We have carried out time resolved stroboscopic diffraction experiments on standing surface acoustic waves (SAWs) of Rayleigh type on a LiNbO3 substrate. A novel timing system has been developed and commissioned at the storage ring Petra III of Desy, allowing for phase locked stroboscopic diffraction experiments applicable to a broad range of timescales and experimental conditions. The combination of atomic structural resolution with temporal resolution on the picosecond time scale allows for the observation of the atomistic displacements for each time (or phase) point within the SAW period. A seamless transition between dynamical and kinematic scattering regimes as a function of the instantaneous surface amplitude induced by the standing SAW is observed. The interpretation and control of the experiment, in particular disentangling the diffraction effects (kinematic to dynamical diffraction regime) from possible non-linear surface effects is unambiguously enabled by the precise control of phase between the standing SAW and the synchrotron bunches. The example illustrates the great flexibility and universality of the presented timing system, opening up new opportunities for a broad range of time resolved experiments.
Surface acoustic waves (SAWs) are an enabling technology for electromechanical frequency filters, micro-fluidic devices1,2 as well as sensor applications.3 Here we show that time resolved diffraction, combining structural resolution on the sub-
The presented timing scheme is not limited to experiments using SAW excitation. It can be applied to a broader range of stroboscopic experiments, using e.g. laser, high voltage or microwave based excitation mechanisms.
II. TIME RESOLVED EXPERIMENTS AT P08 / PETRA III
The basic principle of a stroboscopic time resolved experiment is to match the frequency of a cyclic excitation (pump) with the repetition rate of short probe pulses. A precise variation of the time delay t between pump- and probe pulse allows to sample the dynamics induced by the pump pulse at a high number of time points. The temporal resolution is limited by probe pulse length and the overall jitter of the experimental setup. We start with a brief description the most important timing signals provided by the synchrotron storage ring. We then turn to a more detailed discussion on two distinct experimental configurations for time resolved x-ray experiments.
The revolution frequency of a synchrotron is approximately given by the circumference and the speed of light. More precisely, it depends on the energy E of the relativistic electron (or positron) in the storage ring, which in turn is set by the microwave frequency facc. of the accelerating cavities. Furthermore, the filling mode of the storage ring determines the repetition rate fb of the synchrotron bunches in an actual experiment. For Petra III the circumference of ≈2304 m leads to a revolution frequency of frev. ≈ 130.1 kHz (microwave frequency facc. ≈ 499.564 MHz, positron energy E ≈ 6 GeV) and hence a bunch frequency of fb ≈ 5.2 MHz in the 40-bunch mode. Note that facc. and hence fb are known and stable during each experimental run but may vary by some Hz between succeeding runs (machine alignment / tuning). The temporal structure of Petra III is highly favorable for time resolved experiments since the individual bunches are equally spaced in the the storage ring. Hence no global phase factor or a distinction between individual X-ray pulses is necessary when synchronizing to the synchrotron frequency. Moreover, the temporal spacing of
In order to synchronize the SAW to the synchrotron bunch frequency fb we used a modified version of the Petra III bunchclock. The bunchclock provides three independent and basically arbitrary integral dividers of the microwave frequency which can be used to synchronize pulsed excitation mechanisms (e.g. pulsed lasers) or to gate modern pixel detectors6,7 to the bunch frequency fb with phase stability. The exact frequency values and operating conditions of the storage ring are distributed to the bunchclocks via a modulated 40 MHz signal, see Fig. 1(a). Furthermore, a highly stable 10 MHz reference signal is distributed to every beamline which can be used in order to phase lock any external frequency- or delay generator to the synchrotron. This reference signal is generated from the microwave frequency by a rational divider. These timing capabilities lead to two modes of SAW excitation, see Fig. 2 for representative oscilloscope traces as taken during experiments in both modes:
In the first mode, standing SAWs (cw operation) at any integral multiple frequency fSAW = n · fb of the bunch frequency are generated by phase locking a frequency generator (SG384, Stanford Research Systems) to the 10 MHz reference, see Fig. 2(a). This allows for time resolved diffraction experiments without the need for a reduction of the effective X-ray frequency and hence flux by gated detectors6 or high speed choppers.8 The effective time delay t between SAW pump- and X-ray probe pulses can be set as a phase ϕ between the synchrotron pulses at fb and the SAW signal at fSAW directly on the frequency generator, time delay t and phase ϕ are related via
In the second mode, short SAW pulses at a repetition rate of
Both pump and gate pulses are referenced to a common trigger signal. In order to allow the pump pulse to be temporally ahead of the gate pulse, a pretrigger has to be generated. The gate pulse has hence been delayed by ≈990 μs on channel one of DG1 and fine tuned so that the gated intensity on the Pilatus detector is maximal, see Fig. 1(c). For a desired time delay of t the pump pulse is now delayed by 990 μs − t on channel two of DG1 so that the excitation pulse arrives effectively ahead of the gate pulse.
