Download Brewster angles for magnetic media

International Journal of Infrared and Millimeter Waves, VoL 6, No. 3, t985
BREWSTER ANGLES FOR MAGNETIC MEDIA
C. Lee Giles* and Walter J. Wild**
Optical Sciences Center
Universityof Arizona
Tucson, A r i z o n a 85721
Received November 22, 1984
Brewster angles (angles for which there is no reflection
of an incident electromagnetic plane wave) will exist for
either
incident polarization
at a planar
interface separating two media
provided that either or both
of these
media are magnetic.
Except at certain angles (normal and
grazing
incidence)
and specific
material
properties,
a
Brewster
angle cannot exist simultaneously for both incident polarizations, but always exists for one polarization
or the other
for real values of permittivity
and permeability.
The existence
conditions for the Brewster angle
are readily
formulated in terms of relative
media permittivity and permeability, or relative media refractive index
and electromagnetic impedance.
When the ratio of permeabilities
and permittivities
is not unity,
the Brewster
angle
is no longer
orthogonal to the
transmitted
beam
angle.
From the viewpoint of the dipole oscillator model,
the contributions of the electric and magnetic
dipoles are
such that they do not independently dominate
the Brewster
angle
existence; rather the ratio of media permittivities
and/or
permeabilities determine when a Brewster
angle may
exist.
*Present address:
Naval Research Laboratory, Optical
Sciences Division, Washington, D.C. 20375.
**Also at IIT Research Institute, 10 W. 35th St., Chicago,
Illinois 60616
187
0195-9271/85/0300-0187504,50/0 zg 1985 Plenum Publishing CorporalJon
188
Giles and Wild
INTRODUCTION
Recently,
interesting
and unusual
properties
of the
r e f l e c t i o n a n d scattering of electromagnetic plane waves in
magnetic materials have been reported.
It has been shown
that for a planar
interface and the special case
of equal
refractive
indices for both magnetic media,
the amplitude
reflection coefficient is constant, and independent
of the
angle
of incidence
and incident polarization I .
For the
scattering of a plane wave
from a magnetic
sphere, the
scattered wave
can be zero for various
scattering angles
for certain specific
ratios of permittivity
and permeability. 2 This paper describes another interesting property
of nonabsorbing magnetic
materials, the existence
of a
Brewster
angle (incident
angle of
total
transmission or
zero
reflection) for either incident
polarization
of an
electromagnetic plane wave. This variation of the Brewster
angle with polarization
is not well-known. 3
This
is
because
most studies
of the Fresnel
reflection equations
begin with
the assumption
of unity permeability
which is
true for the visible and infrared regions of the spectrum.
In the submillimeter to radio-frequency parts of the spectrum one cannot necessarily ignor the permeability; indeed,
the permeability may be complex.4,5, 6
For the case where
dielectrics are considered, most
treatments
which incorporate the permeability usually go no further than a simple
statement of derived equations
(such as for the Brewster
angles)
and
there
is no detailed
discussion
of the
influence of the permeability or the physical
interpretation. 7,8
We expect
that some of the results
discussed
here can be verified experimentally with materials
such as
the Yttrium Iron Garnets. 6
We describe the material conditions that determine which
incident polarization
has a Brewster angle and show that
the Brewster angle cannot exist
simultaneously
for both
incident polarizations, except at normal and grazing incidence.
We also show that whether or not the Brewster angle
exists
for either
polarization
is determined
by the
relative strengths of the ratios of permittivity and permeability or ratios of refractive index and bulk impedance.
Further, in general the angle between the Brewster angle
and the transmitted wave deviates from
90 ~ whenever the
ratio of permittivities and permeabilities
is different
from unity.
ANALYSIS
Brewster Angles for Magnetic Media
t89
Consider the reflection and transmission of an incident
electromagnetic plane wave at a planar interface separating
two media, either or both of which m a y be magnetic.
