3-D Analysis of Terahertz Frequency Multiplier Excited Due to
Interaction of Convection Electron Beam and Surface Waves
(Smith-Purcell Effect)
Alireza Tavousi
1
, Ali Rostami
1,2
, Ghassem Rostami
2
and Mahboubeh Dolatyari
2
1
Photonic and Nanocrystal Research Lab. (PNRL), Faculty of Electrical and Computer Engineering,
University of Tabriz, Tabriz 5166614761, Iran
2
OIC Research Group, School of Engineering-Emerging Technologies, University of Tabriz, Tabriz 5166614761, Iran
Keywords: Convection Electron Beam, Terahertz Wave Generation, Smith-Purcell Effect, Surface Wave, Frequency
Multiplier.
Abstract: In this paper, we present a three dimensional numerical study on a terahertz frequency multiplier. The
process of frequency multiplication is proposed via study of spontaneous Smith-Purcell (SP) radiation.
Through introducing a single electron bunch perturbation which is passing over above a metallic grating,
and due to interaction of electron bunch with surface wave, the evanescent wave radiates from the ends of
the grating and the SP radiation is generated in the span of 0.466~1.1 THz. We found that the maximum of
SP radiation amplitude is distributed around 90° with the centre frequency 0f ~0.65 THz. Simulations on the
generation of SP radiation at THz frequencies have been performed with the help of the 3D particle-in-cell
(PIC) finite integral method and results agree very well with analytic ones.
1 INTRODUCTION
The ‘terahertz’ term has been applied in diverse
electromagnetic spectra such as for frequency
coverage of point contact diode detectors
(Kerecman, 1973), frequency of a Michelson
interferometer (Fleming, 1974), frequencies below
the far infrared, the resonant frequency of a water
laser, and now terahertz is applied to sub-millimetre
wavelengths span between 100 and 1000 μm (0.3
THz to 3 THz) (Siegel, 2002). Until recently there
were no appropriate sources and detectors for THz
and thus this portion of the EM spectrum has been
used just about. Among the most advanced terahertz
applications which the scientific research is
concerned about, we bring up the followings:
medical imaging, security, communications,
industrial applications, chemistry and biochemistry
measurements, molecular recognition and protein
folding, and sub-millimetre astronomy (Rostami et
al., 2010).
Smith–Purcell (SP) radiation (Smith & Purcell,
1953) which works based on excitation of “surface
Eigen wave” (Bratman et al., 2009; Cao et al., 2014)
in metallic gratings, is one of the most interesting
methods for generating sub-millimetre waves
especially in THz domain (Bratman et al., 1979;
Bratman et al., 2007; Gover & Sprangle, 1981; Liu
& Xu, 2014; Mizuno et al., 1973; Price et al., 1991;
Rusin & Bogomolov, 1966; Schächter & Ron,
1988). At the frequency harmonics of the surface
wave, electron bunches arise and produce
spontaneous Smith–Purcell radiation. Due to
spontaneous nature of SP radiation as discussed in
the most of theoretical works (Andrews & Brau,
2004) and in experiment (Urata et al., 1998) the
radiated power is low. But by using an additional
cavity, higher power and better selectivity can be
obtained (Bratman et al., 2007). This mechanism
which is conceptually shown in Figure 1 is
frequently studied for electron-beam driven
frequency multiplication and undoubtedly it is a
promising candidate for producing radiation in the
terahertz domain (Andrews & Brau, 2004; Bratman
et al., 2009; Li et al., 2006; Shin et al., 2007).
The grating is assumed to be a perfect conductor
and unlike the experiment (Urata et al., 1998) with
pencil-like electron beams, in the proposed design a
sheet electron beam is used which through
enhancing the interaction of electron beam and
surface wave, increases the operating current and
output power. It is known that single-mode
34
Tavousi A., Rostami A., Rostami G. and Dolatyari M..
3-D Analysis of Terahertz Frequency Multiplier Excited Due to Interaction of Convection Electron Beam and Surface Waves - (Smith-Purcell Effect).
DOI: 10.5220/0005335100340039
In Proceedings of the 3rd International Conference on Photonics, Optics and Laser Technology (PHOTOPTICS-2015), pages 34-39
ISBN: 978-989-758-093-2
Copyright
c
2015 SCITEPRESS (Science and Technology Publications, Lda.)
operation of the surface wave in the case of a fairly
narrow beam is possible (Li et al., 2007). However
when the beam width becomes much larger than the
wavelength of the surface mode, different transverse
parts of the beam would be able to excite different
transverse surface modes which their frequencies
and phases are slightly different (Bratman et al.,
2009). This is already mentioned for SP backward
wave oscillators (BWOs) that they will not operate
with an infinitely wide grating because the optical
beam diffuse (Li et al., 2007).
