6 Fundamentals
From Eq. (1.10) it is obvious that the decomposition of (t)intoω and ϕ(t)
is not unique. The most useful decomposition is one that ensures the smallest
dϕ/dt during the intense portion of the pulse. A common practice is to identify
ω
with the carrier frequency at the pulse peak. A better definition—which is
consistent in the time and frequency domains—is to use the intensity weighted
average frequency:
ω=
∞
−∞
|
˜
E(t)|
2
ω(t)dt
∞
−∞
|
˜
E(t)|
2
dt
=
∞
−∞
|
˜
E
+
()|
2
d
∞
−∞
|
˜
E
+
()|
2
d
(1.18)
The various notations are illustrated in Figure 1.2 where a linearly up-chirped
pulse is taken as an example. The temporal dependence of the real electric field is
sketched in the top part of Fig 1.2. A complex representation in the time domain
is illustrated with the amplitude and instantaneous frequency of the field. The
positive and negative frequency components of the Fourier transform are shown
in amplitude and phase in the bottom part of the figure.
1.1.2. Power, Energy, and Related Quantities
Let us imagine the practical situation in which the pulse propagates as a beam
with cross section A, and with E(t) as the relevant component of the electric field.
The (instantaneous) pulse power (in Watt) in a dispersionless material of refrac-
tive index n can be derived from the Poynting theorem of electrodynamics [1]
and is given by
P(t) =
0
cn
A
dS
1
T
t+T/2
t−T/2
E
2
(t
)dt
(1.19)
where c is the velocity of light in vacuum,
0
is the dielectric permittivity and
A
dS stands for integration over the beam cross section. The power can be
measured by a detector (photodiode, photomultiplier, etc.) which integrates over
the beam cross section. The temporal response of this device must be short as
compared to the speed of variations of the field envelope to be measured. The
temporal averaging is performed over one optical period T = 2π/ω
. Note that
the instantaneous power as introduced in Eq. (1.19) is then just a convenient
theoretical quantity. In a practical measurement T has to be replaced by the
actual response time τ
R
of the detector. Therefore, even with the fastest detectors
available today (τ
R
≈ 10
−13
−10
−12
s), details of the envelope of fs light pulses
cannot be resolved directly.
Characteristics of Femtosecond Light Pulses 7
Electric field
Pulse chirp
Spectral amplitude
Spectral phase
0
0
d
dt
t
t
1
2
ii
Figure 1.2 Electric field, time-dependent carrier frequency, and spectral amplitude of an
upchirped pulse.
8 Fundamentals
A temporal integration of the power yields the energy W (in Joules):
W =
∞
−∞
P(t
)dt
(1.20)
where the upper and lower integration limits essentially mean “before” and “after”
the pulse under investigation.
The corresponding quantity per unit area is the intensity (W/cm
2
):
I(t) =
0
cn
1
T
t+T/2
t−T/2
E
2
(t
)dt
=
1
2
0
cnE
2
(t) = 2
0
cn
˜
E
+
(t)
˜
E
−
(t) =
1
2
0
cn
˜
E(t)
˜
E
∗
(t) (1.21)
and the energy density per unit area (J/cm
2
):
W =
∞
−∞
I(t
)dt
. (1.22)
Sometimes it is convenient to use quantities which are related to photon
numbers, such as the photon flux F (photons/s) or the photon flux density F
(photons/s/cm
2
):
F(t) =
P(t)
ω
and F(t) =
I(t)
ω
(1.23)
where ω
is the energy of one photon at the carrier frequency.
The spectral properties of the light are typically obtained by measuring the
intensity of the field, without any time resolution, at the output of a spectrometer.
The quantity that is measured is the spectral intensity:
S() =|η()
˜
E
+
()|
2
(1.24)
where η is a scaling factor which accounts for losses, geometrical influences,
and the finite resolution of the spectrometer. Assuming an ideal spectrometer,
|η|
2
can be determined from the requirement of energy conservation:
|η|
2
∞
−∞
|
˜
E
+
()|
2
d = 2
0
cn
∞
−∞
˜
E
+
(t)
˜
E
−
(t)dt (1.25)
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