Transverse operator method for wakefields in a rectangular dielectric loaded accelerating structure

Cherenkov radiationgeneratedbya relativistic electronbunch in a rectangular dielectric-loadedwaveguide is analyzed under the assumption that the dielectric layers are inhomogeneous normal to the beam path. We propose a method that uses eigenfunctions of the transverse operator applied to develop a rigorous full solution for the wakefields that are generated. The dispersion equation for the structure is derived and the wakefield analysis is carried out. The formalism developed here allows the direct solution of the inhomogeneous system ofMaxwell equations, an alternative analytic approach to the analysis of wakefields in contrast to the previously used impedance method for rectangular structure analysis. The formalism described here was successfully applied to the analysis of rectangular dielectric-lined structures that have been recentlybeam tested at the Argonne (ANL/AWA) and Brookhaven (BNL/ATF) accelerator facilities.


I. INTRODUCTION
Future high energy charged particle (electron and ion) accelerators will require new technologies to ensure expansion of the energy and intensity frontiers.Future linear colliders will need to be based on new methods since foreseeable normal and superconducting technologies are unavoidably limited in accelerating gradient.Progress in compact x-ray source design based on free electron lasers also requires high gradient nonconventional linear accelerators to be developed.Some of the most promising new methods of charged particle acceleration are laser-plasma [1,2], beam-driven plasma [2,3], and dielectric based accelerators [4][5][6][7].All the advantages and limitations of these schemes as well as their current status in simulations and experimental results can be found in the references.In this paper we focus on dielectric based wakefield acceleration technology (DWA) as one of the most promising new technologies for next generation linear colliders [8] and future x-ray free-electron lasers (FELs) [9].
In this paper we consider dielectric-lined structures that are excited by a high intensity relativistic electron beam to generate high power X-band, mm-wave, or THz radiation [7][8][9].This rf pulse is used to drive another dielectric or conventional accelerating structure (e.g. the CLIC project power extraction structure [10,11]) or accelerate a second electron beam in the same structure.This method of electromagnetic wave generation avoids the need to develop more traditional rf sources and components that are able to sustain and transmit GW-level microwave or THz power into the structure [5][6][7].
A dielectric-loaded accelerating (DLA) or power extraction structure is a dielectric waveguide with an axial vacuum channel for beam propagation and which is surrounded by a conducting metal wall [4][5][6].Recent experiments have shown that dielectric based structures can sustain accelerating gradients in excess of 100 MV=m [12] and GV=m [6] in the upper GHz and THz frequency ranges, respectively.
For example, a high current (up to 100 nC) short (1-2 mm) relatively low energy (15-100 MeV) electron bunch in this type of a structure can generate Cherenkov wakefields with the longitudinal field magnitude up to 100-300 MV=m in the X-Ka band frequency ranges [12].A 3 nC charge and 30 m long bunch of the 23 GeV SLAC/ FACET accelerator generates 1-10 GV=m wakefields in the THz range [6,13].These wakefields are used for accelerating a less intense but high energy electron bunch propagating behind the drive bunch at a distance corresponding to the accelerating phase of the E z field [4,5,14].Dielectric based structures provide, in addition to the high accelerating gradient, the possibility of controlling the frequency spectrum of the structure by introducing nonlinear dielectrics (ferroelectrics) [15,16].There is also the potential of using new manufactured materials (such as diamond and sapphire) with unique dielectric strength and thermoconductive properties [17][18][19].
As a rule, the cylindrical geometry usually proposed for dielectric based structures is required for attaining the highest accelerating gradients as well as for providing the maximal possible shunt impedance of the structure [4,5].Results of analytic mode analyses of cylindrical dielectric accelerating structures are found in [20,21].
At the same time, the dielectric-lined planar or rectangular cross section geometries are also being considered; a minor reduction in performance is acceptable in view of the technological difficulties of fabricating cylindrical structures (related to the stringent tolerance requirements for geometrical parameters and uniformity of the permittivity of the dielectric [3]).Rectangular structures can be used for measurements on new dielectric materials effective in sustaining high acceleration rates and tolerant of pulsed heating of the structure [4].Planar structures are also being studied (along with cylindrical geometries) for generating terahertz radiation in the frequency range 0.5-1.0THz [4] and for energy spread reduction (''silencer'') systems for FEL linacs [22].
