## Abstract

We show that it is possible to produce terahertz wave generation in an open waveguide, which includes a multilayer dielectric plate. The plate consists of two dielectric layers with a corrugated interface. Near the interface, there is a thin semiconductor layer (quantum well), which is an electron-conducting channel. The generation and amplification of terahertz waves occur due to the efficient energy exchange between electrons, drifting in the quantum well, and the electromagnetic wave of the waveguide. We calculate the inhomogeneous electric fields induced near the corrugated dielectric interface by electric field of fundamental mode in the open waveguide. We formulate hydrodynamic equations and obtain analytical solutions for density waves of electrons interacting with the inhomogeneous electric field of the corrugation. According to numerical estimates, for a structure with a plate of quartz and sapphire layers and silicon-conducting channel, it is possible to generate electromagnetic waves with an output power of 25 mW at a frequency of 1 THz.

### Keywords

- terahertz source
- corrugated waveguide
- open waveguide
- drifting electrons
- interaction of electrons with a wave

## 1. Introduction

For past few decades terahertz radiation, which occupies the bandwidth from approximately 0.3–10 THz, have received a great deal of attention. Devices exploiting this waveband are set to become increasingly important in a very diverse range of applications, including medicine and biology [1, 2]. Nevertheless, despite significant progress in the study of terahertz sources in recent years (see, for example, [3, 4] and references therein), this range is mastered much less than its neighboring frequency ranges: the optical range, in which optoelectronic devices are used, and the microwave range, in which electro-vacuum microwave devices are mainly used.

Currently, the key problem lies in the creation of a sufficiently intense, and at the same time compact, terahertz source that can be adapted for a variety of applications.

Here we consider a scheme of a compact terahertz generator, which uses some methods and ideas that have been successfully applied in vacuum microwave generators. In the schemes of terahertz generator discussed here, as in microwave generators (such as backward wave and traveling wave tubes), electrons interact with the waveguide wave and transfer their energy to this wave. However, a straight-forward adaptation of microwave generator circuit for terahertz generators causes serious difficulties. It is known that the most efficient energy transfer between electrons and an electromagnetic wave occurs 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 the electromagnetic wave in a vacuum (light speed), slow-wave structures are used to reduce the wave velocity. The characteristic dimensions of such slow-wave structures should be comparable with the length of the amplified electromagnetic wave. There is a strong absorption of electromagnetic radiation in slow-wave structures with small characteristic dimensions, which are necessary for slowing down terahertz waves. This significantly affects the generation conditions. It is extremely difficult both to slow the wave and to avoid losses in this frequency range.

Here, we offer another phase-matching method. In the proposed terahertz generator, a speed of the electromagnetic wave, which propagate along a waveguide, is close to the speed of light in vacuum, but electrons are still able to effectively transfer their energy to the wave. Such a situation may occur in corrugated waveguides, in which the corrugated dielectric surface is located in the region of a high electric field of an electromagnetic wave, and electrons move over this corrugated surface. The electromagnetic wave causes the polarization of the dielectric in the zone of the corrugation, and this polarization, in turn, induces an alternating electric field near the corrugated dielectric interface. The characteristic scale that determines the induced electric field is determined by the corrugation period. Since the corrugated interface has a periodic structure, the induced electric field near the interface can be regarded as the sum of an infinite set of harmonics. By selecting the size of the period, we can ensure that electrons, moving at a small distance above the corrugation, would effectively interact with the first harmonic of the induced electric field and would give its energy to the wave.

Here we consider a scheme of an open corrugated waveguide, which includes a thin multilayer dielectric plate (Figure 1). The plate consists of two different dielectric layers with a dielectric interface corrugated periodically in the transverse direction of the waveguide (in the direction perpendicular to the direction of wave propagation). The electric field of the transverse electromagnetic wave (TE wave) induces near the corrugation an electric field that is non-uniform in the transverse direction of the waveguide.

