Wavetrains and Coherency. © 2006 Walter Fendt Beats Animation .

Wavetrains and Coherency

© 2006 Walter Fendt

Beats Animation

http://www.walter-fendt.de/ph11e/beats.htm

y(x) = Sin x

© SPK

y(x) = [Sin x + Sin (1.08 x)]/2

© SPK

y(x) = [Sin x + Sin(1.04 x) + Sin (1.08 x)]/3

© SPK

0 < x < 200

y(x) = [Sin x + Sin(1.02 x) + Sin (1.04 x) + Sin(1.06 x) + Sin (1.08 x)]/5

© SPK

0 < x < 400

y(x) = [Sin x + Sin(1.01 x) + Sin (1.02 x)+ Sin(1.03 x) + Sin (1.04 x) + Sin (1.05 x) + Sin (1.06 x) + Sin (1.07 x) + Sin (1.08 x)]/9

© SPK

0 < x < 400

y(x) = [Sin x + Sin(1.01 x) + Sin (1.02 x)+ Sin(1.03 x) + Sin (1.04 x) + Sin (1.05 x) + Sin (1.06 x) + Sin (1.07 x) + Sin (1.08 x)]/9

© SPK

Solitary pulse

x

y2

0 cos

0

pE E k x

For

For

LxL

Lx

-L +L

E0

Fourier integral

0 0

1( ) ( )cos( ) ( )sin( )f x A k kx dk B k kx dk

( ) ( )cos( )

( ) ( )sin( )

A k f x kx dx

B k f x kx dx

0

0

( ) cos( )cos( )

sin( ) sin( )

( ) ( )

L

p

L

p p

p p

A k E k x kx dx

k k L k k LE L

k k L k k L

0)k(B

Lkk

Lkk

p

p

)(

)sin(

Lkk

Lkk

p

p

)(

)sin(

kp=2

0 2 4 6 8 10 12 14 16 18-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

2/L

kp=2

k

L=1.5

Wave packet or Wave group

for kp=k

Lk

2

Frequency Bandwidth

Range of frequency k (or ) in wavetrain

(k) (or

t

t

1

4

xk

xk

1

4

Temporal coherence:

Coherence time:

Coherence length:

vtc 1

cc tcl © SPK

Spectral lines from helium gas tube

I

A characteristic Spectral line

20

cl

0

Spectral line broadening

1. Natural linewidth

2. Doppler broadening

3. Collision broadening

4. Pressure broadening

Red Cadmium = 6438 Å

= 1010

Hz, 30 cm

Yellow Sodium

= 109

3 cm= 5893 Å

He-Ne Laser 6328 Å

300 m= 106 Hz,

cl

v

v

v

cl

cl

Hz,

© SPK

Kr discharge lamp has roughly the following intensity distribution at various wavlengths, (in nm),

02 8

36( )

36 ( 605.616) 10

II

Estimate the coherence length of the Kr source.

vtc 1

•Coherence time

•Coherence length

cc tcl