Another advantage of using a planar geometry is related to the drive beam stability problem for DWAs.The longitudinal electric field magnitude of the wakefield is inversely proportional to the square of the vacuum channel radius; therefore it is critically important to employ small aperture structures.At the same time, reducing the vacuum channel radius increases the severity of beam breakup (BBU) effects [5,7,23].Planar DWA structures can provide a reasonable compromise between the reduced accelerating gradient and the required beam transverse stability [23].The planar option has the additional advantage that a flat electron beam can be generated directly in photoinjectors [24].
Theoretical analyses of dielectric accelerating structures with rectangular geometries have been reported in a number of publications [23,[25][26][27][28][29][30][31].To determine the amplitudes of individual Cherenkov radiation modes excited in a rectangular waveguide with dielectric slabs, the impedance matching technique was used in the earlier papers [23,25,26].When that formalism is used instead of the direct solution of the nonhomogeneous system of Maxwell equations (the standard analytic approach in analysis of wakefields in cylindrical structures [20,21]), the amplitudes of the wakefields needs to be expressed in terms of the shunt impedance (or integrated loss factor) for each mode of the structure [23,25,26].The approach involves certain approximations, while the direct solution of the inhomogeneous system of Maxwell's equations without intermediate approximations is always preferable for analyzing the wakefield generation problem in DWA structures.In this paper, a rigorous direct solution was found for the wakefields in an arbitrary rectangular dielectric-lined waveguide.
In this paper, we applied a transverse operator method that was used previously as an analytical algorithm for wave propagation of modes in a waveguide with complex loading, where the rectangular waveguide is partially filled with dielectric, ferrite, ferroelectric, or any type of anisotropy is present (chiral loading, for example) [28][29][30][31][32][33][34].Some specific version of transverse operator method called the mode basis method was developed and applied to the analysis of wave propagation in conical antennas with dielectric filling [35,36].
The method of the first order transverse operator has been applied to waveguide problems in [32][33][34].The generalized orthogonality relations between the longitudinal magnetic and longitudinal electric modes have been derived in [28,29].However, the bilinear form introduced in [28,29] is not the scalar product, and requires justification of the use of this relation for orthogonality between the components of the electric and magnetic field vectors.In [30], the analysis was performed on the basis of the second order differential equations for the transverse field components (direct solution of the Maxwell equations) of a two-channel rectangular structure with dielectric-lined waveguides.This structure design was proposed for increasing the transformer ratio from the drive beam to the beam being accelerated.However, some of the mathematical aspects of the analysis [28][29][30] have not been rigorously verified and need additional studies.To be specific, there is a problem of sign inversion of the weight function in the definition of the norm, and the completeness of the set of functions over which the expansion was performed was not demonstrated.
Our approach to this problem is the use of a modified transverse operator method.It should be noted that in contrast to the analysis of [28][29][30][31][32][33][34], we consider the Sturm-Liouville second order operator with an alternating sign weight function.It will be shown below that the integral relation derived in [28,29] is a consequence of the bi-orthogonality of the eigenfunctions as well as the similarity of the operator to a self-adjoint operator.This approach makes it possible to obtain a complete analytic solution for the eigenmodes of this system, and thus, to solve in the most general form the problem of Cherenkov wakefield generation in a rectangular accelerating structure with a composite dielectric loading.
Finally, (i) the proposed method does not require any assumptions on the form of the beam's Coulomb field (in contrast to the impedance method [25,26], where it is often assumed that the field of the beam is essentially flat, an assumption that is valid only at ultrarelativistic energies); (ii) we also did not use the mode decomposition for the problem formulation and its subsequent solution; the mode series obtained as a solution emerges in a natural way as a result of solving the problem; (iii) as a consequence of (ii), and the use of the transverse operator formalism the series of modes of the solution is uniformly convergent.This is not obvious for the solution obtained by the impedance method [25,26] and requires additional analysis and proof.

II. THEORETICAL ANALYSIS
We consider the waveguide shown in Fig. 1.We start from the Maxwell system in CGS units (1)-( 4), with the additional constitutive relations: where E, H are the electric and magnetic field vectors respectively; D, B are the electric and magnetic flux vectors respectively; V-average speed of the bunch; n-particle concentration of the bunch; e-electron charge; t-time.
