PHYSICAL REVIEW SPECIAL TOPICS - ACCELERATORS AND BEAMS 16, 051302 (2013)
Transverse operator method for wakefields in a rectangular dielectric loaded accelerating structure
S. S. Baturin,1,* I. L. Sheinman,1 A. M. Altmark,1 and A. D. Kanareykin1,2 1St. Petersburg State Electrotechnical University, St. Petersburg, Russia 2Euclid Techlabs, LLC, 5900 Harper Road, Solon, Ohio 44139, USA (Received 12 February 2013; published 16 May 2013) Cherenkov radiation generated by a relativistic electron bunch in a rectangular dielectric-loaded waveguide 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 inhomoge- neous system of Maxwell 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 recently beam tested at the Argonne (ANL/AWA) and Brookhaven (BNL/ATF) accelerator facilities.
DOI: 10.1103/PhysRevSTAB.16.051302 PACS numbers: 41.75.Lx, 42.82.Et, 84.40. x
I. INTRODUCTION more traditional rf sources and components that are able to sustain and transmit GW-level microwave or THz power Future high energy charged particle (electron and ion) into the structure [5–7]. accelerators will require new technologies to ensure ex- A dielectric-loaded accelerating (DLA) or power extrac- pansion of the energy and intensity frontiers. Future linear tion structure is a dielectric waveguide with an axial vac- colliders will need to be based on new methods since uum channel for beam propagation and which is foreseeable normal and superconducting technologies are surrounded by a conducting metal wall [4–6]. Recent ex- unavoidably limited in accelerating gradient. Progress in periments have shown that dielectric based structures can compact x-ray source design based on free electron lasers sustain accelerating gradients in excess of 100 MV=m [12] also requires high gradient nonconventional linear accel- and GV=m [6] in the upper GHz and THz frequency erators to be developed. ranges, respectively. Some of the most promising new methods of charged For example, a high current (up to 100 nC) short particle acceleration are laser-plasma [1,2], beam-driven (1–2 mm) relatively low energy (15–100 MeV) electron plasma [2,3], and dielectric based accelerators [4–7]. All bunch in this type of a structure can generate Cherenkov the advantages and limitations of these schemes as well as wakefields with the longitudinal field magnitude up to their current status in simulations and experimental results 100–300 MV=m in the X–Ka band frequency ranges [12]. can be found in the references. In this paper we focus A 3 nC charge and 30 m long bunch of the 23 GeV SLAC/ on dielectric based wakefield acceleration technology FACET accelerator generates 1–10 GV=m wakefields in (DWA) as one of the most promising new technologies the THz range [6,13]. These wakefields are used for for next generation linear colliders [8] and future x-ray accelerating a less intense but high energy electron bunch free-electron lasers (FELs)[9]. propagating behind the drive bunch at a distance corre- In this paper we consider dielectric-lined structures that sponding to the accelerating phase of the Ez field [4,5,14]. are excited by a high intensity relativistic electron beam to Dielectric based structures provide, in addition to the high generate high power X-band, mm-wave, or THz radiation accelerating gradient, the possibility of controlling the fre- [7–9]. This rf pulse is used to drive another dielectric or quency spectrum of the structure by introducing nonlinear conventional accelerating structure (e.g. the CLIC project dielectrics (ferroelectrics) [15,16]. There is also the poten- power extraction structure [10,11]) or accelerate a second tial of using new manufactured materials (such as diamond electron beam in the same structure. This method of elec- and sapphire) with unique dielectric strength and thermo- tromagnetic wave generation avoids the need to develop conductive properties [17–19]. As a rule, the cylindrical geometry usually proposed for *[email protected] dielectric based structures is required for attaining the highest accelerating gradients as well as for providing Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License. Further distri- the maximal possible shunt impedance of the structure bution of this work must maintain attribution to the author(s) and [4,5]. Results of analytic mode analyses of cylindrical the published article’s title, journal citation, and DOI. dielectric accelerating structures are found in [20,21].
1098-4402=13=16(5)=051302(15) 051302-1 Published by the American Physical Society BATURIN et al. Phys. Rev. ST Accel. Beams 16, 051302 (2013)
Since 1988 when the first proof of principle experimental specific version of transverse operator method called the results were published [4], studies of cylindrical dielectric mode basis method was developed and applied to the structures have been presented in numerous publications analysis of wave propagation in conical antennas with (see [5–9,11–16]). dielectric filling [35,36]. At the same time, the dielectric-lined planar or rectan- The method of the first order transverse operator has been gular cross section geometries are also being considered; a applied to waveguide problems in [32–34]. The generalized minor reduction in performance is acceptable in view of orthogonality relations between the longitudinal magnetic the technological difficulties of fabricating cylindrical and longitudinal electric modes have been derived in structures (related to the stringent tolerance requirements [28,29]. However, the bilinear form introduced in [28,29] for geometrical parameters and uniformity of the permit- is not the scalar product, and requires justification of the use tivity of the dielectric [3]). Rectangular structures can be of this relation