Waveguides

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A waveguide is a strip of transparent material that confines light and guides it along a path. On a silicon chip, the standard waveguide is a silicon strip 500 nm wide and 220 nm high, surrounded by silicon dioxide [1]. It plays the role that a wire plays in an electronic circuit. This article explains how such a strip holds light and what sets the phase of the light along it.

Guiding light

A beam of light in open space spreads as it travels, and the narrower the beam, the faster it spreads. A waveguide prevents this. It consists of a core of high refractive index surrounded by a cladding of lower index: on a silicon chip, silicon ( $n \approx 3.47$ ) in silicon dioxide ( $n \approx 1.44$ ) [1]. The layers are made from a silicon-on-insulator (SOI) wafer, a silicon substrate carrying a 2 µm layer of oxide and on top of it the 220 nm layer of silicon from which the waveguides are etched [1].

The ray picture: total internal reflection

The simplest description treats light as rays. A ray that meets the boundary between two materials is partly reflected and partly refracted into the other material. Its direction there follows Snell’s law, $n_1 \sin\theta_1 = n_2 \sin\theta_2$ , with the angles measured from the normal to the boundary. For a ray going from a high index $n_1$ to a low index $n_2$ , the refracted ray bends away from the normal. At the critical angle

\sin\theta_\text{c} = \frac{n_2}{n_1} \tag{1}

it runs along the boundary, and for any larger angle no light enters the second material at all: the ray is reflected completely [2]. This is total internal reflection. For silicon in oxide, $\theta_\text{c} = 24.6°$ . A ray inside the core that meets the boundaries at more than this angle is reflected back and forth and stays in the core.

Figure 1 shows a ray at a boundary. How the light divides between the reflected and the refracted ray follows from the Fresnel equations [3]. As the angle grows towards the critical angle, the reflected ray becomes brighter and the refracted one weaker, without a jump, until at the critical angle all the light is reflected [3]. From oxide into silicon, towards the higher index, the ray bends towards the normal, and there is no critical angle. Below the rays, Figure 1 shows the same light as a wave. Beyond the critical angle the field does not stop at the boundary: it reaches a fraction of a micrometre into the second material and decays there, an evanescent wave that carries no power away [3].

the field, pointing out of the drawing (+) or into it (−)−+

From → into

Angle of incidence θ₁15.0 °

reflected24.6 %

refracted75.4 %

angle of the refracted ray θ₂: 38.5 °
critical angle θc: 24.6 °

Angles from the normal (dotted). The brightness of each ray is its share of the power, for light whose field lies along the boundary (s-polarisation, as for the TE modes of the series). Below the rays, the same light as a wave at one instant, 1550 nm: the pattern of crests above the boundary is the incident and the reflected wave together; beyond the critical angle, the field below decays within a fraction of a micrometre. Indices at 1550 nm; air taken as 1.

A ray in silicon meets the boundary to oxide at 15.0°. 24.6 % of the power is reflected; the rest enters the oxide at 38.5°.

Figure A ray of light at the boundary between two materials. Above: the rays; the brightness of the reflected and the refracted ray is their share of the power, for light whose field lies along the boundary. The dashed line marks the critical angle; beyond it, no light enters the second material. Below: the same light as a wave, its field at one instant, after Hecht [3].

The wave picture: modes

Rays are a model with limits. They describe light well when everything it passes is much larger than its wavelength [4]; the core of a silicon waveguide is smaller than the wavelength of the light in it.

In the wave picture, a ray is a plane wave, and a ray zig-zagging through the core is a wave that crosses the core and returns. After one round trip across the core, the wave meets itself. Interference (Interference) decides what happens: unless the wave arrives back in step with itself, its repeated reflections cancel. A pattern survives only if the phase it gains on one round trip, including the phase shifts at the two reflections, is a whole number of cycles, $2\pi m$ [2]. Only certain angles fulfil this condition.

