# Analytical treatment of the wakefields driven by transversely shaped beams in a planar slow-wave structure

###### Abstract

The suppression of transverse wakefield effects using transversely elliptical drive beams in a planar structure is studied with a simple analytical model that unveils the geometric nature of this phenomenon. By analyzing the suggested model we derive scaling laws for the amplitude of the longitudinal and transverse wake potentials as a function of the Gaussian beam ellipticity - . We explicitly show that in a wakefield accelerator application it is beneficial to use highly elliptical beams for mitigating transverse forces while maintaining the accelerating field. We consider two scaling strategies: 1) aperture scaling, where we keep a constant charge to have the same accelerating gradient as in a cylindrical structure and 2) charge scaling, where aperture is the same as in the cylindrical structure and charge is increased to match the gradient.

## I Introduction

Single-bunch beam breakup (BBU) effects stem from the excitation of transverse wakefields driven by off-axis particles in a particle accelerator. One of the suggested methods of mitigating the effects of beam coupling to transverse wakefields is the introduction of a bunched beam with high transverse ellipticity in a rectangular structure with high aspect ratio FLb (). Transverse mode control and suppression is relevant for many accelerator applications, however these effects are particularly urgent when considering advanced accelerator concepts operating at high frequency and gradient. For example, collinear wakefield acceleration driven by intense charged particle beams in dielectric materials, has demonstrated GV/m fields Oshea:2016 () in THz wakefields, and is considered a candidate method to surpass the field gradients of existing radio frequency structures. The practicality of beam-driven wakefield acceleration for high-energy applications depends on the ability to extend the length of the acceleration process, which may be limited by BBU instability Li:2014 (); Qd3 (). The beam-mode coupling can be dramatically reduced for beams with high transverse ellipticity in structures with planar geometry. Although when employing elliptical beams the longitudinal electric field behind the elliptical bunch decreases as , the deflecting force due to transverse fields scales as , leading to an advantageous scenario where the effects of the transverse forces can be heavily suppressed. This effect was described using the direct solution of the Maxwell?s equations for the case of a planar dielectric loaded waveguide in Ref. FLb (). The continued studies on structures with planar geometry are relevant as these structures are commonly used in wakefield acceleration and beam phase space manipulation experiments today Antipov:2012 (); Andonian:2012 (); Gao:2018 (); SlacDC () and provide a natural path forward for more advanced applications.

In this paper, we further investigate the phenomenon of wakefield generation in a rectangular structure driven by an elliptical beam, by using an alternative approach based purely on geometric considerations. We base our explanations on the limiting values of loss and kick factors derived in Reference mySTAB (). In this approach, wakefields in a longitudinally translationally invariant structure lined with layers of an arbitrary impedance material (of dielectric, resistive, or corrugated type) have been developed. The analysis of Ref. mySTAB () yields a derivation of expressions for the wakefields that are based on a conformal mapping method. It has been shown that the limiting value of losses and kicks for a point-like bunch is independent of material properties and depends only on the transverse shape of the bunch and the cross-section shape of the vacuum channel myPRL (); Karl1 (); Karl2 (); BaneDc (). Following the conventions of Reference mySTAB (), we consider a point-like bunch in the longitudinal coordinate with a given distribution in the transverse coordinates. Using this model, the transverse wake potential is calculated for various transverse beam distributions. For charge distributions of varying ellipticity, the results are compiled to construct trade-off curves comparing the relative strengths of the longitudinal and transverse wake potentials. Finally, the transverse wake potentials of the elliptical beam model are directly compared to those obtained in the cylindrically symmetric case using scaled variables with respect to structure aperture and beam charge.

