Abstract

X-ray fluorescence imaging using perfect planar square pore micro-channel plate x-ray optics (MPO) is investigated through the modeling of the MPO point spread function (PSF). A semi-continuous model based on the use of a simplified two parameters reflectivity curve is developed including, in particular, three kinds of contributions. A validation of this model is carried out by calculating variations of several PSF characteristics with the MPO and fluorescence imaging parameters and comparing the results with ray-tracing simulations. A good agreement is found in a large range of x-ray energies; however, it is shown that for the lower values of the working distance a discrete model should be used to take into account the periodic nature of the PSF. Ray-tracing simulated images of extended monochromatic sources are interpreted in light of both the semi-continuous and discrete models. Finally, solutions are proposed to improve the imaging properties of the MPOs.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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 [25] 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 [912], in particular when compared with straight polycapillary optics [13,14] used for 11 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 λ/D, where λ 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 σ. 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 ±0.1rad 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 ls and is equal to li, the distance between the MPO and the image plane (see also Fig. 15). ls and li 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.

 figure: Fig. 1.

Fig. 1. MPO geometrical parameters. (a) Point source imaged in the 11 arrangement. (b) Detail showing the precise position (x0,y0) 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 γc and R¯, which is the reflectivity averaged between 0 and γc. The total reflection critical angle is written as follows:

γc=λre2πNavρmK1.643103λρmK[γc(rad),λ(Å),ρm(g·cm3)]K=2ici(f0i+fi(λ))iciMi,
where λ is the x-ray wavelength, re is the classical electron radius, Nav is the Avogadro number, ρm is the mass density, ci is the element i composition in the reflecting material, f0i+fi(λ) is the element i atomic scattering factor including the anomalous part f, and Mi is the element i atomic mass. For light elements, the K coefficient is close to 1; for heavy elements such as Ir or Pt, it is around 0.8; and, of course, for a mixture of light and heavy elements, K will be between these values.

 figure: Fig. 2.

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 R¯(λ) as

R¯(λ)=1γc(λ)0γc(λ)R(γ,λ)dγ,
where R(γ,λ) is the reflectivity of the material or stack of materials at the grazing angle γ and x-ray wavelength λ. The value of R¯(λ) depends on the surface roughness and on the reflecting material elements’ atomic scattering factors, which themselves depend on energy and can have singular points at x-ray absorption edges. It can be calculated directly by numerical integration of the reflectivity curve. The simplified reflectivity is then given by
R(γ,λ)={1+2(R¯(λ)1)γ/γc(λ)γγc(λ)0γ>γc(λ).
When absorption and roughness are low, R¯(λ) will be close to 1 and the reflectivity model will be close to the model used by Chapman et al. at high energy. For higher absorption, R¯(λ) will be close to 0.5 and the simplified reflectivity will be similar to the model used by Chapman at low energy. In the case of medium absorption and roughness (see, for example, Fig. 2), it should work better than the two asymptotic models.

γc(λ) and R¯(λ) are the two parameters defining a simplified reflectivity curve. In the calculations, however, it will be more convenient to use ΔR(λ) instead of R¯(λ), with ΔR(λ)=2(R¯(λ)1) (Fig. 2). For high absorption and because of the additional effect of roughness, it might occur that Eq. (2) gives a value of R¯ lower than 0.5. In this case, Eq. (3) gives negative values for R when γ is close to γc. To avoid this, the γc value used in the model is replaced by an effective value γceff and R¯ is replaced by R¯eff, as follows:

R¯<0.5{R¯eff=0.5γceff=2R¯γc.
This substitution applies, for example, in the 2 keV–3 keV region in the case of Ir for a rms roughness of 2 nm, as will be shown later in Fig. 6.

It appeared that there was a problem with commonly used f and/or f 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 Ωeff, 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.

 figure: Fig. 3.

Fig. 3. Ray-tracing simulated PSF. The reflecting material is a 25 nm Ir layer on SiO2, with both the surface and interface roughness equal to 2 nm. E=15keV, ls=0.05m, D=20μm, T=26μm, and t=4.8mm. 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 (nx,ny). 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 Ω2d. 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 Ω1d 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 Ω0d as it comes from a very localized region. We have the following relation:

Ωeff=Ω2d+Ω1d+Ω0d.

 figure: Fig. 4.

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

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Ω2d 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/lsγc, the rays entering a particular pore of angular position with respect to the source (θx,θy) [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 nx reflections in the (x,z) plane and ny reflections in the (y,z) plane, will have a dimension δnx(θrx) along x and δny(θry) along y. When the number of reflections is greater than 0, δnx(θrx) and δny(θry) are given by the following expression:

δns(θrs)D={0|θrs|ns1|θrs|(ns1)ns1<|θrs|nsns+1|θrs|ns<|θrs|<ns+10|θrs|ns+1,
where the index s will be either x or y and θrs=θst/D. In the case where ns=0,
δ0(θrs)D={1|θrs|0|θrs|<10|θrs|1.

The corresponding effective collected solid angle corresponding to (nx,ny) reflection in the same single pore is

Ωp(θx,θy)=δnx(θrx)δny(θry)(lst/2)2Rnx(θx)Rny(θy).
For infinitely small pores, the number of pores in the dθxdθy element of the solid angle is
d2N(θx,θy)=η(lst/2)2D2dθxdθy,
where η is the fraction of the MPO entrance surface occupied by pores. Considering a pore size D and a period of T in both x and y directions,
η=D2/T2.

The total effective solid angle corresponding to (nx,ny) reflections, Ω(nx,ny), is then obtained by the following integration:

Ω(nx,ny)=θx=+θy=+Ωp(θx,θy)d2N(θx,θy)=4ηγc2SnxSny
with
Sn={1α0αδn(θrs)DRn(θrst/D)dθrsn012αn=0,
where n is either nx or ny and
α=tγcD.
Equation (10) is valid for any expression of the reflectivity for which the upper bound, above which reflectivity is equal to 0, is the total reflection critical angle γc. It is the case with the simplified reflectivity model of Eq. (3), which is then used to calculate the integral of Eq. (10). Writing ΔR=2(R¯1) and after integration, we obtain the following in the case where n>0:
αSn={0αn1Fn(ΔR)Fn((n1)ΔR/α)n1<α<nFn(nΔR/α)Fn((n1)ΔR/α)+Gn(ΔR)Gn(nΔR/α)nα<n+1Fn(nΔR/α)Fn((n1)ΔR/α)+Gn((n+1)ΔR/α)Gn(nΔR/α)αn+1,
where the functions Fn(U) and Gn(U) are defined as follows:
Fn(U)=(αΔR)2(U(1+U)n+1n+1(1+U)n+2(n+1)(n+2))(n1)αΔR(1+U)n+1n+1,Gn(U)=αΔR(1+U)n+1(αΔR)2(U(1+U)n+1n+1(1+U)n+2(n+1)(n+2)).

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

Ω2d=4ηγc2Sodd2(α,ΔR)
with
Sodd(α,ΔR)=k=0+S2k+1.
Ω2d [Eq. (12)] is proportional to the product of the square of the critical angle and of a function that only depends on α and ΔR. It is important to note that this main contribution to the effective solid angle does not depend on ls (as will be illustrated later in Fig. 9). The reason is that the 1/(lst/2)2 dependence of the single pore effective solid angle [Eq. (7)] is perfectly compensated by the (lst/2)2 dependence of the number of pores involved [Eq. (8)]. Figure 5 shows Sodd(α,ΔR) for a set of values of ΔR going from the maximum value 0 to the minimum value 1 and a range of values of α.

 figure: Fig. 5.

Fig. 5. Plots of Sodd2(α,ΔR) as a function of α for different values of ΔR showing the influence of absorption and roughness induced loss on Ω2d.

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In [11], only the extreme cases are considered, ΔR=0 in the case of high energies and ΔR=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 ΔR and γc 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. ΔR 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 σ. In the 2–3 keV region, it was necessary to use effective values of ΔR and γc in the case of Ir with a 2 nm rms roughness, calculated using Eq. (4).

 figure: Fig. 6.

Fig. 6. External total reflection critical angle γc and reflectivity loss ΔR as a function of energy in the case of Ir reflective material. ΔR is shown for two values of the rms surface roughness σ. The insert shows the 2–3 keV region where effective values γceff [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, ΔR=ΔReff is constant and equal to 1.

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Using the data shown on Fig. 6, it is possible to plot Ω2d 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 Ω2d on the working distance ls, so for a particular reflecting material all the information on Ω2d is in this map.

 figure: Fig. 7.

Fig. 7. Ω2d 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 Ω1d 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

Θ1x(nx)=θx=+η(lst/2)Dδnx(θrx)(lst/2)Rnx(θx)dθx=2ηγcSnx
and an equivalent relation obtained by substitution of x by y for Θ1y(ny) corresponding to the column in the y direction. The resulting total solid angle collected by the row and column and focused in the central PSF spot is
Ω1d=Dls+t/2k(Θ1x(2k+1)+Θ1y(2k+1))
=4Dηγcls+t/2Sodd(α,ΔR)=2Dls+t/2Ω2d.
Ω1d has a lower dependence on x-ray energy than Ω2d because it depends linearly on γc, which is roughly proportional to the x-ray wavelength, and also because Sodd changes with x-ray energy are lower than the changes of Sodd2. Ω1d depends on ls; shorter distances will increase this contribution.

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

Ω0d=D2(ls+t/2)2,
which shows no spectral dependence and a stronger dependence with the working distance ls.

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

Ωeff=4ηγc2Sodd2(α,ΔR)+4Dηγcls+t/2Sodd(α,ΔR)+D2(ls+t/2)2=n=02g(n)[ηγcSodd(α,ΔR)]n(Dls+t/2)2n,
where g(n) is a kind of degree of degeneracy, g(n)=4 for n=1 or n=2 and g(0)=1. It corresponds to the number of equivalent regions for the three different contributions, which are visible on Fig. 4.

Ray-tracing results are compared in Fig. 8 with the model of Eq. (17) in a case where the dominant term in Ωeff is Ω2d, because ls is large (ls=0.1m). For this reason, the plots of Fig. 8 are similar to the ones of Fig. 5. The influence of the two other terms, Ω1d and Ω0d, is visible on Fig. 9. Ω2d does not depend on ls so the ls dependency of Ωeff, clearly visible on this figure, is related to Ω1d and Ω0d.

 figure: Fig. 8.

Fig. 8. Ωeff/γc2 as a function of α=tγc/D. Variations of α 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 σ=2nm, ls=10cm, t=1.2mm, D=20μm, and T=26μm.

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 figure: Fig. 9.

Fig. 9. Ωeff, Ω2d, Ω1d, and Ω0d dependence with ls. Ray-tracing simulations results (symbols) are compared with the model of Eqs. (12), (14), (16), and (17) (continuous lines). Parameters: Ir layer with σ=2nm, t=1.2mm, D=20μm, T=26μm, and E=6400eV.

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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 Ω2d; 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 Ωeff. For this purpose, two functions of x or y have to be calculated as follows:

P1x(x)={ηγcD|ΔR|k=0+[(1U)2(k+1)2(k+1)]Umin(k,x)Umax(k,x)|x|D<10|x|D1P2x(x)={12ls|x|<Dlsls+t/20otherwise,
where
D=11t/(2ls)DUmin(k,x)=(2k+|x|/D)|ΔR|/αUmax(k,x)=min(Um(k,x),max(|ΔR|,Umin(k,x)))Um(k,x)=(2(k+1)|x|/D)|ΔR|/α.
The equivalent functions P1y(y) and P2y(y) are obtained by simple substitution of x by y. The three components—P2d, P1d, and P0d—of the profile Peff are then given by
P2d(x,y)=P1x(x)P1y(y)P1d(x,y)=P1x(x)P2y(y)+P2x(x)P1y(y)P0d(x,y)=P2x(x)P2y(y).
The following equation can be used to determine the total effective solid angle in a small dxdy surface element (a detector pixel for example):
d2Ωeff(x,y)=Peff(x,y)dxdy=(P2d(x,y)+P1d(x,y)+P0d(x,y))dxdy.

