Excitation of wake field and transition radiation in a semi-infinite waveguide by a relativistic electron bunch
The theoretical investigations and simulation are performed for the emission of a relativistic electron bunch
 during the injection into a semi-infinite vacuum/dielectric waveguide. The power and the frequency spectrum of
 high intensity pulse of ultra-wideband transition radiation,...
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| Date: | 2002 |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2002
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| Cite this: | Excitation of wake field and transition radiation in a semi-infinite waveguide by a relativistic electron bunch / V.A. Balakirev, N.I. Onishchenko, D.Yu. Sidorenko, G.V. Sotnikov // Вопросы атомной науки и техники. — 2002. — № 5. — С. 115-117. — Бібліогр.: 16 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860011506651889664 |
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| author | Balakirev, V.A. Onishchenko, N.I. Sidorenko, D.Yu. Sotnikov, G.V. |
| author_facet | Balakirev, V.A. Onishchenko, N.I. Sidorenko, D.Yu. Sotnikov, G.V. |
| citation_txt | Excitation of wake field and transition radiation in a semi-infinite waveguide by a relativistic electron bunch / V.A. Balakirev, N.I. Onishchenko, D.Yu. Sidorenko, G.V. Sotnikov // Вопросы атомной науки и техники. — 2002. — № 5. — С. 115-117. — Бібліогр.: 16 назв. — англ. |
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| container_title | Вопросы атомной науки и техники |
| description | The theoretical investigations and simulation are performed for the emission of a relativistic electron bunch
during the injection into a semi-infinite vacuum/dielectric waveguide. The power and the frequency spectrum of
high intensity pulse of ultra-wideband transition radiation, excited by a finite size electron bunch in a vacuum
waveguide, are calculated numerically. It is shown that in a dielectric waveguide the short pulse of Cherenkov wake
field drifts behind the relativistic bunch with the group velocity.
|
| first_indexed | 2025-12-07T16:41:56Z |
| format | Article |
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EXCITATION OF WAKE FIELD AND TRANSITION RADIATION
IN A SEMI-INFINITE WAVEGUIDE
BY A RELATIVISTIC ELECTRON BUNCH
V.A. Balakirev, N.I. Onishchenko, D.Yu. Sidorenko and G.V. Sotnikov
NSC “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine,
E-mail: onish@kipt.kharkov.ua
The theoretical investigations and simulation are performed for the emission of a relativistic electron bunch
during the injection into a semi-infinite vacuum/dielectric waveguide. The power and the frequency spectrum of
high intensity pulse of ultra-wideband transition radiation, excited by a finite size electron bunch in a vacuum
waveguide, are calculated numerically. It is shown that in a dielectric waveguide the short pulse of Cherenkov wake
field drifts behind the relativistic bunch with the group velocity.
PACS: 52.40.-w; 52.35.-g
1. INTRODUCTION
At present, much attention is focused on the problem
of generation of short intense electromagnetic pulses
whose frequency spectra width is comparable with their
mean frequency, the so-called “ultra-wideband” (UWB)
pulses [1, 2]. The current interest to the problem of
pulsed emission of high-power electromagnetic signals
arises primarily due to their application in UWB
radiolocation.
Along with traditional methods of generation of
short UWB pulses based on the use of UWB antennas
(TEM-horns, spiral and biconical antennas, etc. [3]) the
intense pulsed relativistic electron beams (IREBs) can
be used. For the most efficient generation of
electromagnetic UWB pulses the non-resonant (impact)
mechanisms of IREBs radiation should be used, such as
charging of a rod antenna by REB [4] or impact
excitation of TEM-horn antenna by IREB [5]. High-
power UWB pulses can also be the result of coherent
transition radiation of pulsed IREB [6].
Here we theoretically investigate the excitation of
UWB transition radiation during the injection of a short-
pulsed IREB into a semi-infinite circular cross-section
waveguide whose entrance (through which the beam is
injected) is short-circuited by a conducting diaphragm.
In such a waveguide, the large dispersion of
electromagnetic waves is the peculiarity of the transition
radiation, so that the shape of a UWB pulse of transition
radiation propagating in a waveguide will deform
permanently.
In the case of dielectric filling the intense Cherenkov
wake field excitation can take place. It can be applied
for wake field acceleration of charged particles [7,8] or
for radiation sources [9,10]. In the case of short-pulsed
IREB the interference of big number of radial modes
leads to the essential peaking of the wake field with
formation of narrow spikes of alternative sign [11]. The
spatial structure of excited field is determined by the
spatially limited Cherenkov field and transition
radiation field.
