1. Inroduction

Focusing of high-energy X-rays to small beam sizes is not as developed as for the case of lower energies, where nanofocusing is becoming routine. As the energy increases from conventional hard X-rays (7–30 keV) to high-energy X-rays (50–100 keV), optics for efficient control of the radiation (e.g., monochromatization, analyzers, focusing) become more challenging. In focusing, shorter wavelengths $\lambda$ lead to the requirement of increasingly finer features in fabricated structures, typically in the direction transverse to beam propagation. At the same time, the weakening of the interaction with matter, i.e., the refractive strength decreasing as $\lambda ^{2}$ , lengthens longitudinal dimensions. So depending on the device type, this leads to encountering problematic issues like smaller grazing angles, longer devices, or longitudinally thicker structures, which in conjunction with finer lateral features, present more difficult longitudinal-to-transverse fabrication aspect ratios. Focusing below 1 $\mu$m at high energies ($> 50$ keV) is still non-trivial, having only a few reports, such as with Kirkpatrick-Baez mirrors imaging a small aperture (pseudo-source) on a very long beamline [1], etched Si compound refractive lenses [2], and kinoforms [3].

Closing the focusing capability gap for high-energy X-rays is necessary to exploit their penetration property and availability from high-brilliance synchrotron radiation sources for application to bulk microstructural diffraction studies at much-needed spatial resolutions. Submicron resolutions will enable the imaging of fine, micron-grain-sized polycrystalline materials (under load), to provide better insight into deformation micromechanics: grain boundary phenomena, onset of cracks/failure, and intra-granular (as well as inter-granular) strain distributions.

Saw-tooth refractive lenses (SRLs) are well-suited for high-energy X-rays and have been in routine use for line (1D) and point (2D) focusing at the Advanced Photon Source (APS) 1-ID high-energy X-ray undulator beamline, operating in the 40–140 keV range. These lenses are in-line, tunable in energy (or focal length), effectively parabolic (i.e., having that desired profile in thickness projected along the beam), and have zero attenuation on-axis [4,5]. In 1D focusing applications, very large spatial acceptances of many millimeters are available perpendicular to the focusing direction, enabling long line foci when needed. The practical methodology of implementing SRLs on a beamline is detailed extensively in Ref. [6], with regard to their properties, control, alignment, and focal diagnostics. The current article delves into the achieving of submicron vertical focusing of 60 keV X-rays with Si SRLs, with relevant discussion of aberration contributions, fabrication, and mounting. Aberration correction is addressed for further progress and impact. The next section proceeds to offer insight into SRLs, along with comparisons to other focusing optics, especially the well-known compound refractive lenses (CRLs).

2. SRL features and comparison to other optics

SRLs operate on the principle that a linear, triangular saw-tooth structure, in an overall grazing incidence setting with respect to an X-ray beam, can present a very well-approximated parabolic thickness profile with zero attenuation on-axis [4,5]. A full symmetric parabolic profile requires placement of two such saw-tooth structures face-to-face, canted symmetrically about the optical axis (Fig. 1). Alternatively, individual teeth can be viewed as refractive prisms imparting equal angular deflections to the X-rays. A ray receives a number of prism deflections that is proportional to its distance $y$ off-axis, resulting in all rays being directed to a single focus, whose focal distance depends on the refractive index $1-\delta$, saw-tooth profile geometry, and the grazing angle $\alpha$ of the array. The focal length of an SRL is easily tuned by symmetric adjustment of the two pieces’ taper angles, which alters the vertex curvature-radius $R$ of the effective parabola, through the relation $R=v\sin \alpha$, where $v$ is the tooth height. The focal length is then given by the expression for the single plano-concave refractive lens $f=R/\delta$, giving

 

Fig. 1. X-ray wavefronts are focused by passage through a saw-tooth refractive lens pair, whose material path length profile is parabolic, with sampling by continuous line-segments in uniform intervals $\Delta y = h \sin \alpha \ll v$ over $[-v,v]$.

