# Fiber Nonlinearities

This is a continuation from the previous tutorial - ** Gaussian analysis of afocal lenses**.

### Overview

Highly focused coherent laser light, propagating with low loss through optical fiber over long distances (kilometers), is an ideal breeding ground for nonlinear interaction with the glass material.

Although nonlinear effects were found in early optical transmission work with analog signal delivery (CATV, etc.), much attention lately has been given to resolution of nonlinear problems in long-haul optical communications and high-power operation in specialty fibers.

In particular, new fiber types have been developed to overcome nonlinear impairments. As fiber design introduced dispersion-shifted fibers (DSFs) in the early 1990s, to overcome chromatic dispersion impairments, it was soon found that multiple lightwaves, with different wavelengths, were able to efficiently interact through a four-wave mixing (FWM) process since the coupling waves were well matched in phase and group velocity.

This led to the development of NZDFs that struck a balance between the high chromatic dispersion of standard single-mode fiber and the very low dispersion, at operating wavelengths, of DSFs. With the advent of high-power erbium-doped fiber amplifiers (EDFAs) and high-power laser diodes, many nonlinear issues arose because of the long distance between signal regeneration points and the multiple optical wavelengths that could simultaneously be used.

In particular, stimulated Brillouin scattering became apparent (at 5–10 dBm levels with laser line widths <5 MHz). This required new features in transmitters to broaden the effective source line width. Self- and cross-phase modulation issues were also noted.

Generally, these problems increased with small effective area fibers (such as those often used in specialty applications). In the late 1990s, Raman amplification received renewed attention because of potential noise improvements due to its distributed nature.

This amplifier was based on stimulated Raman scattering of a signal wavelength by a high-powered laser pump in a transmission fiber medium. Here, we outline some of the general principles of these interactions and provide sufficient references to start the interested reader on a course of further study.

### Background

The optical transmission nonlinearities occur as a result of interaction of the traveling lightwave with the doped glass dielectric medium. The wave equation for lightwave transmission in an optical fiber can be written as

\[\tag{2.37}\boldsymbol{\nabla}^2\pmb{E}-\frac{1}{c^2}\frac{\partial^2\pmb{E}}{\partial{t^2}}=\boldsymbol{\mu}_0\frac{\partial^2\pmb{P}}{\partial{t^2}},\text{ where }\pmb{P}\cong\boldsymbol{\epsilon}_0\left\{\boldsymbol{\chi}^{(1)}\pmb{E}+\boldsymbol{\chi}^{(3)}\vdots\pmb{EEE}\right\}\]

Here, the nonlinear polarization due to the dielectric is written in terms of the tensor third-order susceptibility, because the second-order effect is negligible due to molecular symmetry of silica.

Time retardation, which may be important in some nonlinear interactions, is not included. Because the nonlinearity in fibers is weak, a perturbation approximation is normally used in solving Eq. (2.37). That is, the full solution is approximated as a slight modification of the solution with \(\boldsymbol{\chi}^{(3)}=0\).

Changing to a time harmonic representation of the fields, an index of refraction can be related to the susceptibilities:

\[n=n_0+n_2|\pmb{E}|^2\]

where

\[\tag{2.38}n_0=1+\frac{1}{2}\text{Re}\left\{\boldsymbol{\chi}^{(1)}\right\}\quad\text{and}\quad{n_2}=\frac{3}{8n_0}\text{Re}\left\{\boldsymbol{\chi}^{(3)}_{xxxx}\right\}\]

Here, \(\boldsymbol{\chi}^{(3)}_{xxxx}\) is the component of the susceptibility tensor aligned with the wave polarization and \(\text{Re}\{z\}\) indicates the real part of \(z\). In proceeding to derive propagation equations from Eq. (2.37), it is found that the nonlinear effects are proportional to \(n_2/A_\text{eff}\).

