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Abstract
Moiré fringe method in X-ray grating interferometry is characterized by its advantage to obtain multi-contrast data through single-frame imaging. However, how the visibility of the Moiré fringe is influenced by the system parameters, such as the misalignment angle, still lacks investigation, although it closely relates to the signal-to-noise ratio of the image data. In this paper, a simplified model of the Moiré fringe visibility is presented, which reveals that the Moiré fringe visibility can be expressed as the product of a misalignment-angle-related “sinc” function and a relatively independent factor. The following experimental results further suggest that the crosstalk between the detector pixels in the direction perpendicular to the Moiré fringe orientation is another main cause for the visibility reduction of the Moiré fringes.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Talbot-Lau grating interferometry is a newly developed multi-contrast x-ray imaging technique, which, in addition to the conventional absorption contrast image, can acquire two extra physical contrast images [1–4]. These so-called differential phase contrast image and the dark field contrast image can significantly increase the image contrast of low absorption samples, such as biological tissues and industrial polymer materials [5–7].
The generation and detection of the reference fringe pattern is essential to the contrast separation of the imaging data in Talbot-Lau grating interferometry. However, the reference fringe patterns generated by the Talbot-Lau effect of x-ray gratings have a period of several micrometers and usually cannot be directly observed by the detector [8]. Therefore, additional methods have been developed for this detection of the reference fringe pattern. Among them, the two mainstream ones are phase stepping method [2] and Moiré fringe method [3,9].
The phase stepping method requires a series of images with an absorption analyzer grating stepping to several specific positions within one or several periods of the reference fringe pattern (phase-stepping). This method usually has higher visibility, and the spatial resolution of the contrast separation results is independent with the contrast separation algorithm. However, the accuracy of this mechanical stepping has a direct impact on the imaging quality [10], which not only places high requirements on the position adjustment capability, but also limits the imaging speed.
The second one, namely the Moiré fringe method, detects the small-scale reference fringe pattern indirectly by observing the detector-visible Moiré fringe, which is generated by a slight misalignment between the reference fringe pattern and the analyzer grating. One of the most common ways to manually generate Moiré fringe is rotating the analyzer grating slightly in the grating plane on the basis of ideal alignment, which can continuously control the period of the Moiré fringe by adjusting the small rotation angle. In actual applications, this small angle could be generally limited between the ratio of the grating period to the side length of the FOV and the ratio of the grating period to the detector spatial resolution limit.
Compared with the phase stepping method, the Moiré fringe method sacrifices a certain signal to noise ratio, for the visibility of the Moiré fringe is generally less than that of the stepping curves and will decrease as the angle increases. But the Moiré fringe method does not require precision mechanical motions of the analyzer grating in the order of micrometers, and can operate in several modes with great application potential: single-frame imaging mode [11–14], scanning imaging mode [15–18], trochoidal x-ray vector radiography [19] and helical X-ray phase-contrast computed tomography [20].
The visibility of the Moiré fringe and the visibility of the phase-stepping curve are both important parameters for their direct relationship to the signal-to-noise ratio of the imaging results [21]. The visibility of the phase-stepping curve has been investigated for its intuitive connection to the reference fringe pattern [22,23]. However, besides a few studies of the generation mechanism [24], the properties of the Moiré fringe for multi-contrast imaging purpose still lacks detailed formulation and analysis, leading to difficulties in the design and evaluation of such a device.
This article deals with the Moiré fringe generated by rotating the analyzer grating. A detailed formulation of the intensity profile of Moiré fringe is implemented, and a quantitative model of the Moiré fringe visibility is therefore obtained. After that, experimental results are presented and discussed. It is found from the experimental data that the crosstalk of the detector pixels in the direction perpendicular to the orientation of the fringes is another main reason for the reduction of the Moiré fringe visibility. Finally, a convenient method is proposed in discussion for estimating the visibility degradation of the Moiré fringes.
