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Waveguide tapering for FEL in the high gain regime
Optics Communications North-Holland
OPTICS COMMUNICATIONS
100 ( 1993) 13 l-1 36
Waveguide tapering for FEL in the high gain regime A. Dipace,
A. Doria,
G.P.
Gallerano
and E. Sabia
ENEA Area INN, Dipartimento Sviluppo Tecnologie di Punta. CRE-Frascati, 00044 Frascati (Rome), Italy Received
8 September
1992; revised manuscript
received
8 February
1993
A free electron laser (FEL) operation in the far infrared or millimeter wave region of the spectrum requires the use of a waveguide to confine the radiation. A new tapering scheme for the operation in the high gain regime, in the amplifier configuration, can thus be utilized in which the waveguide gap is varied while the undulator magnetic field and period are kept constant. The results of a numerical code are presented.
1. Introduction As FEL devices have demonstrated to be reliable sources of coherent radiation [ 11, the problem of finding new schemes with higher efficiency has been faced by theorists. First FELs worked as oscillators; later experiments have been performed in the amplifier configuration [ 2 1. From the theoretical point of view we can distinguish three different working principles. In the low gain regime electron longitudinal motion is determined by a fixed parameter undulator field and a laser field which is practically constant. Therefore gain is simply calculated as energy lost by electrons. The high gain regime is due to a collective instability of the electron beam [3] which results in the exponential growth of radiation until a saturation regime is reached. It is well known, from the theoretical and experimental point of view [ 41, that efficiency is given by the so called Pierce parameter p whose range goes from a few percent in the microwave and millimeter regions, to about 10v3 in the VUV region. A third regime, designed to improve FEL efficiency after the saturation point in the exponential regime has been reached, is the tapered wiggler regime [ 5 1. This latter operating system is founded on the possibility of maintaining the resonance condition for the electrons, while they lose energy n=(n,/2y2)(l+a:)
0030-4018/93/$06.00
(1) 0 1993 Elsevier
Science Publishers
(where il is the laser wavelength, 2” the undulator period, a,= [&A,)/ (fi 2~mc)] and B is the undulator magnetic field) by varying the period or amplitude of the magnetic field. Owing to experimental constraints, practical tapering schemes have been designed by varying only the undulator magnetic field amplitude [ 6 1. Usually high efficiency FEL undulators are composed of two sections: a constant parameter section for trapping of electrons and exponential gain of radiation, and a tapered one for further energy extraction and efficiency enhancement. FEL operation in the far infrared and millimeter wavelength regions requires radiation confinement by a waveguide structure [ 71. In this case the resonance condition ( 1) is slightly modified according to k,+k-WIc=(~/2cy2)(1+u:), to take into account the waveguide dispersion 02= (ck)2+w:o.
(2) relation (3)
It is interesting to observe that in this case we have one additional parameter which can be chosen, i.e. the cut-off frequency w,,. Therefore a new possible tapering scheme exists, in which the waveguide gap is varied to keep electrons resonant as they lose energy through the FEL interaction, while the undulator parameter a, and the period are kept constant.
B.V. All rights reserved.
131
Volume
100, number
1,2,3,4
OPTICS
COMMUNICATIONS
2. Theoretical model The theoretical approach to the tapering scheme relies on the classical KMR formulation [ 83. It is shown there that, by a series of canonical transformation and well verified approximations, the classical hamiltonian of an electron in the combined wiggler and laser fields, expressed in the canonical variables “energy” y and “phase” y, can be cast in the following form HZ
ku+hk
yr
(hJ)“-
7
(cosv+vsiny/,)
,
dyr -=dz
(4)
(5)
’
a,a,w ~ smv, CYr
The resonant particle is kept as representative of the whole beam, around which all electrons of the real beam execute a synchrotron motion in the potential F(W)=-
oa,a, cy r
(cost+
w siny/,) .
