Extremum seeking x-ray position feedback using power line harmonic leakage as the perturbation

Small x-ray beam sizes necessary for probing nanoscale phenomena require exquisite stability to prevent data corruption by noise. One source of instability at synchrotron radiation x-ray beamlines is the slow detuning of x-ray optics to marginal alignment where the onset of clipping increases the beam’s susceptibility to higher frequency position oscillations. In this article, we show that a $1\mu \mathrm{m}$ amplitude horizontal x-ray beam oscillation driven by power line harmonic leakage into the electron storage ring can be used as perturbation for horizontal position extremum seeking feedback. Feedback performance is characterized by convergence to 1.5% away from maximum intensity at optimal alignment.

Figure 1

Schematic of the electron storage ring, undulator source, and beamline optics. Radio frequency (rf) accelerator cavities (not shown) replenish energy lost to x-rays generated by the bending magnet or undulator sources. The 1.8 kHz power line harmonic leaks through the high voltage rf power supplies driving a $1\mu \mathrm{m}$ amplitude horizontal oscillation in the particle beam. A narrow energy band of x-rays generated in the undulator are selected by the dual crystal monochromator (DCM). The horizontal focusing mirror (HFM) and vertical focusing mirror VFM steer and focus the beam at the sample. The collimating aperture reduces the beam size before passing through the sample. The intensity detector in this demonstration is a silicon PIN junction diode.

Figure 2

The phase reference is extracted from the horizontal beam position monitor (BPM) downstream from the DCM using a phase locked loop (PLL) with the voltage controlled oscillator (VCO) centered at 1.8 kHz. The phase locked signal is then fed into the lock-in amplifier (LIA) external reference input. The LIA detects the phase between the reference and intensity oscillations using a digital mixer (multiplication sign). The resultant signal from the LIA is then digitally integrated using a field programmable gate array (FPGA). The integrated signal out of the FPGA is used to control the HFM piezo actuator, steering the beam towards optimal intensity.

Figure 3

(a) Unencoded HFM piezo steering angle set value and (b) sample intensity convergence times as a function of feedback gain. Curves have been offset for clarity. The convergence time decreases with increasing feedback gain. The 30 Hz oscillations in the diode intensity are due to the HFM mechanical resonance at 30 Hz that is excited by applying a step deflection to the HFM steering angle set value. The black curves are best fits to an exponential.

Figure 4

(a) The power spectrum of the HFM steering angle set value to step deflection is shown in Fig. 3 for varying feedback gain. Curves have been offset for clarity. (b) The 3 dB cutoff increases linearly with feedback electronic gain.

Figure 5

The beam intensity through the $5\mu \mathrm{m}$ aperture as the HFM steering is scanned. The intensity FWHM is $93\mu \mathrm{m}$. The lock-in signal referenced to 360 Hz is simultaneously recorded. The horizontal oscillation amplitude can be extracted by comparing their amplitude ratio. The vertical line crosses through where the lock-in signal is zero.