2. NON-RESONANT WIDEBAND EMISSION
IN A SEMI-INFINITE WAVEGUIDE
We consider a semi-infinite (0 ≤ z < ∞ ) cylindrical
metal waveguide of radius b which is filled with a
homogeneous dielectric with permittivity ε.
The waveguide input end (z = 0) is short-circuited by a
metal wall transparent to relativistic electrons. An
axisymmetric monoenergetic electron bunch is injected
through the metal wall and moves with a constant
velocity ε/cv0 < along the symmetry axis of the
waveguide (the z axis). We start with the determination
of the field of an infinitely short and infinitely thin
charged ring with the charge density
)()(
0
00
00 v
zttrr
vr2
eN −−−−= δδ
π
ρ , (1)
where -e is the charge of an electron, N is the number of
electrons in the ring, v0 is the ring velocity, r0 is the ring
radius, and t0 is the time at which the ring enters the
waveguide. Solving Fourier-transformed Maxwell’s
equations with allowance for the boundary condition Er
= 0 at the end metal wall, we obtain the following
expression for the radial electric field of an
axisymmetric E -wave:
,}{
)(
)/()/(
),,,,(
∑ −×
=
n
n1n2
n
2
1n
n10n0
2
n0
0
00r
II
J
brJbrJ
vb
Ne2rtzrtE
λλ
λλω
επ
(2)
∫
∞
∞− +−
++−
=
))((
)]/(exp[
n0n0
00
n1 ii
vztiti
dI
ωωωω
ωω
ω , (3)
∫
∞
∞− +−
−+−
=
))((
]exp[
n0n0
2
n
2
n2 ii
ii
dI
ωωωω
αωξω τ
ω , (4)
where 0tt −=τ , cz /εξ = , )/( ελα bcnn = ,
)//( 22
00nn0 cv1bv ελω −= , and λn is the n-th root of
the Bessel function J0.
Integral (3) describes the Coulomb field of a charge
moving in an infinite waveguide. Integral (4)
corresponds to free oscillations of a cylindrical
waveguide and describes the transition radiation. The
exact analytic solution of integral similar to (4) was
obtained in [12]. In [13] the saddle point technique was
used in order to obtain the approximate solution of
integral (4) under condition of Cherenkov resonance.
Below we applied the method proposed in [12].
In order to calculate the integral (4) we used a
sequence of substitutions: ωip −= ,
Problems of Atomic Science and Technology. 2002. № 5. Series: Plasma Physics (8). P. 115-117 115
n
2
n
2 pp ααζ /)( −+= , and βζ /−=w , where
)/()( ξτξτβ +−= . After these conform
transformations we passed from the integration along
the real axis to the integration along the closed circular
contour. This allows us to separate the integral form of
the Bessel functions. Finally [14] we obtain the
expression for the total electric field (2) of a thin
annular electron bunch (1) in the form of the
superposition of the Coulomb field of a moving charge
and the transition radiation field:
trans
r
coul
r00r EErtzrtE +=),,,,( , (5)
[ ]{
[ ]})/(exp
)/(exp
)(
)/()/(
/
),,,,(
00n02
00n01
1n n
2
1
n10n0
22
0
2
00
coul
r
vztt
vztt
J
brJbrJ
cv1b
Ne2rtzrtE
−−−+
−−×
×
−
−=
∑
∞
=
ωϑ
ωϑ
λ
λλ
εε
, (6)
coul
r
0m
n1m2
1m2
2
1m2
12
0m
n1m2
1m2
2
1m2
11
n n
2
1
n10n0
22
0
2
00
trans
r
EyJqq
yJqq
J
brJbrJ
cv1b
Ne4rtzrtE
−++
−×
×
−
=
∑
∑
∑
∞
=
+
−−+
∞
=
+
++
})()(
)()({
)(
)/()/(
/
),,,,(
ϑ
ϑ
λ
λλ
εε
, (7)
where 11 =ϑ if pr0 zzz <≤ , else 01 =ϑ ; 12 =ϑ if
0zz0 <≤ , else 02 =ϑ ; 000 vttz )( −= is the position
of the ring-shaped bunch (1), pr0pr vttz )( −= is the
position of the precursor of transition field,
ε/cv pr = is the maximal velocity of the EM
perturbation propagation in the dielectric waveguide;
))(/(
))(/(
,
00
00
21
vccztt
vccztt
q
εε
εε
±+−
−−
=
,
( ) 222
0nn czttbcy /)/( εελ −−= .
The quasistatic (6) and the transition radiation (7)
components are non-zero in the region przz < . For t>t0
neither Coulomb nor transition radiation fields enter the
region przz > . The fastest and the highest-frequency
component of the electromagnetic signal is the
precursor, which propagates with the velocity vpr [15].
