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Photonic Crystal (Maxwell's Equations (unit ([SI unit] (Fraday's…
Photonic Crystal
Maxwell's Equations
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unit
[SI unit]
Fraday's induction law
\(\vec{\nabla}\times \vec{E}(\vec{r},t) =\frac{\partial\vec{B}(\vec{r},t)}{\partial t}\)
generalized Ampere's law
\(\vec{\nabla}\times \vec{H}(\vec{r},t) =\vec{J}(\vec{r},t)+\frac{\partial\vec{D}(\vec{r},t)}{\partial t}\)
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Gaussian uint
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generalized Ampere's law
\(\vec{\nabla}\times \vec{H}(\vec{r},t) =\frac{4\pi}{c}\vec{J}(\vec{r},t)+-\frac{1}{c}\frac{\partial\vec{D}(\vec{r},t)}{\partial t}\)
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Fraday's induction law
\(\vec{\nabla}\times \vec{E}(\vec{r},t) =-\frac{1}{c}\frac{\partial\vec{B}(\vec{r},t)}{\partial t}\)
The Wave equation
source free medium
(rho=0, J=0)
electric field
consider time
Phase
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common symbol
kz ->propagation -> beta
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gruop velocity
contain many k
fourier transform pair
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the case of envelope
group delay
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group velocity
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isotropic medium
plane wave
\(\vec{E}(\vec{r},t)=\vec{E_0} e^{i(\vec{k}\cdot\vec{r}-\omega t)}\)
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\(\nabla(\nabla \cdot \vec{E} - \nabla^2 \vec{E} )= -\frac{\partial}{\partial t} \nabla \times \vec{B}\)
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defect
defect modes
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Slater - Koster modell
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illustrates the formation of localize modes inside the bandgap as a result of defects in the periodic structure
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principle
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the defect is represent by V(p,q)
\(V (p,q) = \int a^*_l (r-p) \Delta \epsilon (r) a_n (r-q) dr\) #
\(\Delta \epsilon\) is the change (or perturbation) of dielectric constant due ti the defect in periodic lattices
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