1. MLLs fabricated by masked deposition

Multilayer Laue lenses (MLLs) are diffractive optics for focusing and imaging of hard X-ray beams of wavelengths $\lambda$ that are typically shorter than 0.1 nm [1]. They are made by the alternating deposition of two materials onto a substrate, forming the diffracting structure. The lens is then sliced from this structure. The layer period decreases with distance $r$ from the optical axis such that the diffraction angle $2\theta (r)$ of rays increases with distance from the axis to ensure that incident collimated rays, for example, are brought to a common focus. Given the optical properties of materials in the X-ray regime, high diffraction efficiency requires that the thickness of the lens along the direction of the optical axis is much greater than the layer periods $d(r)$. In such cases, where the aspect ratio of the layers (equal to the ratio of the optical thickness to the period) is large, the angular acceptance for efficient diffraction may be narrower than the numerical aperture (NA) of the lens. If all layers were parallel to each other then there would only be one layer that could be placed in the Bragg condition (given by $\lambda = 2d(r)\sin \theta (r)$) and only a limited range of layers that could reflect rays with sufficient efficiency, dependent on the rocking-curve width of the diffraction from the layers. This limits the effective NA of the lens to about 0.003 for a wavelength of 0.071 nm [2]. Lenses of higher NA must therefore consist of layers that are tilted and wedged so that rays impinge upon them at their Bragg angle $\theta$. Such lenses are often referred to as wedged MLLs [1]. It can also be seen that for focusing a collimated beam to a focus a distance $f$ from the lens requires periods that vary as

 

Fig. 1. Schematic of the deposition of film by a masked sputtering source.

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A period $d$ obtained without the mask becomes $h(z)\,d$ when the mask is used, and so a multilayer stack $d(r)$ becomes $h(z)\,d(r/h(z))$ [6]. For the MLL generated by the prescription given by Eq. (1), the modulated layer periods are therefore given by $d_m(r,z) = h^2(z)\, \lambda f/r$. The focal length of a lens cut perpendicular to the optical axis (and also to the direction of modulation $z$) is thus $f_m(z) = h^2(z)\,f$ and the deflection of rays at a position $r$ is $2\theta _m(r,z) = r/f_m(z)$. The scaling of the total height of the lens (or the height of a particular layer) can be found from $r_m = \lambda f_m / d_m = \lambda h^2 f / (h d) = h r$ as expected.

For the focusing of a collimated incident beam, the tilt of each layer is equal to the Bragg angle or half the diffraction angle. To first order this is equivalent to the layers being oriented on planes (or cones, in the case of axi-symmetric lenses) that intersect the optical axis a distance $2f_m$ from the lens as shown in Fig. 2. The layers are normal to cylindrical or spherical surfaces which are centered on that intersection point. For a given profile $h(z)$, we can define a “geometrical focal length” $f_h$ equal to half the distance to where the tangent of the profile intersects the axis, which for small angles can be approximated by the height factor $h(z)$ divided by its gradient $h'(z) = \partial h(z)/\partial z$, so that

 

Fig. 2. Configurations for masked deposition onto a flat substrate. (a): A deposition from thick to thin layers, starting with the layer corresponding to the optical axis, which therefore coincides with the substrate (orange dashed line). (b): A deposition from thin to thick layers, ending with the layer corresponding to the optical axis, which therefore is inclined at the tilt of the last layer. (c): Deposition from thick to thin layers, starting with a layer that is some distance above the optical axis. The profiles of layers are shown in thick blue lines. Rays are shown in orange. The optical axis, depicted by the dashed orange line, intersects both the focus and the convergence point of the layers (thin blue lines).

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The resolution of a single (cylindrical) MLL cut at a position $z$ is given by $\delta _m(z) = \lambda _m(z)/(2\mathrm {NA}_m(z)) = \lambda _m(z) f_m(z)/(h(z) P)$, where $P$ is the height of the unshadowed multilayer, $P = \lambda f/\delta$. From Eqs. (6) and (7) the expression for the resolution is

For the case of the error-function shadow of Eq. (2), the operating wavelength of the MLL cut at a position $z$ is thus given by

 

Fig. 3. Plot of the transmission factor $h(z)$ in blue, the normalised focal length $f_m(z)$ in green, and the normalised operating wavelength $\lambda _m(z)$ in orange, for the case of an error-function shadow of width $w$.

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In designing a particular MLL or a range of MLLs to be cut from a deposited structure, one can choose two independent design parameters: the product $\lambda f$ in the deposition prescription, and the shadow width $w$ from the positioning of the mask. To avoid having to deposit a multilayer that is much larger in height than the desired lens, these parameters should be chosen so that the desired focal length occurs for a transmission factor $h$ which is not greatly smaller than $1$. For example, $h(z) = 0.80$ occurs at $z -z_0\approx 0.85 w$, for which $f_m(0.85 w) = 1.44\, w$ and $\lambda _m(0.85 w) = 0.45 \, \lambda \,f/w$.

