Substrate leakage

Substrate leakage is the loss of light from a waveguide into a substrate whose refractive index is higher than the mode's effective index. In silicon on insulator the silicon core sits on a buried oxide (BOX), usually 2–3 µm thick, which in turn sits on a silicon handle wafer of index 3.48. The mode's evanescent tail extends into the BOX, and where it reaches the handle it couples to waves that propagate down into the substrate and do not return. The lower cladding of the core and cladding structure is then a barrier of finite thickness, and the loss falls exponentially as it thickens.

Decay of the evanescent tail

In a cladding of index $n_\text{ox}$ the field of a mode with effective index $n_\text{eff}$ decays as $e^{-\gamma x}$ with

$$\gamma = k_0\sqrt{n_\text{eff}^2 - n_\text{ox}^2}, \qquad k_0 = \frac{2\pi}{\lambda}.$$

For the TE mode of a 220 nm silicon slab waveguide in oxide at 1550 nm ($n_\text{Si}$ = 3.476, $n_\text{ox}$ = 1.444), a slab solution gives $n_\text{eff}$ = 2.848. Then $k_0$ = 4.054 µm⁻¹ and $\gamma$ = 9.95 µm⁻¹: the field falls by $1/e$ every 0.10 µm. The power in the tail falls by $e^{-2\gamma} \approx 2 \times 10^{-9}$ per micrometer of oxide, and by $e^{-2\gamma d} \approx 5 \times 10^{-18}$ across a 2 µm BOX. This is an order-of-magnitude estimate of the coupling to the substrate; the loss itself needs a calculation that includes it.

Computed leakage versus oxide thickness

Solving for the complex effective index of the four-layer slab (silicon substrate, BOX, 220 nm silicon core, semi-infinite oxide above) at 1550 nm gives:

Each additional 0.5 µm of oxide divides the TE loss by about 2 × 10⁴ and the TM loss by about 370, as the decay constants predict. These are slab results; a 450–500 nm wide channel waveguide has a lower effective index and leaks more, and a two-dimensional mode solver with an absorbing boundary gives its value.

What makes it worse

• Lower effective index. Since $\gamma$ grows with $n_\text{eff}^2 - n_\text{ox}^2$, any mode close to the cladding index leaks first. The TM mode of the 220 nm slab has $n_\text{eff}$ = 2.053 and $\gamma$ = 5.92 µm⁻¹, which is why it needs a thicker BOX in the table. Narrow waveguides, modes near cutoff and the expanded mode at an inverse taper tip, which can lose several dB/mm on a 2 µm BOX if the tip is narrow, are affected in the same way.
• Lower core index. In silicon nitride photonics the core index is about 2.0, and the TE mode of a 400 nm nitride slab has $n_\text{eff}$ = 1.743 and $\gamma$ = 3.96 µm⁻¹. The same slab calculation gives 4 × 10⁻³ dB/cm on 2 µm of oxide and 1.5 × 10⁻⁶ dB/cm on 3 µm. For low-loss nitride, where the target is a few dB per meter or less, the lower oxide is usually made thicker than in SOI for this reason.
• Longer wavelength. At 2.0 µm the 220 nm silicon slab has $n_\text{eff} \approx 2.63$ and $\gamma \approx 6.9$ µm⁻¹, against 9.95 µm⁻¹ at 1550 nm. Mid-infrared waveguides therefore use thicker cores and lower claddings, and beyond about 3.6 µm the oxide also absorbs.

Grating couplers and other radiating structures

A grating coupler diffracts light both up toward the fiber and down toward the substrate. The downward part crosses the BOX, partly reflects at the oxide–silicon interface and partly enters the handle, so the BOX thickness sets the phase of the reflected wave and therefore the directionality. Bottom reflectors made of metal or of a distributed Bragg mirror under the grating, or local removal of the substrate, return more of the downward light and raise the coupling efficiency.

Measurement and diagnosis

Leakage adds a length-proportional term to the measured propagation loss that a cutback alone does not separate from sidewall scattering. It shows as a loss that is much higher for TM, rises with wavelength faster than scattering would, and drops on wafers with a thicker BOX.

Common questions

How thick does the buried oxide need to be?

For 220 nm silicon strip waveguides at 1550 nm, 2 µm makes TE leakage negligible; 3 µm adds margin for TM, expanded modes and longer wavelengths.

Why does the TM mode leak more than TE?

Its effective index is lower, closer to the oxide index, so its evanescent tail decays more slowly and reaches the substrate with more amplitude.

Can substrate leakage be removed after fabrication?

Locally, by undercutting the silicon handle beneath a structure with an isotropic etch, as in suspended edge couplers. Globally it is set by the wafer's oxide thickness.

References: B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019); K. Okamoto, Fundamentals of Optical Waveguides, 2nd ed. (Academic Press, 2006); L. Chrostowski and M. Hochberg, Silicon Photonics Design (Cambridge University Press, 2015).