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Nonlinear optics gives rise to a host of optical phenomena:
Frequency mixing processes
In these processes, the medium has a linear response to the light, but the properties of the medium are affected by other causes;
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Frequency mixing processes
Theory
A number of nonlinear optical phenomena can be described as frequency-mixing processes. In general, the dielectric polarization at time in a medium can be written as a power series in the electrical field:
Here, the coefficients are the -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 a broken inversion symmetry. If we write
where c.c. denotes the complex conjugate, the second-order term in will read
Note: in this description, is a scalar. In reality, 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.
Phase matching
The above ignores the position dependence of the electrical fields. In a typical situation, the electrical fields are traveling waves with an electric field
at position , with the wave vector , where is the velocity of light and the index of refraction of the medium at angular frequency . Thus, the second-order polarization angular frequency is
At each position , the oscillating second-order polarization radiates at angular frequency and a corresponding wave vector . Constructive interference, and therefore a high intensity field, will occur only if
The above equation is known as the phase matching condition. Typically, three-wave mixing is done in a birefringent crystalline material (i.e., the index of refraction depends on the polarization and direction of the light that passes through), where the polarizations of the fields and the orientation of the crystal are chosen such that the phase-matching condition is fulfilled.