Tool · PIC design

Waveguide Mode Explorer

Which modes does a slab of given thickness and index contrast guide, what are their effective indices, and at what thickness does the next mode appear? The explorer solves the exact TE and TM dispersion relation of the asymmetric slab and reports the effective and group index, the confinement factor, the evanescent decay lengths and the cutoff thicknesses, with the mode profiles, the n_eff versus thickness map and the propagating field drawn live. Background: effective index, confinement factor, and single-mode versus multimode.

Structure

Platform preset

Indices at 1550 nm, held constant with wavelength: this tool shows waveguide dispersion only.

Core thickness t (nm)

Wavelength λ (nm)

Core index n_f

Substrate index n_s

Cover index n_c

Guided modes need n_eff above both n_s and n_c. The higher of the two sets the cutoff.

Polarization

Guided modes at this point

Select a mode to inspect it in the profile, field view, and readouts.

Learn with it

Four short experiments. Each sets the controls, points at the relevant panel, and asks for a prediction before showing the result.

Mode profile

n_eff versus thickness · the cutoff map

Each curve is one mode. A curve is born at the substrate line (its cutoff) and climbs toward the core index as the guide thickens. The amber marker column is the current thickness.

Field view · propagation

Readouts · selected mode

Checked against

These validation rows run in your browser on every load: the solver's answers are pushed back through the exact dispersion relation and against closed-form limits.

Check	Expected	Computed	Tolerance

The table checks the slab solver against its own exact dispersion relation, the symmetric-slab cutoff at V = π, and limits known in closed form: field continuity, the thick-guide and weak-guide limits, and the mode count, plus two checks on the arithmetic behind the field view. A ± tolerance is the largest absolute difference that still passes, in the units of the Expected column; a percentage is relative to the expected value; “condition” means the Expected entry is itself the test, and “exact” means the outcome must match.

The model

A core of index $n_{f}$ and thickness $t$ lies between a substrate $n_{s}$ and a cover $n_{c}$ , and is taken as infinite in the other transverse direction. For a guided mode of effective index $N$ the field oscillates in the core with transverse wavenumber $κ$ and decays into each cladding with constant $γ_{i}$ :

κ = k_{0} \sqrt{n_{f}^{2} - N^{2}}

γ_{i} = k_{0} \sqrt{N^{2} - n_{i}^{2}}

with $k_{0} = 2 π / λ$ . Matching the fields at both interfaces gives the dispersion relation that the tool solves by bisection for each mode order $m$ :

κ t = m π + \arctan \frac{w_{s} γ_{s}}{κ} + \arctan \frac{w_{c} γ_{c}}{κ}

where $w_{i} = 1$ for TE and $w_{i} = n_{f}^{2} / n_{i}^{2}$ for TM. Mode $m$ is cut off when $N$ falls to the higher cladding index, which sets its cutoff thickness; in a symmetric guide mode $m$ appears at $V = k_{0} t \sqrt{n_{f}^{2} - n_{s}^{2}} = m π$ . The group index is $n_{g} = N - λ d N / d λ$ by central difference with the material indices held fixed, so it contains waveguide dispersion only. The confinement factor is the fraction of power inside the core, integrated from the profile (TM weights $H_{y}^{2}$ by $1 / n^{2}$ ). The model assumes lossless, isotropic, non-dispersive layers and a slab of unlimited width. A real ridge or strip guide is also confined laterally and has a lower effective index than the slab of the same thickness; the slab value is the first step of the effective index method for such guides.

Worked example

The default preset is a 220 nm silicon slab ( $n_{f} = 3.476$ ) clad on both sides by oxide ( $n_{s} = n_{c} = 1.444$ ) at 1550 nm, TE. Then $V = 2.820$ , below $π$ , so only TE0 is guided. Its solution is $N = 2.847782$ , for which $κ t = 1.7775$ and $γ / κ = 1.2315$ ; the two arctangent terms are 0.8888 each and sum to 1.7775, as the relation requires. The tool shows a group index of 3.5768, confinement of 81.0 %, a decay length $1 / γ$ of 101 nm on each side, TE1 appearing at t = 245.1 nm, and a TE0 to TM0 effective index difference of 0.7945 (TM0 has $N = 2.053320$ ).

References: D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed., Academic Press (1991). H. Kogelnik, “Theory of optical waveguides,” in T. Tamir (ed.), Guided-Wave Optoelectronics, Springer (1988). K. Okamoto, Fundamentals of Optical Waveguides, 2nd ed., Academic Press (2006).