It might be useful to give some properties of the nonlinear response function. For simplicity we demonstrate these properties by means of the second-order response function (2.19), which generates a second-order polarization. The properties shown below can be generalized to higher orders of nonlinearity without further ado [41].
We, first, consider symmetries, which are related either to the temporal or the frequency dispersion of the response function. To this end, we introduce our notation for the Fourier transform of an arbitrary time-dependent function f(t)
f˜(ω) = 1 2π
Z
dt f(t) eiωt, (2.20)
whereω denotes the angular frequency. Then the back transform of (2.20) is given by f(t) =
Z
dωf˜(ω) e−iωt. (2.21)
2Note, that we have used the general relation [43] D
A(D)(t)B(D)(t0)E
=D
AB(D)(t0−t)E
=D
A(D)(t−t0)BE to arrive at (2.17) and (2.19).
Due to the response function’s homogeneity in time (c.f. section 2.1) the temporal Fourier transform of the second-order polarization (2.18) yields
P˜i(2)(r, ω) =ε0
X
j,k
Z dω0
Z
dω00 χ(2)ijk
−ω;ω0, ω00
E˜j(r, ω0) ˜Ek(r, ω00) (2.22)
according to (2.20), where we have introduced the second-order susceptibility3 χ(2)ijk(−ω;ω0, ω00) =δ[ω−(ω0+ω00)]
Z dt0
Z
dt00 R(2)ijk(t0, t00) ei(ω0t0+ω00t00). (2.23) A material which exhibits a response relation of the form (2.22) is said to be dispersive in frequency. Since both the polarization and the electric field are measurable quantities they are real. Thus, their Fourier amplitudes obey P(r,−ω) = P∗(r, ω) and E(r,−ω) =E∗(r, ω), respectively, where we demand the frequencies to be real. Hence, the response function ought to be real as well yielding
χ(2)ijk(ω;−ω0,−ω00) =
χ(2)ijk(−ω;ω0, ω00) ∗
.
Of course, the reality of the response function is only one property. Additionally, the response function and its according susceptibility possess certain symmetries [41, 49, 50], some of which we present in the following.
Intrinsic permutation symmetry
One symmetry, which is valid independently of the specific material or the excitation, is the intrinsic permutation symmetry of the nonlinear response function
R(2)ijk(t−t0, t−t00) =R(2)ikj(t−t00, t−t0). (2.24) One can proof this property by assuming that one can split the response function into a sym-metric function Sijk(2) and an anti-symmetric functionA(2)ijk leading to the ansatz
R(2)ijk(t−t0, t−t00) =Sijk(2)(t−t0, t−t00) +A(2)ijk(t−t0, t−t00) (2.25) where we postulate
Sijk(2)(t−t0, t−t00) = Sikj(2)(t−t00, t−t0) (2.26) and
A(2)ijk(t−t0, t−t00) =−A(2)ikj(t−t00, t−t0). (2.27)
3The notation for the susceptibility is chosen according to P. Butcher and D. Cotter [49] and is commonly used in the field of nonlinear optics.
Properties of the second-order response function in dipole approximation 2.2
Hence, the polarization (2.18) can be separated into a symmetric and an anti-symmetric part Pi(2)(r, t) =Pi,s(2)(r, t) +Pi,as(2)(r, t). (2.28) Now we only consider the anti-symmetric part of the polarization, which according to (2.18) is given by
Pi,as(2)(r, t) =ε0
X3 j,k=1
Z dt0
Z
dt00 A(2)ijk(t−t0, t−t00)Ej(r, t0)Ek(r, t00)
=ε0 2
X3 j,k=1
Z dt0
Z dt00
h
A(2)ijk(t−t0, t−t00)Ej(r, t0)Ek(r, t00)
+A(2)OKs(t−t0, t−t00)
| {z }
(2.27)
= −A(2)ijk(t−t00,t−t0)
Ek(r, t0)Ej(r, t00)i
=ε0 2
X3 j,k=1
Z dt0
Z
dt00 A(2)ijk(t−t0, t−t00) h
Ej(r, t0)Ek(r, t00)−Ek(r, t00)Ej(r, t0) i
= 0
Thus, the total polarization equals the the symmetric polarization Pi(2) =Pi,s(2) and it follows that
R(2)ijk(t−t0, t−t00) =Sijk(2)(t−t0, t−t00),
i.e. the second-order response function possesses intrinsic permutation symmetry. Accordingly, the susceptibility χ(2)ijk(−ω;ω0, ω00) is invariant under the pairwise permutation of (j, ω0) and (k, ω00) leading to 2! permutations.
