Electron Ballistics and Electromagnetic Waves in the Ionosphere

UNITED STATES
NAVAL POSTGRADUATE SCHOOL

ELECTRON BALLISTICS
AND
ELECTROMAGNETIC WAVES IN
THE IONOSPHERE

J. G. Chaney

10 November 1965
TECHNICAL REPORT No. 56

UNITED STATES NAVAL POSTGRADUATE SCHOOL
Monterey, California

Rear Admiral E. J. O'Donnell, USN ,
Superintendent

Dr. R. F. Rinehart
Academic Dean

ABSTRACT:

It is shown that ion collisions introduce a slight reduction in the
plasma frequency, along with an exponential decay of transient
electron oscillations. The critical frequencies for penetration of a
homogeneous ionosphere, for both isotropic and anisotropic iono-
spheres, are determined. The characteristic waves, for electromagnetic
propagation within a homogeneous anisotropic ionosphere, are developed
by considering an infinite series of electron velocities, produced by
Ampere's force law, reacting with the electron oscillation produced by
an exciting electric field. The complex indices of refraction are
determined, both from a dispersion equation and from a derivation of
the Appleton equation, which uses an arbitrary selection of the co-
ordinate axes, thus emphasizing the invariance of the Appleton equation.
Vector and tensor algebra is used throughout the analytical developments

This task was supported by:

Navy Department, Naval Missile Center
Work Request No. 62001

Prepared by: J. G. Chaney

Approved by:

C. H. Rothauge
Chairman, Department of
Electrical Engineering

Released by:

C. E. Menneken
Dean of
Research Administration

U. S. Naval Postgraduate School Technical Report No. 56

10 November 1965

UNCLASSIFIED

Table of Contents

Page

1. Plasma Frequency 1

1. 1 Low less case

1. 2 Lossy case

2. Homogeneous Isotropic Ionosphere 2
2 .1 Low loss case

2 . 2 Lossy case

2.21 Critical frequencies

3. Homogeneous Anisotropic Ionosphere 6

3.0 Direction cosines

3.1 Electron current density

3.2 Path of an electron

3.3 The curl H equation

3.31 Cartesian coordinates

3.32 Rotating coordinates

3.321 The complex dielectric tensor

3.322 The complex indices of refraction

3.323 Physical interpretation

3.4 Azimuth component of exciting field

3.5 Horizontally polarized exciting field

4. Dispersion Equation 21

4.1 Maxwell's equations in complex form
4.11 Orientation of field vectors

4.2 The dispersion equation

4 . 21 Special cases

5. Appleton Equation for Arbitrarily Oriented Geodesic Field

5.1 The Appleton equation

5.2 Lorentz conductivity tensor for an exciting wave

5.3 Resistivity tensor for wave in ionosphere

5 . 4 Index of refraction

6. Faraday Rotation 31

6.1 Waves through the ionosphere

6.2 Polarization

7. Conclusion 34

7.1 Equivalence of two points of view

7.2 Inhomogeneous anisotropic ionosphere

ii

Symbols

E 0 = Vector magnitude of incident field

B 0 = Geodesic field vector

)c0= o)/c = Free space wave number

H ' = Relative permeability constant

c '= Complex relative dielectric constant

e '= Complex relative dielectric tensor

n = Complex vector index of refraction

y- ) k0 n = Complex vector propagation constant

-
t 4>= Angle between yand B0

S = Cos-1 aQ . az

X , y, Z = Cartesian coordinates

r, d,ip= Spherical coordinates

£= Small displacement along X-axis

F = Force vector

e = Positive magnitude of electron charge

m = Mass of electron

- e/m = -1.77 x 1011 coul/Kg

v= Average number of electron-ion collisions per second

r = Radius vector

V = Vector velocity of an electron

N = Density of electrons per cubic meter

_7
ju0 = 4 7T x 10 h/m = Permeability of free space

1 -9

c0 « 1 x 10 ' fd/m = Permitivity of free space

3 6rr
a? = / Ne * = Resonant angular frequency, lossless plasma
P m Co

OJ = eB0 = Cyclotron, or gyro, frequency
m

E, H = Electromagnetic field components

iii

Symbols (Continued)

D = Electric flux density vector

i = Current density in amps, per sq. m.

p = Surface charge density in coul. per sq. m.

p = Charge density in coul. per cubic m.

X = wp2/ co2

Y=ub/u=JY2 +y2 +y2

x y z

Y = Y cos 6, Y = Y sin 9 cos <p, Y = Y = Y sin Q sin ^= Y cos $

YT=<V+ YZ =K/ Y2 - Y2 =Ysin0

X y L

r = l-j v/w

M2 = n2 - 1 = n . n -1

a . = Unit vectors

i

A = a x a = a, sin f
1 6 z 1 ^

A2 = (a0x aj az=a2 sin £

x = i - r x

2 2

r - y

Y = X Y

X

r2 -x2
Z= 1 - x_

r

E =^2jr2 ' , <a =tan~lE-
Xy E + E vy "T

x y E

co = Critical frequency
c

to = Ordinary wave critical frequency

Co

CO , = Left circularly polarized wave critical frequency
co D - Right circularly polarized wave critical frequency

CK

a = Lorentz conductivity tensor
p = Lorentz resistivity tensor

IV

Symbols (Continued)

I = Identity tensor

s = Length of path through the ionosphere

o

c pa 3 x 10 m/sec = Speed of light in a vacuum

V = Group speed

g

V = Phase speed
P

a = Attenuation constant

jS = Phase constant

90 = Angle of incidence

6 = Angle of refraction

R = Measure of polarization

n.. = n, n. = Components of refractive tensor
ij i j

v

1. PLASMA FREQUENCY
1. 1 Lossless Case

Consider a lossless plasma having a density N (^*f c )

IWS

D

mv

~Ps~~^ei of free electrons. If the electrons are compressed

.X laterally through a small displacement, £ , along the

X=0 y = i

X-axis, a surface charge density

Figure 1 ^T ...

"Ps = "Ne £ (1)

may be considered to exist at y = £ . Then, from the definition of the

electric displacement vector, from y= o to y= £ ,

Dx=fv <2>

Hence, D = c0 E = Ne £ , (3)

or the force F is given by,

X F = -eE = - Ne' fc (4)

X X ~ —

Since force is given by the product of mass with acceleration,

I

1

2
m d ^ + Ne2 4=0. (5)

dt

2 2

Substituting a) = Ne into (5), the harmonic equation for free oscill-

m e 0
ations of the electrons is obtained, namely,

L- + ^p C- 0. (6)

The solution of (6) is sinusoidal with an angular frequency a) . Thus co

is called the resonant angular frequency for the lossless plasma.

1. 2 Lossy Case

Suppose there are an average number, v , of collisions per second
between electrons and positive ions. This introduces an average loss of
momentum, or an equivalent fluid resistance term, into equation (5), which

equation becomes,

2 2

m d £ + vm d 4 + Ne t - g. (7)

dt2 dt €0

Pt
Dividing by m, and using the operational method of substituting £=£0e

into equation (7), the determinantal equation is obtained,

(p2 + UP+ co ) 4= 0. (8)

Equation (8) yields the characteristic roots, .

p = -y/2 ± j a) V 1 - ( ^/(2 co^ (9)

From equation (9), it may be concluded that ionic collisions in the
plasma introduce a time decay constant of 2/v, and reduce the plasma
frequency by a factor V ■, / UAn .\2 . For a low loss plasma, the
latter factor usually is ignored.

