Beaming of High-Order Harmonics Generated from Laser-Plasma Interactions

Beam divergences of high-order extreme ultraviolet harmonics from intense laser interactions with steep plasma density gradients are studied through experiment and Fourier analysis of the harmonic spatial phase. We show that while emission due to the relativistically oscillating mirror mechanism can be explained by ponderomotive surface denting, in agreement with previous results, the divergence of the emission due to the coherent wake emission mechanism requires a combination of the dent phase and an intrinsic emission phase. The temporal dependence of the divergences for both mechanisms is highlighted while it is also shown that the coherent wake emission divergence can be small in circumstances where the phase terms compensate each other.

Figure 1

Spectrometer configurations. (a) Separation of harmonic signal into on- and off-axis components. On-axis emission apertured to a 0.25 mrad cone while off-axis emission at angles between 35 and 50 mrad was collected onto the CCD using a gold focusing mirror. (b) No apertures or focusing optics to obtain the pure spectrally resolved angular distribution while the CCD was rotated so as to observe a larger cone angle.

Figure 2

Transition from the lower intensity ($\approx {10}^{17}\mathrm{W}\mathrm{c}{\mathrm{m}}^{-2}$) CWE mechanism to the relativistic ROM ($\approx 2\times {10}^{19}\mathrm{W}{\mathrm{cm}}^{-2}$) using the setup in Fig. 1a. Target placed (a) $200\mu \mathrm{m}$ from best focus and (b) at best focus. The CWE cutoff harmonic (19th) is indicated by the extended dashed black line.

Figure 3

Fourier analysis of the different divergence behavior for CWE and ROM for a Gaussian laser focus with a FWHM of $2.5\mu \mathrm{m}$, a pulse duration of 30 fs, and a peak intensity of $2\times {10}^{19}\mathrm{W}{\mathrm{cm}}^{-2}$. Normalized angular distributions for each laser cycle for the 19th harmonic (CWE) (a), with both the intrinsic and dent phase, and for the 20th harmonic (ROM) (b), with only the dent phase considered. The white dots indicate the $1/e^2$ beam radius at each laser cycle. (c) Balancing of phase terms at where the beam radius is a minimum for the 19th harmonic. The normalized laser intensity is also shown. (d) Temporally summed angular distributions for the 19th harmonic (solid curve) and the 20th harmonic (dashed curve). Also shown is the 19th harmonic with the dent phase excluded (dot-dashed curve). The shaded region corresponds to that collected in the experiment as the off-axis emission.

Figure 4

(a) Divergence for the 19th harmonic calculated from the model for different intensities. Divergence values determined from Gaussian fits. At low intensities the intrinsic phase dominates, but as the intensity increases the denting phase can compensate and produce a narrower beam. At very high intensities the surface dent begins to dominate. (b) The $1/e^2$ beam radii obtained from Fourier analysis for (from the right) harmonics 17, 19, 20, 27, and 40 at different peak intensities. Laser parameters are the same as for Fig. 3. The black dotted line shows the divergence for flat phase (diffraction limited) for the modeled Gaussian sources. The dashed blue lines indicate the upper and lower error bounds for the experimentally measured harmonic divergence (as previously reported in 8).