lecture 13 v2 - mycourses.aalto.fi

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


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


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


Single interfaceAirHalf space

DielectricHalf space

E

ΓE

E

ΓE

𝜺𝒓𝟏


Finite thickness layer AirHalf space


Layer 1

𝜺𝒓𝟏 𝜺𝒓𝟐

E

ΓE

E

ΓE

ℓ𝟏


Two layersAirHalf space


Layer 1 Layer 2

𝜺𝒓𝟏 𝜺𝒓𝟐 𝜺𝒓𝟐

E

ΓE

E

ΓE

ℓ𝟏 ℓ𝟐


N LayersAirHalf space


Layer 1 Layer 2 Layer N

𝜺𝒓𝟏 𝜺𝒓𝟐 𝜺𝒓𝐍

E

ΓE

E

ΓE

ℓ𝟏 ℓ𝟐 ℓ𝑵


Today: Normal incidence, single layerAirHalf space


Layer 1

𝜺𝒓𝟏 𝜺𝒓𝟐

E

ΓE

Γ𝜌 𝜌 𝑒 ℓ

1 𝜌 𝜌 𝑒 ℓ


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


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


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


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



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



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


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



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



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



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


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


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


Reflection coefficient propagation

η

E

E

ρ , τ

η k η

E

E

E

E

E

ρ , τ


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


Matching and propagation matrices (1/2)

η

E

E

ρ , τ

η k η

E

E

E

E

E

ρ , τ


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


Matching and propagation matrices (2/2)

η

E

E

ρ , τ

η k η

E

E

E

E

E

ρ , τ


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


Transmission

η

E

E

ρ , τ

η k η

E

E

E

E

E

ρ , τ


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


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

lecture 13 v2 - mycourses.aalto.fi

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