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Depending on the period $h$ and the taper angle $\alpha$ with respect to the beam, the effective parabola is approximated in a very fine, piecewise-linear, but continuous fashion (with discontinuities in the derivative, Fig. 1), assuming the teeth narrow down to ideal edges. The fineness of this sampling interval

Tunability, in this discussion, is the ability to vary independently the energy and focal length. This enables preserving working (focal) distance or demagnification, while changing energy, which is important for a beamline that offers continuous energy tunability over a wide range. Offering a suite of complementary techniques using line-focused, point-focused, and unfocused beams delivered to the same sample/instrument location makes in-line focusing optics critical, aside from their stability benefits. As mentioned, SRLs satisfy the simultaneous requirements of tunable and in-line operation, whereas Kirkpatrick-Baez optics, Fresnel-zone-based optics, and kinoforms do not.

So, to compare with SRLs, one is left with CRLs, which achieve tunable focusing in an in-line configuration by the cumulative refractive action of many concave lens elements in a linear array, through their $1/f$-values being additive. By varying the number of elements, tunability is attained in a discrete way, which may be viewed as effectively continuous only if the resulting increments in focal distance are within the depth of focus. For high energies, given that the number of elements increases quadratically with energy, actuator-based mechanical systems that facilitate varying their number [9–11] can be bulky, especially if both 1D and 2D focusing capabilities are desired, as they are implemented by different types of elements. Etched arrays of 1D focusing CRLs on Si wafers [2] can be compact. However, their small (sub-millimeter) etch depths (in the direction perpendicular to focusing) preclude providing long (few millimeters) line foci, as needed for certain techniques, such as near-field high-energy diffraction microscopy [12].

Figure 2 compares the acceptance apertures of SRLs and CRLs by plotting their transmission profiles as a function of departure $y_{in}$ from the axis. Being parabolic devices, their transmissions are gaussian. All devices correspond to $f=1.3$ m for 60 keV X-rays. The CRL parameters assumed are based on commercial products of stacked, biconcave Al elements, individually available down to $R=100\,\mu$m and $R=50\,\mu$m for 1D and 2D focusing capability, respectively, requiring stacks of $N=260$ and $N=130$ elements. A residual on-axis wall thickness of $25\,\mu$m in each element results in the CRL stacks having significantly less than 100% transmission on-axis. Although each element’s parabolic form is fabricated to extend radially out to a few hundred microns, the gaussian aperture of the stack makes the transmission small beyond $y_{in} > 100\,\mu$m. In contrast, the SRL profiles start on-axis at 100% transmission, giving them higher overall efficiency. However, their transmission profiles are truncated at $y_{in} =100\,\mu$m, which arises from the tooth height $v = 100\,\mu$m chosen for the SRLs in this comparison, beyond which the parabolic profile ceases. So the maximum spatial acceptance of a single SRL lens piece is the tooth height $v$, provided the piece is long enough to accommodate the beam footprint $v/\sin \alpha$ at the operating grazing incidence; otherwise it is limited to the length $\times \sin \alpha$. For the two-piece SRL, the acceptance is doubled, the maximum possible being $2v$. Despite this hard limit to the aperture, the throughput of the SRL is very marginally affected. In the overall comparison with the CRL, the on-axis transmission factor is more significant.

 

Fig. 2. Comparison of transmission profiles (over half-aperture) of saw-tooth refractive lenses (SRL) and compound refractive lenses (CRL).

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The solid-line profile in Fig. 2 corresponds to the parameters of the Si SRLs tested here ($h=141\,\mu$m, $v=100\,\mu$m, and 90 mm device lengths), requiring operation at grazing angle $\alpha =0.1^{\circ }$ for $f=1.3$ m at 60 keV. At this inclination angle, realizing the full tooth height aperture requires a lens piece of minimum length $v/\sin \alpha =57$ mm. This is exactly twice the 28.5 mm length of the equivalent plano-concave parabolic lens of 100 $\mu$m half-aperture and vertex curvature $R=0.175\,\mu$m, since the most off-axis ray going through an SRL traverses equal amounts of material and empty space. Attempting to increase the spatial aperture of an SRL by increasing $v$ might not only yield gains of just marginal value due to attenuation, but would be accompanied by a quadratic increase in device length.

