The Spectrum

1
The Spectrum

• We define the spectrum, S(w), of a wave E(t) to
  be

This is the measure of the frequencies present in
a light wave.
2
7. Usando el teorema de Rayleigh, calcular
3
The Pulse Width

• There are many definitions of the "width" or
  length of a wave or pulse.
• The effective width is the width of a rectangle
  whose height and area are the same as those of
  the pulse.
• Effective width Area / height

(Abs value is unnecessary for intensity.)
Advantage Its easy to understand. Disadvantages
The Abs value is inconvenient. We must
integrate to 8.
4
The rms pulse width

• The root-mean-squared width or rms width

The rms width is the second-order moment.
Advantages Integrals are often easy to do
analytically. Disadvantages It weights wings
even more heavily, so its difficult to use for
experiments, which can't scan to )
5
The Full-Width-Half-Maximum

• Full-width-half-maximum is the distance between
  the half-maximum points.

Advantages Experimentally easy. Disadvantages
It ignores satellite pulses with heights lt 49.99
of the peak!
Also we can define these widths in terms of
f(t) or of its intensity,f(t)2. Define spectral
widths (Dw) similarly in the frequency domain (t
w).
6
The Uncertainty Principle

• The Uncertainty Principle says that the product
  of a function's widths
• in the time domain (Dt) and the frequency domain
  (Dw) has a minimum.

Define the widths assuming f(t) and F(w) peak at
0
Combining results
or
(Different definitions of the widths and the
Fourier Transform yield different constants.)
7
The Time-Bandwidth Product

• For a given wave, the product of the time-domain
  width (Dt) and
• the frequency-domain width (Dn) is the
• Time-Bandwidth Product (TBP)
• Dn Dt º 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."

8
The Time-Bandwidth Product is a measure of the
pulse complexity.

• The coherence time (tc 1/Dn)
• indicates the smallest temporal
• structure of the pulse.
• In terms of the coherence time
• TBP Dn Dt Dt / 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.

9
Temporal and Spectral Shapes
10
Parsevals Theorem

• Parsevals Theorem says that the energy is the
  same, whether you integrate over time or
  frequency
• Proof

11
Parseval's Theorem in action
The two shaded areas (i.e., measures of the light
pulse energy) are the same.
12
The Pulse Width

• There are many definitions of the "width" or
  length of a wave or pulse.
• The effective width is the width of a rectangle
  whose height and area are the same as those of
  the pulse.
• Effective width Area / height

(Abs value is unnecessary for intensity.)
Advantage Its easy to understand. Disadvantages
The Abs value is inconvenient. We must
integrate to 8.
13
The rms pulse width

• The root-mean-squared width or rms width

The rms width is the second-order moment.
Advantages Integrals are often easy to do
analytically. Disadvantages It weights wings
even more heavily, so its difficult to use for
experiments, which can't scan to )
14
The Full-Width-Half-Maximum

• Full-width-half-maximum is the distance between
  the half-maximum points.

Advantages Experimentally easy. Disadvantages
It ignores satellite pulses with heights lt 49.99
of the peak!
Also we can define these widths in terms of
f(t) or of its intensity, f(t)2. Define
spectral widths (Dw) similarly in the frequency
domain (t w).
15
The Uncertainty Principle

• The Uncertainty Principle says that the product
  of a function's widths
• in the time domain (Dt) and the frequency domain
  (Dw) has a minimum.

Define the widths assuming f(t) and F(w) peak at
0
(Different definitions of the widths and the
Fourier Transform yield different constants.)
Combining results
or
16
The Uncertainty Principle
For the rms width, Dw Dt ½ Theres an
uncertainty relation for x and k Dk Dx ½
17
Calculating the Intensity and the Phase
Its easy to go back and forth between the
electric field and the intensity and phase. The
intensity
I(t) E(t)2
The phase
E(ti)
Im
vI(ti)
-f(ti)
Equivalently,
Re
f(t) -Im lnE(t)?
18
Intensity and Phase of a Gaussian

• The Gaussian is real, so its phase is zero.

Time domain Frequency domain
A Gaussian transforms to a Gaussian
So the spectral phase is zero, too.
19
The spectral phase of a time-shifted pulse
Recall the Shift Theorem
Time-shifted Gaussian pulse (with a flat phase)
So a time-shift simply adds some linear spectral
phase to the pulse!
20
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 j(w) 0 Note that this wave-form
sees constructive interference, and hence peaks,
at t 0. And it has cancellation everywhere
else.
w1 w2 w3 w4 w5 w6
0
21
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 w2 w3 w4 w5 w6
j(w1) 0
j(w2) 0.2 p
j(w3) 0.4 p
j(w4) 0.6 p
j(w5) 0.8 p
j(w6) p
t
22
The spectral phase distinguishes a light bulb
from an ultrashort pulse.
23
Complex Lorentzianand its Intensity and Phase
a
Real
component
0
0
Imaginary
component
w
Real part
Imag part
24
Intensity and Phase of a decaying exponential and
its Fourier transform
Time domain Frequency domain
(solid)
25
Light has intensity and phase also.
We usually extract out the carrier frequency.
A light wave has the time-domain electric field
Intensity
Phase
The minus signs are just conventions.
(neglecting the negative-frequency component)
Equivalently, vs. frequency
Spectral
Spectrum
Phase
Knowledge of the intensity and phase or the
spectrum and spectral phase is sufficient to
determine the light wave.
26
Fourier Transform with respect to space
If f(x) is a function of position,

• F f(x) F(k)

We refer to k as the spatial frequency. Everyth
ing weve said about Fourier transforms between
the t and w domains also applies to the x and k
domains.
27
The Shah Function

• The Shah function, III(t), is an infinitely long
  train of equally spaced delta-functions.

t
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.
28
The Fourier Transform of the Shah Function
III(t)

• If w 2np, where n is an integer, the sum
  diverges otherwise, cancellation occurs. So

29
The Shah Function and a pulse train
An infinite train of identical pulses (from a
laser!) can be written
where f(t) is the shape of each pulse and T is
the time between pulses.
Set t /T m or t mT
30
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

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 dw 2p/T or dn 1/T.
31
The Fourier Transform of a Finite Pulse Train

• A finite train of identical pulses can be written

where g(t) is a finite-width envelope over the
pulse train.
32
Laser Modes
A lasers frequencies are often called
longitudinal modes. Theyre separated by 1/T
c/2L. Which modes lase depends on the gain
profile.
Here, additional narrowband filtering has yielded
a single mode.
Intensity
Frequency
33
The 2D generalization of the Shah function The
Bed of Nails function
We wont do anything with this function, but I
thought you might like this colorful image Can
you guess what its Fourier transform is?