lecture -- transfer matrices · 2 days ago · concept of transfer matrices (6 of 15) slide 9 11...

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Advanced Computation:

Computational Electromagnetics

Transfer Matrices

Outline

• Concept of transfer matrices

• Wave vector components

• Calculating transfer matrices

• Stability of transfer matrices

Slide 2

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Slide 3

Concept of Transfer Matrices

Concept of Transfer Matrices (1 of 15)

Slide 4

Suppose it is desired to simulate four dielectric slabs.

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Slide 5

The process starts by defining the field at the input face of the device.

0

0

,

,

E x y

H x y


Slide 6

1 0

1 0

, ,?

, ,

E x y E x y

H x y H x y

How can the field at the first interface inside the device be calculated?

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Slide 7

The first transfer matrix 𝑇 is calculated by analyzing Layer 1.

1 0

1 0

, ,?

, ,

E x y E x y

H x y H x y


Slide 8

1 111 121 0

1 11 021 22

, ,

, ,

t tE x y E x y

H x y H x yt t

The transfer matrix 𝑇 is used to calculate the field at the first interface inside the device.

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Slide 9

1 111 121 0

1 11 021 22

, ,

, ,

t tE x y E x y

H x y H x yt t

Now the field at the first interface inside the device is known.


Slide 10

The second transfer matrix 𝑇 is calculated by analyzing Layer 2.

1 111 121 0

1 11 021 22

, ,

, ,

t tE x y E x y

H x y H x yt t

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Slide 11

2 2 1 111 12 11 122 0

2 2 1 12 021 22 21 22

, ,

, ,

t t t tE x y E x y

H x y H x yt t t t

The second transfer matrix 𝑇 is used to calculate the field at the second interface inside the device.


Slide 12

2 2 1 111 12 11 122 0

2 2 1 12 021 22 21 22

, ,

, ,

t t t tE x y E x y

H x y H x yt t t t

Now the field at the second interface inside the device is known.

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Slide 13

The third transfer matrix 𝑇 is calculated by analyzing Layer 3.

2 2 1 111 12 11 122 0

2 2 1 12 021 22 21 22

, ,

, ,

t t t tE x y E x y

H x y H x yt t t t


Slide 14

3 3 2 2 1 111 12 11 12 11 123 0

3 3 2 2 1 13 021 22 21 22 21 22

, ,

, ,

t t t t t tE x y E x y

H x y H x yt t t t t t

The third transfer matrix 𝑇 is used to calculate the field at the third interface inside the device.

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Slide 15

3 3 2 2 1 111 12 11 12 11 123 0

3 3 2 2 1 13 021 22 21 22 21 22

, ,

, ,



Now the field at the third interface inside the device is known.


Slide 16

3 3 2 2 1 111 12 11 12 11 123 0

3 3 2 2 1 13 021 22 21 22 21 22

, ,

, ,



The fourth transfer matrix 𝑇 is calculated by analyzing Layer 4.

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4 4 3 3 2 2 1 111 12 11 12 11 12 11 124 0

4 4 3 3 2 2 1 14 021 22 21 22 21 22 21 22

, ,

, ,

t t t t t t t tE x y E x y

H x y H x yt t t t t t t t


Slide 17

The fourth transfer matrix 𝑇 is used to calculate the field at the output face of the device.

4 4 3 3 2 2 1 111 12 11 12 11 12 11 124 0

4 4 3 3 2 2 1 14 021 22 21 22 21 22 21 22

, ,

, ,




Slide 18

Finally the field at the output face the device is known.

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4 4 3 3 2 2 1 111 12 11 12 11 12 11 124 0

4 4 3 3 2 2 1 14 021 22 21 22 21 22 21 22

, ,

, ,



The Global Transfer Matrix

Slide 19

Alternatively, all of the intermediate transfer matrices can be multiplied together to calculate the global transfer matrix that directly relates the field at the input and output faces.

Global Transfer Matrix

𝑇 𝑇 𝑇 𝑇 𝑇

Slide 20

Wave Vector Components

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Calculation of the Wave Vector Components

Slide 21

The components kx and ky are determined by the incident wave and are equal throughout the entire device. The kz component is different in each layer and is calculated from the dispersion relation in that layer.

0 r,inc r,inc

0 r,inc r, c

,inc

,inc in

sin cos

sin sin

x

y

x

y

k k

k

k

k k

2 2 2, 0 r, r,z i i i x yk k k k

Layer #i

kx and ky are Continuous Throughout Device

Slide 22z

xinck

kx

refk -kz,air

kxkz,air

22 2,air 0 airz xk k n k

,inc 0 ai

,inc a

r

0 ir

cos sin

cosz

x

k k n

k k n

kx

kz,11k

22 2,1 0 1z xk k n k

1n

2n

3n

kx

kz,22k

22 2,2 0 2z xk k n k

kx

kz,33k

22 2,3 0 3z xk k n k

kx

kz,airtrnk

22 2 2,trn 0 air ,airz x zk k n k k

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Slide 23

Calculating Transfer Matrices

Geometry of an Intermediate Layer

Slide 24

Layer i Layer i+1Layer i-1

0iψ

1 0 1i ik L ψ

iL

0i ik Lψ

1 0iψ

1iL 1iL

1icic1ic

i izψ

is a local z‐coordinate inside the ith layer that starts at zero at the layer’s left side.

