Chromatic Aberration

So far, only the wavefront of the entire XUV spectrum has been considered, which corresponds to an average of all harmonic contributions. Papers II and III reveal that the process of HHG creates an intrinsic variation of the harmonic wavefronts.

This leads to a spread of the harmonic foci (virtual or real) around the generation medium, thus also creating a spread of the focus position along the propagation axis when the harmonics are refocused.

The variation of the wavefront curvature can be understood by studying the effect of the dipole phase on the spatial phase of the generated harmonic beam, which can be written for the q^thharmonic as:

Φq(r, z) = qϕ(r, z) + Φ^s,l(r, z). (2.20) The last term Φ^s,l(r, z)corresponds to the dipole phase described in equation 2.11, where the spatial dependence is introduced using I = I(r, z). Further, the phase of the fundamental can be written as

ϕ(r, z)∝ kr²

2R(z), (2.21)

where k is the wavevector, r is the radial distance from the center of the beam and R(z) is the radius of curvature. Figure 2.16 (a) illustrates the curvature due to the fundamental field qϕ(r, z) in black and the contribution introduced by the dipole phase Φsfor shot trajectories in green. As can be seen, the fundamental curvature varies through the focus from negative to positive, as expected from a Gaussian beam. The dipole phase contribution to the curvature on the other hand is constant over the propagation direction z, which means that the combined harmonic curvature is flat at a position different from the fundamental focus, as indicated by the blue dashed line. Given the fact that the dipole phase contribution is intensity and order dependent, the focus position of the harmonics therefore varies as a function of order. As a consequence, the high­order harmonics are generated with an intrinsic chromatic aberration.

Assuming a Gaussian fundamental beam profile I(r, z) = I0e^−2r2/w2(z), with the radial width w(z) at 1/e², the radius of curvature of the harmonic field can be approximated using a Taylor expansion around the center of the beam for the long and short trajectories:

1

R^s,l(z) = 1

R(z)−4α^s,lI₀w₀²c

w⁴(z)Ω +4γ^s,l(Ω− Ωp)²c

I0w₀²Ω (2.22)

The width of the harmonic beam at the generation position is approximated to be half of the fundamental width, which is confirmed by TDSE calculations and detailed in paper III. Figure 2.16 (b) plots the radii of curvature for the fundamental beam (black), the long trajectories (red) and the short trajectories (blue) for the 23rd harmonic as a function of the generation position. The focus positions (where the radius of curvature goes to infinity) are quite different for the long and short

-2 -1.5 -1 -0.5 0 0.5 1 -30

-20 -10 10 20

30 Z₊

s

Radius of curvature Ri (in units of z0)

Generation position z (in units of z₀)

-1/

0

(a)

(b)

Figure 2.16: (a) Contributions to the harmonic wavefront created by the fundamental (black) and the dipole phase (green) at different generation positions z. (b) Radius of curvature for the fun-damental infrared (black), the generated short (blue) and long (orange) trajectories of the 23^rdharmonic as a function of the generation position z relative to the fundamental focus.

The position of the infrared focus is indicated by the vertical black dashed line and the point where two wavefront contributions add up to a flat wavefront is marked by the blue dashed line. Figure adapted from paper III.

trajectories as well as for the fundamental beam. As a result, also the divergence of the generated harmonic field varies and is minimal when the total radius of curvature is infinity, i.e., when the harmonics are generated with a flat wavefront.

For the high­order harmonics generated at the IXB, these properties affect the refo­

cusing in the application chamber. For a high intensity in the focus, all harmonics need to be located at the same position. However, the predicted chromatic aberra­

tion introduces a variation of the intensities of different spectral components along the propagation axis and modifies the temporal structure of the attosecond pulses.

In paper II, the model described above was used to estimate the focus positions of the harmonic beam for the experimental parameters of the IXB. Figure 2.17 shows a comparison of the harmonic focus positions in the generation chamber (a,c) as well as after refocusing with the Wolter optics in the application chamber (b,d) for harmonic order 11 (red) to 25 (blue). The top row corresponds to the simula­

tion run without the dipole phase, whereas in the bottom row the dipole phase is included. The x­axis is the position of the medium relative to the fundamental


   







 




   








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Figure 2.17: Simulated focus positions of generated harmonics as a function of the generation position Z, in the generation chamber (a,c) and the application chamber after refocusing (b,d). The simulations were performed without (left column) and with the dipole phase contribution (right column). The color code represent the different harmonic orders from 11 (red) to 25 (blue). Figure adapted from paper II.

focus. As shown in the top row, the refocused harmonics are already displaced in the application chamber when generating within the infrared focus without the dipole phase involved in the process. This is due to the Rayleigh range of the har­

monics, which is longer than the distance between the generation position and the Wolter optics. It is however possible to compensate for this effect and refocus all harmonics to the same position, when generating slightly outside of the infrared focus (at Z ≈ −0.05 m). With the dipole phase included in the simulations, the focus positions vary strongly with the harmonic order, in particular for negative generation positions, corresponding to a generation before the infrared focus.