III. MATERIALS AND METHODS
A. Surface acoustic wave generation
Piezoelectric materials offer the possibility to excite SAWs electrically:9,10 A metallic comb-like finger structure is processed onto the surface by optical lithography and a RF voltage is applied to this so-called interdigital transducer (IDT), see Fig. 1(a). As the RF signal matches the design frequency
The SAW excited on 128° rot. Y LiNbO3 in the direction perpendicular to X is a Rayleigh mode, so surface atoms follow an elliptical motion polarized in the sagittal plane (defined by SAW propagation direction and the surface normal, see Fig. 1(b)). The piezoelectricity of the substrate material induces an electric potential Φ which is linked to the deformation of the crystal. Both the deformation and Φ decay exponentially into the substrate, a detailed derivation of the solutions to the piezoelectric wave equations can e.g. be found in Ref. 10. The surface of 128° rot. Y cut LiNbO3 is parallel to the (1,0,4) crystal plane (space group R3c), leading to a d-spacing of
It will be important to distinguish between the SAW amplitude H1, denoting the instantaneous amplitude at the substrate surface and the amplitude σ ∝ H1, denoting the effective amplitude averaged over the penetration depth of the X-ray beam (corresponding to an effective mean square displacement).
We will refer to the nominal RF-power U after the RF amplifier (given in dBm, see Fig. 1(a)) when giving experimental numbers characterizing the intensity of the SAW, as the SAW intensity is assumed to be proportional to U. The SAW amplitude H1 is proportional to
B. X-ray diffraction setup
The diffraction experiments have been performed at an X-ray energy of 18 keV at the high resolution diffraction beamline P08, Petra III, DESY.13 The sample was mounted vertically on a multi-circle diffractometer (Khozu NZD-3) in a horizontal scattering geometry, the X-ray beam being directed along the sound path of the SAW device. To ensure a perfect overlap of the X-ray beam with the acoustically excited area of the SAW device, the beam was collimated to 55 μm × 290 μm (h,v) using a slit system at a distance of ≈1 m as well as a set of collimating CRLs at a distance of 30 m in front of the sample. The X-ray attenuation length of 28 μm in LiNbO3 for the (1,0,4) Bragg reflection (θ = 7.2306° at E = 18 keV) is well below the penetration length of the SAW, which is on the order of the acoustic wavelength. The horizontal beam size of 55 μm leads to a footprint of 411 μm along the sample surface for the 1,0,4 reflection, hence only the acoustically excited substrate volume is probed by the X-ray beam. The energy bandwidth after the Si111 high heat-load monochromator and the Si311 large offset monochromator was
IV. ULTRAFAST TRANSITION BETWEEN DYNAMICAL AND KINEMATIC DIFFRACTION REGIMES IN LiNbO3 EXCITED BY SAWS
Two kinds of time resolved X-ray measurements have been performed in order to study ultrafast SAW induced atomistic displacements in the LiNbO3 crystal lattice.
