The Spectrum
We define the spectrum, S(w), of a wave E(t) to be:
S (w )  F {E (t )}
2
This is the measure of the frequencies present in a light wave.
7. Usando el teorema de Rayleigh, calcular:


2
sen t
 t 2 dt


T
0 , t  
2


T
T
f t   1 ,   t 

2
2

T
0 ,
t


2


Rayleigh
2
fˆ (w ) dw 


 f (t)
2
dt

T
sen(
w
)
T
2
ˆ
 f (w ) 
2 w T
2
 0 ,
T
T

TT

1 sen(

0
,
t

sen(
w
w
)
)
0
,
t




2
 
T
f

t


2
2
2
2
ˆ
ˆ
 ff 
(t)
dt
)  dt  
f t   
(
w
)
f

(
w

1 2, t , Tt  T 
2
T
T
 2
2

1

ww
 T
2 2
22


The Pulse Width
t
There are many definitions of the
"width" or “length” of a wave or pulse.
t
The “effective width” is the width of a rectangle whose height and
area are the same as those of the pulse.
f(0)
Effective width ≡ Area / height:

teff
1

f (t ) dt

f (0) 
teff
(Abs value is
unnecessary
for intensity.)
Advantage: It’s easy to understand.
Disadvantages: The Abs value is inconvenient.
We must integrate to ± ∞.
0
t
The rms pulse width
t
The “root-mean-squared width”
or “rms width:”
1/ 2
 2

  t f (t ) dt 

  


  f (t ) dt 
 


trms
t
The rms width is the “second-order moment.”
Advantages: Integrals are often easy to do analytically.
Disadvantages: It weights wings even more heavily,
so it’s difficult to use for experiments, which can't scan to ±  )
The Full-WidthHalf-Maximum
1
0.5
tFWHM
“Full-width-half-maximum”
is the distance between the
half-maximum points.
Advantages: Experimentally easy.
Disadvantages: It ignores satellite
pulses with heights < 49.99% of the
peak!
t
tFWHM
t
Also: we can define these widths in terms of f(t) or of its intensity,|f(t)|2.
Define spectral widths (w) similarly in the frequency domain (t w).
The Uncertainty Principle
The Uncertainty Principle says that the product of a function's widths
in the time domain (t) and the frequency domain (w) has a minimum.
Define the widths
assuming f(t) and
F(w) peak at 0:

1
t 
f (t ) dt

f (0) 


1
w 
F (w) dw

F (0) 

1
1
F (0)
t 
f
(
t
)
dt

f
(
t
)
exp(

i
[0]
t
)
dt



f (0) 
f (0) 
f (0)


1
1
2 f (0)
w 
F
(
w
)
d
w

F
(
w
)
exp(
i
w

d
w



F (0) 
F (0) 
F (0)
Combining results:
w t  2
f (0) F (0)
F (0) f (0)
or:
w t  2
 t  1
(Different definitions of the widths and the Fourier Transform yield
different constants.)
The Time-Bandwidth Product
For a given wave, the product of the time-domain width (t) and
the frequency-domain width () is the
Time-Bandwidth Product (TBP)
 t  TBP
A pulse's TBP will always be greater than the theoretical minimum
given by the Uncertainty Principle (for the appropriate width definition).
The TBP is a measure of how complex a wave or pulse is.
Even though every pulse's time-domain and frequency-domain
functions are related by the Fourier Transform, a wave whose TBP is
the theoretical minimum is called "Fourier-Transform Limited."
The Time-Bandwidth Product is a
measure of the pulse complexity.
The coherence time (tc = 1/)
indicates the smallest temporal
structure of the pulse.
In terms of the coherence time:
tc
A complicated
pulse
I(t)
t
time
TBP =  t = t / tc
= about how many spikes are in the pulse
A similar argument can be made in the frequency domain, where the
TBP is the ratio of the spectral width to the width of the smallest
spectral structure.
Temporal
and
Spectral
Shapes
Parseval’s Theorem
Parseval’s Theorem says that the
energy is the same, whether you
integrate over time or frequency:

f (t )
2
dt 

 1
 
 2
 



1
2
1
2


F (w )


f (t ) f *(t ) dt 


1

2

1
f ( t ) dt 
2
2



Proof:




