Phase-contrast microscopy at high x-ray energy with a laboratory setup

Marco Endrizzi; Fabio A. Vittoria; Paul C. Diemoz; Rodolfo Lorenzo; Robert D. Speller; Ulrich H. Wagner; Christoph Rau; Ian K. Robinson; Alessandro Olivo

doi:10.1364/OL.39.003332

Hard x-ray radiation has a high penetration power in matter, thus enabling nondestructive inspection of the inner structure of samples. At the same time, this power is also a weakness because the contrast arising from differences in the absorption of radiation tends to vanish when partially transparent samples are illuminated. X-ray phase-contrast imaging (XPCi) [1] overcomes this problem because it is sensitive to the phase shifts imparted to the x-ray wave when traversing the sample. Several methods [2–15] have been developed for performing XPCi. Approaches have also been proposed for high-resolution XPCi, including free-space propagation, Zernike phase contrast, and grating-based methods [16–22]. Here we show the design, modelling, and realization of a laboratory system based on the edge-illumination [10,15] principle and implemented through its area-imaging counterpart, sometimes referred to as the coded-aperture [23] method. These are noninterferometric methods that do not use the Talbot self-imaging effect or Moire patterns [24]; contrast is generated by fine angular selection, in analogy with analyzer-based imaging which uses the rocking curve of a crystal [25]. We aim to push the current resolution limits of a few tens of micrometers [26] toward microscopic resolution while still performing quantitative phase-contrast imaging at high x-ray energies. A magnified projection geometry was used in order to achieve high spatial resolution while being able to efficiently detect the radiation. The ability of the method to be quantitative and its spatial resolution are experimentally demonstrated and numerically simulated while the potential of the technique in terms of image quality is illustrated through images of a complex wood sample.

The experimental setup consists of a microfocus transmission target x-ray tube, two apertured masks, and a detector (Fig. 1). The tungsten target x-ray tube is operated at 80 kVp and has a focus of 3.5 μm. The first mask $M_{1}$ is placed at 13 cm from the focus and the source-to-detector distance is $z_{sd} = 130 cm$ . The sample is positioned at about 14 cm from the focus, with a geometrical magnification factor $G = z_{sd} / (z_{sd} - z_{od})$ of about 9. The first mask $M_{1}$ has a pitch $p_{1} = 20 μm$ and apertures $a_{1} = 3 μm$ , while $p_{2} = 98 μm$ and $a_{2} = 29 μm$ are used for the second mask $M_{2}$ . They are made of gold on a graphite ( $M_{2}$ ) and silicon ( $M_{1}$ ) substrate and were manufactured by Creatv Microtech (Potomac, Maryland) and Microworks GmbH (Karlsruhe, Germany), respectively. The detector is a passive pixel complementary metal-oxide semiconductor flat panel sensor (Hamamatsu Photonics C9732DK), with pixel pitch $p_{3} = 50 μm$ . The signal degradation due to cross talk between neighboring pixels is limited by the use of a line-skipped mask design [27]. The main limiting factor for the field of view is $a_{1}$ , which defines the angular acceptance of the transmitted radiation $θ_{m} \approx a_{1} / t_{M_{1}}$ , where $t_{M_{1}}$ is the thickness of the mask, about 200 μm in this case. For our setup the field of view was $2 mm \times 5 mm$ in the $x$ and $y$ directions, respectively. Bent masks could be used for obtaining a larger field of view with an even more compact setup [28]. Provided that the angular spread of the beam is limited to $θ_{l} < 8 p_{3} / z_{od} \sim 340 μrad$ , the pixels can be considered independent from one another, which allows us to model the image formation for a single aperture only.

Fig. 1. Experimental setup: the x-ray beam, generated by a microfocus x-ray source $S$ , is shaped by the first mask $M_{1}$ , traverses the sample, and is analyzed by the second mask $M_{2}$ before being recorded by the pixels $P$ of a digital detector.

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Let us consider the Fourier transform of the intensity $I_{D} (x) = \int d f \exp (2 π i x f) {\hat{I}}_{D} (f)$ on the detector [29]:

{\hat{I}}_{D} (f) = \int d x^{'} e^{- 2 π i x^{'} f} T (x^{'} - w) T^{*} (x^{'} + w),

where

w = z_{od} λ f / (2 G)

and

T (x) = A (x) \exp [- M (x)] \exp [- i ϕ (x)]

combines the aperture transmission function

A (x)

and the sample absorption and phase shift. If not differently specified, the integration is always carried out between

- \infty

and

\infty

. The signal detected is given by

S = \int_{0}^{a_{2}} d x I_{D} (x) \approx \int_{0}^{\infty} d x I_{D} (x)

assuming

a_{2}

is wider than the beam extension. Using

\int_{0}^{\infty} d x \exp (2 π i x f) = (1 / 2) δ (f) - [1 / (2 π i f)]

, where

δ (f)

represents the Dirac delta function,

S

can be expressed as a sum of two terms,

S = S_{1} + S_{2}

.

