## Abstract

We present an experimental and theoretical evaluation of an x-ray energy filter based on the chromatic properties of a prism-array lens (PAL). It is intended for small-scale applications such as medical imaging. The PAL approximates a Fresnel lens and allows for high efficiency compared to filters based on ordinary refractive lenses, however at the cost of a lower energy resolution. Geometrical optics was found to provide a good approximation for the performance of a flawless lens, but a field-propagation model was used for quantitative predictions. The model predicted a 0.29 Δ*E/E* energy resolution and an intensity gain of 6.5 for a silicon PAL at 23.5 keV. Measurements with an x-ray tube showed good agreement with the model in energy resolution and peak energy, but a blurred focal line contributed to a 29% gain reduction. We believe the blurring to be caused mainly by lens imperfections, in particular at the periphery of the lens.

©2009 Optical Society of America

## 1. Introduction

In recent years, x-ray optics have been suggested for energy filtering in small-scale applications with x-ray tubes. One option is to use Bragg reflection on mosaic crystals [1, 2, 3]. Another approach, which has been pursued in this work, is to use chromatic inline x-ray lenses [6, 7].

A major application of small-scale energy filters is found in medical x-ray imaging because the dose necessary for an x-ray examination varies with the energy spectrum [4]. Although fairly unselective and inefficient, absorption filtering has been the dominant method for optimizing the spectrum and lower the dose to the patient since the dawn of x-ray imaging [5]. Energy filters based on refractive x-ray optics [6, 7] have been shown to improve on energy resolution and thereby lower the dose, but the low efficiency of the lenses is problematic.

Refractive lenses can be manufactured for hard x-rays with reasonable curvatures if the weak refractive effect is divided over a large number of surfaces [8]. One implementation is the multi-prism lens (MPL) [9], which consists of two rows of prisms put on an angle in relation to the optical axis. The parabolic profile of these lenses lead to a rapidly increasing absorption towards the periphery of the lens, which limits the usable aperture, and hence the efficiency.

One way to increase lens transmission is to use Fresnel representations with lens material corresponding to a phase shift of integer steps of 2*π* removed. If this strategy is applied to the MPL, the extremely high aspect ratios that would otherwise be needed for a large-aperture Fresnel lens for hard x-rays are avoided. In the prism-array lens (PAL) [10], each prism in the MPL is exchanged for a column of smaller prisms. It is a generalization of the so-called Clessidra lens [11, 12], in that the successive displacement of the columns is a free parameter and is not limited by the prism height. The approximation of an ideal Fresnel pattern can thus be made better, and the focal length shorter for a given length of the lens. For small-scale applications, particularly the latter option is attractive as the length of the setup is often a limiting parameter.

Previous studies on PAL’s and Clessidras have focused on spatial resolution. In this study, the energy dependence of the PAL is investigated, and we present an energy filter based on the lens. Compared to energy filters based on ordinary refractive lenses, the efficiency has the potential to increase substantially, but the energy resolution might suffer.

Some focusing properties of the PAL were studied at a synchrotron radiation facility, whereas energy filtering was investigated with an x-ray tube in a realistic setup for small-scale applications such as medical imaging. Experimental results and expectations from geometrical optics have been compared to a physical-optics model based on field propagation. Physical-optics models have previously been presented for other types of x-ray optics, including waveguides [13, 19], Fresnel lenses [14], compound refractive lenses [15], and Clessidra lenses [12].

## 2. Geometrical optics approach to the PAL

#### 2.1. PAL focusing

Figure 1 shows the transition from MPL to PAL. In this figure and henceforth, *x* is the optical axis, *y* is the focusing direction, and *z* is the depth of the lens. The projection of the two prism rows of the MPL approximates a parabola with straight line segments [9]. It is a planar lens and therefore focuses radiation into a line focus. In a PAL, each large prism in the MPL is exchanged for a column of smaller prisms with height *h*, base *b*, and prism angle *θ* [10]. The columnar *y*-displacement increases in steps of *d* along the *x*-axis, and the focal lengths of the two lenses are the same if *θ* and *d* are equal. The projection of the PAL approximates a Fresnel lens, superimposed on a linear profile with slope *K*=*γb*/2*h* for *γ*>1. A large *γ*=*h/d* indicates a good approximation of the ideal Fresnel pattern, however at the cost of a steep linear profile and high absorption.

