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Abstract
We report on the efficient collimation of soft X-rays with an energy of 277 eV by a halved polycapillary lens (PCL), made of borosilicate glass. Using electron-excited, micro fluorescence emission in the focus of the PCL, experiments reveal an angular divergence of (6.9 ± 0.2) mrad in the far field of the emitted beam. For a source of $\approx 5\,\mathrm{\mu}\textrm{m}$ in size, that result is confirmed by simulations, obtained with a newly developed ray tracing code. An analytical fit model is proposed and applied to characterize the evolution of the measured as well as calculated, three-dimensional (3-D) intensity distribution. The photon flux density in a free-space propagation distance of (0.4 − 0.9) m from the PCL is enhanced by a factor of ≈(30 − 90) in comparison to the direct, not collimated radiation, as it is detected through a mm-sized transmission slit. Our findings could help to establish the halved PCL as a versatile tool in the table-top metrology of optical elements, such as mirrors and gratings for soft X-rays.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Since their invention [1], polycapillary lenses became increasingly popular not only in neutron optics but especially in X-ray science [2,3]. Several types of such devices have been designed and commercialized [4,5] for various purposes [6,7]. In fluorescence spectroscopy with a typical photon energy of a few keV for instance [8–10], the tapered [11] and halved, collimating PCL is used to grasp the point-like emission from the sample and to redirect a quasi parallel beam to an analyzer crystal in the Bragg geometry [12]. The divergence, which can be as low as $3\,\textrm {mrad}$ or even less, fits well to the narrow rocking curve of the crystal and is determined by the energy-dependent critical angle for total external reflection (TER) inside the capillaries [13].
A sufficiently collimated beam [14] not only supports wavelength-dispersive hard X-ray spectroscopy but is of great advantage also in the sub-keV and extreme ultraviolet (XUV) domain [15]. In a recent work, the properties of a halved PCL were investigated in its focusing mode at an energy of 36 eV and, based on these results, an efficient spectrometer with a reflection zone plate was designed [16]. Later, optical elements like diffraction gratings were probed in the laboratory with a collimating PCL of similar shape, using Carbon $\textrm {K}_{\alpha }$ radiation at 277 eV [17].
However, the properties of narrow X-ray beam collimation by a PCL at low photon energies of a few $10^{2}\,\textrm {eV}$ or less are still widely unexplored. Despite instructive simulations [18–22], promising applications [23] and apart from the specification of PCLs by manufacturers for a “broad spectral bandwidth: 10 eV – 50 keV” [5], experimental data and a consistent theoretical description in the XUV and soft X-ray spectral range are rarely available.
In this paper, we present direct measurements of the beam, collimated by a halved polycapillary lens, at an energy of 277 eV. We characterize them by a universal fit model and consider the achievable gain in the photon flux as a practical implication for use in, e.g., metrology. Section 2 introduces the polycapillary sample under study, mounted in the home-built test bench. In Sect. 3, measured intensity patterns are fitted by the proposed analytical function and compared to the outcome of simulations in Sect. 4. The versatility of the ray tracing code and its limitations are briefly discussed for varied parameters in Sect. 5. With an outlook to perspectives for future experimental and theoretical research as well as to potential applications, we conclude in Sect. 6.
2. Optics specification and setup
The polycapillary optic [4] in use can be described as a solid bundle of $2.9\times 10^{4}$, approximately hexagonal tubes made of borosilicate glass $(\textrm {Si}_{81}\textrm {O}_{215}\textrm {B}_{26}\textrm {Na}_{8}\textrm {K}_{8}\textrm {Al}_{4})$, assembled in a “honeycomb” structure [16] with a thickness of the walls close to $4\,\mathrm{\mu}\textrm{m}$. The tapered geometry of the halved PCL, as shown in Fig. 1, is optimized to collimate the spherical emission of a point-like source with a solid angle of $26.7\,\textrm {msr}$ into an almost parallel beam at the outer aperture, with an estimated net transmission of $50\,{\% }$ at $277\,\textrm {eV}$. This Carbon $\textrm {K}_{\alpha }$ fluorescence is excited by an electron beam at 4.4 keV [17], focused on the target to a spot whose size is estimated to $\varnothing _{\textrm {src.}}\approx 5\,\mathrm{\mu}\textrm{m}$, based on a comparison with simulations (Sect. 4). As sketched in Fig. 2, the collimator is mounted in the focal distance $F$ from the source and aligned to the optical axis, similar to the method in [24]. An evacuated $\left (10^{-5}\,\textrm {mbar}\right )$ tube of variable length allows for free space propagation of the slightly divergent beam along the distance $\Delta x\lesssim 1\,\textrm {m}$ from the outer aperture of the PCL to the CCD camera “ANDOR CCD 47-10”, whose $1024\times 1024$ square pixels of $13\,\mathrm{\mu}\textrm{m}$ in size provide an effective spatial resolution of $\approx 25\,\mathrm{\mu}\textrm{m}$. The device features a dynamic range of $5\times 10^{4}:1$ at a readout noise of $2.0\,\textrm {e}^{-}\,\textrm {(rms)}$. As an option, a transmission slit of adjustable width in the range $0.1\,\textrm {mm}\leq \Delta h\leq 2.5\,\textrm {mm}$ (at a constant height of $10\,\textrm {mm})$ can be inserted into the beam, either near to or far from the PCL, as an entrance pupil for the CCD camera which is placed behind it.
