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Numerical analysis of the open resonator for an eel
445
Nuclear Instruments and Methods in Physics Research A282 (1989) 445-447 North-Holland, Amsterdam
NUMERICAL ANALYSIS OF THE OPEN RESONATOR FOR AN FEL A.I. KLEEV and A.V . TIHOMIROV Institute for Physical Problems of the USSR Academy of Sciences, 117334, Moscow, ul. Kosygma 2, USSR
Integral equations are used to calculate the fields and losses m the open waveguide resonator used in FELS . The parameters of the interaction of the electron beam with the field of the resonator mode are computed .
1. Introduction For the correct analysis of the open resonator for a free-electron laser (FEL), the effect of the metallic electron guide placed between the mirrors (see fig. 1) must be taken into consideration. The observation of
the modes of the resonator for a millimeter-wave FEL in the case when the mirrors are attached to the electron
guide is given in refs . [1,2]. This paper describes the more general situation when the mirrors are positioned
at some distance from the ends of the electron guide. This system is used in the optical region due to its high mode selectivity.
2 . Theory
We consider all dimensions of the resonator to be much more important than the wavelength X, so that a parabolic equation can be used. The field in the circular electron guide, made of conductive materials with conductivity a, is defined by the z-components of the Hertz electric and magnetic vectors IIz and rim which we approximate by the sum of the waveguide modes : IIZ 17m=
2N-1
n=0
where t=r/a, O=z/D, an = Jl(gn)> Nn = J(gn)lgnl A = D/ka 2, k = 2ir/X (we assume a harmonic time variation exp ( - t w T) with w = kv, where v is the velocity of light) . The constants gn are the roots of the equation 2 ( ( ) +1W.T1 -tW J (gn)[gnJl(gn) (9n)] f1 g n ) 2
c
w=ka
which is derived from the boundary condition . Using the symmetry of the structure, the problem of finding the resonator modes can be reduced to the determination of the scalar functions, which are the solution of the system of integral equations :
f01 Km(t , 1 ) 0m(t, 1 )tl
Y_ AnPnJ(gnt) cos $ exp(ikz-ig,22a0), n=0
1
dt1 = Y0m(t, -1),
m=0,2
where K,n(t, t1)=
1+cgJm(1
+g ttll
m XexpI-i2(m+1)+
`4tt«nJl(gnt) sin (p exp(1kz - lgnde) .
2N-1
1/2 w 4Tria
icg (t 2 +t~ 1+ g
c = ka 2/2l, g = 1 - 2l/R and for the unknown func-
tions we propose the following form : 2N-1
n=0
2N-1
1
n=0
BnJo(gnt) exP( - lgnde), BnYn J2 (gn t ) exp(-ig,2,4B),
Btt_ 2gnAn(an-tBn), R Fig . 1 . Geometry of the resonator of an FEL . 0168-9002/89/$03 .50 © Elsevier Science Publishers B .V . (North-Holland Physics Publishing Division)
Yn =
an + & . an-Nn
The same method was used to analyse a resonator with II . FELS AND UNDULATORS
446
A . I Kleev, A . V Tihomtrov / Numerical analysts of an open resonator
a plane electron guide: in this case, the initial problem can be reduced to the determination of one scalar function . To reduce eq . (3) to an algebraic problem, we used the collocation technique. Substituting eq . (5) into eq . (3) and demanding that both parts of eq . (3) are equal in the collocation points t, we get the algebraic eigenvalue problem. As the calculation showed, the optimum distribution of collocation points is the equidistant one. In this case the eigenvalue becomes stable when N in eq . (1) increases. If the alternative distribution of collocation points is used, the eigenvalue calculation procedure may diverge. When calculating the interaction of the field with the electron beam, we assume the width of the beam to be small compared to the other dimensions . In this case, using a linear approach, we can calculate the interaction coefficient, which is given by the following formula: 2
S=f~ 1Ey(t,0)1dt/ fi IEy (0,0)d0 .
(6)
The interaction power is proportional to S- ' . We calculated the normalised coefficient S2 = So/S, where So is the value of the interaction coefficient for the resonator without an electron guide. 3. Results The dependence of the characteristics of the resonator on the parameter d is the most important for the optimisation of its geometry. Fig. 2a shows the dependence of -201g I y I on 4, calculated for a resonator
0 04 0.2 Fig. 2. Losses (a) and S2 (b) as a function of the parameter 4 (c = Tr, g = 0.75) .
20
é1
2 Re w Fig. 3. Losses in the resonator with a planar electron guide versus Re w (c = m, g = 0.75, d = 0.01) . See text for the curve numbers. o.
