Determination of Angle of Refraction in X-Ray Phase-Contrast Imaging Using Geometric Optics Method

Abstract

The accurate calculation of the angle of refraction of X-rays passing through an object is essential in X-ray phase-contrast imaging. While the wave optics-based method is commonly employed to calculate the angle of refraction, it presents several limitations. First, in cases where the object induces significant phase variations, the angle of refraction becomes divergent. Second, the method fails to adequately account for point-source illumination conditions, particularly the influence of the finite X-ray source size on the angle of refraction. In this study, we demonstrate that a geometric optics-based method can effectively simulate propagation-based X-ray phase-contrast imaging with a low-brilliance X-ray source and compute the angle of refraction more accurately than the wave optics-based method. Our studies reveal that the geometric optics-based method can robustly determine the angle of refraction, even under conditions of substantial phase variations within the object. Furthermore, we show that reducing both the X-ray source size and the detector pixel size increases the angle of refraction in both simulations and experiments. Additionally, our results highlight that the angle of refraction is not invariant. Instead, it increases with the system’s total length and as the object moves closer to the light source. For systems with a Fresnel number of N ≥ 1, our method exhibits full compatibility with wave optics methods and can be extended to grating-based X-ray interferometry. The approach offers a robust alternative for calculating the angle of refraction under diverse imaging conditions.

1. Introduction

X-ray phase-contrast imaging (XPCI) is a powerful technique that enables the detection of small angles of refraction of X-rays passing through an object [1,2,3,4,5]. For weakly absorbing objects, XPCI provides significantly enhanced contrast compared to conventional absorption imaging methods [6,7]. Therefore, XPCI has emerged as a critical tool in diverse applications, including cartilage detection [8,9,10], material inspection [11,12,13] and various other scientific and industrial fields [14,15,16]. Currently, five primary methods are employed to achieve XPCI: crystal interferometry [17], propagation-based X-ray phase-contrast imaging (pbXPCI) [18,19,20,21], diffraction-enhanced imaging (DEI) [22], grating-based X-ray interferometry (gbXI) [23,24,25] and edge illumination (EI) [26,27]. Among these, gbXI and pbXPCI have garnered significant attention due to their unique advantages and potential applications. Grating-based X-ray interferometry can quantitatively retrieve the absorption, phase-contrast and scattering information of an object. However, its imaging energy is limited by the absorption grating, and its field of view is limited by the size of the analyzer grating [28,29]. Propagation-based X-ray phase-contrast imaging provides high-resolution images. Nevertheless, a significant challenge lies in the difficulty to completely separate phase information from absorption information [30,31].

In X-ray phase-contrast imaging, utilizing the wave optics-based method to calculate the angle of refraction when X-rays pass through an object is a widely used method and often the only one available. For the case of plane-wave illumination and when the phase of the object changes gradually, the calculated angle of refraction is consistent with experimental results. However, as Wilkins pointed out in the literature [18], in some special cases, such as the position where the phase of the object changes significantly, the angle of refraction calculated by the wave optics-based method is divergent. Moreover, this method assumes that the incident X-rays are plane waves. For the case of point-source illumination, calculating the angle of refraction will be challenging. Furthermore, the impact of X-ray source focal spot size on the angle of refraction is frequently disregarded. As is well known, the angle of refraction changes with the angle of incidence. Therefore, when using an extended X-ray source for illumination, the X-rays emitted from different positions of the source result in different angles of refraction. When calculating the angle of refraction, the weighted average of the detected angles of refraction within a pixel should be considered, with the weighting factors depending on the number of photons and the spectrum. It can be seen that both the X-ray source focal spot size and the detector pixel size will impact the angle of refraction. Another interesting aspect is that in different X-ray phase-contrast imaging systems, the interactions between objects and X-rays are described by different theories. For example, when X-rays pass through an object, it will be diffracted in the propagation-based X-ray phase-contrast imaging system [15], whereas X-rays will be refracted in the grating-based X-ray interferometry [5]. If a unified theory could be developed to describe the interaction between X-rays and objects in both systems, it would advance the development of X-ray phase-contrast imaging.

