1. Introduction

X-ray microscopy using synchrotron radiation is becoming an important tool to analyze matter on the nanoscale in many different disciplines, like material, environmental and life science, to name only a few [1]. For hard x-rays, i.e. energies between 5 – 20 keV, the so-called scanning x-ray microscope or hard x-ray nanoprobe arrangement is commonly used [2]. Here, nano-focusing optics like a zone plate, nanofocusing lens or mirror optic are used to focus an intense x-ray undulator beam onto a sample which is raster-scanned to obtain a 2D image. Tomographic nano-imaging can be performed if 2D images from many different angles are acquired by rotating the sample in the beam. Current development goes towards the implementation of optics and instruments that are capable of x-ray nano-tomography at the 10 nm resolution level [3, 4]. This poses strict stability requirements on the two major components of an x-ray microscope: the optics positioning device and the sample stage. It is especially important for x-ray nano-tomography where acquisition of complete datasets can take rather long time, up to several hours. Here we present a Moiré method capable of measuring directly the relative instability between optics and sample stage in a hard x-ray scanning microscope on the nanometer level.

The Moiré effect in optical physics describes the phenomenon when by overlapping two repetitive structures a new pattern becomes visible that is not present in the original structures. It has been realized that the Moiré phenomenon offers the basis for metrology techniques that can measure displacements between objects. A recent area of application is proximity and nanoimprint lithography where a special arrangement of grating structures with slightly different periods creates Moiré patterns that can be used to align mask and sample with nanometer precision [5–7]. In this case the Moiré pattern arises by overlapping a near-field Talbot image of one grating (called image grating) with a second grating (reference grating). We have adopted this scheme for the instability investigation of x-ray microscope components by introducing a lens between the gratings.

While designs of modern hard x-ray scanning microscopes can be quite different and mainly dependent on the optics used, they all share two major components: optics positioning device and sample stage. These are normally equipped with nanometer-resolution positioners in order to accurately align optics and sample to the x-ray beam and to scan and rotate the sample. Typical working distances between optics and sample are a few centimeters up to tens of centimeters, as defined by the focal length of the used nano-focusing optic. Larger working distances are preferred, since they allow easier design of complex sample environments like cryo-cooling or heating. In order to reach 10 nm spatial resolution in the x-ray images, it is obvious that the positioning stages need high intrinsic stability on the nm level. Moreover, the relative position between optics and sample has to be stable on the same level during a measurement. Fiber laser interferometers and differential laser interferometers are commonly used to measure positional instabilities [4,8,9].

In this paper we present a new method based on Moiré patterns to investigate the relative instability of the optics and sample stages in the two dimensions perpendicular to the x-ray beam. The arrangement comes at low cost since it uses only a simple light emitting diode, two custom optical grating objects, a triplet lens and a low-magnification microscope. Since the first grating object is mounted instead of the x-ray nanofocusing optic and the second grating object instead of the sample, the arrangement simulates the situation of a real x-ray experiment but using visible light. Results show that the relative positional instability in two perpendicular directions can be measured down to few nanometers at repetition rates up to 140 Hz, limited only by the frame rate of the microscope camera. We believe that our method is an important tool to complement measurements from other fiber and differential interferometry channels in order to pre-characterize the stability and performance of x-ray microscopes prior to x-ray experiments, saving valuable x-ray beamtime.

2. Experimental arrangement

Our Moiré measurement technique uses two grating samples, which we call image grating and reference grating (Fig. 1). Each grating sample contains 4 different gratings, two in horizontal (x) and two in vertical (y) direction. The two gratings that belong to the same direction have slightly different grating periods p₁ and p₂, and the position of gratings with the identical period are interchanged between the image and reference grating. Moiré patterns are formed by imaging the image grating with a triplet lens in 1:1 imaging geometry onto the reference grating, resulting in Moiré fringes with a period of

 

Fig. 1 Schematic arrangement of the method.

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We used grating patterns with periods of p₁ = 5 µm and p₂ = 6.1 µm for our experiments. Gratings consisted of a thin 100 nm Chromium layer on glass and were fabricated in-house by e-beam lithography and a wet-etch process. The novelty of our arrangement is that image and reference gratings are not in close contact, but several cm away from each other. This was achieved by a lens placed in between, in our case a commercial Steinheil triplet (Thorlabs), optimized for a 1:1 imaging geometry, with 0.5 inch diameter and a working distance of 35 mm. The resulting distance between image and reference grating was approx. 85 mm, which is a typical value for our scanning hard x-ray microscope using zone plate optics. The triplet lens was mounted on a rigid block of aluminum in order to minimize its vibrations. The arrangement was illuminated by a simple LED at 534 nm wavelength, situated approx. 100 mm in front of the image grating. The position of the LED was adjusted for optimal contrast in all four Moiré patterns. No special collector optics was needed, since light levels were sufficient to saturate the detector and could easily be adjusted by the voltage applied to the LED. We tested also laser diodes as light source but with limited success due to the formation of strong speckles in the image. Moiré patterns were observed with a microscope consisting of a 5× magnification long-working distance objective (Edmund Optics, working distance 34 mm) and a standard Ethernet CMOS camera (Basler Ace), resulting in a resolution of approx. 1 µm/pixel. This made it possible to observe simultaneously both the original gratings and the Moiré fringes in the images.

