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

126 2.1.8 Carrier envelope offset (CEO) [Ref. p. 134
2.1.8 Carrier envelope offset (CEO)
Progress in ultrashort pulse generation [99Ste] has reached a level where the slowly-varyingenvelope
approximation starts to fail. Pulse durations of Ti:sapphire laser oscillators have reached
around 5 fs which is so short that only about two optical cycles of the underlying electric field
fit into the full-width half maximum of the pulse envelope. For such short pulses the maximum
electric-field strength depends strongly on the exact position of the electric field with regards to the
pulse envelope, i.e. the Carrier Envelope Offset (CEO) [99Tel]. In passively mode-locked lasers this
carrier envelope offset is a freely varying parameter because the steady-state boundary condition
only requires that the pulse envelope is the same after one round trip (Fig. 2.1.19). Therefore the
CEO phase may exhibit large fluctuations, even when all other laser parameters are stabilized. We
have discussed the physical origin of these fluctuations before [02Hel2, 02Hel1]. Because nonlinear
laser–matter interaction depends strongly on the strength of the electric field, this CEO fluctuations
cause strong signal fluctuations in nonlinear experiments such as high-harmonic generation
[00Bra], attosecond pulse generation [01Dre], photoelectron emission [01Pau1] etc.
T R
Electric field E
Time t
Fig. 2.1.19. Electric field E(t) of three subsequent
pulses from a mode-locked laser with a pulse repetition
rate of T R =1/f rep. The envelope ±A(t) is
shown as dashed lines. The electric-field patterns of
the pulses experience large pulse-to-pulse fluctuations.
Different techniques have been proposed to stabilize these CEO fluctuations in the time domain
[96Xu1] and in the frequency domain [99Tel]. The frequency-domain technique is based on the
frequency comb generated from mode-locked lasers (Sect. 2.1.2.2) and is much more sensitive and
is the technique that is being used today. Optical frequency comb techniques recently received a
lot of attention and were also an important part of the 2005 Nobel Prize in Physics for Theodor W.
Hänsch and John L. Hall in the area of laser-based precision spectroscopy [05Nob].
In the following we follow the notation and derivation of Helbing et al. [03Hel]. A very stable
frequency comb is generated with a mode-locked laser which follows directly from the time domain:
E train (t) =A (t)exp(iω c t +iϕ 0 (t)) ⊗
+∞∑
m=−∞
δ (t − mT R ) , (2.1.96)
where E train (t) is the electric field of a pulse train (Fig. 2.1.19), ω c is the (angular) carrier frequency
whichisthecenterofgravityofthemode-lockedspectrum,A(t) the pulse envelope, and ϕ 0 (t) the
absolute phase of the pulse. The change of the carrier-envelope offset phase ϕ 0 in (2.1.96) per round
trip is given by:
Δϕ 0 =Δϕ GPO mod 2π . (2.1.97)
The Group-Phase Offset (GPO) only arises from the first-order dispersion of the material along
the beam path:
Δϕ GPO = −ω
∫ L
0
( 1
− 1 ) ∫ L
ω 2
dx =
v g v p 0 c
dn(ω, x)
dx , (2.1.98)
dω
where v g is the group velocity and v p is the phase velocity (Table 2.1.7).
Landolt-Börnstein
New Series VIII/1B1