S_{1} = \int d x^{'} {| T (x^{'}) |}^{2}

and

S_{2} = - \int d x^{'} \int d f \frac{e^{- 2 π i x^{'} f}}{2 π i f} T (x^{'} - w) T^{*} (x^{'} + w) .

The integration of

S_{2}

can be split into two parts,

\int_{- \infty}^{\infty} d f = \int_{- \infty}^{0} d f + \int_{0}^{\infty} d f

, and rearranged into a single integral according to the properties of symmetry possessed by the integrand. The first term

S_{1}

depends only on absorption while the second one

S_{2}

is also related to phase. This can be shown explicitly by assuming constant absorption within the limits defined by the sample aperture and considering

| ϕ (x^{'} + w) - ϕ (x^{'} - w) | ≪ 1

[30]. The following form for

S_{2}

is obtained:

S_{2} = \frac{e^{- 2 M}}{π} \int_{0}^{\frac{a_{1}}{2}} d w \int_{- b_{1}}^{b_{1}} d x^{'} \frac{Δ ϕ_{x^{'} w}}{w} \cos [4 π \frac{x^{'} w G}{λ z_{od}}],

where

Δ ϕ_{x^{'} w} = ϕ (x^{'} - w) - ϕ (x^{'} + w)

,

b_{1} = (a_{1} - 2 w) / 2

, and the limits of integrations have been redefined according to the transmission function of the sample aperture. This expression provides a means to study how the phase signal is modulated by the imaging system. The component of the signal related to the phase effects is given by the weighted sum of all the possible difference quotients within the limits defined by the aperture. The weak phase condition used to derive Eq. (3) is required only to obtain a formulation easier to interpret, without the need to carry out the full integration numerically. In order to visualize this, the

S_{2}

signal was calculated for a sinusoidal phase object, and its maximum value was plotted against the period in Fig. 2. It is interesting to note that the modulation is peaked at 3 μm, which corresponds to

a_{1}

, and also that signals with period smaller than the aperture are detected; for example,

S_{2}

is reduced by about 50% at 1.5 μm.

Fig. 2. Evaluation of Eq. (3) as a function of the period of a sinusoidal phase object.

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The direct comparison between the measured and the theoretical signal from a star pattern test object is carried out by means of a numerical simulation of the whole imaging system [31]. Each monochromatic component was weighted according to the x-ray source spectrum and the detector response as function of energy [32]. In order to perform quantitative retrieval of the absorption and refraction of the sample, we followed the approach based on the knowledge of the translation curve (TC) of the system [25]. The TC describes how the detected intensity changes as a function of the displacement $Δ ξ$ between the two masks. Images were acquired by setting the displacement between $M_{1}$ and $M_{2}$ as equal to $Δ ξ_{1} = - Δ ξ_{2} = 2 μm$ . The resulting images can be expressed as [25]

I_{L, R} = \exp [- \int_{O} \bar{μ} d z] I_{TC} (Δ ξ_{1, 2} - z_{od} {\bar{ϕ}}_{x} / G),

where

\bar{μ}

denotes the linear attenuation coefficient of the sample evaluated at the effective energy of the system [33],

z_{od}

is the sample-to-detector distance,

{\bar{ϕ}}_{x} = \partial \bar{ϕ} / \partial x

is the partial derivative of

ϕ

at the effective energy of the system, and

I_{TC}

is the value of the translation curve. We also introduce the following images,

I_{Σ} = I_{L} + I_{R}

and

I_{Δ} = I_{L} - I_{R}

[25]. It is worth noting that the quantity

I_{Δ} / I_{Σ}

depends only upon

{\bar{ϕ}}_{x}

and the operational parameters of the setup [25]; we will refer to this quantity as the differential phase image. Moreover,

I_{Σ}

depends only on the absorption properties of the sample. The following samples were imaged experimentally: a star pattern, a cylindrical two-material sample, and a bamboo wood sample. The star pattern was etched into a few hundred micrometers of crystalline silicon and provides a good approximation of a pure phase object at the x-ray energies we used. The

I_{L}

image was acquired by using 100 sample scan steps of 0.2 μm, with an exposure time of 70 s per step. The cylindrical sample was composed of two materials, 220 μm diameter of boron with a 14 μm diameter tungsten core, and it was scanned with 16 steps of 1.2 μm, 100 s exposure time each. The bamboo wood was sliced to a thickness of about 500 μm;

I_{L}

and

I_{R}

were acquired by scanning the sample with 44 steps of 0.5 μm, with an exposure time of 100 s per step. For comparison, an image of the bamboo sample was also acquired in free-space propagation XPCi, with monochromatic synchrotron radiation and a high-resolution detector. The image was acquired at the I13 beamline of the diamond light source by using 9.7 keV x-rays and a detector featuring 0.8 μm size pixels. The sample was placed at about 200 m from the undulator source and the detector 30 cm downstream of the sample.