!
with an ideal conductive circular electron guide (w = 0) . Fig. 2b gives the dependence of 2 on ~A, calculated for a resonator with the same parameters as used for fig. 2a . The gaps on the curve illustrating the dependence of 2 on 4 do not coincide with the regions of minimum losses . This fact gives the possibility to choose the range of 4 for practical use. All these results were given for a resonator with an ideal conductive electron guide. In the case where the parameter w in eq . (2) is large, the finite conductivity of the material of the electron guide is important not only for the magnitude of the losses, but for the distribution of the field as well, and, therefore, for the interaction coefficient with the electron beam . Fig. 3 gives the dependence of the losses in the resonator in the plane of the electron guide on Re w . Curve 1 was calculated with the help of the integral equation, and curve 2 by the perturbation technique. The current in the electron guide walls was calculated for the solution of the problem for an ideal conductive structure. The losses of power were defined as the amount of heat produced by the current in the material with finite resistance . Curve 3 presented the losses for the main TEM mode . The influence of the finite resistance on the field distribution is presented in fig. 4, which gives the I E, I
Fig. 4. Field distribution m the electron guide. See text for the curve numbers.
A. I. Kleev, A. V. Tihomtrov / Numerical analysis of an open resonator
distribution for waves propagating in the positive z-direction. The dependence was obtained for B = 1 . The calculations were performed for Re w = 2 (curve 1) and for an ideal conductive electron guide (curve 2) .
447
Acknowledgements The authors would like to express their sincere thanks to S.P . Kapitza, L.A. Wainstein, G.D . Bogomolov and V.V . Zavialov for useful discussions and help .
4. Conclusion The presence of an electron guide placed between the mirrors of the resonator leads to a considerable change in the field distribution due to the difference of the phase velocities of the waveguide modes. The proposed method allows one to calculate the main characteristics of the open resonator with an electron guide placed between the mirrors and can be used for optimisation of the structure.
References [1] L.R . Ehas and J.C . Gallardo, Appl . Phys . B31 (1983) 229. [2] L.R . Elias, G. Ramian, J. Hu and A. Amir, Phys . Rev. Lett. 57 (1986) 424. [3] A.I . Kleev and A.B . Manenkov, Radiotechnica e Electromka 33 (7) (1988) 1387 (m Russian) .
II . FELS AND UNDULATORS
Nuclear Instruments and Methods in Physics Research A282 (1989) 445-447 North-Holland, Amsterdam
NUMERICAL ANALYSIS OF THE OPEN RESONATOR FOR AN FEL A.I. KLEEV and A.V . TIHOMIROV Institute for Physical Problems of the USSR Academy of Sciences, 117334, Moscow, ul. Kosygma 2, USSR
Integral equations are used to calculate the fields and losses m the open waveguide resonator used in FELS . The parameters of the interaction of the electron beam with the field of the resonator mode are computed .
1. Introduction For the correct analysis of the open resonator for a free-electron laser (FEL), the effect of the metallic electron guide placed between the mirrors (see fig. 1) must be taken into consideration. The observation of
the modes of the resonator for a millimeter-wave FEL in the case when the mirrors are attached to the electron
guide is given in refs . [1,2]. This paper describes the more general situation when the mirrors are positioned
at some distance from the ends of the electron guide. This system is used in the optical region due to its high mode selectivity.
2 . Theory
We consider all dimensions of the resonator to be much more important than the wavelength X, so that a parabolic equation can be used. The field in the circular electron guide, made of conductive materials with conductivity a, is defined by the z-components of the Hertz electric and magnetic vectors IIz and rim which we approximate by the sum of the waveguide modes : IIZ 17m=
2N-1
n=0
where t=r/a, O=z/D, an = Jl(gn)> Nn = J(gn)lgnl A = D/ka 2, k = 2ir/X (we assume a harmonic time variation exp ( - t w T) with w = kv, where v is the velocity of light) . The constants gn are the roots of the equation 2 ( ( ) +1W.T1 -tW J (gn)[gnJl(gn) (9n)] f1 g n ) 2
c
w=ka
which is derived from the boundary condition . Using the symmetry of the structure, the problem of finding the resonator modes can be reduced to the determination of the scalar functions, which are the solution of the system of integral equations :
f01 Km(t , 1 ) 0m(t, 1 )tl
Y_ AnPnJ(gnt) cos $ exp(ikz-ig,22a0), n=0
1
dt1 = Y0m(t, -1),
m=0,2
where K,n(t, t1)=
1+cgJm(1
+g ttll
m XexpI-i2(m+1)+
`4tt«nJl(gnt) sin (p exp(1kz - lgnde) .