for orthogonality between the components used for measurements on new dielectric materials of the electric and magnetic field vectors. In [30], the analysis was performed on the basis of the second order effective in sustaining high acceleration rates and tolerant differential equations for the transverse field components of pulsed heating of the structure [4]. Planar structures (direct solution of the Maxwell equations) of a two-channel are also being studied (along with cylindrical geometries) rectangular structure with dielectric-lined waveguides. This for generating terahertz radiation in the frequency range structure design was proposed for increasing the trans- 0.5–1.0 THz [4] and for energy spread reduction former ratio from the drive beam to the beam being accel- (‘‘silencer’’) systems for FEL linacs [22]. erated. However, some of the mathematical aspects of the Another advantage of using a planar geometry is related analysis [28–30] have not been rigorously verified and need to the drive beam stability problem for DWAs. The longi- additional studies. To be specific, there is a problem of sign tudinal electric field magnitude of the wakefield is inversely inversion of the weight function in the definition of the proportional to the square of the vacuum channel radius; norm, and the completeness of the set of functions over therefore it is critically important to employ small aperture which the expansion was performed was not demonstrated. structures. At the same time, reducing the vacuum channel Our approach to this problem is the use of a modified radius increases the severity of beam breakup (BBU) effects transverse operator method. It should be noted that in [5,7,23]. Planar DWA structures can provide a reasonable contrast to the analysis of [28–34], we consider the compromise between the reduced accelerating gradient and Sturm-Liouville second order operator with an alternating the required beam transverse stability [23]. The planar sign weight function. It will be shown below that the option has the additional advantage that a flat electron integral relation derived in [28,29] is a consequence of beam can be generated directly in photoinjectors [24]. the bi-orthogonality of the eigenfunctions as well as the Theoretical analyses of dielectric accelerating structures similarity of the operator to a self-adjoint operator. This with rectangular geometries have been reported in a number approach makes it possible to obtain a complete analytic of publications [23,25–31]. To determine the amplitudes of solution for the eigenmodes of this system, and thus, to individual Cherenkov radiation modes excited in a rectan- solve in the most general form the problem of Cherenkov gular waveguide with dielectric slabs, the impedance match- wakefield generation in a rectangular accelerating structure with a composite dielectric loading. ing technique was used in the earlier papers [23,25,26]. Finally, (i) the proposed method does not require When that formalism is used instead of the direct solution any assumptions on the form of the beam’s Coulomb field of the nonhomogeneous system of Maxwell equations (in contrast to the impedance method [25,26], where it is (the standard analytic approach in analysis of wakefields often assumed that the field of the beam is essentially in cylindrical structures [20,21]), the amplitudes of the flat, an assumption that is valid only at ultrarelativistic wakefields needs to be expressed in terms of the shunt energies); (ii) we also did not use the mode decomposition impedance (or integrated loss factor) for each mode of the for the problem formulation and its subsequent solution; structure [23,25,26]. The approach involves certain approx- the mode series obtained as a solution emerges in a natural imations, while the direct solution of the inhomogeneous way as a result of solving the problem; (iii) as a conse- system of Maxwell’s equations without intermediate ap- quence of (ii), and the use of the transverse operator proximations is always preferable for analyzing the wake- formalism the series of modes of the solution is uniformly field generation problem in DWA structures. In this paper, a convergent. This is not obvious for the solution obtained by rigorous direct solution was found for the wakefields in an the impedance method [25,26] and requires additional arbitrary rectangular dielectric-lined waveguide. analysis and proof. In this paper, we applied a transverse operator method that was used previously as an analytical algorithm for II. THEORETICAL ANALYSIS wave propagation of modes in a waveguide with complex loading, where the rectangular waveguide is partially filled We consider the waveguide shown in Fig. 1. We start with dielectric, ferrite, ferroelectric, or any type of anisot- from the Maxwell system in CGS units (1)–(4), with the ropy is present (chiral loading, for example) [28–34]. Some additional constitutive relations:
