INTRODUCTION

Computational tomography methods–mathematical algorithms for reconstructing the internal structure of an object from a set of projection data–are widely used in today’s research. Typically, projection data is obtained for a thin layer (slice of an object), the internal structure of which is described by a 2D distribution function. This class of objects includes bunches of charged particles circulating in a synchrotron. Indeed, the sensor of the pulsed current of circulating charged particles registers data along the length of the flying bunch, which correspond to the longitudinal intensity profile of the particles. The accelerated bunch is formed by particles undergoing stable synchrotron oscillations near the equilibrium state for a synchronous particle under the conditions of autophasing. Phase trajectories of accelerated particles with energies \(E(t)\), different from the energy of the synchronous particle, \({{E}_{{\text{s}}}}({{t}_{{\text{s}}}})\), are closed curves. Their projections on the time axis set the longitudinal profile of the bunch \(n(t - {{t}_{{\text{s}}}})\) as a data set for intensities in bunch cross sections. Identification of deviations \(E(t) - {{E}_{{\text{s}}}}({{t}_{{\text{s}}}})\) based on observed profiles \(n(t - {{t}_{{\text{s}}}})\) is the goal of reconstructing the 2D particle distribution function \(f(E - {{E}_{{\text{s}}}},\;t - {{t}_{{\text{s}}}})\) in a bunch.

It was shown in [1] that the computational tomography for bunches of charged particles in a synchrotron makes it possible to reconstruct the longitudinal 2D distribution function of particles in a bunch. In studies [2, 3] the tomographic procedure for reconstructing the longitudinal phase portrait of ion bunches was applied at JINR Nuclotron [4]. New publications on the use of the tomographic procedure algorithm are presented in [5, 6]. We describe below a set of processes for recording data on the bunch length and their subsequent mathematical processing to reconstruct the longitudinal 2D distribution function of ions in a bunch accelerated in the JINR superconducting booster synchrotron [7], where commissioning works are carried out at the NICA complex [8].

BASIC CONCEPTS

Observed analog signal \(n(t)\), which is proportional to the longitudinal intensity of ion bunches during their acceleration in the synchrotron, is usually converted into a sequence of digital samples \(n[i] \triangleq n({{t}_{i}})\), which correspond to the moments of time \({{t}_{i}} = i{{T}_{{{\text{clk}}}}}\) with constant sampling period \({{T}_{{{\text{clk}}}}}\). The functional scheme for recording data on bunch intensities for tomographic reconstruction of the longitudinal phase portrait of an ion bunch in the superconducting booster synchrotron at JINR is shown in Fig. 1. This setup, which was successfully used at JINR Nuclotron, is described in detail in [2, 3].

Fig. 1.
figure 1

Functional diagram of data logging.

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During injection, five equidistant bunches are formed on the booster orbit (the RF harmonic number of the accelerating field frequency \({{h}_{{{\text{rf}}}}} = 5\)). The intensity of the bunches is recorded by the fast current transformer FCT. The analog signal from the FCT is supplied to the first input of a two-channel digitizer, where it is converted into a sequence of digital samples \(n[i]\) corresponding to the moments of time \({{t}_{i}} = i{{T}_{{{\text{clk}}}}}\) with constant sampling period \({{T}_{{{\text{clk}}}}}\) = 20 ns and amplitude resolution \({{N}_{{\text{a}}}} = 14\) bit. Each such reading, being the result of measurements, corresponds to the observed signal, taking into account the measurement noise provided the Kotelnikov–Nyquist conditions are fulfilled [9].

The second input of the digitizer is supplied with a harmonic signal for accelerating stations \({{\tilde {V}}_{{{\text{rf}}}}}(t) = \) \({{V}_{0}} + {{V}_{{{\text{rf}}}}}(t)\cos ({{\omega }_{{{\text{rf}}}}}(t) + {{\varphi }_{0}})\), whose cyclic frequency \({{\omega }_{{{\text{rf}}}}}(t)\) is set in accordance with the law of change of the magnetic field in dipoles \(B(t)\) [10].

