1.7 Photonic Crystals
17
1.7.1 Maxwell’s Equations Inside Photonic Crystals
Photonic crystals are all-dielectric structures and do not sustain localized charges
and steady-state currents, and hence the time-harmonic (e
−iωt ) version of Maxwell’s
equations can be written as
∇ · E(r) = 0
(1.55)
∇ · H(r) = 0
(1.56)
∇ × E(r) = iωμH(r)
(1.57)
∇ × H(r) = −iωω 0 (r)E(r)
(1.58)
Using these equation, we obtain the following master equation [37]:
∇ ×
1
(r)
∇ × H(r)
=
ω
c
2
H(r)
(1.59)
This equation is an eigenvalue equation which can be solved to obtain the eigenvalues ω and the corresponding eigenfunctions H(r). Electric field E(r) can then be
determined from H(r) by
E(r) =
i
ωω 0 (r)
∇ × H(r)
(1.60)
One must understand that the eigenfunction of the above master equation is not of
a plane wave type but of a Bloch wave kind, i.e., a plane wave (e
ik·r ) times a periodic
function (u k (r)) which inherits the periodicity of the photonic crystal. Hence,
H(r) = e
ik·r u k (r)
(1.61)
where u k (r) satisfies u k (r) = u k (r + T) if T is the periodicity of the photonic crystal. The master equation can be solved by numerical techniques such as plane wave
expansion method (PWEM) [41, 42], finite difference method (FDM) [43, 44], finite
element method (FEM) [45], etc. [9, 46] to obtain the photonic band diagram. The
photonic bands are dispersion curves (ω versus k plots) for different eigenvalues. All
the photonic band diagrams shown in this book have either been obtained by COMSOL Multiphysics [47] or by MIT’s open-source software package MIT Photonic
Bands (MPB) [48].
1.7.2 Photonic Band Structure
A photonic band structure is a photonic analog of the electronic band structure. As
said above, it is obtained by solving the master equation. Below are the photonic bands
17
1.7.1 Maxwell’s Equations Inside Photonic Crystals
Photonic crystals are all-dielectric structures and do not sustain localized charges
and steady-state currents, and hence the time-harmonic (e
−iωt ) version of Maxwell’s
equations can be written as
∇ · E(r) = 0
(1.55)
∇ · H(r) = 0
(1.56)
∇ × E(r) = iωμH(r)
(1.57)
∇ × H(r) = −iωω 0 (r)E(r)
(1.58)
Using these equation, we obtain the following master equation [37]:
∇ ×
1
(r)
∇ × H(r)
=
ω
c
2
H(r)
(1.59)
This equation is an eigenvalue equation which can be solved to obtain the eigenvalues ω and the corresponding eigenfunctions H(r). Electric field E(r) can then be
determined from H(r) by
E(r) =
i
ωω 0 (r)
∇ × H(r)
(1.60)
One must understand that the eigenfunction of the above master equation is not of
a plane wave type but of a Bloch wave kind, i.e., a plane wave (e
ik·r ) times a periodic
function (u k (r)) which inherits the periodicity of the photonic crystal. Hence,
H(r) = e
ik·r u k (r)
(1.61)
where u k (r) satisfies u k (r) = u k (r + T) if T is the periodicity of the photonic crystal. The master equation can be solved by numerical techniques such as plane wave
expansion method (PWEM) [41, 42], finite difference method (FDM) [43, 44], finite
element method (FEM) [45], etc. [9, 46] to obtain the photonic band diagram. The
photonic bands are dispersion curves (ω versus k plots) for different eigenvalues. All
the photonic band diagrams shown in this book have either been obtained by COMSOL Multiphysics [47] or by MIT’s open-source software package MIT Photonic
Bands (MPB) [48].
1.7.2 Photonic Band Structure
A photonic band structure is a photonic analog of the electronic band structure. As
said above, it is obtained by solving the master equation. Below are the photonic bands