Modeling of the beam core in phase space using kernel density estimation

Phase space is a mathematical construct that encompasses particle positions and their corresponding momenta. In general, discrete phase space structures are experimentally measured due to the limited spatial and angular resolutions of individual devices. From the perspective of beam focusing characteristics, beams are discussed in terms of core components and halo components. While the beam core is typically the primary component with low divergence, it may occasionally consist of multiple components with different velocity distributions. Mathematical modeling of the beam core in phase space is essential for accurately quantifying beam divergence and emittance. This paper presents a model that can be applied to beam cores with inner structures and reconstructs the phase space structure using kernel density interpolation. The reconstructed phase space structure is then utilized to determine beam divergence and emittance with greater precision. Additionally, these insights contribute to an enhanced understanding of beam physics.


Introduction
The phase space is a mathematical construct that describes the distribution of particle positions and their corresponding momenta [1].It can be visualized as a multidimensional space in which each particle is represented by a point, and the density of points in the phase space reflects the probability of finding particles in those states.The shape and size of the phase space distribution can provide important information about the quality of the particle beam, such as its emittance (a measure of the spread of particle trajectories) and its energy spread.This is a mathematical concept that is widely used to analyze and model the behavior of charged particle beams, such as those used in accelerators [2], colliders [3], and other particle-based technologies [4].
The construction of a phase space structure often requires the use of a pinhole array and a detector in combination [5].The spatial resolution of this measurement system is determined by the spacing between pinholes, while the angular resolution is affected by both the diameter of each pinhole and the distance between the pinhole plate and the detector.As a result, discrete phase space structures are typically measured experimentally due to the limited spatial and angular resolutions of individual devices [6].Beams are often described in terms of their focusing characteristics, namely their core and halo components [7].The beam core refers to the central, well-focused component, while the beam halo refers to the outer, less-focused components.Recent research has revealed that the core component of a negative ion beam produced by a cesium-seeded negative ion source consists of multiple velocity distribution components that form its internal structure [8].Therefore, reconstructing phase space structures from measurements is highly desirable as it provides critical information about the behavior and trends of individual beams in phase space.This paper presents a model of the beam core in phase space and demonstrates how kernel density estimation can be used to interpolate experimental data and reconstruct the phase space structure as a continuous function [9].This approach offers a powerful tool for analyzing beam dynamics and could have important applications in areas such as particle accelerators and ion implantation [10].

Measuring system
In order to experimentally construct the phase space structure, an emittance meter consisting of a pinhole plate and a Kapton foil has been developed and installed in the research-anddevelopment negative ion source at the National Institute for Fusion Science [11].The negative ion source has a filamented arc discharge ion source with cesium injection, which enhances the surface production of hydrogen negative ions [12].Figure 1 shows the source and the diagnostic beamline.The source is adjusted to ensure that the beam axis (z) is perpendicular to the diagnostic plane.Hydrogen negative ions are accelerated using four electrode grids: a plasma grid, an extraction grid that includes permanent magnets to remove co-extracted electrons, a steering grid, and a grounded grid.The maximum strength of the magnetic field formed by the permanent magnets is 470 G along the y-axis [13].After passing through the electron suppression structure, negative ions are injected into the pinhole array, which subdivides the beam into multiple partial-beamlets.These partial-beamlets pass through each pinhole and reach the surface of the Kapton foil after traveling a fixed distance, creating footprints.By measuring the footprint positions relative to the pinhole positions, the transverse momenta can be obtained by In this expression, the coordinates of the pinhole array are represented by lowercase letters (x mn , y mn ), while the coordinates of the Kapton foil are represented by uppercase letters (X mn , Y mn ).
The distance between the pinhole plate and the Kapton foil is denoted by L. The subscripts m and n indicate the numbering of the pinholes in the vertical (y) and horizontal (x) directions, respectively.The pinholes are arranged in a square grid pattern with a spacing of 3 mm, and each has a diameter of ϕ0.4 mm.In the present case, with L=166 mm, the angular resolution is 2.4 mrad.

