- Email: [email protected]
A generalized reverse projection method for fan beam geometry under partially coherent illumination
Radiation Physics and Chemistry 95 (2014) 280–283
Contents lists available at ScienceDirect
Radiation Physics and Chemistry journal homepage: www.elsevier.com/locate/radphyschem
A generalized reverse projection method for fan beam geometry under partially coherent illumination Z. Wu a, Z.L. Wang a,n, K. Gao a, K. Zhang b, X. Ge a, D.J. Wang a, S.H. Wang a, J. Chen a, Z.Y. Pan a, P.P. Zhu a,b, Z.Y. Wu a,b,nn a b
National Synchrotron Radiation Laboratory, University of Science and Technology of China, Hefei 230029, China Institute of High-Energy Physics, Chinese Academy of Sciences, Beijing 100049, China
H I G H L I G H T S c c c c
Fan beam geometry with geometrical magnification is used with the modified RP method. In the calculation of conjugate projection images, unitary ray is taken into account for fan beam geometry. We have considered the influence of the spatial coherence on the extracted information. The measurement range of the generalized RP method is enlarged by moving the sample close to the light source.
a r t i c l e i n f o
a b s t r a c t
Article history: Received 19 October 2012 Accepted 6 January 2013 Available online 16 January 2013
In this paper, a generalized reverse projection (RP) method for grating-based fan beam phase contrast imaging is presented. Compared to the original RP method, rays rather than projection images are taken into account during the information extraction process. We also discuss the influence of partial coherence on the extracted information. Theoretical derivations and numerical simulations are performed to confirm the validity of the method. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Spatial coherence Phase contrast imaging RP method Fan beam geometry
1. Introduction Conventional X-ray absorption imaging is a power tool in medical diagnosis, industrial non-destructive testing and security inspection. However, its poor contrast to low-Z elements limits its applications in soft tissue imaging. If we write the complex refractive index of an object as n¼ 1 d ib, its real part determines the phase shift of the wave front while its imaginary part is associated to the linear attenuation coefficient. In the hard X-ray regime and for low-Z elements, d is about 1000 times greater than b. Therefore, the phase shift is about three orders of magnitude of the absorption. As a consequence different phase contrast imaging methods have been proposed (Bravin et al., 2013; Chen et al., 2010; Davis et al., 1995a; Ge et al., 2011; Momose, 2005; Wang et al., 2010a, 2010b; Zhou and Brahme, 2008), such as, interferometric methods (Momose et al.,
n
Corresponding author. Tel.: þ86 551 6360 2017. Corresponding author at: National Synchrotron Radiation Laboratory, University of Science and Technology of China, Hefei 230029, China. Tel.: þ86 551 6360 2077; fax: þ 86 551 6514 1078. E-mail addresses: [email protected] (Z.L. Wang), [email protected] (Z.Y. Wu). nn
0969-806X/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.radphyschem.2013.01.006
1995), crystal analyzer-based methods (Davis et al., 1995b; Yuan et al., 2006), propagation-based methods (Gureyev et al., 2009; Wilkins et al., 1996; Snigirev et al., 1995) and grating-based methods (David et al., 2002; Momose et al., 2003; Pfeiffer et al., 2006, 2008; Weitkamp et al., 2005). Grating-based methods are considered the most promising phase contrast imaging methods, thanks to their less stringent requirements regarding the light source. In the phase contrast imaging, phase and absorption information are collected together thus information separation is inevitable. So far, three primary phase extraction methods have been introduced: the Fourier harmonic method (Wen et al., 2008), the phase stepping (PS) method (Weitkamp et al., 2005) and the reverse projection (RP) method (Zhu et al., 2010). The first one is the earliest method and quite fast due to only one required projection image at each projection position. Compared with the Fourier harmonic method, the PS method collects several images at each projection angle but provides higher quality results. It is widely adopted in the laboratory. Nevertheless, because of the long exposure time, the large radiation dose is unacceptable for practical applications, especially for medical imaging. On the contrary, the RP method requires lower dose and is a simpler and faster phase extraction method. It takes advantage of the anti-symmetry (i.e., with a rotation of 1801, you can collect an image with the same absorption
Z. Wu et al. / Radiation Physics and Chemistry 95 (2014) 280–283
but opposite differential phase information), thus reduces the number of the required images by extending the scanning angle range from 1801 to 3601. However, the RP method has been mainly introduced to run experiments with parallel beam geometrical configurations. In this contribution, we present a generalized RP method for fan beam geometry. We also discuss the effects of partially coherent illumination on the extraction information.