The hard x-ray scanning microscope prototype we investigated was developed in a collaboration between the synchrotron radiation facilities SOLEIL (France) and MAXIV (Sweden). The goal was to build a modular x-ray microscope from commercial components only but with the ability to perform 3D nano-tomography experiments on the 10 nm resolution level. The system consists of two independent positioning stages: one for the sample and one for the focusing optics.

The optics stage is designed for stacking of two Fresnel zone plates, which is the type of optics used for focusing X-rays. It features a flexure-based kinematic design with a total of 10 stick-slip piezo stages (SmarACT SLC1720) that allows manipulating both optical elements with 5 degrees of freedom with nanometer increments. Positions of the two zone plates are monitored with 8 fiber laser interferometers (Attocube FPS3010), enabling active feedback loops on the positions and orientation of each zone plate. The image grating was mounted at the position of one of the zone plates.

The sample stage consists of the linear stage for fast continuous scanning (Aerotech ANT95-L) at the bottom of the stack, two multi-feet piezo stages (PI LPS-65) for sample positioning and step-scanning, a stick-slip piezo rotary stage (Xeryon XRTA) and two additional multi-feet piezo stages (Nanos LPS30-30) for sample positioning. Such stack of different stages allows for nanometric sample positioning and fast continuous scanning for high resolution 3D tomographic imaging. With current x-ray imaging techniques resolution is mostly limited by the positioning errors that occur due to vibrations and drifts of the sample during the scans. Vibrations can originate from external sources or can be generated by the internal control loops of the positioning stages. Typical frequencies for the vibrations are in the range of few hundred Hz to few kHz. Slower drifts are usually of thermal origin, where temperature changes of the mechanically active parts lead to thermal expansion or contraction. Temperature variations are also result of external factors, such as ambient room temperature as well as internal heat dissipation from the stages and encoders. In our set-up the thermal drifts were reduced down to 5 mK/h using water cooling of the base plate hosting the mechanical parts. Additionally, air turbulence was reduced by a plastic enclosure on top of the granite block on which the prototype was situated. The granite block itself was insulated from ambient air using thermal insulation materials. A more detailed description of the sample stage and its performance can be found in Ref. [10].

In order to monitor the stability of the sample with respect to the focusing optics, the reference grating was placed on top of the sample holder. For the experiments described in the next section, no active interferometric feedback was used. Instead, we tested the passive stability of the system, relying on the thermal and mechanical stability. All stages operated in closed-loop mode using internal optical encoders. We then measured the relative displacements between the sample stage and the optics using the Moiré method while keeping the stages in place, as well as performing step movements. Additionally, we investigated the dynamic performance of the method by recording the transient response after a mechanical impact.

Figure 2 shows a photograph of the complete experimental arrangement with the zone plate, the sample stage and the triplet lens in between.

 

Fig. 2 Photograph of the experiment with (from right to left): LED, zone plate stage with image grating, triplet lens, sample stage with reference grating and long-working distance microscope objective.

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3. Results

A typical Moiré pattern obtained on the detector is shown in Fig. 3 a). It shows that contrast in the fringes was good, consequently we used raw images for further data processing. The first step was to integrate each of the four patterns in the direction parallel to the grating lines. The resulting vector was multiplied with a window function before performing a discrete Fourier transform. As an example, the absolute value of the transform for the upper horizontal grating (marked with a white rectangle in a)) can be seen in Fig. 3 b). Main frequency peaks indicate the two grating periods and the Moiré frequency, marked with crosses. The phase angle of the same data is plotted in Fig. 3 c), and the phase values were extracted at the position of the Moiré frequency determined in the previous step. This was done for all four patterns and the relative displacements were then calculated with Eq. 2 for the horizontal (Δx) and vertical (Δy) directions. In order to simplify and accelerate data analysis, special analysis software was written in MATLAB that automatically analyzed camera images and calculated displacements. Effectively, the evaluation of a large dataset with a few 1000 images took below one minute.

