The experiment was performed at the X-ray Pump Probe (XPP)29 instrument at LCLS. The photon energy of the FEL was calibrated near the absorption edge of Ni K shell and set to 8.34 keV, with a bandwidth of ~30 eV full width. A float zone Si(400) crystal and a synthetic single crystal diamond (400) crystal were chosen as the Bragg optics. The grazing incidence mirror has a length of 300 mm and was coated with Pt. The cavity was initially aligned with monochromatic beam at 8.350 keV for calibration11. At this photon energy the Bragg angle of the Si crystal M1 is 33.14°; the diamond crystal M2 has a Bragg angle of 56.36°. The starting angle of the grazing incidence mirror M3 is at 0.5°. From this starting point the energy scans were executed in the negative direction from 8.35 to 8.34 keV.
The Bragg crystals are mounted onto three-axis piezo mirror mounts (Newport PSM2SG-D) that hosts both the X-ray crystal and the mirror for the interferometer IR laser beam. The piezo motors are mounted on stepper motor driven goniometer stage stack with larger range rotational motions to steer the beam reflection in both horizontal (yaw, θ1,2) and vertical (pitch, χ1,2). In addition, M1 can translate in in z direction (z1) along the incoming beam direction. M2 can translate in both the beam direction (z2) and the transverse horizontal direction (x2). The grazing incident mirror M3 has stepper motor driven motion to adjust the incidence angle (θ3) and translate in transverse horizontal direction (x3). Pre-programmed motion trajectory executing an Energy and Delay scan have the following motion vectors:
The laser interferometer is routed parallel but above the X-ray beam, at an elevation 25 mm higher. It’s in plane angle closely follows that of the X-ray except a higher grazing angle was used on the grazing incidence X-ray mirror M3 by 0.5 to reduce clipping by the mirror edges. The optical mirrors and X-ray crystals on M1/M2 motion stack were rigidly connected mechanically to minimize angular relative drift. The layout of the laser interferometer is shown in Fig. 3a. It is in a heterodyne Mach–Zehnder configuration operating at wavelength of \(\lambda =1064\,{\rm{nm}}\). The optical beams are in-coupled into the vacuum vessel via polarization preserving optical fibers and couplers. For the heterodyne interferometry the laser light from one fiber has a frequency offset of 4096 Hz as compared to the light from the other fiber. When the two beams are merged and interfered with one another, a beat note at the heterodyne frequency is observed. A single element diode is used to measure the length change \(\Delta l\) of the interferometer from the phase change \(\Delta \varphi\) of the interferometric beat note as
The data processing in the CDS is illustrated in Fig. 2b and based on the digital two quadrature demodulation described in ref. 33.
The beam delivery via two optical fibers can introduce differential phase noise into the measurement system, which was tracked and compensated. In Fig. 2a the optical layout of the laser measurement system is shown. The yellow and blue areas highlight the detection of the two interferometers, in yellow the reference interferometer and in blue the measurement interferometer. The reference interferometer is used to measure phase noise of the laser preparation and delivery system. This measurement is subtracted from the measurement interferometer readout to remove common mode phase noise. In addition, a fiber stretcher is used to cancel the differential path length noise in a feedback control loop. The phase noise and therefore length measurement is fully differential and does not track absolute path lengths.
