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Photonic Crystal_eigen (Maxwell's equation (The problem of waveguide…
Photonic Crystal_eigen
Maxwell's equation
The problem of waveguide and resonator problem
formula eigenvalue problem
condition
linear vector space
topic(path)
linear vector spaces
Inner product space
Hibert space
Hermitian operator for EM field
linear vector space
condition
condition 1
(a+b)+c=a+(b+c)
a+0=0+a=a
0 : zero vector
a+(-a)=(-a+a)=0
a+b=b+a
condition 2
x(ya)=(xy)a
1*a=a
x(a+b)=xa+xb
(x+y)a=xa+ya
in linear space property
linearly dependent
condition
If two vector
a=(a1,a2,a3...); x=(x1,x2,x3...)
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Inner product space
properties
if a,b,c is nature number vector, and x is complex number vector
\( < xa / b > = x < a/b > \)
\( < a/a > \le 0 \) and \( < a/a > = 0 \)
if and only if a=0
norm(or length) of a vector
\( | a | = \sqrt{ < a/a > }\)
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\( < a/b > = < b/a > \)
\( < a+b /c > = < a/c > + < b/c > \)
inner product
\( < g/f > = \int_a^b \vec{g}^* (x) \vec{f} (x) dx\)
a vector space with an inner product
Hilbert space
completness
sequence
if it has no "holes" in it
Hibert space=inner produce space + complete
If Hilbert space is norm
Lz space
basis is unique
hermitian operator
condition
source free region
The Maxwell curl equation for time harmonic field
\( \nabla \times \vec{E} = -i \omega \mu \vec{H}\\ \nabla \times \vec{H} = -i \omega \epsilon \vec{E} \)
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properties
appropriate boundary condition
The eigenvales of hermitian operator are real
The eigenfuctions of an Hermitian operator are orthoonal
The eigenfunctions of an Hermitian operator form a complete set
introduction
a vector in an abstract infinite dimensional linear vector space
Polarization
origin
free space
electric field
\(\nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0}\)
spatical inversion
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B field
spatical inversion
\(\nabla \times \vec{E} = \frac{\partial \vec{B}}{\partial t}\)
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a slab waveguide
condision
propagation along z
symmetry of the waveguide
reflection operator: O_M
because the waveguide is physically unchanged under this mirror reflection operation operation x-y plane reference
its solutions to be eigenfuciotn of O_My
E field
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H filed
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conclusion
two type
eigenvectorof O_My =1
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eigenvectorof O_My =-1
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can seperate them
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\(\hat{O}_{My}\)
\(\hat{O}_{My} = \begin{pmatrix}1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{pmatrix}\)
periodic condition
condition
dielectric structure
two D
Bravais lattce
infinite array of discreye point
same
arrangement
orientation
a lattice vector
\(\vec{R} = n_1 \hat{a} _1 n_2 \hat{a} _2\)
The types Barais lattice
square
oblique
rectangular
centered rectangular
hexagonal
the dash line regions are the primitive unit cell
no unique
choice bavais lattice
Winger-Seitz
drawinng lines connecting the point to all other in the lattices
periodic function
f
Fourier Transform
\(f(\vec{r}) = \int f (G) e^{i \vec{G} \cdot \vec{r}} d \vec{G}\)
combine last equation
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epsilon
\(\epsilon (\vec{r}) = \epsilon (\vec{r} + \vec{R})\)
Plane Wave Expansion Method
Eigenvalue problem
conception
Process
\(\nabla\times(\frac{1}{\epsilon(\vec{r})}\nabla\times\vec{H}(\vec{r}))=\omega^2 \vec{H}(\vec{r})\)
H(r) is a Bloch state
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Conclusion
The Hermitian operator O the periodic boundary condition
has a complete set of Eigenvectors u_k(r)
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Solve
Conception
similar to finding electronic band structure for solid
defferent
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condition
two dimensional photonic crystal lattice
dielectric rods
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TM mode
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TM mode
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Solve
EM field component
{Ex , Ey , Ez , Hx , Hy , Hz}
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base on
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Propertys
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Case
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