2. HOMOGENEOUS ISOTROPIC IONOSPHERE
2.1 Low Loss Cas_e

Consider a small collision frequency for a homo-

geneous isotropic ionosphere of electron density
N(^L) . Let a uniform plane wave front, E0ge
be incident upon the ionosphere at an angle of

Figure 2 incidence 90. If r is a radius vector describing the

path of motion of an electron having a velocity V, the force F upon an
electron is given by,

or

F =

^ j2- . -
-eE = m d r +ymdr

at2 dt

d V

+ v V = - e E .

dt

m

Substituting V e into (11) and solving for the steady state velocity,

V=-eE = -eE

(10)
(ID

(12)

m (jco+ v) jm( to -j v)
The equivalent current density due to the motion of electrons then becomes,

7=pV = -NeV =N£Ll— . (13)

m [i>+ jco)

Equations (12) and (13) show that the electron path, velocity, and current

density are linear in the direction of E.

In order to see the effects upon the electromagnetic wave, Maxwell's
equations in the steady state phasor form will be used.

The Maxwell's equations are, div B = 0, div D = 0, curl E = - _d b '

3t
curl H = 7 + 5D , B = jli' ju 0 H , 5 = e' c0 E . (14)

dt
The corresponding complex form is,

v. H=0, v.E = 0, vxE = -jco/i0H,
V x H = i + j a) € 0 E .

Substituting from equation (13),

2 -

(15)

vx H = Ne E + jco c0E = jcoe0 *
m {v + j co)

[1 + p? -—] E ,

(16)

or,

Jcoe0 m(y + Jco)

VX H = JCO €0[ 1 +

C0p2 (l/- jco)

iuiv2 + co2 )

E =

JCO £,

1-J* )-,^2

i/ + co2 tod^ + CO2)

(17)

2 2

in which co = Ne was also substituted in the right member of equation

P m c 0
(17).

obtained,

From equation (17), an equivalent complex dielectric constant c is

c

COr

< ^£ '"j e" = (1-

c 2^ 2

CO + y

J -j JLHL

(18)

2 2
coz+ vl

Returning to euqations (15), the wave equation will be determined

by elimination.

2 - 2 -

Vx ( vx E) sv( V . E) - V E = -V E = -jc0jUo*

V x H .
Using y = a+ j 0 = jk0 V e ' , and k0 = coV jli0 e0 = co/c , upon

substituting for vx H, equation (19) becomes,

(19)

For v f»o, n =N

V2 E + k? € ' E = 0.

will be evaluated.
1/2

(2 0)

n=(€Hc')1/Z=V-

,/ 1/2
( 1-J _J_)

e'

2 e'

-j

2/7"

(21)

and hence,

y = j knn » k* g +i k^AT" .

From equation (18) with i/p»0,

y«s y k0

2777

_ + j k0<yi - ( Wp) 2

Vl- ( ^P ) 2 CO

CO

The phase velocity is,

V = to = co/ k„
P

^ V 1- ( Wp)2 -/l- ( cop)2

CO

CO

and the group velocity is,

V = _d_co__ = c yi- (cop )2

(22)

(23)

(24)

(25)

CO

2 .2 Lossy Case

For the lossy case, the exact value of n = JJ7" should be used. For

-^-"^ > -1

this purpose, consider ^ ± . ^ , j^>i>-^[/- , B = tan b/a .

^T7r=1/l2T7e±iB/2

cos B/2

= M + cos B = 1 /

1 + 7-»

JTNL+J^n

sin B/2 =7r ^=2 .

b + a

Substituting from equation (18) into (2 6),

2

n=

1

(1

.<*

CO

2 + ^

2 cop4^2
) +

CO

+ (1-

2 + 2
CO V

)L I/2 -J '

7T.

Mi- "p2

1/2

co„ c

(1- _£

co2 + ,2

2/ 2^ 2,2
co (co + v )

2 co 4 v2

) + P

(26)
1/2

1/2

co2( co2+ i^2)2

(27)

2 2

co + y

Thus, y = k0:Jm [n]and 0 = k0Re [n]. (28)

The Re [ n] is also spoken of as the refractive index for the ordinary wave.

2. 21 Critical Frequency

From equation (28), the phase velocity within the ionosphere is,

1

V =

i — o

jLioe'e 0 Re [n]

Designating the angle of refraction by 0 , from Snell's law,

sin 8 = sin v °

r Re [n]

(2 9)

(3 0)

If 8r < tr/2, the wave enters the ionosphere, is refracted, and passes
through the ionosphere, unless the right member of (3 0) is greater than
unity.

Figure 3

If the right member of (3 0) is greater than unity, no real angle 0 exists and

the wave is reflected. Designating the critical angular frequency by to ,

this frequency is determined by 6 = tt/2. Thus, for any angle of incidence,

r

the critical frequency is determined from,

s in Q0 = Re I" n ] .
For the low loss case, from equations (23) and (31),

s:

:in * = V 1- ( _^J2 ,

W_

or

co = to sec 9 o
c p °

Equation (33) is known as the secant law, and yields to = to for

c p

normal incidence. Substituting the numerical values of e, m, and c0 into
(33),

(31)
(32)
(33)

f = 94~N~sec 9
c

o '

(34)

in which f is in cycles per second. If the electron density is given per
c

cubic centimeter, the f will be in kilocycles per second.

c

At a given angle of incidence, frequencies f > f will pass through

the ionosphere, and frequencies f < f will be reflected.

c

The preceding analysis has ignored the earth's geodesic field. All
waves acting accordingly are termed ordinary waves.

3. HOMOGENEOUS ANISOTROPIC IONOSPHERE

3 . 0 Direction Cosines

In order to take into consideration the effects of the geodesic field

upon an electromagnetic wave propagating in the ionosphere, various

vector directions must be taken into consideration. Hence, a table of

direction cosines is necessary. Such a table may be formulated either

by projections or by spherical trigonometry, or by a combination of

both methods. Referring to the figure:

cos ij) = cos Qcost/2 + sin 9 x
sin t/2 cos ( tt/2 -<p)

cos \p= sin 0 sin <p

a . a = cos j/f sin 9 sin tp

X-* * - a , . a = sin \b

* J f

- a , . a = -cos ib sin tf

4> _y

•* - a , .a = -cos ibcos Q'

a, ij) x

cos 9= cos 0cos tt/2 + sin ip x
sin tt/2 cos 0/

cos 9' = cos Q

sin j/j

Figure 4

sm e'=h- cos2 9 =

' sin 0

Vl-sin2Q sin 2(p-cos^ q =
' sin2 0

sin 9cos cp

sin ip

sin 0= Vl- sin2 flsin <p

From the above, the following table may be compiled:

a
X

a
y

a

z

"a /

0

ar

cos ©

sin $ cos (p

sin 6 sin cp

0

a6

-sin 6

cos 9cos (p

cos 0 sin (p

cos Ssin c/D

sin ij)

a

0

-sin <p

cos <p

cos cp

sin ib

-sin 9cos
sin ib

6s in (p

-sin^ 0sinocos cp

sin ^

1

sin 0

(35)

3 . 1 Electron Current Density

Assume the Z-axis of a coordinate system to lie along the geodesic
field in the ionosphere, and assume an electromagnetic field, E = E0 a« ,
to be incident at angle 90 upon the ionosphere. Equation (13) gives the
current density, due to the presence of the electromagnetic field along
with the free electrons. However, it ignores the presence of the geodesic
field, the geodesic field being B0 a

The zero order component, i0 of equation (13), reacts with B to
produce a new component l. , of the current density, normal to both
i0 and B . This component lies within the XY - plane. The i , component,
in turn, produces another component, u , normal to both l and B , and

hence also lies within the XY - plane, but it is rotated it/2 from (,,. The
new component, i , produces still another component, u , normal to
I within the XY - plane. Since 1 has been rotated if from f , it is in

the same direction as ti / but of the opposite sense. Also, the component
I , produced by i , and is also of the opposite sense. Continuing, an
infinite alternating series is obtained for the two directions lying within
the XY - plane. These series will converge under certain conditions.
They will be obtained analytically in the following analysis.

