- Email: [email protected]
Experimental and theoretical investigations of diffraction enhanced imaging
ARTICLE IN PRESS
Nuclear Instruments and Methods in Physics Research A 580 (2007) 803–807 www.elsevier.com/locate/nima
Experimental and theoretical investigations of diffraction enhanced imaging Junyue Wanga,b, Peiping Zhua,, Qingxi Yuana, Wanxia Huanga, Hang Shua,b, Bo Chena, Enrong Lia,b, Yijin Liua, Tiandou Hua, Ziyu Wua a
Beijing Synchrotron Radiation Facility, Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, PR China b Graduate University of the Chinese Academy of Sciences, Beijing 100049, PR China Available online 8 May 2007
Abstract This contribution investigates the effect on the imaging contrast of the small angle scattering and of the rocking curve in the entire angular range. We show that based on the small angle scattering properties, the reflectivity of the crystal analyzer and the rocking curve of the monochromator–analyzer, in a diffraction enhanced imaging (DEI) experimental setup the contrast of the image collected at the top of the rocking curve is always higher than that of the apparent absorption image. Moreover, our experimental data confirm that the quality of a refraction image is superior to a refraction-like image. In order to understand the observed behavior we introduce and discuss the contribution of a new term in the classical DEI equation. r 2007 Elsevier B.V. All rights reserved. PACS: 87.59.e; 07.85.Qe Keywords: Phase contrast imaging; Synchrotron radiation
1. Introduction Diffraction enhanced imaging (DEI) is one of the most effective and practical X-ray phase contrast imaging methods that exhibits significant advantages when compared to conventional X-ray absorption imaging techniques, widely applied to the investigation of biological and biomedical systems [1–16]. However, in the interaction of X-rays with matter the distribution angles of the small angle scattering are mainly confined in a small cone and the refraction angles, in particular at the edge of a sample, are larger than Darwin’s width of the crystal analyzer. As a consequence, in DEI methods their contributions have to be clearly understood. Several combinations of images can be considered using data taken at different positions of the rocking curve. For example, the ‘‘apparent absorption’’ image obtained from adding images taken at two shoulders of the rocking curve is different from the ‘‘refraction-like’’ image obtained as the difference of the two previous images Corresponding author.
E-mail address: [email protected] (P. Zhu). 0168-9002/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2007.05.004
or the ‘‘refraction image’’ first discussed by Chapman and obtained as the ratio between the ‘‘refraction-like’’ image and the ‘‘apparent absorption’’ image. Our recent imaging experiments addressed two interesting results: the first associated to the so-called ‘‘top image’’, e.g., data collected at the top of the rocking curve are characterized by a contrast higher than the ‘‘apparent absorption’’ image; the second regarding the contrast of the ‘‘refraction’’ image that is higher than that of the ‘‘refraction-like’’ image. Here, an analysis of several DEI images collected on different samples will be presented and discussed in the framework of a mathematical procedure for image reconstruction. 2. Setup of the DEI experiments Fig. 1 shows a schematic layout of the experimental setup available at BSRF and used for DEI. An Si(1 1 1) single crystal is used as the monochromator while a second crystal of the same kind is placed downstream and is used as the analyzer. The samples to be investigated are set
ARTICLE IN PRESS J. Wang et al. / Nuclear Instruments and Methods in Physics Research A 580 (2007) 803–807
804
Fig. 1. Layout of the experimental setup of the DEI technique.
