Week 8: Nonlinearity

Welcome back to my blog! Previously, I used S4 to replicate a photonic slab structure with relative success. Today, I’ll return to the idea of nonlinear optics

Introducing Nonlinearity

Consider your everyday atomic structure. After applying an electric field, there’ll be an induced polarization, quantified by the “polarization vector” P. Normally, this quantity is directly proportional or linear to the magnitude of applied electric field. A stronger electric field will induce a stronger dipole, which is proportional to the polarization vector. In nonlinear optics, this gives the constitutive relation of:

The “exponents” of the chi terms actually denote an index, not an exponent. These are the susceptibility tensors. Linear optics considers only the chi (1) term; all higher order terms are eliminated. This is the region of geometric optics, ray tracing, and the sort. Nonlinear optics considers the higher order chi terms. For example, second harmonic generation occurs after including second order terms. The Kerr effect occurs after including the third order terms.

Kerr Effect

The Kerr effect states the index of refraction grows linearly with the intensity of light.

According to [1], third order effects can be modeled by an effective linear permittivity tensor. With this simplification, it suffices to run linear FMM. This will be exactly what I’ll try to do in the next blog. See you then!

[1] Subhajit Bej, Jani Tervo, Yuri P. Svirko, and Jari Turunen, “Modeling the optical Kerr effect in periodic structures by the linear Fourier modal method,” J. Opt. Soc. Am. B 31, 2371-2378 (2014)