 1
F (w  exp(iwt ) d w  
  2



1
F (w )
2



F (w )
1
2







F *(w ')  exp(i[w  w '] t ) dt  d w ' d w
 




F *(w ') [2 w  w ')] dw ' d w




F *(w  exp( iw t ) d w  dt



F (w ) F *(w ) d w 
1
2



F (w ) d w
2
2
dw
Parseval's Theorem in action
Time domain
Frequency domain
F(w)
f(t)
w
t
|F(w)|2
|f(t)|2
t
w
The two shaded areas (i.e., measures of the light pulse energy) are
the same.
The Pulse Width
t
There are many definitions of the
"width" or “length” of a wave or pulse.
t
The effective width is the width of a rectangle whose height and
area are the same as those of the pulse.
f(0)
Effective width ≡ Area / height:

teff
1

f (t ) dt

f (0) 
teff
(Abs value is
unnecessary
for intensity.)
Advantage: It’s easy to understand.
Disadvantages: The Abs value is inconvenient.
We must integrate to ± ∞.
0
t
The rms pulse width
t
The root-mean-squared width or
rms width:
1/ 2
 2

  t f (t ) dt 

  


  f (t ) dt 
 


trms
t
The rms width is the “second-order moment.”
Advantages: Integrals are often easy to do analytically.
Disadvantages: It weights wings even more heavily,
so it’s difficult to use for experiments, which can't scan to ±  )
The Full-WidthHalf-Maximum
1
0.5
tFWHM
Full-width-half-maximum
is the distance between the
half-maximum points.
Advantages: Experimentally easy.
Disadvantages: It ignores satellite
pulses with heights < 49.99% of the
peak!
t
tFWHM
t
Also: we can define these widths in terms of f(t) or of its intensity, |f(t)|2.
Define spectral widths (w) similarly in the frequency domain (t w).
The Uncertainty Principle
The Uncertainty Principle says that the product of a function's widths
in the time domain (t) and the frequency domain (w) has a minimum.
Define the widths
assuming f(t) and
F(w) peak at 0:


1
t 
f (t ) dt

f (0) 

1
w 
F (w) dw

F (0) 

1
1
F (0)
t 
f (t ) dt 
f (t ) exp(i[0] t ) dt 


f (0) 
f (0) 
f (0)


1
1
2 f (0)
w 
F
(
w
)
d
w

F
(
w
)
exp(
i
w

d
w



F (0) 
F (0) 
F (0)
Combining results:
f (0) F (0)
w t  2
F (0) f (0)
(Different definitions of the widths and the
Fourier Transform yield different constants.)
or:
w t  2
 t  1
The Uncertainty Principle
For the rms width, w t ≥ ½
There’s an uncertainty relation for x and k: k x ≥ ½
Calculating the Intensity and the Phase
It’s easy to go back and forth between the electric field and the
intensity and phase.
The intensity:
I(t) = |E(t)|2
The phase:
 Im[ E (t )] 
 (t )   arctan 

 Re[ E (t )] 
Equivalently,
(t) = -Im{ ln[E(t) ]
}
Im
E(ti)
(ti)
Re
Intensity and Phase of a Gaussian
The Gaussian is real, so its phase is zero.
Time domain:
A Gaussian
transforms
to a Gaussian
Frequency domain:
So the spectral phase
is zero, too.
The spectral phase of a time-shifted pulse
Recall the Shift Theorem: F
Time-shifted
Gaussian pulse
(with a flat phase):
So a time-shift
simply adds some
linear spectral
phase to the
pulse!
 f (t  a)  exp(iw a)F (w)
What is the spectral phase anyway?
The spectral phase is the abs phase of each frequency in the wave-form.
All of these
frequencies have
zero phase. So
this pulse has:
w1
w2
j(w) = 0
w3
Note that this
wave-form sees
constructive
interference, and
hence peaks, at
t = 0.
w4
w5
w6
0
t
And it has
cancellation
everywhere else.
Now try a linear spectral phase: j(w) = aw.
By the Shift Theorem, a linear spectral phase is just a delay in time.
And this is what occurs!
w1
j(w1) = 0
w2
j(w2) = 0.2 
w3
j(w3) = 0.4 
w4
j(w4) = 0.6 
w5
j(w5) = 0.8 
w6
j(w6) = 
t
The spectral phase distinguishes a light
bulb from an ultrashort pulse.
Complex Lorentzian
and its Intensity
and Phase
a
Real
component
0
0
Imaginary
component
w
Real part
Imag part
1
1 a  iw
a
w
L(w ) 

