Numerical analysis of partially coherent radiation at soft x-ray beamline

Xiangyu Meng; Chaofan Xue; Huaina Yu; Yong Wang; Yanqing Wu; Renzhong Tai

doi:10.1364/OE.23.029675

1. Introduction

Coherence optics become more and more important since the third-generation synchrotron radiation facilities [1, 2] and the x-ray free-electron lasers [3–5] have been developed rapidly around the world. The x-ray with good coherence expands new insights in many fields including biology, particle physics and condensed matter physics. Coherent light is important in many x-ray technologies, such as x-ray photon correlation spectroscopy [6], x-ray interference lithography [7], x-ray holography [8] and coherent x-ray diffraction imaging [9], etc.

It is necessary to analyze the coherence propagation through synchrotron beamlines to generate x-ray with good coherence. Many models are developed to satisfy the requirement in recent years. Sánchez et al extended the traditional ray tracing models into a hybrid ray-tracing wave-optics approach in SHADOW3, which can deal with the partial coherence by changing the phase correlation of the source [10–12]. Chubar et al presented Synchrotron Radiation Workshop (SRW) model [13, 14] to simulate partially coherent wavefront propagation from a finite emittance electron beam. SRW calculated the spontaneous emission of electrons through undulators using the retarded potential [15], simulated the wavefront propagation based on Fresnel-Kirchoff equation. A hybrid model for partially coherent source simulation combining geometric ray-tracing and wavefront propagation had been reported recently by Shi et al [16]. The hybrid model used ray-tracing to simulate the geometric effects and implemented the wavefront propagation to calculate the distribution of diffraction. The results from the diffraction and geometric effects were integrated together by numerical convolution and ray re-sampling.

In this paper a new model is established based on propagation of four dimensional coherence function to analyze the partial coherence in synchrotron beamlines. Wavefront is described by the Mutual Optical Intensity (MOI), which is a four dimensional function of space coordinates. One can get the spot size and coherence length from MOI of the wavefront. In our model the wavefront is separated into many small elements to realize the numerical calculation of MOI propagation through free space. Combining with the local stationary phase approximation, the MOI propagation through reflecting mirrors and gratings are realized. With the model we analyzed the coherence of x-ray beam in the Scanning Transmission X-ray Microscopy (STXM) beamlines at Shanghai Synchrotron Radiation Facility (SSRF). Furthermore, we carried out the Fresnel diffraction experiment of a single slit at the endstation and obtained the transverse coherence length by simulating the pattern, which was in good agreement with the theoretical result.

2. Establishment of numerical calculation on propagation of partially coherent light

2.1 Description of partially coherent wavefront

Mutual coherence function plays a fundamental role in the theory of partial coherence. It represents the correlations between two complex scalar values of the electric field at different place and different time [17]. Its definition can be expressed as

Γ_{12} (P_{1}, P_{1}, τ) = < u (P_{1}, t + τ) u^{*} (P_{2}, t) >,

where

u (P_{1}, t + τ)

and

u (P_{2}, t)

are the field values at two positions

P_{1}

and

P_{2}

,

t

and

τ

represent different time, the brackets <···> denote the time average. In most cases the synchrotron beam is monochromatic at the endstaion, therefore we could consider only the monochromatic light in the following analysis. In this case

τ

is 0 and the mutual coherence function becomes mutual optical intensity

Γ_{12} (0) = J (P_{1}, P_{2}) = < u (P_{1}, t) u^{*} (P_{2}, t) > .

The four-dimensional function

J (P_{1}, P_{2})

describes the optical information of the wavefront, including the distribution of electric field and correlations between every two points. The propagation of MOI from wavefront P to wavefront Q is represented by equation [18]

J (Q_{1}, Q_{2}) = \iint \iint J (P_{1}, P_{2}) e^{- i \frac{2 π}{λ} (r_{2} - r_{1})} \frac{χ (θ_{1})}{λ r_{1}} \frac{χ (θ_{2})}{λ r_{2}} d S_{1} d S_{2} .

The degree of coherence between two points is defined as

γ_{12} = \frac{| J_{12} |}{\sqrt{I_{1} I_{2}}} .

where

I_{1}

and

I_{2}

are the intensities at Q₁ and Q₂, respectively.

