X-Ray Optics for Imaging Ideal Mirror Real Mirror Outline (A)

X-Ray Optics for Imaging ideal mirror real mirror Outline (a)

l a -500 1.0 l a -500 t t e r e r ►► Introduction + History

al -250 al -250 a a xis, 0 0.5 xis, 0 ►► Reflective Optics z z in nm 250 in nm 250 0.0 ► Diffractive Optics 500 500 ► -1.0 0.0 1.0 -1.0 0.0 1.0 Hercules Specialised Course 19 x in mm ►► Refractivex in mm Optics (b) Phase divided by plane wave

l a -500 π l a -500 t t e r Markus Osterhoff e r ►► Waveguides al -250 al -250 a a

University of Göttingen xis, xis, 0 0 0 ►► Discussion z z Grenoble, May 2017 in nm 250 in nm 250 -π feature articles 500 500 -1.0 0.0 1.0 -1.0 0.0 1.0 x in mm x in mm 2010), with controlled mode structure spot sizes down to sub- (c) Focus cut 10 nm. The mode and coherence ﬁltered waves are ideally 1.0 +π 1.0 +π suited for quantitative holographic image recording. Mode ﬁltering minimizes wavefront distortions and artifacts encountered in many other hard X-ray focusing optics. This

0.5 0 phase in 0.5 0 phase in enables quantitative reconstruction of the object by robust i n i n

t t phase-retrieval algorithms. By selecting the waveguide-to- ensity ensity

r r sample distance, objects can be imaged at a single distance in a ad ad 0.0 -π 0.0 -π -200 0 200 -200 0 200 full-field configuration without scanning. Robust and quickly lateral axis, z in nm lateral axis, z in nm converging iterative reconstruction schemes can be applied to invert the holographic near-field diffraction patterns. Weakly scattering biological specimen can thus be phased even without exact knowledge of the illumination function. The method proved to be very dose-efficient providing images of cells at doses below 105 Gy, and takes photon noise effects into account quantitatively (Giewekemeyer et al., 2011). The Figure 6 tomographic extension provides quantitative three-dimen- Holographic imaging with the waveguide probe. Scale bar: 585 mm (detector plane). (a) Hologram of a test pattern after division by the sional density reconstructions of biological cells and tissues empty beam. Scale bar: 6 mm (detector plane). (b) Raw data of a (Bartels et al., 2012; Olendrowitz et al., 2012). To probe the hologram with three Deinococcus radiodurans cells, showing the smooth three-dimensional structure of a larger specimen, the full-field line-shape and tails of the waveguide probe. (c) Phase reconstruction of the hologram shown in (a), with 22.4 nm pixel size. Scale bar: 2 mm. (d) holographic technique is of particular advantage, since it Rendered three-dimensional density distribution of a Deinococcus avoids the problem of overheads in detector readout which is radiodurans bacteria (Bartels et al., 2012), reconstructed from wave- pertinent in scanning three degrees of freedom (two transla- guide-based holographic tomography. tions, one tomographic rotation). In contrast to scanning SAXS or diffraction microscopy (ptychography), extended empty beam corrected hologram in (a) shows the holographic specimens from several cells and multicellular organisms to fringes with respect to the flat background (after empty beam tissues up to the organ level of small animals can be covered in division and raw data corrections), recorded with a test one or a few exposures, eventually enlarged by stitching. Using pattern milled by focused ion beam into a 200 nm-thick gold zoom magnification by defocus variation, the magnification layer on 200 nm-thick Si3N4 over a total accumulation time of and the FOV can easily be adapted and combined. At the 30 s at 7.9 keV photon energy, waveguide-to-object distance same time the optical scheme is ideally suited for high-flux

z1 = 17.55 mm, and detector distance z1 z2 = 5.13 m. Inter- ptychographic phasing and scanning SAXS/WAXS applica- ference fringes extend all the way toþ the corners of the tions, and a fast switch in imaging modality on the same diffraction pattern indicating a high-quality hologram. The sample is supported by the optical design. The advanced phase reconstructed from this hologram is shown in Fig. 6(c), detection systems already tested for the KB nano-probe based on the algorithms presented by Giewekemeyer et al. (Giewekemeyer et al., 2014; Wilke et al., 2014), but also the (2011) and Bartels et al. (2015), which enforces compact object Eiger detector (Guizar-Sicairos et al., 2014), can cope with the support as well as measured intensity values. Importantly, the high ﬂux density. support information required for this algorithm is retrieved As a future direction, the described optics and imaging from a deterministic holographic reconstruction and thus does scheme for all imaging modalities is compatible with pink not need prior information such as in CDI. The raw data of a beam operations, which would increase ﬂux by one to two holographic recording is illustrated in Fig. 6(b) for bacteria, orders of magnitude. For scanning nano-diffraction in SAXS before empty beam correction. Waveguide-enhanced holo- or even WAXS mode as long as weakly ordered systems are graphic imaging can readily be extended to tomography, as considered, the intrinsic undulator bandpath of Á�=� 0.006 demonstrated for Deinococcus radiodurans cells by Bartels would not be prohibitively large, and hence even’ weakly et al. (2012). A corresponding rendering of the three-dimen- scattering samples such as the hydrated cell in Fig. 4 could be sional electron density distribution is shown in Fig. 6(d). The acquired with dwell times of 10 ms. Scanning with contin- quantitative two-dimensional and three-dimensional phase uous motor movement and� pixel detector technology for retrieval reveals dense structures which may be associated frame rates of 100 Hz is already available, warranting a with DNA rich bacterial nucleoids. straightforward implementation� of such a fast nano-diffraction mode. Concerning radiation damage, further studies should investigate whether a ‘diffract-and-destroy’ strategy can be 6. Conclusion and outlook adopted based on increased scanning speed to outrun diffu- A unique aspect of the compound mirror–waveguide system sion-limited reactions of free radicals. For ptychography, pink discussed here is the delivery of quasi-spherical wavefronts beam operation could bring about higher robustness and emanating from a two-dimensional X-ray waveguide exit reconstruction quality, by avoiding any vibrations associated (Salditt et al., 2008; Giewekemeyer et al., 2010; Kru¨ger et al., with monochromator cooling. For small spot sizes, i.e. in the

J. Synchrotron Rad. (2015). 22, 867–878 Tim Salditt et al. � Coherent imaging and nano-diffraction 875 Introduction and History Advantages of Hard X-Rays Outline ►► high penetration length ►► Introduction + History looking into the volume, ►► Reflective Optics 3D tomography ►► Diffractive Optics ►► element specific access to ionisation state ►► Refractive Optics absorption, fluorescence, Auger ►► Waveguides ►► short wavelength ►► Discussion matching length scales in crystals, glasses ►► scattering from electron density Disadvantages ►► beam damage ►► weak interaction with optics ►► expensive X-Ray Optics, Index of Refraction What we want ►► small focus, i.e. high numerical aperture ►► high efficiency ►► long working distance, i.e. high angles of convergence ►► multiple contrasts ►► weak interaction with sample; low dose, but strong signal? X-Ray Optics, Index of Refraction What we want What we have ►► small focus, ►► index of refraction i.e. high numerical aperture n = 1 – d + ib ►► high efficiency ►► d: dispersive part, yields phase shift ►► long working distance, i.e. high angles of convergence ►► b: absorption ►► multiple contrasts ►► typically: d ~ 10-5, b ~ 10-7 (hard x-rays) ►► weak interaction with sample; ►► weak interaction with optics, low dose, but strong signal? x-rays cannot be bent so easily … X-Rays and Optics – the First 50 Years Not so much has happend … ►► discovery of X-Rays by C.W. Röntgen in 1895 ►► no interaction with lenses found ►► particles? waves? something else? ►► what are crystals? arrays of atoms? ►► lattice spacing + wavelength agree

William J. Morton, American Technical Book Co., 1896 X-Rays and Optics – the First 50 Years Not so much has happend … Generation of X-rays … ►► discovery of X-Rays ►► only by traditional X-ray tubes by C.W. Röntgen in 1895 ►► parasitic use of synchrotron radiation ►► no interaction with lenses found ►► dedicated synchrotron radiation sources ►► particles? waves? something else? ►► ESRF as 3rd generation source ►► what are crystals? ►► 4th generation to come, arrays of atoms? x-ray free electron lasers ►► lattice spacing + wavelength agree ►► brilliance increase

Optics for (hard) X-rays: ►► started about 50 years after Röntgen’s discovery Pioneers of X-Ray Optics

Kirkpatrick and Baez G. Schmahl and coworkers A. Snigirev and his team 1950ies: reflecting mirrors 1980ies: Fresnel Zone Plates 1990ies: Refractive Lenses

Kirkpatrick, Baez, JOSA 1948 Schmahl et al, Ultramicroscopy, 2000 Snigirev, Lengeler et al, Nature, 1996 Mirrors and Multilayer Mirrors Outline ►► Introduction + History ►► Reflective Optics ►► Diffractive Optics ►► Refractive Optics ►► Waveguides ►► Discussion Principle and Geometry

(a) (b) Synchrotron orbit VFM p+q = const HFM KB Ellipse: ►► the distance from S to F via any point on the ellipse is constant ►► constant distance = constant phase ►► waves interfere constructively, leading to a focus Requirements: ►► high reflectivity of the surface ►► good surface quality ►► 3D shape or 2 crossed 2D shapes ideal mirror real mirror (a)

l a -500 1.0 l a -500 t t e r e r

Principle and Geometry al -250 al -250 a a xis, 0 0.5 xis, 0 Measured height deviation proﬁles (a) (b) Synchrotron orbit z HVM z in nm 250 VFM in nm 250 VFM 8 heighr deviation in nm 0.0 500 500 p+q = const HFM -1.0 0.0 1.0 -1.0 0.0 1.0 x in mm x in mm KB 0 (b) Phase divided by plane wave

l a -500 π l a -500 t t e r Ellipse: e r

al -250 al -250 a a ►► the distance from S to F via any point -8 xis, 0 0 xis, 0

-50 -25 0 z 25 50 z on the ellipse is constant mirror coordinate, s in in nm mm250 in nm 250 -π typical figure errors, 2010: 500~ nm over 10 cm 500 -1.0 0.0 1.0 -1.0 0.0 1.0 ►► constant distance = constant phase x in mm x in mm

ideal mirror real mirror (c) Focus cut (a)

l a -500 1.0 l a -500 t ► waves interfere constructively, t e r ► e r 1.0 +π 1.0 +π al -250 al -250 a a xis, 0 0.5 xis, 0

leading to a focus z z in nm 250 in nm 250 0.0 500 500 -1.0 0.0 1.0 -1.0 0.0 1.0 x in mm x in mm 0.5 0 phase in 0.5 0 phase in Requirements: (b) Phase divided by plane wave

l a -500 π l a -500 i n i n t t e r e r t t al -250 al -250 ensity ensity a a xis, 0 0 xis, 0 r r z z ad ad ►► high reflectivity of the surface in nm in nm 250 250 -π 0.0 -π 0.0 -π 500 500 -1.0 0.0 1.0 -1.0 0.0 1.0 -200 0 200 -200 0 200 x in mm x in mm lateral axis, z in nm lateral axis, z in nm ►► good surface quality (c) Focus cut wave-optical1.0 +π simulations1.0 +π of an ideal and real ► 3D shape or 2 crossed 2D shapes ► 0.5 0 phase in 0.5 0 phase in

mirror;i n focal spoti n size <100 nm (point-source) t t ensity ensity r r ad ad 0.0 -π 0.0 -π Osterhoff, PhD thesis, 2011 -200 0 200 -200 0 200 lateral axis, z in nm lateral axis, z in nm MR47CH02-Macrander ARI 6 February 2017 13:37

The second crystal of such a DCM can be bent around an axis parallel to an X-ray beam to focus sagittally (3), i.e., in a plane perpendicular to the diffraction plane. Unless special designs are implemented, such bending will cause changes in the Bragg spacing. This effect is known as anticlastic bending, which can be avoided with special optical designs (4, 5). The weak-link DCM is a double-bounce monochromator made from a monolithic crystal, but the two parts are connected by a narrow neck that can be easily bent by external forces to adjust the relative orientation between the two parts. But the main disadvantage of this kind of DCM is that bending-induced strains can still propagate from the neck to the diffraction parts. As in the case of a single-bounce monochromator, the angular acceptance and bandwidth of a

DCM can be adjusted by using asymmetric reﬂections. As shown in Figure 4b,c, both �θ Dw and �EDw of the channel-cut crystals again decrease with the asymmetric factor b1 of the ﬁrst crystal TotalC1, similar to what Reflection, is shown in Figure 3d,e. Coating

3. X-RAY MIRRORS (a) KB mirror far-field (b) Fresnel reflectivity X-ray mirrors rely on total external reflection. The real part of the index of refraction is given by integration 1.0 integrated line cut (6, 7) 1.0 reflectivity n 1 δ 1 λ2r ρ/2π. 2. = − = − e 5 Here λ is the wavelength; r is the classical electron radius, which equals 2.814 10− ; and ρ e × is the electron density. The result is that n is less than unity so that, according to Snell’s law, 0.5 5 there is total external reflection, and because δ is very small ( 10− ), reflection occurs only at low ∼ 0.7

grazing angles. The reflectivity of smooth surfaces can be computed from Fresnel formulas (8). reflectivity The critical grazing angle above which total external reflection does not occur is given by θ c = √(2δ). Figure 5 shows calculated (9) specular reflectivity curves at 8 keV for a Si mirror without any coating as well as for one coated with Pd. At extremely grazing angles, an evanescent wave 0.0 to a depth of λ/(2πθ c) propagates just below the surface. The thickness of the coating should be plot direction 0.4 chosen to be much larger than the penetration depth of the evanescent wave, which is 3 nm in the -40 0 40 case of a Pd mirror. surface coordinate, s in mm

1 ►► total external reflection for grazing incidence Annu. Rev. Mater. Res. 2017.47. Downloaded from www.annualreviews.org

0.1 Pd ►► typical angles: 0.5° – few milli radian Access provided by WIB6055 - University of Goettingen on 05/14/17. For personal use only.

Reflectivity Si ►► proportional to electron density × wavelength

0.01

0.0 0.2 0.4 0.6 0.8 1.0 Grazing angle (°) Figure 5 TheoreticalMacrander, X-ray reﬂectivity Huang: at 8 keV Annual from a ﬂat Si Review substrate and of after Material coating it with Research, a thick layer of Pd. ‘17 Osterhoff, PhD thesis, 2011

2.6 Macrander Huang ·

Changes may still occur before final publication online and in print equal to ␪ϩarctan͑dh͑zm͒͞dz͒ due to the large source distance and low grazing angle. The intensity reflectivity R͑zm͒ ϭ r͑zm͒r*͑zm͒, where the superscript asterisk represents complex conjuga- tion, is unity for ␪z Ͻ␪c and the incident x rays are totally reflected from the surface. While the boundary conditions obtained from Maxwell’s equations stipu- late the continuity of fields across the boundary, there is no net energy flow into the material. In a real experiment, the incident wave is bounded both in space and time, and a wave field within the material initiated at the beginning of the reflection process manifests as an evanescent wave whose amplitude Fig. 1. Schematic of elliptical mirror geometry. The semimajor axis a, semiminor axis b, and the eccentricity e are defined in terms decreases exponentially with increasing depth into the material; the time average of the component of of the source distance s1 ϭ SO, focal length s2 ϭ OD, and the grazing incidence angle ␪ (described in the text). The distance the Poynting vector normal to the surface vanishes, between the foci of the ellipse is 2F and the center of the mirror and the net energy flow is parallel to the surface in 1,38 length L is located at the coordinates ͑z0, x0͒. the plane of incidence. Since the reflectivity coefficient is of unit magnitude and takes the form of a complex number divided the grazing angle ␪ and the distances s1 and s2 from by its conjugate, Eq. (8) can be written in the form 37 i␾͑zm͒ the center of the mirror to the foci, i.e., the semi- r͑zm͒ ϭ e . The quantity ␾͑zm͒ represents the phase 1 change of the components of the reflected wave and major axis a ϭ 2͑s1 ϩ s2͒ and semiminor axis b ϭ ͓a2͑1 Ϫ e2͔͒1͞2, where the eccentricity is given by depends upon the local angle of incidence and the 2 2 1͞2 e ϭ 1͞2a͓s1 ϩ s2 Ϫ 2s1s2 cos͑␲Ϫ2␪͔͒ . polarization of the radiation. For the aforementioned The design parameters for a mirror ͑s1, s2, ␪͒ are case of ␴-polarized x rays, the phase change on total tailoredGouy to a Phase particular beamline or application. For reflection example, a vertically oriented nanofocusing mirror 2 used at the APS beamline 34-ID is designed for a Ϫͱ2␦Ϫsin ␪z ␾͑zm͒ ϭ 2 arctan (9) sourcemir distanceror n < 1 s1 ϭ 60 m, focal length s2 ϭ 0.06 m, ͩ sin ␪z ͪ 13 and grazingvacuum n incident= 1 angle ␪ϭ0.0032 rad.transmittedThese beam θ1 parameters are adopted for the simulationsθ described varies in the range ͓0, Ϫ␲͔ as shown in Fig. 2. 2 ►► evanescent wave enters below (Subsection 3.A). As1 a consequence of the low In experiments utilizing a beam of finite cross sec- incoming beam reflected beam even under total reflection incidence angle, the longitudinal roughness has a dom- tion, the variation of ␾ for different components in the (a) (b) Simulated focus inant effect on the images downstream of the mirror, angularreal n spectrum►► surface: results phase in shift interference gradient causing an which4.5 show well-pronounced one-dimensional2.0 1.0 fringes apparentcomplex n shift of the reflected beam with respect to the lo 18 ►► shift of focus position perpendicularc to the incidence plane. Thus, the scat- reflection predicted by geometrical optics (the Goos– al angle, θ in m tering surface can be considered as one dimensional 38–41 Hänchen effect).►► reducedThis intensity interpretation, which arises (1D), which is equivalent to assuming that a given 4.0 1.5 0.5 in ray opticsdue loses to someabsorption significance in the case of a mirror is a segment of an elliptical cylinder continuousi n large beam approximating a plane wave, i.e., when t in the y direction perpendicular to the opticalensity axis. the surface►► isbut completely no impact illuminated on focus size, by coherent ra- r ad Surface3.5 height fluctuations are measured1.0 as ordinates0.0 no aberrations h͑z ͒ from-40 the perfect 0 elliptical 40 surface, where-100 z 0 100 m mirror’s surface, s in mm lmateral axis, y in nm represents the position along the mirror.