The
constitutive relation between the electric field E and its
displacement field D is linear, isotropic,
homogeneous and
time-independent, i.e. D = EE where e is the medium permittivity. By magnetic, we mean that the constitutive relation
between the magnetic field
H and its induction field
B is
linear, isotropic, homogeneous and time-independent, i.e. B
= pH where
p is the medium permeability,
we assume
that
for either or both media the permeability p is not equal to
Po, the permeability of free space and that both e and p
are real and positive.
Note that materials
which closely
approximate such properties exist at specific frequencies
in the far infrared, millimeter,
and microwave spectral
regions.
I
El ,F1
E2F 2
\
B ~
\\,
!
O2
PLANEOF
INCIDENCE
%L
Fig. I. Incident polarizations, media permittivities
and permeabilities,
angles of incidence
and refraction, and Brewster angle are defined.
190
Giles and Wild
The reflection coefficients for either
p o l a r i z a t i o n are
easily derived. I0,11
We let r• and rll denote the electric
field reflection coefficients for the polarizations respectively p e r p e n d i c u l a r
and parallel
to the plane
of incidence.
See figure I for further clarification.
Define el,
UI and e2,
U 2 as the permittivity, p e r m e a b i l i t y
pairs for
the incident
and reflecting
media.
Further analysis
is
simplified
if we define dimensionless ratios
for permittivity and p e r m e a b i l i t y such that e = e2/~ I and U = U2/UI,
i.e. e is
the ratio
of p e r m i t t i v i t y
of
the
reflecting
medium to the p e r m i t t i v i t y of the incident medium and similarly for U.
Now e and U are independent of the choice of
units and can
be interpreted as the relative
strengths of
the
electric and magnetic dipole
contributions
of
each
media. 12 The reflection coefficients have the form8:
r•
rll
(~)
cos @I - cos 82
({U/e)
cos 81 + cos
=
(I)
@2
cos
81 - ( ~ )
cos 82
cos
@I + (U~7~) cos 82
=
(2)
where 81 and 82 are the angles of incidence and refraction.
The value of the reflection angle 81 for which r• or rll is
zero is defined as the Brewster angle for that p o l a r i z a t i o n
(all polarizations thus denoted are for the electric field)
and will be defined as 81 or 811. Using Snell's law, sin 81
= e ~ sin @2, and
setting equations
(1) and (2) equal to
zero, we
find the
following expressions for
the Brewster
angle:
polarization
El :
tan 2 81 = U (e-p)
l-ep
polarization
Ell
tan2
:
811
= E (pqe)
I-EU
(3)
(4)
Another form expresses the Brewster angles in terms
of the
relative
refractive index m and relative
bulk electromagnetic impedance z, where m ~ ~
and z ~ ~ / ~ .
~quations
(3) and (4) may now be written as
Brewster Angles for Magnetic Media
191
1-Z 2
polarization
E1 :
(5)
tan 2 81 = m 2
I -m 2
polarization
Ell
:
tan2 81t =
m2
z2-I
-z2
1-m 2
(6)
From these
equations, certain special cases for
the Brewster angle
are i m m e d i a t e l y obvious and reassuring.
Equations (3) and (4) yield
tan2 8jl
= g
81 does not exist,
le # I, H = 11
(7)
tan 2 8• = P
811 does not exist.
As is well-known, the Brewster angle for d i e l e c t r i c s
(Pl =
P2 = P = 1)exists only for the p a r a l l e l p o l a r i z a t i o n .
However, the a n a l o g o u s m a g n e t i c
case (g = I) has
a Brewster
angle only
for the p e r p e n d i c u l a r
p o l a r i z a t i o n . Inspection
of equations
(5) and (6)
s h o w that
for z =
I (impedance
matching), Brewster angles occur for both p o l a r i z a t i o n s at
normal incidence.
For m = I (refractive index matching),
Brewster
angles occur
for both p o l a r i z a t i o n s
at grazing
incidence. I However,
the existence
of the latter
case is
d i f f i c u l t to show
except in
the limit
of
one r e f r a c t i v e
index a p p r o a c h i n g the other.
General
c o n d i t i o n s for
the existence
of
the Brewster
angle
for
either or b o t h p o l a r i z a t i o n s
can be
readily
d e r i v e d from equations (3) and (4) or equations (5) and (6)
by requiring the right side of these equations to be positive and real.
