Construction of self-consistent longitudinal matches in multipass energy recovery linacs

Any proposal for an accelerator facility based upon a multipass energy recovery linac (ERL) must possess a self-consistent match in longitudinal phase space, not just transverse phase space. We therefore present a semianalytic method to determine self-consistent longitudinal matches in any multipass ERL. We apply this method in collider scenarios (embodying an energy spread minimizing match) and FEL scenarios (embodying a compressive match), and discuss the consequences of each. As an example of the utility of the method, we prove that the choice of common or separate recirculation transport determines the feasibility of longitudinal matches in cases where disruption, such as synchrotron radiation loss, exists. We show that any high energy multipass ERL collider based upon common recirculation transport will require special care to produce a self-consistent longitudinal match, but that one based upon separate transport is readily available. Furthermore we show that any high energy multipass ERL FEL driver based upon common recirculation transport requires a larger resultant rf beam load than the one based on separate transport, favoring the separate transport designs.

Figure 1

Simplest racetrack ERL configurations, blue cylinders represent the linacs, and the spiked ball represents the interaction region. (a) common transport and (b) separate transport, with solid and dashed lines indicating the arcs traversed during acceleration and deceleration respectively.

Figure 2

Classification of longitudinal matches for ERLs whose feasibility will be studied in this paper. Labels I, IA, II and IIA refer to the different sections of this paper where these examples are developed.

Figure 3

Example I: Compressive match. Sequence of longitudinal phase space manipulations maximizing bunch current at interaction point. From top left: Initial, accelerated, compressed, decompressed and decelerated charge distributions. Bottom right shows the total rf load in the complex plane with rf phase choices during acceleration (black), deceleration (red), and resulting beam load (blue). To achieve zero chirp after deceleration we must decompress with opposite sign $R_{56}$ to that of the compression.

Figure 4

Example IA: Compressive match with SR loss compensation. Image sequence as per Fig. 3. Choosing decompressing $R_{56}$ of equal and opposite sign to compressing now results in finite residual chirp as we must move decelerating phase further off-trough to account for SR energy loss. rf phase choices during acceleration (black), deceleration (red), and x 10 resulting beam load (blue) for clarity.

Figure 5

Longitudinal phase spaces of accelerating (solid) and decelerating (dashed) bunches in an intermediate arc. The required energy acceptance for common transport corresponds to the height of the black arrow. The required energy acceptance in the corresponding separate transport configuration corresponds to only the height of the red and blue arrows for the accelerating and decelerating arcs respectively.

Figure 6

rf beam load plots for different phase configurations in common transport longitudinal matches that minimize beam energy spread. Applicable to, e.g., PERLE. rf phase choices during acceleration (black), deceleration (red), and resulting beam load (blue) of each linac independently. Number labels indicate the ordering of the rf passes. (a) minimal phase changes, (b) multiple phase changes and (c) rf imbalance between linacs.

Figure 7

Energy spread minimization with common transport. Sequence of longitudinal phase space manipulations with two linearizing arcs with opposite sign of $T_{566}$. Beam load plots for the two linacs are shown underneath with labels indicating the order of each of the rf passes during acceleration (black), deceleration (red), and resulting beam load (blue) of each linac independently.

Figure 8

Energy spread minimizing matches with nonisochronous intermediate arcs. rf beam load plot shows rf phase choices during acceleration (black), deceleration (red), and resulting beam load (blue). (a) Common transport, (b) separate transport. Different angle highlights in the rf load plots correspond to different magnitudes off-crest or off-trough.

Figure 9

Arc 1 centroid energy difference between accelerating and decelerating beams for a range of accelerating and decelerating phases. Black dashed contour lines represent peak energy in GeV.

Figure 10

Energy spread minimizing match with SR loss compensation using separate transport. Shown longitudinal phase spaces correspond to the example beam at the exit of the element specified. Applicable to, e.g., LHeC. rf beam load plot shows phase choices during acceleration (black), deceleration (red), and resultant (blue).

Figure 11

Sequences of compressive common transport longitudinal matches with different choice of linearizing arcs. Longitudinal phase spaces at the exit of the specified elements. The remaining arcs have a nonzero $T_{566}$ of their natural sign. (a) uses arc 3 to linearize, (b) uses arc 1 to linearize. Note the change of scale in the later stages of (b).

Figure 12

Arc path length as a function of relative momentum deviation with only second order longitudinal dispersion nonzero. Red and blue displaced axes highlight the path length dependence on momentum for off-momentum beams with an effective nonzero $R_{56}$.

Figure 13

Arc path length as a function of relative momentum deviation where we choose reference momentum, first, second and third order longitudinal dispersions. This allows independent control over path length and $R_{56}$ of two off-momentum beams. Example shows for two beams at $\delta =\pm 0.4\%$ the path length difference is 5 mm with effective $R_{56}=0\mathrm{mm}$ and $-10\mathrm{mm}$ respectively, illustrated by the dashed orange lines.

Figure 14

Equivalent of example IA shown in Fig. 4 but bunch decompression does not occur at top energy. The bunch undergoes decompressions during acceleration and deceleration resulting in a bunch longer at the dump than at the injector. Black dashed lines indicate initial bunch length, orange dashed line indicates maximum bunch length, green dashed line indicates final bunch length, red dashed line indicates bunch energy spread at top energy and blue dashed line indicates adiabatic growth of energy spread of the decelerating fully compressed bunch. rf beam load plot shows phase choices during acceleration (black), deceleration (red), and resultant (blue).

Figure 15

Equivalent of Example IA shown in Fig. 4 but with utilization of a parasitic crossing and deceleration on opposite side of rf trough in order to remove linear chirp. This results in a large imaginary resultant rf load. We see residual curvature as the natural $T_{566}$ value of the arcs add to the rf curvature for deceleration on falling side of trough. rf beam load plot shows phase choices during acceleration (black), deceleration (red), and resultant (blue).

Figure 16

Change in curvature as an example bunch (black) undergoes a linear compression (solid to dashed). The same compression acting on a linearized bunch is shown in red.

Figure 17

Longitudinal match solutions using parasitic crossings to cancel rf curvature in common transport. rf beam load plot (bottom right) shows phase choices during acceleration (black), deceleration (red), and resultant (blue).

Figure 18

Compressive longitudinal match with energy spread growth at the interaction point. Longitudinal phase spaces during acceleration (top) and during deceleration (bottom) with a parasitic crossing. rf beam load plot (bottom right) shows phase choices during acceleration (black), deceleration (red), and resultant (blue) with highlighted angles representing matching magnitudes.