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

82 2.1.4 Loss modulation [Ref. p. 134
R tot = F ∫
out Iout (t)dt
= ∫
F in Iin (t)dt =1− 2 ∫
F in
q (t) I in (t)dt. (2.1.9)
This determines the total absorber loss coefficient q p , which results from the fact that part of the
excitation pulse needs to be absorbed to saturate the absorber:
R tot =e −2qp ≈ 1 − 2q p . (2.1.10)
From (2.1.9) and (2.1.10) it then follows
q p = 1 ∫
∫
q (t) I in (t)dt = q (t) f (t)dt, (2.1.11)
F in
where
f (t) ≡ I in (t)
F in
= P in (t)
E p,in
with
∫
f (t)dt = 1
F in
∫
I in (t)dt =1. (2.1.12)
We then distinguish between two typical cases: a slow and a fast saturable absorber.
2.1.4.2.1 Slow saturable absorber
In the case of a slow saturable absorber, we assume that the excitation pulse duration is much
shorter than the recovery time of the absorber (i.e. τ p ≪ τ A ). Thus, we can neglect the recovery
of the absorber during pulse excitation and (2.1.7) reduces to:
dq (t)
dt
q (t) P (t)
≈− . (2.1.13)
E sat,A
This differential equation can be solved and we obtain for the Self-Amplitude Modulation (SAM):
⎡
⎤
q (t) =q 0 exp ⎣−
E ∫t
p
f (t ′ )dt ′ ⎦ . (2.1.14)
E sat,A
0
Equation (2.1.11) then determines the total absorber loss coefficient for a given incident pulse
fluence F p,A :
∫
F
(
)
sat,A
q p (F p,A )= q (t) f (t)dt = q 0 1 − e −Fp,A/Fsat,A . (2.1.15)
F p,A
It is not surprising that q p does not depend on any specific pulse form because τ p ≪ τ A .Itis
useful to introduce a saturation parameter S ≡ F p,A /F sat,A . For strong saturation (S >3), we
have q p (F p,A ) ≈ q 0 /S (2.1.15) and the absorbed pulse fluence is about F sat,A ΔR.
2.1.4.2.2 Fast saturable absorber
In the case of a fast saturable absorber, the absorber recovery time is much faster than the pulse
duration (i.e. τ p ≫ τ A ). Thus, we can assume that the absorption instantaneously follows the
absorption of a certain power P (t) and (2.1.7) reduces to
0=− q (t) − q 0
τ A
−
q (t) P (t)
E sat,A
. (2.1.16)
Landolt-Börnstein
New Series VIII/1B1