1. INTRODUCTION

Elemental imaging using x-ray fluorescence is a nondestructive technique capable of bringing out important information in many fields [1]—from materials science to cultural heritage [2–5] and planetary surface analysis [6]. A first class of methods by which x-ray fluorescence images can be obtained consists of the two-dimensional scanning of a beam on the object and collection of the fluorescence x-rays at every point of the map [7,8]. It requires a focusing device and a minimum of a two-axis motorized scanner. A second class of methods is based on direct x-ray fluorescence imaging with no moving parts. These direct imaging methods are more suited to situations where mechanical simplicity is critical. Among the different imaging devices that can be used, the square-pore micro-channel plate x-ray optics (MPO), sometimes referred to as “square multi-channel plate optics,” “multi-pore optics,” or “lobster-eye optics,” is one of the most attractive because of its efficiency given by the corner cube effect [9–12], in particular when compared with straight polycapillary optics [13,14] used for $1:1$ imaging. The planar MPOs, to which this study will be restricted, offer a magnification of one to cover a surface area commensurate with the size of the detector and provide an additional degree of freedom compared to the spherical MPOs, with the possibility of changing the working distance without changing the magnification.

The understanding of MPO properties relies first on the modeling of perfect structures for which the 1991 Chapman et al. publication [11] is a reference. The intention of the present article is to bring several improvements to this model and examine the effect of short distances, which are not considered in the work of Chapman et al. The effect of defects related to the manufacturing technology is beyond the scope of our article with the exception of the modification of reflectivity by surface roughness.

In Chapman et al. [11], two approximations of the reflectivity curve are used—depending on the x-ray energy—and are defined with a single parameter. We will show that a single simplified description might be used with two parameters: the first corresponding to the real part of the reflecting material refractive index and the second related to both absorption (imaginary part of the refractive index) and surface roughness. Using this new simplified curve, the behavior of the MPO is much better described in a large range of energies, in particular for heavy reflecting materials like platinum or iridium.

In addition, the Chapman et al. model is a continuous model where the pore size is considered as small enough to replace summations by integrals. For this reason, the only contributions to the central spot of the point spread function (PSF) considered by Chapman et al. are the ones corresponding to an odd number of reflections in each of the two perpendicular planes containing the pore faces. It will be shown here that two additional contributions cannot be neglected when the source-MPO distance is decreased. The first additional contribution is related to reflections on the central row and column of pores relative to the optical axis and is visible at medium distance. The second contribution comes from direct transmission by the central pores closest to the optical axis and becomes important only at very small working distances. A new semi-continuous model, based on the use of the two-parameter reflectivity curve and taking into account the three contributions mentioned above, will be described in the first part of this article. Using this model, the influence of geometrical parameters and x-ray energy on the PSF central spot integrated intensity and profile is shown. The modifications of other features of the PSF, such as the two characteristic perpendicular wings and the pseudo-background related in part to the direct transmission through the pores, are also investigated.

In the second part, it will be shown that the PSF is not unique; it is a periodic function of the source coordinates reflecting the periodicity of the planar MPO itself. It will be shown that the influence of this periodicity is enhanced at small distances and can be reproduced in most cases using a discrete model. The semi-continuous model can, however, be used in the case of a modified PSF where a small square with a size (defined as the side length) equal to the pore periodicity is used as a source instead of a point. Finally, in the third part, the trends shown in part two for the PSF are illustrated by images simulated using ray tracing and improvements to the standard MPO are proposed.

Throughout this article, ray-tracing simulations are used to support our modeling because the analysis carried out in this paper is based on geometrical optics. Using the very crude criterion of an angular diffraction contribution on the order of $\lambda /D$, where $\lambda$ is the x-ray wavelength and D is the pore size, we believe that it should be valid in the energy range of the fluorescence of most of the elements and for pore sizes of tens of microns. However, it might be necessary to use a wave theory, taking into account wave-guide effects, diffraction, eventual interference between pores, and the effect of partial coherence for lower energy fluorescence and smaller pore sizes, but it is beyond the scope of this paper.

2. RAY TRACING

Ray-tracing simulations (sometimes referred to as Monte Carlo simulations) are used to produce PSFs and quantities that are eventually compared with the results of analytic models. These simulations include the calculation of x-ray reflectivity, which is done using the standard matrix method [15]. Anomalous scattering and absorption are taken into account through the calculation of the atomic scattering factor as well as surface and interface roughness through a static Debye–Waller factor with a single parameter, which is the root mean square (rms) roughness $\sigma$. Most of the ray-tracing simulations are carried out with a monochromatic point source to obtain the PSF. The remainder use two-dimensional object emitting monochromatic x-rays as a source. The results shown do not depend on the number of rays used except for the shot noise. The initial number of rays (between 10 million and one billion) was, for each kind of simulation, large enough to make the shot noise negligible or at least not cumbersome. Emission was considered isotropic in the maximum $\pm 0.1\mathrm{rad}$ range used in the simulations.

