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Electromagnetics:
Electromagnetic Field Theory
Dispersion Relation
Lecture Outline
•Dispersion relation• Index ellipsoids•Material properties explained by index ellipsoids
Slide 2
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Slide 3
Dispersion Relation
Derivation in LHI Media (1 of 2)
Slide 4
Start with the wave equation.
220 0E k n E
Plane wave solution
jk rE r Pe
Solve wave equation
Finish Derivation
220 0jk r jk rPe k n Pe
220 0jk r jk re k n e
220 0jk r jk rk e k n e
220 0k k n
Substitute plane wave solution into wave equation.
Divide both sides by 𝑃.
Calculate the Laplacian.
Divide both sides by 𝑒 · ⃗.
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Derivation in LHI Media (2 of 2)
Slide 5
Starting from the previous slide
220k k n Move 𝑘 𝑛 to the right‐hand side.
220 0k k n
2
22 2 2 20
0x y z
nk k k k k n
c
Recall that 𝑘 𝑘 𝑘 𝑘 and 𝑘 𝜔𝑛 𝑐⁄ , the dispersion relation is
The dispersion relation relates frequency to wave number k. For LHI media, it fixes the magnitude of the wave vector to be a constant for all wave directions.
Slide 6
Index Ellipsoids
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The Index Ellipsoid
Slide 7
22 2 20x y zk k k k n
From the previous slide, the dispersion relation for a LHI material was:
This is an equation for a sphere of radius 𝑘 𝑛.
2 2 2 2x y z r
Pick a point on the surface.
The vector from the origin to
this point is the wave vector 𝑘.
𝑘
Index Ellipsoids Convey Refractive Index
Slide 8
0k k n
If frequency 𝑘 is known, the magnitude of the wave
vector 𝑘 conveys refractive index 𝑛.
Based on this, the index ellipsoid can be interpreted as a map of the refractive index that a wave experiences as a function of direction of that wave.
𝑘
𝑛 𝑘
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Index Ellipsoid for Isotropic Materials
Slide 9
2 2 2 2 20x y zk k k k n
z
Index Ellipsoid for Uniaxial Materials
Slide 10
2 2 2 2 2 22 20 02 2 2
o e o
0x y z z y zk k k k k k
k kn n n
𝑛
𝑛
𝑛
𝑛
𝑛
𝑛
𝑛
𝑛
𝑛
𝑛
Ordinary WaveSphere
Extraordinary WaveEllipsoid
Negative Uniaxial Medium𝑛 𝑛
Positive Uniaxial Medium𝑛 𝑛
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Index Ellipsoid for Biaxial Materials
Slide 11
22 2
2 2 22 2 2 2 2 20 0 0
1yx z
x y z
kk k
k k n k k n k k n
Convention𝑛 𝑛 𝑛
Slide 12
Material Properties Explained by Index
Ellipsoids
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Direction of Phase and Power
Slide 13
Isotropic Materials
k
x
y
k
Anisotropic Materials
x
y
Power propagates in the direction of ℘ which is normal to the surface of the index ellipsoid.
Phase propagates in the direction of 𝑘.
Illustration of 𝑘 Not Same as ℘
Slide 14
Negative refraction
𝑘℘
𝑘℘
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Double Refraction in Anisotropic Materials
Slide 15
Isotropic Materials
y
1k
x
2k
Anisotropic materials have two index ellipsoids – one for each polarization. Wave power refracts in directions at the same time, producing double refraction.
y
1k
x
ok
Anisotropic Materials
ek
Self‐Collimation in Photonic Crystals
Slide 16
“flat” underside
large angular
span of 𝑘
direction of power flow
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