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 = Vector magnitude of incident field
B = Geodesic field vector
)c = 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 ) k n = Complex vector propagation constant

t 4> = Angle between yand B
S = Cos 1 a Q . a z
X , y, Z = Cartesian coordinates
r, d,ip= Spherical coordinates
£= Small displacement along Xaxis
F = Force vector
e = Positive magnitude of electron charge
m = Mass of electron
 e/m = 1.77 x 10 11 coul/Kg
v= Average number of electronion collisions per second
r = Radius vector
V = Vector velocity of an electron
N = Density of electrons per cubic meter
_7
ju = 4 7T x 10 h/m = Permeability of free space
1 9
c « 1 x 10 ' fd/m = Permitivity of free space
3 6rr
a? = / Ne * = Resonant angular frequency, lossless plasma
P m Co
OJ = eB = 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 = w p 2 / co 2
Y=u b /u=J Y 2 +y 2 +y 2
x y z
Y = Y cos 6, Y = Y sin 9 cos <p, Y = Y = Y sin Q sin ^= Y cos $
Y T = <V+ Y Z =K/ Y 2  Y 2 = Ysin0
X y L
r = lj v/w
M 2 = n 2  1 = n . n 1
a . = Unit vectors
i
A = a x a = a, sin f
1 6 z 1 ^
A 2 = (a x aj a z =a 2 sin £
x = i  r x
2 2
r  y
Y = X Y
X
r 2 x 2
Z= 1  x_
r
E =^2 jr 2 ' , <a =tan~ lE 
Xy E + E vy "T
x y E
co = Critical frequency
c
to = Ordinary wave critical frequency
C o
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
9 = 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 )
I WS
D
mv
~P s ~~^ e i of free electrons. If the electrons are compressed
.X laterally through a small displacement, £ , along the
X=0 y = i
Xaxis, a surface charge density
Figure 1 ^ T ...
"P s = "Ne £ (1)
may be considered to exist at y = £ . Then, from the definition of the
electric displacement vector, from y= o to y= £ ,
D x =f v < 2 >
Hence, D = c 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 ^ + Ne 2 4=0. (5)
dt
2 2
Substituting a) = Ne into (5), the harmonic equation for free oscill
m e
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)
dt 2 dt €
Pt
Dividing by m, and using the operational method of substituting £=£ e
into equation (7), the determinantal equation is obtained,
(p 2 + 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, E ge
be incident upon the ionosphere at an angle of
Figure 2 incidence 9 . 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
at 2 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 =p V = 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 H , 5 = e' c E . (14)
dt
The corresponding complex form is,
v. H=0, v.E = 0, vxE = jco/i H,
V x H = i + j a) € E .
Substituting from equation (13),
2 
(15)
vx H = Ne E + jco c E = jcoe *
m {v + j co)
[1 + p? —] E ,
(16)
or,
Jcoe m(y + Jco)
VX H = JCO € [ 1 +
C0 p 2 (l/ jco)
iuiv 2 + co 2 )
E =
JCO £,
1J* ),^ 2
i/ + co 2 tod^ + CO 2 )
(17)
2 2
in which co = Ne was also substituted in the right member of equation
P m c
(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
co z + v l
Returning to euqations (15), the wave equation will be determined
by elimination.
2  2 
Vx ( vx E) sv( V . E)  V E = V E = jc0jU o *
V x H .
Using y = a+ j = jk V e ' , and k = coV jli e = co/c , upon
substituting for vx H, equation (19) becomes,
(19)
For v f»o, n =N
V 2 E + k? € ' E = 0.
will be evaluated.
1/2
(2 0)
n=(€H c') 1/Z =V
,/ 1/2
( 1J _J_)
e'
2 e'
j
2/7"
(21)
and hence,
y = j k n n » k * g +i k^AT" .
