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ELEC-E4130Lecture 13: Multilayer (Stratified Media)
One layer, normal incidence
ELEC-E4130 / Taylor
Nov. 01, 2021
ELEC-E4130 / TaylorLecture 13
Multiple layers mirrors
Tune layer permittivity and thickness to enhance constructive interference of multiple reflections at the air dielectric interface
Dielectric surface quality superior to metallic surface quality
ELEC-E4130 / TaylorLecture 13
Multiple layer filters
Tune layer permittivity and thickness to enhance destructive interference of multiple reflections at input the air-dielectric interface
This enhances constructive interference at the output dielectric-air interface
Sometimes impossible to suppress natural periodic behavior
ELEC-E4130 / TaylorLecture 13
Angle dependent behavior
Transmission and reflection are defined by steady state constructive and destructive interference
Interference controlled by optical path length within the dielectric
Optical path is increased going from normal to obliques incidence angle
ELEC-E4130 / TaylorLecture 13
Single interfaceAirHalf space
DielectricHalf space
E
ΓE
E
ΓE
𝜺𝒓𝟏
ELEC-E4130 / TaylorLecture 13
Finite thickness layer AirHalf space
DielectricHalf space
Layer 1
𝜺𝒓𝟏 𝜺𝒓𝟐
E
ΓE
E
ΓE
ℓ𝟏
ELEC-E4130 / TaylorLecture 13
Two layersAirHalf space
DielectricHalf space
Layer 1 Layer 2
𝜺𝒓𝟏 𝜺𝒓𝟐 𝜺𝒓𝟐
E
ΓE
E
ΓE
ℓ𝟏 ℓ𝟐
ELEC-E4130 / TaylorLecture 13
N LayersAirHalf space
DielectricHalf space
Layer 1 Layer 2 Layer N
𝜺𝒓𝟏 𝜺𝒓𝟐 𝜺𝒓𝐍
E
ΓE
E
ΓE
ℓ𝟏 ℓ𝟐 ℓ𝑵
ELEC-E4130 / TaylorLecture 13
Today: Normal incidence, single layerAirHalf space
DielectricHalf space
Layer 1
𝜺𝒓𝟏 𝜺𝒓𝟐
E
ΓE
Γ𝜌 𝜌 𝑒 ℓ
1 𝜌 𝜌 𝑒 ℓ
ELEC-E4130 / TaylorLecture 13
Field relations
E z E e E e E z E z
H z1η E e E e
1η E z E z
E z12 E z ηH z
E z E e
E z E e
E z12 E z ηH z
Total E-field
Total H-field
Forward traveling E-field
Reverse traveling E-field
Algebra
E zH z
1 1η η
E zE z
E zE z
1 η1 η
E zH z
EH
1 1η η
EE
EE
1 η1 η
EH
Matrix vector Multiplication
Suppressz-dep.
Right hand rule
E
E
Forward E-field @ z = 0
Reverse E-field @ z = 0
ELEC-E4130 / TaylorLecture 13
Wave impedance and reflection
Z zE zH z
EH
E E1η E E
η1 E
E
1 EE
η1 Γ1 Γ
z-dependent Wave impedance
Γ zE zE z
EE
12 E z ηH z12 E z ηH z
EH ηEH η
Z ηZ η
z-dependent Reflection coefficientSummarize
Γ z𝑍 z η𝑍 z η
Z z η1 Γ z1 Γ z
ELEC-E4130 / TaylorLecture 13
Propagation of reflection coefficient
Γ zE zE z
E eE e
EE e Γ 0 e
The reflection coefficient at any point along z is the reflection coefficient at z = 0 modified by twice the propagation phase kz
Reflection coefficient magnitude is unchanged Only reflection coefficient phase is modified
ELEC-E4130 / TaylorLecture 13
Propagation matrices: two points on z-axis
z
E E
ZΓ
E 𝐻
E E
ZΓ
E 𝐻
ℓz2z1
E e ℓE
E e ℓE
E e ℓE
E e ℓE
“Reverse” propagation
𝛈
“Forward” propagation “Reverse” propagation
“Forward” propagation
ELEC-E4130 / TaylorLecture 13
Propagation matrices: two points on z-axis
z
E E
ZΓ
E 𝐻
E E
ZΓ
E 𝐻
ℓz2z1
𝛈
EE
e ℓ 00 e ℓ
EE Γ
EE
E eE e
Γ e
ELEC-E4130 / TaylorLecture 13
Propagation matrices: two points on z-axis
z
E E
ZΓ
E 𝐻
E E
ZΓ
E 𝐻
ℓz2z1
𝛈
EH
1 1η η
EE
1 1η η
e ℓ 00 e ℓ
EE
EH
1 1η η
e ℓ 00 e ℓ
1 η1 η
EH
ELEC-E4130 / TaylorLecture 13
Matching matrices across interfaces
η η
E
E
E
E
Total fields
η η
E
E ρE
E τE
+ traveling fields
η η
E
- traveling fields