Overall permutation symmetry
For a non-resonant excitation of a lossless nonlinear medium the second-order response function possesses overall permutation symmetry, meaning that all pairs (i, ω), (j, ω0) and (k, ω00) can be interchanged simultaneously, which leads to 3! permutations. Specifically, we have
χ(2)ijk(−ω;ω0, ω00) =χ(2)jki(ω0;−ω0, ω00)
=χ(2)kji(ω00;ω0,−ω).
(2.29) If we consider the intrinsic permutation symmetry in addition to the overall permutation metry (2.29) we obtain a total of 6 possible permutations. Note that overall permutation sym-metry only holds in the approximation of excitations that are far-off any energetic transition within the material, where the light-matter interaction is expressed by the dipole interaction (1.89). To proof the overall permutation symmetry for lossless nonlinear media with non-resonant excitations one needs to calculate the quantum-mechanical perturbative expression
for the second-order susceptibility including phenomenological damping. As the full derivation of the susceptibility goes beyond the scope of this chapter we would like to reference [49]. The reader should also take a look at section 5.1 to get an idea how such calculation is implemented for third-order susceptibilities.
Kleinman symmetry
Suppose the frequencies of the incoming fields are much lower than the smallest resonance frequency of the material. In this case the material responds almost instantaneously to the external field and the material can be regarded as lossless. Therefore, we assume that the susceptibility is nearly independent of the frequencies of the present field. As a consequence all Cartesian indicesi, j, k of the susceptibility can be permuted independently of the frequencies, i.e.
χ(2)ijk(−ω;ω0, ω00) =χ(2)jki(−ω;ω0, ω00) =χ(2)kij(−ω;ω0, ω00)
=χ(2)ikj(−ω;ω0, ω00) =χ(2)jik(−ω;ω0, ω00)
=χ(2)kji(−ω;ω0, ω00)
(2.30)
resulting in 3! permutations. This kind of symmetry is known as Kleinman symmetry [49, 50].
We will use this symmetry later to derive the nonlinear response of Raman-active materials (see section 5.1 and appendix C.1).
Spatial symmetries
So far, we have only considered symmetries with respect to the time dependence of the response function or the frequency dependence of the susceptibility. In addition, a medium can also possess spatial symmetries due to the underlying structure of the material, which will reduce the response/susceptibility tensor to a minimal set of non-zero elements [41, 50]. In the special case of the spatially independent second-order response function (2.19), one can even state that the second-order response will vanish in an infinitely stretched material: Due to the lack of spatial dependence, the response tensor shall be invariant under inversion
χ(2)(t−t0, t−t00) = Γχ(2)(t−t0, t−t00) (2.31) where we have the inversion operator
Γ =
−1 0 0
0 −1 0
0 0 −1
. (2.32)
Then each component of the response tensor is given by χ(2)ijk(t−t0, t−t00) =
X3 a,b,c=1
ΓiaΓjbΓkc χ(2)abc(t−t0, t−t00) =−χ(2)ijk(t−t0, t−t00). (2.33)
Nonlinear optical phenomena 2.3
This can only be true, if the overall order response tensor vanishes. In this case second-order nonlinear processes are suppressed. However, this only holds for bulk materials, where the response function either exhibits no spatial dependence at all or only depends on the absolute values |r−r0| and |r−r00|, i.e. centrosymmetric materials. For small particles, where the system’s surface plays a significant role, second-order nonlinear effects contribute significantly to the overall nonlinear response. Furthermore, in the case of a fluid, which – if unperturbed – can be seen as homogeneous and isotropic, the spatial symmetry of the material is heavily influenced by the external electromagnetic field[51]. Hence, the situation is not as simple as for a centrosymmetric material with a fixed crystalline structure. We will see in section 4.5 how the second-order signal looks like for a small hydrodynamic particle, where the system’s response is not only nonlinear but also nonlocal.