## Abstract

We discuss misalignment-induced aberrations in a pair of crossed multilayer Laue lenses used for achieving a nanometer-scale x-ray point focus. We thoroughly investigate the impacts of two most important contributions, the orthogonality and the separation distance between two lenses. We find that misalignment in the orthogonality results in astigmatism at 45° and other inclination angles when coupled with a separation distance error. Theoretical explanation and experimental verification are provided. We show that to achieve a diffraction-limited point focus, accurate alignment of the azimuthal angle is required to ensure orthogonality between two lenses, and the required accuracy is scaled with the ratio of the focus size to the aperture size.

© 2017 Optical Society of America

## 1. Introduction

Scanning hard x-ray microscopy (SHXM) offers a suite of analytical tools via absorption-, phase-, fluorescence- and diffraction-contrast for imaging of structural, elemental and chemical variations at the micro- or nano-scale. The multimodality imaging capability makes SHXM a unique nondestructive tool applicable in many areas of science, providing information not accessible through other microscopy methods. To achieve high spatial resolution in SHXM, a focusing optic that can efficiently focus hard x-rays down to a small spot is critical. Owing to the advances in nanofabrication and metrology techniques, developments of hard x-ray nanofocusing optics such as K-B mirrors [1–3], Fresnel Zone plate [4–7], compound refractive lens [8] and multilayer Laue Lens [9,10] have made rapid progress, and the focus size has been steadily reduced well below 20 nm. Among these optics, MLL attracted a particular interest for SHXM applications due to its nanometer focusing capability and high efficiency [11]. Along MLL’s decade-long development path, large apertures [12,13], aberration-free lens [14] and wedged MLLs [15,16] have been realized, paving the way from a novel concept toward realistic optics for scientific applications [17–20].

There are always two sides to every story: using MLL optics in a scanning system has certain disadvantages. Because MLL is a linear focusing optic (a cylindrical lens), one needs two of them to produce a point focus, and in total eight degrees of freedom are required to perform full alignment. This not only poses challenges on the design of MLL manipulator [21,22], but also increases the complexity of the alignment process. Therefore it is very important to understand how individual degrees of freedom affect the focusing performance and what the accuracy requirements are. Here we discuss two most important factors, orthogonality and separation distance between two optics. Profound effects of these two alignment parameters on the focusing performance is investigated both theoretically and experimentally. We show that the angular error in the orthogonality alone can result in an astigmatism effect at 45°, and cause astigmatism at other inclination angles when coupled with the misalignment in the separation distance. At first glance, this result is counterintuitive. We provide a theoretical explanation and derive a set of equations that describe the phenomenon. Experimental results are presented and compared with the theoretical predictions. The goal of this paper is to understand the effect of misalignment on the focusing performance of two crossed MLLs, and define guidelines for the design of future MLL-based imaging instrument.

The presented analysis is generic and applicable to other types of cylindrical lenses as well. It is well known that the orthogonality error between two Kirkpatrick-Baez (K-B) mirrors will broaden the focus size and cause astigmatism [23–26], similar to what we observe for MLLs. The effect of orthogonality error on the focusing of K-B mirrors, to our knowledge, is mostly studied using geometrical ray-tracing simulation, which is not a good approximation near diffraction limit. In comparison, the adopted methodology in this paper is based on the more rigorous wave-optical simulation. Therefore it can be applied to investigate the non-orthogonality induced aberration for K-B mirrors with better accuracy.

