300 Ultrashort Sources I: Fundamentals
Dispersion
Next z
z
Propagation
∆z
∆z
FT
IFT
Gain
Lensing
f
nl
w(z)
R(z)
P(z)
w(z∆z)
R(z∆z)
P(z∆z)
Figure 5.11 Successive calculations to be made to propagate a pulse through each slice z of a
gain crystal. FT and IFT stand for Fourier transform and inverse Fourier transform and indicate that
the treatment of dispersion can conveniently be done in the frequency domain.
˜
E(z , r, t) = E(z, r, t) exp(ϕ(z, r, t)). At the end of each slice we obtain
˜
E(z +z, r, t) which acts as the input for the next slice, z +z → z.
To study the switch on dynamics of a fs laser one starts from noise. The noise
bandwidth is roughly given by the width of the fluorescence spectrum of the
amplifier while its magnitude corresponds to the light emitted spontaneously into
the solid angle defined by the cavity modes. In most cases the particular noise
features vanish after few round-trips, and the final results are independent of the
field originally injected. Figure 5.12 shows as an example the development of
the pulse envelope, instantaneous frequency, and energy as a function of round-
trips completed after the switch on of the laser mode-locked with a slow saturable
absorber. Obviously the pulse parameters become stationary after several hundred
round-trips, which amounts to several microseconds.
5.2.4. Elements of an Analytical Treatment
A number of approximate analytical procedures has been developed by
New [18,19] and Haus [20] to describe the steady-state regime. The problem often
reduces to finding a complex pulse envelope
˜
E(t) that satisfies the steady-state
condition [20]:
˜
E(t + h) =
N
.
i=1
T
i
˜
E(t). (5.28)
Circulating Pulse Model 301
x10
4
0
3.
2.
1.
0.2
0.
0.2
6.
4.
2.
200 400
W
sg
T
1g
τ
p
Round trips
T
2g
(-
ᐉ
)
Time
5
50
500
Round trip number
Intensity, frequency
W
(a) (b)
Figure 5.12 (a) Evolution of pulse envelope (solid line) and instantaneous frequency (dashed line)
after switch on of the laser, and (b) corresponding steady-state pulse energy (solid line), pulse duration
(dashed line), and frequency (dotted line). The active media were described by the density matrix
equations introduced in Chapter 3. All pulse parameters including the average frequency develop as
a result of the interplay of resonator elements. No extra frequency selective element was necessary
to limit the pulse duration (normalized to the spectral width of the gain transition 2/T
2g
). (Adapted
from Petrov [17].)
Equation (5.28) states that the pulse envelope reproduces itself after each round-
trip, except for a temporal translation h including a constant pase shift. The main
challenge is to find appropriate operator functions for the different resonator
elements that are amenable to an analytical evaluation of Eq. (5.28). A con-
venient approximation is to assume that the modification introduced by each
resonator element is small, which allows one to terminate the expansion of the
corresponding operator functions after a few orders. Another consequence of that
approximation is that the order of the resonator elements is no longer relevant.
We will briefly describe this approach here with a small number of possible
resonator elements and processes. Some of the most frequently used operators
representative of resonator elements are derived below.
The transformation of the pulse envelope by a saturable loss–gain can be
expressed as
˜
E
out
(t) =
1 +
1
2
a
(0)
a
˜
L
1 −
W(t)
W
sa
+
1
2
W(t)
W
sa
2
˜
E
in
(t). (5.29)
302 Ultrashort Sources I: Fundamentals
Equation (5.29) is the rate equation approximation (T
2
→ 0) of Eq. (3.79).
The expansion parameters are the small signal absorption coefficient (a
a
< 0)
and the ratio of pulse energy density to the saturation density of the transition.
Equation (5.29) applies to a gain medium with the substitutions a
a
→ a
g
and
W
sa
→ W
sg
.
The transfer function of a GVD element can be derived from Eq. (1.172)
and reads
˜
E
out
(t) =
1 + ib
2
d
2
dt
2
˜
E
in
(t). (5.30)
For this expansion to be valid, the dispersion parameter b
2
has to be much smaller
than τ
2
p
. In the case of a transparent medium of thickness d, b
2
= k
d/2.
A corresponding expression for the action of a Kerr medium of length d is
˜
E
out
(t) =
1 − i
k
n
2
d
n
0
|
˜
E
in
(t)|
2
˜
E
in
(t) (5.31)
which can easily be derived from Eq. (3.146).
A linear loss element, which for example represents the outcoupling mirror,
can be modeled according to
˜
E
out
(t) =
1 −
1
2
γ
˜
E
in
(t) (5.32)
where γ is the intensity loss coefficient (transmission coefficient), for which
γ 1 is assumed.
Each resonator contains frequency selective elements which can be used to
tune the frequency. Such elements are for example prisms, Lyot filters, and mir-
rors with a certain spectral response. Together with the finite gain profile, they
ultimately restrict the bandwidth of the pulse in the laser. Let us assume a
Lorentzian shape for the filter response in the frequency domain
˜
H =[1 +i( −
ω
)/ω
F
]
−1
where the FWHM ω
F
is much broader than the pulse spectrum.
After expansion up to second order and retransformation to the time domain:
˜
E
out
(t) =
1 −
2
ω
F
d
dt
+
4
ω
2
F
d
2
dt
2
˜
E
in
(t). (5.33)
If all passive elements are chosen to have an extremely broad frequency response,
the finite transition profiles of the active media act as effective filters. This can
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