Fabry-Perot Etalon (Lossy Dispersive)

1 Fabry-Perot Etalon (Lossy Dispersive)

The Fabry-Perot etalon has previously been covered. However, introduction of dispersive materials changes the expressions and add complexity to the refelction and transmisssion spectra.

2 Theory

For a material to have gain or loss the refractive index must be complex in nature (\(\tilde{n}=n'-jn''\)). Note that Optiwave uses the engineering sign convention which results in the use of the negative in the refractive index expression. The reflectance (\(\mathcal{R}\)) for the case of a lossy dispersive Fabry-Perot (FP) resonator filled with material \(n_2\), with a background medium of \(n_1\), and a length of L is calculated as [1] \[\begin{equation} \mathcal{R}=\frac{R\left(1-2e^{\alpha L}cos(\phi)+e^{2\alpha L}\right)}{1-2Re^{\alpha L}cos(\phi)+R^{2}e^{2\alpha L}} \end{equation}\] where \(R=\frac{(n_2^{'}(\omega)-n_1)^{2}+n_2^{''2}(\omega)}{(n_2^{'}(\omega)+n_1)^{2}+n_2^{''2}(\omega)}\) is the reflectivity and the absorption coefficient is calculated as \(\alpha=-\frac{4\pi n_2^{''}}{\lambda}\) [2-3]. The parameter \(\phi=\frac{4\pi n^{'}L}{\lambda}\) represents the phase difference as a result of a round-trip between two successive outgoing waves. Within OptiFDTD, a dispersive material \(n(\omega)\) is achieved through fitting the permittivity with the Drude model along with a number of Lorentz resonances i.e. : \[\begin{equation} \varepsilon_{r}=1+\sum_{m=0}^{M}\frac{G_{m}\Omega_{m}^{2}}{\omega_{m}^{2}+i\Gamma_{m}\omega-\omega^{2}} \end{equation}\] where \(\sum_{m=1}^{M}G_{m}=1\). The coefficient \(G_{m}\) corresponds to oscillator strength. \(\Omega_{m}\), \(\Gamma_{m}\) and \(\omega_{m}\) represent the plasma frequency, damping factor and the m-th resonance frequency, respectively. For further details, see the Lorentz-Drude article.

3 Design

The permittivity of Gallium Arsenide (GaAs) can be described adequately by Lorentz-Drude model with six resonance frequencies and \(\varepsilon_{\infty}\) set to unity. The real (\(n^{'}\)) and imaginary (\(n^{''}\)) parts of the complex refractive index of GaAs are shown in Fig. 1. The general layout for the simulation of a FP etalon is shown in Fig. 2. For other settings of the design e.g.  dimensions of the simulation region, the boundary conditions and the input plane features see the article Fabry-Perot Etalon.

4 Results

Figure 3 shows the reflectance spectra of lossy dispersive FP resonators made of GaAs with different lengths i.e. 2 \(\mu\)m, 3 \(\mu\)m and 5 \(\mu\)m. As can be seen, the reflectance for all three lengths of the FP resonator oscillates with increased attenuation for shorter wavelengths resulting from the dispersion in both the real part of the refractive index and propagation loss (\(\alpha\)).