Data reducation
The phase space measurement of the hydrogen negative ion beam was performed using the following parameters: a gas pressure of 3 mTorr, a discharge power of 90 kW, a density of hydrogen negative ions in the source plasma of 3×10 17 m −3 , an extraction voltage of 2.8 kV, an acceleration voltage of 45 kV, and an extracted negative ion current of 9 mA.Footprints were observed on a Kapton foil after 15 s of beam irradiation, as shown in Fig. 2, which displays the intensity distribution of a large number of beamlets passing through each pinhole in the irradiated area.The observed footprints were digitized using a scanner with an 8-bit gray scale depth and a spatial resolution of 5.2 µm/pix.After background subtraction, the increase in darkness of the Kapton foil was determined.It has been previously confirmed that the degree of darkness is proportional to the integrated beam current density while keeping the beam energy constant [14].Three sets of beamlets are subdivided by individual pinholes.To replicate each peak accurately, we assume that a specific velocity distribution component maintains a consistent interval between neighboring peaks along the horizontal direction, which is perpendicular to the magnetic field generated by the electron deflection magnets.The fitting function employed for the peak profile is modeled using three sets of multi-Gaussian distributions; . ( This expression is presented with notations u, c, and l denoting the "upper", "center", and "lower" distributions in phase space (x, x ′ ), respectively.The subscripts m and n represent the numbering of the peaks for each component in the vertical and horizontal directions, respectively.The parameters u mn , c mn , and l mn represent the amplitudes of the mth-nth peak, where c 00 corresponds to the amplitude of the tallest peak in the center component.The positions of the mth-0th peak are denoted as (X um0 , Y um0 ), (X cm0 , Y cm0 ), and (X lm0 , Y lm0 ), where (X c00 , Y c00 ) corresponds to the position of the tallest peak in the center component.The horizontal intervals between neighboring peaks for the upper, center, and lower components are denoted as λ um , λ cm , and λ lm respectively.The e-folding half-widths (σ x , σ y ) are common for all peaks.Figure 3 shows the spatial distributions (X mn , Y mn ) of the three components, which are determined by fitting peak profiles using Eq. ( 2).By substituting the spatial distributions of the partial beams passing through each pinhole into Eq.( 1), it becomes possible to experimentally evaluate the transverse momenta (x ′ mn , y ′ mn ).Consequently, the preparations for reconstructing the phase space structure of the beam are complete.In order to reconstruct the phase space structure (x, x ′ ) for m = 0, a fitting function is defined as a superposition of three elliptical distributions; .
( shows the reconstructed phase space structure (x, x ′ ) obtained by superimposing the three ellipses.Although the three components are horizontally separated, each of them exists within the beam core with a small divergence of less than 10 mrad.It is noteworthy that a similar structure can be observed in the phase space between the upper and lower components that surround the inner small component.This information is crucial for understanding the behavior and trends of negative ion beams.The reconstruction of the phase space structure (y, y ′ ) for n = 0 has been performed with a fitting function defined as In this expression, J u , J c , and J l are the amplitudes; a yu , a yc , and a yl are the horizontal widths;  The geometric emittance is defined as the area in the phase space plane occupied by beam particles, divided by π.The normalized emittances in the (x, x ′ ) and (y, y ′ ) planes are defined as follows: In these expressions, β represents the velocity of beam particles along the z-axis relative to the speed of light in a vacuum, and γ is the relativistic mass correction factor [15].In the present case, γ is approximately equal to unity, with β being roughly 1.0 × 10 −2 .The emittances of the individual components have been well quantified based on the mathematical model using each ellipse.Table 1 presents a list of parameters described in this paper.Normalized emittance in phase space (y, y ′ ) mm mrad 0.15 0.19 0.17

Conclusion
This paper presents a mathematical model of the beam core in the four-dimensional phase space (x, x ′ , y, y ′ ), providing a detailed reproduction of the beam core.In terms of the hydrogen negative ion source, the beam core can be effectively described by the superposition of three ellipses in both the (x, x ′ ) and (y, y ′ ) planes.This discovery, addressed for the first time in the present work, may be closely related to the fundamentals of the production and extraction processes of negative ions.Despite the beam being composed of multiple components with different velocity distributions, the combination of ellipses in phase space proves to be an effective method for reproducing the continuous phase space structure.

Figure 1 .
Figure 1.Conceptual view of the ion source and diagnostic beamline configuration for phase space analysis of a single negative ion beam.

Figure 2 .
Figure 2. The footprints on the Kapton foil created by the partial beamlets passing through each pinhole.

Figure 3 .
Figure 3.The spatial distributions of the three components within the single hydrogen negative ion beam.
) Here, I u , I c , and I l represent the amplitudes; a xu , a xc , and a xl denote the horizontal widths; b xu , b xc , and b xl correspond to the vertical widths of each ellipse; ϑ u , ϑ c , and ϑ l indicate the inclination angles relative to the x-axis.Figure 4(a) b yu , b yc , and b yl are the vertical widths of each ellipse; φ u , φ c , and φ l are the inclination angles relative to the y-axis.The reconstructed phase space structure (y, y ′ ) has been obtained by superimposing the three ellipses, as shown in Fig. 4(b).

Figure 4 .
Figure 4. (a) The reconstructed phase space structure in the (x, x ′ ) plane, obtained by superimposing the three ellipses described by Eq. (3).(b) The reconstructed phase space structure in the (y, y ′ ) plane, obtained by superimposing the three ellipses described by Eq. (4).

Table 1 .
Summary of the parameters obtained from the present work.The subscript i refers to the individual components (u, c, and l) that are observed in the single negative ion beam.