Estimation of angular sensitivity for X-ray interferometers with multiple phase gratings
Jianwei Chen, Jiecheng Yang, Peiping Zhu, Ting Su, Huitao Zhang, Hairong Zheng, Dong Liang, Yongshuai Ge
aa r X i v : . [ phy s i c s . i n s - d e t ] J un Estimation of angular sensitivity for X-ray interferometers with multiple phasegratings
Jianwei Chen, Jiecheng Yang, Peiping Zhu,
2, 3
Ting Su, Huitao Zhang, HairongZheng, Dong Liang,
1, 5, 2, a) and Yongshuai Ge
1, 5, 2, b)1)
Research Center for Medical Artificial Intelligence, Shenzhen Institutes ofAdvanced Technology, Chinese Academy of Sciences, Shenzhen, Guangdong 518055,China. University of Chinese Academy of Sciences, Beijing, 100049,China. Institute of High Energy Physics, Chinese Academy of Sciences, Beijing, 100049,China. School of Mathematical Sciences, Capital Normal University, Beijing, 100048,China. Paul C Lauterbur Research Center for Biomedical Imaging,Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences,Shenzhen, Guangdong 518055, China. (Dated: 25 June 2020)
Recently, X-ray interferometers with more than one phase grating have been devel-oped for differential phase contrast (DPC) imaging. In this study, a novel frameworkis developed to predict such interferometers’ angular sensitivity responses (the mini-mum detectable refraction angle). Experiments are performed on the dual and triplephase grating interferometers, separately. Measurements show strong consistencywith the predicted sensitivity values. Using this new approach, the DPC imagingperformance of X-ray interferometers with multiple phase gratings can be furtheroptimized for future biomedical applications. a) Electronic mail: [email protected]. b) Electronic mail: [email protected].
1s one of the many X-ray phase contrast imaging techniques, the grating-based DPCimaging interferometer has attracted a lot of attention due to its high compatibility withthe conventional laboratory X-ray tube and flat panel detector assembly. Over the past twodecades, the so called Talbot and Talbot-Lau interferometers have been widely investi-gated, and some prototype biomedical imaging systems have already been developed. Forinstance, the lung imaging system , the mammography system , and so on. When designingthese systems, two critical factors need to be taken into account to obtain the best DPCimaging performance: one is the angular sensitivity of the interferometer, and the other isthe number of X-ray photons falling into the detector plane. The importance of angularsensitivity relies on the fact that it reflects the minimum detectable refraction angle of acertain interferometer system. Thus, its value is always expected to be high from the signaldetection prospective. Meanwhile, the latter factor determines the noise level of the DPCimage, and thus is always expected to be low. Aimed at achieving either super high sensitiv-ity or better X-ray radiation dose efficiency, recently, novel X-ray interferometers with morethan one phase grating have been investigate experimentally . By using the transparent(almost no absorption of X-rays) phase gratings, it is possible to improve the radiation doseefficiency by a factor of about two (compared with the Talbot-Lau system with an analyzergrating). However, so far it is still not very clear how to predict and optimize the DPC signalstrength (i.e., angular sensitivity) of such novel interferometers in theory yet, especially forthe ones containing three phase gratings.Mathematically, the angular sensitivity of an X-ray DPC imaging interferometer is definedvia the following equation: φ = S × ϕ = S × λ π ∂ Φ ∂x . (0.1)In Eq. (0.1), S denotes the angular sensitivity, λ denotes the X-ray beam wavelength, theterm ϕ = λ π ∂ Φ ∂x represents the tiny X-ray beam refraction induced by the object along x -axis,and φ corresponds to the measured DPC signal from the interferometer diffraction fringes.The work done by Yan et. al. has offered one possible solution to estimate the angularsensitivity of the dual phase grating interferometer system. In their analyses , it is assumedthat the final diffraction fringes are formed due to the beating effect between the fringesgenerated by the two phase gratings. Therefore, the sensitivity of the dual phase gratinginterferometer was estimated in an overlapping way, especially when estimating the sensi-2ivity behind the second phase grating. More specifically, the sensitivity behind the secondphase grating has two contributors: one comes from the first phase grating G1, and the othercomes from the second phase grating G2. Despite of its feasibility, it is obvious that suchestimation method for three phase grating system would become much more complicated.In addition, the sensitivity between the two phase grating is not explicitly provided in thatwork.The aim of this study is to establish a new general framework to estimate the sensitivityof X-ray interferometers which contain