Phase shifters

4 min readBuilds on Phase and WaveguidesOpen in the lab

A phase shifter is a section of waveguide whose effective index can be changed on demand, so that a circuit can set the phase of the light that leaves it. On silicon chips the most common phase shifter is a metal heater above the waveguide: warmer silicon has a higher refractive index. This article explains how a heater sets the phase, what that costs, and what the faster alternatives are.

Why circuits need phase shifters

The phase of light at the end of a waveguide is fixed by its length and effective index, $\varphi = 2\pi n_\text{eff} L/\lambda$ (Waveguides). Fabrication sets both only approximately: over 1 mm of waveguide, an error in the effective index of 0.001 shifts the phase by 0.65 cycles. A circuit that relies on interference therefore needs a way to set phases after it has been made, both to correct such errors and to switch the light between outputs deliberately. A phase shifter provides it.

The thermo-optic effect

The refractive index of silicon rises with temperature. This is the thermo-optic effect; for silicon near 1550 nm,

\frac{\mathrm{d}n}{\mathrm{d}T} = 1.87 \times 10^{-4}\ \text{K}^{-1} \tag{1}

[1] [2]. The oxide around the waveguide responds 6.3 times less [1]. A heater is a thin strip of metal placed above the waveguide, in one example titanium nitride 1.5 µm above the silicon, far enough that the metal does not absorb the light [3]. An electrical current warms the strip, and the heat spreads down into the waveguide.

The heater changes the speed of light in the waveguide, not the amount of light. The light leaving a heated waveguide has the same power as before; only its phase has changed. The change becomes visible in power only when this light is made to interfere with light from another path.

How much heating

Heating a length $L$ of waveguide by $\Delta T$ raises its effective index by about $\Delta n_\text{eff} \approx (\mathrm{d}n/\mathrm{d}T)\,\Delta T$ and changes the phase of the light by

\Delta\varphi = \frac{2\pi}{\lambda}\, \Delta n_\text{eff}\, L \approx \frac{2\pi}{\lambda}\, \frac{\mathrm{d}n}{\mathrm{d}T}\, \Delta T\, L. \tag{2}

For a heater 100 µm long, a length used in a modelling example of the textbook [1], a phase change of $\pi$ , half a cycle, requires $\Delta n_\text{eff} = \lambda/(2L) \approx 0.008$ and therefore a temperature rise of about 41 K. The estimate uses silicon’s coefficient for the whole mode, although part of the mode travels in the oxide, and assumes a uniform temperature under the heater; real heaters need somewhat more.

Figure 1 shows two equal waveguides with a heater over one of them. The phasor clock compares the light at the end of the heated waveguide with that of the unheated one.

heated (solid) against unheated (dashed)

Temperature rise ΔT0.0 K

Heater length

index change Δn: 0
phase change Δφ: 0 = 0.00 cycle

Wavelength 1550 nm. The heated waveguide's index is assumed to change like bulk silicon's, dn/dT = 1.87 × 10⁻⁴ per kelvin: an estimate (APX-016).

A heater 100 µm long warms one waveguide by 0.0 K. Its phase changes by 0 relative to the unheated waveguide.

Figure Two equal waveguides, one of them under a heater. The phase change of the heated waveguide relative to the other, estimated from Eq. (2), grows in proportion to the temperature rise and to the heated length.APX-016

A longer heater needs a smaller temperature rise for the same phase: the product of length and temperature rise for a phase of $\pi$ is fixed by Eq. (2), about $4\,\mathrm{mm\,K}$ . Half a cycle over 500 µm requires about 8 K.

The cost of heat

Heaters are simple and add almost no optical loss, but heat is slow and expensive. In a switch chip with 32 inputs and 32 outputs, a phase change of $\pi$ took about 18 mW of electrical power per heater, and the whole chip consumed 1.9 W [3]. In silicon, such heaters respond in 10–100 µs, and heat spreading from one heater changes the phase of neighbouring waveguides, an effect called thermal crosstalk [4]. The thermo-optic effect is therefore used mainly to tune a circuit to its working point, for example to set the state of an interferometer [2], and for switches that change their state rarely.