(7)
Once wiggler parameters are fixed, eqs. (5) and (6) give the evolution of the resonant parameters: on the other hand it is possible to consider eqs. (5) and (6) as design equations for the tapering scheme, in which the wiggler parameters a, and k, are determined so that the required values for yr, I,v~,a, are achieved. Some points are worth stressing. First of all, changing the wiggler magnetic field is equivalent to changing the laser wavenumber k in the resonance condition (5 ). Because the first scheme has been theoretically and experimentally tested, also the second scheme is expected to work. The second point is that we are faced with a choice in the simulation of the dynamics of the system (laser field + wiggler parameters + e-beam): either we fix, a priori, the form 132
of the wiggler parameters to solve electron and laser field motion equations, or we close our set of equations by some kind of self consistent conditions. Numerical experience has shown that, in the magnetic field tapering scheme, the second approach is much better in that it gives much more power with the same undulator length, or a much reduced undulator length if a given amount of output power is desired. This is especially important in the design of long undulators, where undulator field random errors can cause degradation of FEL performance.
r
where k, is the undulator wave number, o and k are laser field frequency and wavenumber, 6k= k- o/c, a, and a, are dimensionless wiggler field and laser field potentials, 6y= y-yyr, and yr, wr are resonant electron energy and phase. They are defined by the equations w( 1 +a:) I”= 2c(k,+6k)
I July 1993
3. Numerical model A numerical code has been developed to test the hypothesis of efficiency enhancement by tapering the waveguide gap. We have considered a single pass FEL amplifier, well suited for high gain and high extraction efficiency regimes. We have assumed for our model a linear undulator and a rectangular waveguide with major dimension a and minor dimension b. The tapering acts only on the minor dimension h keeping the undulator field amplitude and period constant. The equations for electron and laser field dynamics depend strongly on the form assumed for the laser field. If a general expression E,_=&_(x, y, z, t) is assumed we have to solve a PDE problem for the laser field evolution [ 91. A code so conceived is then complicated and CPU time consuming. Because our aim is simply to show that waveguide tapering, in principle, works we look for a simplified expression for the laser field. A very accurate study on RF field profiles in tapered waveguide and resonators has been performed in ref. [ 10 1. It is shown there that the most general expression for RF field is a superposition of TE and TM modes. Differential equations for mode coupling coefficients are given, and one of the results is that mode coupling is negligible for taper angles of only few degrees as it is shown below for our case. It is therefore appropriate to make the code as simple as possible for a demonstration of the principle, and then to choose the simplest generalization of the TEo,, mode of constant cross section waveguides. Another strong reason to retain this approximation is that even in an overmoded waveguide, where the laser field could be a superposition of more
Volume
100, number
OPTICS
1,2,3,4
than one mode, only very few or even one can be synchronous and amplified. Our ansatz on the fields is then
J
ay,
2 mc2 e=pe,coSb x e
e=O y
0
JZm O,,bpp e sin ~ h,= P 7ce kZo 0 b
,
’
1 July 1993
COMMUNICATIONS
(8)
with a dependence on the longitudinal coordinate z and on the time t of the form exp[i(J k dz-wt)], where b= bo+ EZ, E is a very small parameter and where /I,= ck/o is the group velocity of the mode. It can be verified that, taking into account the smallness of 6, this mode approximately satisfies the dispersion relation: (o/c)*=k(z)*+ [n/b(z)]*. The field equations, written in the paraxial approximation, are obtained by projecting laser field equation on the TEo,i mode giving the following equations [ 9 ] :
