- Email: [email protected]
On the generation of electromagnetic waves in the terahertz frequency range
Physics Letters A 375 (2011) 2759–2766
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
On the generation of electromagnetic waves in the terahertz frequency range V.A. Namiot a,∗ , L.Yu. Shchurova b a b
Microelectronic Division, Research Institute of Nuclear Physics, Moscow State University, Vorobyovy Gory, 119992 Moscow, Russia Department of Solid State Physics, P.N. Lebedev Physical Institute of Russian Academy of Science, Leninsky Prospect 53, 119991 Moscow, Russia
a r t i c l e
i n f o
Article history: Received 25 February 2011 Accepted 7 March 2011 Available online 7 June 2011 Communicated by V.M. Agranovich
a b s t r a c t It is shown that a thin dielectric plate, which can act as an open dielectric waveguide, it is possible to produce amplification and generation of electromagnetic waves with frequencies in the terahertz range. For this purpose, we propose using a dielectric plate with a corrugated surface, in which case the electric field of the transverse electromagnetic wave in the waveguide has a periodic spatial structure in the local area near to the corrugation. Terahertz electromagnetic waves are excited by a beam of electrons moving in vacuum along the dielectric plate at a small distance from its corrugated surface. Corrugation period is chosen in order to ensure the most effective interaction of the electron beam with the first harmonic of the electric field induced by the corrugation. Amplification and generation of electromagnetic waves propagating in a dielectric waveguide is realized as a result of deceleration of the electron beam by a wave electric field induced by a corrugated dielectric surface in the zone near the corrugation. We discuss possible ways to create electron beams with the desired characteristics. We offer a way to stabilize the beam position above the plate, avoiding the bombardment of the plate by electrons. It is shown that it is possible to significantly increase the efficiency of the device through the recovery of energy that remains in the electrons after their interaction with the wave. © 2011 Elsevier B.V. All rights reserved.
1. Introduction and problem description Currently, most experts believe that the age of vacuum electronic radio devices has largely passed, and they have lost ground in almost all areas to solid state electronics. However, technology advances in recent years, that have been successfully used in the manufacturing of modern solid state electronic devices (such as microlithography), can, in principle, change the situation. This may enable creation of electrovacuum devices that can solve problems that cannot be solved, or are difficult to deal with using current methods. These are devices that generate electromagnetic waves at frequencies ranging from one to several terahertzes. It is this range of frequencies that has recently come into focus of great interest because of its promising applications in medical, biological and industrial research, radar, space science, etc. [1]. Unfortunately, despite significant progress in the study of terahertz radiation in recent years [2–5], this range is mastered much less than others and the parameters of such systems are inferior to parameters systems of other frequency ranges. However, according to our estimates, the parameters of discussed here electrovacuum devices, continuous-wave terahertz generators, could be significantly better
*
Corresponding author at: Microelectronic Division, Research Institute of Nuclear Physics, Moscow State University, Vorobyovy Gory, 119992 Moscow, Russia. E-mail address: [email protected] (V.A. Namiot). 0375-9601/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2011.05.061
than the existing ones. The power of such devices could reach the level of watts, and efficiency may exceed (50–80)%. Traditionally, the electron beam is created in the electrovacuum microwave generator. That electron beam interacts with the electromagnetic wave and transfers its energy to this wave. The electrons moving along the direction of electromagnetic waves interact effectively with this wave when the electron velocity is close to the phase velocity of the wave. However, since the electron velocity is much less than the speed of light, slow-wave structures are used to reduce the wave velocity. This ensures the most effective interaction of the electron beam with an electromagnetic wave and, as a result, amplification and generation of electromagnetic waves. However, the size of slow-wave structures should be comparable with wavelength. There is a high absorption of electromagnetic radiation in small slow-wave structures that are required to slow the terahertz waves. This significantly affects the generation condition. It is extremely difficult both to slow down a wave and to avoid losses in this frequency range. In this Letter, we propose a scheme for a device in which the generation of terahertz electromagnetic waves is the result of the interaction of the electron beam with an electromagnetic wave, as it is in electrovacuum devices. However, the proposed scheme does not require a slow-wave structure. As will be shown below, it is possible that the speed of the electromagnetic wave is close to the speed of light in vacuum, but the electrons are still able to effectively transfer their energy to it.
2760
V.A. Namiot, L.Yu. Shchurova / Physics Letters A 375 (2011) 2759–2766
The proposed here scheme of the device includes bordering the vacuum thin dielectric plate that can act as an open waveguide in which electromagnetic waves in the terahertz frequency range can propagate. In this case, the electromagnetic field partially penetrates the wall of the dielectric plate in the surrounding space, decaying exponentially with distance from the insulator. So, the electromagnetic wave field is focused on the region which includes the plate itself and some area near this plate. Terahertz electromagnetic waves are excited by a beam of electrons moving in vacuum along the corrugated surface of the dielectric plate at a small distance from it. A beam of electrons moving in the transverse direction of the waveguide (in the direction perpendicular to the direction of wave propagation) can interact with the electric field of transverse electromagnetic wave (H-wave). However, the electric field of the transverse wave is uniform (does not contain spatial harmonics) for the dielectric plate with a flat surface, so the effective interaction between this wave and the electron beam should be absent. It is therefore proposed to create a waveguide with a corrugated surface. It is obvious that the wave will change in this case, compared with the unperturbed wave propagating in a plate with a smooth surface. Indeed, the unperturbed wave causes the polarization of the dielectric in the zone of the corrugation, and this polarization, in turn, induces a secondary field which is superimposed on the unperturbed. And this secondary field penetrates into the region outside the plate. The characteristic scale, which changes the secondary field, is determined by the corrugation period. Since the corrugation at the boundary of the dielectric plate and the vacuum has a periodic structure, the induced electric field near the boundary can be regarded as the sum of an infinite set of harmonics. By selecting the size of the period, we can ensure that the electron beam, flying over the corrugated surface at sufficiently low altitudes, would effectively interact with the first harmonic of the electric field near the corrugation and would give its energy to the wave. It is important that in such a scheme, the interaction of the electron beam with the electromagnetic wave is sufficiently effective without slowing down the wave. In the proposed scheme the electromagnetic wave propagating in the waveguide can be regarded as strictly transverse. Therefore, a beam of electrons interacting with the electric field wave should move in a perpendicular direction. As a result of the interaction, the amplitude of the electric field of the transverse electromagnetic wave increases, while the wave itself propagates along the axis of the waveguide at a speed close to the speed of light in vacuum. The characteristic scale on which the secondary field penetrates into the region above the plate is also determined by the corrugation period. Therefore, in order to have an efficient energy transfer from electrons to the wave, it is necessary that the beam would be, firstly, thin enough and, secondly, that it moved at a very low altitude over the plate. In order to fulfill these requirements, it is necessary, firstly, to propose a method to create the thin beam with the desired characteristics, and, secondly, to propose a method for stabilizing the beam position above the plate, avoiding the bombardment of the plate by electrons. Both of these problems are considered in this Letter. In addition, in this Letter we consider the efficiency of the proposed device. It is shown, how to bring its value to unity due to recovery of the electronic energy, which remained after the interaction of electrons with the wave. 2. Wave electric field in dielectric plate with corrugated surface The proposed here scheme of the generator includes bordering the vacuum thin dielectric plate with the thickness 2a and permittivity ε (Fig. 1). This dielectric plate can act as an open waveguide [6, p. 133] in which electromagnetic waves, including waves in the
Fig. 1. Dielectric plate with a flat surface (a) and with a corrugated surface (b).
terahertz frequency range ω , can propagate. For sufficiently thin plate a c /ω and for not too large values of ε , the electromagnetic wave energy is mostly concentrated in the region outside the plate, so these waves propagate with the velocity, that close to the speed of light, and have the wave vector k ≈ ω/c. We will consider the special case of a magnetic wave (H-wave), propagating in the z-direction. The electric field of this wave is directed along the x-axis. The beam of electrons, interacting with the electric field wave, should move also in the x-direction. Precisely the H-wave is of interest to us because an antinode of the H-wave electric field is in the center of the cross-section of this plate. In our problem the electric field of this wave quite well penetrates into the region above the plate, where the electrons move. For a plate with a flat surface (Fig. 1(a)), the electric field of this wave can be written as
E(r, t ) =
Aex cos(α y ) exp(ikz − i ωt ),
| y| a
Bex exp(−β| y |) exp(ikz − i ωt ),
| y| a
(1)
here: ex is a unit vector being directed along an axis x, A ≈ B , α ≈
2 (ε − 1)1/2 ωc , β ≈ (ε−c12)ω a.
