- Email: [email protected]
Superradiance driven by coherent spontaneous emission in the Cherenkov maser
15 May 1999
Optics Communications 163 Ž1999. 203–207
Superradiance driven by coherent spontaneous emission in the Cherenkov maser B.W.J. McNeil, G.R.M. Robb, D.A. Jaroszynski Department of Physics & Applied Physics, UniÕersity of Strathclyde, Glasgow, G4 0NG, UK Received 21 December 1998; accepted 8 March 1999
Abstract An analysis of the Cherenkov maser instability operating with electron pulses is carried out. A model which can describe the effects of Coherent Spontaneous Emission ŽCSE. is derived. The results from this model are compared with previous work which neglect the effects of CSE. The model suggests that CSE can drive and significantly enhance superradiant emission from the electron pulses, offering a potential source of high intensity, short pulse radiation. This mechanism should also be present in other devices such as the Free Electron Laser. q 1999 Published by Elsevier Science B.V. All rights reserved.
1. Introduction The emission of radiation from ensembles of free electrons forming an electron beam, or pulse, interacting with both static andror electromagnetic fields, forms the basis of many successful sources of high power coherent, electromagnetic radiation, such as the cyclotron resonance maser, the free electron laser ŽFEL. and the Cherenkov maser. Superradiant emission from these sources, when driven by electron pulses, has been the subject of much recent interest w1–9x. Such emission is of practical interest as a means of generating very short, intense, pulses of radiation. These pulses have many potential industrial and academic applications. In the microwave to mm-wavelength range, they are of considerable interest for use, for example, as plasma probes and in high-gradient particle acceleration. At shorter wavelengths they would have applications in such subjects as nonlinear spectroscopy, and studies of chemical and biological systems. The nonlinear interaction between an electron pulse and electromagnetic wave in these superradiant amplifiers may be initiated by ‘shot noise’ emission, due to the random position of electrons within the electron pulse, or by a small resonant seed field injected into the interaction region. When the radiation develops in an amplifier from noise, the process has been called self-amplified sponta-
neous emission ŽSASE.. However, another emission initiation process can occur called ‘coherent spontaneous emission’ ŽCSE., arising from the initial current profile, or ‘shape’, of the electron pulse. This has been the subject of a number of recent studies w10,11x. Emission of intensities several orders of magnitude greater than that due to shot noise alone may occur when the current profile of the electron pulse changes significantly on the scale of a radiation wavelength. In this communication, we present an analysis of the Cherenkov maser amplifier initiated by CSE and make comparison with a previous method of analysis where CSE is absent. We find that when CSE effects are included, the CSE acts as a substantial seed from which a high-intensity superradiant radiation pulse rapidly grows. The growth of these pulses is enhanced over that due to the presence of only shot noise or a c.w. seed field. Our results suggest that the generation of superradiant radiation pulses may be greatly enhanced by suitable ‘shaping’ of the electron pulse. The similarity between the Cherenkov interaction with other free-electron interactions suggests that CSE driven superradiance will also be present in other devices such as the free electron laser. This may have important implications for FELs operating in the SASE regime. These consequences will be discussed in a subsequent publication.
0030-4018r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 9 . 0 0 1 3 5 - 2
204
B.W.J. McNeil et al.r Optics Communications 163 (1999) 203–207
2. Theory In this analysis of the Cherenkov maser, the evolution of the interaction between the electron pulse and the radiation field is described using a set of coupled non-linear partial differential equations. The electrons’ positions and momenta evolve under the action of the electromagnetic fields of the waveguide mode as described by the Lorentz force equation. The electron motion constitutes a current which acts as a source of the electromagnetic field as described by Maxwell’s wave equation. To date, most numerical studies of the Cherenkov maser have involved the averaging and discretisation of the wave equation over an interval equal to or greater than a radiation period w12–14x. This restricts the description of the field evolution to situations where the current profile of the electron pulse changes negligibly on the scale of the radiation period. Here we perform no such average, allowing for significant variations in current over a radiation period to be described. We begin with a brief description of our model. We consider the interaction between an annular pulse of electrons co-propagating with a single TM 0 n mode of a dielectric-lined waveguide. As shown in Fig. 1, the annular electron pulse, of inner radius r 1 and outer radius r 2 , is represented by a distribution of N rigid annuli which are assumed to be uniformly distributed on entering the interaction region at z s 0 and which are free to move only in the z-direction. The charge distribution on each annulus is uniform over its transverse area A s p Ž r 22 y r 12 ., the surface charge density of the jth annulus Ž j s 1, . . . , N . being given by yq j sj s A where q j is the total charge magnitude of the jth annulus. An electron pulse of varying current may then be modelled
by assigning the q j appropriately. From the Lorentz force equations, the motion of the jth annulus is governed by the equations d tj
1
Ž1.
s
dz
Õz j
dÕz j
2p e sy
dz
Õz j g j3 m A
Hr
r2
rEz Ž r , z j ,t . d r
Ž2.
1
y1r2
is the relativistic factor, where g s Ž 1 y Õz2rc 2 . yerm is the charge to mass ratio of an electron and Ez is the axial component of the electric field of the waveguide mode. For an azimuthally symmetric TM 0 n mode, E z can be defined as I0 Ž a r .
