5.4 Nonlinearities
161
5.4 Nonlinearities
While many standard characterisation efforts rely on linear spectroscopy or imaging
modes, nonlinear optical techniques can provide a unique access to nonlinear material properties else inaccessible to the basic measurement techniques. How some of
these nonlinear methods have been employed in this work is briefly outlined in the
following with corresponding examples. For introductory purposes, the basic principle behind nonlinear optics based on the nonlinear polarisation is highlighted, the
details of which will be beyond the scope of this work.
It shall be recalled that the complex dielectric function and the susceptibility
χ(ω) are linked by
= 1 + χ(ω),
(5.5)
which sets in relation a polarisable medium’s dielectric displacement D = 0
and an applied electric field E. The light-field induced polarisation P i for a field
with electric field amplitude E i and oscillation frequency ω i can be expressed, with
χ(ω i , E i ) = (ω i , E i ) − 1, as a power series of the incident field amplitudes (see
[69] and references therein):
P i = 0
⎡
⎣
j
χ
(1)
i j E j +
j,k
χ
(2)
i jk E j E k +
j,k,l
χ
(3)
i jkl E j E k E l + ...
⎤
⎦ ,
(5.6)
whereas oscillating electric fields can be represented as E i = eE i exp (−iω i t) + c.c.
(c.c. denotes the complex conjugated part,
13 and e the unit vector of the field’s direction). While linear optics neglects the nonlinear terms due to their often negligible
influence for small fields (thus, linear optics, i.e. in the regime of the harmonic
oscillator), for strong fields (or high intensities), the higher-order terms can play an
important role. These higher-order terms matter for instance for frequency conversion
(e.g., χ
(2) processes), Kerr lensing or two-photon absorption (e.g., χ
(3) processes).
For equal frequencies ω i = ω, a similar expansion is given for the complex refractive
index.
˜
n(ω, I ) = ˜
n 0 (ω) + ˜
n 2 (ω)I + ...,
(5.7)
which is the basis for nonlinear lensing and absorption effects (χ
(3) nonlinearities) in
the regime of high intensities (I = |E|
2 ). Note that the first term is the linear optics
term, which is not intensity dependent. The effect of the second term is for instance
evidenced in Z-scan experiments described below [108–110] (see e.g., Fig. 5.14).
Concerning frequencies, without going into many details, one can discriminate
two basic situations. First, the case with equal frequencies, which leads to a P
(2)
(t) =
0 χ
(2) E
2
(t) that results in an expression such as
13 That is with complex conjugated amplitudes E ∗
i .
161
5.4 Nonlinearities
While many standard characterisation efforts rely on linear spectroscopy or imaging
modes, nonlinear optical techniques can provide a unique access to nonlinear material properties else inaccessible to the basic measurement techniques. How some of
these nonlinear methods have been employed in this work is briefly outlined in the
following with corresponding examples. For introductory purposes, the basic principle behind nonlinear optics based on the nonlinear polarisation is highlighted, the
details of which will be beyond the scope of this work.
It shall be recalled that the complex dielectric function and the susceptibility
χ(ω) are linked by
= 1 + χ(ω),
(5.5)
which sets in relation a polarisable medium’s dielectric displacement D = 0
and an applied electric field E. The light-field induced polarisation P i for a field
with electric field amplitude E i and oscillation frequency ω i can be expressed, with
χ(ω i , E i ) = (ω i , E i ) − 1, as a power series of the incident field amplitudes (see
[69] and references therein):
P i = 0
⎡
⎣
j
χ
(1)
i j E j +
j,k
χ
(2)
i jk E j E k +
j,k,l
χ
(3)
i jkl E j E k E l + ...
⎤
⎦ ,
(5.6)
whereas oscillating electric fields can be represented as E i = eE i exp (−iω i t) + c.c.
(c.c. denotes the complex conjugated part,
13 and e the unit vector of the field’s direction). While linear optics neglects the nonlinear terms due to their often negligible
influence for small fields (thus, linear optics, i.e. in the regime of the harmonic
oscillator), for strong fields (or high intensities), the higher-order terms can play an
important role. These higher-order terms matter for instance for frequency conversion
(e.g., χ
(2) processes), Kerr lensing or two-photon absorption (e.g., χ
(3) processes).
For equal frequencies ω i = ω, a similar expansion is given for the complex refractive
index.
˜
n(ω, I ) = ˜
n 0 (ω) + ˜
n 2 (ω)I + ...,
(5.7)
which is the basis for nonlinear lensing and absorption effects (χ
(3) nonlinearities) in
the regime of high intensities (I = |E|
2 ). Note that the first term is the linear optics
term, which is not intensity dependent. The effect of the second term is for instance
evidenced in Z-scan experiments described below [108–110] (see e.g., Fig. 5.14).
Concerning frequencies, without going into many details, one can discriminate
two basic situations. First, the case with equal frequencies, which leads to a P
(2)
(t) =
0 χ
(2) E
2
(t) that results in an expression such as
13 That is with complex conjugated amplitudes E ∗
i .