In the presence of the geodesic field, the force equation (11), for each
electron, must be modified in accordance with ampere's force law to
become,

m

d V
dt

+ ymV+eVxB = -eE .

(3 6)

Equation (36) is to be solved by iteration. That is, the solution V0 of
equation (12) will be substituted into e V x B and a correction term V ,
obtained. The V term will then be substituted into e V x B to obtain another
correction term V , etc. Before substitution, the successive cross products
will be formulated. From table (35),

Figure 5

But from table (3 5),

Hence ,
and

aj-a sin 8+ a cos 0cos <o+ a cos 0

e x y z

= cos

-1

sin <p, and

A = a xa = a, sin f , F
1 8 z 1 v. 'i*

(V*z> '

A, =

x y z

- sin 0 cos 9 cos ip cos 0 sin ip
0 0 1

a cos Q cos en + a sin 0

x y

(37)

(38)

A

( a x a )xa = a _ s in T =

O -7 "7 / J

2 v "9" " z' " "z

a a a

X Y z

cos 8 cos cpsin 0 0

0 0 1

cos 0 cos <p

= a sin 0 -a

y

(3 9)

E = -E0 sin 8 , E = EQ cos 0cos cp, E =

x y z

E0 cos 0 sin cp .

A = a E - a E

i x y y X

A_ = - (a E + a E ).

2 x x y y

(40)
(41)

(42)

From equation (38) and (39), it may be seen that both A and A lie
within the XY - plane, and that their slopes are negative reciprocals. That

is to say, A is rotated ff/2 from A.. Furthermore, since each unit vector

a. , 1 = 1, 2 , , is within the XY - plane , each a . . = a . x a also is

I 1 + 1 1 z

within the XY - plane, and a. . is in space quadrature with a..

i+I 1

Incidentally, the unit vectors a , a and a are also coplanar, with

Z Z a

a9 at an angle 7/2 + £with a .

Z Q

For the iteration, from equations (12) and (10),

V = ~ eE° g A . , F =. e V0 x B = jm (co -Ji|V.. (43)

j m ( co- jv) 1 z *

Thus, upon defining the gyro frequency,

aD = 6Bz (44)

m

V = e2 goe x Bz = -e E0 cob Aj =

[jm ( u)-jy) J m ( to - jv)

-e cobE0sinC ^ # (45)

m ( co- j y)2

Repeating the procedure for

F =-eV.xB = j m (co- \v ) V. , (46)

2 1 z 2

j m ( cd--jy)3

2
-j e co, E0 sin £

m

2 ( co - j^)3

For

a2

(47)

' to- w) v ,

(48)

"a3 '*3=*V

(49)

V3= e cob3 E0sin £
m ( co- j^)4

C°ntinUing' V =Jecob4E0sinC - - _ - f5Q)

4 b a a --a2, (50)

m ( co - )v)°

Vc = -e co, E0 sin f ,cn

5 b ° > a_ , a =a. , (51)

m ( co- jlO b

— fi

V_= - j e co, E0 sin f - - - . .

6 b ° * ac , a = a0 . (52)

j b b z

m ( CO- }V)
The above procedure may be continued indefinitely.

Upon summing the corrective components (43) through (52) ad infinitum,

factoring - j we0, and substituting co , the velocity vector for an electron

becomes ,

V = + J CO€0E0

co.

+ J a

Ne
2 5

co ( co- j 0) '

CO2 CO 6

_P b_

co ( co - )v)7

B co ( co- \v)

co 2 co, to 2 CO3

p "b + P d

sin £ - a,

^co( co- jv)2 co( co- jv)
2

p b

CO2 CO 2

+

2 4
^p wb

+

CO

( co- )v) 3 CO ( co - ji/) 5

or,

V = + j CO € nE
Ne

sin

CO

(53)

CO l co

a0 p_

+ j a,

CO ( CO- jl/)'

to 4
+ b +

,4

( CO - jv)

CO 4

b

CO ( CO- j!/)

sin f - a_ % "b*

CO

1 +

( co - j v)'

co( co - j^)3

( co- jvY

sin

1+ Wb

+

( co - j v)

(54)

The infinite series within the brackets of equation (54) converges for

cob2

2 2

CO + v

< 1 ,

that is, for

co > cob N 1- ( i/ cob)

(55)

(56)

Hence, for the frequency range of convergence, the brackets become,

2 ■ 1 2

1 ^ ^ =( co- jv)

2

1+ CO,

( CO- j v)2

J

1 - cou2/ ( co- jv) (co- jf) - CO

(57)

and hence,

CO

^ 2

V= + j cocnEn J a __p_
Ne

C0i_ CO sin £

+ b P ^ y

~V~_ ... 2 ~

CO(cO-jy) C0[( CO- jv)Z - ttj^]

jara2

CO

co - JV

(58)

10

3.2 Path of An Electron

For an examination of the path followed by an electron, it is preferable
to eliminate to ^ in equation (58), and integrate in time to obtain a radius
vector, r , whose terminus describes the path. Accordingly,

_1 + ^b /l- cos_2 9 sin Z\p

co( co- iv) 2 2 -

CO [ ( 05- J V) " COb ]

jai"a2

*b

to- j V

in which, from equation (37), sin r has been substituted in the form

(59)

= A^

sin £ = J 1 - cos 9 sin tp . (60)

Assuming v % 0, equation (59) shows that as the electron attempts to
vibrate in a path parallel with the exciting field, it has an additional
elliptical component within a plane normal to the geodesic field, and
resonates at to = to . For frequencies to< to^ , the ionosphere acts as
a conducting medium, for the current density produced by the magnetic
field, rather than as a dielectric medium.

The electron path, as it spirals about the direction of the exciting
electric field, is somewhat like a trochoidal epicycloid. The electron
current density, i = - NeV, may be considered as a source of radiation,
a vector potential formulated, and the resulting fields computed there-
from in accordance with Huygen's principle. Thus, the electric fields
are no longer confined to the path of the driving field. It will be shown
later that D, H, and y constitute an orthogonal system, with E not
necessarily orthogonal to y .

3.3 The Curl H Equation

The current density will be formulated from equation (58), by mul-
tiplication with - Ne, and substituted into curl H in order to formulate
the dielectric tensor. The dielectric tensor subsequently will be used
to determine the indices of refraction. The resulting curl equation is,

11

V xH = j 0)c0EJa

9

1- P

60 (tO - )V)

s ^b2 V sin^

- ja.

co, co sin £

b p *

2 2

co [ (co- j v ) - COb ]

2 2

co(to- jy)[ ( co- iv) - ca ]

(61)

The wave corresponding to the first term of the right member of equation
(61) is sometimes called the ordinary wave, and the wave corresponding to
the second and third terms is companionably called the extraordinary wave.