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.

 figure: Fig. 10.

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=0.25μm2, Ir layer with σ=2nm, ls=0.1m, D=20μm, T=26μm, and E=6400eV.

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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 α for different energies. For α>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 α. 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.

 figure: Fig. 11.

Fig. 11. Central spot x direction profile relative integral breadth W/D as a function of t represented as a function of α=tγc/D for three different energies: ray-tracing simulations results (symbols) and model of Eq. (20) (continuous lines). Parameters: Ir layer with σ=2nm, ls=0.1m, D=20μm, and T=26μm.

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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: Ωarms, the integrated cross arms intensity; Ωarms0, 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 Ω1d [Eq. (14)]. The two first quantities can be calculated as follows using Eqs. (9)–(11):

Ωarms=8ηγc2Seven(α,ΔR)Sodd(α,ΔR)Ω1d,
Ωarms0=4ηγc2αSodd(α,ΔR)Ω1d.
In these two expressions, Ω1d is subtracted from the cross arms intensity because it is considered a part of the PSF central spot. This subtraction can be neglected, if lsDΩ2d. Sodd and Seven are defined as follows by Eqs. (13) and (23), respectively,
Seven(α,ΔR)=k=0+S2k.
The following part is restricted to the case in which lsDΩ2d. To evaluate the cross arms intensity relative to the intensity of the central spot, we can calculate the ratio
ΩarmsΩeffΩarmsΩ2d=2SevenSodd,
Ωarms0ΩeffΩarms0Ω2d=1αSodd.

Figure 12(a) shows that, for α>2, the ratio of the cross arms intensity over the central spot intensity is roughly constant, while it increases sharply when α is reduced below 1. Given fixed values of t/D and a particular reflecting material, α and ΔR 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 Ω2d changes are also minimized as was shown on Fig. 7.

 figure: Fig. 12.

Fig. 12. Ωarms/Ω2d: (a) as a function of α for values of ΔR 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 lsDΩ2d is written as follows:

Ωp.backgroundΩeffΩp.backgroundΩ2d=Seven2Sodd2=14(ΩarmsΩ2d)2.
Large values of Ωarms/Ω2d will correspond to even larger values of Ωp.background/Ω2d as this ratio is always greater than one.

The reach of the cross and pseudo background is

Wcross=max(4lsD/t,4lsγc).

It is proportional to ls 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.

 figure: Fig. 13.

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 ls DISTANCES

For decreasing values of ls, 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 ls values, the periodic nature of the PSF is more visible as shown in an extreme case in Fig. 14.

 figure: 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=6400eV, ls=1.2mm, D=20μm, T=26μm, and t=1.2mm.

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Let us consider the pore (i, j) at a distance (xi,yj) from the optical axis, with xi=x0+iT and yj=y0+jT. x0 and y0 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 θi and angular aperture Δθ can be written as

θi=|xi|lst/2Δθ=Dlst/2.
The minimum and maximum angles of the rays entering in the pore i are, as illustrated in Fig. 15 and in the case where i0,
θmin(i)=θiΔθ/2θmax(i)=θi+Δθ/2.
The case of i=0 is a little bit more complicated. If |x0|D/2, the formula above applies. In the case where |x0|<D/2, the optical axis is inside the pore, and we split the pore into two parts noted with the exponents + and −, with
θmin+(0)=0θmax+(0)=θi+Δθ/2θmin(0)=0θmax(0)=θi+Δθ/2.

 figure: Fig. 15.

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

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From θmin(i) and θmax(i), the minimum and maximum number of reflections of rays entering the pore i can be calculated as

nmin(i)=θmin(i)tDnmax(i)=θmax(i)tD+1,
where the symbols indicate the floor integer value. For a particular position of pore xi and a particular number of reflections n0 in the (xz) plane, the effective source angular range going through the pore can be written as
Δθeff(n,xi)=θ1θ2Rn(λ,γ)dγ=γcΔR[(1+U)n+1n+1]ΔRθ1/γcΔRθ2/γc,
where
θ1,θ2={min(θmin,γc),min(θxi(nmin+1),γc)n=nminmin(θxi(n),γc),min(θxi(n+1),γc)nmin<n<nmaxmin(θxi(nmax),γc),min(θmax,γc)n=nmax
with
θxi(n)=1ls+t/2[|xi|+(n1/2)D].

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 yi 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:

Σx,odd=i=+k=0+Δθeff(2k+1,xi)Σy,odd=j=+k=0+Δθeff(2k+1,yj).
In each of the sums over the pores (indices i and j), the eventual splitting of the central pore into two parts along the x and/or y direction has to be taken into account.

Two other quantities are necessary for the full calculation corresponding to the part of the direct transmission that reaches the image plane at (Δx,Δy) from the optical axis with |Δx|D and |Δy|D, as follows:

Σx,0=k,nminx=0[min(θxk(1),D/(2ls))min(θminxk,D/(2ls))]Σy,0=k,nminy=0[min(θyk(1),D/(2ls))min(θminyk,D/(2ls))].
The sums are carried out over the pores for which the minimum number of reflections in each plane is equal to 0, and again taking into account the eventual splitting of the central pore into two or four parts. The effective collected solid angle then becomes
Ωeffective,d=(Σx,odd+Σx,0)(Σy,odd+Σy,0).

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 x0 with y0 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.

 figure: Fig. 16.

Fig. 16. Ωeff 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, energy=6400eV, σ=2nm, ls=5mm, t=2.4mm, D=20μm, and T=26μm.

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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 ls 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 ls=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 ls, 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.

 figure: Fig. 17.

Fig. 17. Ωeff as a function the x-ray energy in the case of a short ls 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, ls=2.5mm, σ=2nm, t=2.4mm, D=20μm, and T=26μm.

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A. Validity of the Semi-Continuous Model at Short Distance ls 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 Ns is the total number of photons emitted by this source in the 4π solid angle, the number of photons emitted in the solid angle element dΩ by the surface element of the source dS is

d4n=Ns4πT2dSdΩ=Nsη4πD2dSdΩ.
Because of the source size, for any source-MPO distance ls and any pore size D, a full pore will be illuminated in any direction (θx,θy). In this direction, the element of surface of the source providing photons undergoing (nx,ny) reflections will be exactly dS=δnx(θx)δny(θy), which can be calculated using Eqs. (5) and (6). At the exit of the MPO, the number of photons incoming in the solid angle dΩ=dθxdθy at angles (θx,θy) and having experienced (nx,ny) reflections, is given by
d2n=δnx(θrx)δny(θry)Rnx(θx)Rny(θy)Nsη4πD2dθxdθy.
If we calculate this quantity for NS/(4π) equal to 1, we find the product Ωp(θx,θy)d2N(θx,θy) of the two quantities given by Eqs. (7) and (8). We obtain the important result that even for a short working distance, Eq. (17) remains valid with an extended square source with the size of the MPO period T. The local variations of the PSF can only be predicted by a discrete model, but the average over an MPO period can be calculated using the semi-continuous model.

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 ls 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.

 figure: Fig. 18.

Fig. 18. Ray-tracing simulations of a Siemens star imaged at different energies using an MPO for a relatively large distance ls=li=100mm. MPO parameters: Ir with a 2 nm rms roughness as reflecting material, pore size D=20μm, period T=26μm, 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 ls=li 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.

 figure: Fig. 19.

Fig. 19. Ray-tracing simulations of the MPO imaging of a Siemens star for different distances ls=li. MPO parameters: Ir with a 2 nm rms roughness is the reflecting material, x-ray energy is 6400 eV, pore size D=20μm, period T=26μm, 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.

 figure: Fig. 20.

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=20μm, and period T=26μm. 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 η 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.

 figure: Fig. 21.

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: ls=li=100mm, Ir with a 2 nm rms roughness is the reflecting material, pore size D=20μm, period T=26μm, 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.

 figure: Fig. 22.

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=20μm, period T=26μm, 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.

Funding

National Aeronautics and Space Administration (NASA) (MatISSE16 2-0005).

REFERENCES

1. K. Tsuji, T. Matsuno, Y. Takimoto, M. Yamanashi, N. Kometani, Y. C. Sasaki, T. Hasegawa, S. Kato, T. Yamada, T. Shoji, and N. Kawahara, “New developments of x-ray fluorescence imaging techniques in laboratory,” Spectrochim. Acta Part B 113, 43–53 (2015). [CrossRef]  

2. M. Alfeld, K. Janssens, J. Dik, W. de Nolf, and G. van der Snickt, “Optimization of mobile scanning macro-XRF systems for the in situ investigation of historical paintings,” J. Anal. At. Spectrom. 26, 899–909 (2011). [CrossRef]  

3. I. Reiche, K. Müller, M. Albéric, O. Scharf, A. Wähning, A. Bjeoumikhov, M. Radtke, and R. Simon, “Discovering vanished paints and naturally formed gold nanoparticles on 2800 years old Phoenician ivories using SR-FF-MicroXRF with the color x-ray camera,” Anal. Chem. 85, 5857–5866 (2013). [CrossRef]  

4. L. Bertrand, L. Robinet, M. Thoury, K. Janssens, S. X. Cohen, and S. Schöder, “Cultural heritage and archaeology materials studied by synchrotron spectroscopy and imaging,” Appl. Phys. A 106, 377–396 (2012). [CrossRef]  

5. P. Walter, P. Sarrazin, M. Gailhanou, D. Hérouard, A. Verney, and D. Blake, “Full-field XRF instrument for cultural heritage: application to the study of a Caillebotte painting,” X-Ray Spectrom. (2018). [CrossRef]  

6. D. Blake, P. Sarrazin, T. Bristow, R. Downs, M. Gailhanou, F. Marchis, D. Ming, R. Morris, V. A. Solé, K. Thompson, P. Walter, M. Wilson, A. Yen, and S. Webb, “The mapping x-ray fluorescence spectrometer (MapX),” in 3rd International Workshop on Instrumentation for Planetary Missions (2016), p. 4006.