2.2 Propagation of MOI

Equation (3) can be solved if the optical layout matches Fraunhofer or Fresnel conditions. However, in most synchrotron beamlins the two approximation conditions can’t be matched since the wavelength is very short in x-ray range. There is no analytical solution for Eq. (3). Therefore, we establish a numerical method to calculate the propagation of MOI. First, the wavefront is separated into many elements. It is assumed that in every element the beam has full coherence and constant complex amplitude, which is reasonable as long as the dimension of element is much smaller than the coherent length and beam spot size. Second, the propagation of MOI for every element can be calculated with Fraunhofer or Fresnel approximations. Finally, the total MOI after propagation can be obtained from sum of the contribution of all elements. The propagation model is shown in Fig. 1, where the object (marked with P) and image wavefront (marked with Q) are both separated into many elements.

Fig. 1 The sketch for propagation of MOI from wavefront P to wavefront Q in free space. Wavefronts are separated into many elements, r₁ and r₂ are the distances of MOI propagation between different pionts, θ₁ and θ₂ are inclination angles.

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The wavefront is separated equally into M × N elements. Every element is denoted by hj (1≤h≤M, 1≤j≤N) or lm (1≤l≤M, 1≤m≤N). The integration in the total wavefront is the sum of the integration in every element. The Eq. (3) can be written in the following

J (Q_{1}, Q_{2}) = \sum_{l m} \iint e^{i \frac{2 π}{λ} r_{P_{2} Q_{2}}} \frac{χ (θ_{2})}{λ r_{P_{2} Q_{2}}} (\sum_{h j} \iint J (P_{1}, P_{2}) e^{- i \frac{2 π}{λ} r_{P_{1} Q_{1}}} \frac{χ (θ_{1})}{λ r_{P_{1} Q_{1}}} d S_{h j}) d S_{l m},

where

r_{P_{1} Q_{1}}

is the distance between points

P_{1}

and

Q_{1}

, the integration range for

d S_{h j}

is the hj element. Since the complex amplitude is assumed to be constant in every element, the

J (P_{1}, P_{2})

can be taken out from the integration. We can get the following equation

J (Q_{1}, Q_{2}) = \sum_{l m} J (P_{h j}, P_{l m}) \iint e^{i \frac{2 π}{λ} r_{P_{l m} Q_{2}}} \frac{χ (θ_{2})}{λ r_{P_{l m} Q_{2}}} (\sum_{h j} \iint e^{- i \frac{2 π}{λ} r_{P_{h j} Q_{1}}} \frac{χ (θ_{1})}{λ r_{P_{h j} Q_{1}}} d S_{h j}) d S_{l m},

where

r_{P_{h j} Q_{1}}

is the space vector in the local coordination system in the hj element. The integration within the bracket in Eq. (6) is through the hj element. The integration is defined as

A_{h j Q_{1}} = \iint e^{- i \frac{2 π}{λ} r_{P_{h j} Q_{1}}} \frac{χ (θ_{1})}{λ r_{P_{h j} Q_{1}}} d S_{h j} .

Equation (7) can be calculated numerically with Fresnel or Fraunhofer approximation as long as the element size is small enough. Therefore the MOI propagating through free space is calculated numerically

J (Q_{1}, Q_{2}) = \sum_{l m} (A_{l m Q_{2}}^{*} \sum_{h j} J (P_{h j}, P_{l m}) A_{h j Q_{1}}) .

Regarding all elements in wavefront Q, we can change the Eq. (8) into matrix form

J_{Q Q} = {(A_{P Q})}^{T *} J_{P P} A_{P Q} .

2.3 Through mirror and grating

We implement the local stationary phase approximation to simulate the MOI propagating through the reflecting mirrors [19–21]. In this approximation the local propagation of the electric field is assumed to be along the geometrical rays [2, 20, 22].

Two wavefronts are chosen for the MOI propagation. One is before the optics and the other after the optics. The two wavefronts are both perpendicular to the beam axis and close to the optics. The complex amplitude transmission function $t (P)$ describes the relationship between the electric fields in the incident and reflecting wavefronts. $t (P)$ can be assumed to be constant for the reflecting mirror in synchrotron beamline. One can get the amplitude spread function of the reflecting mirror [23]

K (Q, P) = t (P) \exp [i \frac{2 π}{λ} Γ (Q, P)] δ [P_{} - \tilde{P} (Q)],

where

\tilde{P} (Q)

is scalar function describing the transformation of coordinates for points in transverse wavefronts before and after the reflecting mirror, and

