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

Ref. p. 134] 2.1 Ultrafast solid-state lasers 93
Table 2.1.6. Sellmeier equations for different materials. The wavelength λ is given in units of μm.
Material Defining Sellmeier equation Constants
Fused quartz n 2 =1+ Aλ2 + Bλ2 + Cλ2
λ 2 − λ 2 1 λ 2 − λ 2 2 λ 2 − λ 2 3
A =0.6961663
λ 1 =0.0684043
B =0.4079426
λ 2 =0.1162414
C =0.8974794
λ 3 =9.896161
SF10 glass
a 0 =2.8784725
n 2 = a 0 + a 1λ 2 + a2
λ + a3
2 λ + a4
4 λ + a5
a 1 = −0.010565453
6 λ 8 a 2 =3.327942 × 10 −2
a 3 =2.0551378 × 10 −3
a 4 = −1.1396226 × 10 −4
a 5 =1.6340021 × 10 −5
Sapphire n 2 =1+ a1λ2
+
a2λ2
+
λ 2 − b 1 λ 2 − b 2
a 1 =1.023798
a 2 =1.058264
a3λ2
a 3 =5.280792
λ 2 − b 3
b 1 =0.00377588
b 2 =0.0122544
b 3 = 321.3616
Here, the derivatives are evaluated at ω 0 . ∂φ/∂ω is the group delay T g , ∂ 2 φ/∂ω 2 the Group Delay
Dispersion (GDD), ∂ 3 φ/∂ω 3 the Third-Order Dispersion (TOD). The GDD describes a linear
frequency dependence of the group delay and thus tends to separate the frequency components of
a pulse: For positive GDD, e.g., the components with higher frequencies are delayed with respect
to those with lower frequencies, which results in a positive “chirp” (“up-chirp”) of the pulse.
Higher orders of dispersion generate more complicated distortions. Material dispersion is normally
described with Sellmeier equations for the refractive index as a function of the wavelength, i.e.
n (λ). With the Sellmeier equations (Table 2.1.6) all the necessary dispersive quantities can be
calculated (Table 2.1.7). An example is given in Table 2.1.8.
The broader the bandwidth of the pulse (i.e., the shorter the pulse duration), the more terms
of this expansion are significant. GDD which acts on an initially unchirped Gaussian pulse with
Full-Width-at-Half-Maximum (FWHM) pulse duration τ 0 , increases the pulse duration according
to [86Sie]
√
( )
τ p (z)
4ln2d2 φ/d ω
= 1+
2 2
, (2.1.20)
τ 0
τ 2 0
where it is assumed that the incident pulse is transform-limited, i.e. the time–bandwidth product of
the Gaussian pulse is τ 0 Δ ν p =0.4413, where Δ ν p is the Full-Width-at-Half-Maximum (FWHM)
spectral width of the pulse intensity (Fig. 2.1.15). Only second-order dispersion (i.e. ∂ 2 φ/∂ ω 2 )
and higher orders are broadening the pulse. The first-order dispersion gives the group delay, i.e.
the delay of the peak of the pulse envelope. It is apparent that the effect of GDD becomes strong
if GDD >τ 2 0 . Similarly, TOD becomes important if TOD >τ 3 0 . It is important to note that
for dispersive pulse broadening (which is in the linear pulse propagation regime) the spectrum
of the pulse remains unchanged, only the spectral content of the pulse is redistributed in time.
With positive dispersion the long-wavelength part of the spectrum is in the leading edge of the
pulse and the short-wavelength part in the trailing edge of the pulse, i.e. “red is faster than blue”
(Fig. 2.1.15). In the regime of strong pulse broadening, i.e. d 2 φ/d ω 2 ≫ τ 2 0 , we can reduce (2.1.20)
Landolt-Börnstein
New Series VIII/1B1