Soliton (optics)
This has application in graded-index fibers: the change in the refractive index introduces a focusing effect that can balance the natural diffraction of the field.
Spatial solitons are based on the same principle: the Kerr effect introduces a self-phase modulation that changes the refractive index according to the intensity: if
has a shape similar to the one shown in the figure, then we have created the phase behavior we wanted and the field will show a self-focusing effect.
If the field creates a fiber and it is the mode of such a fiber at the same time, it means that the focusing nonlinear and diffractive linear effects are perfectly balanced and the field will propagate forever without changing its shape (as long as the medium does not change and if we can neglect losses, obviously).
The optical waveguide the soliton creates while propagating is not only a mathematical model, but it actually exists and can be used to guide other waves at different frequencies[citation needed].
The condition to be solved if we want to generate a fundamental soliton is obtained expressing N in terms of all the known parameters and then putting
In the picture of various solitons, the spectrum (left) and time domain (right) are shown at varying distances of propagation (vertical axis) in an idealized nonlinear medium.
This shows how a laser pulse might behave as it travels in a medium with the properties necessary to support fundamental solitons.
The first experiment on spatial optical solitons was reported in 1974 by Ashkin and Bjorkholm[6] in a cell filled with sodium vapor.
The field was then revisited in experiments at Limoges University[7] in liquid carbon disulphide and expanded in the early '90s with the first observation of solitons in photorefractive crystals,[8][9] glass, semiconductors[10] and polymers.
During the last decades numerous findings have been reported in various materials, for solitons of different dimensionality, shape, spiralling, colliding, fusing, splitting, in homogeneous media, periodic systems, and waveguides.
[11] Spatials solitons are also referred to as self-trapped optical beams and their formation is normally also accompanied by a self-written waveguide.
[14] The main problem that limits transmission bit rate in optical fibres is group velocity dispersion.
It is because generated impulses have a non-zero bandwidth and the medium they are propagating through has a refractive index that depends on frequency (or wavelength).
This effect is represented by the group delay dispersion parameter D; using it, it is possible to calculate exactly how much the pulse will widen: where L is the length of the fibre and
The approach in modern communication systems is to balance such a dispersion with other fibers having D with different signs in different parts of the fibre: this way the pulses keep on broadening and shrinking while propagating.
Considering higher frequencies, linear dispersion will tend to let them propagate faster, while nonlinear Kerr effect will slow them down.
In 1973, Akira Hasegawa and Fred Tappert of AT&T Bell Labs were the first to suggest that solitons could exist in optical fibres, due to a balance between self-phase modulation and anomalous dispersion.
In 1987, P. Emplit, J.P. Hamaide, F. Reynaud, C. Froehly and A. Barthelemy, from the Universities of Brussels and Limoges, made the first experimental observation of the propagation of a dark soliton, in an optical fiber.
Pump lasers, coupled to the optical amplifiers, activate the erbium, which energizes the light pulses[citation needed].
Anyway, using a representation in the Fourier domain, we can replace the convolution with a simple product, thus using standard relationships that are valid in simpler media.
We make a small approximation, as we did for the spatial soliton: replacing this in the equation we get simply: Now we want to come back in the time domain.
We decide to study the impulse shape, i.e. the envelope function a(·) using a reference that is moving with the field at the same velocity.
Thus we make the substitution and the equation becomes: We now further assume that the medium where the field is propagating in shows anomalous dispersion, i.e.
Adding small perturbations to the equations and solving them numerically, it is possible to show that mono-dimensional solitons are stable.
Working close to this saturation level makes it possible to create a stable soliton in a three-dimensional space.
Unfortunately any medium introduces losses, so the actual behaviour of power will be in the form: this is a serious problem for temporal solitons propagating in fibers for several kilometers.
Experiments have been carried out to analyse the effect of high frequency (20 MHz-1 GHz) external magnetic field induced nonlinear Kerr effect on Single mode optical fibre of considerable length (50–100 m) to compensate group velocity dispersion (GVD) and subsequent evolution of soliton pulse ( peak energy, narrow, secant hyperbolic pulse).
Once soliton pulse is generated it is least dispersed over thousands of kilometres length of fibre limiting the number of repeater stations.
) we can use the following closed form expression: It is a soliton, in the sense that it propagates without changing its shape, but it is not made by a normal pulse; rather, it is a lack of energy in a continuous time beam.