1. Reflectivity Kewish et al, Applied Optics, 2007 Total external reflection occurs for hard x rays inci- Osterhoff, PhD thesis, 2011 dent on a material at grazing angles below the critical Ϫ5 angle ␪c Ϸ ͱ2␦, where ␦ is the decrement (ϳ10 to 10Ϫ6) from unity of the real part of the refractive index n ϭ 1 Ϫ␦ϩi␤. For perpendicularly (␴) polarized x rays the Fresnel reflectivity coefficient1 at each point on the surface is, in the absence of absorption, defined by

2 sin ␪z Ϫ ͱsin ␪z Ϫ 2␦ r zm ϭ , (8) ͑ ͒ 2 sin ␪z ϩ ͱsin ␪z Ϫ 2␦ where the local angle ␪z between the incident wave Fig. 2. Phase change for ␴-polarized x rays on total reﬂection, for vector ͑k ϭ 2␲͞␭͒ and the surface is approximately grazing angles of incidence ␪z below the critical angle ␪c.

10 April 2007 ͞ Vol. 46, No. 11 ͞ APPLIED OPTICS 2013 ARTICLE IN PRESS

100 Ch. Morawe, M. Osterhoff / Nuclear Instruments and Methods in Physics Research A 616 (2010) 98–104

the outgoing waves produce constructive interference and high 3.2. Ray tracing calculations intensity in the focal spot. So far the RML was treated as a single reflecting surface. In order to investigate the impact of the beam penetration into the 3. Theoretical models ML structure, both numerical and analytical ray tracing simulations were developed [16–18]. An analytical approach is shown in 3.1. Geometrical approach Fig. 5 for the case of a parabolic RML. It can be shown that the elliptic case is virtually identical for the high eccentricities that Quantitative estimates of the achievable numerical aperture of occur in synchrotron applications. The ML stack is approximated by a single thick layer of average optical index n=1 d. Multiple Maximum Aperture – Minimum Focus Sizeboth TRM and RML can be obtained by calculating the opening � angle 2e for each case [9]. reflections are neglected, and refraction at the first interface P is In the TRM case one assumes a full opening of up to half of the treated in linear approximation. The aim is to calculate the critical angle intersection points of the ray that penetrates into the structure and is reflected by the second interface P0. An ensemble of parallel 4eryC 7 ð Þ incoming beams would form an envelope curve of enhanced leading to a diffraction limit of intensity (caustic) near the ideal focus F. 1:76 l p Ddiff TRM 1:76 8 nice, but ð Þ� p ¼ r0 r ð Þ 2 d rffiffiffiffiffiffiffiffiffiffie not here :( where r0 is theffiffiffiffiffiffiffi classical electron radius and re the electron density of the material. This means that the focusing power of a TRM is entirely limited by material properties. For a Pt mirror one obtains a minimum spot size of about 25 nm FWHM. A similar estimate can be performed for a RML. Applying the Bragg law without refraction correction l sin y 9 � 2 L ð Þ which is a good approximation for small d-spacings, the diffraction limit can be calculated by 0:88 Ddiff RML 10 ð Þ� 1=L2 1=L1 ð Þ ð Þ�ð Þ

Here L1,2 indicates the ML d-spacings at the corresponding edges of the mirror. The resolution is therefore limited only by the Fig. 5. Ray propagation through a curved layered structure. Rays reflected from lateral d-spacing gradient of the RML. For a short period RML with the upper ML surface reach the ideal focus F. Rays reflected at P0 are refracted twice when crossing P. strong gradient (L=2–3 nm) focal spots of about 5 nm FWHM Harke, Hell etappear al, Optics realistic. Express, Eq. (10) 2008 can beMorawe, compared Osterhoff, with the corresponding NIM A, 2010 formula for a linear Fresnel zone plate (FZP) D FZP 0:88 DR 11 diff ð Þ¼ ð Þ with outermost zone width DR. Considering RML as an equivalent 1 to half of a linear zone plate (2 FZP) with outermost repetition period L2=2DR and setting the largest ML d-spacing L1 to infinity (half of central FZP zone) one obtains

1 Ddiff FZP 0:88 L2 Ddiff RML : 12 ð2 Þ¼ ¼ ð Þ ð Þ The above equation shows that, within the given approximation, both optical elements are different representations of the same focusing conﬁguration (Fig. 4). For applications it is interesting to note that none of the above equations contains any explicit energy dependence.

Fig. 4. RML and FZP as different representations of the same focusing conﬁgura- Fig. 6. Ensemble of rays (straight arrows) emerging from a RML under variable tion. grazing angles and constant s. The lower curve indicates the corresponding caustic. PHYSICAL REVIEW LETTERS week ending VOLUME 91, N UMBER 20 14 NOVEMBER 2003

(a) Incident mode TE # (b) Incident mode TE # an optical lens system. Here, is the numerical 0 1 2 3 4 5 6 0 1 2 3 4 5 6 NA 0.15 1.5 aperture in object space, which in our case equals � . It 8 8 c 7 7 is the very small effective of all x-ray optics that 6 6 NA 0.10 1.0 sets an ultimate limit to the spatial resolution of 10 nm. 5 5 4 4 Beam sizes down to 1 nm, as desired for, e.g., single-atom 3 3 x-ray ﬂuorescence experiments [16], therefore seem to be 0.05 2 0.5 2 1 1 out of reach. Exit mode TE # Exit mode TE # 0 0 Nonetheless, a physical aperture of size W is still Exit angle (degrees) Exit angle (degrees) c 0.00 0.0 0.00 0.01 0.02 0.03 0.0 0.1 0.2 0.3 quite small and a challenge to make. However, our analy- Incident angle (Degrees) Incident angle (Degrees) sis of the focusing performance of a wedge-shaped wave- (c) Incident mode TE # (d) Incident mode TE # guide demonstrated how mode mixing and interference 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0.15 .015 8 8 can be exploited to achieve beam sizes down to near Wc 7 7 without the need for extremely small exit widths. In 6 6 0.10 .010 5 5 principle, such multimode interference effects can also 4 4 be used to advantage in circular capillaries. 3 ARTICLE3 INWe PRESS are indebted to D. Bilderback for an inspiring 0.05 2 .005 2 1 1 discussion which led to this work. Discussions with

100 Ch. Morawe, M. Osterhoff / Nuclear InstrumentsExit mode TE # and Methods in Physics Research A 616 (2010) 98–104 0 0

Exit angle (degrees) M. J. Zwanenburg and A. Dolocan are gratefully ac- 0.00 .000 0.00 0.01 0.02 0.03 0.000 0.001 0.002 0.003 knowledged. The measurements were performed at the outgoingIncident waves angle produce (Degrees) constructiveIncident interference angle (Degrees) and high 3.2. Ray tracing calculations intensity in the focal spot. beam line ID10 of the ESRF, Grenoble. This work was funded by the Swiss National Science Foundation (Project FIG. 3. Logarithmic density plots of the intensity I � ;� So far the RML was treated as a single reflecting surface. In i e No.order 2100-63738). to investigate the impact of the beam penetration into the diffracted from the exit of a planar SiO wedge, with � 3. Theoretical models 2 ML structure, both numerical and analytical ray tracing simula- 0:093 nm: (b)–(d) are calculations for geometries similar to the tions were developed [16–18]. An analytical approach is shown in respective panels of Fig. 2; (a) shows the measured intensity 3.1. Geometrical approach Fig. 5 for the case of a parabolic RML. It can be shown that the distribution, to be compared with (c), at � 0:94 (no data were elliptic case is virtually identical for the high eccentricities that taken for �i < 0:004 and the lower left corner of the plot). occur in synchrotron applications. The ML stack is approximated Quantitative estimates of the achievable numerical aperture of *Present address: Cavendish Laboratory, University of bothNote TRM that and the RML intensity can be distributions obtained by along calculating the vertical the opening lines by a single thick layer of average optical index n=1 d. Multiple Maximum Aperture – Minimum Focus Size Cambridge, Madingley Road, Cambridge� CB3 0HE, anglelabeled 2e asfor incident each case mode[9]. #2 are the diffracted fields for the TE2 reflections are neglected, and refraction at the first interface P is inputIn the mode TRM as case shown one in assumes Figs. 2(b) a full–2(d). opening of up to half of the treatedUnited in linear Kingdom. approximation. The aim is to calculate the critical angle intersection[1] B. Lengeler pointset of al. the, Appl. ray that Phys. penetrates Lett. 74, into 3924 the (1999). structure and[2] is C. reflected David, by B. the No¨ secondhammer, interface and E.P Ziegler,0. An ensemble Appl. Phys. of parallel Lett. 4eryC 7 diameter of 0:67Wc. Optimal phase matching of the in-ð Þ incoming79, 1088 beams (2002). would form an envelope curve of enhanced leadingcident beamto a diffraction to the lowest limit of mode is achieved by prefocus- intensity[3] K. Yamauchi (caustic) nearet al. the, J. ideal Synchrotron focus F. Radiat. 9, 313 (2002). ing a Gaussian beam onto the entrance opening and [4] Y.P. Feng et al., Phys. Rev. Lett. 71, 537 (1993). 1:76 l p Daligningdiff TRM it with the1:76 cone axis. The analogy with the8 [5] F. Pfeiffer et al., Science 297, 230 (2002). nice, but ð Þ� p2 ¼ r0 r ð Þ time-dependentd Schro¨rdingerffiffiffiffiffiffiffiffiffiffie equation still applies, but [6] J. H. H. Bongaerts et al., J. Synchrotron Radiat. 9, 383 not here :( ffiffiffiffiffiffiffi wherenow withr0 is two the spatial classical dimensions. electron radius and re the electron (2002). densityThe of beam the material. size limit This meansW , derived that the here focusing for waveguid- power of a [7] D. H. Bilderback, S. A. Hoffman, and D. J. Thiel, Science TRM is entirely limited by materialc properties. For a Pt mirror one ing geometries, also applies to other x-ray focusing de- 263, 201 (1994). obtains a minimum spot size of about 25 nm FWHM. [8] M. J. Zwanenburg et al., Phys. Rev. Lett. 82, 1696 (1999). vices. For example, the spot size achievable with a Fresnel A similar estimate can be performed for a RML. Applying the [9] M. Abramowitz and I. A. Stegun, Handbook of Braggzone plate law without is usually refraction taken correction to be the outermost zone width Mathematical Functions (Dover, New York, 1965), �r for the main focus, and �r=n for nth order diffraction, l 2 Chaps. 9.1–9.2. ►sin►yindex of refraction scales as l 2 9 i.e.,� a2 smallerL spot size but at the cost of 1=n efficiency.ð Þ [10] J. Als-Nielsen and D. McMorrow, Elements of Modern Volume gratings can increase the efficiency at higher X-Ray Physics (Wiley, New York, 2001), p. 63. (a) Focus size vs. parameter scaling (b) Zoom-in ►which► numerical is a good approximationaperture vanishes for small d-spacings, the mirror length diffractionorders [14,15], limit can and be calculatedso at first by sight it appears that the [11] See, for example, F. Schwabl, Quantum Mechanics 104 55 critical angle photon energy 50 spot size could be made arbitrarily small using small (Springer, New York, 1995), Chap. 3.4. angle of incidence ►► fundamental0:88 focus limit (!/?) Dzonediff RML widths and/or high order diffraction. However,10 as [12] The symbol TEn stands for the transverse electric mode f f ocus si 1/focus distance ocus si ð Þ� 1=L2 1=L1 ð Þ 103 40 we have seen,ð Þ�ð on exitingÞ the zone plate the wave is having n nodes, excluding the ones at the boundaries. [13] M. J. Zwanenburg et al., Physica (Amsterdam) 283B, 285 HereeffectivelyL1,2 indicates smeared the ML out d-spacings laterally at over the corresponding a length scale edges of z z e, Δ in nm e, Δ in nm 30 102 of the mirror. The resolution is therefore limited only by the Fig. 5. Ray(2000). propagation through a curved layered structure. Rays reflected from Wc. This again sets a lower limit of Wc to the focal spot lateral d-spacing gradient of the RML. For a short period RML with the[14] upper D. ML Hambach, surface reach G. the Schneider, ideal focus F and. Rays E. reflected M. Gullikson, at P0 are refracted Opt. size, regardless of the order of diffraction. This is in line twice when crossing P. strong gradient (L=2–3 nm) focal spots of about 5 nm FWHM Lett.. 26, 1200 (2001). 1 20 10 Harke, Hell etappearwith al, Optics the realistic. empirical Express, Eq. (10) spot2008 can be sizeMorawe, compared limit Osterhoff, with of 13 the nm corresponding NIM found A, 2010 in 10-2 10-1 1 101 1 2 3 4 5 6 [15] A. N. Kurokhtin and A.V. Popov, J. Opt. Soc. Am. A 19, parameter scaling, p parameter scaling, p Osterhoff,formularecent fornumerical PhD a linear thesis, Fresnel studies 2011 zone [15] plateBergemann for (FZP) zone plates et al, PRL, made 2003 of 315 (2002). nickel. D FZP 0:88 DR 11 [16] D. H. Bilderback and R. Huang, AIP Series of Con- diff ð Þ¼ ð Þ Note that Eq. (4) for Wc is analogous to the expression ference Proceedings, SRI Conference, San Francisco, with�=2 outermostfor the zone diffraction width DR. Considering limited spatial RML as resolution an equivalent of 2003 (to be published). 1 to halfNA of a linear zone plate (2 FZP) with outermost repetition period L2=2DR and setting the largest ML d-spacing L1 to infinity (half of central FZP zone) one obtains 204801-4 204801-4 1 Ddiff FZP 0:88 L2 Ddiff RML : 12 ð2 Þ¼ ¼ ð Þ ð Þ The above equation shows that, within the given approximation, both optical elements are different representations of the same focusing configuration (Fig. 4). For applications it is interesting to note that none of the above equations contains any explicit energy dependence.

(a) Confocal ellipses (b) Elliptical coordinates θ g t in s s r r n a 0 1 o e c r S F = c 2c n s i t t = const s increasing ML mirror as nested confocal ellipses (a) Kinematical theory (b) Ray-tracing (c) Dynamical theory Incoming wave field Reflected wave field, Reflected wave field, Incoming wave field Reflected wave field, Reflected wave field, Bragg condition modified Bragg condition Bragg condition``cylindrical wave’’ modified Bragg condition ``cylindrical wave’’ Laplacian in elliptical Laplacian coordinates in elliptical coordinates rules of (the same, but (the same, but differentiation with instead of ) with ) rules of (the same, but (the same, but differentiation with instead of ) with )

(the same, but approximation: with ) neglect second (the same, but approximation: derivative of with ) neglect second field envelope derivative of field envelope additional terms from modification approximation: additional terms note the sign neglect from modification (reflected wave) note the sign approximation: vs. Multilayer Coatings, Takagi-Taupin neglect (reflected wave) (a) Confocal ellipses (b) Elliptical coordinates vs. θ g t in s s r r n a 0 1 o e c r S F = c 2c n s i t reorder terms, t = const s increasing introduce ML mirror as nested confocal ellipses reorder terms, abbreviations: (a) Kinematical theory (energy(b) Ray-tracing conservation)(c) Dynamical theory introduce , , abbreviations: (energy conservation) , , , , , , models of increasing complexity: single reflection; (change of propagation coefficient) multiple reflection; (changecoupled ofwave-theory direction) (change of propagation coefficient) (change of direction) final form Takagi-Taupin description in elliptical geometry: incoming wave and reflected wave final form are coupled by quasi-periodic susceptibility Osterhoff, Morawe, Ferrero, Guigay: Optics Letters, 2012 Incoming wave field Reflected wave field, Reflected wave field, Incoming wave field Reflected wave field, Reflected wave field, Bragg condition modified Bragg condition Bragg condition``cylindrical wave’’ modified Bragg condition ``cylindrical wave’’ Laplacian in elliptical Laplacian coordinates in elliptical coordinates rules of (the same, but (the same, but differentiation with instead of ) with ) rules of (the same, but (the same, but differentiation with instead of ) with )

(the same, but approximation: with ) neglect second (the same, but approximation: derivative of with ) neglect second field envelope derivative of field envelope additional terms from modification approximation: additional terms note the sign neglect from modification (reflected wave) note the sign approximation: vs. Multilayer Coatings, Takagi-Taupin neglect (reflected wave) (a) Confocal ellipses (b) Elliptical coordinates (a) 12.4 keV, 20 layers (b) 12.4 keV, 50 layers 0.75 flat 0.75 40 mm vs. θ 4 mm 120 mm g t in 0.50 0.50 s s r r n a 0 1 o e c r S F = c 2c n s i t t = co R 0.00 0.00 reorder terms, nst -1.0 0.0 1.0 2.0 -1.0 0.0 1.0 2.0 s increasing rocking angle, ∆θ in mrad rocking angle, ∆θ in mrad introduce (a) 12.4 keV, 20 layers (b) 12.4 keV, 50 layers 4 mm ML mirror as nested confocal ellipses 0.75 0.75 40 mm reorder terms, (a) Kinematical theory (b) Ray-tracing (c) Dynamical theory 120 mm abbreviations: (energy conservation) reflectivity R 0.50 0.50 introduce , , abbreviations: (energy conservation) 0.00 0.00 , , -1.0 0.0 1.0 2.0 -1.0 0.0 1.0 2.0 , , modification factor, f modification factor, f , , simulated reflectivity of curved ML mirror; models of increasing complexity: single reflection; top: without, bottom: with (change of propagation coefficient) multiple reflection; (changecoupled ofwave-theory direction) correction for refraction inside ML (change of propagation coefficient) (change of direction) final form Takagi-Taupin description in elliptical geometry: incoming wave and reflected wave final form are coupled by quasi-periodic susceptibility Osterhoff, Morawe, Ferrero, Guigay: Optics Letters, 2012 Inspection Report for Repair of the KB mirror set for P10 Beam line

Tangential Figure error of HFM

UPSTREAM

Mark Line 1 Center Line 2

No. Line1 Center Line2 AV E. HFM (nm PV) 0.915 0.896 0.845 0.885 Table 2-5: Tangential Figure error of HFM.

Recent Progress

KBmirrors(HFM) 1 Line1 ► smooth surfaces 0.8 ► Center 0.6 Line2 sub-nm figure error 0.4 0.2 0 -0.2 FigureError(nm) -0.4 -0.6 -0.8 -1 0 20 40 60 80 100 Position(mm)

Figure 2-3: Tangential Figure error profiles of HFM

JTEC, Osaka Inspection Report for Repair of the KB mirror set for P10 Beam line

Tangential Figure error of HFM

UPSTREAM

Mark Line 1 Center Line 2

No. Line1 Center Line2 AV E. HFM (nm PV) 0.915 0.896 0.845 0.885 Table 2-5: Tangential Figure error of HFM.