any number of phase gratings with an object beinginserted at any position between the source and the detector. The core of this new method isbased on our thin lens imaging theory developed for the dual phase grating interferometer .In the previous theory, the role of an X-ray phase grating was considered as a thin opticallens. As a result, the formed diffraction fringe with a large period in the dual phase gratinginterferometer can be explained as a magnified image of the source image (formed betweenthe two phase gratings), see Fig. 1(b). Essentially, the entire imaging procedure is separatedinto two cascaded stages: the first stage is related with the G1 phase grating imaging ofthe X-ray source; the second stage is related with the G2 phase grating imaging of thefringe image formed by the G1 phase grating. Obviously, our explanation to the dualphase grating DPC imaging procedure is different to the theory offered by Yan et. al .Therefore, it is necessary to provide a new angular sensitivity estimation framework for theX-ray interferometers having multiple phase gratings. In this study, we mainly focus ondiscussing the dual phase grating interferometer and the triple phase grating interferometer,see Fig. 1(b)-(c).Intuitively, the sensitivity of different grating interferometers, either the conventionalTalbot or Talbot-Lau interferometer, or the ones containing multiple phase gratings, shouldshare the similar expressions and interpretation mechanisms in physics. Motivated by thisidea, an X-ray optical module (XOM) is assumed to facilitate the following discussions. Bydefinition, an XOM contains three components: the illumination source, the phase grating,and the image of the source (diffraction fringe). Be aware that the illumination source couldbe either a real X-ray tube source or the diffraction fringe formed behind the phase grating.Thus, it is easy to see that the conventional Talbot-Lau interferometer only contains onesingle XOM, see Fig. 1(a). Moreover, there are two cascaded XOMs in the dual phasegrating interferometer system, see Fig. 1(b). Finally, there are three cascaded XOMs in a3 ! G " ! st imageSource (a) st imageG1 ! $ ! Source ! G2 $ % " % nd image(b) 1 st imageG1 ! $ ! Source ! G2 $ " & " rd image % nd image G3 $ % $ & (c) FIG. 1. Schemes of different X-ray interferometers with: (a) one phase grating, (b) two phasegratings, and (c) three phase gratings. The X-ray optical modules (XOM) are highlighted withcertain colors (color available online). The focal length f i of the i -th phase grating, and thecorresponding image formed after that grating are all depicted. An arrayed source is assumed, andno object is added. triple phase grating interferometer system, see Fig. 1(c).As illustrated in Fig. 2, assuming the object is positioned between the grating and thediffraction image of the source, the refraction of primary X-ray beam AD at the objectedge (point A) causes a certain diffraction fringe shift, denoted as ∆, from point D to pointC on the detector plane . According to the well-known phase stepping (PS) model , themeasured beam intensity can be expressed as follows I ( k ) = I + I × cos " π × x ps p ′ + φ obj + φ bkg . (0.2)In it, I is the mean intensity of the fringe, I is the fringe amplitude, φ obj corresponds to the4 ! " Grating Image
Source $ % & ’ & ( & ) *+ , FIG. 2. Illustration of the object-induced X-ray refraction with one single XOM. Dashed ray ADshows the primary beam which is refracted to beam AC at the object edge. The object-inducedrefraction angle is denoted as ϕ . On the fringe image plane, the corresponding diffraction fringeshift is denoted as ∆. object induced DPC signal, φ bkg corresponds to the reference DPC signal without object, x ps is the PS length of the phase grating along x axis, and p ′ = p/n ( n is equal to 2 for π phase grating, and is equal to 1 for 0 . π phase grating). According to Fig. 2, we have φ obj = 2 π × x g p ′ , (0.3)where x g denotes the equivalent PS length of the phase grating corresponding to the ∆fringe shift, and is equal to x g = d d d + d + d ϕ. (0.4)Substituting Eq. (0.4) into Eq. (0.3), and compare with Eq. (0.1), it is easy to demonstratethat the sensitivity S is equal to S = 2 π × d p ′′ . (0.5)Herein, the p ′′ represents the effective self-imaging period modified by the fan-beam effect, p ′′ = p ′ × d + d + d d , (0.6)The d corresponds to the source-to-grating distance, d corresponds to the grating-to-objectdistance, and d corresponds to the object-to-image distance. When the object is positionedbetween the source and the grating, the sensitivity can be determined according to theprinciple of reversibility. It is S = 2 π × d p ′′ × d d + d , (0.7)5nd p ′′ = p ′ × d + d + d d + d . (0.8)Herein, d is the source-to-object distance, d is the object-to-grating distance, and d is thegrating-to-fringe-image distance. When performing the above calculations, the correspond-ing