3.31 Cartesian Coordinates

Using E= a E + a E +a E along with equations (41) and (42),

0xxyvzz

equation (61) may be re-written,
1

v x H = a J E
j coe„ Xl X

x

i CO

1 - ~p_

CO ( co- iv)

-JE ^b "p

y —

CO

(co- iv) -cob

2 2

co( co - iv) [ ( co- iv) - cob .

JE "b "p2
X

- E

2 2

co[(co- iv) - cob ]

co (co - ji^) [ (co- iv) - cob ]

+ a E
z z

1-

w.

co (co- ]v)

Now let

X = 1 -

co.

2 2-

CO, CO

b P

2 2

co(co- )v) co(co - iv) [ ( co- iv) - cob ]

1 -

CO 2 (co - jl^ )

co [(to- iv)z-co*J >

Y- = "b "p

u>[( co- iv)1 - cob2]

, z =1 -

CO

co (co - jy )

and substitute into (62) to obtain,

(62)

(63)
(64)

— V x H = a (EX-jE Y) + a (E X+jE Y ) + a E Z .

j coc0 xx y yy x zz

(65)

12

For cross reference to literature employing the standard URSI symbols,
X=oj2 Y=^h r= 1- j W CO , (66)

0)Z CO

the symbols in (63) and (64) are equivalent to,

X = 1- X r , Y = X Y , Z = 1 - X . (67)

2 2 2 2

r - y r - y r

The symbols (67) will be used later, but for the present it is more conven-
ient to retain the symbols (63) and (64).

3.32 Rotating Coordinates

The a and a components of equation (65) will be broken into left
X Y

and right hand circular components. Re-arranging equation (65),

1 v x H = a (E X-jE Y ) + j a (EY -jE X ) + a E Z ,

T^7I x x y y x y z z (68)

or,

J V x H = E ( a X + j a Y ) - j E (a Y+ja X). (69)

ja)€o xx v y x y

Now let,

a X + ja Y= A, ( a +ja ) + A. ( a -ja) , (70)

x y i x y 2 x y

and solve for the undetermined constants A. and A ,

~ Y + V

A, + A. = X A, =

1 2 1 2

A -A = Y A_ = X - Y

12 2 ?

(71)

also let,

a Y+ja X = B.(a + j a ) + B0 ( a -ja ), (72)

x y i x y 2 x y

and obtain,

(73)

Bl+B2=? B1 = 2L^=A1

BrB2 = * B2=^- = -A2

Substituting from (71) and (73) into equation (69),
± V x H = (E - j E ) 2Li_JL (a + j a ) + (E +jE)x

j o3e0 x y 2 x y x y

X - Y (a -ja ) + jt E Z. (74)

r- x y z z

1 13

From equations (63) and (64), 9

•= a?p2 (to- w)t g?b cop2 x _ ^ C(m- jy) =f a*b]

u>[ ( w- jv) - wb2]

X ±Y

1 - X

r ± y

upon letting ,
E

W[(W- jl>) + 0)b][(w- jl/) - 03b]

(75)

= 1/2 VE2 + E 2

xv x y

-1 E
= tan _y_

Xy E

(76)

equation (74) may be written,

v" x H = j to €

: e " J 0XY (I" — ) (a + j a ) + E

xy r + Y X y

i-a>

xy

XY

( 1 - X ) (a -ja))+ a E ( 1 - X_ )

jzy xv-- r

(77)

3.321 The Complex Dielectric Tensor

Designating the left hand rotating unit vector by a and the right hand

J-j

vector by a , that is ,
R

aT = a + j a

l x y

a = a - j a
R X Y

(78)

and letting ,

6/ =1 - x , ei = 1- x

L r+ y r - y

1 - x

(79)

equation (77) becomes,
V x H = j (ji c0

0 £ 0

0 0

- J

E e

xy

E e

xy

xy

xy

■" -1

a

L

•

*R

a
z

L^> s-J

= JtoD. (80)

3.322 The Complex Indices of Refraction

Equation (80) yields three characteristic waves propagating in a
homogeneous anisotropic ionosphere. There are two circularly polarized
waves within the XY - plane rotating in opposite senses, and a linear
wave parallel to the Z-axis. These waves have three distinct indices
of refraction.

14

Rationalizing the denominators of equation (79),

0>z + v

- J

CO

to (to+ cob)

to(co2 + i^2)^

2 2
to[(co + CO J + V ]

1 - ^p2 (a) - "b}

J

-

- )

- )

V ^P

to[(co + cob) + i/2]
2

1.

i; 0)

to [(to- wb)2 + v1

2 2
to [ (to - u>b) + V J

(81)
(82)

(83)

The corresponding indices of refraction will be obtained by extracting the
square roots of equations (81), (82) and (83). The ordinary index of refrac-
tion, n0 , is identical with that of equation (27). Upon applying equations
(2 6) to equations (81) and (82), the left and right hand indices become,

/ / - CO C 1 03 ± 03, '

n

L
R

= 1

V2

2 2

CO V

p

%2 {Ui± "b} ) J I ! . < (W ±Wb)'

? 2

to[(to ± wbr + v ]

/A - "p2 (m

VL w [(to ±

1/2'

to[ (to ic^)2 + v2~\

2 / 2

. w (to ±to,
1 - p b

2 2

CO [(tO ± tOu) + V ]

1/2

CO

p

cob)2 + y1
\_ top2 (to ± tob)

I tor(to±cob)+^]
2\l/f

t2 2
co[ (to ^i^b) + v 1

It should be kept in mind that equation (80) is normalized, and hence
indices (84) apply only for propagation of these characteristic waves. For
wave combinations other than the characteristic waves, there will be
coupling elements within the dielectric tensor, and a dispersion equation
will be required for determining the indices. Which of these waves appear
depends upon the direction of incidence as well as upon the frequency.

3.323 Physical Interpretation

The critical angles of reflection, corresponding to the real parts of
the indices of refraction in equation (84), may be determined by equation
(30). This will be done for a low loss ionosphere, v pa 0, and a physical
interpretation formulated.

15

Setting v = 0 in (84) and squaring, is equivalent to setting y = 0 in
equations (82) and (83). Hence, for applying equation (30), set

1 - E.

tO ( tO ± tO^)

sin 80 /

or, 2 ? 2

to ± 0!b 05 - Wp sec a0= 0 .

(85)

(86)

Taking the positive sign in equation (86) for to T /

w t = " ^ + V ( u>n sec 80)2 + ( co,
cL P

k 2

W

(87)

and taking the negative sign for to

cR '

tocR = + _^b_ + V (top sec a;)2 + (Ub_f .

(88)

Since to determined by equation (81) is identical with to determined
by equation (33) ,

Wc = Wc = Wp S6C 9° '

while,

/ 2

cL C

w

to

and f 2 . ,2

W_r, = ^ OJ + \ Wb ] tO^

cR

V

2 2

For frequencies sufficiently high such that (toc) >> (to^^'

Uo

[2 to)

v c

1/2 ■ -o

2
8^2

*— —J

"■

Hence,

WcL

* toc ,

rw

8 to

(89)

= to + Hs + . (90)

C 8 to

(91)

and,

to

cR

tOh2 ui, toK

to + P + D ?« to + b

8 to 2

c

(92)

Since the three characteristic waves have distinct indices of refraction,
they propagate within the ionosphere along different ray paths, and have

16

different attenuations. The attenuations spoken of above are those due
to $m [n] . There are other attenuations due to numerous anamolies of
the ionosphere. Because of the different critical frequencies, the right
hand wave may be reflected at a frequency for which the other waves
pass through the ionosphere.

