Field geometries

In this sub-section, the mathematical description of a bichromatic field is presented. In general, a bichromatic field is realized when two different laser frequencies are used in an experiment. In the case of a femtosecond laser, the spectrum of a bichromatic field comprises two separated spectral

contributions like e.g. the fundamental of the Ti:Sa and its second har-monic. Combining a laser field with its second harmonic generated inside a birefringent crystal expands the available field geometries. The experiments described herein are based on the superposition of two laser pulses with a fixed frequency ratio. The effective electric field can be written in the dipole approximation in the form E_x,y(t) = A⁽¹⁾x,ycos(ωt) +A⁽²⁾x,ysin(2ωt+ϕ) with the amplitudesA⁽¹⁾x,y and A⁽²⁾x,y containing the information about the temporal pulse form. The relative phase between the two colors is defined as ϕ. The temporal phase of each pulse is contained in its amplitude. The amplitudes are two component vectors (x- and y- components) containing information on the polarization state of the pulse (see appendix 7.6).

In the following, the field geometries used in the experiments are ex-plained in detail. The field geometries are plotted for a fixed frequency ratio assuming monochromatic laser fields for simplicity. Both fields overlap in time to create bichromatic field geometries.

Both fields linearly polarized along the same axis The easiest geo-metry based on combination of two linearly polarized fields extends the field geometries to shapes, which can in general not be accessed by a single mono-chromatic laser field. In figure 5.2, the effective field geometry (green) when both the blue and the red field are linearly polarized along the same axis is plotted. The calculation assumes 25 fs pulses for both blue and red and a fixed frequency ratio of 2:1 between the monochromatic fields, where the red field has a wavelength of 790 nm (oscillation period of about 2.65 fs). In the upper row of figure 5.2, the resulting sum of both electric fields is shown for different relative phases ϕ between the two frequency components. In the bottom row of the same figure, a magnified view covering two oscillation periods of the blue field (one cycle of the red field) is plotted. For a phase of π/2(0.33 fs), both red and blue field maxima overlap at t= 0, and result in a strong asymmetry along the polarization axis. For a phase of zero the in-terference of both fields leads to vanishing asymmetry along the polarization axis. The asymmetry for π/2 (0.33 fs) is reversed for a phase of 3π/2 (0.99 fs), where the fields interfere destructively at t = 0, whereas each maximum of the red field in the negative direction overlaps with the maximum of the blue field.

This field geometry has been successfully used to find the temporal over-lap of both colors in the experiment by choosing the polarization axis such

that it lies in the detector plane. Scanning the phase between both pulses leads to a change in asymmetry along the polarization axis, which can be observed on the detector.

Figure 5.2: (Top row) Two color field obtained for different phasesϕbetween both colors (purple) or equivalently, delay (orange). Both colors are linearly polarized along the same axis (the inset of the first figure in the middle row shows the polarization). In the experiments, the polarization was along the y-axis. Three cases are selected for a delay of 0 fs, 0.33 fs and 0.99 fs, where a magnified view in the two-cycle representation of the blue field is shown in the bottom row. The calculation is done for pulses of 25 fs pulse duration and same amplitudes. A representation with larger time window is shown in figure5.13.

When one of the fields is much stronger than the other one, the interfer-ence is reduced leading to minor asymmetry. This effect can be enhanced when a multi-photon ionization is driven and the contributions of the fields are scaled according to their respective power laws.

Both fields linearly polarized in orthogonal axes When both fields are linearly polarized along mutually orthogonal directions, the field geo-metry resembles a Lissajous-type curve (see figure 5.3). The laser field para-meters are equal to the ones used in the previous paragraph, although the polarization of both fields is orthogonal. Depending on relative phase ϕ, dif-ferent field geometries are realized. For a phase of π/2 (equivalent delay of 0.33 fs) and 3π/2 (0.99 fs), the field geometry resembles a parabola, which has its open side towards upper or lower half of the graphs (see figure 5.3).

