**2. Basic theory of microwave plasma interaction**

The study on the propagation and interactions of the microwave with the plasma is important to optimize the performance of any plasma devices like the microwave ion sources. The microwave propagation in the plasma is affected by the dielectric prosperities of the plasma medium. The dielectric property, i.e., the permittivity or the refractive index of the plasma, depends on the external magnetic field distribution that is used to confine the plasma particles and also the electrostatic fields that are present in the plasma. Therefore, the microwave while propagating in different directions within the plasma encounters different values of the refractive index as well as the permittivity that makes the magnetized plasma to be anisotropic and inhomogeneous, respectively. To generate a high plasma density, which is one of the primary requirements in some microwave plasma devices, viz., the microwave discharge ion source (MDIS) or electron cyclotron resonance ion sources (ECRIS), an optimum coupling of the microwave energy through the different interaction mechanisms to the plasma medium is necessary. The microwave propagation and the coupling mechanisms are also influenced by the boundary conditions present in the plasma devices and the geometrical shape of the plasma device. In most cases, the dimension of the plasma reactor used for the purpose of ion sources lie in the comparable range of the launched microwave wavelength. This means the microwave electromagnetic field propagation within the ion source reactor (or cavity) is guided by the boundary conditions and the geometrical shape of the ion source cavity. So, the microwave electromagnetic field coupling to the plasma is affected if the cavity geometry is perturbed. Due to the modification of the cavity geometry, the resonating properties of the cavity resonator are no longer dominated by the

*Evolution of Microwave Electric Field on Power Coupling to Plasma during Ignition Phase DOI: http://dx.doi.org/10.5772/intechopen.92011*

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 power coupling to the plasma.

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

$$f = \frac{1}{2\pi} \sqrt{\frac{r^2 \pi^2}{l^2} + h^2} \tag{1}$$

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 solution of the equations gives by:

$$
\nabla\_t^2 H\_{x0} + h^2 H\_{x0} = 0 \tag{2}
$$

$$
\nabla\_t^2 E\_{x0} + h^2 E\_{x0} \tag{3}
$$

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

$$f\_{nml}^{\text{TE}} = \frac{c}{2\pi\sqrt{\mu\_r\varepsilon\_r}}\sqrt{\left(\frac{P\_{nm}'}{r}\right)^2 + \left(\frac{l\pi}{d}\right)^2} \tag{4}$$

$$f\_{nml}^{\text{TE}} = \frac{c}{2\pi\sqrt{\mu\_r\varepsilon\_r}}\sqrt{\left(\frac{P\_{nm}}{r}\right)^2 + \left(\frac{l\pi}{d}\right)^2} \tag{5}$$

Here, *Pnm* and *P*<sup>0</sup> *nm* are, respectively, the zeros of order m of the Bessel functions of order n and its first derivative. The indices represent the three-dimensional electromagnetic field patterns for each cavity resonant mode.

#### **2.1 Microwave propagation in plasmas**

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

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

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

The study on the propagation and interactions of the microwave with the plasma is important to optimize the performance of any plasma devices like the microwave ion sources. The microwave propagation in the plasma is affected by the dielectric prosperities of the plasma medium. The dielectric property, i.e., the permittivity or the refractive index of the plasma, depends on the external magnetic field distribution that is used to confine the plasma particles and also the electrostatic fields that are present in the plasma. Therefore, the microwave while propagating in different directions within the plasma encounters different values of the refractive index as well as the permittivity that makes the magnetized plasma to be anisotropic and inhomogeneous, respectively. To generate a high plasma density, which is one of the primary requirements in some microwave plasma devices, viz., the microwave discharge ion source (MDIS) or electron cyclotron resonance ion sources (ECRIS), an optimum coupling of the microwave energy through the different interaction mechanisms to the plasma medium is necessary. The microwave propagation and the coupling mechanisms are also influenced by the boundary conditions present in the plasma devices and the geometrical shape of the plasma device. In most cases, the dimension of the plasma reactor used for the purpose of ion sources lie in the comparable range of the launched microwave wavelength. This means the microwave electromagnetic field propagation within the ion source reactor (or cavity) is guided by the boundary conditions and the geometrical shape of the ion source cavity. So, the microwave electromagnetic field coupling to the plasma is affected if the cavity geometry is perturbed. Due to the modification of the cavity geometry, the resonating properties of the cavity resonator are no longer dominated by the

microseconds after the MW launch (t = 0 s) into the cavity [6–11].

*Selected Topics in Plasma Physics*

(t = 0 s) can be understood from the behaviors of electric fields.

**2. Basic theory of microwave plasma interaction**

**34**

For the un-magnetized plasma case in which the plasma is considered to be isotropic and the condition, *vthermal* ≪ *vphase* is satisfied, the plasma fluid equation is written as:

$$\frac{\partial \tilde{w}}{\partial t} = q \tilde{E} \tag{6}$$

<sup>∇</sup><sup>~</sup> � *<sup>B</sup>*<sup>~</sup> <sup>¼</sup> *<sup>μ</sup>*<sup>0</sup> <sup>~</sup>*<sup>J</sup>* <sup>þ</sup> *<sup>ε</sup>*<sup>0</sup>

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

*<sup>D</sup>*<sup>~</sup> <sup>¼</sup> *<sup>ε</sup>*<sup>0</sup> *<sup>I</sup>*

*mi ∂*~*v ∂t*

> � � � � � � � �

*ϵ* ¼ ¼ *ϵ*<sup>0</sup>

> *pe ω*2 *ωc <sup>ω</sup>* ( <sup>1</sup> 1�*ω<sup>c</sup>*

electric field by following the standard procedure that shows

The above equation is rewritten by assuming the *e<sup>i</sup>*

~ *<sup>k</sup>* � <sup>~</sup>

*<sup>S</sup>* � *<sup>N</sup>*<sup>2</sup> *cos* <sup>2</sup>*<sup>θ</sup>* �*iD N*<sup>2</sup>

*N*2

<sup>∇</sup><sup>~</sup> � <sup>∇</sup><sup>~</sup> � *<sup>E</sup>*<sup>~</sup> ¼ � *<sup>∂</sup>*∇<sup>~</sup> � *<sup>B</sup>*<sup>~</sup>

tivity tensor is related with the current as *σ*

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

¼

obtained as:

as *ϵ*<sup>0</sup> *I* ¼ <sup>þ</sup> *<sup>i</sup> <sup>ε</sup>*0*<sup>ω</sup> σ*

where the symbol *I*

also be written as *<sup>D</sup>*<sup>~</sup> <sup>¼</sup> *<sup>ε</sup>*<sup>0</sup> *<sup>ϵ</sup><sup>r</sup>*

term is considered as:

dielectric tensor *ϵ*

*<sup>S</sup>* <sup>¼</sup> <sup>1</sup> � *<sup>ω</sup>*<sup>2</sup>

field as:

**37**

*pe ω*2

1 1�*ω<sup>c</sup>* h i � � , *<sup>D</sup>* = [� *<sup>ω</sup>*<sup>2</sup>

*∂E*~ *∂t* � �

Now if the plasma motion follows the *eiω<sup>t</sup>* dependence and the plasma conduc-

¼

¼ þ *i <sup>ε</sup>*0*<sup>ω</sup> <sup>σ</sup>* ¼

approach. Therefore, the fluid equation in case of 'cold plasma' approximation is rewritten by neglecting the collisional and pressure term and the magnetic field

¼ *μ*<sup>0</sup>

represents an identity tensor. In short, the parameter *D*~ can

<sup>¼</sup> *E*~, where the effective dielectric tensor *ϵ*

<sup>¼</sup> � �. The plasma conductivity tensor is evaluated from the fluid plasma

The solution to this equation brings out the relation between the velocity (~*v*) and the electric field of the microwave. As <sup>~</sup>*<sup>J</sup>* <sup>¼</sup> *ne*~*v*, the final expression for the effective

> � � � � � � � �

� *ϵ*<sup>0</sup> *ϵ<sup>r</sup>*

*pe ω*2 .

*c*2 *ϵ* ¼ *: ∂*2 *E*~

~

*S* �*iD* 0 *iD S* 0 0 0 *P*

where *S*, *D* and *P* can be written in terms of three different kinds of frequencies:

The wave equation can be derived from the Maxwell's curl equation for the

*<sup>k</sup>* � *<sup>E</sup>*<sup>~</sup> <sup>þ</sup> *<sup>ω</sup>*<sup>2</sup>

the wave vector and magnetic field to be '*θ*' and denoting the vector *<sup>N</sup>*<sup>~</sup> <sup>¼</sup> *<sup>c</sup>*

*iD S* � *<sup>N</sup>*<sup>2</sup> <sup>0</sup>

*cosθsin<sup>θ</sup>* <sup>0</sup> *<sup>P</sup>* � *<sup>N</sup>*<sup>2</sup> *sin* <sup>2</sup>

*c*2 *ϵ*

The equation can be represented in matrix form by assuming the angle between

*cosθsinθ*

*θ*

2 6 4

*Ex Ey Ez* 3 7

Þ� and *<sup>P</sup>* = 1 � *<sup>ω</sup>*<sup>2</sup>

*<sup>∂</sup><sup>t</sup>* ¼ � <sup>1</sup>

<sup>¼</sup> can be shown in a determinant form as:

*∂D*~

*<sup>∂</sup><sup>t</sup>* (11)

<sup>¼</sup> is represented

. *<sup>E</sup>*<sup>~</sup> <sup>¼</sup> <sup>~</sup>*J*, the relation between *<sup>D</sup>*<sup>~</sup> and *<sup>E</sup>*<sup>~</sup> is

� �*E*<sup>~</sup> (12)

<sup>¼</sup> *<sup>e</sup> <sup>E</sup>*<sup>~</sup> <sup>þ</sup> <sup>~</sup>*<sup>v</sup>* � *<sup>B</sup>*<sup>~</sup> � � (13)

<sup>¼</sup> (14)

*<sup>∂</sup>t*<sup>2</sup> (15)

*<sup>k</sup>:*~*<sup>r</sup>* dependence of the electric

*ω* ~ *k*.

<sup>5</sup> <sup>¼</sup> 0 (17)

<sup>¼</sup> *:E*<sup>~</sup> <sup>¼</sup> <sup>0</sup> (16)

If the microwave electric field (*E*~) and the velocity (~*v*) are assumed to be varying with *ei<sup>ω</sup><sup>t</sup>* , the plasma dielectric constant can be easily shown as:

$$\varepsilon = \varepsilon\_0 \left( 1 - \frac{a\_{pe}^2}{a^2} \right) \tag{7}$$

where the *ωpe* is the electron plasma frequency that is also obtained from the electron density by the relation:

$$
\rho\_{p\epsilon} = \sqrt{\frac{n\_{\epsilon}e^2}{c\_0 m\_{\epsilon}}} \tag{8}
$$

where *me*, e and *ϵ*<sup>0</sup> are the electron mass, charge and electrical permittivity in vacuum condition, respectively. So the electrical permittivity in plasma is affected by the plasma density. As the electrical permittivity has to be positive in ideal case, the plasma density has to be lower than the corresponding microwave frequency that is launched externally to energize the plasma particles. Therefore, in case of homogeneous and isotropic plasma, it is not possible to raise the plasma density beyond certain level that is known as the critical density, which is written as *ncritical* <sup>¼</sup> *<sup>ω</sup>*<sup>2</sup> *pe ϵ*0*me nee*<sup>2</sup> � �. The density below and above the critical density is referred to be underdense and overdense plasma. So the critical density is the main limitation in the un-magnetized plasma for the microwave propagation. The dispersion for the wave in magnetized plasma is written as:

$$
\alpha^2 - \alpha\_{p\epsilon}^2 = k^2 c^2 \tag{9}
$$

In the overdense plasma, if *ω*< *ωpe*, the propagation *k*-vector <sup>0</sup> *k*<sup>0</sup> becomes imaginary that means the complete reflection of the incoming microwave from the plasma. Therefore, if the microwave propagation is assumed to be propagating in the *x*-direction in the overdense plasma, the electric field of the microwave can be written as:

$$
\tilde{E} = \tilde{E}\_0 e^{(ik\cdot\vec{r} - at)} = \tilde{E}\_0 e^{(ikx - at)} = e^{\frac{\pi}{6}} e^{-iat} \tag{10}
$$

So, before encountering the overdense plasma, the electric field becomes an evanescent wave as its magnitude decays exponentially within a distance of approximately the skin depth value; (*δ*) = *c=* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *ω*<sup>2</sup> *pe* � *ω*<sup>2</sup> �q ).

Under the externally applied magnetic field, the dielectric constant for the anisotropic plasma becomes a tensor quantity. It means the microwave propagation becomes dependent on the plasma dielectric properties while propagating in various directions with respect to the externally applied magnetic field. If the magnetic field is oriented axially (*B*<sup>~</sup> <sup>¼</sup> *<sup>B</sup>*0~*z*) in a cylindrical plasma chamber, the plasma can act as a dielectric with current ~*J* and also the Maxwell's equations can be rewritten as:

*Evolution of Microwave Electric Field on Power Coupling to Plasma during Ignition Phase DOI: http://dx.doi.org/10.5772/intechopen.92011*

$$
\tilde{\nabla} \times \tilde{B} = \mu\_0 \left( \tilde{J} + \varepsilon\_0 \frac{\partial \tilde{E}}{\partial t} \right) = \mu\_0 \frac{\partial \tilde{D}}{\partial t} \tag{11}
$$

Now if the plasma motion follows the *eiω<sup>t</sup>* dependence and the plasma conductivity tensor is related with the current as *σ* ¼ . *<sup>E</sup>*<sup>~</sup> <sup>¼</sup> <sup>~</sup>*J*, the relation between *<sup>D</sup>*<sup>~</sup> and *<sup>E</sup>*<sup>~</sup> is obtained as:

$$
\tilde{D} = \varepsilon\_0 \left( \stackrel{=}{\bar{I}} + \frac{\dot{\imath}}{\varepsilon\_0 \alpha} \stackrel{=}{\bar{\sigma}} \right) \tilde{E} \tag{12}
$$

where the symbol *I* ¼ represents an identity tensor. In short, the parameter *D*~ can also be written as *<sup>D</sup>*<sup>~</sup> <sup>¼</sup> *<sup>ε</sup>*<sup>0</sup> *<sup>ϵ</sup><sup>r</sup>* <sup>¼</sup> *E*~, where the effective dielectric tensor *ϵ* <sup>¼</sup> is represented as *ϵ*<sup>0</sup> *I* ¼ <sup>þ</sup> *<sup>i</sup> <sup>ε</sup>*0*<sup>ω</sup> σ* <sup>¼</sup> � �. The plasma conductivity tensor is evaluated from the fluid plasma approach. Therefore, the fluid equation in case of 'cold plasma' approximation is rewritten by neglecting the collisional and pressure term and the magnetic field term is considered as:

$$m\_i \frac{\partial \vec{v}}{\partial t} = e \left(\ddot{E} + \ddot{\nu} \times \ddot{B}\right) \tag{13}$$

The solution to this equation brings out the relation between the velocity (~*v*) and the electric field of the microwave. As <sup>~</sup>*<sup>J</sup>* <sup>¼</sup> *ne*~*v*, the final expression for the effective dielectric tensor *ϵ* <sup>¼</sup> can be shown in a determinant form as:

$$
\bar{\bar{c}} = \epsilon\_0 \begin{vmatrix} \mathcal{S} & -iD & \mathbf{0} \\ iD & \mathcal{S} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & P \end{vmatrix} \equiv \epsilon\_0 \begin{array}{ll} \bar{\bar{c}} \\ \bar{\bar{c}} \end{array} \tag{14}
$$

where *S*, *D* and *P* can be written in terms of three different kinds of frequencies: *<sup>S</sup>* <sup>¼</sup> <sup>1</sup> � *<sup>ω</sup>*<sup>2</sup> *pe ω*2 1 1�*ω<sup>c</sup>* h i � � , *<sup>D</sup>* = [� *<sup>ω</sup>*<sup>2</sup> *pe ω*2 *ωc <sup>ω</sup>* ( <sup>1</sup> 1�*ω<sup>c</sup>* Þ� and *<sup>P</sup>* = 1 � *<sup>ω</sup>*<sup>2</sup> *pe ω*2 .

