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Abstract
A new, to the best of our knowledge, figuring method for a multilayer Laue lens (MLL) in the hard X-ray region is proposed in this paper. Theoretical simulation at 20 keV shows that the figuring method can compensate for the structure error. The phase errors of the first-order diffracted wave decrease from 0.85π to 0.26π after figuring. The spatial resolution changes from 45 nm to 26 nm after figuring, which is almost the same as that of the ideal MLL with a spatial resolution of 24 nm. The figured MLL can achieve 36% of the ideal MLL's first-order diffraction efficiency. Such method may reduce the requirements for the fabrication of the MLLs, and may make it possible to manufacture the larger numerical aperture MLL with the longer working distance in the future.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
During recent years, the X-ray microscopy community has seen tremendous advances in nano-focusing optics for the hard X-rays in the energy range of several keV and above. To date, to focus hard x-ray to below 10 nm has been achieved by multilayer Laue lens (MLL) [1–8]. MLL can be fabricated through the means of alternately depositing multilayers onto a flat substrate, then sectioning and thinning, different layers’ thickness of multilayer obeys the zone plate law [9]. Theoretical simulations have shown that the tilted and the wedge MLLs can achieve a focusing spot below 10 nm and 1 nm with ideal structure [10,11]. However, structure defects are inevitable, and will be generated in every stage during the process of fabrication of the MLLs [12,13]. The structure error can be divided into two main types: interface roughness or interdiffusion and layer placement error, poor deposition condition will lead to rough interfaces or interdiffusion between the adjacent layers, poor calibration or especially instability of the growth rate will lead to intolerable levels of layer-placement error [12].
Interface roughness and diffusion will cause a considerable decrease in the local diffraction intensity. No noticeable broadening effects on the focus due to interface roughness and diffusion is observed. The layer-placement error will lead to deformed MLL’s wavefront, which will compromise the focus. Layer-placement error includes high-frequency errors and low-frequency errors [14]. In the high-frequency error, interface of each layer is found to diverge from its theoretical position by a random error that manifests a normal distribution. In low-frequency error, the spatial frequency of the MLL structure does not follow the linear relationship with the position of n-th layer but a higher-order polynomial, because of the drift of deposition rate [14].
For the high-frequency error, the major effect is the reduced peak intensity while the original focus size is still maintained. However, the low-frequency layer-placement error leads to the decreased peak intensities, significant broadening of the focus size, and the generation of strong interference fringes on one side of the main peak, thereby destroying the focus.
According to the previous study about structure error, only the low-frequency layer-placement error produces dramatic effects on the focus. And it might be quite a challenge to deposit thousands of ultrathin layers without generating obvious imperfections. This means the layer-placement errors exist in real MLL structures. So, it is necessary to compensate for wavefront aberrations in order to achieve nanofocusing. Generally, phase plates (PP) are commonly employed to compensate for phase aberrations of optical systems [15–17]. Typically, the phase plate was placed upstream the MLL, the phase plate induces a phase profile to the collimated wavefront, which is propagated to the MLLs and ideally compensates for wavefront aberrations of the MLLs [18].
In this paper, we propose a new figuring method to compensate for the MLLs layer-placement error. First, the figuring method was described in detail. Then, we apply it to correct the low-frequency layer placement errors, and compare the figuring-compensation results with ideal result and phase plates-compensation result respectively, the simulations were performed using the beam propagation method (BPM) described below. It is revealed that figuring-compensation method is better than the phase plates-compensation method, indicating that our method may serve as an efficient way to compensate for the MLLs without any additional optics like phase plates.
2. Theory
2.1 Figuring method
The coordination system of our simulations in this paper is presented in Fig. 1.
Fig. 1. Schematic of MLL and the coordinate system used in the article and schematic of figuring method for MLL.
Exit wavefront of MLL can be written as [9]:
Figuring of MLL is realized by introducing figuring parameters to MLL: entire MLL is divided into a series of sub-MLLs, each sub-MLL is corresponding to its own figuring parameter. When the figuring parameter is 0, it corresponds to no etching for the sub-MLL, when the figuring parameter is 1, what it corresponds is that the entire sub-MLL is etched away. MLL was divided into a series of sub-MLLs by two methods: with the same number of layers or with equal thickness. For the equal number subdivision method (ENM), each sub-MLL contains two layers, for the equal thickness subdivision method (ETM), each sub-MLL’s thickness is equal to two times of the thickness of the outmost layer. Due to the huge total number of layers which is about 10,000 layers and the mutual coupling strength between each two layers, the figuring parameters cannot be optimized manually, the genetic algorithm is proposed to optimize the figuring parameters. The fitness function used in the simulation is defined as the wave-front difference between the figured MLL and ideal MLL.
2.2 BPM method
The beam propagation method (BPM) is an approximation technique for simulating the propagation of electromagnetic waves in various optical elements, in waveguides, for instance [19–21]. It is essentially the same as the multislice method. BPM was first introduced in the 1970s. Nowadays it has also been successfully applied to simulations of other X-ray optics, for instance: X-ray optics optimization in synchrotron radiation beamlines and X-ray optics damage calculations [22,23].
