Soliton (optics)


In optics, the term soliton is used to refer to any optical field that does not change during propagation because of a delicate balance between nonlinear and dispersive effects in the medium. There are two main kinds of solitons:
  • spatial solitons: the nonlinear effect can balance the dispersion. The electromagnetic field can change the refractive index of the medium while propagating, thus creating a structure similar to a graded-index fiber. If the field is also a propagating mode of the guide it has created, then it will remain confined and it will propagate without changing its shape
  • temporal solitons: if the electromagnetic field is already spatially confined, it is possible to send pulses that will not change their shape because the nonlinear effects will balance the dispersion. Those solitons were discovered first and they are often simply referred as "solitons" in optics.

    Spatial solitons

In order to understand how a spatial soliton can exist, we have to make some considerations about a simple convex lens. As shown in the picture on the right, an optical field approaches the lens and then it is focused. The effect of the lens is to introduce a non-uniform phase change that causes focusing. This phase change is a function of the space and can be represented with, whose shape is approximately represented in the picture.
The phase change can be expressed as the product of the phase constant and the width of the path the field has covered. We can write it as:
where is the width of the lens, changing in each point with a shape that is the same of because and n are constants. In other words, in order to get a focusing effect we just have to introduce a phase change of such a shape, but we are not obliged to change the width. If we leave the width L fixed in each point, but we change the value of the refractive index we will get exactly the same effect, but with a completely different approach.
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. If the two effects balance each other perfectly, then we have a confined field propagating within the fiber.
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. In other words, the field creates a fiber-like guiding structure while propagating. 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. In order to have a self-focusing effect, we must have a positive, otherwise we will get the opposite effect and we will not notice any nonlinear behavior.
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. This way it is possible to let light interact with light at different frequencies.

Proof

An electric field is propagating in a medium showing optical Kerr effect, so the refractive index is given by:
We recall that the relationship between irradiance and electric field is
where and is the impedance of free space, given by
The field is propagating in the direction with a phase constant. About now, we will ignore any dependence on the y axis, assuming that it is infinite in that direction. Then the field can be expressed as:
where is the maximum amplitude of the field and is a dimensionless normalized function that represents the shape of the electric field among the x axis. In general it depends on z because fields change their shape while propagating.
Now we have to solve the Helmholtz equation:
where it was pointed out clearly that the refractive index depends on intensity. If we replace the expression of the electric field in the equation, assuming that the envelope changes slowly while propagating, i.e.
the equation becomes:
Let us introduce an approximation that is valid because the nonlinear effects are always much smaller than the linear ones:
now we express the intensity in terms of the electric field:
the equation becomes:
We will now assume so that the nonlinear effect will cause self focusing. In order to make this evident, we will write in the equation
Let us now define some parameters and replace them in the equation:
  • , so we can express the dependence on the x axis with a dimensionless parameter; is a length, whose physical meaning will be clearer later.
  • , after the electric field has propagated across z for this length, the linear effects of diffraction can not be neglected anymore.
  • , for studying the z-dependence with a dimensionless variable.
  • , after the electric field has propagated across z for this length, the nonlinear effects can not be neglected anymore. This parameter depends upon the intensity of the electric field, that's typical for nonlinear parameters.
The equation becomes:
this is a common equation known as nonlinear Schrödinger equation. From this form, we can understand the physical meaning of the parameter N:
  • if, then we can neglect the nonlinear part of the equation. It means, then the field will be affected by the linear effect much earlier than the nonlinear effect, it will just diffract without any nonlinear behavior.
  • if, then the nonlinear effect will be more evident than diffraction and, because of self phase modulation, the field will tend to focus.
  • if, then the two effects balance each other and we have to solve the equation.
For the solution of the equation is simple and it is the fundamental soliton:
where sech is the hyperbolic secant. It still depends on z, but only in phase, so the shape of the field will not change during propagation.
For it is still possible to express the solution in a closed form, but it has a more complicated form:
It does change its shape during propagation, but it is a periodic function of z with period.
For soliton solutions, N must be an integer and it is said to be the order or the soliton. For an exact closed form solution also exists; it has an even more complicated form, but the same periodicity occurs. In fact, all solitons with have the period. Their shape can easily be expressed only immediately after generation:
on the right there is the plot of the second order soliton: at the beginning it has a shape of a sech, then the maximum amplitude increases and then comes back to the sech shape. Since high intensity is necessary to generate solitons, if the field increases its intensity even further the medium could be damaged.
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 :
that, in terms of maximum irradiance value becomes:
In most of the cases, the two variables that can be changed are the maximum intensity and the pulse width.
Curiously, higher-order solitons can attain complicated shapes before returning exactly to their initial shape at the end of the soliton period. In the picture of various solitons, the spectrum and time domain are shown at varying distances of propagation 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. In practice, in order to reach the very high peak intensity needed to achieve nonlinear effects, laser pulses may be coupled into optical fibers such as photonic-crystal fiber with highly confined propagating modes. Those fibers have more complicated dispersion and other characteristics which depart from the analytical soliton parameters.

Generation of spatial solitons

The first experiment on spatial optical solitons was reported in 1974 by Ashkin and Bjorkholm in a cell filled with sodium vapor. The field was then revisited in experiments at Limoges University in liquid carbon disulphide and expanded in the early '90s with the first observation of solitons in photorefractive crystals, glass, semiconductors 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. Spatials solitons are also referred to as self-trapped optical beams and their formation is normally also accompanied by a self-written waveguide. In nematic liquid crystals, spatial solitons are also referred to as nematicons.

Transverse-mode-locking solitons

Localized excitations in lasers may appear due to synchronization of transverse modes.
In confocal laser cavity the degenerate transverse modes with single longitudinal mode at wavelength mixed in nonlinear gain disc and saturable absorber disc of diameter are capable to produce spatial solitons of hyperbolic form:
in Fourier-conjugated planes and.

Temporal solitons

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. 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 is the bandwidth in terms of wavelength. 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. With temporal solitons it is possible to remove such a problem completely.
Image:Temporal soliton explanation.svg|thumb|300px|right|Linear and nonlinear effects on Gaussian pulses
Consider the picture on the right. On the left there is a standard Gaussian pulse, that's the envelope of the field oscillating at a defined frequency. We assume that the frequency remains perfectly constant during the pulse.
Now we let this pulse propagate through a fibre with, it will be affected by group velocity dispersion. For this sign of D, the dispersion is anomalous, so that the higher frequency components will propagate a little bit faster than the lower frequencies, thus arriving before at the end of the fiber. The overall signal we get is a wider chirped pulse, shown in the upper right of the picture.
Image:Self-phase-modulation-en.svg|thumb|left|325px|effect of self-phase modulation on frequency
Now let us assume we have a medium that shows only nonlinear Kerr effect but its refractive index does not depend on frequency: such a medium does not exist, but it's worth considering it to understand the different effects.
The phase of the field is given by:
the frequency is given by:
this situation is represented in the picture on the left. At the beginning of the pulse the frequency is lower, at the end it's higher. After the propagation through our ideal medium, we will get a chirped pulse with no broadening because we have neglected dispersion.
Coming back to the first picture, we see that the two effects introduce a change in frequency in two different opposite directions. It is possible to make a pulse so that the two effects will balance each other. Considering higher frequencies, linear dispersion will tend to let them propagate faster, while nonlinear Kerr effect will slow them down. The overall effect will be that the pulse does not change while propagating: such pulses are called temporal solitons.