lecture 1e -- preliminary topics - empossible · 12/9/2019 · computational electromagnetics...
TRANSCRIPT
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Advanced Computation:
Computational Electromagnetics
Preliminary Topics
Outline
• Review of Linear Algebra
• Complex Wave Vector
• TE & TM Polarization
• Index Ellipsoids
• Electromagnetic Behavior at an Interface• Phase matching at an interface
• The Fresnel equations• Visualization
• Image Theory
Slide 2
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Slide 3
Review of Linear Algebra
Matrices Represent Sets of Equations
Slide 4
11 12 13 14 1
21 22 23 24 2
31 32 33 34 3
41 42 43 44 4
a w a x a y a z b
a w a x a y a z b
a w a x a y a z b
a w a x a y a z b
11 12 13 14 1
21 22 23 24 2
31 32 33 34 3
41 42 43 44 4
a a a a bw
a a a a bx
a a a a by
a a a a bz
A set of linear algebraic equations can be written in “matrix” form.
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Interpretation of Matrices
Slide 5
11 12 13 1a x a y a z b
21 22 23 2a x a y a z b
31 32 33 3a x a y a z b
11 12 13 1
21 22 23 2
31 32 33 3
a a a x b
a a a y b
a a a z b
11 12 13
21 22 23
31 32 33
a a a
a a a
a a a
Equation for x
Equation for y
Equation for z
EQUATION FOR… RELATION TO…
11 12 13
21 22 23
31 32 33
a a a
a a a
a a a
From a purely mathematical perspective, this interpretation does not make sense. This interpretation will be highly useful and insightful because of how we derive the equations.
Compact Matrix Notation
Slide 6
Ax b
11 12 13 14
21 22 23 24
31 32 33 34
41 42 43 44
a a a a
a a a a
a a a a
a a a a
A
w
x
y
z
x
1
2
3
4
b
b
b
b
b
Matrices and vectors can be represented and treated as single variables.
A x b
or
square matrixcolumnvector
columnvector
11 12 13 14 1
21 22 23 24 2
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41 42 43 44 4
a a a a bw
a a a a bx
a a a a by
a a a a bz
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Matrices Require Special Algebra
Slide 7
AB BA
A B B A
Commutative Laws
AB C A BC
A B C A B C
Associative Laws
A B A B
AB A B A B
Multiplication with a Scalar
A B C AB AC
A B C AC BC
Distributive Laws
11 1
1
?
n
m mn
a a
a a
A
I A
Addition with a Scalar
AB BA
1 11 1 1
1 1
T T TT T T T T
TT
A A AB B A
A A A B A B AB B A
A A
Matrix Inverses and Transposes
Manipulating Matrix Equations
Slide 8
1 1
1
A B C D E
A B CC D E C
A B D E C
Example #1: Dividing both sides by a matrix on the right
Starting equation
Post‐multiply both sides by C‐1
Recall that CC‐1=I
1 1
1
C A B D E
C C A B C D E
A B C D E
Example #2: Dividing both sides by a matrix on the left
Starting equation
Pre‐multiply both sides by C‐1
Recall that C‐1C=I
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1
1 1 1
1
C A D BC D
C A BC
A C BC
A CC BCC
A B
Example #3: Simplify an expression
Starting equation
Subtract D from both sides
Recall inverse of a product rule
Post‐multiply both sides by C‐1
Recall that C‐1C=I
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The Zero and the Identity Matrices
Slide 9
Zero Matrix
0 0
0 0
0
Identity Matrix
1
1
0
0
I 1 1
I A A I A
A A AA I
0 A A 0 0
0 A A 0 A
A A 0
Matrix Division
Slide 10
It is very rare to calculate the inverse of matrix because it is very computationally intensive to do this.
At first glance, matrix division appears to require calculating the inverse of a matrix, but highly efficient algorithms exist that do not need to do this.
Pre‐Division: 1A B C A = B\C;A = inv(B)*C;
Post‐Division: 1A CB A = C/B;A = C*inv(B);
Despite that both of these equations are dividing by B, pre‐ and post‐division do NOT lead to the same result.
1 1 B C CB
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Matrix Division With More than Two Terms
Slide 11
Suppose the following matrix expression is to be evaluated.
1A BC D
Would the following MATLAB code work?