Recent Progress

KBmirrors(HFM) 1 Line1 ► smooth surfaces 0.8 ► Center 0.6 Line2 sub-nm figure error 0.4 0.2 0 ►► simulations to -0.2 FigureError(nm) -0.4 single nano metre -0.6 -0.8 -1 0 20 40 60 80 100 Position(mm)

Figure 2-3: Tangential Figure error profiles of HFM

7 (a) UPBL04, horizontal ML peak intensity 1.0 peak F

ocus si intensity 6 10-1

e (FWHM), Δ in nm 5 0.5 5 sinc-ﬁt 10-2 limit std. dev. of phase 4 0.0 -0.5 0.0 1.0 2.0 2.5

(b) UPBL04, vertical ML peak 8.5 1.0 intensity peak intensity std. dev. F

ocus si of phase 7.5 10-1 z e (FWHM), Δ in nm 0.5

6.5 sinc-ﬁt 10-2 JTEC, Osaka limit 5.5 0.0 Osterhoff, PhD thesis, 2011 -0.5 0.0 1.0 2.0 2.5 f Inspection Report for Repair of the KB mirror set for P10 Beam line

Tangential Figure error of HFM

UPSTREAM

Mark Line 1 Center Line 2

No. Line1 Center Line2 AV E. HFM (nm PV) 0.915 0.896 0.845 0.885 Table 2-5: Tangential Figure error of HFM.

Recent Progress

KBmirrors(HFM) Letter 1 Vol. 4, No. 5 / May 2017 / Optica 494 Line1 ► smooth surfaces 0.8 ► Center 0.6 Line2 sub-nm figure error 0.4 0.2 0 ►► simulations to -0.2 FigureError(nm) -0.4 single nano metre -0.6 -0.8 -1 ►► ML mirrors down to 0 20 40 60 80 100 Position(mm) 12 nm, above 20 keV Figure 2-3: Tangential Figure error profiles of HFM

7 (a) UPBL04, horizontal ML peak intensity 1.0 12.0 nm × peak F

ocus si intensity 6 10-1 12.6 nm z

e (FWHM), Δ in nm 5 0.5 (FWHM), 5 sinc-ﬁt 10-2 limit std. dev. of phase ≥ 20 keV 4 0.0 -0.5 0.0 1.0 2.0 2.5 Fig. 3. X-ray wavefront characterization and focus size. (a) Vertical

(b) UPBL04, vertical ML peak cross-section of the propagation of the wavefront. (b) Horizontal 8.5 1.0 intensity peak cross-section of the propagation of the wavefront. In (a) and (b), the sam- intensity std. dev. F

ocus si of phase ple is at position 0 and the focus is indicated by the dashed yellow line. 7.5 10-1 (c) Beam at the focus position. The inset displays a zoomed-in view of the z e (FWHM), Δ in nm 0.5 central part. (d) Horizontal and vertical profiles of the intensity of 6.5 the beam at the focus position plotted together with a simulated profile sinc-fit 10-2 of an ideal optical system. The FWHM values calculated by JTEC, Osaka limit 2 2 log 2 σ were 12.0 nm for the horizontal focus and 12.6 nm for Osterhoff, PhD thesis, 2011 5.5 0.0 -0.5 0.0 1.0 2.0 2.5 the vertical focus. f pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi da Silva, Cloetens et al, Optica, 2017

XRF experiment in the focus position on the same gold structure from Fig. 2. We measured a flux of 6 × 109 photons per second. The beam shape dominates the point-spread function in the XRF imaging. Therefore, by performing the deconvolution of the im- Fig. 2. Reconstructed object and x-ray beam at the sample position age using the experimentally obtained beam shape in focus, we obtained from ptychographic measurements with a 200-nm-thick gold could demonstrate a substantial improvement of the sharpness lithographic structure using high-energy x-rays (33.6 keV). (a) Phase- of the XRF image shown in Fig. 4. Figure 4(a) shows the results contrast image. (b) Amplitude image. Panels (c)–(e) show three orthogonal modes decomposed from the probe. The dominant mode is (c) with of the XRF experiment and Fig. 4(b) shows the result after the (d) and (e) in decreasing order. Panels (f)–(h) show the respective Fourier deconvolution. This clearly indicates that the beam shape pre- transforms of the probes in panels (c)–(e). (i) Intensity contribution of sented in Fig. 3 corresponds to the beam in the focus during the principal mode. (j) Intensity contributions of the other modes. the XRF experiment. The insets in Fig. 4 highlight a region where (k) Incoherent sum of the intensities of all modes. The white scale bars we can better resolve the separation of the 50-nm-wide lines. represent 500 nm. We devised a nanoprobe that confines 6 billion high-energy x-ray photons per second in an area of 12 nm × 13 nm. This focal spot size is diffraction-limited and breaks a barrier of focal spot This corresponds to the intensity distribution of the XRF edge size and brightness at energies above 20 keV. Such a nanoprobe scans in the focus position. By fitting the dominant peak using enables better resolved imaging where the use of high-energy a Gaussian function [see Fig. 3(d)], we show that most of the x-rays is necessary. It is routinely available for XRF and phase- photons are concentrated in an area of about 12 nm × 13 nm contrast nanoscience-related studies at the ID16A nanoimaging in FWHM. A simulated intensity profile of an ideal optical beamline of the ESRF. The high flux is essential to be able to system is shown in Fig. 3(d) alongside the data, demonstrating a very good agreement. Those values are, within experimental errors, in good match with the diffraction limit calculated for aperture defining slits of 107 μm × 250 μm (H × V ), which is 13 nm × 13 nm (H × V ). Although there are some side lobes in the obtained beam at the focus, the main peak carries most of the total intensity and can, therefore, be representative of the beam size. Such additional side lobes, in addition to the im- perfection of the sharp-edge test object, may explain the previous higher values obtained by edge scans. Fig. 4. Deconvolution of an XRF image of the L-line fluorescence In order to verify that the beam shape retrieved by ptychog- of a 200-nm-thick gold lithographic structure using high-energy x-rays raphy followed by Fresnel propagation corresponds to the real (33.6 keV). (a) Original image. (b) Deconvolved image. The insets to shape of the beam at the focus position, we performed an the bottom right highlight a region for comparison. Fresnel Zone Plates, ML Laue Lenses, ML Zone Plates Outline ►► Introduction + History ►► Reflective Optics ►► Diffractive Optics ►► Refractive Optics ►► Waveguides ►► Discussion MR47CH02-Macrander ARI 6 February 2017 13:37

Aspect ratios for the zones in standard FZPs made by means of photolithography are limited to less than 20. This limit has been improved by a factor of 2 in recent years by means of ∼ a zone-doubling technique (59). The basic limit is the result of the processing steps. Electrons scatter during the exposure and produce secondary electrons so that a mask feature is significantly blurred after development. The net effect for practical zone plates is to limit resist thicknesses to values of a few hundred nanometers. However, to operate at hard-X-ray energies, much thicker FZP plate thicknesses are needed. For example, one absorption length in Au, a typical material for opaque zones, at 8 keV is 2.5 μm. For a material to be reasonably opaque at 8 keV, at least three absorption lengths, or 7.5 μm, in thickness is ideal. For an outermost zone width of 50 nm, this thickness of three absorption lengths leads to a needed aspect ratio of 7,500/50, or 150. Similarly, for a phase zone plate of Si/air at 8 keV, the Si zones would need to be at least 10 μm thick to be three absorption lengths. A requirement of a phase shift by π upon transmission through the phase FZP requires an aspect ratio of 600. The situation only worsens for energies above 8 keV. A new class of zone plates was developed to address this situation by using magnetron sputtering to deposit thin zones with thicknesses according to Equation 11 (60). Photolithography and its limitations are avoided entirely with this new class of zone plates. The thickness of the thinnest zone is instead limited by the thinnest layer that can be deposited, and for magnetron sputtering, this zone thickness can be below 1 nm (52). Historically, zone plates in this class were first deposited onto a wire layer by layer, after which the deposited structure was sliced along the wire into individual zone plates. This so-called jelly roll structure has been realized by several research groups (61, 62) but has not resulted in the hoped-for resolution. The primary reason for this failure has been that wires cannot be obtained with surfaces that are sufficiently circular and smooth. If one instead deposits a linear zone plate structure onto a flat substrate wafer, linear lenses can be formed by cutting the wafer into thin slices. Such lenses are depicted in Figure 10. Useful MR47CH02-Macrander ARI 6 February 2017 13:37 lenses can be obtained, albeit after significant processing steps, to polish and thin to a desired thickness (63). In this case, the aspect ratio is not limited by lithography and can be any desired value. This aspect of MLLs makes such lenses especially useful at high X-ray energies. A linear FZP results if one places such a lens tranverse to the optical axis. However, when such Principle and Geometry a lens is slightly tipped, i.e., tilted by θ n instead of being transverse to the incident X-ray beam, a Annu. Rev. Mater. Res. 2017.47. Downloaded from www.annualreviews.org Access provided by WIB6055 - University of Goettingen on 05/14/17. For personal use only.

Figure 2 Figure 10 A set of concentric ellipses along with a depiction of a multilayer Laue lens. A schematic wedged multilayer Laue lens in which each interface is slanted to achieve a Bragg condition.

2.14 Macrander Huang tilted and should conform to an elliptical shape. A wedged MLL is an approximation to· this ideal MLL wherein planar, instead of elliptically figured, multilayer interfaces are progressively tilted to increasing angles as one proceeds to larger radii. In this case, a Bragg condition must also be individually satisfied for each zone; that is, rays emanating from the source point should be incident on individual zones such that Bragg’s law for that zone thickness is satisfied (seeChanges Section may 5). still occur before final publication online and in print

2. X-RAY MONOCHROMATORS AND CRYSTAL-BASED X-RAY OPTICS Except for a few white-beam X-ray applications (e.g., white-beam diffraction or topography), most X-ray experiments require a monochromatic incident beam that has a narrow spectral bandwidth Macrander, Huang: Annual Review of Material Research, ‘17 �Ebw, typically of a few electron volts (eV). However, most X-ray sources are polychromatic sources (e.g., X-ray tubes that generate multiple characteristic emission lines together with a continuous bremsstrahlung background spectrum or continuous-wavelength synchrotron light sources). Therefore, utilizing monochromators as ﬁlters to select a given bandwidth of photons from the polychromatic spectrum is a basic, indispensable step for most X-ray applications. X-ray monochromatization is typically realized by Bragg diffraction of X-rays from perfect single crystals (e.g., Si and Ge). As shown in Figure 3a, the Bragg law underlying X-ray diffraction is λ 2d sinθ ,1. = B where d is the spacing between the diffracting lattice planes (also termed the Bragg planes), θ B is the Bragg angle, and λ is the X-ray wavelength.

Annu. Rev. Mater. Res. 2017.47. Downloaded from www.annualreviews.org The single-bounce crystal with a symmetric Bragg reﬂection (i.e., where the Bragg planes are parallel to the crystal surface) depicted in Figure 3a is the simplest X-ray monochromator. But if

Access provided by WIB6055 - University of Goettingen on 05/14/17. For personal use only. the incident beam is divergent, different wavelengths will be diffracted along different directions. Consequently, the wavelength spread due to the incidence divergence �θ is �λ 2d cosθ �θ in = m based on Equation 1. To achieve the desired degree of monochromatization, therefore, one has to accordingly collimate the incident beam to the required divergence, typically using slits. Even if the incident beam is collimated with no divergence, a Bragg reﬂection still has a ﬁnite

intrinsic bandwidth �EDw (usually termed the Darwin bandwidth) due to the dynamical diffraction effects. Similarly, for an ideally monochromatic beam, a Bragg reﬂection still has a ﬁnite intrinsic

rocking curve width �θ Dw [termed the (angular) Darwin width, which is the angular acceptance of

the Bragg reflection]. Here one can use asymmetric reflection to adjust �θ Dw and �EDw. Figure 3b shows the asymmetric reflection geometry with the Bragg planes inclined from the surface by an asymmetry angle ϕ, corresponding to an asymmetric factor b sin(θ φ)/sin(θ φ), where θ = B − B + B is the Bragg angle. (Unlike in the conventional treatments of dynamical X-ray diffraction, here we

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2.14 Macrander Huang tilted and should conform to an elliptical shape. A wedged MLL is an approximation to· this ideal MLL wherein planar, instead of ellipticallyZone figured, Plate multilayer Laws interfaces are progressively tilted to increasing angles as one proceeds to larger radii. In this case, a Bragg condition must also be individually satisfied for each zone; that► is,► raysresolution emanating from given the source by pointoutermost should be incident zone width on individual zones such that Bragg’s law for that zone thickness is satisfied (seeChanges Section may 5). still occur before final publication online and in print ►► zone width limited by fabrication 2. X-RAY MONOCHROMATORS►► efficiency AND CRYSTAL-BASED limited by X-RAYoptical OPTICS thickness Except for a few white-beam X-ray applications (e.g., white-beam diffraction or topography), most X-ray experiments require a monochromatic►► thin incident and beam long that zones: has a narrow volume spectral bandwidth diffraction; “Bergemann-Limit” Macrander, Huang: Annual Review of Material Research, ‘17 �Ebw, typically of a few electron volts (eV). However, most X-ray sources are polychromatic sources (e.g., X-ray tubes that generate multiple characteristic emission lines together with a continuous bremsstrahlung background spectrum or continuous-wavelength synchrotron light sources). Therefore, utilizing monochromators as filters to select a given bandwidth of photons from the polychromatic spectrum is a basic, indispensable step for most X-ray applications. X-ray monochromatization is typically realized by Bragg diffraction of X-rays from perfect single crystals (e.g., Si and Ge). As shown in Figure 3a, the Bragg law underlying X-ray diffraction is λ 2d sinθ ,1. = B where d is the spacing between the diffracting lattice planes (also termed the Bragg planes), θ B is the Bragg angle, and λ is the X-ray wavelength.

rocking curve width �θ Dw [termed the (angular) Darwin width, which is the angular acceptance of

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Changes may still occur before final publication online and in print 1980ies: Soft X-Ray Microscopy

Water Window ►► high contrast between C (cell) and O (water) ►► soft x-rays, good optics

http://www.atto.ethz.ch/research/timeresolved.html 1980ies: Soft X-Ray Microscopy

Water Window ►► high contrast between C (cell) and O (water) ►► soft x-rays, good optics

http://www.atto.ethz.ch/research/timeresolved.html Niemann, Rudolph, Schmahl: NIM, 1983 D. Wei et al. / Ultramicroscopy 84 (2000) 185±197 187

Fig. 1. The normalized point spread function IPSF=I0 (cf. Eq. (2)) near the geometric focal point of a zone plate objective as a function of radius r and of defocus z for (a) monochromatic radiation with l 2:4 nm and (b) narrow-bandwidth radiation with a mean wavelength l0 2:4 nm and a monochromaticity l0=Dl 200, for a zone plate with an outermost zone width drn=40 nm and a numerical aperture a=f=0.03. The z-axis has been condensed by a factor of 10. 1980ies: Soft X-Ray Microscopy lens, and k 2s 1 s u Uk u; v 1 Jk 2s v 4 s 0 À v X are Lommel functions, with Jk v the Bessel function of the kth order. The Cartesian coordinates x; y; z are relative to the geometric focal point, with the z-axis being the optical axis. In the TXM, a combination of a condenser zone plate and a pinhole is used to monochromatize the polychromatic synchrotron radiation and to illu- minate the object [7]. Raytracing calculations show Water Window that the spectral distribution of the illuminating intensity is approximately Gaussian. For the ►► high contrast experiments described in this work, this distribu- between tion is characterized by a mean wavelength l0=2.4 nm and a full-width at half-maximum of C (cell) and Dl 0:012 nm, yielding a monochromaticity of O (water) l0=Dl 200. To calculate the eective point spread function ►► soft x-rays, for this kind of narrow-bandwidth illumination, the monochromatic PSF of Eq. (2) is weighted good optics with the spectral distribution of the illuminating intensity, and integrated over the wavelength l. http://www.atto.ethz.ch/research/timeresolved.htmlSince Fresnel zone plates are chromatic objectives, Fig. 6. SeveralWeiß, organelles Schmahl of the et specimenal: Ultramicroscopy, identi®ed by watershed 2000 the focal length f in Eq. (3) is a function of segmentation of the reconstructed linear absorption coecient wavelength: f l 2adrn=l, where drn is the width (LAC): chloroplast (green), pyrenoid (blue), spherical vesicles of the outermost zone of the zone plate. Because (red), and ¯agellar roots (brown). One quadrant of the the Cartesian coordinates of Eq. (3) are relative to chloroplast has been cut away to reveal the pyrenoid. Also displayed is one slice of the reconstructed linear absorption the geometric focal point f l , the integration over coecient. The diameter of the alga is approx. 7.4 mm. the wavelength l results in an elongation of the Visualization using the Amira system developed at ZIB (http:// PSF (Fig. 1). Assuming incoherent image amira.zib.de). Volume Diffraction – Multilayer Zone Plates BRIEF REPORTS PHYSICAL REVIEW B 74, 033405 ͑2006͒

