Optical Mode Solver

Optical modes describe how light is guided or confined within layered devices such as organic LEDs, solar cells, and perovskite stacks. A mode is a self-consistent field pattern that propagates along the device with a characteristic propagation constant β, or equivalently an effective refractive index \( n_\text{eff} = \beta/k_0 \), where \( k_0 = 2\pi/\lambda \).

TE and TM polarization

Two families of solutions are supported:

• TE modes (transverse electric): the electric field is perpendicular to the propagation plane. These modes are usually simpler to calculate and represent the “s” polarization at interfaces.
• TM modes (transverse magnetic): the magnetic field is perpendicular to the propagation plane. These modes are more sensitive to material boundaries and are especially important in plasmonic or strongly refractive-index–modulated systems.

Governing equations

In a two-dimensional cross-section (x–y plane), the solver finds solutions to the scalar Helmholtz-type equations:

• TE: \(\nabla_\perp^2 E + (k_0^2 n^2 - \beta^2)E = 0\)
• TM: \(\nabla_\perp\cdot\!\left(\frac{1}{n^2}\nabla_\perp H\right) + \left(k_0^2 - \frac{\beta^2}{n^2}\right)H = 0\)

Here \(n(x,y)\) is the spatially varying refractive index. The unknown fields E or H correspond to the mode profile.

Numerical method

The solver discretizes the equations on a rectangular grid using finite-difference methods. This produces a large sparse matrix whose eigenvalues correspond to the allowed propagation constants \(\beta\). Modes are identified by searching for values of β that satisfy the eigencondition, and then iteratively refining until convergence is reached.

The outcome is a set of guided or leaky modes, each with an associated field distribution and effective index. These modes describe how light propagates inside the device.

Outputs and interpretation

• Effective index (\(n_\text{eff}\)): tells how strongly the mode is confined relative to the surrounding medium.
• Field distributions: the 2D electric or magnetic field intensity pattern, showing how light is confined to active layers or leaks into claddings.
• Photon density: the normalized energy density associated with the mode, useful for linking to device performance metrics.
• Confinement factor (Γ): fraction of the optical mode overlapping with a specific region (e.g. the active layer of a solar cell).