2.1 Ultrafast solid-state lasers - ETH - the Keller Group

112 2.1.6 Mode-locking techniques [Ref. p. 134
For an analytical solution we have to expand the exponential function up to the second order, i.e.
e x ≈ 1+x + x2
2 :
⎡
⎤
q (t) =q 0 exp ⎣−
σ ∫ t
A
|A (t ′ )| 2 dt ′ ⎦
A A hν
−∞
⎡
⎛
⎞
⎢
≈ q 0 ⎣1 −
σ ∫ t
A
|A (t ′ )| 2 dt ′ σ 2 ∫ ⎤ t
A
+
⎝
A A hν
2(A A hν) 2 |A (t ′ )| 2 dt ⎠2 ′ ⎥ ⎦ (2.1.60)
−∞
and analogous for g (t). In this case we can obtain an analytical solution with a sech 2 -pulse shape:
( ) t
A (t) =A 0 sech
(2.1.61)
τ
and a FWHM pulse duration
τ p ≈ 1.76 × 4 1
, (2.1.62)
π Δ ν g
where Δ ν g is the FWHM gain bandwidth of the laser. In (2.1.62) the conditions (2.1.56)–(2.1.58)
are assumed and in addition E sat,L ≫ E sat,A and E p ≫ E sat,A (i.e. a fully saturated absorber).
For the example of Rhodamin 6G and DODCI (Table 2.1.12) we then obtain for a gain bandwidthofΔν
g ≈ 4 × 10 13 Hz a pulse duration of about 56 fs (2.1.62). Pulses as short as 27 fs have
been demonstrated [85Val]. However, it was recognized early on that SPM together with negative
dispersion results in soliton formation and further reduces pulse duration by about a factor of 2
in dye lasers [84Mar, 85Mar]. This would explain the difference in the theoretical prediction of
(2.1.62) from the experimentally demonstrated 27 fs. However, at that time an analytic solution
for the pulse-shortening effect was not presented.
For semiconductor lasers we typically observe strongly chirped pulses [95Jia]. Therefore, we
would have to include dispersion and self-phase modulation in the rate equation. This is not so
easy because we would have to also include the refractive-index change that occurs during gain
saturation.
−∞
2.1.6.5 Passive mode-locking with a fast saturable absorber
In passive mode-locking the loss modulation is obtained by Self-Amplitude Modulation (SAM),
where the pulse saturates an absorber, for example. In the ideal case, the SAM follows the intensity
profile of the pulse. This is the case of an ideal fast saturable absorber. In this case, SAM produces
a much larger curvature of loss modulation than in the sinusoidal loss modulation of active modelocking,
because the mode-locked pulse duration is much shorter than the cavity round-trip time.
Therefore, we would expect from the previous discussion of active mode-locking, that we obtain
much shorter pulses with passive mode-locking. This is indeed observed.
In the fast saturable absorber model no dynamic gain saturation is required and the short netgain
window is formed by a fast recovering saturable absorber alone (Fig. 2.1.5b). This was initially
believed to be the only stable approach to passively mode-lock solid-state lasers with long upperstate
lifetimes. Additive Pulse Mode-locking (APM) was the first fast saturable absorber for such
solid-state lasers (Sect. 2.1.4.4.2). However, APM required interferometric cavity-length stabilization.
Kerr-Lens Mode-locking (KLM) [91Spe] (Sect. 2.1.4.4.3) was the first useful demonstration of
an intracavity fast saturable absorber for a solid-state laser and because of its simplicity replaced
Landolt-Börnstein
New Series VIII/1B1