Recalling that the series in equation (54) diverges for frequencies
less than the cyclotron frequency (or gyro frequency), it may be con-
cluded that for to < to,

the extraordinary wave confronts a conducting boundary and is reflected

as in path c (Fig. 6). For frequencies such that to, < CO < CO / the
— be

wave enters the ionosphere, suffers a refraction, and is refracted back

to the earth, as in path b (Fig. 6). On the other hand, for frequencies

CO > CO D / the wave is slightly refracted but passes through the
cR

ionosphere as in path a (Fig. 6). The three waves do not necessarily
all exist simultaneously, depending upon the angle of the ray path
with the geodesic field.

3 .4 Azimuth Component of Exciting Field

In the preceding analysis, the exciting field was considered
vertically polarized. However, a wave incident upon the ionosphere
from a horizontally polarized antenna will have a horizontal component
in addition to a vertically polarized component. Hence, it becomes
necessary to consider the azimuth component of an exciting field.

Re-writing equation (3 6) for convenience,

F = -e E = m d V + ymV+eVxB , i = -NeV ,
"dT"
and considering the exciting field in the form,

(36)

17

E = En (- a sincp+a cos m) ,
cp X z

the iteration follows analogously to that for E . Accordingly,

8

following steps (43), etc.,
V, = - E. e a <p
j m (oj- jv)

F. = - eV0 x B = j m (oj - ]v) V
1 z 1

V =e2 Ena -x B

1

<P_

[j m(oj- ji/)]

_ eE0av (-a sin <p + a cos o) x a .

m

(tO " j^)2

e E0 to, sin (p a

m (« - j^)2
2

o E0 B to, sin <P a xa < r. 2

°„ z b ; y z = -jeE0 tob sin <pay

e E0 B to

0 z .

j mz ( to - j v)6

m

( co - jv)3

V = j e2 E°Bz a5b2 sin ^a yx az = eE0 tob3 sin <p a

V

j m2 (to - iv )

m ( to - jv)4

-e2 E0 Bz tob3 sin pax ag = -j e E0 tob sin cpay

) m

T( to- ji/)5

m (to - ]v)b

(93)

(94)

(95)

(96)
(97)

(98)

Substituting the above into curl H , using i = -NeV , and
considering E0 a = a_ E_ + a_ E_ , curl H =

o

y y

z z

J co e(

w.

E°%[1-

, 2

co (to - jl^)

] - ja E.

03

x y "

OJ

[(to- M2

(to - J^)4 (co - jv )6

CO,

+ wb

a1

+

a1

(co - jt^)3

(99)

(to - jv)5 (co - \vjf
Summing the series over the range of frequencies for which the series
converges ,

curl H = j to e0 «

1
L_ * bZ
(co - ]v)

J

r

« a En
cp °

L

[1--P2 1

( 2
-ja E ap

%

a< to- jv)

x y co(to-

1

)z

-5 E *p wb
y y 'a- (u-ji/)3

2

1- Wb

^ (co- \v)L

/

(100)

18

or,

a E0 [l-_£

05,

] - jE aP "b

y a<[ (w- jv)2 - o\z ]

[a - j a b

Substituting for a E0 of equation (93), and using E and E ,

flOl)

curl H = j o.< c,

E o> ui,

"J a Y P b

os.

2 2

P b

05 [( 05 ~ jl/) - 0), ]

+ a E J \- P

7 Y / 05 (0>- ]V)

05 (05 ~ jl^) £( 05- )V) " 05bZ ]

+ a E ri-
z z u

05 (oj - )v)'

(102)

Let

Substituting from equations (63) and (64),

curl H=jo>e0/-jE (a Y+ja X ) + a E Z

] y x y z z

a Y+ja X = A. ( a +ja ) + A_ ( a -ja ),

x y i x y 2 x y

(103)
(104)

and determine the arbitrary coefficients A. and A ,

A1 + A2=Y
Aj-A2=X ,
Thus, (104) becomes,

1

A = 1/2 (Y + X)
A2 = 1/2 (Y - X)

(105)

curl H = - j E (X+Y)(a+ja)+jE (X - ?) x
J O!€0 _L X y _£

2 2

(a - j a ) + a E Z .

x v z z

(106)

Changing to the URSI symbols by substituting from equations (67), (75)
and (78),

curl H = j oJCoi ~ J E„ (1- X ) a + j E^ (1- X ) a +

-f T + Y L -f r-Y R

E (1- X ) a

z r 2

Using equations (79) ,_

durl H = j oj c,

0

0

-i E/2

^5l"

0

€R

0

•

j E/2

•

SR

0

0

1
Co

Ez

a
z_

= j 05 D .

(107)

£.08)

19

Hence, the same characteristic waves appear as in the case of the
vertically polarized exciting wave. Which of these waves appear
depends upon the angle of incidence and direction of incidence/
as well as upon the frequency.

3 . 5 Horizontally polarized exciting field

Suppose an exciting electric field is incident upon the ionosphere
from a horizontally polarized antenna, the incident field being in the

form ,

- _ j wt-y. r

E " ~_E°_ e a0 , (109)

r

- a , being shown in figure 4. It may be seen that - a . is composed of

two orthogonal components in the a and a directions.

If, in equations (40), E0 is replaced with Eofl , and if in equation

(93), E0 is replaced with E0 , then from table (35), E0of equation (109)

<P

(110)

cos cp

may be expressed by

/

^E0

e c

e

+

a E

with,

= E0

cos

e

sin (0

'EV

Jl-

sin^

0 sin <p

E =

Also, in
J E

equations

2 +E 2
X

(40)

= E,

<

v 1- cos

2 esin2

(p

and,

E

= y

E
X

tan

0

xy

- cot e

cos <p .

vl- sin2 9 sin2 <p . (Ill)

(112)
(113)

Therefore, for the horizontally polarized exciting electric field vector,

equations (77) and (107) may be combined intflt .

curl H = j o)C0 I 1/2 [e,.' %Y~- *^ a + e xy e^ aR ] / 1- sin Bcos cp
+ a Co cos 8 sin <p y cos 8 sin o E0

♦ mX\ 1/2 £cL'5 - ( ' ; i sin „+ i f;cos J ?™mT2-

[_ _J -7 1- sin 8 sin^fp

To determine the critical frequencies involved in equation (114), it would

20

seem preferable to consider individually the various normalized components.

4. THE DISPERSION EQUATION
4. 1 Maxwell's Equations in Complex Form
The instantaneous Maxwell equations,

div B = 0, div D= p , curl E = - 5B , curl H = d 5

St Bt'

I = cj E = p V =- NeV , 5 = e ' e 0 E , B =(j'(i0H ,

at at (1I5>

are possibly more useful for steady state time harmonic cases when con-
sidered in the form,

div B = 0,div 5=0, curl E = - j w/iH, curl H = 1+ j w(E . (116)

To show that div D = 0, first formulate the equation for the continuity
of charge by taking div curl H ,

div curl H - 0 = div j, + a_ div D = div i + 9 p ... .

at at ' U '

(118)

(119)
+ 2_ p= 0 (120)

at c

The solution of (120) may be written

p=p0e'lf/^t . (121)

From equation (121), a charge density within a lossy medium must vanish
with time, and hence must vanish within a steady state condition. Of
course, since a dielectric is considered to be free of charges, div D also
vanishes within a dielectric.

Now consider the case of a spherical wave front given by,

- - jwt-y.f (122)

E = E0(a, (p) e,

r

which is sufficiently remote from the source for,

or

div l = - 3p

a t .

