Results: Simulated Data

Chapter 5. The Physics of Transient Diffraction with Ultrafast Streak Cameras

lower row contains the diffraction images at the corresponding times that were reconstructed using Eq.(5.19). From visually inspecting these images it is evident that the technique developed in this chapter does an extremely good job at recovering the underlying dynamics.

Comparing the two sets of images confirms that the shape, position, and intensity of the diffraction spot is fairly accurately reproduced, perhaps with a few small artifacts. The fidelity of the recovered diffraction images to the input, simulated images persists both before and after t = 0, which is indicative of the spatially-varying deconvolution algorithm’s ability to catch the onset of the dynamics.

t = 0

Figure 5.6.: Comparison between the time-dependent amplitude of a simulated diffraction spot recovered by the spatially-varying deconvolution algorithm and the tradi-tional approach to analyze streaked diffraction (by taking the intensity along the streaked image)

The successful recovery is even more apparent in Fig. 5.6, which plots the average diffraction spot intensity as a function of the time delay taken from the sequence of images in Fig. 5.5.

Qualitatively, the amplitude of the recovered diffraction pattern very closely matches the true (input) dynamics. In fact, the mean error between the two is only ∼ 1 %. This result reassures that the spatially-varying deconvolution algorithm introduced in this chapter is capable of retrieving the underlying dynamics.

Also shown in Fig.5.6is the diffraction peak amplitude measured by the traditional approach of taking the intensity profile along the streaked image (which, in this case, is indicated by the dashed line in the streaked image shown over the figure). The traditional method both severely underestimates the amplitude and the period of the oscillations, as well as overesti-mating the decay time of the dynamics. Thus using this approach could lead to erroneous conclusions about the time-scales of the dynamics measured with streaked diffraction

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5.5 Results: Simulated Data

niques. The novel analysis tool presented in this chapter successfully and accurately recovers the true dynamics, and so it is important to emphasize that it is the correct and only way that streaked diffraction patterns should be analyzed. This highlights the importance of using Eq.(5.19) as opposed to traditional, na¨ıve approaches.

To investigate the applicability of Eq.(5.19) to different types of dynamics, two further sim-ulations were performed: one of a diffraction spot that oscillates in size (due to, for instance, acoustic phonons), and another of a diffraction spot that changes its position (due to, for in-stance, lattice contractions/expansions). The results of these two cases are shown in Fig.5.7 and Fig. 5.8, respectively. The top portion of these figures show the streaked diffraction image, and the bottom shows a comparison between some of the instantaneous diffraction patterns. In the case where the spot width changes, it is evident that again the spatially-varying deconvolution approach does a decent job at recovering the underlying dynamics.

There appears to be a bit of asymmetry in the recovered diffraction spot shape that has a slight time-dependence, but the overall spot width closely matches the input data. A similar result is observed for the situation in which the diffraction spot changes its position. Although the time-dependent position of the peak is fairly well reproduced, the recovered diffraction pattern appears to match more closely along one axis than the other. It in unclear if this is a systematic discrepancy or is an artifact of the data input here. This result needs further investigation into how to address the issue and what effects it has on the interpretation of the time-dependent diffraction pattern.

Simulated Recovered

t = 0

Figure 5.7.: Simulated streaked image of a diffraction spot undergoing oscillations in its width after t = 0. The lower panel compares the simulated and recovered instantaneous diffraction patterns taken at selected times throughout the dy-namics.

One final case was simulated, involving two small diffraction peaks with overlapping streaked trajectories. After t = 0 the two peaks oscillate in intensity. In previous experiments, this is a situation that has been carefully avoided since with the traditional analysis technique it is impossible to separate the dynamics of the two spots individually. However, as UED is being applied to samples with more complicated crystal structures, overlapping diffraction spots might be unavoidable in future streaking experiments. Because of this, it is interest-ing to see how the spatially-varyinterest-ing deconvolution of this chapter performs on such cases.

Chapter 5. The Physics of Transient Diffraction with Ultrafast Streak Cameras

Simulated Recovered

t = 0

Figure 5.8.: Simulated streaked image of a diffraction spot undergoing a time-dependent shift in its position after t = 0. The lower panel compares the simulated and recovered instantaneous diffraction patterns taken at selected times throughout the dynamics.

Fig. 5.9 shows the results of applying the algorithm to the streaked data shown in the upper portion of the figure. The recovered instantaneous diffraction patterns indicate that that the algorithm presented here is powerful enough to consider overlapping diffraction spots. The presence of two diffraction spots are successfully reproduced, and qualitatively it appears that the time-dependent amplitudes of the two peaks corresponds to the input dynamics.

Although preliminary, the results suggested by this figure indicate that the spatially-varying deconvolution opens up the streaked diffraction technique to a new broad class of samples.

In all of the cases presented above the regularization parameter computed by the discrep-ancy principle was very small, i.e. µ ≈ 0.001 or smaller, which indicates that very little regularization is required to obtain an acceptable reconstruction.

Simulated Recovered

t = 0

Figure 5.9.: Simulated streaked image of two diffraction spots with trajectories which over-lap during streaking. After t = 0, the spots begin to oscillate in intensity. The lower panel compares the simulated and recovered instantaneous diffraction patterns taken at selected times throughout the dynamics.

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