FIG. 1. Focusing geometry for a thick Fresnel zone plate with FIG. 2. First-order focus of a parabolically tilted zone plate tilted zones. ͑solid line͒, an equivalent ideal thin zone plate ͑dotted line͒, and a straight ͑not tilted͒ zone plate ͑dash-dotted line͒. The inset shows considered in this numerical study. To be exact, however, in the beam intensity distribution around the focus along the optical the following, we consider the focusing of a plane wave with axis for the parabolically tilted zone plate. a parabolically tilted zone plate. For hard x rays—e.g., with an energy E=20 keV ͑␭=0.62 Å͒—focusing to nanometer dimensions involves ͉␺z͘ = ͚ Cn͑z͒exp − ik ͵ ␭n͑z͒dz ͉␾n͑z͒͘. ͑2͒ Efficiencysmall angular deviations fromfor the opticalhard axis, allowingx-rays? one n ͭ ͮ to use scalar wave equations in paraxial approximation. Free- The evolution of the coefficients C ͑z͒ can be determined by space propagation of the x rays is modeled by the Fresnel- n inserting Eq. 2 into Eq. 1 , leading to the system of ordi- Kirchhoff equation that is the paraxial solution to the free- ͑ ͒ ͑ ͒ nary differential equations ►space► vanishing Helmholtz equation. 20absorptionInside the zone plate, ץ ץ propagation along the optical ͑z͒ axis is approximated by the 18,21 parabolicno wave absorption-based equation zoneC kplate=−͚ Cn ␾kͯ ␾n exp − ik ͵ ͑␭n − ␭k͒dz , ͮ ͭ ʹ zץ z n ͳץ ͑3͒ 2uץ 2uץ uץ ► small phase2 2 shift per µm ► 2ik + 2 + 2 + k ͓n ͑x,y,z͒ −1͔u = 0, ͑1͒ which can be solved numerically, if the transition matrix ͘ ץ yץ xץ zץ z ␾n and the eigenvalues and eigenfunctions areץ ͉ ␾k͗ optically thick, “long” zoneknown asplates a function of neededz. In the case of a tilted zone plate where k=2␲/␭ is the wave vector of the x rays and ͑cf. Fig. 1͒, the eigenvalues and functions can be calculated numerically for different positions along the optic. These cal- n͑x,y,z͒=1−␦͑x,y,z͒+i␤͑x,y,z͒ is the refractive index in- culations show that the eigenvalues follow a scaling law and side► theaspect optical element. ratio Equation ͑ 1~͒ is formally1:1000 equivalent and more ► the eigenfunctions are nearly unchanged for a small range of to the Schrödinger equation, when the z coordinate is ͓ a z with respect to coordinates rescaled by a z . In this interpretedoutermost as time. Attenuation zone inside the materialwidth of the vs.͑ ͔͒optical thickness ͑ ͒ optical component is modeled by the imaginary part i␤ way, the eigensystem can be interpolated during the integra- of the refractive index and makes the optical potential com- tion of Eq. ͑3͒. plex, generating a sink for the amplitude u along the propa- Consider now aSchroer, tilted one-dimensional Phys zoneRev plate B, made 2006 of gation direction. With attenuation included, the operator Ni ͑separated by vacuum͒ as shown in Fig. 1 with N=400 2 2 zones r 0 =40 nm . In this geometry, the outermost zone ͔ ͒ 1͑ ͓ 2 2 ץ ץ ˆ H= 2 + 2 +k ͓n ͑x,y,z͒−1͔ is non-Hermitian. y has a width of ⌬rM͑0͒=1.00 nm. The aperture of this optic isץ xץ The solution to this equation can be easily constructed for 1.60 ␮m. This very small zone plate was chosen to be able to optical structures that are invariant along the optical axis— perform a full parabolic wave equation calculation for one -z=0—such as the regular Fresnel zone plate withץ/nץ ,.i.e transverse dimension ͑one-dimensional focusing͒ at 2 Å out tilted zones. In that case, Eq. ͑1͒ does not explicitly de- resolution.22 The material Ni was chosen to make d small pend on z. The incident wave field ␺0͑x,y͒ can be expressed enough with respect to f. In a more practical situation, a zone in a complete set of transverse ͑forward͒ eigenmodes plate with the same numerical aperture would be made of an ͉␺0͘=͚nCn͉␾n͘, where Cn =͗␾n ͉␺0͘. Propagation through the appropriate multilayer system and have many more zones, a optic is given by multiplication of each eigenmode ͉␾n͘ with much larger aperture, and a larger focal distance.5 an appropriate phase factor exp͕−ik␭nd͖, where ␭n is the The transmission of a hard x-ray plane wave ˆ complex eigenvalue of ͉␾n͘ with respect to H. The wave field ͑E=20 keV͒ through the tilted zone plate with a thickness of exiting the optic is given by ͉␺d͘=͚nCn exp͕−ik␭nd͖͉␾n͘. In d=8 ␮m was calculated by numerically integrating Eq. ͑3͒ case the structures of the optic are not invariant along the using a Runge-Kutta method.23 Behind the optic, the result- optical axis—e.g., the tilted zone plate in Fig. 1—the eigen- ing wave field was propagated to the focal plane using the system changes as a function of z. In that case, the wave field Fresnel-Kirchhoff equation. Figure 2 shows the transverse at a position z can be expressed as beam profile in the focus for a tilted thick zone plate tuned to

033405-2 Volume Diffraction – Multilayer Zone Plates BRIEF REPORTS PHYSICAL REVIEW B 74, 033405 ͑2006͒

(a) MZP fabrication pulsed laser deposition multilayer zone plate dr

MZP focus b

t fofocusedcuse ion beam f = 50 µm FIG. 1. Focusing geometry for a thick Fresnel zone plate with FIG. 2. First-order focus of a parabolically tilted zone plate tilted zones. (b) experimental setup ͑solid line͒, an equivalent ideal thin zone plate ͑dotted line͒, and a defocus straight ͑not tilted͒ zone plate ͑dash-dotted line͒. The inset shows considered in this numerical study. To be exact, however, in KB mirror pair 2 mm beamstop the beam intensity distribution around the focus along the optical the following, we consider the focusing of a plane wave with axis for the parabolically tilted zone plate. a parabolically tilted zone plate. For hard x rays—e.g., with an energy E=20 keV KB focus ͑␭=0.62 Å͒—focusing to nanometer dimensions involves ͉␺z͘ = ͚ Cn͑z͒exp − ik ͵ ␭n͑z͒dz ͉␾n͑z͒͘. ͑2͒ Efficiencysmall angular deviations fromfor the opticalhard axis, allowingx-rays? one n ͭ ͮ to use scalar wave equations in paraxial approximation. Free- The evolution of the coefficients C ͑z͒ can be determined by space propagation of the x rays is modeled by the Fresnel- n inserting Eq. 2 into Eq. 1 , leading to the system of ordi- detector Kirchhoff equation that is the paraxial solution to the free- ͑ ͒ ͑ ͒ nary differential equations ►space► vanishing Helmholtz equation. 20absorptionInside the zone plate, ץ ץ Fig. 1. (a) Schematic fabrication process: Pulsed laser deposition of W and Si multilayer propagation along the optical ͑z͒ axis is approximated by the parabolicno wave absorption-based equation18,21 zoneC plate=− C ␾ ␾ exp − ik ͑␭ − ␭ ͒dz , onto a rotating wire according to the Fresnel zone plate law. Focused ion beam fabrication k ͚ n kͯ n ͵ n k ͮ ͭ ʹ zץ z n ͳץ of the MZP by cutting a slice out of the coated wire, placing it onto a sample holder and ͑3͒ 2uץ 2uץ uץ polishing it down to the optimal optical thickness of 0.7 µm. (b) Experimental setup of the ► small phase2 2 shift per µm synchrotron experiment: The MZP is positioned 2 mm downstream of the KB focus. ► 2ik + 2 + 2 + k ͓n ͑x,y,z͒ −1͔u = 0, ͑1͒ which can be solved numerically, if the transition matrix ͘ ץ yץ xץ zץ z ␾n and the eigenvalues and eigenfunctions areץ ͉ ␾k͗ optically thick, “long” zoneknown asplates a function of neededz. In the case of a tilted zone plate Compared to a single lens, this primary beam has a larger divergence due to the pre-focusing where k=2␲/␭ is the wave vector of the x rays and ͑cf. Fig. 1͒, the eigenvalues and functions can be calculated numerically for different positions along the optic. These cal- n͑x,y,z͒=1−␦͑x,y,z͒+i␤͑x,y,z͒ is the refractive index in- and thus beam stop alignment is easier. Also, since the beam is spread out among more pixels, culations show that the eigenvalues follow a scaling law and side► theaspect optical element. ratio Equation ͑ 1~͒ is formally1:1000 equivalent and more single-photon counting detectors could be used improving the efficiency measurements. We ► the eigenfunctions are nearly unchanged for a small range of to the Schrödinger equation, when the z coordinate is ͓ a z with respect to coordinates rescaled by a z . In this would like to emphasize that the divergence of the diffracted beam is larger by one to two interpretedoutermost as time. Attenuation zone inside the materialwidth of the vs.͑ ͔͒optical thickness ͑ ͒ orders of magnitude, so the beamstop does not affect the MZP far-field. optical component is modeled by the imaginary part i␤ way, the eigensystem can be interpolated during the integra- of the refractive index and makes the optical potential com- tion of Eq. ͑3͒. 10From nm the numerical multi-slice simulations summarized in Fig. 2, the optical thickness of Consider now a tilted one-dimensional zone plate made of the MZP was deducedDöring, as follows. Osterhoff In Fig. 2(a), the et simulated al, Optics intensity Express, inside a slice 2013 through the plex, generating a sink for the amplitude u along the propa- Schroer, Phys Rev B, 2006 gation direction. With attenuation included, the operator Ni ͑separated by vacuum͒ as shown in Fig. 1 with N=400 2 2 zones r 0 =40 nm . In this geometry, the outermost zone ͔ ͒ 1͑ ͓ 2 2 ץ ץ ˆ MZP is shown colour coded, for the case of plane wave-irradiance. The “optimal” thickness for H= 2 + 2 +k ͓n ͑x,y,z͒−1͔ is non-Hermitian. y has a width of ⌬r ͑0͒=1.00 nm. The aperture of this optic isץ xץ a phase-shifting MZP would be around 2.1 µm, but due to volume diffraction effects the optical The solution to this equation can be easily constructed for M thickness was decided to be only 0.7 µm. Numerically propagated intensity along the optical 1.60 ␮m. This very small zone plate was chosen to be able to optical structures that are invariant along the optical axis— perform a full parabolic wave equation calculation for one -z=0—such as the regular Fresnel zone plate withץ/nץ ,.i.e axis is shown in Fig. 2(b), including higher order foci. We emphasize that the underlying KB transverse dimension ͑one-dimensional focusing͒ at 2 Å beam is several orders of magnitude fainter in intensity. The optimal MZP focus as simulated out tilted zones. In that case, Eq. ͑1͒ does not explicitly de- resolution.22 The material Ni was chosen to make d small for plane wave illumination and perfect layers is shown in Fig. 2(c), promising a focus size pend on z. The incident wave field ␺0͑x,y͒ can be expressed enough with respect to f. In a more practical situation, a zone in a complete set of transverse ͑forward͒ eigenmodes of about 4.4 nm. In Fig. 2(d), the simulated KB beam in the (KB) defocus position is shown; plate with the same numerical aperture would be made of an ͉␺0͘=͚nCn͉␾n͘, where Cn =͗␾n ͉␺0͘. Propagation through the appropriate multilayer system and have many more zones, a the red shaded areas depict the approximate position of the active zones. Note that the two- optic is given by multiplication of each eigenmode ͉␾n͘ with much larger aperture, and a larger focal distance.5 dimensional intensity distribution is rather homogeneous, as it can be approximated by a tensor an appropriate phase factor exp͕−ik␭nd͖, where ␭n is the The transmission of a hard x-ray plane wave ˆ product of the two curves (horizontal and vertical). complex eigenvalue of ͉␾n͘ with respect to H. The wave field ͑E=20 keV͒ through the tilted zone plate with a thickness of exiting the optic is given by ͉␺d͘=͚nCn exp͕−ik␭nd͖͉␾n͘. In d=8 ␮m was calculated by numerically integrating Eq. ͑3͒ 23 3. Fabrication of the multilayer zone plate case the structures of the optic are not invariant along the using a Runge-Kutta method. Behind the optic, the result- optical axis—e.g., the tilted zone plate in Fig. 1—the eigen- ing wave field was propagated to the focal plane using the In the following we will give more details on the production process by the combination of system changes as a function of z. In that case, the wave field Fresnel-Kirchhoff equation. Figure 2 shows the transverse PLD and FIB. at a position z can be expressed as beam profile in the focus for a tilted thick zone plate tuned to

033405-2

#191774 - $15.00 USD Received 11 Jun 2013; revised 29 Jul 2013; accepted 29 Jul 2013; published 7 Aug 2013 (C) 2013 OSA 12 August 2013 | Vol. 21, No. 16 | DOI:10.1364/OE.21.019311 | OPTICS EXPRESS 19314 Recent Progress

►► Multilayer Zone Plate for high x-ray energies designed for 60 keV, tested also at 100 keV ►► outermost zones: 10 nm ►► optical thickness: 30 µm Eberl, Soltau, Osterhoff et al (work in progress) Recent Progress www.nature.com/scientificreports/

Figure 1. (a) A wedged multilayer Laue lens of focal length f is constructed from layers whose spacing follows the zone-plate condition. To achieve high efficiency the lens must be thick, in which case diffraction is a volume effect described by dynamical diffraction. In this case the layers should be tilted to locally obey Bragg’s law, which places them normal to a circle of radius 2f. (b) SEM image of the 2750-bilayer wedged MLL used in this study. The regions corresponding to the multilayered materials and the Si substrate are indicated. The white scale bar is 20 μm and the inset shows a magnified TEM image of the layered materials.

Scientific RepoRts | 5:09892 | DOi: 10.1038/srep09892 2 Recent Progress www.nature.com/scientificreports/

high-aspect ratio prevents a simple thin-mask description of X-ray diffraction. In particular, such structures are akin to planes in a crystal, in which X-rays only reflect when they are tilted at the Bragg angle θ (given by sin θ = λ/(2 d), where d is the zone period). This comparison is indeed very apt and provides ►► Multilayer Zone Plate for high x-ray energies the insight into constructing an efficient hard X-ray lens of high resolution which ideally consists of reflecting confocal parabolic layers (for an incident plane wave) spaced apart such that each period intro- designed for 60 keV, tested also at 100 keV duces an additional wavelength of path for the rays arriving at the focus4. That is, the lens is composed of layers that simultaneously follow the zone-plate condition and are oriented to obey Bragg’s law across the entire lens aperture. The lens performance is described by dynamical diffraction, and as such the optical thickness of the lens should be set at half a pendellosung period to direct most of the incident ►► outermost zones: 10 nm beam into the diffracted (focused) beam, giving much higher efficiency than could be achieved with a thin zone plate (which is limited by equally partitioning the beam into positive and negative orders). A method to fabricate volume zone plates of high aspect ratios was introduced a decade ago5–7. Called multilayer Laue lenses (MLLs)8, these structures are fabricated by layer deposition, using technologies ►► optical thickness: 30 µm developed for making multilayer mirrors9. Layer periods thinner than 1 nm are achievable by magnetron Eberl, Soltau, Osterhoff et al (work in progress) sputtering10. Lenses are made by alternately depositing two (or more) materials with layer periods that follow the Fresnel zone-plate condition and then slicing the structure approximately perpendicular to the layers to the desired optical thickness. Lenses fabricated to date have consisted of parallel layers in a one-dimensionalFig. 4. Picture (1D) of thestack ANL deposited Z2-37 onto precision a flat substrate. alignment Two-dimensional apparatus focusing for intermediate-field can be achieved withstacking crossed of 1D up stacks to six6, 13 zone or by plates.depositing In-house a multilayer fabricated on a thin arrays wire ofto create Fresnel a circular zone platesmultilayer are 11, 12 zonemounted plate on . diamond In the former holders. case Theeach apparatuslens must be is tilted shown relative as integrated to the incident to the X-ray microprobe beam to setup. max- Bajt et al, Scientific Reports, 2015 imize the region of the lens that satisfies Bragg’s law. Even so, the NA of the lens will depend on the rocking-curve[D. Shu, J. Liu, width S. of C. the Gleber, Laue reflection J. Vila-Comamala, (which unfortunately B. Lai, J. Maser,becomes C.narrower Roehrig, as the M. thickness J. Wojcik, of Gleber et al, Optics Express, 2014 theand lens S. and Vogt, efficiency U. S. Patent of the application Laue reflection in progressis increased for or ANL-IN-13-092] as the layer period is reduced). A tilted MLL consisting of parallel layers was used to focus 12 keV X-rays to a spot of 11.2 nm (FWHM) with 15% efficiency14. When the NA of the lens exceeds the Darwin width of the reflection at any part of the lens then the lens focus will be significantly apodised and the effective NA will be limited by the diffraction the scintillatorefficiency. crystal Only by wherevarying FZPsthe tilt andof the other layers componentsthroughout the canstack, be so seenthat Bragg’s in the law transmitted and the zone X-ray beam.plate The condition focal spot are simultaneously determines fulfilled the sample for every position layer, is whichit possible is to within construct a heliuma large enough chamber. NA The to focus X-rays to nanometer spots. Such a structure is referred to as a wedged MLL, and is schematically fluorescenceillustrated detector in Fig. 1 (a). (Vortex ME-4, SII NanoTechnology USA Inc., Northridge, CA, USA) is at 90 degreeNumerical to the modelling incident of beam. MLLs has Three been chips carried of out FZPs using as methods described such aboveas coupled were wave mounted theory, to the apparatusthe beam on the propagation diamond method holders (equivalent (Fig. 3). to The the multislice motors weretechnique), controlled and dynamical via the diffraction software packageof distorted lattices. In the latter case, it was predicted that efficient wedged MLLs with NAs as high as 0.1 EPICSshould (Argonne be achievable; National that Laboratory,is, focal spots smaller Argonne, than IL,1 nm USA) should andbe possible the long using travel wedged ranges MLLs15 allowed. for theUntil straight now, however, forward a wedged alignment MLL has of not all been three realized matching experimentally, FZPs as due described to difficulties in the in controlling following. material deposition to the necessary precision both in the direction of the film growth and transverse In Fig.to this 6, direction. two steps of the alignment of three FZP with the Z2-34 alignment apparatus is shown. InRecently, the transmission we solved the manufacturing image from problem the CCD, of wedged on MLLs each by FZP depositing chip the the layer dedicated materials FZPby was 16 identifiedmagnetron and aligned sputtering to onto a common a substrate referenceshadowed by point, a straight-edged marked bymask the. The green required cross layer hair period visible on and layer angle was achieved in the penumbra of the mask where the deposition rate changes with dis- the CCDtance image. in a direction While perpendicular all FZP chips to the are mask already edge. Here in the we incidentpresent the X-ray measured beam, one-dimensional first the selected upstream(1D) andfocusing center performance FZPs get of a aligned high-NA (Fig.wedged 6(a)). MLL made The centralin this fashion, stop isand visible compare in this this perfor alignment- mance to calculations based on the beam propagation method (see Methods). We find that the diffrac- as darktion spot efficiency also centered is nearly touniform the reference across the point.entire pupil Once of the those lens, two which FZPs had a are NA aligned, of 0.006 at the 22 keV selected FZP onphoton the downstreamenergy. The depth FZP of the chip lens gets(the thickness moved in into the thedirection line of of the stacked optical axis) FZPs was (Fig.6.5 μm, 6(b)). which In the picture shown, the downstream FZP is only close to alignment, so Moire´ fringes are visible. Scientific RepoRts |Those 5:09892 | fringesDOi: 10.1038/srep09892 are an indication for the alignment, and with all 3 FZPs aligned, the2 fringes disappear and the transmission image of the stack looks like the one with the 2 FZPs aligned (Fig. 6(a)). For this alignment, the relatively large travel range of the stacking apparatus of several millimeters for each FZP chip was crucial. It allowed for subsequently moving each FZP chip into the beam and identifying the specific FZP for stacking within the array as the entire array could be scanned. Misalignment of one FZP in the stack becomes not only visible in the form of Moire´ fringes in the transmission image (Fig. 6), but also in a decrease of focused flux (measured as decrease of counts in the downstream ion chamber) as the focusing efficiency decays with misalignment. To measure the influence of misalignment, two data sets were acquired. The influence of lateral misalignment was measured moving the central zone plate (CZP) out of alignment in horizontal (x) direction. The step size was 10 nm in the region close to alignment and 50 nm in the region of more than 350 nm away from alignment. For each point the focused flux was measured and converted to FZP efficiency. The resulting plot of FZP efficiency as a function of the lateral alignment of the stacked FZPs is shown in blue in Fig. 7(a). According to the lateral alignment

#212663 - $15.00 USD Received 5 Jun 2014; revised 8 Aug 2014; accepted 13 Aug 2014; published 5 Nov 2014 (C) 2014 OSA 17 November 2014 | Vol. 22, No. 23 | DOI:10.1364/OE.22.028142 | OPTICS EXPRESS 28148 Compound Refractive Lenses Outline ►► Introduction + History ►► Reflective Optics ►► Diffractive Optics ►► Refractive Optics ►► Waveguides ►► Discussion

Seiboth et al, APL, 2014 Principle and Geometry

“let’s drill holes into Aluminium, and use that as a lens” … really? yes! ►► focal length: radius / 2 N d ►► with radius ~ 200 µm, d ~ 10-5: f ~ 10 m / N ►► with ≥ 50 lenses, f ≤ 0.2 m possible

Snigirev, Lengeler et al, Nature, 1996

Snigirev, Lengeler et al, Nature, 1996 First Results ►► 30 holes questions for following decades: ►► 300 µm radius each ►► better fabrication? ►► X-ray energy: 14 keV ►► best material? ►► focal length: 1.8 m ►► nanometre possible? ►► line focus: 8 µm ►► flexibility?

