166 Light–Matter Interaction
Generally, numerical integration of the density matrix and wave equations is
required to deal with pulse–matter interaction in the presence of varying popu-
lation number changes. For the limiting case of small saturation [s < ( < ) 1],
a small absorption–gain coefficient [|a|< (<) 1], and pulse durations being still
longer than T
2
, we may utilize a perturbation approach [7]. This gives for the
pulse amplitude at the output of such an absorber–amplifier
˜
E(t, z) =
1 +
1
2
a
(0)
˜
L
1 − 2σ
01
¯
W
0
(t) +
1
2
2σ
01
¯
W
0
(t)
2
+ T
2
˜
L(2σ
01
)F
0
(t)
−
1
2
a
(0)
˜
L
2
6
1 − 2σ
01
¯
W
0
(t)
7
T
2
d
dt
+
1
2
a
(0)
˜
L
3
T
2
2
d
2
dt
2
˜
E
0
(t) (3.79)
where a
(0)
= σ
(0)
01
N
(e)
z is the absorption–gain coefficient at the resonance
frequency of the transition. For T
2
→ 0, we obtain a relation which corresponds
to Eq. (3.55) if we expand it up to terms linear in a and quadratic in (2σ
01
¯
W(t)).
The additional terms in Eq. (3.79) come into play if T
2
(d/dt)
˜
E
0
(t) is not van-
ishingly small, that is if the pulse duration is of the same order of magnitude
as T
2
. Then the medium not only remembers the number of absorbed–amplified
photons but also the phase of the electric field over a time period T
2
.
3.3. NONLINEAR, NONRESONANT OPTICAL
PROCESSES
3.3.1. General
Nonresonant optical processes are particularly useful in femtosecond pheno-
mena because they can lead to conversion of optical frequencies with minimum
losses. Nonlinear nonresonant phenomena are currently exploited to make use of
the most efficient laser sources, which are only available at few wavelengths, to
produce shorter pulses at different wavelengths (nonlinear frequency conversion
and compression) and amplify them (parametric amplification). In contrast to
the previous section where the interaction was dominated by a resonance, we
will be dealing with situations where the light frequency is far away from opti-
cal resonances. Nonlinear crystals lend themselves nearly ideally to frequency
conversion with ultrashort pulses because their nonlinearity is electronic and typ-
ically nonresonant from the near UV through the visible to the near IR spectral
region. Therefore, the processes involved respond (nearly) instantaneously on the
time scale of even the shortest optical pulse. There appears to be no limit in the
palette of frequencies that can be generated through nonlinear optics, from dc
Nonlinear, Nonresonant Optical Processes 167
(optical rectification) to infrared (difference frequency generation and optical
parametric generation and amplification), to visible and UV (sum frequency gen-
eration). The shorter the pulse, the higher the peak intensity for a given pulse
energy (and thus the more efficient the nonlinear process).
For cw light of low intensity, a medium with a nonresonant nonlinearity
appears completely transparent and merely introduces a phase shift. For pulses,
as discussed in Chapter 1, dispersion has to be taken into account, which can lead
to pulse broadening and shortening depending on the input chirp, and to phase
modulation effects. The light–matter interaction is linear, i.e., there is a linear
relationship between input and output field, which results in a constant spectral
intensity. A typical example is the pulse propagation through a piece of glass.
The situation becomes much more complex if the pulse intensity is large, which
can be achieved by focusing or/and using amplified pulses. The high electric field
associated with the propagating pulse is no longer negligibly small as compared
to typical local fields inside the material such as inner atomic (inner molecular)
fields and crystal fields. The result is that the material properties are changed
by the incident field and thus depend on the pulse. The induced polarization
which is needed as source term in the wave equation is formally described by
the relationship
P =
0
χ(E)E =
0
χ
(1)
E +
0
χ
(2)
E
2
+
0
χ
(3)
E
3
+···+
0
χ
(n)
E
n
+ …
= P
(1)
+ P
(2)
+···+P
(n)
+ …. (3.80)
The quantities χ
(n)
are known as the nonlinear optical susceptibilities of n
th
order
where χ
(1)
is the linear susceptibility introduced in Eq. (1.70). The ratio of two
successive terms is roughly given by
P
(n+1)
P
(n)
=
χ
(n+1)
E
χ
(n)
≈
E
E
mat
(3.81)
where E
mat
is a typical value for the inherent electrical field in the material.
For simplicity we have taken both E and P as scalar quantities. Generally,
χ
(n)
is a tensor of order (n + 1) which relates an n-fold product of vector com-
ponents E
j
to a certain component of the polarization of n
th
order,
3
P
(n)
; see,
for example, [2,8,9].
3
Note that this product can couple up to n different input fields depending on the conditions of
illumination.
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