Importance of phase in f(w)…review
f (t ) from adding closely spaced cos(wi t   (wi ))
Im[ f (w )]
 (w )  tan
Re[ f (w )]
1
 (w )  aw
 (w )  0
11 waves
What phase function makes a
shift? Why?
Linear phase dependence:
iaw
g (w )  e g (w )
101 waves
Dt is the same.
Pulse is shifted
Importance of phase in f(w)… review
f (t ) from adding closely spaced cos(wi t   (wi ))
 (w )  0
11 waves
101 waves
Quadratic dependence:
g (w )  e
iaw 2
 (w ) 
g (w )
Dt is bigger,
2
aw lower frequencies are ahead
Plane waves and Fourier
We’ve studied FT at one point in space (e.g.
E (t ) 
1
2


r 0
)
E (w )e iwt d w

The pulse at all positions is (if it moves in z direction):

1
E (t , z ) 
2

 E (w, 0)e
i  k (w ) z wt 

1
2

 E (w, z)e

 iwt
dw
dw
Assume Ew is large only near some wo (typical for a pulse)
1
2
n(w)  n(wo )  n '(wo )(w  wo )  n ''(wo )(w  wo )  ...
2
2

k
1 k
kz   k w 
w  wo  

2
o
w
2
w

wo
 w  wo 
wo
2

 ... z

 wo / v p
1   vg 


2 
2vg  w 
wo
1
E (t , z ) 

2
Put into FT-1:


E (w , 0)e
i  k (w ) z wt 
dw 


 z

 z

E (w , 0) exp  i wo  exp  i w  wo   
 vp

 vg

2 




1

carrier wave

2

2
 exp iz w  wo 

e iwt d w

e iwt d w
moves the envelope rigidly
 z

exp  i wo  
 vp

 z




E (w , 0) exp  i w  wo   
 vg

2




broadens the envelope
 exp iz w  wo 
These products become convolutions in time
Pulse broadens and is “chirped”: lowest frequencies
are ahead.
Broadening of pulses,
from a photonics website
Output pulse duration versus initial pulse duration for dispersive pulse
broadening with different levels of group delay dispersion (GDD). Note that
shorter pulses are increasingly sensitive to dispersion. Substantial
broadening occurs when the square of the pulse duration is smaller than
the group delay dispersion.
Apparent faster-than-light pulses
Apparent faster-than-light pulses
Group velocity changes with w,
so the pulse changes shape.
c
vg 
n  w  dn / dw 
If the pulse’s w band is in the
box on the figure, the portions of
light in the pulse moving fastest
(highest group velocity) are :___
a) higher frequencies
b) lower frequencies
Apparent faster-than-light pulses
The portion of the spectrum that is
absorbed most is ______ frequencies
a) higher
b) lower
Apparent faster-than-light pulses
In absorbing media:
pulse if no absorption
pulse if no material
Peak has “moved” faster than speed of light
simply because of absorption. But no
information has traveled faster than c.
http://optics.byu.edu/animations.aspx
Apparent faster-than-light pulses
Absorbing light from
the middle of the
spectrum can also
reshape light pulse
Amplification of light due to stimulated emission
q
r   r  wo r  Eo cos(wt ) What changed in the
m
Lorenz model?
2
n flips about n=1; k simply becomes negative!
If the pulse’s w band is in the
box on the figure, the portions of
light in the pulse moving fastest
(highest group velocity) are :___
a) higher frequencies
b) lower frequencies
c
vg 
n  w  dn / dw 
Can also get apparent faster-than light pulses
with amplification from stimulated emission
Peak can “move” faster than
speed of light simply because of
amplification. But no information
has traveled faster than c.