Patterned Border Template 1 Purdue University School of Mechanical Engineering ME 587 Engineering Optics Lecture 16: Interference Prof. Robert P. Lucht

Patterned Border Template 1

Purdue University

School of Mechanical Engineering

ME 587Engineering Optics

Lecture 16: Interference

ME 587Engineering Optics

Lecture 16: Interference

Prof. Robert P. Lucht

Room 86, Mechanical Engineering BuildingSchool of Mechanical Engineering

Purdue UniversityWest Lafayette, Indiana

[email protected], 765-494-5623 (Phone)

October 25, 2005


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Optical InterferenceOptical Interference

This Lecture• Two-Beam Interference• Young’s Double Slit Experiment• Virtual Sources• Newton’s Rings• Film Thickness Measurement by Interference

Last Lecture• Conduction Current in Metals• EM Wave Propagation in Metals• Skin Depth• Plasma Frequency


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Two-Beam InterferenceTwo-Beam Interference

1 2

1 01 1 1

2 02 2 2

:

, cos

, cos

Consider two waves E and E that have the same frequency

E r t E k r t

E r t E k r t


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1 2

1 2

2

20 0 1 2 1 2

2 20 1 2 1 2

1 2 12

:

, , ,

/ :

2

p

p

p

At a given position r the wave E is the superposition of E and E

E r t E r t E r t

The irradiance W m is given by

I c E c E E E E

c E E E E

I I I I

Hecht, Optics, Chapter 9.


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2 21 0 1 2 0 2

12 0 1 2

1 2 01 02 1 1 2 2

1 2 01 02 1

1 2 :

2

cos cos

cos cos cos sin sin

cos

The irradiances for beams and are given by

I c E I c E

The interference term is given by

I c E E

E E E E k r t k r t

But A B A B A B

E E E E k

1 1 1

2 2 2 2

2 201 02 1 1 2 2 1 1 2 2

1 1 2 2 1 1 2

cos sin sin

cos cos sin sin

cos cos cos sin sin sin

cos sin cos sin sin cos

r t k r t

k r t k r t

E E k r k r t k r k r t

k r k r t t k r k

2 cos sinr t t


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2 2

2 212 01 02 1 1 2 2 1 1 2 2

1 1 2 2

:

1 1cos sin cos sin 0

2 2

2 cos cos cos sin sin sin

cos sin cos sin sin

The time averages are given by

t t t t

The interference term is given by

I E E k r k r t k r k r t

k r k r t t

1 1 2 2

01 02 1 1 2 2 1 1 2 2

01 02 1 1 2 2

12 01 02 1 1 2 2

01 02

1

cos cos sin

cos cos sin sin

cos

cos

1

2

k r k r t t

E E k r k r k r k r

E E k r k r

I E E k r k r

If E and E are parallel then

I

2 20 01 2 0 02 12 1 2

1cos

2c E I c E I I I


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1 2 1 2

max 1 2 1 2 1 2 1 2

:

2 cos

0, 2 , :

2 cos 2 2

.

The total irradiance is given by

I I I I I

There is a maximum in the interference pattern when

I I I I I n I I I I

This is referred to as constructive interference

There is a minimum in the inter

min 1 2 1 2 1 2 1 2

1 2 0

20 0 0 max 0 min

, 3 , :

2 cos 2 1 2

.

2 2 cos 4 cos 4 02

ference pattern when

I I I I I n I I I I

This is referred to as destructive interference

When I I I

I I I I I I I


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Conditions for InterferenceConditions for Interference

Interference Between Light From Two Sources• Sources must be in phase with each other – sources must be coherent• Sources must have the same frequency• Radiation must overlap within the coherence lengths of each source to produce an interference pattern – discharge lamps have coherence lengths on the order of a few mm, some lasers have coherence lengths on the order of a few km• Clearest patterns will be produced when the sources have nearly the same amplitude – maximum contrast


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Conditions for InterferenceConditions for Interference



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Young’s Double Slit Interference ExptYoung’s Double Slit Interference Expt



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2 1

2 1

. :

sin

:

1sin

2

:

Assume that y s and a s The condition for an interference maximum is

S P S P m a

The condition for an interference minimum is

S P S P m a

Relation between

geometric path difference

and phase difference

2


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2 2 20 0 0

20

max

min

sin4 cos 4 cos 4 cos

2

:

sin tan 4 cos

: 0, 1, 2,

12:

On the screen the irradiance pattern is given by

aI I I I

Assuming that y s

y a yI I

s s

m sBright fringes y m

a

mDark fringes y

0, 1, 2,s

ma


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Interference Fringes From 2 Point SourcesInterference Fringes From 2 Point Sources


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Interference Fringes From

2 Point Sources

Interference Fringes From

2 Point Sources



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Interference With Virtual Sources:Fresnel’s Double Mirror

Interference With Virtual Sources:Fresnel’s Double Mirror



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Interference With Virtual Sources:Fresnel’s Biprism

Interference With Virtual Sources:Fresnel’s Biprism



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Interference With Virtual Sources:Lloyd’s Mirror

Interference With Virtual Sources:Lloyd’s Mirror



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Interference in Dielectric FilmsInterference in Dielectric Films


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Analysis of Interference in Dielectric FilmsAnalysis of Interference in Dielectric Films


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0

0

0 sin sin :

sin sin2

sin

sin2 sin

sin

p f

f f

i f t

t t

i

tt

The phase difference due to optical path length

differences for the front and back

reflections is given by

n AB BC n AD

n AE FC n AD n EB BF

But n n

ACAE AG

AD AC

AE AC AD

0

0 2

2

cos 2 cos

i f

f f

p f f

t p f t

nAD

n

n AD n AE n AE FC

n EB BF n EB

EB t n t


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:

2 cos

:

0, 1, 2,

:

1

2

r

f t r

p r

p r

Also need to account for phase differences

due to differences in the reflection process

at the front and back surfaces

n t

Constructive interference

m m

Destructive interference

m

0, 1, 2,m


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Fringes of Equal InclinationFringes of Equal Inclination

2 cos

:

0, 1, 2,

:

1

2

0, 1, 2,

f t r

p r

p r

Fringes arise as Δ varies due to changes

in the incident angle:

n t


m

m


m

m


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Fringes of Equal ThicknessFringes of Equal Thickness

2 cos

:

0, 1, 2,

:

1

2

f t r

p r

p r

When the direction of the incoming light is

fixed, fringes arise as Δ varies due to changes

in the dielectric film thickness :

n t


m m


m

0, 1, 2,m


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Fringes of Equal Thickness: Newton’s RingsFringes of Equal Thickness: Newton’s Rings


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Fringes of Equal Thickness: Newton’s RingsFringes of Equal Thickness: Newton’s Rings

Documents

Patterned Border Template 1 Purdue University School of Mechanical Engineering ME 587 Engineering Optics Lecture 16: Interference Prof. Robert P. Lucht