Ez Ž r , z ,t . s
2
Ž E Ž z ,t . exp Ž i Ž kz y v t . . q c.c. .
in the vacuum region where InŽ x . is an nth order modified Bessel function of the first kind, k is the axial wavenumber of the mode, v is the angular frequency of the mode, a s k 2 y v 2rc 2 is the transverse wavenumber in vacuo, and E Ž z,t . describes a complex field envelope assumed to obey the slowly varying envelope approximation ŽSVEA.. The evolution of the electromagnetic field is described by Maxwell’s wave equation,
(
ž
=2y
1 E2 c2 E t 2
/
H s y= = J
Ž3.
Here, H is the magnetic field intensity of the waveguide mode and J s J z zˆ is the electron pulse current density, where Jz s y
Ž H Ž r y r1 . y H Ž r y r 2 . . A
N
Ý q j Õz d Ž z y z j Ž t . . js1
and H Ž x . is the Heaviside function. Integrating Eq. Ž3. over the waveguide cross-section and applying SVEA, we obtain
EE
q
Ez
1 EE Õg E t
s
2p I
e 0 v k a AQ
N
Ý q j Õz d Ž z y z j Ž t . . js1
=exp Ž yi Ž kz y v t . .
Fig. 1. Schematic diagram of the Cherenkov maser model. The annular electron pulse is modelled by N annuli of inner radius r1 and outer radius r 2 . The charge on the jth annulus is q j , js1, . . . , N. The waveguide is loaded with a liner of dielectric constant e and of inner radius a and outer radius b.
Ž4.
where the radiation group velocity within the interaction region is c2k Q Õg s
v QX
,
B.W.J. McNeil et al.r Optics Communications 163 (1999) 203–207
and Q, QX and I are mode dependent constants, given by 2p
Qs
a
a
2 1
H0 I
2
2pe
Ž a r . rdr q
2 kH
I02
=
the set of Eqs. Ž1. – Ž3. can be reduced to our working set of scaled equations which describe the self consistent evolution of the radiation field and the electron pulse: d z1 j dz
Ž a a.
Ž J 0 Ž k H a . Y0 Ž k H b . y J 0 Ž k H b . Y0 Ž k H a . .
d pj
2
b
Ha Ž J Ž k 1
Hr
. Y0 Ž k H b .
ž
2
yJ0 Ž k H b . Y1 Ž k H r . . r d r
Ž 10.
s yKpj
ž
ž
s y A z , z 1 j exp yi
dz =
205
E
E
Ez
q
E z1
Ž
/
.
A Ž z , z1 . s
z1 j
rK
/ /
Qs
2p
a
H0
a2
I12
2pe
Ž a r . rdr q
nI
Ý x j d Ž z1 y z1 . j
js1
Ž5.
I02 Ž a a .
=
Ž J 0 Ž k H a . Y0 Ž k H b . y J 0 Ž k H b . Y0 Ž k H a . . b
=
Ha Ž J Ž k 1
Hr
2
. Y0 Ž k H b . 2
yJ0 Ž k H b . Y1 Ž k H r . . r d r I s I1Ž a r 2 . r 2 y I1Ž a r 1 . r 1
Ž6. Ž7.
The transverse wavenumber in the liner of dielectric constant e is k Hs ev 2rc 2 y k 2 , and JnŽ x . and YnŽ x . are nth order Bessel functions of the first and second kinds respectively. Although the integrals in Eqs. Ž5. and Ž6. can be evaluated analytically, they lead to long algebraic expressions and have been left in integral form here for brevity. We now introduce the scaled independent variables
(
zskr z,
z1 s k r K Ž Õz 0 t y z . ,
rs
g0 v
ž
2p 2 eIpk I 2
e 0 mk a 2 A 2 Q
1 3
/
.
Ž8.
Here, Ipk is the peak current of the electron pulse. By defining the scaled momentum and field variables pj s
1
r
ž
1y
A Ž z , z1 . s
Õz 0 Õz j
/
r 2 m v Õz 0 g 03 a A
.
Ž 12.
N
1 nI
j
js1
x Ž z1 . N
X
N
X
Ý exp js1
z1
ž / ž /
Ý x j d Ž z1 y z1 . exp i
i
z1 j
rK
rK ,
X
where the sum Ý Njs1 is over the annuli contained within the periodic interval of z 1 centered at the positions where the radiation field and electron pulse has been sampled and averaged, and x Ž z 1 . is the annulus charge weighting function evaluated at the averaging positions. However, despite this ‘averaged model’ describing some aspects of the pulsed, self consistent interaction of radiation and electrons, it cannot describe the effects of CSE.
