Ultrashort Laser Pulses I

Description of pulses

Intensity and phase

The instantaneous frequency and group delay

Zeroth and first-order phase

Prof. Rick Trebino

Georgia Tech The linearly chirped Gaussian pulse
www.frog.gatech.edu
An ultrashort laser I (t )

Electric field X (t)

pulse has an intensity
and phase vs. time.

Neglecting the spatial dependence for

now, the pulse electric field is given by:
Time [fs]

X (t ) ∝ 1
2 I (t ) exp{i [ω0t − φ (t )]} + c.c.

Intensity Phase
Carrier
frequency

A sharply peaked function for the intensity yields an ultrashort pulse.

The phase tells us the color evolution of the pulse in time.
The real and complex
pulse amplitudes I (t )

Electric field X (t)

Removing the 1/2, the c.c., and the
exponential factor with the carrier
frequency yields the complex
amplitude, E(t), of the pulse:

E (t ) ∝ I (t ) exp{−iφ (t )}
Time [fs]

This removes the rapidly varying part of the pulse electric field and
yields a complex quantity, which is actually easier to calculate with.

I (t ) is often called the real amplitude, A(t), of the pulse.

The Gaussian pulse
For almost all calculations, a good first approximation for any
ultrashort pulse is the Gaussian pulse (with zero phase).

E (t ) = E0 exp  −(t / τ HW 1/ e ) 2 
= E0 exp  −2 ln 2 (t / τ FWHM ) 2 
= E0 exp  −1.38(t / τ FWHM ) 2 

where τHW1/e is the field half-width-half-maximum, and τFWHM is the

intensity full-width-half-maximum.

I (t ) ∝ E0 exp  −4 ln 2 (t / τ FWHM ) 2 
2
The intensity is:

∝ E0 exp  −2.76 (t / τ FWHM ) 2 

2
Intensity vs. amplitude

The phase
of this pulse
is constant,
φ(t) = 0,
and is not
plotted
here.

The intensity of a Gaussian pulse is √2 shorter than its real amplitude.

This factor varies from pulse shape to pulse shape.
Calculating the intensity and the phase
It’s easy to go back and forth between the electric field and the
intensity and phase:
Also, we’ll stop writing “proportional
The intensity: to” in these expressions and take E,
I(t) = |E(t)|2 X, I, and S to be the field, intensity,
and spectrum dimensionless
shapes vs. time.

The phase:

 Im[ E (t )]  E(ti)
φ (t ) = − arctan   Im
 Re[ E (t )] 
−φ(ti)
Equivalently,
Re
φ(t) = − Im{ln[E(t)]}
The Fourier Transform
To think about ultrashort laser pulses, the Fourier Transform
is essential.

∞
X% (ω ) = ∫ X (t ) exp(−iω t ) dt
−∞

1 ∞
X (t ) = ∫ X% (ω ) exp(iω t ) dω
2π −∞

We always perform Fourier transforms on the real or complex pulse

electric field, and not the intensity, unless otherwise specified.
 The frequency-domain electric field
The frequency-domain equivalents of the intensity and phase
are the spectrum and spectral phase.
Fourier-transforming the pulse electric field:

X (t ) = 1
2 I (t ) exp{i [ω0t − φ (t )]} + c.c.
yields: Note that φ and ϕ are different!

X% (ω ) = 1
2 S (ω − ω0 ) exp{−i [ϕ (ω − ω0 )]} +
1
2 S (−ω − ω0 ) exp{+i [ϕ (−ω − ω0 )]}
Note that these two terms
are not complex
The frequency-domain electric field has conjugates of each other
positive- and negative-frequency components. because the FT integral is
the same for each!
The complex frequency-domain pulse
field
Since the negative-frequency component contains the same infor-
mation as the positive-frequency component, we usually neglect it.

We also center the pulse on its actual frequency, not zero.

So the most commonly used complex frequency-domain pulse field

is:

X% (ω ) ≡ S (ω ) exp{−i ϕ (ω )}

Thus, the frequency-domain electric field also has an intensity and

phase.
S is the spectrum, and ϕ is the spectral phase.
The spectrum with and without the
carrier frequency
Fourier transforming X (t) and E(t) yields different functions.

E% (ω ) X% (ω )

We usually use just

this component.
The spectrum and spectral phase
The spectrum and spectral phase are obtained from the
frequency-domain field the same way the intensity and phase
are from the time-domain electric field.

S (ω ) = X (ω )
2
%

 Im[X% (ω )] 
ϕ (ω ) = − arctan  
 Re[X (ω )] 
%
or
ϕ (ω ) = − Im {ln[X% (ω )]}
Intensity and phase of a Gaussian
The Gaussian is real, so its phase is zero.