between the monochromator and the analyzer crystals. While the beam intensity is monitored by an ion chamber, a CCD camera or other bidimensional detectors can be used downstream to the crystal analyzer to record sample images. The analyzer can be rotated, so that energy can be tuned or detuned and images can be collected at any angular position of the rocking curve, e.g., at the top of this double crystal system. Absorption, refraction and small angle scattering occurred in a sample so that the imaging contrast is a function of the position along the rocking curve of the crystals and depends on the way the images are combined. 3. The equations of four different kinds of images In the DEI equation [5] I ¼ I R RðyA þ ym Þ
(1)
originally proposed by Chapman and co-workers yA is the setting angle of the analyzer, while ym is the refraction angle in the meridional plane. I R represents the incident beam. The equation has been widely used by scientists in biological and biomedical imaging but also in other image reconstruction procedures. According to Eq. (1) the top image ðI top Þ, the apparent absorption image ðI abs Þ, the refraction-like image ðI sub Þ and the refraction image (ym ) can be expressed as follows: I top ¼ I R Rðym Þ where yA ¼ 0, yD yD þ ym þ R þ ym I add ¼ I R R 2 2 yD yD þ ym R þ ym I sub ¼ I R R 2 2 yD I sub 2 I add yD RðyD =2 þ ym Þ RðyD =2 þ ym Þ ¼ 2 RðyD =2 þ ym Þ þ RðyD =2 þ ym Þ
(2)
(3)
(4)
ym ¼
ð5Þ
where yA ¼ yD =2. 4. Comparison between top and apparent absorption images The image taken at the top of the rocking curve is similar to the image obtained by the sum of the two images collected at the two shoulders of the rocking curve.
However, in some experiments the top image looks better. A comparison between a top image and an apparent absorption image is shown in Fig. 2. From these two images of a bee the top image clearly exhibits a better contrast. The experimental behavior cannot be explained looking only at Eqs. (2) and (3). However, it is well known that the small angle scattering contributions are different for these two types of images. For both the cases, scattered X-rays are characterized by an angular Gaussian like distribution with a width of 2oS . The reflectivity of the analyzer crystal whose width is the so-called Darwin width, yD , has a very narrow angular acceptance. Actually, the width of the distribution (2oS ) of the scattered X-rays is much wider than yD so that always there is a portion of light, proportional to yD =2oS ; refracted by the analyzer that reaches the detector independent of the angular position of thecrystal analyzer. The solid line in Fig. 3 is the Darwin reflectivity curve rðyÞ of the Si(1 1 1) analyzer crystal for X-rays at 15 keV calculated using the X-ray Oriented Programs (XOP) software developed by Manuel Sanchez del Rio at ESRF and Roger J. Dejus at APS while the dotted line is the sum of rðy þ yD =2Þ and rðy yD =2Þ. For the top image, only a small portion of the scattered X-rays are reflected by the analyzer within the angular width of yD while, as shown by the dashed line, X-rays are scattered within an angular range of 2yD in the case of the apparent absorption image, twice larger than that of a top image. Therefore, in an apparent absorption image the small angle scattering contribution introduces a strong background (and hence a higher noise) resulting in a poorer contrast. As a consequence, we introduced into the original DEI equation an additional term taking into account the small angle scattering contribution under the condition of both weak absorption and weak scattering. The modified DEI expression becomes ~ ¼ I 0 ð1 mt wtÞRðyA þ ym Þ þ I 0 wt yD Ið0Þ 2oS
(6)
where I 0 is the incident intensity, m the linear absorption coefficient, w the extinction coefficient and t the thickness of the sample. The first term on the right in Eq. (6) is similar to the Chapman equation [5], the difference being the contribution ð1 mt wtÞ to the rocking curve term, which is not present in Eq. (1). This term takes into account the refraction contribution due to the beam reflected by the analyzer and describes how the incident intensity on the sample I 0 is reduced by a factor mt þ wt, without being absorbed or scattered. The loss of the radiation associated to mtI 0 is due to the absorption mechanism while the light component described by wtI 0 is neither absorbed nor refracted but scattered with an angular distribution of width 2oS . Therefore, this second term describes the small angle scattering contribution accepted by the analyzer when 2oS byD , a condition that is independent of yA , the analyzer angle. According to Eq. (6) the top image and the apparent absorption image
ARTICLE IN PRESS J. Wang et al. / Nuclear Instruments and Methods in Physics Research A 580 (2007) 803–807
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Fig. 2. Comparison of the image of a bee obtained with the DEI technique collected at the peak of the rocking curve (left). An apparent absorption image of the same bee (right).