where the subscript i refers to the single electron and @ is the electric field phase. The waveguide gap b is self-consistently calculated according to the scheme described in ref. [ 111, i.e. by requiring the resonant particle to be held at constant phase v/r, with the additional constraints of constant undulator period A, and magnetic field b,. Resonant particle equations are therefore [ 121 dy, -=dz
y
C
-+[l+a~+a~-2auaSfBcos~r]+~_ r
de,
_
eZ0
dz -z d$ _=~ dz
dz
(9)
where Z,, is the vacuum beam current,
impedance,
I is the electron
Bessel functions, 8= k,z+Jk(z) dz- ox, the symbol ( ) indicates the particle average and en, e, are the real and imaginary part of the electric field e, of eq. (8 ). In eq. (9 ) we have neglected the time derivative, i.e. we assume a continuous electron beam. As a consequence the electron equations, in the “classical” KMR formalism are [ lo]
J,, are the ordinary
dYi -=dz
de-
leQI4
+@) ,
yihsin(0,
-& =k,+k-
:
sinl//, ,
Wr dz=ku+k-E
21 (sinw) a --fnp “ab Yr
21 e& a -fep 2mc2 “ab
Wr =o _ -=-
fB
’
(cosy) esyr
’ ’
(11)
where e, and C$are the laser electric field and phase, respectively, and a, = e,c/w. The code we have implemented is composed by two main blocks: a tapering part which solves the resonant particle equations ( 11)) evaluates the wavevector k and, as a by product, the gap dimension b by the waveguide dispersion relation (3), and a simulation part. The electron beam is modelled by 4096 macroparticles distributed according to the emittance and energy spread of a real beam. The electron betatron motion equations are not coupled with the other blocks of the code and communicate with them at the end of each integration step to update the electrons transverse position, The simulation part solves the system of equations (8) and (9), by using the Adams-Moulton procedure, to evaluate the dynamics of the electrons and the laser field evolution with the waveguide profile b(z) which dispersion relation (3) has been previously determined by the tapering part.
4. Numerical results
(10)
We have chosen, as test case for our code, the sim133
Volume
100, number
Table 1 Parameters
1,2,3,4
for the numerical
OPTICS
test cases of waveguide-gap
Laser wavelength 1 (mm) Undulator period ,I, (cm) Electron beam energy E (MeV) Relative energy spread AE JE Normalized horizontal emittance c, (mm mrad) Normalized vertical emittance tp (mm mrad) Electron beam current I (A) Horizontal waveguide length a (cm) Vertical waveguide length b (cm) Master oscillator power P, (W)
1July 1993
COMMUNICATIONS
tapering.
Case
I Case 2
1 8 8 10-3 80 n 80 R 300 2 0.5 50
1 8 12 1o-3 80 n 80 rr 300 2 0.35 50
ulation of FEL amplifiers in the millimeter spectral region to compare the calculated efficiency of our FEL scheme with typical efficiency of gyratrons in the same spectral region [ 131. The main parameters of our simulation are summarized in table 1. We have performed simulations with two different
sets of parameters. In the first case (Case 1) the electron beam and waveguide parameters have been chosen to obtain a zero-slippage condition between the electrons and the laser field at the beginning of the interaction, and in Case 2 to make the left hand side of relation (2 ) as small as possible so that higher y is required. The aim of the second simulation is to test the possibility of obtaining higher extraction efficiency from the higher initial electron beam energy. It is important to stress that the code is only a first simulation test. A certain number of simplifying assumptions have been made to keep its structure as simple as possible. (i) The laser mode has been chosen to be the TEo,, mode; this is a good approximation only for a slowly changing waveguide profile. (ii) Only the TEo,i mode has been included in the simulation, to take into account the lack of time variable in our equations, Higher modes have dif-
1(
c
f
7
6
5
4
3
2
6; 0
1
0
1
2
3
4
,,/,,,,~,,~,,,~I~~
,I,
,/,,,,,I,,
,’
,,I,,_
0
2 [Ml Fig. 1. Results of the numerical code for Case 1 of the table 1. (a) Evolution Waveguide gap profile versus the undulator length.
134
of the radiated
power versus the undulator
length.
