Zoom optics in fourth-generation synchrotron radiation: Design and simulation

Abstract

Brilliance of the fourth-generation synchrotron radiation sources are increased in the order of magnitude, which further emphasizes the coherent applications. The zoom system of traditional optics can realize coherence regulation while achieving the target size of focus spots at designated position. This paper develops the design method of zoom system to fully exploit partially coherent fields. According to the first-order optics and imaging theory, the design method is reasonably simplified. The flux-optimization acceptance-angle ratio approximately linearly varies with the coherent fraction, which contributes to the slit-aperture determination. In order to validate the design method, wave-optics simulations are conducted in this paper.

Graphical abstract

Keywords

Fourth-generation synchrotron radiation sources / zoom optics / design optimization

1 Introduction

The fourth-generation synchrotron radiation sources are constructed with the multi-bend achromatic design [1, 2], which compresses the ring emittance to pm\cdotrad and further enhances brilliance and coherence of X-ray radiation. Due to enhanced performance, synchrotron radiation sources have been considered for coherent applications, including coherent X-ray diffraction imaging, X-ray photon correlation spectroscopy, and so on [3-6]. Simultaneously, synchrotron radiation sources serve as a user-supported platform, conventional experiments such as X-ray absorption spectroscopy and X-ray scattering, are still remain fundamental to ensuring scientific advantages.

Zoom optics has been extensively studied in beamline design [7-11]. The advantages of zoom optics encompass: (i) achieving coherence regulation of partially coherent fields, (ii) reducing the effect of errors and instability from pre-stage white-beam optics by the slit, (iii) obtaining focal spots of required sizes at designated positions for samples, (iv) enhancing efficiency through the addition of more experimental stations, which is similar to the X-ray free electron laser (XFEL) light sources. Nevertheless, the design approach of zoom optics has not been clearly established for synchrotron radiation sources considering partial coherence characterization.

In the fourth-generation synchrotron sources, diffraction effect can not be ignored, hence complicating beamline design. Traditional optics serves as a fundamental basis for design process. The zoom photographic lens system involves a series of lenses whose powers are adjusted by altering the number of lenses [12]. Traditional zoom optical system generally comprises three lenses. In mechanically compensated zoom optics, to change the power of the whole system, the middle lens is moved along the axis with simultaneously moving either the front or the rear lens by an in-and-out camera to keep the focal position remained. Due to the limitation of moving distance, the power of X-ray optics serves as a more appropriate variable for beamline design. This study introduces the design method for zoom optics, which relies on investigations of the partially coherent characteristics of the fourth-generation synchrotron radiation sources.

2 First-order optics design

The general model of zoom optics is illustrated in Fig.1, with Fig.1(a) depicting the relay focus and Fig.1(b) representing the secondary focus. The two types of focus are differentiated by the presence of a real image between the lenses. y_1, y_2 are incident positions of the beam on lenses. {\phi }_1, {\phi }_2 are powers of the lenses. L_1, L_2, L_3 are distances from the source to the first lens, from the first lens to the second lens, and from the second lens to the sample, respectively. By applying paraxial optics, the correlation between design parameters and focal features is established.

Suppose that the complementary angle corresponding of the incident angle on the first lens is u, the incident position y_1 is

(1)y_1=L_{1}u .

The lens changes the angle to

(2)u_1^{\prime }=u -y_1{\phi }_1= (1- L_1{\phi }_1)u.

After propagating through free space, the incident position on the second lens is

(3)y_2=y_1+ L_2{u}_1^{\prime }=(L_1+L_2- L_1{L}_2{\phi }_1)u.

The second lens changes the angle to

(4)u_2^{\prime }=u_1^{\prime }-y_2{\phi }_2=[(1-L_1{\phi }_1)(1-L_2{\phi }_2)-L_1{\phi }_2]u.

System magnification M and L_3 can be expressed as

(5)L_3=-{\displaystyle \frac{y_2}{u_2^{\prime }}}=-{\displaystyle \frac{L_1+L_2-L_1{L}_2{\phi }_1}{(1-L_1{\phi }_1)(1-L_2{\phi }_2)-L_1{\phi }_2}},

(6)M={\displaystyle \frac{u_2^{\prime }}{u}}=(1-L_1{\phi }_1)( 1-L_2{\phi }_2)-L_1{\phi }_2.

The powers of lenses can be expressed by

(7){\phi }_1={\displaystyle \frac{L_1+L_2+M{L}_3}{L_1{L}_2}},

(8){\phi }_2={\displaystyle \frac{-M+1-L_1{\phi }_1}{L_1+L_2-L_1{L}_2{\phi }_1}}.

3 Coherence feature of the fourth-generation synchrotron radiation

Radiation characteristics serve as the foundation for beamline design. As a typical fourth-generation synchrotron radiation source, High Energy Photon Source (HEPS) is chosen as the example for simulation in this study. Tab.1 presents parameters of the storage ring and undulator. To enhance computational performance, MPI parallel computation is integrated with the Fresnel Chirp-Z method within the XRT program for wave-optics simulations [13-16].