Each allowed pattern is a mode: a distribution of the field across the waveguide that travels along it without changing its shape. A thin core allows only one angle and therefore one mode; such a waveguide is single-mode. A thicker core allows several, which travel at different speeds.

Figure 2 shows the modes of the series’ standard waveguide, a silicon strip 220 nm high, as their field in the cross-section. They are computed numerically from the wave equation, since a rectangular core has no exact solution. The field of a mode oscillates in time, but its pattern keeps its shape, and a higher mode has more places where its field changes sign. Across the core a mode is a standing wave, the sum of two plane waves that cross the core in opposite directions, the zig-zagging ray of the ray picture; along the waveguide it travels [4].

The modes come in two polarisations. In a TE mode the electric field points mainly across the width of the strip, in a TM mode mainly along its height [1]. The 500 nm strip guides one TE and one TM mode; Figure 2 shows the TE modes, the polarisation most silicon devices use [1]. A second TE mode is barely guided and is lost by scattering at the sidewalls, so the strip works as a single-mode waveguide for TE light [1]. Strictly, a strip 220 nm high has a single TE mode only below a width of 440 nm, and above 660 nm it also has a second TM mode [1].

field E_x, across the width−+

Width of the strip500 nm

guided TE modes: …
height: 220 nm

Silicon in oxide at 1550 nm, computed on a grid of 20 nm with one field component per mode, which puts n_eff about 1.5 % above a full calculation (2.443 for the 500 nm strip).

Computing the modes.

Figure The guided modes of a silicon strip 220 nm high in oxide at 1550 nm, each as its field in the cross-section: red where the field points one way, blue where it points the other, and the background where it is zero. The outline is the silicon. Drawn are the TE modes, with the field component across the width; the TM modes are not shown. Under each mode: its effective index.APX-019

The figure also shows that a mode is not confined to the core. Its field reaches into the oxide and decays there within a few hundred nanometres; for the TE mode of the 500 nm strip, about a tenth of the field lies outside the silicon [1]. This is the evanescent wave of Figure 1. Two waveguides placed close together overlap through these tails, which the article Splitters and combiners uses.

The effective index

Along the waveguide, a mode travels as if it were a plane wave in a uniform material with the index $n_\text{eff}$ , its effective index [2]. For a guided mode, $n_\text{eff}$ lies between the index of the cladding and that of the core [2]: part of the mode travels in the oxide, and the zig-zag path covers less distance along the waveguide than the ray travels. The 220 nm slab has $n_\text{eff} = 2.845$ [1]; the 500 nm × 220 nm strip, which also confines the light sideways, has $n_\text{eff} = 2.443$ [1].

The effective index takes the place that the refractive index has in the article Phase. Along a waveguide of length $L$ the light gains the phase

\varphi = \frac{2\pi}{\lambda}\, n_\text{eff} L = \beta L, \qquad \beta = \frac{2\pi n_\text{eff}}{\lambda}, \tag{2}

where $\beta$ is the propagation constant. In the standard strip waveguide at 1550 nm, the phase advances by one cycle every 0.63 µm, and a waveguide 100 µm long holds 158 cycles.

Equation (2) also shows how sensitive the phase is. A change of the effective index by 0.001 changes the phase over 1 mm of waveguide by 0.65 cycles. Such changes arise because fabrication never reproduces the designed width and thickness exactly [1]. Circuits that rely on phase differences need a way to adjust them after fabrication, which the article Phase shifters describes.

Modes of an optical fibre

An optical fibre is a waveguide too: a round core of glass in a cladding of glass with a slightly lower index. Its modes follow from the same wave equation, written for a round core. Inside the core the field is a Bessel function $J_l$ of the distance from the axis, outside it a decaying function $K_l$ , and around the axis it varies as $\cos l\varphi$ [4]. Figure 3 shows the patterns these modes form, rings and petals, labelled $\mathrm{LP}_{lm}$ .

