Introduction

Large-angle deflection of high-current, ultra-short electron bunches is essential for achieving higher luminosity and enhanced photon brightness in recirculating linacs and X-ray free-electron lasers [1-7]. However, maintaining bunch quality during deflection is of paramount importance. Coherent synchrotron radiation (CSR) during deflection can severely degrade bunch quality, especially when charged particles traverse a curved trajectory. In this process, the bunch emits coherent radiation that interacts with the constituent particles, significantly impacting their dynamics and, as a result, affecting overall beam quality and performance.

When a relativistic bunch traverses a dipole magnet, CSR is generated if the bunch length is comparable to the radiation wavelength. This radiation induces energy modulation between the bunch head and tail, resulting in distortion of the longitudinal phase space. Additionally, the coupling between transverse and longitudinal directions can lead to degradation of transverse bunch quality [8, 9].

Methods for suppressing the CSR effect in compressed multi-bend structures have been extensively studied. The research focuses on achieving both CSR suppression and bunch length compression in compressors [10-16]. Unlike the optimization objective of the compressor, non-compressed multi-bend structures need to maintain the longitudinal profile of bunches while suppressing the CSR effect. Since non-compressed multi-bend structures are often used to bend high-intensity bunches, the intensity of the CSR effect in the isochronous arc region, under the same bending angle, is greater than in the compressor. This can lead to degradation of bunch length, micro-bunching structure, and other properties [17-20].

The triple-bend achromat (TBA) is a relatively simple structure that can adjust the R₅₆, and it is commonly used for beam transport and deflection where preservation of the longitudinal profile is required. Research has also shown that, under certain special designs, the TBA structure can deflect high-brightness bunches while suppressing the CSR effect. The beam envelope matching method minimizes projected emittance growth through Twiss function adjustment, while the R-matrix method analyzes CSR effects in achromatic cells and aligns the undisturbed beam distribution in the transverse phase space [21, 22]. The R-matrix method was modified in [23] with the objective of analyzing the horizontal displacement caused by CSR in one dipole magnet, which is called the kick-point method. The C-S formalism analysis [11] was also modified and combined with the kick-point method to effectively reduce the transverse emittance growth caused by the CSR effect. Using the kick-point method, Huang derived the general conditions for eliminating CSR effects in a single TBA cell and obtained a set of numerical solutions that could be applied in TBA design [24-26]. Another set of numerical solutions was demonstrated [27, 28], which not only suppressed CSR effects but also achieved optical stability, enabling the serial connection of multiple cells to increase the deflection angle. Moreover, longitudinal control such as R₅₆ and T₅₆₆ were applied to maintain the longitudinal profile of the bunch after a 60° deflection, effectively suppressing the gain of microbunching instability (MBI). The integral method achieves CSR suppression through minimizing the integrated (i=1, 2) under small angle approximation in TBA cells [29].

In this study, two methods were employed to optimize a beamline for large-angle deflection to mitigate the effects of CSR. The first method builds on the work in [29, 30]. Our enhanced integral optimization method eliminates the small-angle approximation, enabling accurate CSR suppression even at larger deflection angles. This approach significantly simplifies numerical optimization processes and extends its applicability to various complex lattice designs. The second approach, our optimized I-matrix method, systematically cancels both steady-state (SS CSR) and transient CSR (TR CSR) kicks through precise matrix conditions and second-order dispersion optimization. The beamline based on this method effectively suppresses residual CSR kicks and preserves beam quality even after multiple deflection cells, making it particularly suitable for large-angle deflection beamlines. In Sect. 2, the physics model of emittance degradation caused by CSR is introduced, and we further simplify the integral method. In Sect. 3, a single TBA cell is studied to meet multiple requirements, and a multi-objective optimization algorithm (MOGA) is combined with the integral method to design an isochronous TBA structure. In Sect. 4, the I-matrix method is used to design a double TBA cell, with further optimizations enabling the bunch to be deflected at a larger angle. Additionally, a brief discussion of the SXFEL upgrade plan and preliminary experimental results is provided. Sect. 5 concludes the study.