Because nonlinear effects depend on the lightwave intensity, proportional to \(|\pmb{E}|^2\), these effects are reduced as the lightwave propagates into the fiber because the intensity drops by conventional fiber attenuation. Thus, the effective length, \(L_\text{eff}\), of fiber over which the nonlinearity is important depends both on the physical fiber length, \(L\), and the optical power attenuation coefficient, \(\alpha\):

\[\tag{2.39}L_\text{eff}=\frac{1}{\alpha}[1-e^{-\alpha{L}}]\]

Clearly for high attenuation, \(\alpha\), the \(L_\text{eff}\sim1/\alpha\) and for small attenuation \(L_\text{eff}\sim{L}\), and the effective length equals the physical fiber length.

The formalism based on the nonlinear index is appropriate for describing the intensity-dependent phase change seen in self-phase modulation (SPM), cross-phase modulation (XPM) and FWM.

Modifications are needed to describe scattering phenomena such as stimulated Brillouin scattering (SBS) and stimulated Raman scattering (SRS). These involve the material dynamics of acoustic waves and molecular vibrations.

Parametric processes, such as FWM and parametric gain, are caused by mixing of lightwaves of different frequencies through the nonlinear fiber index. Phase matching is required for these effects to become significant. The following provides a rapid overview of these nonlinearities.

SPM occurs when varying signal intensity changes the phase through the signal pulse, causing new frequencies to develop. This effect can be encapsulated in the ‘‘nonlinear phase’’:

\[\tag{2.40}\Phi(z)=\int{k\text{d}z},\qquad\Phi_\text{NL}=\frac{2\pi{n_2}PL_\text{eff}}{\lambda{A_\text{eff}}}\]

which is obtained by integration of the optical wavenumber, \(k\), (\(k=n\omega/c\)) over the fiber length in which nonlinear effects have consequence (\(L_\text{eff}\)). Here, the nonlinear relation for \(n\), given by Eq. (2.2), is used and \(P\) and \(\lambda\) are the optical power and wavelength, respectively.

The time derivative of the optical phase results in a frequency, so \(\Phi_\text{NL}\) results in frequency changes on amplitude modulated signals (where \(\text{d}P/\text{d}t\ne0\)). This gives spectral broadening and pulse distortion. For a signal pulse, the leading edge is up-shifted in wavelength and the trailing edge is downshifted. Fiber chromatic dispersion can act on this wavelength spectrum to cause pulse broadening.

XPM occurs when varying signal intensity on one channel causes phase change in other channels. Again, Eq. (2.40) applies, but in this case the power, P, is that of the interfering channel.

The frequency broadening is twice that of the SPM effect because of two field combinations that can occur in Eq. (2.37) for the two distinct signals. Despite that there can be many channels simultaneously interfering in a WDM system due to XPM, chromatic dispersion prevents continual coincidence of these channels as the signals propagate down the fiber.

In addition, as pulses pass through each other (because of their different group velocities), the XPM frequency shifts due to the leading and trailing edges will average to zero in the ideal case of a lossless fiber.

Stimulated scattering comprises another class of nonlinear fiber effects. An early analysis, which is still highly relevant, is the paper by Smith. This class includes scattering via the Raman and Brillouin interactions.

Both occur spontaneously when light scatters off of vibrational modes of the glass, exchanging one vibrational quanta of energy, thus shifting the frequency of the lightwave. The scattered waves become large when the interaction becomes stimulated, that is, where the pump and scattered waves self-consistently generate vibrations or acoustic waves.

SBS gives backward radiation in fibers, a low-intensity threshold, and is narrow band (40 MHz). The interaction consists of a forward traveling, high-power ‘‘pump’’ lightwave, a co-propagating acoustic wave, and a backscattered frequency-downshifted lightwave. Using the dispersion relations for the three waves, and requiring momentum conservation, we find the relation between the acoustic and lightwave frequencies and wave numbers:

\[\tag{2.41}\omega_\text{A}=\frac{2n_1v_\text{A}}{c}\omega_\text{P},\quad\text{and}\quad{k_\text{A}}\cong2k_\text{P}\]

where the \(\omega_\text{A}\), \(v_\text{A}\), and \(k_\text{A}\) are the frequency, velocity, and wave number of the acoustic wave, while \(\omega_\text{P}\) and \(k_\text{P}\) refer to the pump lightwave.

