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Generalized Panofsky–Wenzel theorem and hybrid coupling
Nuclear Instruments and Methods in Physics Research A 469 (2001) 21–23
Generalized Panofsky–Wenzel theorem and hybrid coupling A.V. Smirnov*,1 DULY Research Inc., 1912 Mac Arthur St., Rancho Palos Verdes, CA 90275, USA Received 23 January 2001
Abstract The Panofsky–Wenzel theorem is reformulated for the case in which phase slippage between the wave and beam is not negligible. The extended theorem can be applied in analysis of detuned waveguides, RF injectors, bunchers, some tapered waveguides or high-power sources and multi-cell cavities for dipole and higher order modes. As an example, the relative contribution of the Lorentz’ component of the deflecting force is calculated for a conventional circular diskloaded waveguide. # 2001 Elsevier Science B.V. All rights reserved. PACS: 29.27.a; 29.27.Bd; 41.75.i Keywords: Eigenmode; Wakefield; BBU instability; Phase slippage; Synchronism; Hybrid mode; Photoinjector; Multi-cell cavity
Transverse mode coupling causes phase space dilution [1] and instabilities [2,3] that can seriously limit the performance of resonant accelerators. The Panofsky–Wenzel theorem [4] was formulated as a relationship between the transverse and longitudinal wake functions for ultrarelativistic beams and synchronous waves having no phase slippage with respect to the beam. However, in many circumstances there is desynchronism between the beam and at least one partial wave that interacts with the beam. We can encounter this situation, for example, in considering the regenerative beam break-up instability (BBU) in RF injectors, photoinjectors and standing-wave linacs including superconducting ones. In specially detuned waveguides this slippage should be also taken into account. *Corresponding author. Tel.: +1-310-548-0290. E-mail address: alexei [email protected] (A.V. Smirnov). 1 On leave from RRC ‘Kurchatov Institute’, Moscow 123182, Russian Federation
We begin by writing the fields excited by the beam in a multi-cell cavity in the following form: ! ! Er ðr; tÞ ¼ ReCr ðtÞ ðrÞeior t ; Hr mod e r ! ! E 0?r sinðhr zÞ E ?r ; ¼ H ?r iH 0?r cosðhr zÞ ! ! 0 Ezr cosðhr zÞ Ezr ¼ ; 0 iHzr Hzr sinðhr zÞ E H
ð1Þ
where index r corresponds to different cavity modes, including those having different polarization; or ¼ o0r io00r is the modal complex frequency; o00r ¼ o0r =2QrL ; QrL is the loaded Q- factor of the resonator structure for the mode r, and Cr ðtÞ is a slowly varying amplitude that is defined by the beam current (see, e.g., theory of excitation of closed structures established by Condon [5] and developed by Vainshtein [6]).
0168-9002/01/$ - see front matter # 2001 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 0 1 ) 0 0 7 0 8 - 2
22
A.V. Smirnov / Nuclear Instruments and Methods in Physics Research A 469 (2001) 21–23
In the presence of phase slippage effect the transverse motion of the particle is described by a complicated function [7]. This is the typical situation seen in regenerative BBU studies and/or for low-energy beams. Therefore, a description of the detailed transverse dynamics requires one to determine the transverse force F ? ðz; tÞ function along the structure rather than just the transverse wake function w? that is convenient for ‘‘rigid’’, high-energy beams. We can express this force due to the rth mode in terms of longitudinal field components. From the inhomogeneous Maxwell equations and eigenfunction properties directly follows: X g2 F ?r ¼ ReiCr r e 2hr
0 0 1 bbr r? Ezr ðb br Þez r? Hzr Z0
e
iðhr zor tÞ
þ ðb br ÞZ0 j ? ;
ð2Þ
pffiffiffiffiffiffiffiffiffiffiffi where Z0 ¼ mo =eo , b ¼ n=c; br ¼ o0r =chr ; qffiffiffiffiffiffiffiffiffiffiffiffiffi gr ¼ 1= 1 b2r ; ez is unit vector along OZ axis, n is particle velocity, e is particle charge. One can see from Eq. (2), that for relatively short cavities with few-cells the backward wave along with its Lorentz’ component can contribute to the transverse dynamics. For the waveguide mode s described as ! ! E 0s Es ðr; tÞ ¼ Re expðiðhs z otÞÞ; Hs H 0s where E 0s ; H 0s are slowly varying functions of z; t; we obtain analogously (2) F ?s g2 ¼ Rei s e hs 0 0 1 bbs r? Ezs þ ðb bs Þ ez r? Hzs Z0