2. Theory

In X-ray phase-contrast imaging, only a few studies have been conducted using synchrotrons or free-electron lasers. Most research relies on more readily available X-ray sources, such as X-ray tubes. In ordinary laboratories, phase-contrast imaging typically employs low-brightness X-ray tubes and detectors with pixel sizes exceeding ten micrometers, with imaging distances usually less than 3 m. For hard X-rays with wavelengths less than 0.1 nm, the diffraction effect is extremely weak. In this study, we focus on cases with the Fresnel number of N ≥ 1, a condition commonly encountered in ordinary laboratories. Under these conditions, it is feasible to apply a geometric optics-based method, specifically Snell’s law, to describe the interaction between X-rays and objects [32].

As illustrated in Figure 1, an X-ray point source S ($x_0$, $y_0$) is located on the negative half of the X-axis, and X-rays illuminate a sphere with a radius of r centered at the origin of the Cartesian coordinates. A ray emitted by the X-ray point source is incident on point A ($x_1$, $y_1$) on the sphere. The angle between the line connecting point S and point A and the X-axis is denoted as $a_0$.

$$a_0=-\arctan \left({\displaystyle \frac{y_1-y_0}{-sqrt\left(r^2-y_1^2\right)-x_0}}\right)$$

(1)

The angle between the tangent to the sphere at point A and the X-axis is denoted as ${\alpha }_1$.

$${\alpha }_1=-\arctan \left({\displaystyle \frac{\sqrt{r^2-y_1^2}}{y_1}}\right)$$

(2)

The angle of incidence $a_1$ can be calculated by the following formula:

$$a_1=90°-{\alpha }_1+a_0$$

(3)

Then, the angle of refraction $a_2$ of the X-ray inside the sphere can be calculated by using Snell’s law, that is

$$a_2=\arcsin \left({\displaystyle \frac{n_1}{n_2}}\sin a_1\right)$$

(4)

where $n_1$ and $n_2$ are the refractive index of the air and the sphere, respectively. For hard X-rays, the refractive index of the air $n_1$ is approximately equal to 1. Consequently, the difference between the refractive index of the object and that of air is of the order of 10⁻⁶. As the angle of refraction $a_2$ will be slightly larger than the angle of incidence $a_1$, the difference between the two angles is denoted as follows:

$$a_{21}=a_2-a_1$$

(5)

After the X-ray beam is transmitted inside the sphere, it reaches the exit point B. The slope of the line between A and B is given by

$$k_2=\tan \left(180°-a_{21}-a_0\right)$$

(6)

The intercept of the line between point A and point B is

$$b_2=y_1-k_2{x}_1$$

(7)

At this time, the abscissa of the exit point B can be calculated by the equation for line AB and the equation of the sphere:

$$x_2={\displaystyle \frac{-k_2{b}_2+\sqrt{k_2^2{r}^2+r^2-b_2^2}}{k_2^2+1}}$$

(8)

Then, the tangent slope $k_3$ at point B can be calculated by using the abscissa $x_2$ of the exit point B and the equation of the sphere:

$$k_3={\displaystyle \frac{x_2}{\sqrt{r^2-x_2^2}}}$$

(9)

The angle ${\alpha }_2$ between the tangent and the X-axis can be calculated by using the tangent slope $k_3$:

$${\alpha }_2=\arctan k_3$$

(10)

According to the positions of points A and B, as well as the direction of the incident X-rays, the angle of incidence $a_3$ of X-ray at point B can be obtained:

$$a_3={90}^0-{\alpha }_2-{\alpha }_1-a_0$$

(11)

Once again, the angle of refraction $a_4$ can be calculated by using Snell’s law:

$$a_4=\arcsin \left({\displaystyle \frac{n_2}{n_1}}\sin a_3\right)$$

(12)

According to the geometric relationship between the angles in Figure 1, the angle of refraction $a_5$ of the X-ray passing through the sphere can be calculated by the following:

$$a_5=a_3-a_4+a_{21}$$

(13)

The angle $a_6$ between the X-ray emitted from point B and the X-axis is

$$a_6=a_5+a_0$$

(14)

The slope $k_4$ of the X-ray emitted from point B is

$$k_4=\tan ({180}^0-a_6)$$

(15)

The intercept of the X-ray emitted from point B is

$$b_3=y_2-k_4{x}_2$$

(16)

where $y_2$ is the ordinate of the exit point B. The ordinate $y_5$ of the intersection of the exit X-ray and the detector D at ($x_z$, 0) can be calculated by the following:

$$y_5=k_4{x}_z+b_3$$

(17)

As a comparison, the formula for calculating the angle of refraction ${\alpha }_f$ by the wave optics-based method is as follows [18]:

$${\alpha }_f={\displaystyle \frac{2\Delta \delta y}{\sqrt{r^2-y^2}}}$$

(18)

where $\Delta \delta$ is the refractive index difference between the inside and outside of the sphere.