Intensity and Phase

Time domain:

A Gaussian
transforms
to a Gaussian Spectrum and Spectral Phase

Frequency domain:

So the spectral phase

is zero, too.
The spectral phase of a time-shifted pulse
Recall the Shift Theorem: Y { f (t − a)} = exp(−iω a) F (ω )

Intensity and Phase

Time-shifted
Gaussian pulse
(with a flat phase):

Spectrum and Spectral Phase

So a time-shift
simply adds some
linear spectral
phase to the
pulse!
What is the spectral phase?
The spectral phase is the phase of each frequency in the wave-form.

All of these
ω1 frequencies have
zero phase. So
ω2 this pulse has:

ϕ(ω) = 0
ω3
Note that this
wave-form sees
ω4 constructive
interference, and
ω5 hence peaks, at
t = 0.
ω6 And it has
cancellation
0 t everywhere else.
Now try a linear spectral phase: ϕ(ω) = aω.
By the Shift Theorem, a linear spectral phase is just a delay in time.
And this is what occurs!

ϕ(ω1) = 0
ϕ(ω2) = 0.2 π
ϕ(ω3) = 0.4 π
ϕ(ω4) = 0.6 π
ϕ(ω5) = 0.8 π
ϕ(ω6) = π

t
Transforming between wavelength and
frequency
The spectrum and spectral phase vs. frequency differ from
the spectrum and spectral phase vs. wavelength.
2π c
ω=
The spectral phase is
ϕ λ (λ ) = ϕ ω (2π c / λ ) λ
easily transformed:

To transform the spectrum, note that the energy is the same, whether
we integrate the spectrum over frequency or wavelength:
∞ ∞

∫−∞
S λ (λ ) d λ =
∫
−∞
Sω (ω ) dω
∞
−∞
−2π c 2π c
Changing
variables:
=
∫∞
Sω (2π c / λ )
λ 2
dλ = ∫ −∞
Sω (2π c / λ )
λ 2 dλ

d ω −2π c 2π c
= ⇒ S λ ( λ ) = Sω (2π c / λ )
dλ λ2 λ2
The spectrum and spectral phase vs.
wavelength and frequency
Example: A Gaussian spectrum with a linear spectral phase vs.
frequency

vs. Frequency vs. Wavelength

Note the different shapes of the spectrum and spectral phase

when plotted vs. wavelength and frequency.
Bandwidth in various units
In frequency, by the Uncertainty Principle, a 1-ps pulse has bandwidth:

δν = ~1/2 THz using δν δt ∼ ½

In wave numbers (cm-1), we can write:

c
ν= ⇒ δν = c δ (1/ λ ) ⇒ δ (1/ λ ) = δν / c
λ
So δ(1/λ) = (0.5 × 1012 /s) / (3 × 1010 cm/s) or: δ(1/λ) = 17 cm-1

−1
In wavelength: δ (1/ λ ) = δλ ⇒ δλ = λ 2 δ (1/ λ )
λ2
Assuming an δλ = (800 nm)(.8 ×10−4 cm)(17 cm −1 )
800-nm
wavelength: or: δλ = 1 nm
The Instantaneous frequency
The temporal phase, φ(t), contains frequency-vs.-time information.
The pulse instantaneous angular frequency, ωinst(t), is defined as:

dφ
ωinst (t ) ≡ ω 0 −
dt

This is easy to see. At some time, t, consider the total phase of the
wave. Call this quantity φ0:
φ0 = ω 0t − φ (t )
Exactly one period, T, later, the total phase will (by definition) increase
to φ0 + 2π:
φ0 + 2π = ω0 ⋅ [t + T ] − φ (t + T )
where φ(t+T) is the slowly varying phase at the time, t+T. Subtracting
these two equations:
2π = ω 0T − [φ (t + T ) − φ (t )]
Instantaneous frequency (cont’d)
Dividing by T and recognizing that 2π/T is a frequency, call it ωinst(t):

ωinst(t) = 2π/T = ω0 – [φ(t+T) – φ(t)] / T

But T is small, so [φ(t+T) – φ(t)] /T is the derivative, dφ /dt.

So we’re done!

Usually, however, we’ll think in terms of the instantaneous

frequency, νinst(t), so we’ll need to divide by 2π:

νinst(t) = ν0 – (dφ /dt) / 2π

While the instantaneous frequency isn’t always a rigorous quantity,

it’s fine for ultrashort pulses, which have broad bandwidths.
Group delay
While the temporal phase contains frequency-vs.-time information,
the spectral phase contains time-vs.-frequency information.