distances should be altered accordingly, see Fig. 2.So far, the sensitivity S at any arbitrary position within an individual XOM can bepredicted in theory immediately, as long as the imaging geometry and grating specificationsare provided. Notice that this new theory does not need to know the period of the formeddiffraction fringe image. Because every XOM works independently, as a result, this developedtheory is suitable to perform sensitivity predictions for interferometers which having anynumber of XOMs (i.e., any number of phase gratings), as shown in Fig. 1(b)-(c). Becausethe sensitivity for single phase grating based Talbot-Lau interferometer system has beenwell studied , this work mainly focuses on investigating the sensitivity of the dual-phasegrating and the triple-phase grating interferometer systems.Experiments were performed on our benchtop to validate the developed sensitivity esti-mation theory, particularly for the dual-phase grating and the triple-phase grating interfer-ometer systems. The X-ray imaging systems included a micro-focus X-ray tube (L9421-02,Hamamatsu Photonics, Japan) with 7 . µ m focal spot size. The micro-focus tube wasoperated at a tube voltage of 40.00 kV and a tube current of 190 µ A. The X-ray CCDdetector (OnSemi KAI-16000, XIMEA GmbH, Germany) has a native element dimensionof 7.40 µ m and an effective imaging area of 36.00 mm × µ m, with a duty cycle of 0.50. These gratings generated π phase shiftsfor 17.00 keV X-ray photons. Their specifications were provided by the manufacturer (Mi-croworks GmbH, Germany). The G1 grating was moved laterally by seven times with astep length of 0.40 µ m. For each phase stepping position, the X-ray exposure period was300 seconds. A homogeneous PMMA rod a diameter of 2.46 mm was imaged at a couple ofdifferent positions during the experiments.With the acquired PS data, the standard signal retrieval method was first implementedto obtain the DPC images. Then, the extracted DCP images were 10 ×
10 rebinned withthe purpose to reduce signal noise. To determine the corresponding sensitivity at a certainposition, the DPC signal was numerically simulated by inserting the PMMA rod phantom on6 (68.0 cm ) G2 (70.7 cm ) Source (0.0 cm ) Detector (138.7 cm ) . (cid:215) ! st image (69.3 cm ) (cid:311) (cid:312) (cid:313) (cid:314)(cid:315) (cid:316) (cid:317) (cid:318) * * * * * ** " * Measured ! = 6.09 (cid:215) 10 $ = 0.9999 Measured ! = 1.85 (cid:215) 10 $ = 0.9997 " = 67.3 cm " = 69.7 cm Measured ! = 6.09 (cid:215) 10 $ = Measured ! = 3.08 (cid:215) 10 $ = 0.9996" = 71.4 cm = 104.7 cm (a) (b) (c) (d) (e) Theory * Experiment
FIG. 3. The sensitivity response of a dual phase grating interferometer system. (a) Comparisonresults of theoretical predictions and the experimental measurements at 8 different positions. TheDPC images, vertically averaged DPC signal profile (dotted line), and the theoretical DPC profile(solid line) for positions 2 (cid:13) , 5 (cid:13) - 7 (cid:13) , are shown in (b)-(e), correspondingly. The scale bar denotes5.0 mm. the light path with the assumption of a point X-ray source (see the supplementary materialfor numerical simulation details). The goodness-of-fit between the experimental data andthe numerical results was analyzed using the R method.The sensitivity plot in Fig. 3(a) shows the strong consistency between the theory and7 st image (72.0 cm ) G3 (88.8 cm ) Source (0.0 cm ) Detector (160.8 cm )1.85 (cid:215) 10 G2 (80.4 cm ) (cid:311) (cid:312) (cid:314) (cid:315) (cid:316) (cid:317)(cid:318) * * * ! * * * (cid:319) (cid:320) * Measured = ! " = 0.99 Measured $ ! " = 0.99 % = 74.8 cm = 77.7 cm Measured ! = 1.70 (cid:215) 10 " $ = ! = 6.15 (cid:215) 10 ! " = 0.9 cm cm (b) (c) (d) (e) (a) ** * (cid:313) Theory * Experiment ! G1 (72.0 cm ) nd image (72.0 cm ) FIG. 4. The sensitivity response of a triple phase grating interferometer system. (a) Comparisonresults of theoretical predictions and the experimental measurements at 10 different positions. TheDPC images, vertically averaged DPC signal profile (dotted line), and the theoretical DPC profile(solid line) for positions 3 (cid:13) - 5 (cid:13) , 7 (cid:13) , are shown in (b)-(e), correspondingly. The scale bar denotes5.0 mm. the experimental measurements for the dual phase grating interferometer. As expected, theentire sensitivity curve between the X-ray source and the detector has the “M” shape. Inparticular, there are two sensitivity peaks with a valley ( S = 0) between them. Interestingly,the peaks appear exactly at the the phase grating planes, and the sensitivity valley appears8ight at the image plane of the X-ray source after the first phase grating. In addition, thesensitivity plot in Fig. 4(a) also shows the good consistency between the predictions andthe experimental measurements for the triple phase grating interferometer. In this case, theentire sensitivity curve has three peaks and two valleys. Same as