Since the bunch propagation velocity is v0 < vpr, the
field overtakes the bunch. A qualitative pattern of the
propagation of a transition radiation pulse is illustrated
in Fig. 1. The bunch generates an UWB transition
radiation pulse, whose shortest wavelength components
are the components of the precursor. The oscillation
amplitude in the pulse decreases toward the precursor,
so that the total field vanishes at the point przz = .
For simulations we chose an electron bunch of
radius a and length Lb with Gauss density profile. The
characteristics of the transition radiation signal near the
waveguide entrance are illustrated in Fig. 2.
a
b
c
Fig.1. The first harmonic: a – total, b – coulomb, c –
transition field: 10btc =/ , 1br =/ , 50cv0 ./ = , 1=ε
The electromagnetic pulse near the waveguide input end
is characterized by a large field amplitude (15 kV/cm),
high peak power (about 33 MW), and short duration
(less than 1 ns). The spectrum of this signal is
broadband, it has sharp narrow peaks at frequencies
close to the critical frequencies )/( επλ b2cf nn = of
the waveguide. The presence of the low-frequency (f <
f1) part of the spectrum can be explained by the fact that
the transition radiation signal has propagated only a
short distance and, therefore, did not have enough time
to form completely.
a
b
116
c
Fig.2. The characteristics of the transition radiation:
a – field, b – power, c – spectrum. cm4z = , cm1r = ,
cm4b = , 1=ε , 90cv0 ./ = , cm2Lb = , cm50a .=
3.WAKE-FIELD EXCITATION IN A SEMI-
INFINITE DIELECTRIC WAVEGUIDE
In the case of Cherenkov resonance ( ε/cv0 > )
the field, excited by the thin charged ring (1) can be
written [16], similar to [13], in the form of superposition
of spatially limited Cherenkov radiation field and
transition radiation field:
trans
z
cher
z00z EErtzrtE +=),,,,( , (9)
[ ])/(~cos),,(
)(
)/()/(
),,,,(
00n00gr
n n
2
1
n00n0
200
cher
z
vzttzzz
J
brJbrJ
b
Ne4rtzrtE
−−×
×= ∑
ωϑ
λ
λλ
ε ,
(10)
})]()~~()()()[,,(
)()~~()(),,({
)(
)/()/(),,,,(
∑
∑
∑
∞
=
−
∞
=
+−++
+−−×
×=
1m
nm2
m2
2
m2
1
m
n0gr
1m
nm2
m2
2
m2
1
m
prgr
n n
2
1
n00n0
200
trans
z
yJqq1yJz0z
yJqq1zzz
J
brJbrJ
b
Ne4rtzrtE
ϑ
ϑ
λ
λλ
ε
(11)
1zzz 21 =),,(ϑ if 21 zzz <≤ , else 0zzz 21 =),,(ϑ ;
gr0gr vttz )( −= is the position of the group wavefront,
0
2
gr vcv ε/= is the group velocity of the resonance
wave; 2
n0
2
n0 ωω −=~ , 2
21
2
21 qq ,,
~ −= .
a
b
Fig. 3. The topography of the field EZ: (a) – the level
curves of the field EZ, (b) – the respective profile of EZ
at r=0. Level curves are drawn with a step of 0.4 kV/cm
in the range from –4 kV/cm to 4 kV/cm. Dash-dot lines
in figure (b) mark the limits of this range: tc/b=10, b=4
cm, ε=2.6, Lb=1 cm, a=0.5 cm, v0/c=0.9798, Qb=1.6 nC
In Fig.3(a) with the help of level curves the 2D (in the
plane z-r) picture of distribution of longitudinal electric
field excited by relativistic electron bunch with gauss
profile is represented. The position of “group
wavefront” is zgr = 15.6 cm, the coordinate of precursor
is zpr = 24.8cm, and the bunch coordinate is z0 38.7 cm.
In the region zpr<z<z0 the intense Cherenkov wake wave
exists. Structure of this wave is formed as a result of
periodic reflections of Cherenkov cone from the
sidewalls of waveguide. In the region 0<z<zpr the
transition radiation field superimposes on the
Cherenkov field. Weak transition oscillations in the
precursor region 20 cm < z < 23 cm still can be noticed
against the intense Cherenkov field. Behind zgr the field
is small and its structure is different from the one of
Cherenkov wave. Note the high amplitude and small
width of field spikes in Fig.3(b). Ez is maximal at the
waveguide's axis, where the waves, reflected from
sidewalls, are focusing.