The electron bunch is moving in the vacuum channel along the waveguide axis.The wakefield is generated by the bunch when the Cherenkov condition v=c ¼ > " À1=2 is satisfied.Here c is the speed of light and v the speed of the moving particle (or bunch).
For a point charge q, moving along waveguide's z axis the charge distribution is given by nðx; y; z; tÞ ¼ 4Nðz À vtÞðx À x 0 Þðy À y 0 Þ; (6) where x, y are transverse coordinates, z is the longitudinal coordinate, and x 0 , y 0 are the transverse coordinates of the beam, N number of electrons in the bunch, bunch charge q ¼ ÀeN.
We consider a rectangular waveguide with symmetric dielectric layers placed parallel to the OZX plane and separated by a vacuum gap.In this case the dielectric is inhomogeneous along the y axis, so that " ¼ "ðyÞ, ¼ ðyÞ.Next we evaluate the Maxwell system (1)-( 4) with (5) for this case.From (4) we obtain applying the curl operator to (1) and using (7), we get where we define and By inspection of Eq. ( 8), only the equation for the E y component does not include additional terms depending on other field components.In other words, only the E y equation is independent.
The equation for the E y component of the electric field given by ( 8) is The terms of the operator on the left side of the equation containing only derivatives with respect to y can be simplified as We introduce the new variable ¼ z À vt and we will look for a solution in the form E y ðx; y; z; tÞ ¼ E y ðx; y; z À vtÞ ¼ E y ðx; y; Þ.The equation for E y then becomes We introduce the operator We can then rewrite (13) as Following the same steps for the magnetic part of the Maxwell system we obtain where The velocity vector V has only a z component, so We should mention here that for the magnetic vector the situation is similar to the electric case in that the only equation that does not depend on other field components is the one for H y .Thus combining ( 16)-( 18), we can write the equation for the H y component as As before, we seek a solution in the form H y ðx; y; z; tÞ ¼ H y ðx; y; z À vtÞ ¼ H y ðx; y; Þ: We introduce the operator We then can rewrite (19) as We should stress here that Eqs. ( 15) and ( 22) are independent.Each equation corresponds to a specific type of propagating wave, (15) to longitudinal section magnetic (LSM) modes, (22) to longitudinal section electric (LSE) modes.

A. Transverse operator structure and transverse eigenfunctions
If we assume that the charge distribution is a finite function inside the vacuum gap and then the boundary conditions for the y components of the magnetic and electric fields on the conducting boundaries are and on the dielectric boundaries We consider the eigenvalue equations for the operators TE and TH [( 26) and ( 27)]: Then we can introduce a series expansion ansatz for the transverse eigenfunctions É E ðx; yÞ and É H ðx; yÞ: The eigenvalue problem for the X functions is Because of the self-adjointness of the operator defined by (30), the sets x xÞg n (31) define complete orthogonal systems with k n x ¼ n=w.This is of course the familiar Fourier series.
Substituting (28) into (26) and taking into account (25) we can write the equations for the transverse electric eigenfunctions as (suppressing the n index for simplicity) Similarly, substituting (29) into ( 27) and taking into account (25) we get a set of analogous expressions for the transverse magnetic eigenfunctions: Because the material layers are assumed to be placed symmetrically relative to the OZX plane, the electric and magnetic eigenfunctions of ( 32) and ( 33) can be separated further into symmetric and asymmetric functions.
We can the write the solution to (32) as follows.