If no-one has done it, maybe it’s not possible. If no-one has tried yet, maybe it is possible. Snigirev, Lengeler et al, Nature, 1996 PHYSICAL REVIEW LETTERS week ending PRL 94, 054802 (2005) 11 FEBRUARY 2005

Focusing Hard X Rays to Nanometer Dimensions by Adiabatically Focusing Lenses

C. G. Schroer1 and B. Lengeler2 1HASYLAB at DESY, Notkestrasse 85, D-22607 Hamburg, Germany 2II. Physikalisches Institut, Aachen University, D-52056 Aachen, Germany (Received 13 October 2004; published 10 February 2005) We address the question of what is the smallest spot size that hard x rays can be focused to using refractive optics. A thick refractive x-ray lens is considered, whose aperture is gradually (adiabatically) adapted to the size of the beam as it converges to the focus. These adiabatically focusing lenses are shown to have a relatively large numerical aperture, focusing hard x rays down to a lateral size of 2 nm (FWHM), well below the theoretical limit for focusing with waveguides [C. Bergemann et al., Phys. Rev. Lett. 91, 204801 (2003)].

DOI: 10.1103/PhysRevLett.94.054802 PACS numbers: 41.50.+h, 07.85.Qe

Many experimental techniques at synchrotron radiation In this Letter, we consider a refractive x-ray lens that can sources, such as x-ray scanning microscopy [1], x-ray overcome this limit by gradually (adiabatically) following photon and fluorescence correlation spectroscopy [2], and with its aperture the size of the beam as it converges to the coherent scattering techniques [3] for the study of nano- focus. In this way, the refractive power per unit length particles, require small and intensive hard x-ray microbe- increases inside the lens toward its exit approaching a ams. Great instrumental advances have been made in the singularity. The resulting numerical aperture can exceed last few years towards generating ever smaller microbe- p2�, allowing one to focus hard x rays down to 2 nm. ams, in particular, at synchrotron radiation sources of the While this is not a hard physical limit, it seems difficult to third generation, such as the Advanced Photon Source in improve the focal size down to atomic dimensions by this Argonne, Illinois. Today, several optics are available that scheme. In the following, we derive the properties of these can generate x-ray beams with a lateral size in the range of adiabatically focusing lenses (AFLs) and verify them nu- 100 nm and below [4,5]. The current limitations to focus- Adiabaticmerically by wave propagation. Lenses ing to even smaller focal sizes are of a technological nature and may be overcome in the future. The question arises, however, whether there is a physical limit to efficient focusing of hard x rays due to their weak interaction with matter and what, if any, this limit is [6]. Bergemann et al. have determined this limit for waveguides [7]. They find that the effective numerical aperture of waveguides is limited by �c p2�, the critical angle of total external reflection for a� material with refractive index n 1 � i�. The critical angle also limits the numerical� aperture� of� mirror optics and capillaries [8]. Bergemann et al. believe that the same limitations to focusing hold for all x-ray optics, limiting the focal size to slightly below 10 nm [7,9]. First introduced in 1996 [10], refractive x-ray lenses have quickly advanced to an efficient tool in x-ray microscopy and microanalysis [11,12]. In these optics, the weak refraction of hard x rays in matter is compensated by stacking behind each other a large number of strongly curved individual lenses to form a lens assembly, in which the beam is gradually focused. Nanofocusing refractive x- ray lenses (NFLs) currently generate microbeams with a lateral size of about 100 nm [5]. The minimal spot size achievable with these optics lies below 20 nm and is limited by a maximal numerical aperture of NA p2� that is FIG. 1. Adiabatically focusing x-ray lens. The lens is com- the same as for waveguides [7]. It is limited by� the constant posed of a large number of individual (aspherical) refractive refractive power per unit length inside the NFL with con- lenses, whose aperture is matched to the converging beam size, stant aperture. increasing the refractive power per unit length along the lens. Schroer, Lengeler: PRL, 2005

0031-9007=05=94(5)=054802(4)$23.00 054802-1  2005 The American Physical Society PHYSICAL REVIEW LETTERS week ending PRL 94, 054802 (2005) 11 FEBRUARY 2005

Focusing Hard X Rays to Nanometer Dimensions by Adiabatically Focusing Lenses

DOI: 10.1103/PhysRevLett.94.054802 PACS numbers: 41.50.+h, 07.85.Qe

Many experimental techniques at synchrotron radiation In this Letter, we consider a refractive x-ray lens that can PHYSICAL REVIEW LETTERS week ending sources, such as x-ray scanning microscopy [1], x-ray overcome this limit byPRL gradually94, 054802 (2005)(adiabatically) following 11 FEBRUARY 2005 photon and ﬂuorescence correlation spectroscopy [2], and with its aperture the size of the beam as it converges to the The lens design proposed in this Letter is shown in R R R focus. In this way, the refractive power per unit length f 0i ;H 0f � 0i : (4) coherent scattering techniques [3] for the study of nano- Fig. 1. A large number of thin lenses (numbered by the � R0i � R0i 4�0 log 4�0 log particles, require small and intensive hard x-ray microbe- increases inside theindex lensj) toward is stacked its behind exit each approaching other along a common a R0f R0f optical axis as for previous refractive lens designs. To q�� q�� ams. Great instrumental advances have been made in the singularity. The resulting numerical aperture can exceed Here, H is measured from the exit of the lens. Integration avoid spherical aberration, each individual lens is shaped R aspherically. In paraxial approximation, the parabolic of Eq. (3) yields the overall length L 0i last few years towards generating ever smaller microbe- p p4�0 2�, allowing one to focus hard x rays down to 2 nm. R =R � � shape is ideal, as each individual lens is thin compared to 0f 0i 1 1=2 R0i 1 dx logx � p� p of the lens, approximately ams, in particular, at synchrotron radiation sources of the While this is not a hard physical limit, it seems difﬁcult to 4�0 its focal distance fj Rj=2�. Here, Rj is the radius of � � � �� independent of R0f for R0f R0i. Taking the attenuation curvature at the apex of the lens. For a parabolic lens, it is R �� third generation, such as the Advanced Photon Source in improve the focal size down to atomic dimensions by this inside the lens material into account,�� the effective aperture related to other parameters, such as its length l , aperture Argonne, Illinois. Today, several optics are available that scheme. In the following, we derive the properties of thesej D of the lens system can be calculated in analogy to that R , and minimal thickness d by l d R2 =R eff 0j j j � j � 0j j of a thin refractive x-ray lens [12] and is given by can generate x-ray beams with a lateral size in the range of adiabatically focusing(cf. lenses Fig. 1). (AFLs)As a result, and the refractive verify power them per nu- length of

to be the same for all individual lenses. The first integral of 0 dimensionally focusing diamond lens, a focal spot size of curved individual lenses to form a lens assembly, in which 1 2 2 this second order differential equation is R I/I 4000 5:8 4:7 nm would be expected, compared to the diffrac- the beam is gradually focused. Nanofocusing refractive x- 2 � 0� � 10 2�0 log R0 E, where E is a constant determined by tion limited spot size of 4:3 nm. The intensity gain in this the initial� �� conditions. For a parallel beam incident on the geometry would be about 7 106, with of the order of ray lenses (NFLs) currently generate microbeams with a first lens, the initial conditions are R 0 R and 20001 109 ph=s in the focus. As the focal distance of AFLs is 0 0i Li lateral size of about 100 nm [5]. The minimal spot size R 0 0 (cf. Fig. 1), yielding the first order� �� differential extremely short, e.g., f 1:6 mm, demagnifications of the 0 [µm] Be � �� a order of 104 are possible even at short beam lines. achievable with these optics lies below 20 nm and is lim- equation 0.1 0 B We thank A. Snigirev, I. Snigireva, and J. Maser for the -10 -5 0 5 C (graphite) 10 ited by a maximal numerical aperture of NA p2� that is FIG. 1. Adiabatically focusing x-ray lens. The lens is com- C* (diamond) discussion about the limitations to the focusing of hard R 0.01 r [nm] � R 4� log 0i: (3) Si x rays. the same as for waveguides [7]. It is limited by the constant posed of a large number of individual0 (aspherical)0 refractive Ni � s��R0 refractive power per unit length inside the NFL with con- lenses, whose aperture is matched to the converging beam size, FIG. 5. Lateral beam profiles for a nanofocusing lens, an adiabatically focusing2 4 lens, 68 and two2 kinoform 4 68 AFLs with24 three Figure 1 shows the course of R0 z as a solution of Eq. (3). 1 10 100 stant aperture. increasing the refractive power per unit length along� � the lens. and five segments, respectively. Schroer, Lengeler: PRL, 2005 For a given exit aperture R0f, the lens properties can be E [keV] [1] Synchrotron Radiation Instrumentation, edited by T. calculated, such as focal distance, position, and effective as Warwick, J. Arthur, H. A. Padmore, and J. Sto¨hr, AIP p well as numerical aperture. The focal distance f and sec- FIG.Leaving 2. Characteristic all other parameters aperture a unchanged,4 �0 as a function the aperture of x-ray of Conf. Proc. No. 705 (AIP, Melville, NY, 2004). 0031-9007=05=94(5)=054802(4)$23.00 054802-1  2005 The American Physical Society these lenses can be increased by� ap kinoform��0 lens design as ondary principal plane H are (see Fig. 1) energy for different lens materials. �� [2] J. Wang et al., Phys. Rev. Lett. 80, 1110 (1998). shown in Fig. 4(a) [12,16]. It yields a�� significant increase in [3] I. K. Robinson et al., Opt. Express 11, 2329 (2003); G. J. aperture growing with the square root of the number of lens Williams, M. A. Pfeifer, I. A. Vartanyants, and I. K. 054802-2segments. Figure 4(a) shows the intensity distribution in- Robinson, Phys. Rev. Lett. 90, 175501 (2003). side a kinoform lens with five segments for an x-ray wave [4] W. Yun et al., Rev. Sci. Instrum. 70, 2238 (1999); O. front coming from a distant point source. At the border Hignette et al., in X-Ray Micro- and Nano-Focusing: between two segments, the wave front is distorted and Applications and Techniques II, edited by I. McNulty, fringes are formed. However, as the borders of the seg- Proc. SPIE Int. Soc. Opt. Eng. Vol. 4499 (SPIE- International Society for Optical Engineering, ments follow the contracting beam adiabatically, the re- Bellingham, WA, 2001), pp. 105–116; K. Yamauchi sulting disturbance is minimal. The intensity distribution et al., J. Synchrotron Radiat. 9, 313 (2002). behind the lens is depicted in Fig. 4(b). A focus with a [5] C. G. Schroer et al., Appl. Phys. Lett. 82, 1485 (2003); lateral size of 2:21 nm is obtained at the distance of C. G. Schroer et al., in Ref. [1], pp. 740–743. 11:6 �m behind the lens. [6] Inefficient focusing of hard x rays to atomic dimensions Figure 5 shows the beam profiles at the focal position could be achieved, for example, using a focusing mirror obtained by wave propagation for a NFL, an AFL, and two reflecting at large angles. Such a mirror would focus the x rays very inefficiently, but could reach numerical aper- kinoform AFLs with three (AFLk3) and five (AFLk5) segments, respectively. The intensity is normalized to that tures up to one. However, Compton scattering would incident on the lens, giving the gain for one-dimensional strongly occult this focus. focusing. For the AFL with one segment, more than 90% of [7] C. Bergemann, H. Keymeulen, and J. F. van der Veen, Phys. Rev. Lett. 91, 204801 (2003). the transmitted radiation is focused. The segment bounda- [8] D. Bilderback, S. A. Hoffman, and D. Thiel, Science 263, ries of the kinoform lens introduce disturbances in the 201 (1994). wave field. As a consequence, only about 51% of the [9] M. J. Zwanenburg et al., Physica (Amsterdam) 283B, 285 transmitted radiation is focused. In both cases, Compton (2000). scattered photons add to the background. To test the toler- [10] A. Snigirev, V. Kohn, I. Snigireva, and B. Lengeler, Nature ance of nanometer focusing to shape errors, the thickness (London) 384, 49 (1996). of each lens was modulated by a distortion Aj cos �=R0jr . [11] B. Lengeler et al., Appl. Phys. Lett. 74, 3924 (1999); B. Lengeler et al., J. Synchrotron Radiat. 9, 119 (2002). For Aj up to 10% of lj, no significant deterioration of the focus was observed. This alleviates the requirements for [12] B. Lengeler et al., J. Synchrotron Radiat. 6, 1153 (1999). [13] V.V. Protopopov and K. A. Valiev, Opt. Commun. 151, the fabrication of these optics. We are currently developing 297 (1998). fabrication techniques for one-dimensionally focusing [14] R. W. James, The Optical Principles of the Diffraction of lenses. X-rays (Cornell University Press, Ithaca, NY, 1967). The efficient use of adiabatically focusing lenses re- [15] M. Born and E. Wolf, Principles of Optics (Cambridge quires the high brilliance of a modern synchrotron radia- University Press, Cambridge, 1999). tion source. As an example for focusing of an extended [16] V. Aristov et al., Appl. Phys. Lett. 77, 4058 (2000).

054802-4 PHYSICAL REVIEW LETTERS week ending PRL 94, 054802 (2005) 11 FEBRUARY 2005