2. Analytical investigation
2.1 Physical modeling the Moiré effect in grating interferometry
A Talbot-Lau grating interferometer consists of an x-ray source, a source grating G0, a phase grating G1, an analyzer grating G2 and an x-ray detector (Fig. 1). The grating planes are parallel to each other, and the grating lines have the same or nearly the same orientation according to the detailed imaging method used. The direction of the x coordinate is identified as the periodic direction in the grating plane, and the direction perpendicular to it is the y coordinate direction. The z coordinate direction is perpendicular to the grating plane.
The source grating G0 is an x-ray absorption grating, which can split the radiation from common x-ray sources into a series of slit sources to increase the lateral coherence of x-rays. Phase grating G1 causes specific phase shift differences between the two parts of x-ray that passes and not through the grating grid lines. When x-ray with sufficient lateral coherence passes through the phase grating, there are intensity fringes similar to the phase grating appearing at a series of positions downstream of the phase grating (Talbot effect), and these intensity fringes are called G1 self-imaging patterns or Talbot fringes. There are slight misalignments between the G1 self-imaging patterns generated by each slit sources, for these sub sources are spatially dislocated for a source grating period. The analyzer grating is placed at the position behind the phase grating where the dislocation between the G1 self-images of two adjacent slit sources is exactly one analyzer grating period, so that their intensity can be superimposed (Lau effect). In Moiré fringe method, the G2 grating and this superimposed pattern (the reference fringe pattern) are in the same plane with a small angle between them in the z direction, which generates detector-visible intensity distribution behind the G2 grating.
From the above principle analysis, it can be seen that the reference fringe pattern, the G2 grating and the detector are three core factors that generate Moiré fringe. The individual modeling approximation about them are as follows:
- a) The reference fringe pattern is supposed to be uniformly distributed in one direction and a sinusoidal distribution in the other direction. There have been comprehensive studies on the properties of reference fringe pattern, and here we choose to directly cite as the basis of this study [8,9,22,25].
- b) The G2 grating is approximated as a slice under paraxial conditions, the transmittance function of which is uniform in one direction and rectangular in the other direction.
- c) The detector is assumed to be grid-shaped with no crosstalk between the grids, and the response of each pixel is uniform within the grid, in order to get a general conclusion. This assumption differs from the actual detector. The discussion and revision of this assumption will be further analyzed in the experimental part and discussion part later.
- d) The period of the reference fringe is assumed to be much smaller than the pixel size of the detector. This can usually be met for grating interferometers with conventional geometry or high working x-ray energy point.
Additional explanations are needed for the small angle that lead to the Moiré effect. There are actually two independent angles between these three factors: the angle between the reference fringe pattern and the G2 grating, which is one of the core parameters for this study; the angle between the detector and the G2 grating (or the reference fringe pattern), which slightly affect the x-ray collection area (intensity integration area).
The angle between the detector and the G2 grating is different but equivalent in the experiment part (Fig. 1) and modeling part (Fig. 2) in this study. Only the G2 grating is rotated in experiment, for the adjustment simplification and the experiment repeatability. As a result, the orientation of the detector in the experiment is the same as that of the reference fringe pattern (Fig. 1). Meanwhile, the detector is assumed to have the same orientation as the G2 grating in modeling for calculation simplification (Fig. 2), which means that the detector and G2 grating are rotated synchronously.
Fig. 2. Moiré fringe generation and relative positions (${\varphi _G}$, ${\varphi _D}$) in modeling.
There are two factors that guarantee the equivalence of these two choices of the angle between the detector and the G2 grating. The most important factor is that the detector orientation is a secondary factor relative to the detector size and response, which is confirmed by experiment. Theoretically, this small angle has a negligible impact on the integration area. Secondly, from the perspective of sampling the Moiré fringe, the results of sampling under the two orientation are nearly consistent. Since the orientation of the Moiré fringe is on the angle bisector of the two orientations theoretically, which, in other words, the angles between the two orientations and the Moiré orientation are the same.