(b)
1July 1993
OPTICS COMMUNICATIONS
Volume 100, number 1,2,3,4
b [MM1
WV
ld"
/
I
I
4
10
/
(4
10’r 10" r
10'
r
6
10" F
ld 10' r
10"
I”
r
0
2
4
6
6
2
0
'0 _
4
6
44
Fig. 2. Results of the numerical code for Case 2 of the table 1. (a) Evolution Waveguide gap profile versus the undulator length.
ferent phase velocity so that they cannot be properly simulated. (iii) The possibility of combining magnetic field and waveguide tapering to improve efficiency has not been considered. (iv) Space charge effects have been neglected. The results of Case 1 and 2 are shown in figs. 1 and 2, respectively. Figures 1a and 2a show the evolution of the radiated power versus the undulator length, and figs. 1b, 2b the waveguide gap tapering along the undulator. The analogy with the “classical” undulator tapering is evident: the waveguide gap tapering technique increases the amount of energy which can be extracted from the electron beam. To maintain the validity of approximation in the worse case (Case 1) we can stop the gap tapering at a level of 300/6 of variation. It is worth noticing that the proper choice of the cut-off frequency affects the energy extraction efflciency. In Case 1 an efficiency q= 10% can be safely
of the radiated power versus the
0
10
ZIMI undulator length. (b)
reached, while in Case 2 we can obtain an efficiency q z 30%.
5. Conclusions We have implemented a numerical code to simulate the behaviour of an FEL amplifier in the millimeter region as a first check to the hypothesis of extraction efficiency enhancement by tapering the waveguide profile. The code is only a first simulation test and although predicted efficiency can be optimistic, results clearly show that the principle of waveguide tapering works well. Further theoretical and experimental investigations are worthwhile to be performed to achieve practical schemes of very high efficiency for FELs in the millimeter region.
135
Volume 100, number
1,2,3,4
OPTICS
COMMUNICATIONS
References [l] D.A.G. Deacon, L.R. Elias, J.M.J. Madey, G.J. Ramian, H.A. Schwettman and T.I. Smith, Phys. Rev. Lett. 38 ( 1977) 892. [ 21 T.J. Orzechowski. B.R. Anderson, W.M. Fawley, D. Prosnitz, E.G. Sharlemann and S.M. Yarema, Nucl. Instr. Meth. A 250 (1986) 144; J.M. Watson, Nucl. Instr. Meth. A 250 ( 1986) 1. [ 31 R. Bonifacio, C. Pellegrini and L. Narducci, Optics Comm. 50 (1984) 373. [4] R. Bonifacio, F. Casagrande, G. Cerchioni, L. De Salvo Souza, P. Pierini and N. Piovella, Riv. Nuovo Cimento 13 (1990) 9. [ 5 ] D. Prosnitz, A. Szoke and V.K. Niel, Phys. Rev. A 24 ( 198 1) 1436. [ 61 T.J. Orzechowski, B.R. Anderson, J.C. Clark, W.M. Fawley, A.C. Paul, D. Prosnitz, E.T. Sharlemann, S.M. Yarema, D. Hopkins, A.M. Sessler and J.S. Wurtele, Phys. Rev. Lett. 57 (1986) 2172.
136
1July 1993
[ 71 A. Doria, G.P. Gallerano and A. Renieri, Optics Comm. 80 (1991) 417. [8] N.M. Kroll, L.P. Morton and M.N. Rosenbluth, IEEE J. Quantum Electron. QE17 (1981) 1436. [9] W.M. Fawley, D. Prosnitz and E.T. Sharlemann, Phys. Rev. A 30 ( 1984) 2472. [lo] E. Borie and 0. Dumbrajs, Int. J. Elect. 60 (1986) 143; A.W. Fliflet and M.R. Read, Int. J. Elect. 5 1 ( 198 1) 475. [ 111 A. Dipace, A. Doria, G.P. Gallerano and E. Sabia, in: Proc. European Particle Accelerators Conference (EPAC92), eds. H. Henke, H. Homeyer and Ch. Petit-Jean-Genaz, Berlin March 1992, p. 614. [ 121 T.J. Orzechowski, E.T. Sharlemann, B. Anderson, V.K. Neil, W.M. Fawley, D. Prosnitz, S.M. Yarema, D.B. Hopkins, A.C. Paul, A.M. Sessler and J.S. Wurtele, IEEE J. Quantum Electron. QE 21 (1985) 831. [ 131 K.E. Kreisher and R.J. Temkin, Phys. Rev. Lett. 59 (1987) 547.