2N-1
1/2 w 4Tria
icg (t 2 +t~ 1+ g
c = ka 2/2l, g = 1 - 2l/R and for the unknown func-
tions we propose the following form : 2N-1
n=0
2N-1
1
n=0
BnJo(gnt) exP( - lgnde), BnYn J2 (gn t ) exp(-ig,2,4B),
Btt_ 2gnAn(an-tBn), R Fig . 1 . Geometry of the resonator of an FEL . 0168-9002/89/$03 .50 © Elsevier Science Publishers B .V . (North-Holland Physics Publishing Division)
Yn =
an + & . an-Nn
The same method was used to analyse a resonator with II . FELS AND UNDULATORS
446
A . I Kleev, A . V Tihomtrov / Numerical analysts of an open resonator
a plane electron guide: in this case, the initial problem can be reduced to the determination of one scalar function . To reduce eq . (3) to an algebraic problem, we used the collocation technique. Substituting eq . (5) into eq . (3) and demanding that both parts of eq . (3) are equal in the collocation points t, we get the algebraic eigenvalue problem. As the calculation showed, the optimum distribution of collocation points is the equidistant one. In this case the eigenvalue becomes stable when N in eq . (1) increases. If the alternative distribution of collocation points is used, the eigenvalue calculation procedure may diverge. When calculating the interaction of the field with the electron beam, we assume the width of the beam to be small compared to the other dimensions . In this case, using a linear approach, we can calculate the interaction coefficient, which is given by the following formula: 2
S=f~ 1Ey(t,0)1dt/ fi IEy (0,0)d0 .
(6)
The interaction power is proportional to S- ' . We calculated the normalised coefficient S2 = So/S, where So is the value of the interaction coefficient for the resonator without an electron guide. 3. Results The dependence of the characteristics of the resonator on the parameter d is the most important for the optimisation of its geometry. Fig. 2a shows the dependence of -201g I y I on 4, calculated for a resonator
0 04 0.2 Fig. 2. Losses (a) and S2 (b) as a function of the parameter 4 (c = Tr, g = 0.75) .
20
é1
2 Re w Fig. 3. Losses in the resonator with a planar electron guide versus Re w (c = m, g = 0.75, d = 0.01) . See text for the curve numbers. o.
!
with an ideal conductive circular electron guide (w = 0) . Fig. 2b gives the dependence of 2 on ~A, calculated for a resonator with the same parameters as used for fig. 2a . The gaps on the curve illustrating the dependence of 2 on 4 do not coincide with the regions of minimum losses . This fact gives the possibility to choose the range of 4 for practical use. All these results were given for a resonator with an ideal conductive electron guide. In the case where the parameter w in eq . (2) is large, the finite conductivity of the material of the electron guide is important not only for the magnitude of the losses, but for the distribution of the field as well, and, therefore, for the interaction coefficient with the electron beam . Fig. 3 gives the dependence of the losses in the resonator in the plane of the electron guide on Re w . Curve 1 was calculated with the help of the integral equation, and curve 2 by the perturbation technique. The current in the electron guide walls was calculated for the solution of the problem for an ideal conductive structure. The losses of power were defined as the amount of heat produced by the current in the material with finite resistance . Curve 3 presented the losses for the main TEM mode . The influence of the finite resistance on the field distribution is presented in fig. 4, which gives the I E, I
Fig. 4. Field distribution m the electron guide. See text for the curve numbers.
A. I. Kleev, A. V. Tihomtrov / Numerical analysis of an open resonator
distribution for waves propagating in the positive z-direction. The dependence was obtained for B = 1 . The calculations were performed for Re w = 2 (curve 1) and for an ideal conductive electron guide (curve 2) .
447
Acknowledgements The authors would like to express their sincere thanks to S.P . Kapitza, L.A. Wainstein, G.D . Bogomolov and V.V . Zavialov for useful discussions and help .
4. Conclusion The presence of an electron guide placed between the mirrors of the resonator leads to a considerable change in the field distribution due to the difference of the phase velocities of the waveguide modes. The proposed method allows one to calculate the main characteristics of the open resonator with an electron guide placed between the mirrors and can be used for optimisation of the structure.
References [1] L.R . Ehas and J.C . Gallardo, Appl . Phys . B31 (1983) 229. [2] L.R . Elias, G. Ramian, J. Hu and A. Amir, Phys . Rev. Lett. 57 (1986) 424. [3] A.I . Kleev and A.B . Manenkov, Radiotechnica e Electromka 33 (7) (1988) 1387 (m Russian) .
II . FELS AND UNDULATORS