Diffraction of random waves propagated in homogeneous anisotropic mediu

Yeh, K. C.

English

NASA Technical Reports Server

- .  
Diffraction of Random Waves in a 
Homogeneous Anisotropic Medium 
by 
F. C. Yeh 
i 
(THRUI 1 
NsG 24-59 
April, 1965 
Sponsored by: 
National Aeronautics and Space Administration 
Washington 25, D. C. 
Electrical Engineering Research Laboratory 
Engineering Experiment Station 
University of Illinois 
Urbana, Illinois 
https://ntrs.nasa.gov/search.jsp?R=19650015406 2020-03-16T23:56:07+00:00Z
Abstract 
4 This paper considers the diffraction of random waves in a homogeneous 
anisotropic medium. These random waves are produced, for example, by 
reflection from a rough surface such as the moon or by transmitting through 
the ionosphere containing irregularities. Dut to anisotropy both depolariza- 
tion effect and the modification of spectral density functions of the wave 
by the medium may occur. Both effects may be important in certain iono- 
spheric applications. In the forward scatter approximation it has been 
found that the sum of spectra of orthogonally polarized waves is uninfluenced 
by the medium. 
. 
Table of Contents 
1. Introduction 
2. Brief Review 
3. Diffraction of Random Waves 
4. Dependence of Rotation Angle 
5. Correlation Functions 
6. Discussions 
References 
Page 
1 
2 
6 
9 
11 
12 
13 
I 
D i f f r a c t i o n  of Random Waves i n  a 
Homogeneous Aniso t ropic  Medium 
1. In t roduc t ion  
The s tudy of wave propagat ion i n  t h e  magneto-ionic medium i s  concerned 
* w i t h  so lv ing  the  wave equat ion  of,the form 
e 
where f is  t h e  r e l a t i v e  i n v e r s e  d i e l e c t r i c  t enso r .  I n  (1) t h e  e l e c t r i c  
displacement  v e c t o r  D i s  used r a t h e r  t han  t h e  usual  electric f i e l d  i n t e n s i t y  
E s i n c e  i t  is  known t h a t  D i s  t r ansve r se  whi le  i? is, i n  genera l ,  no t  [ 1 3 .  -L -c 
The p lane  wave s o l u t i o n  of (1) has  been s t u d i e d  q u i t e  ex tens ive ly ,  e s p e c i a l l y  
i t s  r e f r a c t i v e  i n d i c e s  known as t h e  Appleton-Hartree formula.  
The p resen t  i n v e s t i g a t i o n  is concerned wi th  t h e  s tudy  of propagat ion 
of random waves i n  such a medium. S p e c i f i c a l l y  t h e  s t a t i s t i c a l  p r o p e r t i e s  
of t h e  wave are g iven  a t  an i n i t i a l  plane.  Through d i f f r a c t i o n  t h e  f i e l d s  
propagate  i n  the  ha l f  space.  The s ta t i s t ica l  p r o p e r t i e s  of t h e  wave a t  
a plane p a r a l l e l  t o  t h e  i n i t i a l  plane are des i r ed .  The method used is  
s i m i l a r  t o  t h a t  used by o t h e r s  who assumed t h e  medium t o  be i s o t r o p i c  [ 2 ] .  
Hence t h i s  s tudy  r e p r e s e n t s  a s l i g h t  g e n e r a l i z a t i o n  of t h e  ear l ier  work. 
However, it i s  found t h a t  due t o  an i so t rop ic  na tu re  of t h e  medium t h e  
a d d i t i o n a l  effects  such as depo la r i za t ion  of t he  wave and the  modi f ica t ion  
of s p e c t r a l  d e n s i t y  of t h e  f i e l d  by t h e  medium w i l l  appear.  
* 
exp jot t i m e  dependence is  assumed throughout .  A l l  q u a n t i t i e s  are i n  
r a t i o n a l i z e d  MKS u n i t s .  
c 
where I is the identity matrix. The immediate conclusion when (3) is substi- 
tuted into (4) is the vanishing of D hence the electric displacement is 
2’ 
polarized entirely in the plane transverse to k .  With this simplification 
4 
the remaining equations expressed in the matrix form are 
2 
jXYL - -  1 P O E O  - 
(1-X12 
2 
k 1 - x  I 
L jXY 
2 
(1-X) I 2 1 P O E O  - -  k2 1 - x  
X 
D 
D 
Y 
= o  (5  I 
In order for the equation to have non-trivial solution the determinant of 
the coefficient matrix must vanish. 
are ignored the following expressions for the refractive indices are obtained. 