Theory analysis for emittance degradation caused by CSR

The effect of CSR on bunches is often investigated using a 1D projected model that neglects vertical influences and considers only longitudinal interactions. Within this approximation, the CSR effect depends exclusively on the natural coordinates s and the longitudinal coordinate of particle z. As the particles are deflected in a beam transport line, the transverse displacements x and x’ of the observation point s_f along the beam line can be expressed as:(1)where and are the oscillation motions of the particles constrained by the magnetic elements and are solely dependent on the initial coordinates of the particles in the six-dimensional phase space and lattice parameters. The transverse offsets x_csr and arise due to the coupling between the longitudinal and transverse directions, caused by the energy spread generated by the CSR effect; the energy spread induced by the CSR effect is represented by , at any point si, the variation in energy spread is denoted as Δδ_csr,i. In the beamline with non-zero dispersion, the coupling terms between the transverse and longitudinal directions lead to additional transverse offsets Δx_csr,i and , which are produced by Δδ_csr,i generated at point si. The transverse offsets Δx_csr,i and generated at observation point s_f are:(2) (i=1, 2) represents elements R₁₆ and R₂₆ in the transfer matrix from any point si to observation point s_f. The total transverse offset generated by the CSR effect from the initial point s₀ to the observation point s_f is obtained by summing the transverse offsets Δx_csr,i and produced at each point along the path. This sum can be represented in the form of an integral, given as(3)We define and as the CSR kick at point z in the particle longitudinal coordinate. According to the discussion in Sect. A, the CSR-induced energy spread variance for arbitrary longitudinal profiles can be approximated as:(4)σz(s) is the RMS bunch length at point s, is the normalized longitudinal position, and the formula of Γ(u) is presented in Eq. (A.2), which is only dependent on the longitudinal profile of the bunches. In this study, we aim to deflect a beam with high peak current while maintaining the distribution of the longitudinal profile. Therefore, two conditions must be satisfied: i) the lattice must be isochronous, and ii) the deflection angle of a single bend must be kept small to prevent excessive local effects R₅₆. Larger local R₅₆ can cause significant changes in the longitudinal profile of the bunch, potentially leading to irreversible beam degradation. Once the above two conditions are satisfied, the variance in bunch length can be neglected in all regions of the beamline. From the Eq. (B.6) in Sect. B, the emittance at the exit of the beamline is(5)εx, the geometric emittance at the entrance and exit of the beamline; βx, αx and γx the Twiss functions at the exit of the lattice. The definition of Ii and I_csr can be found in Sect. B. when the effects of SS CSR are neglected; , when only SS CSR is considered. Therefore, the emittance growth stems from in Eq. (5). Minimizing this term can suppress the emittance variation induced by SS CSR in the design process. Moreover, for the lattice with periodic solutions, the can be further simplified to(6) the periodic solution of TBA cell. Consequently, in this study, an objective function for the lattice with solutions can be defined for evaluating the impact of SS CSR:(7)By optimizing the transfer matrix elements as shown in Eq. (7), the integral can be effectively minimized, thereby mitigating the impact of SS CSR. This result is valid for any bunch distribution. The computation of the integral can be achieved either by fitting the evolution of within the bends during the optimization process or by uniformly sampling points along the bends and summing their contributions. In practical applications, particularly for an asymmetric lattice without periodic solutions, a more universally applicable approach involves deriving all elements of the matrix or adjusting the Twiss functions at the exit of the beamlines as variables to search for the optimal solution.