3. SEMI-CONTINUOUS MODEL

The model developed below will be called “semi-continuous” because at some places sums over the pores of the MPO are replaced by integrals (i.e., pores are considered infinitely small), while at other places the discrete aspect of the MPO is taken into account.

A. Parameters

Figure 1 shows the main geometrical parameters used in the model. In the following, $D$, $T$, and $t$ are the pore size (square side length), the period of the MPO square lattice, and the MPO pore length, respectively. The distance between the source and the MPO is $l_s$ and is equal to $l_i$, the distance between the MPO and the image plane (see also Fig. 15). $l_s$ and $l_i$ are measured with respect to the plane, which is half way from the MPO entrance plane and the MPO exit plane. The MPO in-plane dimensions and, as a result, the total number of pores, are considered to be infinite; edge effects will not be discussed. Additional material parameters are used through the inner pore coating reflectivity properties.

 

Fig. 1. MPO geometrical parameters. (a) Point source imaged in the $1:1$ arrangement. (b) Detail showing the precise position $(x_0,y_0)$ of the optical axis with respect to the nearest pore center.

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B. Reflectivity

In order to enable analytical calculations, a two-parameter simplified (linearized) reflectivity curve is used instead of the reflectivity curve calculated using the standard matrix method (Fig. 2). These two parameters are the total reflection critical angle ${\gamma }_c$ and $\overline{R}$, which is the reflectivity averaged between 0 and ${\gamma }_c$. The total reflection critical angle is written as follows:

 

Fig. 2. Example of the simplified version of the reflectivity curve used for analytic calculations (blue line) compared with the simulated reflectivity of an Ir layer on a silica substrate at 6400 eV (red line).

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We define $\overline{R}(\lambda )$ as

${\gamma }_c(\lambda )$ and $\overline{R}(\lambda )$ are the two parameters defining a simplified reflectivity curve. In the calculations, however, it will be more convenient to use $\mathrm{\Delta }R(\lambda )$ instead of $\overline{R}(\lambda )$, with $\mathrm{\Delta }R(\lambda )=2(\overline{R}(\lambda )-1)$ (Fig. 2). For high absorption and because of the additional effect of roughness, it might occur that Eq. (2) gives a value of $\overline{R}$ lower than 0.5. In this case, Eq. (3) gives negative values for $R$ when $\gamma$ is close to ${\gamma }_c$. To avoid this, the ${\gamma }_c$ value used in the model is replaced by an effective value ${\gamma }_{c\mathrm{eff}}$ and $\overline{R}$ is replaced by ${\overline{R}}_{\mathrm{eff}}$, as follows:

It appeared that there was a problem with commonly used $f^{\prime }$ and/or $f^{\prime \prime }$ tables, such as Henke or Cromer–Liberman tables, in the region of M edges of heavy elements such as Ir or Pt eventually giving negative values of the real part of the atomic scattering factor. For these elements, the values published by Chantler in 2000 [16] were used. Finally, we would like to point out that the simplified analytical approximation of the x-ray reflectivity discussed in this paragraph might be used in grazing incidence applications beyond the context discussed in this article.

C. Point Spread Function Central Spot Integrated Intensity

A typical simulated MPO PSF is shown on Fig. 3. The main features are a central spot, which is the desirable part for image formation, two perpendicular lines forming a cross, and weaker intensity in the quadrants delimited by the cross. All these characteristic parts can be found in experimental PSFs as well [12]. In this section, we will calculate ${\mathrm{\Omega }}_{\mathrm{eff}}$, the effective solid angle acceptance of the MPO, which multiplied by the source intensity per unit solid angle gives the intensity in a square twice the size of the MPO period, i.e., $2T$. The intensity outside this square is not taken into account because it is not properly focused in the image plane.

 

Fig. 3. Ray-tracing simulated PSF. The reflecting material is a 25 nm Ir layer on ${\mathrm{SiO}}_2$, with both the surface and interface roughness equal to 2 nm. $E=15\mathrm{keV}$, $l_s=0.05\mathrm{m}$, $D=20\mathrm{\mu m}$, $T=26\mathrm{\mu m}$, and $t=4.8\mathrm{mm}$. Normalized intensity in log scale.