The wave equation can be derived from the Maxwell's curl equation for the electric field by following the standard procedure that shows

$$
\tilde{\nabla} \times \tilde{\nabla} \times \tilde{E} = -\frac{\partial \tilde{\nabla} \times \tilde{B}}{\partial t} = -\frac{\mathbf{1}}{c^2} \frac{\bar{\varepsilon}}{\bar{c}} \cdot \frac{\partial^2 \tilde{E}}{\partial t^2} \tag{15}
$$

The above equation is rewritten by assuming the *e<sup>i</sup>* ~ *<sup>k</sup>:*~*<sup>r</sup>* dependence of the electric field as:

$$
\tilde{\vec{k}} \times \tilde{\vec{k}} \times \tilde{\vec{E}} + \frac{\alpha^2}{c^2} \stackrel{=}{\bar{c}} \cdot \tilde{\vec{E}} = \mathbf{0} \tag{16}
$$

The equation can be represented in matrix form by assuming the angle between the wave vector and magnetic field to be '*θ*' and denoting the vector *<sup>N</sup>*<sup>~</sup> <sup>¼</sup> *<sup>c</sup> ω* ~ *k*.

$$
\begin{bmatrix} S - N^2 \cos^2 \theta & -iD & N^2 \cos \theta \sin \theta \\\\ iD & S - N^2 & 0 \\\\ N^2 \cos \theta \sin \theta & 0 & P - N^2 \sin^2 \theta \end{bmatrix} \begin{bmatrix} E\_x \\ E\_y \\ E\_z \end{bmatrix} = \mathbf{0} \tag{17}
$$

For the un-magnetized plasma case in which the plasma is considered to be isotropic and the condition, *vthermal* ≪ *vphase* is satisfied, the plasma fluid equation is

> *∂*~*v ∂t*

, the plasma dielectric constant can be easily shown as:

If the microwave electric field (*E*~) and the velocity (~*v*) are assumed to be varying

*pe ω*2 !

> ffiffiffiffiffiffiffiffiffiffi *nee*<sup>2</sup> *ϵ*0*me*

. The density below and above the critical density is referred to be

ð Þ *ikx*�*ω<sup>t</sup>* <sup>¼</sup> *<sup>e</sup>*

�q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *ω*<sup>2</sup> *pe* � *ω*<sup>2</sup>

*x δ e*

).

<sup>2</sup> (9)

*k*<sup>0</sup> becomes

�*iω<sup>t</sup>* (10)

*<sup>ϵ</sup>* <sup>¼</sup> *<sup>ϵ</sup>*<sup>0</sup> <sup>1</sup> � *<sup>ω</sup>*<sup>2</sup>

where the *ωpe* is the electron plasma frequency that is also obtained from the

s

where *me*, e and *ϵ*<sup>0</sup> are the electron mass, charge and electrical permittivity in vacuum condition, respectively. So the electrical permittivity in plasma is affected by the plasma density. As the electrical permittivity has to be positive in ideal case, the plasma density has to be lower than the corresponding microwave frequency that is launched externally to energize the plasma particles. Therefore, in case of homogeneous and isotropic plasma, it is not possible to raise the plasma density beyond certain level that is known as the critical density, which is written as

underdense and overdense plasma. So the critical density is the main limitation in the un-magnetized plasma for the microwave propagation. The dispersion for the

> *pe* <sup>¼</sup> *<sup>k</sup>*<sup>2</sup> *c*

imaginary that means the complete reflection of the incoming microwave from the plasma. Therefore, if the microwave propagation is assumed to be propagating in the *x*-direction in the overdense plasma, the electric field of the microwave can be

So, before encountering the overdense plasma, the electric field becomes an evanescent wave as its magnitude decays exponentially within a distance of

Under the externally applied magnetic field, the dielectric constant for the anisotropic plasma becomes a tensor quantity. It means the microwave propagation becomes dependent on the plasma dielectric properties while propagating in various directions with respect to the externally applied magnetic field. If the magnetic field is oriented axially (*B*<sup>~</sup> <sup>¼</sup> *<sup>B</sup>*0~*z*) in a cylindrical plasma chamber, the plasma can act as a dielectric with current ~*J* and also the Maxwell's equations can be rewritten as:

*<sup>ω</sup>*<sup>2</sup> � *<sup>ω</sup>*<sup>2</sup>

ð Þ *ik:*~*r*�*ω<sup>t</sup>* <sup>¼</sup> *<sup>E</sup>*~0*<sup>e</sup>*

In the overdense plasma, if *ω*< *ωpe*, the propagation *k*-vector <sup>0</sup>

*<sup>E</sup>*<sup>~</sup> <sup>¼</sup> *<sup>E</sup>*~0*<sup>e</sup>*

approximately the skin depth value; (*δ*) = *c=*

*ωpe* ¼

<sup>¼</sup> *qE*<sup>~</sup> (6)

(7)

(8)

written as:

with *ei<sup>ω</sup><sup>t</sup>*

*ncritical* <sup>¼</sup> *<sup>ω</sup>*<sup>2</sup>

written as:

**36**

*pe ϵ*0*me nee*<sup>2</sup> � �

wave in magnetized plasma is written as:

electron density by the relation:

*Selected Topics in Plasma Physics*

The solution to these equations can exist if the determinant is zero. This condition brings out the dispersion relation of the microwave in the plasma. By making the determinant to be zero, two solutions are obtained that are written as:

$$N\_{O,X}^2 = \frac{2\varkappa(1-\varkappa)}{2(1-\varkappa) - \jmath^2 \sin^2\theta \pm \sqrt{\jmath^4 \sin^4\theta \pm 4\jmath^2(1-\varkappa)^2 \cos^2\theta}}\tag{18}$$

To visualize the cut-offs and resonances for the different types of microwaves in a

The polarization of the electric field of the microwaves also plays important role on the damping of microwave power into the plasma. From Eq. (17), the relation between the *x*- and *y*-components of the electric field can be found out to understand the polarization in the plane perpendicular to the magnetic field. The relation

<sup>¼</sup> *<sup>N</sup>*<sup>2</sup> � *<sup>S</sup>*

Eq. (20) indicates that the waves become circularly polarized and linearly polarized when the cut-off condition (*N*<sup>2</sup> <sup>¼</sup> 0) and the resonance condition (*N*<sup>2</sup> <sup>¼</sup> <sup>∞</sup>) are satisfied respectively. The waves are circularly and linearly polarized, respectively. However, it is confirmed that the ordinary waves are left hand polarized and the extraordinary waves are right hand polarized. The electric field of the right hand polarized waves and the electrons gyrating across the magnetic rotate in the same direction. Therefore, if the microwave-launched frequency matches the electron gyration frequency, electrons observe a constant electric field at the resonance location and are resonantly accelerated by the electric field. It is known as electron

*The CMA diagram shows the propagation of microwave launched from high and low magnetic field side. The arrow bend implies the cut-off region and the mode conversion region near the upper hybrid resonance location.*

*<sup>D</sup>* (20)

better way, the dispersion plots are shown in a single diagram, also known as Clemmow-Mullaly-Allis (CMA) diagram. **Figure 1** is applicable to the 'cold plasma' approximation case as discussed before. In the CMA diagram, the *x*- and *y*-axes represent the plasma density and the magnetic field, respectively. It means that for a given launch microwave frequency, a point on the diagram dictates the experimental situation that is characterized by a particular plasma density and the magnetic field. The diagram is divided into two regions: underdense and overdense plasma regions that are based upon the launch microwave frequency. The boundaries shown in the diagram correspond to the cut-off and resonances for the different types of microwaves.

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

*iEx Ey*

is obtained from Eq. (17) as:

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

**Figure 1.**

**39**

where the symbols '*O*' and '*X*' correspond to the ordinary microwave and extraordinary microwave, respectively. The notations *x* and *y* in Eq. (18) represent the electron density scale and magnetic field scale respectively and are denoted as: *<sup>x</sup>* <sup>¼</sup> *<sup>ω</sup>*<sup>2</sup> *pe <sup>ω</sup>*<sup>2</sup> <sup>∝</sup>*ne* and *<sup>y</sup>* <sup>¼</sup> *<sup>ω</sup><sup>c</sup> <sup>ω</sup>* ∝*B*0*:*

From the solutions, the microwave propagation and damping properties can be explained considering the values of the electron density, magnetic field and the angle of propagation. By the definition, the refractive index ð Þ *N* has to be positive for enabling the microwave to propagate in plasma. The two different values for the refractive index *N* = 0 and *N* = ∞ imply the cut-off and the resonance conditions for the microwave, respectively. From Eq. (18), three different types of cut-offs can be obtained by putting *N* = 0 that depends on the plasma parameters. Three cut-offs are: (1) *ω* ¼ *ωpe*; this cut-off is also known as 'O' cut-off; (2) *ω* ¼ *ω<sup>R</sup>* ¼ 0*:*5 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *ω*2 *<sup>c</sup>* þ 4*ω*<sup>2</sup> *pe* <sup>q</sup> þ *ωc*; this cut-off is also called upper or R cut-off frequency that occurs both above *ωpe* and *ωc*; and (3) *ω* ¼ *ω<sup>L</sup>* ¼ 0*:*5 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *ω*<sup>2</sup> *<sup>c</sup>* þ 4*ω*<sup>2</sup> *pe* <sup>q</sup> � *ωc*; it is also called the lower cut-off or L cut-off for the microwave. Based on the local plasma density values and magnetic field values in the plasma volume, "ordinary waves" have 'O' cut-off region, and extra-ordinary waves have two cut-offs, R cut-off, and L cut-off regions, from where corresponding waves are reflected back. To satisfy resonance condition (*N* = ∞), it is found from Eq. (18) that the term *Pcos*<sup>2</sup> *<sup>θ</sup>* <sup>þ</sup> *S sin* <sup>2</sup> *θ* has to be zero. It gives the resonance angle in terms of the plasma parameters that are given by:

$$
\cos^2 \theta = \frac{\varkappa + \jmath^2 - 1}{\varkappa \jmath^2} \tag{19}
$$

It is clear that the wave can resonate at the resonance angle (*θ*) that is decided by the angle of microwave propagation with respect to the magnetic field. From Eq. (18), it is also confirmed that only the extraordinary wave can have the resonances. From Eq. (19), the resonance angles for the extraordinary wave are determined by the plasma density and the magnetic field. For example, if the resonance angle is set to be zero, it is verified that the resonance takes place at *ω* ¼ *ωRF* ¼ *ωc*. This means the microwave matches the larmour frequency of the electrons that are gyrating around the magnetic field lines. It is also known as electron cyclotron resonance (ECR). Again, for the resonance angle, *θ* = 90°, another type of resonance can occur when *<sup>x</sup>* <sup>þ</sup> *<sup>y</sup>*<sup>2</sup> � 1 = 0, and this condition yields the resonance frequency to be *ω* ¼ *ωRF* ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *ω*2 *<sup>c</sup>* þ *ω*<sup>2</sup> *pe* <sup>q</sup> . This type of resonance is known as upper hybrid resonance (UHR) applicable to the wave (e.g., extraordinary wave) that is propagating perpendicularly with respect to the magnetic field. The resonance of the ordinary wave can also be understood from Eq. (18). It can be seen from one of the solutions obtained from Eq. (19) that another resonance can exist in the region where *x*, *y* ≥ 1. This is the forbidden region for the ordinary wave, which is the 'O' cut-off region. Therefore, the ordinary-type microwave can reach this resonance only by tunneling through this cut-off region.

*Evolution of Microwave Electric Field on Power Coupling to Plasma during Ignition Phase DOI: http://dx.doi.org/10.5772/intechopen.92011*

To visualize the cut-offs and resonances for the different types of microwaves in a better way, the dispersion plots are shown in a single diagram, also known as Clemmow-Mullaly-Allis (CMA) diagram. **Figure 1** is applicable to the 'cold plasma' approximation case as discussed before. In the CMA diagram, the *x*- and *y*-axes represent the plasma density and the magnetic field, respectively. It means that for a given launch microwave frequency, a point on the diagram dictates the experimental situation that is characterized by a particular plasma density and the magnetic field. The diagram is divided into two regions: underdense and overdense plasma regions that are based upon the launch microwave frequency. The boundaries shown in the diagram correspond to the cut-off and resonances for the different types of microwaves.