7. R. Alberti, T. Frizzi, L. Bombelli, M. Gironda, N. Aresi, F. Rosi, C. Miliani, G. Tranquilli, F. Talarico, and L. Cartechini, “Crono: a fast and reconfigurable macro x-ray fluorescence scanner for in-situ investigations of polychrome surfaces,” X-Ray Spectrom. 46, 297–302 (2017). [CrossRef]  

8. F. P. Romano, C. Caliri, P. Nicotra, S. Di Martino, L. Pappalardo, F. Rizzo, and H. C. Santos, “Real-time elemental imaging of large dimension paintings with a novel mobile macro x-ray fluorescence (MA-XRF) scanning technique,” J. Anal. At. Spectrom. 32, 773–781 (2017). [CrossRef]  

9. J. R. P. Angel, “Lobster eyes as x-ray telescopes,” Astrophys. J. 233, 364–373 (1979). [CrossRef]  

10. G. J. Price, G. W. Fraser, J. F. Pearson, J. P. Nussey, I. B. Hutchinson, A. D. Holland, K. Turner, and D. Pullan, “Prototype imaging x-ray fluorescence spectrometer based on microchannel plate optics,” Rev. Sci. Instrum. 75, 2314–2319 (2004). [CrossRef]  

11. H. N. Chapman, K. A. Nugent, and S. W. Wilkins, “X-ray focusing using square channel-capillary arrays,” Rev. Sci. Instrum. 62, 1542–1561 (1991). [CrossRef]  

12. P. Sarrazin, D. Blake, M. Gailhanou, F. Marchis, C. Chalumeau, S. Webb, P. Walter, E. Schyns, K. Thompson, and T. Bristow, “MapX: 2D XRF for planetary exploration—image formation and optic characterization,” J. Instrum. 13, C04023 (2018). [CrossRef]  

13. H. N. Chapman, K. A. Nugent, and S. W. Wilkins, “X-ray focusing using cylindrical-channel capillary arrays. I. Theory,” Appl. Opt. 32, 6316–6332 (1993). [CrossRef]  

14. F. P. Romano, C. Caliri, L. Cosentino, S. Gammino, D. Mascali, L. Pappalardo, F. Rizzo, O. Scharf, and H. C. Santos, “Micro x-ray fluorescence imaging in a tabletop full field-x-ray fluorescence instrument and in a full field-particle induced x-ray emission end station,” Anal. Chem. 88, 9873–9880 (2016). [CrossRef]  

15. J. Daillant and A. Gibaud, eds., “Chap. 3: Specular reflectivity from smooth and rough surfaces,” in X-Ray and Neutron Reflectivity, Theory and Applications (Springer, 1999).

16. C. T. Chantler, “Detailed tabulation of atomic form factors, photoelectric absorption and scattering cross section, and mass attenuation coefficients in the vicinity of absorption edges in the soft x-ray (z = 30–36, z = 60–89, e = 0.1 kev–10 kev), addressing convergence issues of earlier work,” J. Phys. Chem. Ref. Data 29, 597–1056 (2000). [CrossRef]  

17. R. Willingale, G. W. Fraser, and J. F. Pearson, “Optimization of square pore optics for the x-ray spectrometer on Bepi-Columbo,” Proc. SPIE 5900, 590012 (2005). [CrossRef]  

18. R. Willingale, J. F. Pearson, A. Martindale, C. H. Feldman, R. Fairbend, E. Schyns, S. Petit, J. P. Osborne, and P. T. O’Brien, “Aberrations in square pore micro-channel optics used for x-ray lobster eye telescopes,” Proc. SPIE 9905, 99051Y (2016). [CrossRef]  

References

  • View by:

  1. K. Tsuji, T. Matsuno, Y. Takimoto, M. Yamanashi, N. Kometani, Y. C. Sasaki, T. Hasegawa, S. Kato, T. Yamada, T. Shoji, and N. Kawahara, “New developments of x-ray fluorescence imaging techniques in laboratory,” Spectrochim. Acta Part B 113, 43–53 (2015).
    [Crossref]
  2. M. Alfeld, K. Janssens, J. Dik, W. de Nolf, and G. van der Snickt, “Optimization of mobile scanning macro-XRF systems for the in situ investigation of historical paintings,” J. Anal. At. Spectrom. 26, 899–909 (2011).
    [Crossref]
  3. I. Reiche, K. Müller, M. Albéric, O. Scharf, A. Wähning, A. Bjeoumikhov, M. Radtke, and R. Simon, “Discovering vanished paints and naturally formed gold nanoparticles on 2800 years old Phoenician ivories using SR-FF-MicroXRF with the color x-ray camera,” Anal. Chem. 85, 5857–5866 (2013).
    [Crossref]
  4. L. Bertrand, L. Robinet, M. Thoury, K. Janssens, S. X. Cohen, and S. Schöder, “Cultural heritage and archaeology materials studied by synchrotron spectroscopy and imaging,” Appl. Phys. A 106, 377–396 (2012).
    [Crossref]
  5. P. Walter, P. Sarrazin, M. Gailhanou, D. Hérouard, A. Verney, and D. Blake, “Full-field XRF instrument for cultural heritage: application to the study of a Caillebotte painting,” X-Ray Spectrom. (2018).
    [Crossref]
  6. D. Blake, P. Sarrazin, T. Bristow, R. Downs, M. Gailhanou, F. Marchis, D. Ming, R. Morris, V. A. Solé, K. Thompson, P. Walter, M. Wilson, A. Yen, and S. Webb, “The mapping x-ray fluorescence spectrometer (MapX),” in 3rd International Workshop on Instrumentation for Planetary Missions (2016), p. 4006.
  7. R. Alberti, T. Frizzi, L. Bombelli, M. Gironda, N. Aresi, F. Rosi, C. Miliani, G. Tranquilli, F. Talarico, and L. Cartechini, “Crono: a fast and reconfigurable macro x-ray fluorescence scanner for in-situ investigations of polychrome surfaces,” X-Ray Spectrom. 46, 297–302 (2017).
    [Crossref]
  8. F. P. Romano, C. Caliri, P. Nicotra, S. Di Martino, L. Pappalardo, F. Rizzo, and H. C. Santos, “Real-time elemental imaging of large dimension paintings with a novel mobile macro x-ray fluorescence (MA-XRF) scanning technique,” J. Anal. At. Spectrom. 32, 773–781 (2017).
    [Crossref]
  9. J. R. P. Angel, “Lobster eyes as x-ray telescopes,” Astrophys. J. 233, 364–373 (1979).
    [Crossref]
  10. G. J. Price, G. W. Fraser, J. F. Pearson, J. P. Nussey, I. B. Hutchinson, A. D. Holland, K. Turner, and D. Pullan, “Prototype imaging x-ray fluorescence spectrometer based on microchannel plate optics,” Rev. Sci. Instrum. 75, 2314–2319 (2004).
    [Crossref]
  11. H. N. Chapman, K. A. Nugent, and S. W. Wilkins, “X-ray focusing using square channel-capillary arrays,” Rev. Sci. Instrum. 62, 1542–1561 (1991).
    [Crossref]
  12. P. Sarrazin, D. Blake, M. Gailhanou, F. Marchis, C. Chalumeau, S. Webb, P. Walter, E. Schyns, K. Thompson, and T. Bristow, “MapX: 2D XRF for planetary exploration—image formation and optic characterization,” J. Instrum. 13, C04023 (2018).
    [Crossref]
  13. H. N. Chapman, K. A. Nugent, and S. W. Wilkins, “X-ray focusing using cylindrical-channel capillary arrays. I. Theory,” Appl. Opt. 32, 6316–6332 (1993).
    [Crossref]
  14. F. P. Romano, C. Caliri, L. Cosentino, S. Gammino, D. Mascali, L. Pappalardo, F. Rizzo, O. Scharf, and H. C. Santos, “Micro x-ray fluorescence imaging in a tabletop full field-x-ray fluorescence instrument and in a full field-particle induced x-ray emission end station,” Anal. Chem. 88, 9873–9880 (2016).
    [Crossref]
  15. J. Daillant and A. Gibaud, eds., “Chap. 3: Specular reflectivity from smooth and rough surfaces,” in X-Ray and Neutron Reflectivity, Theory and Applications (Springer, 1999).
  16. C. T. Chantler, “Detailed tabulation of atomic form factors, photoelectric absorption and scattering cross section, and mass attenuation coefficients in the vicinity of absorption edges in the soft x-ray (z = 30–36, z = 60–89, e = 0.1 kev–10 kev), addressing convergence issues of earlier work,” J. Phys. Chem. Ref. Data 29, 597–1056 (2000).
    [Crossref]
  17. R. Willingale, G. W. Fraser, and J. F. Pearson, “Optimization of square pore optics for the x-ray spectrometer on Bepi-Columbo,” Proc. SPIE 5900, 590012 (2005).
    [Crossref]
  18. R. Willingale, J. F. Pearson, A. Martindale, C. H. Feldman, R. Fairbend, E. Schyns, S. Petit, J. P. Osborne, and P. T. O’Brien, “Aberrations in square pore micro-channel optics used for x-ray lobster eye telescopes,” Proc. SPIE 9905, 99051Y (2016).
    [Crossref]

2018 (1)

P. Sarrazin, D. Blake, M. Gailhanou, F. Marchis, C. Chalumeau, S. Webb, P. Walter, E. Schyns, K. Thompson, and T. Bristow, “MapX: 2D XRF for planetary exploration—image formation and optic characterization,” J. Instrum. 13, C04023 (2018).
[Crossref]

2017 (2)

R. Alberti, T. Frizzi, L. Bombelli, M. Gironda, N. Aresi, F. Rosi, C. Miliani, G. Tranquilli, F. Talarico, and L. Cartechini, “Crono: a fast and reconfigurable macro x-ray fluorescence scanner for in-situ investigations of polychrome surfaces,” X-Ray Spectrom. 46, 297–302 (2017).
[Crossref]

F. P. Romano, C. Caliri, P. Nicotra, S. Di Martino, L. Pappalardo, F. Rizzo, and H. C. Santos, “Real-time elemental imaging of large dimension paintings with a novel mobile macro x-ray fluorescence (MA-XRF) scanning technique,” J. Anal. At. Spectrom. 32, 773–781 (2017).
[Crossref]

2016 (2)

F. P. Romano, C. Caliri, L. Cosentino, S. Gammino, D. Mascali, L. Pappalardo, F. Rizzo, O. Scharf, and H. C. Santos, “Micro x-ray fluorescence imaging in a tabletop full field-x-ray fluorescence instrument and in a full field-particle induced x-ray emission end station,” Anal. Chem. 88, 9873–9880 (2016).
[Crossref]

R. Willingale, J. F. Pearson, A. Martindale, C. H. Feldman, R. Fairbend, E. Schyns, S. Petit, J. P. Osborne, and P. T. O’Brien, “Aberrations in square pore micro-channel optics used for x-ray lobster eye telescopes,” Proc. SPIE 9905, 99051Y (2016).
[Crossref]

2015 (1)

K. Tsuji, T. Matsuno, Y. Takimoto, M. Yamanashi, N. Kometani, Y. C. Sasaki, T. Hasegawa, S. Kato, T. Yamada, T. Shoji, and N. Kawahara, “New developments of x-ray fluorescence imaging techniques in laboratory,” Spectrochim. Acta Part B 113, 43–53 (2015).
[Crossref]

2013 (1)

I. Reiche, K. Müller, M. Albéric, O. Scharf, A. Wähning, A. Bjeoumikhov, M. Radtke, and R. Simon, “Discovering vanished paints and naturally formed gold nanoparticles on 2800 years old Phoenician ivories using SR-FF-MicroXRF with the color x-ray camera,” Anal. Chem. 85, 5857–5866 (2013).
[Crossref]

2012 (1)

L. Bertrand, L. Robinet, M. Thoury, K. Janssens, S. X. Cohen, and S. Schöder, “Cultural heritage and archaeology materials studied by synchrotron spectroscopy and imaging,” Appl. Phys. A 106, 377–396 (2012).
[Crossref]

2011 (1)

M. Alfeld, K. Janssens, J. Dik, W. de Nolf, and G. van der Snickt, “Optimization of mobile scanning macro-XRF systems for the in situ investigation of historical paintings,” J. Anal. At. Spectrom. 26, 899–909 (2011).
[Crossref]

2005 (1)

R. Willingale, G. W. Fraser, and J. F. Pearson, “Optimization of square pore optics for the x-ray spectrometer on Bepi-Columbo,” Proc. SPIE 5900, 590012 (2005).
[Crossref]

2004 (1)

G. J. Price, G. W. Fraser, J. F. Pearson, J. P. Nussey, I. B. Hutchinson, A. D. Holland, K. Turner, and D. Pullan, “Prototype imaging x-ray fluorescence spectrometer based on microchannel plate optics,” Rev. Sci. Instrum. 75, 2314–2319 (2004).
[Crossref]

2000 (1)

C. T. Chantler, “Detailed tabulation of atomic form factors, photoelectric absorption and scattering cross section, and mass attenuation coefficients in the vicinity of absorption edges in the soft x-ray (z = 30–36, z = 60–89, e = 0.1 kev–10 kev), addressing convergence issues of earlier work,” J. Phys. Chem. Ref. Data 29, 597–1056 (2000).
[Crossref]

1993 (1)

1991 (1)

H. N. Chapman, K. A. Nugent, and S. W. Wilkins, “X-ray focusing using square channel-capillary arrays,” Rev. Sci. Instrum. 62, 1542–1561 (1991).
[Crossref]

1979 (1)

J. R. P. Angel, “Lobster eyes as x-ray telescopes,” Astrophys. J. 233, 364–373 (1979).
[Crossref]

Albéric, M.