281
slope by its first order Taylor term. Consequently formulae for information retrieval can be given as follows: Z
1
-1
yr ðxr , f,zÞ ¼
2S xg =R2 I0 Iðxr , f,zÞ þ Iðxr , f þ p þ 2a,zÞ
mðx,y,zÞdr ¼ ln
1 Iðxr , f,zÞIðxr , f þ p þ2a,zÞ kC Iðxr , f,zÞ þ Iðxr , f þ p þ 2a,zÞ
ð3Þ
2. The RP method in fan beam geometry Based on the setup shown in Fig. 1 and the scale factor of fan beam geometry (Donath et al., 2009; Engelhardt et al., 2007), the intensity recorded by a virtual detector at the rotation axis in the paraxial approximation can be written as: Z 1 x Iðxr , f,zÞ ¼ I0 exp mðx,y,zÞdr S g þ kyr , R2 1 ( R0 =R1 0 oR0 oR1 k¼ ð1Þ ðR1 þR2 -R0 Þ=R2 R1 o R0 o R1 þ R2 where f is the projection angle and I0 is the intensity at the position of phase grating G1 without the sample. m denotes the linear attenuation coefficient of the sample and the function S is the shifting curve. R0, R1 and R2 are the distances respectively from the source to the rotation axis of the sample, between the source and the phase grating G1 and from G1 to the analyzer grating G2. xg refers to the displacement of G2 along the direction perpendicular to the groove of grating. yr is the refraction angle. (xr,yr,z) is a stationary coordinate system while (x,y,z) is a rotational one fixed on the sample shown in Fig. 1. Their origins of coordinates are set at the intersection of the rotation axis and the optical axis. We know that the RP method for parallel beam geometry is based on the anti-symmetric characteristic. However, under fan beam illumination, we can see from Fig. 2(a) that the anti-symmetry between the mutually reverse projection images is no longer fulfilled. In the following, a modified approach considering unitary ray (see Fig. 2(b)) is introduced to find the conjugate projection images. It can be seen from Fig. 2 that the anti-symmetry between mutually reverse rays still exists in fan beam geometry. If we consider a single ray I(xr,f1,z) and its reverse ray I( xr,f2,z), from the geometrical relationship exhibited in Fig. 2(b), we can obtain: x f2 ¼ f1 þ p þ2arctan r ð2Þ R0 When the refraction angle yr od2/4kR2, with d2 the period of G2, the shifting curve can be linearly approximated at its half
Fig. 1. Schematic diagram of the grating-based interferometer in fan beam geometry: the sample is located (a) before the phase grating (0 o R0 o R1) and (b) after the phase grating (R1 o R0 o R1 þR2).
Fig. 2. The realization of the RP method in fan beam geometry: (a) mutually reverse projections and (b) single ray is considered as unit.
282
Z. Wu et al. / Radiation Physics and Chemistry 95 (2014) 280–283
where C ¼
dSðxg =R2 Þ 1 dy Sðxg =R2 Þ
is a constant and a ¼ arctan Rxr0 . Substi-
tuting k ¼1 and R0 ¼N into Eq. (3), we will obtain the formulae for parallel beam geometry. In the Talbot interferometer, the phase grating G1 acts as the beam splitter. When it is illuminated by a fully coherent source in fan beam geometry, the shifting curve at the plane of G2 can be expressed as ! 2 X xg R1 1 4 þ cos 2 p nx =d ¼ ð4Þ Sfully g 2 2 n ¼ 1,3,5, p2 n2 R2 R1 þ R2 In the case of a partially coherent illumination, if the intensity distribution can be expressed as a Gaussian function with a standard deviation ss, the shifting curve Spartially is given by the convolution of the projection source profile f xg ¼ x2g 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp with Sfully: 2 pR2 ss =R1 2ðR2 ss =R1 Þ Spartially
xg R2
¼ f xg Sf ully ¼
R1 R1 þ R2
2
X 1 nR2 ss 2 exp 2p2 þ d2 R1 2 n ¼ 1,3,5
!
