42 Fundamentals
At given material parameters and carrier frequency, shorter pulses always lead
to a dominant pulse spreading. For T
2
= 10
−10
s (typical value for a single
electronic resonance), and a detuning ωT
2
= 10
4
, we find for example:
τ
rel
W
rel
≈ W
rel
1200 fs
τ
G0
4
. (1.163)
To summarize, a resonant transition of certain spectral width 1/T
2
influences short
pulse (pulse duration < 1ps) propagation outside resonance mainly because of
dispersion. Therefore, the consideration of a transparent material (
i
≈ 0) with
a frequency dependent, real dielectric constant (), which was necessary to
derive Eq. (1.93), is justified in many practical cases involving ultrashort pulses.
1.3. INTERACTION OF LIGHT PULSES WITH
LINEAR OPTICAL ELEMENTS
Even though this topic is treated in detail in Chapter 2, we want to discuss
here some general aspects of pulse distortions induced by linear optical elements.
These elements comprise typical optical components, such as mirrors, prisms,
and gratings, which one usually finds in all optical setups. Here we shall restrict
ourselves to the temporal and spectral changes the pulse experiences and shall
neglect a possible change of the beam characteristics. A linear optical element of
this type can be characterized by a complex optical transfer function
˜
H() = R()e
−i()
(1.164)
that relates the incident field spectrum
˜
E
in
() to the field at the sample
output
˜
E()
˜
E() = R()e
−i()
˜
E
in
(). (1.165)
Here R() is the (real) amplitude response and () is the phase response.
As can be seen from Eq. (1.165), the influence of R() is that of a frequency
filter. The phase factor () can be interpreted as the phase delay which a
spectral component of frequency experiences. To get an insight of how the
phase response affects the light pulse, we assume that R() does not change over
the pulse spectrum whereas () does. Thus, we obtain for the output field from
Eq. (1.165):
˜
E(t) =
1
2π
R
+∞
−∞
˜
E
in
()e
−i()
e
it
d. (1.166)
Linear Optical Elements 43
Replacing () by its Taylor expansion around the carrier frequency ω
of the
incident pulse
() =
∞
n=0
b
n
( − ω
)
n
(1.167)
with the expansion coefficients
b
n
=
1
n!
d
n
d
n
ω
(1.168)
we obtain for the pulse
˜
E(t) =
1
2
˜
E(t)e
iω
t
=
1
2π
Re
−ib
0
e
iω
t
+∞
−∞
˜
E
in
()
× exp
−i
∞
n=2
b
n
( − ω
)
n
e
i(−ω
)(t−b
1
)
d. (1.169)
By means of Eq. (1.169) we can easily interpret the effect of the various expansion
coefficients b
n
. The term e
−ib
0
is a constant phase shift (phase delay) having no
effect on the pulse envelope. A nonvanishing b
1
leads solely to a shift of the
pulse on the time axis t; the pulse would obviously keep its position on a time
scale t
= t − b
1
. The term b
1
determines a group delay in a similar manner
as the first-order expansion coefficient of the propagation constant k defined
a group velocity in Eq. (1.97). The higher-order expansion coefficients produce
a nonlinear behavior of the spectral phase which changes the pulse envelope and
chirp. The action of the term with n = 2, for example, producing a quadratic
spectral phase, is analogous to that of GVD in transparent media.
If we decompose the input field spectrum into modulus and phase
˜
E
in
() =
|
˜
E
in
()|exp(i
in
()), we obtain from Eq. (1.165) for the spectral phase at the
output
() =
in
() −
∞
n=0
b
n
( − ω
)
n
. (1.170)
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