051302-2 TRANSVERSE OPERATOR METHOD FOR WAKEFIELDS ... Phys. Rev. ST Accel. Beams 16, 051302 (2013) 2E E 2E @ E y @" @ þ x;y þr " 2 2 @z2 " @y c @t n @n V @ ¼ e r þ þ ; (8) " c@t jVj c@t where we define 0 1 H @ B z @y C @ 0 A; (9) @ Hx @y FIG. 1. A rectangular dielectric-loaded accelerating structure; and (1) vacuum, (2) dielectric, (3) conducting walls. 2 2 @ @ x;y 2 þ 2 : (10) @B @x @y r E ¼ ; (1) c@t By inspection of Eq. (8), only the equation for the Ey component does not include additional terms depending @D enV on other field components. In other words, only the Ey r H ¼ ; (2) c@t c equation is independent. The equation for the Ey component of the electric field r B ¼ 0; (3) given by (8)is 2 @ E @" @2E @ n @ Ey y y þ x;yEy þ " 2 2 ¼ e : r D ¼ en; (4) @z2 @y " @y c @t @y " (11) B H; D E ¼ ¼ " ; (5) The terms of the operator on the left side of the equation containing only derivatives with respect to y can be sim- where E, H are the electric and magnetic field vectors D B plified as respectively; , are the electric and magnetic flux vectors respectively; V—average speed of the bunch; @2E @ E @" @ 1 @ y þ y ¼ ½"E : (12) n—particle concentration of the bunch; e—electron @y2 @y " @y @y " @y y charge; t—time. The electron bunch is moving in the vacuum channel We introduce the new variable ¼ z vt and we along the waveguide axis. The wakefield is generated will look for a solution in the form Eyðx; y; z; tÞ¼ by the bunch when the Cherenkov condition v=c ¼ Eyðx; y; z vtÞ¼Eyðx; y; Þ. The equation for Ey then >" 1=2 is satisfied. Here c is the speed of light and v becomes the speed of the moving particle (or bunch). @2E @2E @2E E For a point charge q, moving along waveguide’s z axis y 1 2 y y @ y @" e @ n 2 ð " Þþ 2 þ 2 þ ¼ : the charge distribution is given by @ @x @y @y " @y "0 @y " (13) nðx; y; z; tÞ¼4 N ðz vtÞ ðx x0Þ ðy y0Þ; (6) We introduce the operator where x, y are transverse coordinates, z is the longitudinal 2 coordinate, and x0, y0 are the transverse coordinates of the 1 @ E @ 1 @ T^ E ¼ y þ ½"E : (14) beam, N number of electrons in the bunch, bunch charge E y ð1 " 2Þ @x2 @y " @y y q ¼ eN. We consider a rectangular waveguide with symmetric We can then rewrite (13)as dielectric layers placed parallel to the OZX plane and 2 e @ n separated by a vacuum gap. In this case the dielectric is @ Ey ^ 2 þ TEEy ¼ 2 : (15) inhomogeneous along the y axis, so that " ¼ "ðyÞ, ¼ @ "0ð1 " Þ @y " ðyÞ. Next we evaluate the Maxwell system (1)–(4) with Following the same steps for the magnetic part of the (5) for this case. From (4) we obtain Maxwell system we obtain E E y @" en ; H r @ðr EÞ @ e r ¼ þ (7) H þr ¼ þ r ðnVÞ; " @y " c@t c@t c applying the curl operator to (1) and using (7), we get (16)
051302-3 BATURIN et al. Phys. Rev. ST Accel. Beams 16, 051302 (2013) @E where 0; 0; y 0; 0 1 Eyjx¼0 ¼ Eyjx¼w ¼ ¼ E @" @y y¼ c z @y (24) B 0 C @H @H @ A: (17) y 0; y 0; 0; @" ¼ ¼ Hyjy¼ c ¼ Ex @y @x x¼0 @x x¼w The velocity vector V has only a z component, so and on the dielectric boundaries 0 1 @n Eyjy¼ b 0 ¼ "1Eyjy¼ b 0; B @y C V @ @n A @E @E r ðn Þ¼v : (18) y y ; @x ¼ 0 @y y¼ b 0 @y y¼ b 0 ; (25) We should mention here that for the magnetic vector the Hyjy¼ b 0 ¼ 1Hyjy¼ b 0 situation is similar to the electric case in that the only @H @H y y equation that does not depend on other field components ¼ : @y y¼ b 0 @y y¼ b 0 is the one for H . y ^ Thus combining (16)–(18), we can write the equation for We consider the eigenvalue equations for the operators TE ^ the Hy component as and TH [(26) and (27)]: 2 ^ H @ H T ðx;yÞ¼ ðx;yÞ; ð0;yÞ¼0; @ y @ y @n E E E E E Hy þ " 2 2 ¼ e : (19) @y @y c @t @x ð Þ 0; @ E x;y 0; Eðw;yÞ¼ ¼ @y y¼c As before, we seek a solution in the form Hyðx; y; z; tÞ¼ Hyðx; y; z vtÞ¼Hyðx; y; Þ: @ Eðx;yÞ ¼ 0 (26) @y @2H @2H @ 1 @ y¼ c y ð1 " 2Þþ y þ ½ H ¼ ev@n: 2 2 y @x @ @x @y @y ð Þ ^ @ H x; y 0; (20) TH Hðx; yÞ¼ H Hðx; yÞ; ¼ @x x¼0 ð Þ We introduce the operator @ H x; y 0; 0; ¼ Hðx; cÞ¼ (27) 2 @x 1 @ H @ 1 @ x¼w T^ H ¼ y þ ½ H : (21) 0 H y ð1 " 2Þ @x2 @y @y y Hðx; cÞ¼ : We then can rewrite (19)as Then we can introduce a series expansion ansatz for the transverse eigenfunctions Eðx; yÞ and Hðx; yÞ: @2H ev @n y þ T^ H ¼ : (22) X1 2 H y 1 2 n n @ " @x Eðx; yÞ¼ XEðxÞYEðyÞ; (28) n¼1 We should stress here that Eqs. (15) and (22) are inde- pendent. Each equation corresponds to a specific type of X1 n n propagating wave, (15) to longitudinal section magnetic Hðx; yÞ¼ XHðxÞYHðyÞ: (29) (LSM) modes, (22) to longitudinal section electric (LSE) n¼1 modes. The eigenvalue problem for the X functions is 2 A. Transverse operator structure and @ 2 0 0; 2 XE;HðxÞ¼ kxXE;HðxÞ;XEð Þ¼ transverse eigenfunctions @x (30) @XHð0Þ @XHðwÞ If we assume that the charge distribution is a finite XEðwÞ¼0; ¼ 0; ¼ 0: function inside the vacuum gap and @x @x 1;y2½ b; b Because of the self-adjointness of the operator defined "ðyÞ¼ by (30), the sets "1;y2½ c; bÞ[ðb; c ; fX ðxÞg ¼fsinðknxÞg ; fX ðxÞg ¼fcosðknxÞg (31) 1;y2½ b; b (23) E n x n H n x n ðyÞ¼ define complete orthogonal systems with kn ¼ n=w. This 1;y2½ c; bÞ[ðb; c ; x is of course the familiar Fourier series. then the boundary conditions for the y components of the Substituting (28) into (26) and taking into account (25) magnetic and electric fields on the conducting boundaries we can write the equations for the transverse electric are eigenfunctions as (suppressing the n index for simplicity)
051302-4 TRANSVERSE OPERATOR METHOD FOR WAKEFIELDS ... Phys. Rev. ST Accel. Beams 16, 051302 (2013) 1 @ 1 @"Y ðyÞ k2 @Y ð cÞ @Y ðcÞ E x Y ðyÞ¼ Y ðyÞ; E ¼ 0; E ¼ 0; ð1 " 2Þ @y " @y ð1 " 2Þ E E E @y @y (32) 0 0 ; @YE @YE YEð b Þ¼"1YEð b Þ ¼ : @y y¼ b 0 @y y¼ b 0 Similarly, substituting (29) into (27) and taking into account (25) we get a set of analogous expressions for the transverse magnetic eigenfunctions: 1 @ 1 @ Y ðyÞ k2 H x Y ðyÞ¼ Y ðyÞ;Yð cÞ¼0; Y ðcÞ¼0: ð1 " 2Þ @y @y ð1 " 2Þ H H H H H 0 0 ; @YH @YH (33) YHð b Þ¼ 1YHð b Þ ¼ : @y y¼ b 0 @y y¼ b 0 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.— 8 cosh E > ðkchbÞ E > E cos½k ðc yÞ ;b y c > "1 cos½k ðc bÞ md <> md E Y sðy; Þ¼As coshðk yÞ; b y b (34) E; E > ch > cosh E : ðkchbÞ cos E cos E ½kmdðc þ yÞ ; c y b: "1 ½kmdðc bÞ
