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Full text of "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 = 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 X-axis 

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 electron-ion 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 = l-j 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 

X-axis, 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 £, 



1-J* )-,^ 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 
( 1-J _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 l-sin 2 Q sin 2(p-cos^ 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 Z-axis 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 



aj-a 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 -Ji|V.. (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 ( cd--jy) 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(cO-jy) 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 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) -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 (EX-jE 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. 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 ) + 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 



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 



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^> 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+ 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. 

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 = 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 


[1--P 2 1 


( 2 
-ja E a p 


% 


a< to- jv) 




x y co(to- 

1 




) z 


-5 E *p w b 
y y 'a- (u-ji/) 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 
Aj-A 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 r-Y 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 wt-y. 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, 

- - jwt-y.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 



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 - 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 

€ -n-n 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, re-write 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 

1-n 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 



i-rx 



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 





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 







? 2 

r - y z 
i-n 2 - r x 
r 2 -x 2 



n sin 9 cos 9 
Expanding, 



1-n 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 1-n^ 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 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 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 

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 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 re-arrangement 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 non-trivial 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 


r-x 



= 



(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< 



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(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 
1-J V/Ui - u_ 



b cos 

CO 2 



172 



2(1-0) 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 


-(r-x) 




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) 



(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, 



r Y 2 
_J 

r- x 




(r-x) 



= -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 


i-rx 
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 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 



la. TOTAL NO. OF 'ttll 



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12 SPONSORING MILITARY ACTIVITY 

United States Naval Missile Center 



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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Ionosphere 
Electron ballistics 
Electromagnetic waves 



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42 



Mr. John M. Goodman 
Search Radar Branch 
Radar Division 
Naval Research Laboratory 
Washington, D. C. 20390 

Dr. G. P. Ohman 

High Resolution Branch 

Radar Division 

Naval Research Laboratory 

Washington, D. C. 2 0390 

Mr. Allan Oster 

Space Research Div. (Code N223.3) 

Astronautics Dept. 

Naval Missile Center 

Point Mugu, California 

Mr. ferald Leish 

Avionics Division (Code 3 222) 

Systems Dept. 

Naval Missile Center 

Pt. Mugu, California 

Mr. Edward Keller 
Bureau of Naval Weapons 
Code (RTOS 311) 
Washington, D. C. 

Dr. A. B. Dember 
Astronautics Dept. (Code N22) 
Naval Missile Center 
Point Mugu, California 

Mr. W. A. Eberspacher 

Space Research Div. (Code N223) 

Astronautics Dept. (18 copies) 

Naval Misfeile Center 

Point Mugu, California 

Mr. D. Sullivan 
Technical Director (Code N01) 
Naval Missile Center 
Point Mugu, California 



Capt. C. O. Holmquist 
Commander, Naval Missile Center 
(Code N00) 
Point Mugu, California 

Mr. A. Solferino 

Space Systems Dev.Div. 

(Code N222) Astronautics Dept. 

Naval Missile Center 

Point Mugu, California 

Dr. Leonard Porcello 
Radar Laboratory 
Institute of Science & Techn. 
University of Michigan 
Ipsilanti, Michigan 

Dr. William Brown 

Radar Laboratory 

Institute of Science & Techn. 

University of Michigan 

Ipsilanti, Michigan 

Mr. S. Weisbrod 

Smyth Research Association 

3555 Aero Court 

San Diego, California 92123 

Mr. J- Smyth 

Smyth Research Association 

3555 Aero Court 

San Diego, California 92123 

Mr. Robert Anderson 

Space Research Div. (Code N223.4) 

Naval Missile Center 

Point Mugu, California 

Mr. William Titus 
Avionics Div. (Code 3222) 
Systems Dept. 
Naval Missile Center 
Point Mugu, California 



43 



Mr. Edward Ornstein 
High Resolution Branch 
Radar Division 
Naval Research Laboratory- 
Washington, D. C. 20390 

Dr. Irving Page 

Radar Division 

Naval Research Laboratory 

Washington, D. C. 20390 

Mr. A. Conley 
Bureau of Naval Weapons 
Code (RTOS 41) 
Washington, D. C. 

CDR J. R. High 

Director, Astronautics (Code N2) 

Naval Missile Center 

Point Mugu, California 

Chief of Naval Operations 
Codes OP-76, OP-321 (2 copies) 
Washington, D. C. 

Chief, Bureau of Naval Weapons 
Code RT 
Washington, D. C. 



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