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The central spot intensity is usually considered as coming from an odd number of reflections in the two (x,z) and (y,z) planes. At a long working distance, this contribution is the most significant; however, there are two other contributions to the central spot. Figure 4 shows the distribution in the entrance plane of the MPO of the rays contributing to the PSF central spot as a function of the number of reflections in the (x,z) and (y,z) planes noted as $(n_x,n_y)$. In this figure, the MPO parameters and the energy of x-rays are such that a maximum of one reflection can occur in each plane. The (odd,odd) reflections—(1,1) in the figure—come from four equivalent two-dimensional regions of the MPO, so this contribution will be named ${\mathrm{\Omega }}_{2\mathrm{d}}$. Note that these four regions correspond to the four regions visible on Fig. 6 of the 1991 Chapman et al. publication [11]. The second and newly considered contribution is coming from (odd,0) and (0,odd) reflections [(1,0) and (0,1) in the figure] and corresponds to the one or two rows and columns of pores, which are the closest to the optical axis. This contribution will be noted ${\mathrm{\Omega }}_{1\mathrm{d}}$ as the MPO regions from which it is coming are linear. The last contribution, related to the one to four pores, which are the closest to the optical axis, is noted ${\mathrm{\Omega }}_{0\mathrm{d}}$ as it comes from a very localized region. We have the following relation:

 

Fig. 4. Distribution of rays, contributing to the PSF central spot, at the entrance surface of the MPO. In the notation $(n_x,n_y)$, $n_x$ is the number of reflections in the xz plane, and $n_y$ is the number of reflections in the yz plane. Because of the parameters used here, $n_x$ and $n_y$ are either 1 or 0. Parameters: $D=20\mathrm{\mu m}$, $T=26\mathrm{\mu m}$, $t=1.2\mathrm{mm}$, $l_s=10\mathrm{cm}$, $E=6400\mathrm{eV}$, and reflective layer: Ir.

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${\mathrm{\Omega }}_{2\mathrm{d}}$ is first calculated with a method close to, and largely inspired by, the method of Chapman et al. [11] using small angle approximations. If $D/l_s\ll {\gamma }_c$, the rays entering a particular pore of angular position with respect to the source $({\theta }_x,{\theta }_y)$ [see Fig. 1(a)] are considered parallel and their direction is defined by the same angles as the pore angular position. The beam entering this particular pore, which is submitted to $n_x$ reflections in the (x,z) plane and $n_y$ reflections in the (y,z) plane, will have a dimension ${\delta }_{\mathrm{nx}}({\theta }_{rx})$ along x and ${\delta }_{\mathrm{ny}}({\theta }_{ry})$ along y. When the number of reflections is greater than 0, ${\delta }_{\mathrm{nx}}({\theta }_{rx})$ and ${\delta }_{\mathrm{ny}}({\theta }_{ry})$ are given by the following expression:

The corresponding effective collected solid angle corresponding to $(n_x,n_y)$ reflection in the same single pore is

The total effective solid angle corresponding to $(n_x,n_y)$ reflections, $\mathrm{\Omega }(n_x,n_y)$, is then obtained by the following integration:

Finally, as this contribution to the central PSF spot corresponds to odd numbers of reflections in both the xz and yz planes,

 

Fig. 5. Plots of $S_{\mathrm{odd}}^2(\alpha ,\mathrm{\Delta }R)$ as a function of $\alpha$ for different values of $\mathrm{\Delta }R$ showing the influence of absorption and roughness induced loss on ${\mathrm{\Omega }}_{2\mathrm{d}}$.

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In [11], only the extreme cases are considered, $\mathrm{\Delta }R=0$ in the case of high energies and $\mathrm{\Delta }R=-1$ for low energies. The refinement brought by the new two parameters reflectivity curve separates the effect of absorption/roughness from the effect of the energy dependent critical angle. In the case of heavy reflecting materials, for example, the absorption is neither low nor high in a wide energy range; hence, using this two-parameter reflectivity curve is a necessity.

When considering a particular reflecting material, the dependence of $\mathrm{\Delta }R$ and ${\gamma }_c$ with the x-ray energy can be calculated. The example of Ir is shown on Fig. 6. The refractive index has a simple $1/E$ dependence with the x-ray energy, except for the energy region of the M absorption edges. $\mathrm{\Delta }R$ is a more complex function of the x-ray energy—it is much more sensitive to the presence of absorption edges, such as the Ir L edges in the 11–13 keV range and the M absorption edges in the 2–3 keV range. It is also clearly dependent on the surface rms roughness $\sigma$. In the 2–3 keV region, it was necessary to use effective values of $\mathrm{\Delta }R$ and ${\gamma }_c$ in the case of Ir with a 2 nm rms roughness, calculated using Eq. (4).