Fig. 1. Schematic of the polycapillary collimator in use, drawn to scale. Focal distance and capillary length are specified to $F=34.5\,\textrm {mm}$ and $L=27\,\textrm {mm}$, respectively. The measured angular divergence is denoted as $\Delta \beta _{\textrm { {exp.}}}^{\textrm { {(o)}}}$ (Sect. 3). Inner and outer diameters of the full aperture and single capillaries are listed as $\varnothing _{\textrm { {aperture}}}^{\textrm { {(in,out)}}}$ and $\varnothing _{\textrm { {capillary}}}^{\textrm { {(in,out)}}}$ on the right.
Fig. 2. Experimental setup. The point-like $\left (\varnothing _{\textrm {src.}}\right )$ X-ray emission (277 eV, red star) and the PCL collimator with a focal distance $F$ and geometrical length $L$ are located in the source chamber. The quasi parallel beam propagates along the optical axis (red dashed line) and is detected by the CCD ($13.3\times 13.3\,\textrm {mm}^{2}$) in an adjustable distance $\Delta x$ to record the intensity patterns as shown in Fig. 3. Alternatively, a slit of width $\Delta h$ can be inserted at two positions and the CCD behind measures the transmitted flux.
3. Experimental results and fit model
After eye-guided alignment of the focus to the source (Fig. 1) under criteria of axial symmetry [24] and minimal divergence $\Delta \beta _{\textrm { {exp.}}}^{\textrm { {(o)}}}$, the two-dimensional (2-D) intensity distribution of the almost parallel beam is measured by the CCD camera in a variable distance $374.8\,\textrm {mm}\leq \Delta x\leq 897.8\,\textrm {mm}$ from the exit aperture of the PCL. At an exposure time of $(10-40)\,\textrm {s}$ per image, six patterns are recorded, as shown in Fig. 3. The star-like shape, which is prominent for $\Delta x\leq 538.8\,\textrm {mm}$ in particular, can be attributed to reflections at opposing walls of the hexagonal capillaries, unlike helical propagation [18]. Neglecting that 6-fold symmetry for simplicity from now on, the radial intensity distribution $I_{\perp }(r)$ on the CCD in a given distance $\Delta x$ can be modeled universally as
Fig. 3. Experimental intensity profiles of the collimated beam in a series of $y-z$ planes at an increasing free space propagation distance $\Delta x$, measured from the exit aperture of the PCL. The color code represents the number of counts. White contours indicate $30\,{\% }$ and $50\,{\% }$ of the specific peak intensity in each CCD image with $940\,\times \,940$ pixels.
Fig. 4. Values (black dots) of the fit parameters in Eq. (1), used to model the measured intensity profiles from Fig. 3. The peak intensity $c_{1}$ (left) is described by an exponential, Gaussian-like decay, whereas polynomial and linear functions are drawn as a guide to the eye (red curves) for the exponent $p$ (center) and base level $c_{0}$ (right), respectively.
We rate the goodness of the fit by Eq. (1) with the “coefficient of determination,” which is computed to $R^2=(99.73\pm 0.11)\,{\% }$ in average over the six samples from Fig. 3. That value close to 1 implies that the model in Eq. (1) explains the CCD data accurately, and residuals are mainly caused by pixel-to-pixel fluctuations, i.e., the statistical noise: the standard deviation of the relative difference between measured pixel values and the fit $I_{\perp }(r)$, averaged over the six CCD planes in Fig. 3, is calculated to $\sigma =\pm \, 8.5\,{\% }$ within a circular region of 940 pixels in diameter and further reduced to $\pm \, 5.8\,{\% }$ inside the central disk of 470 pixels with higher intensity. In all, the radial intensity distributions of the measured data nearly coincide with their fits, both in one dimension (1-D), i.e., for the central pixel row and column, and two dimensions (2-D).
