# Fiber Nonlinearities

As seen in the nonlinear optics section in particular, intense light propagating in a dielectric medium will promote a nonlinear material response from the atoms in the material. As we saw in the CEO model (in the “Classical Optics” and “Lasers” sections of this website in case you did not in fact see this), the atom can be represented by an electron on a spring and the application of a linear electric field promotes a linear i.e. harmonic oscillatory motion. The effect of an intense field is to promote anharmonic motion, or a nonlinear response. This is mathematically stated through the macroscopic polarization term:

$$P = \epsilon_0[\chi^{(1)}\cdot{E}+\chi^{(2)}\cdot(E\cdot{E})+\chi^{(3)}\cdot(E\cdot{E}\cdot{E})+ …]$$

$\chi^{(j)}$ is a tensor of rank $j+1$, as we had in the “Light Propagation in Crystals”, or LPIC because acronyms are the brain juice of science, section of this website. This to account for the potential effects of birefringence in the fiber. The first order susceptibility $\chi^{(1)}$ is the same susceptibility that we then turned into a dielectric tensor in the LPIC section, so it should hopefully be familiar to you already.(If not, as a shameless self-promotion, check out the LPIC section!) Then, $\chi^{(2)}$ is responsible for second-harmonic generation and sum-frequency generation, as is discussed in the “Nonlinear Optics” section in much further detail. Since this effect is only nonzero for media that lack an inversion symmetry, the second-order nonlinear effect is essentially negligible for silica fibers that have a symmetric $SiO_2$ constituency. There can be some second order nonlinear effects due to electric quadrupoles and magnetic-dipoles (the other material responses briefly mentioned in the “Lasers” section), but the magnitude of these is fairly small. The rest of this subsection, and the majority of the rest of this section in general, thus devote attention almost entirely to the third order nonlinear response.

NONLINEAR REFRACTION

Without the $\chi^{(2)}$ contribution, the $\chi^{(3)}$ dominates and is responsible for effects such as third-harmonic generation, four-wave mixing, and nonlinear refraction. However, since both four-wave mixing and third-harmonic generation require considerable phase-matching efforts, the main effect of note arising from $\chi^{(3)}$ is nonlinear refraction. Nonlinear refraction is basically an intensity dependence of the refractive index such that the refractive index in the presence of a sufficiently intense beam becomes:

$$n(\omega,|E|^2) = n(\omega) + n_2|E|^2$$

where $n(\omega)$ is the linear refractive index we all know and love that arises from the Sellmeier equation, $|E|^2$ is the optical intensity in the fiber, and $n_2$ is the nonlinear-coefficient that arises from $\chi^{(3)}$ as:

$$n_2 = \frac{3}{8n}Re(\chi_{xxxx}^{(3)})$$

where “Re” stands for the real part of the susceptibility (as the imaginary part is what governs atomic absorption/gain), and the tensor of $\chi_{xxxx}^{(3)}$ can affect the polarization of optical beams through nonlinear birefringence.

This intensity dependance of the refractive index manifests itself in two main phenomena of study: self-phase modulation and cross-phase modulation. Self-phase modulation, or SPM (again, brain juice), is phase shift experienced by a pulse propagating in a fiber due to the pulse itself. The total phase shift induced by self-phase modulation can be given as:

$$\phi = nk_0L = (n+n_2|E|^2)k_0L$$

where $k_0 = 2\pi/\lambda$ and $L$ is the fiber length. SPM will be discussed in further detail in subsections 4 and 5 of this “Fiber Optics” section.

The other effect, cross-phase modulation, or XPM, is the nonlinear phase shift induced by a copropagating field at a different wavelength. We can understand this by noting that if the total electric field for two co-propagating fields is given as:

$$E = \frac{1}{2}\hat{x}[E_1e^{-i\omega_1{t}}+E_2e^{-i\omega_2{t}} + c.c.]$$

then the total nonlinear phase shift of the field at $\omega_1$ is given as:

$$\phi_{NL} = n_2k_0L(|E_1|^2+2|E_2|^2)$$

STIMULATED INELASTIC SCATTERING

The nonlinearities arising from the nonlinearity of the susceptibility are considered elastic in the traditional mechanical sense of the word that no energy is transferred between the medium and the field. There is another field of nonlinear optics that arises due to inelastic interactions, or stimulated inelastic scattering, in which the field transfers part of its energy to the nonlinear medium. The two main effects occuring in optical fibers are stimulated Raman scattering (SRS) and stimulated Brillouin scattering (SBS). The main difference between these two types of scattering are that optical phonons participate in SRS while it is acoustic phonons that participate in SBS. The physical mechanisms that create SRS and SBS are discussed in further detail later in this section as well as in even greater detail in the “Nonlinear Optics” section. The reason these effects are considered nonlinear is because they require a certain intensity of light in order to occur, and the presence of that intensity of light promotes a nonlinear response from the material. Note that they do not, however, result from the $\chi$ nonlinearity.

IMPORTANCE OF NONLINEAR EFFECTS

The nonlinear index coefficient of fibers is in the range of 2.2-2.4$\times 10^{-20}~m^2/W$, depending on the composition of the fiber and whether or not the polarization is preserved. In comparison to other gain media, this value of the nonlinear coefficient is actually a couple of orders of magnitude lower. In a similar manner, the measurements of Raman and Brillouin- gain coefficients in silica media are also two orders of magnitude lower than other gain media. The main difference between the nonlinearities in fibers compared to other gain media, though, is that in fibers high powers are confined to small areas i.e. on the order of $2-4~\mu{m}^2$ for single mode fibers. This then inherently increases the intensity of the light, thereby inducing nonlinear effects with much lower intensities than in other media.