 

Fig. 6. External total reflection critical angle ${\gamma }_c$ and reflectivity loss $\mathrm{\Delta }R$ as a function of energy in the case of Ir reflective material. $\mathrm{\Delta }R$ is shown for two values of the rms surface roughness $\sigma$. The insert shows the 2–3 keV region where effective values ${\gamma }_{c\mathrm{eff}}$ [Eq. (4)] replace, in the case of the 2 nm roughness (dotted line), the critical angle value used in the case of a perfect flat surface (continuous line). In this region and in the case of the 2 nm rms roughness, $\mathrm{\Delta }R=\mathrm{\Delta }{R}_{\mathrm{eff}}$ is constant and equal to $-1$.

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Using the data shown on Fig. 6, it is possible to plot ${\mathrm{\Omega }}_{2\mathrm{d}}$ as a function of the x-ray energy on one axis and the ratio $t/D$ on the other axis. The map obtained in the case of an Ir layer with a rms roughness of 2 nm is shown in Fig. 7. This map can be a guide for the choice of the MPO $t/D$ ratio, which will depend on the spectral band of interest. For example, large values might be chosen for higher fluorescence energies. As already mentioned, there is no dependence of ${\mathrm{\Omega }}_{2\mathrm{d}}$ on the working distance $l_s$, so for a particular reflecting material all the information on ${\mathrm{\Omega }}_{2\mathrm{d}}$ is in this map.

 

Fig. 7. ${\mathrm{\Omega }}_{2\mathrm{d}}$ as a function of x-ray energy and $t/D$ in the case of an Ir top layer with a 2 nm rms roughness. Discontinuities induced by M and L absorption edges of Ir are clearly visible.

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The ${\mathrm{\Omega }}_{1\mathrm{d}}$ contribution to the PSF central spot coming from the row and column of pores is shown in dark blue in Fig. 4. It corresponds to an odd number of reflections in one direction and a direct transmission in the other. A one-dimensional integration is carried out in this case, giving for the row in the x direction

The last contribution is related to the pinhole-like transmission by the central pore

Finally, the total effective solid angle collection of the MPO is given by

Ray-tracing results are compared in Fig. 8 with the model of Eq. (17) in a case where the dominant term in ${\mathrm{\Omega }}_{\mathrm{eff}}$ is ${\mathrm{\Omega }}_{2\mathrm{d}}$, because $l_s$ is large ($l_s=0.1\mathrm{m}$). For this reason, the plots of Fig. 8 are similar to the ones of Fig. 5. The influence of the two other terms, ${\mathrm{\Omega }}_{1\mathrm{d}}$ and ${\mathrm{\Omega }}_{0\mathrm{d}}$, is visible on Fig. 9. ${\mathrm{\Omega }}_{2\mathrm{d}}$ does not depend on $l_s$ so the $l_s$ dependency of ${\mathrm{\Omega }}_{\mathrm{eff}}$, clearly visible on this figure, is related to ${\mathrm{\Omega }}_{1\mathrm{d}}$ and ${\mathrm{\Omega }}_{0\mathrm{d}}$.

 

Fig. 8. ${\mathrm{\Omega }}_{\mathrm{eff}}/{\gamma }_c^2$ as a function of $\alpha =t{\gamma }_c/D$. Variations of $\alpha$ are obtained for each energy by changing the MPO thickness $t$, with D being constant. Ray-tracing simulations results (symbols) are compared with the model of Eq. (17) (continuous lines). Parameters: Ir layer with $\sigma =2\mathrm{nm}$, $l_s=10\mathrm{cm}$, $t=1.2\mathrm{mm}$, $D=20\mathrm{\mu m}$, and $T=26\mathrm{\mu m}$.

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Fig. 9. ${\mathrm{\Omega }}_{\mathrm{eff}}$, ${\mathrm{\Omega }}_{2\mathrm{d}}$, ${\mathrm{\Omega }}_{1\mathrm{d}}$, and ${\mathrm{\Omega }}_{0\mathrm{d}}$ dependence with $l_s$. Ray-tracing simulations results (symbols) are compared with the model of Eqs. (12), (14), (16), and (17) (continuous lines). Parameters: Ir layer with $\sigma =2\mathrm{nm}$, $t=1.2\mathrm{mm}$, $D=20\mathrm{\mu m}$, $T=26\mathrm{\mu m}$, and $E=6400\mathrm{eV}$.