As stated above, the collimated beam divergence is minimized in case of an optimal alignment as sketched in Fig. 5, with the point-like source in the focal position of the halved PCL. For the data and the linear regression labeled by $\Delta \beta _{\textrm { {exp.}}}^{\textrm { {(o)}}}$ on the left in Fig. 5, the diameter $\varnothing _{\textrm { {FWHM}}}^{\textrm { {(p)}}}\left (w_{0}\right )$ is averaged over the respective fit values $w_{0}$ in Eq. (1) at each position $\Delta x$ from 1-D cross sections $I_{H}(y,0)$ and $I_{V}(0,z)$ as well as the radial 2-D fit $I_{\perp }(r)$, accounting for slight H / V asymmetries in the measured intensity profiles (Fig. 3). The error bars represent the standard deviation around this arithmetic mean. We find an angular divergence $\Delta \beta _{\textrm { {exp.}}}^{\textrm { {(o)}}}=(6.9\pm 0.2)\,\textrm {mrad}$. For an axial defocus of the optic by either $\delta x = -5\,\textrm {mm}$ or $\delta x = +4\,\textrm {mm}$ in the coordinate system of Fig. 2, the divergence increases to $(17.6\pm 0.8)\,\textrm {mrad}$ – within enlarged error margins, since the beam expands and blurs. The formal extrapolation of the fit $\varnothing _{\textrm { {FWHM}}}(\Delta x)$ on the left in Fig. 5 intersects the abscissa at $\Delta x_{0}=(72\pm 2)\,\textrm {mm}$ in the aligned case and at $\Delta x_{0}=(256\pm 12)\,\textrm {mm}$ under defocus. This offset implies a diminished divergence in the “near field,” relatively close to the outer PCL aperture. The phenomenon has been observed in simulations [20] and measurements [21] and is associated with the predominance of central rays in that near field, passing through almost straight capillary channels near the optical axis without TER and thus at a particularly low divergence [21]. Around a certain, short distance $\Delta x$, they are crossed by the widely deflected rays from the outer aperture, leading to an enhanced on-axis intensity and in effect, to a local reduction of the averaged beam divergence. Though possible in principle, we did not measure the beam diameter for $\Delta x < 0.37\,\textrm {m}$ in our experiment – because of technical constraints but also due to the “spiky” intensity distribution which impedes an approximately homogeneous illumination of optical elements under test, as required for their valid characterization [17].
Fig. 5. Left: experimental beam diameter (FWHM, dots with error bars) as a function of the free space propagation distance $\Delta x$ and the linear fit (red line) with extrapolation (black dashed) and standard deviation (light gray), whose slope yields the divergence $\Delta \beta _{\textrm { {exp.}}}^{\textrm { {(o)}}}$. Data for defocus are displayed as rectangles (dark blue) and their fit is shown in gray. Right: photon flux through a slit of width $\Delta h$, measured using the PCL and related to the transmitted flux without PCL. This gain is evaluated at two slit positions $\Delta x$. Data (black dots) are fitted with $2^{\textrm { {nd}}}$ order polynomials (red) as a guide to the eyes.
In an application-oriented experiment we return to the aligned, in-focus PCL collimator with its minimal divergence $\Delta \beta _{\textrm { {exp.}}}^{\textrm { {(o)}}}$. The photon flux through a 1-D slit of width $\Delta h$, centered around the optical axis and located at a distance $\Delta x$ of either $374.8\,\textrm {mm}$ or $897.8\,\textrm {mm}$ from the exit aperture of the PCL (Fig. 2), is compared to the count rate within the same slit aperture but without PCL. With an only slight dependence on $\Delta h$, we observe an increase by a factor of $(26.6 - 29.7)\,\times$ for the short propagation length and by $(92.3 - 95.9)\,\times$ in the long range case, as plotted on the right in Fig. 5. Since all other parameters like source size and bandwidth are unchanged, this “gain” implies a $(5-10)$-fold improved signal-to-noise ratio in, e.g., the metrology of optical elements.
4. Ray tracing code and simulations
A customized Monte Carlo code is written in the programming language Mathematica/Optica Software, adapted to the presumed ellipsoidal shape of a halved PCL, like that used in a previous [16] as well as in the present work. In a local Cartesian coordinate system with its origin at the geometrical center of the PCL, the geometry of a single capillary or “channel,” identified by its index $m\,\epsilon \,\mathbb {N}$, is sketched in Fig. 6. The circular, bent and tapered surface $\vec {s}_{m}\left (x,\psi _{m},\xi \right )$ can be described as the revolution $\vec {\rho }\left (x,\xi \right )=\rho _{\textrm { {cap.}}}^{\textrm { {(o)}}}\left [\cos \xi \vec {e}_{y}+\sin \xi \vec {e}_{z}\right ]\mathcal {F}(x)$ around the local optical axis $\vec {a}_{m}\left (x,\psi _{m}\right )=x\vec {e}_{x}+r_{\textrm { {aper.}}}^{\textrm { {(m)}}} \left [\cos \psi _{m}\vec {e}_{y}+\sin \psi _{m}\vec {e}_{z}\right ]\mathcal {F}(x)$, using the outer radii $r_{\textrm { {aper.}}}^{\textrm { {(m)}}}\leq \varnothing _{\textrm { {aperture}}}^{\textrm { {(out)}}}/2$ within the full aperture and $\rho _{\textrm { {cap.}}}^{\textrm { {(o)}}}\equiv \varnothing _{\textrm { {capillary}}}^{\textrm { {(out)}}}/2$ for an individual capillary. The real function $\mathcal {F}(x)\equiv [1-a_{S}^{-2}(L/2-x)^{2}]^{1/2}$ is specified by $a_{S}=40.9202$ for the semi-major axis, adjusted to match the focal length from Fig. 1. In closed terms, we write the surface vector $\vec {s}_{m}$ in the form
Fig. 6. A single capillary in the ray tracing model. Each point on the surface (brown) is defined by the vector sum $\vec {s}_{m}\left (x,\psi _{m},\xi \right )=\vec {a}_{m}\left (x,\psi _{m}\right )+\vec {\rho }\left (x,\xi \right )$, drawn as red straight lines. $\xi$ denotes the local azimuth angle within the channel. The sketch is not to scale.