3. Coherent spontaneous emission
, ep I
rK
Here, n I is the number of annuli per unit z 1 at z s 0 and the annulus charge weighting parameter x j s I Ž z s 0, z1 j .rIpk , where I Ž z s 0, z 1 . is the electron pulse current on entering the interaction region. In deriving Eqs. Ž10. – Ž12. it has been assumed that the interaction is in the low efficiency limit of r < 1. This corresponds to there being only small relative changes in the electrons’ energies during the interaction so that changes in the relativistic factors of the electrons Dg j < g 0; j. Eqs. Ž10. – Ž12. may be solved using the finite element method w15,16x to describe the field. This method allows the successful modelling of CSE and the resultant interaction. In previous works w12–14x, the wave Eq. Ž12. has been averaged over at least one radiation period, here corresponding to an interval in z 1 of w0,2p K x, within which the charge weightings x j are assumed equal. In this case the field is an averaged variable, AŽ z, z 1 . ™ AŽ z, z 1 ., and the source term of Eq. Ž12. reduces to
™
where K s ÕgrŽ Õz 0 y Õg ., Õz 0 is the mean of the electrons’ velocities on entering the interaction region and r is a generalised gain parameter, defined as 1
z1
ž /
2
2 kH
Ž 11 .
N
1
=exp i X
q c.c.
E Ž z , z1 .
Ž9.
We consider an electron pulse with a rectangular current profile, so that the electrons are distributed uniformly over the interval 0 - z1 - l e at z s 0 and with x j s 1;j. To demonstrate the significant influence of CSE on the
206
B.W.J. McNeil et al.r Optics Communications 163 (1999) 203–207
Cherenkov interaction, we solve both the averaged and unaveraged equations, and compare their predictions. For the averaged equations the above electron distribution corresponds to a x Ž z 1 . s 1 within the interval 0 - z 1 - l e and x Ž z 1 . s 0 outwith. To obtain any interaction for the averaged model it is necessary to have a non-zero initial field amplitude, which can be assumed constant Ž A 0 s AŽ z s 0, z 1 .., andror to include the effects of the electron shot-noise, otherwise the system is in Žunstable. equilibrium. Here we choose an initial field of A 0 s 10y2 and ignore the effects of shot-noise. For the non-averaged equations however, the CSE alone acts as an initial radiation source which is then subsequently amplified. When solving these equations we have no shot-noise and set the initial field A 0 s 0. To evaluate the evolution of the superradiant pulse on its own, we detuned the electron energy away from reso3
'
nance by setting pj Ž z s 0. ) dth , where dth s 27r4 is a threshold value above which there is no steady-state Cherenkov instability, and hence no exponential growth of the radiation. This is in close analogy with the FEL instability w17x. However, because a pulsed interaction is being considered, there is always a region of the interaction which does not evolve as the steady state, or c.w., limit. This region is called the ‘slippage region’ of the pulse within which there is no threshold on pj Ž z s 0. for exponential instability, as in FEL theory. In fact, it is within this region that superradiant emission occurs and the radiation intensity scales as the number of emitters squared, in contrast to the four thirds scaling in the steady-state. Fig. 2 shows the scaled field intensities plotted as a function of z 1 when z s 1, obtained from solutions to
Fig. 2. The scaled field intensities Ža. < A < 2 excluding CSE and Žb. < A < 2 including CSE, plotted as functions of z1 when z s1, for a rectangular electron pulse current profile of scaled length l e s 2. In both cases d s 4, r s 0.01 and K s 3.5. The initial fields were < A 0 < 2 s10y4 and < A 0 < 2 s 0.
Fig. 3. The scaled field intensities Ža. < A < 2 excluding CSE and Žb. < A < 2 including CSE, plotted as functions of z1 when z s16, for a rectangular electron pulse current profile of scaled length l e s16. In both cases d s 4, r s 0.01 and K s 3.5. The initial fields were < A 0 < 2 s10y4 and < A 0 < 2 s 0.
both the averaged and non-averaged equations, for a scaled electron pulse length of l e s 2. For the solution of averaged equations, in the absence of CSE, there is no evolution observable on this scale as the scaled intensity has increased by only f 20%. It can be seen from the unaveraged CSE model that a series of radiation pulses develop at each end of the electron pulse, initially at z 1 s 0 and z1 s l e , and propagate in the direction of positive z 1. These pulses are one radiation cycle in duration and arise from a coherent interference of the radiation emitted by each electron. Because this is an effect of constructive interference only, this emission is spontaneous in nature. This phenomenon, arising from the propagation and interference of the radiation emitted from different regions of the electron pulse, may be characterised by the current profile of the electron pulse. In the case of the rectangular pulse used here, the important characteristics are the discontinuities of the electron density, initially at z 1 s 0 and z 1 s l e , which act as a source of CSE. Note that while the CSE for z1 ) l e of Fig. 2 propagates into free space, the CSE from the slippage region propagates into the electron pulse, allowing it to be amplified by further interaction with the electrons. The result of this interaction is shown in Fig. 3, which shows the radiation intensity as a function of z 1 at z s 16 and for a longer electron pulse of l e s 16 than that of Fig. 2. It can be seen that the CSE pulse has been strongly amplified to form a high intensity spike of radiation propagating through the electron pulse. Comparison with the solution to the averaged equations of Fig. 3, clearly demonstrates the importance of CSE in the evolution of the radiation field. The CSE within the slippage region acts as a strong seed field from which a high intensity spike of radiation quickly develops and propagates. Although in the
B.W.J. McNeil et al.r Optics Communications 163 (1999) 203–207
207
shown that CSE could act to drive the generation of strong superradiant pulses of Cherenkov radiation. The growth of these CSE-driven superradiant pulses was noticeably greater than that due to incoherent shot noise or a small input seed field alone. These results are particularly significant for Cherenkov maser experiments at longer wavelengths where variations of the electron pulse on the scale of the radiation wavelength are relatively easy to obtain.