Symmetric eigenfunctions. - Asymmetric eigenfunctions. - Here we have defined the wave numbers in the vacuum channel and dielectric respectively as The dispersion relations for the symmetric and asymmetric functions can then be written as Eigenfunctions of the magnetic problem (33) also separate into symmetric and asymmetric parts.Symmetric eigenfunctions are given by Here we define The dispersion relations for the symmetric and asymmetric functions respectively are given by TRANSVERSE OPERATOR METHOD FOR WAKEFIELDS . . .Phys.Rev. ST Accel.Beams 16, 051302 (2013)

B. Expansion method
Let us next examine the operator TE (14).We consider the Hilbert space H E ¼ L 2 f½Àc; c; j"ð1 À " 2 Þjg Â L 2 ð½0; wÞ, where ¼ j"ð1 À " 2 Þj is the weight function.The scalar product in this space is given by ðc where ', c 2 H E .The metallic (24) and dielectric boundary conditions (25) lead to 'ð0; yÞ ¼ 'ðw; yÞ ¼ 0; 'ðx; AEb Ç 0Þ ¼ " 1 'ðx; AEb AE 0Þ; @ y 'ðx; AEb Ç 0Þ ¼ @ y 'ðx; AEb AE 0Þg: (41) [DomðXÞ indicates the domain of the operator X.] The adjoint operator TE Ã can be found using the definition This leads to , which means that the operators TE and TH are not self-adjoint in H E and H H , respectively, so the eigenfunctions are not guaranteed to form a complete orthogonal set.In turn this would imply that all solutions for É E (É H ) are not necessarily given by ( 28), (34), and ( 35) [( 29), (38), and (39)].However, it can be proved that the eigenfunctions of the tasks ( 26) and ( 27) along with the conditions (25) Here fÈ E g are eigenfunctions of TÃ E and fÈ H g are eigenfunctions of TÃ H .Because " and are bounded on ½Àc; c and ", 2 L 1 ð½Àc; cÞ, metrics on H E and H H are equivalent to the metrics on L 2 ð½Àc; cÞ Â L 2 ð½0; wÞ.Hence, the expansions (47) hold for f, g 2 L 2 ð½Àc; cÞ Â L 2 ð½0; wÞ.However, we should stress here that the conditions (25) hold only for finite functions on the intervals ½Àc; Àb, ½Àb; b, ½b; c; in other words the excitation charge distribution should vanish at material boundaries.
The functions fÉ E g, fÈ E g and fÉ H g, fÈ H g satisfy the biorthogonality conditions Because of the fact that which corresponds to a result from [29] of the so-called mode orthogonality condition

III. SOLUTIONS FOR E y AND H y
Assuming a point charge passing through the structure at transverse coordinates ðx 0 ; y 0 Þ, Fourier transformation of ( 15) with respect to leads to T E ½ Ẽy ðx; y; kÞ À k 2 Ẽy ðx; y; kÞ We introduce the normalized eigenfunctions We then expand the transverse electric field (assuming Substituting (55) into (53), one can write for the region Àb < y 0 < b, (56) Performing the inverse Fourier transform for < 0 (field behind the bunch) leads to a solution in the form where þ E refers to the positive eigenvalues ( E > 0) of TE and À E to the negative eigenvalues ( E < 0) of TE .By a similar process applied to (22) we obtain the solution for H y .For the case < 0, Àb < y 0 < b we can write

IV. ACCELERATING FIELD AND DEFLECTING FORCE
By Fourier transformation with respect to of the Maxwell equations ( 1) and (2), we can express the transverse fields in terms of the longitudinal fields as Combining the equations for Ẽy and Hy we obtain Substitution of @ 2 Ẽy @y 2 from (63) and Hx from (61) into (64) with Ẽy from (60) gives ñ À ik @ Hy @x : (65) Now we consider the Ẽy Helmholtz equation: Integrating this equation with respect to y, we have Multiplication of this equation by ik and substitution into (65) leads to Using the boundary condition on the conductors Ẽz ðAEcÞ ¼ 0, we find the constant of integration C ¼ 0.
For the boundary conditions on the sides of the structure, E z ðx¼0Þ¼0, E z ðx¼aÞ¼0, we can expand Ẽz in the series Taking into account that where Y Ã E is the conjugate eigenfunction of the adjoint problem (32), and where Y Ã H is the conjugate eigenfunction of the adjoint problem (33).Substitution of ( 69)-( 71) into (68) gives Here we define and denote for brevity the integral by the operation I: Using the inverse Fourier transformation the Cherenkov radiation field (only real poles contributing for < 0 and only positive eigenvalues of TE and TH ) can be found as The transverse force inside the channel can be found using the Panofsky-Wenzel theorem: Hence, Substitution of E z leads to the full expressions for the transverse force components:

V. COMPARISON WITH NUMERICAL SIMULATIONS AND EXPERIMENTAL RESULTS
The formalism presented above allows obtaining a rigorous full solution for the wakefields generated by charged particle bunches passing along a rectangular accelerating structure loaded with symmetric dielectric slabs.Note that this analysis allows the permittivity in one slab to be an arbitrary function of y as long as the distribution in the other slab is mirrored across the central x-z plane.Comprehensive wakefield analysis and optimization of the structure parameters require electric and magnetic field magnitudes and spectra to be evaluated and used in the expressions for the accelerating gradient.The forces deflecting the electron bunch and the amplitudes of the electric and magnetic fields on the inner surfaces of the structure are also of importance for any practical accelerator based on this technology.To solve this problem using the formalism described above, the code RECTANGULAR was developed, and the results of computations based on the code using the analytic approach developed here are both in good agreement with the results of analyses based on traditional numerical codes like CST TM [37].