The backscattered lightwave is frequency downshifted by about 11 GHz for a 1550-nm pump. The acoustic wave is formed by density variations created by electrostriction. Because the electrostriction causes high density everywhere there is high light intensity, it is clear that the wave number of the acoustic wave will be twice that of the pump wave.

The interaction line width is set by the decay time of the acoustic wave, leading to line widths of less than 40 MHz. The frequency spectra of the backscattered lightwave may consist of multiple peaks due to the presence of various radial acoustic modes in the fiber.

Figure 2.19 shows an NZDF fiber with this type of structure, along with some theoretical predictions. The peak gain is given by

\[\tag{2.42}g_\text{B}=\frac{2\pi{n^7}p_{12}^2}{c\lambda_p^2\rho_0v_\text{A}\Delta\nu_\text{B}}\]

where \(p_{12}\) is a stress-optic coefficient, \(\lambda_p\) is the pump wavelength, \(\rho_0\) is the material density, and \(\Delta\nu_\text{B}\) is the width of the dominant Brillouin spectral peak.

This equation holds for pump line widths much less than the \(\Delta\nu_\text{B}\). The gain drops for broader pump line widths. When the gain is higher than the fiber loss, the backscatter signal builds up along the length of the fiber toward the lightwave source. The effect is stronger for longer fibers and comes at the expense of optical power which would otherwise be transmitted.

SBS has largely receded into the background as an impairment for systems deployed over cabled communications fiber because phase dithering of the transmitter can almost always raise the SBS threshold high enough for practical applications. SBS remains relevant for high-power device applications such as high-power amplifiers or fiber lasers.

SRS gives forward or backward scattered waves, has a higher intensity threshold than SBS, and is wideband (~12 THz). The spontaneous Raman interaction consists of an input optical signal generating up- and down-frequency shifted lightwaves (anti-Stokes and Stokes waves) as the waves exchange energy due to transitions between molecular vibration levels.

The new frequencies are sum and difference frequencies from the original lightwave and the vibrational transition frequencies. For a large pump signal, the anti-Stokes wave is suppressed and the Stokes wave grows. For maximal gain, the polarizations of signal and pump must be aligned.

The primary current interest in SRS in the telecommunications world is its application for Raman amplification. The advantages are several.

First, the optical signal-to-noise ratio is improved when amplification is distributed throughout the transmission fiber, so the signal is not allowed to attenuate to a very low level.

Second, because the Raman gain spectrum (Fig. 2.20) is approximately 13 THz below the pump frequency, one can obtain gain at any wavelength if appropriate pump wavelength and power are available. The bandwidth of a Raman amplifier can be expanded by multiplexing pumps at optimally chosen wavelengths.

Figure 2.20 shows the measured Raman gain curve of a matched clad standard single-mode fiber when pumped at 1453 nm.

The Raman gain efficiency \(C_R\), measured consistently with IEC Technical Report 62324, can be used to calculate the on–off gain by the relationship \(G_\text{on-off}=\exp[C_RP_\text{pump}L_\text{eff}]\),where \(P_\text{pump}\) is the Raman pump power.

This relation holds in the regimen where the signal power is low compared to the Raman pump power \(P_\text{pump}\), so depletion of the pump power is negligible.

Multiple copies of the gain shape in Fig. 2.20 can be superposed to form flat gain across a band by multiplexing Raman pumps at appropriately chosen wavelengths and powers.

The output from a single Raman pump laser may be several hundred milliwatts; the total power for multiplexed pumps may approach 500 mW or higher, depending on design and eye safety targets.

The next tutorial discusses about ** materials and fabrication technologies in optical fiber manufacturing**.