Frequency vs. Time: Chirp

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Frequency vs. Time Chirp

• Chirp and its definition
• The Linearly Chirped Gaussian Pulse
• The Instantaneous Frequency vs. Time
• The Fourier Transform of a Chirped Pulse
• The Group Delay vs. Frequency

Prof. Rick Trebino Georgia Tech www.physics.gatech
.edu/frog
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A light wave has intensity and phase vs. time.
Neglecting the spatial dependence for now, the
pulse electric field is given by
Intensity
Phase
Carrier frequency
The phase tells us the color evolution of the
pulse in time.
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The Instantaneous frequency
The temporal phase, ?(t), contains
frequency-vs.-time information. The pulse
instantaneous angular frequency, ?inst(t), is
defined as
This is easy to see. At some time, t, consider
the total phase of the wave. Call this quantity?
?0 Exactly one period, T, later, the total
phase will (by definition) increase to ?0 2p
where ?(tT) is the slowly varying phase at
the time, tT. Subtracting these two equations
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Instantaneous frequency (contd)
Dividing by T and recognizing that 2p/T is a
frequency, call it ?inst(t)
?inst(t) 2p/T ?0 ?(tT) ?(t) / T But
T is small, so ?(tT)?(t) /T is the
derivative, d? /dt. So were done! While the
instantaneous frequency isnt always a rigorous
quantity, its fine for most cases, especially
for waves with broad bandwidths.
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The Chirped Pulse

• A pulse can have a frequency that varies in time.

This pulse increases its frequency linearly in
time (from red to blue). In analogy to bird
sounds, this pulse is called a chirped pulse.
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The Chirped Pulse(continued)

• We can write a linearly chirped Gaussian pulse
  mathematically as

Carrier wave
Chirp
Gaussian amplitude
Note that for b 0, when t partially cancel, so the phase changes slowly
with time (so the frequency is low). And when t
0, the terms add, and the phase changes more
rapidly (so the frequency is larger).
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The Instantaneous Frequency vs. time for a
Chirped Pulse

• A chirped pulse has

where
The instantaneous frequency is
which is
So the frequency increases linearly with
time. More complex phases yield more complex
frequencies vs. time.
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The Negatively Chirped Pulse

• We have been considering a pulse whose frequency
  increases
• linearly with time a positively chirped pulse.
• One can also have a negatively
• chirped (Gaussian) pulse, whose
• instantaneous frequency
• decreases with time.
• We simply allow b to be negative
• in the expression for the pulse
• And the instantaneous frequency will decrease
  with time

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Nonlinearly Chirped Pulses

• The frequency of a light wave can also vary
  nonlinearly with time.
• This is the electric field of aGaussian pulse
  whose frequency varies quadratically with time
• This light wave has the expression
• Arbitrarily complex frequency-vs.-time behavior
  is possible.

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The Fourier Transformof a Chirped Pulse

• Writing a linearly chirped Gaussian pulse as
• or
• Fourier-Transforming yields
• Rationalizing the denominator and separating the
  real and imag parts

A Gaussian with a complex width!
A chirped Gaussian pulse Fourier-Transforms to
itself!!!
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The Group Delay vs. Frequency

• The frequency-domain quantity that is analogous
  to the
• instantaneous frequency vs. t is the group delay
  vs. w.
• If the wave in the frequency domain is
• then the group delay is the derivative of the
  spectral phase
• The group delay is also not always the actual
  delay of a given
• frequency. It is only an approximate quantity.

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The Group Delay vs. w for a Chirped Pulse

• The group delay of a wave is the derivative of
  the spectral phase
• For a linearly chirped Gaussian pulse, the
  spectral phase is
• So
• And the delay vs. frequency is also linear.
• When the pulse is long (a 0), then
• which is just the inverse of the instantaneous
  frequency vs. time.

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Spectral-Phase Taylor Series

• Its common practice to expand the spectral phase
  in a Taylor Series

What do these terms mean? ?0 Absolute
phase ?1 Delay ?2 Quadratic phase (linear
chirp) ?3 Cubic phase (quadratic chirp)
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Spatio-temporal distortions
Ordinarily, we assume that the pulse-field
spatial and temporal factors (or their
Fourier-domain equivalents) separate
where the tilde and hat mean FTs with respect to
t and x, y, z, respectively.
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Spatial chirp is a spatio-temporal distortion in
which the color varies spatially across the beam.