Wakefields in rectangular dielectric-loaded structures have been experimentally studied [14,[38][39][40], Table I.In [38,39] a wakefield measurement using a single crystal diamond-loaded rectangular accelerating structure was reported.Diamond was proposed as a material for dielectricloaded accelerating (DLA) structures a few years ago [18,19].It has a very low microwave loss tangent, the highest available coefficient of thermal conductivity, and high rf breakdown field [17].In [38,39], the high charge beam from the argon wakefield accelerator (AWA) linac ($ 70 nC, z ¼ 1:5-2:5 mm) was transported through a rectangular diamond-loaded resonator and induced an intense wakefield behind the bunch.
The expressions derived above were used for analyzing the wakefields generated by a Gaussian relativistic electron bunch with the parameters of the Argon Wakefield AcceleratorAWA in this structure-w ¼ 8 mm, b ¼ 2 mm, c ¼ 3:19 mm, and " 1 ¼ 5:7.The dependence of the E z component as a function of the distance ¼ z À vt behind it is shown in Fig. 2. The bunch is located at point x 0 ¼ w=2, y 0 ¼ 0, 0 ¼ 8 cm; the observation point is x ¼ w=2, y ¼ 0. The results of computations based on the theory developed here (dotted curve computed using the RECTANGULAR code) show good agreement with the results of numerical calculations of fields in the same structure using the CST PARTICLE STUDIO TM code (solid) [37].Small difference between curves in Fig. 2 is connected with limitation of mode quantity and taking into account here only the Cherenkov part of radiation.
In [40], wakefield acceleration of a relativistic electron beam in a dielectric-lined slab-symmetric THz structure was reported.The high energy tail of a $60 MeV electron beam was accelerated by $150 keV in a 2 cm long, slabsymmetric quartz (" ¼ 3:8) structure.This split-bunch distribution was verified by a longitudinal reconstruction analysis of the emitted coherent transition radiation.The dielectric waveguide structure was characterized by spectral analysis of the Cherenkov radiation at THz frequencies.The spacing of a relativistic bunch train was selectively tuned to excite the second longitudinal mode LSM 21 [40].The structure parameters are b ¼ 120 m, c ¼ 360 m, structure width w ¼ 2 cm, structure length L ¼ 2 cm, bunch charge Q ¼ 90 pC, bunch length z ¼ 120 m, x ¼ 5 mm, y ¼ 0; bunch energy W ¼ 59 MeV. Figure 3(a) shows the E z field spectrum and Fig. 3(b) shows the E z magnitude behind the bunch with the structure parameters [40].It should be noted that the only TABLE I. DLA structure geometries and electron beam parameters used for the comparison of the theory developed in this paper and wakefield experiments carried out at the Argonne Wakefield Accelerator at ANL [38,39] and the Accelerator Test Facility at BNL [14,40]  ; the dotted curve is computed using the RECTANGULAR code [38,39].
