44 Fundamentals
It is interesting to investigate what happens if the linear optical element is chosen
to compensate for the phase of the input field. For Taylor coefficients with n ≥ 2:
b
n
=
1
n!
d
n
d
n
in
()
ω
. (1.171)
A closer inspection of Eq. (1.169) shows that when Eq. (1.171) is satisfied, all
spectral components are in phase for t−b
1
= 0, leading to a pulse with maximum
peak intensity, as was discussed in previous sections. We will come back to this
important point when discussing pulse compression. We want to point out the
formal analogy between the solution of the linear wave equation (1.74) and
Eq. (1.165) for R() = 1 and () = k()z. This analogy expresses the fact
that a dispersive transmission object is just one example of a linear element.
In this case we obtain for the spectrum of the complex envelope
˜
E(, z) =
˜
E
in
(,0)exp
−i
∞
n=0
1
n!
k
(n)
( − ω
)
n
z
(1.172)
where k
(n)
= (d
n
/d
n
)k()
|
ω
.
Next let us consider a sequence of m optical elements. The resulting transfer
function is given by the product of the individual contributions
˜
H
j
()
˜
H() =
m
.
j=1
˜
H
j
() =
⎛
⎝
m
.
j=1
R
j
()
⎞
⎠
exp
⎡
⎣
−i
m
j=1
j
()
⎤
⎦
(1.173)
which means an addition of the phase responses in the exponent. Subsequently,
by a suitable choice of elements, one can reach a zero-phase response so that
the action of the device is through the amplitude response only. In particular,
the quadratic phase response of an element (e.g., dispersive glass path) leading
to pulse broadening can be compensated with an element having an equal phase
response of opposite sign (e.g., grating pair) which automatically would recom-
press the pulse to its original duration. Such methods are of great importance for
the handling of ultrashort light pulses. Corresponding elements will be discussed
in Chapter 2.
1.4. GENERATION OF PHASE MODULATION
At this point let us briefly discuss essential physical mechanisms to produce a
time-dependent phase of the pulse, i.e., a chirped light pulse. Processes resulting
Generation of Phase Modulation 45
in a phase modulation can be divided into those that increase the pulse spectral
width and those that leave the spectrum unchanged. The latter can be attributed
to the action of linear optical processes. Any transparent linear medium, or
spectrally “flat” reflector, can change the phase of a pulse, without affecting
its spectral amplitude. The action of these elements is most easily analyzed in
the frequency domain. As we have seen in the previous section, the phase mod-
ulation results from the different phase delays that different spectral components
experience on interaction. The result for an initially bandwidth-limited pulse,
in the time domain, is a temporally broadened pulse with a certain frequency
distribution across the envelope, such that the spectral amplitude profile remains
unchanged. For an element to act in this manner its phase response () must
have nonzero derivatives of at least second order as explained in the previous
section.
A phase modulation that leads to a spectral broadening is most easily discussed
in the time domain. Let us assume that the action of a corresponding optical
element on an unchirped input pulse can be formally written as:
˜
E(t) = T(t)e
i(t)
˜
E
in
(t) (1.174)
where T and define a time-dependent amplitude and phase response,
respectively. For our simplified discussion here let us further assume that
T = const., leaving the pulse envelope unaffected. Because the output pulse
has an additional phase modulation (t) its spectrum must have broadened dur-
ing the interaction. If the pulse under consideration is responsible for the time
dependence of , then we call the process self-phase modulation. If additional
pulses cause the temporal change of the optical properties we will refer to it
as cross-phase modulation. Often, phase modulation occurs through a temporal
variation of the index of refraction n of a medium during the passage of the pulse.
For a medium of length d the corresponding phase is:
(t) =−k(t)d =−
2π
λ
n(t)d. (1.175)
In later chapters we will discuss in detail several nonlinear optical interaction
schemes with short light pulses that can produce a time dependence of n.
A time dependence of n can also be achieved by applying a voltage pulse at an
electro-optic material for example. However, with the view on phase shaping of
femtosecond light pulses the requirements for the timing accuracy of the voltage
pulse make this technique difficult.
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