that the better contrast of a top image is associated to its better rejection of the scattered radiation. 5. Comparison between a refraction image and a refractionlike image
Fig. 3. Comparison of the Darwin reflectivity curves associated to the image at the peak of the rocking curve (solid line) and to the apparent absorption image (dashed line).
can be rewritten as yD I~top ¼ I 0 ð1 mt wtÞRðym Þ þ I 0 wt (7) 2oS yD yD þ ym þ R þ ym I~add ¼ I 0 ð1 mt wtÞ R 2 2 yD þ I 0 wt ð8Þ oS and taking into account the general property of the rocking curve yD yD Rð0Þ R þR (9) 2 2 when ym ¼ 0, and yD yD Rðym ÞoR þ ym þ R þ ym 2 2
(10)
when ym a0. From the comparison among Eqs. (7)–(9) it is now clear why a top image exhibits a higher contrast than that of an apparent absorption image. In fact, when the typical DEI condition 2oS byD is fulfilled, the acceptance angle rA ðyÞ is half of rA ðyD =2 þ yÞ þ rA ðyD =2 þ yÞ and the photons first scattered and then reflected by the analyzer are much less than those collected in an apparent absorption image. This condition determines also the strong background of the apparent absorption image so
Fig. 4 shows the comparison between a refraction image of a mouse paw and its corresponding refraction-like image. Looking at Fig. 4 it is evident that the contrast of the refraction image is higher than that of the refraction-like image. A possible explanation is certainly associated to higher contrast of the refraction image at the edges of the sample where X-rays are scattered with large refraction angles. In Fig. 5 (left) the experimental rocking curves at 16 keV, RðyD =2 þ ym Þ and RðyD =2 þ ym Þ, are drawn, respectively, for the two Si(1 1 1) crystals. According to Eqs. (4) and (5) the two curves in the right of Fig. 5: yD yD þ ym R þ ym R (11) 2 2 and yD yD yD þ ym R þ ym þ ym R R 2 2 2 yD þR þ ym 2
ð12Þ
are the intensity–angle dependence for the two image types. Within the linear slope range the contrast is substantially the same, but at large refraction angles, the curve of the refraction-like image falls down rapidly while the curve of the refraction image decays slowly. Therefore, it is evident that more radiation is refracted in a refraction image determining a high contrast. Although considering Eqs. (4) and (5) the behavior can be understood we would like to introduce and discuss the more general DEI formula: Eq. (6). According to Eq. (6) the expression for a refraction-like image may be written as ~I sub ¼ I 0 ð1 mt wtÞ R yD þ ym R yD þ ym . 2 2 (13) Because the small angle scattering noise due to rA ðyD =2 þ yÞ and rA ðyD =2 þ yÞ is partially superimposed to the image
ARTICLE IN PRESS J. Wang et al. / Nuclear Instruments and Methods in Physics Research A 580 (2007) 803–807
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Fig. 4. Refraction image of a mouse paw obtained with the DEI technique (left). Refraction-like image of the same mouse paw (right).
Fig. 5. On the left comparison between experimental rocking curves: RðyD =2 þ ym Þ (solid line) RðyD =2 þ ym Þ (dashed line). The intensity–angle dependence for a refraction-like image (solid line) and for a refraction image (dashed line).