But, for conductors,

div i = crdiv E = cr
or, . f

div D

=_cr p

3 E

0

o , a e0 « o ,

ae ap

21

to hold locally. Also assume y = a y . Then,
div (E

'o e

> = -J— ^-2-

= a . E,

- --y ■ r( = - v E.

i_ y . i. e

r^ sin 9/ Br

- yr e ' + e

-y. r

EA r sin 8 - y . r -
e . a

y

- y . r

(123)

But, the wave is assumed to be a radiated wave such that the inverse
square terms vanish. Hence, in practice, for a radiated wave,

div

E0 (e , <p) -y . r

e

y. L ( 9 , (g) e "y ' r
r

Likewise, consider

curl (J. e "y ' F) = A.

r sin 9

r
^3r

r a

9
d/39

XJT-y. r rE

o9 e- y . r

(124)

r sin 9 a

a/a<p

o

- y . r

r sin 8 r e

- - ^O

cos 0 - y. r r

E0cp e - aft r sin 6 , ->

9 E (-ye ) +

r sin 9 ^

= a rz sin 9
r

-y . r

<P

E°8 ( -ye

) .

(125)

Again, assuming a radiated field for which the inverse square terms^vanish,

e cp

*) =ar 0 -aQ (-y E0 ^ fi" "V • r)

- y . r ) .

(126)

But,

yxE0 =

y
e.

o

°9 E°

o

a 0- a v E0 + a y E
r 9 ° o <p '

(127)

22

Therefore, in practice,

curl

[^

• <o)

-y . r

= - y x E0 (9 , (p)

r

-y . r

(128)
(129)

Also, in case the relative constant is a tensor, e', then

D = € J ' . E .

Therefore, the complex form of Maxwell's equations for radiated fields
may be written,

y . H= 0, y . eoc'. E = ,0, I = - NeV, y xS ^ j u>jl0 H, -yxH =

-NeV + j o)€0 E = j OJ e07 ' . E , (130)

in which, c 'is defined by

1 ' . E = - NeV/ joJCo + E , (131)

The wave equation becomes,

yx(yxE)-k0"i'. 2 = 0.

4.11 Orientation of field Vectors

(132)

From equations (129) and (130), the relative orientation of the field
vectors may be obtained. For this purpose, consider

y . 5 = 0,"y . H = 0 , (133)

from which it may be concluded that y is normal to both D and H.
Then consider,

-yxH = +jo)D, (134)

from which it appears that D is normal to both y and H. Therefore, the
three vectors D, H, and y constitute an orthogonal triple. Finally
consider

y x E = j a) jLl0 H , (135)

from which it is seen that H is also normal to the plane containing yand E.

Assuming y to bqat an arbitrary angle i/j with the geodesic field B0 ,
the relative orientation of the unit vectors is then illustrated in figure 7.

Figure 7

23

Thus, the TEM incident wave becomes a complex TM wave within the
ionosphere.

4. 2 The Dispersion Equation

For y = j k0 n , the wave equation reduces to,

nx(nxE)+?.E=0. (136)

upon expanding the cross products and providing for all components of the
dielectric tensor, equation (133) may be expressed as,

, 2 2.

-(n + n )

y z

n n
y X

n n
z X

n n
X Y

n n
X z

-(n2 + n2) n n

n n
z y

y

-(n + n )

x y

y

t

xx xy

y x yy

€
Z X

zy

X*

yz

zz

y

= o

(13 7)

Combining the tensors,

€ ' -n 2 -n 2 ^ +nn

xx y z \y x y

2,

e +n n

y y x

e' + n n
z X z x

e' -n2 -n2

yy x z

c' + n n
zy z y

c + n n
Xz X z

€7 + n n
yz y z

c> -n 2 -n

zz

y

= o

(138)

Equations (13 6) constitute three homogeneous linear equations defin-
ing the components of E. A necessary and sufficient condition that a
solution other than the trivial exists, is that the determinant of the
coefficient matrix vanishes.. The resulting characteristic equation is
called the dispersion equation, as its eigenvalues determine the indices
of refraction and coefficients of attenuation. That is, the dispersion
equation is ,

c' -n 2 - n 2
X X y z

C' + n n

y x y x

c' + n n
z X z X

e + n n

xy x y

c + n n
Xz X z

/ 2 2 € ' L n n

€ -n-n yz+yz

,YY V Z , 2 2

c +nn c - n — n

zy y y zz y x

= 0 .

(139)

24

4.21 Special Cases

As an indication of the application of the dispersion equation, some

special cases will be considered. For this purpose, re-write equations

(68) and (103), respectively, in the following forms. For E ,

9

- y x H = j o)€

1-

x r
r2 - yz

-iXY
TZ -y2

0

j

X Y

rz - y2

l- x r
rz - y2

0

0

o

1- X

and for E

- y x H = j ud €c

Let

"E 1

X

•

E

y

E
z

_ _

(140)

0 -j X Y

r2 -y2
o -l- x r

r2 -y2
o o

0

0

1- X

y

(141)

y = ya = j k0n = j k0n (a cos 9+ a sin 0 cos cp+ a sin 9 cos (p)

(142)

Using the dielectric tensor from equation (140) along with the dispersion
equation (139), and also substituting from equation (142), for 6= tt/2,

1- n2 sin2 6 - TX - j X Y

■p ^ _ y2 y2^_ v2

n sin 9 cos 9

j X Y
r2 - Y2

1 - n'

n sin 9 cos 9

p2 - Y2

0

2 2

1-n cos 9 - X

= 0 (143)

and for <p = 0,

i~ rx _ n2 sin
r2 -y2

2 0 " J X Y - + n2 sin 9 cos 9 0

r2 - x2

i X Y 2

t^ \r2 + n sin 9 cos 9 1- TX - n2 cos 2 9

1 ' Y r2 - Y2

0

0

1- X

-n

= 0

(144)

25

Equation (143) is the dispersion equation for a vertically polarized
wave propagating in a longitudinal direction, whereas equation (144)
is for a vertically polarized wave propagating transverse to the geodesic
field.

Now, in equation (143), let 0 = ir/2,

1- n2 - r X

-j X Y

r2 - y2 r2 - x'

i X Y

r - x

1 - n - r x

2_X2

i-rx

r2 - x2

Thus,

( i - r x

and

Y - Y

n - 1 - X

' n2)2 - (XY)2 = 0,

r2 - yz

r± y

For equation (144) with 8 = tt/2,

1 - n - r X

r - yz

1 X Y

-j X Y

r2-v2
i - rx

2 2

r - y

r2 _ y2

1- X

- n

= 0

(145)

(146)

(147)

(148)

one solution is ,
n2 = 1 - X

(149)

For the other solution,

(1 - TX -n2) (1- rx ) - ( X Y ) = 0,

or,

2 2

r - y

r2-Y2 r2-Yz

n2 = 1 - r x

x2 y2

rz-Y2

(r2 -rx- y2) (r2 -y2)

n2 = 1 - X

r- y

r-x

26

(15 0)

(151)
(152)

Solution (147) is for a vertically polarized wave in the longitudinal
direction whose exciting field is normal to the geodesic field, and
solutions (149) and (152) are analogous solutions for the transverse case.