## Ii Theoretical model of a planar structure

The theoretical basis for our wakefield model, where the loss and kick factors are derived, has been explored in detail in Ref.mySTAB () and is summarized here. The expressions for the upper limits of the longitudinal electric field, , and of the transverse component of the Lorentz force, , as a function of generalized complex coordinates are

(1) |

(2) |

Here is the charge of the particle generating the wakefield, is the charge of the test particle, is the size of the structure aperture, is the longitudinal distance behind the particle and the test particle, and is the Heaviside function which ensures that the field is non-vanishing only behind the particle, as dictated by causality. The cross-section of the the structure is described as a complex plane with ; in this analysis and is the conformal mapping function that transforms the cross-section of interest onto a circle such that the point corresponds to the center of a circle. Here the use of asterisks denote complex conjugation. Following Ref.mySTAB (), these expressions can be utilized as Green’s function for summing over the transverse bunch distributions.

Now let us consider the cross-section of a planar structure (Fig.1 first left panel). First we introduce a change of the coordinates to rotate the strip by in angle, . Then we build a conformal map of the strip onto a circle, as diagrammed in Fig.1. The plane is mapped on a circle of radius with the function

(3) |

The point of the bunch location , is mapped to a point . Then, we map a new circle on this circle such that the point corresponds to the center of the last circle. This mapping is accomplished using the function

(4) |

Using equtions (1) and (2) we can calculate the transverse part of the Lorentz force and longitudinal electric field in plane. Combining (3) and (4) we arrive at

(5) |

First we calculate

(6) |

here prime denotes total derivative by .

Consequently

(7) |

Combining (6) and (7) with (1) we have an expression of the longitudinal field in the plane

(8) |

Now, we examine the transverse force component and note that

(9) |

and consequently

(10) |

Making a substitution in (8) we arrive at the transverse dependence of in plane

(11) |

Taking into account that with the substitution and (II) we arrive at

(12) |

Since we are only interested in the properties of the distributions, we introduce scaled wake potentials per unit length in the form and . With this we finally obtain the longitudinal and transverse wake potentials,

(13) |

(14) |

It is noteworthy that the results for the longitudinal and transverse wake potentials for a planar structure in (13) and (II) agree with previously derived results for a rectangular corrugated structure BaneDc (). In Ref. BaneDc (), a different approach of surface impedances, developed in StupSrf (), was employed. Both approaches mySTAB () and StupSrf (); BaneDc () predict the same interesting result, namely, the limiting value of the loss and kick factors are independent of the properties of the retarding material, and therefore the transverse dependences are properties of the geometry only. For a more detailed explanation of these concepts we refer the reader to the original works myPRL (); mySTAB (); StupSrf (); BaneDc ().

## Iii Mechanism for the transverse wakefield damping

The effects of the transverse wake potential are manifested in transverse forces that may lead to the growth of the BBU instability. There are many methods proposed to stabilize the growth of this effect, such as using external magnetic focusing elements superimposed on the accelerating channel Qd1 (); Li:2014 (); Qd3 () to exploit BNS damping BNS (). In this section, we explore an alternative method described in FLb (); Mihalcea:2012 (), whereby the coupling to the transverse wake is mitigated by employing highly elliptical drive beams in rectangular structures.

First, we plot field lines for the transverse wake potential (II) for the case when a point particle is travelling along the axis of the planar structure (Fig.2 left panel), and second, for an off-axis point particle displaced by (Fig.2 right panel). From Fig.2 it is apparent that the transverse wake potential has a vortex-like structure away from the origin, while near the origin, the field can be described as ”quadrupole-like”, because the field focuses along -axis and defocuses along -axis. In the case of the off-axis particle, the vortex-like structure actually leads to focusing in both and directions, along the displacement line (dashed line on Fig.2 right panel) for .

For simplicity in further analysis, we assume that the -dimension of the drive bunches is much smaller then the aperture , and we consider small displacements from the center. Using this approximation we decompose (13) and (II) in a Taylor series at the point keeping only linear terms in and . Thus, for the longitudinal wake potential we have

(15) |

and for the -component of the transverse wake potential we have

(16) | |||

Formulas (15) and (16) are approximate transverse Green’s functions for the longitudinal wake potential and the component of the transverse wake potential respectively, that are valid for bunches with and small displacements in .