OPTICS COMMUNICATIONS
100 ( 1993) 13 l-1 36
Waveguide tapering for FEL in the high gain regime A. Dipace,
A. Doria,
G.P.
Gallerano
and E. Sabia
ENEA Area INN, Dipartimento Sviluppo Tecnologie di Punta. CRE-Frascati, 00044 Frascati (Rome), Italy Received
8 September
1992; revised manuscript
received
8 February
1993
A free electron laser (FEL) operation in the far infrared or millimeter wave region of the spectrum requires the use of a waveguide to confine the radiation. A new tapering scheme for the operation in the high gain regime, in the amplifier configuration, can thus be utilized in which the waveguide gap is varied while the undulator magnetic field and period are kept constant. The results of a numerical code are presented.
1. Introduction As FEL devices have demonstrated to be reliable sources of coherent radiation [ 11, the problem of finding new schemes with higher efficiency has been faced by theorists. First FELs worked as oscillators; later experiments have been performed in the amplifier configuration [ 2 1. From the theoretical point of view we can distinguish three different working principles. In the low gain regime electron longitudinal motion is determined by a fixed parameter undulator field and a laser field which is practically constant. Therefore gain is simply calculated as energy lost by electrons. The high gain regime is due to a collective instability of the electron beam [3] which results in the exponential growth of radiation until a saturation regime is reached. It is well known, from the theoretical and experimental point of view [ 41, that efficiency is given by the so called Pierce parameter p whose range goes from a few percent in the microwave and millimeter regions, to about 10v3 in the VUV region. A third regime, designed to improve FEL efficiency after the saturation point in the exponential regime has been reached, is the tapered wiggler regime [ 5 1. This latter operating system is founded on the possibility of maintaining the resonance condition for the electrons, while they lose energy n=(n,/2y2)(l+a:)
0030-4018/93/$06.00
(1) 0 1993 Elsevier
Science Publishers
(where il is the laser wavelength, 2” the undulator period, a,= [&A,)/ (fi 2~mc)] and B is the undulator magnetic field) by varying the period or amplitude of the magnetic field. Owing to experimental constraints, practical tapering schemes have been designed by varying only the undulator magnetic field amplitude [ 6 1. Usually high efficiency FEL undulators are composed of two sections: a constant parameter section for trapping of electrons and exponential gain of radiation, and a tapered one for further energy extraction and efficiency enhancement. FEL operation in the far infrared and millimeter wavelength regions requires radiation confinement by a waveguide structure [ 71. In this case the resonance condition ( 1) is slightly modified according to k,+k-WIc=(~/2cy2)(1+u:), to take into account the waveguide dispersion 02= (ck)2+w:o.
(2) relation (3)
It is interesting to observe that in this case we have one additional parameter which can be chosen, i.e. the cut-off frequency w,,. Therefore a new possible tapering scheme exists, in which the waveguide gap is varied to keep electrons resonant as they lose energy through the FEL interaction, while the undulator parameter a, and the period are kept constant.
B.V. All rights reserved.
131
Volume
100, number
1,2,3,4
OPTICS
COMMUNICATIONS
2. Theoretical model The theoretical approach to the tapering scheme relies on the classical KMR formulation [ 83. It is shown there that, by a series of canonical transformation and well verified approximations, the classical hamiltonian of an electron in the combined wiggler and laser fields, expressed in the canonical variables “energy” y and “phase” y, can be cast in the following form HZ
ku+hk
yr
(hJ)“-
7
(cosv+vsiny/,)
,
dyr -=dz
(4)
(5)
’
a,a,w ~ smv, CYr
The resonant particle is kept as representative of the whole beam, around which all electrons of the real beam execute a synchrotron motion in the potential F(W)=-
oa,a, cy r
(cost+
w siny/,) .