 

Fig. 3 a) Example of a Moiré image obtained from the detector. The white rectangular marks one area that is used for analysis, and the coordinate system is indicated in the top right corner. b) Discrete Fourier transform (absolute value) of the signal in the white rectangular area after integration in vertical y direction. The Moiré frequencies are marked with crosses. c) Phase angle representation of the same data. The phase values at the Moiré frequencies that are used for displacement calculations are indicated with crosses.

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We performed four different experiments to test the method and evaluate instabilities in our microscope setup. These are summarized in the graphs of Fig. 4. For the first measurement series we recorded images at 40 frames/s for a total of 0.6 sec without moving anything. Since on this short time scales thermal drifts were minimal, we probed the intrinsic steady-state instabilities of optics and sample stage due to high frequency vibrations. Results are shown in Fig. 4 a). The positional instabilities are ±2.5 nm in the horizontal and ±5 nm in the vertical direction. This confirms early measurements of vibration instabilities of optics and sample stage done with fiber interferometers.

 

Fig. 4 Examples of different types of measurements performed with Moiré interferometry. a) Steady-state vibration level measured at 40 Hz is within ±5 nm. b) “Pyramid” scan step test of the linear bottom stage performing 10 nm steps along x. No parasitic motion along y indicates good quality of the drive mechanics. c) Response to a strong punch applied in horizontal direction to the underlying granite block. Measurements show irreversible displacement of 5.8 µm and oscillations at the frequency of 34 Hz. in x direction d) Long term thermal drifts measured during 10 hours correlate along x with changes of the ambient temperature by 60 mK.

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The measurement accuracy of the method was estimated by an error analysis: a separate MATLAB script took all experimental parameters as input as well as contrast and noise levels of measured images. It then simulated 30 identical Moiré patterns with the same contrast and noise levels but with random noise for each individual image. These were analyzed with the same software routines as for the measured images, and the resulting variations in the calculated relative positions according to Eq. 2 were taken as an error estimate. These resulting errors were smaller than 1 nm, well below the instability levels of our microscope.

In the second measurement a “pyramid” scan was performed in the horizontal direction with 10 nm steps (Fig 4 b)). The step movements are clearly visible in the horizontal direction, while the relative position in the vertical direction stays within short-term vibration levels.

The third measurement tested at which maximum repetition rate we could measure. For this we reduced the readout area on the detector to cover approx. half the Moiré patterns, and exposure time was minimized. In this way we could achieve a maximum camera readout frequency of 140 Hz. We used this mode to investigate the transient response of the microscope to a fast mechanical impact. This was done by a quick but strong punch on the side of the granite block on which the system was situated. The result is shown in Fig. 4 c). The relative position was affected by several micrometers, but the difference between the two directions was substantial. While in vertical direction the system seemed to come back close to its initial position, a permanent position jump of 5.8 micrometers was observed in horizontal direction. This was followed by a sinusoidal damped oscillation of the signal until position stabilized again. The initial jump might be explained by a permanent change in position of one of the piezo actuators due to the heavy impact. The sinusoidal signal indicates that we were able to measure a transient shock wave that traveled through the granite.

The last measurement looked at the long-term instability of the system, which is mainly induced by thermal drifts, i.e., temperature variations. Data were taken over night during a period of 10 hours and in time intervals of 2 sec. Fig 4 d) displays the results together with a measurement of the temperature in the base plate. During the whole period the temperature dropped by approx. 60 mK. Interestingly, the position curve for the horizontal direction followed largely the slope of the temperature, which indicates that there is a clear correlation between position drift and temperature drift. On the other hand, the position data for the vertical direction does not show an obvious connection. We think the reason for this is the close thermal contact of the horizontal sample stage (Aerotech ANT95-L) with the base plate. The vertical stages on top of it are less coupled and consequently follow thermal variations with lower amplitude. The same is true for the optics stages, since they are not in direct contact with the base. The remaining different dynamics in vertical direction most probably originate from ambient air temperature variations. Finally, it has to be noticed that long term position errors can be measured during x-ray experiments with laser interferometers and compensated for by translation of according positioners [4,8,9]. Unfortunately, our current x-ray microscope prototype does not have this possibility, but future development should implement this in order to reach best-possible resolution levels.

4. Conclusion

Modern hard x-ray nano-tomograhy is approaching the 10 nm resolution level thanks to the development of new highly-brilliant synchrotron radiation facilities and new and better designs of scanning hard x-ray microscopes. It is of paramount importance to be able to verify the performance of newly developed microscopes with respect to stability conveniently and preferentially without x-ray beam. Our Moiré method can do exactly this, using just simple optical components and visible light. We were able to investigate both short-term and long term instabilities of an x-ray microscope prototype with nanometer precision. Time resolution was limited mainly by the camera technology, rather than the measurement principle, opening the way to faster data acquisition. With better cameras the measurement speed could potentially be increased into the kHz regime. We think the method can be become an important tool in future instrument development.