From equations
(3) and
(4), we have
the
following existence conditions:
81
exists
if
H ~ e > I/p
or
p ~ e ~ I/p,
BII
exists
if
e > ~ > 1/e
or
e ~ ~ r I/e;
(8)
and from equations
(5) and
(6):
192
Giles and Wild
8•
exists if
[: i> ] ~
or
[: ~ I]
8,} exists if
[zm I> ] ~
or
[: <111>
(9)
These inequalities are graphically portrayed as decision
regions in figures 2 and 3.
Equations (8) and (9) show
that for real values of m and z or e and ~, a Brewster
angle always exists for one polarization. Note that many
Z)o
M.
)IELECTRIC)
L=I
I
0
i
E
Fig. 2.
Decision regions are illustrated for Brewster angles for different polarizations as determined
by the relative permittivity e and relative permeability ~.
The regions denoted by 81 define values
of e and ~ where the Brewster angle exists for the
electric field polarization perpendicular to the
plane of incidence.
Similar regions denoted by 8{{
are for the parallel polarization. The solid lines
indicate where a Brewster angle exists for both
polarizations. The dielectric case is denoted by the
dotted line.
Brewster Angles for Magnetic Media
193
values
of
(el,e 2) and
(~I,~2) could
represent
a single
p o i n t (c,~) in
figure 2. A similar
situation
exists for
(m,z) in figure 3.
In each figure, the dotted
line represents the d i e l e c t r i c (~I = ~2 = U = I) case. A c t u a l l y U = I
does not mean that both m e d i a
have to be d i e l e c t r i c s . In
fact, both m e d i a may be m a g n e t i c but
their p e r m e a b i l i t i e s
m u s t be equal. These m a t e r i a l s are then m a g n e t i c a l l y equivalent
but
e l e c t r i c a l l y dissimilar.
A similar
situation
occurs for e =
I. In figure 3, the p o i n t m = z =
I (both
media electromagnetically
identical) has a B r e w s t e r angle
~
0
_
0
o
I
m
Fig. 3.
Decision regions
are i l l u s t r a t e d for Brewster angles for d i f f e r e n t p o l a r i z a t i o n s as d e t e r m i n e d
by
the
relative
r e f r a c t i v e index
m and
relative
impedance
z.
The
regions
denoted
by
81 define
values of m and z where the Brewster angle exists for
the electric field p o l a r i z a t i o n p e r p e n d i c u l a r
to the
plane of incidence.
Similar regions
d e n o t e d by
8H
are
for the p a r a l l e l p o l a r i z a t i o n . The
solid lines
indicate
where
a B r e w s t e r angle
exists
for
both
p o l a r i z a t i o n s . The d i e l e c t r i c case is denoted
by the
dotted line.
194
Giles and Wild
for all angles.
A similar
p o i n t exists at
e = ~ = I in
figure 2.
For both p o l a r i z a t i o n s the critical angle 8c is defined
by
sin 0 c = m = e / ~ and exists if m < I or e~ ~
I. From
figures 2 and 3 we
see that
a c r i t i c a l angle
can coexist
w i t h a Brewster angle
p r o v i d i n g the above
c o n d i t i o n s are
satisfied. From the d e f i n i t i o n of the c r i t i c a l and Brewster
angles, we have the relations:
tan 81 =
{(1-~/e)
tan @c
tan 811 =
{(1-e/~)
tan 0c .
Since the e x i s t a n c e
c o n d i t i o n s for
81 and
8ff i m p l y that
respectively
~/e < I and
e/~ < I,
the Brewster
angle is
always smaller than the critical angle.
DISCUSSION
With figures 2 and 3, we can i n t e r p r e t the m a t e r i a l conditions which
d e t e r m i n e the existence of a Brewster angle
for
a particular polarization.
Figure 3 implies that if
both the r e f r a c t i v e index and impedance of
the r e f l e c t i n g
m e d i u m are j o i n t l y greater than or less than the refractive
index and
i m p e d a n c e of
the i n c i d e n t medium,
the Brewster
angle exists only for the p e r p e n d i c u l a r p o l a r i z a t i o n .