One might question why Eq. (1) has no dependence on the tooth spacing $h$ and/or the tooth-tip angle (which, with refractive index, governs the single-prism deflection angle). This is clarified by noting that the equation assumes, as done here, a saw-tooth profile that descends all the way to sharp valleys (i.e, corners, and not flat bottoms—in which case $v$ for Eq. (1) would be taken as the depth to the extrapolated sharp valleys). One then cannot vary separately the parameters $h$, $v$, and the tooth-tip angle. Fixing $v$ establishes a mutually constrained relationship between $h$ and the tooth-tip angle. This remaining freedom can be viewed as a longitudinal stretching of the saw-tooth pattern with fixed $v$, which leaves $f$ invariant, because the sparsening of the teeth by increasing $h$ is compensated by stronger refraction through more obtuse tooth-tips. However, the choice of $h$ does affect the discreteness aberration (Eq. (2)).

3. Focal size contributions/geometrical aberrations

To interpret results presented later, this section discusses the focal spot size contributions for the specific Si SRL case tested ($h=141\,\mu$m, $v=100\,\mu$m, $\alpha =0.1^{\circ }$ for $f=1.3$ m at 60 keV). For Si at 60 keV, $\delta =1.34\times 10^{-7}$, giving a $0.19\,\mu$rad prism deflection from each tooth. Like all refractive optics, SRL focusing has contributions from source size, diffraction limit, and chromatic aberration, here having FWHM values 489 nm, 254 nm, and 95 nm, respectively (Table 1). With the focus in the APS 1-ID-E experimental station at 70 m from the source, a distance ratio of (1.3 m)/(68.7 m) demagnifies the vertical undulator source size $2.35\,\sigma _{y}=26\,\mu$m. Focusing vertically is a more stringent test of lens quality, since the horizontal source size $2.35\,\sigma _{x}=635\,\mu$m is much larger. The diffraction limit is obtained from the standard fourier transform calculation on the one-sided, truncated $0 < y_{in} < 100\,\mu$m solid-line transmission profile in Fig. 2, since only a single, upright Si SRL piece was used in the vertical focusing test. The chromatic aberration is based also on this transmission profile and on the $9.2\times 10^{-4}$ relative energy bandwidth delivered at 60 keV by liquid-nitrogen-cooled, fixed-vertical-offset, bent double-Laue monochromator [13] that preserves source size and divergence to a high degree [14].

Table 1. Expected FWHM vertical focus size contributions and convolved totals for the single-piece (tested) and two-piece SRL cases.

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For efficiency and economy in conducting comparative performance assessment of lenses from various SRL fabrication processes and parameters, only an upright lens piece is tested. There is no need to test a combined upright/inverted SRL pair, as that entails the uninformative additional steps of optimizing the focus from a second piece (in addition to fabricating a duplicate prototype) and steering the foci from the two pieces to coincide, a straightforward alignment procedure described in Ref. [6]. The SRL pair, however, does have a smaller focusing diffraction limit contribution (139 nm for this case) than the single piece, due to its doubled aperture, leading to marginally smaller final spot sizes. All the other focal size contributions, i.e., aside from the diffraction limit, are the same for the full-aperture (symmetric pair) versus the half-aperture (single-asymmetric) lens arrangement.

Turning attention to two geometrical aberrations unique to SRLs, Fig. 3(a) depicts their causes. The contribution to the geometrical optics point-spread-function from the refractive discreteness of the teeth has already been mentioned (discreteness aberration ). Here, the incident rays can be partitioned into $\Delta y$-intervals (Eq. (2)), where the rays within a given interval (Fig. 3(a) upper sketch) experience the same number and schedule of prism deflections. Hence, they propagate though the lens and reach at the focal plane with an aberration due to preservation of the intra-interval spread $\Delta y$. Neighboring intervals, however, reach the focal plane almost exactly superimposed due to the steering compensation of incremental teeth. But there is a gradually increasing departure from this inter-interval coalescence in focusing as one moves off-axis, manifesting the length aberration. This arises from the SRL not being a zero-length device, but one whose refractive deflections from the teeth are distributed longitudinally over a distance that might not be negligible compared to the focal distance. Equation (1) is based on the assumption that a ray receives its refractions all-at-once, in unison with all rays, and is strictly valid only for the near-axis teeth (i.e., paraxial rays). In reality, rays more off-axis are subjected to premature deflections, resulting in them being over-steered in focusing, as indicated in the red trajectory in the lower sketch in Fig. 3(a). This aberration has also been raised in a different manner, by indicating the non-adherence to Fermats principle of invariant optical path length in SRL focusing trajectories [15].