A = B*C\D;No! Remember the order of operations.MATLAB will first multiply B and C.
A BCIt will then backward divide on D.
1A BC DSo what is the correct code?
A = B/C*D;A = B*(C\D);
It might be a good idea to time these two calculations to see which is faster.
Proper Notation for Matrix Division
Slide 12
It is almost never correct to write matrix division as a fraction.A
BWhy?
Does this represent predivision or postdivision?
1 1 or ??? B A AB
It is impossible to say, thus fraction notation is incorrect for matrices.
One exception When both A and B are diagonal matrices, both pre‐ and post‐division will give the same answer.
1 1
11 11 11 11 11 11
22 22 22 22 22 22
MM MM MM MM MM MM
b a a b a b
b a a b a b
b a a b a b
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Ax = b Vs. Ax = x
Slide 13
Standard Linear Problem
Ax b
• Has only one answer• Requires a source
Eigen‐Value Problem
Ax x
• Has an infinite number of answers• Modes• No source
Slide 14
Complex Wave Vector
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The Complex Wave Number 𝑘
Slide 15
A wave travelling the 𝑧 direction can be written in terms of the complex wave number kas jkzE z Pe
k k jk Substituting this into the wave solution yields
k z jk zPe e j k jk zE z Pe
attenuation oscillation
attenuation & oscillation
Waves with Complex k
Slide 16
Purely Real k Purely Imaginary k Complex k
•Uniform amplitude•Oscillations move power• Considered to be a propagating wave
•Decaying amplitude•Oscillations move power• Considered to be a propagating wave (not evanescent)
•Decaying amplitude•No oscillations, no flow of power• Considered to be evanescent
This implies that these are the only 2.5 configurations that electromagnetic fields can take on.
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2D Waves with Doubly Complex 𝑘
Slide 17
Real yk
Imaginary yk
Complex yk
Real xk Imaginary xk Complex xk
xy
xy
xy
xy
xy x
y
xy
xy
xy
Slide 18
TE & TM Polarization
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TE and TM
Slide 19
We use the labels “TE” and “TM” when we are describing 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.
TM/parallel/p – the electric field is polarized parallel to the plane of incidence.
Calculating the Polarization Vectors
Slide 20
inc 0 inc
sin cos
sin sin
cos
k k n
0
ˆ 0
1
n
incTE
inc
ˆ 0
ˆˆ 0
ˆ
ya
n ka
n k
inc TETM
inc TE
ˆˆ
ˆ
k aa
k a
Incident Wave Vector Surface Normal Unit Vectors of Polarization Directions
Composite Polarization Vector
TE TMTE TMˆ ˆP p p aa z
x
y
Right‐handedcoordinate system
1P
In CEM, we usually make
ˆ ˆ ˆx y za a a
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Slide 21
Phase Matching at an Interface
Illustration of the Dispersion Relation
Slide 22
y
x
k
xk
yk
2 22 2
0x yk k k k n
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.
Index ellipsoid
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Index Ellipsoid in Two Different Materials
Slide 23
2 22 2
,1 ,1 1 0 1x yk k k k n
Material 1 (Low n) Material 2 (High n)
2 22 2
,2 ,2 2 0 2x yk k k k n
1 2n n
Phase Matching at the Interface Between Two Materials Where n1 < n2
Slide 24
2 22 2
,1 ,1 1 0 1x yk k k k n
Material 1
Material 2
2 22 2
,2 ,2 2 0 2x yk k k k n
1 2n n
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Summary of the Phase Matching Trend for n1 < n2
Slide 25
2 22 2
,1 ,1 1 0 1x yk k k k n
Material 1 Material 2
2 22 2
,2 ,2 2 0 2x yk k k k n 1 2n n
1 2 3
4 5 6
Properly phased matched at the interface.
Phase Matching at the Interface Between Two Materials Where n1 > n2
Slide 26
Material 2
2 22 2
,2 ,2 2 0 2x yk k k k n
1 2n n
inc c inc c 2 22 2
,1 ,1 1 0 1x yk k k k n
Material 1
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Summary of the Phase Matching Trend for n1 > n2
Slide 27
2 22 2
,1 ,1 1 0 1x yk k k k n
Material 1 Material 2
2 22 2
,2 ,2 2 0 2x yk k k k n 1 2n n
3
Properly phased matched at the interface.
