Derivation of the Optical Mode Equations

Optical modes are the allowed electromagnetic field distributions that can propagate through a photonic structure. These modes arise naturally from solutions to Maxwell’s equations together with the material boundary conditions imposed by the refractive-index distribution.

This page derives the optical mode equations used in waveguide and photonic simulations, beginning from Maxwell’s equations and progressing to the finite-difference matrix eigenvalue problem solved numerically by the optical mode solver.

Maxwell’s equations

In source-free, non-magnetic dielectric materials, Maxwell’s equations may be written as:

\[ \nabla \cdot \mathbf{D}=0 \]

\[ \nabla \cdot \mathbf{B}=0 \]

\[ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \]

\[ \nabla \times \mathbf{H} = \frac{\partial \mathbf{D}}{\partial t} \]

For isotropic dielectric materials:

\[ \mathbf{D}=\epsilon \mathbf{E} \]

\[ \mathbf{B}=\mu \mathbf{H} \]

Assuming harmonic time dependence:

\[ e^{-i\omega t} \]

the curl equations become:

\[ \nabla \times \mathbf{E} = i\omega\mu \mathbf{H} \]

\[ \nabla \times \mathbf{H} = -i\omega\epsilon \mathbf{E} \]

Deriving the wave equation

Taking the curl of Faraday’s law gives:

\[ \nabla \times \left( \nabla \times \mathbf{E} \right) = i\omega\mu \left( \nabla \times \mathbf{H} \right) \]

Substituting Ampère’s law:

\[ \nabla \times \left( \nabla \times \mathbf{E} \right) = \omega^2\mu\epsilon\mathbf{E} \]

Using the vector identity:

\[ \nabla\times(\nabla\times\mathbf{E}) = \nabla(\nabla\cdot\mathbf{E}) - \nabla^2\mathbf{E} \]

and assuming source-free dielectric materials where:

\[ \nabla\cdot\mathbf{E}\approx0 \]

gives the vector Helmholtz equation:

\[ \nabla^2\mathbf{E} + k_0^2 n^2 \mathbf{E} = 0 \]

where:

\[ k_0=\frac{2\pi}{\lambda} \]

is the free-space wavevector and:

\[ n^2=\frac{\epsilon}{\epsilon_0} \]

is the refractive index distribution.

Guided mode ansatz

Optical modes propagating along the longitudinal \(z\) direction are assumed to take the form:

\[ \mathbf{E}(x,y,z) = \mathbf{E}(x,y)e^{i\beta z} \]

where:

• \( \mathbf{E}(x,y) \) is the transverse mode profile.
• \( \beta \) is the propagation constant.

Substituting this form into the Helmholtz equation gives:

\[ \nabla_\perp^2\mathbf{E} + \left( k_0^2n^2-\beta^2 \right)\mathbf{E} = 0 \]

This equation defines an eigenvalue problem in which the allowed values of \( \beta \) correspond to the supported optical modes of the structure.

Effective refractive index

The propagation constant is often expressed using the effective refractive index:

\[ n_{\mathrm{eff}} = \frac{\beta}{k_0} \]

The effective index provides a convenient measure of optical confinement. Modes with larger \(n_{\mathrm{eff}}\) are generally more strongly confined within high-index regions.

Transverse electric (TE) modes

For TE polarization, the electric field is transverse to the propagation plane. The governing equation reduces to:

\[ \nabla_\perp^2 E + \left( k_0^2 n^2 - \beta^2 \right)E = 0 \]

This scalar Helmholtz equation is commonly used to calculate optical modes in slab waveguides and multilayer dielectric structures.

Transverse magnetic (TM) modes

For TM polarization, the magnetic field is transverse to the propagation plane. The spatial variation of the dielectric function modifies the governing equation:

\[ \nabla_\perp \cdot \left( \frac{1}{n^2} \nabla_\perp H \right) + \left( k_0^2 - \frac{\beta^2}{n^2} \right)H = 0 \]

TM modes are generally more sensitive to refractive-index discontinuities and are particularly important in plasmonic and high-index-contrast systems.

Finite-difference discretization

Analytical solutions are only available for relatively simple structures. Most realistic photonic devices therefore require numerical methods.

The optical mode solver discretizes the transverse differential operators using finite-difference approximations on a rectangular mesh.

For example, the second derivative in the \(y\) direction may be approximated using:

\[ \frac{\partial^2E}{\partial y^2} \approx \frac{ E_{y-1} - 2E_y + E_{y+1} }{ \Delta y^2 } \]

Equivalent expressions are constructed for the \(x\) direction. In nonuniform meshes, local left and right mesh spacings are used to construct asymmetric finite-difference coefficients.

Matrix eigenvalue problem

After discretization, the optical mode equations become a sparse matrix eigenvalue problem:

\[ A\psi = \beta^2\psi \]

where:

• \(A\) contains the finite-difference Laplacian and refractive-index terms.
• \( \psi \) is the discretized optical field.
• \( \beta^2 \) is the eigenvalue.

The sparse matrix structure arises because each mesh point only couples to neighboring points in the finite-difference stencil.

Physical interpretation

The resulting eigenvectors correspond to the supported optical modes of the photonic structure, while the eigenvalues determine their effective refractive indices.

Different eigenmodes represent different field distributions and confinement characteristics inside the device. Some modes remain strongly confined, while others leak energy into surrounding layers or radiate into free space.