Splitters and combiners

7 min readBuilds on Interference and WaveguidesOpen in the lab

A splitter divides the light of one waveguide between two; a combiner joins the light of two waveguides. On silicon chips both functions are performed by the same few structures, most often the Y-branch and the directional coupler. A splitter is simple to describe: it divides the power in a fixed ratio. A combiner is not, because the light arriving in its two inputs interferes, and where it goes depends on the phase difference between the inputs. This article describes both, and the phase relation that makes a coupler work.

Ports

A component’s ports are the ends of its waveguides, where light enters or leaves. This series names them as the lab does: inputs in0, in1 and outputs out0, out1, numbered from the top. Light is not restricted to one direction; a port called an output can also receive light.

The Y-branch

In a Y-branch, one waveguide divides into two branches that bend apart, like the letter Y. Because the two branches are mirror images of each other, the light divides equally: each branch receives half the power, and its field is the input field divided by $\sqrt{2}$ [1]. An optimised Y-branch in silicon loses less than 0.3 dB [1]. Its splitting ratio is fixed by its symmetry and does not depend on the wavelength.

The directional coupler

A directional coupler consists of two waveguides that run close to each other over a length $L$ , for example with a gap of 200 nm between two standard strip waveguides. It is the most common way to split and combine light in photonic systems [1]. The mode of each waveguide reaches into the oxide (Waveguides), and where the waveguides are close, the tail of one mode overlaps the other waveguide. Light entering one waveguide gradually crosses over to the other, and if the waveguides stay together long enough, it crosses back.

The fraction of the power that has crossed after a length $L$ is

\kappa^2 = \sin^2\!\left(\frac{\pi}{2}\,\frac{L}{L_x}\right), \tag{1}

where $L_\text{x}$ , the cross-over length, is the length after which all the light has crossed [1]. For two strip waveguides with a 200 nm gap at 1550 nm, $L_\text{x} = 37.5\,\mu\mathrm{m}$ [1]. The fraction $\kappa^2$ is the coupler’s power coupling; the rest, $t^2 = 1 - \kappa^2$ , stays in the first waveguide. A coupler of half the cross-over length, 18.75 µm, divides the light equally: it is a 50:50 coupler.

The output straight across from an input is the bar output of that input; the output in the other waveguide is its cross output. For light entering in0, out0 is the bar and out1 the cross output.

Figure 1 shows a directional coupler from above. The light is drawn inside the coupling section as it crosses over.

Light enters

Coupling length L18.8 µm

out050 %

out150 %

field at out0

field at out1

Silicon strip waveguides 500 nm wide with a 200 nm gap, 1550 nm: all light crosses over after 37.5 µm. The light inside the section is drawn from the coupler's model (APX-008); powers are percentages of the total input.

Light enters in0 of a coupler 18.8 µm long. 50 % leaves at out0 and 50 % at out1.

Figure A directional coupler of two silicon strip waveguides with a 200 nm gap. The brightness of the light shows its power along the coupler; the clocks show the fields at the two outputs.APX-001, APX-004, APX-008

Why the light crosses over becomes clear when the two waveguides are treated as one structure. Together they guide two modes that extend over both, the supermodes: a symmetric one, whose field has the same sign in both waveguides, and an antisymmetric one, whose field changes sign and passes through zero in the middle of the gap [1]. Light entering the upper waveguide excites both equally, in such a way that they add in the upper waveguide and cancel in the lower one. The two supermodes have slightly different effective indices, so they travel at slightly different speeds and fall out of step. After the cross-over length they are half a cycle apart: now they cancel in the upper waveguide and add in the lower one, and all the light has crossed [1]. For the 200 nm gap, their effective indices differ by $\lambda/(2L_\text{x}) = 0.021$ . Figure 2 shows the two supermodes and their sum at any place along the coupler.

Cross-section at z = 18.8 µm, upper waveguide left

field−+

Place along the coupler z18.8 µm

upper: symmetric + antisymmetric

lower: symmetric − antisymmetric

upper waveguide50 %

lower waveguide50 %

The supermodes are computed with one field component (E_x) on a grid of 20 nm; the light at z is their sum with the phases of the clocks, at one instant: the field in the lower waveguide is a quarter cycle ahead of the upper one. Δn = λ/(2Lₓ) = 0.0207 for strips 500 nm wide with a 200 nm gap at 1550 nm. The arrows are drawn relative to the mean phase of the two supermodes.

At 18.8 µm along the coupler the two supermodes are 0.50π apart in phase. Their sum puts 50 % of the light in the upper waveguide and 50 % in the lower one.

Figure The coupling section of Figure 1 over two cross-over lengths. Top: the light from above. Middle: the cross-section at the chosen place, coloured as the field (red one way, blue the other): the symmetric and the antisymmetric supermode, computed as in the article Waveguides, compare Chrostowski and Hochberg [1], and the light, their sum at one instant. Bottom: the light in each waveguide as the sum or the difference of the two supermodes’ phasors.APX-008, APX-019

A quarter cycle between the outputs

The two outputs of a coupler differ not only in power but in phase. For light entering in0, the light that has crossed to out1 is a quarter cycle, $\pi/2$ , ahead of the light that stayed in out0, at every coupling length [1]. The two arrows in Figure 1 are at right angles.

Energy conservation requires this phase relation of every lossless splitter, not only of silicon couplers. If light enters both inputs, each output receives a sum of two contributions, and power must be conserved for every phase difference between the inputs. When one output is bright through constructive interference, the other must be dark through destructive interference. For a symmetric splitter, this is possible only if the crossing and the straight paths differ in phase by a quarter cycle [2].

