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

Ref. p. 134] 2.1 Ultrafast solid-state lasers 127
Intensity I
rep
CEO
0
Frequency
Fig. 2.1.20. Equidistant frequency comb of a
mode-locked laser. The comb lines are spaced
by the repetition rate f rep and exhibit a nonvanishing
offset frequency f CEO at zero frequency
unless the electric field pattern exactly reproduces
from pulse to pulse.
We define the (angular) carrier-envelope offset frequency as
2π f CEO = ω CEO =Δϕ 0 f rep = Δϕ 0
≡ Δϕ (
GPO
∂
mod 2π ≡ GPO)
T R T R
∂t ϕ mod 2π . (2.1.99)
Taking into account the varying carrier-envelope offset phase the electric field of the pulse train
becomes:
E train (t) =A (t)exp(iω c t +iω CEO t) ⊗
+∞∑
m=−∞
δ (t − mT R ) . (2.1.100)
The Fourier-transformation of (2.1.100) then results in the frequency comb as shown in Fig. 2.1.20:
Ẽ train (f) =Ã (f − f c) ·
+∞∑
m=−∞
δ (f − mf rep − f CEO ) . (2.1.101)
The whole equidistant frequency comb is shifted by f CEO due to the per round trip carrier-envelope
offset phase shift of Δϕ 0 . Therefore, a mode-locked laser generates an equidistant frequency comb
with a frequency step given by the pulse repetition frequency f rep which can be measured and
stabilized with very high precision [86Lin, 86Rod, 89Rod, 89Kel]. The uniformity of the modelocked
frequency comb has been demonstrated to a relative uncertainty below 10 −15 [99Ude]. The
timing jitter in the arrival time of the pulses (i.e. variations in f rep ) produces a “breathing” of
the fully equidistant frequency comb. The additional freedom of the frequency offset to DC of this
frequency comb is given by the CEO frequency f CEO (Fig. 2.1.20) and every “tick of the frequency
ruler” then can be described by [99Tel]
f m = mf rep + f CEO (2.1.102)
with m being an integer number. The timing jitter of the CEO results in a translation of the
full frequency comb. Note that the equidistant frequency-comb spacing is given by the round-trip
propagation time of the pulse envelope (i.e. by the group velocity and not the phase velocity). This
means that the axial modes of a mode-locked laser are not the same as the ones from a cw laser
for which the phase velocity determines the axial mode spacing.
Even though a measurement of the repetition rate is straightforward, it is virtually impossible to
access the CEO-frequency directly, as the laser spectrum contains no energy close to zero frequency.
One therefore has to use an indirect way to measure the second comb parameter. Depending on
the available optical bandwidth different techniques have been proposed by Telle et al. [99Tel] such
as f-to-2f, 2f-to-3f heterodyne techniques and others. Selecting some low-frequency portion of
the comb and frequency doubling it gives rise to the comb
Landolt-Börnstein
New Series VIII/1B1