As a result, two digital signals are formed \(n{\kern 1pt} [i]\) and \({{\tilde {V}}_{{{\text{rf}}{\kern 1pt} }}}[i]\), which are sent to a remote computer, where all the necessary signal processing and computer tomography procedures are carried out. First (see [11, 12]) these digital signals are jointly processed to transform a 1D data array \(n{\kern 1pt} [i]\) into a 3D array of data by samples \(i\) within the bunch with the number \(j\) on the revolution \(k\). The base level of the FCT signal is shifted in such a way that, within the bunch, the measured data correspond to a positively defined finite function. Then a 3D data array is converted into projection data \(f[i,j,k]\) for \({{h}_{{{\text{rf}}}}}\) bunches needed for a tomographic procedure starting at time \({{t}_{{{\text{init}}}}}\). And, finally, data for intensity profiles \({{f}_{j}}{\kern 1pt} [i,j,k]\) are generated for a specific bunch \(j\) for which the computed tomography procedure is performed.

The described procedure for proceedings from a digital signal \(n{\kern 1pt} [i]\) for the pulse intensity of circulating bunches to the functions \({{f}_{j}}{\kern 1pt} [i,j,k]\), which are a characteristic of the differential distribution law of particles in bunch \(j\) on revolution \(k\), makes it possible to identify the projection data needed for the computer tomography algorithm. The correctness of this procedure is controlled visually by constructing a 3D plot for the obtained profiles. The delay value of the digital signal \({{\tilde {V}}_{{{\text{rf}}}}}[i]\) is selected in such a way that all profiles for \(n{\kern 1pt} [i]\) consistently follow one after the other. This procedure is additionally checked using a plot for the number of particles in the bunch [11], which should correspond to a constant (or decreasing in the case particles are lost) function.

To reconstruct the phase portrait from the data for the longitudinal profiles of a particular bunch, study [1] used the iterative procedure ART [13]. It can be applied at the time \({{t}_{{{\text{init}}}}}\) for ions circulating along an orbit with a perimeter \({{C}_{0}} = 2\pi {{R}_{0}}\) with rest mass \(m\) and charge \(q\) if the magnetic field induction in dipoles \(B\) and the rate of its change \(\dot {B}\), the accelerating field \({{V}_{{{\text{rf}}}}}\), the rate of its change \({{\dot {V}}_{{{\text{rf}}}}}\), the RF harmonic number of the accelerating field \({{h}_{{{\text{rf}}}}}\), radius of curvature of the trajectory in dipoles \(\rho \), and the relativistic factor for the critical energy \({{\gamma }_{{{\text{tr}}}}}\) are known. The period of synchrotron oscillations \({{T}_{{\text{s}}}}\) expressed in terms of the particle circulation period \({{T}_{{{\text{rev}}}}}\) increases with an increase in the oscillation amplitude [10]. The more compact the bunch, the more accurate procedure [13] applied in [1]. Therefore, the result of the phase portrait reconstruction should correspond to a convergent iterative procedure, for which a criterion is specified in [13]: the estimate \({{D}^{q}}\)characterizing the discrepancy between the measured and calculated projections as a function of the number of iterations.

A feature of the observed signal \(n(t)\) when xenon ions 124Xe28+ were accelerated in the booster during commissioning operations was its low intensity against the background of measuring noise. To distinguish the differential distribution function of particles in a bunch against the noise background, we used the phase selection procedure [2, 3]. Accelerating voltage \({{V}_{{{\text{rf}}}}}\) corresponds to the height of the separatrix for phase trajectories (see Fig. 2), which is proportional to the maximum allowable deviation of the particle energy \(\Delta {{E}_{{{\text{max}}}}}\) from equilibrium [10]. The longitudinal distribution function of particles in a bunch usually corresponds to a Gaussian distribution. Consequently, the number of particles in the space of phase trajectories outside the trajectory with height \(\Delta E\) near the trajectory for \(\Delta {{E}_{{{\text{max}}}}}\) is not large. Therefore, it is possible to limit the array of processed data to the phase trajectory \(\Delta E < \Delta {{E}_{{\max }}}\). As a result of applying this procedure, a positive-definite finite function is formed, which has non-zero values only inside a given phase trajectory \(\phi \in [{{\phi }_{{2m}}},{\kern 1pt} {{\phi }_{{1m}}}]\) (see Fig. 2).

Fig. 2.
figure 2

Scheme of the phase selection procedure.