If,
on the other
hand, the
relative refractive index
and the
relative i m p e d a n c e
are not
j o i n t l y greater than u n i t y or
less than unity,
the Brewster
angle exists
only
for the
parallel polarization.
For the special cases of z = I and
m = I, the Brewster angle is r e s p e c t i v e l y at 8 = 0~ and 8 =
90 ~ for both p o l a r i z a t i o n s .
If we can d e s c r i b e the m a t e r i a l p r o p e r t i e s of e and ~ to
be
due
primarily
to
the electric
and m a g n e t i c
dipole
moments, 12 figure 2 offers some unusual
i n t e r p r e t a t i o n s of
the existence conditions.
The relative
m a g n i t u d e s of the
e l e c t r i c or m a g n e t i c dipoles do not
independently dominate
the Brewster
angle existence,
c o n t r a r y to what
one m i g h t
expect.
The s i t u a t i o n is more complicated.
For example,
one c o n d i t i o n for the e x i s t e n c e of 81 is that
the relative
m a g n e t i c dipole
s t r e n g t h of
the r e f l e c t i n g m e d i u m
to the
i n c i d e n t m e d i u m is greater than the similar
relative electric
dipole strength,
plus the a d d i t i o n a l
c o n d i t i o n that
the p r o d u c t
of the electric and magnetic
dipole strengths
of
the
reflecting
medium
is greater
than
the
similar
Brewster Angles for Magnetic Media
195
p r o d u c t for the i n c i d e n t medium.
Replace greater by lesser
in the p r e c e d i n g
a r g u m e n t and we have the
other e x i s t e n c e
c o n d i t i o n for
~•
C o n t r a s t this
to a
similar e x i s t e n c e
condition
for
Ell where
the
relative
magnetic
dipole
s t r e n g t h of the
incident medium
is also greater
than the
relative e l e c t r i c dipole strengths but the
d i p o l e strength
p r o d u c t of the r e f l e c t i n g
m e d i u m is less than that
of the
incident
medium.
In summary, the m a t e r i a l p r o p e r t i e s that
d e t e r m i n e the e x i s t e n c e of a Brewster angle for an i n c i d e n t
E 1 (TE)
:
Eli ( T ~ ) : ~ -
i\',\
i
I
l!o
~z~~ \
/
/
I
o
;o
~
iI
/
~176
#.
90 ~
6o ~ /
- " ",,~ / / "
/
./o{,,
60 %#
",, ~
(DIELECTRIC)
~
12oL..._
', "~-..,8oo
~
.. 120 ~
Fig. 4.
Curves of
constant ~ ~ 8 +
82 are
illustrated as a
function of e and p.
When either e = I
or p = I, we
have the w e l l - k n o w n c o n d i t i o n e = 90 ~ .
If e = p (impedance m a t c h i n g ) , ~ = 8 = 0 ~ (Brewster's
angle at normal incidence); and if e =
I/p
(refractive
index m a t c h i n g ) , e
= 180 ~ (Brewster's
angle at
g r a z i n g incidence).
Giles and Wild
196
p o l a r i z a t i o n are more complex that might be expected 9
But
simple i n e q u a l i t i e s
b e t w e e n the relative
p e r m i t t i v i t y and
the
relative p e r m e a b i l i t y or between the
relative refractive
index and
the relative impedance
provide d e f i n i t i v e
e x i s t e n c e conditions.
It is also i n t e r e s t i n g to consider how
the p e r m e a b i l i t y
changes the o r t h o g o n a l i t y c o n d i t i o n
between
the Brewster
angle
and
t r a n s m i t t e d beam
angle, i.e.,
how
does
~+82
b e h a v e (see figure I) for general
e and ~?
R e f e r r i n g to
e q u a t i o n s (3) and (4), we easily see that
polarization
El :
cos(~+@ 2) =
{(~/e)
I-l~
(I0)
I-1-I
polarization
Eli .-
cos(8+@ 2) =
~(e/~)
1-e
Figure 4 i l l u s t r a t e s a family
of curves
of e E 8+82 for
v a r i a b l e e,~. We see that e = 90 ~ only when e = I or ~ = I.
