PCF with a solid glass core and the cladding composed of a hexagonal air holes lattice is considered (see Fig. 1). Assuming that the geometry of the fiber does not change along a fiber’s principal axis, the problem can be reduced to a vector two-dimensional one (V2D), solvable at a plane transverse to the propagation with an analytically imposed longitudinal phase shift (Gwarek et al. 1993):
$$\begin{aligned} \beta _f =\frac{2\pi }{\lambda _0 }n_{ eff }, \end{aligned}$$
(1)
where \(\lambda _{0}\) stands for the wavelength in free space and \(n_{ eff }\) is an effective refractive index of the considered mode.
Consequently, computational effort of the EM analysis can be substantially reduced. In addition, since the analysis is restricted to the fundamental mode with a priori known symmetries, the FDTD model can be reduced to a quarter of the fiber’s cross-section with electric and magnetic symmetry conditions imposed, as indicated in Fig. 1. The model is truncated with a perfectly matched layer (PML) surrounding the fiber (Berenger 1996). Before the analysis is started, material properties need to be determined and properly represented with the models available in FDTD. The fibers investigated in this paper are made of Schott glasses, the refractive index of which can be represented by the Sellmeier equation:
$$\begin{aligned} n^{2}\left( \lambda \right) =1+\frac{B_1 \lambda ^{2}}{\lambda ^{2}-C_1 }+\frac{B_2 \lambda ^{2}}{\lambda ^{2}-C_2 }+\frac{B_3 \lambda ^{2}}{\lambda ^{2}-C_3 }, \end{aligned}$$
(2)
where \(n\) denotes the refractive index of the material, \(\lambda \) is wavelength in microns, \(B_{1,2,3}\) and \(C_{1,2,3}\) are experimentally determined Sellmeier coefficients.
Since the Sellmeier equation does not have its direct implementation in the FDTD method, a triple-pole Lorentz model is applied instead:
$$\begin{aligned} \varepsilon _r \left( \omega \right)&= \varepsilon _\infty +A_1 \frac{\left( {\varepsilon _s -\varepsilon _\infty } \right) 2\pi f_{p1}^2 }{2\pi f_{p1}^2 +j\omega 2\pi v_{c1} -\omega ^{2}} \nonumber \\&+A_2 \frac{\left( {\varepsilon _s -\varepsilon _\infty } \right) 2\pi f_{p2}^2 }{2\pi f_{p2}^2 +j\omega 2\pi v_{c2} -\omega ^{2}}+A_3 \frac{\left( {\varepsilon _s -\varepsilon _\infty } \right) 2\pi f_{p3}^2 }{2\pi f_{p3}^2 +j\omega 2\pi v_{c3} -\omega ^{2}}, \end{aligned}$$
(3)
where \(\varepsilon _{\infty }\) is optical relative permittivity, \(\varepsilon _{S}\) is static relative permittivity, \(f_{p1,2,3}\) stand for poles’ frequencies, \(v_{c1,2,3}\) represents relaxation frequencies, and \(A_{1,2,3}\) denote weight coefficients of the corresponding dispersive poles.
As it can be noticed, relaxation frequencies \(v_{c1,2,3}\) occurring in the Lorentz model contribute to a non-zero imaginary part of the permittivity model as in (3). It seems to be in contrast with (2), which is purely real. However, according to the Kramers-Kronig theorem (Landau and Lifshitz 1960), frequency dispersion of a real part of a complex function imposes an imaginary part to be non-zero as well. Although imaginary components are neglected in the Sellmeier equations, it is not allowed to do so in FDTD, which explicitly solves Maxwell curl equations in any causal system, such as the one represented with the Lorentz permittivity model as in (3). Otherwise, an FDTD simulation would have become unstable. For those reasons, relaxation frequencies are set non-zero but small enough (\(v_{c1,2,3} = 10^{-4} \quad f_{p1,2,3}\)) to assure that the loss factor is negligible. Table 1 shows Sellmeier coefficients and the corresponding Lorentz parameters of two glasses, namely TWNN16 and PBS517 (Stępień et al. 2011).
Table 1 Sellmeier and Lorentz coefficients
Once the material properties are properly represented, total fiber dispersion can be determined. Total fiber dispersion is comprised of material dispersion and waveguide dispersion. For proper modeling of the fiber both parameters have to be taken into account. Several works previously studied this task (Zhu and Brown 2002; Koshiba and Saitoh 2001; Ferrando et al. 1999).
For total dispersion calculations, several FDTD simulations have to be carried out for a varying longitudinal phase shift \(\beta _{f}\) as in (1), imposed in the V2D FDTD algorithm. In each simulation run, the structure shown in Fig. 1 is excited with a point source driven with the Kronecker delta to evenly cover the investigated spectrum. Subsequently, the Fourier transform of a current injected by the source, as shown in Fig. 2a, is computed to detect resonances indicating guided modes. Those frequencies (or wavelengths) allow determining effective permittivity \(n_{ eff }\) of the modes for the imposed \(\beta _{f}\) imposed in (1). Figure 2b shows an electric field distribution of the fundamental mode occurring at \(f = 300\) THz for \(\beta _{f} = 2{,}1692\) rad/m.
Once the spectrum of the effective refractive index \(n_{ eff }\) is computed, in the subsequent step group velocity dispersion can be computed as:
$$\begin{aligned} D=\frac{\lambda }{c}\frac{d^{2}n_{ eff }}{d\lambda ^{2}}. \end{aligned}$$
(4)
A similar procedure has to be called if the photonic crystal fiber is supposed to be investigated with the FDFD method (Zhu and Brown 2002). A major difference in respect to FDTD is that a single FDFD simulation provides the solution for a single frequency only, so it has to be executed several times, thus, increasing computational effort.