ρ, τ ρ , τ ρ, τ ρ , τ ρ, τ ρ , τ
E ρ E
E τ E
E E E
H H H
E E E
H H H
Total fields
BoundaryConditions
E E
H H
Lossless dielectric No free charge TEM wave at normal incidence All fields tangential
Tangential field continuity
ELEC-E4130 / TaylorLecture 13
Matching matrices across interfaces
η η
E
E
E
E
Total fields
η η
E
E ρE
E τE
+ traveling fields
η η
E
- traveling fields
ρ, τ ρ , τ ρ, τ ρ , τ ρ, τ ρ , τ
E ρ E
E τ E
EE
1τ1 ρρ 1
EE
EE
1τ
1 ρρ 1
EE
E E E E
H H H H BoundaryConditions
ELEC-E4130 / TaylorLecture 13
Matching matrices across interfaces
η η
E
E
E
E
Total fields
η η
E
E ρE
E τE
+ traveling fields
η η
E
- traveling fields
ρ, τ ρ , τ ρ, τ ρ , τ ρ, τ ρ , τ
E ρ E
E τ E
ρη ηη η
τ2ηη η
𝛈 → 𝛈
ρη ηη η
τ2ηη η
𝛈 → 𝛈 Relations
τ 1 ρ
τ 1 ρ
ρ ρ
ττ 1 ρ
ELEC-E4130 / TaylorLecture 13
Matching matrices across interfaces
η η
E
E
E
E
Total fields
η η
E
E ρE
E τE
+ traveling fields
η η
E
- traveling fields
ρ, τ ρ , τ ρ, τ ρ , τ ρ, τ ρ , τ
E ρ E
E τ E
Impedance continuity
ZEH
EH Z η
1 Γ1 Γ η
1 Γ1 Γ
Slide 12
Algebra
Γρ Γ1 ρΓ
Γρ Γ1 ρ Γ
ELEC-E4130 / TaylorLecture 13
What if there only an incident field from left?
η η
E
E ρE
E τE
+ traveling fields
ρ, τ ρ , τ
E 0
ΓEE 0
Z η1 01 0 η
Z Z η
Γρ 01 ρ0 ρ
EE
1τ1 ρρ 1
E0
E ρE
Plug it in
E τE
solve This is the final condition for the dielectric slab Once energy enters the half space it does not
reflect back
ELEC-E4130 / TaylorLecture 13
Single Dielectric Slab
η
E
E
ρ , τ
η k η
E
E
E
E
E
ρ , τ
ℓ𝟏Interface 1 Interface 2
Z ZΓ Γ Γ Γ
ρη ηη η
τ 1 ρ
ρη ηη η
τ 1 ρ
Half space of 𝜂 and half space of 𝜂 separated by a slab of dielectric, ℓ𝟏 thick with 𝜂
Field incident from the right
Find total reflection: Γ
ELEC-E4130 / TaylorLecture 13
Reflection coefficient propagation
η
E
E
ρ , τ
η k η
E
E
E
E
E
ρ , τ
ℓ𝟏Interface 1 Interface 2
Z ZΓ Γ Γ Γ
Γρ Γ1 ρ Γ
From slide 19
Γ Γ e ℓFrom slide 14
Γρ Γ e ℓ
1 ρ Γ e ℓ
Γ ρ
From slide 20
Γρ ρ e ℓ
1 ρ ρ e ℓ
ELEC-E4130 / TaylorLecture 13
Matching and propagation matrices (1/2)
η
E
E
ρ , τ
η k η
E
E
E
E
E
ρ , τ
ℓ𝟏Interface 1 Interface 2
Z ZΓ Γ Γ Γ
From slide 17
EE
1τ
1 ρρ 1
EE
From slide 14
EE
1τ
1 ρρ 1
e ℓ 00 e ℓ
EE
From slide 17
EE
1τ
1 ρρ 1
e ℓ 00 e ℓ
1τ
1 ρρ 1
EE
ELEC-E4130 / TaylorLecture 13
Matching and propagation matrices (2/2)
η
E
E
ρ , τ
η k η
E
E
E
E
E
ρ , τ
ℓ𝟏Interface 1 Interface 2
Z ZΓ Γ Γ Γ
From slide 20: 𝐄𝟐 𝟎EE
1τ
1 ρρ 1
e ℓ 00 e ℓ
1τ
1 ρρ 1
E0
Multiply through
Ee ℓ
τ τ 1 ρ ρ e ℓ E
Ee ℓ
τ τ ρ ρ e ℓ Eratio
ΓEE
ρ ρ e ℓ
1 ρ ρ e ℓ
ELEC-E4130 / TaylorLecture 13
Transmission
η
E
E
ρ , τ
η k η
E
E
E
E
E
ρ , τ
ℓ𝟏Interface 1 Interface 2
Z ZΓ Γ Γ Γ
From slide 21: 𝐄𝟐 𝟎EE
1τ
1 ρρ 1
e ℓ 00 e ℓ
1τ
1 ρρ 1
E0
TEE
τ τ e ℓ
1 ρ ρ e ℓ
Ee ℓ
τ τ 1 ρ ρ e ℓ E
ELEC-E4130 / TaylorLecture 13
Conclusions and next time Boundary conditions (field continuities) allow for a solution to the steady state time
harmonic total fields E+(z), E−(z), and Γ(z) have simple propagation matrices but complicated relations
across the interface E(z), E(z), and Z(z) have complicated propagation matrices but simple relations
across the interface The matrix from provides a natural form for recursion and enables solution of the total
transmitted field Next time we’ll Solve for the reflection via geometric series (ray tracing) Extrapolate to multiple layers Explore oblique incidence angle
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