FIMMPROP

Modelling a passive optical cavity (VCSEL, DFB)

3D simulations with FIMMPROP software

FIMMPROP can be used to model the passive modes of an optical cavity; it is particularly efficient to model cavities of VCSEL (vertical cavity surface-emitting laser) or DFB (distributed feedback) lasers. FIMMPROP can model such devices very efficiently and accurately:

• The Cavity Mode Calculator allows the user to launch light into the middle of a cavity and calculate the cavity modes that exist.

• Different cylindrical mode solvers optimised for cylindrical geometries can be used to calculate the modes in each section of the VCSEL, allowing you to perform a benchmark between different independent methods. FIMMPROP can also model non-cylindrical cavities using Cartesian mode solvers.

• For this device we relied on FIMMPROP's EigenMode Expansion (EME) algorithm to model each grating. FIMMPROP's EME uses a scattering matrix approach, which makes FIMMPROP extremely efficient to simulate periodic structures: the modes and scattering matrix for the periods are only calculated once, and the scattering matrix of the period is then multiplied by itself N times to obtain the scattering matrix of a periodic section with N periods. 
  
  We could have also used FIMMPROP's Rigorous Coupled Mode Theory (RCMT) algorithm for modelling the gratings; RCMT is particularly efficient for modelling long planar gratings (DBR or DFB).

• The fully bidirectional algorithm will inherently account for all internal reflections, without having to use iterative techniques.

• FIMMPROP can handle arbitrary features along the longitudinal axis of the cavity e.g. lensed waveguides, free space regions, contacts etc.

Modelling optical cavities with EME

The cavity mode is given by the eigenmode of the scattering matrix (Rl*Rr) describing the propagation through one loop of the cavity, where Rl, Rr are the reflection matrices of the left and right halves of the cavity. Tools are provided with FIMMPROP to find the eigenvalues and eigenvectors of (Rl*Rr). 

Geometry of the design

For this example we considered a cavity based on a cylindrical symmetry waveguide, which could be the underlying structure for a VCSEL or a DFB laser.

We considered two designs:

• a fully fully periodic grating

• a quarter-wavelength shifted grating; this is identical to the periodic structure except for a quarter-wavelength shift introduced in the centre of the grating, which will allow us to obtain a resonance at the Bragg wavelength.

The geometry of the quarter-wavelength shifted grating is shown schematically below; you can see that the high-index half-period (shown in yellow) in the centre of the cavity is twice the length of the high-index half-periods in the gratings either side. In the fully periodic design, the central period is identical to the rest of the grating.

Schematic view of the design; the number of periods shown here is smaller than in the actual design

The design parameters are summarised in the table below.

FIMMPROP can be used to find cavity modes based on either a single 2D mode of the cross-section, or on a combination of modes. In this case we solved the eigenproblem for the fundamental mode in the VCSEL (this would be the HE11 mode in a fibre-based DFB laser cavity). This mode is shown below for the high-index waveguide.

Waveguide cross-section and mode list, showing the intensity profile for the fundamental HE11 mode (high-index)

Simulation results

The final cavity mode profile can be seen below. The Cavity Mode Calculator is able to launch the light into the middle of the cavity. The Calculator finds many modes across the given wavelength spectrum and the peak efficiency is found close to 1.1µm.

The cavity mode is shown above. It is efficiently confined within the cavity.

You can find below a plot of the real part (blue) and imaginary part (green) of the eigenvalue of the solution of the round-trip scattering matrix for the fundamental mode. The resonances are shown in red lines. They correspond to wavelengths for which the imaginary part of the eigenvalue is zero and the real part of the eigenvalue is positive, which means that the beam is in phase with itself after a round trip.

This first plot shows the resonances for the periodic grating; there are multiple resonances away from the Bragg wavelength, with significant round-trip losses ranging from 9.4% to 96.4%.

Spectrum of the eigenvalues for the periodic grating: real part in blue and imaginary part in green.
The resonant wavelengths correspond to the dashed red lines.

This second plot shows the same data for the quarter-wavelength shifted grating. Here you can see a single resonance at the Bragg wavelength, with a much lower round-trip loss of 0.02%.

Spectrum of the eigenvalues for the quarter-wavelength shifted grating: real part in blue and imaginary part in green.
There is a single resonance (dashed red line) at the Bragg wavelength.

You can see below the Ex field profile of the cavity mode plotted versus Z and measured in the centre of the fibre. This plot was measured at resonance for the quarter-wavelength shifted design. The variations in refractive index of the grating are shown underneath in red.

Ex field profile (blue) and core index (red) plotted against position along the cavity in microns