In this case an asymmetry in field strength is generated along the polariz-ation direction of the blue field, which can be explained by calculating the effective field strength for different positions along the curve. For the two extreme points lying in the upper half of the plot, the combined field strength

|E|=√

2is higher than in the point, where the red field is zero. By orienting the field geometry inside the VMIspectrometer accordingly, this asymmetry along the polarization axis of the blue field can be used to investigate the relative phase between the two colors. Measurements performed with this field geometry are shown in the appendix (see section 7.6). Setting the phase to either 0 orπ (0.66 fs) results in a field geometry looking like the symbol for infinity (see figure 5.3). The sense of rotation is indicated by the arrows when the light field is propagating towards a spectator, where all fields are shown in the single-cycle representation.

This type of field shows different sense of rotation in the two halves along the red field polarization. In the case of photoionization, the spec-tator is typically a molecule and the convention used throughout this thesis is the so-called optical convention. The field comprises different sense of rotation within a full optical period of the red field. It was demonstrated theoretically^[102,103]that using this type of field, a chiral response when pho-toionizing chiral molecules can be observed. In addition, experimental results were reported.^[104] The chiral interaction in this case is fundamentally differ-ent as compared to the case of usual PECD. In the case of PECD with circularly polarized light, the chiral target transfers the photon spin angu-lar momentum onto the emission direction of the photoelectron. Using a tailored bichromatic field, the sense of rotation of the field is transferred to the photoelectron emission direction by the chiral target. This field geometry allows to simultaneously measure the effect of different sense of rotation on the photoelectron emission within a single measurement. The photoelectrons experience different sense of rotation of the field in the two hemispheres along the red field polarization direction.

Figure5.3:(Toprow)Generationofdelay-dependentLissajous-typefieldgeometrieswhenbothcolorsare linearlypolarizedalongmutuallyorthogonaldirections.(Middlerow)Threeexamplesoffieldgeometries andcorrespondingPAD-imagesinanartists’pointofview(bottomrow).Thefieldgeometryinthe experimentisshownontherighthandside.

The photoelectrons originating from both hemispheres are imaged onto different halves on the detector by the VMI (the experimental geometry is shown as inset in figure 5.3). The experiments are performed such that the delay between both colors is scanned and VMI PADs are recorded for every delay.

Changing the delay between both colors results in scanning across the different Lissajous-type field geometries as shown in figure 5.3. As can be seen in the VMI PADs shown in figure 5.3, the expectation is that a for-ward/backward asymmetry along the laser propagation direction is gener-ated. This asymmetry is in opposite direction on both sides of the PAD along the y-axis. For a phase of 0 or π (0.66 fs), this asymmetry is maxim-ized, whereas for a phase of π/2 (0.33 fs), the asymmetry is very weak. This asymmetry is opposite for the two halves of the VMI PADs along the red field polarization axis.

Choosing a different amplitude ratio between the fields stretches the field geometry along the direction of the stronger field.

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Figure 5.4: Examples of field geometries generated by overlapping red and blue pulses, when both fields are circularly polarized. The field amplitude is set to 1 for each color and the field geometry is shown in the single-cycle representation. The two field geometries differ by using either a co- (a) or a counter-rotating (b) sense. In both images, the overlap of both blue and red fields results in a purple color of the circle.

Both fields circularly polarized When both fields are circularly polar-ized, two main possibilities are given: The fields can rotate in the same or in the opposite direction. Both fields can in principle evoke a chiral re-sponse when both have circular polarization. The field geometry for the case of counter- or co-rotating circularly polarized fields is shown in figure 5.4.

Changing the amplitude of one of the fields for the counter-rotating case, the ’cloverleaf’ (shown in figure 5.4(b)) changes its shape. If the red field has higher amplitude (as shown in figure 5.5(a)) than the blue field, the

’cloverleaf’ type field approaches a triangular shape. For slightly different amplitudes, the electric field trajectory does no longer cross through zero.

For higher amplitude of the blue field, the combined field approaches a circle with three small lobes (see figure 5.5(b)). In this case the combined electric field does not cross through zero. In that case the effective field is close to circular polarization. In the case of co-rotating fields, the effective field ap-proaches circular polarization if one of the fields is dominant over the other one.

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Figure 5.5: Field geometries for different amplitude ratios A_red/A_blue of 2.5 (a) and 0.4 (b). Please note that trajectories of electrons driven by fields with unequal amplitude ratio do no longer cross through zero, when the amplitude ratio deviates from 1.