Khac An Dao, Dong Chung Nguyen

,

Diep Dao

ESTIMATION

OF THE SPECIFIC REAL

PHASE AND GROUP REFRACTIVE

INDEXES BY THE ALTITUDE

IN THE

EARTH’S

IONIZED REGION

USING THE

FIRST ORDER

APPLETON

–

HARTREE

EQUATIONS

Khac An Dao

∗

1,2

, Dong C

hung

Nguyen

3

, and Diep Dao

4

1

Instituite of

Theoretical and Applied Research (ITAR), Duy Tan University, Ha Noi 100000,

Vietnam

2

Faculty of Electrical and

Electronic

Engineering, Duy Tan University, Da

N

ang 550000, Vietnam

3

Institute of Research and Development, Duy Tan University, Da

N

ang 550000, V

ietnam

4

Department of Geography and Environmental Studies,

University of Colorado,

Colorado

Springs,U.S.

A

1

Abstract

:

The specific phase and group refractive

indexes concerning the

specific

phase and group velocities

of single an

d packet electromagnetic waves

contain all

interactions between the electromagnetic wa

ves and the

propagating medium.

T

he

determination of the sp

ecific

refractive indexes vs. altitude is

also

a

challenging

and

complicated problem.

B

ased on

t

he

first

–

order Appleton

–

Hartree equations

and

the

values of

free electron density by

altitude

,

t

his paper

outlined

the numerical estimated results

of the specific real phase, group refractive indexes vs. the

altitude from 100 km up to 1000 km in the ionized

region.

The specific real phase refractive index has

a

value smaller

than 1, corresponding to this value, the spe

cific phase

velocity

is

larger than the light speed (

c

)

meanwhile

the

value of the specific real group refractive index is larger

than 1, the specific group velocity will always be smaller

than light speed (

c

). These estimated results are agreed with

the t

heory and forecasted

model

predicted

.

These

results

could be applied for both the experiment and theoretical

researches, especially for application in finding the

numerical solution of mathematics problems of Wireless

Information and Wireless Power Transmi

ssions.

Keywords

:

Specific real phase and group r

efractive

index

es

by altitude,

The

First order

Appleton

–

Hartree

equations

,

the Earth’s

ionized region,

Microwave

propagati

on

.

Co

r

responding Author: Khac An D

ao

Email:

Sending to Journal:

9/2020;

Revised

: 11/2020;

Accepted

:

12/2020.

I.

INTRODUCTION

The developments of the theoretical aspects

of

the

refractive indexes

concerning

the electromagnetic waves

(EMW)

propagation in the Earth’s ionized region

always

have been

studying

up today

. The

refractive index of the

EMW

is

an essential concept that reflects

the interactions

between the EMW and a given medium. Depending on the

features of

a

given

propagating

medium

and the forms of

EMWs

, the refractive index

is changed

and it has

be

en

discussed and

formulated

in

different

forms

,

such as by

Sellmeyer formula

and

Lorentz formul

a

[2

–

5].

During the

time f

rom

1927

to

1932, the essential formula for the

refractive index of

the Earth

‘s

atmosphere’s

ionized

region

i

n a magnetic field

h

as

been

developed and

called

by

the

name

of

the

Appleton

–

Hartree

equation.

This

equation

describes

generally

the

refractive

index

for

EMW

propagation in a cold magnetized

plasma

region

–

the ionosphere region

.

Since then there

were

m

any

aspects concerning th

is

refractive index expression

that

have been stud

ied

and published in Literat

ure

, for example

:

the determination of constants being in the Appleton

Hartree equation [4, 5];

the study of effect of electron

collisions on the formulas by magneto

–

ionic theory; the

development of theory, mathematical formulas concerning

the

complex

refractive indices of an ionized medium

[4, 5

and 7]; the conditions and the validity of some

ESTIMATION OF THE SPECIFIC REAL PHASE AND GROUP REFRACTIVE INDEXES BY

….

approximations related to the refractive index have also

been studied including the high order ionosphere effects on

the global positioning system observables a

nd means of

modeling [6, 7, 8 and 9]; the proposed model and predicted

values of the refractive index in the different layers of the

earth atmosphere medium [10]; the scattering mechanisms

of EMW [11]; the variation of the ionosphere conductivity

with diff

erent solar and geomagnetic conditions [12]; the

ionosphere absorption in vertical propagation [13]; the

atmospheric influences on microwaves propagation[14];

the stochastic perception of refractive index variability of

ionosphere [16]; and a lot of other

aspects have been

studied in references [15

–

19].

Recently

there are also

many works

continuing to

stud

y

deeply different problems

such as

determin

ation of

the

specific phase and group refractive indexes in different

propagating

environments

,

the calculation of the discrete

refractive indexes based on some conditions

,

the

calculation

of

the refractive index at

F

region altitudes

based on the global network of

Super Dual Auroral Radar

Network (SuperDARN)

[17

–

21]

.

