Second-Order Nonlinearities

The second-order susceptibility χ⁽²⁾ is responsible for second harmonic generation, sum and difference frequency generation, and optical parametric amplification.
The above-mentioned second-order nonlinear effects are produced by two waves, which interact to produce a third wave. Conservation of momentum and conservation of photon energy is always required in these processes. The optical fields of the three waves are coupled to one another through the second-order susceptibility. The coupling provides the mechanism for the exchange of energy among the interacting fields. The second-order nonlinear susceptibility in a Cartesian coordinate system can be written as

.......(1)

The tensor χ⁽²⁾_{i jk} in general has 27 independent coefficients. By taking into account symmetry conditions, the second-order susceptibility can be expressed by a two-dimensional 3×6 tensor, commonly known as the d-tensor.

Most nonlinear crystals are described by only a few d-coefficients. For example, the nonlinear crystal KDP has only three nonvanishing terms (d₁₄, d₂₅, d₃₆). Furthermore, for those crystals of main interest there is usually one predominant coefficient associated with a beam propagation direction which yields maximum harmonic power. In centrosymmetric crystals the χ⁽²⁾ tensor is actually zero; therefore second-order nonlinear processes are only possible in crystals that lack inversion symmetry. Using the d-coefficients, the nonlinear polarization is given by

......(2)

where d has the dimension of meter per volt.