1 Introduction

Supercontinuum (SC) generation refers to coherent white light generated by spectral
broadening of an injected spectrum and the generation of new frequency components
within the pulse spectrum propagating in a nonlinear medium [^{5}^{,}^{4}]. SC generation has recently attracted a great
deal of attention because of its wide range of applications [^{4}^{,}^{8}]. For that reason, it has been widely studied
as a complex process introducing a variety of experimental and theoretical
challenges in different waveguide types [^{4}^{,}^{8}^{,}^{6}].

The theoretical study of the SC generation in optical dielectric waveguides has been
possible with the use of the generalized nonlinear Schro-dinger equation (GNLSE)
[^{1}]. Broadly speaking,
the GNLSE appropriately describes the coherent spectral broadening and the
generation of new frequency components within the spectrum of high power optical
pulses propagating in nonlinear dispersive waveguides [^{1}^{,}^{3}].

However, in the context of photonic nanowires, i.e., waveguides with sub-micron
transversal dimensions, the use of the GNLSE as an accurate and feasible method to
describe the SC generation is questionable because it does not take into account, on
one hand, the effect of the large longitudinal field component of the propagating
electromagnetic waves generated due to the strong optical confinement and high
optical powers [^{11}^{,}^{10}], and on the other hand, the effect of the complete
strong dispersion implicit in the propagation constant // of the propagating mode
inside the spectral body of the optical pulses [^{1}].

The importance of consider the effect of the large longitudinal electric field
component of the propagating modes along photonic nanowires is because it can
enhance the waveguide nonlinearity through the nonlinear parameter γ [^{7}]. In this regard, nonlinear
effects like SC generation can be modified along the photonic nanowires. This can be
crucial in nano-photonic devices based on optical nonlinearities [^{2}]. Therefore, to analytically
study the SC generation in photonic nanowires is required to readapt the
conventional GNLSE from the vectorial form of Maxwell's equations and does not
neglect the longitudinal electric field.

Moreover, in the derivation of the conventional GNLSE the following approximation is
used *β* + *β*
_{0} ≈ 2*β*
_{0}, where *β*
_{0} is the wave number of the propagating optical pulse [^{1}]. This approximation simplify
the GNLSE but it is only valid when the optical field is assumed to be
quasi-monochromatic. Therefore, when short pulses are launched into photonic
nanowires, with large enough power such that supercontinuum generation is generated,
that approximation is questionable.

In this theoretical approach, we despise before approximation and derive a new
nonlinear pulse propagation equation considering the vectorial nature of Maxwell's
equations and the longitudinal electric field component of the propagating mode. We
investigate how the complete strong dispersion implicit in *β* of the
propagating mode inside the spectral body of the optical pulses affects the SC
generation in air-silica photonic nanowires waveguides.

2 Nonlinear Propagation Equation

Let us begin by considering an air-silica cylindrical photonic nanowire of core
radius *τ* and length *L*. The photonic nanowire is
initially pumped with an optical pulse, at the carrier frequency *ω*
_{0}, such that it excites the fundamental mode. To describe the propagation
of the optical pulse along the photonic nanowire, we use the Maxwell
frequency-domain wave equation given by:

where *ω* is the angular
frequency, *μ*
_{0} is the vacuum permeability, and *n* is the linear part
of the refractive index profile.

The Fourier transform of

where *β*
_{0} is the wavenumber. Here *N* is related to the spectral power of the
pulse, and its obtained using the Poynting vector [^{1}].

Substituting Eq. (2) into (1) and applying the slowly varying envelope approximation, we obtain after associating terms:

where:

Here *F*
_{z} is the longitudinal component and **F**
_{
T
} is the transverse part of the normalized electric field. Using the fact that
the fundamental mode distribution satisfies the equation: *β*(*ω*) is the propagation
constant of that mode, we obtain after substitute the last relation on the last term
of the right-hand side of Eq. (3) and associating terms:

Here the usual way to simplify this equation is to use the following approximation: ^{1}].
However, before approximation is only valid when the optical field is assumed to be
quasi-monochromatic. Therefore, when femtosecond pulses are launched into photonic
nanowires with large enough power such that supercontinuum generation is generated,
that approximation is questionable. In this theoretical approach, we despise that
approximation and derive a new nonlinear propagation equation.

Multiplying Eq. (4) with **F**
^{*}, using the vectorial identity

Considering that ^{9}^{,}^{11}], we obtain:

From Eq. (6) and its complex conjugate, we obtain that

If we consider the generalized orthogonality condition given by *a*(*z, t*) can be
written as:

where we have expanded *β*(*ω*) in a Taylor series
about *ω*
_{0} and replaced by *ω* by

where *t* because the nonlinear response function
*R*(*t*-*π*) must be zero for
*τ*>*t* to ensure the causality [^{1}]. Therefore by substituting the
Fourier transform of Eq. (2) into Eq. (9), and using it in Eq. (8), we obtain the
following time-domain propagation equation:

where *β*
_{1} term by assuming that *T* represents time in a reference
frame moving at the group velocity of the input pulse. In obtaining Eq. (10), we
have using the relation ^{1}], where
*c* is the speed of light in vacuum and *n*
_{2} is the nonlinear refractive index. Comparing Eq. (10) with the
conventional GNLSE [^{1}], we
can notice the appearance of additional shock terms in the dispersion and nonlinear
terms. The nonlinear coefficient *γ* is given by:

Figure 1 shows how the nonlinear parameter
changes with the core radius of an air-silica photonic nanowire. For our
calculations, we chose the following values: the nonlinear refractive index
*n*
_{2} = 2.6 × 10^{-20} m^{2} W^{-1}, the wavelength λ
= 800 nm, and the relative core-cladding index difference Δ = 0.312.

To examine the main correction that our nonlinear pulse propagation model does to the
SC generation in air-silica photonic nanowires, we compare in Fig. 2 the SC generation produced by the conventional GNLSE and
by our new nonlinear propagation equation. Figure
2(a) and (b) shows the SC generation
produced by the conventional GNLSE and Fig.
2(c) and (d) shows the SC generation
produced by our new pulse propagation model. For this purpose, we have used the
following value of the nonlinear parameter: γ = 0.182 W^{-1} m^{-1}.
We numerically simulate the SC generation in an air-silica photonic nanowire with
length *L* = 35 cm, and core radius *r* = 900 nm. The
input sech pulse has a central wavelength λ = 800 nm, peak power *P*
_{0} = 1 kW, and *T*
_{0} = 50 fs. Figure 2 shows that the
additional shock terms in our new propagation equation (Eq. (10)) have the effect of
reducing the long-wavelength edge of the supercontinuum spectrum.

Figure 3 shows that our new propagation model has more correction on the conventional model when the length of the photonic nanowire waveguide is larger, i.e., for higher legths there is a higher reduction of the extreme long-wavelength edge of the supercontinuum spectrum and therefore the spectrum becomes narrower. This result would be expected to be crucial for accurate comparison of simulations with experiments. In fact, our results

3 Conclusion

We have derived a new comprehensive evolution equation to describe the nonlinear
propagation of high power optical pulses through photonic nanowires. (ur basic
formulation part from the vectorial form of Maxwell's equations and take into
account the effect of the complete strong dispersion implicit in the propagation
constant *β* of the propagating mode inside the spectral body of the
optical pulses.

Applying our new nonlinear propagation equation in air-silica photonic nanowires, we have showed evidence of additional effects that perturb the supercontinuum, reducing its extreme long-wavelength edge and making it narrower.