1. Wave Incidence at Oblique angles
Dr. S. Cruz
Wave Incidence at Oblique angles
Sandra Cruz-Pol
ECE Dept. UPRM
Wave incidence at oblique angles

2. Quiz VIERNES Uno de los problemas asignados en el prontuario
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3. Wave Incidence Normal incidence Wave arrives at 90o from the surface
Dr. S. Cruz
Wave Incidence
Normal incidence
Wave arrives at 90o from the surface
Standing waves
Oblique incidence (lossless)
Wave arrives at an angle
Snell’s Law and Critical angle
Two types: Parallel or Perpendicular
Brewster angle
Wave incidence at oblique angles

4. Oblique incidence y First, we need to define:
Dr. S. Cruz
Oblique incidence
First, we need to define:
Expression for uniform plane wave traveling in any direction
The Normal , an
Plane of incidence
Angle of incidence x
Medium 1 : e1 , m1
Medium 2 : e2, m2 qr qt qi y z
z=0
Wave incidence at oblique angles

5. Oblique incidence Uniform plane wave in general form
Dr. S. Cruz
Oblique incidence
Uniform plane wave in general form
the radius or position vector
the wave number or propagation vector
For lossless unbounded media, k = b
Wave incidence at oblique angles

6. Oblique Incidence Medium 1 : e1 , m1 Medium 2 : e2, m2 kr kt qr qt y
Dr. S. Cruz
Oblique Incidence x
Medium 1 : e1 , m1
Medium 2 : e2, m2 kr kt qr qt z y qi
z=0 ki
Wave incidence at oblique angles

7. Look at the direction of travel
Dr. S. Cruz
Look at the direction of travel x
Medium 1 : e1 , m1
Medium 2 : e2, m2 kr kt qr qt z y qi
kix ki
z=0
kiz
Wave incidence at oblique angles

8. Expression for Waves kiz kix ki qi Dr. S. Cruz
Wave incidence at oblique angles

9. Tangential E must be Continuous
Dr. S. Cruz
Tangential E must be Continuous
From this we know that frequency is a property of the wave. So is color.
So 700nm is not always red!!
Give example of bathing suit.
Wave incidence at oblique angles

10. Snell Law Equating, we get or
Dr. S. Cruz
Snell Law
Equating, we get or
where, the index of refraction of a medium, ni , is defined as the ratio of the phase velocity in free space (c) to the phase velocity in the medium.
Wave incidence at oblique angles

11. Critical angle, qc …All is reflected
Dr. S. Cruz
Critical angle, qc …All is reflected
When qt =90o, the refracted wave flows along the surface and no energy is transmitted into medium 2.
The value of the angle of incidence corresponding to this is called critical angle, qc.
If qi > qc, the incident wave is totally reflected.
Wave incidence at oblique angles

12. Dr. S. Cruz
Fiber optics
Light can be guided with total reflections through thin dielectric rods made of glass or transparent plastic, known as optical fibers.
The only power lost is due to reflections at the input and output ends and absorption by the fiber material (not perfect dielectric).
Wave incidence at oblique angles

13. Dr. S. Cruz
Optical fibers have cylindrical fiber core with index of refraction nf, surrounded by another cylinder of lower, nc < nf , called a cladding.
For total reflection:
[Figure from Ulaby, 1999]
Acceptance angle
Wave incidence at oblique angles

14. Exercise: optical fiber
Dr. S. Cruz
Exercise: optical fiber
An optical fiber (in air) is made of fiber core with index of refraction of 1.52 and a cladding with an index of refraction of 1.49.
Find the acceptance angle:
Answer: 17.5 degrees
Wave incidence at oblique angles

15. Parallel polarization
Dr. S. Cruz
Parallel polarization
It’s defined as E is || to incidence plane x
Medium 1 : e1 , m1 Er Et kr kt qr qt z qi y z Ei
z=0 ki
Medium 2 :
e2, m2
Wave incidence at oblique angles

16. Equating for continuity, the tangent fields
Dr. S. Cruz
Equating for continuity, the tangent fields
Which components are tangent to the interface between two surfaces?
y and x
At z = 0 (interface):
Wave incidence at oblique angles

17. Reflection and Transmission Coefficients: Parallel Incidence
Dr. S. Cruz
Reflection and Transmission Coefficients: Parallel Incidence
Reflection
Wave incidence at oblique angles

18. Reflection and Transmission Coefficients: Perpendicular Incidence
Dr. S. Cruz
Reflection and Transmission Coefficients: Perpendicular Incidence
Wave incidence at oblique angles

19. Java Animation http://www.amanogawa.com/archive/Oblique/Oblique-2.html
Dr. S. Cruz
Java Animation
Wave incidence at oblique angles

20. Exercise A plane wave in air with
Dr. S. Cruz
Exercise
A plane wave in air with
Is incident upon planar surface of nonmagnetic dielectric material with er=4, on z>0, Find
The polarization of the incident wave
The angle of incidence
The time-domain expressions for the reflected electric and magnetic fields.
The average power density carried by the wave in the dielectric medium.
Wave incidence at oblique angles

21. Exercise A plane wave in free space with
Dr. S. Cruz
Exercise
A plane wave in free space with
Is incident upon planar surface of nonmagnetic lossless dielectric material with er=4, on z>0, Find
The polarization of the incident wave
The angle of incidence and transmission
The reflection and transmission coefficients.
The E field in reflected and in dielectric
Brewster angle
Wave incidence at oblique angles

22. Brewster angle, qB * qB is known as the polarizing angle
Dr. S. Cruz
Brewster angle, qB
Is defined as the incidence angle at which the reflection coefficient is 0 (all transmission).
The Brewster angle does not exist for perpendicular polarization for nonmagnetic materials.
* qB is known as the polarizing angle
Wave incidence at oblique angles

23. Reflection vs. Incidence angle.
Dr. S. Cruz
Reflection vs. Incidence angle.
Reflection vs. incidence angle for different types of soil and parallel or perpendicular polarization.
Wave incidence at oblique angles

24. Exercise (Brewster angle)
Dr. S. Cruz
Exercise (Brewster angle)
A wave in air is incident upon the flat boundary of a nonmagnetic soil medium with er=4, at qi=50o.
Find reflection and transmission coefficients for both incident polarizations, and the Brewster angle.
Answers:
EMAG
Wave incidence at oblique angles

25. Summary Property Normal Incidence Perpendicular Parallel
Dr. S. Cruz
Summary
Property
Normal Incidence
Perpendicular
Parallel
Reflection coefficient
Transmission coefficient
Relation
Power Reflectivity
Power Transmissivity
Snell’s Law:
Wave incidence at oblique angles