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PHASE PORTRAIT OF THE BUNCH IN THE PROCESS OF BEING EXTRACTED FROM THE BOOSTER

To fulfill the conditions for the transfer of charged particles from the booster to Nuclotron on the intermediate plateau of the magnetic field cycle in the booster, five bunches are ungrouped and the expanded beam is then grouped into a single bunch [7]. Plots of \(n{\kern 1pt} [i]\) and \({{\tilde {V}}_{{{\text{rf}}}}}[i]\) on the output plateau are shown in Fig. 3a (a fragment is reproduced in the range \(t \in [0.580007,\;0.580015]{\kern 1pt} \)s for \(k = 6\) revolutions). Fitting an array of measured data at k = 7767 revolutions (see Fig. 3b, \(t = \in [0.586,0.596]{\kern 1pt} \)s) for the period of revolution of a single bunch on the output plateau yields a statistical estimate of the value \({{T}_{{{\text{rev}}}}} = {{T}_{{{\text{rf}}}}} = 1.23066\,{\kern 1pt} \mu {\text{s}}\) (systematic error \( \pm 20{\kern 1pt} \)ns in accordance with \({{T}_{{{\text{clk}}}}}\)). The magnetic field induction in dipoles on the output plateau was \(B = 0.683903{\kern 1pt} \)T. For the design length of the orbit in the booster C0 = 210.96 m according to the data for \({{T}_{{{\text{rev}}}}}\) the radius of curvature of the ion trajectory in the dipole was calculated to be ρ= 14.0 × 1.0051 m. The resulting value of \(\rho \) is somewhat more than the design value \(\rho = 14.0{\kern 1pt} \) m.

Fig. 3.
figure 3

(a) \(n{\kern 1pt} [i]\) (solid line) and \({{\tilde {V}}_{{{\text{rf}}}}}{\kern 1pt} [i]\) (dashed line); (b) \({{T}_{{{\text{rev}}}}}{\kern 1pt} [k],\;k \in [0,7767]\).

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Figure 4a displays a 3D plot of the evolution of the longitudinal profiles of the FCT signal in the time interval \(t \in [586,\,587]{\kern 1pt} \) ms. A projection of 3D plot onto the \((k \times \tau )\) plane is shown in Fig. 4b. The 2D plot makes it possible to check the correctness of the procedures applied for synchronizing signals from the FCT and from the accelerating section: on the magnetic field plateau \(\dot {B} = 0\) and synchronous phase \({{\varphi }_{{\text{s}}}} = 0\). Consequently, the bunch center should be located at \(\tau = {{{{T}_{{{\text{rf}}}}}} \mathord{\left/ {\vphantom {{{{T}_{{{\text{rf}}}}}} 2}} \right. \kern-0em} 2}\). A plot displayed in Fig. 4b confirms that the synchronization conditions are fulfilled. A 3D plot in Fig. 4a indicates oscillations in the bunch shape, which makes it possible to estimate the value \({{{{T}_{{\text{s}}}}} \mathord{\left/ {\vphantom {{{{T}_{{\text{s}}}}} {{{T}_{{{\text{rev}}}}}}}} \right. \kern-0em} {{{T}_{{{\text{rev}}}}}}} \approx 1600\) and hence the effective accelerating field \({{V}_{{{\text{rf}}}}}\).

Fig. 4.
figure 4

(a) 3D plot and (b) 2D plot of an intensity profile.

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Thus, all the quantities necessary for the application of the tomographic procedure were determined by analyzing the data on the evolution of longitudinal profiles for the bunch intensities, taking into account the design specifications of the booster.

Figure 5c shows the phase portrait of the bunch under study. This result was obtained taking into account the application of the phase selection procedure at \(\Delta E = 0.98{\kern 1pt} \Delta {{E}_{{\max }}}\). At \(\Delta E = \Delta {{E}_{{\max }}}\) the specific weight of the peripheral components of the data array for the longitudinal profiles is significant due to the measurement noise. As a result, the central region for small amplitudes of synchrotron oscillations is poorly reconstructed. Thus, the use of the phase selection procedure (limitation of the reconstructed longitudinal energy spread in the presence of a noise component in the processed digital signal) makes it possible to provide the necessary reconstruction of the longitudinal phase portrait of the ion bunch for small amplitudes of synchrotron oscillations.

Fig. 5.
figure 5

(c) phase portrait and its corresponding distribution functions (a) in time and (d) in energy; (b) plot of \({{D}^{q}}\) vs the number of iterations.