From equation (18) with i/p»0,
y«s y k
2777
_ + j k <yi  ( Wp) 2
V l ( ^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 (co p )2
(22)
(23)
(24)
(25)
CO
2 .2 Lossy Case
For the lossy case, the exact value of n = JJ 7 " 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»
JT NL+ J^n
sin B/2 = 7r ^=2 .
b + a
Substituting from equation (18) into (2 6),
2
n=
1
(1
.<*
CO
2 + ^
2 co p 4 ^ 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 v 2
) + P
(26)
1/2
1/2
co 2 ( co 2 + i^ 2 ) 2
(27)
2 2
co + y
Thus, y = k :Jm [n]and = k Re [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 Re [n]
Designating the angle of refraction by , 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 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 Q = 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 c 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 . 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 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 /
cos 9' = cos Q
sin j/j
Figure 4
sm e'=h cos2 9 =
' sin
V lsin 2 Q sin 2(pcos^ q =
' sin 2
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 /
a r
cos ©
sin $ cos (p
sin 6 sin cp
a6
sin 6
cos 9cos (p
cos sin (p
cos Ssin c/D
sin ij)
a
sin <p
cos <p
cos cp
sin ib
sin 9cos
sin ib
6s in (p
sin^ 0sinocos cp
sin ^
1
sin
(35)
3 . 1 Electron Current Density
Assume the Zaxis of a coordinate system to lie along the geodesic
field in the ionosphere, and assume an electromagnetic field, E = E a« ,
to be incident at angle 9 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 B a
The zero order component, i of equation (13), reacts with B to
produce a new component l. , of the current density, normal to both
i 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 V 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
aja sin 8+ a cos 0cos <o+ a cos
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 cos 9 cos ip cos sin ip
1
a cos Q cos en + a sin
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
1
cos cos <p
= a sin a
y
(3 9)
E = E sin 8 , E = E Q cos 0cos cp, E =
x y z
E cos 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
a 9 at an angle 7/2 + £with a .
Z Q
For the iteration, from equations (12) and (10),
V = ~ eE ° g A . , F =. e V x B = jm (co JiV.. (43)
j m ( co jv) 1 z *
Thus, upon defining the gyro frequency,
a D = 6B z (44)
m
V = e 2 g oe x B z = e E co b Aj =
[jm ( u)jy) J m ( to  jv)
e co b E sinC ^ # (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 ( cdjy) 3
2
j e co, E sin £
m
2 ( co  j^) 3
For
a 2
(47)
' to w) v ,
(48)
" a 3 '*3 = *V
(49)
V 3 = e co b 3 E sin £
m ( co j^) 4
C ° ntinUing ' V =Jeco b 4 E sin C   _  f5Q)
4 b a a a 2 , (50)
m ( co  )v)°
V c = e co, E sin f , cn
5 b ° > a_ , a =a. , (51)
m ( co jlO b
— fi
V_=  j e co, E sin f    . .
6 b ° * a c , a = a . (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 we , and substituting co , the velocity vector for an electron
becomes ,
V = + J CO€ E
co.
+ J a
Ne
2 5
co ( co j 0) '
CO 2 CO 6
_P b_
co ( co  )v) 7
B co ( co \v)
co 2 co, to 2 CO 3
p "b + P d
sin £  a,
^co( co jv) 2 co( co jv)
2
p b
CO 2 CO 2
+
2 4
^p w b
+
CO
( co )v) 3 CO ( co  ji/) 5
or,
V = + j CO € n E
Ne
sin
CO
(53)
CO l co
a 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+ W b
+
( co  j v)
(54)
The infinite series within the brackets of equation (54) converges for
co b 2
2 2
CO + v
< 1 ,
that is, for
co > co b N 1 ( i/ co b )
(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  co u 2 / ( co jv) (co jf)  CO
(57)
and hence,
CO
^ 2
V= + j coc n E n J a __p_
Ne
C0i_ CO sin £
+ b P ^ y
~V~_ ... 2 ~
CO(cOjy) C0[( CO jv) Z  ttj^]
ja r a 2
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 ]
ja i" a 2
*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)c EJa
9
1 P
60 (tO  )V)
s ^b 2 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),
0xxyv zz
equation (61) may be rewritten,
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) co b
2 2
co( co  iv) [ ( co iv)  co b .