Since figuring will compromise the local grating structure, so coupled wave theory (CWT) cannot be used. Considering the flexible modeling method of BPM, so BPM is used for the simulation in this article. Here we will give a brief description of the BPM method.
In the ideal MLL, the n-th layer radius xn is given by the zone-plate equation:
The distribution of the susceptibility function constant in the MLL along x direction (for a given z) can be defined as a complex function χ (xi, z), where xi denotes location of the points where the susceptibility function constant changes. Function χ (xi, z) is given by Eq. (2) and Eq. (3). When considering layer-placement errors, Eq. (2) will take into account the alteration of the thickness of the layer caused by introduced errors.
Assume that the MLL consists of many thin slices with thickness △z. If the wavefront at the incident surface of the first slice, ${E_{inc}}({x,z} )$ is known, then the zero-order approximation of the wavefront at its exit surface ${E_{exi}}({x,z + \Delta z} )$ can be given by free-space propagation:
Then the propagation of the wavefront due to the presence of the optic can be written as:
The simulation of the entire MLL is described by the following steps: First, compute the exit wave from the first slice: the incident plane wavefront is propagated through the first slice of the MLL based on Eq. (5) and Eq. (6). Next, the exit wavefront of the first slice will be considered as the incident wavefront of the second slice, then propagate this wavefront to the next slice. Continue this propagation process through subsequent slices until the exit-wave for the last slice has been reached. Finally, the wavefront exiting the MLL’s exit-surface is then propagated in the free space directly to the focus plane, where the intensity near the focus is acquired by a fine z scan.
3. Simulation result and discussion
Here a flat MLL (WSi2/Si) with the following parameters is considered in the simulation: the MLL with a lens aperture size of 15 µm and outermost zone width of 10 nm. MLLs are aligned at tilt angle of θ = 12 mrad at 20 keV. At this angle, the Bragg condition is satisfied for the −1st diffraction order at 12 µm. The optimum thickness of the MLL was defined as the MLL thickness corresponding to the maximum focusing efficiency. A lens thickness of 12 µm corresponds to the optimum thickness of the MLL.
3.1 MLL with ideal structure and layer-placement error
For the ideal structure, the spatial frequency of the structure 1/dn should follow a linear relationship with the radius xn, according to Eq. (3), since in this case the total radius, 15 µm, is much smaller than the focal length, 6.2 mm. The spatial frequency of the structure for a MLL with layer-placement error, can be express as a higher-order polynomial. Figure 2 shows the ideal structure and error structure curves of the spatial frequency vs. radius which we used in the resolution.
Fig. 2. The spatial frequency as a function of radius for ideal structure and structure with layer placement error.
Figure 3(a) and Fig. 3(b) shows the intensity contour map calculated near the focal spot by BPM with ideal structure and with structural error; Fig. 4 shows the intensity profiles on the best focal plane.
Fig. 3. The wavefield intensity distributions near the focus of ideal MLL and MLL with low frequency error.
Fig. 4. Intensity profiles on the best focal plane with normalized intensity. The FWHM of central peak of the ideal MLL was 24 nm, the FWHM of central peak of the ideal MLL was 45 nm.
As shown in Fig. 4, though the layer-placement error deviated the MLL structure from the zone-plate equation, it still focuses X-rays to a size of around 45 nm, which is larger than the diffraction limit of the ideal structure of 24 nm. But, interference fringes around the diffraction peak on the graph were observed, but it was only observed on just one side of the diffraction peak. The Strehl ratio of the MLL with low frequency error is 0.5. (Strehl ratio is defined as the ratio of diffraction peak intensity in the best focal plane of the figured MLL to that of the ideal MLL). The interference fringes may be generated by the subtle phase effect as a result of the quadratic error term which will not appear if the MLL structure is perfect.
3.2 Figuring result
For MLL figured by the equal layer number and equal thickness subdivision methods, the wavefield intensity distributions near the focal spot are shown in Fig. 5(a) and Fig. 5(b). Figure 6 shows the intensity profiles on the best focal plane of two different subdivision methods.
Fig. 5. (a) Wavefield intensity distributions near the focus figured by ETM. (b) Wavefield intensity distributions near the focus figured by ENM.
Fig. 6. Intensity profiles on the best focal plane with normalized intensity. The FWHM of central peak of the ENM and ETM figuring result was 30 nm and 26 nm.
The phase error of the -1st order diffracted wave was 0.85π when the structural error is generated to the MLL. The phase error for the figured MLL by equal layer number and equal thickness subdivision methods is respectively 0.38π and 0.26π.
As shown in Fig. 6, the FWHM of the central peak of two different subdivision methods differ significantly, so does the side lobes around the central peak. The FWHM of the equal thickness subdivision method is almost the same as the ideal MLL, but with lower efficiency and some side lobes.
