Analysis of X-ray multilayer Laue lenses made by masked deposition

: Multilayer Laue lenses are diffractive optics for hard X-rays. To achieve high numerical aperture and resolution, diffracting structures of nanometer periods are required in such lenses, and a thickness (in the direction of propagation) of several micrometers is needed for high diffracting efficiency. Such structures must be oriented to satisfy Bragg’s law, which can only be achieved consistently over the entire lens if the layers vary in their tilt relative to the incident beam. The correct tilt, for a particular wavelength, can be achieved with a very simple technique of using a straight-edge mask to give the necessary gradient of the layers. An analysis of the properties of lenses cut from such a shaded profile is presented and it is shown how to design, prepare, and characterize matched pairs of lenses that operate at a particular wavelength and focal length. It is also shown how to manufacture lenses with ideal curved layers for optimal efficiency.


MLLs fabricated by masked deposition
Multilayer Laue lenses (MLLs) are diffractive optics for focusing and imaging of hard X-ray beams of wavelengths λ 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 2θ(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 λ = 2d(r) sin θ(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 θ. 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 The requisite tilting of layers in wedged multilayer Laue lenses can be achieved by using extended sputtering targets that are shadowed with complex masks to give a spatial variation in the rate of material deposited onto a substrate [3,4], or by depositing material from an extended sputtering target onto a substrate that is shadowed by a straight edge [5] as shown in Fig. 1. Alternating layers are then obtained by moving the substrate through the atomic plumes of different sputtering targets. In the latter case, the mask is rigidly connected to the substrate, but is located at some stand-off distance from the substrate so that the penumbra of the source provides a profile of the rate of accumulated material. The straight-edge mask thus moves with the substrate through the plumes causing a modulation of the deposition rate at a distance z perpendicular to the edge by a factor h(z). For a Gaussian angular distribution of atoms arriving at the plane of the substrate over the course of its trajectory through each plume, the straight-edge mask yields a profile where w is a width of the penumbra that depends on the mask height and the sputtering source characteristics, z 0 is the position of 50 % shading, and the function Φ rises monotonically from 0 to 1 according to its definition in terms of the error function (erf): The derivative of this function with respect to the argument x is the standard normal distribution, Complex masks that shadow the sputter targets have been made to create a linear profile, h(z) = z/w [4]. Our analysis is general and can be applied to any profile, but we primarily illustrate it with the error-function profile of the straight-edge mask for comparison with experimental results in Sec. 2. 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) λf /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 2θ 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 = λf m /d m = λh 2 f /(hd) = hr 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) = ∂h(z)/∂z, so that This geometrical focal length is that obtained if rays were to specularly reflect from the layers (as in a capillary array optic [7]). For the parameter equal to the product λ f used to design the layer periods d(r) according to Eq. (1), the focal length f m (z) given by diffraction of a lens cut at a particular location z will match the geometric length f h (z) generally only at one or two positions, if any, in the shadow profile. However, we note that, given Eq. (1), the focal length of the lens is inversely proportional to the wavelength of the wave-field, λ m , with and thus for any z there will be a particular wavelength λ m (z) for which the focal length f m = f h . From Eqs. (5) and (6) this wavelength is This operating wavelength λ m is the wavelength at which the Bragg condition is satisfied over the entire lens aperture. The resolution of a single (cylindrical) MLL cut at a position z is given by where P is the height of the unshadowed multilayer, P = λf /δ. From Eqs. (6) and (7) the expression for the resolution is and Thus cutting shorter lenses from the shadowed region of the deposition produces lenses of higher resolution, since the height reduces faster than the focal length. 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 with a corresponding focal length Plots of these functions are given in Fig. 3. The focal length increases with increasing h (or z) since the axial intercept of the tangent of h(z) increases with h. The operating wavelength λ m is described by a skewed Gaussian function that reaches a maximum approximately equal to 0.487 λ f /w at z − z 0 = 0.506 w where h(0.506w) = 0.694. Some implications of this are discussed below. 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 λf 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 ≈ 0.85w, for which f m (0.85w) = 1.44 w and λ m (0.85w) = 0.45 λ f /w.