(i) Rocking measurements around the LiNbO3 (1,0,4) Bragg reflection have been recorded for different phases ϕ between the standing wave and the synchrotron bunches, corresponding to different time delays
(ii) Time resolved intensity traces I(U, ϕ) of the LiNbO3 (1,0,4) Bragg reflection have been recorded as a function of ϕ for a given SAW RF-amplitude U at a fixed angle of incidence (Bragg angle).
Standing SAWs (cw excitation) have been applied in all cases, diffraction signals have been recorded stroboscopicly without the need for high speed gating or individual pulse selection, see section II. A typical oscilloscope trace as taken during phase locked stroboscopic experiments is depicted in Fig. 2(a). While time resolved rocking measurements are a valuable tool to disentangle the complex interplay between dynamical, kinematic and possible sample related effects, time resolved intensity traces I(U, ϕ) over a broader range of SAW intensities U help to prove the correctness of these interpretations and to further illustrate the experimental capabilities enabled by the timing scheme introduced in this paper.
While we will refer to the phase ϕ between the standing SAW at fSAW = n · fb and the synchrotron bunches at fb as the generic temporal coordinate for cw waves throughout the discussion, let us just shortly emphasize the close relation of ϕ to the temporal delay
A. General remarks on expected dynamical and kinematic diffraction effects
The interpretation of the observed intensity variations is based on scattering related effects, in particular on a SAW induced transition from dynamical to kinematic scattering regimes. Alternative explanations based e.g. on non-linear acoustic effects (i.e. higher acoustic harmonic generation) or temperature induced modifications of the SAW resonator have been carefully studied, but can not explain our experimental findings in full detail. To therefore facilitate the understanding of the complex interplay between diffraction related and sample related effects, three main remarks on x-ray diffraction from a SAW distorted crystal lattice will be given, before we turn to the discussion of the experimental data in sections IV B and IV C. In particular the need for dynamical diffraction theory will be motivated, providing a phenomenological description of extinction related effects. For LiNbO3 both extinction length and absorption length are of the same order of magnitude, as detailed below, so that the dynamical scattering of the perfect crystal can be modified by small changes in real structure (mosaicity) resulting in an increase of diffraction intensity. We will, however, treat this issue in rather simple terms. A detailed description of dynamical and kinematic scattering in terms of a unified and complete theory, for example as presented in Ref. 14 is beyond the scope of this paper which focuses on a proof of principle of phase stabilized time-resolved diffraction.
The instantaneous Debye Waller factor:
For standing waves I(U, ϕ) can in a first basic approach be understood in terms of the instantaneous Debye Waller factor
Kinematic approximation – Satellite reflections corresponding to a harmonically distorted crystal lattice:
Depending on the acoustic wavelength of the SAW, diffraction satellites arising from the SAW modulated substrate15 contribute to the resulting phase and amplitude dependent intensity traces I(U, ϕ). In case of the fifth harmonic SAW frequency (
Dynamical diffraction effects have to be taken into account for diffraction experiments on highly oriented LiNbO3 single crystals.16,17 Extinction effects strongly influence the shape and intensity of individual Bragg reflections since the linear attenuation length 1/μx ≈ 27.3 μm18 is more than one order of magnitude larger than the extinction length χ ≈ 1.2 μm for the undistorted LiNbO3 (1,0,4) reflection. Furthermore the Darwin width ΔθD = 18.6 μrad19 of the (1,0,4) Bragg reflection is significantly smaller than the horizontal beam divergence of Δαi ≈ 28 μrad, an angular integration is therefore performed even without sample rotation. It is then straightforward to show that a small distortion, or an increase in mosaicity and hence acceptance angle ΔθB, leads to an increase of the integrated intensity (see Fig. 3) rather than a decrease with respect to a perfectly ordered crystal lattice. In other words, the increase in extinction length in the dynamical theory of x-ray diffraction20 over-compensates the intensity decrease due to the Debye Waller factor in the kinematic approximation.