CSR effects in single Isochronous TBA cells with periodic optics function

In this study, we focused on the isochronous TBA cells. And the following conditions need to be met to completely eliminate the SS CSR kick in a single TBA cell: 1) and 2): isochronous and achromatic, 3) and 4) I₁ and I₂ equal to zero, 5) . In the case of symmetric TBA configurations with three identical bends, no structure can simultaneously satisfy all the specified conditions [27, 29]. This is due to the fact that the number of constraints exceeds the number of degrees of freedom. As a result, the ratio between the middle bend and the side bends in the symmetric TBA structure is introduced as an additional degree of freedom. An analytical investigation incorporating this additional degree of freedom is presented in Sect. C. In this investigation, it is described that the value of decreases with the reduction of the absolute value of the ratio k when conditions (1) to (4) are met; however, its minimum value still reaches . Section C revealed the absence of an exact solution that satisfies all of these conditions, necessitating the search for an asymptotic solution for the symmetric TBA. The Non-dominated Sorting Genetic Algorithm II (NSGA2)[31], a versatile multi-objective optimization algorithm, was employed for this task. It utilizes non-dominated sorting and genetic operators to efficiently generate diverse and effective solutions in a single run. The accelerator physics simulation program BMAD [32] was used to simulate beam dynamics with CSR effects, and to ensure accuracy, the simulation outcomes were validated using the accelerator physics program ELEGANT [33]. All results presented in this paper are based on BMAD simulations and have undergone comparative validation. The parameters of the bunch used in the simulation are listed in Table 1. The bunch in the first column of the table is a Gaussian bunch with a peak current of 2000 A. The second column represents the actual case, with the non-ideal bunch coming from the Shanghai Soft X-ray Free-Electron Laser (SXFEL)[34, 35], the first X-ray FEL facility in China. Within the TBA design, the angular ratio between the middle bend and the side bends is defined as k, resulting in a total deflection angle of 15°. The TBA assembly comprises six quadrupole magnets and four sextupole magnets, intricately arranged to refine both the first- and second-order transfer matrices. The strategic placement and calibrated strengths of these magnets exhibit a symmetrical distribution with respect to the central axis of the TBA cell.

The asymptotic solution must lie within the optically stable region of the isochronous TBA structure. By combining the achromatic, isochronous, and symplectic conditions of the transfer matrix, it is known that for any given k, the two optical stability regions of the isochronous TBA structure are denoted as regions S1 and S2 in Fig. 1. The S1 region is defined within the interval , which aligns precisely with the curves in Fig. 13, where and . Notably, within the S1 region, the magnitudes of I₁ and I₂ are observably lower than those within the S2 region. Furthermore, the was calculated within the S1 region which is defined in Eq. (6). When m₂₂ approaches -k or –k/2, as shown in Fig. 2, becomes smaller than in other cases in region S1, resulting in a lesser impact of SS CSR on the transverse quality of the bunch. It is worth mentioning that the TBA structures corresponding to m₂₂=-k and m₂₂=-k/2 have with a non-zero , which indicates that these TBA cells do not have a periodic solution for the β function. Simultaneously, the periodic solutions corresponding to the β function are large in the adjacent areas of m₂₂=-k and m₂₂=-k/2, although theoretical calculations indicate that the SS CSR does not have a significant impact in these cases. Therefore, when considering the impact of TR CSR at the entrance and exit of the bend, x_csr and no longer conform to the form described in Sect.5.2. Furthermore, when m₂₂=-k or m₂₂=-k/2, or no longer equals zero. As known from Eq. (5), a larger βx and γx coupled with the residuals and will lead to a significant increase in emittance. Therefore, in the optimization process of this study, the β function corresponding to the periodic solution was constrained to lie within the range to effectively mitigate this effect.

Fig. 1Full-screen View

(Color online) Optical stability region for a symmetric isochronous TBA structure

Fig. 2Full-screen View

(Color online) in S1, the two boundaries of this region are respectively m₂₂=-k/2 and m₂₂=-k

