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4/27/2015
Instructor
Dr. Raymond Rumpf
(915) 747‐6958
[email protected]
EE 4395/5390 – Special Topics
Computational Electromagnetics
Lecture #3
Electromagnetic Principles
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Lecture 3
Slide 1
Outline
•
•
•
•
Wave vectors
Polarization
Index Ellipsoids
Electromagnetic behavior at an interface
– Phase matching at an interface
– Critical angle and Brewster’s angle
– Reflection and transmission
• Image theory
• Lenses
Lecture 3
Slide 2
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Wave Vectors
Lecture 3
Slide 3

Wave Vector k
A wave vector conveys two pieces of information at the same time. First, its orientation describes the direction of the wave. It is perpendicular to the wave front. Second, its magnitude conveys the spatial period of the wave. It is 2
divided by the spatial period of the wave (wavelength).

 

E  r   E0 exp jk  r



r  xxˆ  yyˆ  zzˆ
position vector
 2 2 n
k 


0

k  k x xˆ  k y yˆ  k z zˆ
Lecture 3

k
If the frequency of the wave is known and constant, and it usually is, the magnitude of the k vector conveys the refractive index of the material the wave is in. Slide 4
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The Complex Wave Vector
A wave travelling the +z direction can be written in terms of the wave number k as


E  z   E0 e jkz
k  k   jk 
Substituting this back into the wave solution yields

 j k  jk  z  jk z  k z
E  z   E0 e 
 E0 e e
oscillation
growth/decay
Lecture 3
Slide 5
 and 
A wave travelling the +z direction can also be written in terms of a propagation constant  and an attenuation coefficient  as

  z j  z
E  z   E0 e
e
oscillation
growth/decay
We now have physical meaning to the real and imaginary parts of the wave vector.
k    j
Lecture 3
k’ = Re[k]  phase term

2


2 n
0
k’’ = Im[k]  attenuation term
Slide 6
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1D Waves with Complex k
Purely Real k
• Uniform amplitude
• Oscillations move power
• Considered to be a propagating wave
Purely Imaginary k
• Decaying amplitude
• No oscillations, no flow of power
• Considered to be evanescent
Complex k
• Decaying amplitude
• Oscillations move power
• Considered to be a propagating wave (not evanescent)
Lecture 3
Slide 7
2D Waves with Doubly Complex k
Real k x
Real k y
Imaginary k x
x
x
x
y
Imaginary k y
Complex k x
x
y
y
y
x
x
y
y
Complex k y
x
Lecture 3
y
x
y
x
y
Slide 8
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Polarization
Lecture 3
Slide 9
What is Polarization?
Polarization is that property of a radiated electromagnetic wave which describes the time‐varying direction and relative magnitude of the electric field vector.
Linear Polarization (LP)
Circular Polarization (CP)
Left‐Hand Circular Polarization (LCP)
Lecture 3
Slide 10
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Handedness Convention
As Viewed From Source
As Viewed From Receiver
Polarization is taken as the time‐varying electric field view with the wave moving away from you. Primarily used in engineering and quantum physics.
source
Polarization is taken as the time‐varying electric field view with the wave coming toward you. Primarily used in optics and physics.
RCP
RCP
source
receiver
receiver
LCP
LCP
source
source
receiver
receiver
Lecture 3
http://en.wikipedia.org/wiki/Circular_polarization
Slide 11
Linear Polarization
A wave travelling in the +z direction is said to be linearly polarized if: 

E  x, y, z   Pe jk z z

P   sin   xˆ   cos   yˆ
P is called the polarization vector.
aˆ
For an arbitrary wave,
 
 
E  r   Pe  jk r

P   sin   aˆ   cos   bˆ

aˆ  bˆ  k
bˆ
All components of P have equal phase.
Lecture 3

k

k
Slide 12
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Circular Polarization
A wave travelling in the +z direction is said to be circularly polarized if: 

