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 φ ( x ) {\displaystyle \varphi (x)} , 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:
φ ( x ) = k 0 n L ( x ) {\displaystyle \varphi (x)=k_{0}nL(x)}
where L ( x ) {\displaystyle L(x)} is the width of the lens, changing in each point with a shape that is the same of φ ( x ) {\displaystyle \varphi (x)} because k 0 {\displaystyle k_{0}} 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 n ( x ) {\displaystyle n(x)} 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:
φ ( x ) = k 0 n ( x ) L = k 0 L [ n + n 2 I ( x ) ] {\displaystyle \varphi (x)=k_{0}n(x)L=k_{0}L[n+n_{2}I(x)]}
if I ( x ) {\displaystyle I(x)} 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 (as long as the medium does not change and if we can neglect losses, obviously). In order to have a self-focusing effect, we must have a positive n 2 {\displaystyle n_{2}} , 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 (this is impossible in linear media).
Proof An electric field is propagating in a medium showing optical Kerr effect, so the refractive index is given by:
n ( I ) = n + n 2 I {\displaystyle n(I)=n+n_{2}I}
We recall that the relationship between irradiance and electric field is (in the complex representation)
I = | E | 2 2 η {\displaystyle I={\frac {|E|^{2}}{2\eta }}}
where η = η 0 / n {\displaystyle \eta =\eta _{0}/n} and η 0 {\displaystyle \eta _{0}} is the impedance of free space, given by
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