There is a quasi-two-dimensional conducting channel (quantum well) in one of the dielectric layers. In the quantum well located near the corrugation, electrons drift in the transverse direction of the waveguide, and interact with the transverse electric field of the induced surface electromagnetic wave. Phase matching (*L* (which defines wave number *L* ∼ 0.1–1 μm (structures with such parameters are achievable at the present technological level). The drift velocity of electrons in a conducting channel can have a value of the order of

We propose a generator circuit of an open waveguide, which includes a thin multilayer dielectric plate. In such waveguide, the field of an electromagnetic wave is focused in a region that includes both the plate itself and some region near the plate. Ohmic losses of electromagnetic waves are only into the dielectric plate, while wave energy is concentrated mainly outside the plate. So, in open waveguides, it is possible to reduce the energy loss of an electromagnetic wave (and, thus, to improve the generation conditions).

In our works [5, 6], a similar synchronization scheme was proposed for a open corrugated waveguide, in which electrons move ballistically in vacuum above a corrugated plate surface and interact with the non-uniform electric field induced near the corrugation. Such electro-vacuum terahertz generators can have an output power of the order of watts, and efficiency up to 80% [5].

However, in such schemes it is necessary to stabilize the electron beam position above the plate, avoiding the bombardment of the plate by electrons. In [5], we proposed a method for stabilizing the electron beam, but this greatly complicated the scheme of the terahertz generator.

In the scheme considered here, the position of the electrons is already stabilized, since the electrons drift in the quantum well, and the conducting channel can be located close to the corrugated dielectric interface. Such a terahertz generator is compact and, moreover, quite simple to produce. However, the parameters of such a generator are low compared with the parameters of the generator described in [5]. In the terahertz generator scheme considered here, electrons drift in a semiconductor quantum well and experience a large number of collisions. As a result, there are significant losses in Joule heat in such a waveguide. According to our estimates, in such a scheme, it is possible to generate terahertz waves with an output power of tens and hundreds of milliwatts and efficiency of the order of 1%.

A brief description of the terahertz generator, based on the interaction of electrons in a quantum well with an electromagnetic wave of a corrugated waveguide, was previously presented in our works [7, 8].

In the second section of this chapter, we present calculations of the induced inhomogeneous electric field of the wave near the corrugated dielectric interface. In the third section, we use a hydrodynamic approach to describe a system of charged carriers, which drift in a quantum well and interact with the inhomogeneous electric field of the wave in the zone of the corrugation. We define the parameters of the system under which the amplification of the electromagnetic field is the most effective. In the fourth section, we give a brief conclusion.

## 2. Wave electric fields in an open waveguide with a plate consisting of two dielectric layers with a corrugated interface

The generator circuit considered here is an open waveguide, which includes a thin dielectric plate consisting of two dielectric layers with a dielectric interface, periodically corrugated in the transverse direction of the waveguide. An electron conducting channel is included in one of the dielectric layers of the plate close to the corrugated dielectric interface (Figure 2), and the electrons drift in the conducting channel in the transverse direction of the waveguide (in the direction perpendicular to the direction of wave propagation).

We assume that the conducting channel is sufficiently thin (its thickness is much smaller than the corrugation amplitude). In this case, the problem of calculating of electric fields in the waveguide, and the problem of calculating of interactions of electrons with electromagnetic waves can be solved separately. In this section, we solve the problem of calculating of electric fields without electrons. In the next section, we introduce electrons, which interact with wave electric fields, into the picture.

Electromagnetic waves, including waves in the terahertz frequency range, can propagate in the proposed open waveguide. The waveguide contains a dielectric plate of thickness 2*a* consisting of two dielectric layers with different permittivities *ε*_{1} and *ε*_{2} (*ε*_{1} and *ε*_{2} (

Let us consider the case of a TE wave, in which there is an antinode of the electric field in the center of the cross section of the plate (where the dielectric plate is) [7] (Figure 1).

