I-BEAT: Ultrasonic method for online measurement of the energy distribution of a single ion bunch

The shape of a wave carries all information about the spatial and temporal structure of its source, given that the medium and its properties are known. Most modern imaging methods seek to utilize this nature of waves originating from Huygens’ principle. We discuss the retrieval of the complete kinetic energy distribution from the acoustic trace that is recorded when a short ion bunch deposits its energy in water. This novel method, which we refer to as Ion-Bunch Energy Acoustic Tracing (I-BEAT), is a refinement of the ionoacoustic approach. With its capability of completely monitoring a single, focused proton bunch with prompt readout and high repetition rate, I-BEAT is a promising approach to meet future requirements of experiments and applications in the field of laser-based ion acceleration. We demonstrate its functionality at two laser-driven ion sources for quantitative online determination of the kinetic energy distribution in the focus of single proton bunches.


Detector Setup
An ionoacoustic detector relies on the detection of the acoustic signal that is generated due to thermal heating of an ion bunch dissipating its kinetic energy in water. Since operation in vacuum was desired, we chose a KF40 vacuum pipe with 10 cm length as water container. A hole of 1 cm diameter at the front plate is covered with an 11 µm thick titanium foil that is airtight and waterproof and functions as an entrance window for the ion bunch. The transducer was attached to the rear flange and positioned in the water sample. We chose a focusing transducer (focal length of 25.4 mm) to enhance the signal.
Signals generated in focal distance will have the best temporal resolution while the resolution drops off out of focus. Supplementary Fig. 1c shows the geometry of the detector used at the Laboratory of Extreme Photonics (LEX Photonics). A picture of the transducer is given in Supplementary Fig. 1d and the used amplifier (60 dB, HVA-10M-60-B, FEMTO Messtechnik GmbH) in Supplementary Fig. 1e. Note that all parts in the electronic chain influence the signal response and have to be included in the calibration.
Since the motivation for I-BEAT was its implementation in a laser-plasma ion accelerator a setup-picture is shown in Supplementary Fig. 1f. The picture is taken at LEX Photonics in Garching near Munich. It shows the implementation of I-BEAT inside the vacuum chamber. The detector was modified at the experiment at the Draco laser. The length of the tube and thus the distance of the source of the generated sound signal was shortened and now positioned directly in the focal plane of the transducer.
This improves the signal-to-noise ratio and the temporal resolution of the detector. Supplementary Fig.   1h is a picture of the setup at the Draco laser.
Supplementary figure 1 | Setup of the ionoacoustic detector. a, is a picture of the tube accommodating the water sample (front view). b, shows the detector and depicts the physical process of I-BEAT. c is a sketch emphasizing the dimensions of the used detector at LEX-Photonics. d, is a picture of the transducer that was used during the experiments. It is a focusing transducer with a mean frequency of 10 MHz. e, is the Voltage amplifier used in the experiment. f, shows the setup at the laboratory of extreme photonics in Garching. This shows the implementation of I-BEAT directly in the vacuum chamber. The laser was focused with the off-axis parabola (OAP) onto the target. Two permanent magnet quadrupoles were used to focus the proton bunch into the water volume. g, shows the modified detector that was used at the Draco laser. The detector was shortened in order to accommodate the Bragg peak in the focus of the transducer. h, is a picture of the setup at the Draco laser. This part describes a more detailed derivation of equation (1) 2,3 . Also the derivation of the reflection coefficient is explained in detail.
By solving the wave equation ) .
In the setting relevant for our case < 10 and 0 = 11µ , 0 0 < 0.13. The phase term of the reflectivity can thus be neglected and the reflection = −1 corresponds to that of a fixed end such that the polarity of the reflected wave packet is inverted. ( (1) For a quantitative calibration the measurements were also used to estimate the number of protons per , where is the average current that was delivered from the Tandem to the water volume, and = 5 kHz is the bunch repetition rate. We estimated, considering the pinhole size of the detector entrance and the spot size of the beam, that 60% of the bunch enters the detector such that current was estimated = 0.6 × 7 = 5.2 .
( ) is thus quantitatively connected to the ideal pressure trace ( ) for an arbitrary ion energy distribution ( ) (predicted by equation (1)) via