Acknowledgements The authors would like to thank the EPSRC and the Royal Society of Edinburgh for support of BM c N and GRMR respectively. Fig. 4. The peaks of the scaled field amplitudes, Ža. < A pk < excluding CSE and Žb. < A pk < including CSE, for a rectangular electron pulse current profile of scaled length l e s16. These are the peak amplitudes at position z1 that occur within the interaction interval 0 - z -16.
absence of CSE a superradiant spike does evolve, its growth rate is reduced and its evolution significantly retarded. These high intensity spikes, both with and without the effects of CSE, are superradiant in nature, their intensity being proportional to n2e , where n e is the electron density of the electron pulse. Furthermore, these superradiant pulses are self-similar in nature and in the absence of second order effects do not saturate Žsee e.g. w17x.. In order to demonstrate the superradiant scaling, we plot the peak value of the magnitude of the scaled field amplitude < A < pk for fixed values of z 1, that occurs within the interaction interval of z. Hence, if the radiation spike has propagated through a position z 1, then < A < pk will be the maximum scaled amplitude of the spike at that value of z 1. If this scaled amplitude of the spike has a linear dependence with z 1 then, from the scaling of Eqs. Ž8. and Ž9., it is simple to show that the real intensity of the spike is superradiant in nature, i.e. it scales as n 2e . Fig. 4 shows this graph as calculated from a numerical solution to both the averaged and unaveraged equations. It can be seen from this graph that < A < pk A z 1, for both the averaged and unaveraged solutions, confirming the superradiant nature of the spikes. The growth of the spike for the unaveraged solution, which includes the effects of CSE, is seen to be significantly greater than that of the averaged solution where CSE is absent.
4. Conclusions In conclusion, we have presented a one-dimensional analysis of a Cherenkov maser amplifier in the low efficiency limit of r < 1, taking into account the effect of CSE on the electron-radiation field interaction. It was
References w1x R. Bonifacio, B.W.J. McNeil, Nucl. Instr. Meth. A 272 Ž1988. 280. w2x R. Bonifacio, C. Maroli, N. Piovella, Optics Comm. 68 Ž1988. 369. w3x R. Bonifacio, B.W.J. McNeil, P. Pierini, Phys. Rev. A 40 Ž1989. 4467. w4x D.A. Jaroszynski, D. Oepts, G.M.H. Knipples, A.F.G. van der Meer, H.H. Weits, P. Chaix, N. Piovella, Phys. Rev. Lett. 78 Ž1997. 1699. w5x N.S. Ginzburg, I.V. Zotova, A.S. Sergeev, I.V. Konoplev, A.D.R. Phelps, A.W. Cross, S.J. Cooke, V.G. Shpak, M.I. Yalandin, S.A. Shunailov, M.R. Ulmaskulov, Phys. Rev. Lett. 78 Ž1997. 2365. w6x N.S. Ginzburg, A.S. Sergeev, Optics Comm. 91 Ž1992. 140. w7x G.R.M. Robb, B.W.J. McNeil, A.D.R. Phelps, Proc. 19th Int. Conf. on Infrared and Millimeter Waves, Sendai, Japan, JSAP Report No. AP941228, 155 Ž1994.. w8x P. Aitken, B.W.J. McNeil, G.R.M. Robb, A.D.R. Phelps, Phys. Rev. E 59 Ž1999. Žin press.. w9x M.I. Yalandin, S.A. Shunailov, V.G. Shpak, N.S. Ginzburg, I.V. Zotova, A.S. Sergeev, A.D.R. Phelps, A.W. Cross, P. Aitken, Tech. Phys. Lett. 23 Ž1997. 948. w10x F. Ciocci, A. Doria, P. Gallerano, I. Giabbai, M.F. Kimmitt, G. Messina, A. Renieri, J.E. Walsh, Phys. Rev. Lett. 66 Ž1991. 699. w11x D.A. Jaroszynski, R.J. Bakker, A.F.G. van der Meer, D. Oepts, P.W. van Amersfoort, Phys. Rev. Lett. 71 Ž1993. 3798. w12x J.S. Choi, K.J. Kim, M. Xie, Nucl. Instr. Meth. A 331 Ž1993. 587. w13x H.P. Freund, A.K. Ganguly, Phys. Fluids B 2 Ž1990. 2506. w14x E.P. Garate, J.E. Walsh, IEEE Trans. on Plasma Sci. PS 13 Ž1985. 524. w15x E.A. Huebner, E.A. Thornton, T.G. Byrom, The finite element method for engineers, Wiley, New York, 1995. w16x C. Johnson, Numerical solution of partial differential equations by the finite element method, Cambridge University Press, Cambridge, 1995. w17x R. Bonifacio, F. Casagrande, G. Cerchioni, L. De Salvo, P. Pierini, N. Piovella, Riv. Nuovo Cimento 13 Ž1990. 9 and references therein.
