Electromagnetic Wave Propagation in Ionospheric Plasma

Ionosphere physics is related to plasma physics because the ionosphere is, of course, a weak natural plasma: an electrically neutral assembly of ions and electrons [1]. The ionosphere plays a unique role in the Earth’s environment because of strong coupling process to regions below and above [2]. The ionosphere is an example of naturally occurring plasma formed by solar photo-ionization and soft x-ray radiation. The most important feature of the ionosphere is to reflect the radio waves up to 30 MHz. Especially, the propagation of these radio waves on the HF band makes the necessary to know the features and the characteristics of the ionospheric plasma media. Because, when the radio waves reflect in this media, they are reflected and refracted depending on their frequency, the frequency of the electrons in the plasma and the refractive index of the media and thus, they are absorbed and reflected by the media. The understanding of the existence of a conductive layer in the upper atmosphere has been emerged in a century ago. The idea of a conductive layer affected by the variations of the magnetic field in the atmosphere has been put forward by the Gauss in 1839 and Kelvin in 1860. Newfoundland radio signal from Cornwall to be issued by Marconi in 1901, at the first experimental evidence of the existence of ionospheric, respectively. In 1902, Kennely and Heaviside indicated that the waves are reflected from a conductive layer on the upper parts of the atmosphere. In 1902, Marconi stated that changing the conditions of night and day spread. In 1918, high-frequency band has been used by aircraft and ships. HF band in the 1920s has increased the importance of expansion. Then, put forward the theory of reflective conductive region, has been shown by experiments made by Appleton and Barnet. The data from the 1930s started to get clearer about the ionosphere and The Radio Research Station, Cavendish Laboratory, the National Braun of Standards, the various agencies such as the Carnegie Institution began to deal with the issue. In the second half of the 20th century, the work of the HF electromagnetic wave has been studied by divided into three as the fullwave theory, geometrical optics and conductivity. Despite initiation of widespread use of satellite-Earth communication systems, the use of HF radio spectrum for civilian and military purposes is increasing. Collapse of communication systems, especially in case of emergency situations, communication is vital in this band. In the ionosphere, a balance between photo-ionization and various loss mechanisms gives rise to an equilibrium density of free electrons and ions with a horizontal stratified


Introduction
Ionosphere physics is related to plasma physics because the ionosphere is, of course, a weak natural plasma: an electrically neutral assembly of ions and electrons [1]. The ionosphere plays a unique role in the Earth's environment because of strong coupling process to regions below and above [2]. The ionosphere is an example of naturally occurring plasma formed by solar photo-ionization and soft x-ray radiation. The most important feature of the ionosphere is to reflect the radio waves up to 30 MHz. Especially, the propagation of these radio waves on the HF band makes the necessary to know the features and the characteristics of the ionospheric plasma media. Because, when the radio waves reflect in this media, they are reflected and refracted depending on their frequency, the frequency of the electrons in the plasma and the refractive index of the media and thus, they are absorbed and reflected by the media. The understanding of the existence of a conductive layer in the upper atmosphere has been emerged in a century ago. The idea of a conductive layer affected by the variations of the magnetic field in the atmosphere has been put forward by the Gauss in 1839 and Kelvin in 1860. Newfoundland radio signal from Cornwall to be issued by Marconi in 1901, at the first experimental evidence of the existence of ionospheric, respectively. In 1902, Kennely and Heaviside indicated that the waves are reflected from a conductive layer on the upper parts of the atmosphere. In 1902, Marconi stated that changing the conditions of night and day spread. In 1918, high-frequency band has been used by aircraft and ships. HF band in the 1920s has increased the importance of expansion. Then, put forward the theory of reflective conductive region, has been shown by experiments made by Appleton and Barnet. The data

Conductivity for ionospheric plasma
One of the main parameters affecting the progression of the electromagnetic wave in a medium is the conductivity of the media. Conductivity of the media statement is obtained from motion equation of the charged particle that is taken account in the total force acting on charged particle in ionospheric plasma. Accordingly, the forces acting on charged particle is given as follows [6].
The plasma approach, which the thermal motion due to temperatures of particles, are neglected, is called cold plasma. In this study, because of the cold plasma approximation, any force does not effect on charged particles due to the temperature and therefore the term of pressure changes is ignored [7]. Likewise, the gravitational force from the gravity is negligible due to so small according to the electric and magnetic forces. In addition, due to the electron mass (m e ) is very small according to the mass of the ion (m i ) (m e <<m i ) and thus the electron collision frequency is greater than the ion collision frequency, the movement of ions in the plasma is ignored near the electron movement. Accordingly, Equation (1) This expression is called the Langevin equation [8]. Where, the electron collision frequency e ν is sum of the electron-ion collision frequency ( ) ei ν and electron-neutral particle collision ( ) en ν frequency.

