† Corresponding author. E-mail:

Project supported by the Fundamental Research Funds for the Central Universities, China (Grant Nos. 2012LZD07 and 2014kJJCA03) and the National Natural Science Foundation of China (Grant Nos. 11375027 and 11075017).

A geometrical description of square polycapillary x-ray optics and the basic theory of the transmission of x-rays are presented. A method of numerical calculation is developed based on ray-tracing theory. The method simulates the intensity distribution of x-rays propagating through slice square polycapillary x-ray optics. The simulation results are compared with the experimental results.

Slice square polycapillary x-ray optics has been studied for more than a decade^{[1–4]} and has attracted increasing attention recently.^{[5,6]} Square pores arrange regularly in the slice square polycapillary x-ray optics. In flat slices, rays undergoing reflections at grazing incidence from two orthogonal walls will have both their direction cosines reversed, which enables point-to-point focusing of a source, which means that the source distance and the image distance are equal, i.e., *f*_{1} = *f*_{2}. As to the slumped slices, this kind of optic can be used to convert the diverging beam from a point-like source at its focus into a quasi-parallel output beam, or conversely, converge the parallel beam. Usually, this kind of optics is used in the area of x-ray imaging, so its focal spot size is not taken into consideration in this paper. If there are no channel misalignments happening during the manufacturing process, the shape of the focal spot is a cross with no defect. So the shape of the focal spot is considered as a criterion to evaluate the quality of the arrangement of the channels in the slice square polycapillary x-ray optics.

In this paper, a simulation model is presented, a method is presented to establish a geometrical model of the slice square polycapillary x-ray optics. Simulation results based on this model are compared with the experimental results, and the influence of the channel misalignments is also demonstrated.

The test optic is flat slice square polycapillary x-ray optics, which has a large 20 mm×20 mm planar format, comprised of square-packed channel multifibers. The channel side-length is 20 μm, and the mean thickness of the channel wall is 3.91 μm. X-rays are generated by a Cu target x-ray tube with a 25-kV source voltage and its focal spot size is 50 μm.

As the slice square polycapillary x-ray optics in this experiment is flat, it will focus x-rays in a point-to-point manner. It is thus mounted in a two-axis carriage at the midpoint of the beam line as shown in Fig. *f*_{1} = *f*_{2} = 500 mm.

An image (Fig.

One kind of misalignment is called “tilting”, some channel layers tilt to a small angle in their manufacturing process (see Fig.

An x-ray is a kind of electromagnetic wave which has relatively high energy; it can be reflected from the surface of a material if its glancing incidence angle is less than a value that is called the critical angle *θ*_{c}. Because the energy loss of this type of reflection is very small, the reflection can be regarded approximately as total reflection.

The critical angle is a function of photon energy and material itself; usually, the critical angle for total reflection of x-rays can be written as^{[7]}

*E*is the photon energy (in unit keV).

_{k}Assume that and are unit vectors in the directions of the incident and reflected beams, respectively, and is the unit vector normal to the inner wall of channel. As shown in Fig. *θ* = sin^{–1}(* n* ×

*). If, the x-ray is totally reflected, and the reflected vector can be expressed as = –2sin*r

*θ*. Otherwise, the x-ray will be absorbed by the inner wall.

If the tangents to the edges of all the channels point to a fixed point on the central axis of the slice, then the optic can focus rays from infinity (see Fig. *f _{n}*(

*z*) =

*a*

_{n}z^{2}+

*b*+

_{n}z*c*, and this kind of optics is called “slumped slice”. If the center line of each channel is parallel to the center line of the slice (see Fig.

_{n}*a*= 0,

_{n}*b*= 0, and this kind of optics is termed “flat slice”. Each channel in the slice consists of four walls. To realize the ray-tracing process, equations to express these walls must be established.

_{n}Cross-section of the slice square polycapillary x-ray optics can be seen in Fig. *i* = 0, *j* = 0), there are in total *n* layers in the square lens. Assume that the external wall of the outermost layer of channels can be fit by the function *f*(*z*) = *a*_{0}*z*^{2} + *b*_{0}*z* + *c*_{0}. It is not difficult to show that for the channel in row *j* and column *i*, the external walls can be expressed by four equations. The top wall is expressed as

The thickness of the channel wall cannot be ignored; therefore we also impose the condition that the ratio between internal and external side lengths of the channel is fixed to be 0.8 (see Fig.