1.1 Scaling of aberrations

We have previously considered wavefront aberrations in MLLs caused by a change in the rate of the deposition over the course of depositing the entire multilayer structure [6]. This leads both to an error in the period $d$ of the layers (changing the phase gradient of the wavefront of the focused beam due to a change in the diffraction angle) as well as an error in the positions of layers (changing the phase of the wavefront at a particular position in the lens arising from the $2\pi$ accumulation of phase per period). For example, a linear relative change in the deposited thickness per unit thickness of material, $\beta$, leads to layers scaled according to $(1+\beta r) d(r-\beta r^2/2)$, and a wavefront error (or optical path difference) given by $\textrm {OPD}(r) = -\beta r^3/f$ (see Sec. 6.1 of [6]). Assuming the change in rate $\beta$ of the deposited material is due to a change in the emission rate of the sputtering sources, then the effect of the mask is to give a change in deposition per unit thickness of material of $\beta _m(z) = \beta /h(z)$. The mask also leads to a coordinate scaling of the OPD according to

The dependence of wavefront aberrations on the position $z$ will be different if there is a contribution to the deposited layer period that does not scale with the transmission factor $h$. Such would be the case for a contraction or expansion of the layer period due to intermixing of the layer materials, causing a change $\Delta d$ of every layer period, independent of that period and therefore also independent of $h(z)$. The $n$th layer will accumulate a placement error of $n\,\Delta d = r^2 \Delta d/(2 f \lambda )$. Comparing this with the placement error of the same layer for the linear change in deposition rate, which is $\beta _m r^2/2$ [6], we can infer a wavefront aberration of

1.2 Reverse deposition

Often, MLLs are manufactured by depositing the layers in reverse order—that is, starting from the layers of smallest period. This is done to reduce the accumulation of roughness and sputtering errors from affecting the more sensitive finer layers. In such a case the finest layers are in contact with the flat substrate, and the thicker layers take on the form $h(z)$ due to the shadowing of the straight-edge mask used in the deposition. As illustrated in Fig. 2(b) the optical axis therefore lies above the substrate, and is tilted to the substrate. Since all rays parallel to the optical axis and which are incident on the (ideal) MLL are efficiently reflected and directed to the focus, an incident ray that Bragg-reflects from the layers at the substrate is also parallel to the optical axis, which is to say the optical axis is tilted to the substrate by this Bragg angle, $\lambda /(2 d(r=0))$. Given a height $T$ of the optical axis from the substrate for the unshadowed multilayer, the prescription for the layer periods in that structure is, to first order, $d(r)=\lambda f/(T-r)$. The multilayer usually terminates before reaching the optical axis, $P < T$, but this does not necessary need to be the case. With the deposition mask, the multilayer stack is again modified to $h(z)\,d(r/h(z))$ to give

At the design wavelength $\lambda$ the focal length of the MLL cut at the position $z$ can be seen from Eq. (14) to again be expressed as $f_m(z) = h^2(z)\,f$. This is also consistent with the condition that the Bragg angle at $r=0$, multiplied by the focal length $f_m(z)$ is equal to $T\,h(z)/2$ as expected from Eq. (14), $\theta _m(0) = \lambda /(2d_m(0)) = T/(2 h(z)\,f)$, so $\theta _m(0) \, h^2(z) f = T\,h(z)/2$. Furthermore, if used at a different wavelength $\lambda _m$, the focal length is still given by Eq. (6), and so, in turn, Eq. (7) remains valid as the condition for the optimum wavelength for an MLL, $\lambda _m(z)$. It also holds that these equations are valid for MLLs made by depositing thicker layers first but starting the deposition at some position above the optical axis, such that $d(r) =\lambda f/(T+r)$. In that particular case $T$ is the distance of the optical axis below the substrate.

1.3 Manufacture of curved MLLs

For MLLs with large numerical apertures, where the resolution $\delta$ approaches the wavelength, the Bragg condition will not necessarily be obeyed throughout the optical thickness of the lens [1]. As shown by Yan et al. [10], constructive interference of all reflected rays at the focal point requires layers that follow a set of nested confocal parabolas or ellipses for an incident plane wave or spherical wave. Yan et al. [10] refer to this type of lens as a curved MLL. Such a structure necessarily requires that the focal length at a position $z$ throughout the thickness of the MLL increases with $z$, or that $\partial f_m(z)/\partial z = 1$. From Eq. (6),

As discussed above and shown in Fig. 3, the example of a shadow of a straight edge from a Gaussian source indeed provides a curved MLL at the position where $\lambda _m(z)$ is maximised at $z-z_0=0.506 w$, with $h(z) = 0.694$. The focal length at this position is $f_m(0.506 w)=0.988 w$. Thus, a lens of a desired focal length can be made from a deposition with a particular mask to substrate distance to achieve the necessary value of $w$. This lens will only operate efficiently for a single wavelength as mentioned above, which can be set independent to the focal length by the choice of the product $\lambda f$ in the layer deposition schedule. More generally, we see from Eq. (7) that a parabolic profile $h(z) = (z/w)^{1/2}$ gives $\lambda _m(z) = \lambda f /w$ which is independent of $z$, and $f_m(z) = z$. Such a profile could be achieved using the same approach of depositing in the shadow of a straight-edged mask, but with an inverse square root angular distribution of the source instead of a Gaussian, or by using a complex mask of the targets, akin to those demonstrated by Huang et al. [4]. A linear profile, $h=z/w$, cannot give the condition $\partial \lambda _m(z)/\partial z = 0$ and as a corollary cannot provide two lenses of different focal lengths for a common wavelength.

