1

Interference in One DimensionInterference in One DimensionWe will assume that the waves have the same frequency, and travel to the right along the x-axis.

source

202022

101011

sin

sin

tkxEE

tkxEE

2

202101

202101

0201202

2010

0

21

0

102202022

101101011

coscos

sinsintan

cos2

sin

,

sinsin

sinsin

EE

EE

EEEEE

tEE

EEE

kx

tEkxtEE

tEkxtEE

3

The Phase DifferenceThe Phase Difference1 1 10kx t

2 2 20kx t

1 2 1 10 2 20

1 2 10 20 0

( ) ( )

( ) ( ) 2

kx t kx t

xk x x

Path-length difference Inherent phase difference

If the waves are initially in-phase

00

22kxnx

21 xxnxn The optical path difference (OPD)

If 0 is constant waves are said to be coherent.

4

5

1 1

2 2sin sin 2cos ( ) sin ( )

2

sin2

cos2 01

xxkt

xkEE

0, 00201

0201

so

EEif

If x << λ constructive interference

If x = λ/2 destructive interference

6

The Phase DifferenceThe Phase Difference

00

For constructive interference:

2 2 or 2 2

x xm m

The condition of being in phase, where crests are aligned with crests and troughs with troughs, is that

= 0, 2, 4, or any integer multiple of 2.

For identical sources, 0 = 0 rad , maximum constructive interference occurs when x = m ,

Two identical sources produce maximum constructive interference when the path-length difference is an integer number of wavelengths.

7

The Phase DifferenceThe Phase Difference

10 2

102

For destructive interference:

2 2( ) or

2 2

xm

xm

The condition of being out of phase, where crests are aligned with troughs of other, that is,

=, 3, 5 or any odd multiple of .

For identical sources, 0 = 0 rad , maximum constructive interference occurs when x = (m+ ½ ) ,

Two identical sources produce perfect destructive interference when the path-length difference is half-integer number of wavelengths.

8

9

The Superposition of Many WavesThe Superposition of Many WavesThe superposition of any number of coherent harmonic waves having a given frequency and traveling in the same direction leads to a harmonic wave of that same frequency.

i

N

ioi

i

N

ioi

N

ijioj

N

ij

N

ioii

N

iii

E

E

EEEE

tEtEE

cos

sintan

cos2

coscos

1

1

1 1

20

20

01

0

10

The Mathematics of Standing The Mathematics of Standing WavesWaves

The Mathematics of Standing The Mathematics of Standing WavesWaves

tkxatkxaDDtxD LR sinsin,

tkxaDR sin

A sinusoidal wave traveling to the right along the x-axis

An equivalent wave traveling to the left is

tkxaDL sin

sincoscossinsin

tkxa

tkxtkxatkxtkxatxD

cos)sin2(

)sincoscos(sin)sincoscos(sin,

txAtxD cos)(),( Where the amplitude function A(x) is defined as

kxaxA sin2)(

tkxEtxE cossin2),( 01

11

NotesNotes

txAtxD cos)(),(

kxaxA sin2)( The amplitude reaches a maximum value Amax = 2a at points where sin kx =1.

The displacement is neither a function of (x-vt) or (x+vt) , hence it is not a traveling wave.

The cos t , describes a medium in which each point oscillates in simple harmonic motion with frequency f= /2.

The function A(x) =2a sin kx determines the amplitude of the oscillation for a particle at position x.

12

txAtxD cos)(),(

The amplitude of oscillation, given by A(x), varies from point to point in the medium.

The nodes of the standing wave are the points at which the amplitude is zero. They are located at positions x for which

0sin2)( kxaxA

That is true if ,3,2,1,02

mmx

kx mm

Thus the position xm of the mth node is

,3,2,1,02

mmxm

Where m is an integer.

13

Example:Cold Spots in a Microwave Oven

Example:Cold Spots in a Microwave Oven

“Cold spots”, i.e. locations where objects are not adequately heated in a microwave oven are found to be 1.25 cm apart.

