Soliton (optics) - Temporal Solitons - Proof

Proof

An electric field is propagating in a medium showing optical Kerr effect through a guiding structure (such as an optical fiber) that limits the power on the xy plane. If the field is propagating towards z with a phase constant, then it can be expressed in the following form:

where is the maximum amplitude of the field, is the envelope that shapes the impulse in the time domain; in general it depends on z because the impulse can change its shape while propagating; represents the shape of the field on the xy plane, and it does not change during propagation because we have assumed the field is guided. Both a and f are normalized dimensionless functions whose maximum value is 1, so that really represents the field amplitude.

Since in the medium there is a dispersion we can not neglect, the relationship between the electric field and its polarization is given by a convolution integral. 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 Fourier-transform the electric field using the following definition:

using this definition, a derivative in the time domain corresponds to a product in the Fourier domain:

the complete expression of the field in the frequency domain is:

Now we can solve Helmholtz equation in the frequency domain:

we decide to express the phase constant with the following notation:

where we assume that (the sum of the linear dispersive component and the non linear part) is a small perturbation, i.e. . The phase constant can have any complicated behavior, but we can represent it with a Taylor series centered on :

where, as known:

we put the expression of the electric field in the equation and make some calculations. If we assume the slowly varying envelope approximation:

we get:

we are ignoring the behavior in the xy plane, because it is already known and given by . We make a small approximation, as we did for the spatial soliton:

beta^2 (omega) - beta_0^2 = = approx 2 beta_0 Delta beta (omega)

replacing this in the equation we get simply:

now we want to come back in the time domain. Expressing the products by derivatives we get the duality:

we can write the non linear component in terms of the amplitude of the field:

for duality with the spatial soliton, we define:

and this symbol has the same meaning of the previous case, even if the context is different. The equation becomes:

We know that the impulse is propagating among the z axis with a group velocity given by, so we are not interested in it because we just want to know how the pulse changes its shape while propagating. 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 assume the medium where the field is propagating to show anomalous dispersion, i.e. or in term of the group delay dispersion parameter . We make this more evident replacing in the equation . Let us define now the following parameters (the duality with the previous case is evident):


L_d = frac{T_0^2}{|beta_2|}; qquad
tau=frac{T}{T_0}; qquad
zeta = frac{z}{L_d} ; qquad
N^2 = frac{L_d}{L_{nl}}

replacing those in the equation we get:

that is exactly the same equation we have obtained in the previous case. The first order soliton is given by:

the same considerations we have made are valid in this case. The condition becomes a condition on the amplitude of the electric field:

or, in terms of intensity:

or we can express it in terms of power if we introduce an effective area defined so that :

Read more about this topic:  Soliton (optics), Temporal Solitons

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