A coupler is therefore described completely by a $2 \times 2$ matrix acting on the complex amplitudes $a_0$ , $a_1$ at its inputs:

\begin{pmatrix} b_0 \\ b_1 \end{pmatrix} = \begin{pmatrix} t & i\kappa \\ i\kappa & t \end{pmatrix} \begin{pmatrix} a_0 \\ a_1 \end{pmatrix}, \qquad t = \sqrt{1 - \kappa^2}. \tag{2}

The factor $i$ is the quarter cycle on the crossing paths. The lab uses this matrix for every directional coupler (conventions of this series).Some books write $-i$ or place the factor elsewhere. The choice depends on the sign convention for phases; the powers are the same in every convention.NoteSome books write $-i$ or place the factor elsewhere. The choice depends on the sign convention for phases; the powers are the same in every convention.

Joining two waves

When light enters both inputs, the coupler becomes a combiner. For a 50:50 coupler, $t = \kappa = 1/\sqrt{2}$ , and with equal inputs $a_0 = 1$ , $a_1 = e^{i\Delta\varphi}$ , Eq. (2) gives the output powers

P_0 = \tfrac12\,|1 + i e^{i\Delta\varphi}|^2 = 1 - \sin\Delta\varphi, \qquad P_1 = 1 + \sin\Delta\varphi. \tag{3}

Two equal waves that enter in phase, $\Delta\varphi = 0$ , leave through both outputs equally. If the wave at in1 is a quarter cycle behind, $\Delta\varphi = -\pi/2$ , all the light leaves at out0; a quarter cycle ahead, all of it leaves at out1. The coupler sends the light to the output where the contributions of the two inputs interfere constructively. Which output that is depends only on the phase difference between the inputs; the total power, 2, is always conserved.

A Y-branch used backwards combines differently. Its single output waveguide carries one mode, and the two branches can feed only the part of their light that is in phase into it: the sum of the two fields divided by $\sqrt{2}$ [1]. The out-of-phase part leaves the junction as light that is no longer guided, radiating into the chip [1]. Two equal beams in phase pass completely, two beams half a cycle apart are radiated completely, and light entering only one branch leaves the output with half its power [1]. No design avoids this loss: a device with one output waveguide cannot combine two beams of arbitrary phase without losing light. A combiner that keeps all the light needs two outputs, as the directional coupler has.

The multimode interference coupler

A third structure, the multimode interference (MMI) coupler, joins the waveguides to a wide section of silicon in which several modes travel at different speeds. Their interference forms images of the input field at certain lengths: a single image, or two images of half the power each, which makes a splitter [3]. MMI couplers tolerate variations of wavelength and fabrication better than directional couplers [3]; the Toolbox page on MMIs describes them.

Key idea

A combiner sends light to the output at which the light from its two inputs interferes constructively; which output that is depends on the phase difference between the inputs.

How do we model itOptional · the two supermodes, and why the matrix must contain i

Two coupled waveguides together guide two modes that extend over both: a symmetric and an antisymmetric supermode, with effective indices $n_1 > n_2$ [1]. Light entering one waveguide excites both supermodes equally, in phase in that waveguide and out of phase in the other. The two travel with the propagation constants $\beta_{1,2} = 2\pi n_{1,2}/\lambda$ and fall out of step. After the length where $(\beta_1 - \beta_2) L = \pi$ , they add in the second waveguide and cancel in the first: all the light has crossed. This gives

\begin{gathered} L_x = \frac{\lambda}{2\,\Delta n}, \qquad \Delta n = n_1 - n_2, \\ \kappa^2 = \sin^2\!\left(\frac{\pi \Delta n}{\lambda}\, L\right) \end{gathered}

[1]. With the time convention of this series, the crossed light is ahead of the light that stays by $\pi/2$ at every length.

A lossless coupler conserves power for every input, $|b_0|^2 + |b_1|^2 = |a_0|^2 + |a_1|^2$ , which requires its matrix $M$ to be unitary, $M^\dagger M = I$ . For a symmetric matrix $\begin{pmatrix} t & c \\ c & t\end{pmatrix}$ with real $t$ , the off-diagonal element of $M^\dagger M$ is $t^* c + c^* t = t\,(c + c^*) = 2t\,\operatorname{Re} c$ , which vanishes only if $c$ is purely imaginary: $c = \pm i\kappa$ . The quarter cycle follows from energy conservation alone. The article Circuits as matrices uses this property for whole circuits.

In short

A Y-branch splits light equally by symmetry; a directional coupler exchanges light between two close waveguides, with a split set by its length.
In a lossless coupler, the light that crosses is a quarter cycle ahead of the light that stays; energy conservation requires it.
As a combiner, a coupler sends two input waves to one output or the other depending on their phase difference; a Y-branch keeps only their in-phase part.

Next · FoundationsThe Mach-Zehnder interferometerA circuit that splits light into two paths and recombines it, so that the phase difference between the paths sets how the power divides between two outputs.Read next Build it in the labMZI switchSelect a directional coupler and change its coupling; the lab draws the light crossing over inside it.Open the lab

References

1Chrostowski, L., Hochberg, M. (2015). Silicon Photonics Design. Cambridge University Press. doi:10.1017/CBO9781316084168
2Saleh, B. E. A., Teich, M. C. (2019). Fundamentals of Photonics (3rd edition). Wiley. link
3Soldano, L. B., Pennings, E. C. M. (1995). Optical multi-mode interference devices based on self-imaging: principles and applications. Journal of Lightwave Technology 13(4), 615–627. doi:10.1109/50.372474

Approximations used on this page: APX-001 lossless components by default, APX-004 wavelength-independent coupling ratio, APX-008 power exchange drawn inside couplers, APX-011 radiated light drawn as a fan, APX-019 semi-vectorial modes on a grid.