E  x, y, z   Pe jkz z

P  xˆ  jyˆ

P is called the polarization vector.
For an arbitrary wave,
j
LCP
 
  jk r
E  r   Pe

P  aˆ  jbˆ

aˆ  bˆ  k
RCP
j

The two components of P have equal amplitude and are 90 out of phase.

k

k
Lecture 3
Slide 13
Summary
In general, a wave travelling in the +z direction can be written as

j
E  x, y, z    Ex e jx xˆ  E y e y yˆ  e  j  z



1
j
H  x, y, z    Ex e jx xˆ  E y e y yˆ  e  j z



 


LPx

 

LPy
RCP
LCP
Elliptical
Ex = ER + EL
Ey = ER – EL
RH: ER>EL
LH: ER<EL
Ex and Ey
Ey = 0
Ex = 0
Ex=Ey=ER
Ex=Ey=EL
x
0
0
0
0
y
0
0
-/2
+/2
Lecture 3
Slide 14
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LPx + LPy = LP45
A linearly polarized wave can always be decomposed as the sum of two orthogonal linearly polarized waves that are in phase.
Lecture 3
Slide 15
LPx + jLPy = CP
A circularly polarized wave is the sum of two linearly polarized waves that are 90° out of phase.
Lecture 3
Slide 16
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RCP + LCP = LP
A linearly polarized wave is can be expressed as the sum of two circularly polarized waves. The phase between the CPs determines the tilt of the LP.
Lecture 3
Slide 17
Why is Polarization Important?
• Different polarizations can behave differently in a device
• Orthogonal polarizations will not interfere with each other
• Polarization becomes critical when analyzing devices on the scale of a wavelength
Lecture 3
Slide 18
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Poincaré Sphere
The polarization of a wave can be mapped to a unique point on the Poincaré sphere.
RCP
Points on opposite sides of the sphere are orthogonal.
‐45° LP
90° LP
See Balanis, Chap. 4.
+45° LP
0° LP
LCP
Lecture 3
Slide 19
TE and TM
We use the labels “TE” and “TM” when we are trying to describe the orientation of a linearly polarized wave relative to a device.
TE/perpendicular/s – the electric field is polarized perpendicular to the plane of incidence.
Lecture 3
TM/parallel/p – the electric field is polarized parallel to the plane of incidence.
Slide 20
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Calculating the Polarization Vectors
Incident Wave Vector
sin  cos  

kinc  k0 ninc  sin  sin  
 cos  
Right‐handed
coordinate system
x
y
z
Surface Normal
0
nˆ  0
1 
Unit Vectors Along Polarizations
 aˆ y


aˆTE   nˆ  kinc

 nˆ  k
inc


aˆ  k
ˆaTM  TE inc
aˆTE  kinc
  0
  0
Composite Polarization Vector

P  pTE aˆTE  pTM aˆTM
In CEM, we make

P 1
Lecture 3
Slide 21
Index Ellipsoids
Lecture 3
Slide 22
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Dispersion Relations
The dispersion relation for a material relates the wave vector to frequency. Essentially, it tells us the refractive index as a function of direction through a material.
It is derived by substituting a plane wave solution into the wave equation.
For an ordinary linear homogeneous and isotropic (LHI) material, the dispersion relation is:
ka2  kb2  kc2  k02 n 2
This can also be written as:
ka2  kb2  kc2
 k02  0
2
n
Lecture 3
Slide 23
Index Ellipsoids
From the previous slide, the dispersion relation for a LHI material was:
ka2  kb2  kc2  k02 n 2
This defines a sphere called an “index ellipsoid.”
cˆ
The vector connecting the origin to a point on the sphere is the k‐vector for that direction. Refractive index is calculated from this.