At 277 eV, the refractive index of borosilicate glass with a density of $2.23\,\textrm {g}\,\textrm {cm}^{-3}$ is calculated to $0.99531 + 0.00151\,i$, using the CXRO database [25]. The corresponding single bounce reflectivity at the capillary walls is averaged over the squared absolute values of the complex s- and p-polarized Fresnel coefficients. A surface roughness of $\pm 1.8\,\textrm {arcsec}$ (rms) is assumed, following estimations on a formerly used [16], similar PCL from the same manufacturer [4], and implemented as a normal distribution of geometrical slope errors in our simplified albeit sufficiently realistic scheme. In fact, “a random factor” is introduced “to the surface normals of one or more optical surfaces in the component,” as described in the user manual of Optica Software. In a vivid picture, each propagating ray that impinges on the inner capillary wall under the local grazing angle $\theta _{i}$ is reflected to $\theta _{i}\pm \delta \theta$, modified by the small random angle $\delta \theta$, which follows a Gaussian probability distribution with the standard deviation from above. Nevertheless, the moderate surface roughness as assumed in our work turns out as a less critical parameter, whereas geometry and refractive index affect the performance of the PCL significantly.
Emitted from an incoherent, 2-D Gaussian source, $6.6\times 10^{4}$ rays are traced through random (2-D normal distribution, FWHM $\lesssim$ entrance aperture) samples of channels $\{r_{\textrm { {aper.}}}^{\textrm { {(m)}}},\psi _{m}\}$ with $1\leq m\lesssim 9.5\times 10^{3}$, as defined by Eq. (2), corresponding to seven rays per channel. Multiple reflections at the glass walls reduce the transmission per channel to $(85.9\pm 8.7)\,{\% }$, depending on its radial location $r_{\textrm { {aper.}}}^{\textrm { {(m)}}}$; and the stochastic nature of the photon transport along various optical paths leads to an accidental blur of the phase [16], like in a multi mode fiber.
In its current implementation, the ray tracing engine takes around 18 hours net running time to process the presented or similar examples (Sect. 5) on an 8-core Xeon workstation (4.6 GHz, 128 GB RAM) in the parallel computing mode. As a benefit, the statistical precision competes with the experimental accuracy in terms of the simulated “counts” per pixel.
Beam shape and angular divergence, both derived from a fit of the ray tracing data based on Eq. (1), vary moderately within the region of interest $\varnothing _{\textrm {src.}}\leq 20\,\mathrm{\mu}\textrm{m}$ for micron sized sources. The predicted far field divergence is as low as $4.3\,\textrm {mrad}$ in the limit of an ideal point source. With its expansion, the beam spread tends to grow as well. Within some statistical uncertainty due to the finite number of traced rays per run, a source of $\varnothing _{\textrm {src.}}\approx (5\pm 3)\,\mathrm{\mu}\textrm{m}$ yields an acceptable match between experiment and simulation in terms of the beam parameters, especially the divergence.
For a representative example, the free space propagation, depicted as cross sections in Fig. 7, is analyzed in the same way as the measured data in Sect. 3. With $R^{2}=(99.5\pm 0.2)\,{\% }$, the model in Eq. (1) applies to the simulated intensity distribution as well. Aside from $c_{0}=0$ (no out-of-plane scattering), the fit yields similar values for the exponent $0.99\leq p\leq 1.23$ and the diameter $\propto w_{0}$, which falls below the experimental results for $\varnothing _{\textrm { {FWHM}}}^{\textrm { {(p)}}}\left (w_{0}\right )$ by $(8.7\pm 7.8)\,{\% }$. In order to illustrate the suitability of Eq. (1) to model the simulated beam profile, and for a direct comparison with the respective fit of the experimental data (Fig. 3) using the parameters from Fig. 4, we plot the radial intensity distributions in Fig. 8. The diameter (FWHM) of the 3-D intensity profile, as marked in Fig. 7, grows from one plane to the next. The beam width can be characterized by a “waist-like” minimum of $2.3\,\textrm {mm}$ (and locally zero divergence) near the first CCD plane at $\Delta x = 374.8\,\textrm {mm}$, a quadratic growth between the first and third plane, and a linear expansion for $\Delta x\geq 538.8\,\textrm {mm}$ – similar to the behavior of a Gaussian beam within the evaluated range for $\Delta x$. Regardless of that difference to the experiment, the simulated far field divergence $\Delta \beta _{\textrm { {sim.}}}^{\textrm { {(o)}}}=(7.0\pm 0.1)\,\textrm {mrad}$ coincides with the measured result (Fig. 5) within error margins (rms). Moreover, its extrapolation in analogy to Sect. 3 intersects the abscissa at $\Delta x_{0} = 154\,\textrm {mm}$ – in qualitative agreement with our experiments and published results [20].