Now, we assume a transverse bunch distribution of the form

(17) |

We plot the dependence of and for the point particle with . The peak longitudinal field is accompanied by a peak in the transverse wake at =0, as expected, with a tailing off for higher values of . However, the -component of the transverse wake potential in Fig.3 , has a region where it is negative. This implies that instead of being deflected, particles that are located in this region will be attracted back to the -axis. The length of the defocusing region is , where is the zero crossing in , and the location of the focusing maximum is , as derived in Appendix A. There are no such features on the longitudinal wake potential, .

The existence of this region in the -dependence of wake potential and absence of this feature in the longitudinal wake potential allows one to benefit from stretching the beam in direction. Increasing the beam size in diminishes both and , however, due to the narrower peak in the pattern, the reduction for will be more pronounced. Furthermore, partial cancelation of is possible due to the focusing regions.

In order to illustrate this feature, we consider a simple example of two synchronous drive particles placed at a distance (we assume ), equal to the distance from the origin to the minimum of (See Appendix A). This selected arrangement minimizes the amplitude of the combined transverse wake potential of the two particles. From Fig.4 we see that due to decoherence, the longitudinal wake amplitude was reduced by only , whereas the transverse wake potential acting on each bunch was reduced by due to the interference effect. Moreover, it is worth mentioning that if a witness particle will be placed at the point after the two driver bunches considered in this example, the maximum kick that the witness bunch will experience is reduced by compared to standard scheme when a single driver with the same total charge is placed at , as it is seen from the Fig.4.

This simple illustration unveils the mechanism behind the idea of transverse wakefield suppression by a highly elliptical drive bunch that was introduced earlier in literature FLb (). In particular, the coherence length in the direction is greater for the longitudinal wake potential then for the -components of the transverse wake potential. In addition, partial cancelation is possible for the -component of the transverse wake potential due to the existence of the focusing regions in the -dependence of the Green’s function .

We would like to stress that the analysis above is purely dependent on structure and beam geometry, and independent of the material in the structure, thus is valid for dielectric-lined structures, metallic corrugated structures, resistive wall and the recently considered photonic planar structures Andonian:2014 (); Hoang:2018 ().

## Iv Tradeoff curves

In this section we further analyze the tradeoff between the loss of the amplitude of the longitudinal wake potential compared to the suppression of the transverse wake potential for varying beam distributions. We consider a Gaussian distribution in and calculate both longitudinal and transverse wake potentials using approximate equations (15) and (16) as

(18) | ||||

(19) | ||||

In Fig.5 we plot the normalized wake potentials and as a function of the bunch flatness, which we define as . When the bunch flatness is increased both longitudinal and transverse wake potentials are reduced. However, the rate of decrease for the transverse wake potential is significantly greater, as also predicted in Refs. FLb (); Mihalcea:2012 ().

To introduce a figure of merit of how fast both wake potentials decrease we calculate asymptotes for relatively large bunch flatnesses () for amplitudes of both longitudinal and transverse wake potentials as a function of

(20) |

(21) |

For convenience we introduced the notations , . Derivation of these expressions is found in Appendix B and Appendix C respectively.

The first terms in the brackets of Equation (20) reduces as , while the leading terms in the brackets of Equation (21) reduces as . This scaling demonstrates the favorable tradeoff in longitudinal to transverse wake effects for flat bunches, and is consistent with the predictions in FLb ().

Until this point, we have only considered the component of the transverse potential. For completeness, we derive the component of the wake potential, taking equation (II) and decomposing the real part in a Taylor series at the point keeping only terms linear in and and arrive at,

(22) |

It is immediately apparent that the focusing force is independent of the source particle transverse displacement in and the witness particle transverse displacement. This implies that the effect is present even for a perfectly aligned beam on-axis. For this reason, it is often referred to as the quadrupole wakefield in the literature Chao ().

Next, we consider a bunch with Gaussian distribution in and calculate inside this bunch as

(23) |

As it is shown in Appendix D for the case when and for (the point where focusing force is maximal within the bunch core ) is given by

(24) |

On Fig.6 we plot normalized wakefields and as a function of the bunch flatness. One can see that in contrast to the case of when bunch flatness increased first increases up to and only then goes down. We also notice that decrease rate for is higher then for but lower then for .