! 4 cos 2 p nx =d g 2 p2 n2
ð5Þ where is the convolution operator. The Fourier transform and its inverse transform have been utilized here for this derivation. After a simple analysis, we can recognize that the spatial coherence has no effect on the extracted absorption information. Hereafter, we only investigate the influence of the spatial coherence on the extracted phase information. The shifting curves for different source sizes are compared in Fig. 3. (All system parameters are given in pffiffiffiffiffiffiffiffi ffi the next section). The full width at half maximum given by 2 ln 4ss is usually defined as the size of Gaussian light source. We can see in Fig. 3, that the gradient and linear range of the shifting curve near the half maximum become smaller, when a larger size light source is used. In other words, increasing the source size, both the sensitivity and the measurement range of the RP method decrease. Furthermore, from Eq. (1), we can
predict that the measurement range of the RP method is k times the linear range of the shifting curve. In the next section, numerical simulations are presented to discuss the influence of the spatial coherence on the extracted phase information.
3. Simulations and discussion The simulated sample was an ideal cylinder with a diameter of 2.56 cm, composed of Mylar (C10H8O4) with a density of 1.4 g/cm3. The simulations were carried out at the X-ray energy of 25 keV, and the decrement of the real part and the imaginary part of the refractive index are d ¼ 4.841 10 07 and b ¼1.793 10 10 respectively. The parameters of the imaging system were designed as follows: the p/2 phase grating G1 with a pitch of 4 mm was situated at 80.64 cm after the light source. The analyzer grating G2 with a pitch of 5 mm was located at the first fractional Talbot distance from G1 (i.e. R2 ¼20.16 cm). The detector was set just behind G2. Two different positions of the sample were considered: they are at the optical axis respectively 40 cm and 60 cm away from the light source. In Fig. 4, theoretical and extracted values for a dimensionless and 8 mm light source are shown when the sample is respectively set at the positions of 40 cm (a) and 60 cm (b). Comparing the curves with the same sample position, we can find that the accurate measurement range with ideal point source is larger than that with 8 mm source for both positions. Comparing the curves with the same source size, the accurate measurement range with the sample at the position of 40 cm is larger than that with the sample at the position of 60 cm and both of them are larger than the linear range of shifting curve shown in Fig. 3. This is because of the factor k in the extraction formulae [Eq. (3)]. It greatly expands the measurement range by moving the sample to the position of source, though sacrificing the field of view (FOV) of the imaging system. As it can be seen in the figure, the extracted value is smaller than the theoretical value but maintain the relationship of size, when the refraction angle is beyond the accurate measurement range and smaller than 4kd2R2 . Consequently, the measurement range may be larger in experiment when certain error is accepted. In the simulations, planar gratings which limit the FOV at higher energies are employed for the current grating manufacturing technology. Properly shaped non-planar grating imaging system will be presented and discussed in a forthcoming work.
4. Conclusion In this manuscript, we propose a successful generalization of the RP method to fan beam illumination. Thanks to the geometrical relationship of fan beam geometry, the measurement range of the RP method is k times the linear region of the shifting curve. Data show that the measurement range will increase, when the sample close to the light source. Moreover, we investigate the influence of the spatial coherence on the information extracted by the modified method. From the simulations, we may conclude that the measurement range and sensitivity of the RP method for phase information extraction decrease when the source size increases. Hence, we can find that the requirement of coherence can be relaxed by moving the sample close to the light source. This work also presents a practical approach to identify typical source sizes of the X-ray grating interferometer using the RP method.
Fig. 3. The shifting curves as a function of xg =R2 for different source sizes (0 mm means an ideal point light source). The straight line, the tangent of shifting curve at its half maximum with an 8 mm source size, clearly shows the linear range is about 1.8 mrads.
Acknowledgments This work was partly supported by the Knowledge Innovation Program of the Chinese Academy of Sciences (KJCX2-YW-N42), the
Z. Wu et al. / Radiation Physics and Chemistry 95 (2014) 280–283
283
Fig. 4. Comparison between the extracted phase information and the theoretical value: the sample is located respectively at 40 cm (a) and 60 cm (b) from the light source. Panel (c) and (d) are respectively the magnified view of the rectangular region in panel (a) and (b).