The refractive focal length of the PAL equals the MPL focal length, and has been derived previously as *F*
_{ref}=*dh/δ b* [10], where *δ* is the decrement of the real part of the refractive index from unity; *n*=1-*δ*+*iβ*. *F*
_{ref} has restricted validity as diffractive effects in the repetitive lens structure are not considered, and, instead, each prism column can be regarded a blazed phase grating [11]. Consider a monochromatic ray emanating from a point source on the optical axis and incident on a thin lens. At a location on the lens aperture *jd*<*y*<(*j*+1)*d*, a total of *j* gratings intersect the ray as it passes through the lens. The incidence and diffraction angles (*α _{i}* and

*α*

_{d}) at each point on the lens are related according to the grating equation [16],

*jmλ*=(sin

*α*

_{i}+sin

*α*

_{d})

*h*=(

*jd*/

*s*

_{o}+

*jd/s*)

_{i}*h*, where

*m*is an integer corresponding to the diffraction order and

*λ*is the wavelength, inversely proportional to the x-ray photon energy (

*E*).

*s*

_{o}and

*s*

_{i}are the source and image distances, and a diffractive focal length at

*m*=1 can be defined as

*s*

_{i}when

*s*

_{o}→∞;

If *b* corresponds to an x-ray phase shift of 2*πq*, where *q* is an integer, the blazing condition is fulfilled, and *kbδ*=2*πq* for a wave with propagation number *k*=2*π/λ*. The deflection of the first diffraction order then equals the refractive deflection of a single prism, and *F*
_{diff}=*F*
_{ref} [12]. Henceforth, *q*=1 is assumed, and parameters that fulfill the blazing condition are indicated with an asterisk, in particular the design energy of the lens (*E**).

If the blazing condition is violated, *F*
_{diff}≠*F*
_{ref}, and the question arises which one will be dominating. In accordance with [16], the fraction radiation in the mth diffraction order of a blazed saw-tooth grating can be shown to be proportional to [(2*πm*-*kbδ*)^{2}+(*kbβ*)^{2}]^{-1} for given *k, b, δ*, and *β*. If *kbδ*=2*π*, virtually all of the deflected intensity is found in the first diffraction order because (*kbβ*)^{2} is small. In fact, as long as *π*<*kbδ<*3π, or ^{2}/3*E**<*E*<2*E** for *δ*∝*E*
^{-2} and *b*=*b**, the peak intensity in the focal plane is found at *m*=1. If we define the focus to be at the peak intensity, the focal length of the PAL is equal to *F*
_{diff} in the stated interval. Note, however, that the intensity in the focus decreases in favor of secondary maxima when the blazing condition is violated, and the focal line is blurred along the optical axis.

#### 2.2. PAL energy filtering

Because the PAL is chromatic, a slit placed at *s**_{i} will transmit *E**, whereas other energies from a polychromatic incident beam are out of focus and preferentially blocked. This energy filtering scheme is outlined in Fig. 2, and is henceforth referred to as the PAL filter. It is similar to previously presented MPL filters [6, 7], but the energy resolution of the two filters can be expected to differ because of different transmission functions and focal length energy dependence. To predict the PAL filter performance in a qualitative fashion, we have set up a geometrical (GM) model, based on the thin lens approximation and the assumption from the previous section that *F* ∝ *E*.