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PHASE PORTRAIT OF A BUNCH ON THE INJECTION PLATEAU

The ion beam is injected from the linear accelerator on the plateau of magnetic field injection in the booster with the RF system turned off. As a result, the injected bunch is ungrouped. After about 135 ms, an accelerating voltage is applied to the accelerating system, and 5 bunches are formed (the RF harmonic number of the accelerating field \({{h}_{{{\text{rf}}}}} = 5\)). Figure 6 shows 2D plots of the evolution of the longitudinal profiles of formed bunches (\(t \in [120,{\kern 1pt} 200]{\kern 1pt} \)ms) with the electron cooling system turned off and on [14]. The tomographic procedure is applied to the last 10 ms of the digital data array, since when the accelerating RF system is turned on, the bunch formation is accompanied by some beam losses.

Fig. 6.
figure 6

Electron cooling system is (a) off and (b) on.

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Figure 7 shows 2D plots of the evolution of the longitudinal profiles of formed bunches prepared for the application of the tomographic procedure (\(t \in [190,{\kern 1pt} 200]{\kern 1pt} \)ms). To do this, with the electron cooling system turned off, for detuning from noise, the phase selection procedure was used to select phase trajectories lying inside the trajectory \(\Delta E = 0.92{\kern 1pt} \Delta {{E}_{{\max }}}\) (see Fig. 7a). In addition to the phase selection procedure, a rebinning procedure was applied in accordance with the algorithm presented in [1]. This procedure is reduced to averaging data for longitudinal profiles over several revolutions \({{k}_{{{\text{av}}}}}\), which somewhat improves the phase portrait resolution (\({{k}_{{{\text{av}}}}} = 2\) for the profiles displayed in Fig. 7a).

Fig. 7.
figure 7

Profiles for tomography: the electron cooling system is (a) off and (b) on.

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With the electron cooling system turned on, the bunch is more compact. Therefore, it is sufficient to use the phase selection procedure to select phase trajectories that are located inside the trajectory \(\Delta E = 0.98{\kern 1pt} \Delta {{E}_{{\max }}}\) (see Fig. 7b).

It should be noted that the driving frequency supplied to the accelerating station and the rate of switching on the accelerating voltage were not chosen very carefully. Therefore, particles were grouped into bunches with significant coherent synchrotron oscillations (see Fig. 6). However, the graphic dependences displayed clearly demonstrate the operability of the electron cooling system. The measurements were carried out on January 25, 2023, at 16:07:07 (electron cooling system turned off) and at 16:10:19 (electron cooling system turned on). Graphic dependences presented in Fig. 7 confirm that the conditions for synchrotron motion with respect to the parameter \({{{{T}_{{\text{s}}}}} \mathord{\left/ {\vphantom {{{{T}_{{\text{s}}}}} {{{T}_{{{\text{rev}}}}}}}} \right. \kern-0em} {{{T}_{{{\text{rev}}}}}}}\) are identical.

The results of reconstructing the phase portraits of the ion bunch are shown in Figs. 8. With the electron cooling system being turned on, the bunch is compact. However, when the cooling system is turned off, the particles are distributed along almost the entire length of the bunch, which is consistent with the data on the longitudinal profiles shown in Fig. 7a. The root-mean-square spread in the energies of the formed bunch is approximately \({{\sigma }_{{\text{E}}}} = 4.1 \times {{10}^{{ - 6}}}\) when the electron cooling system is turned off, which is significantly smaller than the same estimate for the expanded beam obtained in [15]. This result is in agreement with the observed losses of charged particles during the formation of five bunches on the injection plateau.

Fig. 8.
figure 8

Phase portrait of a bunch: the electron cooling system is (a) off and (b) on.

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CONCLUSIONS

The method of tomographic reconstruction of the longitudinal phase portrait of an ion bunch in the JINR superconducting booster synchrotron has been successfully applied.

It has been established that the use of the phase selection procedure (limitation of the reconstructed longitudinal energy spread in the presence of a noise component in the processed digital signal) makes it possible to provide the necessary reconstruction of the longitudinal phase portrait of an ion bunch for small amplitudes of synchrotron oscillations.

Examples are given of reconstructed phase portraits of a xenon ion bunch after acceleration in the booster and after injection, also with the electron cooling system turned on.