3. Results

3.1. Comparison of Intensity Distributions

Figure 2a is the X-ray intensity distributions of a sphere simulated by Fresnel–Kirchhoff diffraction [33].

$$f\left(y_5;x_z\right)={\left({\displaystyle \frac{i}{\lambda x_z}}\right)}^{1/2}\exp \left(-ik{x}_z\right){\displaystyle \int q\left(y\right)}\exp \left({\displaystyle \frac{-ik{\left(y_5-y\right)}^2}{2x_z}}\right)dy$$

(19)

where $\lambda$, $k$ and $q(\mathrm{y})$ are the wavelength, the wave number and the complex transmission, respectively. Figure 2b shows the simulation result obtained using Snell’s law. Simulation and numerical calculations were performed using Matlab R2021b. The object used in the simulation was a PMMA (polymethyl methacrylate) sphere with a diameter of 3 mm. The distance between the X-ray source and the detector was 1.5 m, while the distance between the X-ray source and the object was 0.5 m. The incident X-ray was a monochromatic plane wave with an energy of 28 keV, and the detector pixel size was 20 μm. Figure 2c illustrates the cross-section profiles denoted by the dark line in Figure 2a, and Figure 2d is the zoomed-in view at the ellipse indicated in Figure 2c. It can be seen that the intensity distribution of the sphere obtained by Snell’s law also exhibits the edge enhancement effect [32]. This is because the angle of refraction near the edge of the sphere is very large, causing a lot of photons to be deflected outside the sphere’s edge. This results in a decrease in intensity inside the edge of the sphere, making the intensity outside the edge higher than the background intensity. The results obtained by the wave optics-based method differ slightly from those obtained by the geometric optics-based method, mainly due to the unavoidable diffraction effect present in the experiment. Except for the slight differences at the edges of the sphere, the two intensity distributions almost completely overlap.

3.2. Calculation of Angles of Refraction and Object Contrast

Figure 3 illustrates the angles of refraction of X-rays passing through different positions of a sphere under plane-wave illumination, calculated using Snell’s law (Formula (13)) and the wave optics-based method (Formula (18)). The incident X-ray was a monochromatic plane wave with an energy of 28 keV, and the object-to-detector distance was 1 m. The blue and red curves represent the angles of refraction calculated by Snell’s law and the wave optics-based method, respectively. In cases where the calculated region is not very close to the sphere’s edge (submicron scale), the two methods yield nearly identical results, with an average error of only 1.3 × 10⁻¹¹ rad, causing the curves to overlap almost completely (Figure 3a). However, near the edge of the sphere (nanoscale), the wave optics-based method produces divergent results, while Snell’s law remains finite (Figure 3b). Therefore, we deduce that the angle of refraction can be more efficiently calculated using Snell’s law if compared with existing techniques.

From Figure 3, it can be seen that for the case of plane-wave illumination, using Formula (18) to calculate the angle of refraction is straightforward and accurate in the regions with gradual phase changes. In the calculation of the angle of refraction, achieving nanometer-level accuracy, as illustrated in Figure 3b, is not necessary. Figure 3b employs an exaggerated example to highlight one of Formula (18)’s limitations.

From the blue curve in Figure 3b, it can be seen that although the angle of refraction calculated by Snell’s law is finite at position A, the maximum value is about 1.5 mrad, which is three orders of magnitude higher than the actual measurement results. Given that the number of photons refracted to position A is relatively small, their contribution to the overall intensity distribution is negligible, making it difficult to detect the maximum angle of refraction experimentally. On the contrary, at the position with the highest intensity at the edge of the object, a large number of photons can be easily detected. Therefore, the angle of refraction at this position is defined as the maximum detectable angle of refraction in this study. In the subsequent discussion on the effect of system parameters on the angle of refraction, the maximum detectable angle of refraction is considered instead of the maximum angle of refraction.