For the azimuth, or horizontal component of the exciting field, the
dispersion equation for <p = tt /2 , that is, for longitudinal propagation,
becomes ,

9 2

-rr sin 9 -j XY

n ' sin 9 cos 9

0

? 2

r - yz
i-n2 - r x
r2-x2

n sin 9 cos 9
Expanding,

1-n2 cos2 9 - X_

r

= o

(153)

i X Y (0) + (1- r x _ n2)

r 2 - y2 r2 - x2

from which a solution is obtained,
n2 = 1 - r X

- sin 9 n sin 9 cos 9

cos 9 1-n^ cos^ Q- X

= 0, (154)

r2 - Y2

(155)

For the other factor,

-(1- X ) sin 9 + n sin 9 cos 9 - n sin 9 cos 9= -(1- X sin Q ) ,

r r

which is constant. Therefore, solution (155) is the sole solution for this
case.

For transverse propagation, <p= 0, and the exciting field is parallel
with the geodesic field. Hence, in this case, the ordinary solution

holds, namely
n2 = 1- X

(15 6)

5. APPLETON EQUATION FOR ARBITRARILY ORIENTED GEODESIC FIELD
5 . 1 The Appleton Eugation

The Appleton equation, sometimes referred to as the Appleton-Hartree
equation, is an equation for determining the complex index of refraction in

27

a homogeneous, anisotropic ionosphere. It is customarily derived by
choosing one of two coordinate planes as being determined by the y
and B0 vectors. It will be derived herein with an arbitrarily oriented
geodesic field so that a greater leeway in the selection of the
coordinate axes is permissible for applications. Refer to figure 8.

t a,

+ £

fxj'A;/n

Figure 8
The following symbols will be used,

ub=JL^

m

2 2

, jw = Ne

m c0

, r --■- i •■ j y/w / x = u> 2

2 2 2

Y + Y + Y

x y z

CO

Y = ^s

CO

(157)

Y = Y cos 9 / Y = Y sin 9 cos <n , Y = Y sin 9 sin <p =

x y z

Y cos 0 = Y

YT = Y sin 0

L '

) =y/i-

2^-2
sin 9 sin 0

e Bo
j CO m

= - jy

The geodesic field is taken as,

B0 = B0 (a cos 9+ a sin 8 cos <p + a sin 9 sin cp) ,

— x y z

and y as,

r

= az y = jk(

(158)
(159)

(160)
(161)

5 . 2 Lorentz Conductivity Tensor for An Exciting Wave

The exciting field will be postulated to vary as J e

jcot - y . r .

Hence, Maxwell's equations may be written in the form,

y .H=0,y .€oe'.E = 0,yxE =jco/J0H,-yxH=f +
j co €0 E , i = - N eV .

(162)

28

Solving for i ,

L = -y x H - J6l3C0 E = - [j o)C0 E + J v x (y x E) ] = - jtoe0<*

(163)

[ E + - k 2 n x (n x E ) ] .

- k

Now,

nx(nxE) = nx[nx(a E + a E + a E)] = n2

XX y y z zJ

a x(a E -

z y X

a E ) ,

x y

or,

2 ,-

nx(nxE) = -n (a E +a E +a 0).

XX y y z

(164) •
(165)

Therefore ,

L = -j W€0 [ E + n x ( n x E)J= - j we0 C E - n2 (a E +a E +a 0)],

J X x y y z

(166)

and hence,

L = J W€

n - 1 0 0

0 n2 - 1 0

0 0-1

(167)

The Lorentz conductivity, a , is defined by,
L = a . E .

(168)

2 2 2

Hence, if M is defined by M = n - 1, the Lorentz conductivity tensor

becomes ,
o = j oo e0

"n2-l

? °

O n -1 O

O O -1

= J w-

M
O

O

-^

o

vl2

o
o

O

-II

M2

1 o

0

O 1

0

O 0

-1

Mil

(169)

A corresponding Lorentz resistivity tensor, p , may be defined by,

p. l=p.<J. E = f . E = E, (170)

with "l being the identity tensor.

29

The resistivity tensor may be found from equation (3 6), which will be
repeated here for convenience ,

m d V + v m V+eV xB0= -eE.
d t

In the steady state phasor form, equation (36) is,
-eE =(j ojm + ym) V + e V x B0 0

Using equation (160) for B0in V x B0,
V x B„ = a a a

x y

V V V

x y z

cos 9 sin 9 cos o sin B sin cp

(36)

(171)

or,

(172)

VxB = B.[a (V sin0 sin co- V sin 0 cos o) + a (V cos 9 - V

0 x y z y z 3

sin 8 sin cp) + a ( V sin 9 cos <p- V cos 8 ) ] .

z x y

(173)

Substituting equation (173) into equation (171), factoring j u) m from
the first two terms of (171) and using the symbol T , the resulting equation
is,

-e E = j aim T(a V+a V +a V) + e B n[ a ( V sin 9 sin <p-

xxyy zz xy

V sin 9 cos ip ) + a (V cos 9 - V sin 9 sin cp ) + a (V sin 9 cos (p
z v/yz x ZX

- V cos 9) ] .

y
Substituting symbols (157) into the re-arrangement of equation (174) as
a tensor equation, the equation may be written,

= - ] usm
e

r

jYz

y

jY

|Y
z

jV

V
X

r

-jY
X

•

V

y

T

X

r

V
z

= -1

j O) c0X

r

"jY

r

jY

JY.

■jY

r~ 1

y

I
X

X

Ly

lz

—

^ —^

(175)

X

with the velocity matrix converted to the current density matrix by

30

multiplying and dividing by - Ne.

From equations (170) and (175), the resistivity tensor may be taken
to be ,

= - 1

J we, X

r -jyv jy

z y

JY r -JY

z X

-jy jy r
y x

(176)

Consider equation (17 0), and into it, substitute from equation (168).

The result is ,

E = p .f= p . <J • E , I . E - p . ct . E = 0, (177)

or,

[ p. Gr- I ] . E = 0 . (178)

Equations (178) constitute a set of the linearly homogeneous equations

for the three components of E. Hence, for a non-trivial solution to exist,

p . 5 -1 =0 . (179)

Multiplying (169) by (176) and substituting into equation (179), the

characteristic equation becomes,
r6

- M

X3

r+ x

M<

JY.

JY

JY_

■JY

y

Expanding the determinant,

y
m2

r + x
772-

"iY^

M

M2

JY

r-x

= 0

(180)

M

(r+ X )2 (r-X) +]Y Y Y - jY Y Y - Y '' ( T+ X ) -

~^r x y z x y z y

M<

y2 ( r+ _x_ ) - Yz ( r -x) = 0,

X M2

or,

(r+ _x ,2 - yt2 (r + x_) -y2 =0,

2 2 2

in which substitutions YT = Y and Ym = Y, , + Y were made.

l z t x y

(181)
(182)

31

Solving for r+ X

(18 3)

or, transposing ,

(184)

Taking the reciprocal, multiplying by X, and eliminating M

X

M2 = n2 -1 = -

Y '

r- t

2(r-x)

+ Y,

(185)

4 (r-x)2

Solving for n and substituting from equations (157) and (159),

w „2 / "2

w

n = 1- -

2 2

, / (o-v/^) sin 0
1-J V/Ui - u_

b cos 0

CO2

172

2(1-0) 2/ 05Z - j!//u>)

o\ Vur sin i/j

4(1- a> 2/tal - jv/u)2

(186)

Equation (186) is the well known Appleton equation with 0 being the angle
between the geodesic field and the direction of propagation. It is not
restricted to any coordinate system. That is, it is a mathematical
invariant.

6 . FARADAY ROTATION

6. 1 Waves Through the Ionosphere

Waves passing through the ionosphere are, generally speaking, broken
into two or three distinct waves having distinct indices of refraction, and
they travel by distinct ray paths. If the waves are attenuated, they are
unlikely to recombine into linearly polarized waves.