(7)
Once wiggler parameters are fixed, eqs. (5) and (6) give the evolution of the resonant parameters: on the other hand it is possible to consider eqs. (5) and (6) as design equations for the tapering scheme, in which the wiggler parameters a, and k, are determined so that the required values for yr, I,v~,a, are achieved. Some points are worth stressing. First of all, changing the wiggler magnetic field is equivalent to changing the laser wavenumber k in the resonance condition (5 ). Because the first scheme has been theoretically and experimentally tested, also the second scheme is expected to work. The second point is that we are faced with a choice in the simulation of the dynamics of the system (laser field + wiggler parameters + e-beam): either we fix, a priori, the form 132
of the wiggler parameters to solve electron and laser field motion equations, or we close our set of equations by some kind of self consistent conditions. Numerical experience has shown that, in the magnetic field tapering scheme, the second approach is much better in that it gives much more power with the same undulator length, or a much reduced undulator length if a given amount of output power is desired. This is especially important in the design of long undulators, where undulator field random errors can cause degradation of FEL performance.
r
where k, is the undulator wave number, o and k are laser field frequency and wavenumber, 6k= k- o/c, a, and a, are dimensionless wiggler field and laser field potentials, 6y= y-yyr, and yr, wr are resonant electron energy and phase. They are defined by the equations w( 1 +a:) I”= 2c(k,+6k)
I July 1993
3. Numerical model A numerical code has been developed to test the hypothesis of efficiency enhancement by tapering the waveguide gap. We have considered a single pass FEL amplifier, well suited for high gain and high extraction efficiency regimes. We have assumed for our model a linear undulator and a rectangular waveguide with major dimension a and minor dimension b. The tapering acts only on the minor dimension h keeping the undulator field amplitude and period constant. The equations for electron and laser field dynamics depend strongly on the form assumed for the laser field. If a general expression E,_=&_(x, y, z, t) is assumed we have to solve a PDE problem for the laser field evolution [ 91. A code so conceived is then complicated and CPU time consuming. Because our aim is simply to show that waveguide tapering, in principle, works we look for a simplified expression for the laser field. A very accurate study on RF field profiles in tapered waveguide and resonators has been performed in ref. [ 10 1. It is shown there that the most general expression for RF field is a superposition of TE and TM modes. Differential equations for mode coupling coefficients are given, and one of the results is that mode coupling is negligible for taper angles of only few degrees as it is shown below for our case. It is therefore appropriate to make the code as simple as possible for a demonstration of the principle, and then to choose the simplest generalization of the TEo,, mode of constant cross section waveguides. Another strong reason to retain this approximation is that even in an overmoded waveguide, where the laser field could be a superposition of more
Volume
100, number
OPTICS
1,2,3,4
than one mode, only very few or even one can be synchronous and amplified. Our ansatz on the fields is then
J
ay,
2 mc2 e=pe,coSb x e
e=O y
0
JZm O,,bpp e sin ~ h,= P 7ce kZo 0 b
,
’
1 July 1993
COMMUNICATIONS
(8)
with a dependence on the longitudinal coordinate z and on the time t of the form exp[i(J k dz-wt)], where b= bo+ EZ, E is a very small parameter and where /I,= ck/o is the group velocity of the mode. It can be verified that, taking into account the smallness of 6, this mode approximately satisfies the dispersion relation: (o/c)*=k(z)*+ [n/b(z)]*. The field equations, written in the paraxial approximation, are obtained by projecting laser field equation on the TEo,i mode giving the following equations [ 9 ] :