The beam pointing deviation is measured via DWS and DPS with a QPD. DPS is the normalized position measurement on a segmented photodiode and measured as
$${{\rm{DPS}}}_{x,y}=\frac{{P}_{\text{right},\text{top}}-{P}_{\text{left},\text{bottom}}}{\sum \,P}$$
Calibration to position in \(x\) and \(y\) on the photodiode is related to the beam size \({\omega }_{{\rm{m}}}\) of the measurement beam on the diode as well as the power \({P}_{{\rm{m}},{\rm{r}}}\) of the measurement and reference beam respectively. Following ref. 34, the position on the diode is measured to be
$$x=\sqrt{\frac{{\rm{\pi }}}{8}}\cdot {{\rm{\omega }}}_{{\rm{m}}}\cdot \frac{{P}_{{\rm{m}}}+{P}_{{\rm{r}}}}{{P}_{{\rm{m}}}}\cdot {{\rm{DPS}}}_{x}$$
$$y=\sqrt{\frac{{\rm{\pi }}}{8}}\cdot {{\rm{\omega }}}_{{\rm{m}}}\cdot \frac{{P}_{{\rm{m}}}+{P}_{{\rm{r}}}}{{P}_{{\rm{m}}}}\cdot {{\rm{DPS}}}_{y}$$
The calibration factor is
$${c}_{{\rm{DPS}}}=\sqrt{\frac{{\rm{\pi }}}{8}}\cdot {{\rm{\omega }}}_{{\rm{m}}}\cdot \frac{{P}_{{\rm{m}}}+{P}_{{\rm{r}}}}{{P}_{{\rm{m}}}}$$
The DWS signal is measured in relation to the interferometric phase measurement \({\rm{\varphi }}\) as
$${{\rm{DWS}}}_{\varphi ,{\rm{yaw}},{\rm{pitch}}}={1/2{\rm{\varphi }}}_{\text{right},\text{top}}-1/2{{\rm{\varphi }}}_{\text{left},\text{bottom}}$$
Following ref. 28, the tilt \({\rm{\alpha }}\) of the wavefront is related to the DWS measurement in yaw as
$${{\rm{DWS}}}_{\varphi ,{\rm{yaw}}}\left({\rm{\alpha }},x\right)=\sqrt{\frac{2}{{\rm{\pi }}}}\cdot k\cdot {{\rm{\omega }}}_{\text{eff}}\cdot \left({\rm{\alpha }}-\frac{x}{{R}_{{\rm{m}}}}\right)\cdot F\left({\rm{\sigma }}\right)+O\left({{\rm{\alpha }}}^{2},{x}^{2}\right)$$
With the wavenumber \(k\), the effective beam radius \({\omega }_{\text{eff}}\), the beam tilt against the reference beam \(\alpha\), the horizontal displacement on the QPD \(x\), the radius of curvature of the wavefront of the measurement beam \({R}_{{\rm{m}}}\), and
$$F\left({\rm{\sigma }}\right)=\frac{1}{\sqrt{2}}\sqrt{\frac{1+\sqrt{1+{{\rm{\sigma }}}^{2}}}{1+{{\rm{\sigma }}}^{2}}},{\rm{\sigma }}=\frac{k{{\rm{\omega }}}_{\text{eff}}^{2}}{4{R}_{\text{rel}}},\frac{1}{{R}_{\text{rel}}}=\frac{1}{{R}_{\text{r}}}-\frac{1}{{R}_{\text{m}}},\frac{1}{{{\rm{\omega }}}_{\text{eff}}^{2}}=\frac{1}{{{\rm{\omega }}}_{\text{r}}^{2}}-\frac{1}{{{\rm{\omega }}}_{\text{m}}^{2}}$$
\({R}_{\text{r},\text{m}}\) and \({{\rm{\omega }}}_{\text{r},\text{m}}\) are the radius of curvature and beam radius at the QPD of the reference and measurement beam respectively. The equivalent equation for pitch and vertical displacement holds true. The DWS signal can be corrected with the DPS signal to turn it into a pure angular measurement. We calibrated the DPS correction signal by displacing the laser beam on the diode by \(x\) in the horizontal direction, without introducing a tilt \({\rm{\alpha }}\), which results in
$${{\rm{DWS}}}_{\varphi ,{\rm{yaw}}}\left(x\right)=-\sqrt{\frac{2}{{\rm{\pi }}}}\cdot k\cdot {{\rm{\omega }}}_{\text{eff}}\cdot \frac{x}{{R}_{m}}\cdot F\left({\rm{\sigma }}\right)$$
This is subtracted and the remaining DWS signal only depends on the angle \(\alpha\)
$${{\rm{DWS}}}_{\varphi ,{\rm{yaw}}}\left({\rm{\alpha }}\right)=\sqrt{\frac{2}{{\rm{\pi }}}}\cdot k\cdot {{\rm{\omega }}}_{\text{eff}}\cdot {\rm{\alpha }}\cdot F\left({\rm{\sigma }}\right)$$
The angle is measured as
$${\rm{\alpha }}=\sqrt{\frac{\pi }{2}}\cdot \frac{1}{{k\cdot {\rm{\omega }}}_{\text{eff}}\cdot F\left({\rm{\sigma }}\right)}\cdot {{\rm{DWS}}}_{\varphi ,{\rm{yaw}}}$$
In our signal processing chain, the phase is converted to an equivalent length change before the DWS signal is calculated, hence \({k\cdot {\rm{DWS}}}_{l,{\rm{yaw}},{\rm{pitch}}}={{\rm{DWS}}}_{\varphi ,{\rm{yaw}},{\rm{pitch}}}\). The angle measurement in our signal processing chain is calculated as
