OD GroupPhaseVelocity1

7/29/2019 OD GroupPhaseVelocity1

1/8

When two waves of different frequency

interfere, they produce beats.

0 1

1

0 0

2

0

2

21

0( ) Re{ }

2 2

( ) Re{ exp ( ) exp ( )}

Re{ exp( )[exp(

exp

) exp( )]}

exp( ( )

)

ave

ave

ave

tot

tot ave

E iE t

E t E i t t E i t t

E i t i t i t

i tt E

Let and

So :

0

0

Re{2 exp( )cos( )}

2 cos( ) cos( )

ave

ave

ave

E i t t

E t t

Summing waves of two different frequencies yields the product

of a rapidly varying cosine ( ) and a slowly varying cosine ( ).

TakeE0 to be real.

2

1 ave

ave


2/8

When two waves of different frequency

interfere, they produce "beats."

Indiv-

idual

waves

Sum

Envel-

ope

Irrad-

iance:


3/8

When two light waves of different frequency

interfere, they also produce beats.

0 2 2

2

0 1 1

1 2

2 21 1

0

1

0

( , ) Re{ }

2 2

2 2

( , ) Re{ exp ( ) ex

exp ( )

p

ex ( )

(

ptot

t

ave

ave

avot e ave ave

E i k x t

k k

E x t

k

E x

E i k x t

kk

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

k

Let and

Similiarly, and

So:

0

0

0

)}

Re{ exp ( ) exp[ ( )] }

Re{2 cos( )}

exp ( )

exp ( )

cos( )2 cos( )

ave

ave ave

ave ave

ave ave

i k x t

i k x

t t

E i kx t i kx t

E kx t

E kx t

t

k x t

TakeE0 to be real.

For a nice demo of beats, check

out:

http://www.olympusmicro.com/pri

mer/java/interference/


4/8

Group velocity

vg d /dk

Light-wave beats (continued):

Etot(x,t) = 2E0 cos(kavexavet) cos(kxt)

This is a rapidly oscillating wave: [cos(kavexavet)]

with a slowly varying amplitude: [2E0 cos(kx

t)]

The phase velocity comes from the rapidly varying part: v = ave / kave

What about the other velocitythe velocity of the amplitude?

Define the group velocity: vg /k

In general, we define the group velocity as:

carrier wave

amplitude


5/8

Group velocity is not equal to phase velocity

if the medium is dispersive (i.e., n varies).

0 1 0 2

1 1 2 2

1 2

0 01 2

1 2

1 2

v

, v

g

g

k

c k c k

n k n k k k

c ck k

n n n n k k n

For our example,

where and are the k - vector magnitudes in vacuum.

If phase velocity

1 2

, vg

n n c If


6/8

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.


7/8

vg d/dk

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

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

vg c0 / ( n 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 velocity

1

v /g dk d

v v / 1g

dn

n d


8/8

0

0

2

0 0 0 00 0

2 20 0 0

0

00

2 22 /

(2 / ) 2

v / 1

2

v / 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:

2

0

0 0 02

dn

n d c

or :

Calculating 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 00 0

0 0

v / 1 /gc dn dn

c nn n d d

Documents

OD GroupPhaseVelocity1