Fig. 7. Simulated 2-D beam profiles in analogy to the sequence in Fig. 3. The color code represents the intensity relative to the maximum for $\Delta x=374.8\,\textrm {mm}$ at $(y,z)=(0,0)$. Contours mark $30\,{\% }$ and $50\,{\% }$ of the fitted peak value in each plane. From the original size of $(34\times 34)\,\textrm {mm}^{2}$, all images are cropped to $47\,\times \,47$ pixels of $(0.53\times 0.53)\,\textrm {mm}^{2}$.
Fig. 8. Radial intensity distribution of the 2-D beam cross sections from Fig. 3 and Fig. 7 for each CCD plane, normalized to the peak value at $\Delta x=374.8\,\textrm {mm}$. The displayed ray tracing data (black dots) are averaged over the central pixel row $(y,z\approx 0)$ and column $(y\approx 0,z)$; whereas the symmetric fit of the simulated, full 2-D profile in Fig. 7 by Eq. (1) is drawn in red. For the plot of the experimental intensity distribution (black, dashed line), the – very similar – 1-D and 2-D fit parameters are averaged (see Sect. 3).
At last, an increased spread of $8.4\,\textrm {mrad}$ is evaluated for a widely extended emission with an FWHM of $20\,\mathrm{\mu}\textrm{m}$. Notably, Eq. (1) remains valid in each case, though the numerical values of the function variables vary with the source size more or less, reflecting a gradual change of the 3-D profile from a “spiky,” axial beam to a relatively smooth or even Gaussian-like distribution.
5. Variations of energy and geometry
The simulation model and its implementation as a ray tracing code, as described in Sect. 4, are not restricted to the PCL from Fig. 1 and the setup in Fig. 2 but may be rather used at different XUV or X-ray energies and under various experimental conditions.
$\bullet$ If reversed and illuminated by a parallel beam of the diameter $\varnothing _{\textrm { {apert.}}}^{\textrm { {(out)}}}$, the optic works as a focusing lens. The ray tracing scheme clearly shows the footprint of multiple reflections inside the capillaries. In the focal distance $F=34.5\,\textrm {mm}$, the X-rays with an energy of $277\,\textrm {eV}$ are concentrated to an – approximately – Gaussian spot of $367\,\mathrm{\mu}\textrm{m}$ (FWHM) in diameter. The focal depth of field can be investigated by additional, defocused detection planes (“screens”), inserted in the propagation path. We compare the simulated focus size with the value as expected from the literature on (hard) X-ray capillary optics: according to Eq. (8) in [2], the single capillary aperture of diameter $\varnothing _{\textrm { {capill.}}}^{\textrm { {(in)}}}$ is convolved with the spread of rays in the observation plane, characterized by its spatial dimension $F{\cdot }\Delta \beta _{\textrm { {theor.}}}^{\textrm { {(o)}}}$, where $\Delta \beta _{\textrm { {theor.}}}^{\textrm { {(o)}}}\approx 1.3\,\theta _{c}$ [2,3]. The “theoretical” focus size calculated in this way exceeds our simulated spot diameter by a factor of $12$, confirming similar, experimentally verified results for another PCL in the XUV domain [16].
$\bullet$ At an energy of $8\,\textrm {keV}$ ($\textrm {Cu K}_{\alpha }$), with an adapted refractive index (CXRO) but other things being equal, the code can be applied as well. The – slightly less accurate ($R^{2}=98.4\,{\% }$) – fit using Eq. (1) reveals a narrow beam, associated with reduced transmission of about $42\,{\% }$ due to absorption losses. The simulation yields an angular divergence of $2.6\,\textrm {mrad}$ in this hard X-ray domain, somewhat below but in the same order of the critical angle $\theta _{c}=3.8\,\textrm {mrad}$ for total external reflection.
$\bullet$ Not only the photon energy but also the geometrical design parameters of the PCL are modified in the case of the XUV collimator, operated at an energy of 36 eV [16]. With moderately enlarged inner and outer capillary diameters of $35\,\mathrm{\mu}\textrm{m}$ and $59\,\mathrm{\mu}\textrm{m}$, respectively, ray tracing using a source of $5\,\mathrm{\mu}\textrm{m}$ leads to an asymptotic far field divergence of $7.4\,\textrm {mrad}$. This result is in reasonable agreement with the predicted value of $7.8\,\textrm {mrad}$ [16] – and much smaller than the critical angle at this low energy. As a peculiarity, the beam cross section is evaluated at 24 equidistant positions $0\leq \Delta x\leq 1.4\,\textrm {m}$ in this version of the ray tracing program, unveiling the signature of a distinct waist near $\Delta x\approx 0.5\,\textrm {m}$, as it is reported in the literature [20,21].