Two phase parameters (${\varphi _G}$: G2 grating - reference fringe pattern;${\varphi _D}$: detector - reference fringe pattern) are used to characterize the relative position of the three factors, for they are all assumed to be periodic functions of an infinite region.
2.2 Derivation of intensity distribution function of Moiré fringes
Firstly, it is assumed here that the reference fringe pattern ${I_S}({x,y} )$ is uniformly distributed in the y-direction and a sinusoidal distribution of a period ${p_2}$ in the x-direction [25].
$A$ represents the amplitude and k describes the background intensity (DC component). The reference fringe pattern is rotated by the angle of $\theta$ to generate the misalignment angle with the G2 grating, which can be expressed asAn ideal model for the G2 grating is employed, the transmittance function of which is uniform in the y direction and rectangular in the x direction (Fig. 3). This ideal grating has a period of ${p_2}$, a duty cycle of $\gamma$ and grid line transmittance of $\tau$. A phase parameter ${\varphi _G}$ is used to characterize the relative position of the reference fringe pattern and the G2 grating, given that both of them are assumed to be infinite planes.
The size of the detector pixel is assumed to be over an order of magnitude larger than the period of the analyzer grating (Fig. 2), which can usually be met for interferometers applicable to the Moiré fringe method. Then, the intensity distribution of the moiré fringes (discretized by detector pixels) can be calculated through the integration of the intensity of self-imaging pattern after blocking by the G2 grating within a pixel size range.
Another phase parameter ${\varphi _D}$ is used to characterize the relative position of intensity integration area and the G2 grating. The intensity integration area is assumed to be rectangular (${D_x} \times {D_y}$) and has the same orientation as the G2 grating.
According to the description above, the x-ray intensity distribution after G2 grating can be calculated as
When the side length of the integration region in the x direction is much larger than the period of the G2 grating (${D_x} \gg {p_2}$), this integration in x-direction can be reduced to the product of $({{{{D_x}} / {{p_2}}}} )$ and the integral result within one analyzer grating period, as
An expression of ${I_M}({\theta ,\varphi ,n} )$ in ‘sine’ form is given after substituting ${I_S}({x,y} )$ into ${I_M}({\theta ,\varphi ,n} )$ as
The phase-shift variable $\Phi ({\varphi ,n} )$ is composed of two parts. One part is determined by the relative positions, which not only implies the phase-stepping method but also reminds us that the phase-stepping method can also be used for experimental Moiré fringe study. The other part is determined by the position of the integration area in the y-direction when misalignment angle does not equal to zero, which tells us that there is a periodic signal in the y-direction. This additive relationship of these two parts implies that the property of the Moiré fringe pattern will be affected by the change of the reference fringe pattern, making it possible to indirectly detect the latter through the observation of the former.
$R(\theta )$ contains all the sub functions of ${A_M}(\theta )$ that are relevant to $\theta$. Considering that $R(\theta )$ has multiple parameters and the detailed expressions are tedious, its analysis is one of the major difficulties and highlights in this study.
2.3 Derivation and analysis of Moiré fringe visibility
The visibility of the Moiré fringe is given by the ratio of the amplitude ${A_M}(\theta )$ and the DC component ${k_M}(\theta )$, which can be written as the product of $V(\textrm{0} )$ and $R(\theta )$, thereby separating the effect of the misalignment angle $\theta$ from the rest factors.
