Light Interference Continued…

Superposition

t

+1

-1

t

+1

-1

t

+2

-2

+

Constructive Interference

In Phase

5

Superposition

t

+1

-1

t

+1

-1

t

+2

-2

+

Destructive Interference

7

Out of Phase

180 degrees

Superposition

+Different f

1) Constructive 2) Destructive 3) Neither

-1.5

-1

-0.5

0

0.5

1

1.5

-1.5

-1

-0.5

0

0.5

1

1.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

10

Interference Requirements

• Need two (or more) waves

• Must have same frequency• Must be coherent (i.e. waves must have definite

phase relation)

12

Interference for Sound …For example, a pair of speakers, driven in phase, producing a tone of a single f and :

l1 l2

But this won’t work for light--can’t get coherent sources

hmmm… I’m just far enough away that l2-l1=/2, and I hear no sound at all!

15

Observe Laser Light Through…

One Slit:

Two Slits:

Multiple Slits:

Observe Laser Light Through…

One Slit: Broad Central Maximum…

Two Slits: Central Bright Spot with

symmetric dark fringes.

Multiple Slits: Central Bright Spot. Narrowor

bright spots, brighter maximums, darker

minimums.

Single Slit Diffraction

Double Slit

Interference Only Interference + Diffraction

Five Slit Diffraction Grating (Inteference & Diffraction)

How do we predict the locations of the bright and dark fringes produced by a single slit? double slit? Multiple slit?

Young’s Double Slit #1

Screen a distance L from slits

Single source of monochromatic light

d

2 slits-separated by d

A. Constructive

B. Destructive

C. Depends on L

The rays start in phase, and travel the same distance, so they will arrive in phase.

L

23

Light waves from a single source travel through 2 slits before meeting on a screen. The interference will be:

Young Double Slit #2

Screen a distance L from slits

Single source of monochromatic light

d

2 slits-separated by d

1) Constructive

2) Destructive

3) Depends on L

The rays start out of phase, and travel the same distance, so they will arrive out of phase.

L

25

½ shift

The experiment is modified so that one of the waves has its phase shifted by ½ . Now, the interference will be:

Young’s Double Slit Concept

Screen a distance L from slits

Single source of monochromatic light

d

2 slits-separated by d

L

At points where the difference in path length is 0, ,2, …, the screen is bright. (constructive)

27

At points where the difference in path

length is

the screen is dark. (destructive)

2

5 ,

23 ,

2

Young’s Double Slit Key IdeaL

30

Two rays travel almost exactly the same distance. (screen must be very far away: L >> d)

Bottom ray travels a little further.

Key for interference is this small extra distance.

d

Path length difference =

d

Young’s Double Slit Quantitative

Destructive interference dsin (m

12)

Constructive interference dsin m

where m = 0, or 1, or 2, ...

d sin

32Need < d

d

Destructive interference dsin (m

12)

Constructive interference dsin m

where m = 0, or 1, or 2, ...

Young’s Double Slit Quantitative

y

sin() tan() = y/L

dLm

y

d

Lmy

21

33

L

A little geometry…

d

L

Young’s Double Slit #3

y

When this Young’s double slit experiment is placed under water. The separation y between minima and maxima

1) increases 2) same 3) decreases

Under water decreases so y decreases35

d

Path length difference d

Double Slit #4

L

= d sin

8

2) dsin (m

12)

1) dsin m

where m = 0, or 1, or 2, ...

Which condition gives destructive interference? d sin()

d

Path length difference 1-2

Multiple Slits: (Diffraction Grating – N slits with spacing d)

L

= d sin

13 dsin mConstructive interference for all paths when

d

Path length difference 1-3= 2d sind

3

Path length difference 1-4= 3d sin

4

1

2

N slits with spacing d

Constructive Interference Maxima are at:

sin m

d

* screen VERY far away

Diffraction Grating

Same as for Young’s Double Slit !

3

23

2 19

3

23

2

dsin

Three slit interference

I0

9I0

For many slits, maxima are still at sin m

d

Region between maxima gets suppressed more and more as no. of slits increases – bright fringes become narrower and brighter.

10 slits (N=10)

dsin

inte

nsi

ty

0

2 slits (N=2)

dsin

inte

nsi

ty

0

Multiple Slit Interference (Diffraction Grating)

22

Peak location depends on wavelength!

Single Slit Interference?!

Wall

Screen with opening (or obstacle without screen)

shadow

bright

This is not what is actually seen!

Diffraction Rays

26

•

•

Diffraction/ HuygensEvery point on a wave front acts as a source of tiny wavelets that move forward.

We will see maxima and minima on the wall.

Light waves originating at different points within opening travel different distances to wall, and can interfere!

30

1st minima

Central maximum

W

w2

sin

W2

1

1

Rays 2 and 2 also start W/2 apart and have the same path length difference.

2

2

1st minimum at sin = /w

When rays 1 and 1 interfere destructively.

w2

sin 2

Under this condition, every ray originating in top half of slit interferes destructively with the corresponding ray originating in bottom half.

Single Slit Diffraction

33

w

Rays 2 and 2 also start w/4 apart and have the same path length difference.

2nd minimum at sin = 2/w

Under this condition, every ray originating in top quarter of slit interferes destructively with the corresponding ray originating in second quarter.

Single Slit Diffraction

4w

1

1

)sin(4w

2

2

When rays 1 and 1 will interfere destructively.

2)sin(

4w

35

Condition for quarters of slit to destructively interfere

sin m

w

(m=1, 2, 3, …)

Single Slit Diffraction SummaryCondition for halves of slit to destructively interfere w

)sin(

w 2)sin(

Condition for sixths of slit to destructively interfere w

3)sin(

38

THIS FORMULA LOCATES MINIMA!!

Narrower slit => broader pattern

All together…

Note: interference only occurs when w >

Recap.• Interference: Coherent waves

– Full wavelength difference = Constructive– ½ wavelength difference = Destructive

• Multiple Slits– Constructive d sin() = m m=1,2,3…)– Destructive d sin() = (m + 1/2) 2 slit only– More slits = brighter max, darker mins

• Huygens’ Principle: Each point on wave front acts as coherent source and can interfere.

• Single Slit:– Destructive: w sin() = m m=1,2,3…)– Resolution: Max from 1 at Min from 2

50

op

posi

te!