 

Fig. 3. (a) Origins of two geometrical aberrations in SRLs and their effects shown in (b) ray-trace simulations tracking entrance ray position $y_{in}$ to focal plane destination $y_{foc}$. Curves A, B are for uncorrected Si lenses, as tested here. Curve C is for a lens corrected for the length aberration, as proposed in Sec. 6.

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Ray-trace calculations are presented in Fig. 3(b), mapping the off-axis coordinate $y_{in}$ at entrance to the arrival $y_{foc}$ at the destination focal plane positioned at distance $f_{0}$ from the end of a single-piece SRL. Parallel incident rays are assumed. Curve A corresponds to the investigated plane at the nominal value $f_{0} =1.3000$ m prescribed by the SRL parameters through Eq. (1). It appears as a thick band, due to a triangular oscillatory fine-structure detailed in the inset, indicating the discreteness aberration of 250 nm. The curvature of the band, which begins flat on-axis, reveals the length aberration. To assess the extent of the length aberration, one needs to consider only up to a certain range on the abscissa, since the ray-trace does not reflect any suppressed weighting of off-axis rays. Based on the Si SRL’s attenuation profile in Fig. 2 (solid-line) extending out to $60\,\mu$m (at half-maximum), the length aberration in this simulation can be estimated as 800 nm (examining the $0<y_{in}<60\,\mu$m region in curve A, where the range is $-800$ nm$\,< y_{foc} < 0$). One can also arrive at this estimate by simple geometry. The nominally peripheral ray entering at $y_{in}=60\,\mu$m interacts with a length of the SRL spread over $60\,\mu$m$/\sin \alpha = 34$ mm from the on-axis end, along which it receives an cumulative deflection of $60\,\mu$m$/1.3$ m$\,=46\,\mu$rad for focusing. This uniformly distributed deflective action produces a ballistic trajectory, equivalent to an abrupt $46\,\mu$rad deflection imparted halfway, i.e., at 17 mm from the end. Subjected to this early steering, the ray crosses the axis 17 mm before the nominal focal point at 1.3 m, landing meridionally off-focus by (17 mm)$\,\sin (46\,\mu \mbox {rad})=780$ nm, in agreement with the ray-trace simulation.

The $\sim 800$ nm length aberration, combined with the other four contributions (refractive discreteness, source size, diffraction limit, and chromatic), would seem to preclude attainment of a submicron focus in the considered configuration. However, the length aberration analysis so far, displayed in curve A of Fig. 3(b), assumes a candidate focal plane at the nominal $f_{0}=1.3000$ m, appropriate for paraxial focusing. One might ask if the length aberration can be mitigated by investigating compromise distances $f_{0}$ that are slightly closer, where the over-steered rays are crossing the axis. The ray-trace simulations show that this is indeed the case, with an optimal result at $f_{0}=1.2825$ m (curve B). This distance flattens the overall curvature of the band over $0<y_{in}<60\,\mu$m, reducing the length aberration to about $200$ nm, evidenced by the range $0<y_{foc}<200$ nm. This is more favorable, resulting in all five contributions convolving to a combined $640$ nm FWHM focal size for a single-piece SRL (Table 1). For an upright/inverted pair, one calculates 610 nm, due to the improved diffraction limit. The values tabulated for the pair-case are also valid for ideally fabricated CRLs, except that the two geometrical point-spread-function aberrations do not apply, resulting in a combined 520 nm.