Γ (Q, P)

is the function describing the optical path between the two wavefronts [21]. The MOI at the reflecting plane can be expressed as

J_{r} (Q_{1}, Q_{2}) = \iint \iint J_{i} (P_{1}, P_{2}) K (Q_{1}, P_{1}) K^{*} (Q_{2}, P_{2}) d S_{1} d S_{2},

where

J_{i} (P_{1}, P_{2})

and

J_{r} (Q_{1}, Q_{2})

are incident and reflecting MOI, respectively. Taking Eq. (10) into Eq. (11) one can get the MOI at the reflecting plane

J_{r} (Q_{1}, Q_{2}) = J_{i} (\tilde{P} (Q_{1}), \tilde{P} (Q_{2})) t (\tilde{P} (Q_{1})) t^{*} (\tilde{P} (Q_{2})) \exp {i \frac{2 π}{λ} [Γ (Q_{1}, \tilde{P} (Q_{1})) - Γ (Q_{2}, \tilde{P} (Q_{2}))]} .

From Eq. (12), we can numerically calculate the MOI propagation through a reflecting mirror.

Grating is a special reflecting optics if we consider only one diffraction harmonics. It is similar to the reflecting mirrors except that the diffraction should be considered instead of reflection. Thus Eq. (12) can be applied for the MOI propagation of grating as long as the reflection path is replaced with diffraction path. The grating MOI propagation can be expressed as

J_{r} (Q_{1}, Q_{2}) = J_{i} (\frac{\cos α}{\cos β} Q_{1}, \frac{\cos α}{\cos β} Q_{2}) t (\frac{\cos α}{\cos β} Q_{1}) t^{*} (\frac{\cos α}{\cos β} Q_{2}) \exp {i \frac{2 π}{λ} [Γ (Q_{1}, \tilde{P} (Q_{1})) - Γ (Q_{2}, \tilde{P} (Q_{2}))]} .

where α and β are the incident and diffraction angles, respectively.

Finally, the MOI propagation is realized through free space, reflecting optics and grating. From the given MOI of source one can calculate the MOI anywhere in the beamline. Next we analyze the coherence of the STXM beamline at SSRF with the established model.

3. Propagation of the coherence through the STXM beamline

The layout of the STXM beamline [23–25] is shown in Fig. 2. The source is an Elliptical Polarized Undulator (EPU). A four-blade-aperture is placed at 20m to define the divergence of the beamline. The beam is collimated vertically by a cylindrical mirror. A SX700-type monochromator locates at 32m including a plane mirror and two plane gratings. There is a toroidal mirror after the monochromator to focus the beam at the exit slit both horizontally and vertically. The endstation locates 2m downstream the exit slit [25]. In the following we calculate the MOI propagation after every optics and analyze the coherence. The energy is chosen to be 486eV in the calculation.

Fig. 2 Schematic layout of the BL08U beamline, the SX700-type monochromator consists of Plane Mirror and Grating Mirror.

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3.1 MOI at the source

The source from an undulator has a narrow angular divergence, which is small enough to match the inequality $\frac{1}{η_{x, y}^{2}} ≪ \frac{2 π^{2}}{λ^{2}}$ [26]. Thus it is reasonable to assume the MOI at the source to be Gaussian Schell Model (GSM) [26–28]. It should be pointed out that the GSM is not necessary for our model. Any MOI distribution can be used for the source. GSM is a good choice in this case due to its simplicity. The space distribution of the intensity and the degree of coherence are Gaussian functions as shown in the following

D (x, y) = \sqrt{I_{0}} \exp (- \frac{x^{2}}{2 σ_{x}^{2}} - \frac{y^{2}}{2 σ_{y}^{2}}),

J (x_{1}, y_{1}, x_{2}, y_{2}) = D_{1} D_{2} \exp (- \frac{{(x_{1} - x_{2})}^{2}}{2 ξ_{x}^{2}} - \frac{{(y_{1} - y_{2})}^{2}}{2 ξ_{y}^{2}}),

where

σ_{x}

and

σ_{y}

describe the source size,

ξ_{x}

and

ξ_{y}

are the coherence length of the source. The parameters of a GSM source satisfy the following equation [26, 29]

\frac{1}{η_{x, y}^{2}} = \frac{1}{{(2 σ_{x, y})}^{2}} + \frac{1}{ξ_{x, y}^{2}} .

