4/18/2020

Electromagnetics: Electromagnetic Field Theory Dispersion Relation

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Lecture Outline

• Dispersion relation • Index ellipsoids • Material properties explained by index ellipsoids

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Dispersion Relation

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Derivation in LHI Media (1 of 2)

Start with the wave equation. Plane wave solution 2   2EknE 0 jkr 0 Solve wave equation Er   Pe

Finish Derivation  2 jk r2  jk r Substitute plane wave solution into wave equation. Pe k0 n Pe 0  2 jk r2  jk r ekne 0 0 Divide both sides by 𝑃.  2 jk r2  jk r Calculate the Laplacian. ke kn0 e 0 2 2 ·⃗ kkn0 0 Divide both sides by 𝑒 .

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Derivation in LHI Media (2 of 2)

Starting from the previous slide

2 2 kkn0 0

2 2 𝑘 𝑛 kkn 0 Move to the right‐hand side.

Recall that 𝑘 𝑘 𝑘 𝑘 and 𝑘𝜔𝑛𝑐⁄ , the dispersion relation is

2 The dispersion relation relates frequency  to wave 2222 2 n number k. For LHI media, it fixes the magnitude of kkkkxyz kn0  c the wave vector to be a constant for all wave 0 directions.

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Index Ellipsoids

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The Index Ellipsoid

From the previous slide, the dispersion relation for aLHI material was: 222 2 kkkxyz kn0

This is an equation for a sphere of radius 𝑘𝑛. x2222y zr 𝑘 Pick a point on the surface.

The vector from the origin to this point is the wave vector 𝑘.

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Index Ellipsoids Convey Refractive Index

If frequency 𝑘 is known, the magnitude of the wave vector 𝑘 conveys refractive index 𝑛.  kkn 0 𝑛 𝑘

Based on this, the index ellipsoid can be 𝑘 interpreted as a map of the refractive index that a wave experiences as a function of direction of that wave.

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Index Ellipsoid for Isotropic Materials

z 222 22 kkkknxyz0

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Index Ellipsoid for Uniaxial Materials

Negative Uniaxial Medium kkk222 kk 22  2 𝑛 xyz22 zykz 𝑛 𝑛 kk0 22200 nnnoeo 𝑛 𝑛 Ordinary Wave Extraordinary Wave Sphere Ellipsoid

𝑛

𝑛 𝑛

𝑛 𝑛

Positive Uniaxial Medium 𝑛 𝑛 𝑛 𝑛 Slide 10 10

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Index Ellipsoid for Biaxial Materials

222 kkxzk y 2221 22 22 22 kknkknkkn000xyz

Convention 𝑛 𝑛 𝑛

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Material Properties Explained by Index Ellipsoids

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Direction of Phase and Power

Isotropic Materials Anisotropic Materials  y  y     k k x x

Phase propagates in the direction of 𝑘.

Power propagates in the direction of ℘ which is normal to the surface of the index ellipsoid.

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Illustration of 𝑘 Not Same as ℘

℘ 𝑘

℘ Negative refraction 𝑘

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Double Refraction in Anisotropic Materials

Isotropic Materials Anisotropic Materials y y

  k1 k1 x  x ke   k2 ko

Anisotropic materials have two index ellipsoids –one for each polarization. Wave power refracts in directions at the same time, producing double refraction. Slide 15 15

Self‐Collimation in Photonic Crystals

large angular span of 𝑘

“flat” underside

direction of power flow

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