Asymmetric eigenfunctions.— 8 sinh E > ðkchbÞ E > E cos½k ðc yÞ ;b y c > "1 cos½k ðc bÞ md <> md E Y asðy; Þ¼Aas sinhðk yÞ; b y b (35) E; E > ch > sinh E : ðkchbÞ cos E cos E ½kmdðc þ yÞ ; c y b: "1 ½kmdðc bÞ Here we have defined the wave numbers in the vacuum channel and dielectric respectively as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E 1 2 2 E 2 1 2 kch ¼ ð Þ E þ kx;kmd ¼ ð"1 1 Þ E kx: (36) The dispersion relations for the symmetric and asymmetric functions can then be written as E tanh E E tan E 0 E coth E E tan E 0 "1kch ðkchbÞ kmd ½kmdðc bÞ ¼ ;"1kch ðkchbÞ kmd ½kmdðc bÞ ¼ : (37) Eigenfunctions of the magnetic problem (33) also separate into symmetric and asymmetric parts. Symmetric eigenfunctions are given by 8 cosh H > ðkchbÞ H > H sin½k ðc yÞ ;b y c > 1 sin½k ðc bÞ md <> md H Y sðy; Þ¼Bs coshðk yÞ; b y b (38) H; H > ch > cosh H : ðkchbÞ sin H sin H ½kmdðc þ yÞ ; c y b: 1 ½kmdðc bÞ
Asymmetric eigenfunctions are given by 8 sinh H > ðkchbÞ H > H sin½k ðc yÞ ;b y c > 1 sin½k ðc bÞ md <> md H Y asðy; Þ¼Bas sinhðk yÞ; b y b (39) H; H > ch > sinh H : ðkchbÞ sin H sin H ½kmdðc þ yÞ ; c y b: 1 ½kmdðc bÞ Here we define qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H 1 2 2 H 2 1 2 kch ¼ ð Þ H þ kx;kmd ¼ ð"1 1 Þ H kx: The dispersion relations for the symmetric and asymmetric functions respectively are given by
051302-5 BATURIN et al. Phys. Rev. ST Accel. Beams 16, 051302 (2013)
H tanh H H cot H 0 Dom ^ H 0; 1kch ðkchbÞþkmd ½kmdðc bÞ ¼ ; ðTHÞ¼f’ 2 H:’ðx; cÞ¼ H tanh H H tan H 0 0 0; kmd ðkchbÞþ 1kch ½kmdðc bÞ ¼ : ’ð ;yÞ¼’ðw; yÞ¼
’ðx; b 0Þ¼ 1’ðx; b 0Þ; 0 0 B. Expansion method @y’ðx; b Þ¼ @y’ðx; b Þg: (46) ^ Dom ^ Dom ^ Let us next examine the operator TE (14). We consider It immediately follows that ðTEÞ ðTEÞ and H 1 2 ^ ^ the Hilbert space E ¼ L2f½ c; c ; j"ð " Þjg Domð Þ Domð Þ 2 TH TH , which means that the operators L2ð½0;w Þ, where ¼j"ð1 " Þj is the weight func- ^ ^ TE and TH are not self-adjoint in H E and H H, respec- tion. The scalar product in this space is given by tively, so the eigenfunctions are not guaranteed to form a Z w Z c complete orthogonal set. In turn this would imply that all c 1 2 c ð ;’ÞE ¼ j"ð " Þj ’dxdy; (40) solutions for E ( H) are not necessarily given by (28), 0 c (34), and (35)[(29), (38), and (39)]. However, it can be where ’, c 2 H . proved that the eigenfunctions of the tasks (26) and (27) E H The metallic (24) and dielectric boundary conditions along with the conditions (25) form Riesz bases in E and H (25) lead to H, respectively. This means that for the sets f Eðx; yÞg and f Hðx; yÞg there exist bounded and boundedly inverti- ^ ^ ^ ^ 1 DomðTEÞ¼f’ 2 H E:@y’ðx; cÞ¼0; ble operators XE and XH such that eE ¼ XE Eðx; yÞ and e ¼ X^ 1 ðx; yÞ, where fe g and fe g are orthogonal ’ð0;yÞ¼’ðw; yÞ¼0; H H H E H bases in H E and H H, respectively. (The proof of this is ’ðx; b 0Þ¼"1’ðx; b 0Þ; rather involved; we provide an outline of the proof in the @ ’ðx; b 0Þ¼@ ’ðx; b 0Þg: (41) Appendix. A paper containing the complete proof will be y y submitted to a journal with a more mathematical focus.) H H [DomðXÞ indicates the domain of the operator X.] Hence, any f 2 E and any g 2 H can be expanded as The adjoint operator T^ can be found using the X X E f ¼ ð ;fÞ ;g¼ ð ;fÞ : (47) definition E E E H H H
c ^ ^ c 0 ^ ð ; TE’ÞE ðTE ;’ÞE ¼ : (42) Here f Eg are eigenfunctions of TE and f Hg are ei- genfunctions of T^ . Because " and are bounded on This leads to H ½ c; c and ", 2 L1ð½ c; c Þ, metrics on H E and H Dom ^ H 0; H are equivalent to the metrics on L2ð½ c; c Þ ðTEÞ¼f’ 2 E:@y’ðx; cÞ¼ L2ð½0;w Þ. Hence, the expansions (47) hold for f, g 2 ’ð0;yÞ¼’ðw; yÞ¼0; L2ð½ c; c Þ L2ð½0;w Þ. However, we should stress here that the conditions (25) hold only for finite functions on the ’ðx; b 0Þ¼ "1’ðx; b 0Þ; intervals ½ c; b , ½ b; b , ½b; c ; in other words the @y’ðx; b 0Þ¼ @y’ðx; b 0Þg: (43) excitation charge distribution should vanish at material boundaries. ^ Following the same steps for the operator TH (21), The functions f Eg, f Eg and f Hg, f Hg satisfy the bi- we introduce the Hilbert space H H ¼ L2f½ c; c ; orthogonality conditions 2 j ð1 " Þjg L2ð½0;w Þ, with the scalar product Z ð n ; mÞ ¼ j"ð1 " 2Þjð n Þ mdxdy ¼ An;m ; Z w Z c E E E E E E n;m 2 ðc ;’ÞH ¼ j ð1 " Þjc ’dxdy; (44) Z 0 c n m 1 2 n m n;m ð H; HÞH ¼ j ð " Þjð HÞ Hdxdy ¼ AH n;m: where ’, c 2 H H. Applying the boundary conditions (24) and (25) we get (48) ^ Because of the fact that DomðTHÞ¼f’ 2 H H:’ðx; cÞ¼0; ^ ^ ’ð0;yÞ¼’ðw; yÞ¼0; E ¼ J E; H ¼ J H; (49)
’ðx; b 0Þ¼ 1’ðx; b 0Þ; where @ ’ðx; b 0Þ¼@ ’ðx; b 0Þg: (45) 1;y2½ b; b y y J^ ¼ (50) 1;y2½ c; bÞ[ðb; c ; Using the definition of the adjoint (42), the adjoint opera- tor’s domain is found to be one can write
051302-6 TRANSVERSE OPERATOR METHOD FOR WAKEFIELDS ... Phys. Rev. ST Accel. Beams 16, 051302 (2013)
Z X E n ^ m 1 2 n m n;m @½ ðx0;y0Þ ð E;J E ÞE ¼ "ð " Þð EÞ E dxdy ¼ AE n;m; E ðx;y; Þ¼2 q E c E ðx;yÞ y @y E Z E n ^ m 1 2 n m n;m 2 qffiffiffiffiffiffiffi 3 ð H;J HÞ¼ ð " Þð HÞ Hdxdy ¼ AH n;m; pffiffiffiffiffiffiffiffiffiffi sin þ expð j jj jÞ E j j 4 pffiffiffiffiffiffiffiffiffiffiE 2 qffiffiffiffiffiffiffi 5 (51) ; (57) j j þ E E which corresponds to a result from [29] of the so-called þ 0 ^ mode orthogonality condition where E refers to the positive eigenvalues ( E > )ofTE 0 ^ and E to the negative eigenvalues ( E < )ofTE. Z By a similar process applied to (22) we obtain the "ð" 2 1ÞE E dxdy ¼ LSM ; ym;n yp;q m;n mp nq solution for Hy. For the case <0, b 051302-7 BATURIN et al. Phys. Rev. ST Accel. Beams 16, 051302 (2013) ~ k Hn ~ Ey @" ~ @ 2 2 ~ ~ 2 n H E x y E þik ik H k ½1 " E Hy ¼ q x Xn ðxÞXn ðx0Þ 2 ; x;y z x z k " @y @y H (71) n 2qik H ¼ Y ðy0ÞY ðyÞ; ¼ ½1 " 2 n:~ (64) y H H " Y 2 ~ where H is the conjugate eigenfunction of the adjoint @ Ey ~ 2 problem (33). Substitution of (69)–(71) into (68)gives Substitution of @y from (63) and Hx from (61) into (64) ~ with E from (60) gives n 2 2 n 2 y 2qX ðx0Þ ik ½1 " k þðk Þ En ¼ E x 2 ~ ~ ~ ~ z n 2 2 2 @ E 2 @Ey Ey @" 2 @Hy ðkx Þ þk E k kyðyÞ z k E~ ¼ ik þ þik n~ ik : 2 z @y " @y " @x 2 2 n n 2 @x n q XEðx0Þ ikðkx Þ n I½Ey ½kyðyÞy þ n 2 2 2 Hy : (72) (65) ðkx Þ þk H k ~ Now we consider the Ey Helmholtz equation: Here we define 2 1 ~ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @ @ @ ~ 2 2 ~ @ n þ ½" E ½1 " k E ¼ 2 : 1 2 n 2; @x2 @y " @y y y @y " kyðyÞ¼ ½ " E þðkx Þ (73) (66) and denote for brevity the integral by the operation I: Integrating this equation with respect to y, we have Z d Z 2 I½f ðyÞ fðyÞdy; I½f ðyÞ¼fðyÞ: (74) @ 1 @ n~ dy ½1 " 2 k2 E~ dy þ ½"E~ þC ¼ 2 : @x2 y " @y y " Using the inverse Fourier transformation the Cherenkov (67) radiation field (only real poles contributing for <0 ^ ^ Multiplication of this equation by ik and substitution into and only positive eigenvalues of TE and TH) can be (65) leads to found as X 2 ~ Z 2 n n @ E @ E ðx;y; Þ¼4 q X ðx0ÞX ðxÞ z 2 ~ ¼ ½1 2 2 ~ z E E 2 k Ez ik 2 " k Eydy n @x @x X n pffiffiffiffiffiffi ~ kyI½Ey ½kyðyÞy @Hy cosð j jÞ ik : (68) þðknÞ2 E @x E E x X ðknÞ2Hn pffiffiffiffiffiffiffi Using the boundary condition on the conductors 2 x y cos ~ 2 ð Hj jÞ : (75) Ezð cÞ¼0, we find the constant of integration C ¼ 0. n H þðkx Þ For the boundary conditions on the sides of the structure, H 0 0 0 ~ Ezðx¼ Þ¼ , Ezðx¼aÞ¼ , we can expand Ez in the series The transverse force inside the channel can be found using X the Panofsky-Wenzel theorem: ~ E n E z ¼ Xn ðxÞEz ðyÞ: (69) @F? ¼ er?Ez: (76) Taking into account that @z n E @Y ðy0Þ Hence, ~ 2 E E y n E Ey ¼ qXn ðxÞXn ðx0Þ 2 ;Ey ¼ YEðyÞ; Z Z E k @y @E @E F ¼ z dz; F ¼ z dz: (77) (70) x @x y @y where YE is the conjugate eigenfunction of the adjoint Substitution of Ez leads to the full expressions for the problem (32), and transverse force components: pffiffiffiffiffiffi pffiffiffiffiffiffiffi X n X n sin X n 2 n sin n @XEðxÞ kyI½Ey ½kyðyÞy ð Ej jÞ 2 ðkx Þ Hy ð Hj jÞ F ðx; y; Þ¼4 q X ðx0Þ pffiffiffiffiffiffi pffiffiffiffiffiffiffi ; (78) x E @x þðknÞ2 þðknÞ2 n E E x E H H x H pffiffiffiffiffiffi pffiffiffiffiffiffiffi X X 2 n sin X n 2 n sin n n ðkyÞ Ey ð Ej jÞ 2 ðkx Þ @Hy ð Hj jÞ F ðx; y; Þ¼4 q X ðx0ÞX ðxÞ pffiffiffiffiffiffi pffiffiffiffiffiffiffi : (79) y E E þðknÞ2 þðknÞ2 @y n E E x E H H x H 051302-8 TRANSVERSE OPERATOR METHOD FOR WAKEFIELDS ... Phys. Rev. ST Accel. Beams 16, 051302 (2013) 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]. Experiment/Ref. f, GHz, LSM11 "w,cm b,mm c, mm Energy, MeV z;mm Q ANL/AWA [38,39] 25 5.7 0.8 2 3.19 15 1.5–2.5 70 nC BNL/ATF [40] 170 3.8 2 0.12 0.36 59 0.12 90 pC BNL/ATF [14] 295 5:7=1:8 0.1 0.10 0.25 57 0.10 70 pC V. COMPARISON WITH NUMERICAL SIMULATIONS AND EXPERIMENTAL RESULTS the RECTANGULAR code) show good agreement with the results of numerical calculations of fields in the TM The formalism presented above allows obtaining a rig- same structure using the CST PARTICLE STUDIO orous full solution for the wakefields generated by charged code (solid) [37]. Small difference between curves in particle bunches passing along a rectangular accelerating Fig. 2 is connected with limitation of mode quantity and structure loaded with symmetric dielectric slabs. Note that taking into account here only the Cherenkov part of this analysis allows the permittivity in one slab to be an radiation. arbitrary function of y as long as the distribution in the In [40], wakefield acceleration of a relativistic electron other slab is mirrored across the central x-z plane. beam in a dielectric-lined slab-symmetric THz structure 60 MeV Comprehensive wakefield analysis and optimization of was reported. The high energy tail of a electron 150 keV the structure parameters require electric and magnetic field beam was accelerated by in a 2 cm long, slab- ¼ 3 8 magnitudes and spectra to be evaluated and used in the symmetric quartz (" : ) structure. This split-bunch distribution was verified by a longitudinal reconstruction expressions for the accelerating gradient. The forces analysis of the emitted coherent transition radiation. The deflecting the electron bunch and the amplitudes of the dielectric waveguide structure was characterized by spec- electric and magnetic fields on the inner surfaces of the tral analysis of the Cherenkov radiation at THz frequen- structure are also of importance for any practical accelera- cies. The spacing of a relativistic bunch train was tor based on this technology. To solve this problem using selectively tuned to excite the second longitudinal mode the formalism described above, the code RECTANGULAR LSM21 [40]. The structure parameters are b ¼ 120 m, was developed, and the results of computations based on c ¼ 360 m, structure width w ¼ 2cm, structure length the code using the analytic approach developed here are L ¼ 2cm, bunch charge Q ¼ 90 pC, bunch length 120 m 5mm 0 both in good agreement with the results of analyses based z ¼ , x ¼ , y ¼ ; bunch energy TM on traditional numerical codes like CST [37]. W ¼ 59 MeV. Wakefields in rectangular dielectric-loaded structures Figure 3(a) shows the Ez field spectrum and Fig. 3(b) have been experimentally studied [14,38–40], Table I.In shows the Ez magnitude behind the bunch with the struc- [38,39] a wakefield measurement using a single crystal ture parameters [40]. It should be noted that the only diamond-loaded rectangular accelerating structure was re- ported. Diamond was proposed as a material for dielectric- 200 loaded accelerating (DLA) structures a few years ago [18,19]. It has a very low microwave loss tangent, the 100 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 0 ( 70 nC, z ¼ 1:5–2:5mm) was transported through a (MV/m Ez rectangular diamond-loaded resonator and induced an in- -100 tense wakefield behind the bunch. The expressions derived above were used for analyzing -200 the wakefields generated by a Gaussian relativistic electron 8 10 12 14 16 18 bunch with the parameters of the Argon Wakefield Distance behind the bunch (cm) AcceleratorAWA in this structure—w ¼ 8mm, b ¼ FIG. 2. Longitudinal electric field behind the bunch in a 2mm, c ¼ 3:19 mm, and "1 ¼ 5:7. The dependence of 25 GHz diamond based rectangular accelerating structure with the E component as a function of the distance ¼ z vt z w ¼ 8mm, a ¼ 2mm, b ¼ 3:19 mm, and "1 ¼ 5:7. The bunch behind it is shown in Fig. 2. The bunch is located at point has a Gaussian charge distribution, energy W ¼ 15 MeV, charge ¼ 2 ¼ 0 ¼ 8cm x0 w= , y0 , 0 ; the observation point is Q ¼ 100 nC, and length z ¼ 1:5mm. The solid curve is TM x ¼ w=2, y ¼ 0. The results of computations based on obtained using the PARTICLE STUDIO CST code [37]; the dotted the theory developed here (dotted curve computed using curve is computed using the RECTANGULAR code [38,39]. 