where the subscript i refers to the single electron and @ is the electric field phase. The waveguide gap b is self-consistently calculated according to the scheme described in ref. [ 111, i.e. by requiring the resonant particle to be held at constant phase v/r, with the additional constraints of constant undulator period A, and magnetic field b,. Resonant particle equations are therefore [ 121 dy, -=dz
y
C
-+[l+a~+a~-2auaSfBcos~r]+~_ r
de,
_
eZ0
dz -z d$ _=~ dz
dz
(9)
where Z,, is the vacuum beam current,
impedance,
I is the electron
Bessel functions, 8= k,z+Jk(z) dz- ox, the symbol ( ) indicates the particle average and en, e, are the real and imaginary part of the electric field e, of eq. (8 ). In eq. (9 ) we have neglected the time derivative, i.e. we assume a continuous electron beam. As a consequence the electron equations, in the “classical” KMR formalism are [ lo]
J,, are the ordinary
dYi -=dz
de-
leQI4
+@) ,
yihsin(0,
-& =k,+k-
:
sinl//, ,
Wr dz=ku+k-E
21 (sinw) a --fnp “ab Yr
21 e& a -fep 2mc2 “ab
Wr =o _ -=-
fB
’
(cosy) esyr
’ ’
(11)
where e, and C$are the laser electric field and phase, respectively, and a, = e,c/w. The code we have implemented is composed by two main blocks: a tapering part which solves the resonant particle equations ( 11)) evaluates the wavevector k and, as a by product, the gap dimension b by the waveguide dispersion relation (3), and a simulation part. The electron beam is modelled by 4096 macroparticles distributed according to the emittance and energy spread of a real beam. The electron betatron motion equations are not coupled with the other blocks of the code and communicate with them at the end of each integration step to update the electrons transverse position, The simulation part solves the system of equations (8) and (9), by using the Adams-Moulton procedure, to evaluate the dynamics of the electrons and the laser field evolution with the waveguide profile b(z) which dispersion relation (3) has been previously determined by the tapering part.
4. Numerical results
(10)
We have chosen, as test case for our code, the sim133
Volume
100, number
Table 1 Parameters
1,2,3,4
for the numerical
OPTICS
test cases of waveguide-gap
Laser wavelength 1 (mm) Undulator period ,I, (cm) Electron beam energy E (MeV) Relative energy spread AE JE Normalized horizontal emittance c, (mm mrad) Normalized vertical emittance tp (mm mrad) Electron beam current I (A) Horizontal waveguide length a (cm) Vertical waveguide length b (cm) Master oscillator power P, (W)
1July 1993
COMMUNICATIONS
tapering.
Case
I Case 2
1 8 8 10-3 80 n 80 R 300 2 0.5 50
1 8 12 1o-3 80 n 80 rr 300 2 0.35 50
ulation of FEL amplifiers in the millimeter spectral region to compare the calculated efficiency of our FEL scheme with typical efficiency of gyratrons in the same spectral region [ 131. The main parameters of our simulation are summarized in table 1. We have performed simulations with two different
sets of parameters. In the first case (Case 1) the electron beam and waveguide parameters have been chosen to obtain a zero-slippage condition between the electrons and the laser field at the beginning of the interaction, and in Case 2 to make the left hand side of relation (2 ) as small as possible so that higher y is required. The aim of the second simulation is to test the possibility of obtaining higher extraction efficiency from the higher initial electron beam energy. It is important to stress that the code is only a first simulation test. A certain number of simplifying assumptions have been made to keep its structure as simple as possible. (i) The laser mode has been chosen to be the TEo,, mode; this is a good approximation only for a slowly changing waveguide profile. (ii) Only the TEo,i mode has been included in the simulation, to take into account the lack of time variable in our equations, Higher modes have dif-
1(
c
f
7
6
5
4
3
2
6; 0
1
0
1
2
3
4
,,/,,,,~,,~,,,~I~~
,I,
,/,,,,,I,,
,’
,,I,,_
0
2 [Ml Fig. 1. Results of the numerical code for Case 1 of the table 1. (a) Evolution Waveguide gap profile versus the undulator length.
134
of the radiated
power versus the undulator
length.