Optics Communications 163 Ž1999. 203–207
Superradiance driven by coherent spontaneous emission in the Cherenkov maser B.W.J. McNeil, G.R.M. Robb, D.A. Jaroszynski Department of Physics & Applied Physics, UniÕersity of Strathclyde, Glasgow, G4 0NG, UK Received 21 December 1998; accepted 8 March 1999
Abstract An analysis of the Cherenkov maser instability operating with electron pulses is carried out. A model which can describe the effects of Coherent Spontaneous Emission ŽCSE. is derived. The results from this model are compared with previous work which neglect the effects of CSE. The model suggests that CSE can drive and significantly enhance superradiant emission from the electron pulses, offering a potential source of high intensity, short pulse radiation. This mechanism should also be present in other devices such as the Free Electron Laser. q 1999 Published by Elsevier Science B.V. All rights reserved.
1. Introduction The emission of radiation from ensembles of free electrons forming an electron beam, or pulse, interacting with both static andror electromagnetic fields, forms the basis of many successful sources of high power coherent, electromagnetic radiation, such as the cyclotron resonance maser, the free electron laser ŽFEL. and the Cherenkov maser. Superradiant emission from these sources, when driven by electron pulses, has been the subject of much recent interest w1–9x. Such emission is of practical interest as a means of generating very short, intense, pulses of radiation. These pulses have many potential industrial and academic applications. In the microwave to mm-wavelength range, they are of considerable interest for use, for example, as plasma probes and in high-gradient particle acceleration. At shorter wavelengths they would have applications in such subjects as nonlinear spectroscopy, and studies of chemical and biological systems. The nonlinear interaction between an electron pulse and electromagnetic wave in these superradiant amplifiers may be initiated by ‘shot noise’ emission, due to the random position of electrons within the electron pulse, or by a small resonant seed field injected into the interaction region. When the radiation develops in an amplifier from noise, the process has been called self-amplified sponta-
neous emission ŽSASE.. However, another emission initiation process can occur called ‘coherent spontaneous emission’ ŽCSE., arising from the initial current profile, or ‘shape’, of the electron pulse. This has been the subject of a number of recent studies w10,11x. Emission of intensities several orders of magnitude greater than that due to shot noise alone may occur when the current profile of the electron pulse changes significantly on the scale of a radiation wavelength. In this communication, we present an analysis of the Cherenkov maser amplifier initiated by CSE and make comparison with a previous method of analysis where CSE is absent. We find that when CSE effects are included, the CSE acts as a substantial seed from which a high-intensity superradiant radiation pulse rapidly grows. The growth of these pulses is enhanced over that due to the presence of only shot noise or a c.w. seed field. Our results suggest that the generation of superradiant radiation pulses may be greatly enhanced by suitable ‘shaping’ of the electron pulse. The similarity between the Cherenkov interaction with other free-electron interactions suggests that CSE driven superradiance will also be present in other devices such as the free electron laser. This may have important implications for FELs operating in the SASE regime. These consequences will be discussed in a subsequent publication.
0030-4018r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 9 . 0 0 1 3 5 - 2
204
B.W.J. McNeil et al.r Optics Communications 163 (1999) 203–207
2. Theory In this analysis of the Cherenkov maser, the evolution of the interaction between the electron pulse and the radiation field is described using a set of coupled non-linear partial differential equations. The electrons’ positions and momenta evolve under the action of the electromagnetic fields of the waveguide mode as described by the Lorentz force equation. The electron motion constitutes a current which acts as a source of the electromagnetic field as described by Maxwell’s wave equation. To date, most numerical studies of the Cherenkov maser have involved the averaging and discretisation of the wave equation over an interval equal to or greater than a radiation period w12–14x. This restricts the description of the field evolution to situations where the current profile of the electron pulse changes negligibly on the scale of the radiation period. Here we perform no such average, allowing for significant variations in current over a radiation period to be described. We begin with a brief description of our model. We consider the interaction between an annular pulse of electrons co-propagating with a single TM 0 n mode of a dielectric-lined waveguide. As shown in Fig. 1, the annular electron pulse, of inner radius r 1 and outer radius r 2 , is represented by a distribution of N rigid annuli which are assumed to be uniformly distributed on entering the interaction region at z s 0 and which are free to move only in the z-direction. The charge distribution on each annulus is uniform over its transverse area A s p Ž r 22 y r 12 ., the surface charge density of the jth annulus Ž j s 1, . . . , N . being given by yq j sj s A where q j is the total charge magnitude of the jth annulus. An electron pulse of varying current may then be modelled
by assigning the q j appropriately. From the Lorentz force equations, the motion of the jth annulus is governed by the equations d tj
1
Ž1.
s
dz
Õz j
dÕz j
2p e sy
dz
Õz j g j3 m A
Hr
r2
rEz Ž r , z j ,t . d r
Ž2.
1
y1r2
is the relativistic factor, where g s Ž 1 y Õz2rc 2 . yerm is the charge to mass ratio of an electron and Ez is the axial component of the electric field of the waveguide mode. For an azimuthally symmetric TM 0 n mode, E z can be defined as I0 Ž a r .