However, the electric field of the H-wave is uniform in the flat dielectric plate, so the effective interaction between this wave and the electron beam should be absent. Therefore, we propose to create a dielectric plate with a corrugated surface. Then additional spatial modes are induced by a corrugated dielectric surface in the zone near the corrugation. The electron beam may interact with these modes. Let the plate thickness varies under a periodic law along the x-axis (Fig. 1(b)). Take y = a + R cos(k x) as an instance, and as it will be clear further, it is worth to consider k R ∼ 1 and to accept the condition:
k k
(2)
under which the interaction of electromagnetic waves with electrons and amplification of this wave would be the most effective. This plate can also serve as a waveguide for the H-wave, and its propagation velocity is still be close to c. However, in this case the electric field of this wave is inhomogeneous along the x-axis. The electric field of the wave E (r, t ) can be presented as E (r, t ) = E(r, t ) + F(r, t ), where the function E(r, t ) is determined by formula (1). And the electric field F(r, t ) is comparable with E(r, t ) only in a very narrow area (with a width of about (k )−1 ) near to y = a + R. But F(r, t ) is significantly less than E(r, t ) outside of this area. Owing to that the corrugated surface is described by a periodic function of x the induced electric field F(r, t ) can be written as the sum of an infinite set of harmonics:
F(r, t ) = G(x, y ) exp(ikz − i ωt )
=
∞ n=−∞
Fn ( y ) exp ik nx exp(ikz − i ωt )
(3)
V.A. Namiot, L.Yu. Shchurova / Physics Letters A 375 (2011) 2759–2766
2761
Let us assume that we have created the electron flux moving with a velocity v 0 along the x-axis at a height of l ∼ (k )−1 above the top edge of the corrugated surface. (That is electrons move in a perpendicular direction to the propagation of electromagnetic waves.) We can define the corrugation period (the corrugation period determines the value of k ) that the condition
k v 0 = ω + η
(4)
(where ω |η|) is satisfied. In this case, the effective interaction of electrons with the first harmonic of the electric field is achieved. Indeed, according to (3) the electron flux interacts effectively with that part of an electric field of the electromagnetic wave which is described by a term
g1 (x, y ) exp(ikz − i ωt ) ≡ F1 ( y ) exp ikz + ik x − i ωt
Fig. 2. The boundary between the dielectric plate and vacuum: 1 is dielectric with permittivity ε , 2 is vacuum. The axis O is defined by a condition y = a + R, axis O is exactly on the tops of the corrugation.
As we have v 0 c (in real electro-vacuum devices), Eq. (2) follows directly from (4). (Thus, it does not matter any more that kv 0 = ω .) Thus, in our problem the electromagnetic wave propagates in the waveguide having the shape of a thin dielectric plate with a thickness that varies periodically in the transverse direction. This wave may be amplified by the electron beam, also moving in the transverse direction to the wave. Efficiency of energy transfer from electrons to the wave is determined by F1 ( y ). Let’s proceed to calculation of the electric field F(r, t ) induced by the corrugated surface. Due to the smallness of the corrugation size (typical size of the corrugations (k )−1 ) is compared to the electromagnetic wavelength, the electric field of the corrugation can be considered as a potential and can be described it the terms of a scalar potential φ(x, y ). It means that the function G(x, y ) can be presented as
G(x, y ) = −∇φ(x, y ) = −∇
∞
Fig. 3. 1 is the corrugation described by the equation y = R cos(k x), and 2 are ellipsoids modeling the corrugated surface in the region h.
pn ( y ) exp ik nx
n=−∞
The function φ(x, y ) can be presented as φ1 (x, y ) inside the plate, and φ2 (x, y ) outside the plate. These functions satisfy the Laplace equation φ1 (x, y ) = 0 and φ2 (x, y ) = 0 within their ranges of definition. Then the boundary conditions are as follows: the tangential electric fields are continuously and the normal electric fields differ by the factor ε .
E0 − ∇φ1 (x, y ) τ (x, y ) = E0 − ∇φ2 (x, y ) τ (x, y ) ε E0 − ∇φ1 (x, y ) n(x, y ) = E0 − ∇φ2 (x, y ) n(x, y )
−1
|E0 | exp −k ( y − a − R )
(6)
(7)
The coefficient q1 (k R ) depends only on the dimensionless parameter k R, and q1 (k R ) can be expressed in terms of the Fourier component of the function φ2 (x, a + R ):
(k )2 q1 k R = 2 π |E 0 | i
2
π /k
φ2 (x, a + R ) exp −ik x dx
0
q1 k R =
(5)
Here E0 = A cos(αa)ex ≈ Aex is the electric field of an electromagnetic wave (without facient exp(ikz − i ωt )) on the plate boundary, if no corrugated surface; τ (x, y ) is a tangent vector to the curve y = R cos(k x) in a point with coordinates (x, y ), and n(x, y ) fulfills a condition n(x, y )τ (x, y ) ≡ 0. This situation is shown in Fig. 2. The function F1 ( y ) is expressed in terms of p 1 ( y ). The function p 1 ( y ) in the area y (a + R ), where electrons move, can be written as
p 1 ( y ) = iq1 k R k
In fact, the of the coefficient q1 (k R ), multiplied by exp(−k l), determines the interaction of the electron beam with an electromagnetic wave. With an increase in the absolute value of this product the interaction becomes more effective. Let us consider a case of k R 1 (this case corresponds to a very low corrugation).
(8)
ε−1 kR ε+1
(9)
In the opposite situation, when k R 1 (a very high corrugation), we can use the following method. The induced by corrugated surface electric field along the x-axis is defined, in this case, approximately by the region R − (k )−1 y R. In this region the corrugations can be approximated by a periodic sequence of ellipsoids as shown in Fig. 3. In order to simulate the corrugated surface by the ellipsoids in an area of a size h not far from the axis O (the condition h ∼ (k )−1 should be fulfilled for h), these ellipsoids could be described by the equation
x2 a21
+
y2 b21
+
z2 c 12
=1
where a1 = (k )−1 b1 = R and c 1 → ∞. The problem of the electric field induced by an ellipsoid in a medium has the analytical solution (see [6, p. 52]). Using this solution and taking into account that k R 1 we obtain the equation
q1 k R =
ε−1 (2π )1/2 ε (k R )1/2
(10)
2762
V.A. Namiot, L.Yu. Shchurova / Physics Letters A 375 (2011) 2759–2766
Fig. 4. Scheme, that includes “high-frequency potential” to stabilize the electron motion: 1 is a dielectric plate, 2 is an electronic beam moving above the plate with a velocity v 0 , 3 is a conducting layer under the earth potential, L is a distance between 1 and 3, 4 are conducting layers under a potential ϕ1 with respect to earth, 5 are conducting layers under a potential ϕ2 with respect to earth. No contact between conducting layers 4 and 5.
Approximation of (9) and (10), which is true in the ranges both large and small k R is
q1 k R ≈
k R ε−1 (2π )1/3 ε2/3 ε+1 1+ 2 3/4 2/3 k R
(11)
( ε +1 )
This approximation is not only possible. Nevertheless, we hope that we can use this expression for the intermediate k R at least as a rough estimate. Let ε 1. The expression (11) has a maximum q1 (k R ) ≈ 0.3 for k R ≈ 0.9. A similar approximation for the corrugated surface of the meander form is
q1 k R ≈
8(ε − 1)
π 2 (ε + 1)
arctg
π 2
k R
(12)
This expression has the maximum q1 (k R ) ≈ 1.2. As it follows from the expressions (11) and (12), it is meaningless in using large (k R ) to increase q1 (k R ). It is possible to restrict by the condition (k R ) 1 in this case. Thus, we have the expression for the x-component of the electric field of the first harmonic in the vacuum at a distance l from the top of the corrugation:
F 1x = q1 k R exp −lk E 0 exp ikz + ik x − i ωt
We have shown the electron beam, flying with the velocity v 0 ≈ ω/k over the corrugated surface at a distance l (k )−1 , can interact effectively with the first harmonic of the electric field induced by the corrugation with the amplitude R and period ∼ (k )−1 ((k R ) ≈ 1). 3. Beam movement stabilization above the plate The condition l (k )−1 must be fulfilled for effective interaction of the electromagnetic wave with an electron beam. Even at
high enough velocity v 0 = 6 × 107 m/s (that corresponds to the electron energy of about ten keV) and the frequency ω = 6.28 × 1012 s−1 (that corresponds to one terahertz) we have (k )−1 ≈ 10 μm. Thus, the beam should move very close to the plate surface. Therefore, even a relatively small y-component of velocity or an electric field accelerating the electrons in the same direction leads to the plate bombardment by the beam. It should be also noted that the ratio |ex F1 (x, y )| = |e y F1 (x, y )| takes place (where e y is a unit vector being directed along the y-axis). So with increasing interaction of the beam with a field of electromagnetic wave, the displacement along the y-axis also increases. Therefore, in order to avoid the plate bombardment by electrons, it is necessary to reduce an energy transfer from the beam to the wave. Either the plate width could be reduced that electrons have no time to reach the plate for the flight time, or the electron movement along the y-axis could be stabilized somehow to avoid the surface bombardment by electrons. It is possible to reduce transit time, but it leads to decreasing the generation efficiency. It seems that the movement of electrons along the y-axis could be stabilized by the easiest way with the help of a magnetic field being directed along the x-axis. However the required field magnitude that is defined by the condition (k )−1 rl , where rl is a Larmor radius, is unacceptable high-significantly more than one tesla in this case. Creating such fields is extremely difficult in real devices. No magnetic fields are required, if to consider another opportunity to stabilize the beam along the y-axis. The movement of electrons can be stabilized by using the so-called “high-frequency potential” [7]. Taking into account that it is rarely mentioned in the literature for recent years and the application of this “potential” is specific to the considered problem, an independent conclusion of formulas describing this stabilization is worthwhile to present. Let’s consider a problem about stabilization of the electron beam movement along y in the following scheme (Fig. 4). Let an electron moves with a velocity v 0 in parallel to the flat dielectric
