X-Ray Second Harmonic Generation

We report clear experimental evidence for second harmonic generation at hard x-ray wavelengths. Using a 1.7 Å pumping beam generated by a free electron laser, we observe second harmonic generation in diamond. The generated second harmonic is of order 10 times the background radiation, scales quadratically with pump pulse energy, and is generated over a narrow phase-matching condition. Of importance for future experiments, our results indicate that it is possible to observe nonlinear x-ray processes in crystals at pump intensities exceeding ${10}^{16}\mathrm{W}/{\mathrm{cm}}^2$.

Figure 1

Schematic of the SHG experimental setup. X-ray pulses are incident to a silicon (111) channel cut monochromator. The monochromator selects a 7.3 keV pump beam with 1 eV bandwidth while rejecting a substantial fraction of the second harmonic contamination. A pair of Kirkpatrick Baez mirrors focuses the beam to about $1.5\mu \mathrm{m}$ while reducing the third harmonic contamination. The nonlinear medium is a (111) cut diamond crystal placed in the focus and set for phase matching as indicated in the phase matching diagram (bottom). The phase matching is achieved by using the (220) atomic planes, ${\overrightarrow{k}}_{\omega }$ and ${\overrightarrow{k}}_{2\omega }$ are the wave vectors of the pump and the generated second harmonic, respectively. $\overrightarrow{\mathrm{G}}$ is the reciprocal lattice vector orthogonal to the (220) atomic planes. We control the intensity of the pump by using a set of Al filters before the sample. The second harmonic is detected using a scintillator crystal and photomultiplier tube with additional filters used to reduce the probability of detecting multiple 7.3 keV photons per pulse.

Figure 2

Photon-energy histograms of the measured signal at the peak of the phase-matching condition. The pump-photon energy is 7.3 keV and the SHG photon energy is 14.6 keV. The energy resolution of the detector is $\sim 30\%$. The data in part (a) are taken with no attenuation. The data in part (b) are taken with a 0.025 mm Al filter before the diamond crystal (the transmission through the filter is 0.43 and 0.9 at 7.3 and 14.6 keV, respectively).

Figure 3

Scaling of second harmonic rate with average pump pulse energy and angular deviation. (a) Detected second harmonic signal as a function of input pulse energy and measured at angular deviation of $-57\mu \mathrm{rad}$ from the SHG phase-matching angle. The pulse energy is varied by inserting thin Al filters before the diamond crystal. Each of the measured data points represents an average over 6000 pulses. The dashed blue curve is calculated from the solutions Eq. (2) with a Gaussian chirp model and scaled vertically by a factor of 1.48. (b) Normalized second harmonic signal as a function of the angular deviation of the diamond crystal from the phase matching angle. The data in part (b) are obtained with no attenuation before the diamond crystal. The dashed curve is a Gaussian fit to the experimental data. The FWHM of the rocking curve is $180\mu \mathrm{rad}$. The peak point represents the average signal measured over 2400 pulses. The other points represent the average signal measured over 6000 pulses. The vertical error bars indicate the counting statistics.

Figure 4

Detected second harmonic rate as a function of the pulse energy of the pump at the phase matching condition. The pulse-energy dependence is obtained from the pulse-energy fluctuations of the XFEL beam. The dashed line is the polynomial fit to the experimental data as described in the text. The vertical error bars indicate the counting statistics. The horizontal error bars are the standard deviation of the pulse energy within a bin. The inset shows the histogram of the pulse energy of the XFEL beam.