$\bullet$ Of practical relevance might be an (accidental) tilt of the device relative to the optical axis, due to imperfect alignment. Coarse, semi-quantitative simulations for the same XUV collimator [16] have been performed, changing the orientation of the PCL vs. the $x$-axis gradually. Typical intensity distributions, evaluated in a distance up to $\Delta x\lesssim 1.4\,\textrm {m}$, indicate a tolerance in the order of $\pm \, 0.1^{\circ }$ without any notable effect on the beam profile. For tilts of $\gtrsim 0.5^{\circ }$, blurring is observed, before the characteristic deformation to an annular shape [18] comes up for $\lesssim 1.5^{\circ }$.
All those examples as discussed so far are based on an incoherent, point-like source distribution, traced through an optic of halved ellipsoidal shape. Nonetheless, other versions of polycapillary assemblies, such as point-to-point focusing lenses [2] or cylindrical collimators [20] can be modeled in principle as well, at various photon energies within the validity of ray tracing.
6. Conclusion
The operation of a halved PCL as an efficient beam collimator is demonstrated experimentally at a photon energy of 277 eV, and the 3-D intensity distribution of the propagating beam can be described by a universal, exponential function in the transverse, radial coordinate. With an angular spread $\Delta \beta _{\textrm { {exp.}}}^{\textrm { {(o)}}}=(6.9\pm 0.2)$ mrad, the radiation is confined to a far field divergence well below the critical angle of reflection $\theta _{c}=5.5^{\circ }$ [25] under grazing incidence on the capillary walls, made of borosilicate glass (CXRO).
As it is widely accepted, plausible and proven in medium and hard X-ray experiments, “the divergence ${\ldots }$ is mainly determined by the critical angle of total reflection” [13]. In a first, naive approximation, the “divergence ${\ldots }$ is generally supposed to be a constant of about two critical angles of total reflection,” neglecting “fiber-to-fiber misalignment” in a “modern manufacturing technique” [21]. More specifically, the empirical rule $\Delta \beta _{\textrm { {theor.}}}^{\textrm { {(o)}}}\approx 1.3\,\theta _{c}$ [2,3] is valid in the range of several keV. However, our findings suggest the need for an extension of this relation toward the soft X-ray and XUV [16] domain with its enlarged values for $\theta _{c}$, since the observed divergence at 277 eV is $18\,\times$ smaller than expected $(\Delta \beta _{\textrm { {theor.}}}^{\textrm { {(o)}}}\approx 126\,\textrm {mrad})$ from that rule [2,3]. The discrepancy might be coarsely explained via the reflectivity $R(\theta )$. In the hard X-ray domain, we have $R(\theta )\approx 1$ for $\theta <\theta _{c}$ and $\approx 0$ for $\theta >\theta _{c}$, whereas in the XUV and soft X-ray range, this reversed Heaviside function is smeared out and $R(\theta )$ decays smoothly from 1 to 0, with $R\left (\theta _{c}\right )=0.5$. After $n$ reflections, $R^{n}(\theta )\lesssim 1$ still holds for all rays with an angle up to $\theta _{c}$ in the hard X-ray case. At low energies in contrast, $R^{n}(\theta )$ would drop down rapidly to 0 for all rays except those with $\theta \ll \theta _{c}$. Hence, $\theta _{c}$ can no longer set an upper bound to the divergence. Instead, the capillaries with a mean aperture of $\langle \varnothing _{\textrm { {capill.}}}^{\textrm { {(i,o)}}}\rangle \approx 38\,\mathrm{\mu}\textrm{m}$ may guide the rays by a few $(n_{g}\lesssim 2)$ reflections under grazing angles $\theta _{g}\approx \langle \varnothing _{\textrm { {capill.}}}^{\textrm { {(i,o)}}}\rangle /L_{f}$ with a free length $L_{f}=L/(n_{g}+1)$, such that $\Delta \beta _{\textrm {geo.}}^{\textrm { {(o)}}}\lesssim 2\theta _{g}\approx 8\,\textrm {mrad}$.
An editable, “transparent” simulation program is developed and applied, to unveil the physical mechanisms which are responsible for the superior collimation in that low energy range. Notably, evidence for a mainly geometrical interpretation of the photon transport inside the channels [12], extended only by Fresnel reflectivity and Gaussian slope errors to imitate the surface roughness, is provided by the good agreement of the measured with the simulated beam profile (Fig. 8) and the divergence $\Delta \beta _{\textrm { {sim.}}}^{\textrm { {(o)}}}=(7.0\pm 0.1)\,\textrm {mrad}$, the latter obtained via Monte Carlo ray tracing for a typical source size in the order of $\approx (5\pm 3)\,\mathrm{\mu}\textrm{m}$. In other words, under given conditions, the low divergence may be explained simply as an effect of (multi mode) fiber-like collimation in 3-D.