for the dual phase gratingsystem, the sensitivity reaches to the peak at the phase grating planes, and decreases tothe valley at the corresponding image planes. For our triple phase grating interferometersystem, unfortunately, we did not observe the first and third sensitivity peaks because thesensitivities at these two locations are two orders less than the middle peak.The PMMA rod DPC imaging results and the quantitative DPC signal profiles (both themeasurements and numerical fittings) at different object positions are shown in Fig. 3(b)-(e) and Fig. 4(b)-(e). Overall, the measured DPC signals match well with the theoreticalpredictions, including both the shape and the value. Several factors may contribute errorsto the numerical calculations, and thus leading to errors of the measured sensitivity S .For instance, the simulated X-ray beam spectrum, the estimated energy response of theinterferometer, and the finite focal spot size. In this study, we put all these potentialuncertainties together as measurement errors. Nevertheless, the measured sensitivity atdifferent positions show strong consistency with the theoretical predictions, demonstratingthe robustness of the developed theory for calculating interferometer’s sensitivity, especiallyfor the systems containing more than one phase gratings.When using the dual π phase grating system, it is recommended to put the object in frontof the first grating or put it behind the second grating to avoid the sensitivity valley locatedbetween the two gratings. This is especially important if the object has a relatively largedimension. However, when using the triple π phase grating system, it would be beneficialto put the object close to the second phase grating (either before it or behind it). For othertriple phase grating setups, it might be possible to put the object before the first grating orbehind the third grating (depending on the system design).Both the theory and experiments show that the triple phase grating interferometer hasbetter angular sensitivity performance, i.e., higher detected DPC signal values, than thedual phase grating interferometer, assuming the total system length and the phase gratingpitches are similar. In addition, reducing the phase grating periods can help to boost thesystem sensitivity. Compared with the conventional Talbot-Lau interferometer, however,the less flexible system configurations may still impede the wide applications of the dual9hase grating and triple phase grating interferometers. Therefore, careful system design andoptimization are required.For an individual XOM, we compared the theoretical sensitivity expressions with thepreviously published results , and confirmed the validity of the derived sensitivity resultsfor the assumed XOM of this work in a more general sense: First, our analyses are performedfor the defined XOM in this work, instead of the Talbot-Lau interferometer; Second, theX-ray source in an XOM could be the diffraction fringe formed by the phase grating, ratherthan a real X-ray tube source; Third, the last component in XOM could be the diffractionfringe image, instead of the analyzer grating.We have measured the sensitivity performance of the triple phase grating interferometerfrom experiments for the first time. Even though the imaging theory developed by Miao et. al. was employed to determine the system configurations, we realized that it mightbe possible to interpret such a special interferometer system by generalizing our previoustheory for the dual phase grating system. The main idea is illustrated in Fig. 1(c). Dueto the space limit, rigorous theoretical analyses will be presented in another work in future.In summary, this paper develops a novel framework to estimate the sensitivity of X-rayinterferometers with multiple phase gratings. Validation experiments that are performed oninterferometers with both two phase gratings and three phase gratings show good consistencybetween the predictions and the measurements. In future, this proposed approach cangreatly facilitate the optimization and design of the multiple phase grating interferometersystems to achieve the best biomedical X-ray DPC imaging performance.The authors would like to thank Dr. Zhicheng Li for providing the phase gratings.This project is supported by the National Natural Science Foundation of China (GrantNo. 11804356, 11535015, 61671311).The data that supports the findings of this study are available within the article [and itssupplementary material]. REFERENCES A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, H. Takai, and Y. Suzuki, “Demon-stration of x-ray Talbot interferometry,” Jpn. J. Appl. Phys., Part 2 , 866–8 (2003).10 T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Zeigler,“X-ray phase imaging with a grating interferometer,” Optics Express , 6296–304 (2005). F. Pfeiffer, T. Weitkamp, O. Bunk, and C. 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