Pulse Shaping with Resonant Particles 161
for a saturable medium. Their integration yields
ln
F(z, t)
F
0
(t)
+
F(z, t) −F
0
(t)
F
s
= a (3.63)
and
ln
I(z, t)
I
0
(t)
+
I(z, t) −I
0
(t)
I
s
= a, (3.64)
where a = αz, positive for amplifying media. Equation (3.63) contains the
unknown F(z, t) and I(z, t) implicitly. To get an insight into the pulse distortion
we will assume |a|1, from which we expect F = F
0
+F with |F/F
0
|1.
Inserting this into Eq. (3.63) and expanding the logarithmic function gives the
following relation for the flux at the output of the medium of length z:
F(z, t) = F
0
(t)
1 +
a
1 + F
0
(t)/F
s
. (3.65)
The absorption (or amplification) term becomes time dependent, where its value
is now controlled by the instantaneous photon flux density rather than by the
energy density. A characteristic quantity of the medium is now F
s
or I
s
. The
result of this kind of saturation is that the pulse peak where the intensity takes
on a maximum is less absorbed (amplified) than the wings, as illustrated in
Figure 3.5.
3.2.3. Phase Modulation by Quasi-Resonant
Interactions
According to our previous discussion, the change of the occupation numbers
(saturation) results in a change of the refractive index and we expect a phase
modulation to occur. Using Eqs. (3.48) and (3.52) the time-dependent frequency
change
δω(t) =
∂ϕ
∂t
can be written as
δω(t) =−
(ω
− ω
10
)T
2
2
σ
(0)
01
|
˜
L|
2
z
0
∂
∂t
Ndz =−
(ω
− ω
10
)T
2
2
∂
∂t
ln
F(z, t)
F
0
(t)
.
(3.66)
162 Light–Matter Interaction
Normalized intensity
0.0
0.5
1.0
1.5
2.0
Normalized time
–3
–1
0
1
2
–4
–2
3
4
(1)
(2)
(3)
Figure 3.5 Pulse shaping in a saturable absorber (1) and depletable amplifier (2) for τ
p
T
1
,
a =∓3 and F
0
/F
s
= 1. 5 according to Eq. (3.63). For the input pulse (3) we assumed F
0
(t) =
cosh
−2
(1. 76t/τ
p
).
The sign of δω depends on the sign of (ω
−ω
10
)T
2
, i.e., on whether the interaction
takes place above or below resonance, and on the sign of ln(F/F
0
). The latter is
positive (negative) for F > (<) F
0
which is true for an amplifier (absorber).
For τ
p
T
1
the pulse energy controls the dynamics of N, and by means of
Eq. (3.55), δω(t) becomes
δω(t) =−(ω
− ω
10
)T
2
σ
01
e
−a
− 1
e
−a
− 1 +e
2σ
01
¯
W
0
(t)
F
0
(t), (3.67)
or equivalently, in terms of the intensity and saturation energy density W
s
:
δω(t) =−
(ω
− ω
10
)T
2
2
e
−a
− 1
e
−a
− 1 +e
W(t)/W
s
I(t)
W
s
. (3.68)
The optically thin medium approximation (|a|1) of Eq. (3.67) is:
δω(t) (ω
− ω
10
)T
2
ae
−2σ
01
¯
W
0
(t)
σ
01
F
0
(t). (3.69)
The other limiting case (τ
p
T
1
) is associated with a δω(t) given by
δω(t) = (ω
− ω
10
)T
2
σ
01
T
1
∂
∂t
[
F(t, z) − F
0
(t)
]
, (3.70)
Pulse Shaping with Resonant Particles 163
which for small |a| results in
δω(t) =
(ω
− ω
10
)T
2
a
2
1
(1 + F
0
(t)/F
s
)
2
∂
∂t
F
0
(t)
F
s
, (3.71)
as can easily be verified by inserting Eq. (3.65) into Eq. (3.70). Figure 3.6 shows
the time-dependent frequency change described by Eq. (3.67) for an amplifier.
Equation (3.71) indicates a frequency change toward the leading part of the pulse
with increasing saturation. A similar behavior occurs if the pulse passes through
an absorber. As a result the frequency change across the FWHM of the pulses
becomes maximum at a certain level of saturation.
To get an idea about the order of magnitude of the frequency change let us
estimate δ ¯ω = δω(t =−τ
p
/2) −δω(t = τ
p
/2) for a sech
2
pulse using Eq. (3.69)
for an absorber. For
¯
W
0,∞
/
¯
W
s
= W
0
/W
s
= 1wefind
τ
p
δ ¯ω (ω
− ω
10
)T
2
a (3.72)
which for a =−0. 1 and (ω
−ω
10
)T
2
= 1 yields τ
p
δ ˜ω ≈−0. 1. If we compare
this with the pulse duration–bandwidth product, cf. Table 1.1, τ
p
ω = 2. 8,
we hardly expect that the induced frequency change is of importance. This is
usually true for a single passage; however, it can play a significant role in lasers
where the pulse passes through the media many times before it is coupled out [4].
Normalized chirp
0
1
Time
–3
–1 1
–5
3
Figure 3.6 Normalized chirp τ
p
δω versus time after passage through an amplifier according to
Eq. (3.67) for a sech
2
input pulse, e
|a|
= 100, s = 0. 5, …, 4 (s = 0. 5, from right to left),
(ω
− ω
10
)T
2
=−1.
164 Light–Matter Interaction
Signal (arb. un.)
–0.75
–0.50
–0.25
0.00
0.25
0.50
0.75
1.00
t/T
After amplifier
Input
After absorber
Pulse
Chirp
After absorber
After amplifier
–2
0
2
–4
4
Figure 3.7 Normalized chirp and pulse shapes at the output of a saturable absorber and a
depletable amplifier for τ
p
/T
1
1 [Eqs. (3.63) and (3.71)] and a sech
2
(t/T) input pulse. e
|a|
= 100,
s = 1, (ω
− ω
10
)T
2
= 1.
Chirp introduced by a fast (T
1
τ
p
) absorber and amplifier is shown in Figure 3.7
for comparison. It should be noted that our discussion can easily be expanded
to include effects of multilevel systems [5]. This, for example, is necessary to
model dye molecules more accurately.
For completeness let us now briefly discuss an inhomogeneously broadened
sample, i.e., a distribution of particles having different resonance frequencies
ω
10
. We assume that the preconditions for applying the rate equation approxi-
mations are fulfilled. The source terms in the propagation equations for the field
components E, ϕ, cf. Eqs. (3.47), and (3.48), have to be integrated over the
inhomogeneous distribution function g
inh
. This yields
∂
∂z
E =
1
2
E
∞
0
σ
(0)
01
|
˜
L(ω
− ω
0
)|
2
N
(t, ω
10
)dω
10
=
1
2
E
∞
0
σ
01
(ω
− ω
10
)g
inh
(ω
10
)
¯
N[ρ
11
(ω
10
) − ρ
00
(ω
10
)]dω
10
(3.73)
and
∂
∂z
ϕ =−
1
2
∞
0
(ω
− ω
10
)T
2
σ
01
(ω
− ω
10
)g
inh
(ω
10
)
¯
N
×[ρ
11
(t, ω
10
) − ρ
00
(t, ω
10
)]dω
10
. (3.74)
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