DC conductivity
The first application of the Langevin equation is steady flow caused by of the electric field. For this, e 0 =  V . Accordingly, the Langevin equation for electrons can be defined as follows: For the current density, the following equation is obtained: Taking into consideration that σ =⋅ JE , the generalized Ohm Law can be defined by, Where, is called OHM law for the plasma. The geometry of the velocity and electric field and magnetic field is as shown in Figure 1, namely , the e × VB term in Equation (7) is being as follows: Here, the conductivity tensor is called as dc conductivity tensor. As can be seen in this expression,  , the variation of the conductivities is being as shown in Figure 3. As shown in Figure 3, after the ce e ω 1  ν 1 σ term decreases rapidly. Thus, the current perpendicular to the magnetic field is very small [8].

AC conductivity
There is no steady-state in the ac conductivity. Taking into consideration that the fields vary as it e −ω , the Equation (2) is transformed as follows: According to the geometry given in Figure 1, this expression in terms of current density can be defined as follows: Thus, current density components are obtained as follows: Considering the ω ce cyclotron frequency, the components x J and y J can be obtained as follows: If the expressions (20)-(22) in the tensor form and the conductivity tensor ′ σ are defined as follows, The conductivities 0 ′ σ , 1 ′ σ and 2 ′ σ will be ( ) , these equations can be written as , respectively. Where, as in dc conductivity 0 σ ′ is the conductivity of the magnetic field direction, 1 σ ′ is the conductivity perpendicular to the magnetic field and 2 σ ′ is the conductivity perpendicular to both of the electrical and the magnetic fields, respectively.
, the ac conductivity tensor transforms to the dc conductivity tensor. Example: ac conductivity tensor according to the geometry given in Figure 4 is obtained as in Equation (25): If the magnetic field is only xB = B direction, ac conductivity tensor becomes as follows:  Example: If the magnetic field has two-dimensional geometry as given in Figure 5 ( ) yz yB zB y BCosθ z BSinθˆˆ=+= + B , the conductivity tensor are defined as in Equation (27)

Earth's magnetic field and ionospheric conductivity
In this section, the Langevin equation defined by Equation (16) for ac conductivity will be discussed. However, the true magnetic field in the Earth's northern half-sphere given in Figure 6 will be dealt with. Accordingly, in the selected Cartesian coordinate system, the xaxis represents the geographic east, the y-axis represents the the geographic North and z-axis represents the up in the vertical direction, the magnetic field can be defined as follows [9]: x BCosISinD y BCosICosD -z BSinÎˆ=+ B Where, I is the dip angle and D is the declination angle (between the magnetic north and the geographic north). Thus, the term of e × VB in Equation (16) can be obtained as in Equation (29).
If this expression is written in Equation (16) and the necessary mathematical manipulations are made, three equations are obtained for x, y and z directions as follows: www.intechopen.com

Behaviour of Electromagnetic Waves in Different Media and Structures
Here, the coefficients matrix is named as A, by taking the determinant of this matrix, the Equation (37) can be obtained. This tells us that one solution of the equation. According to Cramer solution is as follows: Here, the terms of 12 3 A, A a n d A are defined as follows: 0x 0x 0x 10 y 20 y 30 y 0z 0z 0z In this statement, the x direction current density is resolved as in Equation (40).
For the a, b and c, if the instead of the expressions given above are written, the x J term is obtained as follows: www.intechopen.com

Behaviour of Electromagnetic Waves in Different Media and Structures 200
By the same way, y direction the current density is resolved as follows:  iωω Cos ICos D iωω Likely, z direction the current density is resolved as follows: The conductivity tensor can defined as in Equation (47) and tensor components can be achieved in a simpler as follows [9,10]: www.intechopen.com

Dielectric constant for ionospheric plasma
Dielectric constant for ionospheric plasma could be founded by using Maxwell equations.
it is obtained as follow 00 0 According to the latest's equation, the dielectric constant of any medium In which, because of ( σ ) the tensorial form 1 ∼ is the unit tensor. 100 10 1 0 001 Generally, the expression of ionospheric conductivity σ′′ given in Equation (47)

The refractive index of the cold plasma
The refractive index (n) determines the behavior of electromagnetic wave in a medium and refractive index of the medium is founded by using Maxwell equations. If the curl ( ) × ∇ of the Equation (50) is taken, term in this expression is re-written in (60), ( ) If the wave vector k in this equation is written in terms of refractive index n, the Equation (63) can be defined in terms of refractive index as follows: If the necessary procedures are used, the Equation (66) is obtained as follows: By writing the conductivity tensor σ′′ in the Equation (47) instead of the conductivity in the Equation (66), a relation for the cold plasma is obtained as follows: 11 12 13 Here, since the electric fields do not equal to zero, the determinant of the matrix of the coefficients equals to zero. So: 11 12 13 www.intechopen.com