$R(\theta )$ approximately equals to ${f_\textrm{1}}({\theta ,{Q_y}} )$ in the small angle range (see Appendix). The ${f_\textrm{1}}({\theta ,{Q_y}} )$ is an ‘sinc’-like $\theta$-related function which is greatly affected by a compound parameter ${Q_y}$ defined by the quotient of the y-direction size of the integration area ${D_y}$ and the period of the G2 grating ${p_2}$. In general, ${f_\textrm{1}}({\theta ,{Q_y}} )$ is a symmetric function about x=0, and it has two fixed points, that are $({\theta ,{f_\textrm{1}}({\theta ,{Q_y}} )} )\textrm{ = }({0,1} )$ and $({\theta ,{f_\textrm{1}}({\theta ,{Q_y}} )} )\textrm{ = }({\textrm{9}{\textrm{0}^ \circ },0} )$. The larger ${Q_y}$ takes, the faster ${f_\textrm{1}}({\theta ,{Q_y}} )$ decreases.
When $\theta$ is small, the zero positions of ${f_\textrm{1}}({\theta ,{Q_y}} )$ can be given by $\arcsin ({{n / {{Q_y}}}} )\approx {n / {{Q_y}}}$ where $n$ is an integer, which physically means that the y-direction size of integration area is equal to an integer of multiple Moiré fringe period. The sign of ${f_\textrm{1}}({\theta ,{Q_y}} )$ is related to $\varphi$. So the absolute value of ${f_\textrm{1}}({\theta ,{Q_y}} )$ (Fig. 4) is supposed to be more suitable for subsequent experimental research, since $\varphi$ also changes when adjusting $\theta$ mechanically.
As for $V(\textrm{0} )$, it takes the similar expression with the visibility of the intensity curve derived and explained in other studies [22]. It reaches the maximum value ${\textrm{2} / \pi }$ when the G2 grating transmittance is 0 in the case where the duty cycle takes 0.5.
Now, we can see that $R(\theta )$ actually represents the reduction of the Moiré fringe visibility relative to its optimal value $V(\textrm{0} )$ as the misalignment angle $\theta$ increases. So, the maximum visibility can be taken as the approximate value of $V(\textrm{0} )$ in experiment, for the ${f_\textrm{1}}({\theta ,{Q_y}} )$ changes relatively slowly around $\theta \textrm{ = 0}$.
3. Experiment
3.1 Setup and phase stepping strategy
All the data in this publication are acquired with a Talbot-Lau interferometer. The setup is described in Tables 1 and 2. The x-ray source is a tungsten target x-ray tube with a focal spot size of 400µm. The detector uses a flat-panel detector with a 127µm pixel size, CsI screen x-ray converter and MTF > 48% @ 1 lp/mm. The area of the G2 grating is 50×50mm2. All the rotation dimensions of the three gratings are separately controlled and adjusted by electronically controlled rotary stages to ensure the accuracy and stability of the experiment. The rotation dimension of the G2 grating in the grating plane is controlled by a precise motorized goniometer with repeatability of 0.0043° (7.50E-5 rad).
All visibility data are obtained by a 7-steps phase-stepping method with an 2s exposure time for each step. Two x-ray tube voltages (45kVp and 60kVp) are selected to verify the relative independence of $R(\theta )$ to the grid line transmittance $\tau$.
Table 1. Distances between gratings.
Table 2. Parameters of the gratings.
Slightly different from the modeled assumptions, the bars of the gratings are generally not simple straight lines, but have periodic support structures and gaps across the lines (Fig. 5). The analyzer grating G2 used in this experiment has a gap with a period of 9.82 µm.
There are two considerations that make this non-ideality-introduced experimental error within acceptable limits. On one hand, the ratio of gap to lines is relatively low, less than 6%. And it could have been taken into account to a certain extent by the grid line transmittance in the model. On the other hand, the orientations of the gap-characterized periodic structure and the model-adopted periodic structure are perpendicular to each other. In other words, the gap-characterized periodic structure and the reference fringe are almost vertical during the experiment, which means there is nearly no moiré effect between them.
3.2 Experimental extraction of the Moiré fringe visibility data
The theoretically derived Moiré fringe intensity distribution tells us that the local Moiré fringe visibility data can be obtained by phase stepping method. However, there are still two other problems to be solved in the experimental extraction of visibility data: one is the experimental results of which detector pixels are reasonable to be collected as research samples; the other is how to determine the ‘zero-angle’ state.