In order to confirm the simulation presented in figure 2.17, a knife­edge scan around the focus in the application chamber was performed, as described in detail in paper II. A knife­edge was placed on a translation stage, making it possible to insert the knife horizontally into the beam for different displacements along the propagation direction, as shown in figure 2.18(a). As a first step, a scan close to the focus was performed for three positions: before, after and in the focus. While inserting the knife, the spectrum was measured by the XUV spectrometer. Fig­

ure 2.18(b) shows the spectrum without any knife insertion. When the knife is inserted before the focus, a shadow appears on the opposite side visible in all har­

monic orders, as shown in (c). Inversely, when inserted after the focus (e), the shadow appears on the same side as the knife. For a knife insertion in the focus shown in (d), the shadow appears from the right side for the low­order harmonics,











 









Z HHG chamber

Application chamber

Al filter FS plate

DM

Iris Focusing mirror

Wolter optics

Gas cell Knife

Spectrometer MCP To IR wavefront sensor

(b) (c) (d) (e)

(a)

Figure 2.18: (a) Schematic of the experimental setup used for the knife-edge measurements. The knife is placed around the focus in the application chamber in a zk-xktranslation stage. (b) Measured XUV spectra for different knife edge positions: no insertion (b), knife inserted before (c), after (e) and in the focus (d). The insets indicate the position of the knife for (c) and (e). Figure adapted from paper II.

i.e., the knife is positioned before the focus, while for the high­order harmonics the shadow is on the left side, which means that the knife is placed behind the focus. For the mid­order harmonics, the signal on the spectrometer is reduced without much change in shape, which indicates that the knife is inserted in or very close to the focus. This experimental result confirms the predictions from the above­mentioned simulations: under certain generation conditions, the generated harmonic orders are refocused at different position due to the intrinsic chromatic aberration caused by the dipole phase.

In order to quantify our findings in more detail, a slightly modified knife­edge scan was performed. The knife was now inserted at four positions outside the Rayleigh range. The beam width at each position was extracted by fitting an error function to the harmonic intensity as a function of the knife insertion. A linear fit to the four extracted beam widths as a function of the knife position in the propagation direction zkdetermines the focus position as the point where the fit crosses zero. This is done for all detected harmonic orders on the spectrometer, while the position of the fundamental infrared focus is varied by changing the curvature on the deformable mirror.

The resulting focus positions of the harmonic beam are shown as functions of the harmonic order in figure 2.19. Each plot corresponds to a different position ∆Z of the generation medium with respect to the infrared focus, indicated in the insets of each figure. The solid black line shows the measured values with the corresponding errors extracted from the fitting procedure and the dashed lines are the simulated values using the model described above. As can be seen, the focus positions are much closer together when the harmonic beam is generated after the fundamental infrared focus (∆Z > 0). In (a) and (b) the XUV foci are separated by almost


 






 

 






 








 


 




 







 







(a) (b) (c) (d) (e)

Figure 2.19: Measured (solid line) and simulated (dashed line) focus positions as a function of the harmonic order for five different generation positions. The position of the generation medium (green) in relation to the infrared focus (red dotted line) is indicated in the insets. Figure adapted from paper II.

1.5 mm, which is a large difference given the short Rayleigh range due to the tight focusing geometry in the application chamber.

Paper II gives a simple estimation of the spatio­temporal properties of the focused pulses. The peak intensity of the resulting XUV pulses is evaluated as a function of the generation position. We show that the spread of the focus position of different harmonics for ∆Z < 0 leads to a drastic drop in intensity compared to the pulse energy. Additionally, due to the time­frequency coupling, the temporal structure of the attosecond pulses varies along the focal region. As a consequence, it is crucial to take the chromatic aberration of the generated harmonic fields into account, when performing experiments that require high intensity and short pulses.

In document Atomic and Molecular Dynamics Probed by Intense Extreme Ultraviolet Attosecond Pulses Peschel, Jasper (Page 54-59)