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Todays summary

A new look at propagation and phase delays

Description of plane & spherical waves in terms of phase delay

Interference

Interferometers

Michelson Mach-Zehnder

Young

MIT 2.71/2.710 Optics

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Optical path delay

t= 0

z

z= 2.875

t= T/4

z


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Optical path delay


10/18/04 wk7-a-3

z

t= 00E 0

2

2E

z= 2.875

t= T/4

z

02

2E

0

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Optical path delay

t= T/2


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z

z

t= 3T/4

0E0

2

2E

02

2E

0

z= 2.875

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Optical path delay

z

t ( )tE cos0

+

4cos0

tE

z= 2.875

t


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Optical path delay

z

t ( )tE cos0

+

4

7cos0

tE

z= 2.875

t

( )

z

iEtkzE 2expcos 00In general,phasor due

to propagation

(path delay)MIT 2.71/2.710 Optics

10/18/04 wk7-a-6

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Plane wave propagation

z=0

path delay increases

linearly withx

x

x

z

plane of observationMIT 2.71/2.710 Optics

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zz=0

path delay increases

linearly withx

x

x

plane of observation

+

cos2

sin2exp0

z

i

xiE


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x

path delay

at fixed z

2

3

2

3

2 33 2

sin2slope

( )

cos2sin2whereexp0zx

iE +=


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Spherical wave propagation


10/18/04 wk7-a-10

z=0


x path delay increases

withxx as

z

( )

z

x

zxz

2

222

+

forx

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z=0


x path delay increases

withxx as

z

( )

z

x

zxz

2

222

+

quadraticquadratic

near the zaxis

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z=0


x

z

+

z

i

z

xiE

2

exp2

0

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path delay

at fixed z

2

3

2 33 2 x

2

3

( )z

yxziE

22

0 2whereexp+

+=


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Optical path delays matter

path delay in

material of

index n :


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L

L

nn

nL2

compare with

free space

propagation :

L2

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Optical path delays matter


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L

L

nn path

differencedifference:

( )

Ln 1

2

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Phase delays matter

Can we measure them and how?

Direct measurement does not work: light waves

oscillate too fast for any instrument to follow

We need an indirect method

Solution: interferometersinterferometers map phase onto light

intensity which can be measured directly


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Interference


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Wave interference

12

observation screen


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Interference: extreme cases

+ =

Waves in-phase

Constructive

interference

+ =

Waves out-of-phase

Destructive

interference


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Interference vs phase delay & contrast

Intensity

I

average

intensity

fringe

visibility

0 2 3

0I

m aka

contrast

( )

( )

+=

=

ta

ta

cos2Field

cos1Field

2

1

2

2

2

1

212

aa

aam+=

( ) ( ) cos1

0mII +=

2

2

2

10 aaI +=

relative phase delay


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Interference vs phase delay & contrast

a1a2imperfect contrast

a1=a2perfect contrastIntensity Intensity


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Intensity

a1

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Polarization and interference

-polarized waves

interfere

I

+


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-polarized wavesdo not interfere

+

I

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Michelson interferometer

l1

l2

path difference: incominglaserbeam

( )2122

ll =


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Michelson with variable phase-delay

l1


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incominglaser

beam

l1

Gas cell

( ) cell122

tn =

n

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Mach-Zehnder interferometerbright

fringe

(matched

paths)


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incominglaser

beam

incoming

laser

beam

interference

pattern

spatial

period~

sin

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Young interferometer

incoming

laser

beamopaque

screen

x

d1

d2

a

l


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Young interferometerx

incoming

plane waveopaque

screen


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d1

d2

intensity

a

a

l=x=a/2

x=a/2

l

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Two point sources interfering: mathParaxial analysis:

x

intensity

a

l=

( ) ( )

( ) ( )

.2cos12

cos4

cos42exp2

),(),(:Intensity

.cos42exp2

22exp

22exp42exp

1

2/exp

2/exp2exp

1),(

:Amplitude

2

2

2

2

22

2

2

22

2

22

2

2222

+=

=

=

+++==

+++=

+

++

+=

=

++

+

+

=

x

l

a

ll

xa

l

l

xa

l

y

a

xi

li

liyxeyxI

l

xa

l

yaxi

li

li

l

xai

l

xai

l

ya

xi

li

li

l

yaxi

l

yaxi

li

liyxe


10/18/04 wk7-a-28

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