According to Eq. (16) the coherence length of the source can be calculated from the beam size and divergence from SPECTRA [30]. The source parameters are shown in Table 1.

Table 1. Source parameters in STXM beamline.

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3.2 MOI after every optical element

The wavefront is chosen to be at the four-blade-aperture. The MOI at this wavefront is calculated with free space model in section 2.2. The wavefront at the source and four-blade-aperture are separated into 500 × 500 equal elements. The element size in source is 2.45 µm × 0.24 µm (H × V), at which the integration Eq. (8) matches the Fresnel approximation. The intensity and coherence distribution at the four-blade-aperture are shown in Fig. 3(a) and 3(b), where the degree of coherence $γ_{12}$ is between the central point and every other point. It can be seen that the distribution of intensity and coherence are both perfect Gaussian profile. The source sizes is 746 µm × 358 µm (H × V) and coherence length is 54.3 µm × 800 µm (H × V) at the four-blade-aperture listed in the Table 2, where the values all indicate σ in Gaussian fitting. The source size calculated from the MOI propagation is the same as those from the geometric optics, since the propagation is in free space and the beam is not limited.

Fig. 3 The distribution of the intensity and the degree of coherence at the optical elements, the degree of coherence between central point and other points. (a), (b) The intensity and the degree of coherence distribution at the four-blade-aperture. (c), (d) The intensity and the degree of coherence distribution at the cylindrical mirror. (e), (f) The intensity and the degree of coherence distribution at the exit slit. (g), (h) The intensity and the degree of coherence distribution at the endstation.

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Table 2. MOI parameters through STXM beamline.

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The MOI at the four-blade-aperture is directly used as the second source for the propagation to the next optics. The size at the four-blade-aperture is 1.6 mm × 1.6 mm, which would limit the beam to partial Gaussian profile. A cylindrical mirror (CM) is placed 10m downstream the four-blade-aperture and collimates the beam vertically. The intensity and coherence distribution at the CM are shown in Fig. 3(c) and 3(d). One can see that the center region of intensity are well Gaussian profile in horizontal direction. There are two minor peaks close to the sharp edges due to the diffraction through the aperture. However, the intensity distribution in vertical direction has many apparent stripes since the coherence is much better than horizontal. The coherence length is (80.5 ± 0.05) µm × (1187 ± 96) µm (H × V), the edge oscillations of vertical coherence originates from the aperture diffraction. The errors following each value are the standard errors of σ in the Gaussian fitting. After the cylindrical mirror, the beam changes from a spherically divergent field to a plane filed in vertical direction, and keeps divergent in horizontal direction.

We calculate the MOI propagation through the plane mirror, grating, and toroidal mirror, and get the MOI at the exit slit. The intensity and coherence distribution at the exit slit are shown in Fig. 3(e) and 3(f). The vertical coherence is better than horizontal one. There are apparent oscillations in the coherence distribution relative to the center both in horizontal and vertical directions. The vertical oscillations are much higher so that the coherence distribution is not a good Gaussian profile. The coherence length is calculated from the Gaussian fitting of the central peak. Table 2 shows the beam size and coherence length from MOI at every optics in the beamline. The beam size from geometric optics are also shown in Table 2 for comparison. It can be seen from Table 2 that the beam size from MOI is coincided with the geometric one at every optical element.

Finally the MOI at the endstation is calculated from the propagation of MOI at the exit slit. The analysis of the MOI propagation through the exit slit is more complicated due to the energy dispersion by the grating. The beam with different energy has a linear distribution in vertical direction at the exit slit. In our case the vertical exit slit opening is 100 µm, much larger than the monochromatic beam vertical size 11.1 µm. Thus the MOI propagation from the exit slit should include the beam with different energy. The total MOI is the numerical sum of a series of MOI with different energy, which can be written as $J (Q_{1}, Q_{2}) = \sum J (Q_{1}, Q_{2}, E) .$ The energy range is 486 ± 0.04 eV when the exit slit size is 100 µm × 100 µm. The MOI at the endstation is calculated and the results are shown in Fig. 3(g) and 3(h). The beam size is (303 ± 0.7) µm × (67.8 ± 0.2) µm (H × V) and the coherence length is (30.7 ± 0.6) µm × (31 ± 5.3) (H × V) µm. The geometrical beam size at the endstation is also calculated and it is 315 µm × 62.6 µm. The results from our model are in good agreement with the geometrical ones, which means the model is applicable to the analysis of the beamline. Besides, one can get the coherence distribution anywhere in the beamline and the evolution of coherence through every optics.