Selection of sampling pixels is worth studying, because of the grating defects and the coupling of angle adjustment and grating displacement. Some grating defects can be directly observed from the images of $\theta \approx \textrm{0}$, which should be avoided when selecting sampling pixels. The coupling of angle adjustment and grating displacement means the adjustment of $\theta$ will slightly change the relative position between G2 grating and others. In this experiment, the rotation center of the adjustment device is below the imaging area.
The center point of G2 grating is chosen as a reference basis of the sampling pixels, which is determined by an image recognition program and will slightly move with the adjustment of $\theta$. Additionally, the intensity distributions of the G1 self-images have all been homogenized at the detector pixel scale using the image captured before adding the G2 grating into the optical path as a reference.
The identification of the ‘zero-angle’ state is critical for the experiment, because it will determine the basis for calculating the angle between the G1 self-imaging pattern and the G2 grating. In theory, there are two methods for judgment after other system parameters being ideally adjusted. One is directly observing the distribution of moiré fringes in the field of view. When the field of view tends to be uniform (the period of the moiré fringes tends to infinity), the grating interferometer tends to be at the ‘zero-angle’ state. The other basis for judgment is that, according to the results of the aforementioned theoretical derivation, when the grating interferometer is in a zero-angle state, its moiré fringes visibility reaches the maximum.
The uniform field of view is chosen for the ‘zero-angle’ state identification in experiment for convenience. However, images at $\theta \approx \textrm{0}$ in Fig. 6 are not ideally uniform, which is difficult to further improve through the adjustment of this one rotation dimension alone. Nevertheless, since the central area has intensity changes of less than one period in the y direction, this deviation should be very small (less than $({{{{p_2}} / {5.0mm}}} )= 6E - 4\textrm{ }rad$). From the curve results of $R(\theta )$ in subsequent experiments, it can be seen that the accuracy of this judgment method is acceptable.
Fig. 6. Moiré fringe patterns and the stepping curves of the center point obtained in experiments (45kVp).
The left part of Fig. 6 shows Moiré fringe images at two different misalignment angles and two stepping positions. It can be seen directly that the Moiré fringe in the field of view becomes denser as the angle increases, and the step position will periodically change the intensity of certain pixel without changing the Moiré fringe density.
The right part shows the stepping data of the pixel at the center point of G2 grating projection on detector. Reliable visibility data are extracted after fitting these data by a least squares fitting program in the form of a sine function. Parameter ‘Ratio’ refers to the $R(\theta )$ in formula (16), which clearly shows the decrease in visibility as the angle increases.
3.3 Experimental results and pixel merging
There are two series of data presented in this section. The first series only extracted the data of one pixel under two different tube voltages, verifying the relative independence of $R(\theta )$. The deviation between the experimental results and the theoretical expectations is then discussed, which is mainly attributed to severe crosstalk between adjacent single pixels in experiment. The latter series uses a method of merging several pixels in one dimension to bridge the deviation between the experimental conditions and the theoretical model.
The visibility data were extracted with misalignment angle $\theta$ changing from −0.04° to 1.2° and the x-ray tube voltage switching between 45kVp and 60kVp. The visibility reduction ratio $R(\theta )$ of single pixel are shown on Fig. 7. The function curve of ${f_\textrm{1}}({\theta ,{Q_y}} )$ is also drawn with ${Q_y}$ of 42.3 calculated by the ratio of pixel size 127µm and period of the G2 grating 3.0µm. The change of x-ray tube voltage seems to have no effect on the $R(\theta )$. Meanwhile, the maximum visibility $V(\textrm{0} )$ actually decreases from 0.21 to 0.13 as the voltage increased from 45kVp to 60kVp. This confirms the theoretical prediction that $R(\theta )$ and G2 grating transmittance parameters $\tau$ are relatively independent at small angle range.