Figure 1: Scheme of an electron-beam driven THz range
frequency multiplier which is self-excited due to surface
waves of open grating (Bratman et al., 2009).
In this paper, we perform a three-dimensional finite
integral based particle-in-cell simulation for the
evaluating the incoherent spontaneous SP radiation
using CST studio suite, a code which simulates the
situations in which the space charge and
electromagnetic fields are concurrently interacting.
2 SIMULATION MODEL
Here in this paper, we consider a metallic grating
slab having a period L and thus a wave number of K
= 2π/L (the length of the grating is set differently for
each particular simulation case). Generally, an
evanescent wave can travel above the surface of
such grating in the direction in which the slab is
periodic (perpendicular to the grooves). The phase
velocity is v
φ
= ω/k = cβ
φ
, and the group velocity is
v
g
= dω/dk = β
g
c where c is the speed of light, ω is
the frequency of the traveling evanescent wave and k
is the wave number. It is known that (Andrews et al.,
2006) the dispersion relation is periodic in k space
and for each Brillion zone the dispersion curve is
symmetric about k/K = 0.5 (π-type surface mode).
Depending on the position of the synchronous point
which may be located on the right-hand or left-hand
side of the Bragg point k/K=0.5 (for which the group
velocity vanishes), the operating characteristics of
the device changes fundamentally. For k/K > 0.5, the
group velocity is negative and for k/K < 0.5 the
group velocity is positive. Here we are interested in
the former case in which the operating point lies in
the negative group velocity regions. If we assume
that there is no gain or losses in the grating due to an
electron beam then we may use Floquet’s theorem in
order of solving Maxwell equations on top of the
grating and the dispersion relation D
0
(ω, k) = 0 is
obtained for the evanescent waves.
With respect to the operating point, our Smith-
Purcell based THz source is a backward wave
oscillator in which, if a certain beam current
threshold known as start current is reached, the
optical intensity grows to saturation even if no
feedback mirrors are employed. In this device, the
moving electron beam must interact significantly
with the fundamental surface mode which is
confined very close to the grating. To describe the
interaction between the surface mode and the
moving electron beam, we need to numerically solve
the coupled Maxwell-Lorentz equations. For this
purpose, we used the CST studio suite software
which solves these coupled equations via finite
integral method. The radiation wavelength
λ
observed at the angle
θ
measured from the direction
of moving electron beam is given by:
11
(cos),
||Ln

(1)
where n is the diffraction order. In order to compare
our 3-D simulation results with experiment results
and 2-D simulations, we chose to use grating
parameters of Urata et al (Urata et al., 1998)
summarized in Table 1. Since we have performed
our simulations in a 3-D scheme, a width of 800 µm
is used for grating. The grating length is arranged to
be 20, 40, 60, 80, and 100 periods. The main
simulations are performed using a grating with
length Ng=100 periods and then the results are
extended and compared width grating lengths of
Ng=20, 40, 60, and 80 periods.
Table 1: Grating profile used in the experiments of Urata
et al (Urata et al., 1998).
Grating period 173µm
Groove width 62µm
Groove depth 100µm
2 SIMULATION RESULTS
The dispersion diagram of the first three modes of a
grating with parameters from Table 1 is shown in
Figure 2. The 40 Kev beam line intersect with the
first mode in negative group velocity region i.e.
k/K>0.5, thus the synchronous point frequency
3-DAnalysisofTerahertzFrequencyMultiplierExcitedDuetoInteractionofConvectionElectronBeamandSurface
Waves-(Smith-PurcellEffect)
35
would be calculated as 0.466 THz. For a device
operating in positive group velocities, higher beam
energies is required. For example in this figure we
have shown the 250 Kev beam line which is
intersecting the evanescent mode in k/K<0.5 region.
To study the incoherent (spontaneous) SP
radiation, first we let the length of the grating to be
N
g
=100L and then we perform the simulation with a
single electron bunch which last for 0.1 ps length
and carry a charge equal to q=0.048 pC. Since this
bunch is short enough compared to the longest
radiation wavelength, we can assume that the
radiation is coherent. Our focus is on the first order
SP radiation (|n|=1). As shown in Figure 3a-f, by
recording the temporal behaviour of the radiated
signal which we have detected them at different
detection points of
θ
=70º, 90º, 125º, 135º, 150º, and
170º
(and 5.5 mm distance from the centre of the
grating) and taking fast Fourier transform, we find
out that two clear radiation peaks exists in the
spectrum. Figure 4a shows that the one which
peaked at 525 GHz is the SP radiation, while the
other peak at 466 GHz is the evanescent wave (for
ease of visualization, only FFT of Figure 3c at
θ
=135º is shown in Figure 4a).
