Study of the Absorption Properties of Terahertz Wave in the Plasma Medium

2.1 The Calculation Method of Electromagnetic Power Absorption

Figure 1 is the computational model. The model structure consists of three parts. The outside part is the air layer. The middle is plasma layer. The inner layer is an ideal metal. When parallel polarised wave impacts on the magnetic plasma with the different incidence angle θ, the incident wave acts on the plasma from the air and is attenuated. The incident wave is fully reflected back into the plasma by the metal layer, absorbed by the attenuation again. According to this physical model, the electromagnetic wave absorption coefficient can be calculated. Because the plasma model is a layer structure, the reflection coefficients of each layer can be calculated by recursive method. The total reflection coefficient of the incident location is obtained by superposition. In Figure 1, suppose the metal layer is the first layer, the plasma medium is the second layer, and the air is the third layer. For the parallel polarised wave, the computational formula of reflection coefficient of the first and second layers is as follows [19], [20], [21], [22]:

Figure 1:

Computational model.

Here, θ₁ is the incidence angle between layers 1 and 2, φ₁ is the refraction angle on the interface, and Z_i (i = 1, 2, 3…) is the wave impedance of each layer. According to the same theory, the reflection coefficient of the interface between mediums 2 and 3 is as follows:

In (2), Z_eff is the equivalent wave impedance of the interface between layers 2 and 3. The calculation method of Z_eff is determined by (3).

Here, k₂ is electromagnetic wave propagation constant of the second layer, and d₂ is the medium thickness of the same layer. For multilayered medium, the calculation method is the same formula and so on.

In the discussion, we write k=ωcεr1/2 as the wave vector, and ε_r is the relative permittivity of plasma. The effective wave impedance Z_eff at the interface of 1 and 2 can be obtained by substituting (3). Finally, the total reflection coefficient of the three-layered structure Γ_2,3 can be calculated by (2). As layer 1 is perfect electric conductor, the electromagnetic wave at the interface between the first layer and the second layer will be reflect totally, and Γ_1,2 = −1. Therefore, the power absorption coefficient of electromagnetic wave in the plasma is written as

The dielectric constants of magnetised plasma can be given by the Appleton formula [23], [24]:

When it is vertical incidence wave, the angle of θ = 0, (5) can be reduced to the following simple form:

Here, ω_p is plasma angular frequency, v_ce is plasma collision frequency, ω is incident wave angular frequency, and ω_b is cyclotron angular frequency caused by external magnetic field. The symbols of ω_b correspond to the left when you pick up the +; it will cause left circular polarisation, and when you pick up the −, it will cause right circular polarisation. When the magnetic plasma in the external magnetic field is zero, ω_b = 0. Its dielectric constant expressions degrade consistent with unmagnetised plasma forms, the method can also be used for the research of unmagnetised plasma.

When the electromagnetic wave is oblique incidence wave, due to the dielectric constant of the plasma being a complex number, direct application of Fresnel law will generate refracting angle for the complex number, which is caused by the imaginary unit of the plasma collision frequency. Refraction angle is a complex number; it can be resolved through the calculation of equivalent refractive index of plasma. Generally, when the collision frequency is small, such as ν_ce < 0.1ω [25], the imaginary part is relatively small for the real part of the refraction. When the angle is small, the impact on the calculation can be ignored.

Next, the absorption coefficient is calculated when the THz wave acts on the plasma slab. Parameters used in this example are as follows: The plasma thickness d = 15 cm, collision frequency is ν_ce = 1.6 × 10¹¹ rad/s, and electron density is n_e = 7.9 × 10¹⁸/m³. From Figure 2, we can see that the calculated results in this article are in good agreement with the FDTD simulation and also agreement with the literature [26]. Result of Jamison and colleagues’ [16] experiment is slightly higher than the peak value, but the overall trend is consistent. In the experiment, the plasma was generated by helium discharge by high-voltage pulse excitation. Helium is encapsulated in a 15-cm tube with a pressure of 24 mbar. Because the uniformity of plasma density and the effect of the tube wall of bound plasma are not considered in the calculation process in our article, this may be the reason why the calculated results are higher than the experimental results, while the positions of absorption peaks are basically the same. Therefore, it is feasible to study THz wave absorption by plasma using this method.

Figure 2:

Results compared with the literature.