JE "b "p 2
X
 E
2 2
co[(co iv)  co b ]
co (co  ji^) [ (co iv)  co b ]
+ 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)  co b ]
1 
CO 2 (co  jl^ )
co [(to iv) z co*J >
Y = "b "p
u>[( co iv) 1  co b 2 ]
, z =1 
CO
co (co  jy )
and substitute into (62) to obtain,
(62)
(63)
(64)
— V x H = a (EXjE Y) + a (E X+jE Y ) + a E Z .
j coc xx y yy x zz
(65)
12
For cross reference to literature employing the standard URSI symbols,
X=oj 2 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. Rearranging equation (65),
1 v x H = a (E XjE 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 ) + B ( a ja ), (72)
x y i x y 2 x y
and obtain,
(73)
B l +B 2 =? B 1 = 2L ^ =A 1
B r B 2 = * B 2 = ^ =  A 2
Substituting from (71) and (73) into equation (69),
± V x H = (E  j E ) 2Li_JL (a + j a ) + (E +jE)x
j o3e 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? p 2 (to w)t g? b co p 2 x _ ^ C(m jy) =f a* b ]
u>[ ( w jv)  w b 2 ]
X ±Y
1  X
r ± y
upon letting ,
E
W[(W jl>) + 0) b ][(w jl/)  03 b ]
(75)
= 1/2 VE 2 + E 2
xv x y
1 E
= tan _y_
X y E
(76)
equation (74) may be written,
v" x H = j to €
: e " J X Y (I" — ) (a + j a ) + E
xy r + Y X y
ia>
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
Jj
vector by a , that is ,
R
a T = a + j a
l x y
a = a  j a
R X Y
(78)
and letting ,
6/ =1  x , e i = 1 x
L r+ y r  y
1  x
(79)
equation (77) becomes,
V x H = j (ji c
£
 J
E e
xy
E e
xy
xy
xy
■" 1
a
L
•
*R
a
z
L^> sJ
= 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 Zaxis. These waves have three distinct indices
of refraction.
14
Rationalizing the denominators of equation (79),
0> z + v
 J
CO
to (to+ co b )
to(co 2 + i^ 2 )^
2 2
to[(co + CO J + V ]
1  ^p 2 (a)  "b }
J

 )
 )
V ^P
to[(co + co b ) + i/ 2 ]
2
1.
i; 0)
to [(to w b ) 2 + v 1
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, n , 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 ±W b )'
? 2
to[(to ± w b r + v ]
/A  "p 2 (m
VL w [(to ±
1/2'
to[ (to ic^) 2 + v 2 ~\
2 / 2
. w (to ±to,
1  p b
2 2
CO [(tO ± tOu) + V ]
1/2
CO
p
co b ) 2 + y 1
\_ to p 2 (to ± to b )
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 = in (84) and squaring, is equivalent to setting y = in
equations (82) and (83). Hence, for applying equation (30), set
1  E.
tO ( tO ± tO^)
sin 8 /
or, 2 ? 2
to ± 0! b 05  Wp sec a = .
(85)
(86)
Taking the positive sign in equation (86) for to T /
w t = " ^ + V ( u> n sec 8 ) 2 + ( co,
cL P
k 2
W
(87)
and taking the negative sign for to
cR '
to cR = + _^b_ + V (to p sec a;) 2 + (Ub_f .
(88)
Since to determined by equation (81) is identical with to determined
by equation (33) ,
W c = W c = W p S6C 9 ° '
while,
/ 2
cL C
w
to
and f 2 . ,2
W_r, = ^ OJ + \ W b ] tO^
cR
V
2 2
For frequencies sufficiently high such that (to c ) >> (to^^'
U o
[2 to)
v c
1/2 ■ o
2
8^2
*— —J
"■
Hence,
W cL
* to c ,
rw
8 to
(89)
= to + Hs + . (90)
C 8 to
(91)
and,
to
cR
tO h 2 ui, to K
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.
