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When ultrashort pulses (on the order of picoseconds or femtoseconds) propagate in optical fibers, they experience pulse width and spectral variations, mainly due to dispersion (Section 10) and nonlinear effects (Section 11). Some overviews of important influences are given here.

**Dispersive pulse stretching and compression**

If the optical peak power is low enough, nonlinear effects are negligible, but we still consider the effect of dispersion. In Part 10 of this tutorial, we have discussed the origin of dispersion. Here we look at some consequences of pulse transmission.

The spectral bandwidth of ultrashort pulses is limited, that is, it consists of different frequency components. The fundamental effect of dispersion is that these frequency components propagate at different group velocities. If they are all brought together by the injected pulse (i.e. we inject a “transform-limited pulse” with the smallest possible pulse width within a given bandwidth), the faster frequency components will appear faster at the fiber end, followed by slower frequency components. For normal dispersion (positive group velocity dispersion), the longer wavelength component is fastest; then an upward chirp of the pulse, i.e. a rise in instantaneous frequency, can be observed.

For example, consider a Gaussian-shaped initial pulse with an energy of 1 pJ, a pulse width of 100 fs, and a center wavelength of 1000 nm. The bandwidth of this pulse is ≈0.44/100 fs = 4.4 THz (the time-bandwidth product of a transform-limited Gaussian pulse is about 0.44). It is injected into a single-mode fiber with a group velocity dispersion of 10,000 fs 2 /m, then a pulse bandwidth of 4.4 THz corresponds to 2 π 4.4 THz 10,000 fs 2 /m = 276 fs/ The transmission time range of m. Therefore, we guess that a significant widening of the 1-meter fiber is expected.

A useful rule of thumb is actually: if the total group delay dispersion (group velocity dispersion multiplied by fiber length) reaches the square of the pulse duration, then dispersive pulse broadening will be significant. For non-transform-limited pulses, the spectral broadening is stronger.

Calculating the pulse broadening effect is not difficult:

Performs a Fourier transform on the input pulse shape.

Apply a frequency-dependent phase change in terms of dispersion: simply multiply the complex amplitude of each frequency component by exp(i(β2/2)(ω−ω0)2), where β2 is the group velocity dispersion and ω0 The angular frequency at the center of the spectrum.

Convert to the time domain and you have pulses in the time domain. Plot its power, instantaneous frequency, or other time-dependent variables as desired.

Of course, it’s handy to use a software tool like RP Fiber Power for these things; you just describe the input pulse, the fiber, and whatever else you want to draw. Figure 1 shows the time evolution of the power and instantaneous frequency of the 1 m and 10 m fiber output pulses.

Figure 1: Optical power and instantaneous frequency of a 1-pJ pulse in 1 m and 10 m fibers.

It can be seen that a linear upward chirp of the pulse occurs, both cases spanning the same frequency range – the optical bandwidth of the pulse. A 10 m fiber has a lower chirp slope than a 1 m fiber because the same frequency range is scanned over a longer period of time. Therefore, if the chirp is quantified in GHz/ps, then the statement “the greater the dispersion, the greater the chirp” is simply false; this is only true at the beginning of the scaling process.

If we do the same simulation for a pulse energy of 1 nJ instead of 1 pJ, we also get significant nonlinear effects (this is numerically harder to simulate; a typical approach is to use the split-step Fourier method). Self-phase modulation now increases the spectral width, so dispersion broadening is stronger than before. Figure 2 gives a wider time frame:

Figure 2: Optical power and instantaneous frequency of 1-nJ pulses after 1 m and 10 m of optical fiber. Here, nonlinear effects become important.

The spectral width is now increased from 4.4 THz (14.7 nm) to 18.8 THz (62.7 nm).

If the nonlinear effects are negligible, the time broadening can be easily recovered by simply using the dispersion of the opposite sign. Thus, dispersion pulse compression can be obtained by sending up-chirped pulses into fibers with anomalous dispersion or down-chirped pulses into fibers with normal dispersion. If the nonlinear effects are not negligible, they can in principle be compensated by a suitable dispersion distribution – however, this is difficult in practice. Therefore, fiber optics as pulse compressors are in most cases limited to pulses with relatively small peak powers. Furthermore, pulse compression is easiest if the spectrally constant second-order dispersion is large enough, i.e., the spectral phase varies with the square of the frequency shift.

**Intermodal dispersion**

Multimode fiber has another type of dispersion: intermodal dispersion. This means that different modes (see Section 2) have different group velocities; we have already discussed intermodal dispersion in Section 4. If an ultrashort pulse is injected into the fiber, multiple modes will be excited, and the corresponding modes will arrive at the output at different times.

Figure 3 shows the output power versus time for a 200-fs pulse passing through a 50 cm step-index fiber. The spatial profile of the input pulse is Gaussian, but it is not perfectly aligned with the fiber axis. The schema comes first because it is the fastest. About 1.7 ps later is the LP 11 mode—actually it is a superposition of two modes in different directions. (These modes are degenerate, assuming the fiber is perfectly symmetrical and straight.) More modes will appear later.

Figure 3: Output power versus time for a fiber with 200-fs input pulses.

Figure 4 will show the results as an animated graphic. It’s nice to see how the fiber breaks down the input into patterns and presents those patterns at different times.

Figure 4: Time-dependent output beam profile of a 200-fs input pulse (animated graphic). Animated graphics were created using RP Fiber Power software.

