5.4 Nonlinearities
163
n(I 0 ) = n 0 + n 2 I 0
(5.11)
(an approximation neglecting higher-order terms). n(I 0 ) is the total refractive index,
n 0 the intensity independent refractive index and I 0 the optical intensity, e.g., at
the peak power of an ultrashort pulse. Such definition of n 2 comprises both noninstantaneous contributions, i.e. those induced by free-carrier nonlinearities (FCN),
and instantaneous contributions, i.e. from the bound-electron Kerr effect (BEKE). In
principle, the relative contributions from FCN and BEKE to the nonlinear refractive
index could be unraveled with the help of time-resolved investigations.
Similarly, an intensity-dependent absorption term can be defined (in similar
approximation neglecting higher-order terms) with a nonlinear absorption coefficient β by the relationship
α(I 0 ) = α 0 + β I 0 ,
(5.12)
with α 0 the linear absorption coefficient. Thus, α(I 0 ) is the total absorption coefficient. In contrast to the effective nonlinear refractive index n 2 , β only comprises
ultrafast χ
(3) effects. This is understandable by the fact that carrier-related absorption
would lead to saturation and thus to a more complex relation on incident intensity.
To perform the Z-scan measurement, the sample is step-wise moved finely in the
beam direction through the focal spot of the focused 100-fs-pulsed probe beam (see
Fig. 5.14a). The simultaneously (behind a beam splitter) recorded sample-transmitted
as well as sample-reflected signal on an open-aperture and closed-aperture power
meter detector are then used to obtain a nonlinear absorption trace (c) as well as an
absorption-corrected nonlinear lensing trace (b), respectively, as shown for example
in Fig. 5.14. A typical setup is employed and described for instance in [108].
One practical example of the determination of the effective n 2 for a sample is
given by a study of a VECSEL chip motivated by the observation of self-modelocking behaviour reminiscent of that in a Ti:sapphire laser [108, 109]. This clearly
indicated nonlinear lensing as a possible driving mechanism for mode-locking. In
a follow-up work to the one without charge-carrier excitation [108], even close-tooperation conditions with an optically-pumped chip were investigated [109]. The
spectroscopic part enters the investigation, when the dependence on the probe wavelength is analysed (in both works). A microcavity effect can for instance drastically
shape the wavelength-dependent nonlinear response of such a semiconductor disk
laser, as discussed in [108]. Moreover, Z-scans readily deliver information about the
nonlinear absorption coefficient β and reveal to which extent intensity-dependent
absorption can act as a possible loss mechanism.
In fact, such technique can be also used to study 2D materials. Endeavours to
achieve clear results by the author’s team have not been completed yet and the
limiting factor remain the size of the specimen and insufficient signal-to-noise ratio
when probing sub-nm-thin microscale flakes with a strong optical focus (particularly
for displacements away from the focal spot). Thus, further improvements on the side
of the sample production and experiment are generally required for such studies. On
the other hand, bulk crystals such as a single-crystalline perovskite material were
163
n(I 0 ) = n 0 + n 2 I 0
(5.11)
(an approximation neglecting higher-order terms). n(I 0 ) is the total refractive index,
n 0 the intensity independent refractive index and I 0 the optical intensity, e.g., at
the peak power of an ultrashort pulse. Such definition of n 2 comprises both noninstantaneous contributions, i.e. those induced by free-carrier nonlinearities (FCN),
and instantaneous contributions, i.e. from the bound-electron Kerr effect (BEKE). In
principle, the relative contributions from FCN and BEKE to the nonlinear refractive
index could be unraveled with the help of time-resolved investigations.
Similarly, an intensity-dependent absorption term can be defined (in similar
approximation neglecting higher-order terms) with a nonlinear absorption coefficient β by the relationship
α(I 0 ) = α 0 + β I 0 ,
(5.12)
with α 0 the linear absorption coefficient. Thus, α(I 0 ) is the total absorption coefficient. In contrast to the effective nonlinear refractive index n 2 , β only comprises
ultrafast χ
(3) effects. This is understandable by the fact that carrier-related absorption
would lead to saturation and thus to a more complex relation on incident intensity.
To perform the Z-scan measurement, the sample is step-wise moved finely in the
beam direction through the focal spot of the focused 100-fs-pulsed probe beam (see
Fig. 5.14a). The simultaneously (behind a beam splitter) recorded sample-transmitted
as well as sample-reflected signal on an open-aperture and closed-aperture power
meter detector are then used to obtain a nonlinear absorption trace (c) as well as an
absorption-corrected nonlinear lensing trace (b), respectively, as shown for example
in Fig. 5.14. A typical setup is employed and described for instance in [108].
One practical example of the determination of the effective n 2 for a sample is
given by a study of a VECSEL chip motivated by the observation of self-modelocking behaviour reminiscent of that in a Ti:sapphire laser [108, 109]. This clearly
indicated nonlinear lensing as a possible driving mechanism for mode-locking. In
a follow-up work to the one without charge-carrier excitation [108], even close-tooperation conditions with an optically-pumped chip were investigated [109]. The
spectroscopic part enters the investigation, when the dependence on the probe wavelength is analysed (in both works). A microcavity effect can for instance drastically
shape the wavelength-dependent nonlinear response of such a semiconductor disk
laser, as discussed in [108]. Moreover, Z-scans readily deliver information about the
nonlinear absorption coefficient β and reveal to which extent intensity-dependent
absorption can act as a possible loss mechanism.
In fact, such technique can be also used to study 2D materials. Endeavours to
achieve clear results by the author’s team have not been completed yet and the
limiting factor remain the size of the specimen and insufficient signal-to-noise ratio
when probing sub-nm-thin microscale flakes with a strong optical focus (particularly
for displacements away from the focal spot). Thus, further improvements on the side
of the sample production and experiment are generally required for such studies. On
the other hand, bulk crystals such as a single-crystalline perovskite material were