The spatial resolution of the laboratory-based system is evaluated by means of the star pattern. The image of the star is shown in Fig. 3(a) from which the intensity profile shown in Fig. 3(b) was extracted (dashed line). The experimental data are in good agreement with the numerical simulation, shown as a solid line in the plot in Fig. 3(b). As the absorption is negligible in this case, we note from Eq. (4) that $I_{L}$ depends only on ${\bar{ϕ}}_{x}$ and on the operational parameters of the setup. The changing visibility of the phase-contrast features around the star pattern are a property of the geometry of the setup. Let us consider the phase shift of the star pattern as a replica at various angles $φ$ of a step-like feature; the detected signal is modulated by a cosine function, $S_{2} \propto \cos (φ) \partial_{u} ϕ (u, v)$ , where $(u, v)$ are the $(x, y)$ coordinates rotated by the angle $φ$ . The smallest resolved separation between a dark and a bright fringe is 1.5 μm; the sample used hardly allowed testing finer resolutions. The quantitativeness of the method is demonstrated by retrieving differential phase and amplitude of the boron fiber with a tungsten core and comparing the extracted (dashed line) values to the expected ones (solid line) in Fig. 4(a). The theoretical ${\bar{ϕ}}_{x} (x)$ was calculated assuming a cylindrical shape of the boron and tungsten fiber. The analytical profile was convolved with a Gaussian function, representing the spatial resolution of the system, of width 1.5 μm. The refractive indices of the materials were calculated by using the Xraylib library [34]. The profile was then obtained according to $I_{Δ} / I_{Σ} = (z_{od} {\bar{ϕ}}_{x} / G) / (I_{TC}^{'} (Δ ξ) / I_{TC} (Δ ξ))$ [25] by numerically differentiating $\bar{ϕ}$ . The amplitude and differential phase images are also shown in Figs. 4(c) and 4(b), respectively. The horizontal line noise that can be noticed in Figs. 3(a), 4(b), and 4(c) is a result of the brick-like structure used to manufacture $M_{1}$ . The laboratory images of the bamboo sample are shown in Fig. 5, where the contributions from phase [Fig. 5(a)] and absorption [Fig. 5(b)] are separated. For a better appreciation of the image quality, the image acquired under nearly ideal conditions (with monochromatic synchrotron radiation and high-resolution detector) is also shown for comparison in Fig. 5(d). We note that the higher contrast of this image is expected due to the much lower x-ray energy (9.7 keV). The detail shown in the picture is similar to but not exactly the same as in the laboratory image, due to difficulty in aligning the sample to the same region of interest.

Fig. 3. Star pattern phase object. (a) Differential phase-contrast experimental image (scale bar is 50 μm). (b) Simulation (solid line) compared against the experimental profile (dashed line) extracted from the highlighted region in the center of the star.

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Fig. 4. Boron fiber with a tungsten core sample. (a) Comparison of the experimental intensity profile [highlighted region in (b) sector] against the theoretically expected one for a single fiber. (b) Differential phase and (c) amplitude images of two fibers at an angle.

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Fig. 5. Image of the bamboo sample. (a) Differential phase and (b) amplitude images, quantitatively separated by using the laboratory setup (scale bar 100 μm). (c) Zoom of the highlighted $120 μm \times 120 μm$ regions of (a) and (b), top and bottom, respectively. (d) Free-space propagation image acquired with a high-resolution detector and monochromatic synchrotron radiation.

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In summary, we described a hard x-ray, laboratory-based, phase-contrast microscope, obtained through the appropriate design of a high-magnification edge-illumination XPCi system. The experimental setup was built using commercially available, off-the-shelf instrumentation, and it is currently producing high-quality amplitude and differential phase-contrast images. Micrometric spatial resolution was experimentally measured, theoretically described, and numerically simulated by using a star pattern test object, with good agreement between simulation and experiment. By acquiring two separate images at different configurations, amplitude and differential phase can be quantitatively retrieved. The quantitativeness of the method was tested against theory on a two-material sample of known shape and composition. The instrument uses broadband and hard radiation, merging high resolution with the high penetration power of x-rays in a laboratory setup.

The authors are grateful to Nikon Corporation and X-Tek, part of Nikon Metrology, for their support with the experimental setup. This work was supported by the UK Engineering and Physical Sciences Research Council (Grants EP/I022562/1 and EP/I021884/1). M. Endrizzi and P. C. Diemoz are supported by Marie Curie Career Integration Grants within the Seventh Framework Programme of the European Union, PCIG12-GA-2012-334056 and PCIG12-GA-2012-333990.

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Phase-contrast microscopy at high x-ray energy with a laboratory setup

Abstract

References

Cited By

Figures (5)

Equations (4)

Optics Letters