The polarization of the electric field of the microwaves also plays important role on the damping of microwave power into the plasma. From Eq. (17), the relation between the *x*- and *y*-components of the electric field can be found out to understand the polarization in the plane perpendicular to the magnetic field. The relation is obtained from Eq. (17) as:

$$\frac{iE\_\chi}{E\_\chi} = \frac{N^2 - S}{D} \tag{20}$$

Eq. (20) indicates that the waves become circularly polarized and linearly polarized when the cut-off condition (*N*<sup>2</sup> <sup>¼</sup> 0) and the resonance condition (*N*<sup>2</sup> <sup>¼</sup> <sup>∞</sup>) are satisfied respectively. The waves are circularly and linearly polarized, respectively. However, it is confirmed that the ordinary waves are left hand polarized and the extraordinary waves are right hand polarized. The electric field of the right hand polarized waves and the electrons gyrating across the magnetic rotate in the same direction. Therefore, if the microwave-launched frequency matches the electron gyration frequency, electrons observe a constant electric field at the resonance location and are resonantly accelerated by the electric field. It is known as electron

#### **Figure 1.**

*The CMA diagram shows the propagation of microwave launched from high and low magnetic field side. The arrow bend implies the cut-off region and the mode conversion region near the upper hybrid resonance location.*

The solution to these equations can exist if the determinant is zero. This condition brings out the dispersion relation of the microwave in the plasma. By making

> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>y</sup>*<sup>4</sup> *sin* <sup>4</sup>*<sup>θ</sup>* � <sup>4</sup>*y*2ð Þ <sup>1</sup> � *<sup>x</sup>*

2 *cos* <sup>2</sup>*θ*

<sup>q</sup> (18)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *ω*<sup>2</sup> *<sup>c</sup>* þ 4*ω*<sup>2</sup> *pe*

� *ωc*; it is also called

*θ* has to be

*<sup>θ</sup>* <sup>þ</sup> *S sin* <sup>2</sup>

*xy*<sup>2</sup> (19)

the determinant to be zero, two solutions are obtained that are written as:

*θ* �

(1) *ω* ¼ *ωpe*; this cut-off is also known as 'O' cut-off; (2) *ω* ¼ *ω<sup>R</sup>* ¼

condition (*N* = ∞), it is found from Eq. (18) that the term *Pcos*<sup>2</sup>

*cos* <sup>2</sup>

where the symbols '*O*' and '*X*' correspond to the ordinary microwave and extraordinary microwave, respectively. The notations *x* and *y* in Eq. (18) represent the electron density scale and magnetic field scale respectively and are denoted as:

From the solutions, the microwave propagation and damping properties can be explained considering the values of the electron density, magnetic field and the angle of propagation. By the definition, the refractive index ð Þ *N* has to be positive for enabling the microwave to propagate in plasma. The two different values for the refractive index *N* = 0 and *N* = ∞ imply the cut-off and the resonance conditions for the microwave, respectively. From Eq. (18), three different types of cut-offs can be obtained by putting *N* = 0 that depends on the plasma parameters. Three cut-offs are:

þ *ωc*; this cut-off is also called upper or R cut-off frequency that

the lower cut-off or L cut-off for the microwave. Based on the local plasma density values and magnetic field values in the plasma volume, "ordinary waves" have 'O' cut-off region, and extra-ordinary waves have two cut-offs, R cut-off, and L cut-off regions, from where corresponding waves are reflected back. To satisfy resonance

zero. It gives the resonance angle in terms of the plasma parameters that are given by:

*<sup>θ</sup>* <sup>¼</sup> *<sup>x</sup>* <sup>þ</sup> *<sup>y</sup>*<sup>2</sup> � <sup>1</sup>

It is clear that the wave can resonate at the resonance angle (*θ*) that is decided by

the angle of microwave propagation with respect to the magnetic field. From Eq. (18), it is also confirmed that only the extraordinary wave can have the resonances. From Eq. (19), the resonance angles for the extraordinary wave are determined by the plasma density and the magnetic field. For example, if the resonance angle is set to be zero, it is verified that the resonance takes place at *ω* ¼ *ωRF* ¼ *ωc*. This means the microwave matches the larmour frequency of the electrons that are gyrating around the magnetic field lines. It is also known as electron cyclotron resonance (ECR). Again, for the resonance angle, *θ* = 90°, another type of resonance can occur when *<sup>x</sup>* <sup>þ</sup> *<sup>y</sup>*<sup>2</sup> � 1 = 0, and this condition yields the resonance frequency to

nance (UHR) applicable to the wave (e.g., extraordinary wave) that is propagating perpendicularly with respect to the magnetic field. The resonance of the ordinary wave can also be understood from Eq. (18). It can be seen from one of the solutions obtained from Eq. (19) that another resonance can exist in the region where *x*, *y* ≥ 1. This is the forbidden region for the ordinary wave, which is the 'O' cut-off region. Therefore, the ordinary-type microwave can reach this resonance

q

. This type of resonance is known as upper hybrid reso-

*<sup>O</sup>*,*<sup>X</sup>* <sup>¼</sup> <sup>2</sup>*x*ð Þ <sup>1</sup> � *<sup>x</sup>*

2 1ð Þ� � *<sup>x</sup> <sup>y</sup>*<sup>2</sup> *sin* <sup>2</sup>

*<sup>ω</sup>* ∝*B*0*:*

occurs both above *ωpe* and *ωc*; and (3) *ω* ¼ *ω<sup>L</sup>* ¼ 0*:*5

*N*2

*Selected Topics in Plasma Physics*

*<sup>ω</sup>*<sup>2</sup> <sup>∝</sup>*ne* and *<sup>y</sup>* <sup>¼</sup> *<sup>ω</sup><sup>c</sup>*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *ω*2 *<sup>c</sup>* þ 4*ω*<sup>2</sup> *pe*

*<sup>x</sup>* <sup>¼</sup> *<sup>ω</sup>*<sup>2</sup> *pe*

0*:*5

q

be *ω* ¼ *ωRF* ¼

**38**

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *ω*2 *<sup>c</sup>* þ *ω*<sup>2</sup> *pe*

only by tunneling through this cut-off region.

q

cyclotron resonance heating (ECR). It is to be noted that only in the ECR heating mechanism the energy is transferred directly from the microwave to the gaseous electrons. As the electrons gain sufficient energy by resonance, they ionize the neutral gases and thus form plasma. There are also other mechanisms for energy transfer from the microwave to the plasma. But the direct energy transfer is not possible in these cases. In fact, the energy is shared among the microwave and the plasma oscillation modes by means of collisional absorption or non-linear phenomenon. The plasma oscillation modes are excited by the '*O*' and '*X*' modes of microwaves. A detailed description of the energy damping mechanisms is discussed below.

Here, *ω<sup>c</sup>* denotes the collision frequency. Then the dispersion relation is

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

*θ* �

modified dispersion equation of the wave propagation. This collisional term limits the attainable energy during the ECR heating unlike the collision-less case, as

Although the cold plasma approximation is not valid for the wave dispersion in the region very close to the cut-off, the effect of warm plasma condition cannot be ignored. This is because the wavelength in the latter case is not negligible compared to the scale length of the plasma parameters [14]. At the resonance, where the refractive index is infinity, the wavelength becomes equivalent to the electron larmor radius. So, the finite larmor parameter effect is not negligible and has to be

denotes the perpendicular component of the wave vector with respect to the magnetic field (*Z*-direction). For the non-relativistic case assuming the

> *n* X¼∞ *n*¼�∞

> > 0

Maxwellian distribution function, the distance of frequency from the *n*th cyclotron harmonic resonance in terms of the Doppler shift unit is written as *ξ<sup>n</sup>* ¼

> 2 6 4

Here *Zn* <sup>¼</sup> *<sup>Z</sup> <sup>ξ</sup><sup>n</sup>* ð Þ and *In* <sup>¼</sup> *<sup>e</sup>*�*<sup>ѓ</sup>In*ð Þ*<sup>ѓ</sup>* where *In* is the modified Bessel's function of

This dielectric tensor coming from warm plasma approximation has few new features that affect the wave propagation unlike the cold plasma approximation case. It can be seen from Eq. (23) that the dielectric tensor is not only a function of *ωpe* and *ω* but also a function of the plasma temperature and the *k*-vector components. So, the dispersion relation yields a new kind of solution that is called

Usually, the electric field of an electrostatic wave does not change with time. This fact is known from the derivation of the electrostatic field from a scalar

*th*. Here *kz* represents the *z*-component of the wave vector. The

*nZn*; *ϵxz* ¼ *ϵzx* ¼ �*n*

<sup>2</sup><sup>ѓ</sup> <sup>p</sup> *In*ð Þ <sup>1</sup> <sup>þ</sup> *<sup>ξ</sup>nZn* ; *<sup>ϵ</sup>zz* <sup>¼</sup> <sup>2</sup>*ξnIn*ð Þ <sup>1</sup> <sup>þ</sup> *<sup>ξ</sup>nZn :*

*ϵxx ϵxy ϵxz ϵyx ϵyy ϵyz ϵzx ϵzy ϵzz*

q

*<sup>ω</sup>*. It can be observed that there is collisional term present in the

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>y</sup>*<sup>4</sup> *sin*4*<sup>θ</sup>* � <sup>4</sup>*y*2ð Þ <sup>1</sup> <sup>þ</sup> *iZ* � *<sup>x</sup>*

> ⟂*v*2 *th*Þ*=ω*<sup>2</sup>

3 7

> ffiffi 2 ѓ q

� � <sup>=</sup> <sup>1</sup>ffiffi

*<sup>π</sup>* <sup>p</sup> <sup>Ð</sup> <sup>∞</sup> �∞ *e*�*<sup>s</sup>* 2 *s*�*ξ <sup>j</sup>* . 2 *cos* <sup>2</sup>*θ*

*<sup>c</sup>* where *k*<sup>⟂</sup>

<sup>5</sup> (23)

*In*ð Þ 1 þ *ξnZn* ; *ϵyy* ¼

(22)

*<sup>O</sup>*,*<sup>X</sup>* <sup>¼</sup> <sup>1</sup> � <sup>2</sup>*x*ð Þ <sup>1</sup> <sup>þ</sup> *iZ* � *<sup>x</sup>* 2 1ð Þ <sup>þ</sup> *iZ* ð Þ� <sup>1</sup> � *<sup>x</sup> <sup>y</sup>*<sup>2</sup> *sin* <sup>2</sup>

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

**2.3 Wave propagation in warm plasma condition**

considered. The larmor radius can be written as <sup>ѓ</sup> <sup>¼</sup> <sup>0</sup>*:*5ð*k*<sup>2</sup>

*ϵr* <sup>¼</sup> <sup>¼</sup>*<sup>I</sup>* ¼ þ *ω*2 *pe ω*2

<sup>ѓ</sup> *InZnϵxy* ¼ �*ϵyx* ¼ *inI*

*n*th order and Z is a function that is written as [15] *Z ξ <sup>j</sup>*

modified as:

*N*2

Here, *<sup>Z</sup>* <sup>¼</sup> *<sup>ω</sup><sup>c</sup>*

discussed before.

ð Þ *<sup>ω</sup>* <sup>þ</sup> *<sup>n</sup>ω<sup>c</sup> <sup>=</sup>*j j *kz <sup>v</sup>*<sup>2</sup>

where *ϵxx* = *<sup>n</sup>*<sup>2</sup>

electrostatic wave.

**2.4 Electrostatic wave**

0 *n*

� �*Zn*; *<sup>ϵ</sup>yz* <sup>¼</sup> *<sup>ϵ</sup>zy* ¼ �*<sup>i</sup>* ffiffiffiffi

*n*2 <sup>ѓ</sup> *In* � 2 ѓ*I*

**41**

dielectric tensor is expressed as:

#### **2.2 Electron cyclotron resonance (ECR)**

As discussed above, the ECR heating is an attractive tool in direct energy transferring to the plasma specifically in the ion source applications. The working principle of ECR mechanism is based on the frequency matching condition in which the microwave frequency with same polarization matches with electron cyclotron motion. It looks pretty much simpler in a qualitative approximation. But in case of quantification, it appears to be a non-deterministic method. It means along with the frequency, the phase difference between the microwave and electron motion also plays important role in energy transferring through ECR mechanism. Under frequency matching condition, if the phase difference between the microwave and the electron motion is in the same phase, then the electrons are accelerated by the microwave electric field. On the other hand, if it is 180° out of phase, the electrons are decelerated. Practically, the temporal phase difference between the microwave and the electron motion is a random phenomenon. So, it becomes necessary to take an average energy gain temporally of the electrons for several microwave periods. It is demonstrated by several groups worldwide that the average temporal energy gain of the electrons has positive value if the averaging calculation is performed irrespective of the phase difference between the microwave and the electron cyclotron motion. It has been proved that the net energy gain is related non-linearly to the microwave electric field (∝*E*<sup>2</sup> ). In ECR heating-based plasma, the collisions in plasma can play important role on the confinement of the plasma particles, thermalizing the plasma particles, and also on the ionization rate. The collision properties of plasma are determined by two physical parameters: mean free path (*λmean* <sup>¼</sup> <sup>1</sup> *<sup>n</sup><sup>σ</sup>*) and collision frequency (? = *nσv*). Here, *σv* is the product between cross-section and the velocity of plasma particles that is determined from the velocity distribution function (Maxwellian type). The particle confinement time is inversely proportional to the collision frequency. A contradiction arises from these two relations. The plasma can have a temperature even in collision-less case and follow the Maxwellian distribution. In collision-less case, the particle does not collide with each other and so there is no thermalization. But in ECR case, the kinetic energy of electrons increases, but there would be low temperature of the plasma. It suggests that there must be other mechanisms that are involved for the solution of this contradiction.

In most of the ECR ion source plasmas, the plasma is collision-less. But in other ion sources operating at microwave discharge, where the temperature of the plasma is comparatively lower and the pressure is also higher, a complete collision-less approximation is not valid. The other heating mechanisms that are involved in the collisional absorption condition must be taken into account. Considering the collisional term in the equation of motion,

$$m\_{\epsilon} \frac{\partial \tilde{\boldsymbol{v}}}{\partial t} = e \left( \tilde{\boldsymbol{E}} + \tilde{\boldsymbol{v}} \times \tilde{\boldsymbol{B}} \right) - m\_{\epsilon} \boldsymbol{o}\_{\epsilon} \tilde{\boldsymbol{v}} \tag{21}$$

*Evolution of Microwave Electric Field on Power Coupling to Plasma during Ignition Phase DOI: http://dx.doi.org/10.5772/intechopen.92011*

Here, *ω<sup>c</sup>* denotes the collision frequency. Then the dispersion relation is modified as:

$$N\_{O,X}^2 = 1 - \frac{2\mathbf{x}(1+i\mathbf{Z}-\mathbf{x})}{2(1+i\mathbf{Z})(1-\mathbf{x}) - y^2\sin^2\theta \pm \sqrt{y^4\sin^4\theta \pm 4y^2(1+i\mathbf{Z}-\mathbf{x})^2\cos^2\theta}}\tag{22}$$

Here, *<sup>Z</sup>* <sup>¼</sup> *<sup>ω</sup><sup>c</sup> <sup>ω</sup>*. It can be observed that there is collisional term present in the modified dispersion equation of the wave propagation. This collisional term limits the attainable energy during the ECR heating unlike the collision-less case, as discussed before.

