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High­order Harmonic and Attosecond Pulse Generation

2.5 Spatio­Temporal Aspects

2.5.1 Harmonic Wavefront

polynomials, called pBaseX [60].

In addition to the momentum distribution, the time­of­flight spectrum of the detected ions can be extracted from the MCP using a decoupling circuit. Hence, the mass­over­charge ratio of the resulting ionization products can be determined.

The VMIS is designed with two extracting electrodes and two flight tubes in op­

posite directions to each other, such that ions and electrons are imaged simul­

taneously making use of their opposite charge. The correlation of the respective momentum distributions is of great interest for the understanding of the studied processes. Due to the comparably low repetition rate, a conventional coincidence analysis [61] is not practical. However, the high event rate per shot enables the in­

vestigation of correlations using a covariance scheme, further described in chapter 4 and papers Iv and vI.

Figure 2.10: Working principle of the XUV wavefront sensor. The beam is diffracted through an array of holes, called Hartmann mask. The detected positions on the screen behind the mask depend on the wavefront when passing through the holes. The displacement, referenced to the position for a flat wavefront, is referred to as the Hartmann vector ⃗h. Figure adapted from paper Ix.

A well­established technique for measuring wavefronts is to use a Hartmann wave­

front sensor (WFS). A sketch of the working principle is shown in figure 2.10. The beam passes through an array of diffracting optics, which guide it towards the direction orthogonal to the tangent of the local wavefront. In the infrared and vis­

ible range, such arrays usually consist of microlenses, whereas in the XUV regime a small hole is sufficient to create enough diffraction. A screen placed behind the mask is utilized to detect the displacement behind each hole/lens referenced to the position that would be obtained for a plane wave. The difference between the dis­

placed point and the reference point is called the Hartmann vector ⃗h, which length is proportional to the derivative of the local wavefront. In order to reconstruct the wavefront, two different algorithms are commonly used: following the decompo­

sition in Zernike polynomials the modal reconstruction fits a gradient field of each polynomial to the vector field created by the measured Hartman vectors and thus allows the reconstruction of the wavefront [63]. The zonal reconstruction algo­

rithm on the other hand calculates the wavefront by a numerical integration of all

Aberrant

wavefront Corrected

wavefront

Deformable mirror

Actuated surface

Figure 2.11: The working principle of the deformable mirror, which compensates wavefront aberration.

A set of actuators adjusts the surface of a mirror such that the reflected wavefront is flat.

a) )

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Iris aperture [mm]

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0 1 2 3

Iris aperture [mm]

Pulse energy [μJ] Pulse energy [μJ]

0 1 2

(a) 3 (b)

Figure 2.12: Generated XUV pulse energy as a function of the beam aperture introduced by the iris in the generation chamber. The scan was performed with and without aberration correction (a,b respectively). The insets on the top show the measured beam profile for four different apertures.

the local slopes [64].

Measuring the infrared laser wavefront allows us to compensate the aberrations acquired along its path through the beamline. A common way to do so is to use a deformable mirror (DM), consisting of an array of piezoelectric actuators individ­

ually attached to the surface of a reflective optic. Each actuator changes the local curvature of the surface, which then compensates the wavefront of the reflected beam, as illustrated in figure 2.11. In a closed­loop circuit, an optimization algo­

rithm applies a hill­climb procedure to the actuators of the DM while monitoring the changes to the wavefront. The quality of the wavefront is hereby measured as the root mean square (RMS) of the deviation to a flat wavefront in units of the central wavelength, which typically lies around λ/40 after the aberration correc­

tion.

The implementation of the deformable mirror, correcting the aberrations of the infrared beam, led to a significant improvement of the spatial shape of the XUV beam, both in the far and the near field. Figure 2.12 shows the variation of the XUV pulse energy and the far­field beam profile as a function of the aperture of the iris, which is located before the generation medium, both with and without the aberration correction introduced by the DM. The aperture introduced by the iris affects the pulse energy of the driving infrared field and the shape of the fo­

cal spot in the generation medium. A smaller aperture creates a larger focal spot size and changes the shape of the focal spot towards a super­gaussian, which is favorable for phase­matching considerations. As can be seen in figure 2.12 (a), the

14 16 18 20 22 0.15

0.2 0.25 0.3 0.35

14 16 18 20 22

1 2 3 4 5

−1000 0 1000 2000

0.1 0.2 0.3

53 42

76 133

53

76 101

164

−1000 0 1000 2000 0

2 4

20 25 30 35

0 0.2 0.4 0.6 0.8 1

20 25 30 35 0

1 2 3 4 5

50 100 150 200 250

0.1 0.15 0.2 0.25 0.3 0.35

50 100 150 200 2500

1 2 3 4 5

Back pressure (mbar)

APT energy (nJ)

APT energy (nJ)APT energy (nJ) APT energy (nJ)

(a) (b)

(c) (d)

Wavefront RMS ( )Wavefront RMS ( ) Wavefront RMS ( )Wavefront RMS ( )

GDD (fs2)

IR pulse energy (mJ) Iris diameter (mm)

Figure 2.13: Measured wavefront error (black) and generated pulse energy (blue) as a function of four different generation parameters: (a) the fundamental IR pulse energy, (b) the iris diameter, (c) the backing pressure of the gas cell and (d) the GDD introduced by translating the distance between the two gratings in the compressor varying the pulse duration of the infrared pulses.

Figure adapted from paper vIII.

generated pulse energy increases for an increasing aperture until reaching a maxi­

mum at around 29 mm. The beam profile, measured in the far field on the Andor XUV camera, remains round and of similar size until 29 mm. For higher aperture values the beam profile starts to distort, due to a too high intensity in the genera­

tion medium. The resulting ionization leads to a modified refractive index in the medium, which compromises the phase matching and introduces guiding effects to the beam. Without the aberration correction, the beam profile is elongated already for smaller apertures and breaks apart for apertures larger than around 25 mm, as shown in figure 2.12 (b). Interestingly, the generated pulse energy is in a similar range as with the aberration correction and increases with the iris aperture, even though the beam profile is of very poor quality. Nevertheless, it is certain that the intensity available for further experiments is much higher for the beam with the aberration correction, due to its superior beam profile.

Paper vIII presents an in­depth study of the effect of different generation parame­

ters on the XUV wavefront. The wavefronts were measured on a single­shot basis to avoid averaging effects due to pointing instabilities and wavefront variations.

Figure 2.13 shows the retrieved wavefront error in units of the central wavelength as a function of (a) the IR pulse energy, (b) the iris aperture, (c) the backing pres­

sure of the gas cell and (d) the compressor GDD. The generated pulse energy is

shown in blue. The optimal XUV wavefront is not necessarily generated under the same conditions as the highest XUV pulse energy and it is hence important to take into account the wavefront, when performing experiments that require focusing the radiation on a small focal spot to reach high intensities.

Paper vIII also shows that the XUV wavefront is affected by the infrared beam in two ways: the spatial phase is equal to the phase of the infrared field multiplied by the process order, as in any regular up­conversion process, plus a contribution proportional to the laser intensity profile, due to the intensity dependence of the dipole phase. The intensity distribution across the infrared beam profile, which is approximately a two­dimensional Gaussian distribution, leads to a larger dipole phase contribution in the center compared with the outer parts of the beam. This implies the generation of harmonic orders with varying radii of curvature. Also, if the infrared beam profile is asymmetric, it leads to an astigmatic XUV wavefront.

These effects play an important role when refocusing the generated harmonic beam for further experiments and led to a series of further studies presented in the fol­

lowing sections.