4. Measurement and simulation of the coherence at the endstation

In last section we analyzed the MOI propagation through the STXM beamline and got the MOI of the beam at the endstation with the established model. An experiment of single window diffraction was performed to measure the beam size and coherence length at the endstation for comparison with the theoretical results. A rectangular window with the sizes of 110 µm × 110 µm was placed at the endstation and perpendicular to the beam. The material of the window was 1mm-thick silicon. A PMT scintillant detector located 26 mm after the window. The distance was short due to the limited space in the STXM chamber. There was a hole with 0.5µm diameter clinging to the PMT, to make the PMT only receive the beam through the hole. Since the sample stage has a better movement resolution than the detector stage, the rectangular window was scanned instead of the detector to get better spatial resolution of the pattern measurement. The resolution of the window scanning was 0.25µm.

The diffraction pattern was simulated with our model. The MOI distribution at the endstation was assumed to be GSM. The MOI propagation through the single slit was calculated and the intensity pattern at the detector was obtained. From the simulation the spot size and coherence length at the endstation were acquired.

The diffraction patterns from experiment and simulation are shown in Fig. 4. It can be seen that there are many clear interference fringes in the pattern. The pattern is obviously Fresnel type since the PMT is close to the single slit. The simulated pattern of intensity is shown in Fig. 4(b), which coincides well with the experimental result. The experimental fringes are blurred in the center region because of the noise and insufficient spatial resolution of the PMT detector. Figure 4(c) shows the degree of coherence between the central point and other points at the PMT. It is approximately Gaussian type and one can get the coherence length by Gaussian fitting. The degree of coherence between every two points on axis is shown in Fig. 4(d)-4(g). There is a bright band in each figure, which means high degree of coherence. The width of the band indicates the coherence length. It can be seen from the figures that the horizontal coherence length at the source is smaller than the vertical one. However, the coherence lengths in the two directions are almost the same at the endstation.

Fig. 4 Diffraction pattern of a single slit. (a) Experimental result of the diffraction pattern at the PMT; (b) Simulation of the diffraction pattern; (c) The degree of coherence between the central point and every other point calculated from the simulation; (d), (e) The degree of coherence between every two points at EPU source on horizontal and vertical axis, respectively; (f), (g) The degree of coherence between every two points at the endstation on horizontal and vertical axis, respectively.

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The intensity profiles on axis are shown in Fig. 5(a) and 5(b). It can be seen that the simulation has a good agreement with the experimental results. The profiles are Fresnel type due to the short distance between the slit and PMT. The beam size and coherence length at the slit were obtained from the simulation. The beam size is (300 ± 16) µm × (100 ± 6.1) µm. The coherence length is (31 ± 3.0) µm × (25 ± 2.1) µm. The errors following every value are the standard errors of fitting parameters in the simulation. In section 3.2, the MOI propagating from the source to the endstation is calculated. The beam size and coherence length from the calculation are (303 ± 0.7) µm × (67.8 ± 0.2) µm and (30.7 ± 0.6) µm × (31 ± 5.3) µm, respectively, shown in Table 2. One can see that the horizontal size and coherence length from the experiments coincide with the theoretical results. However, there are some deviation in vertical direction. In theoretical results the size is smaller and the coherence length is larger. The reason might be due to the slope error of the optics. The slope errors of the optics in STXM beamline vary from 0.8 µrad to 3 µrad, which would increase the beam size and decrease the coherence. Here we give a quantitative explanation on how the slope error will influence the beam size and coherence. The slope error of the optics in the beamline will damage the wavefront and cause random deviation in the propagation direction in different position in the wavefront. This will weaken the coherence of the beam, especially at the focus point which is the exit slit in our case. Weakened coherence will cause larger divergence angle after the exit slit. Thus the beam size at the endstation will increase after 2m propagation from the exit slit. The influence of the slope errors on horizontal direction is not as high as vertical direction, because of large beam size and bad coherence in horizontal direction. However, the vertical coherence is better than horizontal one and thus more sensitive to the slope errors.