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 2π accumulation of phase per period). For example, a linear relative change in the deposited thickness per unit thickness of material, β, leads to layers scaled according to (1 + βr)d(r − βr 2 /2), and a wavefront error (or optical path difference) given by OPD(r) = −βr 3 /f (see Sec. 6.1 of [6]). Assuming the change in rate β 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 β m (z) = β/h(z). The mask also leads to a coordinate scaling of the OPD according to The wavefront aberration of the lens is usually measured in terms of the angular pupil coordinate ρ = r m /f m (see for example, [8]), and the imaging properties such as the point spread function depend specifically on the normalised coordinate ρ. From Eq. (12), OPD m (ρ) = −β f 2 ρ 3 /h, which shows that the wavefront aberration increases in magnitude as the mask transmission factor h is reduced, due to the diminished height of the lens deposited over the time that deposition rate changes by a given amount. However, the phase error, ∆ϕ m (ρ) = (2π/λ m ) OPD m (ρ) scales inversely with the operating wavelength. As seen in Fig. 3 this wavelength may be larger or smaller than a particular design. For the case of the error-function profile the minimum phase error occurs at z − z 0 = 0.765w, at which h = 0.778. This is not necessarily the position that provides the lens with the smallest spot size, which will also depend on the NA (or resolution, as given in Eq. (8)) and the operating wavelength λ m . 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 ∆d of every layer period, independent of that period and therefore also independent of h(z). The nth layer will accumulate a placement error of n ∆d = r 2 ∆d/(2f λ). Comparing this with the placement error of the same layer for the linear change in deposition rate, which is β m r 2 /2 [6], we can infer a wavefront aberration of Here, the wavefront aberration expressed as a function of the normalised pupil coordinate is independent of the transmission profile h of the mask. Both processes-the linear consumption β of the target materials producing the deposited structures and the contraction or expansion of layers upon mixing by a value ∆d-lead to a similar effect on the lens, which is to cause an aberration that varies with the cube of the pupil coordinate. As an example, consider a lens with f = 2 mm and NA = 0.02 at a wavelength of 0.071 nm (the Mo K edge) to give a diffraction-limited resolution of 1.8 nm. A typical contraction of ∆d = −0.05 nm found for WC/SiC multilayers would lead to an OPD that varies as ρ 3 between 0 and 11 nm across the pupil of this lens. After removing the best-fit piston, tilt, and defocus terms, the peak to valley of the OPD would be 1.1 nm or a root mean square (RMS) error of 0.21 nm. This is an RMS error of 3 waves or 6π radian, but it could be offset by modifying the deposition recipe in the fabrication of the MLL to account for an apparent drift of the deposition rate by β = 3.5 × 10 −4 µm −1 , at least for one value of h. However, since the magnitude of these errors scales differently with h, it should be possible to determine the relative contributions of the two processes by measuring the aberrations of lenses cut at different positions z from a particular deposited structure, and then account for that in subsequent depositions. Currently, the contraction factor can be calibrated to a precision of about 0.01 nm and the deposition drift to less than 1×10 −4 µm −1 , so such a procedure may be expected to give a residual aberration of about a wave. Any remaining aberration could be corrected using a refractive phase plate [9].

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, λ/(2d(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) = λf /(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( As expected, the distance of the optical axis to the substrate is reduced to T h(z). At the position z where the MLL is cut, to achieve high efficiency, the optical axis must coincide with the tangent of T h(z) and thus the tilt of the optical axis must equal the gradient T h ′ (z). For small tilts of the optic axis, the optical axis will intersect the substrate (r = 0) a distance that is twice the geometrical focal length from the MLL, and so 2f h (z) = T h(z)/(T h ′ (z)), and thus Eq. (5) still holds. At the design wavelength λ 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), used at a different wavelength λ 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, λ 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) = λf /(T + r). In that particular case T is the distance of the optical axis below the substrate.

Manufacture of curved MLLs
For MLLs with large numerical apertures, where the resolution δ 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 ∂f m (z)/∂z = 1. From Eq. (6), and thus the condition of the curved MLL is met only when ∂λ m (z)/∂z = 0.
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 λ m (z) is maximised at z − z 0 = 0.506w, with h(z) = 0.694. The focal length at this position is f m (0.506w) = 0.988w. 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 λf in the layer deposition schedule. More generally, we see from Eq. (7) that a parabolic profile h(z) = (z/w) 1/2 gives λ m (z) = λf /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 ∂λ m (z)/∂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.

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 ∂λ m (z)/∂z = 0, a confocal pair of 1D lenses can be cut from one structure by finding two positions which share the same wavelength, λ 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 λ 2 is obviously restricted to be less than the maximum wavelength of 0.487 λf /w that occurs at z − z 0 = 0.506w. For example, choosing λ 2 = 0.47λf /w, the two solutions to λ m (z) = λ 2 are z − z 0 = 0.291w and z − z 0 = 0.724w, which have transmission factors h = 0.615 and 0.765 and focal lengths f m = 0.804w and 1.246w, 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 λf , 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 λ m (z A ; w A ) = λ m (z B ; w B ), using Eq. (10). While an analytical expression for z in terms of λ 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 λ 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 λ m = 0.425λf /w A can be cut at two positions: z A = 0.087w A and z A = 0.936w A , giving focal lengths of f m = 0.673w A and f m = 1.603w 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 (λ m = 0.425λf /w A ) at w B /w A = 1. The deposited multilayer under a mask edge with w B = 1.1w A , for example, will give a plot of λ m as a function of z B which is broader but lower than that shown in Fig. 3, and it intersects λ m = 0.425λf /w A at z B = 0.303w A and z B = 0.814w A , for which the focal lengths are f m = 0.871w A and f m = 1.396w 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. 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 λ 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.

