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In optical fibers, light is confined to a small lateral area, in which case even moderate power levels produce high light intensities, and light can travel long distances in optical fibers. For the above reasons, nonlinear effects caused by fiber nonlinearity often have substantial influence. This is especially the case for short pulses transmitted in optical fibers and in pulsed fiber amplifiers.

Kerr effect

The simplest and most common nonlinear effect in optical fibers is the Kerr effect. Essentially, as the light intensity increases, the phase delay in the fiber increases. This can be described by the refractive index difference being proportional to the light intensity:

This increase can often be considered to occur instantaneously, even though this is not entirely true in reality. Below we also consider nonlinear responses that do not occur instantaneously. The nonlinear coefficient n 2 in silica fiber is quite small—at a wavelength of 1.5 μm, it is about 2.7 · 10 -16 cm 2 /W. In numerical simulation, its value is usually determined according to the central light frequency, that is, the nonlinear coefficient of light waves with shorter wavelengths is larger. Other glasses, such as fluoride or sulfide glasses, generally have a stronger nonlinear response. At high intensities, electrostriction causes some reduction in fiber diameter, but it takes some time. For nanosecond pulses, electrostriction increases its nonlinear coefficient, but not for picosecond or femtosecond pulses. This is one of the reasons why measurements of nonlinear coefficients vary in the literature. the

self phase modulation

One of the effects caused by the Kerr effect is self-phase modulation (SPM). This means that the light beam in the fiber experiences a nonlinear phase delay caused by its own intensity as it propagates. For a fiber mode, the phase change caused by unit optical power per unit length can be described by the following proportionality constant

(in rad / (W m)) where A eff is the effective mode field area. For a Gaussian model with beam radius w, this value is only half of the value for a Gaussian beam in a homogeneous medium, and here we only consider the on-axis value. In fiber, the phase variation is lower at locations away from the fiber axis, and the overall nonlinear phase delay is only half of the peak value. (Despite the presence of the SPM, the wavefronts of the modes in the fiber remain approximately planar; the modes are “kept together” by diffraction and balancing of waveguides, and nonlinear phase changes are “spread” across the entire beam profile.) As the pulse travels through the fiber, The Kerr effect imposes a time-varying phase delay on the pulse due to the time-varying light intensity variation. Due to this effect, an initially unchirped pulse becomes a chirped pulse, that is, a pulse whose frequency changes instantaneously in the time domain. As shown in Figure 1:

Figure 1: The instantaneous frequency of an initially unchirped pulse after undergoing self-phase modulation. Upward chirping in the center of the pulse

For ultrashort pulses with picosecond or femtosecond pulse duration and high peak power, the instantaneous frequency can fluctuate in the range of several terahertz. This results in a substantial broadening of the spectrum, ie an increase in spectral bandwidth. But the spectrum (Fourier spectrum) doesn’t just reflect the range of instantaneous frequencies. Usually, the spectrum also has a strong oscillation if the self-phase modulation is strong and the dispersion effects (see Section 10) are weak. Figure 2 shows a case:

Figure 2: Spectra of an unchirped pulse with an initial pulse width of 1 ps after undergoing strong self-phase modulation (maximum phase change up to 20 rad). Its spectrum (or spectrum) oscillates strongly.

As explained in our encyclopedia article on self-phase modulation, these oscillations can be understood as an interference effect, related to each value reached by the instantaneous frequency at two times. As a rule of thumb, if the nonlinear phase shift exceeds 2 π , significant broadening begins. For example, for a standard single-mode fiber with a mode area of 75 μm 2 , the peak power of a 1-m-long fiber is several kilowatts. Among them, a peak power of 1 kW means only about 1 nJ of pulse energy for a 1-ps pulse; it is easy to reach several kilowatts with ultrashort pulses. Dispersion (see Section 10) often also has a large effect on ultrashort pulses. The overall effect of Kerr nonlinearity and dispersion is strongly related to the dispersion properties. For example, soliton pulses can be formed in fibers with anomalous dispersion. For fundamental-order optical solitons, self-phase modulation and dispersion work together to make it no longer broaden in time domain or spectrum. For high-order optical solitons, there is a rather complex periodic evolution process. In the normal dispersion range, the dispersion causes time-domain broadening, which reduces the nonlinear phase shift, and strongly chirped pulses may appear.

self-focusing effect

Another effect caused by the Kerr effect is the non-linear self-focusing effect. Taking this into account, the fiber modes can be calculated (see Section 2). When the optical power reaches or even exceeds 1 MW, the mode shrinks significantly. As shown in Figure 3:

Figure 3: Numerically calculated mode field area versus optical power in a silica fiber. The nonlinear refractive index is taken as 2.2 10−20 m2/W. The red line gives the critical power for self-focusing.

At a peak power of 5 MW (in a silica fiber), severe self-focusing effects occur: shrinking the intensity profile further increases the self-focusing effect, which in turn leads to more pinches and finally a complete collapse of the profile. Extremely high intensities then destroy the fiber; one ultrashort pulse is enough. Self-focusing can also destabilize the higher-order modes of multimode fibers.

Figure 5 shows an example where a beam of power 4 MW is propagating in a fiber in LP 11 mode (calculated without considering fiber nonlinearity). After traveling 10 mm, the beam becomes a mixed mode of LP01 and LP11.

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