In the optimization process, a set of 12 variables is used. These include three quadrupole magnet strengths (ranging from [-10, 10]), two sextupole magnet strengths (ranging from [-500, 500]), one angular ratio k (between the middle and side bends (ranging from [0.2, 2]), and six drift lengths between magnetic elements (each ranging from [0.1, 2]). The optimization is formulated as a multi-objective problem with two goals:(8)where f_obj1 represents the impact of the SS CSR on the transverse dimension. As discussed in Sect. 2, it denotes the minimum growth in emittance when αx=0 at the exit of the TBA. denotes the relative change in bunch length due to passage through the TBA while accounting for second-order effects. f is an additional coefficient that equals 1 when the TBA satisfies the following conditions: (1) ; (2) (3) ; otherwise, it is a large number related to these conditions. Condition (1) ensures that the TBA structure is achromatic. Conditions (2) and (3), on the other hand, are designed to enforce the existence of a periodic solution in the TBA cell, characterized by low values of the β and γ functions. The value of I₁ and I₂ within f_obj1 were calculated by uniformly distributing points across the lattice. The parameter n is the number of points in TBA cell. When n=1000, the error between the statistical result and the actual result is within 5%. It is worth mentioning that when Ii is significantly smaller than the maximum value of of the lattice, then this method may lose accuracy. The results can be made more precise by either increasing the number of points or employing curve fitting method. After evolving over 500 generations, the optimization results converged. Figure 3 illustrates the objective functions and key parameters of the TBA in the final generation. As shown in Fig. 3(a), the solutions converge toward a working point near (the dashed line in the figure), which is consistent with the results in Fig. 2. However, due to errors introduced by the small-angle approximation and constraints imposed by the periodic conditions of the beta functions, the solutions do not converge indefinitely toward this boundary. Then two notable cases are marked: case A () and case B (), which correspond to the emphasis on preserving the longitudinal profile and suppressing the emittance growth caused by the SS CSR effects, respectively.

Fig. 3Full-screen View

(Color online) Distribution of solutions in the final generation. (a) Relationship between the bending ratio k and the transfer matrix element m₂₂ as defined in Eq. (C.2). (b) Values of |I₁| and |I₂| computed without the small-angle approximation. (c) Sextupole strengths k₂ for the two sextupoles in the TBA cell. (d) Pareto front illustrating the two objective functions. Red and black asterisks denote the selected solutions corresponding to Case A and Case B

To thoroughly evaluate the suppression capability of SS CSR in Case A (point A) and Case B (point B), we altered the beta function at the midpoint of the TBA under the condition and employed the transfer matrix from the midpoint to the endpoint in conjunction with Eq. (5) to theoretically predict the emittance variation at the TBA exit under symmetric optical conditions (as shown by the curve in Fig. 4). Simulation results across different values reveal that a larger β function leads to more gradual emittance growth, consistent with theoretical predictions, as indicated by the circular markers in Fig. 4. Furthermore, for practical implementation involving the serial connection of multiple TBA cells, the periodic solution—denoted by the cross markers in Fig. 4—deserves special attention. When using a Gaussian bunch as specified in Table 1, the emittance growth due to SS CSR in both Case A and Case B remains below 1%. However, the bunch length variation in Case B is only 16% of that observed in Case A. For the transport of high peak current beams, preserving the longitudinal beam profile is as critical as minimizing emittance growth. Therefore, Case B is selected to ensure both the transverse and longitudinal beam qualities are well maintained across multiple connected TBA cells.

Fig. 4Full-screen View

(Color online) For both TBA cases, the emittance growth induced by SS CSR is illustrated: the solid curve represents the theoretical analysis, the circular markers (o-markers) denote the simulation results obtained using BMAD, and the cross markers (x-markers) correspond to the cases where the optics satisfy the periodic solution

In the simulation work, to better reflect realistic beam dynamics, the effects of TR CSR from components upstream and downstream of the bend were also taken into account. The default order of magnetic elements was set to three. Simulation results for a single TBA cell are summarized in Table 2. As the bunch propagates through a TBA cell, the variation in bunch length for both bunch profiles remains below 0.1%. When only SS CSR is considered, the emittance growth is limited to less than 1%, which is in good agreement with the theoretical predictions from Eq. (5). However, when TR CSR is included, the emittance increases by approximately 3% compared to simulations neglecting CSR effects.

In the simulation of the multi-cell TBA structure, six TBA cells were connected in series to achieve a total deflection angle of 90°. In this section, the values of I₁ and I₂ at the exit of each cell were computed and compared with the results from SS CSR simulations. In the multi-cell configuration, emittance variation is influenced not only by , but also by the transfer matrix of each individual TBA cell, as I₁ and I₂ are not exactly zero. Consequently, their values differ at the exits of various cells. To determine these values, the R₅₁ and R₅₂ curves were extracted from BMAD for the cascaded cell configuration. Furthermore, to maintain statistical accuracy, the number of sampling points was scaled proportionally to the number of connected TBA cells. For example, when calculating I₁ and I₂ at the exit of the sixth TBA cell, the parameter was set to 6000. As illustrated in the top panel of Fig. 5, the theoretical analysis based on Eq. (5) aligns well with the simulation results for the multi-cell TBA scenario. When including both SS CSR and TR CSR effects, the emittance growth for both bunch types is presented in Fig. 5. Simultaneously, the bunch length was well preserved throughout the structure, as depicted in Fig. 6, with less than 1% variation observed for both bunch profiles.