Fig.2 illustrates the variation of flux and the coherent fraction with the acceptance angle. The coherent fraction is determined by the coherent mode decomposition [17-19]. Apparently, as the region of interest expands, flux increases while the coherent fraction decreases. Red and white lines in Fig.2 correspond contours of flux and the coherent fraction, respectively. The coherent fraction contours are clearly distinguishable from one another. In comparison to the source with a coupling constant of 1, the contours of the coherent fraction for HEPS are elongated vertically and compressed horizontally. Therefore, restricting the horizontal acceptance angle is essential for achieving radiation fields with reasonably good coherence. As illustrated in Fig.2 the coherent fraction is less than 50% when \mathrm{\Delta }{\theta }_x > 10 μrad, despite \mathrm{\Delta }{\theta }_y is only about 1 μrad. The observation that coherent fraction contours can be tangent to flux contours implies that the maximum flux under certain coherence requirement can be achieved with appropriate acceptance angles.

Fig.3(a) presents the 90% coherent fraction contour and the corresponding flux, which confirms that the flux optimization acceptance angle exists. Fig.3(b) shows the flux optimization acceptance angles of different coherent fractions, which provides basic parameters for design optimization. Considering the extensive calculation, interpolation and decomposition methods are applied in data processing, which is demonstrated in appendix. Based on the results of Fig.3, Fig.4 illustrates the ratio \frac{\mathrm{\Delta }{\theta }_y}{\mathrm{\Delta }{\theta }_x} of flux optimization. As depicted in Fig.4, the ratio increases as the coherent fraction increases. Additionally, the correlation coefficient and Pearson value of linear fitting are close to 1 and 0, respectively, indicating that the variation is nearly linear. The first order coefficient of the linear fitting function decreases as the photon energy increases. The two fitting functions intersect at the red point where the coherent fraction is about 1.01 and the ratio is about 3.76. The ratio of electron sizes is about 3.83 which is close to 3.76, indicating that when the coherent fraction is 1, the ratio reflects the property of storage rings which is independent of photon energies.

4 Beamline design

4.1 Focal spot size

According to Refs. [20, 21], focal spot size is the convolution of three components determined by geometry optics, diffraction and residual wavefront errors:

(9){\sigma }_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}=\sqrt{{\sigma }_{\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{y}}^2+{\sigma }_{\mathrm{w}\mathrm{a}\mathrm{v}\mathrm{e}\mathrm{f}\mathrm{r}\mathrm{o}\mathrm{n}\mathrm{t}}^2+{\sigma }_{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}^2} .

Considering that the low wavefront distortion in beamlines can be assured by advanced beamline techniques for coherent experiments of the fourth-generation sources, this study mainly concerns about {\sigma }_{\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{y}} and {\sigma }_{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}. To derive {\sigma }_{\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{y}}, the source size is the convolution of electron beam size and Tanaka model of single-electron radiation [22]:

(10){\sigma }_{\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{y}}=M{\sigma }_{\mathrm{s}\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{c}\mathrm{e}}.

{\sigma }_{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}} depends on the image distance q, the radiation wavelength \lambda, the acceptance range D (D=p\mathrm{\Delta } \theta ) with the object distance p and a constant C which is equal to 0.4 for single-slit diffraction:

(11){\sigma }_{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}=qC{\displaystyle \frac{\lambda }{D}}.

Taking into account the characteristics of HEPS, Fig.5 illustrates the correlation between focal spot sizes and acceptance angles in both horizontal and vertical directions. \mathrm{\Delta }\sigma is defined as | \frac{\sigma -{\sigma }_{{\mathrm{s}\mathrm{i}\mathrm{m}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}_{}}}{{\sigma }_{\mathrm{s}\mathrm{i}\mathrm{m}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}}|, where \sigma may represent {\sigma }_{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}, {\sigma }_{\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{y}} and \sqrt{{\sigma }_{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}^2+{\sigma }_{\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{y}}^2}. As shown in Fig.5, \mathrm{\Delta } \sigma of {\sigma }_{\mathrm{s}\mathrm{i}\mathrm{m}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}} and \sqrt{{\sigma }_{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}^2+{\sigma }_{\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{y}}^2} maintains smaller than 10% in horizontal direction, while larger than 10% in vertical direction when the acceptance angle is above 5 μrad. This is because that the vertical electron emittance is smaller than \frac{\lambda }{4\pi } for 10 keV radiation field, rendering HEPS a diffraction-limited source [23, 24] and causing the radiation to deviate further from a Gaussian distribution.

4.2 Design optimization

The goal of optimization is to minimize the merit function by specifying the requirement of optics precision and the system layout parameters. As Refs. [20, 21] mentioned, the optimization is simplified under limiting cases where the system is only contributed by geometry optics or diffraction effect. As depicted in Fig.5, when the acceptance angle is small or large enough, the focal spot size approximates to {\sigma }_{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}} or {\sigma }_{\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{y}}. For diffraction-limited focusing systems, following steps of design optimization are adopted.

　1) Set the sample position L_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}, then define L_3 from L_1 and L_2.