B. Time (i.e. phase ϕ) resolved rocking measurements
The two characteristic length scales of the SAW, namely its wavelength L and its amplitude H1, can be extracted from time (i.e. phase ϕ) resolved rocking curves around the (1,0,4) Bragg reflection, see Fig. 3. As shown in Ref. 21, the transverse acoustic amplitude
Let us briefly discuss the rocking curve measurements in terms of dynamical and kinematic diffraction effects. Without SAW or at nodes of the standing waves the LiNbO3 single crystal is perfect enough to be described by dynamical diffraction.16,17 When the amplitude of the SAW increases, the extinction length χ is extended by the strain field of the SAW, equivalent to an increased mosaicity and hence acceptance angle ΔθB, see Fig. 3(d). The prolonged extinction length leads to an increase of the peak intensity as well as a decrease of the rocking width in the dynamical diffraction regime. A transition from mostly dynamical scattering to regimes dominated by kinematic diffraction occurs as soon as χ becomes comparable to the linear attenuation length 1/μx. This transition is marked by the minimum of the rocking scan width as well as the maximum of the peak intensity at ϕ ≈ 20° and ϕ ≈ 160° in Fig. 3(b). Since the SAW amplitude oscillates as a standing wave, this transition occurs at 4fSAW (for increasing / decreasing amplitudes as well as positive and negative displacements).
Let us now identify the nodes of the standing wave, denoted as ϕ0, on the basis of the data depicted in Fig. 3(b). Nodes of the standing SAW strictly occur at a phase difference of Δϕ = 180°, therefore the minima of the FWHM curve, separated by Δϕ ≈ 140°, can not correspond to nodes of the standing wave, in line with the interpretation as a soft transition from dynamical to kinematic diffraction regimes. Instead, ϕ0 ≈ 0° can be identified with the minimum of the integrated intensity in Fig. 3(b). This interpretation is justified by the observation that, in contrast to the peak intensity, the integrated intensity follows the SAW excitation with a sinusoidal response, which is expected since it corresponds to the sum over N ⩾ 5 experimentally unresolved side reflections (satellites). The intensity of these satellites increases with the instantaneous SAW amplitude H115 and can be described in kinematic approximation. The further interpretation of the intensity traces I(U, ϕ) below is based on this identification of ϕ0 and the assumption that a mild increase of SAW amplitude leads to an increase of the peak intensity due to a reduction of extinction effects (dynamical diffraction theory) whereas high SAW amplitudes lead to a decrease of diffraction intensity due to the Debye-Waller factor in kinematic diffraction theory. This view is further supported by the U dependence of I(U, ϕ) as discussed in section IV C.
C. Intensity traces I (U, ϕ)
Let us now turn to the I(U, ϕ) dataset for the 1st harmonic frequency (
It was shown that the strain field induced by standing SAWs leads to a seamless transition from dynamical to kinematic diffraction regimes for a highly ordered LiNbO3 crystal lattice on the 100 ps timescale. An exact description of the observed effects in terms of detailed dynamical and kinematic diffraction equations (and especially a unified theory as e.g. derived in Ref. 14) is beyond the scope of this paper, but it becomes evident that the accurate control of phase between the standing acoustic wave and the synchrotron bunches has proven to be absolutely indispensable for the data interpretation. It has to be concluded that, independent of the underlying most likely scattering related mechanism, time resolved X-ray diffraction experiments serve as an unique tool to directly probe the atomistic displacements induced by SAWs. Measurement artifacts evoked by any electronic or mechanical effects in the IDTs can be excluded. The timing scheme of Petra III has been proven to be highly accurate and stable, its great flexibility will be highly beneficial for a broad range of time resolved X-ray experiments at Petra III.
We thank Dr. Oliver Seeck and Kathrin Pflaum for excellent support prior to as well as during the synchrotron experiment at P08, Petra III. Furthermore we thank Hans-Thomas Duhme and Jens Klute (MSK group, DESY) for their fast and uncomplicated modification of the Petra III bunchclock, as well as Dr. Karl Lautscham for important support in electronics. Support by the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG) through SFB 937 as well as the Courant Research Centre “Nano-Spectroscopy and X-Ray Imaging” is gratefully acknowledged. Part of this work has been sponsored by the German Initiative of Excellence under the ‘Nanosystems Initiative Munich (NIM)”.