Thanks everybody already for the answers!
I guess I need to clarify a few things:
As KT88 and Marco mentioned already a few times, the solar cell is the DUT. The measurement technique is called transient photovoltage (TPV). There is a continuous-wave laser shining on the solar cell to provide the background voltage and then the pulsed 1ns laser to provide the small signal pulse that I am interested in. One wants to measure at specific cw light intensities and will get different voltages depending on the quality of the cell and then one adjusts the pulsed laser intensity to get a pulse of about 20 mV. Typically, people look at the decay of this pulse but I want to study the rise. We do not apply any external voltage/electric field, this makes it different from a time-of-flight experiment. The whole measurement is done in the hope to understand the physics better that happen in the solar cell (recombination, diffusion, ...).
We cannot use the 50 Ohm impedance because this would extract a lot of current from the solar cell but we want to test it at an open-circuit situation, the physics change in between these cases. That's why I suggested the unity gain buffer (UGB) because it seems that it can have a very high impedance at the entrance (generating the open-circuit situation) and a 50 Ohm exit, allowing to use the scope with 50 Ohm as well. If we could put the UGB close to the sample we could avoid most of the reflections? And the UGB apparently does not change the voltage signal, making it possible to measure the actual voltage values which is important as well (or at least be able to calculate it easily).
I'm not sure how you are using a LEMO cable, but sticking with BNC (assuming the actual cable is a suitable 50-ohm type like RG-58) a 15cm cable gives you about 0.75ns each way. So we'd have to see your ringing to see if that might correlate. Otherwise, the inductance of the wiring at the cells combined with any capacitance in the system might cause ringing as well. And some small amount of ringing won't destroy your measurements--what you have doesn't look that terrible at first glance.
I am using a LEMO cable just by inserting two LEMO to BNC adapters. I was quite wrong about the length though... The LEMO cable is 30 cm and the shortest BNC is 50 cm. I tried the same measurement with different BNC cables lengths and attached the scope screenshots below. At least one of the cables (the 100 cm one) was an RG-58, about the others I don't have any information. The frequency and amplitude of the oscillations does change. I also attached an image where I blocked the pulsed laser to just see the background with very small oscillations. Not sure if this give you any info.
In general, the ringing doesn't look too bad in the images and we could probably live with this but it would be nice to be able to reduce it further.
Edit: Just to be clear, I was explaining what an active probe would accomplish, not specifically recommending one for your application. Noise may be an issue, for example. What you are trying to do is fairly challenging based on the signal level and bandwidth required.
But could you maybe recommend one? :D I understand that this would be probably better than using a BNC or maybe SMA cable (I heard that these are supposed to be good for HF) for us?
Regarding the measurement setup, I realized that I didn't explain it very well and that probably a lot of problems come from the whole sample box. I am still not able to explain it properly because I don't know the exact wiring. But I made some pictures, so that you get an better idea. Each substrate of our DUT has 4 pixels on it (each with size 4x4 mm2) and is placed on some pins for electrical contacting (see Measurement setup 3.jpg), all having a common ground. The pins are mounted on a PCB which is wired to a 14 pin plug (Measurement setup 4.jpg). From there, a cable (not sure what kind of cable) goes to another box which is made to choose the pixel under test (just 1 of the 4 is illuminated) with a rotary switch (Measurement setup 2.jpg). In principle, one could apply a voltage to the cell here ("wire" connection) but we just measure and connect to the "sense" the cable (here BNC-to-Lemo and Lemo cable) which goes to the scope (Measurement setup 1.jpg).
It is clear that all these electrical elements cannot help. So we thought about making a new box in which the substrates lies still in the pins, but the pins from each pixel are directly wired to a SMA connector which can go directly to the scope. Does this make sense?
We cannot get rid of the box because we need to measure under inert atmosphere (N2). And the problem is that this will take some weeks to be built...