These values of e will vary from 0 ~ to
180 ~ , d e p e n d i n g on
e and ~.
For the case where
e = ~ we have
the Brewster
angle
at normal incidence. 4
For the case where
e = I/~
(discussed at length in reference I), the Brewster angle is
at extreme grazing incidence; this may be a singular situation
if e1~ I < c2~ 2 (or
e < I/~) since a c r i t i c a l angle
also
exists--the
Brewster
angle
always
precedes
the
c r i t i c a l angle but if e = I/~ each "coexists" at 81 =
8 =
90 ~
All curves pass through the p o i n t ~ = ~ = I; this is
not a p a t h o l o g i c a l
situation, rather,
no e l e c t r o m a g n e t i c
i n t e r f a c e exists for these m a t e r i a l parameters.
Realizing
that m a g n e t i c
dipoles will
radiate the
same
as electric
dipoles, the c o n t r i b u t i o n of each (parameterized by
~ and
e) will lead to the
results shown in figure 4.
We should
also
point
out
that
for a n o n l i n e a r
optical
material
w h e r e b y e and/or ~ can be e l e c t r i c a l l y (or thermally) m o d u lated, a form
of m i l l i m e t e r
wave optical "switch"
can be
devised.
This, however, can only be realized if
suitable
m a t e r i a l s can be located or fabricated.
We believe that this work will i n t e r e s t m i l l i m e t e r wave
researchers concerned
with the d e v e l o p m e n t
and c h a r a c t e r ization
of m a g n e t i c
materials.
Further a p p l i c a t i o n s
of
this and p r e v i o u s studies
include
interesting scattering
properties
of m a g n e t i c
a e r o s o l s for m i l l i m e t e r
and
FIR
lasers, and the a p p l i c a t i o n to Brewster
angle m e a s u r e m e n t s
BrewsterAnglesforMagneticMedia
~97
for the determination of the optical properties of magnetic
media (similar to that for dielectrics in the visible). 2,8
ACKNOWLEDGEMENT
We would like to acknowledge
A. Lohmann.
useful
suggestions
made by
REFERENCES
I.
C.L.
Giles and W. J. Wild,
"Fresnel reflection
and
transmission at a planar boundary from media
of equal
refractive
indices," Appl.
Phys.
Lett. 40 (3), 210
(1982).
2. M. Kerker, D-S. Wang and C. L. Giles, "Electromagnetic
scattering by magnetic spheres," J. Opt. Soc.
Am. 73,
765 (1983).
3. R. F. Harrington, Time-Harmonic Electromagnetic Fields
(McGraw-Hill, New York, 1961) p. 59.
4. G.T.
Ruck,
D.E.
Barrick,
W.D.
Stuart,
and
C.K.
Kirchbaum,
Radar Cross Section Handbook,
(Plemum, New
York, 1970), p. 624.
5. A. Sommerfeld, Optics (Academic Press, New York, 1949)
p. 18.
6. B. Lax and K.J. Button, Microwave Ferrites and Ferrimagnetics, (New York, McGraw-Hill, 1962) Chap. 7.
7. W . T .
Doyle, "Graphical Approach to Fresnal's Equation
for Reflection and Refraction of Light," Am.
J. Phys.,
48(8), 643 (1980).
8. M.
Zahn,
Electromagnetic Field Theory:
A Problem
Solving Approach, (Wiley) p. 540, 2nd Ed. (1979).
9. R. Chiarelli,
S. Faetti, and L. Frounaoui,
"Determination of the Molecular Orientation at the Free Surface
of Liquid
Crystals from Brewster
Angle Measurements,"
Opt. Comm., 46 (I), 9 (1983).
10. E. Hecht and A. Zajac,
Optics
(Addison-Wesley,
Reading, 1979) p. 73.
11. R. K. Wagsness, Electromagnetic Fields 2nd Ed. (Wiley,
New York, 1975) pp. 464, 469.
12. M. Born and E. Wolf,
Principles of Optics
5th Ed.
(Pergamon Press, New York, 1975) p. 84.