Soliton Generation

Solitons are self-reinforcing waves that are studied widely in physics. Femtosecond time scale soliton pulses have been generated in silica micro resonators. The solitons are emitted from the microresonator in periodic streams at a repetition frequency that is readily detected.

In quantum mechanics we learn to solve the Schrodinger equation for a variety of particle-in-a-well potentials, V(x).

Where ψ is the particle wave function and m is the mass of the particle. The solutions of this equation can be classified broadly into bound and unbound states. The bound states are spatially confined to the vicinity of the potential well.

Imagine a situation in which the potential V(x) is not provided by a box, a harmonic potential, a Coulomb potential or any of the familiar cases studied in first year quantum mechanics, but is rather determined by the particle itself. Suppose that this potential well is stronger (i.e., a deeper well) when there is higher probability of finding the particle. In such a case, we could modify the Schrodinger equation in the following way,

where the potential well is "self generated" by the particle and is proportional to the probability density |ψ|2 through a constant α. This so-called nonlinear Schrodinger equation has a self-confining or self-reinforcing solution. The waves that solve the nonlinear Schrodinger equation belong to a class of solutions of wave equations in which a nonlinearity balances wave dispersion. The solutions are called solitary waves or simply solitons.

Even though we have motivated the existence of solitons using a quantum system, the first observation of a soliton occurred in fluid mechanics when John Scott Russell observed what he later called a "translating wave." He observed this solitary wave when a boat stopping abruptly in a water channel thereby causing a displacement of water moving down the channel.  Lord Rayleigh later provided a mathematical model for the effect citing Russell's observations [1].

Today, solitons are important in a wide range of physical systems. The most famous solitons (perhaps only second to Russell's first account of a water-wave soliton) are optical in nature. In fact, using lasers that emit short pulses of light, researchers at Bell laboratories in the 1980s were able to generate solitons in optical fiber [2]. Here, the nonlinearity is caused by the dependence of refractive index on optical intensity (the Kerr nonlinearity – also see discussion of frequency combs). Essentially, when propagating in a dielectric such as glass fiber an optical wave will have a phase velocity that is slower as the intensity of the wave is larger (refractive index increases with intensity). This is a nearly univeral effect, but it is very weak. However, using very short pulses of light that feature high peak power levels, the Kerr nonlinearity provides a significant modification to the phase velocity of the light pulse. Under the right conditions this nonlinearity can balance the forces that cause waves to experience dispersion in dielectrics. These conditions include: (1) the dispersive forces must have the right sign (they must be anomalous) so that the Kerr nonlinearity is able to provide balance. Also, the waveform, must have the right shape. This shape creates an effective potential well that confines the waveform self consistently.

Solitons in optical fiber can travel great distances without changing shape. Also, even though the soliton wave equation in optical fiber does not include optical attenuation or optical amplification terms, it has been found experimentally that solitons are robust with respect to the weak attenuation in optical fiber; moreover, they also can be amplified using fiber optic based optical amplifiers. For all of these reasons, optical solitons were once considered an ideal way to send information over great distances [3]. Even though many tests of this idea were conducted in the 1980s and 1990s, other methods to compensate dispersion were eventually realized. Nonetheless, fiber-optic based soliton research continues and has found other important applications including lasers based on optical fiber that can be "mode locked" by generation of solitons [4]. Also, solitons are important in understanding the process of continuum generation [5].

A new type of soliton called a dissipative Kerr soliton has recently been observed in optical fiber [6] and even more recently in optical microcavities [7,8.9]. Dissipative Kerr solitions, as implied by their name, actually include dissipative effects in their non-linear propagation equation. Also, unlike normal solitons, these new solitons are able to leverage the Kerr nonlinearity to both balance wave dispersion (i.e., maintain their shape in the presence of dispersion), and provide parametric gain [10,11] to balance the optical dissipation. What this means is that the dissipative solitons can be self-sustained both in shape and energy inside a microcavity.

Solitons emitted from microcavities are intense, short pulse trains of light that are periodic (the period is the round trip time in the microcavity). As a result, the optical spectrum created by the solitons is a series of regularly spaced optical frequencies that are locked in phase by the soliton pulse. This phase locking means that the solitons provide a convenient way to create frequency combs (see frequency combs). Moreover, the pulse-like nature of the soliton makes it well suited for spectral broadening (see optical continuum). As a result, the soliton mode locked microcavities are now considered a possible way to achieve a self-referenced frequency microcomb.

A new type of soliton called a dissipative Kerr soliton has recently been observed in optical fiber [6] and even more recently in optical microcavities [7,8.9]. Dissipative Kerr solitions, as implied by their name, actually include dissipative effects in their non-linear propagation equation. Also, unlike normal solitons, these new solitons are able to leverage the Kerr nonlinearity to both balance wave dispersion (i.e., maintain their shape in the presence of dispersion), and provide parametric gain [10,11] to balance the optical dissipation. What this means is that the dissipative solitons can be self-sustained both in shape and energy inside a microcavity.

Solitons emitted from microcavities are intense, short pulse trains of light that are periodic (period is the round trip time in the microcavity). As a result, the optical spectrum created by the solitons is a series of regularly spaced optical frequencies that are locked in phase by the soliton pulse. This phase locking means that the solitons provide a convenient way to create frequency combs (see frequency combs). Moreover, the pulse-like nature of the soliton makes it well suited for spectral broadening (see optical continuum). As a result, the soliton mode locked microcavities are now considered a possible way to achieve a self-referenced frequency microcomb.