ð3Þ eiðhs zotÞ þ ðb bs ÞZ0 j ? : In the particular case of a waveguide with a relativistic beam b ! 1; br ! 1 and j ? ¼ 0 this relationship gives (after integration over the structure length) the Panofsky–Wenzel theorem [8] in the frequency domain. Note, in some devices
the transverse component of the current j ? 6¼ 0 plays an important role. Examples include ubitrons, Free Electron Masers and Free Electron Lasers with oversized waveguides. To calculate properly the transverse forces and threshold currents in cases when the (partial) wave phase velocity does not coincide with the particle longitudinal velocity one must know the contribution of the Lorentz force to the net deflection. Let us postulate a special value Xs (it can be a tensor) reflecting hybrid character of HOM: 0 0 ez r? Hzs Z0 ¼ Xs r? Ezs :
ð4Þ
For a disk-loaded cylindrical structure we can apply the studies [9–12]. The fields in the interaction region that are close to synchronism can be presented in the form of a superposition of TE and TM modes: 0 Ezs ¼ As Im ðps rÞcosðmðj þ j0 ÞÞ; 0 ¼ Xs Z01 As Im ðps rÞcosðmjÞÞ; Hzs
ð5Þ
and from Maxwell equations i qE 0 qH 0 0 ¼ 2 k zs hr zs Z0 ; Ejs ps qr qr
ð6Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where m is an angular index, ps ¼ k b2 s 1 is the transverse wavenumber; k ¼ o=c; Im is the modified Bessel function of the first kind of the mth order; j0 is the angular shift of electric wave polarization with respect to the magnetic one: jj0 j5p. To simplify the analysis and to avoid mixing between the different complex numbers reflecting the time dependences and the angular dependences, it is convenient to use four-component complex numbers characterized with imaginary units i2 ¼ 1; j2 ¼ 1; ij 6¼ 1: This scalar algebra of hypercomplex values was used first by Lewin [10] and developed later to apply to BBU analysis [13] in presence of an axial magnetic field. In this new notation we use the fields in the form: ! ! E 0s Es ðr; tÞ ¼ Rei;j expðiðhs z otÞ jmjÞ; Hs H 0s 0 Ezs ¼ Rej As Im ðps rÞexpðjmðj þ j0 ÞÞ; 0 ¼ Xs Z01 As Im ðps rÞexpðjmjÞ; Hzs
ð7Þ
A.V. Smirnov / Nuclear Instruments and Methods in Physics Research A 469 (2001) 21–23
and hence
23
In other, more general, cases this coupling coefficient is a complex value and can be calculated numerically by applying the matching field technique or with 3D codes like GdfidL [17].
i 0 ¼ Rej 2 As Ejs ps jm 0 kXs ps Im ðps rÞ hs Im ðps rÞexpðjmj0 Þ expðjmjÞ: r ð8Þ
To estimate the coupling coefficient Xs we neglect here the azimuthal field penetrating into the space between disks. Strictly speaking it is valid when hs ðD tÞ 1; where D is the period, t is the disk thickness. In this approximation one can apply the boundary condition Ejs r¼a ¼ 0; i.e. the azimuthal electric field is zero on the radius a of iris aperture. Then, we obtain from (8): m 1 Im ðps aÞ : ð9Þ Xs ¼ jexpðjmj0 Þ bs ps a Im0 ðps aÞ Since Xs is, by definition, a real value in j-space of complex numbers, we have jmj0 j ¼ p=2; i.e. the hybrid wave is a sum of TE and TM modes with an angular shift p=2m between their polarization planes. Thus, we confirmed that for every mth asymmetric wave there are always 2m independent hybrid eigenmodes. Extensive experimental ‘‘cold’’ model studies of hybrid dipole eigenmodes were done at SLAC [14]. In particular, these studies indicated that different orientations of polarization are characterized by different eigenfrequencies. Thus, the coupling coefficient between TE and TM modes composing the hybrid mode can be estimated as follows: m 1 Im ðps aÞ Xs ¼ : ð10Þ bs ps a Im0 ðps aÞ The accuracy of this expression is high enough when bs ! 1 (see [12,15]). In particular, for the HEM11 dipole mode we have at ðps aÞ2 1 1 ðps aÞ2 jXs j ¼ 1 1; ð12Þ bs 2 and the estimated accuracy of this expression is about 2% with respect to experimental data [16].
The author is very thankful to Prof. E. Masunov for his advice and his comprehensive guidance during the preparation of author’s MS thesis (1981) at the Moscow Engineering Physics Institute.