Ez Ž r , z ,t . s
2
Ž E Ž z ,t . exp Ž i Ž kz y v t . . q c.c. .
in the vacuum region where InŽ x . is an nth order modified Bessel function of the first kind, k is the axial wavenumber of the mode, v is the angular frequency of the mode, a s k 2 y v 2rc 2 is the transverse wavenumber in vacuo, and E Ž z,t . describes a complex field envelope assumed to obey the slowly varying envelope approximation ŽSVEA.. The evolution of the electromagnetic field is described by Maxwell’s wave equation,
(
ž
=2y
1 E2 c2 E t 2
/
H s y= = J
Ž3.
Here, H is the magnetic field intensity of the waveguide mode and J s J z zˆ is the electron pulse current density, where Jz s y
Ž H Ž r y r1 . y H Ž r y r 2 . . A
N
Ý q j Õz d Ž z y z j Ž t . . js1
and H Ž x . is the Heaviside function. Integrating Eq. Ž3. over the waveguide cross-section and applying SVEA, we obtain
EE
q
Ez
1 EE Õg E t
s
2p I
e 0 v k a AQ
N
Ý q j Õz d Ž z y z j Ž t . . js1
=exp Ž yi Ž kz y v t . .
Fig. 1. Schematic diagram of the Cherenkov maser model. The annular electron pulse is modelled by N annuli of inner radius r1 and outer radius r 2 . The charge on the jth annulus is q j , js1, . . . , N. The waveguide is loaded with a liner of dielectric constant e and of inner radius a and outer radius b.
Ž4.
where the radiation group velocity within the interaction region is c2k Q Õg s
v QX
,
B.W.J. McNeil et al.r Optics Communications 163 (1999) 203–207
and Q, QX and I are mode dependent constants, given by 2p
Qs
a
a
2 1
H0 I
2
2pe
Ž a r . rdr q
2 kH
I02
=
the set of Eqs. Ž1. – Ž3. can be reduced to our working set of scaled equations which describe the self consistent evolution of the radiation field and the electron pulse: d z1 j dz
Ž a a.
Ž J 0 Ž k H a . Y0 Ž k H b . y J 0 Ž k H b . Y0 Ž k H a . .
d pj
2
b
Ha Ž J Ž k 1
Hr
. Y0 Ž k H b .
ž
2
yJ0 Ž k H b . Y1 Ž k H r . . r d r
Ž 10.
s yKpj
ž
ž
s y A z , z 1 j exp yi
dz =
205
E
E
Ez
q
E z1
Ž
/
.
A Ž z , z1 . s
z1 j
rK
/ /
Qs
2p
a
H0
a2
I12
2pe
Ž a r . rdr q
nI
Ý x j d Ž z1 y z1 . j
js1
Ž5.
I02 Ž a a .
=
Ž J 0 Ž k H a . Y0 Ž k H b . y J 0 Ž k H b . Y0 Ž k H a . . b
=
Ha Ž J Ž k 1
Hr
2
. Y0 Ž k H b . 2
yJ0 Ž k H b . Y1 Ž k H r . . r d r I s I1Ž a r 2 . r 2 y I1Ž a r 1 . r 1
Ž6. Ž7.
The transverse wavenumber in the liner of dielectric constant e is k Hs ev 2rc 2 y k 2 , and JnŽ x . and YnŽ x . are nth order Bessel functions of the first and second kinds respectively. Although the integrals in Eqs. Ž5. and Ž6. can be evaluated analytically, they lead to long algebraic expressions and have been left in integral form here for brevity. We now introduce the scaled independent variables
(
zskr z,
z1 s k r K Ž Õz 0 t y z . ,
rs
g0 v
ž
2p 2 eIpk I 2
e 0 mk a 2 A 2 Q
1 3
/
.
Ž8.
Here, Ipk is the peak current of the electron pulse. By defining the scaled momentum and field variables pj s
1
r
ž
1y
A Ž z , z1 . s
Õz 0 Õz j
/
r 2 m v Õz 0 g 03 a A
.
Ž 12.
N
1 nI
j
js1
x Ž z1 . N
X
N
X
Ý exp js1
z1
ž / ž /
Ý x j d Ž z1 y z1 . exp i
i
z1 j
rK
rK ,
X
where the sum Ý Njs1 is over the annuli contained within the periodic interval of z 1 centered at the positions where the radiation field and electron pulse has been sampled and averaged, and x Ž z 1 . is the annulus charge weighting function evaluated at the averaging positions. However, despite this ‘averaged model’ describing some aspects of the pulsed, self consistent interaction of radiation and electrons, it cannot describe the effects of CSE.
