# Soliton Content of Fiber-Optic Light Pulses
## 4 Methods to Verify Soliton Content
### Soliton Radiation Beat Analysis  
Soliton radiation beat analysis (SRBA) is a numerical method to determine the soliton content of a pulse in an optical fiber even if the system is not integrable. In contrast to DST, which requires the pulse profile at a single position in the fiber, SRBA requires knowledge of the evolution over a certain length section of the fiber. This is easily understood: In integrable systems, soliton parameters are invariant so that if they are known in one place, they are known everywhere. SRBA can follow the evolution of soliton parameters in a non-integrable setting; the price to pay is that representative information about it must be introduced into the calculation. As early as in 1974, it was noted [6] that the envelope of a pulse not exactly conforming to a soliton undergoes oscillations. Such a pulse contains a soliton and some radiation; the oscillation arises as a beating between both. The radiation evolves linearly; the soliton contains a power-dependent phase term as shown in Equation (3). The frequency of the beat note therefore allows a direct conclusion about the soliton energy. The SRBA method was first introduced in [88]. The propagation of a sech pulse with the initial form \(A(z=0,t)=N\sqrt{P_{0}}\operatorname{sech}(t/T_{0})\) was simulated numerically. This was repeated while incrementing the soliton parameter \(N\). As the oscillation of the peak power is damped due to dispersive spreading of the radiative part, it was found advantageous to evaluate the peak spectral power, which undergoes undamped oscillation. A windowed Fourier transform was applied to the spatial evolution of the power at the center frequency \(|\widetilde{A}(\Omega=0,z)|^{2}\). This produces a spectrogram of spatial frequencies called the soliton radiation beat pattern. An example is shown in Figure 11 (left). As \(N\) is increased, a soliton radiation beat first appears at its expected threshold near \(N=\
icefrac{{1}}{{2}}\). The figure shows that this beat note also has overtones. Its amplitude (grey scale) vanishes at integer \(N\) (apparent 'white stripes'), which is of course because there, the radiation amplitude vanishes; the beat amplitude is proportional to the product of both amplitudes. If there is more than one soliton involved, the beat pattern will contain the frequencies of beats between all pairs of solitons, as well as that of each soliton beating with radiation. In Figure 11 (left),  
Figure 11: (Left) Soliton radiation beat analysis (SRBA) chart from repeated numerical simulations with increasing soliton number \(N\); (right) corresponding predictions of beat signals and its overtones (from [88]). traces pertaining to the second soliton begin at \(N=\
icefrac{{3}}{{2}}\) as expected and have amplitude nulls at \(N=2\). Difference frequencies are visible as downward-bending curves. In the right panel of Figure 11, analytical results for beat frequencies are plotted for comparison. Shown are all calculated frequencies, up to four overtones, and all combination frequencies of this multitude. It is obvious that all SRBA traces of the left panel can be identified in this way. SRBA provides the additional information that there are notches in the amplitudes of all traces involving radiation at integer \(N\) values. This shows that SRBA reproduces all known features of the soliton energies as known analytically from IST. In [89], this method was shown to work for chirped Gaussian pulses in the NLSE. Going beyond the NLSE to non-integrable situations is also possible as SRBA makes no assumption about integrability. A lossy fiber was treated in [90]. It was shown that a mild, 'adiabatic' loss, if it persists for a long fiber length, eventually turns diabatic because the relevant length scales \(L_{\text{D}}\) and \(L_{\text{NL}}\) also change in the process. Once the loss is diabatic, the soliton decays.