Fields%20and%20Waves%20I

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Fields and Waves I

• Lecture 24
• Plane Waves at Oblique Incidence
• K. A. Connor
• Electrical, Computer, and Systems Engineering
  Department
• Rensselaer Polytechnic Institute, Troy, NY

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These Slides Were Prepared by Prof. Kenneth A.
Connor Using Original Materials Written Mostly by
the Following

• Kenneth A. Connor ECSE Department, Rensselaer
  Polytechnic Institute, Troy, NY
• J. Darryl Michael GE Global Research Center,
  Niskayuna, NY
• Thomas P. Crowley National Institute of
  Standards and Technology, Boulder, CO
• Sheppard J. Salon ECSE Department, Rensselaer
  Polytechnic Institute, Troy, NY
• Lale Ergene ITU Informatics Institute,
  Istanbul, Turkey
• Jeffrey Braunstein Chung-Ang University, Seoul,
  Korea

Materials from other sources are referenced where
they are used. Those listed as Ulaby are figures
from Ulabys textbook.
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Overview

• EM Waves in Lossless Media
• Wave Equation General Solution
• Energy and Power
• EM Waves in Lossy Media
• Skin Depth
• Approximate wave parameters
• Low Loss Dielectrics
• Good Conductors
• Power and Power Deposition
• Wave Polarization
• Linear, circular elliptical
• Reflection and Transmission at Normal Incidence
• Dielectric-Conductor Interface
• Dielectric-Dielectric Interface
• Multiple Boundaries
• Plane Waves at Oblique Incidence

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Example 1 Arbitrary Propagation Angle
The direction of E and of a electromagnetic
wave with 500nm are shown below. The wave
is traveling through air. The electric field has
a magnitude of 30 V/m. What are the E and H
phasors?
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Example 1
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Example 1
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Arbitrary Propagation Angle
In phasor form we have had
We can generalize this with
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Arbitrary Propagation Angle
For propagation in more than the z direction, let
us consider just adding x propagation, since that
is all we will need to do oblique incidence.
where we have left unspecified the unit vectors
for E H
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Oblique Incidence Parallel Polarization
For the first choice, we can assume that the
electric field in directed in the plane of
incidence. This is called parallel polarization
since E is parallel to this plane. Note that H is
only tangent to the boundary while E has both
normal and tangential components.
Ulaby
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Oblique Incidence Perpendicular Polarization
For the second choice, we can assume that the
electric field in directed out of the plane of
incidence. This is called perpendicular
polarization since E is perpendicular to this
plane. Note that E is only tangential while H has
both components.
Ulaby
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Oblique Incidence
With the two possible polarizations, we have two
sets of boundary conditions. Thus, they will
behave differently.
Note also that the combination of the two
polarizations gives us all possible vector
components for E and H.
Now we must apply the boundary conditions to
determine how the incident, reflected and
transmitted waves relate to one another.
VERY IMPORTANT POINT Because of the x-directed
propagation, the phase of the E and H fields vary
along the boundary. Thus, our first task is to
match the phase and then we will match the
amplitudes. The matching of the phase will allow
us to derive one of the most fundamental laws of
optics.
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Oblique Incidence Matching the Phase of the
Electric and Magnetic Fields at a Boundary
The incident electric field
The reflected electric field
To match the phase of the terms at z 0
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Oblique Incidence Matching the Phase of the
Electric and Magnetic Fields at a Boundary
Thus, we have that the angle of incidence equals
the angle of reflection, a result that all of us
have seen before. Now, we need to see what
happens to the transmitted angle.
Ulaby
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Oblique Incidence Matching the Phase of the
Electric and Magnetic Fields at a Boundary
Consider now all three waves incident,
reflected and transmitted
Matching the phases at z 0
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Oblique Incidence Matching the Phase of the
Electric and Magnetic Fields at a Boundary
This is Snells Law
To put it in its more normal form
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Snells Law
There are many useful wave representations
Ulaby
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Snells Law
Using the wave front representation, we can see
that Snells Law is required to match the wave
variations on the two sides of the boundary.
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Applying the Boundary Conditions for Both
Polarizations Gives the Reflection and
Transmission Coefficients
Ulaby
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Applying the Boundary Conditions for Both
Polarizations Gives the Reflection and
Transmission Coefficients
Perpendicular Polarization
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Applying the Boundary Conditions for Both
Polarizations Gives the Reflection and
Transmission Coefficients
Parallel Polarization
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Example 2 Oblique Incidence

• A plane wave described by
  is incident upon a
  dielectric material with 4.
• Write in phasor form.
• b. What are and ?
• c. What are and ?
• d. What are the reflection and transmission
  coefficients?
• e. Write the total electric field phasors in both
  regions.

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Example 2
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Example 2
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Example 2
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Example 2
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Critical Angle
For total reflection
Ulaby
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Example 3 Snells Law and Critical Angle
For visible light, the index of refraction for
water is n 1.33. If we put a light source 1
meter under water and observe it from above the
surface of the water, what is the largest
for which light will be transmitted? How large
will the circle of illumination be?
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Example 3
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Example 4 -- Polarization
Consider the same material properties and
incident angle as Example 2, but assume the
opposite polarization. a. What are the
reflection and transmission coefficients?
Which polarization has a lower reflection
coefficient (magnitude)? b. Now allow to
vary. At what value of is the wave
completely transmitted? (i.e. What's the
Brewster angle?)
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Example 4
Example 2
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Reflection as a Function of Angle
Note that the reflection varies with angle.
Perpendicular reflects more that Parallel. There
is also an angle for which there is no reflection
for parallel polarization.
Ulaby
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Brewsters Angle for Parallel Polarization
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Optical Fibers
Light is guided down the fiber.
Ulaby
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Optical Fibers
Cladding is added to eliminate surface problems
since part of the wave actually propagates
outside the core. Also note that the pulses
spread and decay due to a variety of losses.
Ulaby
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Rensselaer Other Info Sources

• Prof. E. F. Schubert http//www.ecse.rpi.edu/schu
  bert/Light-Emitting-Diodes-dot-org/chap22/chap22.h
  tm
• Prof. D. J. Wagner http//www.rpi.edu/dept/phys/Sc
  IT/
• Prof. F. Ulaby (From his CD)
• http//www.amanogawa.com/index.html
• Movies of Waves from Prof. H. C. Han at Iowa
  State http//www.ee.iastate.edu/7Ehsiu/em_movies.
  html

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From Prof. Schuberts Notes
This is why the sky is blue.
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Power and Energy
Note that power density (the Poynting Vector) is
not necessarily conserved across the boundary.
However, the total power is.
Because of the boundary conditions, the Poynting
Vector is conserved for perpendicular but not for
parallel polarization. All formulas are
summarized in Table 8-2 of Ulaby.
Ulaby