　2) Derive the slit aperture D from Eq. (11). Negative and positive D respond to relay and secondary focus, respectively.

　3) Derive {\varphi }_1 from Eq. (3), then derive {\varphi }_2 from Eq. (7) and Eq. (8).

For system-limited focusing systems, following steps of design optimization are adopted.

　1) Set the sample position L_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}, then define L_3 from L_1 and L_2.

　2) Derive M from Eq. (10). Negative and positive M respond to relay and secondary focus, respectively.

　3) Derive {\varphi }_1 and {\varphi }_2 from Eq. (7) and Eq. (8).

According to Fig.5, the design optimization should be different in horizontal and vertical directions. In horizontal directions, \mathrm{\Delta } \sigma of {\sigma }_{\mathrm{s}\mathrm{i}\mathrm{m}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}} and \sqrt{{\sigma }_{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}^2+{\sigma }_{\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{y}}^2} remains below 10%, indicating that the design optimization can directly apply Eq. (9). However, because \mathrm{\Delta } \sigma in vertical direction of \sqrt{{\sigma }_{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}^2+{\sigma }_{\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{y}}^2} is larger than that of {\sigma }_{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}} or {\sigma }_{\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{y}}, the design optimization is based on the steps above, and the two focusing systems can be approximately distinguished by whether the acceptance angle is larger than the coherence length [25, 26] according to Fig.5(b).

For identical and specified parameters (L_1, L_2, L_3, {\varphi }_1, {\varphi }_2) in both directions, if the system in vertical direction is diffraction-limited, based on flux optimization results of Fig.3(b), the acceptance angle is larger, thus leading to smaller size of focal spot compared to horizontal direction according to Eq. (11). If the system in vertical direction is system-limited, because the source size is smaller, according to Eq. (10), the focal spot size is also smaller than the size in horizontal direction. The design optimization only concerns about the horizontal direction in this case.

　1) Set the sample position L_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}, then define L_3, L_1 and L_2.

　2) Set the horizontal acceptance angle according to results of Fig.3.

　3) Set the system magnification M.

　4) Derive the slit aperture from Eq. (9).

　5) Derive {\varphi }_1 and {\varphi }_2 from Eq. (7) and Eq. (8).

In addition, realistic constraints in beamline design should be considered.

　1) The focusing element powers are positive, {\phi }_1> 0 and {\phi }_2>0.

　2) The first focusing element usually placed outside the shield wall, which means L_1 has the minimum value.

　3) Because of challenges in manufacturing techniques, the size of the optical aperture has a limit.

　4) The working distance L_3 has a minimum value to accommodate support and auxiliary devices.

According to the design optimization steps above, for specified L_1, L_2 and L_3, solution number of {\varphi }_1 and {\varphi }_2 may exceeds one. Fig.6 shows the intensity distribution at sample position of one solution. The acceptance angle is \mathrm{\Delta } {\theta }_x\approx 2.87 μrad and \mathrm{\Delta }{\theta }_y\approx 8.73 μrad which is the flux optimization acceptance angle when the coherent fraction is about 0.8. The design parameters are L₁ = 40 m, L₂ = 38 m, L₃ = 2 m, ϕ₁ ≈ 0.0184 m⁻¹, ϕ₂ ≈ 0.505 m⁻¹. As shown in Fig.6, the focal spot size is about 0.519 μm and 0.345 μm in the horizontal and vertical directions with the target size 0.500 μm, respectively, verifying the feasibility of the design optimization method.

5 Conclusions

The application of the zoom optics in beamline is studied to achieve spot size scaling and coherence regulation, taking into account the partially coherent characteristics of fourth-generation synchrotron radiation sources. Design method combing classical analytical methods, including first-order optics and imaging theory, with wave-optics simulations for zoom optics is employed. On the one hand, to optimize the size of slits, the coherence and flux contours are calculated. The acceptance angle ratio exhibits an almost linear relationship with the coherent fraction, rather than remaining a constant value of \frac{{\sigma }_{\mathrm{s}\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{c}\mathrm{e}-x}}{{\sigma }_{\mathrm{s}\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{c}\mathrm{e}-y}} as is typically presumed. On the other hand, to obtain target size of the focal spot, the relationship between the focal spot size with the acceptance angle is examined. The size of horizontal direction can be estimated by Eq. (9). However, because the difference between the results of simulation and Eq. (9) is obvious in vertical direction, the size is estimated by Eq. (10) or Eq. (11), which is approximately decided by whether the acceptance angle is larger than the coherence length. If the system parameters are identical in horizontal and vertical directions, the focal spot size remains smaller in vertical direction, which simplifies the design optimization. Finally, we perform numerical simulation of HEPS beamline to validate our design method. Nevertheless, the efficacy of the zoom optics relies heavily on the adjustability of the X-ray optics. The constraints of movement distance and the discontinuous variation of focal length significantly limit the practical application of the traditional approach, such as changing the position of Kirkpatrick–Baez mirrors or quantity of compound refractive lens (CRLs) groups; thus, active optics is considered a viable solution.

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