V.A. Namiot, L.Yu. Shchurova / Physics Letters A 375 (2011) 2759–2766
plate 1 with conducting areas 4 and 5 on it (in general, these areas have an extremely high resistance, as no current should be in them). Conducting plate 3 is located at a distance L above the dielectric plate with the potential on the conducting plate being equal to zero. The widths of areas 4 and 5 are the same and equal to π (k )−1 . The potential ϕ1 is applied to the layer 4, and the potential ϕ2 is applied to the layer 5. Let us designate a difference ϕ1 − ϕ2 as ϕ , and a half-sum (ϕ1 +ϕ2 ) as ϕ¯ . The potential ϕ (x, y ) in the area between 1 and 2 3 where the beam moves satisfies to the Laplace equation with the corresponding boundary conditions on layers 4 and 5, and on conducting plate 3 too. Solving the Laplace equation (under the condition of L (k )−1 ) we obtain:
y
ϕ (x, y ) = ϕ¯ 1 − +
m
d2 y
dt 2
πn
1 − (−1)n sin nk x exp −nk y
(13)
∂ ϕ ( v 0 t + δ x(t ), y ) ∂y ∂ ϕ ( v 0 t + δ x(t ), y ) = −e ∂δ x(t )
(14) (15)
(here m and e are a mass and a charge of an electron). Solution (14) can be presented as y ≡ y (t ) = y 0 (t ) + δ y 1 (t ), where y 0 (t ) is a slowly varying in time part of y (t ), and δ y 1 (t ) is a quickly varying in time part of y (t ). Suppose that δ x(t ) v 0 t and δ y 1 (t ) y 0 (t ). In this case δ x(t ) and δ y 1 (t ) satisfy to the equations:
m
d2 δ x(t ) ∞ e ϕ k
1 − (−1)n cos nk v 0 t exp −nk y 0 (t )
π
n =1
m
(16)
d2 δ y 1 (t ) dt 2
=
∞ e ϕ k
π
n =1
∞ n =1
1 − (−1)n sin nk v 0 t exp −nk y 0 (t )
e ϕ k
π m(nk v
δ y 1 (t ) = −
0
)2
1 − (−1)n
(17)
× cos nk v 0 t exp −nk y 0 (t )
∞
e ϕ k
π m(nk v 0 )2 n =1
1 − (−1)n
(18)
(19)
(As follows from (18) and (19) in the limits of large k and v 0 the assumption about the smallness of δ y 1 (t ) in comparison with y 0 (t ) and δ x(t ) in comparison with v 0 t should be fulfilled.) Using (18) and (19) let us write an equation for y 0 (t ):
d2 δ y 0 (t ) dt 2
=
∼ sin2 nk v 0 t + cos2 nk v 0 t = 1
we obtain:
m
d2 δ y 0 (t ) dt 2
= e ϕ¯
1
−
L
∞ (e ϕ k )2 (−nk )
1 − (−1)n
π 2m(nk v 0 )2
n =1
2
∞ e ϕ k (−nk ) n =1
exp −2nk y 0 (t )
Eq. (21) can be rewritten as follows:
m
d2 δ y 0 (t ) dt 2
π
1 − (−1)n
= −e
∂ (ϕ1 + ϕh ) ∂ y0
(22)
y (t )
where ϕ1 = −ϕ¯ 0L , and ϕh is a so-called “high-frequency potential”, which is in the given case as:
ϕh =
∞ e (ϕ )2 n =1
2π
2 mv 2 n2 0
1 − (−1)n
2
exp −2nk y 0 (t )
(23)
The first term of sum (23) actually works. Thus:
ϕh =
2e (ϕ )2
π
2 mv 2 0
exp −2k y 0 (t )
(24)
Consider the sum ϕa ( y ) = ϕ1 ( y ) + ϕh ( y ). If ϕ¯ < 0, it has a minidϕ mum in a point y 2 , in which dya = 0. Hence:
1 2k
ln −
π 2 ϕ¯ mv 20 4L ek (ϕ )2
(25)
At k = 105 m−1 , ϕ = 600 V, v 0 = 6 × 107 m/s, ϕ¯ = −600 V, L = 2 × 10−3 m we have y 2 = 10 μm. Thus, y 2 ∼ (k )−1 . The expand ϕa in a power series in the neighborhood of point y 2
|ϕ¯ | L
( y − y 2 )2
(26)
A displacement of y from y 2 even at 2 microns at the parameters specified here creates a returning electric field of the order of 105 V/m. So, to stabilize the electron beam along y one can simply cover the corrugated surface by very thin conducting layers of a very high resistance (in order not to worsen the electromagnetic wave absorption) and apply a constant voltage to the layers by using the scheme that is presented in Fig. 4. (Some difficulties are connected with preventing an electric breakdown along the dielectric surface, but they are apparently overcomable.) 4. Electron beam interaction with electromagnetic wave and generation condition
× sin nk v 0 t exp −nk y 0 (t )
m
sin nk v 0 t δ y 1 (t ) − cos nk v 0 t δ x(t )
ϕa ( y ) = ϕa ( y 2 ) + k
Let us solve (16) and (17) considering approximately that y 0 (t ) does not depend on time at all:
δ x(t ) =
y2 = −
dt 2
=−
where the over-line denotes time-averaging. Averaging (20) over time, by using the identity
(21)
= −e
dt 2 d2 δ x(t )
(20)
L
(the condition y = 0 corresponds to the bottom dielectric plate on which layers 4 and 5 are located). Let us write movement equations for electrons:
m
× exp −nk y 0 (t )
∞ ϕ n =1
2763
× sin nk v 0 t δ y 1 (t ) − cos nk v 0 t δ x(t )
Let us now consider the problem of interaction of an electromagnetic wave with the electron beam. In our study frame electrons move in the transverse direction (x direction) of the waveguide in a vacuum at the small distance from a corrugated surface, that is, in large nonuniform electric field region. This field is due to the polarization of the corrugated surface of the dielectric in the electric field of the H-wave in waveguide. Since the field’s amplitude of the first spatial harmonic is the largest of the set of all spatial harmonics of the nonuniform electric field, we consider the interaction of electrons with only the
2764
V.A. Namiot, L.Yu. Shchurova / Physics Letters A 375 (2011) 2759–2766
first harmonic. This first harmonic field is part of the H-wave propagating in the waveguide, so it depends on time as well as the of H-wave field by a factor exp(−i ωt ). The most effective interaction between the first harmonic’s wave and the electrons happens under the condition of synchronism, when phase velocity of the wave is close to the electron beam velocity. Due to the change of the electron velocity in the electric field, the electron beam becomes velocity modulated, so electron density wave is formed. In this case, the electrons with a velocity greater than the phase velocity of the wave are decelerated in the electric field wave and give a part of its energy to the wave. The increase in the amplitude of the electromagnetic wave causes, in turn, increases in the amplitude of the electron density wave. Then, an increasing part of the electrons gives its energy to electromagnetic wave, thus amplifying the wave. This leads to an increase in the amplitude of the electron density wave. Thus, there is an oscillation mode. To describe the system of electrons interacting with the electromagnetic field, a coupled system of equations, that take into account both the change of electromagnetic fields, and the change in velocities and densities of the electronic system, is required. The form of a solution describing the electric field of the first harmonic wave is presented above. The contribution of the electric field of this wave is described by a term F1 ( y ) exp[ikz + (ik x − i ωt )] (since the boundary of the insulator and the vacuum is homogeneous along the z-axis we can take z = 0 without loss of generality). The system of electrons interacting with the wave electric field will be described within the hydrodynamic approach. We assume that the beam shift along the y-axis is suppressed, and to describe the electron velocity v (x, t ) and electron density n(x, t ) we use the equation of motion (the Euler equation) and the continuity equation
∂ v (x, t ) ∂ v (x, t ) + v (x, t ) ∂t ∂x e = q1 Rk exp −lk E 0 exp ik x − i ωt
m ∂ n(x, t )
∂t
+
(27)
∂ v (x, t )n(x, t ) =0 ∂x
(28)
Let us represent v (x, t ) and n(x, t ) as v (x, t ) = v 0 + δ v (x, t ) and n(x, t ) = n0 + δn(x, t ), where n0 is the unperturbed electron concentration (i.e. the electron concentration in absence of an electromagnetic wave). Solving (27) and (28) in a linear approximation over δ v (x, t ) and δn(x, t ) without taking into account small-value terms we obtain:
e
δ v (x, t ) ≈ −i
mη
q1 Rk exp −lk
iηx × E 0 1 − exp − exp ik x − i ωt v0
δn(x, t ) ≈ n0
eq1 ( Rk ) exp(−lk ) E 0 iηx k exp − mη v 0 v0
×
i v0
η
exp
iηx v0
(29)
− 1 + x exp ik x − i ωt
(30)
Let us denote by a symbol W the energy transferred by electrons to an electromagnetic wave in a unit of time and per unit area of the plate that is averaged over a distance passed by the beam across the plate. Using (29) and (30) for W , we obtain:
W ≈ J0
es2 k mv 20
q21 Rk exp −2lk | E 0 |2 χ ( z1 )
(31)
Fig. 5. The function χ ( z1 ) determines the efficiency of energy transfer from electron density wave to the electromagnetic wave. The maximum of this function χ ( z1 ) corresponds to the most favorable conditions for the generation of the terahertz ηs waves. Dimensionless argument z1 = v characterizes departure of the electron ve0 locity (on the length of electron flight s) from the phase velocity of electromagnetic wave ω/k (ω is http://www.multitran.ru/c/m.exe?a=110&t=4345104_2_1&sc=867 frequency in the terahertz range, k is the corrugation wavevector) normalized to the average electron velocity v 0 .