As the projection of the vector normal to the upper wall on the *y* axis is positive, it can be deduced that the unit vector normal to the upper wall can be given as

*m*= 0,

_{x}*m*= 1, and

_{y}*m*= –2

_{z}*a*–

_{u}z*b*.

_{u}For the lower wall, the projection of its normal vector on the *y* axis is negative; it is easy to show that *m _{x}* = 0,

*m*= –1, and

_{y}*m*= 2

_{z}*a*+

_{u}z*b*.

_{u}Accordingly, for the left-side wall, we have *m _{x}* = –1,

*m*= 0, and

_{y}*m*= 2

_{z}*a*+

_{u}z*b*.

_{u}For the right-side wall, we have *m _{x}* = 1,

*m*= 0, and

_{y}*m*= –2

_{z}*a*–

_{u}z*b*.

_{u}Let *p*_{1} be a point on the source whose coordinates are (*x*_{s},*y*_{s},*z*_{s}) and *p*_{2} be a point on the channel entrance whose coordinates are (*x _{p}*,

*y*,

_{p}*z*) (see Fig.

_{p}*d*

_{0}is the distance between the two points, which is given by

Another important part in the mathematical model is to find the intersection point of the beam with the channel inner wall. Assuming that the starting point of an incident ray is (*x*_{s}, *y*_{s}, *z*_{s}), and the end point is (*x*_{c}, *y*_{c}, *z*_{c}), it is easy to see that the end point is the intersection point. Then, the line of the incident ray is given by

*z*

_{s}should be considered to be reasonable, and among them, the minimal root is the one corresponding to the intersection point.

As for the channels in other quadrants, the method of rotating the coordinate axis can be used to establish their geometric model and study their transmission characteristics. Once the point (*x*_{s}, *y*_{s}) is chosen, rotate the coordinate axis and transform the channels into the first quadrant (see Fig. *x*_{s}, *y*_{s}). After the iteration process, rotate the coordinate axis back to determine the coordinates of the emergent beam across the imaging plane.

We choose *N* points on the source at random. We then use a ray-tracing method to find the trajectories of rays and determine whether a ray could emerge from the exit. To find the intensity distribution of the emergent beam, the coordinates of beams crossing the imaging plane need to be known. Let the unit vector in the direction of the emergent light be * un*, the last total reflection point be (

*x*

_{sn},

*y*

_{sn},

*z*

_{sn}), and the distance between the exit plane and the imaging plane be

*f*, then the coordinates of the beam across the imaging plane will be

In the manufacturing process, misalignment is inevitable, so in this model misalignment must be taken into account. As to the tilting error, the method of rotating the coordinate axis is used again. Once the point (*x*_{s}, *y*_{s}) is chosen, rotate the coordinate axis until the center line of the channel is parallel to the *z* axis, then calculate the new coordinates of this point (see Fig. *x*_{s}, *y*_{s}) is chosen, shift the coordinate to correct the dislocation (see Fig.

To demonstrate the capability of the model to simulate the beam transmitting in a slice square polycapillary x-ray optics, the simulation results of the intensity distribution of flat slices are shown in Fig. *p*_{1} is (*x*_{s}, *y*_{s}, *z*_{s}) constantly) and x-ray energy *E* = 8.05 keV are used. The parameters of the slice square polycapillary x-ray optics are *L* = 2 mm, *Φ* = 20 mm, *f*_{1} = 500 mm, *f*_{2} = 500 mm, and the number of layers *n* = 500. In Fig.

A broken cross is shown in Fig.

To show the influence of the tilting error on the defect of the light spot, another two simulations are performed, in these two simulations the parameters are as follows: *E* = 8.05 keV, *L* = 2 mm, *Φ* = 20 mm, *f*_{1} = 500 mm, *f*_{2} = 500 mm, and the number of layers *n* = 500. The relationship between the tilting angle and the line defect is shown in Table

With ray-tracing methods and numerical calculations, intensity distributions of x-rays through a slice square polycapillary x-ray optics are simulated. The simulation results show that the real reason which causes the defect in the light spot is the tilting error not the dislocation error. These simulation results can be used for designing and manufacturing slice square polycapillary x-ray optics.

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