4. Fabrication

The strongly crystallographically preferential process of anisotropic etching by KOH in Si is exploited to create the sharp V-groove structures. The process is similar to that used in the first demonstration of Si SRLs [16]. Flat Si(100) wafers are coated with Si$_{3}$N$_{4}$ and resist. The top resist layer is patterned optically through a Cr-on-glass mask, designed in the form of gratings of rectangular strips, defining the desired placement of alternate SRL V-grooves. This allows the Si$_{3}$N$_{4}$ to receive the same patterning by reactive ion etching (RIE), leaving uncoated Si regions at alternate groove locations. KOH-etching produces the grooves with Si(111) sloping walls, which result properly due to the original resist patterning having been imprinted with the appropriate azimuthal orientation with respect to the crystallographic directions. Thermal oxide coating is used to protect this first set of grooves, after which the remaining Si$_{3}$N$_{4}$ is removed, followed by another KOH-etching to create the second set of interstitial grooves. The oxide coating the first set of grooves is then removed.

For optimal focusing, the flatness of the SRL profile is critical. To ensure the coplanarity of the tooth tips in the final devices, various precautions were taken. First, 1 mm and 2 mm wafers were used for stiffness, in addition to more standard 0.6 mm wafers for comparison. Second, the 1 mm and 2 mm wafers were specially polished (being enabled by their thickness) on their processing side to an ultra-flat specification of $<500$ nm peak-to-valley over 95% of their 150 mm diameter. Figure 4 confirms the flatness of a 2 mm thick wafer to within 320 nm. Use of a 350-nm low-stress nitride coating avoided stress-induced plastic deformation of the Si during the LPCVD process by migration of dislocations. The prime quality of the wafers also minimized such dislocations. The final removal of the nitride layer from the back side eliminated the residual stress-induced bow. Considerable care was given to establishing the crystal orientation accurately before the two KOH etchings described above. This was done by lithographically patterning angular rulers [17] close to the wafer edges and conducting pre-test KOH-etching to detect the pits with the best oriented facet walls in the array. The fineness of the arrays ($3\,\mu$m lines and spaces) and statistical evaluation of 20 rulers per wafer allowed the determination of the crystal orientation to better than $10\,\mu$rad. Due to this accuracy, the first main KOH-etching produced SRL grooves with Si(111) sloping walls that are virtually step-free over their 6 mm lengths, with the remaining nitride properly covering the flat areas. Local thermal oxidation through the nitride mask protected this first set of grooves with a relatively thick $1.2\,\mu$m layer that was mostly preserved while completely removing of the nitride by RIE.

 

Fig. 4. Interferogram-measured flatness of a free-standing, 2-mm-thick, 150-mm-diameter, polished wafer.

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Figure 5(a) shows a completely processed Si wafer based on a mask designed to yield 16 SRL pieces of 9 cm and 6 cm lengths within the 150 mm diameter. The individual devices are extracted by laser cutting into $9\times 1$ cm$^{2}$ and $6\times 1$ cm$^{2}$ rectangular dimensions. The saw-tooth structures extend the full lengths of the individual pieces, and 6 mm across, leaving 2 mm wide side-flats running along the two long edges (Fig. 5(b)). These narrow, flat regions provide contact points for mounting an inverted SRL piece (i.e., the upper one) for vertical focusing and side-facing pieces for horizontal focusing. In the tests with a single upright piece, strain-free mounting is achieved by 3 supporting balls as shown. An identical mount can support an inverted lens piece that is overturned so that the balls contact the side-flats.

 

Fig. 5. (a) Fully processed 150-mm-diameter Si wafer yielding 9 (plus 7) lenses of 9 cm (and 6 cm) lengths. Laser cutting releases individual lenses. (b) 9-cm-long devices have the form sketched, with the saw-tooth structure displayed in the SEM images. In operation, the lens piece is supported by 3 balls to preserve flatness.

