Index contrast

Index contrast is how much higher the refractive index of a waveguide core is than that of the material around it. It is quoted either as the plain difference $\Delta n = n_1 - n_2$ or as the relative index difference $\Delta$ defined below. Standard single-mode fiber has a core only about 0.005 above its silica cladding, $\Delta \approx 0.36\%$. A silicon nitride waveguide ($n \approx 2.0$) clad in silica ($n \approx 1.44$) has $\Delta n = 0.56$ and $\Delta \approx 24\%$; a silicon wire ($n \approx 3.48$) in silica has $\Delta n = 2.04$ and $\Delta \approx 41\%$. Nearly every property of a waveguide's guided modes, from their size to how tightly the guide can bend, follows from this one number together with the core dimensions.

Definitions

The relative index difference used in fiber optics is

$$\Delta = \frac{n_1^2 - n_2^2}{2 n_1^2} \approx \frac{n_1 - n_2}{n_1},$$

where the approximation holds when the contrast is small. The same quantity sets the numerical aperture,

$$\mathrm{NA} = \sqrt{n_1^2 - n_2^2} = n_1\sqrt{2\Delta},$$

and through it the V-number $V = 2\pi a\,\mathrm{NA}/\lambda$. For single-mode fiber with $n_2 = 1.444$ and $\Delta = 0.36\%$, the core index is 1.4492 and the NA 0.123. For silicon and silicon nitride the small-contrast approximation fails badly: for silicon in oxide, $(n_1 - n_2)/n_1 = 0.59$ while $\Delta = 0.41$. Papers and datasheets use $\Delta$, $\Delta n$ and $\Delta n/n$ interchangeably under the same name, so the definition should be checked before comparing numbers.

Measurement

For fiber, the index profile is measured directly with the refracted near-field method, which scans a focused spot across a cleaved end and records how much light escapes the cladding at each point; the maximum step gives $\Delta$. A far-field measurement of the NA gives an indirect value that depends on the angle criterion used. For thin films on a chip, the core and cladding indices are measured separately by ellipsometry or a prism coupler, and the contrast is their difference. In a rib waveguide the contrast that confines the mode sideways is between the effective indices of the thick and thin slab regions, which is much smaller than the material contrast; the strip vs rib waveguide entry compares the two geometries.

What it sets

Mode size. The single-mode core size scales as $\lambda/\mathrm{NA}$. A symmetric slab is single mode for thickness $d < \lambda/(2\,\mathrm{NA})$, which at 1550 nm is 245 nm for silicon in oxide and 558 nm for silicon nitride. A step-index fiber with NA 0.123 is single mode at 1550 nm for core diameters below 9.6 µm. The resulting mode field diameter is about 10.4 µm in fiber and a few hundred nanometers in a 500 × 220 nm silicon strip.

Bend radius. A mode in a bend must keep pace with the outer edge, and the field beyond a certain radius radiates. Higher contrast keeps that point further out and lets the guide bend more tightly: standard fiber is normally kept to bend radii of a few centimeters, silicon strips bend with negligible radiation at a few micrometers, and silicon nitride lies between, from tens of micrometers to millimeters depending on the core thickness. The bend loss entry gives the numbers. Tight bends are what make dense photonic circuits and small ring resonators possible on silicon-on-insulator.

Confinement. A high contrast puts most of the mode in the core and shortens the evanescent tail, which raises the confinement factor, allows closely spaced guides with little crosstalk, and makes the mode strongly sensitive to width and thickness.

Scattering loss. Each excursion of an etched wall perturbs the permittivity by $n_1^2 - n_2^2$, so the power scattered by sidewall roughness grows roughly as the square of that step, weighted by the field intensity at the wall. For the same roughness and wall field, silicon in oxide scatters about 27 times more than silicon nitride by this measure. This, together with the weaker field at the wall of a less tightly confined mode, is the main reason low-contrast platforms, from silica planar circuits to thin silicon nitride, reach losses far below those of silicon wires.

Pitfalls

High contrast brings strong polarization dependence, because the boundary conditions for the two field components differ more at a large step; TE and TM modes in a silicon wire have very different effective and group indices. It also makes coupling to fiber harder, since the chip mode is many times smaller than the fiber mode and a converter such as an inverse taper or grating coupler is needed. Low contrast has the opposite costs: large bends, large devices and weak confinement.

Common questions

What index contrast does standard single-mode fiber have?

About 0.36% relative index difference, a core index step of roughly 0.005 above pure silica, giving an index-profile NA near 0.12; datasheets quote 0.14, measured from the far field. Many bend-insensitive fibers keep a similar step and add a depressed-index trench in the cladding around the core.

Is high index contrast always better?

It depends on the application. It enables compact bends, small rings and dense routing, at the price of higher scattering loss, tighter fabrication tolerances and harder fiber coupling. Long low-loss delay lines and high-Q resonators are usually built on lower-contrast platforms.

References: A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983); 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); F. P. Payne and J. P. R. Lacey, "A theoretical analysis of scattering loss from planar optical waveguides," Optical and Quantum Electronics 26, 977 (1994).