Impact of fiber loss on two-soliton states: Substantial changes in eigenvalue spectrum

The impact of power loss on fiber-optic solitons and soliton compounds has regained interest recently, as coding schemes employing inverse scattering eigenvalues are being discussed. Loss lifts the integrability of the underlying nonlinear Schrödinger equation and has usually been treated by perturbation analysis. Our approach uses localized loss of arbitrary strength. We investigate two-soliton compounds including the $N=2$ soliton and show that loss causes severe qualitative modifications of the eigenvalue spectrum. Peculiar features include power redistribution between solitons so that one of them is actually enhanced by loss and conversion of solitons at rest into a pair with outward velocities. Earlier reports of such features are put into context. We argue that frequency splitting of soliton pairs requires a mechanism that renders the spectrum double-lobed (or multiply lobed) and that the bifurcation is defined by a balance between dispersive and nonlinear forces. Implications for eigenvalue-based communication formats are pointed out.

Figure 1

(a) Evolution of the spectral amplitude $|\tilde{u}\left(\mathrm{\Omega }\right)|$ of an $N=2$ soliton over a whole soliton period ${\xi }_0$. Color indicates values form zero (black) to maximum (white). (b) Corresponding soliton regimes in the $M-\xi$ plane for attenuation of amplitudes by $M$, localized at $\xi$. Labels indicate the number of solitons: “e” indicates energy splitting and “f” frequency splitting. The 2f regime is bounded by points $\mathsf{\text{T}}=(1/2,0.922)$ (top), $\mathsf{\text{B}}=(1/2,0.485)$ (bottom), and $\mathsf{\text{Q}}=(1/2\pm 0.170,0.584)$ (quadruple points). Red dots mark predictions of a lower soliton threshold $M_{\min }$; see at the end of Sec. 3. Vertical dashed lines mark regime of frequency splitting; see Sec. 6.

Figure 2

Soliton eigenvalues when the splice position is at $\xi =0$ (left column) or at $\xi ={\xi }_0/2$ (right column). Upper row, energies in units of $E_{N=1}$; lower row, normalized frequencies.

Figure 3

Soliton energies when $M$ is constant and the splice position $\xi$ is varied. The lower-energy soliton peaks at $1.240E_{N=1}$ ($M=0.950$) and at $1.688E_{N=1}$ ($M=0.922$).

Figure 4

Energy budget for the $N=2$ case with localized loss; compare to Fig. 1. Shown is the radiation energy in units of the $N=1$ soliton energy.

Figure 5

Propagation of an in-phase double soliton amplitude $\left|u\right(\tau \left)\right|$ with an included local loss at $\xi =0$. Three solitons are formed after loss: a blue-shifted soliton $S_{\mathrm{bl}}$, a red-shifted soliton $S_{\mathrm{rd}}$, and the most powerful center soliton $S_{\mathrm{c}}$. Parameters of the initial soliton pair: $\eta =1$, $\sigma =8$; loss: $M=0.6$.

Figure 6

Spectral amplitudes $|\tilde{u}\left(\mathrm{\Omega }\right)|$ of two copropagating sech pulses with different separations $\sigma$ and relative phases $\phi$, normalized to twice the value of a single soliton as specified in the Appendix. Spectral fringes above the soliton threshold (gray region) correspond to individual solitons (marked by blue dots). The soliton content is given in the form $(x·e+y·f)$ with $x$ solitons at center frequency and $y$ frequency-split solitons with same energy. The spectral shape is crucial for the soliton content.

Figure 7

Soliton content of two copropagating sech-pulses with relative phase $\phi =0$ depending on the separation $\sigma$ and the amplitude scaling $M$. Below $M=1/4$ there are no solitons; above $M=3/4$ there are always two energy-split solitons. In between the content depends on the spectral fringes above the soliton threshold. Labels in black circles identify parameters for corresponding examples in Fig. 6.

Figure 8

Soliton content of two copropagating out-of-phase sech-pulses. Plot style corresponding to Fig. 7. Only frequency-split soliton pairs exist. Below $M=1/4$ there are no solitons; above $M=3/4$ and $\sigma =2$ there are always two frequency-split solitons. Labels in black circles identify parameters for corresponding examples in Fig. 6.

Figure 9

Soliton content of an in-phase pair of pulses at a relative velocity of $\mathrm{\Delta }\mathrm{\Omega }=0.5$. The splitting region is reduced to ca. $0.38\le M\le 0.63$. In this regime, a central soliton is always present, while with increasing $\sigma$ more and more frequency-split soliton pairs appear. For more details see text.

Figure 10

Soliton content of in-phase pairs of pulses at relative velocities of $\mathrm{\Delta }\mathrm{\Omega }=0.60,0.75$ and 1.5. With increasing velocity the splitting region (see Fig. 9) narrows, leaving islands of frequency-split soliton pairs. The islands themselves shrink and eventually disappear with growing velocity, as set by $\mathrm{\Delta }\mathrm{\Omega }$.