A further consideration in the manufacture of curved MLLs is the shape of the substrate on which the layers are deposited. The correct curvature of the layers requires that the layer lying on the optical axis (whether or not it is part of the actual structure) must be straight. That is, with a flat substrate, the correct curvature is only obtained if the deposition starts at the optical axis, which necessarily means that it continues with increasing $r$ to thinner layers. For MLLs deposited in the reverse direction of thin to thick layers (Sec. 1.2) or from a position some distance from the optical axis, it is necessary to deposit onto a concave or convex substrate with a parabolic curvature such that the layer formed as $h(z)\,T$ is flat. For a 1D lens, the substrate could be approximated by a cylinder.

1.4 Manufacture of MLL pairs

When MLLs are fabricated on flat substrates, two lenses are needed to create a pair that focus in orthogonal directions. Somewhat similar to finding the position in the shadowed deposition where $\partial \lambda _m(z) / \partial z = 0$, a confocal pair of 1D lenses can be cut from one structure by finding two positions which share the same wavelength, $\lambda _m$, but have different focal lengths $f_m$. The difference in the focal lengths dictates the spacing between the two lenses (oriented orthogonal to each other) to create a common 2D focus. If we again consider the sputtering source with a Gaussian angular distribution, the wavelength $\lambda _2$ is obviously restricted to be less than the maximum wavelength of $0.487\, \lambda f/w$ that occurs at $z-z_0 = 0.506 w$. For example, choosing $\lambda _2 = 0.47 \lambda f/w$, the two solutions to $\lambda _m(z) = \lambda _2$ are $z-z_0=0.291 w$ and $z-z_0=0.724 w$, which have transmission factors $h=0.615$ and $0.765$ and focal lengths $f_m=0.804 w$ and $1.246 w$, respectively. Even though in this case the shorter focal length is about 60 % of the longer, the NA’s of the two lenses do not differ by such a large factor, and in fact the shorter lens has the larger NA, in accordance with Eq. (8). The NA of the short lens scales with $h/f_m = 0.765/w$ versus $0.614/w$ for the long focal length lens, a ratio of $1.25$. Lenses to be bonded in contact to form a pair for 2D focusing [11] could be cut adjacently near the position of maximum wavelength.

Yet more lenses of differing focal length that operate at a common wavelength can be produced in a single deposition run if more straight-edge masks are placed over the substrate. To achieve different focal lengths these should be placed at different heights to give different penumbral widths $w$ of the mask shadows, and the edges should be spaced far enough apart that they do not influence the shadow of the neighboring edge. As seen in Fig. 3, the influence of an edge extends approximately from $-3w$ to $3w$, and so the spacing of edges should be no less than about three times the sum of the widths they produce.

A deposition recipe is constructed for a particular product $\lambda f$, which is obviously common to all the shaded regions in the deposition. Consider two regions with widths $w_A$ and $w_B$. The pairs of positions $z_A$ and $z_B$ which yield a common operating wavelength can be found by ensuring $\lambda _m(z_A;w_A) = \lambda _m(z_B;w_B)$, using Eq. (10). While an analytical expression for $z$ in terms of $\lambda _m$ and $w$ cannot be obtained, Fig. 4 shows a plot of the focal lengths of lenses that operate at a common wavelength, given two mask edges producing widths $w_A$ and $w_B$, as a function of the ratio $w_B/w_A$. This was obtained for particular wavelengths $\lambda _m$ by numerically solving for the position $z_B$ that gives the same wavelength, then evaluating the focal length from Eq. (11). For example, it can be seen from Fig. 3 that the wavelength $\lambda _m = 0.425 \lambda f/w_A$ can be cut at two positions: $z_A =0.087 w_A$ and $z_A = 0.936 w_A$, giving focal lengths of $f_m = 0.673 w_A$ and $f_m = 1.603 w_A$, respectively. (Here, the coordinates $z_A$ and $z_B$ are taken to be relative to the centre of each shadow.) These two values lie on the purple curve in Fig. 4 ($\lambda _m = 0.425 \lambda f/w_A$) at $w_B/w_A = 1$. The deposited multilayer under a mask edge with $w_B = 1.1 w_A$, for example, will give a plot of $\lambda _m$ as a function of $z_B$ which is broader but lower than that shown in Fig. 3, and it intersects $\lambda _m = 0.425 \lambda f/w_A$ at $z_B = 0.303 w_A$ and $z_B = 0.814 w_A$, for which the focal lengths are $f_m = 0.871 w_A$ and $f_m=1.396 w_A$. Interestingly, for the lenses on the branch with longer focal length, which have higher factors $h$, increasing the width $w_B$ provides lenses of shorter focal length. This is because to satisfy the wavelength requirement they must be cut from a position of lower $h$ where the gradient is steeper.