In pbXPCI, as the angle of refraction cannot be measured directly, alternative methods should be considered, such as the edge enhancement effect. The object contrast is used to describe the edge enhancement effect of the pbXPCI, and can be quantified by

$$contrast={\displaystyle \frac{I_{\max }-I_{bg}}{I_{\max }+I_{bg}}}$$

(20)

where $I_{\max }$ is the maximum intensity at the object’s edge and $I_{bg}$ is the background intensity without the object. For a given object and spectrum, a higher contrast indicates a more pronounced edge enhancement effect. Larger angles of refraction result in more photons being refracted outside the object’s edge, enhancing the edge effect. Figure 4a shows the impact of the system length on object contrast under different detector pixel sizes. The X-ray tube voltage was 40 kVp with a focal spot size of 7 μm. The spectrum was simulated by Spektr [34] (Figure 4d) with a 1 keV energy sampling interval. The X-ray tube anode and window materials were tungsten and beryllium, respectively, with no additional filters used. In Figure 4b,c, the object contrast as a function of the system length is shown under different X-ray tube focal spot sizes and spectra. The detector pixel size was 50 μm. The simulation used a 3 mm diameter PMMA sphere positioned 0.75 m from the source.

The simulation results demonstrate that the object contrast decreases with increasing X-ray source focal spot size and detector pixel size, consistent with findings from simulations and experiments [14,19]. Under monochromatic X-ray illumination, the object contrast is higher than under polychromatic illumination, although the difference is not significant. Figure 4 further validates the feasibility of using Snell’s law to simulate the intensity distribution in pbXPCI.

3.3. Factors Affecting the Angles of Refraction

Due to the inability of the wave optics-based method to account for the influence of object and detector positions on the angles of refraction, it is commonly assumed that these angles are independent of such factors. However, calculations using Snell’s law reveal that these factors are, in fact, non-negligible. Figure 5 illustrates the impact of the system length and the distance between the object and the X-ray tube on the angles of refraction obtained using Snell’s law. The simulation used a 3 mm diameter PMMA sphere, with an X-ray tube voltage of 40 kVp and a focal spot size of 7 μm. The X-ray tube anode and window materials were tungsten and beryllium, respectively, and the detector pixel size was 50 μm. Figure 5a indicates that the angles of refraction increase with the increasing system length, while Figure 5b indicates that these angles decrease as the distance between the X-ray source and the object increases.

Figure 6a shows the impact of the X-ray source focal spot size on the angle of refraction, while Figure 6b shows the impact of the distance between the object and the X-ray source on the angle of refraction. The blue and red curves represent the cases where detector pixel sizes are 75 μm and 9 μm, respectively. The simulation used a 5 mm diameter PMMA sphere, with a distance of 1.0269 m between the X-ray source and the detector. The X-ray tube voltage was 50 kVp, and the spectrum used in the simulation is shown in Figure 6c. The X-ray tube anode and window materials were tungsten and beryllium, with no additional filter used. The results indicate that a smaller focal spot size and smaller detector pixel size lead to a larger angle of refraction. This is because the X-ray tube source can be thought of as an array of a large number of point sources, each emitting X-rays that propagate in various directions after being refracted by an object, creating a distribution of the angles of refraction on the detector plane. The propagation directions of X-rays of different wavelengths emitted by the same point source also vary after refraction, producing a distribution of angles of refraction on the detector plane, as shown in Figure 6d. The angle of refraction detected within one detector pixel is the weighted average of these different angles of refraction, where the weighted values are determined by the intensity and spectrum of the point source. Consequently, a smaller X-ray source focal spot size results in a closer distribution of the angle of refraction, leading to a larger detected angle of refraction. Similarly, a smaller detector pixel size will produce a larger detected angle of refraction. As the equivalent pixel size in the object plane increases with increasing distance between the object and the X-ray source, the angle of refraction decreases as this distance increases.

3.4. Experimental Results

Next, we will verify the effectiveness of the theory through experiments. The object was a PMMA rod with a diameter of 3 mm. The X-ray tube voltage was 40 kVp, the focal spot size was 7 μm and the detector pixel size was 50 μm. The X-ray tube anode and window materials were tungsten and beryllium, respectively, with no additional filter used.