Referring to Fig. 2, the PAL focuses radiation onto a slit at *s**_{i}, with a width *d _{s}*. The intensity gain of the focused radiation compared to the incident beam is [6]

where *y*
_{1} refers to the plane of the lens, and the integral is taken over the lens aperture (*D*
_{p}). The lens transmission has been derived previously [10]; *η*
_{t}(*y*
_{1})=exp(-*Kµ*|*y*
_{1}|), where *µ* is the linear attenuation coefficient. *η _{i}* is the part of the image (

*f*

_{i}) from each point on the lens that falls within the slit, i.e.

where *y*
_{2} refers to the plane of the slit. *κ*(*E*)=[*s**_{i}-*s*
_{i}(*E*)]/*s*
_{i}(*E*) accounts for blurring at *s**_{i}, and is zero at *E*
^{*}.

Consider a Gaussian source with a full-width-at-half-maximum (FWHM) *d*
_{0}, and *d _{s}* equal to the FWHM of the image (

*d**

_{i}), i.e.

*d*

_{s}=

*d**

_{i}=

*d*

_{0}

*s**

_{i}/

*s*

_{o}. For an infinitely large lens at

*E*

^{*},

where the factor 0.76 comes from integration in Eq. (3) over the FWHM of the Gaussian. Note that an infinite aperture is often a good approximation because the absorption increases fast towards the periphery. In principle, *G**_{∞} can be increased indefinitely by minimizing *h*, but clearly there are physical constraints. Dependence on lens material is represented by the factor *δ*
^{*}/*µ*
^{*}, which grows with a decreasing atomic number.

With the approximation *µ*(*E*)=*µ*
^{*}, it can be found numerically from Eq. (2) that

*G*
_{∞}(*E*)=0.5*G*
^{*}
_{∞} for ${G}_{\infty}\left(E\right)=0.5{G}_{\infty}^{*}\phantom{\rule[-0ex]{.2em}{0ex}}\mathrm{for}\phantom{\rule[-0ex]{.2em}{0ex}}\genfrac{}{}{0.1ex}{}{\kappa}{{d}_{s}K{\mu}^{*}}=\pm 0.86\phantom{\rule[-0ex]{.2em}{0ex}}\Rightarrow \phantom{\rule[-0ex]{.2em}{0ex}}{F}_{0.5}=\genfrac{}{}{0.1ex}{}{{F}^{*}}{1\pm 2\times 0.76\times 0.86\u2044{G}_{\infty}^{*}},$,

where *F*
_{0.5} is the focal length at *G*
_{∞}(*E*)=0.5*G**_{∞}. Assuming high gain and taking *F* ∝ *E*, we find an approximate expression for the energy resolution of a filter with an infinitely large lens,

where Δ*E* is the FWHM of the gain peak.

Previous studies have derived expressions for gain and energy resolution of an MPL filter, which with slight modifications are similar to Eqs. (4) and (5); *G*
_{MPL}=(2*πs*
^{2}
_{o}
*δ*/*µFd*
^{2}
_{0})^{1/2} and [Δ*E*/*E*]_{MPL}=1.7/*G*
_{MPL} [9, 17]. Hence, the filters are related through

where we have assumed equal lens material. As expected, the PAL energy resolution is lower than that of the MPL. We note that the PAL is advantageous in terms of gain for light materials at high energies. A low *γ* is favorable from an absorption point of view, but there is a tradeoff in representing the Fresnel pattern correctly.

## 3. Physical optics approach to the PAL: field propagation

#### 3.1. The parabolic wave equation

The time independent scalar wave equation can be derived from Maxwell’s equations under the assumptions that the electric and magnetic fields are decoupled, and that the radiation is linearly polarized [18]; *∂*
^{2}
*ψ/∂x*
^{2}
*+∂*
^{2}
*ψ/∂y*
^{2}+*k*
^{2}
*n*
^{2}(*x,y*)*ψ*=0. For a wave *ψ* travelling nearly parallel to the *x*-axis, Δ*x*≫Δ*y*, and the paraxial approximation is applicable. Hence, *ψ*=*ψ*
_{0} exp[-*ikn*(Δ*x*
^{2}+Δ*y*
^{2})^{1/2}]≈*u*(*x,y*)exp(-*ikn*
_{0}Δ*x*) with *u* varying slowly with *x* and *n*
_{0} being a reference refractive index (1 for vacuum) [13]. Insertion of this result into the scalar wave equation yields the parabolic wave equation,

which has been applied to several kinds of x-ray optics, including waveguides [13, 19] and Fresnel lenses [14]. The next two sections describe methods to solve Eq. (7).