What is the frequency of the microwaves?

node 1.25 cm / 2 so 2.50 cmd

810(3.00 10 m/s)

1.20 10 Hz 12.0 GHz(0.0250 m)

cf

14

Beats and ModulationBeats and Modulation

If you listen to two sounds with very different frequencies, you hear two distinct tones.

But if the frequency difference is very small, just one or two Hz, then you hear a single tone whose intensity is modulated once or twice every second. That is, the sound goes up and down in volume, loud, soft, loud, soft, ……, making a distinctive sound pattern called beats.

15

When two light waves of different frequency interfere, they also produce beats.When two light waves of different frequency interfere, they also produce beats.

0 1 1 0 2 2

1 2 1 2

1 2 1 2

0 0

( , ) Re{ exp ( ) exp ( )}

2 2

2 2

( , ) Re{ exp ( ) exp (

tot

ave

ave

tot ave ave ave

E x t E i k x t E i k x t

k k k kk k

E x t E i k x kx t t E i k x kx

Let and

Similiarly, and

So :

0

0

0

)}

Re{ exp ( ) exp ( ) exp[ ( )] }

Re{2 exp ( )cos( )}

2 cos( )cos( )

ave

ave ave

ave ave

ave ave

t t

E i k x t i kx t i kx t

E i k x t kx t

E k x t kx t

Take E0 to be real.

For a nice demo of beats, check out: http://www.olympusmicro.com/primer/java/interference/

16

The group velocity is the velocity of the envelope or irradiance: the math.The group velocity is the velocity of the envelope or irradiance: the math.

0( ) ( v ) exp[ ( v )]gE t E z t ik z t

( ) ( v ) exp[ ( v )]gE t I z t ik z t

And the envelope propagates at the group velocity:

Or, equivalently, the irradiance propagates at the group velocity:

The carrier wave propagates at the phase velocity.

17

vg d /dk

Now, is the same in or out of the medium, but k = k0 n, where k0 is the k-vector in vacuum, and n is what depends on the medium. So it's easier to think of as the independent variable:

Using k = n() / c0, calculate: dk /d = ( n + dn/d ) / c0

vgc0n dn/d) = (c0 /n) / (1 + /n dn/d )

Finally:

So the group velocity equals the phase velocity when dn/d = 0,such as in vacuum. Otherwise, since n increases with , dn/d > 0, and: vg < v

Calculating the Group velocityCalculating the Group velocity

1v /g dk d

v v / 1g

dn

n d

dk

dn

n

kdk

dn

n

kc

n

c

1

2

vv

v

g

g

18

0

0

20 0 0 0

0 0 2 20 0 0

0

00

2 22 /

(2 / ) 2

v / 1

2v / 1

g

g

ddn dn

d d d

d c cc

d c c

c dn

n n d

cc

n

Use the chain rule :

Now, , so :

Recalling that :

we have : 20

0 0 02

dn

n d c

or :

Calculating Group Velocity vs. WavelengthCalculating Group Velocity vs. Wavelength

We more often think of the refractive index in terms of wavelength ,so let's write the group velocity in terms of the vacuum wavelength 0.

0 0

0

v / 1g

c dn

n n d

dk

dn

n

kdk

dn

n

kc

n

c

1

2

vv

v

g

g

19

In regions of normal dispersion, dn/d is positive. So vg < c for these frequencies.

dk

dn

n

kdk

dn

n

kc

n

c

1

2

vv

v

g

g

20

The group velocity is less than the phase velocity in non-absorbing regions.The group velocity is less than the phase velocity in non-absorbing regions.

vg = c0 / (n + dn/d)

In regions of normal dispersion, dn/d is positive. So vg < c for these frequencies.

21

The group velocity can exceed c0 whendispersion is anomalous.

The group velocity can exceed c0 whendispersion is anomalous.

vg = c0 / (n + dn/d)

dn/d is negative in regions of anomalous dispersion, that is, near a resonance. So vg can exceed c0 for these frequencies!

One problem is that absorption is strong in these regions. Also, dn/d is only steep when the resonance is narrow, so only a narrow range of frequencies has vg > c0. Frequencies outside this range have vg < c0.

Pulses of light (which are broadband) therefore break up into a mess.

22

Home WorkHome Work

• 7.5-7.6