Figure 7a displays the experimental result of the PMMA rod in pbXPCI. Figure 7b illustrates a comparison of the cross-sections from the experiment (blue curve), Fresnel diffraction simulation (red curve) and Snell’s law simulation (green curve). In Figure 7b, the maximum error is 1.6% for Snell’s law simulation and 1.0% for Fresnel diffraction simulation compared to experimental results. The largest error occurs at the middle position of the PMMA rod due to the pronounced beam hardening effect associated with its maximum thickness. Although the intensity distribution obtained using Snell’s law is less accurate than that obtained using Fresnel diffraction, its error remains relatively low. It is important to note that the X-ray intensity distribution at the object’s edge in Figure 2c exhibits a peak and a valley, whereas the distribution in Figure 7b only displays a peak. This discrepancy can be attributed to the smaller pixel size used in Figure 2.

Figure 8 illustrates the quantitative comparison of Snell’s law simulations and the experiment results. Figure 8a shows how object contrast changes with the distance between the object and the X-ray source, while Figure 8b shows how object contrast varies with the system length. The simulation using Snell’s law matches the experimental results well, except at positions where contrasts are very low. Overall, Snell’s law can successfully calculate the intensity distribution of the object and the angle of refraction in pbXPCI.

The quantitative influence of the X-ray source focal spot size and detector pixel size on the angle of refraction needs to be verified using grating-based X-ray interferometry. The Talbot–Lau interferometer consists of a source grating G0, a phase grating G1 and an analyzer absorption grating G2. They had periods of $p_0$ = 14.82 μm, $p_1$ = 7.25 μm and $p_2$ = 4.8 μm, respectively, as shown in Figure 9a. The distances between X-ray tube S₂ and G0, G0 and G1, G1 and G2 and G2 and the detector D were 10.6 cm, 65.03 cm, 21.06 cm and 6 cm, respectively. Due to the maximum power of the micro-focus X-ray source being only 8 w, in order to reduce the exposure time, the source grating G0 was removed when using the micro-focus X-ray source, as shown in Figure 9b. The object used in the experiment was a PMMA rod with a diameter of 5 mm, placed 5 cm in front of the G1. There were four types of experimental settings, and the different experimental parameters are shown in

Table 1. Part of experiment parameters in gbXI.

	X-Ray Tube Focal Spot Size (μm)	Detector Pixel Size (μm)	Current (μA)	Exposure Time (s)
Setting 1	7	75	160	11
Setting 2	400	75	4000	1
Setting 3	7	9	160	120
Setting 4	400	9	7000	15

. The experiments were carried out using an X-ray tube (HPX-160-11, Varex Imaging Corporation, Salt Lake City, UT, USA) and a micro-focus X-ray tube (L9421-02, Hamamatsu Photonics K.K., Hamamatsu City, Japan) operated at 50 kVp. The raw images were recorded from six different positions of the phase grating, with two images recorded at each position. The detectors used in the experiments were a flat-panel detector (Dexela 2329, PerkinElmer Inc., Waltham, MA, USA) and an sCMOS detector (16MP_52, Photonic Science Limited, St. Leonards, UK).

Figure 10 presents the phase-contrast images of the PMMA rod and their corresponding cross-sections obtained under different experimental settings. Specifically, experimental setting 3 employed the smallest X-ray source focal spot size and detector pixel size, resulting in the most pronounced phase-contrast signal. This outcome is intuitively demonstrated by Figure 10d and the blue curves in Figure 10f.

Table 2. Quantitative comparison of angles of refraction between different settings.

	Measured Angle of Refraction (Rad)	Angle of Refraction Calculated by Geometric Optics (Rad)	Angle of Refraction Calculated by Wave Optics (Rad)
Setting 1	2.796 × 10−6	5.668 × 10−6	7.648 × 10−6
Setting 2	2.150 × 10−6	3.766 × 10−6	7.648 × 10−6
Setting 3	6.043 × 10−6	10.838 × 10−6	22.065 × 10−6
Setting 4	2.754 × 10−6	4.033 × 10−6	22.065 × 10−6