Focusing Hard X Rays to Nanometer Dimensions by Adiabatically Focusing Lenses

DOI: 10.1103/PhysRevLett.94.054802 PACS numbers: 41.50.+h, 07.85.Qe

100 nm and below [4,5]. The current limitations to focus- merically by wave propagation.the lens is 2 �0L Adiabatic Lenses Deff 2R0i 1 exp ; (5) � � L � � 2 ing to even smaller focal sizes are of a technological nature s��0 � � �� 2 1 2� 2 1 dj=lj � !j � � � ; (1) and may be overcome in the future. The question arises, l f l R R2 where �0 1 dj=lj � and � is the linear attenuation � j j � j j � 0j �� however, whether there is a physical limit to efficient coefficient. Using Eqs. (4) and (5), the numerical aperture ►► absorption limit, NA Deff= 2f for a distant source is focusing of hard x rays due to their weak interaction with growing with decreasing aperture R0j. In order to obtain � � � the optimal refractive power for each individual lens in the use low-Z (Be, Al, C) a R R matter and what, if any, this limit is [6]. Bergemann et al. stack, its aperture R is set to the local beam size, follow- NA p� 4 1 exp 0i log 0i ; (6) 0j � 0 R � � a R have determined this limit for waveguides [7]. They find ing the converging wave field adiabatically to the focus as s��0i � � �� 0f �� ► need phase shift, long as possible. To find the aperture R as a function of p ► that the effective numerical aperture of waveguides is 0j where a 4 �0 is a material specific characteristic aper- position along the optical axis, we make use of ray optics. � p��0 limited by � p2�, the critical angle of total external As the number N of lenses is large and the refraction from ture shown in�� Fig. 2 for different materials. A larger a use high mass density c �� reflection for a� material with refractive index n 1 � each lens is small, the path of a ray r z inside the lens yields a larger numerical aperture, favoring materials with � � � assembly can be described by the second� order� differential low atomic number Z. In addition, the numerical aperture e.g., diamond i�. The critical angle also limits the numerical aperture of equation, is proportional to p�0, favoring lens materials with large mass density �. The largest numerical apertures are there- mirror optics and capillaries [8]. Bergemann et al. believe �� 2 fore expected for high density low Z materials, such as ►► adapt lens diameter d r 2 that the same limitations to focusing hold for all x-ray r00 ! z r; (2) diamond. � dz2 �� optics, limiting the focal size to slightly below 10 nm [7,9]. NA grows with increasing ratio R0i=R0f (cf. Fig. 1). R0f to local beam size should thus be chosen as small as possible.PHYSICAL It is limited REVIEW by LETTERS week ending First introduced in 1996 [10], refractive x-ray lenses that is derived from the transfer matrix formalism [13] in PRL 94, 054802 (2005) 11 FEBRUARY 2005 the continuum limit. Equation (2) holds for each ray, in nanofabrication techniques and ultimately by the granular have quickly advanced to an efficient tool in x-ray micros- (atomic) structure of matter. For the following considera- ► aspherical holes particular, for the outermost ray R0 z that defines the source,► we consider the performance of the optic for � � tions,8000 we choose R large compared to interatomic dis- copy and microanalysis [11,12]. In these optics, the weak aperture of the optic as a function of position along the 0f NFL (14.18nm) 20 keV x rays at L 47 m from a low-� undulator AFL (4.74nm) 1 optical axis (cf. Fig. 1). By inserting Eq. (1) into Eq. (2), tances, e.g., R0f 50 nm. The entrance radius R0i can be source atto the improve European Synchrotron quality Radiation Facility refraction of hard x rays in matter is compensated by � AFLk3 (2.77nm) the differential equation for the aperture R0 z is obtained: 6000 (ESRF) in Grenoble, France. This source has an effective � � AFLk5 (2.21nm) stacking behind each other a large number of strongly R000 2�0=R0. Here, �0 1 dj=lj �, assuming dj=lj 100 lateral size of 125 �m 60 �m (FWHM). For a two- �� ► sub-10 nm possible to be the same for all individual lenses. The first integral of 0 dimensionally► focusing diamond lens, a focal spot size of curved individual lenses to form a lens assembly, in which 1 2 2 this second order differential equation is R I/I 4000 5:8 4:7 nm would be expected, compared to the diffrac- the beam is gradually focused. Nanofocusing refractive x- 2 � 0� � 10 2�0 log R0 E, where E is a constant determined by tion limited spot size of 4:3 nm. The intensity gain in this the initial� �� conditions. For a parallel beam incident on the geometry would be about 7 106, with of the order of ray lenses (NFLs) currently generate microbeams with a first lens, the initial conditions are R 0 R and 20001 109 ph=s in the focus. As the focal distance of AFLs is 0 0i Li lateral size of about 100 nm [5]. The minimal spot size R 0 0 (cf. Fig. 1), yielding the first order� �� differential extremely short, e.g., f 1:6 mm, demagnifications of the 0 [µm] Be � �� a order of 104 are possible even at short beam lines. achievable with these optics lies below 20 nm and is lim- equation 0.1 0 B We thank A. Snigirev, I. Snigireva, and J. Maser for the -10 -5 0 5 C (graphite) 10 ited by a maximal numerical aperture of NA p2� that is FIG. 1. Adiabatically focusing x-ray lens. The lens is com- C* (diamond) discussion about the limitations to the focusing of hard R 0.01 r [nm] � R 4� log 0i: (3) Si x rays. the same as for waveguides [7]. It is limited by the constant posed of a large number of individual0 (aspherical)0 refractive Ni � s��R0 refractive power per unit length inside the NFL with con- lenses, whose aperture is matched to the converging beam size, FIG. 5. Lateral beam profiles for a nanofocusing lens, an adiabatically focusing2 4 lens, 68 and two2 kinoform 4 68 AFLs with24 three Figure 1 shows the course of R0 z as a solution of Eq. (3). 1 10 100 stant aperture. increasing the refractive power per unit length along� � the lens. and five segments, respectively. Schroer, Lengeler: PRL, 2005 For a given exit aperture R0f, the lens properties can be E [keV] [1] Synchrotron Radiation Instrumentation, edited by T. calculated, such as focal distance, position, and effective as Warwick, J. Arthur, H. A. Padmore, and J. Sto¨hr, AIP p well as numerical aperture. The focal distance f and sec- FIG.Leaving 2. Characteristic all other parameters aperture a unchanged,4 �0 as a function the aperture of x-ray of Conf. Proc. No. 705 (AIP, Melville, NY, 2004). 0031-9007=05=94(5)=054802(4)$23.00 054802-1  2005 The American Physical Society these lenses can be increased by� ap kinoform��0 lens design as ondary principal plane H are (see Fig. 1) energy for different lens materials. �� [2] J. Wang et al., Phys. Rev. Lett. 80, 1110 (1998). shown in Fig. 4(a) [12,16]. It yields a�� significant increase in [3] I. K. Robinson et al., Opt. Express 11, 2329 (2003); G. J. aperture growing with the square root of the number of lens Williams, M. A. Pfeifer, I. A. Vartanyants, and I. K. 054802-2segments. Figure 4(a) shows the intensity distribution in- Robinson, Phys. Rev. Lett. 90, 175501 (2003). side a kinoform lens with five segments for an x-ray wave [4] W. Yun et al., Rev. Sci. Instrum. 70, 2238 (1999); O. front coming from a distant point source. At the border Hignette et al., in X-Ray Micro- and Nano-Focusing: between two segments, the wave front is distorted and Applications and Techniques II, edited by I. McNulty, fringes are formed. However, as the borders of the seg- Proc. SPIE Int. Soc. Opt. Eng. Vol. 4499 (SPIE- International Society for Optical Engineering, ments follow the contracting beam adiabatically, the re- Bellingham, WA, 2001), pp. 105–116; K. Yamauchi sulting disturbance is minimal. The intensity distribution et al., J. Synchrotron Radiat. 9, 313 (2002). behind the lens is depicted in Fig. 4(b). A focus with a [5] C. G. Schroer et al., Appl. Phys. Lett. 82, 1485 (2003); lateral size of 2:21 nm is obtained at the distance of C. G. Schroer et al., in Ref. [1], pp. 740–743. 11:6 �m behind the lens. [6] Inefficient focusing of hard x rays to atomic dimensions Figure 5 shows the beam profiles at the focal position could be achieved, for example, using a focusing mirror obtained by wave propagation for a NFL, an AFL, and two reflecting at large angles. Such a mirror would focus the x rays very inefficiently, but could reach numerical aper- kinoform AFLs with three (AFLk3) and five (AFLk5) segments, respectively. The intensity is normalized to that tures up to one. However, Compton scattering would incident on the lens, giving the gain for one-dimensional strongly occult this focus. focusing. For the AFL with one segment, more than 90% of [7] C. Bergemann, H. Keymeulen, and J. F. van der Veen, Phys. Rev. Lett. 91, 204801 (2003). the transmitted radiation is focused. The segment bounda- [8] D. Bilderback, S. A. Hoffman, and D. Thiel, Science 263, ries of the kinoform lens introduce disturbances in the 201 (1994). wave field. As a consequence, only about 51% of the [9] M. J. Zwanenburg et al., Physica (Amsterdam) 283B, 285 transmitted radiation is focused. In both cases, Compton (2000). scattered photons add to the background. To test the toler- [10] A. Snigirev, V. Kohn, I. Snigireva, and B. Lengeler, Nature ance of nanometer focusing to shape errors, the thickness (London) 384, 49 (1996). of each lens was modulated by a distortion Aj cos �=R0jr . [11] B. Lengeler et al., Appl. Phys. Lett. 74, 3924 (1999); B. Lengeler et al., J. Synchrotron Radiat. 9, 119 (2002). For Aj up to 10% of lj, no significant deterioration of the focus was observed. This alleviates the requirements for [12] B. Lengeler et al., J. Synchrotron Radiat. 6, 1153 (1999). [13] V.V. Protopopov and K. A. Valiev, Opt. Commun. 151, the fabrication of these optics. We are currently developing 297 (1998). fabrication techniques for one-dimensionally focusing [14] R. W. James, The Optical Principles of the Diffraction of lenses. X-rays (Cornell University Press, Ithaca, NY, 1967). The efficient use of adiabatically focusing lenses re- [15] M. Born and E. Wolf, Principles of Optics (Cambridge quires the high brilliance of a modern synchrotron radia- University Press, Cambridge, 1999). tion source. As an example for focusing of an extended [16] V. Aristov et al., Appl. Phys. Lett. 77, 4058 (2000).

054802-4 CRL Chips Transfocators ►► zoom capabilities: cartridges of 2n lenses ►► choose combination based on x-ray energy – focal length – beam size ►► developed at ESRF, used at many synchrotrons ►► especially at higher energies, 20 … 100 … keV spot size in micro metre range

Vaughan, Snigirev, Wright et al 101103-3 Patommel et al. Appl. Phys. Lett. 110, 101103 (2017)

focusing lens. Both lenses were adjusted such that they focus A parabolic lens—both nanofocusing4,17 or adiabatically 12 the x rays to a common focal plane about zf f H focusing —has a truncated gaussian aperture function (trans- 0.405 mm behind the exit of the AFL. The overall¼ transmis-� ¼ mission profile), damping the peripheral amplitudes compared sion of the two focusing optics together was about 8% (cf. to those on the optical axis. The FWHM size of the Airy disc 7 1 Table I), resulting in a focused flux of about 2 10 s� . dt (diffraction limited focus) can be associated to an effective � 22 We used ptychography to characterize the nanobeam cre- aperture Deff and with it a numerical aperture NA by ated in this way.14 In this technique, an object is scanned kf k through the focused x-ray nanobeam, recording a far-field dif- (1) 13,14 dt a a ; fraction pattern at each position of the scan. From these ¼ Deff ¼ 2NA data, the object’s complex transmission function and the com- where is the wavelength of the x rays and depends on the plex wave field of the illumination in the object plane is k a aperture function and lies in the interval of 0 75 0 85 reconstructed using iterative phase-retrieval techniques.18,19 : a : for rectangular truncated gaussian apertures. The� effective� The complex wave field in the object plane can be propagated aperture Deff of a truncated gaussian aperture is smaller than numerically along the optical axis, yielding a faithful repre- 22 sentation of the full caustic of the beam.20 In this way, pty- the geometric aperture. The amplitude of the wavefield w Deff=2 at the edge of the effective aperture (in the princi- chography has revolutionized the characterization of x-ray j ð Þj nanofocusing optics and is by now well established.11,21 pal plane) is We recorded a ptychogram using a NTT-AT resolution lL0 chart (model: ATN/XRESO-50HC) as a test object placed c : w Deff=2 = w 0 exp 1 exp ; (2) ðÞ 2 near the focal plane. It was raster scanned perpendicularly to ¼j ð Þj j j¼ � � the beam in 100 100 steps, covering an area of 2 lm 2 lm. times smaller than the amplitude w 0 on the optical axis. � � j ð Þj At each position of the scan, a far-field diffraction pattern was Here, l is the mass attenuation coefficient of the lens mate- recorded by a Pilatus 300k photon counting pixel detector by rial, and L0 is the length of the lens without counting the lens DECTRIS (pixel size: 172 lm) placed 2.22 m behind the sam- material of thickness dj at the apexes of the individual lenses ple (exposure time 0.5 s). From these data, both the complex (cf. Fig. 1). As the beam propagates to the focus, the ray incident wave field and the transmission function of the test with the relative amplitude c as in Eq. (2) follows the numer- object were reconstructed following Ref. 19. ical aperture except in the near-field of the focus. By propagating the complex illumination function along As the amplitude of the complex wave field of the nano- the optical axis, the full caustic of the beam can be recon- focused beam is accessible by ptychography, we can follow structed.20 Fig. 3(a) shows the vertical and Fig. 3(b) horizon- the ray with the relative amplitude c along the caustic and by tal beam profile. The vertical beam profile [Fig. 3(a)] that measure the numerical aperture NA associated with Deff. generated by the nanofocusing lens is nearly gaussian and 3 From the data in Fig. 3(a) NAv 0:48 1 10� is obtained. free of aberrations. The minimal lateral beam size in the ver- ¼ ð Þ� From Eq. (1) follows a diffraction limit d 48.4(10) nm in Recent Progress tical direction is 47.3(2) nm (FWHM). t good agreement with the measured beam size¼ of 47.3(2) nm. The horizontal focus that is generated by the AFL is Focusing hard x rays beyond shown in Figs. 3(b) and 3(c). Ideally, the beam would be similarly gaussian as in Fig. 3(a) but with a significantly higher the critical angle of total numerical aperture. From the reconstruction of the caustic, reflection by adiabatically it becomes apparent that the AFL suffers from aberrations, 101103-2 Patommel et al. Appl. Phys. Lett. 110, 101103 (2017) generating a far from ideal focus. The brightest speckle focusing lenses @ 20 keV [Fig. 3(c)] can be considered the focus and lies in the plane The lens is made of silicon by a plasma-based anisotropic marked by (i) in Fig. 3(c). The horizontal intensity profile in etching process developed at the Institute of Semiconductor- and Microsystems Technology (IHM) of Technische this plane is shown in Fig. 4(i) and shows a focal spot size of Universit€at Dresden, Germany. The resist mask pattern was 18.4 nm (FWHM). About 65% of the radiation fall into this fabricated by optical lithography using an i-line wafer stepper at SAW Components, Dresden. In order to achieve small and precise deeply etched structures with steep side walls and low surface roughness, a semi-continuous deep reactive-ion etching technique (DRIE) was developed. It combines the classical discontinuous DRIE process (also known as Bosch process)15 with a continuous one. The main idea of DRIE is to apply the surface passivation in addition to the reactive ion etching. In discontinuous DRIE, passivation and etching are performed alternatively, while in continuous DRIE, passivation and etching are carried out simultaneously without FIG.interruption. 3. Caustic of The the first nanofocused (discontinuous) beam: amplitude process is of characterized the wave field projected onto (a) the vertical and (b) horizontal plane. The dashed line1 in (a) by very high etch rates in the range of several lm min� . and (b) depicts the sample plane, in which the ptychogram was recorded. FIG. 4. (i) Horizontal intensity distribution in the focal plane of the AFL Furthermore, a high etching selectivity between the mask The semi-transparent lines in (b) delineate the numerical aperture. (c) [line (i) in Fig. 3(c)]. The central speckle has a lateral size of 18.4 nm Intensitymaterial distribution (photoresist) around and the the focus substrate [dashed material area in (silicon)(b)] projected of onto (FWHM). (ii) horizontal intensity distribution at the section with the later- 50 : 1 can be achieved. Passivation by polymer deposition the� horizontal plane. Line (i) in (c) is the section with the highest peak inten- ally smallest speckle [line (ii) in Fig. 3(c)]. The smallest speckle has a lateral sityand (focus), etching line using (ii) is SF a section6 is adjusted through in the such laterally a way, smallest that steep speckle. side size of 14.0 nm (FWHM). walls are created. But the surface roughness is quite high (about 150 nm rms), due to the cyclic changing plasma conditions. In the second (continuous) case, the surface roughness is FIG. 1. (a) Scheme of the adiabatically focusing lens (AFL, adapted from much reduced (to less than 60 nm rms), but the etch rate is too 1 Ref. 12). (b) and (c) Scanning electron micrographs of the AFL. (b) low (<1 lm min� ), the selectivity only amounts to ( 25 : 1), Entrance and (c) exit of the optic on the same scale (magn. 750 ). The � and the tilt angle of the sidewalls lies between 85� and 90�. In Patommel, Schroer et al, APL, 2017 insets show enlarged details of the very first (magn. 3000 ) and last� lenses (magn. 12 000 ) in the stack, respectively. � the semi-continuous DRIE process, etching and non-etching � periods take turns, while passivation is performed continu- ously during the whole process. In this way, the advantages of The aperture radius R0j and with it the radius of curvature the continuous and the cyclic DRIE (Bosch) process are com- Rj of the j-th individual lens in the stack are adapted to the converging x-ray beam at the design energy (Fig. 1). For bined. In addition, the so-called underetching effect (etching practical reasons, the minimal thickness of the individual beyond the margin of the mask) is reduced significantly. We used this AFL for nanofocusing in the hard x-ray lenses was kept constant at dj 1.0 lm (Fig. 1). As a result, ¼ nanoprobe instrument at beamline P06 of the synchrotron radi- the analytical expressions for the optical properties of AFLs 16 given in Ref. 12 do not properly describe the optic consid- ation source PETRA III at DESY in Hamburg, Germany. At ered here. Instead, here, we calculate the properties numeri- P06, the nanoprobe instrument is located approximately 98 m cally by wave propagation. The AFL has an overall length from the undulator source. On the way to the instrument, the x-ray beam is monochromatized (E 20 keV) by a Si (111) of L 3.714 mm and a focal length of f 1.915 mm as ¼ measured¼ from the second principal plane¼ that lies double-crystal monochromator. In order to focus the x-rays in H 1.510 mm before the exit of the lens [cf. Fig. 1(a)]. two dimensions, the AFL was combined with a conventional ¼ nanofocusing refractive x-ray lens4,17 (N 300, R 12.4 lm, Table I summarizes the parameters for this optic. ¼ ¼ R0 19 lm) in a crossed geometry as shown in Figure 2. At the¼ entrance of the nanoprobe, the aperture of the optic is TABLE I. Summary of geometric and optical parameters of both AFL and defined by a pair of high-precision slits that were set to 10 m the nanofocusing lens (NFL). Optical parameters are given for 20 keV. l 50 lm (horiz. vert.). The beam was vertically focused by � � Optic AFL NFL the nanofocusing lens (NFL) with a focal length of fNFL 18.7 mm and then horizontally by the adiabatically Material Si Si ¼ Length L (mm) 3.714 9.0 Number of lenses N 508 300

Entrance aperture 2R0i (lm) 9.4 38

Entrance curvature radius Ri (lm) 2.45 12.4 Exit aperture 2R0f (lm) 2.0 38

Exit curvature radius Rf (lm) 0.5 12.4 Minimal lens thickness d (lm) 1.0 1.0 Transmission 0.27 0.29 Focal length f (mm) 1.915 18.7

Working distance zf (mm) 0.405 14.0 3 3 Numerical aperture NA 1.73 10� 0.48 10� � � FIG. 2. Schematic experimental setup. Waveguides Outline ►► Introduction + History ►► Reflective Optics ►► Diffractive Optics ►► Refractive Optics ►► Waveguides ►► Discussion Principle and Geometry

►► rays reflected back and forth, or: ►► channel size: ~ (20 nm)2 ►► wave trapped inside cavity ►► channel length: ~ 1 mm Analytical and Numerical WG Design

(a) Waveguide modes - amplitude (b) Waveguide modes - intensity

1 n = 0 1.0 n = 1 amplitude

0 0.5 i n t ensity

-1 0.0 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 lateral axis, y / D lateral axis, y / D ►► first and second guided mode, solving eigenvalue equation in cavity ►► for Si: one mode per 20 nm channel width ►► understand WG fundamentals, including curvature, real structure, multiple components Analytical and Numerical WG Design

(a) Waveguide modes - amplitude (b) Waveguide modes - intensity (a) Intensity distribution, θ = 0 (b) Partially coherent intensity -50 -50 1 n = 0 1.0 1.0 n = 1 -25 -25 lateral, y in nm lateral, y in nm amplitude

0 0.5 i n 0 0

t 0.5 ensity 25 25 -1 0.0 50 50 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 0 200 400 600 800 1000 0.0 0 200 400 600 800 1000 lateral axis, y / D lateral axis, y / D axis of propagation, x in µm axis of propagation, x in µm

degree of coherence, | j (c) D = 30 nm degree of coherence, | j (d) D = 50 nm ►► first and second guided mode, ►►1.00simulation inside WG, 1.00 x = 5000 µm solving eigenvalue equation in cavity solvingx = 0 µm parabolic wave equation 0.75 0.75 x = 100 µm st (too noirdsy) ►► for Si: one mode per 20 nm channel width ►►0.50mode beatingx = 1000 of µm 1 and 3 0.50 ►► understand WG fundamentals, including ►►0.25design WGs, including 0.25 curvature, real structure, multiple components 0.00bending, curving and splitting 0.00 0 15 30 45 60 75 90 0 25 50 75 100 125 150 distance from optical axis, d in nm distance from optical axis, d in nm

degree of coherence, | j (e) D = 70 nm (f) Iso-lines of intensity and coherence behind the WG exit 1.00 -4 -3 -3 µm I > 10 I > 10 0.75 I > 10-2 0.50 I > 10-1

WG exit 0 µm 0.25 1 mm 2 mm |j| > 0.9 0.00 |j| > 0.7 0 35 70 105 140 175 210 3 µm distance from optical axis, d in nm |j| > 0.3 |j| > 0.5 Coherence Filtering

