Guided mode

A guided mode is a distribution of the electromagnetic field that travels along a waveguide or optical fiber with a fixed transverse profile, changing only in phase as it propagates. It exists because the core has a higher refractive index than its surroundings, so light that would otherwise spread out by diffraction is held near the core by total internal reflection. Each guided mode has its own effective index $n_\text{eff}$, which always lies between the cladding and core indices. In a 500 × 220 nm silicon strip waveguide in oxide at 1550 nm, the fundamental TE-like mode has $n_\text{eff}$ = 2.446, between 1.44 and 3.48; in standard single-mode fiber the one guided mode has $n_\text{eff} \approx 1.446$, only slightly above the cladding index of about 1.444.

Field form and effective index

A guided mode traveling along $z$ has the form

$$E(x,y,z,t) = E_\text{m}(x,y)\,e^{i(\beta z - \omega t)} ,$$

in the $e^{i(kz-\omega t)}$ convention this site follows, where $E_\text{m}$ is the transverse profile and $\beta = n_\text{eff}k_0$ is the propagation constant. Guidance requires

$$n_\text{clad} < n_\text{eff} < n_\text{core} .$$

For the silicon wire, with $k_0 = 2\pi/\lambda_0$ = 4.054 rad/µm at 1550 nm, this bounds $\beta$ between 5.84 rad/µm (oxide, 1.44) and 14.11 rad/µm (silicon, 3.48). The fundamental mode with $n_\text{eff}$ = 2.446 has $\beta$ = 9.92 rad/µm and a guided wavelength $\lambda_0/n_\text{eff}$ of 634 nm.

Outside the core the field of a guided mode decays as an evanescent wave, with field decay constant

$$\gamma = k_0\sqrt{n_\text{eff}^2 - n_\text{clad}^2} .$$

For the silicon wire, $\gamma$ = 8.00 µm⁻¹, so the field falls to 1/e about 125 nm outside the silicon. This tail couples neighboring waveguides in a directional coupler and scatters from sidewall roughness.

Cutoff and number of modes

A mode exists only while $n_\text{eff}$ stays above the cladding index. As the core shrinks or the wavelength grows, $n_\text{eff}$ of each mode falls, and when it reaches $n_\text{clad}$ the mode reaches cutoff and becomes a radiation mode leaking into the cladding. In a symmetric guide the fundamental mode has no cutoff.

How many modes a guide supports depends on its size relative to the wavelength and on its index contrast. For a step-index fiber this is captured by the V number, $V = 2\pi a\,\text{NA}/\lambda$: below 2.405 only the fundamental LP₀₁ mode is guided, and a large core guides about $V^2/2$ modes. A 50 µm core with NA 0.20 at 850 nm has $V$ = 37 and about 680 modes. For a symmetric slab of thickness $d$, the second TE mode appears when $d = \lambda_0/(2\,\text{NA})$. Silicon in oxide has NA $= \sqrt{3.48^2 - 1.44^2}$ = 3.17, giving 245 nm at 1550 nm, so a standard 220 nm silicon layer guides one TE and one TM slab mode. The cutoff wavelength and single-mode vs multimode entries cover fibers in detail.

Fundamental and higher-order modes

The fundamental mode has a single lobe and the highest effective index. Higher-order modes have nodes across the core, lower $n_\text{eff}$ and more power in the cladding, which makes them more sensitive to bends. Modes of a lossless guide are orthogonal: power stays in a mode unless a bend, taper or grating couples it to another. Because modes have different propagation constants, a field that excites two of them beats along the guide with period $\lambda_0/\Delta n_\text{eff}$; for the TE₀ and TE₁ modes of a 500 nm silicon strip (2.446 and 1.49) that beat length is about 1.6 µm.

Slab waveguides label modes TE and TM, strips TE-like and TM-like; weakly guiding fibers use LP labels, which group exact HE, EH, TE and TM modes of nearly equal $\beta$. The fiber's mode field diameter describes the size of its fundamental guided mode.

Measuring and computing modes

Mode profiles are imaged by focusing the output facet onto a camera; a single-lobed pattern that does not change when the launch position or angle is varied indicates single-mode operation. Effective indices are inferred from Bragg grating wavelengths, grating coupler angles or ring resonance orders. Rectangular waveguides are solved numerically; the Waveguide Mode Explorer computes slab modes and their cutoffs interactively.

Common questions

What is the difference between a guided mode and a radiation mode?

A guided mode has $n_\text{eff}$ above the cladding index and is evanescent outside the core. Radiation modes have $n_\text{eff}$ below the cladding index, oscillate in the cladding, carry power away and form a continuum instead of a discrete set.

Why can a guided mode not have an effective index above the core index?

The field would then have to decay in every region, including the core, and no bounded solution of the wave equation has that form.

Is a guided mode the same as a transverse mode of a laser?

Both are field patterns that keep their shape, and fiber texts often call guided modes transverse modes. A laser's transverse modes are set by its mirrors and gain profile as well as any guiding, so the two coincide only when the laser cavity is itself a waveguide, as in a diode laser or fiber laser.

References: Snyder & Love, Optical Waveguide Theory (Chapman and Hall, 1983); K. Okamoto, Fundamentals of Optical Waveguides 2nd ed. (Academic Press, 2006); Saleh & Teich, Fundamentals of Photonics 3rd ed. 2019; L. Chrostowski and M. Hochberg, Silicon Photonics Design (Cambridge University Press, 2015).