References [1] W. Chao, B. Richter, Y. Chi-Yan, Nucl. Instr. and Meth. A 178 (1980) 1. [2] G.A. Loew, R.H. Helm, H.A. Hoig, R.F. Koontz, R.H. Miller, Proceedings of Seventh International Conference On High Energy Charged Particle Accelerators, USSR, Yerevan, Vol. 2, 1970, p. 229. [3] J. Gareyte, Transverse mode coupling instabilities, Preprint CERN-SL-2000-075-AP. [4] W.K.H. Panofsky, W.A. Wenzel, Rev. Sci. Instr. 27 (1956) 967. [5] E.U. Condon, J. Appl. Phys. 11 (1940) 502. [6] L.A. Vainshtein, Electromagnetic waves, Radio i Svyas, Moscow, 1968 (in Russian). [7] P.M. Lapostolle, A.L. Septier (Eds.), Linear Accelerators, North Holland, Amsterdam, 1970. [8] W.K.H. Panofsky, M. Bander, Rev. Sci. Instr. 39 (1968) 206. [9] E.L. Burshtein, G.V. Voskresensky, Linear Electron Accelerators with Intense Beams, Atomizdat, Moscow, USSR, 1970 (in Russian). [10] L. Lewin, Theory of Waveguides, Newnes-Butterworths, London, 1979. [11] G. Saxon, T.R. Jarvis, T. White, Proc. IEE 110 (8) (1963) 1365. [12] Y. Garalt, Advances in Microwaves, Vol. 5, Academic Press, New York, 1970, p. 188. [13] A.V. Smirnov, Beam Interaction with Fundamental and Asymmetric Modes in Tapered Linac Sections, Ph.D. Thesis, Moscow Engineering Physics Institute, Moscow, MEPhI, 1985 (in Russian). [14] R.B. Neal, D.W. Dupen, H.A. Hogg, G.A. Loew, The Stanford Two-Mile Accelerator, Benjamin, New York, 1968. [15] V.K. Neil, L.S. Hall, R.K. Cooper, Part. Accel. 9 (1979) 213. [16] G. Saxon, T.R. Jarvis, T. White, Proc. IEE 110 (8) (1963) 1365. [17] W. Bruns, Proceedings of Particle Accelerator Conference (PAC’97), Vancouver, BC, Canada 12–16 May 1997, p. 2651.
Generalized Panofsky–Wenzel theorem and hybrid coupling A.V. Smirnov*,1 DULY Research Inc., 1912 Mac Arthur St., Rancho Palos Verdes, CA 90275, USA Received 23 January 2001
Abstract The Panofsky–Wenzel theorem is reformulated for the case in which phase slippage between the wave and beam is not negligible. The extended theorem can be applied in analysis of detuned waveguides, RF injectors, bunchers, some tapered waveguides or high-power sources and multi-cell cavities for dipole and higher order modes. As an example, the relative contribution of the Lorentz’ component of the deflecting force is calculated for a conventional circular diskloaded waveguide. # 2001 Elsevier Science B.V. All rights reserved. PACS: 29.27.a; 29.27.Bd; 41.75.i Keywords: Eigenmode; Wakefield; BBU instability; Phase slippage; Synchronism; Hybrid mode; Photoinjector; Multi-cell cavity
Transverse mode coupling causes phase space dilution [1] and instabilities [2,3] that can seriously limit the performance of resonant accelerators. The Panofsky–Wenzel theorem [4] was formulated as a relationship between the transverse and longitudinal wake functions for ultrarelativistic beams and synchronous waves having no phase slippage with respect to the beam. However, in many circumstances there is desynchronism between the beam and at least one partial wave that interacts with the beam. We can encounter this situation, for example, in considering the regenerative beam break-up instability (BBU) in RF injectors, photoinjectors and standing-wave linacs including superconducting ones. In specially detuned waveguides this slippage should be also taken into account. *Corresponding author. Tel.: +1-310-548-0290. E-mail address: alexei [email protected] (A.V. Smirnov). 1 On leave from RRC ‘Kurchatov Institute’, Moscow 123182, Russian Federation
We begin by writing the fields excited by the beam in a multi-cell cavity in the following form: ! ! Er ðr; tÞ ¼ ReCr ðtÞ ðrÞeior t ; Hr mod e r ! ! E 0?r sinðhr zÞ E ?r ; ¼ H ?r iH 0?r cosðhr zÞ ! ! 0 Ezr cosðhr zÞ Ezr ¼ ; 0 iHzr Hzr sinðhr zÞ E H
ð1Þ
where index r corresponds to different cavity modes, including those having different polarization; or ¼ o0r io00r is the modal complex frequency; o00r ¼ o0r =2QrL ; QrL is the loaded Q- factor of the resonator structure for the mode r, and Cr ðtÞ is a slowly varying amplitude that is defined by the beam current (see, e.g., theory of excitation of closed structures established by Condon [5] and developed by Vainshtein [6]).