If again Y2 and higher order terms 
(6) 
2 2  
k /a poEO = 1 - X - XY COS 8 + 
where 0 is the angle between the propagation vector and the steady magnetic 
field. The wave with the upper sign in (6) as the refractive index is desig- 
nated as the ordinary wave and is circularly polarized in the left handed 
sense; the wave with the lower sign is designated as the extraordinary wave 
and is circularly polarized in the right handed sense. Now (6) is nothing’ 
6 
3 more than the usual expression used in connection with the study of Faraday 
effect and it is valid under the quasi-longitudinal approximation. Fuller 
analysis shows that such approximation is valid for nearly all directions 
rn 
rn of propagation except the region at which the direction of propagation is 
but a few degrees from exactly perpendicular to the steady magnet field. 
Later development assumes (6) is valid f o r  all regions and such an assumption 
4 
8 
makes sense only  i f  t h e  quas i - t ransverse  r eg ion  makes n e g l i g i b l e  c o n t r i -  
bu t ion  t o  t h e  t o t a l  e f f e c t .  
L e t  k(O) and k") denote t h e  propagat ion c o n s t a n t s  of t h e  o rd ina ry  
and t h e  ex t r ao rd ina ry  waves r e s p e c t i v e l y .  Thei r  mean and d i f f e r e n c e  can 
be found as 
I t  is known t h a t  a wave of any p o l a r i z a t i o n  can 
e/ko 
be decomposed i n t o  two 
c h a r a c t e r i s t i c  waves, each w i l l  propagate independent ly  i n  t h e  medium wi th  
i t s  corresponding propagat ion cons tan t  and p o l a r i z a t i o n .  The r e s u l t a n t  i s  
t h e  sum of t h e s e  two waves. I n  t h e  p re sen t  case t h e  r e s u l t a n t  wave i s  
p a r t i c u l a r l y  simple i f  i t  is assumed t h a t  t h e  r e s u l t a n t  is po la r i zed  l i n e a r l y  
a t  some i n i t i a l  p o s i t i o n .  The r e s u l t a n t  wave w i l l  propagate wi th  a propa- 
g a t i o n  cons tan t  equal  t o  t h e  mean of t h e  propagat ion cons t an t s  of t h e  
c h a r a c t e r i s t i c  waves and i t s  p o l a r i z a t i o n  is  kept  l i n e a r  with t h e  plane 
of p o l a r i z a t i o n  r o t a t e d  cont inuously along t h e  d i r e c t i o n  of propagat ion 
(Faraday e f f e c t ) .  As computed i n  (7) t h e  mean of  t h e  propagat ion c o n s t a n t s  
is  j u s t  t h e  propagat ion cons tan t  of t h e  corresponding i s o t r o p i c  medium 
( i * e . ,  i n  t h e  absence of t h e  s teady magnetic f i e l d ) .  The r o t a t i o n  of t h e  
p o l a r i z a t i o n  i s  through an angle  equal t o  one-half of t h e  d i f f e r e n c e  of 
t h e  propagat ion  cons t an t s  mul t ip l i ed  by t h e  d i s t a n c e  of t r a v e l .  For  a 
p lane  wave propagat ing i n  z d i r e c t i o n  wi th  t h e  e lec t r ic  displacement po la r i zed  
i n  y d i r e c t i o n  a t  z = 0 t h e  r e s u l t a n t  t a k e s  t h e  form 
. 
Dx = - D 0 s i n  exp j("t - k 0 ~1 
where t h e  r o t a t i o n a l  angle  i s  
5 
(9) 
I n  gene ra l  i f  t h e  wave i s  assumed t o  be po la r i zed  i n  yz,-plane a t  z = 0 
and t h e  d i r e c t i o n  of propagat ion has a po la r  angle  a and azimuthal angle  
p t h e  f i e l d  is g iven  by 
2 2 2 1/2 - L . &  
= -D ( s i n  p + cos p cos  a) s i n  82 exp J(0t: - ko r )  
DX 0 
- P -  2 2 2 - 1 ' 2  2 
D = D ( s i n  p + cos p cos a) (cosa cos  fi + s i n  a s i n p  cos p s i n G ) e x p j ( w t - k O - r )  Y 0 
+ &  
0 
2 2 2 - 1 / 2  = D ( s i n  p c cos 8 cos a) (cosp fi - s i n p  s i n a  cos  52) exp j ( a t - k  *r) 
DZ 0 
(11) 
For cases  of small  po la r  angles  (11) can be approximated by 
The Faraday r o t a t i o n a l  angle  a appeared i n  (11) and (12) i s  t h e  same a s  (10) 
except  z i s  t o  be rep laced  by r ,  Since fi plays  an important r o l e  i n  t h e  
d i f f r a c t i o n  i t s  d i scuss ion  will. be postponed t o  a l a t e r  s e c t i o n .  