So we can define the group delay vs. frequency, tgr(ω), given by:

tgr(ω) = dϕ / dω

A similar derivation to that for the instantaneous frequency can

show that this definition is reasonable.

Also, we’ll typically use this result, which is a real time (the rad’s
cancel out), and never dϕ/dν, which isn’t.

Always remember that tgr(ω) is not the inverse of ωinst(t).

Phase wrapping and unwrapping
Technically, the phase ranges from –π to π. But it often helps to
“unwrap” it. This involves adding or subtracting 2π whenever there’s
a 2π phase jump.

Example: a pulse with quadratic phase Note the scale!

Wrapped phase Unwrapped phase

The main reason for unwrapping the phase is aesthetics.

Phase-blanking E(ωi)
When the intensity is zero, the phase is Im
meaningless.
When the intensity is nearly zero, the −φ(ωi)
phase is nearly meaningless.
Re
Phase-blanking involves simply not plotting
the phase when the intensity is close to zero.

Without phase blanking With phase blanking

Time or Frequency Time or Frequency

The only problem with phase-blanking is that you have to decide the
intensity level below which the phase is meaningless.
Phase Taylor Series expansions
We can write a Taylor series for the phase, φ(t), about the time t = 0:

t t2
φ (t ) = φ0 + φ1 + φ2 + ...
1! 2!
where

dφ
φ1 = is related to the instantaneous frequency.
dt t =0

where only the first few terms are typically required to describe well-
behaved pulses. Of course, we’ll consider badly behaved pulses,
which have higher-order terms in φ(t).
Expanding the phase in time is not common because it’s hard to
measure the intensity vs. time, so we’d have to expand it, too.
Frequency-domain phase expansion
It’s more common to write a Taylor series for ϕ(ω):

( ω − ω0 )
2
ω − ω0
ϕ (ω ) = ϕ 0 + ϕ1 + ϕ2 + ...
1! 2!

where
dϕ
ϕ1 = is the group delay!
dω ω = ω0

d 2ϕ
ϕ2 = is called the group-delay dispersion.
dω 2 ω =ω0

As in the time domain, only the first few terms are typically required to
describe well-behaved pulses. Of course, we’ll consider badly behaved
pulses, which have higher-order terms in ϕ(ω).
Zeroth-order phase: the absolute phase
The absolute phase is the same in both the time and frequency
domains.
f (t)exp(iφ0 ) ⊃ F( ω )exp(iφ0 )
An absolute phase of π/2 will turn a cosine carrier wave into a sine.
It’s usually irrelevant, unless the pulse is only a cycle or so long.

Different absolute phases Different absolute phases

for a four-cycle pulse for a single-cycle pulse

Notice that the two four-cycle pulses look alike, but the three single-
cycle pulses are all quite different.
 First-order phase in frequency: a shift in time
By the Fourier-transform Shift Theorem, f (t − ϕ1 ) ⊃ F (ω )exp(i ωϕ1 )

Time domain Frequency domain

ϕ1 = 0

ϕ1 = − 20 fs

Note that ϕ1 does not affect the instantaneous frequency, but the
group delay = ϕ1.
 First-order phase in time: a frequency shift
By the Inverse-Fourier-transform Shift Theorem,
F (ω − φ1 ) ⊂ f (t ) exp(−i φ1 t )

Time domain Frequency domain

φ1 = 0 / fs

φ1 = −.07 / fs

Note that φ1 does not affect the group delay, but it does affect the
instantaneous frequency = –φ1.
Second-order phase: the linearly 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.

The Linearly
Chirped Gaussian
Pulse

We can write a linearly chirped Gaussian pulse mathematically as:

E (t ) = E0 exp  −(t / τ G )  exp i (ω 0t + β t ) 

 2
  2

Gaussian Carrier Chirp

amplitude wave

Note that for β > 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).
The instantaneous frequency
vs. time for a chirped pulse

A chirped pulse has:

E(t ) ∝ exp i (ω0t − φ (t ) ) 

where:
φ (t ) = − β t 2

The instantaneous frequency is: ω inst (t ) ≡ ω 0 − dφ / dt

which is: ωinst (t ) = ω 0 + 2 β t

So the frequency increases linearly with time.

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 β to be negative

in the expression for the pulse:

E (t ) = E0 exp − ( t / τ G )  exp i (ω 0t + β t 2 ) 

 2
 
And the instantaneous frequency will decrease with time:

ωinst (t ) = ω 0 + 2 β t = ω 0 − 2 β t