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Definition: pulses with a certain balance of nonlinear and dispersive effects
In general, the temporal and spectral shape of a short optical pulse changes during propagation in a transparent medium due to the Kerr effect and chromatic dispersion. Under certain circumstances, however, the effects of Kerr nonlinearity and dispersion can exactly cancel each other, apart from a constant phase delay per unit propagation distance, so that the temporal and spectral shape of the pulses is preserved even over long propagation distances [1,3]. This phenomenon was first observed in the context of water waves [1], but later also in optical fibers [4].
The conditions for (fundamental) soliton pulse propagation are:
- For a positive value of the nonlinear coefficient n2 (as occurring for most media), the chromatic dispersion needs to be anomalous.
- The temporal shape of the pulse has to be that of an unchirped sech2 pulse (assuming that the group delay dispersion is constant, i.e., there is no higher-order dispersion):
- The pulse energy Ep and soliton pulse duration
have to meet the following condition:
Here, the full-width-at-half-maximum pulse duration is ∼1.76·, γ is the SPM coefficient (in rad per Watt and meter), and GVD is the group velocity dispersion defined as a derivative with respect to angular frequency, i.e., the group delay dispersion per unit length (in s2/m).
Under the mentioned conditions, the pulse can propagate as a (fundamental) soliton (or solitary pulse) with constant temporal and spectral shape. It only acquires a phase shift which is just one half the nonlinear phase shift which the peak of the pulse would experience if only the nonlinearity alone would act on it. This soliton phase shift is constant over time or frequency, i.e., it does not lead to a chirp or to spectral broadening. In most situations, it is thus not relevant.
Figure 1: Solid curve: time-dependent nonlinear phase shift alone (without dispersion), which is proportional to the optical intensity. Dotted curve: overall phase shift, resulting from the combined action of nonlinearity and dispersion on a soliton. The constant phase shift does not modify the temporal or spectral shape of the pulse.
Example: Solitons in Standard Telecom Fiber
As a quantitative example, solitons in standard telecom fibers (single-mode fibers for the 1.5-μm spectral region) can be considered. Assuming the common SMF-28e fiber of Corning, the effective mode area is 85 μm2 at 1550 nm wavelength, resulting in a nonlinear coefficient of 1.43 mrad/(W m). (The nonlinear index is assumed to be 3·10-16 cm2/W.) The chromatic dispersion at 1550 nm is +16.2 ps/(nm km), corresponding to -20660 fs2/m. Using the above formula, we find that 1-ps solitons have to have a pulse energy of 51 pJ, corresponding to a peak power of 45 W.
For ten times shorter solitons with a 100-fs duration, the pulse energy rises tenfold to 510 pJ, whereas the peak power becomes 100 times larger (4.5 kW).
Stability of Solitons
The most remarkable fact is actually not the possibility of the mentioned balance of dispersion and nonlinearity, but rather the fact that soliton solutions of the nonlinear wave equation are very stable: even for substantial deviations of the initial pulse from the exact soliton solution, the pulse automatically "finds" the correct soliton shape while shredding some of its energy into a so-called dispersive wave, a weak background which has too little intensity to experience significant nonlinear effects and temporally broadens as a result of dispersion. Solitons are also very stable against changes of the properties of the medium, provided that these changes occur over distances which are long compared with the so-called soliton period (defined as the propagation distance in which the constant phase delay is π/4). This means that solitons can adiabatically adapt their shape to slowly varying parameters of the medium. Also, solitons can accommodate to some amount of higher-order dispersion; they then automatically adjust their shape to achieve the mentioned balance under the given conditions.
Higher-order Solitons
If the pulse energy is the square of an integer number times the fundamental soliton energy, the pulse is a so-called higher-order soliton. Such pulses do not have a preserved shape, but their shape periodically varies, with the period being the above-mentioned soliton period. However, higher-order solitons can break up into fundamental solitons under the influence of higher-order dispersion and other disturbing effects. They are by far not as stable as fundamental solitons.
Figure 2: Relation between soliton pulse energy and pulse duration in a single-mode fiber. The solid curve applies to fundamental solitons, the dotted curves to higher-order solitons (orders 2, 3, 4).
Importance of Solitons
Fundamental soliton pulses are technically very important, in particular for long-distance optical fiber communications and in mode-locked lasers (→ soliton mode locking). In the latter situation, soliton-like pulses can be formed when the typically lumped pieces of dispersion and nonlinearity in the laser cavity are sufficiently weak per cavity round trip. Solitons are also applied in various techniques for pulse compression using optical fibers; an example is adiabatic soliton compression.
Simulation of Soliton Propagation
Soliton propagation, possibly with additional disturbing effects, can be investigated with numerical simulations (→ pulse propagation modeling). There are also some analytical tools, e.g. soliton perturbation theory, involving equations for small deviations of pulse parameters from those of the ideal soliton.
Spatial Solitons
Apart from the temporal solitons as discussed above, there are also spatial solitons. In that case, a nonlinearity of the medium (possibly of photorefractive type) cancels the diffraction, so that a beam with constant beam radius can be formed even in a medium which would be homogeneous without the influence of the light beam.
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Bibliography
| [1] | J. S. Russell, "Report on waves", in Report of the 14th meeting of the British Association for the Advancement of Science, 331 (1844) |
| [2] | V. E. Zakharov and A. B. Shabat, "Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media", Sov. Phys. JETP 34, 62 (1972) |
| [3] | A. Hasegawa and F. Tappert, "Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion", Appl. Phys. Lett. 23, 142 (1973) |
| [4] | L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, "Experimental observation of picosecond pulse narrowing and solitons in optical fibers", Phys. Rev. Lett. 45 (13), 1095 (1980) |
| [5] | F. M. Mitschke and L. F. Mollenauer, "Discovery of the soliton self-frequency shift", Opt. Lett. 11 (10), 659 (1986) |
| [6] | J. P. Gordon, "Theory of the soliton self-frequency shift", Opt. Lett. 11 (10), 662 (1986) |
| [7] | L. F. Mollenauer et al., "Soliton propagation in long fibers with periodically compensated loss", IEEE J. Quantum Electron. QE-22, 157 (1986) |
| [8] | N. N. Akhmediev et al., "Stable soliton pairs in optical transmission lines and fiber lasers", J. Opt. Soc. Am. B 15 (2), 515 (1998) |
| [9] | V. N. Serkin and A. Hasegawa, "Novel soliton solutions of the nonlinear Schrödinger equation model", Phys. Rev. Lett. 85 (21), 4502 (2000) |
See also: Kerr effect, dispersion, soliton period, sech2-shaped pulses, higher-order solitons, adiabatic soliton compression, pulse compression, Gordon-Haus jitter, pulse propagation modeling
Categories: nonlinear optics, pulses
