162
5 Optical Measurement Techniques
P
(2)
(t) =
1
2
0 χ
(2) E
2
0 (1 − cos (2ωt)) ,
(5.8)
which describes second-harmonic generation, i.e. frequency conversion from ω to
2ω due to the fast oscillating term of the polarisation. Second, one can describe
a more general case with different frequencies, e.g., ω 1 and ω 2 (and their fields
E 1,2 (t) ∝ exp (iω 1,2 t)), for which the P
(2) contains more terms summed up briefly
as the SHG terms for each frequency, an optical rectification term, a sum-frequency
term
P
(2)
(t) ∝ χ
(2) E 1 E 2 exp ((ω 1 + ω 2 )t),
(5.9)
and a difference-frequency term
P
(2)
(t) ∝ χ
(2) E 1 E
∗
2 exp ((ω 1 − ω 2 )t).
(5.10)
The latter term is for instance used for THz generation in periodically- or
aperiodically-poled lithium niobate (nonlinear) crystals placed inside a dual-wavelength laser’s resonator, such as in a VECSEL [111–114] (see e.g., Fig. 5.15).
Note that for efficient frequency conversion, nonlinear crystals with a high nonlinear susceptibility are employed and that phase-matching conditions should be
fulfilled for effective outcoupling, otherwise the energy of the propagating signal
is transferred back to the fundamental modes. Typically, for SHG such as in intensity autocorrelators, or phase-resolving autocorrelation techniques such as FROGs
or GRENOUILLEs [36], the crystal thickness is optimised with regard to signal
strengths, phase-matching and dispersion.
5.4.1 Z-Scans
One effective tool to probe intensity-dependent nonlinear lensing, as well as nonlinear
absorption, in a medium is given by the Z-scan technique. Kerr-lensing, which is a
nonlinear lensing phenomenon, is for instance the pulse forming mechanism in modelocked Ti:sapphire lasers (cf. [115, 116]). Z-scan measurements allow one to directly
measure the nonlinear refractive index changes associated with the Kerr effect.
While the Z-scan technique [117] is the most prominent one due to its high
sensitivity and simplicity, several measurement schemes have been developed in the
past for the characterisation of the nonlinear refractive index [118–120]. With the
rise of novel material classes like graphene and other 2D semiconductors, which
often exhibit a very strong refractive nonlinearity [121–123], the Z-scan technique
continues to be of experimental value.
The effective nonlinear refractive index n 2 of a medium is defined by the relationship
5 Optical Measurement Techniques
P
(2)
(t) =
1
2
0 χ
(2) E
2
0 (1 − cos (2ωt)) ,
(5.8)
which describes second-harmonic generation, i.e. frequency conversion from ω to
2ω due to the fast oscillating term of the polarisation. Second, one can describe
a more general case with different frequencies, e.g., ω 1 and ω 2 (and their fields
E 1,2 (t) ∝ exp (iω 1,2 t)), for which the P
(2) contains more terms summed up briefly
as the SHG terms for each frequency, an optical rectification term, a sum-frequency
term
P
(2)
(t) ∝ χ
(2) E 1 E 2 exp ((ω 1 + ω 2 )t),
(5.9)
and a difference-frequency term
P
(2)
(t) ∝ χ
(2) E 1 E
∗
2 exp ((ω 1 − ω 2 )t).
(5.10)
The latter term is for instance used for THz generation in periodically- or
aperiodically-poled lithium niobate (nonlinear) crystals placed inside a dual-wavelength laser’s resonator, such as in a VECSEL [111–114] (see e.g., Fig. 5.15).
Note that for efficient frequency conversion, nonlinear crystals with a high nonlinear susceptibility are employed and that phase-matching conditions should be
fulfilled for effective outcoupling, otherwise the energy of the propagating signal
is transferred back to the fundamental modes. Typically, for SHG such as in intensity autocorrelators, or phase-resolving autocorrelation techniques such as FROGs
or GRENOUILLEs [36], the crystal thickness is optimised with regard to signal
strengths, phase-matching and dispersion.
5.4.1 Z-Scans
One effective tool to probe intensity-dependent nonlinear lensing, as well as nonlinear
absorption, in a medium is given by the Z-scan technique. Kerr-lensing, which is a
nonlinear lensing phenomenon, is for instance the pulse forming mechanism in modelocked Ti:sapphire lasers (cf. [115, 116]). Z-scan measurements allow one to directly
measure the nonlinear refractive index changes associated with the Kerr effect.
While the Z-scan technique [117] is the most prominent one due to its high
sensitivity and simplicity, several measurement schemes have been developed in the
past for the characterisation of the nonlinear refractive index [118–120]. With the
rise of novel material classes like graphene and other 2D semiconductors, which
often exhibit a very strong refractive nonlinearity [121–123], the Z-scan technique
continues to be of experimental value.
The effective nonlinear refractive index n 2 of a medium is defined by the relationship