 

Fig. 4. Plots of the focal lengths of lenses, normalised by the width $w_A$, operating at a common wavelength $\lambda _m$, cut from a single deposition structure at shaded regions of width $w_B$, plotted against the ratio $w_B/w_A$. The different colored plots correspond to different operating wavelengths $\lambda _m$, normalised by $\lambda f/w_A$. For example the four possible focal lengths of lenses cut from a deposition parameterised by $f\lambda$ for a wavelength $\lambda _m = 0.375 f\lambda /w$ with two shadow widths of $w$ and $1.2w$ are found from the values of the green line at $w_B/w_A = 1$ and $1.2$.

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It is seen in Fig. 4 that for a given deposition, there is a maximum value of $w_B$ that can yield a lens of a particular operating wavelength $\lambda _m$. This corresponds to the maximum wavelength achievable for that shadowed region (as seen in Fig. 3) and is therefore a curved MLL as discussed in Sec. 1.3. Obviously then, with error-function shadows it is only possible to manufacture curved MLLs for a single particular operating wavelength and focal length from a single deposition, no matter how many shadowed regions are used.

2. Measurements

2.1 Characterisation of masked multilayer profiles

Multilayer structures consisting of alternating layers of SiC and WC were fabricated on flat substrates by magnetron sputtering in DESY’s X-ray multilayer deposition laboratory, following the procedures published previously [2,5,12]. The sputtering targets were circular disks of diameter 7.6 cm located 5 cm from the plane of the substrate. During deposition the substrate was moved alternately through the plumes of the two targets while it was spinning about its center to average out any orientational anisotropies of the targets. A straight-edge mask affixed to the spinning substrate holder was used to create the profile in the deposition rate across the substrate. In our earlier work we used a profilometer to measure the profile resulting from a particular distance of the mask edge to the substrate [5]. We find however, for multilayers of total thickness greater than several micrometers, the stress of the film bends the substrate, making it difficult to deduce the profile of the coating alone. Thin surrogate multilayer coatings can be made to characterise the profile, but cutting lenses from the structure at the position that is optimised for a particular wavelength requires an accurate knowledge of the as-deposited profile and its coordinates. We obtain the necessary accuracy of the multilayer thickness profile by making “depth soundings” of the structure at a number of positions [4]. That is, a dual-beam focused-ion-beam scanning-electron-microscope (FIB/SEM) (FEI, Helios) is used to make a cut into the structure that is then imaged at an oblique angle using the scanned electron beam to determine the depth of the film at that point.

An example of the multilayer height profile $p(z) = h(z) P$ obtained this way is given in Fig. 5, along with a plot of its gradient $p'(z)$ determined by a piece-wise cubic fit of $p(z)$. For an error-function profile the gradient would be Gaussian, but we generally find that the gradients of the profiles are slightly skewed with a longer tail towards positions of lower thickness. A fit of a Gaussian to $p'(z)$ in Fig. 5 also suggests this skew. Since the substrate (and mask) spin around their center and transit through the plume of sputtered material from the target it is unlikely that the asymmetry is caused by the geometrical properties of the target. It might instead be caused by the mask, and indeed a mask edge with a finite thickness (i.e. with a rectangular cross sectional shape or rounded knife-edge shape) would cause an asymmetry consistent with that observed. Rays reaching the low-thickness (more shaded) side of the profile will be limited by the bottom edge of the mask whereas rays reaching the high-thickness (less shaded) side will be limited by the top edge. Bringing the edge closer to the substrate casts a sharper shadow, so the gradient will be steeper at the low-thickness side than the high-thickness side.

 

Fig. 5. (a) Measured thickness profile $p(z)$ of a multilayer deposited with a straight-edge mask at a stand-off distance of $y_m = 1.32\,\textrm{mm}$, and (b) a plot of its gradient determined by a piecewise cubic fit. Errors in $z$ are less than 10 µm. The fit of a skewed error function with $\hat {w} = 1.5\,\textrm{mm}$ and $\alpha = -1.7$ is shown as the blue solid lines in (a) and (b), and the fit of a Gaussian to the gradient is shown for comparison as the dashed orange curve. The origin of the coordinate $z$ was chosen after fitting the skewed error function and setting $z_0 = 0$.

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We have already encountered a model of a skewed Gaussian in Eq. (10) formed by multiplying a Gaussian by the error function $\Phi$. More general is the skew normal distribution, formed by multiplying the normal probability distribution by a stretched version of $\Phi$. Its definite integral is given by

Multilayer structures were fabricated for various stand-off distances $y_m$ between the mask and substrate and for different distances between the sputter targets and the substrate. The profiles of these were then measured and fitted with the function $h_s(z)$. For a target to substrate distance of 5 cm and targets of 7.6 cm diameter, we find that $\hat {w} \approx 1.15 y_m$, a relationship that holds to the largest distance we tested of 1 cm. The width $\hat {w}$ becomes smaller for a given $y_m$ when the target to substrate distance is increased, as expected from geometry.

