White n t Wavelength (nm) R t = 400 nm, n = 1.515, = 0.

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white

2

cos16.0 20

II p

o

rpp II

2

0 cos16.0

o

p

otnII

22

0

2cos16.0n t

2

2cos16.0 2

0

o

p

tnII

Wavelength (nm)

R

t = 400 nm, n = 1.515, = 0

0

0.05

0.1

0.15

0.2

400 450 500 550 600 650 700

Eo

Reflection coefficient:

Transmission coefficient:

o

r

EE

r

o

t

E

Et

for internal:o

r

EE

r '

rEo

tEo tr’E

o

tt’r’Eo tt’r’3Eo tt’r’5Eott’r’7Eo

for internal:o

t

E

Et '

nd

Multiple Beam Interference

Amplitude of each reflection:

t

Modify amplitude and phase for each reflected ray:

tioerEE 1

tioeErttE ''2

233 '' ti

oeErttE 35

4 '' tioeErttE

132'' Ntio

NN eErttE

For N >= 2

Phase lag between reflections:

0cos222 t

orp

o

nd

(difference between the first and the rest will be covered by reflection amplitudes r and r’)

2

132''N

Ntjo

NtjoR eErtterEE

Total reflected field:

factor

2

242'''N

NjNjtjoR ererttreEE

2

22'''N

NjjtjoR ererttreEE

combine

...1 2

2

2

xxxN

N

Which converges to:x

11

for |x|<1

This makes a geometric series:

j

jtjoR er

erttreEE 2'11

''

By the way:

21' rtt

'rr Stoke’s Relations

j

jtj

oR erer

eEE 211

…but when using complex notation for fields:

221

RoR cEI

2

21

RoR EcI …which means…*

21

RRoR EcEI

…where * means the complex conjugate ( j -> -j )

j

jtj

j

jtj

ooR erer

eerer

ecEI 22

221

11

11

422

22

21

111rerer

eercEI jj

jj

ooR

1cos2

cos2224

22

21

rrr

cEI ooR

ii ee cos2Use:

221

ooo cEI oR IIR

1cos2cos22

24

2

rrr

R

Double Beam

Multiple Beam

0

0.05

0.1

0.15

0.2

400 450 500 550 600 650 700

R

t = 500 nm, n = 1.515, = 0

Wavelength (nm)

t = 500 nm, n = 1.515, = 0

R

0

0.05

0.1

0.15

0.2

400 450 500 550 600 650 700

Double Beam

Multiple Beam

Add 50% reflective surfaces!

Wavelength (nm)

t = 500 nm, n = 1.515, = 0

00.20.40.60.81

1.2

400 450 500 550 600 650 700

R

00.20.40.60.81

1.2

400 450 500 550 600 650 700

t = 500 nm, n = 1.515, = 0

R

Phasors: Describe amplitude and phase with vector length and direction.

A

B

A+B

constructive destructive

A

B

A+B

A

B

A+B

Phasors: Describe amplitude and phase with vector length and direction.

in between

00.20.40.60.81

1.2

400 450 500 550 600 650 700

destructive:

constructive:

/20 off constructive:

2

2cos16.0 2

o

tnR

/20 off destructive

destructive

00.20.40.60.81

1.2

400 450 500 550 600 650 700

R

tioerEE 1

tioeErttE ''2

233 '' ti

oeErttE 35

4 '' tioeErttE

Double Beam

Multiple Beam

Changing phase, not time

00.20.40.60.81

1.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

R

t = 500 nm, n = 1.515, = 600 nm

0 45 90

i

Multiple beam interference also causes sharp peaks with angle of incidence:

11cos2

cos2224

2

T

rrr

1TR

1cos2cos22

1cos21cos2

24

2

24

24

rrr

rrrr

T

1cos212

24

24

rr

rrT

1cos2124

22

rrr

T

-0.20

0.20.40.60.81

400 450 500 550 600 650 700

t = 500 nm, n = 1.515, = 0

T

Multiple beams interfere constructively and destructively with a much sharper phase dependence than double beam interference since the multiple interfering components dephase at different rates.

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