Despite the ultimate choice of the method solving for nonlinear propagation of the pulse along the PCF (SSFM or FDTD), mode characteristics are usually computed with the aforementioned rigorous numerical techniques (FEM or FDTD). Subsequently, the obtained effective refractive index n
eff
of a given mode can be treated as material dispersion. As a result, S1D-FDTD simulation of the pulse propagating in such artificial (equivalent) material can be undertaken.
Transverse properties
Transverse (mode) properties of the PCF can be computed with the V2D-FDTD method accounting for dispersion of the applied material (Salski et al. 2015). The real part of the refractive index of glass contained in the PCFs is usually represented with a Sellmeier equation, which has to be transformed into a Lorentz model with its representation in the FDTD method (Karpisz et al. 2014) given as:
$$\varepsilon = \varepsilon_{\infty } + \sum\limits_{n = 1}^{N} {\frac{{\left( {\varepsilon_{s,n} - \varepsilon_{\infty } } \right)\,\omega_{p,n}^{2} }}{{\omega_{p,n}^{2} - \omega^{2} + j\,\omega \,\omega_{c,n} }}} ,$$
(3)
where ε
∞
is optical permittivity, ε
S
is static permittivity, ω
p,n
stands for the nth pole frequency, and ω
c,n
represents relaxation frequency of the nth pole.
Longitudinal properties
Once spectral properties of the considered mode, E
t,n
(x, y), are determined, the next step is to represent it as material dispersion in a S1D-FDTD model. The most straightforward way to do that is to fit the aforementioned Lorentz model to the obtained effective refractive index n
eff
of the mode. However, practical realization of that concept is not straightforward as it may look at first glance. First, SC generation is sensitive not only to n
eff
, but even more to dispersion D, which is very vulnerable to fluctuations of n
eff
. Consequently, as it will be shown later in this paper with specific examples, special attention must be paid to represent waveguide dispersion accurately up to its second order derivative with the wavelength.
Consider as an example a PCF fabricated from lead–bismuth–gallium-oxide glass (PBG-08) proposed in (Sobon et al. 2014), which is well-suited for SC generation (see Fig. 1). Solid lines in Figs. 2a and 3 show the characteristics of the corresponding effective refractive index n
eff
and dispersion D, respectively, computed in (Sobon et al. 2014) in the range from 500 nm to 4000 nm. In the first attempt, a triple-pole Lorentz model will be fitted to the aforementioned curves using the Levenberg–Marquardt algorithm executed with no bounds imposed on the Lorentz parameters. According to (Sobon et al. 2014), SC evolves in the considered PCF in the 500–2400 nm range (see Fig. 5 therein), so the fitting has been limited to that range. The obtained Lorentz model (see the 2nd column in Table 1) provides the effective refractive index (dispersion) with a normalized root-mean-square deviation (NRMSD) of 0.047 % (0.777 %) in that spectrum. Due to negligibly small discrepancy, these curves are not plotted in Figs. 2 and 3.
Table 1 Material representation of the PCF as in Fig. 1
It can be noticed that the lack of restrictions imposed on the Lorentz parameters results in a decent agreement with the properties of the fundamental mode computed rigorously. Practically, the third Lorentz pole with a very small resonant frequency f
p,3 = 0.0103 THz and large and negative relaxation frequency f
c,3 = −35.0282 THz can be approximated with a Drude model. Unfortunately, large relaxation frequencies f
c,n
, that have been obtained (see the 2nd column in Table 1), contribute to prohibitively large propagation loss correlated with an imaginary part of (3). It is confirmed by the plot shown in Fig. 2b (Lorentz fit 1). Moreover, the condition ε
s,n
< ε
∞
makes the FDTD analysis numerically unstable. For that reason, it has been assumed in a subsequent step of curve fitting that f
c,n
< 10−6
f
p,n
, thus, imposing substantially reduced losses (see the 3rd column in Table 1). As a result, the NRMSD of the effective refractive index (dispersion) has increased to 0.3363 % (0.7611 %) in the 500–2400 nm range, which is still within reasonable limits.
Several tests undertaken by the authors have shown that the multi-pole Lorentz model cannot accurately represent dispersion-flattened characteristics, which are decreasing with the wavelength at some spectral range. An example of such a scenario is shown in Fig. 3 (solid line) for λ > 2400 nm. Consequently, if that spectral range was essential for SC generation, other representation of mode dispersion should be applied instead. It should be noted, however, that the decrease of D with the wavelength usually coincides with the increase of confinement loss (Stepniewski et al. 2014), so that the evolution of spectrum does not extend beyond that limit.