spot, its subtraction may lead to a very low level of the small angle scattering noise. Although the expressions associated to the low- and high-angle images and derived from Eq. (6) are different from those derived by Eq. (1) [5], the expressions of the refraction-like image derived from them are very similar. The expressions of the refraction image based on Eq. (6) is yD I~sub y~ ref ¼ 2 I~add ¼
R yD =2 þ ym R yD =2 þ ym yD . 2 RðyD =2 þ ym Þ þ RðyD =2 þ ym Þ þ ðwt=ð1 mt wtÞÞyD =oS
ð14Þ When jym jpyD =2, we have yD yD R þ ym þ R þ ym 1 2 2 and when jym j4yD =2, yD yD R þ ym þ R þ ym ¼ 1 Ro1 2 2
Now, according to Eq. (17), when the refraction angles fulfill the condition jym jpyD =2 the contrast of the refraction-like image is always higher than that of a refraction image. In addition, substituting Eq. (16) into Eq. (14), we have yD I~sub yD yD yD y~ ref ¼ R þ ym R þ ym ¼ 1 2 I~add 2 2 2 wt yD ð18Þ R 1 mt wt oS so that according to Eq. (18) because R4ðwt=ð1 mt wtÞÞyD =oS , when the refraction angles jym j4yD =2, the contrast of a refraction-like image is lower than that of a refraction image.
(15) 6. Conclusion (16)
where R is an increasing function of the refraction angle ym . Substituting Eq. (15) into (14), we obtain: yD I~sub y~ ref ¼ 2 I~add yD yD yD R þ ym R þ ym ¼ 1 2 2 2 wt yD . ð17Þ þ 1 mt wt oS
Nowadays, a comparison of DEI with other existing imaging methods indicates that DEI is a practical imaging technique with great possibilities and a really wide range of applications. In this contribution we present a couple of DEI experimental results and discuss the not negligible contribution of the small angle scattering processes to the image quality of this phase contrast imaging method. In addition, adding to the well-known DEI equation a new term that takes into account the small angle scattering processes we introduced a new formalism. The application of this new formalism confirms that, among the many possible images or
ARTICLE IN PRESS J. Wang et al. / Nuclear Instruments and Methods in Physics Research A 580 (2007) 803–807
combination of images, the top image always exhibits the higher contrast, certainly higher than that of the apparent absorption image, while the contrast of the refraction image remains higher than that of the refraction-like image. Acknowledgments This work has been supported by the States Key Project for Fundamental Research (2003CB716900) and the National Center for Nanoscience and Technology of China, the National Outstanding Youth Fund (Project No. 10125523 to Z.W.), the Key Important Project of the National Natural Science Foundation of China (Nos. 10490190, 10490194, 90206032), the National Natural Science Foundation of China (60477006 and 10504033) and the Knowledge Innovation Program of the Chinese Academy of Sciences (KJCX2-SW-N11, KJCX2-SW-H1202). A sincere thank is due to Augusto Marcelli for many fruitful discussions. References [1] T.J. Davis, D. Gao, T.E. Gureyev, A.W. Stevenson, S.W. Wilkins, Nature 373 (1995) 595.
807