$${\rm{\alpha }}=\sqrt{\frac{\pi }{2}}\cdot \frac{1}{{{\rm{\omega }}}_{\text{eff}}\cdot F\left({\rm{\sigma }}\right)}\cdot {{\rm{DWS}}}_{l,{\rm{yaw}}}$$
$${\rm{\beta }}=\sqrt{\frac{\pi }{2}}\cdot \frac{1}{{{\rm{\omega }}}_{\text{eff}}\cdot F\left({\rm{\sigma }}\right)}\cdot {{\rm{DWS}}}_{l,{\rm{pitch}}}$$
The calibration factor is
$${c}_{{\rm{DWS}},l}=\sqrt{\frac{\pi }{2}}\cdot \frac{1}{{{\rm{\omega }}}_{\text{eff}}\cdot F\left({\rm{\sigma }}\right)}$$
All calibration factors have been determined experimentally by rotation and translation of the motion stages.
The beam tilts \(\Delta \alpha\) and \(\Delta \beta\) and length change \(\Delta l\) of the interferometer are related to the coordinate system of the half-cavity experiment, with the mirror tilts M2 yaw (\(\Delta \theta\)) and pitch(\(\Delta \chi\)) and the cavity length \(\Delta z\) as follows:
$$\Delta z=1/2\cdot \Delta l$$
$$\Delta \theta =1/2\cdot \Delta \alpha$$
$$\Delta \chi =1/2\cdot \Delta \beta$$
The data is presented in the coordinate of beam pitch (\(\Delta \beta\)) and yaw (\(\Delta \alpha\)) projected to mirror M2 and interferometer length change (\(\Delta l\)). The X-ray diagnostics does not allow us to measure the length, only the beam tilts.
The data processing is divided into two independent data acquisition and control systems. The LCLS data acquisition (DAQ) system records the data stream related to the X-ray diagnostics, the accelerator condition, as well as the beamline status. It is event-driven, synchronized to the generation of X-ray pulses on a 120 Hz clock. The images from the X-ray camera are captured at this rate. Post processing reduces the data to smaller packets containing fitted beam x-centroid and y-centroid, in synchronized array format with other beamline data such as pulse intensities, motor positions, accelerator parameters, etc. The synchronous motion control was executed using built in functions of the Aerotech Ensemble multi-axis motion controller. An analog signal linear to position within the planned motion trajectory was generated and sent from the controller to the analog input of the LCLS DAQ. This synchronized ADC value can then be used to calculate the equivalent energy change and time delay of a cavity, for each recorded X-ray beam image. Independent of the LCLS DAQ, a second data acquisition stream runs in parallel via the LIGO CDS and is responsible for the interferometer data processing and the real-time control loops. The sampling rate is 65536 Hz, but the data is filtered and stored as proprietary frame files at a rate of 2048 Hz. This data acquisition is not synchronized to the LCLS data acquisition system, but to the network timing protocol (NTP) and therefore to GPS time. Software tools allow quick and easy access to stored and live data. We have exported selected frame data to mat files.
The X-ray and interferometer data were then analyzed in MATLAB. As part of the X-ray data set, the intensity on the detector was recorded and a threshold was used for the X-ray data to exclude weak X-ray images without sufficient signal-to-noise ratio for an accurate centroid position. The centroid data is converted to an equivalent tilt angle of the beam. To measure shot-to-shot variance over the course of the experiment, a moving median of 180 data points was calculated and subtracted from the raw data. Large-scale movements are suppressed and the short-term jitter of the X-ray beam on the detector is shown in the histograms in Fig. 4, normalized to the probability density function (pdf). A fit to the normal distribution gives the standard deviation and is shown in Table 2. For the calculation of the shot-to-shot variance of the open-loop energy scan, data points had to be excluded because there were not enough data points available to calculate the moving median above 4.5 eV.