051302-9 BATURIN et al. Phys. Rev. ST Accel. Beams 16, 051302 (2013) (a) (b) FIG. 3. (a) Ez field spectrum and (b) lineout of the field behind the bunch (black curve) for the LSM1;1, LSM2;1, and LSM3;1 asymmetric modes, x ¼ 5mm, y ¼ 0. Frequencies are in good agreement with [40] 1 ¼ 1800 m, 2 ¼ 600 m, and 3 ¼ 355 m. (For this and subsequent figures, see Ref. [40] and Table I for further details.) asymmetric LSM1;1, LSM2;1, and LSM3;1 modes can be modes are excited for x ¼ 5mm, y ¼ 0 bunch width. It excited by the (relatively wide in X) bunch x ¼ 5mm, is worth mentioning than the mode spectrum consists of an y ¼ 0. These mode frequencies are in full agreement with extremely low frequency component at 7.6 GHz in com- [40] that gives 1 ¼ 1800 m, 2 ¼ 600 m, and 3 ¼ parison with the LSM1;1 mode frequency of 167 GHz but 355 m for the mode wavelengths. Meanwhile, the pro- this low frequency mode is only weakly excited. The posed formalism allows resolving the fine structure of the fine structure of the Fy field spectrum in the range of LSM modes: Fig. 4 shows the Ez field spectrum for the 10–700 GHz is presented in Fig. 8(a) for 40 m offset; wider LSMi;j mode spectrum, where LSM1;1, LSM2;1, and LSM1;j symmetric modes dominate for x ¼ 500 m, LSM3;1 are shown for the narrower bunch width x ¼ y ¼ 0. The transverse deflecting field magnitude behind 500 m 0 , y ¼ . The bar groups near the base LSM1;1, the bunch for a 40 m offset is shown in Fig. 8(b) for the LSM2;1, and LSM3;1 frequencies correspond to higher kx LSM1;j symmetric modes and the bunch width x ¼ values of each mode and can be written as LSM1;2, 500 m, y ¼ 0...... LSM1;3 ; LSM2;2, LSM2;3 ; LSM3;2, LSM3;3 modes A THz structure was recently tested at Brookhaven correspondingly. Figure 5 presents the Ez field spectrum of National Laboratory [14], consisting of a rectangular the symmetric LSEi;j modes for the same bunch width waveguide loaded with polycrystalline chemical vapor x ¼ 500 m, y ¼ 0, where the bar groups now deposition diamond plates as an accelerating structure. In correspond to higher kx values. Figure 6(a) shows the this experiment, a drive (high charge) electron beam was spectrum of the Ez field and Fig. 6(b) presents the Ez field launched through the structure exciting a wakefield follow- for the bunch width x ¼ 500 m, y ¼ 0. The vertical ing behind it. A smaller charge witness beam was launched deflecting force Fy spectrum and magnitude for a 40 m at a variable delay with respect to the drive beam and experienced acceleration or deceleration depending on offset are shown in Fig. 7. The only LSM1;1 and LSM1;2 the delay. This measurement effectively maps the wake- field produced by the drive beam. The work presented FIG. 4. Ez field spectrum for the LSMi;j modes. LSM1;1, LSM2;1, and LSM3;1 modes are shown for 500 m 0 x ¼ , y ¼ . Bar groups near LSM mode frequencies FIG. 5. Ez field spectrum of the LSEi;j modes for x ¼ 500 m 0 correspond to higher kx values for each mode: LSM1;2, LSM1;3; , y ¼ . Bar group near the LSE mode frequencies LSM2;2, LSM2;3; LSM3;2, LSM3;3, respectively. corresponds to higher kx values. 051302-10 TRANSVERSE OPERATOR METHOD FOR WAKEFIELDS ... Phys. Rev. ST Accel. Beams 16, 051302 (2013) 1.6 1.2 0.8 0.4 Ez (MV/m) 0 –0.4 –0.8 0 0.25 0.5 0.75 1 Distance behind the bunch (cm) (a) (b) FIG. 6. (a) Ez field spectrum (a) and Ez field magnitude behind the bunch (b) for the bunch width x ¼ 500 m. 6 4 Ez (KV/m) 2 0 0 0.25 0.5 0.75 1 Distance behind the bunch (cm) (a) (b) FIG. 7. Fy field spectrum (left) and distribution behind the bunch (right) for 40 m offset. Only LSM1;1 and LSM1;2 symmetric modes survive for x ¼ 5mm, 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. in [14] represents the first mapping of the wakefield in a conducting wall of the waveguide was formed by a dielec- dielectric device in the THz regime (Table I). tric glue with permittivity " 1:8. The beam gap was The structure parameters presented in [14] and Table I 2b ¼ 200 m. This structure yields a wakefield domi- provide a good opportunity to use the rigorous solution nated by a LSM11 mode with a 1200 m wavelength. presented above for analytical simulations of the spectrum The ATF drive beam is short enough to excite higher and magnitude of the wakefields excited by the bunch order modes, hence the wake is not a pure sine wave. passing through a dual-layer rectangular DLA structure. Parameter optimization carried out to fit the experimental 75 m thick polycrystalline diamond plates were loaded in data and analytical simulations with the formalism a 6 cm long waveguide. The second outer layer with a presented above allows us to predict the following parame- thickness of 55–75 m between each diamond plate and ters for the data fit of the experiment [14]: b ¼ 100 m, 600 400 200 Ez (KV/m) 0 –200 0 0.25 0.5 0.75 1 Distance behind the bunch (cm) (a) (b) FIG. 8. Fy field spectrum (left) and distribution behind the bunch (right) for a 40 m beam offset. LSM1;j symmetric modes dominate for x ¼ 500 m, y ¼ 0. 