2.2 The Computational Model and Results

The computational model is shown in Figure 1, in which medium 1 is the ideal metal, medium 2 is plasma, and medium 3 is free space. The THz wave acts into the plasma medium from the free space; the incidence angle is θ. The plasma slab thickness is d = 10 cm. The relationship between magnetic field B and cyclotron angular frequency ω_b is ω_b = qB/m, where suitable q is elementary charge, and m is the electron quality. When plasma density is n_e = 5 × 10²¹/m³, plasma collision frequency is v_ce = 0.05 × 10¹² Hz, incidence angle θ = 0, and the plasma slab double-pass absorption power can be obtained and shown in Figures 3 and 4. For comparison, the results of the FDTD method are given below. The FDTD space step dx = 5 μm, and time step dt = dx/2c, with 10 layers of perfectly matched layer on both sides as absorption boundary. The whole computing space is 20,500 meshes. The plasma slab with thickness of 10 cm occupies 20,000 meshes between 400 and 20,400. The program runs 150,000 time steps. Differential Gauss pulse is used as wave source in the form of

Figure 3:

The influence of magnetic field on the L-wave power absorption spectrum.

Figure 4:

The influence of magnetic field on the R-wave power absorption spectrum.

When the magnetic field intensity changes, the absorption coefficient of plasma for the L-wave is shown in Figure 3, as the graph shows that when the magnetic field intensity increases, the THz wave absorption coefficient is smaller, and the absorption peak position is moving to the low-frequency band. This is due to the magnetic field intensity having weakened the electron collision, which results in decreased absorption.

When the magnetic field intensity changes, the absorption coefficient of plasma for the R-wave is shown in Figure 4. We can see from Figure 4, when the magnetic field intensity is zero, there is an absorption peak at 1.5-THz position. When the magnetic field intensity enhances and ω_b = 10 × 10¹² rad/s, the absorption peak divides into two parts, one at the 1-THz point and the other at the 3-THz point. These two absorption peaks are, respectively, corresponding to the magnetic field intensity caused by cyclotron absorption and plasma collision absorption, and when the magnetic field is strengthened, the collision absorption is strengthened as well. The second absorption peak width is larger. There is a strong reflection spectrum that is very narrow between the two absorption peaks. As the field continues to strengthen, both absorption peaks are moving to high frequency and are gaining more bandwidth. The bandwidth basically remains unchanged in the two strong reflection peak areas, but the reflection gradually reduced. When ω_b = 70 × 10¹² rad/s and ω_b = 100 × 10¹² rad/s, the absorption peaks move to a high-frequency field, until removed from the observed frequency range.

When ω_b = 10 × 10¹² rad/s and the other parameters are constant, the incident angle changes that affect the plasma slab power absorption can be examined. The results are shown in Figures 5 and 6. The L-wave power absorption coefficient varies with the change of incident angle, as shown in Figure 5. The incident angle increases lead to the L-wave absorption peak moving towards the high frequencies. When the incidence angle is 30°, the absorption peak splits into two parts, and the first absorption peak is almost not affected by the incident angle, whereas the second absorption peak is moving to high frequency with the increase in angle. This suggests that cyclotron absorption is affected by the magnetic fields. When ω_b is changeless, the cyclotron absorption basically remains unchanged. The collision absorption is influenced by the mutual interaction of the electromagnetic wave and ion. The electromagnetic wave’s propagation wave path is larger in plasma at oblique incidence. Thus, the high-frequency spectrum absorption becomes stronger. It causes the absorption peak to move to the right side. There appears a reflection area between the two absorption peaks.

Figure 5:

The effect of incident angle of L-wave power absorption spectrum.

Figure 6:

The effect of incident angle of R-wave power absorption spectrum.

The R-wave power absorption coefficient varies with the change of incident angle, as shown in Figure 6. There are two absorption peaks when the R-wave incident vertically, as shown in Figure 6. With the increase in incident angle, the frequency bandwidth of the two absorption peaks of the R-wave becomes wider and wider. The reflection is becoming stronger. The absorption of the high-frequency band is becoming stronger. The absorption spectrum is becoming wider.

Now, using the above parameters, when the plasma layer thickness d is changed, we study its influence on the electromagnetic wave absorption power. The results are shown in Figures 7 and 8. The L-wave power absorption coefficient in the different plasma thickness is shown in Figure 7. While the plasma layer thickness d is increased from 5 cm to 25 cm, the peak of power absorption spectrum is almost unchanged, but the absorption spectrum bandwidth grows and mainly increases at high frequencies. For the R-wave, there is a similar change. The position of the peak and the strong reflection area has not changed, and the absorption bandwidth of high frequency increases. This is because the plasma reflects the low-frequency wave more strongly than the high-frequency wave. By increasing the plasma thickness, the wave path of electromagnetic wave in the plasma field will increase; it will lead to collision frequency increase. These reasons generate the results.

Figure 7:

The plasma thickness affects L-wave power absorption spectrum.

Figure 8:

The plasma thickness affects R-wave power absorption spectrum.