2.2 Characterisation of lens parameters

A deposition was carried out on a substrate with two straight-edge masks affixed at stand-off distances of $y_m = 1.77\,\textrm{mm}$ (referred to here as the $A$ profile) and $y_m = 1.32\,\textrm{mm}$ (the $B$ profile). The two edges faced each other across a gap of about 1 cm to produce a coating with largest thickness in the center and profiles with two different widths on each side. After calibrating the deposition rates the prescription of the layer thicknesses was set to create an off-axis lens consisting of 2759 periods with $\lambda f = 0.237\,\textrm {nm} \cdot \textrm {mm}$ for the unshaded structure. That is, the lens cut at a transmission factor of $h=0.8$ would diffract with a focal length of 2.13 mm at $\lambda = 0.071\,\textrm{nm}$ (a photon energy of 17.4 keV) for example, which was set to correspond to the geometrical focal length $f_h$ (Eq. (5)) for the $A$ profile at the height $h = 0.8$. At that postion the periods ranged from a smallest value of 4.31 nm up to a largest period of 7.58 nm, corresponding to distances of 35.2 µm to 20.0 µm from the optical axis and an NA of 0.0036, for a resolution $\lambda _m/(2 \textrm {NA}) = 10\,\textrm{nm}$. The thinnest layers were deposited first, following the scheme illustrated in Fig. 2(b).

Using the method of FIB depth soundings, the profiles of the two penumbras were measured—the $B$ profile is shown in Fig. 6(a). The function $h_s(z)$ was fit to both profiles, giving parameters $\hat {w}_A = 1.99\,\textrm{mm}$, $\alpha _A = -1.7$, $\hat {w}_B = 1.52\,\textrm{mm}$, and $\alpha _B = -1.6$. From these parameters it is possible to determine the coordinates at which to cut lenses to operate at a particular wavelength. The operating wavelength $\lambda _m$ for a particular lens is that for which all periods in the lens are oriented in the Bragg condition for an incident collimated beam of that wavelength, which can be checked by mapping X-ray intensities at diffracting angles $2\theta$, measured on a detector in the far field beyond the focus. If the beam wavelength is not equal to $\lambda _m$ then the layers of different periods throughout the lens will need to be tilted by different amounts to bring them into the Bragg condition. That is, the layers meeting the Bragg condition (and hence providing maximum reflectivity) will shift across the pupil of the lens as the lens is tilted to the beam. Maps of the diffraction efficiency of three different lenses are shown in Fig. 6(b) as a function of $2\theta$ and the tilt angle $\omega$ of the lens, for a probe wavelength of 0.071 nm (Mo K$\alpha$), measured using a collimated beam from a micro-focus X-ray source with a Mo anode (Sigray, California, USA) and monochromatised with a Si 111 channel-cut monochromator. Measurements were recorded with a pixel array detector with a Si sensor and pixels that are 55 µm wide (Lambda detector, X-Spectrum, Hamburg, Germany). The maps can be interpreted by considering a lens with layers parallel to planes that intersect at the optical axis a distance $2f_h$ from the lens. These layers are oriented in the Bragg condition for the operating wavelength $\lambda _m$, as if to reflect rays to the focus at a distance $f_m = \lambda f / \lambda _m = f_h$ from the lens. The ray intersecting the lens a distance $r$ from the optical axis will be deflected by an angle $r/f_m$. If, however, the wavelength of this ray was $\lambda _p$, it would be deflected to a focus at $f_p = f_m \lambda _m/\lambda _p$ by an angle $r/f_p$. The ray would only be efficiently reflected if the layers at a distance $r$ were brought into the Bragg condition, achieved by rotating the lens by half of the difference between the angles of deflection for these two wavelengths, or by $\omega = (r/f_p - r/f_m)/2$. This is equivalent to the condition

 

Fig. 6. (a) Height of the $B$ profile ($\hat {w}_B = 1.52\,{mm}$, $\alpha _B = -1.6$) of the multilayer structure, showing FIB-measured depths (filled circles) and heights of some extracted lenses (open circles). The relative diffraction efficiencies of three of the lenses, labelled $\alpha$, $\beta$, and $\gamma$ are shown in (b), mapped as a function of the diffraction angle $2\theta$ and the tilt of the lens $\omega$ at a probe wavelength of 0.071 nm (17.4 keV). The dashed lines are fits of Eq. (18) to the maximum efficiency, indicating operating wavelengths of 0.060 nm (20.5 keV) ($\alpha$), 0.071 nm (17.4 keV) ($\beta$), and 0.091 nm (13.6 keV) ($\gamma$). The position of another lens with an operating wavelength of 0.071 nm is indicated by the label $\beta '$ in (a).