 2
2 i 2
a  iw
a  iw a  iw
a w
a w2
1
Intensity(w )  2
a  w2
 Im[ L(w )] 
Phase(w )   arctan 
  arctan(w / a)
 Re[ L(w )] 
Intensity and Phase of a decaying
exponential and its Fourier transform
Time domain:
(solid)
Frequency domain:
Light has intensity and phase also.
A light wave has the time-domain electric field:
E (t )  Re

We usually extract out
the carrier frequency.

I (t ) exp i(w0t   (t )) 
Intensity
Equivalently, vs. frequency:
Phase
The minus
signs are just
conventions.
(neglecting the
negative-frequency
component)
E (w )  S (w ) exp  ij (w )
Spectrum
Spectral
Phase
Knowledge of the intensity and phase or the spectrum and spectral
phase is sufficient to determine the light wave.
Fourier Transform with respect to space
If f(x) is a function of position,

F (k )   f ( x) exp(ikx) dx

x
F {f(x)} = F(k)
We refer to k as the “spatial frequency.”
k
Everything we’ve said about Fourier transforms between the t and w
domains also applies to the x and k domains.
The Shah Function
The Shah function, III(t), is an infinitely long train of equally spaced
delta-functions.
t

III(t ) 
  (t  m)
m 
The symbol III is pronounced shah after the Cyrillic character III, which is
said to have been modeled on the Hebrew letter
(shin) which, in turn,
may derive from the Egyptian
a hieroglyph depicting papyrus plants
along the Nile.
The Fourier Transform of the Shah
Function
III(t)



   t  m) exp(iw t )dt
 m 




t
 t  m) exp(iw t )dt
m  


 exp(iw m)
m 
If w = 2n, where n is an integer,
the sum diverges; otherwise,
cancellation occurs. So:
F {III(t )}  III(w 
F {III(t)}
w
2
The Shah Function
and a pulse train
An infinite train of identical pulses
(from a laser!) can be written:
E (t )  III(t / T )  f (t )
where f(t) is the shape of each pulse and T is the time between
pulses.



   (t / T  m) f (t  t) dt
m 
Set t’ /T = m or t’ = mT



m 
f (t  mT )
The Fourier Transform of an Infinite Train of Pulses
An infinite train of identical pulses can be written:
E(t) = III(t/T) * f(t)
where f(t) represents a single pulse and T is the time between pulses.
The Convolution Theorem states that the Fourier Transform of a
convolution is the product of the Fourier Transforms. So:
E (w ) 
III(wT / 2 ) F (w 
If this train of pulses results from a single pulse bouncing back and
forth inside a laser cavity of round-trip time T. The spacing between
frequencies is then w = /T or  = 1/T.
The Fourier Transform of a Finite Pulse Train
A finite train of identical pulses can be written:
E (t )  {III(t / T ) g (t )}  f (t )
where g(t) is a finite-width envelope over the pulse train.
E(w)  {III(wT / 2 )  G(w)}F (w)
Laser Modes
A laser’s frequencies are often called “longitudinal modes.”
They’re separated by 1/T = c/2L.
Which modes lase depends on the gain profile.
Intensity
Here,
additional
narrowband
filtering has
yielded a
single mode.
Frequency
The 2D generalization of the Shah function:
“The Bed of Nails” function
We won’t do anything with this function, but I thought you might like
this colorful image… Can you guess what its Fourier transform is?