#### **2.3 Wave propagation in warm plasma condition**

Although the cold plasma approximation is not valid for the wave dispersion in the region very close to the cut-off, the effect of warm plasma condition cannot be ignored. This is because the wavelength in the latter case is not negligible compared to the scale length of the plasma parameters [14]. At the resonance, where the refractive index is infinity, the wavelength becomes equivalent to the electron larmor radius. So, the finite larmor parameter effect is not negligible and has to be considered. The larmor radius can be written as <sup>ѓ</sup> <sup>¼</sup> <sup>0</sup>*:*5ð*k*<sup>2</sup> ⟂*v*2 *th*Þ*=ω*<sup>2</sup> *<sup>c</sup>* where *k*<sup>⟂</sup> denotes the perpendicular component of the wave vector with respect to the magnetic field (*Z*-direction). For the non-relativistic case assuming the Maxwellian distribution function, the distance of frequency from the *n*th cyclotron harmonic resonance in terms of the Doppler shift unit is written as *ξ<sup>n</sup>* ¼ ð Þ *<sup>ω</sup>* <sup>þ</sup> *<sup>n</sup>ω<sup>c</sup> <sup>=</sup>*j j *kz <sup>v</sup>*<sup>2</sup> *th*. Here *kz* represents the *z*-component of the wave vector. The dielectric tensor is expressed as:

$$\bar{\bar{c}}\_r \stackrel{=}{=} \bar{I} + \frac{\alpha\_{p\epsilon}^2}{\alpha^2} \sum\_{n=-\infty}^{n=\infty} \begin{bmatrix} \mathfrak{e}\_{xx} & \mathfrak{e}\_{xy} & \mathfrak{e}\_{xx} \\ \mathfrak{e}\_{yx} & \mathfrak{e}\_{yy} & \mathfrak{e}\_{yx} \\ \mathfrak{e}\_{xx} & \mathfrak{e}\_{xy} & \mathfrak{e}\_{xx} \end{bmatrix} \tag{23}$$

$$\begin{split} \text{where } \mathfrak{e}\_{\mathbf{x}\mathbf{x}} &= \underline{n}^{2} \overline{I\_{n}} Z\_{n} \mathfrak{e}\_{\mathbf{x}\mathbf{y}} = -\mathfrak{e}\_{\mathbf{y}\mathbf{x}} = i \boldsymbol{\Pi}\_{n}^{\overline{I}} Z\_{n} ; \mathfrak{e}\_{\mathbf{x}\mathbf{z}} = \mathfrak{e}\_{\mathbf{x}\mathbf{x}} = -n \sqrt{\frac{2}{l}} \overline{I\_{n}} (\mathfrak{1} + \mathfrak{f}\_{n} Z\_{n}); \mathfrak{e}\_{\mathbf{y}\mathbf{y}} = \mathfrak{e}\_{\mathbf{y}\mathbf{x}} = -n \overleftarrow{\mathcal{L}\_{n}} \overline{I\_{n}} (\mathfrak{1} + \mathfrak{f}\_{n} Z\_{n}); \mathfrak{e}\_{\mathbf{y}\mathbf{y}} = \mathfrak{e}\_{\mathbf{y}\mathbf{x}} = -n \overleftarrow{\mathcal{L}\_{n}} \overline{I\_{n}} (\mathfrak{1} + \mathfrak{f}\_{n} Z\_{n}); \mathfrak{e}\_{\mathbf{y}\mathbf{y}} = \mathfrak{e}\_{\mathbf{y}\mathbf{x}} = -n \overleftarrow{\mathcal{L}\_{n}} \overline{I\_{n}} (\mathfrak{1} + \mathfrak{f}\_{n} Z\_{n}). \end{split}$$

Here *Zn* <sup>¼</sup> *<sup>Z</sup> <sup>ξ</sup><sup>n</sup>* ð Þ and *In* <sup>¼</sup> *<sup>e</sup>*�*<sup>ѓ</sup>In*ð Þ*<sup>ѓ</sup>* where *In* is the modified Bessel's function of *n*th order and Z is a function that is written as [15] *Z ξ <sup>j</sup>* � � <sup>=</sup> <sup>1</sup>ffiffi *<sup>π</sup>* <sup>p</sup> <sup>Ð</sup> <sup>∞</sup> �∞ *e*�*<sup>s</sup>* 2 *s*�*ξ <sup>j</sup>* .

This dielectric tensor coming from warm plasma approximation has few new features that affect the wave propagation unlike the cold plasma approximation case. It can be seen from Eq. (23) that the dielectric tensor is not only a function of *ωpe* and *ω* but also a function of the plasma temperature and the *k*-vector components. So, the dispersion relation yields a new kind of solution that is called electrostatic wave.

#### **2.4 Electrostatic wave**

Usually, the electric field of an electrostatic wave does not change with time. This fact is known from the derivation of the electrostatic field from a scalar

cyclotron resonance heating (ECR). It is to be noted that only in the ECR heating mechanism the energy is transferred directly from the microwave to the gaseous electrons. As the electrons gain sufficient energy by resonance, they ionize the neutral gases and thus form plasma. There are also other mechanisms for energy transfer from the microwave to the plasma. But the direct energy transfer is not possible in these cases. In fact, the energy is shared among the microwave and the plasma oscillation modes by means of collisional absorption or non-linear phenomenon. The plasma oscillation modes are excited by the '*O*' and '*X*' modes of microwaves. A detailed description of the energy damping mechanisms is discussed below.

As discussed above, the ECR heating is an attractive tool in direct energy transferring to the plasma specifically in the ion source applications. The working principle of ECR mechanism is based on the frequency matching condition in which the microwave frequency with same polarization matches with electron cyclotron motion. It looks pretty much simpler in a qualitative approximation. But in case of quantification, it appears to be a non-deterministic method. It means along with the frequency, the phase difference between the microwave and electron motion also plays important role in energy transferring through ECR mechanism. Under frequency matching condition, if the phase difference between the microwave and the electron motion is in the same phase, then the electrons are accelerated by the microwave electric field. On the other hand, if it is 180° out of phase, the electrons are decelerated. Practically, the temporal phase difference between the microwave and the electron motion is a random phenomenon. So, it becomes necessary to take an average energy gain temporally of the electrons for several microwave periods. It is demonstrated by several groups worldwide that the average temporal energy gain of the electrons has positive value if the averaging calculation is performed irrespective of the phase difference between the microwave and the electron cyclotron motion. It has been proved that the net energy gain is related non-linearly to the microwave

). In ECR heating-based plasma, the collisions in plasma can play

*<sup>n</sup><sup>σ</sup>*) and collision

important role on the confinement of the plasma particles, thermalizing the plasma particles, and also on the ionization rate. The collision properties of plasma are

frequency (? = *nσv*). Here, *σv* is the product between cross-section and the velocity of plasma particles that is determined from the velocity distribution function (Maxwellian type). The particle confinement time is inversely proportional to the collision frequency. A contradiction arises from these two relations. The plasma can have a temperature even in collision-less case and follow the Maxwellian distribution. In collision-less case, the particle does not collide with each other and so there is no thermalization. But in ECR case, the kinetic energy of electrons increases, but there would be low temperature of the plasma. It suggests that there must be other mech-

In most of the ECR ion source plasmas, the plasma is collision-less. But in other ion sources operating at microwave discharge, where the temperature of the plasma is comparatively lower and the pressure is also higher, a complete collision-less approximation is not valid. The other heating mechanisms that are involved in the collisional absorption condition must be taken into account. Considering the colli-

<sup>¼</sup> *<sup>e</sup> <sup>E</sup>*<sup>~</sup> <sup>þ</sup> <sup>~</sup>*<sup>v</sup>* � *<sup>B</sup>*<sup>~</sup> � *meωc*~*<sup>v</sup>* (21)

determined by two physical parameters: mean free path (*λmean* <sup>¼</sup> <sup>1</sup>

anisms that are involved for the solution of this contradiction.

*me ∂*~*v ∂t*

sional term in the equation of motion,

**40**

**2.2 Electron cyclotron resonance (ECR)**

*Selected Topics in Plasma Physics*

electric field (∝*E*<sup>2</sup>

potential (*E*<sup>~</sup> ¼ �∇~*V*), which makes <sup>∇</sup><sup>~</sup> � *<sup>E</sup>*<sup>~</sup> to be zero. However, the time-dependent electric field is sometimes denoted as electrostatic wave as it obeys *<sup>E</sup>*<sup>~</sup> ¼ �∇~*<sup>V</sup>* relation. So a concrete example for calling an electric field to be electrostatic is the k-vector direction of electrostatic wave with respect to its electric field. If the kvector is parallel to the electric field, it is called as an electrostatic wave. It means <sup>∇</sup><sup>~</sup> � *<sup>E</sup>*<sup>~</sup> <sup>¼</sup> <sup>~</sup> *<sup>k</sup>* � *<sup>E</sup>*<sup>~</sup> that makes the *<sup>∂</sup>B*<sup>~</sup> *<sup>∂</sup><sup>t</sup>* component to be zero. Therefore, the electrostatic wave does not have any magnetic field component.

A commonly occurred electrostatic wave in a warm plasma condition is named as the Langmuir wave [16]. The Langmuir wave is the main constituent of unmagnetized plasma that appears together with the ion-acoustic wave (IAW). In case of magnetized plasma, electrostatic waves are also present. In this case, if the electrons are displaced by some force, an electric field builds up to restore the electrons back to their initial position to maintain the plasma quasi-neutrality condition. Due to the very low inertia, the electrons will show an overshoot and oscillate around an equilibrium position. The frequency of oscillations is equivalent to the electron plasma frequency of the plasma. The dispersion relation of the Langmuir wave is written as [16]:

$$
\alpha\_L^2 = \alpha\_{p\epsilon}^2 + \frac{3}{2}k^2 v\_{th}^2 \tag{24}
$$

Here *k*⟂, *vth*, and *In* represent *k*-vector perpendicular component, electron thermal velocity and the Bessel's function, respectively. From Eq. (26), it can be seen

Therefore, the EBW waves can have resonance at the harmonics of the cyclotron resonance frequencies. In microwave ion source, the resonance absorption of the EBW is possible at the harmonics of the cyclotron frequencies of the ECR magnetic

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

In microwave-generated magnetized plasma, the presence of plasma density gradient and the variation of the magnetic field make the wave propagation and its energy absorption unpredictable. It is difficult to estimate the wave trajectory from the simple linear uniform plasma theory [21]. It is natural that the wave would cross the boundaries shown in the CMA diagram by travelling up or down depending on the magnetic field variation and plasma density distribution. Inhomogeneous and anisotropic plasma can exhibit a wide variety of possibilities for the cut-off, resonance, cut-off-resonance and/or the back-to-back cut-off pairs. In inhomogeneous plasma, two or more waves can coexist that propagates in the plasma having density gradient. Although their polarization and propagation vector are different from one another, they can exhibit identical characteristics at some particular plasma regions having particular plasma loading conditions. At those particular scenarios, the waves can remain no longer distinguishable and therefore can convert into another. The mode conversion theory deals with establishing resonance characteristic in inhomogeneous plasma considering two different waves present in the plasma by taking into account the wave reflection, cut-off, resonance and absorption conditions. As the microwaves that are present in the microwave ion source plasma is dominated by the ordinary- and extraordinary-type microwave, the mode conversion theory is mainly focused upon considering the cut-off-resonance pair condition in plasma. The *X* mode microwave is unable to propagate the dense plasma because it is reflected at R cut-off. On the other hand, O-mode microwave is able to propagate in the dense plasma, where it converts into the X mode microwave under certain condition that is obtained from Eq. (18). As per CMA diagram, the *X* mode can have resonance at the UHR region. Therefore, before entering the UHR region, the launched microwave can convert into the *X* mode based on the following

• If the *X* mode microwave is launched from the high magnetic field side, the *X* mode will not see the R cut-off (see **Figure 1**). It will propagate toward the UHR zone where it can convert into an EBW and an ion wave as per the

• Another method in generating the EBW and ion wave is the *O-X-B* mode conversion process. In this process, the *O* mode microwave launched from the vacuum side crosses the R cut-off and converts into a slow *X* mode after tunneling through the evanescent layer. The generated slow *X* mode is then

In microwave ion source plasma under mirror magnetic field configuration, there can coexist two types of components (*O* and *X* modes) of the launched microwave. The ion source cavity acts as a resonator having comparable cavity dimension with respect to the launch microwave wavelength. Therefore, different

literature. This method is known as *X-B* conversion process.

converted into an EBW and an ion wave near the UHR layer.

*MW* <sup>¼</sup> *<sup>n</sup>*2*ω*<sup>2</sup>

*c*.

that the *k*-vector becomes infinite at the cyclotron harmonic, i.e., *ω*<sup>2</sup>

field (*BECR*Þ value.

mechanisms:

**43**

**2.5 Mode conversion theory**

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

As the electron plasma oscillates very fast compared to the massive ions present in plasma, the massive ion motion is considered to be fixed in the GHz frequency scale (Langmuir frequency range). Although the frequency of the massive ion motion is very low compared to the Langmuir wave, the massive ions part will take part in the oscillations due to the electric field build-up. This low-frequency oscillations fall usually in the range of ion-acoustic wave frequency. The ion wave dispersion is obtained from the fluid equation as,

$$\frac{\alpha\_{\rm IAW}}{k\_{\rm ion}} = \left\{ k\_B (T\_\epsilon + \gamma\_i T\_i) / \mathcal{M}\_i \right\}^{1/2} \tag{25}$$

Usually, the plasma oscillations in the ion-acoustic frequency range lie in between few kHz to tens of MHz.

There exists another kind of electrostatic wave in magnetized plasma, which is known as electron Bernstein waves (EBW). EBW exist in warm plasma conditions when the electron temperature has finite value. It is known that the superposition of the static magnetic field with the oscillating electric field of the plasma waves can make the electron orbit to be elliptical [17, 18]. Now, if the magnetic field is increased further, the electron orbit will become a circular one as the Lorentz force dominates the electrostatic component [17, 18]. The presence of EBW makes the electron gyrophase to organize in such a manner that the space charge distribution in plasma obtains a minima and maxima in the direction perpendicular to the externally applied magnetic field. It was shown [19] that the space charge accumulation is periodic. The charge accumulation propagates with a wavelength that is four times the electron larmor radius [19]. As the wavelength of the EBW is much lower than the length of a typical Langmuir probe tip, used for the plasma characterization, the Langmuir probe is unable to detect the EBW wave directly [20]. The dispersion of electron Bernstein wave (EBW) can be written as:

$$\frac{k\_{\perp}^{2}}{2a\_{\rm pe}^{2}/v\_{\rm th}^{2}} = 2a\_{\rm c}^{2} \sum\_{n=1}^{\infty} n^{2} \frac{e^{-b}I\_{n}(b)}{a\_{\rm MW}^{2} - n^{2}a\_{\rm c}^{2}} \tag{26}$$

*Evolution of Microwave Electric Field on Power Coupling to Plasma during Ignition Phase DOI: http://dx.doi.org/10.5772/intechopen.92011*

Here *k*⟂, *vth*, and *In* represent *k*-vector perpendicular component, electron thermal velocity and the Bessel's function, respectively. From Eq. (26), it can be seen that the *k*-vector becomes infinite at the cyclotron harmonic, i.e., *ω*<sup>2</sup> *MW* <sup>¼</sup> *<sup>n</sup>*2*ω*<sup>2</sup> *c*. Therefore, the EBW waves can have resonance at the harmonics of the cyclotron resonance frequencies. In microwave ion source, the resonance absorption of the EBW is possible at the harmonics of the cyclotron frequencies of the ECR magnetic field (*BECR*Þ value.