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5. Focusing test results

Figure 6 displays the vertical focal profiles for three different upright, single Si SRL pieces tested. They were placed at 68.7 m distance from the source, focusing to 70 m. The profiles were measured with L-fluorescence from a fine, 245 nm tall Au wire scanned vertically through the focus [6]. The FWHM values given include the correction (reduction) for the wire width broadening in the profiles. The 60 keV incident beam was apertured with high-precision slits to $300\,\times 100\,\mu$m$^{2}$ (horizontal $\times$ vertical), with the vertical size defined to match exactly the tooth height spatial acceptance. The narrowest focus resulted from the 2 mm thick, ultra-flat polished wafer device (690 nm FWHM, solid-line), followed by the 1 mm thick, ultra-flat device (770 nm, dashed-line). The thinnest 0.6 mm devices, not conforming to stringent flatness specifications, focused to 800 nm, sometimes with an asymmetric tail on one side (dotted-line). With respect to the expected 640 nm focusing calculated (Table 1) from the five contributions, including geometrical aberrations, the best measured focus is in reasonable agreement given possible additional influences from vibration, device imperfections, slight tilts of the highly eccentric elliptical source [6,8,18], and source size perturbation by the bent double-Laue monochromator.

 

Fig. 6. Vertical focus profiles for 60 keV X-rays focused by various Si lenses placed at 68.7 m from the source.

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The three SRL focal profiles in Fig. 6 are normalized to the same peak height to render a visual discernment of their differing widths. If normalized to the incident photon flux instead, the raw profiles have significantly disparate heights, reflecting different flux density gain performances, which can be measured by the ratio of the Au wire fluorescence signal at the focal peak to that in the unfocused beam. In the order of widening focal spots, the device gains are 71, 66, and 45, with the highest gain value being consistent with its associated device collecting 50–60 $\mu$m of radiation and concentrating it into 690 nm. Note that an upright/inverted SRL pair of the best performing device type would have a doubled aperture and flux density gain of $\approx 140$. The theoretical maximum for this arrangement and these SRL parameters is 170, assuming focusing down to 610 nm, a value stated earlier that includes geometrical aberrations and the effect of a lowered diffraction limit contribution in a two-piece SRL (Table 1).

6. Geometrical aberration mitigation

Following care given to Si substrate selection and preparation, nano-fabrication, and final device mounting, the vertical focusing of 60 keV X-rays to $< 700$ nm FWHM has been demonstrated with Si SRLs, in reasonable agreement with expectations. Two geometrical effects discussed, the discreteness aberration and the length aberration, contribute a few hundred nanometers each to the focal size, raising the question of their possible reductions.

The discreteness aberration arises from the SRL thickness profile being a piecewise-linear approximation to the desired parabolic form. In a ray-trace study tracking where rays are focused as their entrance displacement from the axis is monotonically increased (Fig. 3(b)), this is manifested as an oscillation of the destination position. This could perhaps be suppressed by compensating for the deviation of the SRL profile from the parabola with an additional, thin, corrective refractive optic, as has already been demonstrated for imperfections in CRLs [19].

The length aberration arises from the device length being non-negligible relative to the focal distance, highlighting the effect of off-axis rays being subjected to refraction too early compared to the paraxial rays. This can be compensated for by weakening the deflection strength away from the axis, i.e., along the length of the inclined device. Achieving this by varying the tooth-tip angle at fixed period was suggested [15], but this would not be feasible via anisotropic etching. Another approach is to bend the SRL pieces away from the axis, i.e., convex facing the axis. A third method is to reduce the saw-tooth period along the device length, as one moves off-axis. To explore this last approach, one can consider Eq. (1) and recognize that the focal length $f$ of the SRL needs to be adjusted continuously along the device length coordinate $z_{n}$ (depicted in Fig. 3(a), lower sketch, where $n$ also serves as a tooth index label). This would be achieved by varying the tooth height $v$, which is fixed by proportionality to the period $h$, since the prism angles in this case are dictated by the crystallographically guided etching. So transitioning from a fixed to varying focal length $f_{0} \rightarrow f_{0}+z_{n} = (1 + z_{n}/f_{0}) \, f_{0}$ is implemented by imposing a constant-gradient period with respect to length, given by $h \rightarrow (1 + z_{n}/f_{0}) \, h\,$. The placement coordinates $z_{n}$ for the tooth tips are generated by the first one at $z_{0} = 0$ and the recursive relation