## V Comparison to cylindrical geometry: Scaled aperture

Although the mitigating effects of the flat bunch are relatively favorable in regards to optimizing the ratio of the longitudinal to transverse potentials, we must address the fact that there is a reduction to the longitudinal (accelerating) field. In this section, we compare the results derived for the planar structure to the wake potential of a cylindrical structure for a given, i.e. fixed, accelerating gradient. In this case, we keep the drive bunch charge fixed and scale the aperture of the planar structure to match the gradient of the cylindrical structure. The maximum gradient in the cylindrical structure per unit charge per unit length for a point particle is given by

(25) |

Here is the radius of the cylindrical structure. We equate the gradient in a flat structure (20) to the gradient in cylindrical structure (25) and derive formula for the scaled aperture of the planar structure to achieve the same gradient as in the cylindrical:

(26) |

Equation (26) gives the scaling law for the aperture in the planar structure and is valid for the . Next we substitute (26) into the Eq. (21), set , notice that deflecting transverse wakefield for a point particle in the cylindrical structure is given by

(27) |

and arrive at

(28) |

Formula (29) shows the reduction in transverse field in a flat structure with flat beam in comparison to cylindrical structure while maintaining the same accelerating gradient. Ultimately for a very flat beam , this ratio reduces further to the simple form

(29) |

The relation above shows that one can reduce the deflecting wakefield while maintaining an equivalent gradient, by implementing a elliptical beam in a planar structure. The reduction in amplitude, however, is quite modest, scaling as .

Now we consider the component of the wake potential that is a property of planar structure and compare the amplitude of to the amplitude of the deflecting wake potential in a cylindrical structure. With equations (26) and (27) we express as

(30) |

Ultimately for large values of we have

(31) |

From Eq.31, we can set a conditional statement such that for a planar structure be less then of a cylindrical, in the following inequality

(32) |

For an offset in a cylindrical structure of we immediately achieve . This analysis leads to a conclusion that the aperture scaling approach demands a high degree of bunch flatness to achieve both and component amplitudes lower, compared to the deflecting wake in the cylindrical structure.

Indeed, it has been experimentally demonstrated that it is possible to produce beams with very high transverse emittance ratios Piot1 () and thus have a high ellipticity , with more recent work demonstrating emittance ratios and ellipticity. Piot2 (); Piot2c ().

It is worth emphasizing this result: we have explicitly shown theoretically that both the and components of the transverse wake potential in a planar structure could be simultaneously reduced with the aperture scaling strategy, and both and components could be simultaneously lower then the deflecting wake potential in a cylindrical structure with the same amplitude of the longitudinal wake potential.

## Vi Comparison to cylindrical geometry: Scaled charge

The second approach to match the gradient in a planar structure is to increase the charge of the bunch while maintaining the same aperture.

We start from the equation (20) for the amplitude of the longitudinal wake potential and compare it to the expression for the cylindrical structure (25). We notice that for the longitudinal electric fields (gradients) to be equal, charge in the planar structure should be

(33) |

Since the dependence on charge is linear, one can achieve the ratios

(34) |

and

(35) |

by simply increasing and by a factor of . Taking the limit of large we arrive at

(36) |

and

(37) |

From Eqs. (36) and (37) we see that in case of the charge scaling strategy the decrease in the transverse wake potential is more pronounced than in case of the aperture scaling approach. As in the previous section we can set a conditional statement such that for a planar structure be less then of a cylindrical. This leads to the following inequality

(38) |

We repeat the same estimating procedure as in the previous section, from Eq. (35) for an offset in a cylindrical structure we achieve for the to be the smaller then . This value of a beam shape looks very reasonable for practically achievable flatness and could indeed be greater, thus further suppressing to be much smaller then the deflecting force in the cylindrical structure.