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.
We have already encountered a model of a skewed Gaussian in Eq. (10) formed by multiplying a Gaussian by the error function Φ. More general is the skew normal distribution, formed by multiplying the normal probability distribution by a stretched version of Φ. Its definite integral is given by whereŵ is a width parameter, and T O (x, α) is Owen's T function [13] with a scalar stretch factor α, such that The widthŵ is equal to w of Eq. (2) only when α = 0, and in our example below it has slightly different relationships to those discussed in Sec. 1. A fit of this skewed error-function profile h s (z) to the measured profile is shown in Fig. 5(a), where it is found thatŵ = 1.52 mm and α = −1.7. While the difference between the curves obtained from the Gaussian and skewed Gaussian models is small, we find that using the skewed Gaussian model gives a much better prediction of where to cut the structure to obtain a desired operating wavelength. 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 thatŵ ≈ 1.15y m , a relationship that holds to the largest distance we tested of 1 cm. The widthŵ becomes smaller for a given y m when the target to substrate distance is increased, as expected from geometry.

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 mm (referred to here as the A profile) and y m = 1.32 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 λf = 0.237 nm · 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 λ = 0.071 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 λ m /(2NA) = 10 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 parametersŵ A = 1.99 mm, α A = −1.7,ŵ B = 1.52 mm, and α 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 λ 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 2θ, measured on a detector in the far field beyond the focus. If the beam wavelength is not equal to λ 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 2θ and the tilt angle ω of the lens, for a probe wavelength of 0.071 nm (Mo Kα), 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 λ m , as if to reflect rays to the focus at a distance f m = λf /λ 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 λ p , it would be deflected to a focus at f p = f m λ m /λ 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 ω = (r/f p − r/f m )/2. This is equivalent to the condition Thus, the operating wavelength of a lens can be determined from the slope of the maximum intensity mapped in terms of 2θ and ω for a given probe wavelength. Here, the sign of ω is in the same sense as 2θ, which is to say positive ω rotates the lens in the direction of increasing ray deflection angles to bring the thinner layers towards the focus. Measurements of the intensities as a function of 2θ and ω are shown in Fig. 6(b) for three positions on the B profile and labelled α, β, and γ. 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 λ m increases in the progression α, β, γ and that the β lens is obviously optimised to operate at the probe wavelength of 0.071 nm. The operating wavelengths range from λ m = 0.060 nm (20.5 keV) for α to 0.091 nm (13.6 keV) for γ, 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 ≈ 0.5, and where λ 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 β ′ in Fig. 6(a). Compared with the β lens, which has a focal length 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 ∆ 2θ decreases as the operating wavelength increases (in the sequence α, β, γ). 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 the position where the operating wavelength matches the probe wavelength, this range is twice the NA, but generally the range is only accessible by varying the tilt ω of the lens as depicted in Fig. 6. When used at their operating wavelengths, the NA of the lenses will decrease even faster than ∆ 2θ, in agreement with Eq. (9) (noting that the derivative h ′ (z) is decreasing with z where these lenses were cut). 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 withŵ B = 1.52 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 α, β, and γ as shown in Fig. 6(b), maps of diffraction efficiency as a function of 2θ and ω were recorded at positions along this structure using the monochromatised Mo Kα 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 ∆ 2θ(z) as given by Eq. (19). The offset x l0 was determined from the best fit of the function to the data. A plot of the measured operating photon energy E m = hc/λ 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 ofŵ B and α 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 δ 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.

Demonstration of lens pairs
For the multilayer structure produced here with the two mask edges, A and B, creating shadows withŵ A = 1.99 mm andŵ B = 1.52 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 β and β ′ lenses in the B shadow). Here, the larger lens cut from the long-f branch (similar to the β ′ lens) was oriented to focus in the horizontal direction and the lens cut from the short-f branch (similar to the β lens) focused vertically. Even though the vertical lens has a smaller aperture, it has a larger NA (NA = 0.0052 and f = 0.93 mm) than that of the horizontally-focusing lens (NA = 0.0036 and f = 2.31 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 nm × 6.8 nm (h × v) if the lenses were free of wavefront aberrations.  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 caseŵ B /ŵ 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 β ′ lens from the B shadow with NA = 0.0032 and f = 2.70 mm. As noted in Fig. 4, this lens, cut from the shadow with the smaller widthŵ, 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 nm × 11.1 nm and has an aperture with about twice the area of the pair taken both from the A shadow profile.

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)/(2h ′ (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, λf , 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 2θ and the tilt ω of the lens. Of course, if the probe wavelength equals the operating wavelength then there will be one value of ω where the lens diffracts across its entire pupil. Otherwise, the optimum ω for high diffraction efficiency scales linearly with 2θ 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 λf /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 λf . 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, axisymmetric 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.