#### 3.2. The Kirchhoff diffraction integral

Assuming uniform refractive index over the propagation distance (*n*=*n*
_{0}), a Fourier transform of Eq. (7) with respect to *y* yields -2*ikn*
_{0}
*dU/dx-υ*
^{2}
* _{y}U*=0, with

*U*being the Fourier transform of

*u*, and

*υ*the spatial frequency in the

_{y}*y*-direction. The solution of this differential equation is

where *U*
_{0} is the transform of the initial field. The inverse Fourier transform is a convolution, *u*=(*u*
_{0}∗*h*)(*y*), known as the paraxial Kirchhoff diffraction integral. It can be solved directly in the Fourier domain, according to Eq. (8), to speed up the calculation.

#### 3.3. Finite differences

Let the *x*- and *y*-axes within the computational window be divided into *P* and *Q* segments, respectively, so that the electric field is discretized into local fields *u ^{p}_{q}* with

*p*∊ {1,2,…,

*P*} and

*q*∊ {1,2,…,

*Q*}. By using a Taylor expansion, the field and its derivatives at the center of segment

*p*can be expressed as

$$\genfrac{}{}{0.1ex}{}{\partial u}{\partial x}=\genfrac{}{}{0.1ex}{}{1}{2\Delta x}[{u}_{q}^{p+1}-{u}_{q}^{p-1}]+O\left(\Delta {x}^{2}\right),\mathrm{and}$$

which is similar to the Crank-Nicolson scheme of finite differences [20].

Similar to previous studies [13], we assume Dirichlet boundary conditions,

i.e. the field far away from the optical axis is a plane wave. This approximation leads to reflections at the boundary; transparent boundary conditions would be more accurate, but are computationally intense. In case of propagation over a lossy medium, the computational window can instead be increased so that most of the reflected wave is absorbed and can be neglected.

Substituting Eq. (9) into Eq. (7) and given Eq. (11), *u*
_{q+1} can be calculated from *u*
_{q} with a system of linear equations.

#### 3.4. Field-propagation model

In general, a PAL is not a thin lens and *F* ∝ *E* does not hold. Therefore, for more reliable, quantitative predictions than provided by the GM model, we have used the methods described in the last three sections to assemble a field-propagation (FP) model of the PAL filter. Note that the FP model too has restricted validity, mainly because of the paraxial approximation.

Equation (8) was applied outside the lens, where large regions of uniform refractive index are found. Inside the lens, we instead used finite differences because of higher speed and accuracy for a non-uniform refractive index and small steps in *x*. A monochromatic finite x-ray source of energy *E* was treated as a superposition of point sources, incoherent relative to each other, and the intensity at any point in the setup was found from the field *u*
_{E} as *I*
_{E}(*x,y*)=∑|*u*
_{E,y0} (*x,y*)|^{2}, where *y*
_{0} is a point in the plane of the source. Accordingly, a polychromatic x-ray source was regarded a superposition of monochromatic sources over a distribution of energies. G was found by integrating *I* over a slit in the image plane.

In both the GM an FP models, published linear absorption coefficients [21] and semiempirical data on atomic scattering factors [22] were used to calculate the complex refractive index. Compton and Rayleigh scattering were treated as absorption and added after propagation.

## 4. Measurements and experimental setups

#### 4.1. The experimental PAL

A PAL according to Figs. 3(a) and 3(b) was used for the experimental part of this study. Compared to the previously presented PAL [10], the design has been altered by converting the isosceles prisms into right-angled half-prisms. This sparser arrangement is favorable for manufacturing due to fewer narrow corners and acute angles. The discussion in section 2 is still valid, provided tan*θ*′=tan*θ*/2, where *θ*′ is the prism angle of the modified lens. Silicon was chosen as lens material because of readily available manufacturing methods, but, according to Eq. (4), higher gains can be expected for lighter materials, such as plastics.