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1 2
inc c
inc c
inc c
inc c
Slide 28
Reflection and Transmission:The Fresnel Equations
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Reflection, Transmission, and Refraction at an Interface
Slide 29
TE Polarization
2 1 1 2TE
2 1 1 2
2 1TE
2 1 1 2
TE TE
cos cos
cos cos
2 cos
cos cos
1
r
t
r t
TM Polarization
2 2 1 1TM
1 1 2 2
2 1TM
1 1 2 2
2TM TM
1
cos cos
cos cos
2 cos
cos cos
cos1
cos
r
t
r t
Angles
inc ref 1
1 1 2 2sin sinn n
Snell’s Law
1 1, n
2 2, n
Reflectance and Transmittance
Slide 30
Reflectance
The fraction of power R reflected from an interface is called reflectance. It is related to the reflection coefficient r through
Transmittance
The fraction of power T transmitted through an interface is called transmittance. It is related to the transmission coefficient t through
2
TE TER r 2
TM TMR r
2 1 2TE TE
2 1
cos
cosT t
2 1 2TM TM
2 1
cos
cosT t
2
TETE2
TM TM
tT
T t
2
TETE2
TM TM
rR
R r
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Amplitude Vs. Power Terms
Slide 31
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.
ref incrE E trn inctE EWave 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.
2 2
ref incRE E 2 2
trn incTE E
Often, these quantities are expressed on the decibel scale.
1dB 010 logR R 1dB 010 logT T
Conservation of Power
Slide 32
1A R T
When an electromagnetic wave is incident onto a device, it can be absorbed (i.e. converted to another form of energy), reflected and/or transmitted. Without a nuclear reaction, nothing else can happen.
Reflectance, RFraction of power from the applied wave that is reflected from the device.
Transmittance, TFraction of power from the applied wave that is transmitted through the device.
Absorptance, AFraction of power from the applied wave that is absorbed by the device.
Applied Wave
Reflected Wave
Transmitted Wave
Absorbed Wave
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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.
Slide 33
Slide 34
Visualization of Wave Scattering at an Interface
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Longitudinal Component of the Wave Vector
Slide 35
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.
22 2,2 ,2 0 2
2 2,2 0 2 ,2
x y
y x
k k k n
k k n k
0 2 ,2
0 2 ,
We see that will be purely real if .
We see that will be purely imaginary if .
y x
y y z
k k n k
k k n k
Field at an Interface Above and Below the Critical Angle (Ignoring Reflections)
Slide 36
1 2
No critical angle
n n 1 2
1 C
n n
1 2
1 C
n n
1. The field always penetrates material 2, but it may not propagate.2. Above the critical angle, penetration is greatest near the critical angle.3. Very high spatial frequencies are supported in material 2 despite the dispersion relation.4. In material 2, energy always flows along x, but not necessarily along y.
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Simulation of Reflection and Transmission at a Single Interface (n1<n2)
Slide 37
n1=1.0, n2=1.73 B=60°
Simulation of Reflection and Transmission at a Single Interface (n1>n2)
Slide 38
n1=1.41, n2=1.0 C=45°
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Field Visualization for C=45°
Slide 39
inc = 44° inc = 46°
inc = 67° inc = 89°
Electromagnetic Tunneling
Slide 40
If an evanescent field touches a medium with higher refractive index, the field may no longer be cutoff and become a propagating wave.
This is a very unusual phenomenon because the evanescent field is contributing to power flow.
This is called electromagnetic tunneling and is analogous to electron tunneling through thin insulators.
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Slide 41
Image Theory
Image Theory Reduces Size of Models
Slide 42
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.
G. Bellanca, S. Trillo, “Full vectorial BPM modeling of Index‐Guiding Photonic Crystal Fibers and Couplers,” Optics Express 10(1), 54‐59 (2002).
Reduced modelDevice is 75% smaller.Model is 94% smaller.
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Summary of Image Theory
Slide 43
perfect electric conductor (PEC) perfect magnetic conductor (PMC)
Duality
Electric Fields Magnetic Fields Electric Fields Magnetic Fields
image fields
Image Theory Applied to Simulation of an Airplane
Slide 44
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