Optical soliton

In the anomalous dispersion regime ($D>0$ for typical fiber telecom convention), chromatic dispersion and self-phase modulation produce opposing effects:

• Dispersion broadens a pulse by sending different frequency components at different group velocities
• SPM chirps a pulse: the leading edge is red-shifted (lower frequency) and the trailing edge is blue-shifted (higher frequency)

In anomalous dispersion, red frequencies travel slower than blue. The SPM-induced chirp therefore acts to compress the pulse: the red-shifted leading edge slows down while the blue-shifted trailing edge speeds up. At a specific peak intensity, SPM compression exactly balances dispersive broadening. The pulse propagates indefinitely without changing shape.

Fundamental soliton solution of the nonlinear Schrödinger equation:

The soliton energy and pulse width are related:

where ${\beta }_2=-{\lambda }^{2}D\mathrm{/}(2\pi c)$ is the GVD parameter and $\gamma =2\pi n_2\mathrm{/}(\lambda A_{\text{eff}})$ is the fiber nonlinear coefficient.

For SMF-28 at 1550 nm (${\beta }_2\approx -21$ ps²/km, $\gamma \approx 1.3$ /(W·km)): a fundamental soliton with $T_0=10$ ps (FWHM 17.6 ps) requires about 0.16 W peak power.

Higher-order solitons ($N=2,3,\dots$) require power $N^2{P}_{\text{peak}}$ and undergo periodic shape oscillations (soliton breathers) with period $z_0=\pi T_0^2\mathrm{/}(2\mathrm{\mid }{\beta }_2\mathrm{\mid })$.

Applications.

Dissipative and conservative solitons. The soliton of the lossless nonlinear Schrödinger equation is conservative: it keeps its shape without exchanging energy with anything. A dissipative soliton also balances gain against loss, so it exists only where energy is supplied continually: in a laser cavity, or in a microresonator pumped by a continuous-wave laser, whose solitons are called dissipative Kerr solitons. The pulses of mode-locked fiber lasers with normal dispersion are dissipative solitons shaped by gain, loss and spectral filtering as well as by dispersion and nonlinearity. They are strongly chirped, with a spectrum that has steep edges rather than a sech² shape, and they reach much higher pulse energies than the conventional solitons of lasers with anomalous dispersion.

Microresonator solitons became the basis of chip-scale optical frequency combs in the 2010s. Pumping a high-Q microresonator with a narrow-linewidth CW laser at the right detuning produces a single soliton circulating in the ring; its output spectrum is a broad comb, octave-spanning in dispersion-engineered resonators, with line spacing equal to the FSR. Used in compact optical clocks, ranging, and dual-comb spectroscopy.

References: L. F. Mollenauer, R. H. Stolen and J. P. Gordon, "Experimental observation of picosecond pulse narrowing and solitons in optical fibers," Physical Review Letters 45, 1095 (1980). A. Chong, J. Buckley, W. Renninger and F. Wise, "All-normal-dispersion femtosecond fiber laser," Optics Express 14, 10095 (2006). T. Herr et al., "Temporal solitons in optical microresonators," Nature Photonics 8, 145 (2014). G. P. Agrawal, Nonlinear Fiber Optics (Academic Press).