Large phase-stepping approach for high-resolution hard X-ray grating-based multiple-information imaging

Zhifeng Huang; Zhiqiang Chen; Li Zhang; Kejun Kang; Fei Ding; Zhentian Wang; Haozhi Ma

doi:10.1364/OE.18.010222

1. Introduction

High-resolution hard X-ray grating-based multiple-information imaging technology is capable of providing attenuation, refraction (i.e., phase-contrast) and scattering (i.e., dark-field) information synchronously by a phase-stepping approach, and reconstructing the distributions of the linear attenuation coefficient, refractive index gradient and generalized scattering parameter of a tested sample in one computed tomographic scanning process, so it has been the most promising tool for clinical diagnosis, material examination, etc..

X-ray grating-based phase-contrast imaging method firstly realized with highly coherent synchrotron radiation sources in the early 21st century, though similar or related technologies, such as Talbot interferometry [1], shearing interferometry [2], moire deflectometry [3] and Talbot-Lau interferometry [4], have been developed maturely in the visible-light and atom fields since 1970s. David et al applied two phase gratings and a Bragg crystal to generate moiré patterns and retrieved X-ray differential phase-contrast information at the ESRF in 2002 [5]. Momose et al demonstrated Talbot interferometry with a pair of transmission gratings to generate moiré fringes and then adopted a phase-shifting technique to retrieve phase-contrast images at the Spring-8 [6,7]. Subsequently, Weitkamp et al also implemented Talbot interferometry with a phase grating and an absorption grating to obtain phase-contrast information by use of a phase-stepping approach at the ESRF [8,9]. In 2006, Pfeiffer et al presented Talbot-Lau interferometry with 3 gratings to implement differential phase-contrast imaging with low-brilliance X-ray sources [10]. Two years later, they retrieved (ultra) small angle scattering information additionally under the similar systemic conFig. [11]. In their method, a source grating split a large-sized source into an array of partially coherent line X-rays and a Talbot interferometer consisting of a phase grating and an absorption grating measured multiple information. On the other hand, our group proposed an alternative method based on classic optics theory under the incoherent condition to achieve the retrieval of multiple information by use of the phase-stepping approach [12,13]. Furthermore, Donath et al presented an inverse geometry to relax the requirements on the large-scale high resolution grating fabrication [14]. It is an attempt to simplify part of difficulties in actual applications of the X-ray grating-based imaging technology with conventional X-ray sources.

Above methods mostly adopted the phase-stepping (or phase-shifting) approach by moving either of the last two gratings relatively step by step over one grating period to retrieve multiple information by measuring intensity oscillation curves of each pixel on the detector. Nesterets and Wilkins presented another type of the phase-stepping approach by moving both gratings together [15]. Both gratings have very small periods, for example, 2 or 4 $u m$ in Refs [10,11], so the step length in the phase-stepping approach is usually less than 1 $u m$ . That is to say, the relative movement of two gratings requires very high positioning resolution on the order of sub-micrometers. So in the current phase-stepping approach, the positioning resolution and correlative circumstances (i.e., temperature and pressure, etc.) for the grating-based imaging system should have a very high level of stability. The requirements maybe mostly limit actual applications of the X-ray grating-based imaging technology. In this paper, we present an equivalent phase-stepping approach by moving the source grating with lower positioning resolution to solve this problem.

2. Theory

The X-ray grating-based multiple-information imaging method based on either Talbot-Lau interferometry or classic optics theory can be uniformly illustrated by Fig. 1 . It consists of a source grating (G0) and two gratings (G1, G2). In general, the source grating has a large period on the order of tens of micrometers, even over one hundred micrometers, while the last two gratings have very small periods on the order of several micrometers. In the large phase-stepping approach, we fix the last two gratings and only relatively move the source grating over one grating period along the x-axis to achieve the measurement of the intensity oscillation curve of each pixel on the detector.

Fig. 1 The schematic diagram of the X-ray grating-based multiple-information imaging method.

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Suppose $p_{0}$ , $p_{1}$ and $p_{2}$ are the periods of G0, G1 and G2, respectively. L is the distance between G0 and G1. D is the distance between G1 and G2. Their relationship can be expressed by

\frac{p_{0}}{p_{2}} = \frac{L}{D},

\frac{p_{1}}{p_{2}} = \frac{L}{L + D},

in classic optics theory [12],

\frac{p_{2}}{p_{1} / 2} = \frac{L}{L - D_{T}},

in Talbot–Lau interferometry [16], where

D_{T}

is the Talbot distance.

Without loss of generality, the imaging system is supposed to be under the following ideal conditions in the x-z plane in order to simplify the computational complexity:

1) Suppose that the distance between G0 and the X-ray source is zero. The distance between G2 and the detector is zero. The detective efficiency of the detector is 100%. The influence of the noises in the whole imaging system doesn’t be considered.