[2] S.W. Wilkins, T.E. Gureyev, D. Gao, A. Pogany, A.W. Stevenson, Nature 384 (1996) 335. [3] A. Momose, T. Takeda, Y. Itai, K. Hirano, Nat. Med. 2 (1996) 473. [4] K. Hirano, A. Momose, Phys. Rev. Lett. 76 (1996) 3735. [5] L.D. Chapman, W.C. Thomlinson, R.E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, D. Sayers, Phys. Med. Biol. 42 (1997) 2015. [6] G. Margaritondo, G. Tromba, J. Appl. Phys. 85 (1999) 3406. [7] Y. Hwu, H. Hsieh, M.J. Lu, W.L. Tsai, H.M. Lin, W.C. Goh, J.H. Je, C.K. Kim, D.Y. Noh, H.S. Youn, G. Tromba, G. Margaritondo, J. Appl. Phys. 86 (1999) 4613. [8] D. Gao, A. Pogany, A.W. Stevenson, T.E. Gureyev, S.W. Wilkins, Z. Mai, Phys. Sin. 49 (2000) 2357 (in Chinese). [9] F.A. Dilmanian, Z. Zhong, B. Ren, X.Y. Wu, L.D. Chapman, I. Oriont, W.C. Thomlinson, Phys. Med. Biol. 45 (2000) 933. [10] W.L. Tsai, P.C. Hsu, Y. Hwu, C.H. Chen, L.W. Chang, J.H. Je, A. Groso, G. Margaritondo, Nature 417 (2002) 139. [11] W. Huang, Y. Tian, P. Zhu, Z. Mai, X. Hu, Phys. Sin. 51 (2002) 1040. [12] Y. Tian, T. Xiao, P. Zhu, W. Huang, G. Li, Q. Yuan, M. Chen, Nucl. Technol. 27 (2004) 417 (in Chinese). [13] Q. Yuan, Y. Tian, P. Zhu, W. Huang, J. Wang, T. Hu, X. Jiang, Z. Wu, Nucl. Technol. 27 (2004) 725 (in Chinese). [14] P. Zhu, J. Wang, Q. Yuan, Y. Tian, W. Huang, G. Li, T. Hu, X. Jiang, Z. Wu, Phys. Sin. 54 (2005) 58 (in Chinese). [15] W. Huang, Q. Yuan, Y. Tian, P. Zhu, X. Jiang, J. Wang, Phys. Sin. 54 (2005) 677 (in Chinese). [16] P. Zhu, J. Wang, Q. Yuan, W. Huang, H. Shu, B. Gao, T. Hu, Z. Wu, Phys. Lett. 87 (2005) 264101.
Nuclear Instruments and Methods in Physics Research A 580 (2007) 803–807 www.elsevier.com/locate/nima
Experimental and theoretical investigations of diffraction enhanced imaging Junyue Wanga,b, Peiping Zhua,, Qingxi Yuana, Wanxia Huanga, Hang Shua,b, Bo Chena, Enrong Lia,b, Yijin Liua, Tiandou Hua, Ziyu Wua a
Beijing Synchrotron Radiation Facility, Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, PR China b Graduate University of the Chinese Academy of Sciences, Beijing 100049, PR China Available online 8 May 2007
Abstract This contribution investigates the effect on the imaging contrast of the small angle scattering and of the rocking curve in the entire angular range. We show that based on the small angle scattering properties, the reflectivity of the crystal analyzer and the rocking curve of the monochromator–analyzer, in a diffraction enhanced imaging (DEI) experimental setup the contrast of the image collected at the top of the rocking curve is always higher than that of the apparent absorption image. Moreover, our experimental data confirm that the quality of a refraction image is superior to a refraction-like image. In order to understand the observed behavior we introduce and discuss the contribution of a new term in the classical DEI equation. r 2007 Elsevier B.V. All rights reserved. PACS: 87.59.e; 07.85.Qe Keywords: Phase contrast imaging; Synchrotron radiation
1. Introduction Diffraction enhanced imaging (DEI) is one of the most effective and practical X-ray phase contrast imaging methods that exhibits significant advantages when compared to conventional X-ray absorption imaging techniques, widely applied to the investigation of biological and biomedical systems [1–16]. However, in the interaction of X-rays with matter the distribution angles of the small angle scattering are mainly confined in a small cone and the refraction angles, in particular at the edge of a sample, are larger than Darwin’s width of the crystal analyzer. As a consequence, in DEI methods their contributions have to be clearly understood. Several combinations of images can be considered using data taken at different positions of the rocking curve. For example, the ‘‘apparent absorption’’ image obtained from adding images taken at two shoulders of the rocking curve is different from the ‘‘refraction-like’’ image obtained as the difference of the two previous images Corresponding author.