Fig. 5Full-screen View

(Color online) Simulation results of multi-cell TBA cells in BMAD: (top) emittance growth, ‘Gau’ represents the Gaussian bunch, ‘SX’ represents the SXFEL bunch. The left Y-axis represents the case considering only SS CSR and the right Y-axis represents the case considering SS CSR and TR CSR

Fig. 6Full-screen View

(Color online) The longitudinal profile of the bunch at the exit of each TBA cell. (Left) The SXFEL bunch, (Right) The Gaussian bunch, the black line represents the initial bunch and the term ‘cell i’ refers to the bunch located at the exit of the ith TBA cell. The purple box with dash line is the core region of SXFEL

For the Gaussian bunch, the longitudinal profile remained well-preserved throughout the bunch range. At the exit of the sixth TBA cell, there was no significant deviation from the initial bunch profile, as shown by the red and black lines in the right panel of Fig. 6. For the SXFEL bunch, the longitudinal profile in the core region (highlighted within the purple box in the left panel of Fig. 6) was well-maintained. However, degradation in bunch quality was primarily observed at the sides of the bunch. As indicated by the black curve in the left image of Fig. 6, the slices at the edges of the bunch exhibited higher current intensity and steeper gradients in current strength. In practical applications, the core region of the bunch is typically of primary interest. Therefore, the impact of CSR effects on the longitudinal profile of the bunch can be considered negligible within a limited number of cells. For small-angle deflections, the single TBA structure used in this study effectively mitigates beam quality degradation. Compared to the two-cell configuration discussed in Sect. D, the single TBA cell is more practical due to its simpler structure and fewer magnets. However, as the number of cells increased, the projected emittance exhibited a noticeable rise for both bunch types. This increase can be attributed to the neglected CSR effects during the sequence drift and at the entrance of the bends in the design phase. Previous studies have shown that completely canceling CSR effects in the sequence drift of TBA cells is impossible [36].

CSR effects in double Isochronous TBA cells with periodic optics function

For a single isochronous TBA cell, the complete elimination of SS CSR effects remains a challenge. As outlined in Sect. 3, an optimal working point is determined that strikes a balance between suppressing CSR effects and preserving the longitudinal bunch profile. Although the design and optimization process considers only the SS CSR kick, the TBA configuration presented in Sect. 3 proves highly effective in mitigating CSR kick, both in theoretical analysis and simulations. However, as the number of cascaded cells increases, the quality of the bunch progressively degrades. This degradation is primarily due to the cumulative impact of TR CSR effects throughout the beam transport process. Similar to Eq. (5) for SS CSR, the TR CSR kick, in combination with the Twiss functions, contributes to a significant increase in emittance. Consequently, to achieve larger deflection angles while preserving beam quality, it becomes crucial to simultaneously suppress both TR and SS CSR effects.

A -I matrix between bends with identical deflection directions (while an I matrix for bends with opposite deflection directions) is a common approach to suppress SS CSR kicks: For any points s_A and s_B in bends A and B, the meets(9)This results in the complete cancellation of the SS CSR kick. Following the four cases presented in [37], we investigate the energy spread induced by TR CSR. The analysis assumes constant magnetic fields in the bends and negligible bunch length variation. Under these conditions, the energy spread induced by TR CSR depends on two key parameters: the position s’ relative to the entrance of the single n-bend achromatic structure (TBA in this study), and the position x relative to the upstream bend. So, for any s’ and x in double TBA cell, meets(10)The subscript ‘A’ and ‘B’ denote any points in different TBA in double TBA cells. Moreover, similar to Eq. (9), the transfer matrix for the re-lat (detailed description provided in Sect. B) satisfies:(11)L is the distance between the entrances of adjacent TBA cells. Thus, for the beamline consisting of two TBA cells, the re-lat meets(12)

