PHYS 408Applied Optics (Lecture 10)JAN-APRIL 2016 EDITIONJEFF YOUNGAMPEL RM 113

Quick review of key points from last lectureS and M matricies are associated with the transfer of fields across each interface, and their propagation through uniform films.

The matrix elements of S for going across interfaces are obtained from the Fresnel reflection and transmission coefficients.

The matrix elements of S for propagating through a uniform film include diagonal phase accumulation terms only.

The S matricies are straight forward to figure out, and the associated M matricies come from transforming the S matrices using linear algebra.

The net r and t for a stack of thin films is obtained by multiplying all M matricies sequentially to obtain Mnet for the entire structure, and then using linear algebra to either solve for r and t, or using the transformation properties from Mnet to Snet.

Bottom line

Consider Mnet as M, then Snet(A,B,C,D) yields t02,t20,r02,r20

Or don’t be lazy, and just solve for and from

0

2

UU

0

0

UU

02

0

0 UUU

M net

Can anyone think of another way to circumvent transforming from Mnet to Snet?

02

0

0 UUU

M net

What prevents you from, once you find Mnet, putting in values forand just multiplying it by Mnet to get the transmission?

0

0

UU

Would this help?

021

0

0 UM

UU

net

Going “inside” the structureThere is a very significant advantage to this approach.

02

0

0 UUU

M net

02

0

0011112

UUU

MMMor for instance, in our anti-refection example:

021

0

00111 12

UM

UU

MM

What does the right hand side of the following equation give you?

Internal Field Distributions (d2 infinite)

)()(

0)(

)0()0(

11

11121

0

00111 12 dzU

dzUdzUM

zUzU

MM

n1

d1

n2

d2

)(0U

)(0U

)(1U

)(1U )(

2U

z0

?)(?)(

0)(

)0()0(

?

?121111

0

001 12 zU

zUdzUMM

zUzU

M

?)(?)(

0?)(

?

?21111

101 12 zU

zUzUMMM

And what can you do with all the intermediate values?

0)(

)()( 121

11

1112

dzUM

dzUdzU

)(11

)(111#

1111 exp)(exp)(|)( dzikdziklayerinside dzUdzUzE

You should verify this agrees with the previous result:

)()(

0)(

)0()0(

11

11111

121111

0

001 12 dzU

dzUM

dzUMM

zUzU

M

11

11

~

~

11 00din

din

ee

M

Recall

Generalize

n1

d1

n1

d1

n2

d2

n1

d1

n2

d2

n3

d3

n1

d1

n2

d2

… …

nlayers-1 nlayers-1

Uniform periodic multilayer stack

n1

d1

n2

d2

n1

d1

n2

d2

n1

d1

n2

d2

n1

d1

n2

d2

……

Bragg reflection n1=1.3; n2=1.4; n3=n2

d1=400 nm; d2=200 nm; d2=d3

21 periods

0 2000 4000 6000 8000 100000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8X: 6284Y: 0.7877

Wavenumber 1/

Ref

lect

ivity

0 200 400 600 800 1000 12000

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

X: 297.8Y: 1.147e-006

Wavenumber 1/

Ref

lect

ivity

Bragg reflection n1=1.3; n2=1.4; n3=n2

d1=580 nm; d2=20 nm; d2=d3

21 periods

0 2000 4000 6000 8000 100000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

X: 6522Y: 0.1349

Wavenumber 1/

Ref

lect

ivity

0 200 400 600 800 10000

0.01

0.02

0.03

0.04

0.05

0.06

0.07

X: 304.7Y: 7.109e-007

Wavenumber 1/R

efle

ctiv

ity

Bragg reflection n1=1.3; n2=1.4; n3=n2

d1=580 nm; d2=20 nm; d2=d3

200 periods

0 2000 4000 6000 8000 100000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Wavenumber 1/

Ref

lect

ivity

6250 6300 6350 6400 6450 6500 65500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Wavenumber 1/

Ref

lect

ivity

Bragg reflection n1=2; n2=sqrt(12); n3=n2

d1=300 nm; d2=173 nm; d2=d3

10 periods

0 2000 4000 6000 8000 100000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Wavenumber 1/

Ref

lect

ivity

0 2000 4000 6000 8000 100000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Wavenumber 1/

Tran

smis

sion

Add a “defect”

n1

d1

n1

d1

n2

d2

n1

d1

n2

d2

n3

d3

n1

d1

n2

d2

……

What is this? n1=2; n2=sqrt(12); n3=4d1=300 nm; d2=173 nm; d3=0.15*d2

10 periods

0 2000 4000 6000 8000 100000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Wavenumber 1/

Tran

smis

sion

-5 -4 -3 -2 -1 0 1

x 10-4

0

10

20

30

40

50

60

70

Z (cm)

|E|2

-5 -4 -3 -2 -1 0 1

x 10-4

0

0.5

1

1.5

2

2.5

3

3.5

4

X: -0.0004499Y: 1

Z (cm)

|E|2

And this? n1=2; n2=sqrt(12); n3=4d1=300 nm; d2=173 nm; d3=2*d2

10 periods

0 2000 4000 6000 8000 100000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Wavenumber 1/

Tran

smis

sion

-5 -4 -3 -2 -1 0 1

x 10-4

0

10

20

30

40

50

60

70

Z (cm)

|E|2

Cavity Modes! n1=2; n2=sqrt(12); n3=4d1=300 nm; d2=173 nm; d3=10*d2

10 periods

0 2000 4000 6000 8000 100000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Wavenumber 1/

Ref

lect

ivity

-7 -6 -5 -4 -3 -2 -1 0 1

x 10-4

0

5

10

15

20

25

30

35

40

45

50

Z (cm)

|E|2

-7 -6 -5 -4 -3 -2 -1 0 1

x 10-4

0

5

10

15

20

25

30

Z (cm)

|E|2

-7 -6 -5 -4 -3 -2 -1 0 1

x 10-4

0

10

20

30

40

50

60

70

Z (cm)

|E|2

Cavity Modes! n1=2; n2=sqrt(12); n3=4d1=300 nm; d2=173 nm; d3=10*d2

10 periods SYMMETERIZED

3500 4000 4500 5000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Wavenumber 1/

Tran

smis

sion

-7 -6 -5 -4 -3 -2 -1 0 1

x 10-4

0

10

20

30

40

50

60

70

80

90

100

Z (cm)

|E|2

-7 -6 -5 -4 -3 -2 -1 0 1

x 10-4

0

20

40

60

80

100

120

140

160

180

200

Z (cm)

|E|2

-7 -6 -5 -4 -3 -2 -1 0 1

x 10-4

0

5

10

15

20

25

30

35

Z (cm)

|E|2