shows a quantitative comparison of the angles of refraction obtained under different experimental settings. The second column displays the experimental results obtained using gbXI, while the third and fourth columns represent the angle of refraction calculated using the geometric optics-based method and wave optics-based method, respectively. It is evident that the angles of refraction calculated using the wave optics method are consistently larger than those determined by the geometric optics method. This discrepancy arises because the wave optics method assumes that parallel X-rays are incident on the object, whereas the geometric optics method considers the scenario where cone beams are incident. Due to the fact that parallel X-ray illumination can be treated as analogous to light from an ideal point source at infinity, the angle of refraction under parallel illumination is greater than that under cone beam X-ray illumination. Additionally, the wave optics method typically ignores the influence of the X-ray tube focal spot size, despite experimental evidence showing a correlation between the two. The geometric optics method overestimates the angle of refraction because it neglects the impact of gratings, such as phase gratings that alter the propagation direction of X-rays via diffraction effects. However, when the X-ray source focal spot size and the detector pixel size change significantly, the geometric optics method accurately predicts the variation in the angle of refraction, closely matching the experimental results. For example, in setting 3 and setting 2, the measured angles of refraction are 6.043 × 10⁻⁶ radians and 2.150 × 10⁻⁶ radians, respectively, with a ratio of 2.8017. The theoretical calculation for these settings yields angles of 10.838 × 10⁻⁶ radians and 3.766 × 10⁻⁶ radians, respectively, with a ratio of 2.8779. The close agreement between these ratios demonstrates that the geometric optics method effectively captures the variation trend of the angle of refraction under specific conditions, thereby validating its applicability in related scenarios.

4. Discussion

Similarly to the Monte Carlo method, geometric optical approaches also face challenges in deriving analytical expressions for intensity distribution and angles of refraction. Consequently, it becomes difficult to intuitively visualize the impact of parameters such as wavelength, distance, X-ray source focal spot size and detector pixel dimensions on these optical characteristics. Nevertheless, numerical computation results remain effective for quantifying the influence of these factors on angles of refraction. Our research findings demonstrate that in pbXPCI, when the diffraction effects of X-rays are minimal, X-rays undergo refraction as they pass through an object, similar to that observed in gbXI. Calculations using Snell’s law reveal that the angle of refraction is influenced by both the X-ray source focal spot size and the detector pixel size. This challenges the conventional understanding of angular sensitivity in gbXI, which has long been considered independent of the X-ray source focal spot size and detector pixel size [35,36]. Currently, grating-based X-ray phase-contrast imaging faces a major issue: weak phase-contrast signals that make it hard to distinguish objects with low absorption. To address this, enhancing angular sensitivity is a common approach to improve detection. However, our results reveal another method: decreasing the X-ray source focal spot and detector pixel sizes also strengthens the system’s detection ability. For example, in setting 3 and setting 4, the smallest refraction angles are 9.34 × 10⁻⁷ rad and 1.86 × 10⁻⁷ rad, respectively. This indicates that setting 4 has higher angular sensitivity. Setting 3 uses an X-ray source with a smaller focal spot size. Although this raises noise levels and reduces the angular sensitivity, it delivers a larger angle of refraction and better image resolution. Setting 3 yields stronger phase-contrast signals for the same object, showing better detection. Hence, evaluating the system’s detection ability only by angular sensitivity is incomplete. The impact of the X-ray source focal spot and detector pixel sizes on signal strength must also be considered. Interestingly, the measured angles of refraction are consistently smaller than those calculated using the geometric optics-based method in gbXI. This discrepancy is due to the presence of gratings, which appear to reduce the angles of refraction. However, the exact mechanism by which the gratings influence the angle of refraction remains unclear. This indicates that there is still significant room for improvement in the phase-contrast signal in gbXI. Further investigation into this phenomenon is expected to significantly enhance the imaging performance of gbXI and provide deeper insights into the underlying physical processes.

5. Conclusions

In this study, we demonstrate the feasibility of simulating propagation-based X-ray phase-contrast imaging and calculating the angle of refraction using the geometric optics-based method. For the case of plane-wave illumination, the angles of refraction calculated by the geometric optics method and by the wave optics method are almost the same when the object’s phase changes slowly. However, in regions where the phase changes rapidly, geometric optics provides more accurate and reasonable results. The edge enhancement effect of an object is due to the strong refraction of X-rays at the edge of the object, resulting in the redistribution of X-ray photons. Meanwhile, the simulation results using geometric optics demonstrated that a smaller X-ray source focal spot size and detector pixel size lead to a larger angle of refraction. The angle of refraction also changes with the object’s position and the system’s length. Although the angle of refraction calculated using the geometric optics method is larger than the measured value in gbXI, it still provides a new perspective for investigating the influence of gratings on the angle of refraction.