The parameters of the simulation are adjusted for a good match with the experiment. Despite its promising functionality in the frame of the present study, the algorithm must be validated and improved for general usability. For instance, the effect of large sources in the order of $(10^{2}-10^{3})\,\mathrm{\mu}\textrm{m}$ on the collimated output shall be explored, in comparison to experiments. If applicable, the spatial distribution of the traced capillaries across the PCL aperture could be refined, to imitate the real angular acceptance of the optic and its illumination by the micro fluorescence source more closely. Besides, the circular cross section of single capillaries from Eq. (2) might be replaced by the hexagonal structure. Clearly, a more sophisticated model of reflection from the inner surface of capillaries is desirable, e.g., as detailed recently [12]. Additional detection planes (”screens”) can be inserted along the $x$-axis, especially near the outer aperture of the PCL $(\Delta x\gtrsim 0)$, for a better tracking of the beam profile. From a technical point of view, the program code may be translated from Mathematica/Optica Software to a high-speed routine in Fortran or Python, provided that numerical precision is maintained.
Regarding applications, the low divergence implies a $(30-90)\,\times$ gain in the photon flux density, and predestines the halved PCL for testing, e.g., the reflectivity of XUV and soft X-ray mirrors, or the diffraction efficiency of reflection zone plates (RZPs) with a weak laboratory source [17] and a quasi parallel beam in a meter-scaled test stand.
For spectroscopy however, the random phase distribution, caused by a statistical variation of the optical path in the capillaries, should be reduced [16]. Simulations with the code from Sect. 4 indicate a a nearly plane wavefront with a well-defined phase for a PCL whose channels are narrowed to $\sim 4\,\mathrm{\mu}\textrm{m}$, i.e., by a factor of about $10$. In case of a successful experimental verification [26,27], such an optimized PCL might enable high-resolution spectroscopy of low-Z elements like Li [15], using an RZP as the wavelength dispersive element in a compact setup.
Funding
Investitionsbank Berlin (10162112); EuroNorm GmbH (49MF190056).
Acknowledgments
The authors are grateful to Alexei Erko$^{\dagger }$, who initiated and supervised this research. We thank Ying Liu from the University of Science and Technology of China for her support during manuscript revision.
Disclosures
JP and HL: NOB Nano Optics Berlin GmbH (E), MT and SB: Helmut Fischer GmbH (E). The other authors declare no conflicts of interest.
Data availability
Data and the code underlying the results may be obtained from the authors upon request.
References
1. M. A. Kumakhov and V. A. Sharov, “A neutron lens,” Nature 357(6377), 390–391 (1992). [CrossRef]
2. C. A. MacDonald, “Focusing Polycapillary Optics and Their Applications,” X-Ray Opt. Instrum. 2010, 1–17 (2010). [CrossRef]
3. T. Sun and C. A. MacDonald, “Full-field transmission x-ray imaging with confocal polycapillary x-ray optics,” J. Appl. Phys. 113(5), 053104 (2013). [CrossRef]
4. Helmut Fischer GmbH, Institut für Elektronik und Messtechnik, Industriestraße 21, 71069 Sindelfingen, Germany, www.helmut-fischer.com/products/x-ray-optics/collimating-polycapillary (2023).
5. XOS Inc., 15 Tech Valley Drive | East Greenbush, NY 12061, USA, https://www.xos.com (2023).
6. T. Sun, S. Peng, Z. Liu, W. Sun, Y. Ma, and X. Ding, “Performance of polycapillary X-ray optics for confocal energy-dispersive small-angle X-ray scattering,” J. Appl. Cryst. 46(6), 1880–1883 (2013). [CrossRef]
7. K. M. Sowa, A. Last, and P. Korecki, “Grid-enhanced X-ray coded aperture microscopy with polycapillary optics,” Sci. Rep. 7(1), 44944 (2017). [CrossRef]
8. X. Sun, X. Zhang, Y. Wang, S. Shao, Y. Li, S. Peng, Z. Liu, and T. Sun, “Performance of assembled X-ray optics consisted of a polycapillary X-ray optics and a monocapillary X-ray optics for micro X-ray fluorescence spectrometry,” Spectrochim. Acta, Part B 165, 105770 (2020). [CrossRef]
9. W. Malzer and B. Kanngießer, “A model for the confocal volume of 3D micro X-ray fluorescence spectrometer,” Spectrochim. Acta, Part B 60(9-10), 1334–1341 (2005). [CrossRef]
10. A. Erko and I. Zizak, “Hard X-ray micro-spectroscopy at Berliner Elektronenspeicherring für Synchrotronstrahlung II,” Spectrochim. Acta, Part B 64(9), 833–848 (2009). [CrossRef]
11. S. H. Nowak, M. Petric, J. Buchriegler, A. Bjeoumikhov, Z. Bjeoumikhova, J. von Borany, F. Munnik, M. Radtke, A. D. Renno, U. Reinholz, O. Scharf, J. Tilgner, and R. Wedell, “Road to micron resolution with a color X-ray camera–polycapillary optics characterization,” arXiv, arXiv:1705.08939 [physics.ins-det] (2017). [CrossRef]
12. P. Jagodziński, M. Pajek, D. Banaś, A. Kubala-Kukuś, J. Szlachetko, M. Cotte, and M. Salomé, “Properties of polycapillary optics dedicated to low-energy parallel-beam wavelength-dispersive spectrometers for synchrotron-based X-ray fluorescence study,” Opt. Express 29(17), 27193–27211 (2021). [CrossRef]
13. T. Sun and X. Ding, “Measurements of energy dependence of properties of polycapillary x-ray lens by using organic glass as a scatterer,” J. Appl. Phys. 97(12), 124904 (2005). [CrossRef]
14. S. D. Padiyar, H. Wang, M. V. Gubarev, W. M. Gibson, and C. A. MacDonald, “Beam Collimation using Polycapillary X-ray optics for Large Area Diffraction Applications,” Adv. X-ray Anal. 43, 254–259 (2000).