The conclusion that can be derived from attempts to conveniently represent material properties undertaken so far is that suitable fitting of waveguide dispersion can be obtained only if no restrictions are imposed on an imaginary part of the Lorentz model. However, such an approach cannot be applied due to resulting erroneously large propagation loss leading to a rapid attenuation of the signal along the fiber. It seems that these losses may be suppressed if metamaterial representation is applied, where the effective refractive index is the square root of the product of complex permittivity as in (3), and complex permeability given in a similar Lorentz form:
$$\mu = \mu_{\infty } + \sum\limits_{n = 1}^{N} {\frac{{\left( {\mu_{s,n} - \mu_{\infty } } \right)\,\omega_{pp,n}^{2} }}{{\omega_{pp,n}^{2} - \omega^{2} + j\,\omega \,\omega_{cc,n} }}}$$
(4)
where μ
∞
is optical permeability, μ
S
is static permeability, ω
pp,n
stands for the n
th magnetic pole frequency, and ω
cc,n
represents the corresponding magnetic relaxation frequency.
If (3) and (4) are conjugates of each other (ε′ = μ′, ε″ = −μ″), the resulting effective refractive index of such single-negative metamaterial will be purely real:
$$n = \sqrt {\left( {\varepsilon^{{\prime }} - j\varepsilon^{{\prime \prime }} } \right)\;\left( {\varepsilon^{\prime} + j\varepsilon^{\prime\prime}} \right)} = \sqrt {\varepsilon^{{{\prime }2}} + \varepsilon^{{{\prime \prime }2}} } ,$$
(5)
The fourth column in Table 1 depicts the Lorentz parameters of the obtained metamaterial model, which resembles waveguide dispersion of the considered mode. The NRMSD of the effective refractive index (dispersion) is as low as 2.17 % (2.63 %) in the 500–2400 nm range, so there is no significant improvement when compared with the third column in Table 1. However, if the spectrum above the 2400 nm limit was also essential in the analysis, metamaterial representation would be a much better choice than the fitting done with the permittivity function only. Due to negligibly small discrepancy in the whole 500–4000 nm range, the plot of the corresponding metamaterial curves is omitted in Figs. 2a and 3. Unfortunately, tests undertaken by the authors have shown that the FDTD simulation is quickly becoming unstable if such metamaterial is computed. Apparently, instability is due to a large level of the positive imaginary part of (3) (loss), which is suppressed by the negative imaginary part of (4) (gain). It seems that the undertaken attempt at compensating the imaginary part of (5) improves the quality of the fit but it does not suppress numerical noise of the FDTD algorithm. Consequently, the noise is quickly amplified by the FDTD routines associated with the permeability model given by (4). Although that subject goes beyond the main scope of this paper, it can be expected that the presented technique with metamaterial representation of PCF waveguide dispersion may be useful in future studies if the aforementioned stability issues are resolved. In this paper, the fiber will be represented with the already discussed Lorentz permittivity model (see the third column in Table 1).
Nonlinear properties
Dispersive properties of the fiber will be supplemented in the S1D-FDTD model with a third-order nonlinear polarization model derived on the basis of the Born–Oppenheimer approximation (Fujii et al. 2004; Hellwarth 1977):
$$P^{\left( 3 \right)} \left( t \right) = \varepsilon_{0} \;\chi_{0}^{\left( 3 \right)} \;E\;\left( t \right)\;\left[ {g\,\left( t \right) \otimes E^{2} \left( t \right)} \right],$$
(6)
where ε
0 is free-space permittivity, χ
(3)0
is third-order susceptibility, and g(t) stands for a nonlinear response function consisting of both Kerr and Raman contributions. It is assumed that nonlinear properties of glass are not dispersive, which is quite common in EM modelling of SC generation (Agrawal 2007; Travers et al. 2010; Sobon et al. 2014).
The details of the corresponding FDTD model are discussed in (Salski et al. 2015), so only major features will be pointed out hereafter. Equation (3) can be implemented in FDTD with the aid of ADE [14]. However, an FDTD solution of (3) suffers from either low computational efficiency or substantially deteriorated accuracy (Maksymov et al. 2011). In (Salski et al. 2015), a new approach that combines the accuracy of a rigorous FDTD update scheme and the computational speed of an approximate solution has been proposed and successfully validated. In that approach, rigorous FDTD update equations are applied only in those nonlinear cells, where instantaneous amplitude of an electric field is relatively high, while all the remaining nonlinear FDTD cells are updated with approximate update equations. In the case of short femtosecond pulses propagating in a long transmission line, most of FDTD cells are populated with low electric field’s amplitude, and only a small fraction of FDTD cells requires more rigorous computation. As a result, FDTD simulation of nonlinear materials can be substantially speeded up with no deterioration of accuracy. As shown in (Salski et al. 2015) in selected cases the improvement can exceed 80 %.