3.1.

Alignment of the Polarimetry Setup

The most important task for alignment is to find the rotation axis of the light-source chamber and set it as the optical axis of the beamline. For this purpose, a small, compact red laser mounted on a multiaxis holder was first placed inside the light-source chamber near the worm gearing and pointed through the aperture to the sample chamber, as shown in Fig. 2. By carefully adjusting the multiaxis holder, the red laser spot, which was monitored ∼3 m away from the source, shifted within $\pm 1\mathrm{mm}$ when rotating the light-source chamber. This laser beam path was then considered as the rotation axis and the optical axis. Adjacent to this red laser, a compact green laser was installed at the left side outside the sample chamber and pointed along the optical axis toward the source chamber. The green laser beam was adjusted to be in alignment with the red laser beam along the whole optical axis.

Fig. 2

Schematic of the alignment method for the polarimetry setup.

After setting the optical axis, the polarizer mirror was set at a working angle of 45 deg with the aid of a right-angle prism and a plane mirror. The right-angle prism was placed at the center of the light-source chamber, and the prism’s leg face was adjusted to be perpendicular to the incident green laser beam. Therefore, the angle between the incident and exit green laser beam was precisely 90 deg with an accuracy of $\pm {30}^{″}$, which was determined by the angle tolerance of the prism. The exit green beam illuminated the plane mirror placed behind the mounting flange for the x-ray light tube. By adjusting the plane mirror’s holder, the green beam reflected from the plane mirror propagated along the path in the opposite direction to the incoming beam and returned to the green laser emitter. Finally, the right-angle prism was replaced by the Co/C multilayer polarizer, and the polarizer holder was carefully adjusted to ensure that the green laser beam reflected by the plane mirror returned to the laser emitter along the same path. After positioning the polarizer mirror, a soft x-ray light tube was mounted on the flange to ensure that the green laser pointed to the anode center.

In the sample chamber, a twin Co/C multilayer was used as the sample mirror and attached to a dual-axis mirror holder. The mirror holder was mounted on a rotary manipulator, which is not shown in Figs. 1 or 2. The sample mirror was first rotated to be perpendicular to the green laser beam. Then, the mirror holder was adjusted to reflect the green laser back to the emitter. Finally, the rotary manipulator was rotated clockwise by 135 deg to set the incidence angle at 45 deg. The angular resolution of the rotatable manipulator was $\sim 0.1\mathrm{deg}$. The soft x-ray beam reflected from the sample mirror was detected by the CCD detector oriented perpendicular to the optical axis, as shown in Fig. 1.

3.2.

Divergence and Stability of X-Ray Beam

In this polarimetry setup, the light-beam divergence depends on the aperture dimensions, distance between the source and aperture, and spot size of the light source. Because the aperture size and distance were known, the spot size of the light source was measured using a pinhole imaging method in this experiment for estimating the beam divergence. As shown in Fig. 1, a $50\text{-}\mu \mathrm{m}$-diameter pinhole was placed on the shaft of the first linear manipulator. The CCD detector was oriented directly toward the polarizer mirror. The aperture and multilayer sample were also removed without blocking the incident beam. The distance between the pinhole and light source was $\sim 1.3\mathrm{m}$, which was roughly the same as the image distance from the pinhole to the CCD detector. Therefore, the amplification factor of the pinhole imaging is $\sim 1$.

The pinhole image of the soft x-ray light source after exposure for 20 s is shown in Fig. 3(a), in which the light source was operated at a voltage of 5 kV and the x-ray beam current was 0.4 mA. Figures 3(b) and 3(c) show the horizontal and vertical projection data of the pinhole image, respectively. By fitting the experimental data with Gaussian profiles and taking account of the CCD pixel size of $13\mu \mathrm{m}$, the diameter of the x-ray spot was determined to be $\sim 0.43\mathrm{mm}$ (full width at half maximum; FWHM). According to the dimensions of the x-ray light source, we set up a simulation model for this optical system by using Lighttools software. Based on Monte–Carlo ray-tracing simulation results, the soft x-ray light-beam divergence was determined to be $\sim 0.08\mathrm{deg}$ (FWHM), which is much less than the angular bandwidth ($\mathrm{FWHM}\approx 0.7\mathrm{deg}$) of the curve of reflectivity as a function of incidence angle for the Co/C multilayer mirror.

Fig. 3

X-ray pinhole imaging experiment for the light source after alignment. (a) Horizontal projection data of the pinhole image and its fitting curve with a Gaussian profile. (b) Extracted CCD image having $80\times 80\text{pixels}$. (c) Vertical projection data of the pinhole image and its fitting curve with a Gaussian profile.

For polarimetry measurement, it is quite important to have a stable beam during rotation of the light-source chamber. After alignment, the incident polarized beam intensity was measured using the CCD detector mounted on the flange oriented toward the polarizer mirror. The pinhole was moved out of the optical path, and the collimating square aperture was used to reduce the beam divergence. In this measurement, the light-source chamber was rotated from 0 deg to 360 deg in steps of 10 deg. The voltage and beam current were set to have the same values as in the pinhole image experiment, and the CCD acquisition time was 30 s for each image at each step. The acquired image has a square shape similar to the aperture, which is not given here. Figure 4(a) shows the averaged CCD count rate per pixel as a function of the rotation angle. It is found that the intensity variation of the polarized x-ray beam was about 4.5% (relative root mean square error; RRMSE) during rotation of the light-source chamber. In the measurement, the image position was also found to be shifted in the horizontal and vertical directions when changing the rotation angle. The relative position shifts are shown in Fig. 4(b). The x-ray spot position shifted within $\pm 0.28\mathrm{mm}$ in the horizontal direction and $\pm 0.55\mathrm{mm}$ in the vertical direction, and the shifts are smaller than the spot size of $\sim 2\mathrm{mm}\times 2\mathrm{mm}$. These results demonstrate that the intensity and position stability of the rotatable polarized x-ray source are suitable for the polarimetry experiment.