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Measurements of the intensities as a function of $2\theta$ and $\omega$ are shown in Fig. 6(b) for three positions on the $B$ profile and labelled $\alpha$, $\beta$, and $\gamma$. These measurements correspond to the $z$ positions indicated in Fig. 6(a). It is seen from the slope of the maximum intensity mapped by the dashed lines in Fig. 6(b) that the operating wavelength $\lambda _m$ increases in the progression $\alpha$, $\beta$, $\gamma$ and that the $\beta$ lens is obviously optimised to operate at the probe wavelength of 0.071 nm. The operating wavelengths range from $\lambda _m = 0.060\,\textrm{nm}$ (20.5 keV) for $\alpha$ to 0.091 nm (13.6 keV) for $\gamma$, but it is clear that lenses could be extracted from this deposition that span an even greater range of operating wavelengths. These three measurements were made at positions where $h \approx 0.5$, and where $\lambda _m$ increases with $z$. This is the region where $z$ is below the position of maximum wavelength (see Fig. 3). That is, these lenses were cut from the short-focal-length (short-$f$) branch of the plot in Fig. 4. Lenses of matching wavelength but larger size and larger focal length can be cut from $z$ positions above the position of maximum wavelength, corresponding to the long-$f$ branch of the plot in Fig. 4. The location of the long-$f$ lens that operates at the probe wavelength of 0.071 nm is indicated by $\beta '$ in Fig. 6(a). Compared with the $\beta$ lens, which has a focal length of $f = 0.61\,\textrm{mm}$ and a height $p = 7.8\,\mathrm{\mu}\textrm{m}$, the $\beta '$ lens has a focal length of $f = 2.70\,\textrm{mm}$ and a height $p = 17.0\,\mathrm{\mu}\textrm{m}$.

It can also be seen in Fig. 6(b) that the lenses cut from different positions in the profile have different ranges of diffraction angles, and that the range $\Delta \,2\theta$ decreases as the operating wavelength increases (in the sequence $\alpha$, $\beta$, $\gamma$). This follows from the increasing transmission factors $h$, so the maximum and minimum $d$ spacings in each lens also increase, to diffract the probe wavelength at smaller angles according to

For a fuller comparison of the dependence of the operating wavelength with the coordinate $z$ as predicted from Eq. (7) and the fitted profile $h_s(z)$, a long lens structure was cut from the same $B$ shadow profile with $\hat {w}_B = 1.52\,\textrm{mm}$. Lenses are usually cut perpendicular to the coordinate $z$, but this structure was cut at an angle of 26° to perpendicular so that a position $x_l$ along the lens occurs at a coordinate $z = 0.49(x_l + x_{l0})$ where $x_{l0}$ is an offset that was initially unknown. The structure was polished down to a thickness of about 15 µm in the $z$ direction, using a FIB. As was done for the positions $\alpha$, $\beta$, and $\gamma$ as shown in Fig. 6(b), maps of diffraction efficiency as a function of $2\theta$ and $\omega$ were recorded at positions along this structure using the monochromatised Mo K$\alpha$ beam. A pinhole of about 100 µm diameter was placed in the beam to limit the region of the structure exposed. The lens structure was maintained at the angle of 26° to the perpendicular of the beam so that the optical axis of the MLL remained parallel to the incident beam. Figure 7(a) shows a plot of the range of diffraction angles of the lens at 10 different positions, along with the function $\Delta \,2\theta (z)$ as given by Eq. (19). The offset $x_{l0}$ was determined from the best fit of the function to the data.

 

Fig. 7. (a) Plot of $\Delta \,2\theta$ (circles) measured at various positions $x_l$ of the diagonally-cut lens, corresponding to positions $z$ in in the $B$ shadow profile of Fig. 6(a). The line shows the function given by Eq. (19) after fitting the offset $x_{l0}$. (b) Operating photon energies $E_m$ (orange circles) determined from $2\theta -\omega$ scans such as shown in Fig. 6(b), the photon energy (orange line) obtained from Eq. (7), and the focal length (green line) from Eq. (6). The squares in (a) and (b) show the measurements of the $\beta '$ lens, placed at $z=0.40\,\textrm{mm}$.

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A plot of the measured operating photon energy $E_m = hc/\lambda _m$ for positions across the lens structure is shown in Fig. 7(b), along with the prediction obtained from Eq. (7) using the profile $h_s$ (Eqs. (16) and (17)) and the previously fitted values of $\hat {w}_B$ and $\alpha _B$. As can be seen from the plot, the operating wavelength of the lens can be well predicted from the fit of the skewed error function to the thickness profile produced by the straight-edge mask.

Figure 7(b) also shows (in green) the predicted focal lengths of lenses cut at a particular position $z$. The long lens structure only spanned a narrow range of $z$ on the short-$f$ branch of the profile. In practise, lenses are either cut near the position of maximum wavelength or on the long-$f$ branch, since larger lenses make better use of the deposited structure. Although the spatial resolution $\delta _m$ scales directly with $h$ (Eq. (8)), favouring lenses cut from the short-$f$ branch, such lenses can be made in less time and expense on the long-$f$ branch by a suitable adjustment of the deposition parameters and mask stand-off distance.