0168-9002/01/$ - see front matter # 2001 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 0 1 ) 0 0 7 0 8 - 2
22
A.V. Smirnov / Nuclear Instruments and Methods in Physics Research A 469 (2001) 21–23
In the presence of phase slippage effect the transverse motion of the particle is described by a complicated function [7]. This is the typical situation seen in regenerative BBU studies and/or for low-energy beams. Therefore, a description of the detailed transverse dynamics requires one to determine the transverse force F ? ðz; tÞ function along the structure rather than just the transverse wake function w? that is convenient for ‘‘rigid’’, high-energy beams. We can express this force due to the rth mode in terms of longitudinal field components. From the inhomogeneous Maxwell equations and eigenfunction properties directly follows: X g2 F ?r ¼ ReiCr r e 2hr
0 0 1 bbr r? Ezr ðb br Þez r? Hzr Z0
e
iðhr zor tÞ
þ ðb br ÞZ0 j ? ;
ð2Þ
pffiffiffiffiffiffiffiffiffiffiffi where Z0 ¼ mo =eo , b ¼ n=c; br ¼ o0r =chr ; qffiffiffiffiffiffiffiffiffiffiffiffiffi gr ¼ 1= 1 b2r ; ez is unit vector along OZ axis, n is particle velocity, e is particle charge. One can see from Eq. (2), that for relatively short cavities with few-cells the backward wave along with its Lorentz’ component can contribute to the transverse dynamics. For the waveguide mode s described as ! ! E 0s Es ðr; tÞ ¼ Re expðiðhs z otÞÞ; Hs H 0s where E 0s ; H 0s are slowly varying functions of z; t; we obtain analogously (2) F ?s g2 ¼ Rei s e hs 0 0 1 bbs r? Ezs þ ðb bs Þ ez r? Hzs Z0
ð3Þ eiðhs zotÞ þ ðb bs ÞZ0 j ? : In the particular case of a waveguide with a relativistic beam b ! 1; br ! 1 and j ? ¼ 0 this relationship gives (after integration over the structure length) the Panofsky–Wenzel theorem [8] in the frequency domain. Note, in some devices
the transverse component of the current j ? 6¼ 0 plays an important role. Examples include ubitrons, Free Electron Masers and Free Electron Lasers with oversized waveguides. To calculate properly the transverse forces and threshold currents in cases when the (partial) wave phase velocity does not coincide with the particle longitudinal velocity one must know the contribution of the Lorentz force to the net deflection. Let us postulate a special value Xs (it can be a tensor) reflecting hybrid character of HOM: 0 0 ez r? Hzs Z0 ¼ Xs r? Ezs :
ð4Þ
For a disk-loaded cylindrical structure we can apply the studies [9–12]. The fields in the interaction region that are close to synchronism can be presented in the form of a superposition of TE and TM modes: 0 Ezs ¼ As Im ðps rÞcosðmðj þ j0 ÞÞ; 0 ¼ Xs Z01 As Im ðps rÞcosðmjÞÞ; Hzs
ð5Þ
and from Maxwell equations i qE 0 qH 0 0 ¼ 2 k zs hr zs Z0 ; Ejs ps qr qr
ð6Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where m is an angular index, ps ¼ k b2 s 1 is the transverse wavenumber; k ¼ o=c; Im is the modified Bessel function of the first kind of the mth order; j0 is the angular shift of electric wave polarization with respect to the magnetic one: jj0 j5p. To simplify the analysis and to avoid mixing between the different complex numbers reflecting the time dependences and the angular dependences, it is convenient to use four-component complex numbers characterized with imaginary units i2 ¼ 1; j2 ¼ 1; ij 6¼ 1: This scalar algebra of hypercomplex values was used first by Lewin [10] and developed later to apply to BBU analysis [13] in presence of an axial magnetic field. In this new notation we use the fields in the form: ! ! E 0s Es ðr; tÞ ¼ Rei;j expðiðhs z otÞ jmjÞ; Hs H 0s 0 Ezs ¼ Rej As Im ðps rÞexpðjmðj þ j0 ÞÞ; 0 ¼ Xs Z01 As Im ðps rÞexpðjmjÞ; Hzs
ð7Þ
A.V. Smirnov / Nuclear Instruments and Methods in Physics Research A 469 (2001) 21–23
and hence
23
In other, more general, cases this coupling coefficient is a complex value and can be calculated numerically by applying the matching field technique or with 3D codes like GdfidL [17].