Faster phase shifters

To encode data onto light, phase changes have to follow an electrical signal at billions of changes per second. A device that does so is a modulator [2]. In silicon, the fastest phase shifters change the density of free electrical charges in the waveguide. More free charges lower the refractive index and increase the absorption of light: the plasma dispersion effect [2]. A diode of p- and n-doped silicon built across the waveguide adds or removes charges when a voltage is applied [2]. Modulators that remove charges from such a junction are the most common commercial type and the workhorse of silicon transceivers [2], and switches built this way change their state in about a nanosecond [3].

The price is loss: the free charges absorb light, and the amount depends on the phase shift [4]. The Toolbox pages on phase shifters and modulators describe them in detail.

Key idea

A phase shifter changes the speed of light in a section of waveguide and so its phase, not its power; only interference turns the change into a change of power.

How do we model itOptional · the phase shifter as a factor, and the figure of merit

A phase shifter multiplies the complex amplitude of the light by $e^{i\Delta\varphi}$ : the arrow of the phasor clock turns by $\Delta\varphi$ and keeps its length. A positive $\Delta\varphi$ corresponds to a longer optical path, as for a heated silicon waveguide. In matrix form, for a phase shifter on the upper of two waveguides,

\begin{pmatrix} b_0 \\ b_1 \end{pmatrix} = \begin{pmatrix} e^{i\Delta\varphi} & 0 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} a_0 \\ a_1 \end{pmatrix},

and $|b_0|^2 = |a_0|^2$ : the power is unchanged.

From Eq. (2), the length $L_\pi$ and temperature rise $\Delta T_\pi$ for a phase change of $\pi$ satisfy

\begin{aligned} L_\pi\, \Delta T_\pi &= \frac{\lambda}{2\,\mathrm{d}n/\mathrm{d}T} \\ &= \frac{1.55\ \mu\text{m}}{2 \times 1.87 \times 10^{-4}\ \text{K}^{-1}} \approx 4.1\ \text{mm K}. \end{aligned}

Electrical phase shifters are compared by the analogous product of the voltage and the length for a phase change of $\pi$ , $V_\pi L_\pi = V_\pi \lambda / (2\,\Delta n_\text{eff})$ ; a smaller product means a more efficient phase shifter [2].

In short

Heating silicon raises its refractive index by $1.87 \times 10^{-4}$ per kelvin, so a heater over a waveguide changes the phase of the light in it.
Half a cycle over 100 µm of waveguide needs a temperature rise of roughly 40 K; heaters draw milliwatts each and respond in tens of microseconds.
Phase shifters that move free charges in the waveguide respond in nanoseconds, at the cost of optical loss.

Next · FoundationsSplitters and combinersHow light is divided between two waveguides and joined again, and why the result of joining depends on the phases of the incoming light.Read next Build it in the labMZI switchHover over the heater and turn its phase knob; the light moves between the two outputs.Open the lab

References

1Chrostowski, L., Hochberg, M. (2015). Silicon Photonics Design. Cambridge University Press. doi:10.1017/CBO9781316084168
2Reed, G., Thomson, D., Zhang, W., Gardes, F., Mastronardi, L., Li, K., Matsuo, S., Kanazawa, S., Vivien, L., Lafforgue, C., Bowers, J. E., Koos, C., Romagnoli, M., Lončar, M., Zhang, M., Abel, S., Liao, L. (2023). Optical modulators. In: Glick, Liao & Schmidtke (eds.), Integrated Photonics for Data Communication Applications, Elsevier, ch. 3, pp. 69–121. doi:10.1016/B978-0-323-91224-2.00011-4
3Cheng, Q., Yao, C., Calabretta, N., Stabile, R., Suzuki, K., Kawashima, H., Tang, W., Glick, M., Chu, T., Ikeda, K., Matsumoto, R., Namiki, S., Bergman, K., Penty, R. (2023). Photonic switch fabrics in data center/high-performance computing networks. In: Glick, Liao & Schmidtke (eds.), Integrated Photonics for Data Communication Applications, Elsevier, ch. 8, pp. 265–301. doi:10.1016/B978-0-323-91224-2.00003-5
4Bogaerts, W., Pérez, D., Capmany, J., Miller, D. A. B., Poon, J., Englund, D., Morichetti, F., Melloni, A. (2020). Programmable photonic circuits. Nature 586, 207. doi:10.1038/s41586-020-2764-0

Approximations used on this page: APX-016 a heated waveguide's index changes like bulk silicon, APX-007 wavelength-independent phase shift.