### **2.5 Mode conversion theory**

potential (*E*<sup>~</sup> ¼ �∇~*V*), which makes <sup>∇</sup><sup>~</sup> � *<sup>E</sup>*<sup>~</sup> to be zero. However, the time-dependent electric field is sometimes denoted as electrostatic wave as it obeys *<sup>E</sup>*<sup>~</sup> ¼ �∇~*<sup>V</sup>* relation. So a concrete example for calling an electric field to be electrostatic is the k-vector direction of electrostatic wave with respect to its electric field. If the kvector is parallel to the electric field, it is called as an electrostatic wave. It means

A commonly occurred electrostatic wave in a warm plasma condition is named as the Langmuir wave [16]. The Langmuir wave is the main constituent of unmagnetized plasma that appears together with the ion-acoustic wave (IAW). In case of magnetized plasma, electrostatic waves are also present. In this case, if the electrons are displaced by some force, an electric field builds up to restore the electrons back to their initial position to maintain the plasma quasi-neutrality condition. Due to the very low inertia, the electrons will show an overshoot and

oscillate around an equilibrium position. The frequency of oscillations is

*ω*2 *<sup>L</sup>* <sup>¼</sup> *<sup>ω</sup>*<sup>2</sup> *pe* þ 3 2 *k*2 *v*2

equivalent to the electron plasma frequency of the plasma. The dispersion relation

As the electron plasma oscillates very fast compared to the massive ions present in plasma, the massive ion motion is considered to be fixed in the GHz frequency scale (Langmuir frequency range). Although the frequency of the massive ion motion is very low compared to the Langmuir wave, the massive ions part will take

¼ *kB Te* þ *γ* f g ð Þ *iTi =Mi*

There exists another kind of electrostatic wave in magnetized plasma, which is known as electron Bernstein waves (EBW). EBW exist in warm plasma conditions when the electron temperature has finite value. It is known that the superposition of the static magnetic field with the oscillating electric field of the plasma waves can make the electron orbit to be elliptical [17, 18]. Now, if the magnetic field is increased further, the electron orbit will become a circular one as the Lorentz force dominates the electrostatic component [17, 18]. The presence of EBW makes the electron gyrophase to organize in such a manner that the space charge distribution in plasma obtains a minima and maxima in the direction perpendicular to the externally applied magnetic field. It was shown [19] that the space charge accumulation is periodic. The charge accumulation propagates with a wavelength that is four times the electron larmor radius [19]. As the wavelength of the EBW is much lower than the length of a typical Langmuir probe tip, used for the plasma characterization, the Langmuir probe is unable to detect the EBW wave directly [20]. The

Usually, the plasma oscillations in the ion-acoustic frequency range lie in

part in the oscillations due to the electric field build-up. This low-frequency oscillations fall usually in the range of ion-acoustic wave frequency. The ion wave

*<sup>∂</sup><sup>t</sup>* component to be zero. Therefore, the electrostatic

*th* (24)

<sup>1</sup>*=*<sup>2</sup> (25)

<sup>∇</sup><sup>~</sup> � *<sup>E</sup>*<sup>~</sup> <sup>¼</sup> <sup>~</sup>

*<sup>k</sup>* � *<sup>E</sup>*<sup>~</sup> that makes the *<sup>∂</sup>B*<sup>~</sup>

*Selected Topics in Plasma Physics*

of the Langmuir wave is written as [16]:

dispersion is obtained from the fluid equation as,

between few kHz to tens of MHz.

**42**

*ω*IAW *k*ion

dispersion of electron Bernstein wave (EBW) can be written as:

<sup>¼</sup> <sup>2</sup>*ω*<sup>2</sup> *c* X∞ *n*¼1

*<sup>n</sup>*<sup>2</sup> *<sup>e</sup>*�*bIn*ð Þ *<sup>b</sup> ω*2

*MW* � *n*<sup>2</sup>*ω*<sup>2</sup>

*c*

(26)

*k*2 ⟂ 2*ω*<sup>2</sup> *pe=v*<sup>2</sup> *th*

wave does not have any magnetic field component.

In microwave-generated magnetized plasma, the presence of plasma density gradient and the variation of the magnetic field make the wave propagation and its energy absorption unpredictable. It is difficult to estimate the wave trajectory from the simple linear uniform plasma theory [21]. It is natural that the wave would cross the boundaries shown in the CMA diagram by travelling up or down depending on the magnetic field variation and plasma density distribution. Inhomogeneous and anisotropic plasma can exhibit a wide variety of possibilities for the cut-off, resonance, cut-off-resonance and/or the back-to-back cut-off pairs. In inhomogeneous plasma, two or more waves can coexist that propagates in the plasma having density gradient. Although their polarization and propagation vector are different from one another, they can exhibit identical characteristics at some particular plasma regions having particular plasma loading conditions. At those particular scenarios, the waves can remain no longer distinguishable and therefore can convert into another. The mode conversion theory deals with establishing resonance characteristic in inhomogeneous plasma considering two different waves present in the plasma by taking into account the wave reflection, cut-off, resonance and absorption conditions. As the microwaves that are present in the microwave ion source plasma is dominated by the ordinary- and extraordinary-type microwave, the mode conversion theory is mainly focused upon considering the cut-off-resonance pair condition in plasma. The *X* mode microwave is unable to propagate the dense plasma because it is reflected at R cut-off. On the other hand, O-mode microwave is able to propagate in the dense plasma, where it converts into the X mode microwave under certain condition that is obtained from Eq. (18). As per CMA diagram, the *X* mode can have resonance at the UHR region. Therefore, before entering the UHR region, the launched microwave can convert into the *X* mode based on the following mechanisms:


In microwave ion source plasma under mirror magnetic field configuration, there can coexist two types of components (*O* and *X* modes) of the launched microwave. The ion source cavity acts as a resonator having comparable cavity dimension with respect to the launch microwave wavelength. Therefore, different types of propagating modes can coexist in the presence of plasma. This makes it difficult to identify which mechanism is actually allowing the *X* mode to reach the UHR region. However, it is possible to estimate this mode conversion efficiency. For the case of *X* mode microwave, the refractive index is written as:

$$\frac{\sigma^2 k^2}{\alpha^2} = 1 - \frac{\alpha\_{pe}^2}{\alpha^2} \frac{\alpha^2 - \alpha\_{pe}^2}{\alpha^2 - \alpha\_{UHR}^2} \tag{27}$$

UHR region with time due to its propagation from the cut-off region to the UHR region. The accumulated electric field will be able to share some part of its energy through non-linear interactions to the oscillating modes present in the plasma. This non-linear energy coupling between the *X* mode microwave and the oscillation modes is known as parametric instability. It is demonstrated that the *X* mode through this type of instability couples energy to the Bernstein wave and ion wave near the UHR region. The *X* mode normally contains the longitudinal component of the electric field (TM type). Considering the *x*-component of the *X* mode propagation from Eq. (17), the dispersion of *X* mode wave can be written as follows:

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

*ω*2 *peω<sup>c</sup>*

*<sup>ω</sup> Ey* <sup>¼</sup> <sup>0</sup> (32)

<sup>1</sup>*x*<sup>1</sup> ¼ *C*1*x*2*E*<sup>0</sup> cos*ω*0*t* (33)

<sup>2</sup>*x*<sup>2</sup> ¼ *C*2*x*1*E*<sup>0</sup> cos*ω*0*t* (34)

*t* and *X*<sup>2</sup> ¼ *x*<sup>2</sup> cos*ω*00*t*. In the absence of any non-

*ω*<sup>00</sup> ¼ *ω*<sup>0</sup> � *ω*<sup>0</sup> (35)

*ω*<sup>0</sup> ffi *ω*<sup>2</sup> � *ω*<sup>1</sup> (36)

*<sup>k</sup>*<sup>2</sup> � <sup>~</sup> *k*1.

*<sup>k</sup>*<sup>0</sup> <sup>¼</sup> <sup>~</sup>

*k:*~*r*). Following this,

*UHR* ! 0 and

*<sup>ω</sup>*<sup>2</sup> � *<sup>ω</sup>*<sup>2</sup> *UHR Ex* <sup>þ</sup> *<sup>i</sup>*

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

*d*2 *x*1 *dt*<sup>2</sup> <sup>þ</sup> *<sup>ω</sup>*<sup>2</sup>

*d*2 *x*2 *dt*<sup>2</sup> <sup>þ</sup> *<sup>ω</sup>*<sup>2</sup>

simple harmonic oscillator.

Let us assume, *X*<sup>1</sup> ¼ *x*<sup>1</sup> cos*ω*<sup>0</sup>

matching condition is arrived:

**45**

From Eq. (11), as the *<sup>X</sup>* mode approaches the UHR, the term *<sup>ω</sup>*<sup>2</sup> � *<sup>ω</sup>*<sup>2</sup>

so *Ey* would be zero. Therefore, the component *Ex* will remain non-zero at the UHR, which is directed in the direction of propagation of *X* mode. This means the *X* mode at the UHR becomes electrostatic. The *X* mode microwave that becomes an electrostatic wave at the UHR matches with the Bernstein wave and the ion wave. With the two oscillation modes that are coupled with the *X* mode electric field (*E*0), the motion of one of the modes (Bernstein wave) can be expressed in terms of the

Here the motion of the first oscillator (amplitude *x*<sup>1</sup> and the resonant frequency *ω*1) is driven by the time-dependent electric field of the *X* mode and the amplitude (*x*2Þ of the second oscillator. The equation of the motion for the second oscillator *x*<sup>2</sup> is:

linear interaction, it can be expressed as *ω*<sup>0</sup> ¼ *ω*1, *ω*<sup>00</sup> ¼ *ω*<sup>2</sup> and *ω*<sup>0</sup> ¼ *ω*1. But in the presence of non-linear interaction, this is incorrect. For the case of non-linear interaction, as the driving terms cause frequency shifting, the frequency *ω*<sup>2</sup> is approximately equal to *ω*00. However, in the absence of non-linear interaction, the

Now if *ω*<sup>0</sup> is small such that both values of *ω*<sup>00</sup> lie within the bandwidth of oscillator, *x*2, there exist two oscillators: *x*<sup>2</sup> *ω*<sup>0</sup> þ *ω*<sup>0</sup> ð Þ and *x*<sup>2</sup> *ω*<sup>0</sup> � *ω*<sup>0</sup> ð Þ. Under this assumption, solving the coupled Eqs. (33) and (34), the following frequency

The frequency and k-vector matching conditions correspond to the energy conservation and momentum conservation following the quantum mechanics theory. It is proved that [15] the *X* mode generates two waves through parametric decay: a

solution from the system of coupled Eqs. (33) and (34) is obtained as:

As the oscillators are waves, the '?t' term is replaced by '(?t-~

a new matching condition of k-vector is reached, ~

Let us suppose the *X* mode wave sees cut-off at the electron plasma frequency, *ωpe*1. The wave cut-off condition yields

$$\frac{\alpha\_{p\epsilon1}^2}{\alpha^2} = \mathbf{1} - \frac{\alpha\_c}{\alpha} \tag{28}$$

After reflection at the cut-off, the wave propagates in the inhomogeneous and anisotropic plasma in the location where it will find a resonance (refractive index = ∞). The resonance is called upper hybrid resonance where electron plasma frequency is *ωpe*2. The corresponding Eq. (27) becomes then

$$\frac{\alpha\_{pe1}^2}{\alpha^2} = \mathbf{1} - \frac{\alpha\_c^2}{\alpha^2} \tag{29}$$

By dividing Eq. (28) by Eq. (29), one gets

$$\frac{\alpha\_{p\epsilon2}^2}{\alpha\_{p\epsilon1}^2} = 1 + \frac{\alpha\_\epsilon}{\alpha} = \frac{n\_{\epsilon2}}{n\_{\epsilon1}} \approx 1 + \frac{\Delta x}{L} \tag{30}$$

where *Δx* and *L* denote the distance between the cut-off and the upper hybrid resonance layer and the length parameter, respectively. From Eq. (30), one obtains

$$
\Delta \mathbf{x} = \frac{\alpha\_{\mathbf{c}}}{\alpha} L \tag{31}
$$

Now, for the wavelength *λ* ≪ *L*, the *Δx* will be many times the wavelength. So the wave will not be able to reach the upper hybrid resonance point. The mode conversion efficiency is determined by the penetration depth of the wave into the plasma. The effective mode conversion can be achieved when the parameter *L* is comparable to the wavelength, *λ*. In another way, from Eq. (31), it is also seen that mode conversion can be improved if the magnetic field is reduced.

If a strong electromagnetic field is present in the ion source cavity, the plasma particle follows the relation *vE* ≤ *vthe*; *vE* ¼ *eE=mω*, where *vE*, *E*,*e*, *m* and *vthe* are wave phase velocity, electric field intensity, electron charge, electron mass and electron thermal velocity, respectively. This condition leads to plasma parameters to vary with time. In effect, the non-linear effect such as parametric instability comes into play. Off-course, the parametric decay happens above certain threshold value of the electric field and the microwave energy is shared among the plasma waves and the microwave through the non-linear interaction phenomenon.

