17' Frequency vs' Time: Chirp

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17. 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

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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! Usually, however, well think in
terms of the instantaneous frequency, ?inst(t),
so well need to divide by 2? ?inst(t)
?0 d??/dt / 2? 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

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

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

What do these terms mean? ?0 Absolute
phase ?1 Delay ?2 Quadratic phase, i.e.,
linear chirp ?3 Cubic phase, i.e., quadratic
chirp