The Strehl ratio for the figured MLL by equal layer number and equal thickness subdivision methods is respectively 0.26 and 0.36. The side lobes may be caused by the high frequency structure introduced by figuring method. As the simulation suggests, the equal layer number subdivision method not only further lowers the Strehl ratio but also raises the side lobes even higher. According to the simulation result, figuring with equal thickness subdivision method is a better way to compensate for structural error of an MLL.
3.3 Phase plates
Phase plate was employed to correct the phase aberrations of MLLs. Due to the limited focal length, we typically employ the phase plate upstream from the MLLs, the phase plate induces a phase profile to the collimated wavefront, and thus will ideally compensates wavefront aberrations of the MLLs. Silver (Ag) was chosen as phase plate material as it has moderate density and offers a good compromise between the phase plate dimensions and δ/β ratio. The transmission of the phase plate is shown in Fig. 7. The overall transmission of the corrector used is 60%. The design principle of the phase plate is described in Ref. [16].
The wavefield intensity distributions near the focal spot are shown in Fig. 8 after phase plate was placed before MLL. The Strehl ratio for the phase plate compensated MLL is 0.19, and the side lobes are much smaller in comparison with peak intensity.
To compare the best figuring compensation result, phase plate compensation result, and the ideal MLL, the intensity profiles on the best focal plane of figuring result, the phase plate result, and ideal MLL are shown in Fig. 9. Figure 10 shows the Strehl ratio of different compensation methods. It can be seen that figuring can achieve better spatial resolution as well as higher efficiency.
Fig. 9. Intensity profiles on the best focal plane with normalized intensity of phase plate compensation result.
However, the Strehl ratio of the MLL with phase plate is much lower than the transmission of the Ag phase plate. Figure 11 shows the -1st order diffraction efficiency at the exit surface of the ideal MLL and the MLL with phase plate, as shown in the simulation result, the local diffraction intensity drops significantly around x = 6 µm, 10 µm and 12 µm, much lower than that of ideal MLL. Above all, the diffraction intensity at the outmost layer region (from 12 µm to 16 µm) is the lowest among the entire exit surface. Due to this, one can see that not only the lower efficiency, but also the effective numerical aperture (NA) decreases, which explained why the FWHM of the MLL with phase plate is much higher than that of the ideal MLL.
Fig. 11. The -1st order local diffraction intensities at the exit surface of Ideal MLL and MLL with phase plate.
3.4 Discussion
From the simulation results shown in the previous sections, we observe that in terms of FWHM of the focus spot as well as diffraction efficiency, figuring is a better wavefront compensation method than the phase plates. But figuring may cause some side peaks around the focus while the phase plates can keep the shape of the MLL focus perfectly. Side peaks may be generated by the high frequency structural features introduced by the figuring. This could be overcome by dividing MLL into a large number of sub-MLLs, whereby the figuring result will tend to be smoother. Equal thickness subdivision is a better subdivision method which may be attributed to the overall small sub-MLL thickness, for equal layer number method, the sub-MLL’s thickness becomes larger with decreasing radius.
As for the overall low efficiency of the phase-plate compensation result, it can be explained that the phase-plate changes the incident wavefront of the MLL, as a result, it does not satisfy the Bragg condition of the MLL, especially when the layer thickness is small. However, when the layer thickness is large, the local efficiency will not be sensitive to the Bragg condition. This also explains why the efficiency is much lower at thinner layer zones after the phase-plate is introduced.
The figured MLL can only be used in the corresponding energy, i.e., 20 keV in this article. This is due to the figuring parameters for the MLL, which are calculated based on the index of refraction of the material at a specified photon energy. So, the figured MLL can’t be used at different photon energies.
4. Conclusion
In this paper, we proposed a new figuring method to compensate for the MLLs layer-placement error. Simulation result shows that it is capable to compensate for the wavefront for MLLs, in terms of FWHM of the focus spot as well as the diffraction efficiency, figuring is a better wavefront compensation method than the phase plate. The figuring method can be realized by following steps: first, the multilayer structure is measured by the SEM [13,24]; then spatial frequency of the real multilayer structure can be achieved by polynomial fitting [13,25]; last, the figuring process can be performed following the description in the section 2.1. Such method will reduce the requirements for the fabrication of MLLs, namely the total thickness and layer placement accuracy of the multilayer structure, which are the main constraints on the development of the large aperture MLLs, thus making the production of larger NA MLL with the longer working distance possible.
Such method will be of benefit to some experiments, for instance, high NA and longer working distance will make high resolution in-situ XRF possible.
Funding
International Partnership Program of Chinese Academy of Sciences (113111KYSB20160021); National Natural Science Foundation of China (12005250, 22027810).
Acknowledgments
This work was supported by the National Natural Science Foundation of China and the International Partnership Program of Chinese Academy of Sciences; this work was supported by High Energy Photon Source (HEPS), a major national science and technology infrastructure in China, by the cross-research platform projects of Beijing Huairou Science City “Platform of Advanced Photon Source Technology R&D, PAPS”.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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