#### **2.6 Parametric decay**

As the plasma parameters vary with time under the conditions of intense electric field of the microwave, the corresponding velocity becomes close to the electron thermal velocity. It is known that the *X* mode electric field accumulates near the

*Evolution of Microwave Electric Field on Power Coupling to Plasma during Ignition Phase DOI: http://dx.doi.org/10.5772/intechopen.92011*

UHR region with time due to its propagation from the cut-off region to the UHR region. The accumulated electric field will be able to share some part of its energy through non-linear interactions to the oscillating modes present in the plasma. This non-linear energy coupling between the *X* mode microwave and the oscillation modes is known as parametric instability. It is demonstrated that the *X* mode through this type of instability couples energy to the Bernstein wave and ion wave near the UHR region. The *X* mode normally contains the longitudinal component of the electric field (TM type). Considering the *x*-component of the *X* mode propagation from Eq. (17), the dispersion of *X* mode wave can be written as follows:

$$(\alpha^2 - \alpha\_{\text{UHR}}^2)E\_\text{x} + i \frac{\alpha\_{p\epsilon}^2 \alpha\_\ell}{\alpha} E\_\text{y} = \mathbf{0} \tag{32}$$

From Eq. (11), as the *<sup>X</sup>* mode approaches the UHR, the term *<sup>ω</sup>*<sup>2</sup> � *<sup>ω</sup>*<sup>2</sup> *UHR* ! 0 and so *Ey* would be zero. Therefore, the component *Ex* will remain non-zero at the UHR, which is directed in the direction of propagation of *X* mode. This means the *X* mode at the UHR becomes electrostatic. The *X* mode microwave that becomes an electrostatic wave at the UHR matches with the Bernstein wave and the ion wave. With the two oscillation modes that are coupled with the *X* mode electric field (*E*0), the motion of one of the modes (Bernstein wave) can be expressed in terms of the simple harmonic oscillator.

$$\frac{d^2\mathbf{x}\_1}{dt^2} + \alpha\_1^2 \mathbf{x}\_1 = C\_1 \mathbf{x}\_2 E\_0 \cos \alpha\_0 t \tag{33}$$

Here the motion of the first oscillator (amplitude *x*<sup>1</sup> and the resonant frequency *ω*1) is driven by the time-dependent electric field of the *X* mode and the amplitude (*x*2Þ of the second oscillator. The equation of the motion for the second oscillator *x*<sup>2</sup> is:

$$\frac{d^2\mathbf{x}\_2}{dt^2} + \alpha\_2^2 \mathbf{x}\_2 = \mathbf{C}\_2 \mathbf{x}\_1 E\_0 \cos \alpha\_0 t \tag{34}$$

Let us assume, *X*<sup>1</sup> ¼ *x*<sup>1</sup> cos*ω*<sup>0</sup> *t* and *X*<sup>2</sup> ¼ *x*<sup>2</sup> cos*ω*00*t*. In the absence of any nonlinear interaction, it can be expressed as *ω*<sup>0</sup> ¼ *ω*1, *ω*<sup>00</sup> ¼ *ω*<sup>2</sup> and *ω*<sup>0</sup> ¼ *ω*1. But in the presence of non-linear interaction, this is incorrect. For the case of non-linear interaction, as the driving terms cause frequency shifting, the frequency *ω*<sup>2</sup> is approximately equal to *ω*00. However, in the absence of non-linear interaction, the solution from the system of coupled Eqs. (33) and (34) is obtained as:

$$
\alpha'' = \alpha\_0 \pm \alpha'\tag{35}
$$

Now if *ω*<sup>0</sup> is small such that both values of *ω*<sup>00</sup> lie within the bandwidth of oscillator, *x*2, there exist two oscillators: *x*<sup>2</sup> *ω*<sup>0</sup> þ *ω*<sup>0</sup> ð Þ and *x*<sup>2</sup> *ω*<sup>0</sup> � *ω*<sup>0</sup> ð Þ. Under this assumption, solving the coupled Eqs. (33) and (34), the following frequency matching condition is arrived:

$$a o\_0 \cong a o\_2 \pm a o\_1 \tag{36}$$

As the oscillators are waves, the '?t' term is replaced by '(?t-~ *k:*~*r*). Following this, a new matching condition of k-vector is reached, ~ *<sup>k</sup>*<sup>0</sup> <sup>¼</sup> <sup>~</sup> *<sup>k</sup>*<sup>2</sup> � <sup>~</sup> *k*1.

The frequency and k-vector matching conditions correspond to the energy conservation and momentum conservation following the quantum mechanics theory. It is proved that [15] the *X* mode generates two waves through parametric decay: a

types of propagating modes can coexist in the presence of plasma. This makes it difficult to identify which mechanism is actually allowing the *X* mode to reach the UHR region. However, it is possible to estimate this mode conversion efficiency.

> *pe ω*2

Let us suppose the *X* mode wave sees cut-off at the electron plasma frequency,

After reflection at the cut-off, the wave propagates in the inhomogeneous and anisotropic plasma in the location where it will find a resonance (refractive index = ∞). The resonance is called upper hybrid resonance where electron plasma fre-

> *<sup>ω</sup>* <sup>¼</sup> *ne*<sup>2</sup> *ne*<sup>1</sup>

where *Δx* and *L* denote the distance between the cut-off and the upper hybrid resonance layer and the length parameter, respectively. From Eq. (30), one obtains

*<sup>Δ</sup><sup>x</sup>* <sup>¼</sup> *<sup>ω</sup><sup>c</sup>*

Now, for the wavelength *λ* ≪ *L*, the *Δx* will be many times the wavelength. So the wave will not be able to reach the upper hybrid resonance point. The mode conversion efficiency is determined by the penetration depth of the wave into the plasma. The effective mode conversion can be achieved when the parameter *L* is comparable to the wavelength, *λ*. In another way, from Eq. (31), it is also seen that

If a strong electromagnetic field is present in the ion source cavity, the plasma particle follows the relation *vE* ≤ *vthe*; *vE* ¼ *eE=mω*, where *vE*, *E*,*e*, *m* and *vthe* are wave phase velocity, electric field intensity, electron charge, electron mass and electron thermal velocity, respectively. This condition leads to plasma parameters to vary with time. In effect, the non-linear effect such as parametric instability comes into play. Off-course, the parametric decay happens above certain threshold value of the electric field and the microwave energy is shared among the plasma waves

As the plasma parameters vary with time under the conditions of intense electric field of the microwave, the corresponding velocity becomes close to the electron thermal velocity. It is known that the *X* mode electric field accumulates near the

*<sup>ω</sup>*<sup>2</sup> � *<sup>ω</sup>*<sup>2</sup> *pe*

*UHR*

*<sup>ω</sup>* (28)

*<sup>ω</sup>*<sup>2</sup> (29)

*<sup>ω</sup> <sup>L</sup>* (31)

*<sup>L</sup>* (30)

(27)

*<sup>ω</sup>*<sup>2</sup> � *<sup>ω</sup>*<sup>2</sup>

*c*

≈ 1 þ

*Δx*

For the case of *X* mode microwave, the refractive index is written as:

*<sup>ω</sup>*<sup>2</sup> <sup>¼</sup> <sup>1</sup> � *<sup>ω</sup>*<sup>2</sup>

*ω*2 *pe*1 *<sup>ω</sup>*<sup>2</sup> <sup>¼</sup> <sup>1</sup> � *<sup>ω</sup><sup>c</sup>*

*ω*2 *pe*1 *<sup>ω</sup>*<sup>2</sup> <sup>¼</sup> <sup>1</sup> � *<sup>ω</sup>*<sup>2</sup>

<sup>¼</sup> <sup>1</sup> <sup>þ</sup> *<sup>ω</sup><sup>c</sup>*

mode conversion can be improved if the magnetic field is reduced.

and the microwave through the non-linear interaction phenomenon.

**2.6 Parametric decay**

**44**

*c*2*k*<sup>2</sup>

quency is *ωpe*2. The corresponding Eq. (27) becomes then

*ω*2 *pe*2 *ω*2 *pe*1

By dividing Eq. (28) by Eq. (29), one gets

*ωpe*1. The wave cut-off condition yields

*Selected Topics in Plasma Physics*

high-frequency Bernstein wave and a low-frequency ion wave following the frequency matching condition. This leads to the generation of secondary peaks around the launched microwave frequency, which is called as sideband. Therefore, the simultaneous presence of the sideband peaks around the launched frequency signifies the parametric decay near the mode conversion region.

The parametric decay occurs above a certain threshold value, which actually depends on the damping rate of the oscillator. If the damping rates *Γ*<sup>1</sup> and *Γ*<sup>2</sup> for two oscillators *x*<sup>1</sup> and *x*2, respectively, are introduced in Eqs. (33) and (34), one can conclude that the lowest threshold occurs at the exact frequency matching condition. The corresponding threshold electric field is obtained as [19]:

$$\mathbf{C}\_{1}\mathbf{C}\_{2}\mathbf{E}\_{0\\\\thresh}^{2} = 4a\_{1}a\_{2}\boldsymbol{\Gamma}\_{1}\boldsymbol{\Gamma}\_{2} \tag{37}$$

#### **2.7 Damping of electrostatic and electromagnetic energy**

Electrostatic waves generated in the plasma through the parametric decay instability can damp their energy to the plasma particles and thus increase the plasma density. When the phase velocity of the electrostatic wave becomes comparable to the thermal velocity of the plasma particles, the energy is transferred from the wave to the plasma particles and is known as Landau damping mechanism. In microwave ion source plasma, density can be increased 2–3 times more than the ECR heating mechanisms through the off-resonance heating mechanism. For this reason, the offresonance condition is used to create favorable conditions of the upper hybrid resonance heating. Under certain plasma temperature, the electrostatic wave can transfer energy resonantly to the plasma particles if the wave phase velocity matches the plasma particle velocity. In some cases, the plasma particle velocity can be higher than the wave phase velocity. Under this condition, the plasma particle can transfer energy to the wave. The Landau damping mechanism follows the equation written below:

$$I\_m \left(\frac{\alpha}{\alpha\_{p\epsilon}}\right) = -0.2\sqrt{\pi} \left(\frac{\alpha\_{p\epsilon}}{k v\_{th}}\right)^3 e^{-12k^2 \lambda\_D^2} \tag{38}$$

of a microwave coupled reactor, which is a cylindrical plasma cavity. A MW of frequency 2.45 GHz is injected into the microwave coupled reactor through a ridge waveguide port (on the left side of **Figure 2**) to ionize the gaseous particles, thus forming the plasma that is confined under the mirror magnetic field configuration. The reactor has dimension of 107-mm and 88-mm diameter. The microwave power is fed into a cavity resonator through a tapered waveguide. The waveguide is tapered by embedding four ridge sections having different ridge length, ridge gap and ridge width on the inner sidewalls of the waveguide. The ridge dimensions are optimized from the analytical calculation as well as from the electromagnetic simulation. The mirror magnetic field is created by using two pairs of ring magnets that surround the microwave coupled reactor [22]. On the right side of the microwave coupled reactor (**Figure 2**), the ion beam extraction system is attached through a 5-mm hole on the wall of the reactor. The similar computational domain is used in the experimental set-up (Section 5) to validate the simulated data. Here, the finite

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

The MW propagation and the plasma evolution are assumed to be decoupled to each other during the simulation modeling in the temporal scale [10]. The MW electric fields (̃EÞ are averaged for some MW periods before putting their value in the plasma model and the resultant field is given as input to the plasma model. The electron's momentum equation is time integrated along with Maxwell's equations for some MW periods until the MW model of the FEM gets a periodic solution of the equation to transfer an average power to the particles over such a period. The FEM model continues this process until it gets a steady state solution for at least

) MW periods. Here, ω is MW frequency and ν<sup>m</sup> is electron's

momentum transfer frequency. Since the electrons stay in the ECR zone for a very short time (transit time) duration, it causes non-local kinetic effects. This results in the de-phasing between the velocity and field oscillations that becomes very difficult to describe using the fluid model. This problem is resolved by introducing the effective collision frequency ð Þ *ν*eff in the simulation to converge the solution. In the low pressure condition, for the collision-less heating, νeff has to be in the order of

element method (FEM)-based COMSOL model is used [22].

inverse transit time for the electrons [10].

�10<sup>3</sup> (<sup>ω</sup>*=*<sup>ν</sup><sup>m</sup> �10<sup>3</sup>

**47**

**Figure 2.**

*Simulation domain of the MW ion source.*

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

The exponential term on the right hand side represents that the Landau damping will be small for small value of *kλD*. The Landau damping phenomenon is applicable for both case of electrons and ions. The electrostatic wave whose frequency falls in the range of electrons can heat the electrons only in the parallel direction with respect to the magnetic field. In case of electrostatic ion wave, such restriction does not apply. This is because the electrons are strongly magnetized and so will not be able to move across the magnetic field.

In case of a compact microwave plasma device where the microwave wavelength becomes comparable to the device dimension, the cavity resonant mode can also play crucial role in damping the electromagnetic energy to the plasma particles. The presence of multiple cavity modes in the plasma can produce modulated wave due to the interaction between each pairs of the cavity resonant modes. The generated modulated wave propagates in the plasma and damps its energy to the plasma particles where the frequency of the modulated wave matches the local plasma frequency of the plasma particles.

### **3. Simulation modeling of MW interaction in plasma**

For the MW interaction modeling during the plasma evolution, a schematic of the computational domain is shown in **Figure 2**. The computational domain consists *Evolution of Microwave Electric Field on Power Coupling to Plasma during Ignition Phase DOI: http://dx.doi.org/10.5772/intechopen.92011*

**Figure 2.** *Simulation domain of the MW ion source.*

high-frequency Bernstein wave and a low-frequency ion wave following the frequency matching condition. This leads to the generation of secondary peaks around the launched microwave frequency, which is called as sideband. Therefore, the simultaneous presence of the sideband peaks around the launched frequency sig-

The parametric decay occurs above a certain threshold value, which actually depends on the damping rate of the oscillator. If the damping rates *Γ*<sup>1</sup> and *Γ*<sup>2</sup> for two oscillators *x*<sup>1</sup> and *x*2, respectively, are introduced in Eqs. (33) and (34), one can conclude that the lowest threshold occurs at the exact frequency matching condi-

Electrostatic waves generated in the plasma through the parametric decay instability can damp their energy to the plasma particles and thus increase the plasma density. When the phase velocity of the electrostatic wave becomes comparable to the thermal velocity of the plasma particles, the energy is transferred from the wave to the plasma particles and is known as Landau damping mechanism. In microwave ion source plasma, density can be increased 2–3 times more than the ECR heating mechanisms through the off-resonance heating mechanism. For this reason, the offresonance condition is used to create favorable conditions of the upper hybrid resonance heating. Under certain plasma temperature, the electrostatic wave can transfer energy resonantly to the plasma particles if the wave phase velocity matches the plasma particle velocity. In some cases, the plasma particle velocity can be higher than the wave phase velocity. Under this condition, the plasma particle can transfer energy to the wave. The Landau damping mechanism follows the equation written below:

<sup>0</sup>*threshold* ¼ 4*ω*1*ω*2*Γ*1*Γ*<sup>2</sup> (37)

nifies the parametric decay near the mode conversion region.

*Selected Topics in Plasma Physics*

tion. The corresponding threshold electric field is obtained as [19]:

*C*1*C*2*E*<sup>2</sup>

**2.7 Damping of electrostatic and electromagnetic energy**

*Im*

able to move across the magnetic field.

frequency of the plasma particles.