Here J 0 is a beam current per unit length, s is a plate width, z1 = ηs , the function χ ( z1 ) has a form v 0
sin( z1 ) 2 χ (z1 ) = − 3 cos(z1 ) − 1 + 2 z1
z1
The plot χ ( z1 ) is shown in Fig. 5. This figure shows, that in conditions when the electron velocity is strictly equal to the phase velocity of electromagnetic wave ω/k (this corresponds to η = 0), energy transfer from the beam to the wave is absent. The point is that in this case, the wave of the electron density is shifted by exactly π /2 with respect to the wave field interacting with the beam. Thus the maximum electron density falls on the area where the electromagnetic field is zero. In order to transmit energy from the beam to the wave, the electron’s velocity should be slightly higher than ω/k . This excess of velocity should be small enough to preserve the high amplitude of the electron density wave. However, in this situation, the maximum electron density lies in a region where the electromagnetic field is different from zero. Electrons are slowed down in the electromagnetic field wave and may give their energy to the wave. Maximum energy transfer corresponds to the maximum of the function χ ( z1 ), which is attained at z1 = zm ≈ 3. At the maximum we have χ ( zm ) ≈ 0.13. The necessary condition for starting the generation is that W should exceed the energy which is lost by an electromagnetic wave due to absorption in a dielectric. This condition is as follows:
J0
es2 k mv 20
q21 Rk exp −2lk
χ (zm )
ε1 ωa tg(δ) 4π
(32)
where tg(δ) is a loss angle tangent [8]. For example, (tg(δ) ∼ 10−4 ) for a plate made of sapphire or quartz at J 0 ∼ 1 A/m, s ∼ 10−2 m, ω ≈ 6.28 × 1012 s−1 , v 0 ≈ 6 × 107 m/s, k ≈ 105 m−1 , k R ∼ kl ∼ 1, a ≈ 10−5 m, condition (32) is easy fulfilled. At the initial generation stage while the linear approximation is correct, the wave amplitude increases according to the exponential law with an increment γ . Taking into account (31) and (32) an estimate for γ is:
V.A. Namiot, L.Yu. Shchurova / Physics Letters A 375 (2011) 2759–2766
γ≈
2πω2 (ε − 1) J 0 es2 ak mc 2 v 20
q21 k R exp −2k l
χ (zm )
(33)
At the above specified parameters γ is within the limits from 3 × 108 s−1 to 3 × 1010 s−1 , depending on a choice of a corrugated surface form, a magnitude of s, etc. It is meaningful to estimate the maximum value of the amplitude of the wave at the stationary conditions when the exponential growth comes to the end and the system goes to a steady state. Using the inequality | W | < J 0 q1 ( Rk ) exp(−lk )|E0 | we can write:
|E0 | < |Emax | =
4π J 0 q1 (k R ) exp(−k l)
εωa[tg(δ)]
(34)
The estimate of (34) is |Emax | ∼ 106 V/m assuming the same parameters as earlier. Assuming, that |E0 | in the stationary state can reach the magnitudes of order of 10−1 · |Emax |, then the generation power for the total beam current ∼ 10 mA is at the level of few watts. Thus, we have shown that in this system, in principle, lasing is possible, and estimated the parameters of the proposed generator scheme. 5. Production of long thin electron beams with high current densities The electron beam which can be used in the considered generation scheme should possess rather specific characteristics. It should be long enough of the order of 10−2 –10−3 m, but very thin. Its thickness d should not exceed 2–4 μm. Though the total beam current per length unit is rather small ∼ 1 A/m, but the current density is rather high of the order of (2–5) × 105 A/m2 due to small d. Besides the beam should be well collimated, because the velocity variation along the y-axis shouldn’t exceed (2–3) × 105 m/s for the stabilized beam. Creating such beam by using an electron thermo-emission cathode is extremely difficult (if it is probable in principle). The use of cold emission for this purpose is possible in principle, but it’s also quite difficult. Therefore another beam generation scheme, where the electron multiplication phenomenon, is used looks more preferable. In this scheme the initial electron beam is generated from the photocathode illuminated by a light-emitting diode (LED). However, taking into account the existing powers of light-emitting diodes the photon flux is too small for generating the desired electron beam. To increase the beam current the initial electron beam should be multiplied as it is done in photomultipliers. That is, photoelectrons should be directed to the first dynode to which a positive voltage relative to the photocathode is applied. Accelerated photoelectrons knock out secondary electrons from the dynode, whose number exceeds by several times the number of initial electrons. Secondary electrons in their turn bombard the next dynode and are also multiplied on it. This process is repeated until the number of electrons emitted by a next dynode will be sufficient for generating the desired beam current. (It is not difficult to modulate the beam current by controlling the accelerating voltage applied to one of the dynodes, and thereby to control amplitude modulation of the generated wave.) The beam is formed in two stages. First, electrons pass through a collimator forming a beam of a thickness d and separating electrons with velocities out of the required velocity range along y. After that the electrons get in a gap between two electrodes to which the accelerating voltage is applied. Passing through this gap and getting the velocity v 0 on the output from the gap electrons move in a space area above the dielectric plate. Thus both the collimator, and the gap are located in such a way that these electrons move in parallel to the plate at the height l, where a “potential” ϕa (l) is minimal.
2765
6. Recovery of electron energy and generation system efficiency On the face of it, the efficiency of considered system appears to be very low due to the principal restrictions. Indeed, electrons can effectively interact with a wave until their velocities are close to v 0 . As soon as some essential part of the electron energy is transferred to the wave, the electron velocity decreases, and electrons cannot give more their energy to the wave. Therefore the most part of the energy spent for electron acceleration appears unused as a result. If an extra accelerating voltage is applied to the gap where the dielectric plate is located, the electron energy loss can be compensated so that the electron velocity can be kept in the required range. If the value of this accelerating voltage is controlled in appropriate way depending on the generated wave power, the energy transmitted from the beam to the wave could be increased significantly in principle. However electrons which are not be captured by the wave (they can make the most part) are also accelerated by this extra field. As these electrons do not participate in the wave amplification, the energy consumption for the electron acceleration can be also considered as energy losses. However an implicit assumption made above is to be considered in more details and if take it into account one can change the situation significantly and increase the efficiency up to values close to one. It is taken as a given that, if electrons have not transferred their energy to a wave, this energy is lost forever. It should not be actually like this. If electrons are not scattered (i.e. no losses due to collisions), then in principle it is possible that when the energy consumed for electron acceleration is returned to the energy source (minus the energy transferred to the wave). In other words, to have the higher efficiency, the energy which remains for electrons after their interaction with a wave should be recuperated. In the elementary recuperation scheme the beam with the average electron energy E m and the characteristic energy spread δ E passes through the gap under the retarding potential before the beam reaches the anode. The magnitude of this potential is determined by the difference E m − δ E, thus the ratio of the returned energy to the lost one is the order of δ E / E m . However such recuperation scheme would be rather ineffective in the considered case. The matter is that the electron beam interacting with the wave consists of two parts: one of them includes electrons being captured by the wave – they give their energy to the wave, another of them includes electrons avoided the capture – they do not lose their energy. As against the total energy spread in the beam (that can be even comparable with E m in principle), the energy spread inside every part is rather small, the magnitude of it is mainly determined by electron work function from a dynode material A d (where A d E m ). Therefore, these parts of the electron beam should be separated to increase the recovery efficiency. It can be made easy enough: the beam preliminary slowed down to some intermediate energy should be passed through a beamdeflecting system (e.g., through a system of two electrodes with the applied potential between them). Thus, the various parts of the beam with different electron velocities are deflected on different angels and as a result they get to different places, where the independent recuperation is optimized for each part. The beam energy can be further recovered in these independent systems, where each of them has the required retarding potential connected with the energy of the corresponding part. The total recovery efficiency in such a system can be high enough to increase the efficiency of the considered device up to the values close to one. 7. Conclusion In conclusion, a possible scheme of the device is presented in Fig. 6. The device could in principle generate terahertz ra-
2766
V.A. Namiot, L.Yu. Shchurova / Physics Letters A 375 (2011) 2759–2766
Fig. 6. Possible scheme of the terahertz generator, that includes frame (1), dielectric plate with the surface of the sinusoidal form (2), light-emitting diode (3), is a photocathode (4), dynodes of an electron multiplier (5), diaphragm for collimating the electron beam (6), accelerating electrodes (7), electron beam (8), slowing electrodes for preliminary deceleration of the electron beam (9), deflecting electrodes (10), the first anode with an electrode for final slowing down the electron beam that passes through an electrode orifice (11), the second anode with an electrode for final slowing down the electron beam that passes through an electrode orifice (12). (The scheme also includes microelectrodes evaporated on the dielectric plate 2. Potentials applied to the microelectrodes provide stabilization of the beam 8 above the plate and create additional accelerating voltage to compensate the electron degradation due to loss of energy by wave generation. The microelectrodes are not shown to avoid complicating the figure.)
diowaves according to the proposed theoretical estimations. The scheme includes both the system generating an electron beam, and a corrugated dielectric plate along which an electromagnetic wave propagates (electrons move in a cross-section direction above the plate) and a recuperator including two independent devices, each of which works with its own electron part. Various modifications of this scheme are possible in principle, e.g. the scheme that has two corrugated dielectric plates, so the electron beam moves in a gap between two corrugated plates. This modification can even have some advantages. Since the size of some elements in such schemes are small enough, the order of microns, for their manufacturing it is meaningful to use those technical opportunities that are applied now for manufacturing solid-state electronic devices. Terahertz waves could be used in the most different branches, e.g. in medicine and biology. Specifying a rather unexpected application area is worthwhile: if the efficiency of the similar devices could be made rather high, they can be used for energy pumping in particle accelerators and free-electron lasers.
Acknowledgement The work has been supported by grant NSH 3322.2010.2. References [1] X.-C. Zhang, J. Xu, Introduction to THz Wave Photonics, Springer, New York, 2009. [2] K. Reimann, Rep. Prog. Phys. 70 (2007) 1597. [3] B.S. Williams, Nature Photonics 1 (2007) 517. [4] M. Tonouchi, Nature Photonics 1 (2007) 97. [5] N. Yu, Q.J. Wang, M.A. Kats, J.A. Fan, S.P. Khanna, L. Li, A.G. Davies, E.H. Linfield, F. Capasso, Nature Materials 9 (2010) 730. [6] V.V. Batygin, I.N. Toptygin, Problems in Electrodynamics, second ed., Academic Press, London, 1978. [7] M.A. Leontovich (Ed.), Issues of the Plasma Theory, Gosatomizdat, Moscow, 1963, Issue 2, Section “Movement of charged particles in electromagnetic fields” (in Russian). [8] I.S. Grigoryev, E.Z. Mejlikhov (Eds.), Physical Values, Energoatomizdat, Moscow, 1991 (in Russian).