4. CONCLUSION
When a charged bunch enters a semi-infinite
cylindrical waveguide, it generates a transition
radiation. If the Cherenkov resonance condition is not
satisfied, the excited electromagnetic field is a
superposition of the quasistatic field of a moving charge
and the transition radiation field. The fastest component
of the field (the precursor) propagates with the velocity
ε/c , which is higher than the bunch velocity.
The spectrum of the transition radiation signal is
broadband and contains peaks corresponding to several
radial modes. The peaks in the spectrum occur at
frequencies somewhat higher than the corresponding
critical frequencies of the waveguide. The transition
radiation signal near the waveguide entrance is
characterized by high peak power and short duration.
If the Cherenkov resonance condition is satisfied,
the excited field consists of spatially limited Cherenkov
radiation field and transition radiation field. Accounting
of the boundary leads to the appearing of the effect of
wake field’s drift after the leading bunch with the group
velocity of resonance wave. This results in limitation of
intense wake field region in the longitudinal direction.
REFERENCES
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the 12th Int. Conf. on High-Power Particle Beams,
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Haifa, Israel, 1998 (IEEE, Haifa, 1998), Vol. 2, p.
756.
10. T. B. Zhang, T. C. Marshall, J. L. Hirshfield. //
IEEE Trans. Plasma Sci. 26, 787(1998).
11. T. B. Zhang, J. L. Hirshfield, T. C. Marshall, and B.
Hafizi. // Phys. Rev. E 56, 4647 (1997).
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(1951).
13. E. L. Burshtein and G. V. Voskresenskij. // Zh.
Tekh. Fiz. 33, 34(1963).
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118
EXCITATION OF WAKE FIELD AND TRANSITION RADIATION
IN A SEMI-INFINITE WAVEGUIDE
BY A RELATIVISTIC ELECTRON BUNCH
E-mail: onish@kipt.kharkov.ua
References
|
| id | nasplib_isofts_kiev_ua-123456789-79284 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T16:41:56Z |
| publishDate | 2002 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Balakirev, V.A. Onishchenko, N.I. Sidorenko, D.Yu. Sotnikov, G.V. 2015-03-30T09:28:35Z 2015-03-30T09:28:35Z 2002 Excitation of wake field and transition radiation in a semi-infinite waveguide by a relativistic electron bunch / V.A. Balakirev, N.I. Onishchenko, D.Yu. Sidorenko, G.V. Sotnikov // Вопросы атомной науки и техники. — 2002. — № 5. — С. 115-117. — Бібліогр.: 16 назв. — англ. 1562-6016 PACS: 52.40.-w; 52.35.-g https://nasplib.isofts.kiev.ua/handle/123456789/79284 The theoretical investigations and simulation are performed for the emission of a relativistic electron bunch
 during the injection into a semi-infinite vacuum/dielectric waveguide. The power and the frequency spectrum of
 high intensity pulse of ultra-wideband transition radiation, excited by a finite size electron bunch in a vacuum
 waveguide, are calculated numerically. It is shown that in a dielectric waveguide the short pulse of Cherenkov wake
 field drifts behind the relativistic bunch with the group velocity. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Plasma electronics Excitation of wake field and transition radiation in a semi-infinite waveguide by a relativistic electron bunch Article published earlier |
| spellingShingle | Excitation of wake field and transition radiation in a semi-infinite waveguide by a relativistic electron bunch Balakirev, V.A. Onishchenko, N.I. Sidorenko, D.Yu. Sotnikov, G.V. Plasma electronics |
| title | Excitation of wake field and transition radiation in a semi-infinite waveguide by a relativistic electron bunch |
| title_full | Excitation of wake field and transition radiation in a semi-infinite waveguide by a relativistic electron bunch |
| title_fullStr | Excitation of wake field and transition radiation in a semi-infinite waveguide by a relativistic electron bunch |
| title_full_unstemmed | Excitation of wake field and transition radiation in a semi-infinite waveguide by a relativistic electron bunch |
| title_short | Excitation of wake field and transition radiation in a semi-infinite waveguide by a relativistic electron bunch |
| title_sort | excitation of wake field and transition radiation in a semi-infinite waveguide by a relativistic electron bunch |
| topic | Plasma electronics |
| topic_facet | Plasma electronics |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/79284 |
| work_keys_str_mv | AT balakirevva excitationofwakefieldandtransitionradiationinasemiinfinitewaveguidebyarelativisticelectronbunch AT onishchenkoni excitationofwakefieldandtransitionradiationinasemiinfinitewaveguidebyarelativisticelectronbunch AT sidorenkodyu excitationofwakefieldandtransitionradiationinasemiinfinitewaveguidebyarelativisticelectronbunch AT sotnikovgv excitationofwakefieldandtransitionradiationinasemiinfinitewaveguidebyarelativisticelectronbunch |