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It is common to have waveguide components of different widths and potentially thicknesses on a photonic integrated circuit (PIC). Therefore, having a low-loss waveguide size converter becomes a necessity. The most common and simple size converter is adiabatic waveguide tapers. However, to achieve low loss and meet the adiabatic condition, the taper inevitably needs to be very long, which is not ideal in many modern high-density PIC designs. To alleviate this shortcoming of the conventional adiabatic taper, novel designs of compact size converters have emerged.

In this notebook, we aim to simulate different types of size converters and compare their performance. We first simulate linear adiabatic tapers of different lengths. Subsequently, we will demonstrate two compact designs: one based on Luneburg lens and the other based on semi-lens emerged from segment optimization. These novel designs achieve ~-0.5 dB loss while being only about 6$\lambda_0$ in footprint. The linear adiabatic taper can only achieve similar performance while being 30$\lambda_0$ long.

For more integrated photonic examples such as the waveguide crossing, the strip to slot waveguide converter, and the broadband directional coupler, please visit our examples page.

If you are new to the finite-difference time-domain (FDTD) method, we highly recommend going through our FDTD101 tutorials.

To suppress unnecessary warnings, we set the logging level to "ERROR".

Define the simulation wavelength (frequency) range.

All devices simulated in this notebook are based on silicon waveguide on a thick oxide layer. We define both materials as nondispersive since the simulation wavelength range is relatively small.

Linear Taper¶

The most straightforward way to connect two waveguides of different widths is via an adiabatic taper. The taper shape can be linear, hyperbolic, Gaussian, and so on. Here, we demonstrate a linear taper and investigate how the loss scales with taper length. To be specific, we model a taper connecting the input waveguide of 10 $\mu m$ wide to an output waveguide of 500 nm wide. Both waveguides have the same thickness of 110 nm. TE0 mode is launched at the input waveguide.

Since we would like to perform a parameter sweep over the taper length, we will define a function called linear_taper_sim to construct the simulation. This function will be called repeatedly to make a simulation batch. The structure of the taper including the input and output waveguides can be conveniently made using as a PolySlab.

With the linear_taper_sim function defined, we can use it to construct a simulation batch. We aim to simulate taper lengths of 10 $\mu m$, 20 $\mu m$, 50 $\mu m$, and 100 $\mu m$.

To visually verify the simulation setup, we can take a simulation from the batch and use the plot method.

Submit the simulation batch to the server.

After the batch simulation is complete, we can see how the loss scales with the taper length. From the results, we can see that a 10 $\mu m$ taper has an insertion loss of ~-6 dB and a 20 $\mu m$ taper has an insertion loss of ~-3 dB. To achieve low loss, the taper needs to be longer than 50 $\mu m$, which is not sufficiently compact for many applications.

The field distributions can also be plotted. From the field distributions, we can see the leakage of energy at the abrupt changes where the taper meets the straight waveguide. This leakage is reduced for longer tapers and thus the insertion loss is smaller.

Luneburg Lens Size Converter¶

Besides adiabatic tapers, there are many novel size converter designs that can achieve both compactness and low loss. The first example is a taper design based on the principle of a Luneburg lens, which we will demonstrate here.

A Luneburg lens is a spherical graded index lens where its refractive index varies with positions. The most common Luneburg lens employees a refractive index profile of $n(r)=\sqrt{2-(\frac{r}{R})^2}$, where $r$ is the distance to the center of the lens and $R$ is the radius of the lens. The result of the graded index is that light propagating parallel to the lens will be focused on the edge of the lens.

On the other hand, waveguides with different thicknesses have different effective indices. By applying the same principle of a classical Luneburg lens, we can design a spherical waveguide region where the thickness of the waveguide varies such that the effective index follows that of a Luneburg lens. Consequently, the incident light from the input waveguide can be focused into the output waveguide and we achieve a low loss while having a small device footprint. The device design is adapted from S Hadi Badri and M M Gilarlue 2019 J. Opt. 21 125802.

To use this design, we first need to obtain the relationship between the effective index and the waveguide thickness. Here, we ignore the effect of finite waveguide width and only consider the effective index of a slab waveguide. This will leave room for further optimization, which is another topic that we will not cover in this example notebook. We will first use the ModeSolver to solve for the effective index. Note that the effective index here is that of the waveguide, not to be confused with the effective index used in a 2.5D simulation. The simulation here will be performed in full 3D.

To order to obtain the waveguide thickness given an effective index, we perform a polynomial fitting using Numpy's polyfit method to the effective index obtained from the previous step.

We will construct a circular Luneburg lens region where the height of the Si layer changes gradually such that the local effective index follows $n(r)=n_{edge}\sqrt{2-(\frac{r}{R})^2}$, where $n_{edge}$ is the effective index at the edge of the circle. To minimize impedance mismatch at the interface between the input/output waveguides and the Luneburg lens, the edge of the Luneburg lens will have the same thickness as the waveguides.

From the above thickness profile, we can construct the Luneburg lens structure using a series of Cylinders. The radius and length of each cylinder follows the above plot.

Build the input and output waveguides.

Submit the simulation to the server.

After the simulation is complete, we can visualize the field distribution. In the Luneburg lens region, the wave front converges into a focus and the coupling to the output waveguide is efficient.

More importantly, we need to quantitatively examine the insertion loss and compare that to a linear taper. The Luneburg lens is only about 10 $\mu m$ in length. Here, we plot the insertion loss of the Luneburg lens size converter to a 50 $\mu m$ linear taper. From the plot, we can see that the Luneburg lens outperforms the 50 $\mu m$ linear taper while being only 1/5 in size.

Semi-lens Beam Expander¶

The Luneburg lens design introduced in the previous section requires a more complicated fabrication since the thickness of the Si layer needs to vary gradually. This is not desirable in many cases. There are a number of other designs that circumvent this issue. One such example is introduced in Siamak Abbaslou, Robert Gatdula, Ming Lu, Aaron Stein, and Wei Jiang, "Ultra-short beam expander with segmented curvature control: the emergence of a semi-lens," Opt. Lett. 42, 4383-4386 (2017), where a compact waveguide taper is segmented into 6 segments and the particle swarm optimization algorithm is used to optimize the shape of each segment. The result of the optimization is a semi-lens structure that also outperforms the conventional adiabatic taper of the same length by a great margin.

In the previous models, the input waveguide is wider than the output waveguide. In this case, we model a beam expander so the output waveguide is 9 $\mu m$ wide and the input waveguide is 450 nm wide.

The width of each segment of the device is described by $w(x_i)=(w_i-w_{i+1})|\frac{x_i-L_i}{L_i}|^{m_i}$, where $i=1,2,...,6$, $w_i$ is the width at the beginning of each segment, $L_i$ is the length of each segment, and $m_i$ is the curvature of each segment. The optimized values of $w_i$, $L_i$, and $m_i$ are listed above.

The structure of the device will be constructed using a series of PolySlabs. The vertices are calculated from the above equation and the optimized parameters.

The simulation will be defined in a similar way as previous simulations. Since in this model, the output waveguide supports higher order modes, we add an additional ModeMonitor to investigate the mode composition at the output waveguide. Ideally, the fundamental TE mode should be dominant.

Submit the simulation to the server.

Plot the field distribution.

Lastly, plot the insertion loss and the mode composition at the output waveguide. The insertion loss is about -0.5 dB, much better than the linear taper of the same length.

Due to the symmetry, the output power can only excite even TE modes. About 80% of the power is in the fundamental TE mode. The rest of the power is mostly in the TE2 mode.