Fig. 4

(a) Averaged CCD count rate per pixel and (b) relative beam position shifts as functions of the rotation angle.

3.3.

X-Ray Polarimetry Measurement

To test whether this system can generate 100% linearly polarized light, an x-ray polarization detection experiment was performed on a twin Co/C multilayer sample. The multilayer sample was oriented at 45 deg with respect to the incident x-ray beam. The CCD detector was placed at the horizontal plane and oriented perpendicular to the beamline, which means x-ray photons with a polarization vector perpendicular to the horizontal plane will be reflected by the multilayer and enter the detector. The voltage and beam current of the x-ray source were the same as those used in the stability experiment. For the polarimetry experiment, the CCD detector acquired an image for 120-s exposures at each angle when rotating the light-source chamber from 0 deg to 360 deg in steps of 10 deg. Figure 5(a) shows the averaged count rate per pixel as a function of the rotation angle, where a rotation angle of 0 deg (180 deg) indicates a polarization vector perpendicular to the plane of incidence (horizontal plane). As expected, the experimental data were well fitted by a sine wave, as shown by the red solid curve in Fig. 5. At rotation angles of 90 deg and 270 deg, the CCD count rates are almost zero, which implies that the combination of the soft x-ray light source and the multilayer polarizer produces nearly 100% linearly polarized x-rays. Additionally, the linearly polarized light can be rotated from 0 deg to 360 deg in the plane perpendicular to the beam-propagation direction. The count rates, as given in Figs. 4(a) and 5(a), represent the intensity of the incident and reflected beams, respectively. The ratio of the reflected intensity to the incident intensity is the reflectivity of the Co/C multilayer sample, which is shown in Fig. 5(b). It can be seen that the maximum reflectivity is 0.16 for s-polarized light, which is close to the estimated reflectivity of 0.18 according to our previous experimental results.⁷ For p-polarized light, the minimum reflectivity is almost zero, which is in agreement with the theoretical predication. It is noted that the incident and reflected beam intensities were not measured in the same experimental run, because the CCD detector has to be moved to a different flange. But for both measurements, the beam current of the light source was set at $0.4\pm 0.001\mathrm{mA}$. It was also checked that the incident-beam intensity variation was less than 2% when using this beam current.

Fig. 5

Results of the soft x-ray polarimetry experiment on a Co/C multilayer. (a) CCD count rate per pixel as a function of the rotation angle and a sine-wave fitting curve. (b) Reflectivity of the Co/C multilayer sample as a function of the rotation angle and a sine-wave fitting curve.

3.4.

Discussion on the Requirements on Experimental Errors

The figure of merit of a flight polarimeter is characterized by the minimum detectable polarization (MDP),²⁰ which is defined at a 99% confidence level as:

In our measurement, there are several factors which give rise in errors for the polarization and reflectivity measurement. The main systematic error is caused by the fluctuation of the incident beam flux during rotating the light source, which has the RRMSE of 4.5% as mentioned in Sec. 3.2. This error is mainly caused by the misalignment between the rotation axis of the light-source chamber and the optical axis of the beamline. In addition, the slight shift and vibration of the polarizer mirror during rotation may also contribute partly to the flux fluctuation. By combining this systemic error with the stability error of the incident beam intensity ($\mathrm{RRMSE}=2\%$ as mentioned in Sec. 3.3), the total error for the incident beam intensity is about 4.9%. For the reflectivity measurement as shown in Fig. 5(b), it is determined by the reflected beam intensity $I_{\mathrm{R}}$ normalized to the incident beam intensity $I_0$, namely $R=I_{\mathrm{R}}/I_0$. Since the reflected and incident beam intensities were not measured simultaneously, the fluctuations for both beams cannot cancel each other out. Therefore, the experimental error for the reflected beam intensity measurement should also be considered with the same RRMSE of 4.9% as the incident beam. By combining all errors together, the experimental error for the reflectivity is estimated to be about 6.9% (RRMSE). For the modulation factor, the measurement error is estimated using the equation $\mu =\frac{R_{\mathrm{s}}-R_{\mathrm{p}}}{R_{\mathrm{s}}+R_{\mathrm{p}}}$, where $R_{\mathrm{s}}$ and $R_{\mathrm{p}}$ are the reflectivity for the s- and p-polarized lights, namely the maximum and the minimum reflectivity in Fig. 5(b). According to the error propagation, the RRMSE for $\mu$ at the Brewster angle is estimated to be less than 2% because of the very small ratio of $R_{\mathrm{p}}/R_{\mathrm{s}}$. So, all the experimental errors for the reflectivity and the polarization measurements meet the requirements of RRMSE of 10% for the LAMP proposal. In our polarimeter setup, there are some other minor error sources, such as the incident beam position shift, the incidence angle variation, nonuniform response of the CCD detector, and statistic errors for collected counts, which are small and not considered in this paper.