-500 -250 (a) HFM focused intensity γ<0.4 γ<0.4 (b) WG ﬁltered intensity ►► WG is placed in -250 γ<0.4 -125 pre-focus, e.g. KB , y in nm , y in nm 0 γ>0.8 γ>0.8 0 ►► illumination is (c) (d) (e) (f) al axi s al axi s 250 125 only of partial e r e r (+ noise) γ>0.8 t t l a l a 500 250 coherence -1.0 -0.5 0.0 0.5 1.0 0.0 0.5 1.0 1.5 2.0 +0.5 ►► WG accepts 1.0 (c) Focal plane (e) Inside WG 1.0 discrete modes, 0.8 intensity 0.8 ence ence

c which are fully 0.6 ohe r coherence 0.6 ohe r ohe r

0.4 ence 0.4 coherent 0.2 0.2 0.0 intensity 0.0 ee of ee c ee of c ►► further filtering -2 -1 0 1 2 -0.50 -0.25 0.00 0.25 0.50 by absorption of , deg r , , deg r , (d) Focal plane +1.0 mm (f) WG +0.1 mm 1.0 1.0 higher modes 0.8 intensity coherence 0.8 ensity ensity

t 0.6 × 5 × 12.5 0.6 t n 0.4 coherence intensity 0.4 ►► in addition, 0.2 0.2 beam is cleaned 0.0 0.0 norm. i -2 -1 0 1 2 -0.50 -0.25 0.00 0.25 0.50 norm. i n from KB stripes Osterhoff, PhD thesis, 2011 research papers interferometers were realized based on a silicon bilens, i.e. by optimizations resulting in higher transmission values of the a pair of compound refractive lenses (CRL) etched into waveguide channels. A silicon wafer (MaTeck) with a silicon. Instead of the CRL bilens, we here demonstrate the diameter of 10 cm was covered with an electron-beam resist approach of X-ray waveguide beamsplitters. In contrast to (ARP 679.04, Allresist) by spin coating (Optispin SB20, Sister refractive or diffractive optics, X-ray waveguides are essen- Semiconductor Equipment GmbH) the solution at 1 tially non-dispersive optical components, and are thus also 2000 r minÀ for 70 s followed by a post-baking step (453 K for suitable for broader bandpass radiation and for ultra-short 90 s). The beamsplitting patterns were brought into the resist pulses. Furthermore, the beamsplitter can be combined with by electron-beam lithography (e-LiNE, Raith). A develop- coherence (mode filtering) and the direction of the exit beams ment procedure was performed, dipping the wafer into a can be manipulated. Already in Fuhse et al. (2006) we have mixture of methyl isobutyl ketone (MIBK) and 2-propanol shown that two adjacent waveguide beams can be used for off- (IPA) in a ratio of 1:3 at 273 K for 30 s. To stop the removal of axis holographic imaging, but in that work the two channels the resist, the wafer was immersed in pure IPA at room did not emerge from a single input channel as we have now temperature and kept there for 30 s. The electron-beam resist realized in the present work. An advanced fabrication scheme serves as a mask for reactive ion etching (PlasmaLab, Oxford) with improved lithography, etching and wafer bonding steps for 160 s. The structures were transferred into the wafer has now paved the way to realizing X-ray optics on a chip. utilizing 10 sccm:40 sccm (where sccm = standard cubic Towards this goal, efficient X-ray beamsplitters as demon- centimetres per minute) of SF6 and CHF3 for etching and strated here will be indispensable components. sidewall passivation (Legtenberg, 1995). After cleaning the etched wafer employing ammonia (32%), hydrogen peroxide (30%) and deionized water [1:1:5, RCA cleaning procedure 2. Experiments (Kern & Werner, 2008)], a second wafer was manually put on top of the first one, embedding the etched structures (see Fig. 2.1. Fabrication of beamsplitters 1b). The wafers were inspected using an infrared camera to TaperingThe waveguiding structures were fabricated and according toCurvingdetect voids in the contact surface of the wafers. The final the protocol described by Neubauer et al. (2014) with further wafer bonding took place in an oven (L 3/11/P330, Naber-

Figure 1 Experimental geometry. (a) Finite differences simulation of a beamsplitter. (b) Top view SEM image of a splitting structure before wafer bonding. Scale bar denotes 1 mm. (c)–(f) SEM images of the exit side of beamsplitters with different spacings S. Scale bars denote 100 nm. (g) Schematic of the experimental geometry showing the coupling of the focused X-ray beam into the entrance of the beamsplitter, the subsequent guiding in the two channels, the free-space propagation behind the chip, and finally the far-field detector at a distance D. The far-field pattern shows the characteristic double-slit interference pattern, modulated with features of the waveguide modal structure. Arrows mark bifurcations in the interference fringes (fork- shaped structures). Length and angles are not to scale. (h) Enlarged view of the interference pattern with a sinusoidal fit to the intensity oscillations. (i) Scan in the y direction indicating the position of different beamsplitters which have all been defined on the same chip with different geometric parameters, and which can be selected by translating the chip in the FZP focus. Detailed scan profile of a single channel with a width (FWHM) of 282.6 nm giving an upper limit for the beam size in the horizontal direction.

516 Hoffmann-Urlaub and Salditt Miniaturized X-ray beamsplitters Acta Cryst. (2016). A72, 515–522

Hoffmann-Urlaub, Salditt, Acta Cryst A, 2016 research papers interferometers were realized based on a silicon bilens, i.e. by optimizations resulting in higher transmission values of the a pair of compound refractive lenses (CRL) etched into waveguide channels. A silicon wafer (MaTeck) with a silicon. Instead of the CRL bilens, we here demonstrate the diameter of 10 cm was covered with an electron-beam resist approach of X-ray waveguide beamsplitters. In contrast to (ARP 679.04, Allresist) by spin coating (Optispin SB20, Sister refractive or diffractive optics, X-ray waveguides are essen- Semiconductor Equipment GmbH) the solution at 1 tially non-dispersive optical components, and are thus also 2000 r minÀ for 70 s followed by a post-baking step (453 K for suitable for broader bandpass radiation and for ultra-short 90 s). The beamsplitting patterns were brought into the resist pulses. Furthermore, the beamsplitter can be combined with by electron-beam lithography (e-LiNE, Raith). A develop- coherence (mode filtering) and the direction of the exit beams ment procedure was performed, dipping the wafer into a can be manipulated. Already in Fuhse et al. (2006) we have mixture of methyl isobutyl ketone (MIBK) and 2-propanol shown that two adjacent waveguide beams can be used for off- (IPA) in a ratio of 1:3 at 273 K for 30 s. To stop the removal of axis holographic imaging, but in that work the two channels the resist, the wafer was immersed in pure IPA at room did not emerge from a single input channel as we have now temperature and kept there for 30 s. The electron-beam resist realized in the present work. An advanced fabrication scheme serves as a mask for reactive ion etching (PlasmaLab, Oxford) with improved lithography, etching and wafer bonding steps for 160 s. The structures were transferred into the wafer has now paved the way to realizing X-ray optics on a chip. utilizing 10 sccm:40 sccm (where sccm = standard cubic Towards this goal, efficient X-ray beamsplitters as demon- centimetres per minute) of SF6 and CHF3 for etching and strated here will be indispensable components. sidewall passivation (Legtenberg, 1995). After cleaning the etched wafer employing ammonia (32%), hydrogen peroxide (30%) and deionized water [1:1:5, RCA cleaning procedure 2. Experiments (Kern & Werner, 2008)], a second wafer was manually put on top of the first one, embedding the etched structures (see Fig. 2.1. Fabrication of beamsplitters 1b). The wafers were inspected using an infrared camera to The waveguiding structures were fabricated according to detect voids in the contact surface of the wafers. The final week ending Taperingthe protocol described by Neubauer et al.and(2014) with further Curvingwafer bonding took place in an oven (L 3/11/P330, Naber- PRL 115, 203902 (2015) PHYSICAL REVIEW LETTERS 13 NOVEMBER 2015

l λ= 4π =β 3.62 μm is obtained, such that the XWGchip¼ ð Þ of¼ 5 mm length is completely intransparent for radiative modes. According to simple geometric optics, the condition that the incoming beam impinges at an angle of incidence αi ≤ αc with respect to the outer channel wall gives a simple constraint for the radius of curvature 2 2 R ≥ w=2sin αc=2 2πw= λ r0ρ , with r0 the Thompson scattering lengthð andÞ ≃ ρ theð electronÞ density. To describe the mode propagation in a “potential well”

Figure 1 given by the one-dimensional electron density profile Experimental geometry. (a) Finite differences simulation of a beamsplitter. (b) Top view SEM image of a splitting structure before wafer bonding. Scale bar denotes 1 mm. (c)–(f) SEM images of the exit side of beamsplitters with different spacings S. Scale bars denote 100 nm. (g) Schematic of the experimental geometry showing the coupling of the focused X-ray beam into the entrance of the beamsplitter, the subsequent guiding in the two function ρ x , we consider the reduced scalar wave channels, the free-space propagation behind the chip, and finally the far-field detector at a distance D. The far-field pattern shows the characteristic double-slit interference pattern, modulated with features of the waveguide modal structure. Arrows mark bifurcations in the interference fringes (fork- ð Þ 2 2 shaped structures). Length and angles are not to scale. (h) Enlarged view of the interference pattern with a sinusoidal fit to the intensity oscillations. (i) equation ψ 4πr ρ x ψ β k ψ describing propa- Scan in the y direction indicating the position of different beamsplitters which have all been defined on the same chip with different geometric 00 0 − 0 parameters, and which can be selected by translating the chip in the FZP focus. Detailed scan profile of a single channel with a width (FWHM) of − ð Þ ¼ð Þ 282.6 nm giving an upper limit for the beam size in the horizontal direction. gation of guided modes in a planar waveguide, with the 2 2 516 Hoffmann-Urlaub and Salditt Miniaturized X-ray beamsplitters Acta Cryst. (2016). A72, 515–522 mode number M≔β − k0, depending on the solution of the FIG. 1 (color online). (a) Schematic of the channels defined in corresponding eigenvalue problem. For the simple case the XWGchip and the experimental geometry. The individual relevant here, ρ x is a rectangular profile with height ρ channels are selected by focusing the beam into the entrance. ð Þ 0 Guiding of a beam in a curved channel is evidenced by and width w. The dimension of M is that of scattering measurement of the far-field pattern. (b) Finite difference length density. For the example of an air-vacuum channel in 2 simulation of beam propagation in the curved WG channels. Ta, we get 4πr0ρ0 0.144 nm− . At z 0 a plane wave of unit amplitude and 7.9 keV photon How does the situation¼ change for a curved channel? energy¼ impinges on a channel of w 100 nm and R 40 mm in Following Ref. [21], translated to the case of x-ray notation, Hoffmann-Urlaub, Salditt, Acta Cryst A, 2016 ¼ Salditt et al,¼ PRL, 2015 Ta (initial data). The intensity distribution is shown in logarithmic we get scale within a rectangular region of length Δz 0.5 mm and ¼ width Δx 5 μm, along with a zoom of the central region with 2k2 4πr ρ x ¼ 0 − 0 2 2 z 100 μm and x 1.5 μm. The transmission is T 0.842, ψ 00 4πr0ρ x ð Þx ψ β −k ψ; 1 Δ Δ − ð Þþ R ¼ð 0Þ ð Þ after¼ a propagation length¼ of 0.5 mm. The spikes show¼ that � � evanescent waves depart into the cladding at certain locations corresponding to reflections in the geometric optical description. where R is the radius of the curvature. The curvature-induced (c) Fundamental mode of a channel in tantalum (Ta) with w term leads to a distortion of the potential, notably a tilt. We 100 nm and R 10 mm as simulated by solving the wave¼ ¼ can further neglect the ρ dependence in the curvature term equation with a tilted potential well. 2 2 2k − 4πr0ρ x 2k , so that only a linear potential is 0 ð Þ ≃ 0 the first 10% of the propagation in the XWGchip. Intensity added to the intrinsic waveguide potential, is shown in logarithmic scaling to highlight the evanescent 2k2 waves. Spikes are observed departing into the cladding. For 0 2 2 ψ 00 4πr0ρ x x ψ β − k ψ: 2 − ð Þþ R ¼ð 0Þ ð Þ Ta and the given channel width w, 13 bound modes � � copropagate in the channel. The fundamental mode as 2 calculated in the effective tilted potential by Numerov’s For R w 2πw= λ r0ρ , the tilt becomes so strong that the ð Þ¼ ð Þ method [18] is shown in (c). right side of the potential well (inner side wall of the curved Experiments were first carried out at the GINIX nano- channel) has the same height as the cladding potential on the imaging end station [19] of the coherence beam line P10 at left side. Hence, the confining potential has changed from the storage ring PETRAIII (DESY, Hamburg), at photon a square to a triangular well, and the width of the barrier energy of E 7.9 keV. The monochromatic [Si(111) at the base of the well decreases to w. Interestingly, this double crystal]¼ undulator beam was focused with two transition corresponds exactly to the above geometric optical elliptical total reflection mirrors in Kirkpatrick-Baez constraint. However, in wave optics we cannot reduce w (KB) geometry, to a focus of 380 nm × 350 nm (h × v, arbitrarily in order to accommodate higher curvature. FWHM values) with a primary beam intensity of about The minimum reasonable channel width is given by the I 4.1 × 1011 ph=s. The focused beam was coupled to critical waveguide thickness for single-mode propagation 0 ¼ selected channels of the XWGchip, aligned in the focal W≔λ=2αc π=4ρr0. Upon further reduction wHolography

Salditt, Osterhoff et al, Journal of Synchrotron Radiation, 2015 WG Holography feature articles 2010), with controlled mode structure spot sizes down to sub- 10 nm. The mode and coherence filtered waves are ideally suited for quantitative holographic image recording. Mode filtering minimizes wavefront distortions and artifacts encountered in many other hard X-ray focusing optics. This enables quantitative reconstruction of the object by robust phase-retrieval algorithms. By selecting the waveguide-to- sample distance, objects can be imaged at a single distance in a full-field configuration without scanning. Robust and quickly converging iterative reconstruction schemes can be applied to invert the holographic near-field diffraction patterns. Weakly scattering biological specimen can thus be phased even without exact knowledge of the illumination function. The method proved to be very dose-efficient providing images of cells at doses below 105 Gy, and takes photon noise effects into account quantitatively (Giewekemeyer et al., 2011). The Figure 6 tomographic extension provides quantitative three-dimen- Holographic imaging with the waveguide probe. Scale bar: 585 mm (detector plane). (a) Hologram of a test pattern after division by the sional density reconstructions of biological cells and tissues empty beam. Scale bar: 6 mm (detector plane). (b) Raw data of a (Bartels et al., 2012; Olendrowitz et al., 2012). To probe the hologram with three Deinococcus radiodurans cells, showing the smooth three-dimensional structure of a larger specimen, the full-field line-shape and tails of the waveguide probe. (c) Phase reconstruction of the hologram shown in (a), with 22.4 nm pixel size. Scale bar: 2 mm. (d) holographic technique is of particular advantage, since it Rendered three-dimensional density distribution of a Deinococcus avoids the problem of overheads in detector readout which is radiodurans bacteria (Bartels et al., 2012), reconstructed from wave- pertinent in scanning three degrees of freedom (two transla- guide-based holographic tomography. tions, oneSalditt, tomographic Osterhoff rotation). et al, Journal In contrast of Synchrotron to scanning Radiation, 2015 SAXS or diffraction microscopy (ptychography), extended empty beam corrected hologram in (a) shows the holographic specimens from several cells and multicellular organisms to fringes with respect to the flat background (after empty beam tissues up to the organ level of small animals can be covered in division and raw data corrections), recorded with a test one or a few exposures, eventually enlarged by stitching. Using pattern milled by focused ion beam into a 200 nm-thick gold zoom magnification by defocus variation, the magnification layer on 200 nm-thick Si3N4 over a total accumulation time of and the FOV can easily be adapted and combined. At the 30 s at 7.9 keV photon energy, waveguide-to-object distance same time the optical scheme is ideally suited for high-flux z1 = 17.55 mm, and detector distance z1 z2 = 5.13 m. Inter- ptychographic phasing and scanning SAXS/WAXS applica- ference fringes extend all the way toþ the corners of the tions, and a fast switch in imaging modality on the same diffraction pattern indicating a high-quality hologram. The sample is supported by the optical design. The advanced phase reconstructed from this hologram is shown in Fig. 6(c), detection systems already tested for the KB nano-probe based on the algorithms presented by Giewekemeyer et al. (Giewekemeyer et al., 2014; Wilke et al., 2014), but also the (2011) and Bartels et al. (2015), which enforces compact object Eiger detector (Guizar-Sicairos et al., 2014), can cope with the support as well as measured intensity values. Importantly, the high flux density. support information required for this algorithm is retrieved As a future direction, the described optics and imaging from a deterministic holographic reconstruction and thus does scheme for all imaging modalities is compatible with pink not need prior information such as in CDI. The raw data of a beam operations, which would increase flux by one to two holographic recording is illustrated in Fig. 6(b) for bacteria, orders of magnitude. For scanning nano-diffraction in SAXS before empty beam correction. Waveguide-enhanced holo- or even WAXS mode as long as weakly ordered systems are graphic imaging can readily be extended to tomography, as considered, the intrinsic undulator bandpath of Á�=� 0.006 demonstrated for Deinococcus radiodurans cells by Bartels would not be prohibitively large, and hence even’ weakly et al. (2012). A corresponding rendering of the three-dimen- scattering samples such as the hydrated cell in Fig. 4 could be sional electron density distribution is shown in Fig. 6(d). The acquired with dwell times of 10 ms. Scanning with contin- quantitative two-dimensional and three-dimensional phase uous motor movement and� pixel detector technology for retrieval reveals dense structures which may be associated frame rates of 100 Hz is already available, warranting a with DNA rich bacterial nucleoids. straightforward implementation� of such a fast nano-diffraction mode. Concerning radiation damage, further studies should investigate whether a ‘diffract-and-destroy’ strategy can be 6. Conclusion and outlook adopted based on increased scanning speed to outrun diffu- A unique aspect of the compound mirror–waveguide system sion-limited reactions of free radicals. For ptychography, pink discussed here is the delivery of quasi-spherical wavefronts beam operation could bring about higher robustness and emanating from a two-dimensional X-ray waveguide exit reconstruction quality, by avoiding any vibrations associated (Salditt et al., 2008; Giewekemeyer et al., 2010; Kru¨ger et al., with monochromator cooling. For small spot sizes, i.e. in the