Rewriting 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 = E n ( 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. =  eV x B = j m (oj  ]v) V
1 z 1
V =e 2 E n a  x B
1
<P_
[j m(oj ji/)]
_ eE av (a sin <p + a cos o) x a .
m
(tO " j^) 2
e E to, sin (p a
m («  j^) 2
2
o E B to, sin <P a xa < r. 2
°„ z b ; y z = jeE to b sin <pa y
e E B to
z .
j m z ( to  j v) 6
m
( co  jv) 3
V = j e2 E ° B z a5 b 2 sin ^ a y x a z = eE to b 3 sin <p a
V
j m2 (to  iv )
m ( to  jv) 4
e 2 E B z to b 3 sin pax a g = j e E to b sin cpa y
) 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 E 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 M 2
(to  J^) 4 (co  jv ) 6
CO,
+ w b
a 1
+
a 1
(co  jt^) 3
(99)
(to  jv) 5 (co  \vj f
Summing the series over the range of frequencies for which the series
converges ,
curl H = j to e «
1
L _ * b Z
(co  ]v)
J
r
« a E n
cp °
L
[1P 2 1
( 2
ja E a p
%
a< to jv)
x y co(to
1
) z
5 E *p w b
y y 'a (uji/) 3
2
1 W b
^ (co \v) L
/
(100)
18
or,
a E [l_£
05,
]  jE a P "b
y a<[ (w jv) 2  o\ z ]
[a  j a b
Substituting for a E 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) " 05 b Z ]
+ a E ri
z z u
05 (oj  )v)'
(102)
Let
Substituting from equations (63) and (64),
curl H=jo>e /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 ,
A 1 + A 2 =Y
AjA 2 =X ,
Thus, (104) becomes,
1
A = 1/2 (Y + X)
A 2 = 1/2 (Y  X)
(105)
curl H =  j E (X+Y)(a+ja)+jE (X  ?) x
J O!€ _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 rY R
E (1 X ) a
z r 2
Using equations (79) ,_
durl H = j oj c,
i E/2
^ 5 l"
€ R
•
j E/2
•
S R
1
Co
E z
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 wty. r
E " ~_E°_ e a , (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), E is replaced with E ofl , and if in equation
(93), E is replaced with E , then from table (35), E of equation (109)
<P
(110)
cos cp
may be expressed by
/
^E
e c
e
+
a E
with,
= E
cos
e
sin (0
' E V
Jl
sin^
sin <p
E =
Also, in
J E
equations
2 + E 2
X
(40)
= E,
<
v 1 cos
2 esin 2
(p
and,
E
= y
E
X
tan
xy
 cot e
cos <p .
vl sin 2 9 sin 2 <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)C I 1/2 [e,.' %Y~ *^ a + e xy e^ a R ] / 1 sin Bcos cp
+ a Co cos 8 sin <p y cos 8 sin o E
♦ mX\ 1/2 £c L '5  ( ' ; i sin „ + i f ;cos J ?™ mT 2 
[_ _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 E , B =(j'(i H ,
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  = div j, + a_ div D = div i + 9 p ... .
at at ' U '
(118)
(119)
+ 2_ p= (120)
at c
The solution of (120) may be written
p=p e'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,
  jwty.f (122)
E = E (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
o , a e « 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
E A 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
E (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
XJTy. r rE
o9 e  y . r
(124)
r sin 9 a
a/a<p
o
 y . r
r sin 8 r e
  ^O
cos  y. r r
E cp e  a ft r sin 6 , >
9 E (ye ) +
r sin 9 ^
= a r z sin 9
r
y . r
<P
E °8 ( ye
) .
(125)
Again, assuming a radiated field for which the inverse square terms^vanish,
e cp
*) =a r a Q (y E ^ fi " "V • r )
 y . r ) .
(126)
But,
yxE =
y
e.
o
°9 E °
o
a 0 a v E + a y E
r 9 ° o <p '
(127)
22
Therefore, in practice,
curl
[^
• <o)
y . r
=  y x E (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>jl H, yxH =
NeV + j o)€ E = j OJ e 7 ' . E , (130)
in which, c 'is defined by
1 ' . E =  NeV/ joJCo + E , (131)
The wave equation becomes,
yx(yxE)k "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 = , (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) jLl 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 B ,
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 k 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
(n 2 + n 2 ) 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 n 2
yy x z
c' + n n
zy z y
c + n n
X z 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
X z X z
/ 2 2 € ' L n n
€ nn yz+yz
, YY V Z , 2 2
c +nn c  n — n
zy y y zz y x
= .