In addition, presently

many attempts are devoted

to

researches of

The

Wireless Power Transmission (WPT) problems

using

high

power

microwaves and Laser

power b

eams

. During

propagation of

high power

beams,

the Earth atmosphere

region will be ionized

,

this fact

has

generated

some

research

problems

concerning

the propagating theory

development of EMWs power beams with

Gaussian

energy

distributions, the real

interactions

of High power beams

and

the Earth atmosphere this fact brought about

the

modified concepts of

the relative permittivity,

EMWs

velocit

ies

,

and

refractive indexes

[25

–

32, 39, 40

].

So far, it has a few

systematic

data of the specific phase

and group refractive indexes

vs. altitude of the ionized

region published

in the

L

iterature

[10,

27,

28,

37,

42,

43]

.

In our

previous

published work

[28

, 39

]

,

we have

studied

and

outlined

the relative permittivity and the

numerical

data of the

complex phase refractive index

by altitude

based on the

free electron density (

N

e

) distribution

[38].

In

t

his paper

using

t

he

first

–

order Appleton

–

Hartree

equations

by

pass

ing

the imaginary

parts due

to their values

are

very

small, we

estimated and outlined

the systematic numerical

results

of both

kinds of

the real phase and group refractive

indexes (

n

ph

and

n

gr

) vs. the altitude

concerning the single

and packet EMWs forms propagating

in the ionized regions

from 100 km to 1000 km

depending on the frequency range

of from 8 MHz to 5.8 GHz

.

II.

THE

EXPRESSIONS

OF

R

ELATIVE

PERMITTIVITY

AND

REAL

REFRACTIVE

INDEXES EXPRESS

IONS

FOR THE EARTH’S

IONIZED REGION

II.1.

Briefly on electromagnetic waves propagation in the

ionized region

The features of the ionosphere region

strongly influence

microwaves

propagation.

The

m

echanism of refraction

mainly occurs

in

the following ways

: when

t

he EMW

comes to the ionosphere

region

, the electric field

of EMW

forces the free electrons being in the ionosphere into

oscillation with the same frequency as that of the EMW.

Some of the

radio

–

frequency energy is transferred to this

resonant oscillation, and the oscillating electrons will then

either be lost due to recombination or will re

–

radiate the

original wave energy. The total refraction can occur when

the collision frequency of the

ionosphere is less than the

EMW frequency

,

and the electron density in the ionosphere

is high

enough [9,

14,

15,

25].

When the EMW frequency increases

to

higher

values

,

the

number

of reflection decreases and

then

not the

refraction. So there will be

a

defined limiting frequency

(

so

–

called,

critical frequency or

plasma frequency

) where

the signals could pass throu

gh the ionosphere layer

[9,14,

33]

.

If the propagating EMW’s frequency is higher than the

plasma frequency

of the ionosphere, then the free ele

ctrons

cannot respond fast enough, and they are not able to re

–

radiate the signal. The

expression determin

ing

the critical

frequency

has the form

:

f

critical

=9.

√

N

e

.

Herein,

N

e

[m

–

3

]

is

a

free

electron density

being in

the

ionosphere region

.

If we

do not take into account the number collision of ionized

particles (O, N, H…), then

the

effective permittivity (

ε

eff

)

as a function of

critical frequency or

plasma frequency

(

p

)

and EMW’s frequency (

)

that can be written as the

following form [32,

38,

39]:

ε

eff

=

ε

0

(

1

–

ω

p

2

ω

2

)

(a)

;

ω

p

=

√

N

e

e

2

m

ε

0

(b)

(

1

)

Based on this formula, the plasma frequency (

p

) of the

ionized region has been calculated, and its value is about

8 MHz [32

–

34].

II.2.

The expressions of the complex relative

permittivity

in the ionized region

In the ionized region,

the dielectric permittivity has been

accepted as a complex number.

Various processes are

labeled on the imaginary part: ionic and dipolar

relaxation, atomic and electronic resonances at higher

energies. As the response of the ionized region to external

fields

that

strongly depends on the

EMW

frequency

, the

response must always arise gradually after the applied

field

,

which can be represented by a phase difference

leading to the formation of the imaginary part.

The

complex relative permittivity in the ionized region can be

expre

ssed in the following form [

36,

38

,

39

]:

r

(

) =

1

–

4

π

N

e

.

e

2

ε

o

m

e

1

(

ω

2

+

S

2

)

–

i

4

πσ

ω

=

ε

r

‘

(

)

+ i

ε

r

”

(

)

(2)

σ =

N

e

.

e

2

m

e

S

(

ω

2

+

S

2

)

(3)

Herein,

N

e

is the

free

electron density,

ω

is

the

angular

frequency,

m

e

is electron mass,

o

is the

vacuum dielectric

constant,

σ

is the conductivity, and

S

is the

collision

angular frequency of ionized particles

in the ionized

region. The

휀

푟

′

(

휔

)

and

휀

푟

′′

(

휔

)

are

denoted as

the real part

and imaginary part of the relative permittivity,

respectively.

Based on the graphic curves of free electron

density by altitude in the ionized region

,

the different kind

s

of conductivities and relative permittivity vs. the altitude