3. D i f f r a c t i o n  of Random Waves 
6 
i ,- 
The problem assumes t h a t  t h e  s t a t i s t i c a l  p r o p e r t i e s  of t h e  f i e l d s  are 
given a t  some i n i t i a l  plane,  s a y  a t  z = 0. Our i n t e r e s t  i s  i n  l ea rn ing  t h e  
s t a t i s t i c a l  p r o p e r t i e s  of t h e  displacement vec to r  a t  some plane p a r a l l e l  
t o  t h e  i n i t i a l  plane a f t e r  propagating i n  xhe kind of medium discussed  i n  
s e c t i o n  2. This  sugges ts  t h e  use of t h e  n o t a t i o n  D(rL ;z) which stresses 
4-  
t h e  f a c t  t h a t  s t a t i s t i c a l  p r o p e r t i e s  a r e  d e s i r e d  f o r  var ious  v a l u e s  of z 
i n  t h e  two dimensional space rL = (x,y),  when t h e i r  p r o p e r t i e s  a t  z = 0 
are known. The problem is  e x p l i c i t l y  posed when t h e  fo l lowing  (boundary) 
4 
c o n d i t i o n s  are assumed., 
4 
1) D ( r , ; O )  is po la r i zed  i n  yz-plane,  However w e  a r e  most i n t e r e s t e d  
i n  t h e  case  of s m a l l  po l a r  angles ,  i t  w i l l  be assumed t h e  approximate 
expres s ion  (12) is  v a l i d  f o r  a l l  plane waves. This  means t h a t  a t  z = 0, 
t h e  wave can be assumed t o  be polar ized  e n t i r e l y  a long y-axis .  
2)  D ( < ; O )  i s  a complex homogeneous random f i e l d s  [3] with known 
-c 
c o r r e l a t i o n  func t ions  t h a t  are abso lu te ly  i n t e g r a b l e  over  a l l  r, This  
assures t h e  e x i s t e n c e  of t h e  s p e c t r a l  dens i ty  func t ions .  
4) As z -c 00, t h e  r e t a rded  s o l u t i o n  i s  taken ,  
The method used he re  fol lows tha t  used i n  t h e  theory  of t i m e  series 
a n a l y s i s ,  Like t h e  theory  of t i m e  series ana lys i s ,  t h e  Four i e r  t ransform 
of D ( Z  ; O )  does not  i n  gene ra l  e x i s t  because of t h e  cond i t ion  3. 
d i f f i c u l t y  can be overcome as done by Wiener [ 4 ]  by t r u n c a t i n g  D(rL ;O)* 
Thereforej  l e t  
This  
+ 
-b 
D(< ;O) r E R  
0 E R  
DR(<;OI = 1 
-+ 
. 
7 
where R = { x , y \  X/2 < x < - X/2, Y / 2  < y < - Y;2} a *  
square  sense  t h e  amplitude s p e c t r a l  d e n s i t y  funct ion** 
Define i n  t h e  mean 
,. 
I 
R 
Then through Four ie r  i n v e r s i o n  
-c --L 
Note t h a t  s i n c e  D (rL ; O )  i s  assumed t o  have zero  mean, s o  is D (K ; O ) .  
R R 
I ,  The a u t o c o r r e l a t i o n  is  obta ined  by 
where t h e  angular  b racke t s  are used t o  denote t h e  ensemble average.  Since 
K is  assumed t o  be a b s o l u t e l y  i n t e g r a b l e  over t h e  i n f i n i t e  range t h e r e  
e x i s t s  a non-negative s p e c t r a l  dens i ty  f z n c t i o n  de f i zed  by 
X- 
Y- 
4 
* 
Note t h a t  he re  X and Y are s p e c i f i c  va lues  of x ana y r e s p e c i i v e i y ,  not 
t hose  de f ined  by (2) 
**Fourier t ransform p a i r s  a r e  d i s t ingu i shed  by t h e i r  arguments, same symbols 
are used he re .  
8 
By Wiener-Khintchine theorem K ( & q ; O )  and S ( z  ;O) form a Four i e r  t ransform 
p a i r  [ 5 ] .  I n  case  D(rl;O) is a d d i t i o n a l l y  e rgodic  t h e  a u t o c o r r e l a t i o n  
- 
f u n c t i o n  def ined  by (16) can a l s o  be obta ined  through s p a t i a l  average 
wi th  p r o b a b i l i t y  1. 