National Natural Science Foundation of China (NSFC 11179004, 10979055, 11205157), the National Basic Research Program of China (2009CB930804, 2012CB825800), the Fundamental Research Funds for the Central Universities (WK2310000021), and the China Postdoctoral Science Foundation (2011M501064). References Bravin, A., Coan, P., Suortti, P., 2013. X-ray phase-contrast imaging: from preclinical applications towards clinics. Phys. Med. Biol. 58, R1–R35. Chen, G., Zambelli, J., Bevins, N., Qi, Z., Li, K., 2010. X-ray phase sensitive imaging methods: basic physical principles and potential medical applications. Curr. Med. Imaging Rev. 6, 1–10. David, C., Nohammer, B., Solak, H., Ziegler, E., 2002. Differential X-ray phase contrast imaging using a shearing interferometer. Appl. Phys. Lett. 81, 3287–3289. Davis, T., Gao, D., Gureyev, E., Stevenson, A., Wilkins, S., 1995a. Phase-contrast imaging of weakly absorbing materials using hard X-rays. Nature 379, 595–598. Davis, T., Gureyev, E., Gao, D., Stevenson, A., Wilkins, S., 1995b. X-ray image contrast from a simple phase object. Phys. Rev. Lett. 74, 3173–3176. Donath, T., Chabior, M., Pfeiffer, F., Bunk, O., Reznikova, E., Mohr, J., Hempel, E., Popescu, S., Hoheisel, M., Schuster, M., Baumann, J., David, C., 2009. Inverse geometry for grating-based X-ray phase-contrast imaging. J. Appl. Phys. 106, 054703. Engelhardt, M., Baumann, J., Schuster, M., Kottler, C., Pfeiffer, F., Bunk, O., David, C., 2007. High-resolution differential phase contrast imaging using a magnifying projection geometry with a microfocus X-ray source. Appl. Phys. Lett. 90, 224101. Ge, X., Wang, Z., Gao, K., Zhang, K., Hong, Y., Wang, D., Zhu, Z., Zhu, P., Wu, Z., 2011. Investigation of the partially coherent effects in a 2D Talbot interferometer. Anal. Bioanal. Chem. 401, 865–870. Gureyev, T., Mayo, S., Myers, D., Nesterets, Y., Paganin, D., Pogany, A., Stevenson, A., ¨ Wilkins, S., 2009. Refracting Rontgen’s rays: propagation-based X-ray phase contrast for biomedical imaging. J. Appl. Phys. 105, 102005.
Momose, A., 2005. Recent advances in X-ray phase imaging. Jpn. J. Appl. Phys. 44, 6355–6367. Momose, A., Kawamoto, S., Koyama, I., Hamaishi, Y., Takai, K., Suzuki, Y., 2003. Demonstration of X-ray talbot interferometry. Jpn. J. Appl. Phys. 42, L866–L868. Momose, A., Takeda, T., Itai, Y., Hirano, K., 1995. Phase-contrast X-ray computed tomography for observing biological soft tissues. Nat. Med. 2, 473–475. ¨ ¨ Pfeiffer, F., Bech, M., Bunk, O., Kraft, P., Eikenberry, E., Bronnimann, C., Grunzweig, C., David, C., 2008. Hard-X-ray dark-field imaging using a grating interferometer. Nat. Mater. 7, 134–137. Pfeiffer, F., Weitkamp, T., Bunk, O., David, C., 2006. Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources. Nat. Phys. 2, 258–261. Snigirev, A., Snigireva, I., Konh, V., Kuznetsov, S., Schelokov, I., 1995. On the possibility of X-ray phase contrast microimaging by coherent high-energy synchrotron radiation. Rev. Sci. Instrum. 66, 5486–5492. Wang, Z., Liu, X., Zhu, P., Huang, W., Yuan, Q., Li, E., Liu, Y., Zhang, K., Hong, Y., Wu, Z., 2010a. Analysis of partial coherence in grating-based phase-contrast X-ray imaging. Nucl. Instrum. Meth. A 619, 319–322. Wang, Z., Zhu, P., Huang, W., Yuan, Q., Liu, X., Zhang, K., Hong, Y., Zhang, H., Ge, X., Gao, K., Wu, Z., 2010b. Analysis of polychromaticity effects in X-ray Talbot interferometer. Anal. Bioanal. Chem. 397, 2137–2141. Weitkamp, T., Diaz, A., David, C., 2005. X-ray phase imaging with a grating interferometer. Opt. Exp. 13, 6296–62304. Wen, H., Bennett, E., Hegedus, M., Carroll, C., 2008. Spatial harmonic imaging of Xray scattering-initial results. IEEE Trans. Med. Imaging 27, 997–1002. Wilkins, S., Gurevey, T., Gao, D., Pogany, A., Stevenson, A., 1996. Phase-contrast imaging using polychromatic hard x-rays. Nature 384, 335–338. Yuan, Q., Zhu, P., Wang, J., Huang, W., Shu, H., Hu, T., Wu, Z., 2006. A new diffraction enhanced imaging set-up for larger samples at BSRF. Rad. Phys. Chem. 75, 1990–1994. Zhou, S., Brahme, A., 2008. Development of phase-contrast X-ray imaging techniques and potential medical applications. Phys. Med. 24, 129–148. Zhu, P., Zhang, K., Wang, Z., Liu, Y., Liu, X., Wu, Z., McDonald, S., Marone, F., Stampanoni, M., 2010. Low-dose, simple, and fast grating-based X-ray phasecontrast imaging. PNAS 107, 13576–13581.