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
On the generation of electromagnetic waves in the terahertz frequency range V.A. Namiot a,∗ , L.Yu. Shchurova b a b
Microelectronic Division, Research Institute of Nuclear Physics, Moscow State University, Vorobyovy Gory, 119992 Moscow, Russia Department of Solid State Physics, P.N. Lebedev Physical Institute of Russian Academy of Science, Leninsky Prospect 53, 119991 Moscow, Russia
a r t i c l e
i n f o
Article history: Received 25 February 2011 Accepted 7 March 2011 Available online 7 June 2011 Communicated by V.M. Agranovich
a b s t r a c t It is shown that a thin dielectric plate, which can act as an open dielectric waveguide, it is possible to produce amplification and generation of electromagnetic waves with frequencies in the terahertz range. For this purpose, we propose using a dielectric plate with a corrugated surface, in which case the electric field of the transverse electromagnetic wave in the waveguide has a periodic spatial structure in the local area near to the corrugation. Terahertz electromagnetic waves are excited by a beam of electrons moving in vacuum along the dielectric plate at a small distance from its corrugated surface. Corrugation period is chosen in order to ensure the most effective interaction of the electron beam with the first harmonic of the electric field induced by the corrugation. Amplification and generation of electromagnetic waves propagating in a dielectric waveguide is realized as a result of deceleration of the electron beam by a wave electric field induced by a corrugated dielectric surface in the zone near the corrugation. We discuss possible ways to create electron beams with the desired characteristics. We offer a way to stabilize the beam position above the plate, avoiding the bombardment of the plate by electrons. It is shown that it is possible to significantly increase the efficiency of the device through the recovery of energy that remains in the electrons after their interaction with the wave. © 2011 Elsevier B.V. All rights reserved.
1. Introduction and problem description Currently, most experts believe that the age of vacuum electronic radio devices has largely passed, and they have lost ground in almost all areas to solid state electronics. However, technology advances in recent years, that have been successfully used in the manufacturing of modern solid state electronic devices (such as microlithography), can, in principle, change the situation. This may enable creation of electrovacuum devices that can solve problems that cannot be solved, or are difficult to deal with using current methods. These are devices that generate electromagnetic waves at frequencies ranging from one to several terahertzes. It is this range of frequencies that has recently come into focus of great interest because of its promising applications in medical, biological and industrial research, radar, space science, etc. [1]. Unfortunately, despite significant progress in the study of terahertz radiation in recent years [2–5], this range is mastered much less than others and the parameters of such systems are inferior to parameters systems of other frequency ranges. However, according to our estimates, the parameters of discussed here electrovacuum devices, continuous-wave terahertz generators, could be significantly better
*
Corresponding author at: Microelectronic Division, Research Institute of Nuclear Physics, Moscow State University, Vorobyovy Gory, 119992 Moscow, Russia. E-mail address: [email protected] (V.A. Namiot). 0375-9601/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2011.05.061
than the existing ones. The power of such devices could reach the level of watts, and efficiency may exceed (50–80)%. Traditionally, the electron beam is created in the electrovacuum microwave generator. That electron beam interacts with the electromagnetic wave and transfers its energy to this wave. The electrons moving along the direction of electromagnetic waves interact effectively with this wave when the electron velocity is close to the phase velocity of the wave. However, since the electron velocity is much less than the speed of light, slow-wave structures are used to reduce the wave velocity. This ensures the most effective interaction of the electron beam with an electromagnetic wave and, as a result, amplification and generation of electromagnetic waves. However, the size of slow-wave structures should be comparable with wavelength. There is a high absorption of electromagnetic radiation in small slow-wave structures that are required to slow the terahertz waves. This significantly affects the generation condition. It is extremely difficult both to slow down a wave and to avoid losses in this frequency range. In this Letter, we propose a scheme for a device in which the generation of terahertz electromagnetic waves is the result of the interaction of the electron beam with an electromagnetic wave, as it is in electrovacuum devices. However, the proposed scheme does not require a slow-wave structure. As will be shown below, it is possible that the speed of the electromagnetic wave is close to the speed of light in vacuum, but the electrons are still able to effectively transfer their energy to it.
2760
V.A. Namiot, L.Yu. Shchurova / Physics Letters A 375 (2011) 2759–2766
The proposed here scheme of the device includes bordering the vacuum thin dielectric plate that can act as an open waveguide in which electromagnetic waves in the terahertz frequency range can propagate. In this case, the electromagnetic field partially penetrates the wall of the dielectric plate in the surrounding space, decaying exponentially with distance from the insulator. So, the electromagnetic wave field is focused on the region which includes the plate itself and some area near this plate. Terahertz electromagnetic waves are excited by a beam of electrons moving in vacuum along the corrugated surface of the dielectric plate at a small distance from it. A beam of electrons moving in the transverse direction of the waveguide (in the direction perpendicular to the direction of wave propagation) can interact with the electric field of transverse electromagnetic wave (H-wave). However, the electric field of the transverse wave is uniform (does not contain spatial harmonics) for the dielectric plate with a flat surface, so the effective interaction between this wave and the electron beam should be absent. It is therefore proposed to create a waveguide with a corrugated surface. It is obvious that the wave will change in this case, compared with the unperturbed wave propagating in a plate with a smooth surface. Indeed, the unperturbed wave causes the polarization of the dielectric in the zone of the corrugation, and this polarization, in turn, induces a secondary field which is superimposed on the unperturbed. And this secondary field penetrates into the region outside the plate. The characteristic scale, which changes the secondary field, is determined by the corrugation period. Since the corrugation at the boundary of the dielectric plate and the vacuum has a periodic structure, the induced electric field near the boundary can be regarded as the sum of an infinite set of harmonics. By selecting the size of the period, we can ensure that the electron beam, flying over the corrugated surface at sufficiently low altitudes, would effectively interact with the first harmonic of the electric field near the corrugation and would give its energy to the wave. It is important that in such a scheme, the interaction of the electron beam with the electromagnetic wave is sufficiently effective without slowing down the wave. In the proposed scheme the electromagnetic wave propagating in the waveguide can be regarded as strictly transverse. Therefore, a beam of electrons interacting with the electric field wave should move in a perpendicular direction. As a result of the interaction, the amplitude of the electric field of the transverse electromagnetic wave increases, while the wave itself propagates along the axis of the waveguide at a speed close to the speed of light in vacuum. The characteristic scale on which the secondary field penetrates into the region above the plate is also determined by the corrugation period. Therefore, in order to have an efficient energy transfer from electrons to the wave, it is necessary that the beam would be, firstly, thin enough and, secondly, that it moved at a very low altitude over the plate. In order to fulfill these requirements, it is necessary, firstly, to propose a method to create the thin beam with the desired characteristics, and, secondly, to propose a method for stabilizing the beam position above the plate, avoiding the bombardment of the plate by electrons. Both of these problems are considered in this Letter. In addition, in this Letter we consider the efficiency of the proposed device. It is shown, how to bring its value to unity due to recovery of the electronic energy, which remained after the interaction of electrons with the wave. 2. Wave electric field in dielectric plate with corrugated surface The proposed here scheme of the generator includes bordering the vacuum thin dielectric plate with the thickness 2a and permittivity ε (Fig. 1). This dielectric plate can act as an open waveguide [6, p. 133] in which electromagnetic waves, including waves in the
Fig. 1. Dielectric plate with a flat surface (a) and with a corrugated surface (b).
terahertz frequency range ω , can propagate. For sufficiently thin plate a c /ω and for not too large values of ε , the electromagnetic wave energy is mostly concentrated in the region outside the plate, so these waves propagate with the velocity, that close to the speed of light, and have the wave vector k ≈ ω/c. We will consider the special case of a magnetic wave (H-wave), propagating in the z-direction. The electric field of this wave is directed along the x-axis. The beam of electrons, interacting with the electric field wave, should move also in the x-direction. Precisely the H-wave is of interest to us because an antinode of the H-wave electric field is in the center of the cross-section of this plate. In our problem the electric field of this wave quite well penetrates into the region above the plate, where the electrons move. For a plate with a flat surface (Fig. 1(a)), the electric field of this wave can be written as
E(r, t ) =
Aex cos(α y ) exp(ikz − i ωt ),
| y| a
Bex exp(−β| y |) exp(ikz − i ωt ),
| y| a
(1)
here: ex is a unit vector being directed along an axis x, A ≈ B , α ≈
2 (ε − 1)1/2 ωc , β ≈ (ε−c12)ω a.
However, the electric field of the H-wave is uniform in the flat dielectric plate, so the effective interaction between this wave and the electron beam should be absent. Therefore, we propose to create a dielectric plate with a corrugated surface. Then additional spatial modes are induced by a corrugated dielectric surface in the zone near the corrugation. The electron beam may interact with these modes. Let the plate thickness varies under a periodic law along the x-axis (Fig. 1(b)). Take y = a + R cos(k x) as an instance, and as it will be clear further, it is worth to consider k R ∼ 1 and to accept the condition:
k k
(2)
under which the interaction of electromagnetic waves with electrons and amplification of this wave would be the most effective. This plate can also serve as a waveguide for the H-wave, and its propagation velocity is still be close to c. However, in this case the electric field of this wave is inhomogeneous along the x-axis. The electric field of the wave E (r, t ) can be presented as E (r, t ) = E(r, t ) + F(r, t ), where the function E(r, t ) is determined by formula (1). And the electric field F(r, t ) is comparable with E(r, t ) only in a very narrow area (with a width of about (k )−1 ) near to y = a + R. But F(r, t ) is significantly less than E(r, t ) outside of this area. Owing to that the corrugated surface is described by a periodic function of x the induced electric field F(r, t ) can be written as the sum of an infinite set of harmonics:
F(r, t ) = G(x, y ) exp(ikz − i ωt )
=
∞ n=−∞
Fn ( y ) exp ik nx exp(ikz − i ωt )
(3)
V.A. Namiot, L.Yu. Shchurova / Physics Letters A 375 (2011) 2759–2766
2761
Let us assume that we have created the electron flux moving with a velocity v 0 along the x-axis at a height of l ∼ (k )−1 above the top edge of the corrugated surface. (That is electrons move in a perpendicular direction to the propagation of electromagnetic waves.) We can define the corrugation period (the corrugation period determines the value of k ) that the condition
k v 0 = ω + η
(4)
(where ω |η|) is satisfied. In this case, the effective interaction of electrons with the first harmonic of the electric field is achieved. Indeed, according to (3) the electron flux interacts effectively with that part of an electric field of the electromagnetic wave which is described by a term
g1 (x, y ) exp(ikz − i ωt ) ≡ F1 ( y ) exp ikz + ik x − i ωt
Fig. 2. The boundary between the dielectric plate and vacuum: 1 is dielectric with permittivity ε , 2 is vacuum. The axis O is defined by a condition y = a + R, axis O is exactly on the tops of the corrugation.
