Klein, M. Krumrey, R. Müller, M. Richter, F. Scholze, R. Thornagel, and G. Ulm, “A quarter-century of metrology using synchrotron radiation by PTB in Berlin,” Phys. Status Solidi B |

**OCIS Codes**

(050.2770) Diffraction and gratings : Gratings

(340.7480) X-ray optics : X-rays, soft x-rays, extreme ultraviolet (EUV)

**ToC Category:**

X-ray Optics

**History**

Original Manuscript: November 1, 2011

Revised Manuscript: January 1, 2012

Manuscript Accepted: January 2, 2012

Published: January 12, 2012

**Citation**

C. Braig, L. Fritzsch, T. Käsebier, E.-B. Kley, C. Laubis, Y. Liu, F. Scholze, and A. Tünnermann, "An EUV beamsplitter based on conical grazing incidence diffraction," Opt. Express **20**, 1825-1838 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-2-1825

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### References

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- L. G. Parratt, “Surface studies of solids by total reflection of X-rays,” Phys. Rev.95, 359–369 (1954). [CrossRef]
- Lawrence Berkeley National Laboratory’s Center for X-ray optics, Mail Stop 2R0400, 1 Cyclotron Road Berkeley, CA 94720 USA, http://henke.lbl.gov (2011).
- P. Lemaire, “Ultraviolet conical diffraction: a near-stigmatic tandem grating mounting spectrometer,” Appl. Opt.30, 1294–1302 (1991). [CrossRef] [PubMed]
- Y.-Y. Yang, F. Süßmann, S. Zherebtsov, I. Pupeza, J. Kaster, D. Lehr, H.-J. Fuchs, E.-B. Kley, E. Fill, X.-M. Duan, Z.-S. Zhao, F. Krausz, S. L. Stebbings, and M. F. Kling, “Optimization and characterization of a highly-efficient diffraction nanograting for MHz XUV pulses,” Opt. Express19, 1954–1962 (2011). [CrossRef] [PubMed]
- R. K. Heilmann, M. Ahn, E. M. Gullikson, and M. L. Schattenburg, “Blazed high-efficiency x-ray diffraction via transmission through arrays of nanometer-scale mirrors,” Opt. Express16, 8658–8669 (2008). [CrossRef] [PubMed]
- J. Tümmler, G. Brandt, J. Eden, H. Scherr, F. Scholze, and G. Ulm, “Characterization of the PTB EUV reflectometry facility for large EUVL optical components,” Proc. SPIE5037, 265–273 (2003). [CrossRef]
- F. Scholze, B. Beckhoff, G. Brandt, R. Fliegauf, R. Klein, B. Meyer, D. Rost, D. Schmitz, M. Veldkamp, J. Weser, G. Ulm, E. Louis, A.E. Yakshin, S. Oestreich, and F. Bijkerk, “The new PTB-beamlines for high-accuracy EUV reflectometry at BESSY II,” Proc. SPIE4146, 72—82 (2000). [CrossRef]
- B. Beckhoff, A. Gottwald, R. Klein, M. Krumrey, R. Müller, M. Richter, F. Scholze, R. Thornagel, and G. Ulm, “A quarter-century of metrology using synchrotron radiation by PTB in Berlin,” Phys. Status Solidi B246, 1415–1434 (2009). [CrossRef]

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