(b)
1July 1993
OPTICS COMMUNICATIONS
Volume 100, number 1,2,3,4
b [MM1
WV
ld"
/
I
I
4
10
/
(4
10’r 10" r
10'
r
6
10" F
ld 10' r
10"
I”
r
0
2
4
6
6
2
0
'0 _
4
6
44
Fig. 2. Results of the numerical code for Case 2 of the table 1. (a) Evolution Waveguide gap profile versus the undulator length.
ferent phase velocity so that they cannot be properly simulated. (iii) The possibility of combining magnetic field and waveguide tapering to improve efficiency has not been considered. (iv) Space charge effects have been neglected. The results of Case 1 and 2 are shown in figs. 1 and 2, respectively. Figures 1a and 2a show the evolution of the radiated power versus the undulator length, and figs. 1b, 2b the waveguide gap tapering along the undulator. The analogy with the “classical” undulator tapering is evident: the waveguide gap tapering technique increases the amount of energy which can be extracted from the electron beam. To maintain the validity of approximation in the worse case (Case 1) we can stop the gap tapering at a level of 300/6 of variation. It is worth noticing that the proper choice of the cut-off frequency affects the energy extraction efflciency. In Case 1 an efficiency q= 10% can be safely
of the radiated power versus the
0
10
ZIMI undulator length. (b)
reached, while in Case 2 we can obtain an efficiency q z 30%.
5. Conclusions We have implemented a numerical code to simulate the behaviour of an FEL amplifier in the millimeter region as a first check to the hypothesis of extraction efficiency enhancement by tapering the waveguide profile. The code is only a first simulation test and although predicted efficiency can be optimistic, results clearly show that the principle of waveguide tapering works well. Further theoretical and experimental investigations are worthwhile to be performed to achieve practical schemes of very high efficiency for FELs in the millimeter region.
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References [l] D.A.G. Deacon, L.R. Elias, J.M.J. Madey, G.J. Ramian, H.A. Schwettman and T.I. Smith, Phys. Rev. Lett. 38 ( 1977) 892. [ 21 T.J. Orzechowski. B.R. Anderson, W.M. Fawley, D. Prosnitz, E.G. Sharlemann and S.M. Yarema, Nucl. Instr. Meth. A 250 (1986) 144; J.M. Watson, Nucl. Instr. Meth. A 250 ( 1986) 1. [ 31 R. Bonifacio, C. Pellegrini and L. Narducci, Optics Comm. 50 (1984) 373. [4] R. Bonifacio, F. Casagrande, G. Cerchioni, L. De Salvo Souza, P. Pierini and N. Piovella, Riv. Nuovo Cimento 13 (1990) 9. [ 5 ] D. Prosnitz, A. Szoke and V.K. Niel, Phys. Rev. A 24 ( 198 1) 1436. [ 61 T.J. Orzechowski, B.R. Anderson, J.C. Clark, W.M. Fawley, A.C. Paul, D. Prosnitz, E.T. Sharlemann, S.M. Yarema, D. Hopkins, A.M. Sessler and J.S. Wurtele, Phys. Rev. Lett. 57 (1986) 2172.
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[ 71 A. Doria, G.P. Gallerano and A. Renieri, Optics Comm. 80 (1991) 417. [8] N.M. Kroll, L.P. Morton and M.N. Rosenbluth, IEEE J. Quantum Electron. QE17 (1981) 1436. [9] W.M. Fawley, D. Prosnitz and E.T. Sharlemann, Phys. Rev. A 30 ( 1984) 2472. [lo] E. Borie and 0. Dumbrajs, Int. J. Elect. 60 (1986) 143; A.W. Fliflet and M.R. Read, Int. J. Elect. 5 1 ( 198 1) 475. [ 111 A. Dipace, A. Doria, G.P. Gallerano and E. Sabia, in: Proc. European Particle Accelerators Conference (EPAC92), eds. H. Henke, H. Homeyer and Ch. Petit-Jean-Genaz, Berlin March 1992, p. 614. [ 121 T.J. Orzechowski, E.T. Sharlemann, B. Anderson, V.K. Neil, W.M. Fawley, D. Prosnitz, S.M. Yarema, D.B. Hopkins, A.C. Paul, A.M. Sessler and J.S. Wurtele, IEEE J. Quantum Electron. QE 21 (1985) 831. [ 131 K.E. Kreisher and R.J. Temkin, Phys. Rev. Lett. 59 (1987) 547.