2) Suppose that the sizes of three gratings along the x-axis are infinite. The transmission functions of G0, G1, and G2 can be expressed by $T_{0} (x)$ , $T_{1} (x)$ and $T_{2} (x)$ , respectively, that is,

T_{0} (x) = \sum_{n} f_{n} e^{2 π i \frac{n}{p_{0}} x},

T_{1} (x) = \sum_{n} a_{n} e^{2 π i \frac{n}{p_{1}} x},

T_{2} (x) = \sum_{n} t_{n} e^{2 π i \frac{n}{p_{2}} x},

where

a_{n}

,

t_{n}

and

f_{n}

denote the nth order Fourier coefficients.

Let the intensity distribution function of the X-ray source be $S_{0} (x)$ , then the intensity distribution function of the X-ray downstream after G0, $S_{T} (x)$ , is expressed by

S_{T} (x) = S_{0} (x) T_{0} (x) .

It can be regarded as an array of sub-sources from the source grating.

Consider the intensity distribution functions of the X-ray upstream before G2 in two cases: $I_{s} (x)$ without the sample and $J_{s} (x)$ with the sample present in the beam path, respectively,:

I_{s} (x) = I_{1} (x) * S (x),

J_{s} (x) = J_{1} (x) * S (x),

where

I_{1} (x)

denotes the projection image or self-image of G1 illuminated by a point source from the position

z_{0}

to the position

z_{2}

,

J_{1} (x)

denotes the projection image or self-image of both the sample and G1 illuminated by a point source from the position

z_{0}

to the position

z_{2}

.

S (x)

denotes the geometrical projection of

S_{T} (x)

in the position

z_{2}

, that is,

I_{1} (x) = T_{1} (\frac{L}{L + D} x),

J_{1} (x) = T_{1}^{'} (\frac{L}{L + D} x),

S (x) = S_{T} (- \frac{L}{D} x),

where

T_{1}^{'} (x)

denotes the distribution function of the X-rays pass through both the sample and G1 in the position

z_{1}

. Note that

I_{1} (x)

,

J_{1} (x)

and

S (x)

have the same period as

T_{2} (x)

in the position

z_{2}

, so p is utilized to denote their periods for convenience, then Eqs. (10)-(12) can be rewritten to

I_{1} (x) = T_{1} (\frac{L}{L + D} x) = \sum_{n} a_{n} e^{2 π i \frac{n}{p} x},

J_{1} (x) = T_{1}^{'} (\frac{L}{L + D} x) = \sum_{n} a_{n}^{'} e^{2 π i \frac{n}{p} [x + D ϕ (x)]},

S (x) = S_{T} (- \frac{L}{D} x) = S_{0}^{'} (x) T_{0}^{'} (x) = S_{0} (- \frac{L}{D} x) T_{0} (- \frac{L}{D} x),

where

S_{0}^{'} (x) = S_{0} (- \frac{L}{D} x)

,

T_{0}^{'} (x) = T_{0} (- \frac{L}{D} x) = \sum_{n} f_{n} e^{2 π i \frac{n}{p} x}

,

ϕ (x)

denotes the beam deflection angle at the sample caused by refraction and

a_{n}^{'}

contains the attenuation effect of the sample. To simplify the computation, Eq. (15) can also be approximated to

S (x) = S_{0}^{'} (x) T_{0}^{'} (x) = S_{0}^{'} (x) [\sum_{n} f_{n} e^{2 π i \frac{n}{p} x}] \approx \sum_{n} f_{n}^{'} e^{2 π i \frac{n}{p} x},

where

f_{n}^{'}

contains the influence of the distribution of the X-ray source.

Finally, the X-ray intensities are captured by the detector in two cases: $I_{D} (x)$ without the sample and $J_{D} (x)$ with the sample present, respectively, that is,

I_{D} (x) = I_{s} (x) T_{2} (x),

J_{D} (x) = J_{s} (x) T_{2} (x) .

In the large phase-stepping approach, G0 is relatively moved along the x-axis and the last two gratings G1 and G2 are fixed, then the intensity oscillation curve functions can be measured in two cases: $I_{D} (x, χ)$ without the sample and $J_{D} (x, χ)$ with the sample present, respectively, that is,

\begin{array}{l} I_{D} (x, χ) = I_{s} (x, χ) T_{2} (x) = [I_{1} (x) * S (x + χ)] T_{2} (x) \\ = [I_{1} (x) * (S_{0}^{'} (x) T_{0}^{'} (x + χ))] T_{2} (x) \\ = [(\sum_{n} a_{n} e^{2 π i \frac{n}{p} x}) * (S_{0}^{'} (x) \sum_{m} f_{m} e^{2 π i \frac{m}{p} (x + χ)})] (\sum_{k} t_{k} e^{2 π i \frac{k}{p} x}) \\ \approx \sum_{n} a_{n} f_{n}^{'} t_{n} e^{2 π i \frac{n}{p} χ}, \end{array}

\begin{array}{l} J_{D} (x, χ) = J_{s} (x, χ) T_{2} (x) = [J_{1} (x) * S (x + χ)] T_{2} (x) \\ = [J_{1} (x) * (S_{0}^{'} (x) T_{0}^{'} (x + χ))] T_{2} (x) \\ = [(\sum_{n} a_{n}^{'} e^{2 π i \frac{n}{p} [x + D ϕ (x)]}) * (S_{0}^{'} (x) \sum_{m} f_{m} e^{2 π i \frac{m}{p} (x + χ)})] (\sum_{k} t_{k} e^{2 π i \frac{k}{p} x}) \\ \approx \sum_{n} a_{n}^{'} f_{n}^{'} t_{n} e^{2 π i \frac{n}{p} [χ + D ϕ (x)]}, \end{array}

where χ is the displacement of G0.