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D. PSF Central Spot Profile

The general shape of the central spot is a pyramid, taking the intensity as a third dimension. However, this is the shape related specifically to ${\mathrm{\Omega }}_{2\mathrm{d}}$; the two other contributions to the PSF central spot have different profiles. These three profiles can be calculated using the same simplified reflectivity model that was used to determine ${\mathrm{\Omega }}_{\mathrm{eff}}$. For this purpose, two functions of x or y have to be calculated as follows:

The profiles obtained using Eq. (20) in the x direction for $y=0$ are compared with the results of ray-tracing simulations in Fig. 10. This figure shows that the modifications induced in the central spot profile by varying the MPO thickness are well reproduced by the model.

 

Fig. 10. Central spot profiles for a series of MPO thicknesses t. Ray-tracing simulations results (symbols) are compared with the model of Eq. (20) (continuous lines). Parameters: $dS=0.25{\mathrm{\mu m}}^2$, Ir layer with $\sigma =2\mathrm{nm}$, $l_s=0.1\mathrm{m}$, $D=20\mathrm{\mu m}$, $T=26\mathrm{\mu m}$, and $E=6400\mathrm{eV}$.

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Figure 11 shows the integral breadth (the integrated intensity divided by the maximum intensity, close to the full width at half-maximum here) of the central spot in the x (or equivalently y) direction as a function of $\alpha$ for different energies. For $\alpha >1.5$, the central spot integral breadth is almost constant and equal to the pore size D, while for values between 0 and 1, there is a linear increase of the integral breadth with $\alpha$. The consequence is that for a particular MPO thickness, pore size, and material, there will be a critical energy below which the central spot size will be constant, and above which it will decrease. We have, however, to keep in mind that this study focuses on perfect MPO structures, while the central spot profile is sensitive to some MPO defects, in particular the pore orientation dispersion.

 

Fig. 11. Central spot x direction profile relative integral breadth W/D as a function of $t$ represented as a function of $\alpha =t{\gamma }_c/D$ for three different energies: ray-tracing simulations results (symbols) and model of Eq. (20) (continuous lines). Parameters: Ir layer with $\sigma =2\mathrm{nm}$, $l_s=0.1\mathrm{m}$, $D=20\mathrm{\mu m}$, and $T=26\mathrm{\mu m}$.

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E. PSF Cross Arms and Pseudo Background

An important feature of the PSF is the presence of a cross centered on the main spot (Fig. 3). Having detrimental effects on image resolution, it is interesting to see how the cross’s intensity and reach, both of which should be minimized, are influenced by instrumental parameters. It is, however, a complex task because the PSF cross arms result from the concentration of all the rays undergoing an odd number of reflections in one direction (x or y) and their dispersion in the other direction because of an even number of reflections (including 0). Furthermore, the intensity and eventually the reach of the cross arms in the region around the PSF central spot depend on the direct transmission by the MPO, which is a purely geometrical factor, and on the reflectivity, which changes with the material and x-ray energy.

To evaluate the effect of PSF cross arms, we will use three quantities: ${\mathrm{\Omega }}_{\mathrm{arms}}$, the integrated cross arms intensity; ${\mathrm{\Omega }}_{\mathrm{arms}0}$, the intensity of the cross arms close to the PSF central spot corresponding to (odd,0) and (0,odd) reflections; and the local cross arm intensity within the central spot estimated using ${\mathrm{\Omega }}_{1\mathrm{d}}$ [Eq. (14)]. The two first quantities can be calculated as follows using Eqs. (9)–(11):

Figure 12(a) shows that, for $\alpha >2$, the ratio of the cross arms intensity over the central spot intensity is roughly constant, while it increases sharply when $\alpha$ is reduced below 1. Given fixed values of $t/D$ and a particular reflecting material, $\alpha$ and $\mathrm{\Delta }R$ will only depend on the x-ray energy. As a result, it is possible to plot the cross arms relative intensity as a function of the energy. Figure 12(b) shows that in the case of an Ir reflecting surface with a 2 nm rms roughness, a $t/D$ ratio in the 200–400 range will give small changes in this relative intensity over a 3–20 keV range. It is a $t/D$ range where ${\mathrm{\Omega }}_{2\mathrm{d}}$ changes are also minimized as was shown on Fig. 7.

 

Fig. 12. ${\mathrm{\Omega }}_{\mathrm{arms}}/{\mathrm{\Omega }}_{2\mathrm{d}}$: (a) as a function of $\alpha$ for values of $\mathrm{\Delta }R$ in the range 0 to $-1$; (b) as a function of energy for different values of $t/D$, (60,120,240,480), in the case of an Ir layer with a 2 nm surface rms roughness. Continuous lines are obtained using Eq. (24); symbols are results from ray-tracing simulations.