k  k0 n
index ellipsoid
For LHI materials, the refractive index is the
same in all directions.
Think of this as a map of the refractive index as a function of the wave’s direction through the medium.
Lecture 3
bˆ
aˆ
Slide 24
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Index Ellipsoids for Uniaxial Materials
Observations
cˆ
nO
nE
nE
nO
bˆ
aˆ
nO
• Both solutions share a common axis.
• This “common” axis looks isotropic with refractive index n0 regardless of polarization.
• Since both solutions share a single axis, these crystals are called “uniaxial.”
• The “common” axis is called:
o Optic axis
o Ordinary axis
o C axis
o Uniaxial axis
• Deviation from the optic axis will result in two separate possible modes.
 ka2  kb2  kc2
 k 2  k 2

 k02  a 2 b  k02   0

2
n
n

O
E


Lecture 3
Slide 25
Index Ellipsoids for Biaxial Materials
Biaxial materials have all unique refractive indices. Most texts adopt the convention where
na  nb  nc
The general dispersion relation cannot be reduced. 
cˆ
Notes and Observations
• The convention na<nb<nc causes the optic axes to lie in the a‐c plane.
optic axes
aˆ
Lecture 3
• The two solutions can be envisioned as one balloon inside another, pinched together so they touch at only four points.
bˆ
• Propagation along either of the optic axes looks isotropic, thus the name “biaxial.”
Slide 26
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Direction of Power Flow
Isotropic Materials
Anisotropic Materials

P
y
y

k

k
x


P
x

Phase propagates in the direction of k . Therefore, the refractive index derived from k
is best described as the phase refractive index. Velocity here is the phase velocity.

Power propagates in the direction of P which is always normal to the surface of the index ellipsoid. From this, we can define a group velocity and a group refractive index.
Lecture 3
Slide 27


Illustration of k versus P

P

k

P

k
Negative refraction into an electromagnetic band gap material.
We don’t need a negative refractive index to have negative refraction.
Lecture 3
Slide 28
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Phase Matching at an Interface
Lecture 3
Slide 29
Illustration of the Dispersion Relation
2
2
k  k  k   k0 n 
2
x
y
2
y
Index ellipsoid

k
ky
kx
x
The dispersion relation for isotropic materials is essentially just the Pythagorean theorem. It says a wave sees the same refractive index no matter what direction the wave is travelling.
Lecture 3
Slide 30
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Index Ellipsoid in Two Different Materials
Material 1 (Low n)
Material 2 (High n)
2
2
k x2,1  k y2,1  k1   k0 n1 
 2
2
k x2,2  k y2,2  k2   k0 n2 
n1  n2
Lecture 3
Slide 31
Phase Matching at the Interface Between Two Materials Where n1 < n2
n1  n2
Material 1
2
2
k  k y2,1  k1   k0 n1 
2
x ,1
Material 2
k
Lecture 3
2
x ,2
 2
2
 k y2,2  k2   k0 n2 
Slide 32
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Summary of the Phase Matching Trend for n1 < n2
n1  n2
Material 1
2
2
k x2,1  k y2,1  k1   k0 n1 
Material 2
 2
2
k x2,2  k y2,2  k2   k0 n2 
1
2
3
4
5
6
Properly phased matched at the interface.
Lecture 3
Slide 33
Phase Matching at the Interface Between Two Materials Where n1 > n2
n1  n2
inc   c
Material 1
2
2
k  k y2,1  k1   k0 n1 
2
x ,1
inc   c
Material 2
 2
2
k x2,2  k y2,2  k2   k0 n2 
Lecture 3
Slide 34
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Summary of the Phase Matching Trend for n1 > n2
n1  n2
Material 1
2
2
k x2,1  k y2,1  k1   k0 n1 
1
Material 2
 2
2
k x2,2  k y2,2  k2   k0 n2 
2
inc   c
inc   c
3
4
inc   c
inc   c
Properly phased matched at the interface.
Lecture 3
Slide 35
Critical Angle and Brewster’s Angle
Lecture 3
Slide 36
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Longitudinal Component of the Wave Vector
1. Boundary conditions require that the tangential component of the wave vector is continuous across the interface.
Assuming kx is purely real in material 1, kx will be purely real in material 2.
 We have oscillations and energy flow in the x direction.
2. Knowing that the dispersion relation must be satisfied, the longitudinal component of the wave vector in material 2 is calculated from the dispersion relation in material 2.
k x2,2  k y2,2   k0 n2 
2

