Directional coupler

A directional coupler is two waveguides, or two fiber cores, run side by side so closely that their evanescent fields overlap, and power crosses from one to the other and back as the light travels. Light entering one guide has transferred a fraction ${\sin }^2(\kappa z)$ to the other after a length $z$, where $\kappa$ is the coupling coefficient; all of it has crossed at the cross-over length $L_c=\pi \mathrm{/}(2\kappa )$, and half of it at $L_c\mathrm{/}2$, which makes the 3 dB splitter of Mach–Zehnder interferometers. Choosing the length sets any split ratio, so the same structure makes 50:50 splitters, 1% taps and the coupling section of a ring resonator. Its weakness is that $\kappa$ depends on the wavelength and on the gap, so the split drifts across a band and from one wafer to the next.

Coupled-mode formulas

For two identical lossless waveguides with coupling coefficient $\kappa$, light launched into waveguide 1 divides as

$$P_1(z)=P_0{\cos }^2(\kappa z),P_2(z)=P_0{\sin }^2(\kappa z)\mathrm{.}$$

The coupled pair can equally be described by its two supermodes, one even and one odd across the pair, with effective indices $n_e$ and $n_o$. Light launched into one guide excites both equally; they drift in and out of phase, and the power moves to the other guide when they have slipped by half a wavelength:

$$L_c=\frac{\pi }{2\kappa }=\frac{\lambda }{2(n_e-n_o)}\mathrm{.}$$

A mode solver gives $n_e-n_o$ for a given cross-section and gap; the length for a cross-coupled fraction $F$ is then

$$z=\frac{2}{\pi }\arcsin \text{\hspace{0.17em}⁣}\sqrt{F}{L}_c\mathrm{.}$$

Worked numbers for a supermode index difference of 0.05 at 1550 nm:

The bends that bring the waveguides together and take them apart also couple some power, so the straight section of a real coupler is shorter than the table's length, and simulations or test structures include them. The Directional Coupler and MZI Calculator gives the split for any supermode index difference, length and mismatch between the guides, the length for 50:50, and how far in wavelength the split holds.

Sensitivity to the gap and the wavelength

The coupling falls roughly as $e^{-\gamma g}$ with the gap $g$, where $\gamma =(2\pi \mathrm{/}\lambda )\sqrt{n_{\text{eff}}^2-n_{\text{clad}}^2}$ is the decay constant of the evanescent field. For a silicon wire in oxide ($n_{\text{eff}}\approx 2.4$, $n_{\text{clad}}=1.444$) at 1550 nm, $\gamma \approx 7.8\mu {\text{m}}^{-1}$, so a gap 10 nm narrower than designed raises $\kappa$ by about 8% and turns a 50:50 coupler into about 56:44. The wavelength acts the same way: a longer wavelength spreads the mode further into the gap, $\kappa$ grows, and a coupler set to 50:50 at one wavelength drifts away from it on either side, which is why the flat band in the comparison table below is only ±5 to 10 nm. Broadband designs use a 2×2 MMI coupler, an adiabatic coupler, or a curved or asymmetric directional coupler whose wavelength dependences cancel.

Asymmetric couplers

When the two waveguides have different propagation constants, with a mismatch $\mathrm{\Delta }\beta ={\beta }_1-{\beta }_2$ and $\delta =\mathrm{\Delta }\beta \mathrm{/}2$, the transfer is incomplete:

$$\frac{P_2(z)}{P_0}=\frac{{\kappa }^2}{{\kappa }^2+{\delta }^2}{\sin }^2\text{\hspace{0.17em}⁣}\left(\sqrt{{\kappa }^2+{\delta }^2}z\right)$$

A mismatch equal to the coupling ($\delta =\kappa$) caps the transfer at 50%. Because the mismatch depends on wavelength and polarization, asymmetric couplers select: they make wavelength filters and add-drop elements, polarization splitters in which only one polarization is phase-matched and crosses, and mode multiplexers that couple the fundamental mode of a narrow waveguide to a higher-order mode of a wide one.

Typical values

Typical SOI directional couplers (220 nm strip, 1550 nm):

Compared to MMI couplers:

Ring resonators and fiber couplers

Directional couplers are the standard coupling element for ring resonators: the ring and the bus waveguide form a short asymmetric coupler at their tangent point, transferring a controlled fraction of power per round trip, and tuning that coupling decides whether the ring is under-coupled, critically coupled or over-coupled.

In fiber, the same physics makes the fused biconical taper coupler: two fibers are twisted together, heated and drawn until their cores are close enough to couple, and the draw is stopped when the monitored split reaches its target. The standard taps and splitters of fiber optics (50:50, 90:10, 99:1) are made this way, and so are the wavelength-division couplers that combine 980 nm pump light with a 1550 nm signal in an erbium-doped fiber amplifier, which use the wavelength dependence of the coupling on purpose.

Common questions

What is the coupling length of a directional coupler?

The length for full transfer from one waveguide to the other, $L_c=\pi \mathrm{/}(2\kappa )=\lambda \mathrm{/}[2(n_e-n_o)]$. A supermode index difference of 0.05 at 1550 nm gives 15.5 µm; the 50:50 length is half of that.

How do you design a 50:50 directional coupler?

Choose the gap, find $\kappa$ or the supermode indices with a mode solver, and make the coupling region half the cross-over length, counting the coupling in the bends. Then check the split across the band and against gap variation, since both move it.

What is the difference between a directional coupler and an MMI coupler?

A directional coupler exchanges power gradually between two guides and can give any ratio, with very low loss, but narrowband and sensitive to the gap. An MMI coupler splits light by self-imaging in a wide multimode section; it is broadband and tolerant but gives fixed ratios and slightly more loss.

Is an optical directional coupler the same as an RF directional coupler?

They share the name and the four ports, and both divert a set fraction of power to a second line. The RF component samples forward or reflected waves on a transmission line and is specified by coupling, directivity and isolation; the optical one splits light between two waveguides and is specified by split ratio, excess loss and bandwidth.

References: A. Yariv, "Coupled-mode theory for guided-wave optics," IEEE Journal of Quantum Electronics 9, 919 (1973). K. Okamoto, Fundamentals of Optical Waveguides, 2nd ed. (Academic Press, 2006). L. Chrostowski and M. Hochberg, Silicon Photonics Design (Cambridge University Press, 2015).