Behaviour of Electromagnetic Waves in Different Media and Structures 204
The matrix given by Equation (68) includes the information about the propagation of the wave. Since the matrix has a complex structure, it is impossible to solve the matrix in general. However, the numerical analysis can be done for the matrix. Therefore, solutions should be made to certain conditions, in terms of convenience.

k//B condition: plasma oscillation and polarized waves
When the progress vector of the wave (k) is parallel or anti-parallel to the earth's magnetic field or the any components of the earth's magnetic field, two cases can be observed for the ionospheric plasma depending on the refractive index of the medium. For example, if the wave propagates in the z direction as in the vertical ionosondas, the vertical component of the earth's magnetic field effects to the propagation of the wave. In this case, two waves occur from the solution of the determinant given by Equation (68). The first one is the vibration of the plasma defined as follows: The second one is the polarized wave given by Equation (70) [9].
The signs ± in the Equation (70)

k⊥B condition: ordinary and extra-ordinary waves
When the propagation vector of the wave (k) is perpendicular to the earth's magnetic field, two waves occur in the ionospheric plasma [9]. The first wave is the ordinary wave given as follows: This wave does not depend on Earth's magnetic field. However it depends on the collisions. The collisions can change the resonance and the reflection frequencies of the wave. The second wave is the extra ordinary wave depending on the magnetic field. The refractive index of the extra ordinary wave can be defined as follows: www.intechopen.com . This relation is valid for the y direction. So, the magnetic field in the Y is the cyclotron frequency caused by the y component of the earth's magnetic field. By the same way, the wave defined by this relation is also observed in the x direction. For the extra ordinary wave in the x direction, the cyclotron frequency, that is contained in Y, is the cyclotron frequency caused by the x component of the earth's magnetic field.

The Binom expansion ( )
1Z  +≈   and the refractive indices The solutions of the refractive indices mentioned above are complex and difficult. The solutions can be obtained by using the binom expansion. Accordingly, the refractive indices can be obtained by using the Binom expansion and the real part of refractive indices as follows: 1. For the right-polarized wave given by Equation (70): Where, -X X 1Y S i n I ′ = and - 2. For the extra ordinary wave given by Equation (71): 3. For the extra ordinary wave given by Equation (72) , we obtain the differential from of the charge conversation law: Integrating both sides over a closed volume V surrounded by the surface S, and using the divergence theorem, we obtain the integrated The left-hand represents the total amount of charge flowing outwards through the surface S per unit time. The right-hand side represents the amount by which the charge is decreasing inside the volume V per unit time. In other words, charge does not disappear into (or get created out of) nothingness-it decreases in a region of space only because it flows into other regions.

Fig. 7. Flux outwards through surface
Another consequence is that in good conductors, there cannot be any accumulated volume charge. Any such charge will quickly move to the conductor's surface and distribute itself such that to make the surface into an equipotential surface. Assuming that inside the conductor we have =ε⋅ DE and =σ⋅ JE , we obtain σσ ⋅= σ⋅ = ⋅ = ρ εε n J index effects to the phase velocity and the imaginary part of the refractive index is associated with spatial attenuation of the wave.
In accordance with the above information, in order to obtain the relation of the attenuation, amplitude of the wave, such as the electric field strength, is need to be expressed depending on the refractive index of the medium. Accordingly, the electric field strength of the wave can be defined as follows: Here, the wave vector k is written in terms of refractive index n, Equation (92) becomes as follows: The refractive index in the ionospheric plasma is also defined as in equation (93).
where, µ and χ represent the real and the imaginary part of the refractive index, respectively. The collision of the electron with the other particles effects to the real and the imaginary part of the refractive index. In the High Frequency (HF) waves, Z is very smaller than 1 ( ) Z1  . Therefore, Z can be defined as ( ) The part of the damping in the electric field strength in Equation (95) is the exponential term related to second χ in the right side of the equation. Thus, the attenuation of the wave is represented by χ. According to this, the refractive indexes of the polarized, ordinary and extra-ordinary waves given (70)-(72) equations for cold plasma could be showed as the real and imaginary parts as follow.
Accordingly, the electric field strength given in Equation (95) In the Equation (97), signs (+) and (-) in front of the first exponential expression represents the wave propagated to upward (ẑ + ) and downward (ẑ − ) after the reflection, respectively.
Here, by considering the electromagnetic wave that propagated to upward ẑ + , the term The second term in equation (98) represents how much the attenuation of wave in ionospheric plasma.

Conclusion
Ionospheric plasma has double-refractive index and generally weak conductivity in every direction, season and local time. Due to these, the ionospheric parameters such as conductivity, dielectric constant and the refractive index are different in every direction and have the complex structure (both the real and imaginary part). This shows that the diagonal elements of conductivity, refractive index and dielectric tensor which the conductivities dominate have generally higher conductivity than other elements; besides any electromagnetic wave propagating in ionospheric plasma could be sustained attenuation in every direction. So this attenuation results from the imaginary part of the refractive index.