## 2. Theoretical discussion

MLL resembles a one-dimensional Fresnel zone plate, but it utilizes the dynamical diffraction effects to improve its performance [11]. Therefore, an additional angular motion is required to fulfill the Bragg condition. When two MLLs are assembled together, they also have to be orthogonal with respect to each other. This leads to an azimuthal angle alignment between two lenses. Consequently, eight degrees of freedom (five translations and three rotations) are required to completely align two MLLs [27]. As shown in Fig. 1, the first MLL has *x*, y and *z* translational motions, one angular motion *θ*_{x} for the Bragg angle and the other *θ*_{z} for the azimuthal angle alignment. The second MLL has *x* and y translational motions and one angular motion *θ*_{y} for the Bragg angle. The in-plane translations ensure that MLL is fully illuminated by the incident beam. After the independent alignment of two individual lenses, one needs to align the separation distance between two MLLs to eliminate the astigmatism, i. e, foci in two different planes, and align the azimuthal angle, *θ*_{z}, to ensure the orthogonality. Alignment errors in these two degrees of freedom are the most important ones and will be discussed thoroughly in the following.

For the sake of simplicity, we consider a thin-lens model to describe the transfer function of the lens system,

*λ*is the wavelength. Here we ignore any amplitude or phase variations resulted from the diffraction effects. A more rigorous discussion taking into account these effects was reported previously [28]. Nevertheless, it is sufficient to illustrate the misalignment effect which matters for this study. Considering Fresnel diffraction with paraxial approximation, we obtain the wavefield at (

*x*′, y′,

*L*) in a measurement plane that is a distance

*L*from the lens,

*F*represents Fourier transform,

*p*is the pupil function,

*A*is the lens’ aperture size,

*ϕ*has a meaning of optical path difference and accounts for aberrations from the misalignment and the deviation of the measurement plane from the focal plane, and

*C*is a constant of no importance for this study. If MLLs are perfectly aligned, in the common focal plane of the two lenses the optical path function is zero. As a result, the wavefield is simply expressed as a Fourier transform of the pupil function. However, azimuthal misalignment can lead to a very small deviation angle,

*γ*, from the perfect perpendicularity, and there can be a small error in the separation distance, which leads to

*f*

_{1}≠

*f*

_{2}. Consequently, we arrive at (we assume the vertical focusing MLL is slightly rotated about

*z*axis),

In the derivation of Eq. (3), we assume *γ* is very small so that sin^{2} *γ* ≈ 0 and cos^{2} *γ* ≈ 1 Also, for the sake of generality, we leave *λ* in Eq. (3) not canceled out so the aberration function is expressed in units of the wavelength. Dimensionless terms enclosed in the parenthesis can be written into a simpler form,

When Eq. (5) is satisfied, we can see that along the direction, *u* + *bv* = 0, the optical path change is always zero. Therefore we infer that the incoming wave will be focused into a diffraction-limited spot in this specific direction even in the presence of misalignment. Because along other directions *ϕ* is not zero, it will form a line-shape focus. For given *f*_{1}, *f*_{2} and *γ*, there are two solutions of *L* to Eq. (5), indicating that this would occur in two planes,

When *γ* ≠ 0, they are neither focal planes of two lenses. That is, *L* ≠ *f*_{1} and *L* ≠ *f*_{2}.

To illustrate the impact of misalignment on the focusing performance, we conduct a theoretical simulation. Parameters used are *A* = 40 *µ*m and *f* = 4000 *µ*m at 12 keV. First, we consider the azimuthal misalignment only, so that *f*_{1} = *f*_{2} = *f*. In the common focal plane (*L* = *f*), we have *c*_{1} = *c*_{2} = 0 and only the cross term remains. We plot in Figs. 2(a)–(d) the intensity variation in the common focal plane as a function of the misalignment angle. Four angles, *γ* = 0°, 0.01°, 0.05° and 0.1°, which correspond to an optical path function of 0, 0.68*λuv*, 3.4*λuv* and 6.8*λuv*, are considered. As evident in the simulation, at 0.01° we can barely see any difference from the ideal case, but above 0.05° the broadening effect is appreciable. A tolerance on *γ* can be deduced [29],

The constant, *c _{γ}*, depends on the criterion used. It is either 0.5 based on a maximum phase change argument [28], or 0.9 based on a Strehl ratio of 0.8. Interestingly, the orthogonality tolerance is scaled with the ratio of the focus size,

*s*, to the aperture size,

*A*.

