Tool · PIC design

Directional Coupler and MZI Calculator

How long must two waveguides run side by side to split light 50 : 50, how does the split drift with wavelength, and what extinction does a Mach-Zehnder interferometer built from such couplers reach? The calculator solves the coupled-mode equations for a coupler set by its supermode index difference and length, then puts two of them around a pair of unequal arms and plots the interferometer’s response. Background: directional coupler, Mach-Zehnder interferometer, and MMI coupler.

Directional coupler

Wavelength λ (nm)

Coupling Δn, the supermode splitting n_even − n_odd of identical guides

From a mode solver: the even and odd supermode indices of the pair with both guides made identical; it sets κ = πΔn/λ. With a guide mismatch, the actual pair splits by √(Δn² + (n₁ − n₂)²), shown below; enter the identical-guide value here, not that one.

Δn slope (per µm of λ)

Guide mismatch n₁ − n₂

Coupling length L (µm)

Mach-Zehnder interferometer

Couplers

Fixed split, to cross port (%)

Arm difference ΔL (µm)

Effective index n_eff

Group index n_g

Arm loss difference (dB)

Both couplers are the same. Light enters the upper input; the longer arm is the upper one.

Presets

The waveguide numbers in the presets are round examples of the size found in silicon strip waveguides near 1550 nm, not the values of a particular process. Coupled-mode theory with lossless couplers; bend sections, higher-order modes, polarization and reflections are not modeled.

Readouts

Coupler output against length

cross portbar port

Coupler output against wavelength

cross portbar port

MZI transmission

cross portbar port

Learn with it

Three short experiments. Each one sets the inputs, says where to look, and asks for a prediction before it shows the result.

Checked against

These checks run in your browser on every load. The closed forms are compared with values worked out by hand, the coupler solution is compared with a step-by-step numerical integration of the coupled-mode equations, and the couplers are checked for power conservation and the interferometer for its free spectral range and extinction limits.

Check	Expected	Computed	Tolerance

The expected values follow the coupled-mode solution and interferometer relations in Yariv and Yeh, Photonics, and Okamoto, Fundamentals of Optical Waveguides, evaluated by hand for the stated cases. The tolerance is the largest relative difference from Expected that still passes.

The model

Two parallel waveguides exchange power through their overlapping evanescent fields. In coupled-mode theory, with coupling coefficient $κ$ and a mismatch $δ = π (n_{1} - n_{2}) / λ$ between the effective indices of the two guides, light launched into guide 1 leaves guide 2 after a length $L$ with the power fraction

\frac{P_{2}}{P_{0}} = \frac{κ^{2}}{κ^{2} + δ^{2}} \sin^{2} ⁣ (\sqrt{κ^{2} + δ^{2}} L)

For identical guides the coupling follows from the difference $Δ n = n_{e v e n} - n_{o d d}$ between the effective indices of the even and odd supermodes, which a mode solver gives directly:

κ = \frac{π Δ n}{λ}, L_{x} = \frac{π}{2 κ} = \frac{λ}{2 Δ n}

so all the light has crossed after the crossover length $L_{x}$ and half of it after $L_{x} / 2$ . A mismatch makes the transfer faster but incomplete. With a mismatch the input $Δ n$ keeps its meaning as the splitting the pair would have if the guides were identical, which fixes $κ$ ; the supermodes of the mismatched pair are split by $λ \sqrt{κ^{2} + δ^{2}} / π = \sqrt{Δ n^{2} + (n_{1} - n_{2})^{2}}$ , 0.0424 for the mismatched preset. Entering that larger value from a mode solver of the asymmetric pair would overstate $κ$ , by a factor of $\sqrt{2}$ in that preset. The supermode splitting of a real coupler grows with wavelength as the modes spread; the tool takes it as linear in $λ$ with a slope you set. The bends that bring the guides together and apart add coupling that is here folded into $L$ .

A Mach-Zehnder interferometer places two couplers around arms whose lengths differ by $Δ L$ . With the effective index taken to first order in wavelength, the arm phase difference and the free spectral range are

Δ ϕ (λ) = \frac{2 π n_{e f f} (λ) Δ L}{λ}, n_{e f f} (λ) = n_{e f f} - (n_{g} - n_{e f f}) \frac{λ - λ_{0}}{λ_{0}}, F S R = \frac{λ_{0}^{2}}{n_{g} Δ L}

The tool multiplies the two coupler matrices and the arm phases and loss directly. With ideal 50 : 50 directional couplers and equal arms all the light leaves by the cross port; some texts label the ports the other way round. Each output is a sum $u + v e^{- i Δ ϕ}$ of two paths, so its extinction ratio is ${((∣ u ∣ + ∣ v ∣) / (∣ u ∣ - ∣ v ∣))}^{2}$ . Two identical couplers that send a power fraction $k$ to the cross port, a split $k : 1 - k$ (not the coupling coefficient $κ$ ), leave the cross port with a perfect null and limit the bar port to $1 / (1 - 2 k)^{2}$ ; an arm loss difference with field ratio $a$ limits both ports to ${((1 + a) / (1 - a))}^{2}$ . The couplers are lossless and the waveguides single-mode, and polarization, reflections and the wavelength dependence of the arm loss are left out.

The inputs are the effective index and group index of the arms; the fringe spacing is the free spectral range and the null depth the extinction ratio. The supermode index difference comes from a mode solver such as the waveguide mode calculator, and the same phase argument sets the resonances of a ring resonator.

Worked example

A supermode index difference of 0.03 at 1550 nm gives $κ$ = 60.81 rad/mm and a crossover length of 25.83 µm, so a 12.92 µm coupler splits the power 50 : 50. Put two ideal 50 : 50 couplers around arms that differ by 100 µm, with an effective index of 2.44 and a group index of 4.2: the free spectral range is 5.72 nm, or 713.8 GHz. At 1550 nm the arm phase difference is 0.839π modulo 2π, and 6.28 % of the light (−12.02 dB) leaves the cross port and 93.72 % (−0.28 dB) the bar port. Make both couplers 45 : 55 and the bar port’s extinction falls from unlimited to 20.00 dB.

References: A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications, 6th ed., Oxford University Press (2007). K. Okamoto, Fundamentals of Optical Waveguides, 2nd ed., Academic Press (2006). L. Chrostowski and M. Hochberg, Silicon Photonics Design, Cambridge University Press (2015).