3. Coherent spontaneous emission
, ep I
rK
Here, n I is the number of annuli per unit z 1 at z s 0 and the annulus charge weighting parameter x j s I Ž z s 0, z1 j .rIpk , where I Ž z s 0, z 1 . is the electron pulse current on entering the interaction region. In deriving Eqs. Ž10. – Ž12. it has been assumed that the interaction is in the low efficiency limit of r < 1. This corresponds to there being only small relative changes in the electrons’ energies during the interaction so that changes in the relativistic factors of the electrons Dg j < g 0; j. Eqs. Ž10. – Ž12. may be solved using the finite element method w15,16x to describe the field. This method allows the successful modelling of CSE and the resultant interaction. In previous works w12–14x, the wave Eq. Ž12. has been averaged over at least one radiation period, here corresponding to an interval in z 1 of w0,2p K x, within which the charge weightings x j are assumed equal. In this case the field is an averaged variable, AŽ z, z 1 . ™ AŽ z, z 1 ., and the source term of Eq. Ž12. reduces to
™
where K s ÕgrŽ Õz 0 y Õg ., Õz 0 is the mean of the electrons’ velocities on entering the interaction region and r is a generalised gain parameter, defined as 1
z1
ž /
2
2 kH
Ž 11 .
N
1
=exp i X
q c.c.
E Ž z , z1 .
Ž9.
We consider an electron pulse with a rectangular current profile, so that the electrons are distributed uniformly over the interval 0 - z1 - l e at z s 0 and with x j s 1;j. To demonstrate the significant influence of CSE on the
206
B.W.J. McNeil et al.r Optics Communications 163 (1999) 203–207
Cherenkov interaction, we solve both the averaged and unaveraged equations, and compare their predictions. For the averaged equations the above electron distribution corresponds to a x Ž z 1 . s 1 within the interval 0 - z 1 - l e and x Ž z 1 . s 0 outwith. To obtain any interaction for the averaged model it is necessary to have a non-zero initial field amplitude, which can be assumed constant Ž A 0 s AŽ z s 0, z 1 .., andror to include the effects of the electron shot-noise, otherwise the system is in Žunstable. equilibrium. Here we choose an initial field of A 0 s 10y2 and ignore the effects of shot-noise. For the non-averaged equations however, the CSE alone acts as an initial radiation source which is then subsequently amplified. When solving these equations we have no shot-noise and set the initial field A 0 s 0. To evaluate the evolution of the superradiant pulse on its own, we detuned the electron energy away from reso3
'
nance by setting pj Ž z s 0. ) dth , where dth s 27r4 is a threshold value above which there is no steady-state Cherenkov instability, and hence no exponential growth of the radiation. This is in close analogy with the FEL instability w17x. However, because a pulsed interaction is being considered, there is always a region of the interaction which does not evolve as the steady state, or c.w., limit. This region is called the ‘slippage region’ of the pulse within which there is no threshold on pj Ž z s 0. for exponential instability, as in FEL theory. In fact, it is within this region that superradiant emission occurs and the radiation intensity scales as the number of emitters squared, in contrast to the four thirds scaling in the steady-state. Fig. 2 shows the scaled field intensities plotted as a function of z 1 when z s 1, obtained from solutions to
Fig. 2. The scaled field intensities Ža. < A < 2 excluding CSE and Žb. < A < 2 including CSE, plotted as functions of z1 when z s1, for a rectangular electron pulse current profile of scaled length l e s 2. In both cases d s 4, r s 0.01 and K s 3.5. The initial fields were < A 0 < 2 s10y4 and < A 0 < 2 s 0.
Fig. 3. The scaled field intensities Ža. < A < 2 excluding CSE and Žb. < A < 2 including CSE, plotted as functions of z1 when z s16, for a rectangular electron pulse current profile of scaled length l e s16. In both cases d s 4, r s 0.01 and K s 3.5. The initial fields were < A 0 < 2 s10y4 and < A 0 < 2 s 0.
both the averaged and non-averaged equations, for a scaled electron pulse length of l e s 2. For the solution of averaged equations, in the absence of CSE, there is no evolution observable on this scale as the scaled intensity has increased by only f 20%. It can be seen from the unaveraged CSE model that a series of radiation pulses develop at each end of the electron pulse, initially at z 1 s 0 and z1 s l e , and propagate in the direction of positive z 1. These pulses are one radiation cycle in duration and arise from a coherent interference of the radiation emitted by each electron. Because this is an effect of constructive interference only, this emission is spontaneous in nature. This phenomenon, arising from the propagation and interference of the radiation emitted from different regions of the electron pulse, may be characterised by the current profile of the electron pulse. In the case of the rectangular pulse used here, the important characteristics are the discontinuities of the electron density, initially at z 1 s 0 and z 1 s l e , which act as a source of CSE. Note that while the CSE for z1 ) l e of Fig. 2 propagates into free space, the CSE from the slippage region propagates into the electron pulse, allowing it to be amplified by further interaction with the electrons. The result of this interaction is shown in Fig. 3, which shows the radiation intensity as a function of z 1 at z s 16 and for a longer electron pulse of l e s 16 than that of Fig. 2. It can be seen that the CSE pulse has been strongly amplified to form a high intensity spike of radiation propagating through the electron pulse. Comparison with the solution to the averaged equations of Fig. 3, clearly demonstrates the importance of CSE in the evolution of the radiation field. The CSE within the slippage region acts as a strong seed field from which a high intensity spike of radiation quickly develops and propagates. Although in the
B.W.J. McNeil et al.r Optics Communications 163 (1999) 203–207
207
shown that CSE could act to drive the generation of strong superradiant pulses of Cherenkov radiation. The growth of these CSE-driven superradiant pulses was noticeably greater than that due to incoherent shot noise or a small input seed field alone. These results are particularly significant for Cherenkov maser experiments at longer wavelengths where variations of the electron pulse on the scale of the radiation wavelength are relatively easy to obtain.