First Tests at the Tandem accelerator
Before applying I-BEAT to a laser-plasma accelerator, calibrations and first tests were performed at the MLL Tandem accelerator at Garching, using well defined proton bunches of 40 ns duration with 10 MeV

Estimation on the behavior of the detector at higher fluences
The detector is capable of measuring really high particle fluxes. Since this scaling could not be measured so far, the expected temperature increase is estimated in this section, starting with a derivation of Boyle's law: with kappa being the isothermal compressibility and beta the volume expansion coefficient. δp and δT are the changes in pressure and temperature respectively. As we consider only adiabatic heating in ionoacoustics, the volume expansion is neglected and only the transfer from temperature gradient to dynamic pressure is considered. With the use of the specific heat capacity, the following expression can be derived: where Cv is the isochoric specific heat capacity, m the mass of the heated area and the applied energy as heat. This conversion of energy to pressure is material depended and is quantified with the dimensionless Grüneisen parameter Γ: where ρ is the material density, c the speed of sound and Cp the isobaric specific heat capacity. In order to estimate, weather a very high particle number will significantly change the linearity of the energy transfer to dynamic pressure, the expected temperature increase and the change in the Grüneisen parameter is investigated. For liquid water and the temperature T in degrees Celsius, the Grüneisen parameter is well approximated by: Γ ( ) = 0.0043 + 0.0053 .
In independent measurements at the MLL Tandem accelerator, the pressure from 20 MeV protons and 3x10 6 / 2 was measured with a calibrated, broadband needle hydrophone (Precision Acoustics, UK). Measured at different distances from the source, the pressure at source level was extrapolated to 115 Pa, which in this case corresponds to a temperature gradient of 0.14 mK. Assuming a linear dependence, a temperature gradient of 4.2 K can be expected for 10 11 protons/ 2 per bunch.
Based on the approximation for the Grüneisen parameter given above, we derive the following values: Γ (25) = 0.1368, Γ (29) = 0.158, ΔΓ = 0.0212 , which corresponds to a relative change of 15%, in absolute particle numbers, over several orders of magnitude. We can thus say that the detector will also work at much higher particle numbers.

Data Analysis
This section describes the algorithm of simulated annealing 7 in a bit more detail and discusses the influence of different beam diameters onto the retrieval in the case of the laser-accelerated ions.

Supplementary figure 4 | Workflow of simulated annealing. a, The workflow of simulated annealing is shown. b,
The initial spectrum fi can be guessed or started with a flat distribution. c, Small modulation to the initial spectrum is done (fm). d, Acoustic signals are calculated and the performance is compared. e, converges after about 10 4 loops.
The method of simulated annealing relies on varying an initial spectrum ( ) (a little change applies to the estimated spectrum, and both its position and amplitude are decided by pseudo-random generators) to obtain a modified spectrum ( ). As a starting point ( ) was chosen to be zero for all energies. Typically, the maximum of the amplitude modification is set to be smaller than 1 % of the maximum of . With the input of the initial and the modified spectrum in eq. 1 the predicted acoustic signals ( ), ( ) are calculated and compared to the acoustic signal 0 ( ). The residuals is smaller than , the algorithm continues with the modified spectrum as the updated input distribution for the next cycle. For smaller than , with probability = (−( − )/ ), the algorithm continues with the modified spectrum, while, with the probability 1 − , it is rejected and the initial spectrum is taken into the next cycle. This additional random choice prevents from being caught in a local minimum. is the annealing schedule temperature and was set to 1. After a sufficient amount of iterations ( 10 4 ) the temperature during the iteration would become stable around the temperature global minimum ( converges), shown in the insets of Supplementary Fig. 5b, Fig. 5c and Fig. 5d , and the obtained proton energy spectrum is the retrieved spectrum. As explained before the ion bunch standard deviation can be treated as unknown in the retrieval process. In this case the complete process of simulated annealing depicted in Supplementary Fig. 4 is repeated by choosing another . As a result the final residuals ( ) after a sufficient number of steps (when a minimum for ) is found) shows a broad but distinct minimum for a certain bunch diameter (Fig 5a).