**46**

*ω ωpe* � �

**3. Simulation modeling of MW interaction in plasma**

¼ �0*:*<sup>2</sup> ffiffiffi

*<sup>π</sup>* <sup>p</sup> *<sup>ω</sup>pe kvth* � �<sup>3</sup>

The exponential term on the right hand side represents that the Landau damping will be small for small value of *kλD*. The Landau damping phenomenon is applicable for both case of electrons and ions. The electrostatic wave whose frequency falls in the range of electrons can heat the electrons only in the parallel direction with respect to the magnetic field. In case of electrostatic ion wave, such restriction does not apply. This is because the electrons are strongly magnetized and so will not be

In case of a compact microwave plasma device where the microwave wavelength becomes comparable to the device dimension, the cavity resonant mode can also play crucial role in damping the electromagnetic energy to the plasma particles. The presence of multiple cavity modes in the plasma can produce modulated wave due to the interaction between each pairs of the cavity resonant modes. The generated modulated wave propagates in the plasma and damps its energy to the plasma particles where the frequency of the modulated wave matches the local plasma

For the MW interaction modeling during the plasma evolution, a schematic of the computational domain is shown in **Figure 2**. The computational domain consists

*e* �12*k*<sup>2</sup> *λ*2

*<sup>D</sup>* (38)

of a microwave coupled reactor, which is a cylindrical plasma cavity. A MW of frequency 2.45 GHz is injected into the microwave coupled reactor through a ridge waveguide port (on the left side of **Figure 2**) to ionize the gaseous particles, thus forming the plasma that is confined under the mirror magnetic field configuration. The reactor has dimension of 107-mm and 88-mm diameter. The microwave power is fed into a cavity resonator through a tapered waveguide. The waveguide is tapered by embedding four ridge sections having different ridge length, ridge gap and ridge width on the inner sidewalls of the waveguide. The ridge dimensions are optimized from the analytical calculation as well as from the electromagnetic simulation. The mirror magnetic field is created by using two pairs of ring magnets that surround the microwave coupled reactor [22]. On the right side of the microwave coupled reactor (**Figure 2**), the ion beam extraction system is attached through a 5-mm hole on the wall of the reactor. The similar computational domain is used in the experimental set-up (Section 5) to validate the simulated data. Here, the finite element method (FEM)-based COMSOL model is used [22].

The MW propagation and the plasma evolution are assumed to be decoupled to each other during the simulation modeling in the temporal scale [10]. The MW electric fields (̃EÞ are averaged for some MW periods before putting their value in the plasma model and the resultant field is given as input to the plasma model. The electron's momentum equation is time integrated along with Maxwell's equations for some MW periods until the MW model of the FEM gets a periodic solution of the equation to transfer an average power to the particles over such a period. The FEM model continues this process until it gets a steady state solution for at least �10<sup>3</sup> (<sup>ω</sup>*=*<sup>ν</sup><sup>m</sup> �10<sup>3</sup> ) MW periods. Here, ω is MW frequency and ν<sup>m</sup> is electron's momentum transfer frequency. Since the electrons stay in the ECR zone for a very short time (transit time) duration, it causes non-local kinetic effects. This results in the de-phasing between the velocity and field oscillations that becomes very difficult to describe using the fluid model. This problem is resolved by introducing the effective collision frequency ð Þ *ν*eff in the simulation to converge the solution. In the low pressure condition, for the collision-less heating, νeff has to be in the order of inverse transit time for the electrons [10].

In case of MW model, the equations for the electric field are solved in frequency domain while keeping the other parameters in time domain. In the beginning, the FEM model started from the Maxwell's equations in order to justify the modeling approach. The MW electric fields are changing with time at a frequency of <sup>ω</sup>*=*<sup>2</sup>*<sup>π</sup>* . During the plasma evolution, the total electric field (E) value in the μs time zone is regarded as a resultant quantity (or total) that comes from the superposition between the MW and ambipolar-type electric field of the plasma. The modification of the resultant electric field follows the equation that is given below:

$$(\nabla \times \mu^{-1} \left(\nabla \times E\right) - k\_0^2 \left(\varepsilon\_r - \frac{i\sigma}{a\varkappa\_0}\right) E = \mathbf{0} \tag{39}$$

energy mobility, total electric field, electron diffusivity, electron energy density, energy loss/gain from inelastic collisions, electron energy flux, electron energy mobility and electron energy diffusivity, respectively. Some of the terms described above, the electron's diffusivity, mobility and energy diffusivity, are estimated

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

The above written electron transport properties represent full tensor parameters in which the tensor term electron mobility is influenced by the magnetic field. The

netic field is the known parameters. Here, *Nn* represents the neutral density of the gaseous particles. The electron source term in Eq. (40) is obtained from the relevant plasma chemistry that is expressed below with the help of the rate coefficients *Re* =

*<sup>j</sup>*¼<sup>1</sup>*<sup>x</sup> jk jNnne*. The presented symbols *<sup>M</sup>*, *<sup>x</sup> <sup>j</sup>* and *<sup>k</sup> <sup>j</sup>* represent the number of reactions, the mole fraction of a specific species and the rate co-efficient for the reaction j, respectively [10]. The energy loss terms as shown in Eq. (41) is written

loss that occurs from the reaction j. The energy source and the loss terms are calculated inherently in the FEM model. The rate coefficients mentioned in the above expressions are taken from the cross-section data as per the relation kk =

f are represented as the mass of electron, energy of electron, cross-section for the reaction and the electron energy distribution function (Maxwellian) for the

respectively. In addition, the heavy plasma particle losses at the boundary walls and their migration are considered to be originating from only the surface reactions and ambipolar electric field, respectively. A fixed power is maintained that is absorbed by the plasma (Pabsorbed). This is done by re-adjusting the normalization factor (α) during the moment of the plasma evolution (ns to μs). The normalization of the plasma absorbed power follows the relation Pabsorbed ¼ *α*∭ nePset dV*:* Here, Pset �

cavity. This normalization helps in convergence of the solution and avoids any disproportionate absorption of the power by the plasma. Here, a fixed plasma absorbed power of 70 W is chosen to benchmark its results with the experimental findings that are reported taking the boundary conditions [23, 24] similar to the computation

In the MW-plasma simulation model, the instant of MW launch is taken as reference (t = 0 s) when the MW is launched into the cavity. The MW is launched in right (R) hand mode (R mode is extraordinary type) using the four step ridge waveguide. This makes the E-field intensity to be maximum in the center of the ~ cavity that is propagating in parallel to the externally applied magnetic field [25, 26]. As soon as the MW is launched, it continues to interact with the gas particles on their propagation timescales (ns). To understand and visualize the profile modification of the electrical field from the start of MW launch to the steady state plasma generation and also their impact on the power coupling to the plasma

*<sup>t</sup>* is represented as the average set power applied to the cylindrical plasma

The description for the boundary conditions taken during the plasma simulation is as follows. The plasma chamber wall is kept at ground potential. The reflections, secondary emission and also the thermal emission from the electrons are assumed to be negligible at the wall boundaries. In effect of that the electron flux and electron

*De* ¼ *μeTe*; *με* ¼ ð Þ 5*=*3 *μ<sup>e</sup>* and *D<sup>ɛ</sup>* ¼ *μεTe* (42)

� � and with the mag-

*<sup>j</sup>*¼<sup>1</sup>*<sup>x</sup> jk jNnneΔε <sup>j</sup>*. The notation Δε<sup>j</sup> denotes the energy

. In this expression, the symbols me, ε, σ<sup>k</sup> and

<sup>2</sup> ve,thne<sup>Þ</sup> and n*:*Γε = (<sup>5</sup>

<sup>6</sup> ve,thneÞ,

following the relations that are given below:

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

for all the reactions as *R<sup>ɛ</sup>* = P*<sup>M</sup>*

electrons, respectively.

<sup>0</sup> εσkð Þ<sup>ε</sup> <sup>f</sup>ð Þ<sup>ε</sup> <sup>d</sup>*ε*, where <sup>γ</sup> is 2ð Þ *<sup>q</sup>=me* <sup>1</sup>*=*<sup>2</sup>

system and the operating conditions.

energy flux at wall boundary can be written as n*:*Γ<sup>e</sup> = (<sup>1</sup>

P*<sup>M</sup>*

γ Ð <sup>∞</sup>

�<sup>e</sup> <sup>v</sup>~e*:*E<sup>~</sup> � �

**49**

electron mobility without the magnetic field � <sup>1</sup> � <sup>10</sup>25*=Nn*

The notations used in the above equations *k*0, *εr*, *σ and E* are the vacuum wavenumber, relative permittivity, the total electric field and the plasma conductivity in full tensor form, respectively. This plasma conductivity is a function of plasma density, collision frequency and B-field. So, all types of total electric field components are possible to estimate as the FEM model can determine the above parameters for some particular plasma loading conditions during its evolution. The model also considers that the ̃*E*-field evolution in the sub-nanosecond region is well separated from the total E-field quantity. This means the total E-field changing during the μs region is composed of the MW electric field and the ambipolar electric field.

For representing the MW propagation in an infinite space, the perfectly matched layers (PML) are introduced in the computational domain as shown in **Figure 1**. The present FEM model considers the electron transport properties to follow the Boltzmann distribution function. The distribution is an integro-differential equation in phase space (*r*, *u*) that cannot be solved efficiently. For this reason, the FEM model assumes the plasma as a fluid following the drift diffusion approximation. The assumption is adapted from the Boltzmann equation that is multiplied by some weighing function and then integrated the resulted function over the velocity space. This exercise yields completely three-dimensional and time dependent equations [10, 23]. During the modeling, FEM assumes that ion motion is negligible with respect to the electron motion in the timescale (ns) of MW. Additionally, the electron density is constant spatially within the ECR surface. The Debye length is also assumed to be much smaller than the interaction length of the MW. The average value of the electron velocity on the microwave timescale is obtained from the assumption of Maxwellian distribution function and from the first derivative of the Boltzmann equation. The following drift-diffusion equations are shown that are used to compute the electron density and electron energy density:t

$$\frac{\partial n\_{\epsilon}}{\partial t} + \nabla.[-n\_{\epsilon}(\mu\_{\epsilon}.E) - D\_{\epsilon}.\nabla n\_{\epsilon}] = R\_{\epsilon} \tag{40}$$

$$\frac{\partial n\_{\ell}}{\partial t} + \nabla.[-n\_{\ell}(\mu\_{\ell}.E) - D\_{\ell}.\nabla n\_{\ell}] + E.\Gamma\_{\ell} = R\_{\ell} \tag{41}$$

The term in Eq. (42) is *E:Γ<sup>e</sup>* = *eneve* � *Eambipolar* – Π. This is a heating term *E:Γ<sup>e</sup>* that comprises two components. The first component means the electrons are getting energy through the ambipolar field during the plasma evolution. The second component (Π) signifies the MW absorption power (*ne ve* � *<sup>E</sup>*<sup>~</sup> ) that is carried by the electrons in the plasma. The other terms *ne*, *ve*, *Re*, *Γe*, *μe*, *με*, *E*, *De*, *nɛ*, *Rɛ*, *Γε*, *n<sup>ɛ</sup>* and *D<sup>ɛ</sup>* are the electron density, average electron velocity, the source term of the electrons, electron flux, the mobility of the electrons (4 � <sup>10</sup><sup>4</sup> *<sup>m</sup>*<sup>2</sup> /(*Vs*)), electron

*Evolution of Microwave Electric Field on Power Coupling to Plasma during Ignition Phase DOI: http://dx.doi.org/10.5772/intechopen.92011*

energy mobility, total electric field, electron diffusivity, electron energy density, energy loss/gain from inelastic collisions, electron energy flux, electron energy mobility and electron energy diffusivity, respectively. Some of the terms described above, the electron's diffusivity, mobility and energy diffusivity, are estimated following the relations that are given below:

$$D\_{\varepsilon} = \mu\_{\varepsilon} T\_{\varepsilon}; \mu\_{\varepsilon} = (\mathsf{5}/\mathsf{3})\mu\_{\varepsilon} \text{ and } D\_{\varepsilon} = \mu\_{\varepsilon} T\_{\varepsilon} \tag{42}$$

The above written electron transport properties represent full tensor parameters in which the tensor term electron mobility is influenced by the magnetic field. The electron mobility without the magnetic field � <sup>1</sup> � <sup>10</sup>25*=Nn* � � and with the magnetic field is the known parameters. Here, *Nn* represents the neutral density of the gaseous particles. The electron source term in Eq. (40) is obtained from the relevant plasma chemistry that is expressed below with the help of the rate coefficients *Re* = P*<sup>M</sup> <sup>j</sup>*¼<sup>1</sup>*<sup>x</sup> jk jNnne*. The presented symbols *<sup>M</sup>*, *<sup>x</sup> <sup>j</sup>* and *<sup>k</sup> <sup>j</sup>* represent the number of reactions, the mole fraction of a specific species and the rate co-efficient for the reaction j, respectively [10]. The energy loss terms as shown in Eq. (41) is written for all the reactions as *R<sup>ɛ</sup>* = P*<sup>M</sup> <sup>j</sup>*¼<sup>1</sup>*<sup>x</sup> jk jNnneΔε <sup>j</sup>*. The notation Δε<sup>j</sup> denotes the energy loss that occurs from the reaction j. The energy source and the loss terms are calculated inherently in the FEM model. The rate coefficients mentioned in the above expressions are taken from the cross-section data as per the relation kk = γ Ð <sup>∞</sup> <sup>0</sup> εσkð Þ<sup>ε</sup> <sup>f</sup>ð Þ<sup>ε</sup> <sup>d</sup>*ε*, where <sup>γ</sup> is 2ð Þ *<sup>q</sup>=me* <sup>1</sup>*=*<sup>2</sup> . In this expression, the symbols me, ε, σ<sup>k</sup> and f are represented as the mass of electron, energy of electron, cross-section for the reaction and the electron energy distribution function (Maxwellian) for the electrons, respectively.

The description for the boundary conditions taken during the plasma simulation is as follows. The plasma chamber wall is kept at ground potential. The reflections, secondary emission and also the thermal emission from the electrons are assumed to be negligible at the wall boundaries. In effect of that the electron flux and electron energy flux at wall boundary can be written as n*:*Γ<sup>e</sup> = (<sup>1</sup> <sup>2</sup> ve,thne<sup>Þ</sup> and n*:*Γε = (<sup>5</sup> <sup>6</sup> ve,thneÞ, respectively. In addition, the heavy plasma particle losses at the boundary walls and their migration are considered to be originating from only the surface reactions and ambipolar electric field, respectively. A fixed power is maintained that is absorbed by the plasma (Pabsorbed). This is done by re-adjusting the normalization factor (α) during the moment of the plasma evolution (ns to μs). The normalization of the plasma absorbed power follows the relation Pabsorbed ¼ *α*∭ nePset dV*:* Here, Pset � �<sup>e</sup> <sup>v</sup>~e*:*E<sup>~</sup> � � *<sup>t</sup>* is represented as the average set power applied to the cylindrical plasma cavity. This normalization helps in convergence of the solution and avoids any disproportionate absorption of the power by the plasma. Here, a fixed plasma absorbed power of 70 W is chosen to benchmark its results with the experimental findings that are reported taking the boundary conditions [23, 24] similar to the computation system and the operating conditions.

