Effective index (n_eff)

The effective index of a guided mode is the refractive index the mode behaves as if it traveled in: its phase advances along the waveguide as a plane wave's would in a uniform medium of index $n_{\text{eff}}$. It is defined from the mode's propagation constant $\beta$ as

$$n_{\text{eff}}=\frac{\beta }{k_0}=\frac{\beta {\lambda }_0}{2\pi }=\frac{\beta c}{\omega },$$

where $k_0=2\pi \mathrm{/}{\lambda }_0$ is the free-space wavenumber, ${\lambda }_0$ the free-space wavelength and $c$ the speed of light. The phase velocity of the mode is $c\mathrm{/}{n}_{\text{eff}}$ and its wavelength along the guide ${\lambda }_0\mathrm{/}{n}_{\text{eff}}$. For the fundamental TE mode of a 500 × 220 nm silicon strip waveguide at 1550 nm, $n_{\text{eff}}$ = 2.446, so the guided wavelength is 634 nm. A guided mode always has $n_{\text{clad}}<n_{\text{eff}}<n_{\text{core}}$: part of its field travels in the cladding, and the more tightly the mode is confined, the closer $n_{\text{eff}}$ comes to the core index.

What it means physically

In the ray picture of a slab waveguide, light bounces between the two core boundaries at an angle $\theta$ to their normal, and only the component of its wavevector along the guide counts toward the phase advance, so $n_{\text{eff}}=n_{\text{core}}\sin \theta$. Total internal reflection requires $\sin \theta >n_{\text{clad}}\mathrm{/}{n}_{\text{core}}$, which gives the lower bound, and a ray running straight down the guide would give the upper one. In the wave picture, $n_{\text{eff}}$ is roughly an average of the core and cladding indices weighted by where the mode's power lies, which is why a narrower or thinner core, which pushes more of the field into the cladding, lowers it.

Values

Computed values at 1550 nm for the fundamental TE-like mode unless stated, with oxide cladding on all sides:

The silicon and nitride rows were computed with a full-vectorial finite-difference mode solver from the bulk indices of silicon (3.476), silica (1.444) and stoichiometric silicon nitride (1.996) at 1550 nm; measured values for fabricated waveguides differ somewhat, depending on their exact dimensions and sidewall angle. The fiber's effective index is computed for a step-index core with a 0.36% index step, and its group index is Corning's specification for SMF-28.

Width and polarization

The effective index of a silicon strip 220 nm thick depends strongly on its width:

As the strip widens, the TE index approaches the 2.85 of an unbounded 220 nm slab; as it narrows, the index falls toward the oxide's 1.444 and the mode spreads out, which is how an inverse taper enlarges the mode for coupling to fiber. A second TE mode appears once the width passes about 450 nm and is still barely guided at 500 nm, with an effective index of 1.49, so routing waveguides are kept at or below about 500 nm. The TE and TM indices differ by 0.67 at 500 nm, the large birefringence that makes most silicon circuits work in one polarization only.

The steep width dependence also sets fabrication tolerance. Between 450 and 500 nm the TE index rises by 0.0018 per nanometer of width, so a ring resonator made 1 nm wider shifts its resonances by $\lambda \mathrm{\Delta }{n}_{\text{eff}}\mathrm{/}{n}_g$, about 0.7 nm at 1550 nm.

Calculating it

Slab waveguides and round step-index fibers have exact solutions, as the roots of transcendental equations; rectangular waveguides do not, and are solved numerically with finite-difference or finite-element mode solvers. The effective index method is the classic approximation: it replaces the 2D cross-section by a vertical slab, whose effective index becomes the core index of a horizontal slab solved next. For the 500 × 220 nm strip it gives 2.49 against the full solution's 2.45, close enough for trends but not for resonance design. The Waveguide Mode Explorer solves slab waveguides exactly and shows how $n_{\text{eff}}$ moves with thickness, wavelength and polarization.

Which numerical method to use for computing effective indices and simulating devices is covered in Choosing a Photonics Simulation Method.

Where it is used

The effective index enters every condition that depends on the phase of a guided wave:

• Bragg gratings: a grating of period $\mathrm{\Lambda }$ reflects ${\lambda }_B=2n_{\text{eff}}\mathrm{\Lambda }$ (the Bragg condition).
• Grating couplers: light leaves a grating coupler into air at the angle given by $\sin \theta =n_{\text{eff}}-{\lambda }_0\mathrm{/}\mathrm{\Lambda }$.
• Ring resonators: a ring of circumference $L$ resonates at wavelengths where $n_{\text{eff}}L=m{\lambda }_0$ for integer $m$.
• Phase shifters and interferometers: a change $\mathrm{\Delta }{n}_{\text{eff}}$ over a length $L$ shifts the phase by $2\pi \mathrm{\Delta }{n}_{\text{eff}}L\mathrm{/}{\lambda }_0$; a π shift in 1 mm at 1550 nm takes $\mathrm{\Delta }{n}_{\text{eff}}=7.75\times {10}^{-4}$.
• Directional couplers: the length for full power transfer is ${\lambda }_0\mathrm{/}[2(n_e-n_o)]$, set by the effective indices of the even and odd supermodes (directional coupler).

Spacings between resonances, and anything to do with pulses and delay, depend on the group index rather than the effective index: the free spectral range of a ring is ${\lambda }_0^2\mathrm{/}(n_{g}L)$, not ${\lambda }_0^2\mathrm{/}(n_{\text{eff}}L)$.

Common questions

What is the effective refractive index?

The same quantity: the propagation constant of a guided mode divided by the free-space wavenumber. It describes how fast the phase of the mode advances, and it lies between the cladding and core indices.

Why is the effective index lower than the core index?

Because part of the mode's field travels in the lower-index cladding, and in the ray picture because the light zigzags rather than traveling straight along the guide.

What is the difference between effective index and group index?

The effective index sets the phase velocity, $c\mathrm{/}{n}_{\text{eff}}$; the group index sets the speed of a pulse or of energy, $c\mathrm{/}{n}_g$, and includes how $n_{\text{eff}}$ changes with wavelength: $n_g=n_{\text{eff}}-{\lambda }_{0}d{n}_{\text{eff}}\mathrm{/}d{\lambda }_0$. In a silicon strip the two are 2.446 and 4.18.

How is the effective index measured?

Indirectly, from a quantity that depends on it with the other factors known: the reflection wavelength of a Bragg grating of known period, the emission angle of a grating coupler, or the resonance order of a ring together with its group index.

References: K. Okamoto, Fundamentals of Optical Waveguides, 2nd ed. (Academic Press, 2006). L. Chrostowski and M. Hochberg, Silicon Photonics Design (Cambridge University Press, 2015). Z. Zhu and T. G. Brown, "Full-vectorial finite-difference analysis of microstructured optical fibers," Optics Express 10, 853 (2002), for the mode solver. H. H. Li, "Refractive index of silicon and germanium and its wavelength and temperature derivatives," Journal of Physical and Chemical Reference Data 9, 561 (1980). I. H. Malitson, "Interspecimen comparison of the refractive index of fused silica," Journal of the Optical Society of America 55, 1205 (1965). K. Luke et al., "Broadband mid-infrared frequency comb generation in a Si3N4 microresonator," Optics Letters 40, 4823 (2015), for the silicon nitride index.