For a thin plate (

The field

Here, *s* are transverse components of the wave vector in a dielectric and vacuum, respectively. The values *s* are related by

However, the electric field

We propose a scheme of a waveguide with a thin multilayered dielectric plate, and the interface between two dielectric layers is a corrugated surface (Figure 1). The waveguide with this plate can also serve as a waveguide for the TE wave, and the wave propagation speed along the Z-axis is still being close to the speed of light. But in this case, in the vicinity of corrugated dielectric interface, an additional surface wave is induced in a field of the TE wave. The induced inhomogeneous electric field of the surface wave is comparable with the electric field of the volume electromagnetic wave only in a very narrow region near the dielectric interface. And the inhomogeneous electric field decreases exponentially (in Y-direction) with distance from the interface. The induced electric field is inhomogeneous in the Z-direction of the waveguide, and electrons also drift in X-direction in quasi-two-dimensional conductance channel, located in the zone of the corrugation (Figure 2). Such electrons can interact with an inhomogeneous field of the wave.

Let the coordinates of the dielectric interface in the waveguide cross section vary according to a periodic law, for example, to the law *R* is significantly smaller than the plate thickness,

We accept the condition

and, as will be clear from the following, it is worth considering the case

In such waveguide, the electric field of the wave can be represented as *R*) near the dielectric interface. And

Let us proceed to calculation of the electric field

The problem can be described by the scalar potential *ε*_{1}, and as *ε*_{2} [7]. The functions

The conditions (4)–(5) represent the continuity of the tangential electric field and normal component of an electric induction vector in a point with coordinates

Since the corrugated dielectric interface is described by a periodic function of *x*, the electrostatic potential functions

In our problem,

The resulting expression for the x-component

The coefficients *m*th harmonic to the electric field induced above the corrugation, depend on the shape and size of corrugation and, moreover, on the parameters *ε*_{1} and *ε*_{2}. We carried out numerical analysis on an example of a structure with a plate of quartz (*ε*_{1} = 4.5) and sapphire (*ε*_{2} = 9.3). In the case, when the dielectric boundary is described by *kcR*. Figure 3 shows the calculated dependences

Figure 4 shows the first harmonic of electric field

Thus, we have the expression for the x-component of the first harmonic electric field of a wave at

The optimal corrugation amplitude *R* and period *L*, in which the first harmonic amplitude of the non-uniform electric field takes its maximum, are determined by

## 3. Electron interaction with electromagnetic wave

Let us now consider the problem of interaction of an electromagnetic wave with the electrons. In our study frame electrons drift in a quasi-two-dimensional conducting channel in the transverse direction (X direction) of the waveguide in a quasi-two-dimensional conducting channel (Figure 2). The conductive channel is located close to the corrugated dielectric interface, that is, in large inhomogeneous electric field region.

The most effective interaction between the first harmonic of a wave and electrons happens under the condition of synchronism, when phase velocity *L* and applied voltage are selected in such a way as to satisfy the condition

To describe the electrons interacting resonantly with the wave, a self-consistent system of equations is required. The system should take into account the change as electromagnetic wave fields, and electron velocity *n*(*x*, *t*). We assume that the electron density is quite low and the conducting channel is very thin, so that the screening of the corrugation electric field by electrons is negligible. TE-wave of the waveguide includes *Hy* and *Hz* components of the magnetic field. Since the electron collision frequency is much greater than the cyclotron frequency, the influence of magnetic field on the electron motion is negligible. The form of a solution describing the electric field of the first harmonic wave is described by (8). Since the corrugated dielectric interface is homogeneous along the Z-axis we can take *γ*. Then, taking into account (8), the contribution of the first harmonic to the wave’s electric field at

We will consider only the effect of the electric field (directed along the X-axis) on the drifting electron in the quantum well located at

In the hydrodynamic approximation, functions

and the equation of continuity

Here,

Let us assume that

where *D* is diffusion coefficient of the charge carrier system in the quantum well.

We solve the system of Eqs. (9) and (11) for the boundary conditions

Here,

Amplification of electromagnetic waves is achieved due to the fact that drifting charge carriers transfer their energy to the radiation field. Represent the current of carriers, which interact resonantly with the wave, as the sum.