Contents lists available at ScienceDirect
Radiation Physics and Chemistry journal homepage: www.elsevier.com/locate/radphyschem
A generalized reverse projection method for fan beam geometry under partially coherent illumination Z. Wu a, Z.L. Wang a,n, K. Gao a, K. Zhang b, X. Ge a, D.J. Wang a, S.H. Wang a, J. Chen a, Z.Y. Pan a, P.P. Zhu a,b, Z.Y. Wu a,b,nn a b
National Synchrotron Radiation Laboratory, University of Science and Technology of China, Hefei 230029, China Institute of High-Energy Physics, Chinese Academy of Sciences, Beijing 100049, China
H I G H L I G H T S c c c c
Fan beam geometry with geometrical magnification is used with the modified RP method. In the calculation of conjugate projection images, unitary ray is taken into account for fan beam geometry. We have considered the influence of the spatial coherence on the extracted information. The measurement range of the generalized RP method is enlarged by moving the sample close to the light source.
a r t i c l e i n f o
a b s t r a c t
Article history: Received 19 October 2012 Accepted 6 January 2013 Available online 16 January 2013
In this paper, a generalized reverse projection (RP) method for grating-based fan beam phase contrast imaging is presented. Compared to the original RP method, rays rather than projection images are taken into account during the information extraction process. We also discuss the influence of partial coherence on the extracted information. Theoretical derivations and numerical simulations are performed to confirm the validity of the method. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Spatial coherence Phase contrast imaging RP method Fan beam geometry
1. Introduction Conventional X-ray absorption imaging is a power tool in medical diagnosis, industrial non-destructive testing and security inspection. However, its poor contrast to low-Z elements limits its applications in soft tissue imaging. If we write the complex refractive index of an object as n¼ 1 d ib, its real part determines the phase shift of the wave front while its imaginary part is associated to the linear attenuation coefficient. In the hard X-ray regime and for low-Z elements, d is about 1000 times greater than b. Therefore, the phase shift is about three orders of magnitude of the absorption. As a consequence different phase contrast imaging methods have been proposed (Bravin et al., 2013; Chen et al., 2010; Davis et al., 1995a; Ge et al., 2011; Momose, 2005; Wang et al., 2010a, 2010b; Zhou and Brahme, 2008), such as, interferometric methods (Momose et al.,
n
Corresponding author. Tel.: þ86 551 6360 2017. Corresponding author at: National Synchrotron Radiation Laboratory, University of Science and Technology of China, Hefei 230029, China. Tel.: þ86 551 6360 2077; fax: þ 86 551 6514 1078. E-mail addresses: [email protected] (Z.L. Wang), [email protected] (Z.Y. Wu). nn
0969-806X/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.radphyschem.2013.01.006
1995), crystal analyzer-based methods (Davis et al., 1995b; Yuan et al., 2006), propagation-based methods (Gureyev et al., 2009; Wilkins et al., 1996; Snigirev et al., 1995) and grating-based methods (David et al., 2002; Momose et al., 2003; Pfeiffer et al., 2006, 2008; Weitkamp et al., 2005). Grating-based methods are considered the most promising phase contrast imaging methods, thanks to their less stringent requirements regarding the light source. In the phase contrast imaging, phase and absorption information are collected together thus information separation is inevitable. So far, three primary phase extraction methods have been introduced: the Fourier harmonic method (Wen et al., 2008), the phase stepping (PS) method (Weitkamp et al., 2005) and the reverse projection (RP) method (Zhu et al., 2010). The first one is the earliest method and quite fast due to only one required projection image at each projection position. Compared with the Fourier harmonic method, the PS method collects several images at each projection angle but provides higher quality results. It is widely adopted in the laboratory. Nevertheless, because of the long exposure time, the large radiation dose is unacceptable for practical applications, especially for medical imaging. On the contrary, the RP method requires lower dose and is a simpler and faster phase extraction method. It takes advantage of the anti-symmetry (i.e., with a rotation of 1801, you can collect an image with the same absorption
Z. Wu et al. / Radiation Physics and Chemistry 95 (2014) 280–283
but opposite differential phase information), thus reduces the number of the required images by extending the scanning angle range from 1801 to 3601. However, the RP method has been mainly introduced to run experiments with parallel beam geometrical configurations. In this contribution, we present a generalized RP method for fan beam geometry. We also discuss the effects of partially coherent illumination on the extraction information.
