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 (\(\alpha\)).

References:

M. Fopx, Optical Properties of Solids 2nd edition (Oxford Master
Series in Pysics). New York, New York, USA: Oxford University
Press, 2003.

A. Ghatak, Optics 3rd edition. New Delhi, India: Tata
McGraw-Hill Education, 2005.

E. Hecht, Optics 4th edition. San Francisco, Ca, USA:
Addison Wesley, 2002.

M. N. Polyanskiy, Refractive index of GaAs (Gallium Arsenide) –
Aspnes,
https://refractiveindex.info/?shelf=main&book=GaAs&page=Aspnes
(accessed March 01, 2023).