Second, at *γ*= 0.1° we move the measurement plane ±7 *µ*m away from the common focal plane, where we expect to see line foci based on Eq. 6. Figures 2(e) and (f) show the corresponding intensity distributions in these planes. A tilted line focus can be seen on each plane, with an inclination angle of ±45°. Because we have *f*_{1} = *f*_{2}, it is easy to verify that *b* ≡ ±1. Therefore the tilting angle is always 45°, independent of the azimuthal misalignment. It is not the case, however, when there is also an error in the separation distance. In the third case, in addition to an angular misalignment of 0.1° we move the vertical focusing lens 20 m upstream, so that the two lenses do not share a common focal plane. From Eq. (6) we can obtain *L* = 3978 and 4002 *µ*m, respectively. In these two planes line foci are seen [Figs. 2(g) and (h)], but the tilting angle is no longer 45°. Its value can be determined from Eq. (4).

As this theoretical study indicates, the azimuthal misalignment between two linear focusing optics alone can result in astigmatism at 45°. When it is coupled with the mismatch of the focal distance between two lenses, the astigmatism occurs at other inclination angles. When this happens, we expect to see two line foci in planes which are neither of their focal planes.

Aberration theory for circular lens has been well established [30]. To make a connection to the existing theory, we express Eq. (3) in the polar coordinates,

They are Zernike polynomials of number 3, 4 and 5, and are known to cause the shift of the focus and astigmatism, respectively [31]. For rectangular apertures, there is also effort to expand the aberration function into 2D Chebyshev polynomials [32]. In our case, we have,

It is known that the cross term alone, which corresponds to the Chebyshev polynomial of number 4, causes a 45° astigmatism.

## 3. Experiment

In this section we investigate experimentally impacts of the alignment errors in the azimuthal angle and the separation distance. The experiment was conducted at the hard x-ray nanoprobe (HXN) of the National Synchrotron Light Source II, Brookhaven National laboratory. Two MLLs, one with an aperture size of 53 *µ*m and focal length of 5.3 mm and the other with an aperture size of 43 um and focal length of 4.3 mm, were tested on the dedicated nanoscale multimodality imaging instrument (nano-Mii). Both lenses ideally produce a diffraction-limited focus of 10 nm (Rayleigh criterion). The MLL with a larger aperture size was used for vertical focusing and was placed upstream. Monochromatic beam at 12 keV was first shaped by a beam-defining-slits and then incident on MLLs. An order-sorting-aperture was used to block the transmitted beam and high-order foci [33]. The sample was a 10-*µ*m thick Si membrane with Au nanoparticles on the surface, which were formed by annealing a 20 nm Au film at 800°C for 8 hours. The nanoparticle has a size ranging from 100 nm to 500 nm. The sample was placed in the focal plane and was raster-scanned for both fluorescence imaging and ptychography reconstruction [34]. A pixel-array detector (Merline, 512 × 512 pixels, 55 *µ*m pixel size) was placed 0.5 m downstream to record the farfield diffraction pattern from the nanobeam. Ptychography allowed the reconstruction of the complex wavefront of the probe so we could simulate beam propagation along the optical axis and search for the best focus.

In Figure 3 we show the reconstructed results under six different conditions. Figure 3(a) corresponds to the best aligned condition, in this case both alignment errors, i.e. orthogonality and separation distance, are minimized. Ptychography reconstruction produces a sharp point focus in the sample plane, with full-width-of-half-maximum (FWHM) size of 14 and 12 nm in horizontal and vertical directions, respectively. Propagating the wavefront along the optical axis indicates that the sample is right in the focal plane, and there is no astigmatism (see Visualization 1). The reconstructed particle has a hexagonal shape and a size of about 150 nm. The acquired fluorescence image in the same time confirms these observations, showing an Au particle with six facets clearly visible. Figure 3(b) corresponds to the case where the sample still sits in the common focal plane, but with an azimuthal misalignment of 0.3°. The focus is broadened two-dimensionally, as evident from the ptychography reconstruction. The focus size (FWHM) becomes 59 nm and 85 nm. The reconstructed particle remains the same, which confirms high fidelity of the reconstruction result. Because of the much larger focus, the fluorescence image of the same particle is blurred two-dimensionally, and facets are no longer visible.