051302-11 BATURIN et al. Phys. Rev. ST Accel. Beams 16, 051302 (2013) 20 15 10 5 Ez (MV/m) Ez 0 –5 –10 –0.020 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Distance behind the bunch (cm) (a) (b) 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 Ez for the structure parameters and the flattop beam charge distribution, Q ¼ 70 pC and length 320 m, Ref. [14] and Table I. c ¼ 250 m consisting of a 75 m thick diamond plate self-adjoint operator made it possible to obtain a rigorous and 75 m thick dielectric glue layer. Permittivity of the and complete analytic solution for the eigenmodes and to diamond slab is "1 5:7, and for the dielectric glue layer solve in the most general form the problem of Cherenkov "2 ¼ 1:8. Figure 9(a) shows LSM=LSE mode spectrum wakefield generation in a rectangular accelerating structure for the structure parameters presented above and a with a composite dielectric loading. This formalism does Gaussian electron beam with z 100 m and beam not require any assumptions on the self-Coulomb field of charge 70 pC. The lowest frequency corresponds to an the beam, and we did not use the mode decomposition for LSE11 symmetric mode at 273 GHz while the modes at the problem formulation and its subsequent solution. 295 and 845 GHz are LSM11 and LSM12 asymmetric Finally, using the transverse operator method, the modes modes, respectively. Figure 9(b) presents the magnitude series for the solution is uniformly convergent. of the accelerating field Ez for the structure parameters, The formalism described here was successfully applied a flattop beam charge distribution Q ¼ 70 pC and to the analysis of rectangular dielectric-lined structures length 320 m as it was used in the wakefield mapping that were recently beam tested in experiments at the experiment [14]. Argonne (ANL/AWA) and Brookhaven (BNL/ATF) accel- erator facilities. The analytic solutions obtained here were implemented VI. CONCLUSION in the RECTANGULAR code. The use of an analytic solutions We have developed a modified method for the decom- based code makes it independent of the frequency range position of Maxwell’s equations by transverse operator and the geometrical size of the structure. This ensures eigenfunctions and we have applied this method to calcu- fast and effective analysis compared to the use of other lating the Cherenkov radiation fields in a rectangular wave- standard numerical codes in the frequency range from guide. Our technique allows a rigorous full solution of the 10 GHz to up 1 THz, which usually require massive electromagnetic wakefields generated by a relativistic time consuming computations. In the Ka band frequency bunch moving along a rectangular dielectric-lined wave- range, we have carried out analytical simulations of wake- guide with nearly arbitrary dependence of the permittivity fields excited in rectangular dielectric accelerating struc- on the transverse coordinates. tures; using this method for the Argonne/AWA beam A method based on the first order transverse operator parameters, we have analyzed a dielectric structure with has been previously applied to the analysis of Cherenkov a rectangular cross section, in which accelerating gradients radiation in waveguides, but at the same time the applica- higher than 100 MV=m can be attained. In the THz range, tion of this method to transversely inhomogeneous the structures tested at the BNL/ATF accelerator were rectangular waveguides is not obvious because of the considered and compared to experimental results. problems of orthogonality between the components of ACKNOWLEDGMENTS the electric and magnetic field vectors and the definition of the bilinear form in L2 space. This work was supported by the Russian Foundation for In this study we used a modification of the transverse Basic Research and the Ministry of Education and Science operator method, where we considered the Sturm-Liouville of the Russian Federation in the framework of the Federal second order operator with an alternating sign weight Program ‘‘Human Capital for Science and Education in function. Consequently, the bi-orthogonality of the eigen- Innovative Russia’’ for 2009–2013. We give special thanks functions as well as the similarity of the operator to a to P. Schoessow for useful discussions and suggestions. 051302-12 TRANSVERSE OPERATOR METHOD FOR WAKEFIELDS ... Phys. Rev. ST Accel. Beams 16, 051302 (2013) 1 APPENDIX i b1 ¼ : (A4) 1 In this Appendix we demonstrate the main steps of the þ i ^ ^ proof that eigenfunctions of TE and TH operators form One can see that spectrum is purely real but not positive. Riesz basis sets in L2 space. It consists of two branches. The first is purely positive and First of all, because of the symmetry of the system, the ^ ^ the second is purely negative. analysis of the TE and TH operator properties can be (ii) Partial Gram matrix.