k y ,2 
 k0 n2 
2
 k x2,2
We see that k y will be purely real if k0 n2  k x ,2 .
We see that k y will be purely imaginary if k0 n2  k y , z .
Lecture 3
Slide 37
Field at an Interface Above and Below the Critical Angle (Ignoring Reflections)
1.
2.
3.
4.
n1  n2
n1  n2
n1  n2
No critical angle
1  C
1  C
The field always penetrates material 2, but it may not propagate.
Above the critical angle, penetration is greatest near the critical angle.
Very high spatial frequencies are supported in material 2 despite the dispersion relation.
In material 2, energy always flows along x, but not necessarily along y.
Lecture 3
Slide 38
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Simulation of Reflection and Transmission at a Single Interface (n1<n2)
n1=1.0, n2=1.73  B=60°
Lecture 3
Slide 39
Simulation of Reflection and Transmission at a Single Interface (n1>n2)
n1=1.41, n2=1.0  C=45°
Lecture 3
Slide 40
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Field Visualization for C=45°
inc = 44°
inc = 46°




inc = 67°
inc = 89°




Lecture 3
Slide 41
Reflection and Transmission:
The Fresnel Equations
Lecture 3
Slide 42
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Reflection, Transmission, and Refraction at an Interface
Angles
inc   ref  1
n1 sin 1  n2 sin  2
Snell’s Law
TE Polarization
 2 cos 1  1 cos  2
 2 cos 1  1 cos  2
22 cos 1

 2 cos 1  1 cos  2
rTE 
tTE
1  rTE  tTE
TM Polarization
2 cos  2  1 cos 1
1 cos 1  2 cos  2
2 2 cos 1

1 cos 1  2 cos  2
cos  2

tTM
cos 1
rTM 
tTM
1  rTM
Lecture 3
Slide 43
Reflectance and Transmittance
Reflectance
The fraction of power R reflected from an interface is called reflectance. It is related to the reflection coefficient r through
RTE  rTE
2
RTM  rTM
2
r
RTE
 TE
RTM rTM
2
2
Transmittance
The fraction of power T transmitted through an interface is called transmittance. It is related to the transmission coefficient t through
TTE  tTE
Lecture 3
2
1 cos  2
2 cos 1
TTM  tTM
2
1 cos  2
2 cos 1
t
TTE
 TE
TTM tTM
2
2
Slide 44
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Amplitude Vs. Power Terms
Wave Amplitudes
The reflection and transmission coefficients, r and t, relate the amplitudes of the reflected and transmitted waves relative to the applied wave. They are complex numbers because both the magnitude and phase of the wave can change at an interface.
Eref  rEinc
Etrn  tEinc
Wave Power
The reflectance and transmittance, R and T, relate the power of the reflected and transmitted waves relative to the applied wave. They are real numbers bound between zero and one.
Eref
2
 R  Einc
2
2
Etrn  T  Einc
2
Often, these quantities are expressed on the decibel scale
RdB  10 log10  R 
TdB  10 log10 T 
Lecture 3
Slide 45
The Critical Angle (Total Reflection)
Above the critical angle c, reflection is 100%
rTE 
 2 cos  c  1 cos  2
1
2 cos  c  1 cos  2
rTM 
 2 cos  2  1 cos  c
1
1 cos  c  2 cos  2
This will happen when cos(2) is imaginary. These conditions are derived from Snell’s Law.
cos  2  1  sin 2  2  1 
n 
1   c  sin 1  2 
 n1 
Condition for Total Internal Reflection (TIR)
Lecture 3
1
n12
sin 2  c  1
n22
n12
sin 2  c
n22
n1 sin  c  n2 sin  2
n12
sin 2  c  1
n22
 n2 

 n1 
 c  sin 1 
Slide 46
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Brewster’s Angle (Total Transmission)
TE Polarization
rTE 
 2 cos  B  1 cos  2
0
 2 cos  B  1 cos  2