The PAL was fabricated by a commercial vendor (Silex Microsystems, Järfälla, Sweden) using deep reactive ion etching (DRIE) of silicon according to the so-called Bosch process. It is a plasma-based cyclic process, which can be used to attain micro-structures with high aspect ratios [23]. For masking, the surface of the silicon wafer was oxidized, and the oxide was patterned using standard lithographic methods. As can be seen in Fig. 3(a), the lens exhibits structure failures, in particular over etch of the convex prism corners. The nonoptimized manufacturing process also limited the depth of the lens to approximately 160 *µ*m.

The lens was designed with F=178 mm for *E**=23 keV and *γ*=3.75, which yields *b*=59 *µ*m, *d*=1.6 *µ*m, *h*=6 *µ*m, *θ*=5.8°, and *K*=18.4. A physical aperture of *D*
_{p}=194 *µ*m was chosen, resulting in a total of 960 prisms arranged in *N*=63 columns with 2×16 prisms in the first column, and a lens length of *L*≈9 mm. The lens was equipped with support structures, *t*
_{1}=100 *µ*m and *t*
_{2}=8 *µ*m thick, at the entrance and exit of the lens, and for each prism column, respectively, thus adding a constant term of 2*t*
_{1}+*Nt*
_{2} to the linear projection. To facilitate etching, the columns were separated by 14 *µ*m.

#### 4.2. PAL focusing, synchrotron setup

A characterization of some PAL focusing properties was made with synchrotron radiation (SR) at the optics evaluation beamline BM05 at the European Synchrotron Radiation Facility (ESRF). The source was 270 *µ*m FWHM in *y*, and the beam was kept at 23 keV by a double crystal Si(111) monochromator. The lens was mounted on precision stages for lateral translation and tilting at *s*
_{o}=40 m, yielding *s**_{i}≈*F**, and the beam was collimated in y and z by an adjustable slit. As it can be expected that focusing properties vary across the aperture, the slit could be used to restrict the illumination to a small part of the lens.

The beamline was equipped with a FReLoN *x*-ray CCD camera. We scanned the camera in the *x*-direction to find an intensity maximum, and *s*◇_{i} was defined as the distance from the entrance of the lens to the maximum. All measurements at *s*◇_{i} are denoted with a diamond and are ideally equal to the corresponding parameters at *E**. *G*◇ was measured over a *d _{s}*=14 µm distance in

*y*, corresponding to a reasonable slit size in an energy filter. The camera had a pixel size of 0.68

*µ*m, but the line spread function (LSF) was measured to 5.5

*µ*m FWHM from a slanted edge image. This is substantially larger than

*d**

_{i}, and all measurements are subject to blur. To facilitate comparisons, results from the FP model were convolved in the

*y*-direction with the LSF. In the

*z*-direction, the free beam above the lens added an almost constant background to the measurement, and we therefore subtracted from the measured image the mean value in an area far away from the lens focus.

#### 4.3. PAL filtering, small-scale bremsstrahlung setup

The experimental setup depicted in Fig. 3(c) was used to measure the filtering performance of the PAL filter. The bremsstrahlung (BS) source was a tungsten target x-ray tube (Philips PW2274/20) at 33 kVp acceleration voltage. A 3.5(±0.2)° anode angle resulted in a 24.5(±1.5) *µ*m source size in y as specified by the manufacturer. An edge scan confirmed this size and determined the shape to approximately trapezoidal.

Precision stages for mounting the lens were located at *s*
_{o}=585 mm, and an 11 *µ*m 23 keV image of the source can be expected at *s*
_{i}=256 mm for a thin lens. A 200 µm slit collimated the beam in *y* upstream of the lens, and two 50 *µ*m slits on either side of the lens restricted the beam in the *z*-direction.