asymmetric LSM 1;1 , LSM 2;1 , and LSM 3;1 modes can be excited by the (relatively wide in X) bunch x ¼ 5 mm, y ¼ 0. These mode frequencies are in full agreement with [40] that gives 1 ¼ 1800 m, 2 ¼ 600 m, and 3 ¼ 355 m for the mode wavelengths.Meanwhile, the proposed formalism allows resolving the fine structure of the LSM modes: Fig. 4 shows the E z field spectrum for the wider LSM i;j mode spectrum, where LSM 1;1 , LSM 2;1 , and LSM 3;1 are shown for the narrower bunch width x ¼ 500 m, y ¼ 0. The bar groups near the base LSM 1;1 , LSM 2;1 , and LSM 3;1 frequencies correspond to higher k x values of each mode and can be written as LSM 1;2 , LSM 1;3 . . .; LSM 2;2 , LSM 2;3 . . .; LSM 3;2 , LSM 3;3 modes correspondingly.Figure 5 presents the E z field spectrum of the symmetric LSE i;j modes for the same bunch width x ¼ 500 m, y ¼ 0, where the bar groups now correspond to higher k x values.Figure 6 A THz structure was recently tested at Brookhaven National Laboratory [14], consisting of a rectangular waveguide loaded with polycrystalline chemical vapor deposition diamond plates as an accelerating structure.In this experiment, a drive (high charge) electron beam was launched through the structure exciting a wakefield following behind it.A smaller charge witness beam was launched at a variable delay with respect to the drive beam and experienced acceleration or deceleration depending on the delay.This measurement effectively maps the wakefield produced by the drive beam.The work presented in [14] represents the first mapping of the wakefield in a dielectric device in the THz regime (Table I).
The structure parameters presented in [14] and Table I provide a good opportunity to use the rigorous solution presented above for analytical simulations of the spectrum and magnitude of the wakefields excited by the bunch passing through a dual-layer rectangular DLA structure.75 m thick polycrystalline diamond plates were loaded in a 6 cm long waveguide.The second layer with a thickness of 55-75 m between each diamond plate and conducting wall of the waveguide was formed by a dielectric glue with permittivity " $ 1:8.The beam gap was 2b ¼ 200 m.This structure yields a wakefield dominated by a LSM 11 mode with a 1200 m wavelength.
The ATF drive beam is short enough to excite higher order modes, hence the wake is not a pure sine wave.Parameter optimization carried out to fit the experimental data and analytical simulations with the formalism presented above allows us to predict the following parameters for the data fit of the experiment [14]

VI. CONCLUSION
We have developed a modified method for the decomposition of Maxwell's equations by transverse operator eigenfunctions and we have applied this method to calculating the Cherenkov radiation fields in a rectangular waveguide.Our technique allows a rigorous full solution of the electromagnetic wakefields generated by a relativistic bunch moving along a rectangular dielectric-lined waveguide with nearly arbitrary dependence of the permittivity on the transverse coordinates.
A method based on the first order transverse operator has been previously applied to the analysis of Cherenkov radiation in waveguides, but at the same time the application of this method to transversely inhomogeneous rectangular waveguides is not obvious because of the problems of orthogonality between the components of the electric and magnetic field vectors and the definition of the bilinear form in L 2 space.
In this study we used a modification of the transverse operator method, where we considered the Sturm-Liouville second order operator with an alternating sign weight function.Consequently, the bi-orthogonality of the eigenfunctions as well as the similarity of the operator to a self-adjoint operator made it possible to obtain a rigorous and complete analytic solution for the eigenmodes and to solve in the most general form the problem of Cherenkov wakefield generation in a rectangular accelerating structure with a composite dielectric loading.This formalism does not require any assumptions on the self-Coulomb field of the beam, and we did not use the mode decomposition for the problem formulation and its subsequent solution.Finally, using the transverse operator method, the modes series for the solution is uniformly convergent.
The formalism described here was successfully applied to the analysis of rectangular dielectric-lined structures that were recently beam tested in experiments at the Argonne (ANL/AWA) and Brookhaven (BNL/ATF) accelerator facilities.
The analytic solutions obtained here were implemented in the RECTANGULAR code.The use of an analytic solutions based code makes it independent of the frequency range and the geometrical size of the structure.This ensures fast and effective analysis compared to the use of other standard numerical codes in the frequency range from 10 GHz to up 1 THz, which usually require massive time consuming computations.In the Ka band frequency range, we have carried out analytical simulations of wakefields excited in rectangular dielectric accelerating structures; using this method for the Argonne/AWA beam parameters, we have analyzed a dielectric structure with a rectangular cross section, in which accelerating gradients higher than 100 MV=m can be attained.In the THz range, the structures tested at the BNL/ATF accelerator were considered and compared to experimental results.I.
Thus, to ensure that the angle is not equal to zero we should estimate the norm of this projector.This is done by using an additional self-adjoint resolvent of some operator B and considering the difference Pd between the resolvent of operator Â and B. Estimating the norm of one can show that k Pd k < 1, which leads to the boundedness of the projector and nonzero angle .