i 0 ¼ Rej 2 As Ejs ps jm 0 kXs ps Im ðps rÞ hs Im ðps rÞexpðjmj0 Þ expðjmjÞ: r ð8Þ
To estimate the coupling coefficient Xs we neglect here the azimuthal field penetrating into the space between disks. Strictly speaking it is valid when hs ðD tÞ 1; where D is the period, t is the disk thickness. In this approximation one can apply the boundary condition Ejs r¼a ¼ 0; i.e. the azimuthal electric field is zero on the radius a of iris aperture. Then, we obtain from (8): m 1 Im ðps aÞ : ð9Þ Xs ¼ jexpðjmj0 Þ bs ps a Im0 ðps aÞ Since Xs is, by definition, a real value in j-space of complex numbers, we have jmj0 j ¼ p=2; i.e. the hybrid wave is a sum of TE and TM modes with an angular shift p=2m between their polarization planes. Thus, we confirmed that for every mth asymmetric wave there are always 2m independent hybrid eigenmodes. Extensive experimental ‘‘cold’’ model studies of hybrid dipole eigenmodes were done at SLAC [14]. In particular, these studies indicated that different orientations of polarization are characterized by different eigenfrequencies. Thus, the coupling coefficient between TE and TM modes composing the hybrid mode can be estimated as follows: m 1 Im ðps aÞ Xs ¼ : ð10Þ bs ps a Im0 ðps aÞ The accuracy of this expression is high enough when bs ! 1 (see [12,15]). In particular, for the HEM11 dipole mode we have at ðps aÞ2 1 1 ðps aÞ2 jXs j ¼ 1 1; ð12Þ bs 2 and the estimated accuracy of this expression is about 2% with respect to experimental data [16].
The author is very thankful to Prof. E. Masunov for his advice and his comprehensive guidance during the preparation of author’s MS thesis (1981) at the Moscow Engineering Physics Institute.
References [1] W. Chao, B. Richter, Y. Chi-Yan, Nucl. Instr. and Meth. A 178 (1980) 1. [2] G.A. Loew, R.H. Helm, H.A. Hoig, R.F. Koontz, R.H. Miller, Proceedings of Seventh International Conference On High Energy Charged Particle Accelerators, USSR, Yerevan, Vol. 2, 1970, p. 229. [3] J. Gareyte, Transverse mode coupling instabilities, Preprint CERN-SL-2000-075-AP. [4] W.K.H. Panofsky, W.A. Wenzel, Rev. Sci. Instr. 27 (1956) 967. [5] E.U. Condon, J. Appl. Phys. 11 (1940) 502. [6] L.A. Vainshtein, Electromagnetic waves, Radio i Svyas, Moscow, 1968 (in Russian). [7] P.M. Lapostolle, A.L. Septier (Eds.), Linear Accelerators, North Holland, Amsterdam, 1970. [8] W.K.H. Panofsky, M. Bander, Rev. Sci. Instr. 39 (1968) 206. [9] E.L. Burshtein, G.V. Voskresensky, Linear Electron Accelerators with Intense Beams, Atomizdat, Moscow, USSR, 1970 (in Russian). [10] L. Lewin, Theory of Waveguides, Newnes-Butterworths, London, 1979. [11] G. Saxon, T.R. Jarvis, T. White, Proc. IEE 110 (8) (1963) 1365. [12] Y. Garalt, Advances in Microwaves, Vol. 5, Academic Press, New York, 1970, p. 188. [13] A.V. Smirnov, Beam Interaction with Fundamental and Asymmetric Modes in Tapered Linac Sections, Ph.D. Thesis, Moscow Engineering Physics Institute, Moscow, MEPhI, 1985 (in Russian). [14] R.B. Neal, D.W. Dupen, H.A. Hogg, G.A. Loew, The Stanford Two-Mile Accelerator, Benjamin, New York, 1968. [15] V.K. Neil, L.S. Hall, R.K. Cooper, Part. Accel. 9 (1979) 213. [16] G. Saxon, T.R. Jarvis, T. White, Proc. IEE 110 (8) (1963) 1365. [17] W. Bruns, Proceedings of Particle Accelerator Conference (PAC’97), Vancouver, BC, Canada 12–16 May 1997, p. 2651.