1. 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
Spatio-temporal distortions
Prof. Rick Trebino
Georgia Tech

2. A light wave has intensity and phase vs. time.
Neglecting the spatial dependence for now, the pulse electric field is given by:
Intensity
Carrier
frequency
Phase
The phase tells us the color evolution of the pulse in time.

3. The Instantaneous frequency
The temporal phase, (t), contains frequency-vs.-time information.
The pulse instantaneous angular frequency, inst(t), is defined as:
Proof: 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 (t+t) is the slowly varying phase at the time, t+t. Subtracting these two equations:

4. Instantaneous frequency (cont’d)
Dividing by t and recognizing that 2π/t is a frequency, call it inst(t):
inst(t) = 2π /t = 0 – [(t+t) – (t)] / t
But t is small, so [(t+t)–(t)] /t is the derivative, d /dt.
So we’re done!
While the instantaneous frequency isn’t always a rigorous quantity, it’s fine for most cases, especially for waves with broad bandwidths.

5. 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.

6. The chirped pulse (continued)
We can write a linearly chirped Gaussian pulse mathematically as:
Gaussian
amplitude
Carrier
wave
Chirp
Note that for b > 0, when t < 0, the two terms 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).

7. 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.

8. The negatively chirped pulse
We’ve been considering a pulse whose frequency increases
linearly with time: a positively chirped pulse.
We could 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:

9. Nonlinearly chirped pulses
The frequency of a light wave can also vary nonlinearly with time.
This is the electric field of a Gaussian pulse whose frequency varies quadratically with time:
This light wave has the expression:
Arbitrarily complex frequency-vs.-time behavior is possible.

10. The Fourier transform of a chirped pulse
Writing a linearly chirped Gaussian pulse:
or:
Fourier-Transforming yields:
Rationalizing the denominator and separating the real and imag parts:
where
A chirped Gaussian pulse
Fourier-Transforms to itself!!!

11. 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.

12. 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.

13. Spectral-phase Taylor series
It’s 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)

14. 3rd-order spectral phase: quadratic chirp
The reddest and bluest colors coincide in time and interfere.
E-field vs. time
Spectrum and spectral phase
Spectrum
j(w)
tg(w)
3 = 3x10^4 fs^3
Trailing satellite pulses in time indicate positive spectral cubic phase, and leading ones indicate negative spectral cubic phase.

15. 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. When such separations don’t occur, we say the beam has spatio-temporal distortions or spatio-temporal couplings.

16. 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 pair
Input pulse
Spatially chirped output pulse
Prism pairs are inside nearly every ultrafast laser. A third and fourth prism undo this distortion, but must be aligned carefully.

17. How to think about spatial chirp
x0(w9) z x
x0(w8)
x0(w7)
x0(w6)
Suppose we send the pulse through a set of monochromatic filters and find the beam center position, x0, for each frequency, w.
x0(w5)
x0(w4)
x0(w3)
x0(w2)
x0(w1)
where we take x, y, z, and w to be the difference between them and the pulse average position or frequency.

18. 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.

19. Undistorted input pulse
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.

20. Modeling pulse-front tilt
Pulse-front tilt involves coupling between the space and time domains:
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 aren’t aligned properly.

21. 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:
Prism
Input pulse
Angularly dispersed output pulse x z
where kx0(w) is the mean kx vs. frequency w.

22. Angular dispersion = pulse-front tilt!
Pulse-front tilt means that:
where g = dt0 /dx
Fourier-transforming from t to w 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!

23. Spatially chirped input pulse
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.
Dispersive medium
Spatially chirped input pulse
vg(red) > vg(blue)
Spatially chirped pulse with pulse-front tilt, but no angular dispersion
The total pulse-front tilt is the sum of that due to dispersion and that due to this effect.

24. 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.

25. 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.

26. The pulse shaper l(x) x grating grating f f f f f f How it works:
Recall that this geometry maps angle (and hence wavelength) to position at this plane, called the Fourier transform plane!
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.

27. A phase mask selectively delays colors.
Image from Femtosecond Laser Pulses: Linear Properties, Manipulation, Generation and Measurement
Matthias Wollenhaupt, Andreas Assion and Thomas Baumert
An amplitude mask shapes the spectrum.

28. The pulse-shaper Amplitude mask Transmission = t(x) = t(l) Phase mask
Phase delay = j(x) = j(l)
grating
grating f f f f f f
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!

29. A shaped pulse for telecommunications
Ones and zeros…
Andy Weiner web site