(iv) Completeness.-Forcompleteness we need to show that the Riesz integral over the infinite contour À 2 which surrounds both branches is equal to the identity operator.This can again be done using as a supportive construction a self-adjoint resolvent and considering only the difference between the resolvent of operator Â and a self-adjoint operator B. In this case the Riesz integral of the difference should vanish at the infinity of the contour.This check can be made on any dense set of functions, for example on functions that are finite at y ¼ 0. In (iii) and (iv) we suggest using the self-adjoint operator B given by À d 2 f dy 2 À Cf ¼ f; fðÀ0Þ ¼ fðþ0Þ; f 0 ðÀ0Þ ¼ À1=f 0 ðþ0Þ; fðÀ1Þ ¼ fð1Þ ¼ 0: The results of the steps (i)-(iv) prove the statement that eigenfunctions of the problem (A2) form a Riesz basis in L 2 .

FIG. 2 .
FIG. 2. Longitudinal electric field behind the bunch in a 25 GHz diamond based rectangular accelerating structure with w ¼ 8 mm, a ¼ 2 mm, b ¼ 3:19 mm, and " 1 ¼ 5:7.The bunch has a Gaussian charge distribution, energy W ¼ 15 MeV, charge Q ¼ 100 nC, and length z ¼ 1:5 mm.The solid curve is obtained using the PARTICLE STUDIO CST TM code [37]; the dotted curve is computed using the RECTANGULAR code[38,39].
(a) the spectrum of the E z field and Fig. 6(b) presents the E z field for the bunch width x ¼ 500 m, y ¼ 0. The vertical deflecting force F y spectrum and magnitude for a 40 m offset are shown in Fig. 7.The only LSM 1;1 and LSM 1;2 modes are excited for x ¼ 5 mm, y ¼ 0 bunch width.It is worth mentioning than the mode spectrum consists of an extremely low frequency component at 7.6 GHz in comparison with the LSM 1;1 mode frequency of 167 GHz but this low frequency mode is only weakly excited.The fine structure of the F y field spectrum in the range of 10-700 GHz is presented in Fig. 8(a) for 40 m offset; LSM 1;j symmetric modes dominate for x ¼ 500 m, y ¼ 0. The transverse deflecting field magnitude behind the bunch for a 40 m offset is shown in Fig. 8(b) for the LSM 1;j symmetric modes and the bunch width x ¼ 500 m, y ¼ 0.

FIG. 7 .FIG. 6 .FIG. 8 .
FIG.7.F y field spectrum (left) and distribution behind the bunch (right) for 40 m offset.Only LSM 1;1 and LSM 1;2 symmetric modes survive for x ¼ 5 mm, y ¼ 0. It is worth mentioning that this class of modes has an extremely low base frequency (7.6 GHz) and that their amplitude is also small.
Figure 9(a) shows LSM=LSE mode spectrum for the structure parameters presented above and a Gaussian electron beam with z $ 100 m and beam charge 70 pC.The lowest frequency corresponds to an LSE 11 symmetric mode at 273 GHz while the modes at 295 and 845 GHz are LSM 11 and LSM 12 asymmetric modes, respectively.Figure 9(b) presents the magnitude of the accelerating field E z for the structure parameters, a flattop beam charge distribution Q ¼ 70 pC and length 320 m as it was used in the wakefield mapping experiment [14].

FIG. 9 .
FIG. 9. (a) LSM mode spectrum for the structure parameters in [14] and a Gaussian electron bunch with z $ 100 m and beam charge 70 pC; (b) the magnitude of the accelerating field E z for the structure parameters and the flattop beam charge distribution, Q ¼ 70 pC and length 320 m, Ref. [14] and TableI.
form Riesz bases in H E and H H , respectively.This means that for the sets fÉ E ðx; yÞg and fÉ H ðx; yÞg there exist bounded and boundedly invertible operators XE and XH such that e E ¼ XÀ1 E É E ðx; yÞ and e H ¼ XÀ1 H É H ðx; yÞ, where fe E g and fe H g are orthogonal bases in H E and H H , respectively.(The proof of this is rather involved; we provide an outline of the proof in the Appendix.A paper containing the complete proof will be submitted to a journal with a more mathematical focus.)Hence, any f 2 H E and any g 2 H H can be expanded as H ; fÞ H É H : .