E-mail address: [email protected] (P. Zhu). 0168-9002/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2007.05.004
or the ‘‘refraction image’’ first discussed by Chapman and obtained as the ratio between the ‘‘refraction-like’’ image and the ‘‘apparent absorption’’ image. Our recent imaging experiments addressed two interesting results: the first associated to the so-called ‘‘top image’’, e.g., data collected at the top of the rocking curve are characterized by a contrast higher than the ‘‘apparent absorption’’ image; the second regarding the contrast of the ‘‘refraction’’ image that is higher than that of the ‘‘refraction-like’’ image. Here, an analysis of several DEI images collected on different samples will be presented and discussed in the framework of a mathematical procedure for image reconstruction. 2. Setup of the DEI experiments Fig. 1 shows a schematic layout of the experimental setup available at BSRF and used for DEI. An Si(1 1 1) single crystal is used as the monochromator while a second crystal of the same kind is placed downstream and is used as the analyzer. The samples to be investigated are set
ARTICLE IN PRESS J. Wang et al. / Nuclear Instruments and Methods in Physics Research A 580 (2007) 803–807
804
Fig. 1. Layout of the experimental setup of the DEI technique.
between the monochromator and the analyzer crystals. While the beam intensity is monitored by an ion chamber, a CCD camera or other bidimensional detectors can be used downstream to the crystal analyzer to record sample images. The analyzer can be rotated, so that energy can be tuned or detuned and images can be collected at any angular position of the rocking curve, e.g., at the top of this double crystal system. Absorption, refraction and small angle scattering occurred in a sample so that the imaging contrast is a function of the position along the rocking curve of the crystals and depends on the way the images are combined. 3. The equations of four different kinds of images In the DEI equation [5] I ¼ I R RðyA þ ym Þ
(1)
originally proposed by Chapman and co-workers yA is the setting angle of the analyzer, while ym is the refraction angle in the meridional plane. I R represents the incident beam. The equation has been widely used by scientists in biological and biomedical imaging but also in other image reconstruction procedures. According to Eq. (1) the top image ðI top Þ, the apparent absorption image ðI abs Þ, the refraction-like image ðI sub Þ and the refraction image (ym ) can be expressed as follows: I top ¼ I R Rðym Þ where yA ¼ 0, yD yD þ ym þ R þ ym I add ¼ I R R 2 2 yD yD þ ym R þ ym I sub ¼ I R R 2 2 yD I sub 2 I add yD RðyD =2 þ ym Þ RðyD =2 þ ym Þ ¼ 2 RðyD =2 þ ym Þ þ RðyD =2 þ ym Þ
(2)
(3)
(4)
ym ¼
ð5Þ
where yA ¼ yD =2. 4. Comparison between top and apparent absorption images The image taken at the top of the rocking curve is similar to the image obtained by the sum of the two images collected at the two shoulders of the rocking curve.
However, in some experiments the top image looks better. A comparison between a top image and an apparent absorption image is shown in Fig. 2. From these two images of a bee the top image clearly exhibits a better contrast. The experimental behavior cannot be explained looking only at Eqs. (2) and (3). However, it is well known that the small angle scattering contributions are different for these two types of images. For both the cases, scattered X-rays are characterized by an angular Gaussian like distribution with a width of 2oS . The reflectivity of the analyzer crystal whose width is the so-called Darwin width, yD , has a very narrow angular acceptance. Actually, the width of the distribution (2oS ) of the scattered X-rays is much wider than yD so that always there is a portion of light, proportional to yD =2oS ; refracted by the analyzer that reaches the detector independent of the angular position of thecrystal analyzer. The solid line in Fig. 3 is the Darwin reflectivity curve rðyÞ of the Si(1 1 1) analyzer crystal for X-rays at 15 keV calculated using the X-ray Oriented Programs (XOP) software developed by Manuel Sanchez del Rio at ESRF and Roger J. Dejus at APS while the dotted line is the sum of rðy þ yD =2Þ and rðy yD =2Þ. For the top image, only a small portion of the scattered X-rays are reflected by the analyzer within the angular width of yD while, as shown by the dashed line, X-rays are scattered within an angular range of 2yD in the case of the apparent absorption image, twice larger than that of a top image. Therefore, in an apparent absorption image the small angle scattering contribution introduces a strong background (and hence a higher noise) resulting in a poorer contrast. As a consequence, we introduced into the original DEI equation an additional term taking into account the small angle scattering contribution under the condition of both weak absorption and weak scattering. The modified DEI expression becomes ~ ¼ I 0 ð1 mt wtÞRðyA þ ym Þ þ I 0 wt yD Ið0Þ 2oS
