In
optics
Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviole ...
, the term soliton is used to refer to any
optical field
An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical c ...
that does not change during propagation because of a delicate balance between
nonlinear
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...
and linear effects in the medium.
There are two main kinds of solitons:
* spatial solitons: the nonlinear effect can balance the
diffraction
Diffraction is defined as the interference or bending of waves around the corners of an obstacle or through an aperture into the region of geometrical shadow of the obstacle/aperture. The diffracting object or aperture effectively becomes a s ...
. The electromagnetic field can change the
refractive index
In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium.
The refractive index determines how much the path of light is bent, or ...
of the medium while propagating, thus creating a structure similar to a
graded-index fiber
A graded-index or gradient-index fiber is an optical fiber whose core has a refractive index that decreases with increasing radial distance from the optical axis of the fiber.
Because parts of the core closer to the fiber axis have a higher ref ...
. 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
Dispersion may refer to:
Economics and finance
*Dispersion (finance), a measure for the statistical distribution of portfolio returns
*Price dispersion, a variation in prices across sellers of the same item
*Wage dispersion, the amount of variatio ...
. 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
A lens is a transmissive optical device which focuses or disperses a light beam by means of refraction. A simple lens consists of a single piece of transparent material, while a compound lens consists of several simple lenses (''elements''), ...
. 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
The propagation constant of a sinusoidal electromagnetic wave is a measure of the change undergone by the amplitude and phase of the wave as it propagates in a given direction. The quantity being measured can be the voltage, the current in a ci ...
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
In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium.
The refractive index determines how much the path of light is bent, or ...
we will get exactly the same effect, but with a completely different approach.
This has application in
graded-index fiber
A graded-index or gradient-index fiber is an optical fiber whose core has a refractive index that decreases with increasing radial distance from the optical axis of the fiber.
Because parts of the core closer to the fiber axis have a higher ref ...
s: 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
The Kerr effect, also called the quadratic electro-optic (QEO) effect, is a change in the refractive index of a material in response to an applied electric field. The Kerr effect is distinct from the Pockels effect in that the induced index chang ...
introduces a
self-phase modulation Self-phase modulation (SPM) is a nonlinear optical effect of light–matter interaction.
An ultrashort pulse of light, when travelling in a medium, will induce a varying refractive index of the medium due to the optical Kerr effect. This variation i ...
that changes the refractive index according to the intensity:
: