SCIENCE CHINA Technological Sciences

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*Corresponding author (email: Huasi_hu@mail.xjtu.edu.cn)

• Article • July 2015 Vol.58 No.7: 1289–1293

doi: 10.1007/s11431-015-5807-2

Optimal design of coded aperture used for Z-Pinch driven fusion neutron imaging

JIA QingGang1, HU HuaSi1*, ZHENG XiangYang2, ZHANG FengNa1, WANG DongMing1, HU Guang1, ZHANG TianKui3 & WU YueLei2

1 Institute of Nuclear Science and Technology, Xi’an Jiaotong University, Xi’an 710049, China; 2 State Environmental Protection Administration (SEPA), Nuclear and Radiation Safety Centre, Beijing 100082, China;

3 Research Center of Laser Fusion, China Academy of Engineering Physics, Mianyang 621900, China

Received December 3, 2014; accepted March 23, 2015; published online May 4, 2015

Currently, the neutron yield of Z-Pinch is lower than that of laser driven fusion. In the neutron imaging for this facility, the signal to noise ratio (SNR) has a significant influence on the expected spatial resolution of the reconstructed fusion core, espe-cially in the condition of low neutron yield. In this paper, mathematical model is purposed to describe the dependence of aper-ture parameters on the imaging SNR. The investigation shows that the imaging SNR is closely related to the size of contrast boundary on the point spread function. According to this, a novel non-uniform redundancy penumbra apertures array is de-signed. In addition, the imaging performances of this novel coded aperture, penumbra aperture and ring aperture are evaluated and compared by Monte Carlo method. The comparison shows that this novel aperture has significant advantage compared to the penumbra aperture which is commonly used for neutron imaging with low yield. The encouraging results can provide ref-erence for the optimal design of the coded aperture used in the neutron imaging for Z-pinch driven fusion with low neutron yield.

Z-pinch driven fusion, neutron imaging, coded aperture, signal to noise ratio, low neutron yield

Citation: Jia Q G, Hu H S, Zheng X Y, et al. Optimal design of coded aperture used for Z-Pinch driven fusion neutron imaging. Sci China Tech Sci, 2015, 58: 12891293, doi: 10.1007/s11431-015-5807-2

1 Introduction

The project of using Z-Pinch facility to drive the inertial confinement fusion (ICF) has been carried out in China. The neutron imaging as the diagnosis of implosions is very im-portant for obtaining the information about the fusion core during the reaction [1–5]. Currently, to the best of our knowledge, the related reports [6–8] of the neutron imaging for Z-pinch facility are of one-dimensional. Therefore, the relative research should refer to the methods provided by the neutron imaging in the laser-driven ICF experiment.

The primary differences of these two diagnostics are the scale of the field of view (FOV) and the neutron yield. Moreover, the fusion core of Z facility, which is in larger scale, owns lower distribution gradient. For Z-pinch facility, the expected fusion area is 1 cm in diameter, and neutron yield is less than 1012. Therefore, the aperture, suitable for imaging of the large FOV 0.5 cm in diameter and low neu-tron yield 1010–1012, is very important. To obtain high im-aging signal to noise ratio (SNR), penumbra aperture is adopted in the laser driven fusion neutron imaging, and the imaging performance of penumbra aperture is given as well. Furthermore, based on Hadamard transformation, uniform penumbra apertures array is proposed which significantly increases the imaging SNR [9,10] demonstrated by X-ray

1290 Jia Q G, et al. Sci China Tech Sci July (2015) Vol.58 No.7

imaging experiment. While the isoplanaticity of point spread function (PSF) in neutron imaging is not discussed and imaging performance for neutron is not given either.

The main motive of this paper is to develop the SNR model not only for penumbra imaging condition but also suitable for other coded imaging. Based on linear algebra, a new SNR model is proposed which has one simple indicator. In addition, a novel non-uniform compact apertures array is developed to improve the neutron imaging SNR. The opti-mal design of this new apertures array involves the trade off between the PSF isoplanaticity and imaging contrast and the inner radius of each aperture. The simulation given by MCNP shows that this new aperture array significantly im-proves the imaging performance compared to the optimized penumbra aperture.

2 Signal to noise ratio (SNR) model based on linear algebra

In order to investigate the noise in imaging, the equation set presenting the imaging process is given by

SI SJ

1 1

Image( , ) PSF( , , , ) Source( , ),i j

m n i j m n i j

(1)

where Image(1,1) and Image(m,n) stands for the neutron flux in pixel marked (m,n); Source(i,j) is the value of neu-tron flux released from source cell marked as (i,j). PSF(i,j,m,n) is the value of point spread function in Im-age(m,n) for the response of an imaging system to Source (i,j).

It is well known that if the number of independent linear equations is larger than the amount of source variables, the equations can be resolved. While, the reality is that in the coded image, a considerable part of equations is linear cor-relation. Taking the penumbra imaging as an example, the system simulated by MCNP is indicated in Figure 1. In the simulation, 0.5 mm diameter and 10 cm scintillation fiber is employed to build a 100 cm2 square array as the neutron

position sensitive detector, of which the detection efficiency is about 30%.

The fusion neutron source shape is assumed as a plane ‘E’ character. Figure 2 shows the image of the artificial neutron source simulated by MCNP.

The neutron flux distribution in Figure 2 can be roughly divided into three parts, the umbra, the penumbra and back-ground area. The neutron flux value of umbra is the largest (normalized to one), because the neutron released from source area can arrive at the umbra without being attenuated by the aperture. The neutron flux value of the background area is which means the value of the remaining neutron flux when neutron penetrates the same thickness of aperture material. In penumbra area, the neutron flux value is in the range from to one, indicating that part of the neutron is attenuated by the aperture. Therefore, if equation set (1) is applied to describe the penumbra imaging, the equations also can be classified into three kinds according to the three parts of the area in the coded image. The first kind of equa-tions is in the umbra area. Coefficients, namely the PSF, of the equations are the same, equaling one, and all the equa-tions of this kind are linear correlation. The second kind corresponds to background area, the coefficients (PSF) are equal to . The equations are linear correlation as well. The third kind of equations stands for the penumbra area, and their coefficients are in the range between and one. From the view of solution of equation set, namely coded image reconstruction, only the equations that are linearly inde-pendent contain information of the source area and contrib-ute to the coded image reconstruction. It means that only part of the scintillation fiber array plays the role in detecting the useful neutron from source.

To improve the imaging process, the third kind of equa-tions should be increased. The number of equations which are linearly independent is doubled, it behaves as the scin-tillation fiber array obtains the coded image of the source area twice. So a useful simple model for SNR in the coded imaging is described as follows:

SNR CA , (2)

Figure 1 Schematic of the imaging system platform [11].

Jia Q G, et al. Sci China Tech Sci July (2015) Vol.58 No.7 1291

Figure 2 (Color online) The simulated coded image of plane ‘E’ character.

where CA means contrast area, it presents the region of the coded image in which the neutron flux is above the back-ground and below the maximum. In penumbra imaging condition, CA is the scale of penumbra area.

Here, enlarging the CA portion is equal to increasing the portion of the third kind of equations, this can be achieved by creating more holes in the aperture or enlarging the inner radius of the aperture. In the previous report [10], the uni-form penumbra apertures array was adopted for this purpose. While this method may encounter two problems, one is the limitation of the size of scintillation fiber array and the ex-perimental setup; the other is the guarantee of the isopla-naticity of PSF.

3 Optimal design of non-uniform compact pe-numbra apertures array

The optimal design of the coded aperture involves not only imaging SNR, but also a tradeoff between the indictors of imaging performances such as isoplanaticity of PSF. The isoplanaticity of PSF is the prior issue to be considered. Based on the pervious study [11], the aperture thickness is set as 6 cm to make the balance between the imaging SNR and isoplanaticity of PSF. For single aperture, the isopla-naticity of PSF is closely related to the aspect ratio (thick-ness over radius) of aperture. However, this dependence is not precise, because the curve of inner radius varies with aperture depth. If the aperture array is used, besides the in-ner radius and aspect ratio, the arrangement of each aperture has function on the isoplanaticity of PSF. For example, in the view of the aperture with large offset, namely the aper-ture is located at the edge of arrangement matrix, which describes the arrangement of apertures, the source area is shifted from the central area. Therefore, the isoplanaticity design of apertures array is more complex than that of sin-gle aperture.

One thing should be noticed is that if the point source in

the expected source area is eligible for image reconstruction; this apertures array owns higher tolerance against the prob-lem that fusion core is not located in the central of the ex-pected source area (misalignment). Therefore, the arrange-ment of each aperture should be compact, and the inner ra-dius of each aperture should be carefully designed accord-ing to size of apertures and the arrangement of apertures. Although, increasing the number of apertures improves CA and imaging SNR, the aperture number is limited by the size of scintillation fiber array and experiments setup. It means that the CA cannot be improved significantly pro-vided that only the number of aperture is increased without changing the aperture radius. The aperture with small radius is useful to create more CA when there is no space to dig a large radius aperture. In aspect of isoplanaticity design, the inverse radius curvature (IRC) design method [12] is ap-plied for each aperture. Originally this method is proposed to design the penumbra aperture for the laser driven fusion neutron imaging, and can be described as

d d

dd

3 222

20 0

1 ,z

z R zz

(3)

where R0 is the radius of curvature of the aperture at the front face to source area; z0 equals to the distance between aperture front face and source area; z presents the distance from the point of curvature of aperture to source in the z direction; is the radius of aperture curvature at z position.

Here, the R0 as boundary condition can be obtained by the size of the expected source area. Figure 3 shows the PSF of a simple conical aperture and the IRC aperture. These results reveal that the isoplanaticity of the IRC aperture is better than that of conical aperture

Attempts show that non-uniform apertures can be located compactly. One sample of the designed non-uniform aper-tures array is selected which consists of one on-axis mini penumbra and six nearby off-set penumbra apertures, as shown in Figure 4. As the comparison, the result of ring and IRC penumbra aperture is given in Figure 4 as well.

Figure 3 (Color online) Comparison of the offset PSFs obtained by IRC aperture (upper row) and cones aperture (bottom row).

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Figure 4 Comparison of PSF and ‘E’ coded image given by non uniform apertures array ((a), (b)), ring aperture ((c), (d)), and IRC aperture ((e), (f)). (a) PSF; (b) Coded image; (c) PSF; (d) coded image; (e) PSF; (f) coded image.

On the basis of PSF and the coded image, the artifical source area whose shape is ‘E’ character can be reconstructed. To avoid the influence of reconstruction algorithm on the restored image, here, two kinds of algorithm, Wiener filtering and Genetic algorithm, are used, and the reconstructed images can be found in Figure 5. From the view of similarity bewteen the reconstructed image and the source image, the non-uniform apertures owns the best imaging performance, and the IRC penumbra is the worst. In addition, the restored image obtianed by the non-uniform apertures has more smooth and regular source boundray than that of other two apertures. One thing should be noticed that the ring aperture whose outter radius is the same with the inner radius of the IRC penumbra possesses better imaging performance than the penumbra aperture. It means that the imaging SNR increases with CA, and it has no direct relation with neutron flux that passed through the aperture. One proposal may be approved that the non-uniform penumbra apertures in the aperture array should be replaced by ring aperture for higher SNR, while the ring aperture especially the ring apertures array is hard to construct and utilize as designed and expected. For the purpose of evaluating the spatial resolution of the reconstructed source images which are given by different apertures, the artifical line-pair patterns with varies spatial frequences are used as the shape of source area. IRC penumbra aperture and non-uniform apertures array are applied to image the line-pair source, and the reconstructed source image given by R-L algorithm can be found in Fig-ure 6. It is clear that the non-uniform apertures array owns

Figure 5 The reconstructed image obtained by Wiener filtering, see upper row, and by Genetic algorithm bottom row (bracket gives the similarity values between the reconstructed image and the source image).

Jia Q G, et al. Sci China Tech Sci July (2015) Vol.58 No.7 1293

Figure 6 (Color online) The reconstructed images using IRC penumbra aperture ((a)–(c)) and non-uniform apertures array ((d)–(f)); the subcaptions show the spatial frequency /line-pairs·mm1 and neutron yield per mm2. (a) 1.7, 1010; (b) 2.5, 5×109; (c) 2.5, 1.1×1011; (d) 1.7, 1010; (e) 2.5, 5×109; (f) 2.5, 1.1×1011.

better imaing performance. For example, when the neutron yield density is 1010 per mm2, the reconstructed line-pair source using non-uniform apertures array which spatial frequency is 1.7 lp/mm (line-pairs per millimetre) can be separated, while it is hard to distinguish the same line-pair source if IRC penumbra aperture is applied.

Admittedly, the viewer affects whether the reconstructed line-pairs can be separated, but clearly, the spatial resolution of the reconstructed image given by non-uniform apertures array is higher than that by the IRC penumbra aperture. For briefness, the reconstructed line-pairs of different spatial frequencies are not listed and the rough relationship bewteen neutron yield density (per mm2) and spatial resolution of the obtained source are given in Figure 7. Because the recons- tructed image is sensitive to the R-L iterations, the best one is selected when the similarity value reaches the maximum. Due to the influence from reconstruction algorithm, modu-lation transfer function method is not used, another criterion

Figure 7 Rough relationship between neutron yield and spatial resolution of the reconstructed source area, when non-uniform apertures array and IRC penumbra aperture are applied, respectively.

is that if the gray level of dip between line-pairs in the re-constructed image roughly equals to the gray level of back-ground, the line-pairs can be separated, and the correspond-ing spatial resolution can be obtained.

In Figure 7, it can be found that a non-uniform aperture array owns better imaging performance. When the neutron yield density is as low as 6×109 mm2, which corresponds to a 100 mm2 size core area with 6×1011 neutron, 1.7 lp/mm resolution can be achieved using non-uniform aperture array.

4 Conclusion

In this paper, one sample model to describe the imaging SNR is established based on linear algebra. In addition, based on this model, a novel non-uniform apertures array is proposed which owns favorable imaging performance. For example, to obtain the similar spatial resolution, the opti-mized IRC penumbra aperture requires a higher neutron yield, which is roughly over 5 times that required by non-uniform apertures array. The encouraging result can contribute to the initial research of Z-Pinch in China.

This work was supported by the National Natural Science Foundation of China (Grant No. 10975113), and Innovative Research Team in University of Ministry of Education of China (Nuclear Energy Science and Engineer-ing) (Grant No. IRT1280).

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