Fig. 7. Theoretical prediction curve and experimental results of the reduction ratio of fringe visibility $R(\theta )$.
The simplified assumption of the detector response during modeling is regarded to be the main reason for the deviation between the experiment and theory. The actual x-ray intensity collection area corresponding to each detector pixel should be much larger than the pixel area. There is possibly a severe underestimation of the parameter ${Q_y}$ when using the pixel size directly as the characteristic length of the integration area (Fig. 8(a)).
Fig. 8. Different suspicious deviations generated by the crosstalk between pixels used to sample Moiré fringes. Deviation type (a) can only be reduced through 2-D merging. While deviation type (b) can be reduced through 1-D merging in a certain direction. The single orientation of the Moiré fringe indicates that type (b) is closer to reality which is confirmed by the following experimental results.
Pixel merging is used to reduce the effect of the non-uniform detector response on the experimental results. It is worth mentioning that only adjacent pixels in the y direction can achieve the purpose of reducing errors according to the derivation results (Fig. 8).
The experimental results are indeed getting closer to the theoretical curves with merging more pixels in the y direction (Fig. 9). Most of the experimental measurement results are below the theoretical curve, likely due to the method of pixel merging is a method of monotonic approximation from one direction. The experimental data is slightly skewed to the left of the theoretical curve when a large number of pixels are merged, which may reflect a directional system deviation of the ‘zero-angle’ state we discussed before. This could be conversely used to adjust the phase-stepping working state with higher accuracy.
Fig. 9. Functional relationship and experimental results between the reduction ratio of fringe visibility and misalignment angle after merging pixels.
The actual visibility reduction corresponds to a theoretical reduction to 0.5 is evaluated to quantitatively explore the law that the degree of agreement between theoretical predictions and experimental results is affected by the pixel merging (Fig. 10). This value increases rapidly as the number of merged pixels increases, and gradually stabilizes at about 0.45. The interpolation algorithm when estimating the actual visibility reduction and the non-ideal ‘zero-angle’ state may be the two major factors that caused the failure to approach 0.5.
4. Discussions
4.1 Method for estimating Moiré fringe visibility degradation
In order to enable efficient design and operation, a method to evaluate the deterioration degree of the Moiré fringe visibility caused by the misalignment angle is provided here.
From the expression of the phase-shift variable $\Phi ({\varphi ,n} )$, it can be deduced that the pixel number required to sample one period of the Moiré fringe is
The following table gives the $M$ values in the theoretical prediction and experimental tests in the case of specific ${R^ \ast }(M )$ values. The ratio between the two is given as a parameter ${\eta _{exp }}$ to estimate the severity of the y-direction crosstalk of the adjacent pixels. The value of ${\eta _{exp }}$ is usually 4∼7 in Table 3.
Table 3. The number of Moiré fringe sampling pixels required to achieve a certain decrease of visibility.
From another perspective, the value of ${\eta _{exp }}$ corresponding to a specific $R(M )$ value may be used as an empirical parameter to evaluate the crosstalk of the detector in one dimension.
4.2 Suggestions for imaging applications using Moiré fringes
First of all, ensuring that the visibility of the reference fringe pattern is sufficiently high is the most basic requirement in designing and operating a grating interferometer. Since Moiré fringe visibility is definitely lower than the visibility in the phase stepping method, which is further lower than the visibility of the reference fringe pattern.
Secondly, the crosstalk between detector pixels in the y direction needs to be paid enough attention during the selection or development stage of the detector, providing sufficient margins for subsequent optimization. On the one hand, the denser the Moiré fringe is in this dimension, the higher spatial resolution multi-contrast images are supposed to obtain in single-frame Talbot-Lau interferometry. The spatial resolution in the y direction is limited by double the Moiré fringe spacing, according to the Fourier transform method [11]. On the other hand, the dense Moiré fringe means a large misalignment angle, which can lead to severe visibility reduction. The crosstalk is the most important factor affecting the relative optimal value chosen after compromising these two aspects.