*Lz* is the length of the plate. (Usually, *Jv* is much smaller than *Jn*.) The electron current does work on the electromagnetic wave, and gives its energy to the electromagnetic wave. Let P denote the energy transferred by electrons to an electromagnetic wave per unit time and per unit area of the plate, and in addition per unit path of the electron flow averaged over the distance traveled. Using (12) and (13), we obtain:

where the function

characterizes the efficiency of energy transfer from the electron density waves to the electromagnetic wave. From

Figure 5 shows

We carried out numerical estimates for the structure with *p-Si* conducting channel. The semiconductor p-Si has quite high values of the electrical breakdown field strength (

For our estimates, we used the values of

For amplification coefficient

The plate width

Negative energy flow of the electron density wave (represented by the expression (14)) is converted into a positive energy flow of the electromagnetic wave.

The change per unit time of the electromagnetic field energy of the open waveguide is determined by the sum

and the field energy concentrated inside the plate

where

For a structure with a plate of quartz and sapphire with the thickness

To start the electromagnetic oscillations, the gain

which characterizes the attenuation of waves due to dielectric losses. Here,

The carrier density *n*_{0}, which can ensure the energy flow of the charge density wave for generation of terahertz radiation, is determined by the law of conservation: ^{2}.

The formation and amplification of the charge density wave are taking place in the system of the scattering carriers. Let us denote by *Q* the energy that drifting carriers lose in collisions per unit time:

The efficiency of the generator can be estimated from the ratio of the useful energy *P* to the total energy

For the scheme with the above parameters, the generation efficiency is 1%, and the generation power is 25 mW at frequency of

Note that one of the possible ways to improve the output power is to increase the area of the plate. Moreover, the output power can be increased (according to expression (14)) by increasing the average electron concentration in the channel, as well as increasing the electron mobility. However, the mobility cannot be very high, since in the framework of the model described above, condition

## 4. Conclusion

Currently, there are many approaches to the problem of terahertz generator creation, and there are working devices. However, the problem is still not solved completely. Now efforts are focused not on the development of a single device with record-breaking parameters, but on devices that are suitable for wide application.

Here we consider the concept of a compact terahertz generator, for which neither low temperatures nor strong magnetic fields are required. So, it may have some advantages compared other terahertz generator.

We propose a terahertz generator scheme, which is an open waveguide with a thin dielectric plate, since in such a waveguide the ohmic losses of electromagnetic waves can be small. Indeed, in such waveguides, ohmic losses are present only in the dielectric plate, while the wave energy is concentrated mainly outside the plate.

The device plate consists of two dielectric layers with different values of permittivity and dielectric interface, periodically corrugated in the transverse direction of the waveguide. The transverse TE wave propagating in the waveguide causes polarization of the dielectric in the zone of the corrugation, and this polarization, in turn, induces an alternating electric field near the corrugated dielectric interface. An electron conducting channel is included in one of the dielectric layers of the plate close to the corrugated dielectric interface. Electrons drift in the conducting channel in the transverse direction of the waveguide (in the direction perpendicular to the direction of wave propagation). The drifting electrons interact with the inhomogeneous electric field which is induced near the dielectric interface by the TE wave of the waveguide. The corrugation period and the parameters of the electronic system are selected in such a way as to ensure the regime of the most effective interaction of drifting electrons with an electromagnetic wave. The generation and amplification of terahertz waves occurs due to the efficient energy exchange between electrons and the electromagnetic wave of the waveguide [7]. It is important that in the considered scheme, the interaction of the electron flow with the electromagnetic wave is quite effective without slowing down the wave propagating in the waveguide.

One of the advantages of the proposed scheme is its low sensitivity to the spread of heterostructure parameters. Before fabrication the corrugated structure, it is possible to define corrugation parameters, under which the synchronism condition will be satisfied.

We have calculated the induced inhomogeneous electric field of the wave near the corrugated dielectric interface, and we have obtained the optimal values of the corrugation amplitude and period at which the amplitude of the non-uniform electric field takes its maximum.

In the hydrodynamic approximation, we have obtained analytical solutions of the equations taking into account the diffusion and drift of the charge carriers in the quantum well. The solutions describe density wave of charge carriers that interact with the first spatial harmonic wave of an inhomogeneous electric field near the corrugation. We have presented numerical estimates for a structure with a plate of quartz and sapphire and the silicon conductance channel. We have found that it is possible to generate electromagnetic waves at the frequency ^{2} and the average electron concentration in the channel