## 2 Experimental Setup  
The shaping technique presented in this work is based on the fact that a Pockels cell cannot be switched on instantaneously. In contrast, during the switching time \(t\), the voltage \(U(t)\) at the cell increases exponentially, following the charging curve of a capacitor:  
\[U(t)=U_{0}\cdot\left(1-e^{-\frac{t}{\tau}}\right) \tag{1}\]  
Here, \(\tau\) is the time constant defined by the capacity and Ohmic resistivity of the Pockels cell and its driver. As the phase shift between orthogonally-polarized waves \(\Delta\Phi\) introduced by the Pockels cell is proportional to \(U\), it follows the same behavior. If we further assume that the total phase shift shall be \(\frac{\lambda}{2}\) after the Pockels cell is fully switched on, the temporally-dependent transmission \(T(t)\) of the Pockels cell located between crossed polarizers can be written as:  
\[T(t)=\sin^{2}\left(C_{0}\cdot\left(1-e^{-\frac{t}{\tau}}\right)\right) \tag{2}\]  
Figure 1: Basic layout of the laser system. where \(C_{0}\) is a constant. This holds accordingly for the falling edge of the Pockels cell switch. Such a rising edge of a KD*P Pockels cell switch (cell: Leysop Ltd., Basildon, UK; driver electronics: Bergmann Messgerate Entwicklung KG, Murnau, Germany) as it is used in our laser is displayed in Figure 2. The temporal resolution was achieved by sampling with pulses of our Satsuma laser system (Amplitude Systemes, Pessac, France) in delay steps of 100 ps, while the transmission at each point was determined as an average value. The curve was fitted using Equation (2). The 10%-90% rise time was determined to be 7.4 ns. The falling edge, despite not displayed here, was characterized accordingly, resulting in a fall time of 6.1 ns. Using the inverse fitting function, we can directly calculate the necessary delay to achieve a particular transmission of the switch. In the following, we will use the rising edge of the Pockels cell to shape bursts from a continuous pulse train coming from our front-end. Adjusting the time for the individual triggers allows one to achieve a predefined energy of a single pulse as indicated by the green spikes in Figure 2. Here, the rising edge was chosen over the falling edge, since the slower rise ensures a better resolution. The measurements described below were carried out using the high peak power burst laser system, which is described in detail in [9, 13, 14, 15]. An overview of the system is shown in Figure 1. For the sake of simplicity, the system was only used up to the amplifier stage S1 (see Figure 1) in this work, producing a spatially rectangular flat-top beam with up to 800 W average power during the burst. The chirped pulse amplification (CPA) system was designed for the generation and amplification of bursts consisting of up to 1000 pulses, each having a duration of sub-500 fs. The temporal spacing between the pulses is 1 \(\upmu\)s, given by the repetition rate of the seed laser used. The pump laser diodes of the amplifiers are driven at constant current during the burst. An additional pre-pump time before the burst is set to establish the appropriate initial inversion in the laser medium. The burst to burst repetition rate of the amplifier is limited to 10 Hz due to thermal issues. The modulation of the pulse amplitudes was achieved using a Pockels cell following the patent pending technique described in [16]. In the meantime, this scheme was also used by another group for shaping femtosecond laser pulse bursts using a BBO Pockels cell inside a passive cavity [12]. In our case, the application of the so-called alternating voltage mode, as described in the following, allowed us  
Figure 2: Transmission _versus_ time for the rising edge of the Pockels cell. to use a simple KD*P cell at a repetition rate of 1 MHz. KD*P Pockels cells exhibit a larger piezo-optic effect than BBO, which otherwise limits their use at high repetition rates.