1 electromagnetic waves: multiple beam interference friday november 8, 2002

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Electromagnetic waves: Multiple beam Interference

Friday November 8, 2002

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Anti-Reflection coatingsA. A. Determine thickness of filmDetermine thickness of film

filmfilm

glassglass

airairnn11

nn33

nn22nn1 1 < n< n2 2 < n< n33

Thus both rays (1 and 2) are shifted in phase by Thus both rays (1 and 2) are shifted in phase by on reflection. on reflection.

1122

For destructive interference (near normal incidence)For destructive interference (near normal incidence)

2n2n22t=(m+1/2)t=(m+1/2)

Determines the thickness of the filmDetermines the thickness of the film(usually use m=0 for minimum t)(usually use m=0 for minimum t)

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Anti-Reflection coatingsB. Determine refractive index of filmB. Determine refractive index of film

Near normal incidenceNear normal incidence

Amplitude at AAmplitude at A

filmfilm

glassglass

airair

nn11

nn33

nn22

1122

AAA’A’

23

23

23

23

32

2

21

1 22

''

nn

nnE

nn

nn

nn

n

nn

nE

EE

A

A

oA

Since Since ’ ~ 1’ ~ 1

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Anti-reflection coating

12

12' nn

nnEA

Amplitude at A’Amplitude at A’

B. Determine refractive index of filmB. Determine refractive index of film

To get perfect cancellation, we would like ETo get perfect cancellation, we would like EA A = E = E A’A’

21

12

23

23

nn

nn

nn

nn

312 nnn should be index of AR filmshould be index of AR film

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Multiple Beam interference

Thus far in looking at reflectivity from a dielectric layer we have assumed that the reflectivity is small

The problem then reduces to two beam interference Now consider a dielectric layer of uniform thickness d

and assume that the reflectivity is large e.g. || > 0.8 This is usually obtained by coating the surface of the

layer with a thin metallic coating – or several dielectric coatings to give high reflectivity

Or, one can put coatings on glass plates , then consider space between plates

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Multiple beam interferenceLet 12 = 21= ’ 12= 21= ’

nn11

nn22

nn11

’’

AA BB CC DD

EEoo

’’’ ’ EEoo

’ ’ EEoo

((’)’)33’E’Eoo

((’)’)22’E’Eoo ((’)’)66’E’Eoo

((’)’)44’E’Eoo

((’)’)55’E’Eoo

((’)’)77’E’Eoo

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Multiple Beam Interference

Assume a (for the time being) a monochromatic source , ’ small ( < 30o) usually Now || = |’| >> , ’ Thus reflected beams decrease rapidly in amplitude (from

first to second) But amplitude of adjacent transmitted beam is about the

same amplitude Amplitude of successfully reflected beams decreases

slowly (from the second) Thus treat in transmission where contrast should be

somewhat higher The latter is the configuration of most applications

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Multiple Beam Interference

Assume phase of transmitted beam at A is such that,

Now let ’be the phase shift in travelling across and back once, i.e.

tioA eEE

A

'

'cos2' 2

dk

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Multiple Beam interference

At B:

At C:

At D:

tioB eEE

A

'2''

tioC eEE

A

'24''

tioB eEE

A

'36''

etc…etc…

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Multiple beam interference Consider N beams which interfere at infinity We can use a lens and then beams shown interfere in

focal plane of lens The phase difference between adjacent rays outside is,

sin'tan22 111 dkSk

ΔS1

N N-3N-2

N-1

N-4

ΔS1

’

dn2

n1

n1

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Multiple beam interference

Amplitude at point P,

tNiNo

tiNio

tiNio

tiNioP

oNNP

eTE

eeTE

eeTE

eTeEE

EEEE

A

A

A

A

'1)1(2

'234

'22

1

1

'

'

'

12


Amplitude at point P,

]'

''1[')1()1(2

'24'2

1

NiN

ii

tiNioP

e

ee

eTeEEA

i

i

er

er2

'2

'

'

Let

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Multiple beam interference 121 1 NNitio

AP rrreTeEE

NN

NN

rrrS

rrrS

2

121

1

1

NNN rSS 1

This is just a geometric series with r < 1. Thus,

11 NN SrS

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Multiple beam interference111 N

NN rSrS

Nelforrr

rS

N

N arg1

1

1

11

Thus,

rrTEE

and

reTeEE

oAP

NitioAP

1

1

1

1

1

1

222

1

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*2* 1

1

1

1

1

1

rrrrr

cos'2'

'22*

42

ii eerr

r

Evaluate

Thus,

cos1'2'1

1

cos'2'1

1

1

1

1

1

222

222*

rr

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2sin

'1

'41'1

1

1

1

1

1

222

222

*

rr

Now,

2sin

1

41

1

1

,

'

22

2

222

22

R

RR

TEE

and

R

oA

P

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2

2

2

12

1

oAo

oPP

EvI

EvI

Now recall the definition of the intensity of an electromagnetic wave

Thus,

oP I

R

RR

T

I

2sin

1

41

1

22

2

2

is the intensity distribution in the focal plane of the lens.

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'cos2

sin'tan2'cos

2

'

2

12

dkn

dkndkn

o

oo

Fringe pattern

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Maximum intensity when,

mdn

or

mor

o

2'cos4

,

202

sin

2

2

mdn 'cos2 2

1 electromagnetic waves: multiple beam interference friday november 8, 2002

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