154 Light–Matter Interaction
Absorption
Refractive index
10
10
Figure 3.3 Absorption coefficient and refractive index in the vicinity of an optical resonance for
two values of the population inversion ρ = ρ
11
− ρ
00
.
3.2. PULSE SHAPING WITH RESONANT
PARTICLES
3.2.1. General
Let us consider a pulse traveling through a medium of certain length consisting
of particles with number density
¯
N at resonance with the light. How is the
pulse being modified by the medium? In answering this question we assume
that only the resonant particles influence the pulse, i.e., we neglect GVD of the
host material and nonresonant nonlinear contributions. Depending on how the
medium is prepared we expect light to be absorbed or amplified. Absorption
can occur when ρ = (ρ
11
–ρ
00
) < 0, i.e., if the majority of resonant particles
interacting with the pulse are in the ground state. Amplification is expected in
the opposite case where ρ = (ρ
11
–ρ
00
) > 0. To reach this situation the particles
have to be excited by an appropriate pump mechanism.
2
At first glance absorption (amplification) seems only to decrease (increase)
the pulse energy by a certain factor while leaving the other pulse characteristics
unchanged. We will see this to be true only under specific conditions. In the
general case the pulse at the output exhibits a different envelope (shape) as well
as a different time-dependent phase, as compared to its input characteristics. The
leading part of the pulse changes the properties of the medium, which will then act
in a different manner on the trailing part. In terms of amplification and absorption
one can say that these quantities become time dependent because of the change
of the inversion density ρ(t). We expect the pulse to be more heavily absorbed
(amplified) at its leading part which, of course, modifies the envelope shape.
According to our discussion previously, a time-dependent occupation number
means that different pulse parts “see” different optical path lengths, resulting in
2
Other energy levels have to be taken into account to describe the pumping.
Pulse Shaping with Resonant Particles 155
a time-dependent phase change (chirp). To discuss the actual pulse distortion we
need to determine the temporal behavior of the occupation numbers by means
of Eqs. (3.23)–(3.26). In general, numerical methods are required to analyze this
problem. An analytical description can only be made for some limiting cases. The
physics of the pulse–matter interaction depends strongly on the ratio of the pulse
duration and the characteristic response time of the medium, as well as on the
pulse intensity and energy. We will start with the approximation that the phase
relaxation time is much shorter than the pulse duration, generally referred to as
rate equation approximation. Femtosecond pulses do challenge this often used
approximation. Because the duration of fs pulses can be comparable or shorter
than the phase memory of the medium, the polarization oscillations excited by
the leading part of the pulse can interfere coherently with subsequent pulse parts.
We will discuss first this situation approximately as a perturbation of the rate
equation approximation (REA). Phenomena related to the dominant influence of
coherent light–matter interaction will be dealt with in the following chapter.
To illustrate the action of the phase memory, let us inspect the equation of
motion for the polarization. Within the SVEA, and assuming |ω
−ω
10
|ω
10
,
ω
as well as ω
10
T
2
, ω
T
2
1, Eqs. (3.11), (3.23), (3.24), and (3.28) yield a
first-order differential equation for the slowly varying envelope component
˜
P:
d
dt
˜
P +
1
T
2
+ i(ω
− ω
10
)
˜
P = i
¯
Np
2
(ρ
11
− ρ
00
)
˜
E. (3.38)
The solution of Eq. (3.38) can be written formally in the form
˜
P(t) = i
¯
Np
2
t
−∞
˜
E(t
)ρ(t
)e
[i(ω
−ω
10
)+1/T
2
](t
−t)
dt
(3.39)
where ρ = ρ
11
−ρ
00
is the population inversion. Obviously the polarization at
time t depends on values of the electric field (modulus and phase) and population
numbers for all t
≤ t, a dependence weighted by the function e
−(t−t
)/T
2
. The
latter implies that the memory time is T
2
. For later reference we will derive two
other representations of the solution of Eq. (3.38). Continuous application of
partial integration of Eq. (3.39) leads to
˜
P(t) = i
¯
Np
2
T
2
˜
L(ω
− ω
10
)
4
˜
E(t)ρ(t)
+
∞
n=1
˜
L
n
(ω
− ω
10
)
T
2
d
dt
n
[
˜
E(t)ρ(t)]
, (3.40)
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