I. Reiche, K. Müller, M. Albéric, O. Scharf, A. Wähning, A. Bjeoumikhov, M. Radtke, and R. Simon, “Discovering vanished paints and naturally formed gold nanoparticles on 2800 years old Phoenician ivories using SR-FF-MicroXRF with the color x-ray camera,” Anal. Chem. 85, 5857–5866 (2013).
[Crossref]

Alberti, R.

R. Alberti, T. Frizzi, L. Bombelli, M. Gironda, N. Aresi, F. Rosi, C. Miliani, G. Tranquilli, F. Talarico, and L. Cartechini, “Crono: a fast and reconfigurable macro x-ray fluorescence scanner for in-situ investigations of polychrome surfaces,” X-Ray Spectrom. 46, 297–302 (2017).
[Crossref]

Alfeld, M.

M. Alfeld, K. Janssens, J. Dik, W. de Nolf, and G. van der Snickt, “Optimization of mobile scanning macro-XRF systems for the in situ investigation of historical paintings,” J. Anal. At. Spectrom. 26, 899–909 (2011).
[Crossref]

Angel, J. R. P.

J. R. P. Angel, “Lobster eyes as x-ray telescopes,” Astrophys. J. 233, 364–373 (1979).
[Crossref]

Aresi, N.

R. Alberti, T. Frizzi, L. Bombelli, M. Gironda, N. Aresi, F. Rosi, C. Miliani, G. Tranquilli, F. Talarico, and L. Cartechini, “Crono: a fast and reconfigurable macro x-ray fluorescence scanner for in-situ investigations of polychrome surfaces,” X-Ray Spectrom. 46, 297–302 (2017).
[Crossref]

Bertrand, L.

L. Bertrand, L. Robinet, M. Thoury, K. Janssens, S. X. Cohen, and S. Schöder, “Cultural heritage and archaeology materials studied by synchrotron spectroscopy and imaging,” Appl. Phys. A 106, 377–396 (2012).
[Crossref]

Bjeoumikhov, A.

I. Reiche, K. Müller, M. Albéric, O. Scharf, A. Wähning, A. Bjeoumikhov, M. Radtke, and R. Simon, “Discovering vanished paints and naturally formed gold nanoparticles on 2800 years old Phoenician ivories using SR-FF-MicroXRF with the color x-ray camera,” Anal. Chem. 85, 5857–5866 (2013).
[Crossref]

Blake, D.

P. Sarrazin, D. Blake, M. Gailhanou, F. Marchis, C. Chalumeau, S. Webb, P. Walter, E. Schyns, K. Thompson, and T. Bristow, “MapX: 2D XRF for planetary exploration—image formation and optic characterization,” J. Instrum. 13, C04023 (2018).
[Crossref]

P. Walter, P. Sarrazin, M. Gailhanou, D. Hérouard, A. Verney, and D. Blake, “Full-field XRF instrument for cultural heritage: application to the study of a Caillebotte painting,” X-Ray Spectrom. (2018).
[Crossref]

D. Blake, P. Sarrazin, T. Bristow, R. Downs, M. Gailhanou, F. Marchis, D. Ming, R. Morris, V. A. Solé, K. Thompson, P. Walter, M. Wilson, A. Yen, and S. Webb, “The mapping x-ray fluorescence spectrometer (MapX),” in 3rd International Workshop on Instrumentation for Planetary Missions (2016), p. 4006.

Bombelli, L.

R. Alberti, T. Frizzi, L. Bombelli, M. Gironda, N. Aresi, F. Rosi, C. Miliani, G. Tranquilli, F. Talarico, and L. Cartechini, “Crono: a fast and reconfigurable macro x-ray fluorescence scanner for in-situ investigations of polychrome surfaces,” X-Ray Spectrom. 46, 297–302 (2017).
[Crossref]

Bristow, T.

P. Sarrazin, D. Blake, M. Gailhanou, F. Marchis, C. Chalumeau, S. Webb, P. Walter, E. Schyns, K. Thompson, and T. Bristow, “MapX: 2D XRF for planetary exploration—image formation and optic characterization,” J. Instrum. 13, C04023 (2018).
[Crossref]

D. Blake, P. Sarrazin, T. Bristow, R. Downs, M. Gailhanou, F. Marchis, D. Ming, R. Morris, V. A. Solé, K. Thompson, P. Walter, M. Wilson, A. Yen, and S. Webb, “The mapping x-ray fluorescence spectrometer (MapX),” in 3rd International Workshop on Instrumentation for Planetary Missions (2016), p. 4006.

Caliri, C.

F. P. Romano, C. Caliri, P. Nicotra, S. Di Martino, L. Pappalardo, F. Rizzo, and H. C. Santos, “Real-time elemental imaging of large dimension paintings with a novel mobile macro x-ray fluorescence (MA-XRF) scanning technique,” J. Anal. At. Spectrom. 32, 773–781 (2017).
[Crossref]

F. P. Romano, C. Caliri, L. Cosentino, S. Gammino, D. Mascali, L. Pappalardo, F. Rizzo, O. Scharf, and H. C. Santos, “Micro x-ray fluorescence imaging in a tabletop full field-x-ray fluorescence instrument and in a full field-particle induced x-ray emission end station,” Anal. Chem. 88, 9873–9880 (2016).
[Crossref]

Cartechini, L.

R. Alberti, T. Frizzi, L. Bombelli, M. Gironda, N. Aresi, F. Rosi, C. Miliani, G. Tranquilli, F. Talarico, and L. Cartechini, “Crono: a fast and reconfigurable macro x-ray fluorescence scanner for in-situ investigations of polychrome surfaces,” X-Ray Spectrom. 46, 297–302 (2017).
[Crossref]

Chalumeau, C.

P. Sarrazin, D. Blake, M. Gailhanou, F. Marchis, C. Chalumeau, S. Webb, P. Walter, E. Schyns, K. Thompson, and T. Bristow, “MapX: 2D XRF for planetary exploration—image formation and optic characterization,” J. Instrum. 13, C04023 (2018).
[Crossref]

Chantler, C. T.

C. T. Chantler, “Detailed tabulation of atomic form factors, photoelectric absorption and scattering cross section, and mass attenuation coefficients in the vicinity of absorption edges in the soft x-ray (z = 30–36, z = 60–89, e = 0.1 kev–10 kev), addressing convergence issues of earlier work,” J. Phys. Chem. Ref. Data 29, 597–1056 (2000).
[Crossref]

Chapman, H. N.

H. N. Chapman, K. A. Nugent, and S. W. Wilkins, “X-ray focusing using cylindrical-channel capillary arrays. I. Theory,” Appl. Opt. 32, 6316–6332 (1993).
[Crossref]

H. N. Chapman, K. A. Nugent, and S. W. Wilkins, “X-ray focusing using square channel-capillary arrays,” Rev. Sci. Instrum. 62, 1542–1561 (1991).
[Crossref]

Cohen, S. X.

L. Bertrand, L. Robinet, M. Thoury, K. Janssens, S. X. Cohen, and S. Schöder, “Cultural heritage and archaeology materials studied by synchrotron spectroscopy and imaging,” Appl. Phys. A 106, 377–396 (2012).
[Crossref]

Cosentino, L.

F. P. Romano, C. Caliri, L. Cosentino, S. Gammino, D. Mascali, L. Pappalardo, F. Rizzo, O. Scharf, and H. C. Santos, “Micro x-ray fluorescence imaging in a tabletop full field-x-ray fluorescence instrument and in a full field-particle induced x-ray emission end station,” Anal. Chem. 88, 9873–9880 (2016).
[Crossref]

de Nolf, W.

M. Alfeld, K. Janssens, J. Dik, W. de Nolf, and G. van der Snickt, “Optimization of mobile scanning macro-XRF systems for the in situ investigation of historical paintings,” J. Anal. At. Spectrom. 26, 899–909 (2011).
[Crossref]

Di Martino, S.

F. P. Romano, C. Caliri, P. Nicotra, S. Di Martino, L. Pappalardo, F. Rizzo, and H. C. Santos, “Real-time elemental imaging of large dimension paintings with a novel mobile macro x-ray fluorescence (MA-XRF) scanning technique,” J. Anal. At. Spectrom. 32, 773–781 (2017).
[Crossref]

Dik, J.

M. Alfeld, K. Janssens, J. Dik, W. de Nolf, and G. van der Snickt, “Optimization of mobile scanning macro-XRF systems for the in situ investigation of historical paintings,” J. Anal. At. Spectrom. 26, 899–909 (2011).
[Crossref]

Downs, R.

D. Blake, P. Sarrazin, T. Bristow, R. Downs, M. Gailhanou, F. Marchis, D. Ming, R. Morris, V. A. Solé, K. Thompson, P. Walter, M. Wilson, A. Yen, and S. Webb, “The mapping x-ray fluorescence spectrometer (MapX),” in 3rd International Workshop on Instrumentation for Planetary Missions (2016), p. 4006.

Fairbend, R.

R. Willingale, J. F. Pearson, A. Martindale, C. H. Feldman, R. Fairbend, E. Schyns, S. Petit, J. P. Osborne, and P. T. O’Brien, “Aberrations in square pore micro-channel optics used for x-ray lobster eye telescopes,” Proc. SPIE 9905, 99051Y (2016).
[Crossref]

Feldman, C. H.

R. Willingale, J. F. Pearson, A. Martindale, C. H. Feldman, R. Fairbend, E. Schyns, S. Petit, J. P. Osborne, and P. T. O’Brien, “Aberrations in square pore micro-channel optics used for x-ray lobster eye telescopes,” Proc. SPIE 9905, 99051Y (2016).
[Crossref]

Fraser, G. W.

R. Willingale, G. W. Fraser, and J. F. Pearson, “Optimization of square pore optics for the x-ray spectrometer on Bepi-Columbo,” Proc. SPIE 5900, 590012 (2005).
[Crossref]

G. J. Price, G. W. Fraser, J. F. Pearson, J. P. Nussey, I. B. Hutchinson, A. D. Holland, K. Turner, and D. Pullan, “Prototype imaging x-ray fluorescence spectrometer based on microchannel plate optics,” Rev. Sci. Instrum. 75, 2314–2319 (2004).
[Crossref]

Frizzi, T.

R. Alberti, T. Frizzi, L. Bombelli, M. Gironda, N. Aresi, F. Rosi, C. Miliani, G. Tranquilli, F. Talarico, and L. Cartechini, “Crono: a fast and reconfigurable macro x-ray fluorescence scanner for in-situ investigations of polychrome surfaces,” X-Ray Spectrom. 46, 297–302 (2017).
[Crossref]

Gailhanou, M.