How many modes a fibre guides depends on one number, $V = 2\pi a\,\mathrm{NA}/\lambda$ , where $a$ is the radius of the core and $\mathrm{NA} = \sqrt{n_1^2 - n_2^2}$ the numerical aperture. Below $V = 2.405$ only the fundamental mode $\mathrm{LP}_{01}$ is guided [4]. The index step of a fibre is small: in Figure 3 the core’s index is 0.25 % above the cladding’s [4], while silicon’s index is 2.4 times that of oxide. With so small a step, the fibre stays single-mode up to a core radius of 5.8 µm at 1550 nm, a core more than twenty times as wide as the silicon strip.

LP₀₁ · n_eff 1.4464

LP₁₁ · n_eff 1.4456

LP₂₁ · n_eff 1.4445

LP₀₂ · n_eff 1.4441

field−+

Core radius a12.0 µm

V: 4.97
guided LP modes: 4
LP₀₁ in the core: 97 %

A glass fibre with a core index of 1.447, 0.25 % above the cladding, at 1550 nm; the circle is the edge of the core. Modes with l > 0 are drawn in their cos lφ orientation; each also exists turned by a quarter period of the pattern. Single-mode below a radius of 5.8 µm (V = 2.405).

A fibre core of radius 12.0 micrometres has V = 4.97 and guides 4 LP modes: LP01, LP11, LP21, LP02.

Figure The guided modes of a glass fibre at 1550 nm, each as its field in the cross-section, coloured as in Figure 2; the circle is the edge of the core. In

\mathrm{LP}_{lm}

, the field changes sign

2l

times around the axis, and

m - 1

times along a radius inside the core. Under each mode: its effective index.APX-020

Loss and decibels

A waveguide loses light along its length, mainly by scattering at the roughness of its etched sidewalls [1]. Every other component loses a little as well, and on its way through a circuit the light passes many of them. The fractions of power that the components pass multiply: a component that passes half the light followed by one that passes a tenth passes a twentieth. Photonics therefore counts losses on a logarithmic scale, in decibels (dB). A power ratio $P_\text{out}/P_\text{in}$ corresponds to

10\log_{10}\frac{P_\text{out}}{P_\text{in}}\ \mathrm{dB}, \tag{3}

and since the logarithm turns products into sums, the dB values of components in a row add [4]. A loss is quoted as a positive number: a component with a loss of 3 dB has the ratio −3 dB and passes half the power. Table 1 lists the values that come up most often.

Loss	Power that remains
0.1 dB	97.7 %
1 dB	79 %
3 dB	50 %
10 dB	10 %
20 dB	1 %
30 dB	0.1 %

Table Losses in decibels and the share of the power that remains, from Eq. (3).

The propagation loss of a waveguide is given per length. For strip waveguides from advanced processes it is about 2 dB/cm [1], so after 1 cm about 63 % of the power remains. Bends add little: because silicon and oxide differ so much in index, the light stays in the core even where the waveguide bends tightly, and a 90° bend of 5 µm radius has been measured to lose about 0.01 dB [1]. Circuits can therefore be routed in a small area, and for circuits a few millimetres long the loss is small but not negligible.

A power itself can be given on the same scale by comparing it with 1 mW. The unit is the dBm:

P_\text{dBm} = 10\log_{10}\frac{P}{1\,\mathrm{mW}}, \tag{4}

so 1 mW is 0 dBm, 10 mW is 10 dBm and 0.1 mW is −10 dBm [4]. With a power in dBm and losses in dB, following light through a circuit becomes subtraction [4]. A laser that puts 0 dBm into a waveguide 1 cm long (2 dB), followed by a Y-branch that sends half the light into each branch (3 dB) and loses up to 0.3 dB more [1], delivers $0 - 2 - 3 - 0.3 = -5.3$ dBm, about 0.3 mW, to each branch. The two units are not interchangeable: dB is a ratio of two powers, dBm a power. Adding a loss in dB to a power in dBm gives a power, but adding two powers in dBm has no meaning.

Key idea

A waveguide guides light in modes, field patterns that fit the core and travel without changing shape; along the waveguide, a mode behaves like a plane wave in a material with its effective index.