## Vii Conclusions

In this paper, we have presented an analysis for a planar symmetry slow-wave structure for varying transverse beam flatness . The approach is distinct from previous approaches FLb (); Park () in its generality, being derived on the basis of a Green’s function approach that accommodates any impedance. In particular, the analysis is independent of the properties of the retarding material to describe transverse structure of the wake potential. We have explicitly shown that for highly elliptical beams, the transverse wake potential can be dramatically suppressed, and the tradeoff in longitudinal field loss is still favorable for beam ratios greater than 3. Further, we compared the results explicitly to achieve equal accelerating gradients in cylindrical structures by scaling both the aperture and the charge, and propose that charge scaling is a more favorable method. The implications of this work are important for investigating possible designs of the future wakefield accelerators to combat deleterious transverse fields that lead to beam break up. The results are general, depending only on geometric factors, and hold for corrugated, dielectric planar structures and planar structures with the resistive wall and are directly extendable to 1D and 3D photonic-like planar structures that allow for further precision modal control when driven by selectively shaped beams Andonian:2014 (); Hoang:2018 ().

###### Acknowledgements.

This work was supported by the U.S. National Science Foundation under Award No. PHY-1549132, the Center for Bright Beams and under Award No. PHY-1535639; by the U.S. Department of Energy Award No. DE-SC0017648 and by the U.S. Department of Energy Award No. DE-SC0009914. The authors are grateful to Alexander Zholents for useful discussions and suggestions.## Appendix A Derivation of the location of the zeros and coordinate of the minimum of the function

We assume a point particle to be placed at the , in this case equation (16) reads

(39) | |||

Equating wake potential to zero we arrive to the following equation on (zeros of wake potential by )

(40) |

Expanding hyperbolic cosine using well known formula

(41) |

and introducing notation

(42) |

we rewrite (40) as

(43) |

Solution of this equation is

(44) |

Consequently with substitution (42) we have for the zeros of

(45) |

Next let us find location of the minimums of the function . Following standard procedure we equate -derivative of equation (39) to zero and arrive at

(46) |

As far as is the coordinate of maximum for the coordinates of minimums we have an equation

(47) |

We once again expanding hyperbolic cosine as (41) and introducing substitution (42) we arrive at the algebraic equation

(48) |

Solution of this equation is

(49) |

Consequently with substitution (42) we have for the positions of the minimums of

(50) |

## Appendix B Asymptote for the maximal amplitude of the longitudinal wake potential for large ratios

First we consider equation for the longitudinal as given by Eq.(18) and rewrite it in the following form for the point , where function has maximum

(51) |

with .

Assuming we decompose exponent under the integral in Taylor series and keep only first two terms

(52) |

After the substitution of (52) into (51) we have

(53) |

with

(54) | ||||

(55) |

After substitution of variables and integrating by parts in case of the second integral one may show that

(56) |

We combine (53) and (B) and arrive at the final formula for the asymptote

(57) |

## Appendix C Asymptote for the maximal amplitude of the y-component of the transverse wake potential for the large ratios

As in the Appendix B we rewrite amplitude given by

(58) |

for the -component of the transverse wake potential for a Gaussian bunch (19) in the form

(59) | ||||

As before we assume and decompose exponent under the integral in Taylor series. Now we keep first three terms

(60) |

After the substitution of (60) into (59) we have

(61) |

with

(62) | ||||

(63) | ||||

(64) |

After substitution of variables and integrating by parts one may show that

(65) | ||||

We combine (61) and (C) and arrive at the final formula for the asymptote

(66) |

## Appendix D Asymptote for the maximal amplitude of the x-component of the transverse wake potential for the large ratios

We introduce amplitude as

(67) |

for the -component of the transverse wake potential for a Gaussian bunch (23) in the form

(68) |

We assume and decompose exponent under the integral in Taylor series and keep first three terms

(69) |

After the substitution of (69) into (68) we have

(70) |

with

(71) | ||||

(72) | ||||

(73) |

After substitution of variables and integrating by parts one may show that

(74) | ||||

We combine (70) and (D) and arrive at the final formula for the asymptote

(75) |

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