Fabry-Perot Etalon (Non-Dispersive)

1 Fabry-Perot Etalon
(Non-Dispersive)

A Fabry-Perot (FP) resonator is composed of a slab of material with
high refractive index or a pair of parallel reflecting surfaces with
some distance (L). The surfaces can either be mirrors or an interface
between two regions of differing refractive index. The mechanism of
resonance is based on the multiple beam interference as a result of
successive reflections from the two parallel interfaces, see Fig. 1.

2 Theory

The resulting transmitted ((E_{t})) and reflected ((E_{r})) electric fields from an FP
resonator can be obtained as

[begin{eqnarray}
E_{t}&=&E_{i}t_{1}t_{2}left(1+r_{2}^{2}e^{iphi}+r_{2}^{4}e^{i2phi}+…right)\
E_{r}&=&E_{i}left[r_{1}+t_{1}t_{2}r_{2}e^{iphi}left(1+r_{2}^{2}e^{iphi}+r_{2}^{4}e^{i2phi}+…right)right]
end{eqnarray}]

where (E_{i}) is the incident
electric field [1-2]. The coefficients (r_{1}) ((r_{2})) and (t_{1})((t_{2})) are the amplitude reflection and
transmission coefficients from the medium (n_{1})((n_{2})) to (n_{2})((n_{1})), respectively. The parameter (phi=frac{4pi n_{2}L}{lambda}) is the
phase difference as a result of a round-trip between two successive
outgoing waves.

The reflectivity of each interface ((R=r_{1}r_{2})) for the normal angle of
incidence is calculated as

[begin{equation}R=left(frac{n_{2}-n_{1}}{n_{2}+n_{1}}right)^{2}end{equation}]

and the transmittance (the superposition all transmitted waves)
((T)) and reflectance (the
superposition of all reflected waves) ((mathcal{R})) can be obtained as

[begin{eqnarray}
T&=&frac{left|E_{t}right|^{2}}{left|E_{i}right|^{2}}=frac{1}{1+F:mathrm{sin^{2}}left(frac{phi}{2}right)}\
mathcal{R}&=&frac{left|E_{r}right|^{2}}{left|E_{i}right|^{2}}=Ffrac{mathrm{sin^{2}}left(frac{phi}{2}right)}
{1+mathrm{sin^{2}}left(frac{phi}{2}right)}\
F&=&frac{4R}{left(1-Rright)^{2}}
end{eqnarray}]

where (F) is the finesse parameter
indicating how sharp a resonance seems in comparison with the spectral
distance between two consecutive resonances (FSR) [1-2].

For the case of a lossless resonator, the conservation of energy
implies the relation

[begin{equation}mathcal{R}+T=1end{equation}]

The reflectance spectrum of an FP etalon composed of a pair of
parallel mirrors can be obtained analytically, see Fig. 2.

3 Design

The simulation is performed using a 2D design consisting of a
waveguide centered in the domain.

The input plane was configured using a rectangular distribution as
the optical source positioned at z=5.5 (mu)m

The mesh parameters ((Delta)x and
(Delta)z) are chosen as 0.003 (mu)m. The number of time-steps for a
simulation is dependent on the length of an FP resonator. Testing
confirmed that for the FP resonator under test (n=3 and L=3 (mu)m) 35000 time-steps were adequate, see
the article on convergence testing for further details [??]. The
polarization is set to “TE” which is corresponding to components of
(E_{y}), (H_{x}) and (H_{z}).

4 Results

Figure 4 show the simulation results of an FP etalon composed of a
slab with different reflectivity coefficient R and the length L=3 (mu)m. The reflectivity coefficient R=0.1
and 0.3 correspond to the refractive index n=1.92 and 3.42,
respectively. A good agreement between the simulation results and the
analytical ones can be observed. Note that the FSR is not identical for
different values of reflectivity (R) due to different phase-shifts
caused by different refractive indices of the FP resonator.

The structure can be modeled for different lengths of the FP
resonator and the resulting spectra can be seen in Fig. 5. Note that all
reflectance spectra show a zero at (lambda)=1.5 (mu)m due to the even integer ratio
n/(lambda)=2 in Eq. 5.

Convergence testing was completed on the results to ensure suitable
mesh size ((Delta)x and (Delta)z) and number of time-steps were
used.