P. Sarrazin, D. Blake, M. Gailhanou, F. Marchis, C. Chalumeau, S. Webb, P. Walter, E. Schyns, K. Thompson, and T. Bristow, “MapX: 2D XRF for planetary exploration—image formation and optic characterization,” J. Instrum. 13, C04023 (2018).
[Crossref]

P. Walter, P. Sarrazin, M. Gailhanou, D. Hérouard, A. Verney, and D. Blake, “Full-field XRF instrument for cultural heritage: application to the study of a Caillebotte painting,” X-Ray Spectrom. (2018).
[Crossref]

D. Blake, P. Sarrazin, T. Bristow, R. Downs, M. Gailhanou, F. Marchis, D. Ming, R. Morris, V. A. Solé, K. Thompson, P. Walter, M. Wilson, A. Yen, and S. Webb, “The mapping x-ray fluorescence spectrometer (MapX),” in 3rd International Workshop on Instrumentation for Planetary Missions (2016), p. 4006.

Gammino, S.

F. P. Romano, C. Caliri, L. Cosentino, S. Gammino, D. Mascali, L. Pappalardo, F. Rizzo, O. Scharf, and H. C. Santos, “Micro x-ray fluorescence imaging in a tabletop full field-x-ray fluorescence instrument and in a full field-particle induced x-ray emission end station,” Anal. Chem. 88, 9873–9880 (2016).
[Crossref]

Gironda, M.

R. Alberti, T. Frizzi, L. Bombelli, M. Gironda, N. Aresi, F. Rosi, C. Miliani, G. Tranquilli, F. Talarico, and L. Cartechini, “Crono: a fast and reconfigurable macro x-ray fluorescence scanner for in-situ investigations of polychrome surfaces,” X-Ray Spectrom. 46, 297–302 (2017).
[Crossref]

Hasegawa, T.

K. Tsuji, T. Matsuno, Y. Takimoto, M. Yamanashi, N. Kometani, Y. C. Sasaki, T. Hasegawa, S. Kato, T. Yamada, T. Shoji, and N. Kawahara, “New developments of x-ray fluorescence imaging techniques in laboratory,” Spectrochim. Acta Part B 113, 43–53 (2015).
[Crossref]

Hérouard, D.

P. Walter, P. Sarrazin, M. Gailhanou, D. Hérouard, A. Verney, and D. Blake, “Full-field XRF instrument for cultural heritage: application to the study of a Caillebotte painting,” X-Ray Spectrom. (2018).
[Crossref]

Holland, A. D.

G. J. Price, G. W. Fraser, J. F. Pearson, J. P. Nussey, I. B. Hutchinson, A. D. Holland, K. Turner, and D. Pullan, “Prototype imaging x-ray fluorescence spectrometer based on microchannel plate optics,” Rev. Sci. Instrum. 75, 2314–2319 (2004).
[Crossref]

Hutchinson, I. B.

G. J. Price, G. W. Fraser, J. F. Pearson, J. P. Nussey, I. B. Hutchinson, A. D. Holland, K. Turner, and D. Pullan, “Prototype imaging x-ray fluorescence spectrometer based on microchannel plate optics,” Rev. Sci. Instrum. 75, 2314–2319 (2004).
[Crossref]

Janssens, K.

L. Bertrand, L. Robinet, M. Thoury, K. Janssens, S. X. Cohen, and S. Schöder, “Cultural heritage and archaeology materials studied by synchrotron spectroscopy and imaging,” Appl. Phys. A 106, 377–396 (2012).
[Crossref]

M. Alfeld, K. Janssens, J. Dik, W. de Nolf, and G. van der Snickt, “Optimization of mobile scanning macro-XRF systems for the in situ investigation of historical paintings,” J. Anal. At. Spectrom. 26, 899–909 (2011).
[Crossref]

Kato, S.

K. Tsuji, T. Matsuno, Y. Takimoto, M. Yamanashi, N. Kometani, Y. C. Sasaki, T. Hasegawa, S. Kato, T. Yamada, T. Shoji, and N. Kawahara, “New developments of x-ray fluorescence imaging techniques in laboratory,” Spectrochim. Acta Part B 113, 43–53 (2015).
[Crossref]

Kawahara, N.

K. Tsuji, T. Matsuno, Y. Takimoto, M. Yamanashi, N. Kometani, Y. C. Sasaki, T. Hasegawa, S. Kato, T. Yamada, T. Shoji, and N. Kawahara, “New developments of x-ray fluorescence imaging techniques in laboratory,” Spectrochim. Acta Part B 113, 43–53 (2015).
[Crossref]

Kometani, N.

K. Tsuji, T. Matsuno, Y. Takimoto, M. Yamanashi, N. Kometani, Y. C. Sasaki, T. Hasegawa, S. Kato, T. Yamada, T. Shoji, and N. Kawahara, “New developments of x-ray fluorescence imaging techniques in laboratory,” Spectrochim. Acta Part B 113, 43–53 (2015).
[Crossref]

Marchis, F.

P. Sarrazin, D. Blake, M. Gailhanou, F. Marchis, C. Chalumeau, S. Webb, P. Walter, E. Schyns, K. Thompson, and T. Bristow, “MapX: 2D XRF for planetary exploration—image formation and optic characterization,” J. Instrum. 13, C04023 (2018).
[Crossref]

D. Blake, P. Sarrazin, T. Bristow, R. Downs, M. Gailhanou, F. Marchis, D. Ming, R. Morris, V. A. Solé, K. Thompson, P. Walter, M. Wilson, A. Yen, and S. Webb, “The mapping x-ray fluorescence spectrometer (MapX),” in 3rd International Workshop on Instrumentation for Planetary Missions (2016), p. 4006.

Martindale, A.

R. Willingale, J. F. Pearson, A. Martindale, C. H. Feldman, R. Fairbend, E. Schyns, S. Petit, J. P. Osborne, and P. T. O’Brien, “Aberrations in square pore micro-channel optics used for x-ray lobster eye telescopes,” Proc. SPIE 9905, 99051Y (2016).
[Crossref]

Mascali, D.

F. P. Romano, C. Caliri, L. Cosentino, S. Gammino, D. Mascali, L. Pappalardo, F. Rizzo, O. Scharf, and H. C. Santos, “Micro x-ray fluorescence imaging in a tabletop full field-x-ray fluorescence instrument and in a full field-particle induced x-ray emission end station,” Anal. Chem. 88, 9873–9880 (2016).
[Crossref]

Matsuno, T.

K. Tsuji, T. Matsuno, Y. Takimoto, M. Yamanashi, N. Kometani, Y. C. Sasaki, T. Hasegawa, S. Kato, T. Yamada, T. Shoji, and N. Kawahara, “New developments of x-ray fluorescence imaging techniques in laboratory,” Spectrochim. Acta Part B 113, 43–53 (2015).
[Crossref]

Miliani, C.

R. Alberti, T. Frizzi, L. Bombelli, M. Gironda, N. Aresi, F. Rosi, C. Miliani, G. Tranquilli, F. Talarico, and L. Cartechini, “Crono: a fast and reconfigurable macro x-ray fluorescence scanner for in-situ investigations of polychrome surfaces,” X-Ray Spectrom. 46, 297–302 (2017).
[Crossref]

Ming, D.

D. Blake, P. Sarrazin, T. Bristow, R. Downs, M. Gailhanou, F. Marchis, D. Ming, R. Morris, V. A. Solé, K. Thompson, P. Walter, M. Wilson, A. Yen, and S. Webb, “The mapping x-ray fluorescence spectrometer (MapX),” in 3rd International Workshop on Instrumentation for Planetary Missions (2016), p. 4006.

Morris, R.

D. Blake, P. Sarrazin, T. Bristow, R. Downs, M. Gailhanou, F. Marchis, D. Ming, R. Morris, V. A. Solé, K. Thompson, P. Walter, M. Wilson, A. Yen, and S. Webb, “The mapping x-ray fluorescence spectrometer (MapX),” in 3rd International Workshop on Instrumentation for Planetary Missions (2016), p. 4006.

Müller, K.

I. Reiche, K. Müller, M. Albéric, O. Scharf, A. Wähning, A. Bjeoumikhov, M. Radtke, and R. Simon, “Discovering vanished paints and naturally formed gold nanoparticles on 2800 years old Phoenician ivories using SR-FF-MicroXRF with the color x-ray camera,” Anal. Chem. 85, 5857–5866 (2013).
[Crossref]

Nicotra, P.

F. P. Romano, C. Caliri, P. Nicotra, S. Di Martino, L. Pappalardo, F. Rizzo, and H. C. Santos, “Real-time elemental imaging of large dimension paintings with a novel mobile macro x-ray fluorescence (MA-XRF) scanning technique,” J. Anal. At. Spectrom. 32, 773–781 (2017).
[Crossref]

Nugent, K. A.

H. N. Chapman, K. A. Nugent, and S. W. Wilkins, “X-ray focusing using cylindrical-channel capillary arrays. I. Theory,” Appl. Opt. 32, 6316–6332 (1993).
[Crossref]

H. N. Chapman, K. A. Nugent, and S. W. Wilkins, “X-ray focusing using square channel-capillary arrays,” Rev. Sci. Instrum. 62, 1542–1561 (1991).
[Crossref]

Nussey, J. P.

G. J. Price, G. W. Fraser, J. F. Pearson, J. P. Nussey, I. B. Hutchinson, A. D. Holland, K. Turner, and D. Pullan, “Prototype imaging x-ray fluorescence spectrometer based on microchannel plate optics,” Rev. Sci. Instrum. 75, 2314–2319 (2004).
[Crossref]

O’Brien, P. T.

R. Willingale, J. F. Pearson, A. Martindale, C. H. Feldman, R. Fairbend, E. Schyns, S. Petit, J. P. Osborne, and P. T. O’Brien, “Aberrations in square pore micro-channel optics used for x-ray lobster eye telescopes,” Proc. SPIE 9905, 99051Y (2016).
[Crossref]

Osborne, J. P.

R. Willingale, J. F. Pearson, A. Martindale, C. H. Feldman, R. Fairbend, E. Schyns, S. Petit, J. P. Osborne, and P. T. O’Brien, “Aberrations in square pore micro-channel optics used for x-ray lobster eye telescopes,” Proc. SPIE 9905, 99051Y (2016).
[Crossref]

Pappalardo, L.

F. P. Romano, C. Caliri, P. Nicotra, S. Di Martino, L. Pappalardo, F. Rizzo, and H. C. Santos, “Real-time elemental imaging of large dimension paintings with a novel mobile macro x-ray fluorescence (MA-XRF) scanning technique,” J. Anal. At. Spectrom. 32, 773–781 (2017).
[Crossref]

F. P. Romano, C. Caliri, L. Cosentino, S. Gammino, D. Mascali, L. Pappalardo, F. Rizzo, O. Scharf, and H. C. Santos, “Micro x-ray fluorescence imaging in a tabletop full field-x-ray fluorescence instrument and in a full field-particle induced x-ray emission end station,” Anal. Chem. 88, 9873–9880 (2016).
[Crossref]

Pearson, J. F.

R. Willingale, J. F. Pearson, A. Martindale, C. H. Feldman, R. Fairbend, E. Schyns, S. Petit, J. P. Osborne, and P. T. O’Brien, “Aberrations in square pore micro-channel optics used for x-ray lobster eye telescopes,” Proc. SPIE 9905, 99051Y (2016).
[Crossref]

R. Willingale, G. W. Fraser, and J. F. Pearson, “Optimization of square pore optics for the x-ray spectrometer on Bepi-Columbo,” Proc. SPIE 5900, 590012 (2005).
[Crossref]

G. J. Price, G. W. Fraser, J. F. Pearson, J. P. Nussey, I. B. Hutchinson, A. D. Holland, K. Turner, and D. Pullan, “Prototype imaging x-ray fluorescence spectrometer based on microchannel plate optics,” Rev. Sci. Instrum. 75, 2314–2319 (2004).
[Crossref]

Petit, S.