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The last contribution to the PSF is the direct transmission and rays undergoing (even, even) reflections, called pseudo-background for its two-dimensional nature as opposed to the 0-dimensional central spot and one-dimensional cross arms. The ratio between its integrated intensity and the central spot intensity when $l_s\gg D\sqrt{{\mathrm{\Omega }}_{2d}}$ is written as follows:

The reach of the cross and pseudo background is

It is proportional to $l_s$ and depends on the x-ray energy below a critical energy influenced by reflecting material and $D/t$ (see Fig. 13 in the case of Ir). The size of the cross and pseudo-background region tends to be higher at low energy, but their intensity with respect to the central spot intensity tends to be lower. Figure 13 also shows that lower values of the cross arms and pseudo background reaches are obtained with higher values of the $t/D$ ratio.

 

Fig. 13. Relative reach of the cross and pseudo background as a function of x-ray energy for $t/D$ ratios ranging from 60 to 480.

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4. DISCRETE MODEL AND SEMI-CONTINUOUS MODEL IN THE CASE OF SHORT ${\mathsf{l}}_{\mathsf{s}}$ DISTANCES

For decreasing values of $l_s$, some situations result in a limited number of pores contributing to the PSF and, as a consequence, integrations have to be replaced by discrete sums over the pores and integrations within each pore. For these lower $l_s$ values, the periodic nature of the PSF is more visible as shown in an extreme case in Fig. 14.

 

Fig. 14. Ray-tracing simulation showing the enhancement of the periodic nature of the MPO PSF at short distance. (a) Source; (b) image showing a periodic modulation at places where the source is uniform. Reflective material: Ir. $E=6400\mathrm{eV}$, $l_s=1.2\mathrm{mm}$, $D=20\mathrm{\mu m}$, $T=26\mathrm{\mu m}$, and $t=1.2\mathrm{mm}$.

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Let us consider the pore ($i$, $j$) at a distance $(x_i,y_j)$ from the optical axis, with $x_i=x_0+iT$ and $y_j=y_0+jT$. $-x_0$ and $-y_0$ are the coordinates of the optical axis in the plane of the MPO with respect to the center of the nearest pore [see Fig. 1(b)]. First restricting the analysis to the (xz) plane (see Fig. 15), its angular position ${\theta }_i$ and angular aperture $\mathrm{\Delta }\theta$ can be written as

 

Fig. 15. Side view of a channel with the distances and angles used in the discrete model.

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From ${\theta }_{\min }(i)$ and ${\theta }_{\max }(i)$, the minimum and maximum number of reflections of rays entering the pore $i$ can be calculated as

As opposed to the continuous model, the ray angle and the reflectivity changes within a single pore are taken into account. Similar equations can be written for reflections in the (yz) plane and the pore position $y_i$ simply by replacing x with y.

Using the approximated expression of reflectivity of Eq. (3) for the integrals of Eq. (27), the following two quantities are then calculated:

Two other quantities are necessary for the full calculation corresponding to the part of the direct transmission that reaches the image plane at $(\mathrm{\Delta }x,\mathrm{\Delta }y)$ from the optical axis with $|\mathrm{\Delta }x|\le D$ and $|\mathrm{\Delta }y|\le D$, as follows:

This discrete model was tested first by comparing the intensity profile obtained using ray tracing by moving the MPO along the x direction and changing $x_0$ with $y_0$ equal to 0 to the one obtained using Eq. (28) (Fig. 16). Two sets of ray-tracing simulations were carried out; the first set (ray tracing 1 on the figure) using the exact reflectivity curve, while the second set (ray tracing 2 on the figure) used the approximated curve. The points obtained using this second set are very close to the model demonstrating that the quantitative differences observed between the model and the first set are related to the shape of the reflectivity curve used in the model.

 

Fig. 16. ${\mathrm{\Omega }}_{\mathrm{eff}}$ as a function of x for $y=0$. The continuous line was calculated using Eq. (28). The first series of ray-tracing simulations was carried out using the simplified reflectivity curve (circles) and the second series using the real reflectivity (squares). The x profile is quite sensitive to the exact reflectivity profile. Parameters: Ir, $\mathrm{energy}=6400\mathrm{eV}$, $\sigma =2\mathrm{nm}$, $l_s=5\mathrm{mm}$, $t=2.4\mathrm{mm}$, $D=20\mathrm{\mu m}$, and $T=26\mathrm{\mu m}$.