15. T. Ogiwara, T. Kimura, S. Fukushima, K. Tsukamoto, T. Tazawa, and S. Tanuma, “Analysis of ultra light elements with newly developed ultra-soft X-ray spectrometer for electron probe microanalysis,” Microchim. Acta 161(3-4), 451–454 (2008). [CrossRef]
16. C. Braig, A. Sokolov, R. G. Wilks, X. Kozina, T. Kunze, S. Bjeoumikhova, M. Thiel, A. Erko, and M. Bär, “Polycapillary-boosted instrument performance in the extreme ultraviolet regime for inverse photoemission spectroscopy,” Opt. Express 25(25), 31840–31852 (2017). [CrossRef]
17. J. Probst, H. Löchel, C. Braig, C. Seifert, and A. Erko, “Laboratory characterization of soft X-ray optics,” OSA High-brightness Sources and Light-driven Interactions Congress 2020, ETh1A.3 (2020).
18. L. Vincze, K. Janssens, F. Adams, A. Rindby, and P. Engström, “Interpretation of capillary generated spatial and angular distributions of x rays: Theoretical modeling and experimental verification using the European Synchrotron Radiation Facility Optical beam line,” Rev. Sci. Instrum. 69(10), 3494–3503 (1998). [CrossRef]
19. H. J. Sanchez, “Energy dependence of X-ray capillary performance,” Nucl. Instrum. Methods Phys. Res., Sect. B 145(4), 567–572 (1998). [CrossRef]
20. A. Liu, “Simulation of X-ray beam collimation by polycapillaries,” Nucl. Instrum. Methods Phys. Res., Sect. B 234(4), 555–562 (2005). [CrossRef]
21. T. Sun, Z. Liu, Y. Li, X. Lin, P. Luo, Q. Pan, and X. Ding, “Fine structures of divergence of polycapillary X-ray optics,” Nucl. Instrum. Methods Phys. Res., Sect. B 269(23), 2758–2761 (2011). [CrossRef]
22. S. Peng, Z. Liu, T. Sun, K. Wang, L. Yi, K. Yang, M. Chen, and J. Wang, “Simulation of transmitted X-rays in a polycapillary X-ray lens,” Nucl. Instrum. Methods Phys. Res., Sect. A 795, 186–191 (2015). [CrossRef]
23. D. Mazon, C. Liegeard, A. Jardin, R. Barnsley, M. Walsh, M. O’Mullane, A. Sirinelli, and F. Dorchies, “Polycapillary lenses for soft x-ray transmission in ITER: Model, comparison with experiments, and potential application,” Rev. Sci. Instrum. 87(11), 11E302 (2016). [CrossRef]
24. A. Zymaková, K. K. Prasad, A. Picchiotti, W. Blachucki, J. Szlachetko, M. Rebarz, J. Uhlig, and J. Andreasson, “Implementation of a crossed-slit system for fast alignment of sealed polycapillary X-ray optics,” J. Synchrotron Radiat. 27(6), 1730–1733 (2020). [CrossRef]
25. B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-ray interactions: photoabsorption, scattering, transmission, and reflection at E = 50–30000 eV, Z = 1–92,” At. Data Nucl. Data Tables 54(2), 181–342 (1993). [CrossRef]
26. S. V. Kukhlevsky, F. Flora, A. Marinai, K. Negrea, L. Palladino, A. Reale, G. Tomassetti, A. Ritucci, G. Nyitray, and L. Kozma, “Diffraction of x rays in capillary optics,” Appl. Opt. 39(6), 1059–1063 (2000). [CrossRef]
27. A. Bjeoumikhov, S. Bjeoumikhova, H. Riesemeier, M. Radtke, and R. Wedell, “Propagation of synchrotron radiation through nanocapillary structures,” Phys. Lett. A 366(4-5), 283–288 (2007). [CrossRef]