281
slope by its first order Taylor term. Consequently formulae for information retrieval can be given as follows: Z
1
-1
yr ðxr , f,zÞ ¼
2S xg =R2 I0 Iðxr , f,zÞ þ Iðxr , f þ p þ 2a,zÞ
mðx,y,zÞdr ¼ ln
1 Iðxr , f,zÞIðxr , f þ p þ2a,zÞ kC Iðxr , f,zÞ þ Iðxr , f þ p þ 2a,zÞ
ð3Þ
2. The RP method in fan beam geometry Based on the setup shown in Fig. 1 and the scale factor of fan beam geometry (Donath et al., 2009; Engelhardt et al., 2007), the intensity recorded by a virtual detector at the rotation axis in the paraxial approximation can be written as: Z 1 x Iðxr , f,zÞ ¼ I0 exp mðx,y,zÞdr S g þ kyr , R2 1 ( R0 =R1 0 oR0 oR1 k¼ ð1Þ ðR1 þR2 -R0 Þ=R2 R1 o R0 o R1 þ R2 where f is the projection angle and I0 is the intensity at the position of phase grating G1 without the sample. m denotes the linear attenuation coefficient of the sample and the function S is the shifting curve. R0, R1 and R2 are the distances respectively from the source to the rotation axis of the sample, between the source and the phase grating G1 and from G1 to the analyzer grating G2. xg refers to the displacement of G2 along the direction perpendicular to the groove of grating. yr is the refraction angle. (xr,yr,z) is a stationary coordinate system while (x,y,z) is a rotational one fixed on the sample shown in Fig. 1. Their origins of coordinates are set at the intersection of the rotation axis and the optical axis. We know that the RP method for parallel beam geometry is based on the anti-symmetric characteristic. However, under fan beam illumination, we can see from Fig. 2(a) that the anti-symmetry between the mutually reverse projection images is no longer fulfilled. In the following, a modified approach considering unitary ray (see Fig. 2(b)) is introduced to find the conjugate projection images. It can be seen from Fig. 2 that the anti-symmetry between mutually reverse rays still exists in fan beam geometry. If we consider a single ray I(xr,f1,z) and its reverse ray I( xr,f2,z), from the geometrical relationship exhibited in Fig. 2(b), we can obtain: x f2 ¼ f1 þ p þ2arctan r ð2Þ R0 When the refraction angle yr od2/4kR2, with d2 the period of G2, the shifting curve can be linearly approximated at its half
Fig. 1. Schematic diagram of the grating-based interferometer in fan beam geometry: the sample is located (a) before the phase grating (0 o R0 o R1) and (b) after the phase grating (R1 o R0 o R1 þR2).
Fig. 2. The realization of the RP method in fan beam geometry: (a) mutually reverse projections and (b) single ray is considered as unit.
282
Z. Wu et al. / Radiation Physics and Chemistry 95 (2014) 280–283
where C ¼
dSðxg =R2 Þ 1 dy Sðxg =R2 Þ
is a constant and a ¼ arctan Rxr0 . Substi-
tuting k ¼1 and R0 ¼N into Eq. (3), we will obtain the formulae for parallel beam geometry. In the Talbot interferometer, the phase grating G1 acts as the beam splitter. When it is illuminated by a fully coherent source in fan beam geometry, the shifting curve at the plane of G2 can be expressed as ! 2 X xg R1 1 4 þ cos 2 p nx =d ¼ ð4Þ Sfully g 2 2 n ¼ 1,3,5, p2 n2 R2 R1 þ R2 In the case of a partially coherent illumination, if the intensity distribution can be expressed as a Gaussian function with a standard deviation ss, the shifting curve Spartially is given by the convolution of the projection source profile f xg ¼ x2g 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp with Sfully: 2 pR2 ss =R1 2ðR2 ss =R1 Þ Spartially
xg R2
¼ f xg Sf ully ¼
R1 R1 þ R2
2
X 1 nR2 ss 2 exp 2p2 þ d2 R1 2 n ¼ 1,3,5
!
