Controllable Enhancement of Evanescent and Transmitted Waves by a Plasma Sphere

In this study, we present that the plasma sphere can focus and enhance the evanescent and transmitted waves. Electromagnetic waves propagating in a plasma sphere with a positive or negative permittivity, which leads to the enhancement of transmitted and evanescent waves, are analyzed. The intensity of the focused beam can be hundreds of times stronger than that of the incident wave. The enhancement effect is associated with plasma frequency, collision frequency, and incident wave frequency. The results illustrate that the electromagnetic wave can be focused, reflected, and oscillated by controlling the electromagnetic parameters of the plasma sphere. With a strong field enhancement available, it is possible to be used in microwave power amplifiers, plasma antennas, reflectors, etc.


INTRODUCTION
sphere, which is represented by the number of concentric spherical shells, each with a fixed electron density.
The internal and near fields of the dielectric sphere have been studied [15][16][17][18]. Gouesbet et al. [19] and Lock [20] studied the theoretical development of different expanded descriptions of electromagnetic scattering with a homogeneous spherical particle with the generalized Lorenz-Mie theory. Plasma is not only a dispersive medium but also can be controlled electrically [21]. The electromagnetic properties of plasma, such as permittivity, vary due to the plasma frequency, collision frequency, and incident wave frequency [22]. Plasma could evanescent, reflect, or absorb the electromagnetic wave by simply choosing the appropriate electromagnetic properties together with other parameters.
The charged particles in the plasma are capable of interacting with electromagnetic fields, which leads to many phenomena. In this study, we have studied the properties of near field and internal field for the plasma sphere with positive and negative permittivity. The focusing effects of evanescent and transmitted waves by a plasma sphere have been discussed. The influence of the plasma frequency, the collision frequency, and the incident frequency of electromagnetic waves on the intensity distribution of the focused beam is given. With various parameters of the plasma sphere, the different properties of evanescent and transmitted waves are discussed.

THEORIES FOR THE NEAR FIELD OF A PLASMA SPHERE
The plasma is a medium full of free charge carriers and tends to maintain the electric charge neutral under equilibrium conditions. The electromagnetic waves are incident on the plasma, excite, and perturb plasmas. When the plasma is disturbed from the equilibrium condition due to the thermal and electrical disturbances, the internal fields give rise to collective particle motions, which are characterized by a natural frequency of oscillation known as the (electron) plasma angular frequencyω p , given by ω p n e e 2 ε 0 m e The plasma frequency f p is expressed as f p ω p /2π, where m e is the mass of the electron, n e is the density of electrons, ε 0 is the permittivity of free space, and e is the charge of an electron.
We assume that an incident plane wave propagates into a plasma in the direction of the positive z-axis. Considering the effects of electron-neutral collision, the motion equation of the electrons in the plasma under the action of the Lorentz force and collisional forces can be written as [23] m e z u e zt −e E + u e × B − ] c m e u e , where u e is the average electron velocity and ] c is the collision frequency for the electrons and the neutral particles. The average motions of the neutral and ion particles are neglected because they are much more massive than those of the electrons. The variables E i , B i , and u e vary harmonically in space and time. The plasma is a collection of charged and neutral particles moving in their own internal fields and can be treated as a dielectric medium characterized by dielectric permittivity. Combining the Maxwell equation, the relative complex permittivity of plasma medium ε pr can be obtained [24,25] as where ε′ pr and ε′′ pr are the real and imaginary parts of relative permittivity, respectively. The transmission of the electromagnetic waves in the plasma sphere is affected by the plasma parameters. According to Eq. 3, the plasma permittivity depends on the incident frequency f i , the plasma frequency f p , and the collision frequency ] c [26]. The influences of different collision frequencies and plasma frequencies on the plasma permittivity are exemplified with f i 4 GHz in Figure 1.
The real part of plasma permittivity can be positive or negative, as shown in Figure 1A. When f p /f i 1.05, the incident frequency is smaller than the plasma frequency, but the real part of the dielectric constant tends to be positive as ]/f i increases. The imaginary part of plasma permittivity is always negative, as shown in Figure 1B. In this case, the plasma medium behaves as a lossy dielectric medium. As f p /f i increases, the real part and the imaginary part of plasma permittivity decrease.
The dispersion of plasma is caused by the frequencydependent nature of the plasma-wave interaction. The plasma permittivity parameters can be controlled electrically by changing the plasma frequency, the collision frequency, and the incident frequency. The electromagnetic wave may be evanescent, reflected, or absorbed within the plasma medium depending on the appropriate parameters of plasma and the incident wave.
Interactions between a plasma sphere and the plane wave are analyzed based on the Mie theory [27]. In spherical coordinates(r, θ, φ), the internal and scattered fields are expanded in vector spherical harmonics.
The internal field is given as The scattered field is given as Frontiers in Physics | www.frontiersin.org May 2022 | Volume 10 | Article 890213 where A n , B n , C n , and D n are the expansion coefficients of the scattered and internal fields, and M and N are the vector spherical wave functions. The boundary conditions, derived from the continuity of the tangential components of fields on the surface of the sphere can be resolved to obtain all unknown expansion coefficients A n , B n , C n , and D n [28]: The prime denotes derivation with respect to the argument, x is the size parameter of the plasma sphere, and μ p and μ are permeability of plasma and the surrounding medium, respectively. Thus, all fields of the plasma sphere are determined.