In the current high-positioning-resolution phase-stepping approach, G0 is fixed and either of the last two gratings G1 or G2 is relatively moved along the x-axis, then the intensity oscillation curve functions can be measured in two cases: $I_{D}^{'} (x, χ)$ without the sample and $J_{D}^{'} (x, χ)$ with the sample [17–20], respectively, that is,

\begin{array}{l} I_{D}^{'} (x, χ) = I_{s} (x) T_{2} (x, χ) = [I_{1} (x) * S (x)] T_{2} (x + χ) \\ \approx [(\sum_{n} a_{n} e^{2 π i \frac{n}{p} x}) * (\sum_{m} f_{m}^{'} e^{2 π i \frac{m}{p} x})] (\sum_{k} t_{k} e^{2 π i \frac{k}{p} (x + χ)}) \\ \approx \sum_{n} a_{n} f_{n}^{'} t_{n} e^{2 π i \frac{n}{p} χ}, \end{array}

\begin{array}{l} J_{D}^{'} (x, χ) = J_{s} (x) T_{2} (x, χ) = [J_{1} (x) * S (x)] T_{2} (x + χ) \\ \approx [(\sum_{n} a_{n}^{'} e^{2 π i \frac{n}{p} [x + D ϕ (x)]}) * (\sum_{m} f_{m}^{'} e^{2 π i \frac{m}{p} x})] (\sum_{k} t_{k} e^{2 π i \frac{k}{p} (x + χ)}) \\ \approx \sum_{n} a_{n}^{'} f_{n}^{'} t_{n} e^{2 π i \frac{n}{p} [χ + D ϕ (x)]}, \end{array}

Note that Eq. (19) is equivalent to Eq. (21) and Eq. (20) is equivalent to Eq. (22) under the ideal conditions. It means that the large phase-stepping approach by moving the source grating has almost the same intensity oscillation curve functions as the current high-positioning-resolution phase-stepping approach by moving either of the last two gratings. The similar conclusion can also be drawn under actual conditions (for example, the finite sizes of the gratings, the systemic noises, the actual detective efficiency of the detector, unideal grating fabrication, etc), where actual intensity oscillation curve functions are approximately sinusoidal functions [7,8,12].

3. Experiment

An actual experiment on a phantom was carried out at the grating-based multiple-information imaging experimental setup at Tsinghua University in order to validate above deduction. The detailed description of the system can be found in Ref. 8 and 9. In the experiment, the spot size of the X-ray tube was measured about 600 $u m$ when the tube was set around 30kV. The periods of G0, G1 and G2 were 110, 10 and 11 $u m$ , respectively, The distance between G0 and G1 was 1500 $m m$ and the distance between G1 and G2 was 150 $m m$ . The phantom consisted of two cylinders: the outer cylinder made of polymethylmethacrylate (PMMA) and the inner cylinder made of plastic. Both new and current phase-stepping approaches adopted 11 steps, while their step lengths were 10 $u m$ and 1 $u m$ , respectively. Their corresponding multiple images (that is, attenuation, refraction and scattering images) are shown in Fig. 2 . The profile values of each image are plotted to compare the differences between the new and current phase-stepping approaches. It is found that their results are almost the same on the whole, though they aren’t identical completely because of the influences of systemic noises.

Fig. 2 The comparisons of experimental results by use of two phase-stepping approaches. (a), (d) and (g) are the attenuation, refraction and scattering images, respectively, retrieved by the current high-positioning-resolution phase-stepping approach (Approach I). (b), (e) and (h) are the attenuation, refraction and scattering images, respectively, retrieved by the large phase-stepping approach (Approach II). (c) is the comparison of profile values in two attenuation images. (f) is the comparison of profile values in two refraction images. (i) is the comparison of profile values in two scattering images.

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4. Conclusion

In conclusion, both the theoretical deduction and actual experiments prove that the large phase-stepping approach is equivalent to the current high-positioning-resolution phase-stepping approach. The period of the source grating is always much larger than the periods of last two gratings, so the large phase-stepping approach requires at least one order of magnitude lower positioning resolution than the current one. The large phase-stepping approach is applicable to both Talbot-Lau interferometric-based and classic-optics-based grating-based multiple-information imaging methods. It would relax the requirements of high positioning resolution and strict circumstances for future commercial applications of the X-ray grating-based multiple-information imaging technology in practice.

Acknowledgments

The work was supported by a grant from the National Natural Science Foundation of China (NNSFC) grants 10905031 and 10875066 and the Specialized Research Fund for the Doctoral Program of Higher Education grant 20090002120016.

References and links

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Large phase-stepping approach for high-resolution hard X-ray grating-based multiple-information imaging

Abstract

1. Introduction

2. Theory

3. Experiment

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (2)

Equations (22)

Optics Express