Figure 4b compares the analytic expression for
wavelength of SP radiation with the centre
frequencies recorded by sweeping
θ
with 6º step for
all detection angles between 0º<θ<180º. From this
figure we find out that indeed
the longest radiation
wavelength (lowest radiation frequency) is equal to
the wavelength of the evanescent wave. The results
obtained from CST are in good agreement with
those obtained via equation (1). Recording the FFT
amplitude for all swept angles (Figure 5),
we find
out that unlike the evanescent wave frequency
which is angle independent, both SP radiation centre
frequency and amplitude are changing with
detection angle. As shown in this figure the
maximum SP amplitude occurs in angle θ=90º. The
short come of this equation is that it says nothing
about the SP radiation amplitude.
Given the detection angle, period, diffraction
order, and the phase velocity of the electron beam to
the analytic equation (1), one can predict the
Figure 2: Dispersion diagrams of the first three surface
modes of the metallic grating with parameters given in
Table 1. The 40 Kev and 250 Kev beam lines intersect
with the fundamental mode in k/K<0.5 and k/K>0.5
regions respectively.
Figure 3: The temporal behaviour of the radiated signal detected at the angle θ=135º and 5.5 mm distance from the centre of
the grating for Ng=100L.
PHOTOPTICS2015-InternationalConferenceonPhotonics,OpticsandLaserTechnology
36
Figure 4: (a) FFT taken from temporal behaviour of the radiated signal detected at the point θ=135º for Ng=100L. (b)
Comparison of analytic expression for wavelength of SP radiation with the center of SP frequencies swept from θ=0º to
θ=180º.
Figure 5: FFT taken from temporal behaviour of the radiated signal detected at the different points varying from
12º<θ<180º with step of
θ =6º for Ng=100L.
wavelength in which the SP radiation occurs. Figure
6 shows the normalized far field polar plot for the H-
field at different θ angels. These plots show that the
main radiation lobe of each specific frequency is
located in the angle which was predicted by equation
(1). In the 0.466 THz, our result show that radiation
occur near 180º, and as we know from previous, this
frequency belongs to the evanescent wave so
radiates from the upper grating end (near gun). The
0.55 THz, 0.65 THz, and 0.85 THz cases are also
radiating at 120º, 90º, and 40º, respectively.
Figure 7a, and Figure 7b both show the FFT
amplitudes of time signals for different grating
lengths. In Figure 7a, we swept the detection angle
from 0º to 180º and recorded the maximum of FFT
amplitude for gratings with Ng=20, 40, 60, 80, and
100L. We can see that, as the length of grating
increase from 20 to 100 periods, the amplitude
grows nonlinearly. Given this, we are able to
calculate the growth-rate of the produced signal. In
Figure 7b, focusing only on one detection angle in
which the radiation is maximum (i.e.
θ
=90º), we
find out that by increasing the grating length, not
only the FFT amplitude increases but the spectral
resolution of the signal increases too.
a)
b)
3-DAnalysisofTerahertzFrequencyMultiplierExcitedDuetoInteractionofConvectionElectronBeamandSurface
Waves-(Smith-PurcellEffect)
37
Figure 6: Far field polar plots for H-field at different frequencies of a) 0.466 THz, b) 0.55 THz, c) 0.65 THz, and d) 0.85
THz. The plots are normalized.
Figure 7: (a) The max FFT amplitude of time signals versus detection angel for different grating widths of Ng=20L, 40L,
60L, 80L, and 100L (b) The max FFT amplitude of time signals versus detection angel of
θ
=90º for different grating
widths of Ng=20L, 40L, 60L, 80L, and 100L.
a)
b)
PHOTOPTICS2015-InternationalConferenceonPhotonics,OpticsandLaserTechnology
38
3 CONCLUSIONS
In this paper, we presented a three dimensional
numerical study on a terahertz frequency multiplier.
This SP based frequency multiplier can provide
terahertz radiation with a comparatively high level
of output power. The process for frequency
multiplication was proposed by studying the
spontaneous SP radiation. This non-relativistic
device neither requires a high operating voltage nor
a high external magnetic field. It can fill up the need
for convenient compact THz radiation source for
diverse range of applications such as spectroscopy
and diagnostics of different media. Simulations on
the generation of SP radiation at THz frequencies
have been performed with the help of the 3D
particle-in-cell (PIC) finite integral method and
results agree very well with analytic ones.
ACKNOWLEDGEMENTS
The authors would like to thank University of
Tabriz.
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3-DAnalysisofTerahertzFrequencyMultiplierExcitedDuetoInteractionofConvectionElectronBeamandSurface
Waves-(Smith-PurcellEffect)
39