2.1 Ultrafast solid-state lasers - ETH - the Keller Group
2.1 Ultrafast solid-state lasers - ETH - the Keller Group
2.1 Ultrafast solid-state lasers - ETH - the Keller Group
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Ref. p. 134] <strong>2.1</strong> <strong>Ultrafast</strong> <strong>solid</strong>-<strong>state</strong> <strong>lasers</strong> 115<br />
to 30 τ p . Numerical simulations show that this is a reasonable estimate and that with too long<br />
recovery time, <strong>the</strong> pulse does not simply become longer but unstable.<br />
At first this long recovery time might be surprising, because on <strong>the</strong> trailing edge of <strong>the</strong> pulse<br />
<strong>the</strong>re is no shaping action of <strong>the</strong> absorber. There is even net gain, because <strong>the</strong> loss caused by <strong>the</strong><br />
absorber is very small for <strong>the</strong> trailing edge (always assuming a fully saturated absorber), while <strong>the</strong><br />
total loss for <strong>the</strong> pulse is larger and is balanced by <strong>the</strong> saturated gain in steady <strong>state</strong>. Thus, one<br />
might expect that this net gain after <strong>the</strong> pulse would destabilize <strong>the</strong> pulse – but this is not <strong>the</strong><br />
case. The reason is that <strong>the</strong> pulses experience a temporal shift by <strong>the</strong> absorber, which limits <strong>the</strong><br />
time in which noise behind <strong>the</strong> pulse can be amplified. The absorber attenuates mostly <strong>the</strong> leading<br />
wing of <strong>the</strong> pulse, thus shifting <strong>the</strong> pulse center backwards in each cavity round trip. This means<br />
that <strong>the</strong> pulse is constantly moving backward and swallows any noise growing behind itself. An<br />
upper limit on <strong>the</strong> recovery time than follows from <strong>the</strong> condition that this noise cannot grow too<br />
much. Note that weak reflections in <strong>the</strong> laser cavity could generate weak satellite pulses behind<br />
<strong>the</strong> main pulse. These satellite pulses could be stronger than <strong>the</strong> noise level and thus significantly<br />
reduce maximum tolerable recovery time of <strong>the</strong> absorber.<br />
So far we have not included any additional pulse-shaping effects such as SPM or solitons.<br />
Fur<strong>the</strong>r simulations show that SPM alone in <strong>the</strong> positive dispersion regime should always be kept<br />
small because it always makes pulses longer and even destabilizes <strong>the</strong>m, particularly for absorbers<br />
with small modulation depth [01Pas1]. This result might be surprising at first – but again <strong>the</strong><br />
temporal delay caused by <strong>the</strong> absorber gives a simple explanation. SPM (with positive n 2 ,asis<strong>the</strong><br />
usual case) decreases <strong>the</strong> instantaneous frequency in <strong>the</strong> leading wing and increases <strong>the</strong> frequency<br />
in <strong>the</strong> trailing wing. The absorber always attenuates <strong>the</strong> leading wing, thus removing <strong>the</strong> lower<br />
frequency components, which results in an increased center frequency and broader pulses due to<br />
<strong>the</strong> decrease in pulse bandwidth. For strong SPM <strong>the</strong> pulses become unstable. A rule of thumb is<br />
that <strong>the</strong> nonlinear phase shift for <strong>the</strong> peak should be at most a few mrad per 1 % of modulation<br />
depth. It is clear that SPM could hardly be made weak enough in <strong>the</strong> sub-picosecond regime. For<br />
this reason, soliton mode-locking is usually required in <strong>the</strong> sub-picosecond domain, because <strong>the</strong>re<br />
<strong>the</strong> nonlinear phase changes can be much larger.<br />
<strong>2.1</strong>.6.7 Soliton mode-locking<br />
In soliton mode-locking, <strong>the</strong> pulse shaping is done solely by soliton formation, i.e. <strong>the</strong> balance<br />
of <strong>Group</strong>-Velocity Dispersion (GVD) and Self-Phase Modulation (SPM) at steady <strong>state</strong>, with<br />
no additional requirements on <strong>the</strong> cavity stability regime as for example for KLM. In contrast<br />
to KLM we use only <strong>the</strong> time-dependent part of <strong>the</strong> Kerr effect at <strong>the</strong> peak intensity (i.e.<br />
n (t) =n+n 2 I 0 (t), (<strong>2.1</strong>.42)) and not <strong>the</strong> radially dependent part as well (i.e. n (r, t) =n+n 2 I (r, t),<br />
(<strong>2.1</strong>.42)). The radially dependent part of <strong>the</strong> Kerr effect is responsible for KLM because it forms <strong>the</strong><br />
nonlinear lens that reduces <strong>the</strong> beam diameter at an intracavity aperture inside <strong>the</strong> gain medium.<br />
Thus, this nonlinear lens forms an effective fast saturable absorber because <strong>the</strong> intensity-dependent<br />
beam-diameter reduction at an aperture introduces less loss at high intensity and more loss at low<br />
intensity. Such a radially dependent effective saturable however couples <strong>the</strong> mode-locking mechanism<br />
with <strong>the</strong> cavity mode. In contrast, soliton mode-locking does not depend on <strong>the</strong> transverse<br />
Kerr effect and has <strong>the</strong>refore <strong>the</strong> advantage that <strong>the</strong> mode-locking mechanism is decoupled from<br />
<strong>the</strong> cavity design and no critical cavity stability regime is required – it basically works over <strong>the</strong><br />
full cavity stability range.<br />
In soliton mode-locking, an additional loss mechanism, such as a saturable absorber [95Kae1],<br />
or an acousto-optic modulator [95Kae2], is necessary to start <strong>the</strong> mode-locking process and to<br />
stabilize <strong>the</strong> soliton pulse-forming process. In soliton mode-locking, we have shown that <strong>the</strong> netgain<br />
window (Fig. <strong>2.1</strong>.5c) can remain open for significantly more than 10 times longer than <strong>the</strong><br />
ultrashort pulse, depending on <strong>the</strong> specific laser parameters [95Jun2, 96Kae]. This strongly relaxes<br />
Landolt-Börnstein<br />
New Series VIII/1B1