    
sin  B   2  2   1  2 
 1 1    2 1 
 2 1

1  2
We see that as long as 1=2 then there is no Brewster’s angle.
Generally, most materials have a very week magnetic response and there is no Brewster’s angle for TE polarized waves.
TM Polarization
rTM 
2 cos  2  1 cos 1
0
1 cos 1  2 cos  2

    
sin  B   2  2   2  1 
 1 1   1  2 
2 1

1  2
We see that if 1=2 then there is no Brewster’s angle. For materials with no magnetic response, the Brewster’s angle equation reduces to
tan  B 
 2 n2

1 n1
1   2
This is the most well known equation.
Lecture 3
Slide 47
Notes on a Single Interface
• It is a change in impedance that causes reflections
• Law of reflection says the angle of reflection is equal to the angle of incidence.
• Snell’s Law quantifies the angle of transmission as a function of angle of incidence and the material properties.
• Angle of transmission and reflection do not depend on polarization.
• The Fresnel equations quantify the amount of reflection and transmission, but not the angles.
• Amount of reflection and transmission depends on the polarization and angle of incidence.
• For incident angles greater than the critical angle, a wave will be completely reflected regardless of its polarization.
• When a wave is incident at the Brewster’s angle, a particular polarization will be completely transmitted.
Lecture 3
Slide 48
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Image Theory
Lecture 3
Slide 49
Image Theory Reduces Size of Models
When fields are symmetric in some manner about a plane, it is only necessary to calculate one half of the field because the other half contains only redundant information. Sometimes more than one plane of symmetry can be identified. Image theory can dramatically reduce the numerical size of the model being solved.
Reduced model
Device is 75% smaller.
Model is 94% smaller.
G. Bellanca, S. Trillo, “Full vectorial BPM modeling of Index‐Guiding Photonic Crystal Fibers and Couplers,” Optics Express 10(1), 54‐59 (2002).
Lecture 3
Slide 50
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Summary of Image Theory
Electric Fields
Magnetic Fields
Electric Fields
perfect electric conductor (PEC)
Magnetic Fields
perfect magnetic conductor (PMC)
image fields
Duality
Lecture 3
Slide 51
Image Theory Applied to an Airplane
Lecture 3
Slide 52
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Lenses
Lecture 3
Slide 53
Lenses
Lenses are structures that focus electromagnetic waves.
Lenses are also used to collimate a beam or diverge a beam.
Optical Lens
Microwave Lens
W. Chalodhorn, D. R. Deboer, “Use of Microwave Lenses in Phase Retrieval Microwave Holography of Reflector Antennas,” IEEE Trans. Ant. Prop., vol. 50, no. 9, pp. 1274‐
1284, 2002.
Lecture 3
Slide 54
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Ray Tracing
Ray tracing is a graphical technique to determine the direction of a beam that passes through the lens.
Lecture 3
Slide 55
Ray Tracing Definitions
“Thin” lens
focal plane
focal point
Lecture 3
focal plane
optical axis
focal point
Slide 56
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Ray Tracing: Rule #1
The direction of a ray passing through the center of the lens remains unchanged.
Lecture 3
Slide 57
Ray Tracing: Rule #2
A ray parallel to the optical axis will pass through the focal point.
Lecture 3
Slide 58
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Ray Tracing: Rule #3
An arbitrary ray will pass through the focal plane at the same point as a parallel ray passing through the center of the lens.
Lecture 3
Slide 59
Lens’s Makers Formula
The focal length of a thin lens is approximately
1 nlens  n0  1 1 

  
F
n0  R1 R2 
R1
R2
F
Lecture 3
F
Slide 60
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Death Rays From a Skyscraper
The curved glass on a skyscraper in London acts like a lens. In late August / early September, the sun is at just the right angle to focus light down onto the street.
Here, it melted part of an expensive Jaguar.
20 Fenchurch Street, London.
Lecture 3
Slide 61
31