**1. Introduction**

Gaseous particles are ionized to bring them in the form of plasma through the various heating techniques. One of the popular heating techniques is the injection of high frequency microwaves (MW) to a cylindrical cavity that has comparable dimension to the injected MW wavelength. The MW plasma generated by the continuous or pulse feeding of the MW is used in the applications of industrial and accelerator fields for the material science and nuclear applications, respectively. In both of the feeding cases, the plasma is basically produced due to the power absorption by the electrons from the space-time dependent electric field of the MW. The spatio-temporal dynamics and also the steady-state behaviors of the plasma are governed by the ways the MW are coupled to the plasma sustained inside a cavity. The behavioral pattern of the electric field during the plasma evolution can help us to comment on the different MW coupling ways/mechanisms that are involved in the formation of plasma particles and their confinement scenarios. By mastering the basic concepts on those different coupling mechanisms, the coupling efficiency and so the performance of that particular plasma source can be optimized. Performance optimization for this kind of plasma source is indispensable as these are involved in various kinds of applications as mentioned above. One of the important plasma devices is the microwave ion sources that are operated in continuous as well as pulse mode to extract the ion beam during the transient and steady state periods of plasma loading conditions [1–5]. The beam qualities are influenced by the MW coupling mechanisms as they are involved in deciding the plasma parameters during the extraction of a particular instant of the plasma evolution time. Several studies have already reported the electric field evolution during few 10 s of microsecond range when the plasma density was increasing in the very similar plasma device. The electric field was dropped by about more than 50% within a span of few microseconds after the MW launch (t = 0 s) into the cavity [6–11].

fundamental cavity resonant mode. The cavity can resonate with some additional resonating frequencies including the fundamental one. The additional resonating frequencies can lie near to the fundamental one. Due to this reason, if the microwave is launched to the modified cavity, the total microwave field is shared among the cavity resonant modes including the fundamental one and contributes to the

*Evolution of Microwave Electric Field on Power Coupling to Plasma during Ignition Phase*

From the electromagnetic theory of a resonant cavity, only particular cavity resonant modes can exist having fixed frequencies that are given by [12, 13]

s

where *r*, *l* and *h* are the integer, the length of the plasma cavity and the eigenvalues of the cavity, respectively. The eigenvalues are obtained from the

*<sup>t</sup> Hz*<sup>0</sup> <sup>þ</sup> *<sup>h</sup>*<sup>2</sup>

*<sup>t</sup> Ez*<sup>0</sup> <sup>þ</sup> *<sup>h</sup>*<sup>2</sup>

A theoretical calculation for the cavity resonant modes from the eigenvalue equations for the electromagnetic field [12, 13] is performed from the empty and completely closed cavity. By considering a simplest cylindrical cavity of radius r and the length '*d*' that is filled with a dielectric constant *ε<sup>r</sup>* and the relative permittivity *μr*, the resonant frequencies allowed by the cavity that can be determined from the

s

s

of order n and its first derivative. The indices represent the three-dimensional

In case of magnetized plasma, the microwave propagation is influenced by the plasma particle dynamics. As the plasma particle dynamics are represented by the particle velocity, the thermal velocity of the plasma particles should be considered when the microwave propagation in plasma is discussed. If the thermal velocity of the plasma particles is negligible with respect to the phase velocity of the microwave, i.e., *vthermal* ≪ *vphase*, this approximation is useful for determining the dispersion of the microwave in a magnetized plasma. This approximation also known as 'cold plasma' is not applicable to the locations where the microwave encounters resonance in the magnetized plasma. On the other hand, if the temperature of the plasma particles is high enough that makes the velocity of the particle to be relativistic (also called as 'warm plasma'), the influence of the plasma particles cannot be neglected while determining the microwave propagation. In that case, the damping

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

þ *lπ d* � �<sup>2</sup>

*nm* are, respectively, the zeros of order m of the Bessel functions

þ

*lπ d* � �<sup>2</sup>

*P*0 *nm r* � �<sup>2</sup>

*Pnm r* � �<sup>2</sup>

∇2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *r*2*π*<sup>2</sup> *l* <sup>2</sup> <sup>þ</sup> *<sup>h</sup>*<sup>2</sup>

*Hz*<sup>0</sup> ¼ 0 (2)

*Ez*<sup>0</sup> (3)

(1)

(4)

(5)

*<sup>f</sup>* <sup>¼</sup> <sup>1</sup> 2*π*

∇2

eigenvalue equations are shown below from the TE and TM modes:

2*π* ffiffiffiffiffiffiffiffi *μrε<sup>r</sup>* p

2*π* ffiffiffiffiffiffiffiffi *μrε<sup>r</sup>* p

electromagnetic field patterns for each cavity resonant mode.

of the microwave field is greatly influenced by the plasma particles.

*f TE nml* <sup>¼</sup> *<sup>c</sup>*

*f TE nml* <sup>¼</sup> *<sup>c</sup>*

**2.1 Microwave propagation in plasmas**

power coupling to the plasma.

*DOI: http://dx.doi.org/10.5772/intechopen.92011*

solution of the equations gives by:

Here, *Pnm* and *P*<sup>0</sup>

**35**

Since the electric fields can affect the different power coupling mechanisms during the gas ignition moment (ns to μs), the spatio-temporal plasma parameters are influenced significantly especially in the low pressure regime. Many researchers have used the kinetic models like PIC/MCC or even the hybrid fluid/PIC to obtain more precise results in the MW plasma discharge. But they failed to estimate the hot electron dynamics efficiently in lower pressure condition, as these models demand intensive computational hardware due to its particle approach. Therefore, the current chapter presents the electric field evolution and its impact on the plasma parameter build-up during low pressure plasma state. Here, the model used is based on the finite element method (FEM) that gives more appropriate results for the transient plasma parameters through fluid modeling approach and time-dependent, partial differential equation solver (TDPDE) using fewer computer resources [11]. The different MWplasma coupling mechanisms (ECR, UHR and electric field polarity reversal associated with ES wave heating) during the plasma density evolution after the MW launch (t = 0 s) can be understood from the behaviors of electric fields.