In the MW-plasma simulation model, the instant of MW launch is taken as reference (t = 0 s) when the MW is launched into the cavity. The MW is launched in right (R) hand mode (R mode is extraordinary type) using the four step ridge waveguide. This makes the E-field intensity to be maximum in the center of the ~ cavity that is propagating in parallel to the externally applied magnetic field [25, 26]. As soon as the MW is launched, it continues to interact with the gas particles on their propagation timescales (ns). To understand and visualize the profile modification of the electrical field from the start of MW launch to the steady state plasma generation and also their impact on the power coupling to the plasma

In case of MW model, the equations for the electric field are solved in frequency domain while keeping the other parameters in time domain. In the beginning, the FEM model started from the Maxwell's equations in order to justify the modeling approach. The MW electric fields are changing with time at a frequency of <sup>ω</sup>

During the plasma evolution, the total electric field (E) value in the μs time zone is regarded as a resultant quantity (or total) that comes from the superposition between the MW and ambipolar-type electric field of the plasma. The modification

The notations used in the above equations *k*0, *εr*, *σ and E* are the vacuum wavenumber, relative permittivity, the total electric field and the plasma conductivity in full tensor form, respectively. This plasma conductivity is a function of plasma density, collision frequency and B-field. So, all types of total electric field components are possible to estimate as the FEM model can determine the above parameters for some particular plasma loading conditions during its evolution. The model also considers that the ̃*E*-field evolution in the sub-nanosecond region is well separated from the total E-field quantity. This means the total E-field changing during the μs region is composed of the MW electric field and the ambipolar electric

For representing the MW propagation in an infinite space, the perfectly matched layers (PML) are introduced in the computational domain as shown in **Figure 1**. The present FEM model considers the electron transport properties to follow the Boltzmann distribution function. The distribution is an integro-differential equation in phase space (*r*, *u*) that cannot be solved efficiently. For this reason, the FEM model assumes the plasma as a fluid following the drift diffusion approximation. The assumption is adapted from the Boltzmann equation that is multiplied by some weighing function and then integrated the resulted function over the velocity space. This exercise yields completely three-dimensional and time dependent equations [10, 23]. During the modeling, FEM assumes that ion motion is negligible with respect to the electron motion in the timescale (ns) of MW. Additionally, the electron density is constant spatially within the ECR surface. The Debye length is also assumed to be much smaller than the interaction length of the MW. The average value of the electron velocity on the microwave timescale is obtained from the assumption of Maxwellian distribution function and from the first derivative of the Boltzmann equation. The following drift-diffusion equations are shown that are

<sup>0</sup> *<sup>ɛ</sup><sup>r</sup>* � *<sup>i</sup><sup>σ</sup>*

*ωɛ*<sup>0</sup> 

þ ∇*:* �*ne μ<sup>e</sup>* ð Þ� *:E De:*∇*ne* ½ � ¼ *Re* (40)

/(*Vs*)), electron

þ ∇*:* �*n<sup>ɛ</sup> μ<sup>ɛ</sup>* ð Þ� *:E Dɛ:*∇*n<sup>ɛ</sup>* ½ � þ *E:Γ<sup>e</sup>* ¼ *R<sup>ɛ</sup>* (41)

The term in Eq. (42) is *E:Γ<sup>e</sup>* = *eneve* � *Eambipolar* – Π. This is a heating term *E:Γ<sup>e</sup>* that comprises two components. The first component means the electrons are getting energy through the ambipolar field during the plasma evolution. The second component (Π) signifies the MW absorption power (*ne ve* � *<sup>E</sup>*<sup>~</sup> ) that is carried by the electrons in the plasma. The other terms *ne*, *ve*, *Re*, *Γe*, *μe*, *με*, *E*, *De*, *nɛ*, *Rɛ*, *Γε*, *n<sup>ɛ</sup>* and *D<sup>ɛ</sup>* are the electron density, average electron velocity, the source term of the

of the resultant electric field follows the equation that is given below:

<sup>∇</sup> � *<sup>μ</sup>*�<sup>1</sup> ð Þ� <sup>∇</sup> � *<sup>E</sup> <sup>k</sup>*<sup>2</sup>

used to compute the electron density and electron energy density:t

electrons, electron flux, the mobility of the electrons (4 � <sup>10</sup><sup>4</sup> *<sup>m</sup>*<sup>2</sup>

*∂ne ∂t*

*∂n<sup>ɛ</sup> ∂t*

field.

*Selected Topics in Plasma Physics*

**48**

*=*<sup>2</sup>*<sup>π</sup>* .

*E* ¼ 0 (39)

with time, total breakdown time is split into some discrete values such as 10 ns, 67 ns, 158 ns, 452 ns, 630 ns, 2 μs, 2.5 μs, 3 μs, 5 μs, 8 μs, 20 μs, 40 μs, 250 μs and 300 μs. The simulation is exercised with an argon gas. The initial conditions for the calculation of the plasma parameter evolution are plasma density (1 <sup>10</sup><sup>12</sup> <sup>m</sup><sup>3</sup> ) and plasma temperature (4 eV). These parameters are used initially to estimate the plasma conductivity σ. Then the electric field is estimated by putting the above initial conditions in Eq. (39). The estimated field values are later utilized to find out the temporal variation of different plasma parameters and so electric field components'self-consistency.

minimum and maximum size for the magnetic field and microwave-plasma calculations are 0.2 mm and 2.4 mm, respectively. The effect of the different edges that are involved in the computational domain is taken into account by keeping the maximum and minimum mesh element size at 0.5 mm and 0.0055 mm, respectively. The magnetic field estimated is used in the MW-plasma model to estimate the tensor plasma parameters. For the case of MW-plasma simulation, the number

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

Due to the dephasing as discussed above, the current FEM model uses the concept of effective collision frequency (νeff) to describe the sudden phase decoherence. In the dephasing situation, the phase relationship between velocity and the electric field oscillation is destroyed in the temporal scale due to which the electrons experience a large field variation in the ECR surface. The accelerating electric field transfers energy to the electrons that are residing on the ECR surface only for a small time duration in the time range of resonant cyclotron motion of the electrons. The electrons also experience spatial density variations while oscillating across the ECR surface. A large density variation across the ECR surface also generates the radial ambipolar electric field on the resonance surface and so there is a

As the phase de-coherence may happen between electron gyro motion and the

where ve is the electron thermal speed and δ<sup>B</sup> is the gradient of magnetic field. The value of νeff is few orders more than that of actual collisional frequency that an electron encounters with the gaseous particles. However, it is to be noted that the effect of νeff is insignificant on the power absorption profile. Instead it helps in overcoming the numerical instability. For obtaining a steady-state solution, the number of MW periods has to be in the order of ω/νeff. This makes computation less

The evolution of the radial (Er) and axial (Ez) components of the electric fields during the plasma evolution plays an important role in the MW coupling to the plasma and so the plasma parameters as discussed above. To support these facts, the

*Magnetic (B) field contour inside the quadrant section of the plasma chamber, which has cylindrical axis symmetry. The B-field is also simulated using COMSOL software. The narrow contour area near 0.088 T line is the ECR zone (corresponding to 0.0875 T). The span of ECR zone is around z =* �*24 mm and r =* �*28 mm.*

veω*=*δ<sup>B</sup> p ,

of degrees of freedom used for the solution is approximately 47,005.

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

possibility of de-phasing that can happen at the resonance zone.

rigorous than the case of actual collisions, if considered.

**Figure 4.**

**51**

MW oscillatory ̃E-field, the electrons get accelerated and deccelerated assymetrically. This temporal asymmetry of acceleration and decceleration in opposite cycle are responsible for transferring an effective energy to the electrons. The effective collision frequency <sup>ν</sup>eff is estimated using the relation <sup>ν</sup>eff <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

The present FEM model uses different solvers sequentially to compute the magnetic field distribution first throughout the computational domain. Then the solvers related to the frequency-transient analysis are used to calculate the MWplasma parameters. A complete computational flowchart for the magnetic field as well as the MW-plasma is depicted in **Figure 3**. The typical number of degrees of freedom that is used for solving the magnetic field is approximately 54,525. The FEM magnetostatic model uses the equation-based mesh adaptation technique to generate the extremely fine mesh size on the ECR surfaces. The mesh element has

**Figure 3.** *Simulation flowchart of MW plasma interaction in COMSOL Multiphysics.*

#### *Evolution of Microwave Electric Field on Power Coupling to Plasma during Ignition Phase DOI: http://dx.doi.org/10.5772/intechopen.92011*

minimum and maximum size for the magnetic field and microwave-plasma calculations are 0.2 mm and 2.4 mm, respectively. The effect of the different edges that are involved in the computational domain is taken into account by keeping the maximum and minimum mesh element size at 0.5 mm and 0.0055 mm, respectively. The magnetic field estimated is used in the MW-plasma model to estimate the tensor plasma parameters. For the case of MW-plasma simulation, the number of degrees of freedom used for the solution is approximately 47,005.

Due to the dephasing as discussed above, the current FEM model uses the concept of effective collision frequency (νeff) to describe the sudden phase decoherence. In the dephasing situation, the phase relationship between velocity and the electric field oscillation is destroyed in the temporal scale due to which the electrons experience a large field variation in the ECR surface. The accelerating electric field transfers energy to the electrons that are residing on the ECR surface only for a small time duration in the time range of resonant cyclotron motion of the electrons. The electrons also experience spatial density variations while oscillating across the ECR surface. A large density variation across the ECR surface also generates the radial ambipolar electric field on the resonance surface and so there is a possibility of de-phasing that can happen at the resonance zone.

As the phase de-coherence may happen between electron gyro motion and the MW oscillatory ̃E-field, the electrons get accelerated and deccelerated assymetrically. This temporal asymmetry of acceleration and decceleration in opposite cycle are responsible for transferring an effective energy to the electrons. The effective collision frequency <sup>ν</sup>eff is estimated using the relation <sup>ν</sup>eff <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi veω*=*δ<sup>B</sup> p , where ve is the electron thermal speed and δ<sup>B</sup> is the gradient of magnetic field. The value of νeff is few orders more than that of actual collisional frequency that an electron encounters with the gaseous particles. However, it is to be noted that the effect of νeff is insignificant on the power absorption profile. Instead it helps in overcoming the numerical instability. For obtaining a steady-state solution, the number of MW periods has to be in the order of ω/νeff. This makes computation less rigorous than the case of actual collisions, if considered.

The evolution of the radial (Er) and axial (Ez) components of the electric fields during the plasma evolution plays an important role in the MW coupling to the plasma and so the plasma parameters as discussed above. To support these facts, the

#### **Figure 4.**

*Magnetic (B) field contour inside the quadrant section of the plasma chamber, which has cylindrical axis symmetry. The B-field is also simulated using COMSOL software. The narrow contour area near 0.088 T line is the ECR zone (corresponding to 0.0875 T). The span of ECR zone is around z =* �*24 mm and r =* �*28 mm.*

with time, total breakdown time is split into some discrete values such as 10 ns, 67 ns, 158 ns, 452 ns, 630 ns, 2 μs, 2.5 μs, 3 μs, 5 μs, 8 μs, 20 μs, 40 μs, 250 μs and 300 μs. The simulation is exercised with an argon gas. The initial conditions for the calculation of the plasma parameter evolution are plasma density

The present FEM model uses different solvers sequentially to compute the magnetic field distribution first throughout the computational domain. Then the solvers related to the frequency-transient analysis are used to calculate the MWplasma parameters. A complete computational flowchart for the magnetic field as well as the MW-plasma is depicted in **Figure 3**. The typical number of degrees of freedom that is used for solving the magnetic field is approximately 54,525. The FEM magnetostatic model uses the equation-based mesh adaptation technique to generate the extremely fine mesh size on the ECR surfaces. The mesh element has

) and plasma temperature (4 eV). These parameters are used initially to estimate the plasma conductivity σ. Then the electric field is estimated by putting the above initial conditions in Eq. (39). The estimated field values are later utilized to find out the temporal variation of different plasma parameters and so

(1 <sup>10</sup><sup>12</sup> <sup>m</sup><sup>3</sup>

*Selected Topics in Plasma Physics*

**Figure 3.**

**50**

*Simulation flowchart of MW plasma interaction in COMSOL Multiphysics.*

electric field components'self-consistency.

evolution of the different components of the electric fields and correspondingly the plasma parameters is shown in the results and discussion section given below to study their effects on the plasma parameters. To understand the resonance zone in the plasma chamber, the required magnetic field contours are shown in **Figure 4**.

**4.2 Time evolution of plasma with power deposition**

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

one can visualize that the plasma density (1.3 <sup>10</sup><sup>17</sup> <sup>m</sup><sup>3</sup>

is expected to be highest (<sup>5</sup> <sup>10</sup><sup>7</sup> W/m3

t = 630 ns.

**Figure 6.**

**53**

*6.6 <sup>10</sup><sup>4</sup> W/m3*

**Figure 6a** shows that the MW power is being deposited exactly on the ECR (0.0875 T) surface corresponding to the launch MW frequency of 2.45 GHz when the plasma density is low. But as time passes (see **Figure 6b–d**), the power deposition location gets shifted in the off-ECR or upper hybrid resonance (UHR) regime. The UHR zone is a region where the two conditions *ne* < *ncrit* and B < BECR are satisfied [26–28]. The term *ncrit* represents the critical density for the MW frequency, 2.45 GHz that is 7*:*<sup>4</sup> <sup>10</sup><sup>16</sup> <sup>m</sup>3. If one can compare **Figures 6(c)** and **7(a)**,

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

density from 2.5 μs onwards and the plasma density that is above the critical density is denoted as overdense plasma. So as the overdense plasma is achieved, the electrons get accelerated by less amount of MW energy on the ECR surface. Correspondingly, the plasma bulk temperature increases from the start of MW launch (t = 0 s) to the instant of 630 ns. It is evident from **Figure 7a** that the plasma bulk electron temperature increases and becomes steady near a value of 80 eV. Then the plasma bulk temperature decreases in a faster way with further increase of time. Hence, one can conclude that heating through ECR process is being ceased to occur with the further increase in time. The causes to increase the absorbed power density

*Power deposition density at different time steps for 70 W of absorbed power. (a) t = 10 ns, peak power density is*

*, (b) t = 630 ns, peak power density is 1.43 107 W/m<sup>3</sup>*

*5.12 <sup>10</sup><sup>6</sup> W/m<sup>3</sup> and (d) t = <sup>40</sup> <sup>μ</sup>s, peak power density is 1.67 107 W/m<sup>3</sup>*

) crosses the critical

*, (c) t = 2.5 μs, peak power density is*

*.*

) on the ECR surface during the time,