2.3 Demonstration of lens pairs

For the multilayer structure produced here with the two mask edges, $A$ and $B$, creating shadows with $\hat {w}_A = 1.99\,\textrm{mm}$ and $\hat {w}_B = 1.52\,\textrm{mm}$ respectively, the maximum operating wavelengths are 0.086 nm (14.5 keV) and 0.115 nm (10.8 keV). At an operating wavelength of 0.071 nm, which is considerably shorter than the maximum wavelengths, lenses cut from the same shadow will have quite different focal lengths as can be deduced from Figs. 4 and 7(b). Such lenses will also have quite different heights and NA’s. This is demonstrated in Fig. 8(a) depicting the far-field pattern recorded for a crossed pair of lenses that were both cut from the $A$ shadow of the deposition at two positions that produce the desired operating wavelength (similar to the $\beta$ and $\beta '$ lenses in the $B$ shadow). Here, the larger lens cut from the long-$f$ branch (similar to the $\beta '$ lens) was oriented to focus in the horizontal direction and the lens cut from the short-$f$ branch (similar to the $\beta$ lens) focused vertically. Even though the vertical lens has a smaller aperture, it has a larger NA ($\textrm {NA} = 0.0052$ and $f = 0.93\,\textrm{mm}$) than that of the horizontally-focusing lens ($\textrm {NA} = 0.0036$ and $f=2.31\,\textrm{mm}$). These lenses must therefore be separated by 1.38 mm in the direction of the optical axis to create a common focus, which would give a resolution of $10.0\,\textrm{nm} \times 6.8\,\textrm{nm}$ (h $\times$ v) if the lenses were free of wavefront aberrations.

 

Fig. 8. Measured far-field intensities from a pair of lenses cut from the $B$ shadow structure (a) and a pair cut from different shadows $A$ and $B$ (b). All lenses were obtained from a single deposition run.

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Figure 8 shows that both sets of lenses form a matching pair, with high diffraction efficiency across the entirety of their pupils. In previous experiments, while a range of wedged MLLs of different focal lengths were available and which gave extremely uniform diffracting efficiency at their particular operating wavelengths, it was often difficult to obtain two wedged MLLs that match. Lenses with operating photon energies that differ by only about 1 keV sometimes only provide high diffraction efficiency over one quarter of the combined pupil, with a corresponding degradation of throughput and resolution. Mismatched lenses are usually quickly exchanged and not shown in publications, but Fig. 4(c) in the paper of Bajt et al. [14] gives an appreciation of the difficulty. The individual efficiencies of the lenses in Fig. 8 were measured to be 55 %.

Lens pairs of similar NA and sizes can be extracted for a much greater range of operating wavelengths if they are cut from separate shadows, as discussed above and illustrated in Fig. 4. In this case $\hat {w}_B / \hat {w}_A = 0.76$, and the two lenses cut on the long-$f$ branches would differ in focal length by several hundred micrometers, which is enough to space them apart from each other to achieve a common focus. We demonstrated this using the two lenses for 0.071 nm on the long branches of the $A$ and $B$ shadows, as shown in Fig. 8(b). The $A$ lens was the same as used in Fig. 4(a) and, as in that case, it was oriented to focus horizontally. The vertically-focusing lens was the $\beta '$ lens from the $B$ shadow with $\textrm {NA} = 0.0032$ and $f=2.70\,\textrm{mm}$. As noted in Fig. 4, this lens, cut from the shadow with the smaller width $\hat {w}$, has the longer focal length. The two lenses must be spaced by 0.39 mm to focus to a common plane. The far-field pattern of the beam beyond the focus of the crossed and aligned lenses, measured at a wavelength of 0.071 nm is shown in Fig. 8(b). This lens pair would give a resolution of $10.0\,\textrm{nm} \times 11.1\,\textrm{nm}$ and has an aperture with about twice the area of the pair taken both from the $A$ shadow profile.

3. Summary and conclusions

A multilayer Laue lens of high numerical aperture must consist of diffracting layers that not only vary in period to redirect rays by angles that increase with distance away from the optical axis to bring them to a common focus, but also that those layers vary in orientation to ensure that the Bragg condition is always fulfilled. The layers must be tilted as if to reflect the rays towards the focus. Such an MLL is the real-space physical embodiment of the Ewald sphere, and thus only operates efficiently for a single wavelength (or small range of wavelengths). It can be made using a layer deposition process and a straight-edge mask to simply cast a shadow in the atomic plumes to modify the thickness of deposited material with position across the substrate [5]. Dilating the structure of a diffractive optic such as a zone plate modifies its focal length. The correct tilt of layers to meet the Bragg condition also depends on focal length, and thus both the normalised height profile $h(z)$ caused by the shadow mask and the gradient of this profile $h'(z)$ influence the position $z$ in the profile to slice out a lens to operate at a particular wavelength and focal length. In this paper we showed that a lens cut at any position across the shadow profile will be optimised—obeying both the Fresnel zone-plate condition and the Bragg condition—for some particular wavelength. This operating wavelength of the lens scales in proportion to the product of the mask transmission factor $h(z)$ and its derivative $h'(z)$. When used at that wavelength, the focal length of the lens depends purely on geometry and is given by $h(z)/(2 h'(z))$, as to reflect rays towards the focus. A consequence of these scalings is that the resolution (or focal spot size) of a lens used at its operating wavelength is proportional to the factor $h$, but the NA is proportional to the gradient $h'$.