(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, rewrite equations
(68) and (103), respectively, in the following forms. For E ,
9
 y x H = j o)€
1
x r
r 2  y z
iXY
T Z y 2
j
X Y
r z  y 2
l x r
r z  y 2
o
1 X
and for E
 y x H = j ud € c
Let
"E 1
X
•
E
y
E
z
_ _
(140)
j X Y
r 2 y 2
o l x r
r 2 y2
o o
1 X
y
(141)
y = ya = j k n = j k n (a cos 9+ a sin 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 n 2 sin 2 6  TX  j X Y
■p ^ _ y2 y2^_ v2
n sin 9 cos 9
j X Y
r 2  Y 2
1  n'
n sin 9 cos 9
p2  Y 2
2 2
1n cos 9  X
= (143)
and for <p = 0,
i~ rx _ n 2 sin
r 2 y 2
2 " J X Y  + n 2 sin 9 cos 9
r 2  x 2
i X Y 2
t^ \r 2 + n sin 9 cos 9 1 TX  n 2 cos 2 9
1 ' Y r 2  Y2
1 X
n
=
(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 = ir/2,
1 n 2  r X
j X Y
r 2  y 2 r 2  x'
i X Y
r  x
1  n  r x
2_ X 2
irx
r 2  x 2
Thus,
( i  r x
and
Y  Y
n  1  X
' n 2 ) 2  (XY) 2 = 0,
r 2  y z
r± y
For equation (144) with 8 = tt/2,
1  n  r X
r  y z
1 X Y
j X Y
r 2 v 2
i  rx
2 2
r  y
r 2 _ y 2
1 X
 n
=
(145)
(146)
(147)
(148)
one solution is ,
n 2 = 1  X
(149)
For the other solution,
(1  TX n 2 ) (1 rx )  ( X Y ) = 0,
or,
2 2
r  y
r 2 Y 2 r 2 Y z
n 2 = 1  r x
x 2 y 2
r z Y 2
(r 2 rx y 2 ) (r 2 y 2 )
n 2 = 1  X
r y
rx
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
? 2
r  y z
in 2  r x
r 2 x 2
n sin 9 cos 9
Expanding,
1n 2 cos 2 9  X_
r
= o
(153)
i X Y (0) + (1 r x _ n 2)
r 2  y 2 r 2  x 2
from which a solution is obtained,
n 2 = 1  r X
 sin 9 n sin 9 cos 9
cos 9 1n^ cos^ Q X
= 0, (154)
r 2  Y 2
(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
n 2 = 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 AppletonHartree
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 B 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,
u b =JL^
m
2 2
, jw = Ne
m c
, 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 = Y
Y T = Y sin
L '
) =y/i
2^2
sin 9 sin
e Bo
j CO m
=  jy
The geodesic field is taken as,
B = B (a cos 9+ a sin 8 cos <p + a sin 9 sin cp) ,
— x y z
and y as,
r
= a z 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 .€ o e'.E = 0,yxE =jco/J H,yxH=f +
j co € E , i =  N eV .
(162)
28
Solving for i ,
L = y x H  J6l3C E =  [j o)C E + J v x (y x E) ] =  jtoe 0< *
(163)
[ E +  k 2 n x (n x E ) ] .
 k
Now,
nx(nxE) = nx[nx(a E + a E + a E)] = n 2
XX y y z z J
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€ [ E + n x ( n x E)J=  j we C E  n 2 (a E +a E +a 0)],
J X x y y z
(166)
and hence,
L = J W€
n  1
n 2  1
01
(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 e
"n 2 l
? °
O n 1 O
O O 1
= J w
M
O
O
^
o
vl 2
o
o
O
II
M 2
1 o
O 1
O
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 xB = eE.
d t
In the steady state phasor form, equation (36) is,
eE =(j ojm + ym) V + e V x B
Using equation (160) for B in V x B ,
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 cos o) + a (V cos 9  V
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 Z X
 V cos 9) ] .
y
Substituting symbols (157) into the rearrangement of equation (174) as
a tensor equation, the equation may be written,
=  ] usm
e
r
jY z
y
jY
Y
z
j V
V
X
r
jY
X
•
V
y
T
X
r
V
z
= 1
j O) c X
r
"jY
r
jY
JY.