R. Willingale, J. F. Pearson, A. Martindale, C. H. Feldman, R. Fairbend, E. Schyns, S. Petit, J. P. Osborne, and P. T. O’Brien, “Aberrations in square pore micro-channel optics used for x-ray lobster eye telescopes,” Proc. SPIE 9905, 99051Y (2016).
[Crossref]

Price, G. J.

G. J. Price, G. W. Fraser, J. F. Pearson, J. P. Nussey, I. B. Hutchinson, A. D. Holland, K. Turner, and D. Pullan, “Prototype imaging x-ray fluorescence spectrometer based on microchannel plate optics,” Rev. Sci. Instrum. 75, 2314–2319 (2004).
[Crossref]

Pullan, D.

G. J. Price, G. W. Fraser, J. F. Pearson, J. P. Nussey, I. B. Hutchinson, A. D. Holland, K. Turner, and D. Pullan, “Prototype imaging x-ray fluorescence spectrometer based on microchannel plate optics,” Rev. Sci. Instrum. 75, 2314–2319 (2004).
[Crossref]

Radtke, M.

I. Reiche, K. Müller, M. Albéric, O. Scharf, A. Wähning, A. Bjeoumikhov, M. Radtke, and R. Simon, “Discovering vanished paints and naturally formed gold nanoparticles on 2800 years old Phoenician ivories using SR-FF-MicroXRF with the color x-ray camera,” Anal. Chem. 85, 5857–5866 (2013).
[Crossref]

Reiche, I.

I. Reiche, K. Müller, M. Albéric, O. Scharf, A. Wähning, A. Bjeoumikhov, M. Radtke, and R. Simon, “Discovering vanished paints and naturally formed gold nanoparticles on 2800 years old Phoenician ivories using SR-FF-MicroXRF with the color x-ray camera,” Anal. Chem. 85, 5857–5866 (2013).
[Crossref]

Rizzo, F.

F. P. Romano, C. Caliri, P. Nicotra, S. Di Martino, L. Pappalardo, F. Rizzo, and H. C. Santos, “Real-time elemental imaging of large dimension paintings with a novel mobile macro x-ray fluorescence (MA-XRF) scanning technique,” J. Anal. At. Spectrom. 32, 773–781 (2017).
[Crossref]

F. P. Romano, C. Caliri, L. Cosentino, S. Gammino, D. Mascali, L. Pappalardo, F. Rizzo, O. Scharf, and H. C. Santos, “Micro x-ray fluorescence imaging in a tabletop full field-x-ray fluorescence instrument and in a full field-particle induced x-ray emission end station,” Anal. Chem. 88, 9873–9880 (2016).
[Crossref]

Robinet, L.

L. Bertrand, L. Robinet, M. Thoury, K. Janssens, S. X. Cohen, and S. Schöder, “Cultural heritage and archaeology materials studied by synchrotron spectroscopy and imaging,” Appl. Phys. A 106, 377–396 (2012).
[Crossref]

Romano, F. P.

F. P. Romano, C. Caliri, P. Nicotra, S. Di Martino, L. Pappalardo, F. Rizzo, and H. C. Santos, “Real-time elemental imaging of large dimension paintings with a novel mobile macro x-ray fluorescence (MA-XRF) scanning technique,” J. Anal. At. Spectrom. 32, 773–781 (2017).
[Crossref]

F. P. Romano, C. Caliri, L. Cosentino, S. Gammino, D. Mascali, L. Pappalardo, F. Rizzo, O. Scharf, and H. C. Santos, “Micro x-ray fluorescence imaging in a tabletop full field-x-ray fluorescence instrument and in a full field-particle induced x-ray emission end station,” Anal. Chem. 88, 9873–9880 (2016).
[Crossref]

Rosi, F.

R. Alberti, T. Frizzi, L. Bombelli, M. Gironda, N. Aresi, F. Rosi, C. Miliani, G. Tranquilli, F. Talarico, and L. Cartechini, “Crono: a fast and reconfigurable macro x-ray fluorescence scanner for in-situ investigations of polychrome surfaces,” X-Ray Spectrom. 46, 297–302 (2017).
[Crossref]

Santos, H. C.

F. P. Romano, C. Caliri, P. Nicotra, S. Di Martino, L. Pappalardo, F. Rizzo, and H. C. Santos, “Real-time elemental imaging of large dimension paintings with a novel mobile macro x-ray fluorescence (MA-XRF) scanning technique,” J. Anal. At. Spectrom. 32, 773–781 (2017).
[Crossref]

F. P. Romano, C. Caliri, L. Cosentino, S. Gammino, D. Mascali, L. Pappalardo, F. Rizzo, O. Scharf, and H. C. Santos, “Micro x-ray fluorescence imaging in a tabletop full field-x-ray fluorescence instrument and in a full field-particle induced x-ray emission end station,” Anal. Chem. 88, 9873–9880 (2016).
[Crossref]

Sarrazin, P.

P. Sarrazin, D. Blake, M. Gailhanou, F. Marchis, C. Chalumeau, S. Webb, P. Walter, E. Schyns, K. Thompson, and T. Bristow, “MapX: 2D XRF for planetary exploration—image formation and optic characterization,” J. Instrum. 13, C04023 (2018).
[Crossref]

P. Walter, P. Sarrazin, M. Gailhanou, D. Hérouard, A. Verney, and D. Blake, “Full-field XRF instrument for cultural heritage: application to the study of a Caillebotte painting,” X-Ray Spectrom. (2018).
[Crossref]

D. Blake, P. Sarrazin, T. Bristow, R. Downs, M. Gailhanou, F. Marchis, D. Ming, R. Morris, V. A. Solé, K. Thompson, P. Walter, M. Wilson, A. Yen, and S. Webb, “The mapping x-ray fluorescence spectrometer (MapX),” in 3rd International Workshop on Instrumentation for Planetary Missions (2016), p. 4006.

Sasaki, Y. C.

K. Tsuji, T. Matsuno, Y. Takimoto, M. Yamanashi, N. Kometani, Y. C. Sasaki, T. Hasegawa, S. Kato, T. Yamada, T. Shoji, and N. Kawahara, “New developments of x-ray fluorescence imaging techniques in laboratory,” Spectrochim. Acta Part B 113, 43–53 (2015).
[Crossref]

Scharf, O.

F. P. Romano, C. Caliri, L. Cosentino, S. Gammino, D. Mascali, L. Pappalardo, F. Rizzo, O. Scharf, and H. C. Santos, “Micro x-ray fluorescence imaging in a tabletop full field-x-ray fluorescence instrument and in a full field-particle induced x-ray emission end station,” Anal. Chem. 88, 9873–9880 (2016).
[Crossref]

I. Reiche, K. Müller, M. Albéric, O. Scharf, A. Wähning, A. Bjeoumikhov, M. Radtke, and R. Simon, “Discovering vanished paints and naturally formed gold nanoparticles on 2800 years old Phoenician ivories using SR-FF-MicroXRF with the color x-ray camera,” Anal. Chem. 85, 5857–5866 (2013).
[Crossref]

Schöder, S.

L. Bertrand, L. Robinet, M. Thoury, K. Janssens, S. X. Cohen, and S. Schöder, “Cultural heritage and archaeology materials studied by synchrotron spectroscopy and imaging,” Appl. Phys. A 106, 377–396 (2012).
[Crossref]

Schyns, E.

P. Sarrazin, D. Blake, M. Gailhanou, F. Marchis, C. Chalumeau, S. Webb, P. Walter, E. Schyns, K. Thompson, and T. Bristow, “MapX: 2D XRF for planetary exploration—image formation and optic characterization,” J. Instrum. 13, C04023 (2018).
[Crossref]

R. Willingale, J. F. Pearson, A. Martindale, C. H. Feldman, R. Fairbend, E. Schyns, S. Petit, J. P. Osborne, and P. T. O’Brien, “Aberrations in square pore micro-channel optics used for x-ray lobster eye telescopes,” Proc. SPIE 9905, 99051Y (2016).
[Crossref]

Shoji, T.

K. Tsuji, T. Matsuno, Y. Takimoto, M. Yamanashi, N. Kometani, Y. C. Sasaki, T. Hasegawa, S. Kato, T. Yamada, T. Shoji, and N. Kawahara, “New developments of x-ray fluorescence imaging techniques in laboratory,” Spectrochim. Acta Part B 113, 43–53 (2015).
[Crossref]

Simon, R.

I. Reiche, K. Müller, M. Albéric, O. Scharf, A. Wähning, A. Bjeoumikhov, M. Radtke, and R. Simon, “Discovering vanished paints and naturally formed gold nanoparticles on 2800 years old Phoenician ivories using SR-FF-MicroXRF with the color x-ray camera,” Anal. Chem. 85, 5857–5866 (2013).
[Crossref]

Solé, V. A.

D. Blake, P. Sarrazin, T. Bristow, R. Downs, M. Gailhanou, F. Marchis, D. Ming, R. Morris, V. A. Solé, K. Thompson, P. Walter, M. Wilson, A. Yen, and S. Webb, “The mapping x-ray fluorescence spectrometer (MapX),” in 3rd International Workshop on Instrumentation for Planetary Missions (2016), p. 4006.

Takimoto, Y.

K. Tsuji, T. Matsuno, Y. Takimoto, M. Yamanashi, N. Kometani, Y. C. Sasaki, T. Hasegawa, S. Kato, T. Yamada, T. Shoji, and N. Kawahara, “New developments of x-ray fluorescence imaging techniques in laboratory,” Spectrochim. Acta Part B 113, 43–53 (2015).
[Crossref]

Talarico, F.

R. Alberti, T. Frizzi, L. Bombelli, M. Gironda, N. Aresi, F. Rosi, C. Miliani, G. Tranquilli, F. Talarico, and L. Cartechini, “Crono: a fast and reconfigurable macro x-ray fluorescence scanner for in-situ investigations of polychrome surfaces,” X-Ray Spectrom. 46, 297–302 (2017).
[Crossref]

Thompson, K.

P. Sarrazin, D. Blake, M. Gailhanou, F. Marchis, C. Chalumeau, S. Webb, P. Walter, E. Schyns, K. Thompson, and T. Bristow, “MapX: 2D XRF for planetary exploration—image formation and optic characterization,” J. Instrum. 13, C04023 (2018).
[Crossref]

D. Blake, P. Sarrazin, T. Bristow, R. Downs, M. Gailhanou, F. Marchis, D. Ming, R. Morris, V. A. Solé, K. Thompson, P. Walter, M. Wilson, A. Yen, and S. Webb, “The mapping x-ray fluorescence spectrometer (MapX),” in 3rd International Workshop on Instrumentation for Planetary Missions (2016), p. 4006.

Thoury, M.

L. Bertrand, L. Robinet, M. Thoury, K. Janssens, S. X. Cohen, and S. Schöder, “Cultural heritage and archaeology materials studied by synchrotron spectroscopy and imaging,” Appl. Phys. A 106, 377–396 (2012).
[Crossref]

Tranquilli, G.

R. Alberti, T. Frizzi, L. Bombelli, M. Gironda, N. Aresi, F. Rosi, C. Miliani, G. Tranquilli, F. Talarico, and L. Cartechini, “Crono: a fast and reconfigurable macro x-ray fluorescence scanner for in-situ investigations of polychrome surfaces,” X-Ray Spectrom. 46, 297–302 (2017).
[Crossref]

Tsuji, K.