As we have v 0 c (in real electro-vacuum devices), Eq. (2) follows directly from (4). (Thus, it does not matter any more that kv 0 = ω .) Thus, in our problem the electromagnetic wave propagates in the waveguide having the shape of a thin dielectric plate with a thickness that varies periodically in the transverse direction. This wave may be amplified by the electron beam, also moving in the transverse direction to the wave. Efficiency of energy transfer from electrons to the wave is determined by F1 ( y ). Let’s proceed to calculation of the electric field F(r, t ) induced by the corrugated surface. Due to the smallness of the corrugation size (typical size of the corrugations (k )−1 ) is compared to the electromagnetic wavelength, the electric field of the corrugation can be considered as a potential and can be described it the terms of a scalar potential φ(x, y ). It means that the function G(x, y ) can be presented as
G(x, y ) = −∇φ(x, y ) = −∇
∞
Fig. 3. 1 is the corrugation described by the equation y = R cos(k x), and 2 are ellipsoids modeling the corrugated surface in the region h.
pn ( y ) exp ik nx
n=−∞
The function φ(x, y ) can be presented as φ1 (x, y ) inside the plate, and φ2 (x, y ) outside the plate. These functions satisfy the Laplace equation φ1 (x, y ) = 0 and φ2 (x, y ) = 0 within their ranges of definition. Then the boundary conditions are as follows: the tangential electric fields are continuously and the normal electric fields differ by the factor ε .
E0 − ∇φ1 (x, y ) τ (x, y ) = E0 − ∇φ2 (x, y ) τ (x, y ) ε E0 − ∇φ1 (x, y ) n(x, y ) = E0 − ∇φ2 (x, y ) n(x, y )
−1
|E0 | exp −k ( y − a − R )
(6)
(7)
The coefficient q1 (k R ) depends only on the dimensionless parameter k R, and q1 (k R ) can be expressed in terms of the Fourier component of the function φ2 (x, a + R ):
(k )2 q1 k R = 2 π |E 0 | i
2
π /k
φ2 (x, a + R ) exp −ik x dx
0
q1 k R =
(5)
Here E0 = A cos(αa)ex ≈ Aex is the electric field of an electromagnetic wave (without facient exp(ikz − i ωt )) on the plate boundary, if no corrugated surface; τ (x, y ) is a tangent vector to the curve y = R cos(k x) in a point with coordinates (x, y ), and n(x, y ) fulfills a condition n(x, y )τ (x, y ) ≡ 0. This situation is shown in Fig. 2. The function F1 ( y ) is expressed in terms of p 1 ( y ). The function p 1 ( y ) in the area y (a + R ), where electrons move, can be written as
p 1 ( y ) = iq1 k R k
In fact, the of the coefficient q1 (k R ), multiplied by exp(−k l), determines the interaction of the electron beam with an electromagnetic wave. With an increase in the absolute value of this product the interaction becomes more effective. Let us consider a case of k R 1 (this case corresponds to a very low corrugation).
(8)
ε−1 kR ε+1
(9)
In the opposite situation, when k R 1 (a very high corrugation), we can use the following method. The induced by corrugated surface electric field along the x-axis is defined, in this case, approximately by the region R − (k )−1 y R. In this region the corrugations can be approximated by a periodic sequence of ellipsoids as shown in Fig. 3. In order to simulate the corrugated surface by the ellipsoids in an area of a size h not far from the axis O (the condition h ∼ (k )−1 should be fulfilled for h), these ellipsoids could be described by the equation
x2 a21
+
y2 b21
+
z2 c 12
=1
where a1 = (k )−1 b1 = R and c 1 → ∞. The problem of the electric field induced by an ellipsoid in a medium has the analytical solution (see [6, p. 52]). Using this solution and taking into account that k R 1 we obtain the equation
q1 k R =
ε−1 (2π )1/2 ε (k R )1/2
(10)
2762
V.A. Namiot, L.Yu. Shchurova / Physics Letters A 375 (2011) 2759–2766
Fig. 4. Scheme, that includes “high-frequency potential” to stabilize the electron motion: 1 is a dielectric plate, 2 is an electronic beam moving above the plate with a velocity v 0 , 3 is a conducting layer under the earth potential, L is a distance between 1 and 3, 4 are conducting layers under a potential ϕ1 with respect to earth, 5 are conducting layers under a potential ϕ2 with respect to earth. No contact between conducting layers 4 and 5.
Approximation of (9) and (10), which is true in the ranges both large and small k R is
q1 k R ≈
k R ε−1 (2π )1/3 ε2/3 ε+1 1+ 2 3/4 2/3 k R
(11)
( ε +1 )
This approximation is not only possible. Nevertheless, we hope that we can use this expression for the intermediate k R at least as a rough estimate. Let ε 1. The expression (11) has a maximum q1 (k R ) ≈ 0.3 for k R ≈ 0.9. A similar approximation for the corrugated surface of the meander form is
q1 k R ≈
8(ε − 1)
π 2 (ε + 1)
arctg
π 2
k R
(12)
This expression has the maximum q1 (k R ) ≈ 1.2. As it follows from the expressions (11) and (12), it is meaningless in using large (k R ) to increase q1 (k R ). It is possible to restrict by the condition (k R ) 1 in this case. Thus, we have the expression for the x-component of the electric field of the first harmonic in the vacuum at a distance l from the top of the corrugation:
F 1x = q1 k R exp −lk E 0 exp ikz + ik x − i ωt
We have shown the electron beam, flying with the velocity v 0 ≈ ω/k over the corrugated surface at a distance l (k )−1 , can interact effectively with the first harmonic of the electric field induced by the corrugation with the amplitude R and period ∼ (k )−1 ((k R ) ≈ 1). 3. Beam movement stabilization above the plate The condition l (k )−1 must be fulfilled for effective interaction of the electromagnetic wave with an electron beam. Even at
high enough velocity v 0 = 6 × 107 m/s (that corresponds to the electron energy of about ten keV) and the frequency ω = 6.28 × 1012 s−1 (that corresponds to one terahertz) we have (k )−1 ≈ 10 μm. Thus, the beam should move very close to the plate surface. Therefore, even a relatively small y-component of velocity or an electric field accelerating the electrons in the same direction leads to the plate bombardment by the beam. It should be also noted that the ratio |ex F1 (x, y )| = |e y F1 (x, y )| takes place (where e y is a unit vector being directed along the y-axis). So with increasing interaction of the beam with a field of electromagnetic wave, the displacement along the y-axis also increases. Therefore, in order to avoid the plate bombardment by electrons, it is necessary to reduce an energy transfer from the beam to the wave. Either the plate width could be reduced that electrons have no time to reach the plate for the flight time, or the electron movement along the y-axis could be stabilized somehow to avoid the surface bombardment by electrons. It is possible to reduce transit time, but it leads to decreasing the generation efficiency. It seems that the movement of electrons along the y-axis could be stabilized by the easiest way with the help of a magnetic field being directed along the x-axis. However the required field magnitude that is defined by the condition (k )−1 rl , where rl is a Larmor radius, is unacceptable high-significantly more than one tesla in this case. Creating such fields is extremely difficult in real devices. No magnetic fields are required, if to consider another opportunity to stabilize the beam along the y-axis. The movement of electrons can be stabilized by using the so-called “high-frequency potential” [7]. Taking into account that it is rarely mentioned in the literature for recent years and the application of this “potential” is specific to the considered problem, an independent conclusion of formulas describing this stabilization is worthwhile to present. Let’s consider a problem about stabilization of the electron beam movement along y in the following scheme (Fig. 4). Let an electron moves with a velocity v 0 in parallel to the flat dielectric
V.A. Namiot, L.Yu. Shchurova / Physics Letters A 375 (2011) 2759–2766