It is easy to understand that intermodal dispersion can completely mess up electrical signals. When it is strong, the time it takes to send a symbol is considerably longer, and the bitrate is correspondingly lower. However, intermodal dispersion can be minimized by using graded-index fiber with a parabolic index profile, as shown in Section 4 of Multimode Fiber; this can increase bit rates. Another possibility is to use single-mode fiber, which does not have this hazard. So for long-distance transmission, only single-mode fiber needs to be used.

**Combined effects of anomalous dispersion and nonlinearity: soliton pulses**

We have seen that if normal dispersion and nonlinearity (often with a positive nonlinearity exponent) act together, the temporal pulse broadening increases and produces up-chirped pulses. For fibers with anomalous dispersion, that is, fibers with negative group velocity dispersion, the behavior is quite different. Here, dispersion tends to produce a down-chirp, which can counteract the up-chirp produced by the SPM. For a suitable pulse shape and pulse energy, it is even possible that the dispersion and nonlinear effects completely cancel each other out, so that the pulse shape in the time or frequency domain remains unchanged during propagation. (There is only an additional overall phase shift, which is usually not relevant to the application.) Such a pulse is called a fundamental soliton.

Propagation of fundamental soliton pulses in lossless fibers with negligible high-order dispersion provided that the pulses are unchirped sech 2 pulses

And the pulse energy E p and the soliton pulse duration τ satisfy the following conditions:

Here, the full width at half maximum (FWHM) pulse duration is τ p ≈ 1.7627 × τ, γ is the SPM coefficient in rad/(W m), β 2 is the group velocity dispersion, defined as the Derivative, ie group delay dispersion per unit length (in s 2 /m).

Figure 5: Nonlinear and dispersion effects on fundamental soliton pulses. Blue curve: only time-dependent nonlinear phase shift (no dispersion), proportional to light intensity. Red curve: the overall phase shift caused by the combined effect of nonlinearity and dispersion on the soliton.

A constant phase shift does not change the shape of the pulse in the time or frequency domain.

If the pulse energy is N2 times that of the basic soliton (where N is an integer), and the pulse shape is unchanged, then higher-order solitons of order N can be obtained. Here, the pulse shape in the time or frequency domain is not constant but evolves in a periodic manner with a soliton period of:

For example, Fig. 6 shows the time evolution of the third-order soliton spectrogram. Higher-order solitons exhibit more complex behavior.

Figure 6: This animated spectrogram shows how third-order solitons evolve in an optical fiber.

**stimulated Raman scattering**

Stimulated Raman scattering (SRS) can also become very important in cases where strong nonlinear phase shifts occur. It transfers part of the pulse energy to longer wavelengths.

For example, we simulated the pulse evolution in an ytterbium-doped fiber amplifier, where the SRS becomes relative to the fiber end. Figure 7 shows the evolution of the pulse spectrum as it propagates along the fiber. First, a large spectral broadening is obtained by self-phase modulation. At the end of the fiber, the predominantly long-wavelength part of the spectrum is shifted to longer wavelengths by tens of nanometers; this is Raman scattering.

Figure 7: Evolution of the pulse spectrum in a fiber amplifier. Near the right end, a significant portion of the power is shifted into longer wavelengths.

One might wonder why the short wavelength part of the spectrum is apparently not affected. This is because this part propagates at a slightly lower velocity due to group velocity dispersion. As a result, it loses its temporal overlap with the Raman-shifted part, which starts to grow (at a very low level) near the left fiber end. In other cases, such as with less or anomalous dispersion, the results may be quite different.

In the simulated example, the initial pulse is 3 ps long. In this case, the spectrum is very narrow, and Raman gain amplifies a spectral region where there is essentially no optical power — only quantum fluctuations. Therefore, Raman-shifted light exhibits considerable randomness. In other ranges of pulse durations, completely different results can be obtained. Therefore, other cases should also be simulated; the obtained results cannot be safely transferred to other parametric regimes.

**supercontinuum generation**

Strong nonlinear interactions in optical fibers can be exploited for strong spectral broadening of light. However, it is not enough to just inject light with high peak power, because the degree of spectral broadening also depends sensitively on dispersion properties, pump wavelength and pulse duration. Usually, the pump wavelength is selected in the anomalous dispersion region, not too far away from the zero dispersion wavelength of the fiber. By using a photonic crystal fiber with a suitable design, the zero dispersion wavelength can be placed at the wavelength of a suitable __laser __source.

In extreme cases, one obtains a spectral width of more than one octave. More precisely, they have over an octave of optical strength. The full width at half maximum of the spectrum can be significantly smaller.

A variety of effects can contribute to the creation of the supercontinuum. Usually, the injected light in the state of anomalous dispersion has a pulse energy much higher than that of the fundamental soliton pulse. The pulse is then broken down into multiple solitons with different wavelengths. Various additional effects such as four-wave mixing, self-phase modulation and stimulated Raman scattering lead to further broadening until the peak power becomes too low. Figure 8 shows a simulation example where a pulse with a pulse width of 400 fs and a pulse energy of 1–nJ is injected into a photonic crystal fiber.

Figure 8: Evolution of the pulse spectrum in an optical fiber. The color scale used is logarithmic, so it shows relatively weak parts of the spectrum.

Note that the correlation of various effects in fibers strongly depends on various parameters. A complete understanding of this broadening process requires numerical simulations and detailed analysis.

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