We used a cadmium zinc telluride (CZT) solid-state detector (Amptek XR-100T-CZT) with a near 100% detection efficiency, negligible hole tailing, and an energy resolution better than 0.5 keV in the considered energy interval. The image was obtained by differentiating the profile of a 1-*µ*m-step tantalum edge scan. To avoid noise amplification at differentiation, the profile was fitted to an error function with an additional slope and displacement; Φ(*x*)=*p*
_{1} erf[*p*
_{2}(*x*+*p*
_{3})]+*p*
_{4}
*x*+*p*
_{5}. The derivative is a Gaussian with a constant background, which is a good approximation if the ideally trapezoidal image is distorted by lens imperfections.

The edge was scanned in 10 mm steps in *x* to find *s*◇_{i}, and *G*(*E*) was again calculated over a *d*
_{s}=14 *µ*m slit. The energy resolution is Δ*E/E*◇, where *E*◇ is the peak energy.

## 5. Results and discussion

#### 5.1. PAL focusing, synchrotron setup

Results from the measurement in the SR setup are summarized in the first row of Table 1, and results obtained with the FP and GM models are shown in the subsequent three rows. The FP result in row two includes the LSF of the CCD camera, and by comparing to row three it is evident that the limited resolution was responsible for a large portion of the line width, and also degraded the gain. Row four contains results from the GM model, i.e. Eq. (2), and a comparison to row three shows that the two models agree well. The small deviation in *s*◇_{i} can be assigned to the thin lens approximation in the GM model. The measured *s*◇_{i} is 40 mm longer than predicted, but it is biased by a large measurement uncertainty of ±20 mm.

A CCD image at *s*◇_{i} is shown in the top part of Fig. 4(a). The focusing efficiency varies in z, and an intensity maximum is located at approximately 60 *µ*m depth in the lens. A cross section of the focal line at this depth is shown in the lower part of Fig. 4(a), together with results from the FP model. The measured focal line intensity is 40% lower than predicted, and *d*◇_{i} is a factor 2.1 larger. Put together, these deviations yield a 19% lower *G*◇. Figure 4(a) also shows abundant background radiation that extends over a larger area than *D*
_{p}.

The upper part of Fig. 4(b) shows *d*
^{◇}
_{i} as a function of an increasing collimator slit. The measurements correspond well to the FP model predictions for the central 25 *µ*m part of the lens, but *d*
^{◇}
_{i} increases rapidly with the active part of the aperture. The FP model predicts an almost constant *d*
^{◇}
_{i}, close to the LSF of the CCD camera. As can be seen in the lower part of Fig. 4, the increased *d*
^{◇}
_{i} at larger apertures is accompanied by a gain that levels off when *d*
^{◇}
_{i} approaches *d*
_{s}=14 µm. In fact, the peripheral half of the lens contributes only 10% to the total gain, compared to 30% as predicted by the model. This effect is owing to the particular choice of *d*
_{s}, but also to a lower focusing efficiency towards the periphery of the lens.

#### 5.2. PAL filtering, bremsstrahlung setup

The lower part of Table 1 summarizes the measurements and model predictions in the BS setup. The *z*-collimator slits for the measurements were at approximately 60 *µ*m depth in the lens in accordance with the SR measurements. The measured *s*◇_{i} was 33 mm longer than predicted by the FP model, but the exact location of the *x*-ray tube focal spot was only known within ±10 mm, and the scan in *x* to find *s*
^{◇}
_{i} was relatively coarse. *d*
^{◇}
_{i} was 2.1 times wider than predicted, but the comparison was affected by uncertainty and instability of the source size, shape, and lateral position. The two models agreed well in *s*
^{◇}
_{i} with a correction for the lens length, but deviated slightly in *d*
^{◇}
_{i}.

The PAL filtered spectrum with a 14 *µ*m slit is plotted in Fig. 5(a) as a function of energy together with the unfiltered spectrum. These curves yield *G*◇(*E*) by division, with the result shown in Fig. 5(b) along with the model predictions. Compared to the FP model, the measured *G*◇ deviates -29%, *E*
_{p} is in close agreement, and the energy resolution deviates less than 4%. The GM model predicts a slightly lower energy resolution and higher gain, mainly because it does not take blurring of the focal spot at *E*≠*E** into account. Both models predict *E*
_{p}=*E**+0.5, which was not seen in the measurement, but it may be accounted for by the coarse *x* scan.