Acknowledgements The authors would like to thank the EPSRC and the Royal Society of Edinburgh for support of BM c N and GRMR respectively. Fig. 4. The peaks of the scaled field amplitudes, Ža. < A pk < excluding CSE and Žb. < A pk < including CSE, for a rectangular electron pulse current profile of scaled length l e s16. These are the peak amplitudes at position z1 that occur within the interaction interval 0 - z -16.
absence of CSE a superradiant spike does evolve, its growth rate is reduced and its evolution significantly retarded. These high intensity spikes, both with and without the effects of CSE, are superradiant in nature, their intensity being proportional to n2e , where n e is the electron density of the electron pulse. Furthermore, these superradiant pulses are self-similar in nature and in the absence of second order effects do not saturate Žsee e.g. w17x.. In order to demonstrate the superradiant scaling, we plot the peak value of the magnitude of the scaled field amplitude < A < pk for fixed values of z 1, that occurs within the interaction interval of z. Hence, if the radiation spike has propagated through a position z 1, then < A < pk will be the maximum scaled amplitude of the spike at that value of z 1. If this scaled amplitude of the spike has a linear dependence with z 1 then, from the scaling of Eqs. Ž8. and Ž9., it is simple to show that the real intensity of the spike is superradiant in nature, i.e. it scales as n 2e . Fig. 4 shows this graph as calculated from a numerical solution to both the averaged and unaveraged equations. It can be seen from this graph that < A < pk A z 1, for both the averaged and unaveraged solutions, confirming the superradiant nature of the spikes. The growth of the spike for the unaveraged solution, which includes the effects of CSE, is seen to be significantly greater than that of the averaged solution where CSE is absent.
4. Conclusions In conclusion, we have presented a one-dimensional analysis of a Cherenkov maser amplifier in the low efficiency limit of r < 1, taking into account the effect of CSE on the electron-radiation field interaction. It was
References w1x R. Bonifacio, B.W.J. McNeil, Nucl. Instr. Meth. A 272 Ž1988. 280. w2x R. Bonifacio, C. Maroli, N. Piovella, Optics Comm. 68 Ž1988. 369. w3x R. Bonifacio, B.W.J. McNeil, P. Pierini, Phys. Rev. A 40 Ž1989. 4467. w4x D.A. Jaroszynski, D. Oepts, G.M.H. Knipples, A.F.G. van der Meer, H.H. Weits, P. Chaix, N. Piovella, Phys. Rev. Lett. 78 Ž1997. 1699. w5x N.S. Ginzburg, I.V. Zotova, A.S. Sergeev, I.V. Konoplev, A.D.R. Phelps, A.W. Cross, S.J. Cooke, V.G. Shpak, M.I. Yalandin, S.A. Shunailov, M.R. Ulmaskulov, Phys. Rev. Lett. 78 Ž1997. 2365. w6x N.S. Ginzburg, A.S. Sergeev, Optics Comm. 91 Ž1992. 140. w7x G.R.M. Robb, B.W.J. McNeil, A.D.R. Phelps, Proc. 19th Int. Conf. on Infrared and Millimeter Waves, Sendai, Japan, JSAP Report No. AP941228, 155 Ž1994.. w8x P. Aitken, B.W.J. McNeil, G.R.M. Robb, A.D.R. Phelps, Phys. Rev. E 59 Ž1999. Žin press.. w9x M.I. Yalandin, S.A. Shunailov, V.G. Shpak, N.S. Ginzburg, I.V. Zotova, A.S. Sergeev, A.D.R. Phelps, A.W. Cross, P. Aitken, Tech. Phys. Lett. 23 Ž1997. 948. w10x F. Ciocci, A. Doria, P. Gallerano, I. Giabbai, M.F. Kimmitt, G. Messina, A. Renieri, J.E. Walsh, Phys. Rev. Lett. 66 Ž1991. 699. w11x D.A. Jaroszynski, R.J. Bakker, A.F.G. van der Meer, D. Oepts, P.W. van Amersfoort, Phys. Rev. Lett. 71 Ž1993. 3798. w12x J.S. Choi, K.J. Kim, M. Xie, Nucl. Instr. Meth. A 331 Ž1993. 587. w13x H.P. Freund, A.K. Ganguly, Phys. Fluids B 2 Ž1990. 2506. w14x E.P. Garate, J.E. Walsh, IEEE Trans. on Plasma Sci. PS 13 Ž1985. 524. w15x E.A. Huebner, E.A. Thornton, T.G. Byrom, The finite element method for engineers, Wiley, New York, 1995. w16x C. Johnson, Numerical solution of partial differential equations by the finite element method, Cambridge University Press, Cambridge, 1995. w17x R. Bonifacio, F. Casagrande, G. Cerchioni, L. De Salvo, P. Pierini, N. Piovella, Riv. Nuovo Cimento 13 Ž1990. 9 and references therein.