Effective index method

The effective index method (EIM) estimates the effective index and lateral mode shape of a waveguide with a rectangular or stepped cross-section, such as a silicon strip, a rib, or the ridge of a semiconductor laser, by replacing the two-dimensional problem with two one-dimensional slab waveguide problems. Its accuracy is typically within a few percent in effective index for well-guided modes; for the standard 500 × 220 nm silicon strip at 1550 nm it gives 2.49 where a full vectorial solution gives 2.45, an overestimate of 1.7%.

The two-step reduction

The cross-section is divided into vertical columns, each a uniform stack of layers: under a rib, the full film thickness; beside it, the thinner slab or the cladding alone.

1. Each column is solved as a vertical slab, giving an effective index $N_j$ for column $j$. For a quasi-TE mode (electric field mainly horizontal) the vertical slab is solved for TE polarization.
2. The columns are replaced by a horizontal slab whose layers have indices $N_j$ and the column widths. This slab is solved with the opposite polarization, TM for a quasi-TE mode, because the dominant field is now normal to the lateral interfaces.

The effective index of this second slab is the EIM estimate for the channel mode, and its field profile approximates the lateral mode shape.

Worked examples

For a silicon film of index 3.476 between silica layers of index 1.444 at 1550 nm, the 220 nm TE slab has $N$ = 2.848. A strip 500 nm wide is then a horizontal symmetric slab with core index 2.848 and cladding 1.444; solving it for TM gives

$$n_\text{eff} \approx 2.49,$$

the value quoted above. Widening the strip to 600 nm raises the estimate to 2.59, and narrowing it to 400 nm lowers it to 2.32.

For a rib, the outer columns are a 90 nm slab of silicon, whose TE effective index is 2.103. A rib 500 nm wide then has a lateral index step from 2.848 to 2.103, smaller than the strip's step to 1.444, and the EIM gives 2.60; a 1 µm rib gives 2.77. The reduced lateral contrast is what lets ribs stay single-mode at larger widths and with less sidewall scattering, as covered in strip vs rib waveguide.

In a ridge-waveguide laser the same procedure gives the lateral index step: the vertical stack under the ridge and the thinner stack beside it, where the upper cladding has been etched, have effective indices differing by a few parts in a thousand, and that difference sets the lateral mode count and the ridge width at which a second mode appears.

Accuracy and limits

The method assumes the field separates into a product of a vertical and a horizontal function, which is exact only for a uniform slab. Near corners the real field does not separate, and the EIM consistently overestimates lateral confinement for high-contrast guides. The error is largest near cutoff: for the 220 nm silicon strip the EIM predicts a second lateral mode above a width of 316 nm, while a full vectorial solver places that cutoff near 450 nm, and at 500 nm it gives the second mode an index of 1.59 against the solver's 1.49. Single-mode widths taken from the EIM are therefore overly conservative for high-contrast strips.

Other limits:

• Polarization coupling and hybrid modes. The EIM treats TE and TM separately and cannot describe hybrid modes or the polarization conversion in bent or slanted guides; see TE and TM polarization.
• Lateral leakage in shallow ribs. A TM-like mode of a rib can couple to the TE slab mode of the surrounding film and leak sideways at particular widths. The EIM's scalar lateral slab cannot represent this loss.
• Group index and dispersion. These require derivatives of a quantity that is already approximate, so errors in group index are typically larger than those in $n_\text{eff}$.

The weighted index method and Marcatili's approximate analytic treatment of rectangular cores are alternatives that improve on the EIM for some geometries, but the standard practice is to use the EIM for trends and starting dimensions and a numerical mode solver for final design. The article on choosing a photonics simulation method places the EIM among the other options.

Common questions

Which polarization goes in which step?

For a quasi-TE mode, solve the vertical slab for TE and the lateral slab for TM; for a quasi-TM mode, the reverse. Solving the 500 nm strip's lateral slab as TE instead gives 2.63, far from the solver's 2.45.

Is the effective index method accurate enough for ring resonator design?

It is adequate for choosing a waveguide width, coupling gap or bend radius to start from. With a group index of 4.18, a 0.04 error in effective index moves a resonance by about 15 nm at 1550 nm, so resonance wavelengths and free spectral ranges are taken from a full solver and then calibrated against measurement.

References: K. Okamoto, Fundamentals of Optical Waveguides, 2nd ed. (Academic Press, 2006); G. T. Reed and A. P. Knights, Silicon Photonics: An Introduction (Wiley, 2004); E. A. J. Marcatili, "Dielectric rectangular waveguide and directional coupler for integrated optics," Bell Syst. Tech. J. 48, 2071 (1969); B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019).