plate 1 with conducting areas 4 and 5 on it (in general, these areas have an extremely high resistance, as no current should be in them). Conducting plate 3 is located at a distance L above the dielectric plate with the potential on the conducting plate being equal to zero. The widths of areas 4 and 5 are the same and equal to π (k )−1 . The potential ϕ1 is applied to the layer 4, and the potential ϕ2 is applied to the layer 5. Let us designate a difference ϕ1 − ϕ2 as ϕ , and a half-sum (ϕ1 +ϕ2 ) as ϕ¯ . The potential ϕ (x, y ) in the area between 1 and 2 3 where the beam moves satisfies to the Laplace equation with the corresponding boundary conditions on layers 4 and 5, and on conducting plate 3 too. Solving the Laplace equation (under the condition of L (k )−1 ) we obtain:
y
ϕ (x, y ) = ϕ¯ 1 − +
m
d2 y
dt 2
πn
1 − (−1)n sin nk x exp −nk y
(13)
∂ ϕ ( v 0 t + δ x(t ), y ) ∂y ∂ ϕ ( v 0 t + δ x(t ), y ) = −e ∂δ x(t )
(14) (15)
(here m and e are a mass and a charge of an electron). Solution (14) can be presented as y ≡ y (t ) = y 0 (t ) + δ y 1 (t ), where y 0 (t ) is a slowly varying in time part of y (t ), and δ y 1 (t ) is a quickly varying in time part of y (t ). Suppose that δ x(t ) v 0 t and δ y 1 (t ) y 0 (t ). In this case δ x(t ) and δ y 1 (t ) satisfy to the equations:
m
d2 δ x(t ) ∞ e ϕ k
1 − (−1)n cos nk v 0 t exp −nk y 0 (t )
π
n =1
m
(16)
d2 δ y 1 (t ) dt 2
=
∞ e ϕ k
π
n =1
∞ n =1
1 − (−1)n sin nk v 0 t exp −nk y 0 (t )
e ϕ k
π m(nk v
δ y 1 (t ) = −
0
)2
1 − (−1)n
(17)
× cos nk v 0 t exp −nk y 0 (t )
∞
e ϕ k
π m(nk v 0 )2 n =1
1 − (−1)n
(18)
(19)
(As follows from (18) and (19) in the limits of large k and v 0 the assumption about the smallness of δ y 1 (t ) in comparison with y 0 (t ) and δ x(t ) in comparison with v 0 t should be fulfilled.) Using (18) and (19) let us write an equation for y 0 (t ):
d2 δ y 0 (t ) dt 2
=
∼ sin2 nk v 0 t + cos2 nk v 0 t = 1
we obtain:
m
d2 δ y 0 (t ) dt 2
= e ϕ¯
1
−
L
∞ (e ϕ k )2 (−nk )
1 − (−1)n
π 2m(nk v 0 )2
n =1
2
∞ e ϕ k (−nk ) n =1
exp −2nk y 0 (t )
Eq. (21) can be rewritten as follows:
m
d2 δ y 0 (t ) dt 2
π
1 − (−1)n
= −e
∂ (ϕ1 + ϕh ) ∂ y0
(22)
y (t )
where ϕ1 = −ϕ¯ 0L , and ϕh is a so-called “high-frequency potential”, which is in the given case as:
ϕh =
∞ e (ϕ )2 n =1
2π
2 mv 2 n2 0
1 − (−1)n
2
exp −2nk y 0 (t )
(23)
The first term of sum (23) actually works. Thus:
ϕh =
2e (ϕ )2
π
2 mv 2 0
exp −2k y 0 (t )
(24)
Consider the sum ϕa ( y ) = ϕ1 ( y ) + ϕh ( y ). If ϕ¯ < 0, it has a minidϕ mum in a point y 2 , in which dya = 0. Hence:
1 2k
ln −
π 2 ϕ¯ mv 20 4L ek (ϕ )2
(25)
At k = 105 m−1 , ϕ = 600 V, v 0 = 6 × 107 m/s, ϕ¯ = −600 V, L = 2 × 10−3 m we have y 2 = 10 μm. Thus, y 2 ∼ (k )−1 . The expand ϕa in a power series in the neighborhood of point y 2
|ϕ¯ | L
( y − y 2 )2
(26)
A displacement of y from y 2 even at 2 microns at the parameters specified here creates a returning electric field of the order of 105 V/m. So, to stabilize the electron beam along y one can simply cover the corrugated surface by very thin conducting layers of a very high resistance (in order not to worsen the electromagnetic wave absorption) and apply a constant voltage to the layers by using the scheme that is presented in Fig. 4. (Some difficulties are connected with preventing an electric breakdown along the dielectric surface, but they are apparently overcomable.) 4. Electron beam interaction with electromagnetic wave and generation condition
× sin nk v 0 t exp −nk y 0 (t )
m
sin nk v 0 t δ y 1 (t ) − cos nk v 0 t δ x(t )
ϕa ( y ) = ϕa ( y 2 ) + k
Let us solve (16) and (17) considering approximately that y 0 (t ) does not depend on time at all:
δ x(t ) =
y2 = −
dt 2
=−
where the over-line denotes time-averaging. Averaging (20) over time, by using the identity
(21)
= −e
dt 2 d2 δ x(t )
(20)
L
(the condition y = 0 corresponds to the bottom dielectric plate on which layers 4 and 5 are located). Let us write movement equations for electrons:
m
× exp −nk y 0 (t )
∞ ϕ n =1
2763
× sin nk v 0 t δ y 1 (t ) − cos nk v 0 t δ x(t )
Let us now consider the problem of interaction of an electromagnetic wave with the electron beam. In our study frame electrons move in the transverse direction (x direction) of the waveguide in a vacuum at the small distance from a corrugated surface, that is, in large nonuniform electric field region. This field is due to the polarization of the corrugated surface of the dielectric in the electric field of the H-wave in waveguide. Since the field’s amplitude of the first spatial harmonic is the largest of the set of all spatial harmonics of the nonuniform electric field, we consider the interaction of electrons with only the
2764
V.A. Namiot, L.Yu. Shchurova / Physics Letters A 375 (2011) 2759–2766
first harmonic. This first harmonic field is part of the H-wave propagating in the waveguide, so it depends on time as well as the of H-wave field by a factor exp(−i ωt ). The most effective interaction between the first harmonic’s wave and the electrons happens under the condition of synchronism, when phase velocity of the wave is close to the electron beam velocity. Due to the change of the electron velocity in the electric field, the electron beam becomes velocity modulated, so electron density wave is formed. In this case, the electrons with a velocity greater than the phase velocity of the wave are decelerated in the electric field wave and give a part of its energy to the wave. The increase in the amplitude of the electromagnetic wave causes, in turn, increases in the amplitude of the electron density wave. Then, an increasing part of the electrons gives its energy to electromagnetic wave, thus amplifying the wave. This leads to an increase in the amplitude of the electron density wave. Thus, there is an oscillation mode. To describe the system of electrons interacting with the electromagnetic field, a coupled system of equations, that take into account both the change of electromagnetic fields, and the change in velocities and densities of the electronic system, is required. The form of a solution describing the electric field of the first harmonic wave is presented above. The contribution of the electric field of this wave is described by a term F1 ( y ) exp[ikz + (ik x − i ωt )] (since the boundary of the insulator and the vacuum is homogeneous along the z-axis we can take z = 0 without loss of generality). The system of electrons interacting with the wave electric field will be described within the hydrodynamic approach. We assume that the beam shift along the y-axis is suppressed, and to describe the electron velocity v (x, t ) and electron density n(x, t ) we use the equation of motion (the Euler equation) and the continuity equation
∂ v (x, t ) ∂ v (x, t ) + v (x, t ) ∂t ∂x e = q1 Rk exp −lk E 0 exp ik x − i ωt
m ∂ n(x, t )
∂t
+
(27)
∂ v (x, t )n(x, t ) =0 ∂x
(28)
Let us represent v (x, t ) and n(x, t ) as v (x, t ) = v 0 + δ v (x, t ) and n(x, t ) = n0 + δn(x, t ), where n0 is the unperturbed electron concentration (i.e. the electron concentration in absence of an electromagnetic wave). Solving (27) and (28) in a linear approximation over δ v (x, t ) and δn(x, t ) without taking into account small-value terms we obtain:
e
δ v (x, t ) ≈ −i
mη
q1 Rk exp −lk
iηx × E 0 1 − exp − exp ik x − i ωt v0
δn(x, t ) ≈ n0
eq1 ( Rk ) exp(−lk ) E 0 iηx k exp − mη v 0 v0
×
i v0
η
exp
iηx v0
(29)
− 1 + x exp ik x − i ωt
(30)
Let us denote by a symbol W the energy transferred by electrons to an electromagnetic wave in a unit of time and per unit area of the plate that is averaged over a distance passed by the beam across the plate. Using (29) and (30) for W , we obtain:
W ≈ J0
es2 k mv 20
q21 Rk exp −2lk | E 0 |2 χ ( z1 )
(31)
Fig. 5. The function χ ( z1 ) determines the efficiency of energy transfer from electron density wave to the electromagnetic wave. The maximum of this function χ ( z1 ) corresponds to the most favorable conditions for the generation of the terahertz ηs waves. Dimensionless argument z1 = v characterizes departure of the electron ve0 locity (on the length of electron flight s) from the phase velocity of electromagnetic wave ω/k (ω is http://www.multitran.ru/c/m.exe?a=110&t=4345104_2_1&sc=867 frequency in the terahertz range, k is the corrugation wavevector) normalized to the average electron velocity v 0 .
