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Introduction¶

Here we use Tidy3D to simulate a very large dielectric metalens. We base this example off of the recent paper from Khorasaninejad et al. titled Metalenses at visible wavelengths: Diffraction-limited focusing and subwavelength resolution imaging, which was recently published in Science. In this paper, a 2-dimensional array of dielectric structures is used as a lens to focus transmitted light to a single position directly above the device.

Typically, these structures are simulated by simulating each dielectric unit cell individually to compute a phase and amplitude transmittance for each cell. While this approach makes for an approximation of the overall device performance, it would be useful to be able to simulate the entire device as a whole to capture the entire physics. However, a simulation of this scale requires several hours or days to do with a conventional CPU-based FDTD. With Tidy3D, we are able to complete the entire simulation in about 1 minute!

First, we'll show a simple approach for computing fields on the focal plane by simply placing a FieldMonitor at the focal plane to record the fields. However, the focal plane is several wavelengths away from the structure, which means that the above setup will involve simulating fields propagating through lots of empty space. Therefore, we will also show how one can instead record the near fields on a large surface just above the structure, and then project them to the focal plane without actually simulating the fields in the empty region in between, leading to a further improvement in the already-impressive simulation speed.

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

Setup¶

We first perform basic imports of the packages needed.

Define Simulation Parameters¶

We now set the basic parameters that define the metalens. The following image (taken from the original paper) define the variables describing the unit cell of the metalens. The angle of each cell (θ) is chosen for maximum focusing effect. Note that microns are the default spatial unit in Tidy3D.

Create Metalens Geometry¶

Now we can automatically generate a large metalens structure using these parameters.

We will first create the substrate as a td.Box

Then, we will loop through the x and y coordinates of the lens and create each unit cell as a td.PolySlab

Define grid specification¶

We define the grid based on the properties of the geometry. The metalens is quasi-periodic in x and y, in that we have clearly defined unit cells, but each is slightly modified from its neighbors. Such structures are best resolved with a grid that matches the periodicity, which is why we use a uniform grid in x and y. In z, we use the automatic nonuniform grid that will place a higher grid density around the metalens region, and a lower one in the air region away from the metalens. To speed up the auto meshing in the region with the pillars, we put an override box in the grid specification.

Define Sources¶

Now we define the incident fields. We use a normally incident circularly polarized plane wave with Gaussian time dependence centered at our central frequency. The circularly polarized plane wave is created with two perpendicularly linearly polarized plane waves with a 90 degree phase difference. For more details, see the plane wave source documentation and the gaussian source documentation

Define Monitors¶

Now we define the monitors that measure field output from the FDTD simulation. For simplicity, we use measure the fields at the central frequency. We'll get the fields at the following locations:

• The y=0 cross section
• The x=0 cross section
• The z=f cross section at the focal plane
• The central axis, along x=y=0

For more details on defining monitors, see the field monitor documentation.

Create Simulation¶

Now we can put everything together and define a Simulation object to be run.

We get a number of warnings about structures being too close to the PML. In FDTD simulations, this can result in instability, as PML are absorbing for propagating fields, but can be amplifying for evanescent fields. This particular simulation runs without any issues even with PML on the sides, but it is best to heed these warnings to avoid problems. There are two ways that we can fix the simulation: one is to just put some space between the last of the metalens boxes and the PML. The other is to use adiabatic absorbers on the sides, which are always stable. The only downside of the absorbers is that they are slightly thicker than the PML, making the overall simulation size slightly larger. This is why we only put them along x and y, while we leave the PML in z.

The warnings are just letting us know that we are using uniform grid along x and y (per our spec) even though the override structures have some extent in these dimensions. We can ignore as this is intended.

Visualize Geometry¶

Lets take a look and make sure everything is defined properly.

To have a better visualization, we can also plot the simulation in 3D.

Run Simulation¶

Now we can run the simulation over time and measure the results to plot

For reference, the same problem run with FDTD on a single CPU core FDTD is projected to take 40 hours!

Visualize Fields¶

Let's see the results of the simulation as captured by our field monitors.

First, we look at the field intensity along the axis of the lens

We now can inspect the field patterns on the area monitors using the Tidy3D visualization methods.

Field Projection to the Focal Plane¶

Here we'll show how to avoid simulating lots of empty space above the structure by using Tidy3D's field projection functionality.

Run¶

Run the new simulation and note the simulation time recorded in the log.

This approach has the advantage of a faster solver time, cheaper flex credit cost, and about 10x less data that needs to be downloaded compared to the original simulation! Furthermore, only a few seconds (a small fraction of the total time) were spent on projecting the near fields to the focal plane.

Plot Fields¶

Finally, let's plot the fields on the focal plane and compare those recorded previously in the large simulation to those computed via field projections just now. Comparing the actual simulated fields to the projected ones, we notice an excellent match, with a several-fold reduction in compute time.