Section 1.7.2.2. Symmetry properties

Intrinsic permutation symmetry, as already discussed, imposes the condition that the nth order susceptibility be invariant under the permutations of the () pairs as a result of time invariance and causality. Furthermore, the overall permutation symmetry, i.e. the invariance over the permutations of the () and () pairs, may be valid when all the optical frequencies occuring in the susceptibility and combinations of these appearing in the denominators of quantum expressions are far removed from the transitions, making the medium transparent at these frequencies. This property is termed ABDP symmetry, from the initials of the authors of the pioneering article by Armstrong et al. (1962).

Let us consider as an application the quantum expression of the quadratic susceptibility (with damping factors neglected), the derivation of which being beyond the scope of this chapter, but which can be found in nonlinear optics treatises dealing with microscopic interactions, such as in Boyd (1992):where N is the number of microscopic units (e.g. molecules in the case of organic crystals) per unit volume, a, b and c are the eigen states of the system, Ω_ba and Ω_ca are transition energies, is the μ component of the transition dipole connecting states a and b, and is the population of level a as given by the corresponding diagonal term of the density operator. S_T is the summation operator over the six permutations of the (), (), (). Provided all frequencies at the denominator are much smaller than the transition frequencies Ω_ba and Ω_ca, the optical frequencies , , can be permuted without significant variation of the susceptibility. It follows correspondingly that the susceptibility is invariant with respect to the permutation of Cartesian indices appearing only in the numerator of (1.7.2.36), regardless of frequency. This property, which can be generalized to higher-order susceptibilities, is known as Kleinman symmetry. Its validity can help reduce the number of non-vanishing terms in the susceptibility, as will be shown later.

An important consequence of overall permutation symmetry is the Manley–Rowe power relations, which account for energy exchange between electromagnetic waves in a purely reactive (e.g. non-dissipative) medium. Calling W_i the power input at frequency ω_i into a unit volume of a dielectric polarizable medium,where the averaging is performed over a cycle andThe following expressions can be derived straightforwardly:Introducing the quadratic induced polarization P⁽²⁾, Manley–Rowe relations for sum-frequency generation stateSince , (1.7.2.40) leads to an energy conservation condition, namely , which expresses that the power generated at ω₃ is equal to the sum of the powers lost at ω₁ and ω₂.

A quantum mechanical interpretation of these expressions in terms of photon fusion or splitting can be given, remembering that is precisely the number of photons generated or annihilated per unit volume in unit time in the course of the nonlinear interactions.

The tensors or are invariant with respect to (α, β) permutation as a consequence of the intrinsic permutation symmetry. Independently, it is not possible to distinguish the coefficients and by SHG experiments, even if the two fundamental waves have different directions of polarization.

Therefore, these third-rank tensors can be represented in contracted form as matrices and , where the suffix m runs over the six possible (α, β) Cartesian index pairs according to the classical convention of contraction:The 27 elements of are then reduced to 18 in the contracted tensor notation (see Section 1.1.4.10 ).

For example, (1.7.2.35) can be writtenThe same considerations can be applied to THG. Then the 81 elements of can be reduced to 30 in the contracted tensor notation with the following contraction convention:If Kleinman symmetry holds, the contracted tensor can be further extended beyond SHG and THG to any other processes where all the frequencies are different.