Polarization:
The property of a radiated electromagnetic wave describing
the time varying direction and relative magnitude of the
electric field vector.
Consider a plane wave traveling in the -z direction
Linear Polarization:
Circular Polarization:
CW: Clockwise, CCW: Counter clockwise.
Elliptical polarization:
Considerations:
Linear Pol.:
1. Only one component.
2. Two orthogonal linear components that are in time
phase or nx1800 out of phase. (n=integer)
Circular Pol.:
1. Must have two orthogonal linear components.
2. Two components must have the same magnitude.
3. Two components must have a time phase difference of
odd multiples of 900.
Elliptical Pol.:
1. Must have two orthogonal linear components.
2. Two components can have the same or different
magnitude.
3. If not the same magnitude: The phase difference must
not be 00 or nx1800 (linear). If the same magnitude: The
phase difference must not be odd multiples of 900
(circular).
Reflection and Transmission
Normal Incidence, Lossless Media:
Define,
Where Γb and Tb are the reflection and transmission
coefficients.
Also,
Imposing the boundary conditions at the interface,
Solving,
Away from the interface,
Where l1 and l2 are positive distances measured from the
interface to medium 1 and 2 respectively.
Power densities,
Ex:
A uniform plane wave in free space is incident normally
upon a flat lossless medium with er=2.56 (polystyrene).
Determine the reflection and transmission coefficients and
the power densities in each medium. Assume that the
amplitude of the incident wave is 1mV/m.
Ans:
The field expressions,
The swr,
If µ1=µ2,
Oblique Incidence, Lossless Media:
Perpendicular (Horizontal or E-Pol) Polarization:
where
Similarly,
and
Also,
and
From,
Leads to
(Magnitudes of reflection coeff.
and transmission coeff.)
Parallel (Vertical or H-Pol) Polarization:
(Magnitudes of the reflection and transmission coeff.)
Total Transmission (Brewster Angle):
Perpendicular Pol:
or
Using the Snell’s law of refraction,
or
Then,
or
If µ1=µ2,
Since for most mediums µ=µ0. No Brewster angle.
Parallel (Vertical) Pol:
If µ1=µ2,
Brewster angle exists for parallel (vertical pol) pol.
Ex:
A parallel pol. wave radiated from a submerged submarine
impinges upon a water air interface. Assuming the water is
lossless, er=81, and the wave approximates to planar at the
interface, determine the angle of incidence to allow
complete transmission of the energy.
Ans:
Fora ir to water interface,
Hence,
Total Reflection (Critical Angle):
Perpendicular (Horizontal) Pol:
The previous eqn holds for
or
This is known as the critical angle. The condition,
must hold.
If µ1=µ2,
Provided that
Thus, for µ1=µ2 , critical an
gle exists if the wave propagates from a dense to less
dense medium.
Ex:
A perpendicularly polarized wave from a submegred
submarine impinges upon a water to air interface.
Assuming the water being lossless, er=81, and the wave
approximates to a planar, determine the angle of incidence
to allow complete frelection of the energy.
Ans:
What happens to angle of refraction and to the
propagation of the wave when the angle of incidence is
equal to or greater than the critical angle ?
The angle of refraction reduces to
and
Also, the transmitted fields,
(Constant phase and amplitude planes for critical and
above crital incident angles.)
The constant phase planes of the wave are parallel to the z
axis. This wave is referred to as “surface wave”.
The average power density at the critical angle,
When the incident angle is greater than the critical angle,
Which means
No physically realizable angle.
The fields are
where
This is also a surface wave.
Tightly bound slow surface wave.
Ex:
Determine the range of values of the dielectric constant of
a dielectric slab of thickness t, so that when a wave is
incident on it from one of its ends at an oblique angle
0<θi<900, the energy of the wave in the dielectric is
contained within the slab. The geometry of the problem is
shown below.
Ans:
θr>θc.
or
Solving this equation gives
Parallel (Vertical) Pol:
The procedure used to derive the critical angle and to
examine the properties for perpendicular (horizontal) pol.
can be repeated for parallel (vertical) pol.
The only limitation for critical angle is
Lossy Media:
Normal Incidence, Conductor Conductor Interface:
(Fields and current density distribution)
Ex:
A uniform plane wave whose E-field has an x component
with an amplitude at the interface of 10-3V/m, is traveling
in a free space medium, and is normally incident upon a
lossy flat earth as shown in fig. Above. The constitutive
parameters of the earth are e2=9e0, µ2=µ0, σ2=10-1 S/m.
Determine the variation of the conduction current density
in the earth at the frequency of 1MHz.
Ans:
At 1MHz,
On either side of the interface, the total E-field is
where
Thus,
and
The conduction current density at the surface of the earth
is
or
The magnitude of the current density indide the earth
varies as
where