Fig. 5 Intensity profiles on axis through single slit diffraction. Experimental profiles (black dots) and fitting results (red line) in horizontal (a) and vertical (b) direction indicate apparent Fresnel diffraction curves. The insets show enlarged regions of the intensity profiles.

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5. Conclusion

Based on four dimensional coherence function, a new model on numerical analysis of partial coherence in soft x-ray beamline is established in this study. Mutual optical intensity is applied to describe the partial coherence at wavefront. The beam size and coherence length can be obtained from the MOI distribution. The wavefront is separated into many elements to realize the numerical calculation of MOI propagation through free space and optics in beamline. Given the MOI of the source, one can calculate the MOI anywhere in the beamline. The source is not necessary to be Gaussian Schell type. Any MOI distribution can be applied in this model. The STXM beamline at SSRF was analyzed with this model. The beam size and coherence length at the endstation were (303 ± 0.7) µm × (67.8 ± 0.2) µm and (30.7 ± 0.6) µm × (31 ± 5.3) µm, respectively, which coincided with the results from geometry optics. A single slit diffraction experiment was performed to check the theoretical results. The diffraction pattern was simulated with the established model and the beam size and coherence length at the endstation were obtained, which were (300 ± 16) µm × (100 ± 8.1) µm and (31 ± 3.0) µm × (25 ± 2.1) µm, respectively. The experimental results have a good agreement with the theoretical ones. It indicates that the model is applicable in the analysis of partial coherence propagation in soft x-ray beamlines.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (NSFC) (No. 11475251, 11275255 and 11225527) and the Shanghai Academic Leadership Program (No. 13XD1404400). This work was also supported by the Open Research Project of the Large Scientific Facility of the Chinese Academy of Sciences (CAS) Study on Self-assembly Technology and Nanometer Array with Ultra-high Density. We also acknowledge BL08U1A at SSRF for providing the beam time.

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Direction	X	Y
Source size $σ$ (µm)	153.12	15.24
Source divergence $σ^{'}$ (µrad)	37.10	17.87
Coherence length $ξ$ (µm)	10.96	34.11

	Four balde aperture (µm × µm)	Cylindrical mirror (µm × µm)	Toroidal mirror (µm × µm)	Exit slit (µm × µm)	Endstation (µm × µm)
Calculated size H × V	746 × 358	(1107 ± 1.3) × (533 ± 2.4)	(1259 ± 1.5) × (210 ± 1)	(35.8 ± 0.05) × (11.1 ± 0.02)	(303 ± 0.7) × (67.8 ± 0.2)
Geometrical size H × V	742 × 357	1113 × 536	1261 × 214	36 × 10.2	315 × 62.6
coherence length H × V	54.3 × 800	(80.5 ± 0.05) × (1187 ± 96)	(91 ± 0.06) × (480 ± 38)	(4.2 ± 0.1) × (21.6 ± 4.8)	(30.7 ± 0.6) × (31 ± 5.3)

Direction	X	Y
Source size $σ$ (µm)	153.12	15.24
Source divergence $σ^{'}$ (µrad)	37.10	17.87
Coherence length $ξ$ (µm)	10.96	34.11

	Four balde aperture (µm × µm)	Cylindrical mirror (µm × µm)	Toroidal mirror (µm × µm)	Exit slit (µm × µm)	Endstation (µm × µm)
Calculated size H × V	746 × 358	(1107 ± 1.3) × (533 ± 2.4)	(1259 ± 1.5) × (210 ± 1)	(35.8 ± 0.05) × (11.1 ± 0.02)	(303 ± 0.7) × (67.8 ± 0.2)
Geometrical size H × V	742 × 357	1113 × 536	1261 × 214	36 × 10.2	315 × 62.6
coherence length H × V	54.3 × 800	(80.5 ± 0.05) × (1187 ± 96)	(91 ± 0.06) × (480 ± 38)	(4.2 ± 0.1) × (21.6 ± 4.8)	(30.7 ± 0.6) × (31 ± 5.3)

Numerical analysis of partially coherent radiation at soft x-ray beamline

Abstract

1. Introduction

2. Establishment of numerical calculation on propagation of partially coherent light

2.1 Description of partially coherent wavefront

2.2 Propagation of MOI

2.3 Through mirror and grating

3. Propagation of the coherence through the STXM beamline

3.1 MOI at the source

3.2 MOI after every optical element

4. Measurement and simulation of the coherence at the endstation

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (5)

Tables (2)

Equations (16)

Optics Express