At last, specially optimized anisotropy detectors or specially designed imaging processes may be the development direction of scanning imaging. For example, the detector with a series of line-array detection units may be a better choice for scanning imaging than common surface type detectors. Since it can avoid crosstalk and choose a smaller misalignment angle at the same time.
4.3 Relationship and difference with the existing general Moiré effect studies
The Moiré fringe in this study is an angle-induced incoherent-and-multiplicative type 2-D Moiré fringe phenomenon, about which both general theoretical research and application research have gone through a long time. The orientation and period of the Moiré fringe are relatively easy for the research [26,27]. Because the derivation of these general conclusions is sufficient by considering only the phase component. However, it is relatively difficult to derive conclusions that require consideration of Moiré fringe profile formula, as represented by visibility. The analytical solution could only be obtained by defining more of the characteristics of the two periodic structures now, such as assuming that both are binary gratings [28].
The purpose of this research is to analytically derive the visibility which means that the difficulty of deriving Moiré fringe profile must be faced directly. And this research may go at least two steps further. The first step is to consider two common periodic structures: sine distribution and binary distribution. The other step is the attempt to take the detector pixel factor (integration area) into consideration, which is consistent with the rapid development of grid-formed digital x-ray detectors recent years.
These two steps are appropriate for the current Talbot-Lau interferometry research. However, it may require further research when similar derivation being extended to other application or more general situations.
5. Conclusion
A Moiré fringe visibility model is established and analyzed. The visibility of Moiré fringe can be calculated through the product of two parts, namely the intensity curve visibility in ideal phase-stepping method and the visibility reduction factor.
It is experimentally confirmed that the angle between the G1 self-imaging pattern and G2 grating and the crosstalk between the detector pixels in the y-direction are two main reasons for the reduction of the visibility of the Moiré fringes.
The number of pixels used to sample one period of Moiré fringe is an effective parameter to estimate the relative quality of the Moiré fringe, but the crosstalk in y-direction of the detector pixels is still a negligible negative factor, which is supposed to be one of the core problems for further single-frame imaging or scanning imaging studies.
Appendix
The $R(\theta )$ actually consists of three sub-functions. ${f_\textrm{1}}({\theta ,{Q_y}} )$ has been discussed before. ${f_\textrm{2}}({\theta ,\gamma } )$ and ${f_\textrm{3}}({\theta ,\gamma ,\tau } )$ both can be ignored compared to ${f_\textrm{1}}({\theta ,{Q_y}} )$ in the small angle range.
Fig. 11. Function image of ${f_\textrm{2}}({\theta ,\gamma } )$ when the duty cycle $\gamma$ takes different values.
${f_\textrm{3}}({\theta ,\gamma ,\tau } )$ is the sub-function used to characterize the effect of both the duty cycle $\gamma$ and analyzer grating transmittance $\tau$ on the $R(\theta )$, the detail and function image of which is shown below (Fig. 12). No matter how the two parameters change, the value of ${f_\textrm{3}}({\theta ,\gamma ,\tau } )$ within 10 degrees is around 1.0 and doesn’t change much. So it can also be ignored.
Fig. 12. Function image of ${f_\textrm{3}}({\theta ,\gamma ,\tau } )$ when the duty cycle $\gamma$ and grating line transmission $\tau$ take different values.
It is worth mentioning that although the effects of the duty cycle $\gamma$ and analyzer grating transmittance $\tau$ on $R(\theta )$ can be ignored, their effect on the visibility $V(\theta )$ through $V(\textrm{0} )$ is intuitive.
Funding
National Natural Science Foundation of China (51777198); China Academy of Engineering Physics (Civil-military Integration Research Foundation).
Disclosures
The authors declare no conflicts of interest.
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