SIMULATION RESULT OF FOCAL CHARACTERISTICS OF A PLASMA SPHERE
The plane wave is incident on the plasma sphere with a radius R 20 cm in the x-z plane. The internal and near fields of a plasma sphere are calculated. Figure 2 shows the distribution of internal and near fields for a plasma sphere when the plasma frequency f p is 9 GHz, collision frequency ] c is 29 GHz, and incident frequency f i is 2.1 GHz, corresponding to the plasma permittivity ε pr −2.15 − 6.92i. As seen in Figures 2A and B, the incident wave is reflected on the left surface of a plasma sphere, and the electromagnetic wave cannot propagate inside The maximum intensity of the evanescent wave is 70 times more than that of the incident field, as shown in Figure 2B. The result indicates that the plasma sphere with negative permittivity could focus and enhance the field of the evanescent wave. The enhancement effect of the evanescent wave by the plasma sphere is associated with plasma frequency and collision frequency.
For example, when] c 0 GHz, 25 GHz, 31 GHz, 37 GHz, the maximum relative intensities of the evanescent wave on the shadow-side surface of a plasma sphere are 0.3I 0 , 20.82I 0 , 149.56I 0 , 24.30I 0 , respectively, as plotted in Figure 3A where I 0 is the intensity of the incident field. The peak values of the evanescent waves appear on the two sides of the plasma sphere surface and decay rapidly along the z direction. The incident wave, surface wave, and reflected wave are coherent to form an interference phenomenon in the near field [29]. As ] c ≈ 0 GHz in Eq. 3, it could represent a conductor sphere, whose relative permittivity obeys the Drude model [30]. The surface wave of the conductor sphere is much less than that of a plasma sphere. The amplitudes of waves can be grown or damped with different collision frequencies [31]. When collision frequency increases from 0 to 40 GHz, the maximum intensity of the evanescent wave increases when ] c < 32.1 GHz and then decreases as ] c > 32.1 GHz, as shown in Figure 3B.
The influence of different plasma frequencies on the field distribution of the plasma sphere with f i 2.1 GHz and ] c 4 GHz is depicted in Figure 4A. The maximum intensities of the evanescent wave are 2.87I 0 , 44.81I 0 , 8.23I 0 , at     Frontiers in Physics | www.frontiersin.org May 2022 | Volume 10 | Article 890213 5 f p 4.8 GHz, 5.5 GHz, 6.2 GHz, respectively. When the plasma frequency changes from 4.5 to 9 GHz as f p < 5.5 GHz, the enhancement of the evanescent field increases and then decreases as f p > 5.5 GHz, as shown in Figure 4B.
It should also be noted that the enhancement effect of the evanescent wave is not produced in any cases. As shown in Figure 5, most of the incident waves are reflected on the surface of the plasma sphere. But no evanescent wave is enhanced on the shadow-side surface of the sphere, and the electromagnetic wave propagates into the plasma with the intensity slightly increasing along the z direction. This high reflection effect can be achieved by a plasma sphere with negative permittivity.
When the real part of plasma permittivity is positive, the electromagnetic waves propagate inside the plasma in a manner similar to wave propagation in a dielectric medium. The dielectric property of plasma is governed by plasma parameters and incident wave frequency. Figure 6 shows the intensity distribution of plasma sphere with f i 2.1 GHz, f p 2 GHz, ] c 60 GHz, and the corresponding plasma permittivity is ε pr 0.96 − 0.19i.
The plane wave is incident from the left and impinges on the plasma sphere. The focusing field of transmitted waves with narrow and high intensity is observed in the near field of the shadow-side surface. The field is highly confined to the surface along the propagation axis z. The corresponding radial internal and near-external intensity distribution of the plasma sphere along z is shown in Figure 6B. For a given unitary incident plane wave, the intensity maximum outside the plasma sphere is amplified by more than 50 times. On the left side of the main intensity, there are multiple sharp spikes, which have very narrow widths and different intensities. A part of the incident waves is reflected inside the plasma sphere.