We show in the preceding section that the azimuthal misalignment can lead to astigmatism at 45°. If we substitute all numbers to Eq. (6), in two planes ±21 *µ*m away from the focal plane line foci are expected. Figure 3(c) (upstream) and Fig. 3(d) (downstream) depict experimental results in such two planes. As can be seen, the probe becomes a line focus tilted at 45°. A small focus is still achieved in the direction perpendicular to the line. Severe astigmatism is also evident in fluorescence images, where the particle is elongated diagonally. For a better illustration of this phenomenon, a movie showing the variation of the intensity distribution when the nano-beam propagates along the axis is provided (see
Visualization 2).

When the azimuthal misalignment is coupled with the separation distance error, the situation becomes more complicated. In the last two cases, we moved the vertical MLL 21 *µ*m upstream so that there was a mismatch of the focal distance between two lenses. From Eq. (6), in planes 34 *µ*m upstream and 14 *µ*m downstream line foci tilted non-diagonally were expected, and were confirmed by the experimental data shown in Figs. 3(e) and (f). Also, fluorescence images of the same particle acquired simultaneously depicted strong astigmatism effect in corresponding directions.

The strong coupling between these two types of errors complicates the alignment process. If one could be taken out from the equation, the alignment process would become much simpler. A pre-alignment of the azimuthal angle can be done with the far-field pixel-array detector. Because the detector has perfectly aligned square pixels, they can be used as a good reference. If one aligns the line-shaped far-field diffraction pattern from each MLL with the detector pixels, orthogonality can be ensured to a certain degree. The accuracy depends on the spread of the line on the detector, and an accuracy better than 0.1° is achievable. Nevertheless, in order to achieve a truly aberration-free point focus, a few iterations of aligning the azimuthal angle and the separation distance in sequence are required. As the focus size decreases, the required alignment accuracy increases. To show this dependence, we plot the orthogonality tolerance [Fig. 4(a)] and separation distance tolerance [Fig. 4(b)] as a function of the focus size. The former also depends on the aperture size, and the latter is a function of the energy as well.

## 4. Conclusion

In summary, we discussed the required accuracy in aligning two individual MLLs to achieve an aberration-free point focus. We thoroughly investigated two factors that mostly affect the focusing performance. One is the orthogonality and the other is the separation distance. We showed that error in the former could lead to astigmatism at 45°, a phenomenon that was counterintuitive. When the error in the separation distance was present, astigmatism occurred at other inclination angles. A theoretical discussion on the origin of the astigmatism was provided. Experimental data were presented to validate theoretical predictions. Misalignment may be the last obstacle on a way of achieving a diffraction-limited nanometer-scale point focus with two crossed MLLs, and it has to be reduced below the tolerance to avert aberrations that distort the focus. This work reveals how alignment errors in the orthogonality and the separation distance between two crossed MLLs affect the point focusing performance. Present work is not only critical for the alignment of similar systems, but also defines guidelines for the design of future MLL-based instruments that aims at higher spatial resolution.

## Funding

Office of Science, Department of Energy (DE-SC0012704).

## Acknowledgments

The experiment was performed at the beamline 3ID of the National Synchrotron Light Source II, a U.S. Department of Energy (DOE) Office of Science User Facility operated for the DOE Office of Science by Brookhaven National Laboratory under Contract No. DE-SC0012704.

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