Slab waveguide

A slab waveguide, or planar waveguide, is a thin film of higher refractive index sandwiched between two layers of lower index, unbounded in the plane of the film. Light launched along the film is trapped in the thickness direction by total internal reflection at the two boundaries but is free to diffract sideways, so a slab confines light in one dimension only. It is the simplest waveguide with exact solutions and the basis for analyzing channel waveguides. The 220 nm silicon device layer of a silicon-on-insulator wafer, clad by oxide, is a slab that guides a single TE mode at 1550 nm with an effective index of 2.848.

Modes and the single-mode condition

The field in a slab is a set of discrete guided modes, each a standing wave across the film with exponentially decaying evanescent tails in the cladding, traveling along the film with effective index between the cladding and core indices. For a symmetric slab of thickness $d$, core index $n_1$ and cladding index $n_2$, this entry uses the normalized frequency

$$V = \frac{\pi d}{\lambda}\sqrt{n_1^2 - n_2^2} ,$$

defined with the half-thickness, as in the V-number entry. With this convention the $m$-th mode of each polarization reaches cutoff at $V = m\pi/2$, so the slab guides $\lfloor 2V/\pi \rfloor + 1$ TE modes and the same number of TM modes. The fundamental TE0 and TM0 modes of a symmetric slab have no cutoff; the slab is single mode in each polarization when $V < \pi/2$, that is, when

$$d < \frac{\lambda}{2\sqrt{n_1^2 - n_2^2}} .$$

Some texts define $V$ with the full thickness, $V = (2\pi d/\lambda)\sqrt{n_1^2 - n_2^2}$, in which case the single-mode limit reads $V < \pi$; the physical thickness limit is the same.

For silicon ($n_1$ = 3.476) in silica ($n_2$ = 1.444) at 1550 nm, $\sqrt{n_1^2 - n_2^2}$ = 3.162 and the limit is 245 nm. A 220 nm film has $V$ = 1.41 and carries one mode per polarization; a 500 nm film has $V$ = 3.20 and carries three. For silicon nitride ($n$ = 1.996) in silica the same limit is 562 nm, and for low-contrast glass waveguides it is several micrometers, which is how index contrast sets the scale of a photonic platform.

TE and TM modes

A slab's modes divide exactly into two polarizations. TE modes have their electric field parallel to the film; TM modes have their magnetic field parallel to it and an electric field component normal to it, which is discontinuous at the boundaries. The TE and TM polarization entry gives the eigenvalue equations. In a high-contrast slab the two differ strongly: for the 220 nm silicon slab at 1550 nm the TE0 effective index is 2.848 and the TM0 index about 2.05, because the TM field extends much further into the oxide. The TE0 evanescent field decays by $1/e$ over about 0.10 µm in the oxide.

Where slabs appear in practice

Effective index method. A channel waveguide such as a silicon strip or rib has no exact solution. The effective index method treats its cross-section as two slabs in turn: first the vertical slab (the film thickness) gives an effective index for each lateral region, and those values become the core and cladding indices of a horizontal slab whose solution approximates the channel mode. For a 500 × 220 nm silicon strip it gives 2.49 against 2.45 from a full vectorial solver, adequate for trends.

Free-propagation regions. The star couplers of an arrayed waveguide grating and of N × N star splitters are slab regions: light is confined vertically but diffracts freely in the plane, so a waveguide ending at the slab edge radiates a fan of light that the opposite edge collects.

Rib waveguides. In a rib waveguide, the thinner layer left on either side of the rib after a partial etch, for example 90 nm of the 220 nm silicon, is also called the slab. The rib mode sits above a slab whose own effective index forms its lateral cladding.

Lasers and thin films. The double heterostructure of a semiconductor laser is a slab in the growth direction, and prism coupling to the modes of a deposited film gives its index and thickness.

Pitfalls

An asymmetric slab, such as silicon on oxide with air above, has different claddings on the two sides, and then even its fundamental mode has a cutoff thickness; the symmetric formula above does not apply. The single-mode condition gives the onset of the first-order mode, which is weakly guided just above cutoff and may or may not be excited in practice.

Common questions

What is the difference between a slab waveguide and a channel waveguide?

A slab confines light in one transverse direction and is uniform in the other; a channel waveguide, such as a strip, rib or fiber core, confines it in both, so the light cannot spread sideways.

Why is silicon photonics built on 220 nm films?

A 220 nm silicon slab in oxide is below the 245 nm single-mode thickness at 1550 nm, so the film guides one TE mode vertically, while about 80% of the TE0 power stays in the silicon.

References: A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications, 6th ed. (Oxford University Press, 2007); K. Okamoto, Fundamentals of Optical Waveguides, 2nd ed. (Academic Press, 2006); B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019).