The problem assumes t h a t  S ( 3 ; O )  (or  e q u i v a l e n t l y  K ( < p q ; O )  i s  given 
and t h e  expres s ions  of t h e  s p e c t r a l  d e n s i t i e s  and t h e  a u t o c o r r e l a t i o n s  
f o r  some f i n i t e  va lues  of z are des i r ed .  Now i n  t h e  h a l f  space z > 0, 
t h e  f i e l d s  are given by t h e  supe rpos i t i on  of many plane waves of t h e  form 
(12) .  With exp jot f a c t o r  supressed,  t h e s e  are given by 
+ 
where k, = (kx,ky). 
which, s i n c e  52 = 0, (19) reduces to  (15). Th i s  sets U = k L o  The f i r s t  
p rope r ty  of DxR(rL ; z )  and D 
t h e  ensemble, i t  concludes t h a t  both components have ze ro  mean f o r  any 
Above express ions  are c e r t a i n l y  v a l i d  when z = 0 a t  
+ +  
+ * 
(PI ;z) can be obtained by averaging (19) over  
YR 
vlalne of z, A s  i n d i c a t e d  ea r l i e r  t h e  case of i n t e r e s t  is when a l l  component 
waves propagate  i n  t h e  forward d i r e c t i o n  l i k e  t.he cons ide ra t ion  of Fresne l  
d i f f r a c t i o n  problems. In  o t h e r  words, D (k,;O) has  apprec i ab le  va lue  only  
- 
-c 
R 
9 
2 2 
0 
over  t h e  r eg ion  k < k, (corresponding t o  evanescent waves) can  be ignored.  
Therefore ,  t he  elements of t he  s p e c t r a l  ma t r ix  are obta ined  as 
s YY (Z* ; z )  S(’. ;O) cos2  
For t h e  s p e c i a l  case of  i s o t r o p i c  medium (20) shows t h a t  t h e r e  is  no depo la r i -  
z a t i o n  e f f e c t ;  and t h e  s p e c t r a l  dens i ty  and hence a l s o  t h e  a u t o c o r r e l a t i o n  
f u n c t i o n  are independence of z a s  found by o t h e r s  [ 2 ] .  When t h e  medium 
is a n i s o t r o p i c  t h e s e  are no longer  t r u e .  However, i t  i s  i n t e r e s t i n g  t o  
note  t h a t  t h e  sum 
which is  independent of z .  This  has some p r a c t i c a l  importance s i n c e  i t  
means t h a t  t h e  sum of s p e c t r a l  d e n s i t i e s  of or thogonal ly  po la r i zed  waves 
i s  uninf luenced by t h e  medium. 
S ince  t h e  Faraday r o t a t i o n  angle p lays  a dominant r o l e  i n  determining 
t h e  s p e c t r a l  d e n s i t y  func t ions  i t  w i l l  be d iscussed  i n  t h e  next s e c t i o n .  
4.  Dependence of Rota t ion  Angle 
L e t  be t h e  Faraday r o t a t i o n  angle  s u f f e r e d  by a wave propagat ing 
Z 
in z d i rec t ioz .  Prerr! (10) 
* 
w X Y  z P O E O  z 
2k0 
s2 = -  
Z 
10 
For a wave propagat ing i n  a gene ra l  d i r e c t i o n  w i t h  propagat ion cons t an t  
= (z', Jm), t h e  p o l a r i z a t i o n  of t h e  wave w i l l  r o t a t e  i n  t h e  kg 
right-handed sense  from y-axis  through an angle  
* 
-c 
Note t h a t  f o r  a f i x e d  z, fi Hence t h e  dependence on k,of 
t h e  r o t a t i o n a l  angle  is  through t h e  second term i n  (23) .  Because of t h e  
is  a cons tan t .  
