Lecture 21

Superposition and Coherence

Schedule

Week Topic Chapters

Apr 7 Interference Ch. 7 and 9

Apr 14 Diffraction Ch. 9+10

Apr 21 Diffraction/Polarization I Ch. 8, 10-11

Apr 28 Polarization II Ch. 8

May 5 EXAM II/Fourier

May 12 Lasers Ch. 13

May 19 Clean Up FINAL MAY 23

Superposition recall

Recall wave addition

y(x, t) Asin[ t ]

z(x, t) Bsin[ t ]

sin( )y z C t

2 2 2 2 ( )A B ABcosC tanAsin BsinAcos Bcos

Interference2 2 2 2 ( )A B ABcosC

1y(x, t) Asin[ t kx ]

When 2ABcos(-) > 0 Constructive interferenceWhen 2ABcos(-) < 0 Destructive interference

If both A and B are positive then maximum constructive (destructive) interference occurs when -=2m((2m+1))

Example:

2z(x, t) Asin[ t kx ]

Under what conditions does the amplitude of the sum equal zero?

1 2kx kx

1 2 1 2

2 2( ) ( )

o

x x n x x

1 2 )(2on x x

Interference Measurements

Typically when measuring electromagnetic waves, one measuresThe intensity of the light, rather than the amplitude.

2

2oIcE ò

What is the intensity of a beam corresponding to 2 different waves

1 2 1 22totI I I I I cos

21 2( )

2o

tot

cEI E ����������������������������ò

2 21 2 1 2| | | |

22o

tot

cE EI E E

����������������������������ò

Spherical waves

Intensity minimaAre fringes…

Demo

Measuring Interference

I

x

x

Michelson Interferometer

max min

max min

I II

VI

Uses for interferometers

• Measuring index of refraction

• Measuring material thickness

• Measuring material flatness

• Gravitational waves

• Measuring Coherence….

CoherenceIf each wave has a definite phase relationship, then the wave is coherent

Laser

0 50 100 150 200

X Axis

0.0

0.5

1.0

1.5

2.0

YAxis

Coherence lengthThe “length” that a light source is coherent.

x

1 2 1 22totI I I I I cos

1 2kx kx

Characteristic Coherence Lengths

20cm - >107m

0.3mm

900nm

Temporal Coherence

0 2 0 4 0 6 0 8 0 1 0 0

X A x i s

- 1 . 0

- 0 . 5

0 . 0

0 . 5

1 . 0

YAxis

c cx c t

2tc

Sin(x)+sin(1.1x)

Beating

0 20 40 60 80 100

X Axis

-2

-1

0

1

2

YAxis

0 2 0 4 0 6 0 8 0 1 0 0

X A x i s

- 1 . 0

- 0 . 5

0 . 0

0 . 5

1 . 0

YAxis

2 ( )sin( )tot m mEsin k x t xE k t

1 1 1

2 2 2

( )

( )

Esin k x t

E Esin x

E

k t

1 2

1 2

1( )21( )2

k k k

1 2

1 2

1( )21( )2

m

m

k k k

Beating

0 100 200 300 400 500 600 700 800 900 1000-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 100 200 300 400 500 600 700 800 900 1000-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 100 200 300 400 500 600 700 800 900 1000-3

-2

-1

0

1

2

3

0 100 200 300 400 500 600 700 800 900 1000-50

-40

-30

-20

-10

0

10

20

30

40

50

Y=sin(t)Y=sin(t)+sin(1.01t)Y=sin(t)+sin(1.01t)+sin(1.02t)Y=sin(t)+sin(1.01t)+sin(1.02t)+…sin(1.5t)

Fourier0

m mm 1 m 1

Af (x) A cos(mkx) B sin(mkx)

2

Pulse Bandwidth

0 20 40 60 80 100

X Axis

-1.0

-0.5

0.0

0.5

1.0

YAxis

0.00 0.05 0.10 0.15 0.20 0.25 0.30

X Axis

0

2

4

6

8

10

12

14

YAxis

12

v t

Fourier Bandwidth Limit

Equivalent to Heisenberg Uncertainty principle

Example

If we have a pulse that has a frequency bandwidth of 40THz,What is the FWHM delay range in a Michelson interferometer that one could see fringes?

What happens to the beat in a dispersive medium?

Superluminal?

Group velocity vs. Phase Velocity