K. Tsuji, T. Matsuno, Y. Takimoto, M. Yamanashi, N. Kometani, Y. C. Sasaki, T. Hasegawa, S. Kato, T. Yamada, T. Shoji, and N. Kawahara, “New developments of x-ray fluorescence imaging techniques in laboratory,” Spectrochim. Acta Part B 113, 43–53 (2015).
[Crossref]

Turner, K.

G. J. Price, G. W. Fraser, J. F. Pearson, J. P. Nussey, I. B. Hutchinson, A. D. Holland, K. Turner, and D. Pullan, “Prototype imaging x-ray fluorescence spectrometer based on microchannel plate optics,” Rev. Sci. Instrum. 75, 2314–2319 (2004).
[Crossref]

van der Snickt, G.

M. Alfeld, K. Janssens, J. Dik, W. de Nolf, and G. van der Snickt, “Optimization of mobile scanning macro-XRF systems for the in situ investigation of historical paintings,” J. Anal. At. Spectrom. 26, 899–909 (2011).
[Crossref]

Verney, A.

P. Walter, P. Sarrazin, M. Gailhanou, D. Hérouard, A. Verney, and D. Blake, “Full-field XRF instrument for cultural heritage: application to the study of a Caillebotte painting,” X-Ray Spectrom. (2018).
[Crossref]

Wähning, A.

I. Reiche, K. Müller, M. Albéric, O. Scharf, A. Wähning, A. Bjeoumikhov, M. Radtke, and R. Simon, “Discovering vanished paints and naturally formed gold nanoparticles on 2800 years old Phoenician ivories using SR-FF-MicroXRF with the color x-ray camera,” Anal. Chem. 85, 5857–5866 (2013).
[Crossref]

Walter, P.

P. Sarrazin, D. Blake, M. Gailhanou, F. Marchis, C. Chalumeau, S. Webb, P. Walter, E. Schyns, K. Thompson, and T. Bristow, “MapX: 2D XRF for planetary exploration—image formation and optic characterization,” J. Instrum. 13, C04023 (2018).
[Crossref]

P. Walter, P. Sarrazin, M. Gailhanou, D. Hérouard, A. Verney, and D. Blake, “Full-field XRF instrument for cultural heritage: application to the study of a Caillebotte painting,” X-Ray Spectrom. (2018).
[Crossref]

D. Blake, P. Sarrazin, T. Bristow, R. Downs, M. Gailhanou, F. Marchis, D. Ming, R. Morris, V. A. Solé, K. Thompson, P. Walter, M. Wilson, A. Yen, and S. Webb, “The mapping x-ray fluorescence spectrometer (MapX),” in 3rd International Workshop on Instrumentation for Planetary Missions (2016), p. 4006.

Webb, S.

P. Sarrazin, D. Blake, M. Gailhanou, F. Marchis, C. Chalumeau, S. Webb, P. Walter, E. Schyns, K. Thompson, and T. Bristow, “MapX: 2D XRF for planetary exploration—image formation and optic characterization,” J. Instrum. 13, C04023 (2018).
[Crossref]

D. Blake, P. Sarrazin, T. Bristow, R. Downs, M. Gailhanou, F. Marchis, D. Ming, R. Morris, V. A. Solé, K. Thompson, P. Walter, M. Wilson, A. Yen, and S. Webb, “The mapping x-ray fluorescence spectrometer (MapX),” in 3rd International Workshop on Instrumentation for Planetary Missions (2016), p. 4006.

Wilkins, S. W.

H. N. Chapman, K. A. Nugent, and S. W. Wilkins, “X-ray focusing using cylindrical-channel capillary arrays. I. Theory,” Appl. Opt. 32, 6316–6332 (1993).
[Crossref]

H. N. Chapman, K. A. Nugent, and S. W. Wilkins, “X-ray focusing using square channel-capillary arrays,” Rev. Sci. Instrum. 62, 1542–1561 (1991).
[Crossref]

Willingale, R.

R. Willingale, J. F. Pearson, A. Martindale, C. H. Feldman, R. Fairbend, E. Schyns, S. Petit, J. P. Osborne, and P. T. O’Brien, “Aberrations in square pore micro-channel optics used for x-ray lobster eye telescopes,” Proc. SPIE 9905, 99051Y (2016).
[Crossref]

R. Willingale, G. W. Fraser, and J. F. Pearson, “Optimization of square pore optics for the x-ray spectrometer on Bepi-Columbo,” Proc. SPIE 5900, 590012 (2005).
[Crossref]

Wilson, M.

D. Blake, P. Sarrazin, T. Bristow, R. Downs, M. Gailhanou, F. Marchis, D. Ming, R. Morris, V. A. Solé, K. Thompson, P. Walter, M. Wilson, A. Yen, and S. Webb, “The mapping x-ray fluorescence spectrometer (MapX),” in 3rd International Workshop on Instrumentation for Planetary Missions (2016), p. 4006.

Yamada, T.

K. Tsuji, T. Matsuno, Y. Takimoto, M. Yamanashi, N. Kometani, Y. C. Sasaki, T. Hasegawa, S. Kato, T. Yamada, T. Shoji, and N. Kawahara, “New developments of x-ray fluorescence imaging techniques in laboratory,” Spectrochim. Acta Part B 113, 43–53 (2015).
[Crossref]

Yamanashi, M.

K. Tsuji, T. Matsuno, Y. Takimoto, M. Yamanashi, N. Kometani, Y. C. Sasaki, T. Hasegawa, S. Kato, T. Yamada, T. Shoji, and N. Kawahara, “New developments of x-ray fluorescence imaging techniques in laboratory,” Spectrochim. Acta Part B 113, 43–53 (2015).
[Crossref]

Yen, A.

D. Blake, P. Sarrazin, T. Bristow, R. Downs, M. Gailhanou, F. Marchis, D. Ming, R. Morris, V. A. Solé, K. Thompson, P. Walter, M. Wilson, A. Yen, and S. Webb, “The mapping x-ray fluorescence spectrometer (MapX),” in 3rd International Workshop on Instrumentation for Planetary Missions (2016), p. 4006.

Anal. Chem. (2)

I. Reiche, K. Müller, M. Albéric, O. Scharf, A. Wähning, A. Bjeoumikhov, M. Radtke, and R. Simon, “Discovering vanished paints and naturally formed gold nanoparticles on 2800 years old Phoenician ivories using SR-FF-MicroXRF with the color x-ray camera,” Anal. Chem. 85, 5857–5866 (2013).
[Crossref]

F. P. Romano, C. Caliri, L. Cosentino, S. Gammino, D. Mascali, L. Pappalardo, F. Rizzo, O. Scharf, and H. C. Santos, “Micro x-ray fluorescence imaging in a tabletop full field-x-ray fluorescence instrument and in a full field-particle induced x-ray emission end station,” Anal. Chem. 88, 9873–9880 (2016).
[Crossref]

Appl. Opt. (1)

Appl. Phys. A (1)

L. Bertrand, L. Robinet, M. Thoury, K. Janssens, S. X. Cohen, and S. Schöder, “Cultural heritage and archaeology materials studied by synchrotron spectroscopy and imaging,” Appl. Phys. A 106, 377–396 (2012).
[Crossref]

Astrophys. J. (1)

J. R. P. Angel, “Lobster eyes as x-ray telescopes,” Astrophys. J. 233, 364–373 (1979).
[Crossref]

J. Anal. At. Spectrom. (2)

M. Alfeld, K. Janssens, J. Dik, W. de Nolf, and G. van der Snickt, “Optimization of mobile scanning macro-XRF systems for the in situ investigation of historical paintings,” J. Anal. At. Spectrom. 26, 899–909 (2011).
[Crossref]

F. P. Romano, C. Caliri, P. Nicotra, S. Di Martino, L. Pappalardo, F. Rizzo, and H. C. Santos, “Real-time elemental imaging of large dimension paintings with a novel mobile macro x-ray fluorescence (MA-XRF) scanning technique,” J. Anal. At. Spectrom. 32, 773–781 (2017).
[Crossref]

J. Instrum. (1)

P. Sarrazin, D. Blake, M. Gailhanou, F. Marchis, C. Chalumeau, S. Webb, P. Walter, E. Schyns, K. Thompson, and T. Bristow, “MapX: 2D XRF for planetary exploration—image formation and optic characterization,” J. Instrum. 13, C04023 (2018).
[Crossref]

J. Phys. Chem. Ref. Data (1)

C. T. Chantler, “Detailed tabulation of atomic form factors, photoelectric absorption and scattering cross section, and mass attenuation coefficients in the vicinity of absorption edges in the soft x-ray (z = 30–36, z = 60–89, e = 0.1 kev–10 kev), addressing convergence issues of earlier work,” J. Phys. Chem. Ref. Data 29, 597–1056 (2000).
[Crossref]

Proc. SPIE (2)

R. Willingale, G. W. Fraser, and J. F. Pearson, “Optimization of square pore optics for the x-ray spectrometer on Bepi-Columbo,” Proc. SPIE 5900, 590012 (2005).
[Crossref]

R. Willingale, J. F. Pearson, A. Martindale, C. H. Feldman, R. Fairbend, E. Schyns, S. Petit, J. P. Osborne, and P. T. O’Brien, “Aberrations in square pore micro-channel optics used for x-ray lobster eye telescopes,” Proc. SPIE 9905, 99051Y (2016).
[Crossref]

Rev. Sci. Instrum. (2)

G. J. Price, G. W. Fraser, J. F. Pearson, J. P. Nussey, I. B. Hutchinson, A. D. Holland, K. Turner, and D. Pullan, “Prototype imaging x-ray fluorescence spectrometer based on microchannel plate optics,” Rev. Sci. Instrum. 75, 2314–2319 (2004).
[Crossref]

H. N. Chapman, K. A. Nugent, and S. W. Wilkins, “X-ray focusing using square channel-capillary arrays,” Rev. Sci. Instrum. 62, 1542–1561 (1991).
[Crossref]

Spectrochim. Acta Part B (1)

K. Tsuji, T. Matsuno, Y. Takimoto, M. Yamanashi, N. Kometani, Y. C. Sasaki, T. Hasegawa, S. Kato, T. Yamada, T. Shoji, and N. Kawahara, “New developments of x-ray fluorescence imaging techniques in laboratory,” Spectrochim. Acta Part B 113, 43–53 (2015).
[Crossref]

X-Ray Spectrom. (1)

R. Alberti, T. Frizzi, L. Bombelli, M. Gironda, N. Aresi, F. Rosi, C. Miliani, G. Tranquilli, F. Talarico, and L. Cartechini, “Crono: a fast and reconfigurable macro x-ray fluorescence scanner for in-situ investigations of polychrome surfaces,” X-Ray Spectrom. 46, 297–302 (2017).
[Crossref]

Other (3)

P. Walter, P. Sarrazin, M. Gailhanou, D. Hérouard, A. Verney, and D. Blake, “Full-field XRF instrument for cultural heritage: application to the study of a Caillebotte painting,” X-Ray Spectrom. (2018).
[Crossref]

D. Blake, P. Sarrazin, T. Bristow, R. Downs, M. Gailhanou, F. Marchis, D. Ming, R. Morris, V. A. Solé, K. Thompson, P. Walter, M. Wilson, A. Yen, and S. Webb, “The mapping x-ray fluorescence spectrometer (MapX),” in 3rd International Workshop on Instrumentation for Planetary Missions (2016), p. 4006.