The use of I-BEAT at typical conditions for laser ion acceleration experiments
In this section we investigate the functionality of I-BEAT at different condition typically occurring at laser-ion acceleration experiments. We show the performance of I-BEAT measuring a broad band exponential spectrum that is typically obtained close to target without any manipulation of the ion bunch. We also show the performance of I-BEAT in a multi species spectrum, using quadrupoles as charge state separation.

Functionality of I-BEAT close to target measuring a broad energy distribution
In laser ion acceleration typically broad multispecies energy spectra emerging the plasma target 8,9 . We simulated the performance of I-BEAT positioned close to target without any manipulation of the energy distribution (e.g. magnetic quadrupoles). The proton input spectrum was exemplarily taken from 10 . Note that other ion species are typically emitted with significant lower particle numbers and energy and are thus neglected in this consideration. Without the use of charge state separating fields a differentiation of different ion species is not possible. Assuming and opening aperture for the detector with an radius of 3 mm (seems feasible since it supported by the measured data) covers an area of about A = 30 2 and thus the I-BEAT detector was positioned such that 10 9 protons reach the detector. The particle number was chosen to obtain a good signal to noise ratio (knowing the pressure signal of a single proton and the background noise). The measured spectrum provides more than 10 8 protons per msr (all energies summed up). We thus have to cover a steradian Ω of 10 msr. With Ω = 2 and d being the distance to the detector yields d = 50 mm and has thus be positioned close to the target. The given input spectrum was used for a calculation of the expected signal ( Supplementary Fig. 6 b). This signal was then evaluated with the I-BEAT algorithm and thus the spectrum was evaluated. In Supplementary Fig. 6a the original spectrum is compared to the ones evaluated with I-BEAT. The required signal to noise ratio thus sets a limit to the distance (to the source), where the I-BEAT detector can be placed. I-BEAT can thus function as the typical RCF stack that is positioned close to target, admittedly without measuring the beam profile but offering an online evaluation.

Functionality of I-BEAT measuring multiple species
Laser-driven plasmas accelerate not only protons but also other ion species depending on the target material. Especially carbons at different charge states are also emitted from the contamination layer of any target surface. Since the range in the water tank is dependent on the mass (not so much on the initial charge state) and the kinetic energy, I-BEAT is able to assign a certain energy to a certain charge and mass in combination with an energy selective focusing device 11,12 (such as magnetic quadrupoles) and can thus, at least in this configuration also reconstruct the energy distribution of different ion masses and charges in a single shot. An example calculation is presented in the Supplementary Fig. 7.
We assume flat spectra of carbon ions with charge states 4, 5 and 6, as well as protons. The QP-doublet focusses all ions with the same synchrotron radius to the same point as depicted in Supplementary Fig.   7a. A calculation of the signal when such a multispecies ion bunch is measured with I-BEAT is performed and the expected acoustic signal is shown in Supplementary Fig. 7b and 7c. One can clearly distinguish the contributions of the different ions to the acoustic wave form and hence measurement of this waveform will allow for reconstructing the complete information. Of course, the information of the complete ion spectra emitted from the target remain inaccessible (as the QPs filter out ions which are too far of the design energy which is focused). I-BEAT will not replace the currently and also really valuable techniques of characterising the composition of the ion-spray emitted from the target, such as provided by Thomson parabola spectrometers 13,14 . But it will be an additional and complementary option to measure ion energies that will give an experimentalist a new, very powerful, tool for future research at the application site of high flux ion bunches.
Supplementary figure 7| Multispecies in combination with magnetic quadrupoles. a, Multispecies ion energy distribution selected by quadrupoles, set to a design energy of 60 MeV protons. b, Simulated acoustic trace generated by the spectrum in a. c, Enlargement of the central part of b (highlighted with red). The carbon ions do not penetrate far into the water but are still well separated in the oscilloscope trace. Since the peaks are well separated, I-BEAT is capable of reconstructing the complete information.