Experimentally, the deposition profile $h$ produced by a straight-edge mask can be approximated by an error function, for which the gradient $h'$ is a Gaussian. In this case the dependence of the operating wavelength with position, proportional to the product of $h$ and $h'$, follows a skewed Gaussian. (An even better approximation is found using a skewed profile whose gradient is then a skewed Gaussian.) The behaviour of lenses cut from this shaded structure is then determined by two parameters: the width $w$ of the error function (describing the width of the penumbral region and controlled by the distance of the mask to the substrate), and the product of design wavelength and focal length, $\lambda f$, set by the thicknesses of the deposited layers in the unshadowed region.

We demonstrated an accurate procedure to characterise the profile of a deposited multilayer structure by taking “depth soundings” of the structure at various positions using a focused ion beam / scanning electron microscope. The operating wavelength of lenses sliced from the structure can be well predicted from the measured profile, as confirmed by recording the diffraction efficiency of lenses at a particular probe wavelength as a function of the diffraction angle $2\theta$ and the tilt $\omega$ of the lens. Of course, if the probe wavelength equals the operating wavelength then there will be one value of $\omega$ where the lens diffracts across its entire pupil. Otherwise, the optimum $\omega$ for high diffraction efficiency scales linearly with $2\theta$ and the slope of this line indicates the operating wavelength. Such measurements confirmed the dependence of the operating wavelength on position and can be used to select pairs of lenses for 2D imaging.

The skewed Gaussian has a maximum close to the maximum gradient of the profile and so there is a maximum operating wavelength (or minimum photon energy) that a lens can be obtained from a given deposited structure, equal to $0.487\,\lambda f/w$. This also means that there are generally two places in the structure, bracketing the wavelength maximum, that give lenses that are matched in operating wavelength but have different focal lengths. These lenses may be used as an orthogonal pair for two-dimensional focusing, but the aperture height of one of the pair will generally be smaller than half the thickness of the unshaded deposition. The smaller lenses are more susceptible to aberrations caused by variations in deposition rates of materials. In addition, since thick structures may take many days to deposit, it is preferable to create a pair from lenses that are nearly as thick as the maximum deposited height. This can be achieved using two masks of different stand-off distances to the substrate that give different penumbral widths $w$ but the same value of $\lambda f$. We demonstrated both approaches here with measurements of lenses working at an operating wavelength of 0.071 nm.

Matched lenses cut near the position of maximum operating wavelength would be suited for mounting in contact with each other to create a common two-dimensional focus. A lens cut exactly at the maximum operating wavelength possesses curved layers that approach the ideal form for a “curved” MLL [1] in which layers follow elliptical curves. In this case the substrate should ideally conform to the required shape of the layer at its particular distance from the optical axis as defined by the layers. That is, a flat substrate only conforms to the layer profile for the optical axis and so the deposition in that case should start from the central zone. An off-axis curved MLL can be made by depositing layers on a convex substrate of suitable radius if starting with the thicker layers, or on a concave substrate if depositing from thinnest to thickest.

Achieving a matched pair of lenses and an ideal curved MLL only requires that the product $h(z)\,h'(z)$ has a turning point where it reaches a maximum or minimum. This is true for any monotonic profile $h$ which has a maximum gradient, for example. That is, there is no requirement to exactly reproduce an error-function curve. The linear profile $h(z) = z/w$ used by Huang et al. [4] has a constant gradient, giving a linear dependence of the operating wavelength with position $z$ which thus does not have a turning point and can only produce one lens at any particular wavelength.

Although we demonstrated masked-deposition fabrication of MLLs on flat substrates, axi-symmetric lenses (referred to as multilayer zone plates [1,15,16]) could be made using the same principle. In this case the layers should be deposited on a conical substrate that rotates about its central axis and with the shadow mask oriented perpendicular to that axis. Layers must be deposited from thick to thin.

The analysis and methods presented here can be used to fabricate a complement of high NA MLLs to cover a large range of wavelengths, for example to be used at an X-ray microscopy beamline. They are also helpful in the design of lenses with desired aperture sizes and focal lengths. While it might seem difficult to fabricate an MLL with layers that are tilted by angles that are matched to the requirements of the diffraction by those layers, we find that achieving this is quite straightforward and is just a matter of characterising the height profile of the shadowed multilayer structure and slicing a lens at the correct position. Cutting an MLL with the ideal curved layers for optimal diffraction is similarly uncomplicated.