## Abstract

An efficient numerical strategy to compute the higher-order dispersion parameters of optical waveguides is presented. For the first time to our knowledge, a systematic study of the errors involved in the higher-order dispersions’ numerical calculation process is made, showing that the present strategy can accurately model those parameters. Such strategy combines a full-vectorial finite element modal solver and a proper finite difference differentiation algorithm. Its performance has been carefully assessed through the analysis of several key geometries. In addition, the optimization of those higher-order dispersion parameters can also be carried out by coupling to the present scheme a genetic algorithm, as shown here through the design of a photonic crystal fiber suitable for parametric amplification applications.

© 2010 OSA

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### Equations (7)

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(1)
$$\beta (\omega )={\beta}^{(0)}+{\beta}^{(1)}(\omega -{\omega}_{0})+\frac{1}{2}{\beta}^{(2)}{(\omega -{\omega}_{0})}^{2}+\frac{1}{6}{\beta}^{(3)}{(\omega -{\omega}_{0})}^{3}+\text{\hspace{0.17em}}\cdots $$
(2)
$${\beta}^{(i)}={\left(\frac{{\partial}^{i}\beta}{\partial {\omega}^{i}}\right)}_{\omega ={\omega}_{0}}\text{\hspace{1em}}(i=0,\text{\hspace{0.17em}}1,\text{\hspace{0.17em}}2,\text{\hspace{0.17em}}3,\text{\hspace{0.17em}}\dots )$$
(3)
$${\beta}^{(i)}({\omega}_{0})={\left(\frac{{\partial}^{i}{\beta}^{(0)}}{\partial {\omega}^{i}}\right)}_{\omega ={\omega}_{0}}$$
(4)
$${\beta}^{(i)}({\lambda}_{0})=f\text{\hspace{0.17em}}\left[{\left(\frac{{\partial}^{i}{\beta}^{(0)}}{\partial {\lambda}^{i}}\right)}_{\lambda ={\lambda}_{0}},{\left(\frac{{\partial}^{i-1}{\beta}^{(0)}}{\partial {\lambda}^{i-1}}\right)}_{\lambda ={\lambda}_{0}},\mathrm{...}\text{\hspace{1em}},{\left(\frac{\partial {\beta}^{(0)}}{\partial \lambda}\right)}_{\lambda ={\lambda}_{0}}\right]$$
(5)
$$RE=Lo{g}_{10}\left|\frac{{\beta}_{fem}^{(i)}-{\beta}_{ref}^{(i)}}{{\beta}_{ref}^{(i)}}\right|$$
(6)
$$D=-\frac{2\pi c}{{\lambda}^{2}}{\beta}^{(2)}$$
(7)
$$F=\left|{\beta}^{(4)}-{\beta}_{aim}^{(4)}\right|/\sigma +\left|D-{D}_{aim}\right|$$