• Propagation through a prism pair produces a beam
  with no angular dispersion, but with spatial
  dispersion, often called spatial chirp.

Prism pairs are inside nearly every ultrafast
laser. A third and fourth prism undo this
distortion, but must be aligned carefully.
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Spatial chirp is difficult to avoid.

• Simply propagating through a tilted window causes
  spatial chirp!

Because ultrashort pulses are so broadband, this
distortion is very noticeableand often
problematic!
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How to think about spatial chirp
Suppose we send the pulse through a set of
monochromatic filters and find the beam center
position, x0, for each frequency, w.
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Spatial chirp
Why is this expression the way to model this
effect? Typically,
But now
The Gaussian peaks when its argument 0.
So the center position x0 is given by
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Pulse-front tilt is another common
spatio-temporal distortion.
Phase fronts are perpendicular to the direction
of propagation. Because the group velocity is
usually less than phase velocity, pulse fronts
tilt when light traverses a prism.
Angularly dispersed pulse with
pulse-front tilt
Undistorted input pulse
Prism
Angular dispersion causes pulse-front tilt.
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Angular dispersion causes pulse-front tilt even
when group velocity is not involved.
Diffraction gratings also yield pulse-front tilt.
Angularly dispersed pulse
with pulse- front tilt
The path is simply shorter for rays that impinge
on the near side of the grating. Of course,
angular dispersion and spatial chirp occur, too.
Undistorted input pulse
Diffraction grating
Gratings have about ten times the dispersion of
prisms, and they yield about ten times the tilt.
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Modeling pulse-front tilt
Pulse-front tilt involves coupling between the
space and time domains
Usually take
For a given transverse position in the beam, x,
the pulse mean time, t0, varies in the presence
of pulse-front tilt. Pulse-front tilt occurs
after pulse compressors that arent aligned
properly.
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Angular dispersion is an example of a
spatio-temporal distortion.
In the presence of angular dispersion, the
off-axis k-vector component kx depends on w
where kx0(w) is the mean kx vs. frequency w.
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Angular dispersion pulse-front tilt!
Pulse-front tilt means that
where g dt0 /dx
Fourier-transforming with respect to t (to w
w0) yields
using the shift theorem
Fourier-transforming with respect to x, y, and z
yields
using the shift theorem again.
This is just angular dispersion!
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The combination of spatial and temporal chirp
also causes pulse-front tilt.
The theorem we just proved assumed no spatial
chirp, however. So it neglects another
contribution to the pulse-front tilt.
The total pulse-front tilt is the sum of that due
to dispersion and that due to this effect.
Xun Gu, Selcuk Akturk, and Erik Zeek
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A pulse with temporal chirp, spatial chirp, and
pulse-front tilt.
Suppressing the y-dependence, we can plot such a
pulse
where the pulse-front tilt angle is
The height means the intensity color means
instantaneous frequency.
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Spatio-temporal distortions can be useful or
inconvenient.
Bad They usually increase the pulse
length. They reduce intensity. They can be hard
to measure. Good They allow pulse compression
and spectrometers. They help to measure pulses
(tilted pulse fronts). They allow pulse shaping.
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How do we shape a pulse?
We could try to modulate the pulse directly in
time.
Unfortunately, this requires a very fast
modulator, and existing modulators are too slow.
Alternatively, we can modulate the spectrum.
So all we have to do is to frequency-disperse the
pulse in space and modulate the spectrum and
spectral phase by creating a spatially varying
transmission and phase delay.
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The pulse shaper
x
grating
grating
f
f
f
f
f
f
Fourier Transform Plane
John Heritage, UC Davis Andrew Weiner, Purdue

• How it works
• The grating disperses the light, mapping color
  onto angle.
• The first lens maps angle (hence wavelength) to
  position.
• The second lens and grating undo the
  spatio-temporal distortions.

The trick is to place a mask in the Fourier
transform plane.
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The pulse-shaper
Amplitude mask Transmission t(x) t(l)
Phase mask Phase delay j(x) j(l)
Fourier Transform Plane
We can control both the amplitude and phase of
the pulse. The two masks or spatial light
modulators together can yield any desired pulse!
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A phase mask selectively delays colors.
An amplitude mask shapes the spectrum.
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A shaped pulse for telecommunications
Ones and zeros