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As the approximated reflectivity is closer to the real reflectivity when absorption is lower, it is expected that the discrete model works better at higher energies. Figure 17 shows that the two kinds of ray-tracing simulations give almost the same results above 10 keV and are very close to the discrete model. Below that value, there are visible differences. The difference between the discrete model and the ray tracing of type 2 comes from the fact that at low values of $l_s$ and at low energy, other combinations of reflections than the ones considered in the model should be taken into account. At larger distances, the rays undergoing these combinations of reflections are not reaching the central PSF spot. When $l_s=t/2$, every ray that is not absorbed contributes to the central spot, which is actually the whole PSF. For intermediate and short values of $l_s$, a part of these rays will be in the central spot. We did not try to make a model for these specific situations, instead leaving them to ray tracing.

 

Fig. 17. ${\mathrm{\Omega }}_{\mathrm{eff}}$ as a function the x-ray energy in the case of a short $l_s$ distance. RT1 and RT2 are ray-tracing simulations using two kinds of reflectivity curves (see text). These ray-tracing simulations were done using a point source at $x=0$, $y=0$ (square symbols), a point source at $x=T/2$, $y=T/2$ (diamond symbols), or a uniform square source of side length $T$ (circle symbols). The ray-tracing simulations are compared with the discrete and the semi-continuous model (lines). Parameters: Ir, $l_s=2.5\mathrm{mm}$, $\sigma =2\mathrm{nm}$, $t=2.4\mathrm{mm}$, $D=20\mathrm{\mu m}$, and $T=26\mathrm{\mu m}$.

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A. Validity of the Semi-Continuous Model at Short Distance ${\mathsf{l}}_{\mathsf{s}}$ When Using a Square Source

Although simulations with a point source and the discrete model presented above are useful in showing the modulation of the PSF with the source position, it is interesting with respect to fluorescence imaging to know the PSF averaged over the MPO unit cell. It can be obtained using a source emitting uniformly over a square of size $T$ (the MPO period), and it can be shown that it is equivalent to the integration of the semi-continuous model. If $N_s$ is the total number of photons emitted by this source in the $4\pi$ solid angle, the number of photons emitted in the solid angle element $d\mathrm{\Omega }$ by the surface element of the source $dS$ is

The comparison of ray-tracing simulations using a square source (Fig. 17, red circle symbols) with the semi-continuous model, shows a good agreement at a low $l_s$ distance—except at low energy for the reason exposed above in the case of a point source.

5. IMAGING

We have examined in detail and explained the influence of the parameters of an MPO imaging experiment on the MPO PSF using two different models. This section aims to show that the trends outlined in this analysis are visible when imaging extended objects. For that purpose, a Siemens star and a regular grid are used as monochromatic extended x-ray sources in ray-tracing simulations.

Figure 18 shows images of a Siemens star obtained at different energies. The most visible feature is the decrease in intensity when increasing energy, which is predicted by Eq. (12) and illustrated in Fig. 7. At each energy, a cross is visible in the central part of the image; it is directly connected to the PSF cross seen on Fig. 3. The effect of the cross is also noticeable in the region outside of the circle containing the Siemens star, creating a background intensity that does not change substantially when increasing energy. It means that the contrast is decreasing with energy, in agreement with Eq. (24), which predicts an increase with energy in the ratio between the cross integrated intensity and the central spot intensity.

 

Fig. 18. Ray-tracing simulations of a Siemens star imaged at different energies using an MPO for a relatively large distance $l_s=l_i=100\mathrm{mm}$. MPO parameters: Ir with a 2 nm rms roughness as reflecting material, pore size $D=20\mathrm{\mu m}$, period $T=26\mathrm{\mu m}$, and thickness 1.2 mm. The number of rays generated and the intensity scale are the same for the four simulations.

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Figure 19 shows the increase in intensity when decreasing $l_s=l_i$ as predicted by Eq. (17) and illustrated by Fig. 9. For the lower distances, the spatial dependence of the PSF explained by the discrete model [Eq. (28)] is more and more visible.

 

Fig. 19. Ray-tracing simulations of the MPO imaging of a Siemens star for different distances $l_s=l_i$. MPO parameters: Ir with a 2 nm rms roughness is the reflecting material, x-ray energy is 6400 eV, pore size $D=20\mathrm{\mu m}$, period $T=26\mathrm{\mu m}$, and thickness 2.4 mm. The number of rays generated and the intensity scale are the same for the four simulations.