The PAL-filtered spectrum predicted by the FP model is approximately twice as broad as previous results for an MPL filter at similar gain and peak energy [6]. Because of the finite apertures, there is a slight deviation from Eq. (6). The MPL study reports a high gain because of no support structures and epoxy instead of silicon. As a best-case scenario for an equal-focal-length PAL filter in the BS setup, an infinitely large lens made from epoxy (*Z*≈6) and without support structures would have a gain *G**_{∞}=126, which is 6.2 times higher than a corresponding MPL (Eqs. (4) and (6)).

#### 5.3. Discussion of possible lens imperfections

Similar deviations from the FP model are seen in both the SR and BS measurements; *s*
^{◇}
_{i}>*s*
^{*}
_{i}, *d*
^{◇}
_{i}>*d*
^{*}
_{i}, *G*
^{◇}<*G*
^{*}, and measured background radiation that is not predicted by the model. Some of these metrics are coupled; the reduction in gain is an effect of the wider *d*
_{i} and some of the transmitted radiation being deflected into background radiation, and parts of the wider *d*
_{i} is an effect of the larger magnification factor at the longer *s*
_{i}. Apart from the discussed uncertainties for the respective setups, a large part of the deviations are likely caused by lens imperfections.

The varying focusing efficiency in *z* and visual inspection of the lens hint on two partly competing manufacturing problems: (1) over etch in the horizontal direction, which is pronounced at the lens surface (low *z*) because of longer exposure to the etching species, and (2) insufficient etching at large etching depths because of transport problems in the high aspect ratio structure (depletion of etching species and insufficient removal of etching debris). Over etch in the *x*-direction and/or material deposition in *y* could result in systematic figure errors and a longer *F*
_{ref} as *F*
_{ref}∞*h/b*. *F*
_{diff}, on the other hand, is only affected by a change in periodicity. Moderate figure errors therefore cannot move the intensity peak along *x*, but the peak may be flattened and stretched toward higher *x*.

The wider *d*
_{i} and the background radiation may also be partly owing to figure errors. Additionally, scattering caused by random phase errors from surface roughness may play a role; even modest roughness could cause large errors because of the steep prism angle. The declining focusing efficiency at the periphery of the lens is reasonable since a peripheral beam passes a larger number of imperfect prisms and is also in accordance with previous studies [10]. Rayleigh and Compton scattering, on the other hand, are not likely to have caused any of these effects because of large angular spectra.

## 6. Conclusions

We conclude that energy filtering with a PAL works well in small-scale applications, which opens up the possibility to implement the filter for dose reduction in medical imaging.

For a setup based on a silicon PAL and an x-ray tube, the field-propagation model predicted gain and energy resolution of 6.5 and 0.29, respectively, at a peak energy of 23.5 keV. Experimental data showed almost perfect agreement in energy resolution and peak energy, but suggested a two times broader image, which contributed to a gain reduction of 29%. These deviations can be explained by lens imperfections, in particular at the periphery of the lens, and there is a large potential for improvement by an optimized manufacturing process or a lens design with larger prism angles. Additionally, the efficiency of the filter could be improved substantially in future upgrades with a lighter lens material and less support structures.

Predictions by geometrical optics under the thin lens approximation and assuming a focal length proportional to the x-ray energy were in good agreement with the field-propagation model for a flawless lens and can be used for fast calculations. We found that the energy resolution of the PAL filter is at least 50% lower than for filters based on ordinary refractive lenses, but the efficiency of an optimized PAL would be superior.

## Acknowledgments

Our work on prism-array lenses was funded by the Swedish Research Council. We acknowledge the ESRF for provision of synchrotron radiation facilities and we would like to thank L. Peverini, J. Y. Massonnat, and E. Ziegler for assistance in using beamline BM05. We also thank U. Vogt for valuable discussions on field propagation.

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