! 4 cos 2 p nx =d g 2 p2 n2
ð5Þ where is the convolution operator. The Fourier transform and its inverse transform have been utilized here for this derivation. After a simple analysis, we can recognize that the spatial coherence has no effect on the extracted absorption information. Hereafter, we only investigate the influence of the spatial coherence on the extracted phase information. The shifting curves for different source sizes are compared in Fig. 3. (All system parameters are given in pffiffiffiffiffiffiffiffi ffi the next section). The full width at half maximum given by 2 ln 4ss is usually defined as the size of Gaussian light source. We can see in Fig. 3, that the gradient and linear range of the shifting curve near the half maximum become smaller, when a larger size light source is used. In other words, increasing the source size, both the sensitivity and the measurement range of the RP method decrease. Furthermore, from Eq. (1), we can
predict that the measurement range of the RP method is k times the linear range of the shifting curve. In the next section, numerical simulations are presented to discuss the influence of the spatial coherence on the extracted phase information.
3. Simulations and discussion The simulated sample was an ideal cylinder with a diameter of 2.56 cm, composed of Mylar (C10H8O4) with a density of 1.4 g/cm3. The simulations were carried out at the X-ray energy of 25 keV, and the decrement of the real part and the imaginary part of the refractive index are d ¼ 4.841 10 07 and b ¼1.793 10 10 respectively. The parameters of the imaging system were designed as follows: the p/2 phase grating G1 with a pitch of 4 mm was situated at 80.64 cm after the light source. The analyzer grating G2 with a pitch of 5 mm was located at the first fractional Talbot distance from G1 (i.e. R2 ¼20.16 cm). The detector was set just behind G2. Two different positions of the sample were considered: they are at the optical axis respectively 40 cm and 60 cm away from the light source. In Fig. 4, theoretical and extracted values for a dimensionless and 8 mm light source are shown when the sample is respectively set at the positions of 40 cm (a) and 60 cm (b). Comparing the curves with the same sample position, we can find that the accurate measurement range with ideal point source is larger than that with 8 mm source for both positions. Comparing the curves with the same source size, the accurate measurement range with the sample at the position of 40 cm is larger than that with the sample at the position of 60 cm and both of them are larger than the linear range of shifting curve shown in Fig. 3. This is because of the factor k in the extraction formulae [Eq. (3)]. It greatly expands the measurement range by moving the sample to the position of source, though sacrificing the field of view (FOV) of the imaging system. As it can be seen in the figure, the extracted value is smaller than the theoretical value but maintain the relationship of size, when the refraction angle is beyond the accurate measurement range and smaller than 4kd2R2 . Consequently, the measurement range may be larger in experiment when certain error is accepted. In the simulations, planar gratings which limit the FOV at higher energies are employed for the current grating manufacturing technology. Properly shaped non-planar grating imaging system will be presented and discussed in a forthcoming work.
4. Conclusion In this manuscript, we propose a successful generalization of the RP method to fan beam illumination. Thanks to the geometrical relationship of fan beam geometry, the measurement range of the RP method is k times the linear region of the shifting curve. Data show that the measurement range will increase, when the sample close to the light source. Moreover, we investigate the influence of the spatial coherence on the information extracted by the modified method. From the simulations, we may conclude that the measurement range and sensitivity of the RP method for phase information extraction decrease when the source size increases. Hence, we can find that the requirement of coherence can be relaxed by moving the sample close to the light source. This work also presents a practical approach to identify typical source sizes of the X-ray grating interferometer using the RP method.
Fig. 3. The shifting curves as a function of xg =R2 for different source sizes (0 mm means an ideal point light source). The straight line, the tangent of shifting curve at its half maximum with an 8 mm source size, clearly shows the linear range is about 1.8 mrads.
Acknowledgments This work was partly supported by the Knowledge Innovation Program of the Chinese Academy of Sciences (KJCX2-YW-N42), the
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Fig. 4. Comparison between the extracted phase information and the theoretical value: the sample is located respectively at 40 cm (a) and 60 cm (b) from the light source. Panel (c) and (d) are respectively the magnified view of the rectangular region in panel (a) and (b).
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