How do we model itOptional · Snell's law, the mode condition and the slab solution

Critical angle. With Snell’s law $n_1 \sin\theta_1 = n_2 \sin\theta_2$ , the refracted ray runs along the boundary ( $\theta_2 = 90°$ ) for $\sin\theta_1 = n_2/n_1$ . For silicon ( $n_1 = 3.473$ ) in oxide ( $n_2 = 1.444$ ), $\theta_\text{c} = \arcsin 0.416 = 24.6°$ .

Wave equation and propagation constant. In each uniform layer, the field of a wave of vacuum wave number $k_0 = 2\pi/\lambda$ obeys the Helmholtz equation $\nabla^2 E + k_0^2 n^2 E = 0$ [2]. A mode of the slab has the form $E(x)\, e^{i\beta z}$ and, inside the core, a transverse wave number $k_x$ with

k_x^2 + \beta^2 = n_1^2 k_0^2

[2]. The ray angle $\theta$ from the normal satisfies $\beta = n_1 k_0 \sin\theta$ and $k_x = n_1 k_0 \cos\theta$ , so $n_\text{eff} = \beta/k_0 = n_1 \sin\theta$ . In the cladding the field decays as $e^{-\gamma |x|}$ with $\gamma^2 = \beta^2 - n_2^2 k_0^2$ , which is real only for $n_\text{eff} > n_2$ : a guided mode.

The slab’s modes. For a symmetric slab of thickness $d$ , write $u = k_x d/2$ , $w = \gamma d/2$ and $V = (k_0 d/2)\sqrt{n_1^2 - n_2^2}$ . Continuity of the field and its derivative at the surfaces gives, for TE modes,

\begin{gathered} u^2 + w^2 = V^2, \\ w = u \tan u \ \text{(even modes)}, \qquad w = -u \cot u \ \text{(odd modes)}. \end{gathered}

Mode $m$ exists for $V > m\pi/2$ , so the slab is single-mode for $d < \lambda / (2\sqrt{n_1^2 - n_2^2})$ : 245 nm for silicon in oxide at 1550 nm. For $d = 220\,\mathrm{nm}$ they give $n_\text{eff} = 2.845$ [1]; the solver behind Figure 2 reproduces this value for a strip much wider than it is high. The condition $w = u \tan u$ is the ray picture’s round-trip condition written for waves: $2 k_x d$ plus the phase shifts of the two total reflections equals $2\pi m$ [2].

In short

A silicon waveguide holds light because silicon’s refractive index is much higher than that of the surrounding oxide; in the ray picture, the light is totally reflected at the boundaries.
Only field patterns that are in step with themselves after a round trip across the core survive: the modes. A thin core carries a single mode, which extends partly into the cladding.
Along the waveguide the light gains the phase $2\pi n_\text{eff} L/\lambda$ ; for the standard silicon strip at 1550 nm, $n_\text{eff} = 2.443$ , one cycle every 0.63 µm.

Next · FoundationsPhase shiftersHow a circuit sets the phase of light after fabrication, by heating a waveguide or by moving electrical charges in it, and what each method costs.Read next Build it in the labMZI switchDraw a waveguide with a bend and zoom in until the wave appears; its crests are 0.63 µm apart.Open the lab

References

1Chrostowski, L., Hochberg, M. (2015). Silicon Photonics Design. Cambridge University Press. doi:10.1017/CBO9781316084168
2Osgood, R., Jr., Meng, X. (2021). Principles of Photonic Integrated Circuits: Materials, Device Physics, Guided Wave Design. Springer (Graduate Texts in Physics). doi:10.1007/978-3-030-65193-0
3Hecht, E. (2017). Optics (5th edition, global edition). Pearson Education Limited. link
4Saleh, B. E. A., Teich, M. C. (2019). Fundamentals of Photonics (3rd edition). Wiley. link

Approximations used on this page: APX-019 semi-vectorial modes on a grid, APX-020 a weakly guiding fibre.