Wave Equation in Layered Media - San Jose State University
7
Wave Equation in Layered Media Starting with the wave equation we will plug in the dielectric tensor written as a Fourier series with periodicity Λ and an arbitrary wave to get (z )= l l e -i 2πl Λ z E = E 0 (k)e -i(kz+ωt) dk k 2 E 0 (k)e -ikz dk - ω 2 μ l l e -i 2πl Λ z E 0 (k)e -ikz dk =0 ∇× ∇× E + μ ∂ 2 E ∂ t 2 =0 ch 6, 10 corrected
Transcript of Wave Equation in Layered Media - San Jose State University
Ch 6,
Wave Equation in Layered Media
Starting with the wave equation
we will plug in the dielectric tensor written as a Fourier series with periodicity Λ
and an arbitrary wave
to get
ε(z) =∑
l
εle−i 2πl
Λ z
!E =∫
!E0(k)e−i(kz+ωt)dk
∫k2 !E0(k)e−ikzdk − ω2µ
∑
l
εle−i 2πl
Λ z
∫!E0(k)e−ikzdk = 0
22
!∇×(
!∇× !E)
+ µε∂2 !E
∂t2= 0
ch 6, 10corrected
Ch 6,
Wave Equation in Layered Media
Starting with the wave equation
we will plug in the dielectric tensor written as a Fourier series with periodicity Λ
and an arbitrary wave
to get
ε(z) =∑
l
εle−i 2πl
Λ z
!E =∫
!E0(k)e−i(kz+ωt)dk
23
!∇×(
!∇× !E)
+ µε∂2 !E
∂t2= 0
∫k2 !E0(k)e−ikzdk + ω2µ
∑
l
εle−i 2πl
Λ z
∫!E0(k)e−ikzdk = 0
ch 6, 11corrected
Ch 6,
Wave Equation in Layered Media
for all k
Is an infinite set of equations. Consider equations for: kΛ/2π=0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3 …
k2 !E0(k)− ω2µ∑
l
εl!E0(k −
2πl
Λ) = 0
coupled
Let K be the value value of k±2πl/Λ closest to ω2με0 in a series of coupled equations, where ε0 is the zeroth order Fourier coefficient of ε(z).The whole series of equations for -∞<k<∞ can be treated instead as a series of coupled equations for 0<K<2π/Λ. The solution to each set of equations for a value of K only contains terms at k=K±2πl/Λ, thus
K=0K=0.2π/Λ
K=0.4π/Λ
→ 24!E(z) =
∫!E0(k)ei(kz+ωt)dk !E(K, z) =
∑
l
!E0(K − l2π
Λ)ei(K−l 2π
Λ )z−iωtch 6, 12
corrected
Ch 6,
Dispersion Relation in Layered Media
A nontrivial solution to these coupled equations only exists if
k2 !E0(k)− ω2µ∑
l
εl!E0(k −
2πl
Λ) = 0The dispersion relation
for k→K and k→K-2πm/Λ considering l=0 and l=m terms gives
and for ε-m=εm* in a lossless medium, since
(K2 − ω2µεø
)#E0(K)− ω2µεm
#E0(K − 2πm
Λ) = 0
[(K − 2πm
Λ
)2
− ω2µεø
]#E0
(K − 2πm
Λ
)− ω2µε−m
#E0(K) = 0
∣∣∣∣K2 − ω2µεø −ω2µεm
−ω2µε−m
(K − 2πm
Λ
)2 − ω2µεø
∣∣∣∣ = 0
(K2 − ω2µεø
)((
K − 2πm
Λ
)2
− ω2µεø
)−
(ω2µ|εm|
)2 = 0 25
εm =1Λ
∫ Λ
0ε(z)e−i2πmz/Λdz
ch 6, 18corrected
Ch 6,
Dispersion Relation in Layered Media
which can be solved for K, the Bloch wave vector for a wave of frequency ω
graph of dispersion relationship (figure 6.2)
(K2 − ω2µεø
)((
K − 2πm
Λ
)2
− ω2µεø
)−
(ω2µ|εm|
)2 = 0
26
K =mπ
Λ±
√√√√(mπ
Λ
)2+ εøµω2 ±
√
(|εm|µω2)2 +(
2πm
Λ
)2
εøµω2
ch 6, 19corrected
Ch 6,
Bandgaps in Layered Media
When the Bragg condition is met (K-2πm/Λ=ω2ε0 μ)real solutions exist for# # # # # # # and
Solutions for
are complex, this region is called the forbidden band. At the center of the forbidden band where## # # # # # and## # # # # , i.e.
the dispersion relation gives
ω2 <K2
µ (εø + |εm|) ω2 >K2
µ (εø − |εm|)
K2
µ (εø + |εm|) < ω2 <K2
µ (εø − |εm|)
K2 − ω2µεø = 0(K − m
2πΛ
)2≈ ω2µεø ω2 =
(mπ)2
Λ2µεø
K =mπΛ
(1 ± i
|ε1|2εø
)
27
K =mπ
Λ±
√√√√(mπ
Λ
)2+ εøµω2 ±
√
(|εm|µω2)2 +(
2πm
Λ
)2
εøµω2
ch 6, 20corrected
Ch 6,
The propagating waves in the medium are bloch waves with an amplitude that is periodic in Λ and a phase given by Kz, so a bloch wave should obey
requiring
or
with the form# # # # # # # # # # where
giving
Bloch Wave Solutions
(A BC D
) (an
bn
)= eiKΛ
(an
bn
)
eiKΛ =A + D
2± i
√
1−(
A + D
2
)2
eiKΛ =A + D
2±
√(A + D
2
)2
− 1
K =1Λ
cos−1
(A + D
2
)cos ψ =
A + D
2
28
eiKΛ = cos ψ ± i sinψ = e±iψ
ch 6, 44corrected