—The Gram matrix is reduced to a simpler task. We consider operators necessary to demonstrate the linear independence of the expanded in a Fourier series in the x coordinate. eigenfunctions inside each branch. Using the asymptotic This reduces the analysis task from two dimensions eigenfunctions, one can build Gram matrices for each (in the XY plane) to one dimension (only on the y axis) branch of eigenvalues. These matrices are equivalent and with the additional parameter kx. Further, the symmetry only one of them needs to be analyzed. We use the theorem with respect to the origin of the y axis gives us the that if the Gram matrix is invertible then vectors it is built opportunity to consider the reduced task on the interval from are linearly independent. A matrix that is similar to 0 ^ ^ ½ ;c only so that symmetric functions of TE and TH will the Gram matrix of one branch has the form satisfy the Neumann condition at 0 and the corresponding asymmetric functions to the Dirichlet condition at 0. After Gn;m ¼ Tn;m c1Hn;m þ 2; (A5) rescaling the interval, it can be seen that the properties of ^ ^ where Gn;m is the element of the Gram matrix, Tn;m is the the TE and TH operators are the same as the properties of element of the Toeplitz matrix, Hn;m is the element of the the operator Hilbert matrix; 2 ^ d 1 1 A ¼ 2 C ; (A1) T ¼ ;H¼ ; (A6) dy n;m n m n;m n þ m where ¼ 1 for y 2½ 1; 0 and ¼ for y 2ð0; 1 , and 2 is a Hilbert-Schmidt (compact) add-on, which can with the additional conditions on its domain fð 0Þ¼ be ignored, c1 is the constant that depends on initial fðþ0Þ, f0ð 0Þ¼ f0ðþ0Þ. At the points 1 for recon- parameters. ^ ^ Hilbert and Toeplitz matrices are noncompact matrices. structing all parts of the TE and TH operators we need to consider all combinations of conditions: Dirichlet- The matrix is invertible if its spectrum does not include Dirichlet, Neumann-Neumann, etc. However, this does zero. Using the fact that the Gram matrix is a nonnegative not affect the basis properties of eigenfunctions in matrix in finite-dimensional space and then proceeding to principle. For simplicity of calculation we consider here the limit of an infinite dimensional space, one can show the Dirichlet-Dirichlet conditions at the end of the inter- that R 0 val. Here we assume that , , , C 2 , C> G aT; (A7) (although this method of proof will work for complex parameters also). where a>0 is a constant, and T is the Toelplitz The equivalent eigenvalue problem can be formulated as matrix. Using representation of Toeplitz matrix as an op- follows: erator of multiplication by a function on Hardy space, one can show that T>0. This implies the linear indepen- 2 d f 0 0 dence of the eigenfunctions inside one branch or in other 2 Cf ¼ f; fð Þ¼ fðþ Þ; dy (A2) words that each branch is a Riesz basis in its linear envelope. f0ð 0Þ¼ f0ðþ0Þ;fð 1Þ¼fð1Þ¼0: (iii) Angle between eigensubspaces.—The two branches The proof is divided into four steps. are linearly independent if the angle between eigensubspa- (i) Basic analysis and asymptotes.—One can check by ces is nonzero (i.e. the subspaces do not coincide) in the direct calculation using root vectors formulas that all of them limit n; m !1(n and m are the indices of the first and are equal to zero. Thus, the problem (A2) does not have any the second branch). For the proof we use the technique of the Riesz projector and the fact that the Riesz projector is a additional solutions except eigenfunctions that correspond ^ to eigenvalues. Proceeding to the limit of large eigenvalues, parallel projector. We consider the Riesz projector P on one can determine the asymptotic eigenvalues as one eigensubspace calculated over the contour 1 which surrounds only one branch. The inverse relation of the þ 2 ¼ðb1 þ nÞ þ C; projector’s norm bounds the sine of the angle between 2 (A3) the eigensubspaces: ¼ b1 þ þ n C: 2 1 sin j j ^ : (A8) Here b1 is given by kPk 051302-13 BATURIN et al. Phys. Rev. ST Accel. Beams 16, 051302 (2013) Thus, to ensure that the angle is not equal to zero [8] C.-J. Jing, S. P. Antipov, A. Kanareykin, P. Schoessow, we should estimate the norm of this projector. This is M. E. Conde, W. Gai, and J. G. Power, in Proceedings done by using an additional self-adjoint resolvent of of the 2011 Particle Accelerator Conference, NY, USA ^ ^ (IEEE, New York, 2011), p. 2279 [http:// some operator B and considering the difference Pd be- ^ ^ accelconf.web.cern.ch/AccelConf/PAC2011/papers/thp078. tween the resolvent of operator A and B. Estimating the pdf]. norm of [9] C. Jing, A. Kanareykin, J. G. Power, and A. Zholents, in Proceedings of the 2nd International Particle Accelerator Z 1 1 ^ Conference, San Sebastia´n, Spain (EPS-AG, Spain, 2011), P d ¼ d ^ ^ ; (A9) 1 I A I B p. 1485 [http://dx.doi.org/10.1063/1.4790432]. [10] R. Toma´s, Phys. Rev. ST Accel. Beams 13, 014801 ^ (2010). one can show that kPdk < 1, which leads to the bounded- ness of the projector and nonzero angle . [11] C. Jing, S. P. Antipov, A. Kanareykin, P. Schoessow, M. E. (iv) Completeness.—For completeness we need to Conde, W. Gai, J. G. Power, and I. Syratchev, in Proceedings of the 2011 Particle Accelerator Conference, show that the Riesz integral over the infinite contour NY, USA (Ref. [8]), p. 315 [http://accelconf.web.cern.ch/ 2 which surrounds both branches is equal to the identity AccelConf/PAC2011/papers/mop116.pdf]. operator. This can again be done using as a supportive [12] M. Conde, S. Antipov, W. Gai, and R. Konecny, construction a self-adjoint resolvent and considering in Proceedings of the International Particle Accelerator ^ only the difference between the resolvent of operator A Conference, Kyoto, Japan (ICR, Kyoto, 2010), p. 4425 and a self-adjoint operator B^. In this case the Riesz [http://accelconf.web.cern.ch/AccelConf/IPAC10/papers/ integral of the difference should vanish at the infinity thpd062.pdf]. of the contour. This check can be made on any dense set [13] J. Rosenzweig, G. Travish, M. Hogan, and P. Muggli, in of functions, for example on functions that are finite Proceedings of the International Particle Accelerator at y ¼ 0. Conference, Kyoto, Japan (Ref. [12]), p. 3605 [http:// accelconf.web.cern.ch/AccelConf/IPAC10/papers/ In (iii) and (iv) we suggest using the self-adjoint opera- ^ thoamh02.pdf]. tor B given by [14] S. Antipov, C. Jing, A. Kanareykin, J. E. Butler, V. 2 Yakimenko, M. Fedurin, K. Kusche, and W. Gai, Appl. d f 0 0 Phys. 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