iz0

iz

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Enforce the Boundary Conditions

Slide 25

Field inside the ith layer:

,

,

,

,

i i

x i i

y i i zi i i i

x i i

y i i

E z

E zz e

H z

H z

λψ W c

Boundary conditions at the first interface:

Boundary conditions at the second interface:

1 0 1

1 0 1

1 1

0

i i

i i i

k Li i i i

k L

e

λ

ψ ψ

W c Wc

0

0 1

1 1

0

i i

i i i

k Li i i i

k L

e

λ

ψ ψ

W c W c

Must include k0 in the exponential to normalize Li-1 because the parameter i-1 expects to multiply a normalized coordinate.

Note: Must equate the field on either side of the interfaces and not the mode coefficients c.

The Transfer Matrix

Slide 26

The transfer matrix Ti of the ith layer is defined as:

Start with the boundary condition equation from the second interface and rearrange terms…

1i i i c T c

iT0 01

1 1 1 1 i i i ik L k Li i i i i i i ie e

λ λW c W c c W W c

011

i ik Li i ie

λT W W…then read off the transfer matrix.

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Slide 27

Stability of Transfer Matrices

The Multi‐Layer Problem

Slide 28

The diagram below is focused on the ith layer somewhere in the middle of a stack of multiple layers.

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Forward Waves in the ith Layer

Slide 29

Recall that the wave number 𝑘 is in general complex. Waves can oscillate, decay or both.

k k jk decayoscillation

Backward Waves in the ith Layer

Slide 30

Due to reflections at the interfaces, there will be backward waves. These can also oscillate, decay or both.



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Waves According to Transfer Matrices

Slide 31

In terms of signs, transfer matrices treat all waves as forward propagating, even the backward waves.



?

,

,

,

,

i i

x i i

y i i zi i i i

x i i

y i i

E z

E zz e

H z

H z

λψ W c

Pure Transfer Matrices are Unstable

Slide 32

Transfer matrices treat all waves as if they are forward propagating. Backward waves that also decay increase exponentially and become numerically unstable.


k k jk explodebackward

oscillation

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The Fix – Identify Forward and Backward Waves

Slide 33

Clearly, the first part of the fix is to identify forward and backward propagating waves.

This can be accomplished by calculating the Poynting vector ℘ associated with the eigen‐modes and looking at the sign of the z component. Be careful! A normalized magnetic field is being used.

0 0

0

1

z x y y x

y xz x y

z x y y x

E H

E H E H

H HE E

j j

E H E Hj

0.32 0.32 0 0

0 0 0.32 0.32

0 0 0.95 0.95

0.95 0.95 0 0

i i

i i

W

3.0 0 0 0

0 3.0 0 0

0 0 3.0 0

0 0 0 3.0

i

i

i

i

λ

Rearrange the Eigen Modes

Slide 34

Now that it is known which eigen‐modes are forward and backward propagating, they can be rearranged to group them together.

0.32 0.32 0 0

0 0 0.32 0.32

0 0 0.95 0.95

0.95 0.95 0 0

i i

i i

W

3.0 0 0 0

0 3.0 0 0

0 0 3.0 0

0 0 0 3.0

i

i

i

i

λ

0.32 0 0.32 0

0 0.32 0 0.32

0 0.95 0 0.95

0.95 0 0.95 0

i i

i i

W

Also need to adjust the vertical positions of the eigen‐values so that 𝛌 remains a diagonal matrix.

rearrangeeigen‐modes

Original Matrices Rearranged Matrices

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New Interpretation of the Matrices

Slide 35

3.0 0 0 0

0 3.0 0 0

0 0 3.0 0

0 0 0 3.0

i

i

i

i

λ

0.32 0 0.32 0

0 0.32 0 0.32

0 0.95 0 0.95

0.95 0 0.95 0

i i

i i

W

x

y

x

y

E

E

H

H

E E

H H

zz

z

ee

e

λλ

λ

W WW

W W

0

0

The matrices are now partitioned into forward and backward propagating elements.

x

y

x

y

E

E

H

H

3.0 0 3.0 0

0 3.0 0 3.0

i i

i i

λ λNote: For anisotropic materials, all the eigen‐vectors and eigen‐values are in general unique.

Revised Solution to Differential Equation

Slide 36

The matrix differential equation and its original solution was

zdz e

dz

λψ

Ωψ 0 ψ W c

After distinguishing between forward and backward propagating waves and grouping them in the matrices, the solution can be written as

z

E E

zH H

ez

e

λ

λ

0W W cψ

W W c0

There are now separate mode coefficients c+ and c- for forward and backward propagating modes, respectively.

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lecture -- transfer matrices · 2 days ago · concept of transfer matrices (6 of 15) slide 9 11...

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