Z 
appearance of t h e  dot  product it suggests  that  it is advantageous t o  r o t a t e  
t h e  coord ina te s  about z a x i s  so t h a t  y 'z  plane i s  p a r a l l e l  t o  t h e  p lane  
of magnetic meridian and R'z plane perpendicular  t o  i t .  For t h i s  coordi-  
n a t e  system kL Y = k Y Further, c o n s i s t e n t  w i t h  t h e  forward propa- 
g a t i o n  assumption k, can be ignored when compared wi th  k 
i n  an express ion  for t h e  r o t a t i o n a l  angle  which i s  only a f u n c t i o n  of k 
Le t  be t h e  angle  between t h e  displacement vec to r  a t  z = 0 and t h e  magnetic 
meridian plane.  Then 
+ +  
Y '  Y" 
This  r e s u l t s  1 0' 
Y "  - 
n 
L e t  
* where k i s  t h e  s p e c t r a l  w id th  of S(k ;O) i n  t he  plane of magnetic meridian 
b 
plane  (i=e, y s  component). Then (24) can be w r i t t e n  as 
b 
k 
f i (kYl ;z )  = (p + fi + - z - Y '  
'c kb 
c 
11 
(26) i n d i c a t e s  t h a t  i n  t he  reg ion  z << z 
over t h e  s p e c t r a l  width of i n t e r e s t  and the  s p e c t r a l  dens i ty  of t h e  f i e l d  
i s  not a f f e c t e d  apprec iab ly  by t h e  medium. When z has  t h e  same o rde r  of 
magnitude as z or even l a r g e r  than z t h e  s p e c t r a l  d e n s i t i e s  of t h e  f i e l d s  
a r e  expected t o  be s t r o n g l y  a f f e c t e d  by t h e  medium. 
n(ky, ;z) i s  nea r ly  cons tan t  c' 
C C' 
5. Cor re l a t ion  Funct ions 
The elements  of t h e  c o r r e l a t i o n  ma t r ix  can be c a l c u l a t e d  by t ak ing  
t h e  Four i e r  i nve r s ion  of t h e  r e spec t ive  elements  of t h e  s p e c t r a l  matr ix .  
As seen  from (20)' s i n c e  t h e  s p e c t r a l  d e n s i t i e s  are g iven  i n  t h e  product 
form the  convolut ion theorem can be used t o  compute t h e  c o r r e l a t i o n  func t ions .  
Assume t h a t  t h e  approximate express ion  (26) i s  v a l i d  t h e  convolut ion i n t e g r a l s  
can be carried out  e a s i l y ,  ob ta in ing  4 
- j2 (+  + Q  
(27) 
1 Z 
4 
- - K(<',V' - 2z/zckb;O) e 
c . 
12  
By add i t ion ,  it i s  seen  t h a t  K ( < ' , v ' ; z )  + K ([',q';z) = K ( < ' , q 1 ' ; 0 ) .  
S ince  t h e  Four i e r  t ransform is  unique t h i s  is j u s t  t h e  s p e c i a l  case  of (21). 
x ' x '  Y ' Y '  
6. Discussions 
As an example t a k e  f = 20 mc, f = 10 mc ,  fHz = 1 mc,  and k 
P b 
= O . l / m ,  
t h e  c r i t i c a l  d i s t a n c e  comes out  t o  be approximately 1200 km. Hence i t  seems 
t h a t  t h e  e f f e c t  of a n i s o t r o p i c  na ture  of t h e  medium may be important  i n  
ionospher ic  s t u d i e s  of i r r e g u l a r i t i e s  and i n  s t u d i e s  of moon r e f l e c t i o n s .  
As suggested i n  s e c t i o n  3, one poss ib le  way of e l imina t ing  t h e  in f luence  
of a n i s o t r o p i c  na tu re  of t h e  medium i s  through a d d i t i o n  of s p e c t r a  obta ined  
on or thogonal  antennas a s  shown by (21). 
The method desc r ibed  here  can  be extended t o  s tudying  t h e  s c a t t e r i n g  
4 
from i r r e g u l a r i t i e s  and from rough s u r f a c e s  and o t h e r  d i f f r a c t i o n  problems. 
v 
c 
I /  Y 
, c  . 
? c 
References 
L. D. Landan and E. M. Lifshitz, "Electrodynamics of Continuous Media" 
77, Pergamon Press; 1959. 
H. G. Booker, J. A. Ratcliffe and D. H. Shinn, "Diffraction from an 
Irregular Screen with Applications to Ionospheric Problems," Phil. 
Trans. Roy. SOC. A, V o l .  242, pp. 579-607; 1950. J. A. Ratcliffe, 
"Some Aspects of Diffraction Theory 
Ionosphere," Reports on Progress in 
1956. 
A. M. Yaglom, "Theory of Stationary 
Prent ice-Hall, Inc . , 1962. 
and Their Application to the 
Physics, Vol. 19, pp. 188-267; 
Random Functions," Chapter 3, 
N. Wiener, "Extrapolation, Interpolation and Smoothing of Stationary 
Time Series," p. 37, John-Wiley and Sons, 1949. 
D. Middleton, "Introduction to Statistical Communication Theory," 
Chapter 3, McGraw-Hi11 Book Co., 1960. 
13 
c 
h 
c 


Diffraction of random waves in a homogeneous anisotropic medium

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