J. Daillant and A. Gibaud, eds., “Chap. 3: Specular reflectivity from smooth and rough surfaces,” in X-Ray and Neutron Reflectivity, Theory and Applications (Springer, 1999).

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Figures (22)

Fig. 1.
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.
Fig. 2.
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).
Fig. 3.
Fig. 3. Ray-tracing simulated PSF. The reflecting material is a 25 nm Ir layer on SiO 2 , with both the surface and interface roughness equal to 2 nm. E = 15 keV , l s = 0.05 m , D = 20 μm , T = 26 μm , and t = 4.8 mm . Normalized intensity in log scale.
Fig. 4.
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 μm , T = 26 μm , t = 1.2 mm , l s = 10 cm , E = 6400 eV , and reflective layer: Ir.
Fig. 5.
Fig. 5. Plots of S odd 2 ( α , Δ R ) as a function of α for different values of Δ R showing the influence of absorption and roughness induced loss on Ω 2 d .
Fig. 6.
Fig. 6. External total reflection critical angle γ c and reflectivity loss Δ R as a function of energy in the case of Ir reflective material. Δ R is shown for two values of the rms surface roughness σ . The insert shows the 2–3 keV region where effective values γ c 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, Δ R = Δ R eff is constant and equal to 1 .
Fig. 7.
Fig. 7. Ω 2 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.
Fig. 8.
Fig. 8. Ω eff / γ c 2 as a function of α = t γ c / D . Variations of α 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 σ = 2 nm , l s = 10 cm , t = 1.2 mm , D = 20 μm , and T = 26 μm .
Fig. 9.
Fig. 9. Ω eff , Ω 2 d , Ω 1 d , and Ω 0 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 σ = 2 nm , t = 1.2 mm , D = 20 μm , T = 26 μm , and E = 6400 eV .
Fig. 10.
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: d S = 0.25 μm 2 , Ir layer with σ = 2 nm , l s = 0.1 m , D = 20 μm , T = 26 μm , and E = 6400 eV .
Fig. 11.
Fig. 11. Central spot x direction profile relative integral breadth W/D as a function of t represented as a function of α = t γ c / D for three different energies: ray-tracing simulations results (symbols) and model of Eq. (20) (continuous lines). Parameters: Ir layer with σ = 2 nm , l s = 0.1 m , D = 20 μm , and T = 26 μm .
Fig. 12.
Fig. 12. Ω arms / Ω 2 d : (a) as a function of α for values of Δ 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.
Fig. 13.
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.
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 eV , l s = 1.2 mm , D = 20 μm , T = 26 μm , and t = 1.2 mm .
Fig. 15.
Fig. 15. Side view of a channel with the distances and angles used in the discrete model.
Fig. 16.
Fig. 16. Ω 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, energy = 6400 eV , σ = 2 nm , l s = 5 mm , t = 2.4 mm , D = 20 μm , and T = 26 μm .
Fig. 17.
Fig. 17. Ω 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 mm , σ = 2 nm , t = 2.4 mm , D = 20 μm , and T = 26 μm .
Fig. 18.
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 mm . MPO parameters: Ir with a 2 nm rms roughness as reflecting material, pore size D = 20 μm , period T = 26 μm , and thickness 1.2 mm. The number of rays generated and the intensity scale are the same for the four simulations.
Fig. 19.
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 μm , period T = 26 μm , and thickness 2.4 mm. The number of rays generated and the intensity scale are the same for the four simulations.
Fig. 20.
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 μm , and period T = 26 μm . The intensity is normalized for each image.
Fig. 21.
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 mm , Ir with a 2 nm rms roughness is the reflecting material, pore size D = 20 μm , period T = 26 μm , and thickness 1.2 mm. The number of rays generated and the intensity scale are the same for the three simulations.
Fig. 22.
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 μm , period T = 26 μm , and thickness 1.2 mm. The intensity is normalized for each image.

Equations (45)

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γ c = λ r e 2 π N av ρ m K 1.643 10 3 λ ρ m K [ γ c ( rad ) , λ ( Å ) , ρ m ( g · cm 3 ) ] K = 2 i c i ( f 0 i + f i ( λ ) ) i c i M i ,
R ¯ ( λ ) = 1 γ c ( λ ) 0 γ c ( λ ) R ( γ , λ ) d γ ,
R ( γ , λ ) = { 1 + 2 ( R ¯ ( λ ) 1 ) γ / γ c ( λ ) γ γ c ( λ ) 0 γ > γ c ( λ ) .
R ¯ < 0.5 { R ¯ eff = 0.5 γ c eff = 2 R ¯ γ c .
Ω eff = Ω 2 d + Ω 1 d + Ω 0 d .
δ n s ( θ r s ) D = { 0 | θ r s | n s 1 | θ r s | ( n s 1 ) n s 1 < | θ r s | n s n s + 1 | θ r s | n s < | θ r s | < n s + 1 0 | θ r s | n s + 1 ,
δ 0 ( θ r s ) D = { 1 | θ r s | 0 | θ r s | < 1 0 | θ r s | 1 .
Ω p ( θ x , θ y ) = δ n x ( θ r x ) δ n y ( θ r y ) ( l s t / 2 ) 2 R n x ( θ x ) R n y ( θ y ) .
d 2 N ( θ x , θ y ) = η ( l s t / 2 ) 2 D 2 d θ x d θ y ,
η = D 2 / T 2 .
Ω ( n x , n y ) = θ x = + θ y = + Ω p ( θ x , θ y ) d 2 N ( θ x , θ y ) = 4 η γ c 2 S n x S n y
S n = { 1 α 0 α δ n ( θ r s ) D R n ( θ r s t / D ) d θ r s n 0 1 2 α n = 0 ,
α = t γ c D .
α S n = { 0 α n 1 F n ( Δ R ) F n ( ( n 1 ) Δ R / α ) n 1 < α < n F n ( n Δ R / α ) F n ( ( n 1 ) Δ R / α ) + G n ( Δ R ) G n ( n Δ R / α ) n α < n + 1 F n ( n Δ R / α ) F n ( ( n 1 ) Δ R / α ) + G n ( ( n + 1 ) Δ R / α ) G n ( n Δ R / α ) α n + 1 ,
F n ( U ) = ( α Δ R ) 2 ( U ( 1 + U ) n + 1 n + 1 ( 1 + U ) n + 2 ( n + 1 ) ( n + 2 ) ) ( n 1 ) α Δ R ( 1 + U ) n + 1 n + 1 , G n ( U ) = α Δ R ( 1 + U ) n + 1 ( α Δ R ) 2 ( U ( 1 + U ) n + 1 n + 1 ( 1 + U ) n + 2 ( n + 1 ) ( n + 2 ) ) .
Ω 2 d = 4 η γ c 2 S odd 2 ( α , Δ R )
S odd ( α , Δ R ) = k = 0 + S 2 k + 1 .
Θ 1 x ( n x ) = θ x = + η ( l s t / 2 ) D δ n x ( θ r x ) ( l s t / 2 ) R n x ( θ x ) d θ x = 2 η γ c S n x
Ω 1 d = D l s + t / 2 k ( Θ 1 x ( 2 k + 1 ) + Θ 1 y ( 2 k + 1 ) )
= 4 D η γ c l s + t / 2 S odd ( α , Δ R ) = 2 D l s + t / 2 Ω 2 d .
Ω 0 d = D 2 ( l s + t / 2 ) 2 ,
Ω eff = 4 η γ c 2 S odd 2 ( α , Δ R ) + 4 D η γ c l s + t / 2 S odd ( α , Δ R ) + D 2 ( l s + t / 2 ) 2 = n = 0 2 g ( n ) [ η γ c S odd ( α , Δ R ) ] n ( D l s + t / 2 ) 2 n ,
P 1 x ( x ) = { η γ c D | Δ R | k = 0 + [ ( 1 U ) 2 ( k + 1 ) 2 ( k + 1 ) ] U min ( k , x ) U max ( k , x ) | x | D < 1 0 | x | D 1 P 2 x ( x ) = { 1 2 l s | x | < D l s l s + t / 2 0 otherwise ,
D = 1 1 t / ( 2 l s ) D U min ( k , x ) = ( 2 k + | x | / D ) | Δ R | / α U max ( k , x ) = min ( U m ( k , x ) , max ( | Δ R | , U min ( k , x ) ) ) U m ( k , x ) = ( 2 ( k + 1 ) | x | / D ) | Δ R | / α .
P 2 d ( x , y ) = P 1 x ( x ) P 1 y ( y ) P 1 d ( x , y ) = P 1 x ( x ) P 2 y ( y ) + P 2 x ( x ) P 1 y ( y ) P 0 d ( x , y ) = P 2 x ( x ) P 2 y ( y ) .
d 2 Ω eff ( x , y ) = P eff ( x , y ) d x d y = ( P 2 d ( x , y ) + P 1 d ( x , y ) + P 0 d ( x , y ) ) d x d y .
Ω arms = 8 η γ c 2 S even ( α , Δ R ) S odd ( α , Δ R ) Ω 1 d ,
Ω arms 0 = 4 η γ c 2 α S odd ( α , Δ R ) Ω 1 d .
S even ( α , Δ R ) = k = 0 + S 2 k .
Ω arms Ω eff Ω arms Ω 2 d = 2 S even S odd ,
Ω arms 0 Ω eff Ω arms 0 Ω 2 d = 1 α S odd .
Ω p . background Ω eff Ω p . background Ω 2 d = S even 2 S odd 2 = 1 4 ( Ω arms Ω 2 d ) 2 .
W cross = max ( 4 l s D / t , 4 l s γ c ) .
θ i = | x i | l s t / 2 Δ θ = D l s t / 2 .
θ min ( i ) = θ i Δ θ / 2 θ max ( i ) = θ i + Δ θ / 2.
θ min + ( 0 ) = 0 θ max + ( 0 ) = θ i + Δ θ / 2 θ min ( 0 ) = 0 θ max ( 0 ) = θ i + Δ θ / 2.
n min ( i ) = θ min ( i ) t D n max ( i ) = θ max ( i ) t D + 1 ,
Δ θ eff ( n , x i ) = θ 1 θ 2 R n ( λ , γ ) d γ = γ c Δ R [ ( 1 + U ) n + 1 n + 1 ] Δ R θ 1 / γ c Δ R θ 2 / γ c ,
θ 1 , θ 2 = { min ( θ min , γ c ) , min ( θ x i ( n min + 1 ) , γ c ) n = n min min ( θ x i ( n ) , γ c ) , min ( θ x i ( n + 1 ) , γ c ) n min < n < n max min ( θ x i ( n max ) , γ c ) , min ( θ max , γ c ) n = n max
θ x i ( n ) = 1 l s + t / 2 [ | x i | + ( n 1 / 2 ) D ] .
Σ x , odd = i = + k = 0 + Δ θ eff ( 2 k + 1 , x i ) Σ y , odd = j = + k = 0 + Δ θ eff ( 2 k + 1 , y j ) .
Σ x , 0 = k , n min x = 0 [ min ( θ x k ( 1 ) , D / ( 2 l s ) ) min ( θ min xk , D / ( 2 l s ) ) ] Σ y , 0 = k , n min y = 0 [ min ( θ y k ( 1 ) , D / ( 2 l s ) ) min ( θ min yk , D / ( 2 l s ) ) ] .
Ω effective , d = ( Σ x , odd + Σ x , 0 ) ( Σ y , odd + Σ y , 0 ) .
d 4 n = N s 4 π T 2 d S d Ω = N s η 4 π D 2 d S d Ω .
d 2 n = δ n x ( θ r x ) δ n y ( θ r y ) R n x ( θ x ) R n y ( θ y ) N s η 4 π D 2 d θ x d θ y .

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