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
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
82 <strong>2.1</strong>.4 Loss modulation [Ref. p. 134<br />
R tot = F ∫<br />
out Iout (t)dt<br />
= ∫<br />
F in Iin (t)dt =1− 2 ∫<br />
F in<br />
q (t) I in (t)dt. (<strong>2.1</strong>.9)<br />
This determines <strong>the</strong> total absorber loss coefficient q p , which results from <strong>the</strong> fact that part of <strong>the</strong><br />
excitation pulse needs to be absorbed to saturate <strong>the</strong> absorber:<br />
R tot =e −2qp ≈ 1 − 2q p . (<strong>2.1</strong>.10)<br />
From (<strong>2.1</strong>.9) and (<strong>2.1</strong>.10) it <strong>the</strong>n follows<br />
q p = 1 ∫<br />
∫<br />
q (t) I in (t)dt = q (t) f (t)dt, (<strong>2.1</strong>.11)<br />
F in<br />
where<br />
f (t) ≡ I in (t)<br />
F in<br />
= P in (t)<br />
E p,in<br />
with<br />
∫<br />
f (t)dt = 1<br />
F in<br />
∫<br />
I in (t)dt =1. (<strong>2.1</strong>.12)<br />
We <strong>the</strong>n distinguish between two typical cases: a slow and a fast saturable absorber.<br />
<strong>2.1</strong>.4.<strong>2.1</strong> Slow saturable absorber<br />
In <strong>the</strong> case of a slow saturable absorber, we assume that <strong>the</strong> excitation pulse duration is much<br />
shorter than <strong>the</strong> recovery time of <strong>the</strong> absorber (i.e. τ p ≪ τ A ). Thus, we can neglect <strong>the</strong> recovery<br />
of <strong>the</strong> absorber during pulse excitation and (<strong>2.1</strong>.7) reduces to:<br />
dq (t)<br />
dt<br />
q (t) P (t)<br />
≈− . (<strong>2.1</strong>.13)<br />
E sat,A<br />
This differential equation can be solved and we obtain for <strong>the</strong> Self-Amplitude Modulation (SAM):<br />
⎡<br />
⎤<br />
q (t) =q 0 exp ⎣−<br />
E ∫t<br />
p<br />
f (t ′ )dt ′ ⎦ . (<strong>2.1</strong>.14)<br />
E sat,A<br />
0<br />
Equation (<strong>2.1</strong>.11) <strong>the</strong>n determines <strong>the</strong> total absorber loss coefficient for a given incident pulse<br />
fluence F p,A :<br />
∫<br />
F<br />
(<br />
)<br />
sat,A<br />
q p (F p,A )= q (t) f (t)dt = q 0 1 − e −Fp,A/Fsat,A . (<strong>2.1</strong>.15)<br />
F p,A<br />
It is not surprising that q p does not depend on any specific pulse form because τ p ≪ τ A .Itis<br />
useful to introduce a saturation parameter S ≡ F p,A /F sat,A . For strong saturation (S >3), we<br />
have q p (F p,A ) ≈ q 0 /S (<strong>2.1</strong>.15) and <strong>the</strong> absorbed pulse fluence is about F sat,A ΔR.<br />
<strong>2.1</strong>.4.2.2 Fast saturable absorber<br />
In <strong>the</strong> case of a fast saturable absorber, <strong>the</strong> absorber recovery time is much faster than <strong>the</strong> pulse<br />
duration (i.e. τ p ≫ τ A ). Thus, we can assume that <strong>the</strong> absorption instantaneously follows <strong>the</strong><br />
absorption of a certain power P (t) and (<strong>2.1</strong>.7) reduces to<br />
0=− q (t) − q 0<br />
τ A<br />
−<br />
q (t) P (t)<br />
E sat,A<br />
. (<strong>2.1</strong>.16)<br />
Landolt-Börnstein<br />
New Series VIII/1B1