Based on the I matrix method, a large-angle deflection beamline was designed, with a focus on investigating the CSR effects on high-peak-current bunches. A detailed overview of this design is provided in Sect. D. As the bending angles within the TBA cell increase, the local R₅₆ becomes more significant, which invalidates the rigid beam approximation. This leads to substantial and unpredictable growth in the CSR kick. To maintain the validity of the rigid beam approximation, the number of TBA cells in series should be increased proportionally with the bending angle. Additionally, when two double TBA cells with a -I matrix are connected in series, inserting an I matrix between them further enhances the suppression of the CSR kick at the exit of the beamline.

Figure 7 illustrates a simple model for CSR kick error cancellation achieved through an I matrix. While the Double TBA cell theoretically suppresses CSR effects, higher-order dispersion and local bunch length variations result in residual non-zero values of x_csr and at the double TBA cells exit. The erri in Fig. 7 represents the residual values at the i-th double TBA cell. A -I matrix connection between double TBA cells results in aligned err₁ and err₂, leading to cumulative errors with increasing cell numbers (Top insert in Fig. 7). In contrast, an I matrix connection generates opposing errors that mutually cancel each other (Bottom insert in Fig. 7). These errors result in CSR-induced displacements across different slices, ultimately contributing to emittance growth.

Fig. 7Full-screen View

(Color online) The CSR kick errors can be mutually cancelled by introducing an I matrix between double TBA cells with -I matrix

Figure 8 presents the simulation results for both cases, with the bunch parameters provided in Table 1. The I matrix configuration results in smaller displacements, which helps preserve the bunch quality. Conversely, for the double TBA cell described in Sect. D, implementing a -I matrix between two double TBA cells effectively reduces these displacements.

Fig. 8Full-screen View

The CSR-induced displacements along the x-axis at the exit of the second double TBA cell are shown for two configurations: black points represent the case with a -I matrix between the double TBA cells, while red points indicate the case with an I matrix connection. Here, x_bunch refers to the center of the entire bunch, and x_slice represents the center of each individual slice

Furthermore, to achieve larger deflection angles while suppressing CSR-induced emittance and maintaining the longitudinal profile, the second-order matrix cannot be neglected. Similar to , the second-order matrix element couples with the additional energy spread, leading to growth in εi. The elements of the second-order matrix can be expressed as(13)where B_abc is the quadratic coefficient of xb and xc in the dynamical differential equation for xa. It depends solely on the type of elements and remains constant for uniform elements. And(14)To simplify the formula of Eq. (13), we employ Einstein’s summation convention, where repeated indices imply summation, and the summation signs are omitted in Eq. (13). Under the assumption that the matrix between the entrances of two TBA cells is a -I matrix, we have:(15)and(16)In this investigation, we focus on a beamline system characterized by horizontal-plane bending and the absence of x-y coupling in its first-order transfer matrix. A significant consequence of Eqs. (15) and (16) is that Tijk vanishes when the indices (i, j, k) contain an even number (including zero) of occurrences of 5 or 6. This mathematical property leads to an important result: the second-order dispersion terms (specifically T₁₆₆ and T₂₆₆) automatically vanish for the double TBA cell configuration. To effectively control the horizontal emittance growth, four critical second-order matrix elements require careful consideration: T₁₁₆, T₁₂₆, T₂₁₆, and T₂₂₆. Furthermore, in a symmetric beamline configuration, we observe the relationship . For beamlines with either an I or -I transfer matrix, the Twiss functions exhibit periodic behavior under all conditions, with α=0 being characteristic of periodic arc-like sections. The emittance growth at the beamline exit follows a formulation similar to Eq. (6), where β_ps can assume arbitrary values. However, the CSR kick must also include higher-order terms. For β_ps>1, the CSR kick in x’ provides the dominant contribution, with its effect strengthening as β_ps increases. Conversely, when β_ps<1, the CSR kick in x dominates and becomes stronger as β_ps decreases.