The focusing intensity of transmitted waves depends on the plasma parameter. Figure 7 exemplifies the influence of different collision frequencies on the intensity distribution of the plasma sphere with f i 2.1 GHz, f p 2 GHz. It can be seen that the maximum intensity of the focal spot decreases from 106I 0 to 9.5I 0 as the collision frequency increases from 50 to 110 GHz, corresponding to ε pr 0.94 − 0.22i, 0.97 − 0.17i, 0.98 − 0.13i, 0.99 − 0.10i, respectively. All the locations of the focal spot are close to the surface of the plasma sphere. Figure 8 shows the dependence of the maximum intensity of the focal spot on different collision frequencies and plasma frequencies. The maximum intensity decreases as the collision frequency increases at the same plasma frequency. When the plasma frequency is 2.2 GHz, the maximum intensity decreases from 118.21I 0 to 6.9I 0 , while the collision frequency changes from 60 to 158 GHz. As the plasma frequency increases, the peak intensity of the focal spot is getting higher at the same collision frequency.   The characteristics of the intensity distribution of the focal spot also depend on the incident frequency. Figure 9 shows the influence of different incident frequencies on internal and near fields for a plasma sphere with f p 2 GHz, ] c 50 GHz. By increasing the incident frequency from 2.1 to 8.1 GHz, the focal peak shifts far away from the shadow-side of the sphere along the z direction, and the peak value of the intensity decreases. There is a minor narrow peak appearing on the left side of the primary peak.
It should be mentioned that the focus effect of a plasma sphere requires the parameters, such as collision frequency, plasma frequency, incident frequency, and size of the sphere, to meet certain conditions. For different parameters, the electromagnetic wave can be reflected, oscillated, and transmitted by the plasma sphere. Figure 10 shows the three-dimensional intensity pattern of internal and near fields in the x-z plane with f i 2.1 GHz, f p 2 GHz, ] c 6 GHz. The incident wave is reflected on the surface of the plasma sphere along the incident direction, and almost no electromagnetic wave is transmitted through the plasma sphere. The reflected wave interferes with the incident wave, forming interference fringes. In this case, the plasma behaves as a conductor and can be used as a plasma antenna.
With a collision frequency of 6 GHz, a plasma frequency of 2 GHz, and an incident frequency of 3 GHz, the incident electromagnetic waves are divided into three parts by the plasma sphere: partially oscillated, partially reflected, and transmitted, as shown in Figure 11. It is observed that most incident electromagnetic waves are confined inside a plasma sphere and form oscillation.

CONCLUSION
The plasma sphere has focus and enhanced features for transmitted and evanescent waves as proved in simulation results. The intensity of the focus spot can reach one hundred times larger than that of the incident wave. The intensity distributions of the internal and near fields for the plasma sphere are determined by electromagnetic parameters, which are collision frequency, plasma frequency, and incident wave frequency. The plasma sphere focuses on evanescent and transmitted waves by controlling the plasma parameters. The plasma offers solutions for the increasing requirements of electromagnetic field enhancement, such as microwave power amplifiers, plasma antennas, and subwavelength imaging applications.

DATA AVAILABILITY STATEMENT
The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.