Here J 0 is a beam current per unit length, s is a plate width, z1 = ηs , the function χ ( z1 ) has a form v 0
sin( z1 ) 2 χ (z1 ) = − 3 cos(z1 ) − 1 + 2 z1
z1
The plot χ ( z1 ) is shown in Fig. 5. This figure shows, that in conditions when the electron velocity is strictly equal to the phase velocity of electromagnetic wave ω/k (this corresponds to η = 0), energy transfer from the beam to the wave is absent. The point is that in this case, the wave of the electron density is shifted by exactly π /2 with respect to the wave field interacting with the beam. Thus the maximum electron density falls on the area where the electromagnetic field is zero. In order to transmit energy from the beam to the wave, the electron’s velocity should be slightly higher than ω/k . This excess of velocity should be small enough to preserve the high amplitude of the electron density wave. However, in this situation, the maximum electron density lies in a region where the electromagnetic field is different from zero. Electrons are slowed down in the electromagnetic field wave and may give their energy to the wave. Maximum energy transfer corresponds to the maximum of the function χ ( z1 ), which is attained at z1 = zm ≈ 3. At the maximum we have χ ( zm ) ≈ 0.13. The necessary condition for starting the generation is that W should exceed the energy which is lost by an electromagnetic wave due to absorption in a dielectric. This condition is as follows:
J0
es2 k mv 20
q21 Rk exp −2lk
χ (zm )
ε1 ωa tg(δ) 4π
(32)
where tg(δ) is a loss angle tangent [8]. For example, (tg(δ) ∼ 10−4 ) for a plate made of sapphire or quartz at J 0 ∼ 1 A/m, s ∼ 10−2 m, ω ≈ 6.28 × 1012 s−1 , v 0 ≈ 6 × 107 m/s, k ≈ 105 m−1 , k R ∼ kl ∼ 1, a ≈ 10−5 m, condition (32) is easy fulfilled. At the initial generation stage while the linear approximation is correct, the wave amplitude increases according to the exponential law with an increment γ . Taking into account (31) and (32) an estimate for γ is:
V.A. Namiot, L.Yu. Shchurova / Physics Letters A 375 (2011) 2759–2766
γ≈
2πω2 (ε − 1) J 0 es2 ak mc 2 v 20
q21 k R exp −2k l
χ (zm )
(33)
At the above specified parameters γ is within the limits from 3 × 108 s−1 to 3 × 1010 s−1 , depending on a choice of a corrugated surface form, a magnitude of s, etc. It is meaningful to estimate the maximum value of the amplitude of the wave at the stationary conditions when the exponential growth comes to the end and the system goes to a steady state. Using the inequality | W | < J 0 q1 ( Rk ) exp(−lk )|E0 | we can write:
|E0 | < |Emax | =
4π J 0 q1 (k R ) exp(−k l)
εωa[tg(δ)]
(34)
The estimate of (34) is |Emax | ∼ 106 V/m assuming the same parameters as earlier. Assuming, that |E0 | in the stationary state can reach the magnitudes of order of 10−1 · |Emax |, then the generation power for the total beam current ∼ 10 mA is at the level of few watts. Thus, we have shown that in this system, in principle, lasing is possible, and estimated the parameters of the proposed generator scheme. 5. Production of long thin electron beams with high current densities The electron beam which can be used in the considered generation scheme should possess rather specific characteristics. It should be long enough of the order of 10−2 –10−3 m, but very thin. Its thickness d should not exceed 2–4 μm. Though the total beam current per length unit is rather small ∼ 1 A/m, but the current density is rather high of the order of (2–5) × 105 A/m2 due to small d. Besides the beam should be well collimated, because the velocity variation along the y-axis shouldn’t exceed (2–3) × 105 m/s for the stabilized beam. Creating such beam by using an electron thermo-emission cathode is extremely difficult (if it is probable in principle). The use of cold emission for this purpose is possible in principle, but it’s also quite difficult. Therefore another beam generation scheme, where the electron multiplication phenomenon, is used looks more preferable. In this scheme the initial electron beam is generated from the photocathode illuminated by a light-emitting diode (LED). However, taking into account the existing powers of light-emitting diodes the photon flux is too small for generating the desired electron beam. To increase the beam current the initial electron beam should be multiplied as it is done in photomultipliers. That is, photoelectrons should be directed to the first dynode to which a positive voltage relative to the photocathode is applied. Accelerated photoelectrons knock out secondary electrons from the dynode, whose number exceeds by several times the number of initial electrons. Secondary electrons in their turn bombard the next dynode and are also multiplied on it. This process is repeated until the number of electrons emitted by a next dynode will be sufficient for generating the desired beam current. (It is not difficult to modulate the beam current by controlling the accelerating voltage applied to one of the dynodes, and thereby to control amplitude modulation of the generated wave.) The beam is formed in two stages. First, electrons pass through a collimator forming a beam of a thickness d and separating electrons with velocities out of the required velocity range along y. After that the electrons get in a gap between two electrodes to which the accelerating voltage is applied. Passing through this gap and getting the velocity v 0 on the output from the gap electrons move in a space area above the dielectric plate. Thus both the collimator, and the gap are located in such a way that these electrons move in parallel to the plate at the height l, where a “potential” ϕa (l) is minimal.
2765
6. Recovery of electron energy and generation system efficiency On the face of it, the efficiency of considered system appears to be very low due to the principal restrictions. Indeed, electrons can effectively interact with a wave until their velocities are close to v 0 . As soon as some essential part of the electron energy is transferred to the wave, the electron velocity decreases, and electrons cannot give more their energy to the wave. Therefore the most part of the energy spent for electron acceleration appears unused as a result. If an extra accelerating voltage is applied to the gap where the dielectric plate is located, the electron energy loss can be compensated so that the electron velocity can be kept in the required range. If the value of this accelerating voltage is controlled in appropriate way depending on the generated wave power, the energy transmitted from the beam to the wave could be increased significantly in principle. However electrons which are not be captured by the wave (they can make the most part) are also accelerated by this extra field. As these electrons do not participate in the wave amplification, the energy consumption for the electron acceleration can be also considered as energy losses. However an implicit assumption made above is to be considered in more details and if take it into account one can change the situation significantly and increase the efficiency up to values close to one. It is taken as a given that, if electrons have not transferred their energy to a wave, this energy is lost forever. It should not be actually like this. If electrons are not scattered (i.e. no losses due to collisions), then in principle it is possible that when the energy consumed for electron acceleration is returned to the energy source (minus the energy transferred to the wave). In other words, to have the higher efficiency, the energy which remains for electrons after their interaction with a wave should be recuperated. In the elementary recuperation scheme the beam with the average electron energy E m and the characteristic energy spread δ E passes through the gap under the retarding potential before the beam reaches the anode. The magnitude of this potential is determined by the difference E m − δ E, thus the ratio of the returned energy to the lost one is the order of δ E / E m . However such recuperation scheme would be rather ineffective in the considered case. The matter is that the electron beam interacting with the wave consists of two parts: one of them includes electrons being captured by the wave – they give their energy to the wave, another of them includes electrons avoided the capture – they do not lose their energy. As against the total energy spread in the beam (that can be even comparable with E m in principle), the energy spread inside every part is rather small, the magnitude of it is mainly determined by electron work function from a dynode material A d (where A d E m ). Therefore, these parts of the electron beam should be separated to increase the recovery efficiency. It can be made easy enough: the beam preliminary slowed down to some intermediate energy should be passed through a beamdeflecting system (e.g., through a system of two electrodes with the applied potential between them). Thus, the various parts of the beam with different electron velocities are deflected on different angels and as a result they get to different places, where the independent recuperation is optimized for each part. The beam energy can be further recovered in these independent systems, where each of them has the required retarding potential connected with the energy of the corresponding part. The total recovery efficiency in such a system can be high enough to increase the efficiency of the considered device up to the values close to one. 7. Conclusion In conclusion, a possible scheme of the device is presented in Fig. 6. The device could in principle generate terahertz ra-
2766
V.A. Namiot, L.Yu. Shchurova / Physics Letters A 375 (2011) 2759–2766
Fig. 6. Possible scheme of the terahertz generator, that includes frame (1), dielectric plate with the surface of the sinusoidal form (2), light-emitting diode (3), is a photocathode (4), dynodes of an electron multiplier (5), diaphragm for collimating the electron beam (6), accelerating electrodes (7), electron beam (8), slowing electrodes for preliminary deceleration of the electron beam (9), deflecting electrodes (10), the first anode with an electrode for final slowing down the electron beam that passes through an electrode orifice (11), the second anode with an electrode for final slowing down the electron beam that passes through an electrode orifice (12). (The scheme also includes microelectrodes evaporated on the dielectric plate 2. Potentials applied to the microelectrodes provide stabilization of the beam 8 above the plate and create additional accelerating voltage to compensate the electron degradation due to loss of energy by wave generation. The microelectrodes are not shown to avoid complicating the figure.)
diowaves according to the proposed theoretical estimations. The scheme includes both the system generating an electron beam, and a corrugated dielectric plate along which an electromagnetic wave propagates (electrons move in a cross-section direction above the plate) and a recuperator including two independent devices, each of which works with its own electron part. Various modifications of this scheme are possible in principle, e.g. the scheme that has two corrugated dielectric plates, so the electron beam moves in a gap between two corrugated plates. This modification can even have some advantages. Since the size of some elements in such schemes are small enough, the order of microns, for their manufacturing it is meaningful to use those technical opportunities that are applied now for manufacturing solid-state electronic devices. Terahertz waves could be used in the most different branches, e.g. in medicine and biology. Specifying a rather unexpected application area is worthwhile: if the efficiency of the similar devices could be made rather high, they can be used for energy pumping in particle accelerators and free-electron lasers.
Acknowledgement The work has been supported by grant NSH 3322.2010.2. References [1] X.-C. Zhang, J. Xu, Introduction to THz Wave Photonics, Springer, New York, 2009. [2] K. Reimann, Rep. Prog. Phys. 70 (2007) 1597. [3] B.S. Williams, Nature Photonics 1 (2007) 517. [4] M. Tonouchi, Nature Photonics 1 (2007) 97. [5] N. Yu, Q.J. Wang, M.A. Kats, J.A. Fan, S.P. Khanna, L. Li, A.G. Davies, E.H. Linfield, F. Capasso, Nature Materials 9 (2010) 730. [6] V.V. Batygin, I.N. Toptygin, Problems in Electrodynamics, second ed., Academic Press, London, 1978. [7] M.A. Leontovich (Ed.), Issues of the Plasma Theory, Gosatomizdat, Moscow, 1963, Issue 2, Section “Movement of charged particles in electromagnetic fields” (in Russian). [8] I.S. Grigoryev, E.Z. Mejlikhov (Eds.), Physical Values, Energoatomizdat, Moscow, 1991 (in Russian).