■jY
r~ 1
y
I
X
X
L y
l z
—
^ —^
(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 jy v 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 = . (178)
Equations (178) constitute a set of the linearly homogeneous equations
for the three components of E. Hence, for a nontrivial solution to exist,
p . 5 1 =0 . (179)
Multiplying (169) by (176) and substituting into equation (179), the
characteristic equation becomes,
r 6
 M
X3
r+ x
M<
JY.
JY
JY_
■JY
y
Expanding the determinant,
y
m2
r + x
772
" iY ^
M
M 2
JY
rx
=
(180)
M
(r+ X ) 2 (rX) +]Y Y Y  jY Y Y  Y '' ( T+ X ) 
~^r x y z x y z y
M<
y 2 ( r+ _x_ )  Yz ( r x) = 0,
X M 2
or,
(r+ _x , 2  y t 2 (r + x_) y 2 =0,
2 2 2
in which substitutions Y T = Y and Y m = 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
M 2 = n 2 1 = 
Y '
r t
2(rx)
+ Y,
(185)
4 (rx) 2
Solving for n and substituting from equations (157) and (159),
w „ 2 / " 2
w
n = 1 
2 2
, / (ov/^) sin
1J V/Ui  u_
b cos
CO 2
172
2(10) 2 / 05 Z  j!//u>)
o\ Vur sin i/j
4(1 a> 2 /ta l  jv/u) 2
(186)
Equation (186) is the well known Appleton equation with 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 Jk 8 Jn 2 . ds 2 1
Since the ionosphere is homogeneous by hypothesis,
E = E e jk ° r °(e"" Jk ° n l S l + e " jk ° \ S 2 )
(187)
(188)
or,
E = E e
jk(r + Vl +n 2 s 2 )
[ eZJ^.(n 1 s l n 2 s,,) +
e ik^. (n lSl n 2 s 2 ) (189)
in which s and s are the respective path lengths.
Upon multiplying and dividing by 2,
v . 9 P ™ r t / n jk [r +l/2 (ns +n,s )]. (190)
E  2 E cos [ k ( n s  n s )] e 11 2 2
2
Thus, the resultant of the shifted vector is 2E cos [k ( 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 = k o n s
"T
^2 =
21 n 2
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 +
jY z
jY
X
M
2
 jY
r+ x
M 2
JY
jY
X
M
JY .
M 2
M 2
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
M 2
M 2
JY
X
JY
y
x y z
M 2
JY
X
(r  x)
M 2
" jY y
(rx)
M 2
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 =
X
Y z =Y L' and V Y T
This, in effect, rotates the coordinate axes (figure 8) such that B c
lies in the Yz  plane. Now write the ratio,
R = E
r+ x_
M 2
iY x
iY x
<r  x)
jY z
iY x
JY
IT X)
(rx) (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,
r Y 2
_J
r x
(rx)
= J__ (r + x )
jY M 2 "
The other ratios may be expressed
(195)
JY T M
r  x
y
■JY T 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,
[n 2 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 (199)
n = [n ]= [n.i n ] , i, j = x > Y, z
33
Referring to figure 5, and substituting
JY = e B
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
X
•
V
y
r
*
V
^ z _
r jy o
jy r o
o or
~ _
L
X
L
Y
L
z
"1
j w e X
(202)
From equation (202), the Lorentz resistivity tensor p may be written
namely,
T JY
JY r
o o r
=  1
JO) 6oX
(203)
Now, the complex dielectric tensor is given by,
= ' = I + J W€ (p)l
(204)
Thus, the inverse resistivity tensor may be found by customary matrix
algebra ,
= .! = ja?6 x
r(r z  x 2 )
or, = .
p 1 = JOJC c
r x
r
jrY r
o o
jXY
jrY
2
x 2 r 2
^jXY
X
2 2
r  x
rx
2. ^ x z  r z
r 2  y 2
o
X
(205)
(206)
3$
Substituting into equation (2 04) and introducing symbols (63) and
(64),
l r x
X 2  T Z
jXY
x2 r 2
jXY
x 2  r 2
irx
x 2  r
1 X
X
JY
jY
X
(207)
Substituting e' into
curl H = j o)C € 'E ,
yields precisely equation (68), that is, 1
(208)
curl H =
3 &€ e
a (E XJ'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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2a. Ht^ORT SICUKITY CLARIFICATION
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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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