(6)
where I 0 is the incident intensity, m the linear absorption coefficient, w the extinction coefficient and t the thickness of the sample. The first term on the right in Eq. (6) is similar to the Chapman equation [5], the difference being the contribution ð1 mt wtÞ to the rocking curve term, which is not present in Eq. (1). This term takes into account the refraction contribution due to the beam reflected by the analyzer and describes how the incident intensity on the sample I 0 is reduced by a factor mt þ wt, without being absorbed or scattered. The loss of the radiation associated to mtI 0 is due to the absorption mechanism while the light component described by wtI 0 is neither absorbed nor refracted but scattered with an angular distribution of width 2oS . Therefore, this second term describes the small angle scattering contribution accepted by the analyzer when 2oS byD , a condition that is independent of yA , the analyzer angle. According to Eq. (6) the top image and the apparent absorption image
ARTICLE IN PRESS J. Wang et al. / Nuclear Instruments and Methods in Physics Research A 580 (2007) 803–807
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Fig. 2. Comparison of the image of a bee obtained with the DEI technique collected at the peak of the rocking curve (left). An apparent absorption image of the same bee (right).
that the better contrast of a top image is associated to its better rejection of the scattered radiation. 5. Comparison between a refraction image and a refractionlike image
Fig. 3. Comparison of the Darwin reflectivity curves associated to the image at the peak of the rocking curve (solid line) and to the apparent absorption image (dashed line).
can be rewritten as yD I~top ¼ I 0 ð1 mt wtÞRðym Þ þ I 0 wt (7) 2oS yD yD þ ym þ R þ ym I~add ¼ I 0 ð1 mt wtÞ R 2 2 yD þ I 0 wt ð8Þ oS and taking into account the general property of the rocking curve yD yD Rð0Þ R þR (9) 2 2 when ym ¼ 0, and yD yD Rðym ÞoR þ ym þ R þ ym 2 2
(10)
when ym a0. From the comparison among Eqs. (7)–(9) it is now clear why a top image exhibits a higher contrast than that of an apparent absorption image. In fact, when the typical DEI condition 2oS byD is fulfilled, the acceptance angle rA ðyÞ is half of rA ðyD =2 þ yÞ þ rA ðyD =2 þ yÞ and the photons first scattered and then reflected by the analyzer are much less than those collected in an apparent absorption image. This condition determines also the strong background of the apparent absorption image so
Fig. 4 shows the comparison between a refraction image of a mouse paw and its corresponding refraction-like image. Looking at Fig. 4 it is evident that the contrast of the refraction image is higher than that of the refraction-like image. A possible explanation is certainly associated to higher contrast of the refraction image at the edges of the sample where X-rays are scattered with large refraction angles. In Fig. 5 (left) the experimental rocking curves at 16 keV, RðyD =2 þ ym Þ and RðyD =2 þ ym Þ, are drawn, respectively, for the two Si(1 1 1) crystals. According to Eqs. (4) and (5) the two curves in the right of Fig. 5: yD yD þ ym R þ ym R (11) 2 2 and yD yD yD þ ym R þ ym þ ym R R 2 2 2 yD þR þ ym 2
ð12Þ
are the intensity–angle dependence for the two image types. Within the linear slope range the contrast is substantially the same, but at large refraction angles, the curve of the refraction-like image falls down rapidly while the curve of the refraction image decays slowly. Therefore, it is evident that more radiation is refracted in a refraction image determining a high contrast. Although considering Eqs. (4) and (5) the behavior can be understood we would like to introduce and discuss the more general DEI formula: Eq. (6). According to Eq. (6) the expression for a refraction-like image may be written as ~I sub ¼ I 0 ð1 mt wtÞ R yD þ ym R yD þ ym . 2 2 (13) Because the small angle scattering noise due to rA ðyD =2 þ yÞ and rA ðyD =2 þ yÞ is partially superimposed to the image
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Fig. 4. Refraction image of a mouse paw obtained with the DEI technique (left). Refraction-like image of the same mouse paw (right).