32

However, if attenuation is negligible, the circularly polarized
waves may recombine into linearly polarized waves, but they will have
their plane of polarization rotated from the original plane. This is
referred to as Faraday rotation.

To examine this rotation analytically, consider two such waves
emerging from the ionosphere after undergoing different path length
shifts in phases, 0,0, respectively. Write the phasor equations,

E = Eo e "jk/o L "jk° J** , . di , + e -Jk8 Jn2 . ds2 1

Since the ionosphere is homogeneous by hypothesis,
E = E0e-jk°r°(e""Jk°nlSl + e "jk° \ S2 )

(187)
(188)

or,

E = E0 e

-jk(r0 + Vl+n2s2)

[eZJ^.(n1sl-n2 s,,) +

eik^. (nlSl-n2s2) (189)

in which s and s are the respective path lengths.

Upon multiplying and dividing by 2,

v .- 9 P ™ r t / n -jk0[r0+l/2 (ns +n,s )]. (190)

E - 2 E0 cos [ k0 (n s - n s )] e 11 2 2

2

Thus, the resultant of the shifted vector is 2E0cos [k0(ns -ns ],

and_k_2_(n s - n s ) is the angle of the resultant. This may be verified
by referring to figure 9.

^» R = 2 cos

<Pl <P2

<Pi = ko n s
"T

^2 =

21 n2
2

Figure 9
From equation (190), it may be seen that the phase shift of the field
through the medium is determined by the average path length of the two
ray paths .

39

6.2 Polarization

In order to determine a measure of the complex polarization of a
wave propagating within a homogeneous anisotropic ionosphere, it is
desirable to find the ratios of the electric field components as deter-
mined by equation (179). The matrix f»rm is,

M

X

r +

jYz
-jY

X

M

2

- jY

r+ x

M2
JY

-jY

X

M
JY .

M2
M2

r

y

L

= o

From the theory of linear homogeneous equations, the ratio E :

A

E : E may be found from either pair of the three equations, given by
equation (191), by omitting the first, second, and third columns of
coefficients, respectively. The signs are alternately plus, minus,
plus . Thus ,

E : E : E =

r + x

M2

M2

JY
X

JY

y

x y z

M2

JY
X

-(r - x)

M2

"jYy

-(r-x)

M2

jY r+ x

z — 2"

Mr

-jY jY^

•

y

The polarization R is usually defined in terms of the ratio of two
components of the electric field normal to the direction of phase pro-
pagation. To facilitate the algebra, let,

Y = 0
X

Yz=YL'andVYT

This, in effect, rotates the coordinate axes (figure 8) such that Bc
lies in the Yz - plane. Now write the ratio,
R = E

r+ x_

M2

iYx

iYx

-<r - x)

jYz

iYx

-JY

-IT- X)

(r-x) (r+ x )- y^

j (r - X) Y + Y Y

z x y

34

(191)

(192)

(193)

(194)

which, upon substituting from equations (183) and (193), becomes

1

R = _i

2Y,

rY 2
_J

r- x

(r-x)

= -J__ (r + x )

jY M2"

The other ratios may be expressed

(195)

-JYT M

r - x

y

■JYT M

r - x

R,

(196)

7. CONCLUSION

7 . 1 Equivalence of Two Points of View

For the purpose of yielding a better insight into the mechanics involved
when a wave is propagated into the ionosphere, the principles of electron
ballistics were applied to the free electrons. In particular, equation (36)
was solved by an iterative procedure. in the procedure . for deriving
equation (68), the latter equation yielding the dielectric tensor.

The components of the dielectric tensor were used in equation (139)
for determining the indices of refraction. This dispersion equation can be
formally written,

[n2 I - (n + V)] . E = 0, (197)

in which an index tensor n is introduced, with the components of n
being defined as ,

(198)

Equation (36) was also used as a key equation of constraint in
deriving the Appleton equation (186) . Thus equation (36) may be thought
of as a sort of common denominator between the two procedures for
finding the complex indices of refraction.

In fact, equation (68) can be derived much more compactly by
formalized procedures. For this purpose, consider the corresponding
complex form of equation (36),

-eE=(Ju>m+ym)V + e V x B 0 (199)

n = [n ]= [n.i n ] , i, j = x > Y, z

33

Referring to figure 5, and substituting

-JY = e B0
j Ol) m
equation (199) becomes,

-e E = j a) m [( 1 - j v/u) V - j Y V x l^ ]

[TV+jYVa - j Y V a . ]

x y y y

j 03 m x

(200)

(2 01)

Hence,

E

y

E
L Z

•1 lc m
e

r

JY

■JY

r

o o

0

X

0

•

V

y

r

-*

V
^ z _

r -jy o
jy r o
o or

~ _

L

X

L

Y
L

z

"1

j w e0 X

(202)

From equation (202), the Lorentz resistivity tensor p may be written

namely,

T -JY 0

JY r 0

o o r

= - 1

JO) 6oX

(203)

Now, the complex dielectric tensor is given by,

= ' = I + J W€0 (p)-l

(204)

Thus, the inverse resistivity tensor may be found by customary matrix

algebra ,

= .! = -ja?6 x
r(rz - x2 )

or, = .

p -1 = -JOJCc

r x

r
-jrY r

o o

jXY

jrY

2

x2-r2

^jXY

X

0

2 2

r - x

rx

2. ^ xz - rz

r2 - y2

o

0
X

(205)

(206)

3$

Substituting into equation (2 04) and introducing symbols (63) and

(64),

l- r x

X2 - TZ

-jXY

x2-r2

jXY

x2 - r2

i-rx
x2 - r

0 0

0

0

0

1- X

X

-JY

jY

X

0

0

(207)

Substituting e' into
curl H = j o)C0 € 'E ,
yields precisely equation (68), that is, 1

(208)

curl H =

3 &€e

a (E X-J'E Y) + a ( jE Y+E X) + a Z

x x y y x y z

(68)

The above derivation tends to place more confidence in the previous
physical interpretations. It also serves to tie the iterative procedure to
the Appleton equation.

7 . 2 Inhomoqeneous Anisotropic Ionosphere

There is no exact mathematical model for the inhomogeneous aniso-
tropic ionosphere. Many statistical measurements and mathematical
interpretations have been and are being made. A vast amount of lit-
erature exists, but various precise studies remain to be made.

A detailed study of the inhomogeneous ionosphere was entirely
beyond the scope of the time and facilities available for the pre-
paration of this report.

37

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3. RCPORT TITLE

Electron Ballistics and Electromagnetic Waves in the Ionosphere

4. DESCRIPTIVE NOTES (Typa ol raport and inclualra dmtaa)

Technical Report, 1965.

5 AUTHOn(S) (Laat nama. tint naoia, initial)

Chaney, Jesse G.

6 REPORT DATE

10 November 1965

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13. ABSTRACT

It is shown that ion collisions introduce a slight reduction in the
plasma frequency, along wi th an exponential decay of transient electron
oscillations. The critical frequencies for penetration of a homogeneous
ionosphere, for both isotropic and anisotropic ionospheres, are determined.
The characteristic waves, for electromagnetic propagation within a homo-
geneous anisotropic ionosphere, are developed by considering an infinite
series of electron velocities, produced by an exciting electric field. The
complex indices of refraction are determined, both from a dispersion
equation and from a derivation of the Appleton equation, which uses an
arbitrary selection of the coordinate axes, thus emphasizing the invariance
of the Appleton equation. Vector and tensor algebra is used throughout the
analytical developments.

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