From a practical implementation perspective, extremely small initial β functions impose stringent requirements on upstream focusing conditions. To address this, we adopt β_ps>1 in the current design work, with particular emphasis on mitigating the CSR kick effects in the x’-axis. While the calculation of presents significant mathematical challenges, the optimization strategy of minimizing the terms in the double TBA cell configuration proves effective in reducing the CSR effects from the upstream TBA cell. In the design process, three quadrupoles in the TBA cells are optimized to ensure achromatic and isochronous conditions. Matching sections are added to both sides of the TBA to achieve an overall -I transfer matrix while maintaining T₂₂₆=0. Two families of sextupoles are then incorporated into the lattice to minimize T₅₆₆ and T₂₁₆. The beamline consists of six double TBA cells designed to achieve a 180° deflection. Additionally, a matching section between the final two double TBA cells forms an I-matrix for error elimination. Figure 9 illustrates the x’-axis displacements at the beamline exit for both cases with and without T₂₁₆ correction. Relative to the uncorrected case (black dots), reducing |T₂₁₆| (red dots) effectively controls the displacements, thereby suppressing CSR kicks in the x’ axis. This results in reduced growth in horizontal emittance. The simulation results incorporating both TR CSR and SS CSR effects are shown in Figs. 10 and 11, with the initial bunch parameters listed in Table 1. Figure 10 shows the projected emittance at the exit of multiple double TBA cells for different bunches. For the Gaussian bunch, although the projected emittance increases with the number of double TBA cells, the growth remains below 25%. Meanwhile, the longitudinal profile and slice emittance of the Gaussian bunch remain nearly unchanged, as shown in the left insert of Fig. 11. The variation in projected emittance is primarily attributed to centroid offsets. For the SXFEL bunch, the emittance growth reaches approximately 35% at the exit of the sixth double TBA cell. As shown in the right insert of Fig. 11, significant variations occur at the bunch edges due to the dramatic changes in current distribution at both the head and tail regions. Moreover, for quasi-uniform bunches, the CSR effect exhibits a stronger influence at the bunch tail, leading to intense CSR effects in these regions and resulting in projected emittance degradation. Similar phenomena are observed in the simulation results for the single TBA cell, as shown in Fig. 6. However, the emittance degradation in the core region (the purple box in Fig. 6) remains negligible, and the projected emittance growth for this region stays below 20%, which is an acceptable range for the primary radiation region of FELs.

Fig. 9Full-screen View

The CSR-induced displacements along the x’-axis at the beamline exit are shown, where red and black dots represent the cases with and without T₂₁₆ correction

Fig. 10Full-screen View

The normalized project emittance evolution at the exit of multiple double-TBA cells. The “SX Core” denotes the purple box with dashed line in Fig. 6

Fig. 11Full-screen View

(Color online) The longitudinal profiles at the exit of the 6th double TBA cell and the normalized slice emittance of the bunches presented in Table 1

Conclusion

This study introduces two enhanced methods for CSR suppression in isochronous structures, addressing the challenges associated with high-peak-current bunch transport through bending systems.

For single TBA cells, the re-lat design facilitates an effective evaluation of SS CSR effects without relying on small-angle approximations. This approach preserves excellent transverse beam quality and longitudinal profiles, even under the combined SS and TR CSR effects, proving particularly effective for small bending angles. Furthermore, the re-lat optimization method directly determines CSR kicks through accelerator simulations, eliminating the need for additional calculations. This streamlined process not only enhances the efficiency of the design phase but also extends its applicability to more complex structures, such as Multi-Bend Achromats (MBA).

The -I matrix method has been further enhanced through two key improvements: implementing an I matrix between cells and optimizing the higher-order term T₂₁₆ to effectively reduce CSR-induced displacements in the horizontal phase space. This enhanced approach theoretically suppresses both SS and TR CSR effects. For large bending angles, the -I matrix method shows exceptional preservation of bunch quality. Our study demonstrates that combining these improvements allows a configuration of six double TBA cells to achieve a 180-degree deflection while maintaining both transverse emittance and longitudinal profiles for bunches with varying longitudinal distributions. Experimental validation of CSR suppression using the I matrix method is currently underway at SXFEL, with further research planned for future development.