Fig. 5. On the left comparison between experimental rocking curves: RðyD =2 þ ym Þ (solid line) RðyD =2 þ ym Þ (dashed line). The intensity–angle dependence for a refraction-like image (solid line) and for a refraction image (dashed line).
spot, its subtraction may lead to a very low level of the small angle scattering noise. Although the expressions associated to the low- and high-angle images and derived from Eq. (6) are different from those derived by Eq. (1) [5], the expressions of the refraction-like image derived from them are very similar. The expressions of the refraction image based on Eq. (6) is yD I~sub y~ ref ¼ 2 I~add ¼
R yD =2 þ ym R yD =2 þ ym yD . 2 RðyD =2 þ ym Þ þ RðyD =2 þ ym Þ þ ðwt=ð1 mt wtÞÞyD =oS
ð14Þ When jym jpyD =2, we have yD yD R þ ym þ R þ ym 1 2 2 and when jym j4yD =2, yD yD R þ ym þ R þ ym ¼ 1 Ro1 2 2
Now, according to Eq. (17), when the refraction angles fulfill the condition jym jpyD =2 the contrast of the refraction-like image is always higher than that of a refraction image. In addition, substituting Eq. (16) into Eq. (14), we have yD I~sub yD yD yD y~ ref ¼ R þ ym R þ ym ¼ 1 2 I~add 2 2 2 wt yD ð18Þ R 1 mt wt oS so that according to Eq. (18) because R4ðwt=ð1 mt wtÞÞyD =oS , when the refraction angles jym j4yD =2, the contrast of a refraction-like image is lower than that of a refraction image.
(15) 6. Conclusion (16)
where R is an increasing function of the refraction angle ym . Substituting Eq. (15) into (14), we obtain: yD I~sub y~ ref ¼ 2 I~add yD yD yD R þ ym R þ ym ¼ 1 2 2 2 wt yD . ð17Þ þ 1 mt wt oS
Nowadays, a comparison of DEI with other existing imaging methods indicates that DEI is a practical imaging technique with great possibilities and a really wide range of applications. In this contribution we present a couple of DEI experimental results and discuss the not negligible contribution of the small angle scattering processes to the image quality of this phase contrast imaging method. In addition, adding to the well-known DEI equation a new term that takes into account the small angle scattering processes we introduced a new formalism. The application of this new formalism confirms that, among the many possible images or
ARTICLE IN PRESS J. Wang et al. / Nuclear Instruments and Methods in Physics Research A 580 (2007) 803–807
combination of images, the top image always exhibits the higher contrast, certainly higher than that of the apparent absorption image, while the contrast of the refraction image remains higher than that of the refraction-like image. Acknowledgments This work has been supported by the States Key Project for Fundamental Research (2003CB716900) and the National Center for Nanoscience and Technology of China, the National Outstanding Youth Fund (Project No. 10125523 to Z.W.), the Key Important Project of the National Natural Science Foundation of China (Nos. 10490190, 10490194, 90206032), the National Natural Science Foundation of China (60477006 and 10504033) and the Knowledge Innovation Program of the Chinese Academy of Sciences (KJCX2-SW-N11, KJCX2-SW-H1202). A sincere thank is due to Augusto Marcelli for many fruitful discussions. References [1] T.J. Davis, D. Gao, T.E. Gureyev, A.W. Stevenson, S.W. Wilkins, Nature 373 (1995) 595.
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