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Finally, Fig. 20 is an illustration of the dependence of the central spot size—and thus image resolution—with the MPO $t/D$ ratio predicted by Eq. (20) and illustrated by Figs. 10 and 11. Two grid orientations are used: the numerical experiment shows that the grid starts to be visible for higher MPO thicknesses when the grid is parallel to the MPO square array than when it is at 45 deg because of the anisotropy of the PSF. We remind the reader that MPOs are considered perfect in this article and that the relatively high resolution predicted in some cases might be difficult to achieve due to pores’ imperfect orientations or slope errors. Figure 20 is also an illustration of the increase in the ratio between the PSF cross integrated intensity and the PSF central spot that might occur when decreasing the ratio $t/D$.

 

Fig. 20. Ray-tracing simulations of a grid imaged with MPOs of different thicknesses. For each MPO thickness, two images with a different grid orientation (0 deg and 45 deg) are shown. The pitch of the grid is 10 μm; the holes are squares with a side length of 5 μm. MPO parameters: Ir with a 2 nm rms roughness is the reflecting material, x-ray energy is 6400 eV, pore size $D=20\mathrm{\mu m}$, and period $T=26\mathrm{\mu m}$. The intensity is normalized for each image.

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6. PROPOSED IMPROVEMENTS

Two features of the PSF have a negative impact when using MPO for imaging: the cross related to odd/even reflections on adjacent pore sides and the PSF spatial dependence. The cross might be transformed in a more isotropic feature by two different means. The first approach involves an array of square pores with a random orientation of the square cross section. This pore-packing scheme was already proposed by Willingale et al. [17] as well as other pore orientation patterns for the BEPI-COLUMBO spectrometer. As outlined in this reference, the pore open fraction $\eta$ has to be reduced in this case, the consequence being an overall reduction of the intensity of all the parts of the PSF. The second method consists of a rotation of a regular MPO precise enough to achieve a negligible precession of the PSF. Figure 21 shows the PSF resulting from these two kinds of modifications (labeled b and c) compared to the regular MPO PSF (a). In the case of the rotation, the results depend on the position of the rotation axis, but changing this position gives similar results. Both new configurations are quite efficient in making the PSF more isotropic.

 

Fig. 21. Ray-tracing simulations of the MPO PSF. (a) Using a standard MPO with a fixed orientation. (b) Using a modified static MPO with a random orientation of the pores square cross section. (c) A standard MPO with a continuous rotation. The rotation axis is parallel to the pore axis with a position at 150 μm from the PSF center. MPO parameters: $l_s=l_i=100\mathrm{mm}$, Ir with a 2 nm rms roughness is the reflecting material, pore size $D=20\mathrm{\mu m}$, period $T=26\mathrm{\mu m}$, and thickness 1.2 mm. The number of rays generated and the intensity scale are the same for the three simulations.

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The efficiency of the two modified configurations is evaluated by imaging a Siemens star (Fig. 22). It is interesting to see that both configurations have a similar effect at a large distance (100 mm here), but that only the rotation provides a good result at a short distance (5 mm here). As a matter of fact, the image obtained using the randomly oriented squares is not very different at a short distance than the image obtained using a regular MPO. This can be explained by the limited number of channels involved for each point of the extended source, which is also the origin of the periodicity of the PSF discussed above in the case of a regular MPO.

 

Fig. 22. Effect of a random square orientation (central row) or an MPO rotation (bottom row) on imaging for two working distances (100 mm, left column, and 5 mm, right column) compared to the standard MPO (top row). MPO parameters: Ir with a 2 nm rms roughness is the reflecting material, pore size $D=20\mathrm{\mu m}$, period $T=26\mathrm{\mu m}$, and thickness 1.2 mm. The intensity is normalized for each image.

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7. CONCLUSION

Analytical expressions of the main PSF features of the perfect planar MPOs were obtained using a semi-continuous model. This model is based on a two parameters approximation of the x-ray reflectivity curve and takes into account two contributions that were not considered in previous models. It was validated against results of ray-tracing simulations by varying several parameters of an x-ray fluorescence imaging experiment: MPO parameters such as the thickness and ratio between the pore size and the thickness and experimental parameters such as x-ray energy and working distance. The benefit of this analytical model is to evidence the influence of these parameters on the intensity and spatial resolution of an x-ray fluorescence experiment. It was also shown that, for short distances, it is necessary to use a discrete model, which has, however, the disadvantage of clouding the influence of the different parameters.

A series of images of extended sources, such as a Siemens star or a grid, are interpreted in light of the semi-continuous and discrete models we have developed, and solutions are also proposed to improve the quality of these images through an isotropization of the PSF.

Finally, it is worth noting that the real MPOs show imperfections that might add another level of complexity to this analysis and bring important modifications to the behavior of the MPOs [18]. These imperfections were purposely not considered here because it appeared necessary to first establish precise models of a perfect MPO.