Theory
A number of nonlinear optical phenomena can be described as frequency-mixing processes. If the induced dipole moments of the material respond instantaneously to an applied electric field, the dielectric polarization (dipole moment per unit volume) P(t) at time t in a medium can be written as a power series in the electrical field:
Here, the coefficients χ
are the n-th order susceptibilities of the medium. For any three-wave mixing process, the second-order term is crucial; it is only nonzero in media that have no inversion symmetry. If we write
where c.c. denotes the complex conjugate (E1 and E2 being the incident beams of interest), the second-order term in the above expansion will read
where the summation is over
The six combinations (nx,mx) correspond, respectively, to the second harmonic of E1, the second harmonic of E2, the optically rectified signals of E1 and E2, the difference frequency, and the sum frequency. A medium that is thus pumped by the fields E1 and E2 will radiate a field E3 with an angular frequency ω3 = m1ω1 + m2ω2.
Note: in this description, χ(2) is a scalar. In reality, χ(2) is a tensor whose components depend on the combination of frequencies.
Parametric generation and amplification is a variation of difference frequency generation, where the lower-frequency one of the two generating fields is much weaker (parametric amplification) or completely absent (parametric generation). In the latter case, the fundamental quantum-mechanical uncertainty in the electric field initiates the process. ^
:wtf: ^
^and that is how I feel about this^