J. Synchrotron Rad. (2015). 22, 867–878 Tim Salditt et al. � Coherent imaging and nano-diffraction 875 WG Holography feature articles 2010), with controlled mode structure spot sizes down to sub- 10 nm. The mode and coherence filtered waves are ideally suited for quantitative holographic image recording. Mode filtering minimizes wavefront distortions and artifacts encountered in many other hard X-ray focusing optics. This enables quantitative reconstruction of the object by robust phase-retrieval algorithms. By selecting the waveguide-to- sample distance, objects can be imaged at a single distance in a full-field configuration without scanning. Robust and quickly converging iterative reconstruction schemes can be applied to invert the holographic near-field diffraction patterns. Weakly scattering biological specimen can thus be phased even withoutAdvantages exact knowledge of WG-Holography of the illumination function. The method proved to be very dose-efficient providing images of cells►► atfull-field doses below 10method,5 Gy, and takesrobust photon phase-retrieval noise effects into account quantitatively (Giewekemeyer et al., 2011). The Figure 6 tomographic►► coherence extension filtering, provides quantitativeKB artefacts three-dimen- filtering Holographic imaging with the waveguide probe. Scale bar: 585 mm (detector plane). (a) Hologram of a test pattern after division by the sional density reconstructions of biological cells and tissues empty beam. Scale bar: 6 mm (detector plane). (b) Raw data of a (Bartels►► geometricalet al., 2012; Olendrowitz magnificationet al., 2012). To probe the hologram with three Deinococcus radiodurans cells, showing the smooth three-dimensional structure of a larger specimen, the full-field line-shape and tails of the waveguide probe. (c) Phase reconstruction of ►► lower dose than ptychography the hologram shown in (a), with 22.4 nm pixel size. Scale bar: 2 mm. (d) holographic technique is of particular advantage, since it Rendered three-dimensional density distribution of a Deinococcus avoids the problem of overheads in detector readout which is radiodurans bacteria (Bartels et al., 2012), reconstructed from wave- pertinent►► compatible in scanning threewith degrees tomography of freedom (two transla- guide-based holographic tomography. tions, oneSalditt, tomographic Osterhoff rotation). et al, Journal In contrast of Synchrotron to scanning Radiation, 2015 SAXS or diffraction microscopy (ptychography), extended empty beam corrected hologram in (a) shows the holographic specimens from several cells and multicellular organisms to fringes with respect to the flat background (after empty beam tissues up to the organ level of small animals can be covered in division and raw data corrections), recorded with a test one or a few exposures, eventually enlarged by stitching. Using pattern milled by focused ion beam into a 200 nm-thick gold zoom magnification by defocus variation, the magnification layer on 200 nm-thick Si3N4 over a total accumulation time of and the FOV can easily be adapted and combined. At the 30 s at 7.9 keV photon energy, waveguide-to-object distance same time the optical scheme is ideally suited for high-flux z1 = 17.55 mm, and detector distance z1 z2 = 5.13 m. Inter- ptychographic phasing and scanning SAXS/WAXS applica- ference fringes extend all the way toþ the corners of the tions, and a fast switch in imaging modality on the same diffraction pattern indicating a high-quality hologram. The sample is supported by the optical design. The advanced phase reconstructed from this hologram is shown in Fig. 6(c), detection systems already tested for the KB nano-probe based on the algorithms presented by Giewekemeyer et al. (Giewekemeyer et al., 2014; Wilke et al., 2014), but also the (2011) and Bartels et al. (2015), which enforces compact object Eiger detector (Guizar-Sicairos et al., 2014), can cope with the support as well as measured intensity values. Importantly, the high flux density. support information required for this algorithm is retrieved As a future direction, the described optics and imaging from a deterministic holographic reconstruction and thus does scheme for all imaging modalities is compatible with pink not need prior information such as in CDI. The raw data of a beam operations, which would increase flux by one to two holographic recording is illustrated in Fig. 6(b) for bacteria, orders of magnitude. For scanning nano-diffraction in SAXS before empty beam correction. Waveguide-enhanced holo- or even WAXS mode as long as weakly ordered systems are graphic imaging can readily be extended to tomography, as considered, the intrinsic undulator bandpath of Á�=� 0.006 demonstrated for Deinococcus radiodurans cells by Bartels would not be prohibitively large, and hence even’ weakly et al. (2012). A corresponding rendering of the three-dimen- scattering samples such as the hydrated cell in Fig. 4 could be sional electron density distribution is shown in Fig. 6(d). The acquired with dwell times of 10 ms. Scanning with contin- quantitative two-dimensional and three-dimensional phase uous motor movement and� pixel detector technology for retrieval reveals dense structures which may be associated frame rates of 100 Hz is already available, warranting a with DNA rich bacterial nucleoids. straightforward implementation� of such a fast nano-diffraction mode. Concerning radiation damage, further studies should investigate whether a ‘diffract-and-destroy’ strategy can be 6. Conclusion and outlook adopted based on increased scanning speed to outrun diffu- A unique aspect of the compound mirror–waveguide system sion-limited reactions of free radicals. For ptychography, pink discussed here is the delivery of quasi-spherical wavefronts beam operation could bring about higher robustness and emanating from a two-dimensional X-ray waveguide exit reconstruction quality, by avoiding any vibrations associated (Salditt et al., 2008; Giewekemeyer et al., 2010; Kru¨ger et al., with monochromator cooling. For small spot sizes, i.e. in the

J. Synchrotron Rad. (2015). 22, 867–878 Tim Salditt et al. � Coherent imaging and nano-diffraction 875 WG Holo-Tomography ►► Reconstructed asthmatic lung tissue (mouse) ►► Sub-cellular resolution @ organic field of view ►► Mass density in 3D!

Krenkel, Salditt et al, Scientific Reports, 2015 Discussion Outline ►► Introduction + History ►► Reflective Optics ►► Diffractive Optics ►► Refractive Optics ►► Waveguides ►► Discussion ?!? G.J. Price et al. / Nuclear Instruments and Methods in Physics Research A 490 (2002) 276–289 277

2. Microchannel plate Wolter optics

2.1. Conical approximations Fig. 2. Prototype radially packed MCP optic [28]. In X-ray astronomy applications where high throughput is required with moderate spatial the optic may be thermally slumped to a spherical resolution, the hyperboloid and paraboloid of ﬁgure of a particular radius. the ideal Wolter system [13,14] may be replaced Willingale et al. [8] suggested that such radially with simple cones. This conical approximation is packed optics may be used to form a conic usually implemented with lightweight, closely approximation to the Wolter Type I optical nested metal foils [15,16]. There are many treat- system. These authors noted that the radial- ments of the ideal Wolter geometries [17,18] and packing geometry, although lacking the wide ﬁeld their conical approximations [19,20] in the litera- of view of the lobster-eye, had several compensat- ture. G.J. Price et al. / Nuclear Instruments and Methods in Physics Research A 490 (2002) 276–289 277

ing advantages, viz: a radially symmetric point spread function; larger effective/geometric area ratios and smaller grazing angles, allowing operation at higher photon energies than conventional Wolter optics. Owens et al. [9] have proposed the use of an MCP Wolter telescope, with a formidable mass-to- geometric area ratio, for the HERMES X-ray fluorescence imager. This instrument will map the elemental composition of the planet Mercury from the Planetary Orbiter of the European Space Agency (ESA) Bepi Colombo mission. This telescope, with a diameter of 210 mm and a focal length of 1 m has an effective area of 100 cm2 at (a) (b) energies up to 2 keV; comparable with that of the Fig. 1. (a) Packing arrangement around a solid core for a Rosat X-ray telescope ð100 cm2 at 2 keV; 82:5 cm radially packed MCP composed of square multifibres. (b) diameter, 2:4 m focal length [10]). Cross-section through Wolter pair. In this paper, we first describe the focusing mechanism in radially packed MCP structures. X- LobsterG.J. Eye, Price et al. /aka Nuclear Instruments Wolter and Methods Optic in Physics Research aka A 490 (2002)Micro 276–289 Channel277 Plate ray results from both a single planar radially packed optic and a dual-MCP Wolter optic are ing advantages, viz: a radially symmetric point then presented. Both optics were produced as part spread function; larger effective/geometric area of a ESA Technology Research Programme (TRP) ratios and smaller grazing angles, allowing opera- for the development of microchannel plate tech- tion at higher photon energies than conventional nology. Both had previously been tested using Wolter optics. pencil synchrotron X-ray beams by Beijersbergen Owens et al. [9] have proposed the use of an et al. [11,12] but the present work describes the MCP Wolter telescope, with a formidable mass-to- first full aperture illumination of such optics. A geometric area ratio, for the HERMES X-ray Monte Carlo raytrace code is used to model the X- fluorescence imager. This instrument will map the ray response of the optics and identify causes of elemental composition of the planet Mercury from imperfections in the focused images. the Planetary Orbiter of the European Space Agency (ESA) Bepi Colombo mission. This telescope, with a diameter of 210 mm and a focal Fig. 2. Optical micrograph of our MCP. Fig. 4. Transmitted intensity spectrum through 60-␮m Al filter. length of 1 m has an effective area of 100 cm2 at The spectrum was taken to be monoenergetic at an energy of 1.5 2. Microchannel plate Wolter optics (a) (b) energies up to 2 keV; comparable with that of the keV in our analysis. Fig. 1. (a) Packing arrangement around a solid core for a Rosat X-ray telescope ð100 cm2 at 2 keV; 82:5 cm 2.1. Conical approximations radially packed MCP composed of square multifibres. (b) diameter, 2:4 m focal length [10]). was ϳ150 ␮m in diameter. This provides a flux den- Fig. 2. Prototype radially packed MCP optic [28]. 13 4. Results Cross-section through Wolter pair. In this paper, we first describe the focusingsity from the laser on the target surface of order 10 W͞cmIn2 X-rayand produces astronomy a plasma applications that is where an intense high Figure 5 shows the x-ray photograph of the focal mechanism in radially packed MCP structures. X- sourcethroughput of x rays is in required the energy with range moderate 1–1.6 keV spatial with a plane. The checkerboard pattern that is due to ray results from both a single planar radially 14 the optic may be thermally slumped to a spherical conversionresolution, efficiency the hyperboloid as high as 60%. and paraboloid of shading by channel walls in the MCP can be clearly packed optic and a dual-MCP Wolter optic are Images were recorded on Kodak DEF x-ray film. seen. The expected focal distribution is also ob- figure of a particular radius. the ideal Wolter system [13,14] may be replaced If it looks like a Zonethen presented. Plate, Both optics were produced as partThe film was developed and fixed by the procedure served. While there is some flaring along the focal Willingale et al. [8] suggested that such radially with simple cones. This15 conical approximation is of a ESA Technology Research Programme (TRP)given by Henke et al. To measure intensity, a arms, the effect is relatively small. This indicates packed optics may be used to form a conic usually implemented with lightweight, closely that the MCP is extremely well produced. Some for the development of microchannel plate tech-Joyce–Loebls densitometer with matched NA ϭ 0.1 it might be a lobsterapproximation eye to the:) Wolter Type I optical opticsnested was metal used foils to record [15,16]. the There optical are density many of treat- the channel irregularities can also be observed. nology. Both had previously been tested using system. These authors noted that the radial- film.ments The of results the ideal given Wolter by Henke geometrieset al.15 [17,18]were then and Figure 6 shows a one-dimensional intensity trace pencil synchrotron X-ray beams by Beijersbergen packingWolter, geometry, Grundlagen although lacking der theOptik, wide 1956 field usedFig.their 6.to conical calculate One-dimensional approximations the intensity. scan, along [19,20] Aluminium a focal in arm, the foil of litera- intensity was gain.taken alongFig. 7. one Simulated of the focal intensity arms. along The a full focal width arm at overlaid onto the et al. [11,12] but the present work describes theused as a filter to protect the x-ray film from visible half-maximum ͑FWHM͒ of the central focus is 230 of view of the lobster-eye, had several compensat- ture. data in Fig. 6. The agreement is excellent. first fullPrice, aperture Tomaselli illumination et al, NIM of such A, 2002 optics. Alight.Peele, Siegmund et al, Applied Optics, 1996 ␮m compared with 200 ␮m expected for a geometri- Monte Carlo raytrace code is used to model the X- Figure 4 shows the spectrum measured from the cally perfect MCP. If we define the angular resolu- sity of the focused flux to the intensity of the unfo- ray response of the optics and identify causes ofplasma with a flat rubidium hydrogen phthalate crys- tion to be the central focus FWHM divided by the talcused with flux 2d spacing, as the intensity 26.121 Å, gain. and which The has intensity been gainimage distance, the angular resolution is therefore imperfections in the focused images. nonsquareness ͑or twist͒ parameter described below modifiedfigure thus to take calculated into account can the be effect compared of the with 60-␮m the the-very closemimics to that to for some a perfect extent MCP the that effect is of fixed diffuse by scattering, Aloretical filter. It intensity is clear that gain more that than would 90% be of the obtained radi- withthe channeland as size such and the object value distance derived alone. for this We parameter ob- may ationrealistic reaching reflectivity the film is estimates between 1.36 for a and geometrically 1.56 keV, per-serve a resolution of 0.96 mrad, which compares with the absorption edge of Al. In analyzing the data, we 0.84 mradbe for overstated. a perfect MCP. This is the best reso- fect and perfectly smooth MCP. We compare both Rotation of a channel about the long axis of the 2. Microchannel plate Wolter optics made the simplifying assumption that the x-ray en- lution yet reported for an MCP. ergypeak was intensity monochromatic gain and with average an energy intensity of 1.5 keV. gain overA secondchannel prime by indicator a certain of the amount quality has of thean MCP effect of rotating the FWHM of one focal arm. Peak intensity gainis is the efficiencythe flux with distribution which the radiation from that is brought channel to by an equal 2.1. Conical approximations the peak gain recorded in a collection area equalthe to focus.amount. A section Consequently of the film was the exposed focal directly square will tend to Fig. 2. Prototype radially packed MCP optic [28]. the slit size of the densitometer used to record theto the source,become enabling circular us to and estimate the focal the intensity arms will of flare out. In X-ray astronomy applications where high optical density—in our case, 100 ␮m. The averagethe unfocusedThe effect flux. in We measured define the intensity ratio of the is therefore inten- greatest throughput is required with moderate spatial intensity gain over the FWHM of one focal arm is the in the focal arms, causing intensity to drop off dra- the optic may be thermally slumped to a spherical resolution, the hyperboloid and paraboloid of average gain for a one-dimensional scan along one of matically along the arms. In the focal square this figure of a particular radius. the ideal Wolter system [13,14] may be replaced the focal arms with the same collector size as that effect is of less consequence. We characterize rota- Willingale et al. [8] suggested that such radially with simple cones. This conical approximation is used in the peak measurement. tions as normally distributed with a rms rotation packed optics may be used to form a conic usually implemented with lightweight, closely We measure the peak intensity gain in the central equal to ␾. For the purpose of this model we take no approximation to the Wolter Type I optical nested metal foils [15,16]. There are many treat- focus to be 27 Ϯ 4, while the average intensity gain account of the fact that, on observation, channel ro- system. These authors noted that the radial- ments of the ideal Wolter geometries [17,18] and over the FWHM of one focal arm of the focal region is tations appear to be correlated with fiber bundles packing geometry, although lacking the wide field their conical approximations [19,20] in the litera- 21 Ϯ 4. The expected peak intensity gain was cal- within the MCP. of view of the lobster-eye, had several compensat- ture. culated by a Monte Carlo ray-tracing simulation, for Channel tilt about either of the two axes that are a perfect MCP for the experimental parameters, and perpendicular to the long axis of the channel causes was found to be 83. The average intensity gain over the reflected ray to deviate by twice the tilt. This the FWHM of one focal arm for a perfect MCP was 62. Fig. 3. Schematic of the experimental layout. The laser pulse is will appear as a blur in the image. The central focus focused onto the copper target that produces a plasma. The x-ray Fig. 5. Photographwill be particularly of focal distribution affected, produced as by there the MCP are two reflec- emission5. Analysis is then focused by the MCP onto the x-ray film. optic. tions for centrally focused rays. We characterize Obvious departures from the expected intensity dis- tilts along the optic axis as normally distributed with 4422tribution APPLIED for OPTICS a perfect͞ Vol. MCP 35, No. are 22 ͞ reduction1 August 1996 and broad- rms tilt equal to ␰. ening of the central peak and the diminution in If a channel is nonsquare, then the second reflec- intensity along the focal arms. These departures tion will not be orthogonal to the first. Hence the can be explained by a simple model of surface rough- central focus will be broadened and the focal arms ness, channel rotation, channel twisting, and will flare out. Channel twisting will cause the same nonsquareness and channel tilt. effect. This effect is to be distinguished from chan- Surface roughness reduces the intensity in the nel rotation that causes flare and only minor central reflected beam and hence in the focus. Flux is also focus broadening and from channel tilt that causes scattered away from the direction of the specular broadening in both the central and arm foci. We beam because of surface roughness.16 We use the were led to the inclusion of this parameter by an Debye–Waller factor17,18 with ␴ equal to the rms inability to fit the observed intensity distribution roughness to calculate the reduction in intensity in with the other parameters alone ͑although note the the specularly reflected beam. For the purpose of previous comments regarding diffuse scattering͒. this model we take no account of diffusely scattered x We characterize the combined effect of the twist and rays, simply treating such rays as lost. If diffuse nonsquareness by a normally distributed distribution scattering is significant, it would be expected to be in deflection angle with rms equal to ␩. most noticeable as a low-intensity halo about the cen- These parameters were incorporated into a ray- tral peak. In Fig. 7 a small amount of residual tracing simulation that takes account of the real re- broadening at the base of the central peak can in fact flectivity on reflection. The experimentally fixed be observed. It should also be kept in mind that the variables described in Section 3 were included, and

1 August 1996 ͞ Vol. 35, No. 22 ͞ APPLIED OPTICS 4423 Summary and Wrap-Up Mirrors ideal mirror real mirror Outline (a)

l a -500 1.0 l a -500 t t e r ►► totel reflection ≤ 17 keV e r ►► Introduction + History

al -250 al -250 a a ►► multilayers ≥ 15 keV xis, 0 0.5 xis, 0 ►► Reflective Optics z z in nm 250 in nm 250 0.0 ► Diffractive Optics Zone Plates 500 500 ► -1.0 0.0 1.0 -1.0 0.0 1.0 ► Fresnel Zone Plates x in mm ►► Refractivex in mm Optics ► (b) Phase divided by plane wave

for soft x-rays l a -500 π l a -500 t t e r e r ►► Waveguides al -250 al -250 a a

►► Multilayer Laue Lenses, xis, xis, 0 0 0 ►► Discussion z z Multilayer Zone Plates in nm 250 in nm 250 -π feature articles for hard x-rays 500 500 -1.0 0.0 1.0 -1.0 0.0 1.0 x in mm x in mm ►► factor of 20 in energy 2010), with controlled mode structure spot sizes down to sub- (c) Focus cut = 20 years of work 10 nm. The mode and coherence ﬁltered waves are ideally 1.0 +π 1.0 +π suited for quantitative holographic image recording. Mode Lenses Waveguides ﬁltering minimizes wavefront distortions and artifacts encountered in many other hard X-ray focusing optics. This ►► possible, with 100+ lenses ►0.5► as coherence filter0 phase in for holography0.5 0 phase in enables quantitative reconstruction of the object by robust i n i n

t t phase-retrieval algorithms. By selecting the waveguide-to- ensity ensity

J. Synchrotron Rad. (2015). 22, 867–878 Tim Salditt et al. � Coherent imaging and nano-diffraction 875