Interference of Electromagnetic Waves: Wave Properties of Light and Thin Film Interference, Slides of Physics

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Wave

properties

of

light

Interference

of

two

electromagnetic

waves

Interference

of

electromagnetic

waves

in

thin

films

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Webassign

hint:

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Another

example

in

your

text):

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From

previous

lecture:

Diverging

lens:

Converging

lens:

virtual

image

of

face

smaller

virtual

image

of

face

larger

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^

^

^

^

^

 

 

 

 







 

 





  

 





2

cos

1 2

π 2 λ

sin

2

π 2 λ

sin

π 2 λ

sin

) , (

)

(continued

ion

Superposit

max

max

max







vt

x

E

vt

x

E

vt

x

E

t x

E

tot y

  

  



  

  







  

  



  

  







2

cos

2

sin 4

)

sin(

)

sin( :

magnitude

Squared

2

cos

2

sin 2

)

sin(

)

sin(

:

identity

tric

trigonome

the

from

follows

result

that this

Note

2

2

2

B A B A B A

B A B A B A   

  



  

  









2

cos

4

2

cos

2

4

2

1

:

waves

EM

the

of

Intensity

2

1

2

max

0

2

2

max

0

1









I

E

c

I

I

E

c

S

I

tot

avg

Time

average

Constant

in

time

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Example:



=0.

rad

‐‐

plotting

snapshot

of

EM

wave

Example:



=

rad

plotting

snapshot

of

EM

wave

Intensity

as

a

function

of



:



/(



)

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Two

examples

of

superposed

radiation:

Interference

from

refraction

and

reflection

of

thin

films

Young’s

double

slit

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



λ

θ

sin

π

λ

π

2

1

m

d

m

r

r

λ

π

cos

π 2

λ

π

sin

π 2

π^ λ 2

sin

π 2

π^ λ 2

sin

2

1

2

1

max

2

max

1

max

r

r

ft

r

r

E

ft

r

E

ft

r

E

t

P

E

θ

sin

δ

2

1

d

r

r





intensity

maxima

occur

for

Young’s

double

slit

geometry:

Mathematical

analysis

of

bright

fringes:

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λ

θ

sin

π

λ

π

2

1

m

d

m

r

r

Diffraction

pattern

from

a

plane

wave

incident

on

a

double

slit:

λ

π

cos

π 2

λ

π

sin

π 2

π^ λ 2

sin

π 2

π^ λ 2

sin

2

1

2

1

max

2

max

1

max

r

r

ft

r

r

E

ft

r

E

ft

r

E

t

P

E

θ

sin

δ

2

1

d

r

r





intensity

maxima

occur

for

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Summary

of

results:

D^ d

m

y

m

d

λ

screen

on

Image

λ

θ

sin

ce

interferen

ve

Constructi

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Interference

between

reflected

waves:

Recall

‐‐

the

behavior

of

a

plane

‐polarized

electromagnetic

wave

near

the

surface

of

two

materials:

k

1 E

1

B

1

k

k

2

t

E

E

t

E

E

1

1

max

1

1

1

max

1

ω

sin

ω

sin 1 1

r

k

r

k 

 t

E

E

2

2

max

2

ω

sin 2

r

k

2

(^22) ω

v

k

1

(^11)

(^11)

ω

ω

v

k

k

Periodic

waves:

E

’^1

E

2

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k

1 E

1

B

1

k

k E ’^2

1

E

2

Matching

electric

and

magnetic

fields

at

boundary:

n

1 n

2

2

2

1

1

2

2

1

1

max

max

θ

cos

θ

cos

θ

cos

θ

cos

1

1

n

n

n

n

E

E

2

1

1

2

2

1

1

2

max

max

θ

cos

θ

cos

θ

cos

θ

cos

1

1

n

n

n

n

E

E

or

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Multiple

refractions

and

reflections

in

a

thin

film

^

^

^

 

^ 



 

  

^ 





 

 





^ 





 

^ 



 

2

1

2

av

2

1

max

(^22)

max

(^11)

max

λ π

cos^ sin

π 2

λ π

sin cos

2

π 2

π 2 λ

sin

π 2

π λ 2

sin

) , (

r

r

ft

r

r

E

ft

r

E

ft

r

E

t P E

t

r

r

1

2

P

^

^

^



λ

2

at

min

λ

2

at

max

(^2) λ π

sin

;

For

12

2

max

max

1

2

1

1

m

nt

m

nt

t

E

E

n

n









 





     

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Example

of

interference

with

“+”

n

1 n

2 n

3

2

1 2

max

cos

B

A

I

I

peaks

at

π

λ

π

1

2

1 2

m

r

r

B

A

n

1

<

n

< 2

n

3

For

each

surface

with

n

< 1

n

2

E

2

=

‐

E

1

for

reflected

beam

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