1. ENE 428 Microwave Engineering
Lecture 3 Polarization, Reflection and Transmission at normal incidence RS RS

2. Uniform plane wave (UPW) power transmission
from
W/m2 RS

3. field in the slab is given by
Example 6.7: Consider an electric field incident on a copper slab such that the
field in the slab is given by
V/m
We want to find the average power density.
Since copper is a good conductor, we can use
The intrinsic impedance is
Therefore,
At the surface (z = 0) the power density is 300 W/m2. But after only 1 skin
depth, in this case 21 m, the wave’s power density drops to e-2 (13.5%) of
its surface value, or 41 W/m2 in this case. RS

4. Polarization
UPW is characterized by its propagation direction and frequency.
Its attenuation and phase are determined by medium’s parameters.
Polarization determines the orientation of the electric field in a fixed spatial plane orthogonal to the direction of the propagation.
Specifying only the electric field direction is sufficient since magnetic field is readily found from using Maxwell’s equation RS

5. Linear polarization Consider in free space,
At plane z = 0, a tip of field traces straight line segment called “linearly polarized wave” RS

6. Linear polarization
A pair of linearly polarized wave also produces linear polarization
At z = 0 plane
At t = 0, both linearly polarized waves
have their maximum values. RS

7. Linear polarization
The tilt angle  (tau) is the angle the line makes with the x-axis
The axial ratio is the ratio of the long axis of an ellipse to the short axis RS

8. More generalized linear polarization
More generalized of two linearly polarized waves,
Linear polarization occurs when two linearly polarized waves are
Linear polarization is a special case of elliptical polarization that has an infinite axial ratio
in phase
out of phase RS

9. Elliptically polarized wave
Superposition of two linearly polarized waves that
If x = 0 and y = 45, we have RS

10. Circularly polarized wave
occurs when Exo and Eyo are equal and
Right hand circularly polarized (RHCP) wave
Left hand circularly polarized (LHCP) wave
Left and right are referred to as the handedness of wave polarization RS

11. Circularly polarized wave
Phasor forms:
for RHCP,
for LHCP,
from
Note: There are also RHEP and LHEP RS

12. Ex1 Given ,determine the polarization of this waveRS

13. Ex2 The electric field of a uniform plane wave in free space is given by , determine
The magnetic field intensity RS

14. d) Describe the polarization of the waveRS

15. Reflection and transmission of UPW at normal incidenceRS

16. Incident wave
Normal incidence – the propagation direction is normal to the boundary
Assume the medium is lossless, let the incident electric field to be
or in a phasor form
since
then we can show that RS

17. Transmitted wave Transmitted wave
Assume the medium is lossless, let the transmitted electric field to be
then we can show that RS

18. Reflected wave (1) From boundary conditions, At z = 0, we have and
 1 =  are media the same? RS

19. Reflected wave (2) There must be a reflected wave and
This wave travels in –z direction. RS

20. Reflection and transmission coefficients (1)
Boundary conditions (reflected wave is included)
from
therefore at z = 0
(1) RS

21. Reflection and transmission coefficients (2)
Boundary conditions (reflected wave is included)
from
therefore at z = 0
(2) RS

22. Reflection and transmission coefficients (3)
Use Eqns. (1) and (2) to eliminate , we’ll get
Reflection coefficient
Transmission coefficient RS

23. Types of boundaries: perfect dielectric and perfect conductor (1)
From 
Since 2 = 0 then  = -1 and Ex10+= -Ex10-  RS

24. Types of boundaries: perfect dielectric and perfect conductor (2)
This can be shown in an instantaneous form as
Standing wave RS

25. Standing waves (1) When t = m, Ex1 is 0 at all positions.
and when z = m, Ex1 is 0 at all time.
Null positions occur at RS

26. Standing waves (2) Since and , the magnetic field is or .
Hy1 is maximum when Ex1 = 0
So, E and H are said to be 90o out of phase. There will be no power transmission on either side of the media
Poynting vector RS

27. Power transmission for 2 perfect dielectrics (1)
Then 1 and 2 are both real positive quantities and 1 = 2 = 0 
Average incident power densities RS

28. Ex3 Let medium 1 have 1 = 100  and medium 2 have 2 = 300 , given Ex10+ = 100 V/m. Calculate average incident, reflected, and transmitted power densities RS

29. Wave reflection from multiple interfaces (1)
Wave reflection from materials that are finite in extent such as interfaces between air, glass, and coating
At steady state, there will be 5 total waves RS

30. Wave reflection from multiple interfaces (2)
Assume lossless media, we have
then we can show that RS

31. Wave impedance w (1)
Use Euler’s identity, we can show that RS

32. Wave impedance w (2) Since from B.C. at z = -l we may write
Using eqns (1a) and (2a) to eliminate , we’ll get … RS

33. Input impedance in
solve to get RS

34. Power Transmission and Reflection
Pin Pt Pr
The power in region 2 stays constant in steady-state; power leaves
that region to form the reflected and transmitted waves, but is
Immediately replenished by the incident wave (from region 1)
If  = 0, then there’ll be total transmission.
And  = 0 when in = 1 , or the input impedance is matched to that
of the incident medium. So how do we achieve this? RS

35. Half-wave matching method
Suppose
and
Therefore,
and so
Hence, the 2nd region thickness is the multiple “half-wavelength” as measured in that medium
Using eqn:
We’ll get
when
The general effect of a multiple half-wave is to render the 2nd region immaterial to the results on reflection and transmission. Equivalently we have a single interface problem involving 1 and 3 RS

36. RS

37. Refractive index
Under lossless conditions, RS

38. Homework 6.32: Given V/m, find the polarization and handedness.
6.38: Suppose medium 1 (z < 0) is air and medium 2 (z > 0) has r = 16. The trans-
mitted magnetic field intensity is known to be Ht = 12cos(t – β2z)ay mA/m.
Determine the instantaneous value of the incident electric field. (b) Find the
reflected time-averaged power density
6.48: A 100-MHz TE polarized wave with amplitude 1.0 V/m is obliquely incident
from air (z < 0) onto a slab of lossless, nonmagnetic material with r = 25 (z > 0). The angle of incidence is 40o. Calculate (a) the angle of transmission, (b) the reflection and transmission coefficients, and (c) the incident, reflected and transmitted fields. RS