An achromatic metalens is a type of flat optical lens engineered using a nanostructured surface to focus light without the chromatic aberration seen in conventional metalenses. By carefully designing these nanoscale features, it can focus multiple wavelengths to the same point, enabling compact, high-performance imaging systems for applications in microscopy and smartphone cameras. In this example, we present a workflow for designing an achromatic metalens by tailoring the phase slope to achieve achromatic focusing.
Overview
Understand the simulation workflow and key results
To design the achromatic metalens, two different shapes of unit-cells are used, cross and square ring. Using two unit-cell shapes allows the metalens to cover the necessary phase-slope range for achromatic focusing across the target bandwidth. They are both made of Si3N4 on a silica substrate. The parameters that are swept are the width of the cross (\(w_1\)) and the width of the square ring (\(w_2\)). The height, \(h\), of both unit-cells is 1 μm, and the lengths \(l_1\) and \(l_2\) are fixed to 180 nm and 40 nm, respectively.
To focus light at a focal length \(F\), the metalens must impose on an incident plane wave a phase distribution that can be expressed as:
$$φ(x,y,f)-φ(0,0,f)=\frac{2πf}{c}(F-\sqrt{x^2+y^2+F^2})$$
The equation can be re-written as:
$$Δφ(x,y,f)=M(x,y,f) \cdot f$$
with:
$$M(x,y,F)=\frac{2π}{c}(F-\sqrt{x^2+y^2+F^2})$$
The above equation shows that, to meet achromaticity requirements, the phase difference must vary linearly with frequency and have an intercept close to zero.
Step 1: Unit-cell simulations with RCWA
In this step, we sweep the cross and square ring widths to obtain the transmission and phase vs. frequency. The phase slope is calculated, and the results are stored in a .json file which is used as library in the next step.
Step 2: Full lens design with FDTD
In this step, we construct and simulate the full metalens in FDTD, based on the target phase slope profile and the library of phase slope vs. width of the two different unit-cell types from the previous step. While this approach is straightforward, it can pose challenges in terms of memory and simulation time, especially for larger metalens.
Run and Results
Instructions for running the model and discussion of key results
Step 1: Unit-cell simulations with RCWA
In this step, we perform a parameter sweep over the different unit‑cell designs and widths. The wavelength range used for the simulations is 400 nm - 760 nm. Both unit-cell designs are included within the same sweep, with a flag named “design” used to select between them, with 1 corresponding to cross and 2 corresponding to square ring.
- Open the file [[unit_cell.fsp]].
- Open and run the script file [[unit_cell.lsf]].
The script gets the sweep results and calculates the relative phase, defined as the phase of each unit-cell relative to the reference unit-cell. The reference unit-cell that will be placed at the center of the metalens is a cross with \(w_1\) = 140 nm. The relative phase of each unit-cell as a function of frequency exhibits a linear behavior, as shown below.
Next, the slopes of the retrieved phases are calculated using the least-square method. Using the cross unit-cell, the slope ranges from 0 to -4.5e-15 [π/Hz]. This can be extended up to -6e-15 [π/Hz] using the square ring unit-cell.
The intercepts are also computed and are found to have relatively small values, which is a prerequisite for achromaticity.
Finally, the phase slope results vs. design and widths, \(w_1\) and \(w_2\), as well as the frequency points, unit-cell height and period and length \(l_1\) and \(l_2\) are saved in a .json file that will be used in the next step to build the full metalens.
Step 2: Full lens design with FDTD
In this step, the slope vs. unit-cell design and width results are loaded in FDTD and the unit-cells are arranged so that the target phase slope is achieved. The wavelength range for the full metalens simulation is the same as in step 1.
- Open the file [[full_lens.fsp]] and run it.
- Open and run the script file [[full_lens_plot.lsf]] to plot the results (avoid “Image x-z plane” part as it takes too long to do the far field projections).
The target phase is calculated for a focal length of 9.5μm and a metalens radius of 4.32 μm.
After running the simulation, the far‑field results are obtained. The plot below shows the normalized intensity |E| 2 along the propagation z-axis at x=0 and y=0. All wavelengths focus close to the expected focal length. The slight deviation from the design focal length is due to the relatively small size of the metalens.
The normalized far field results in the x-z plane at y=0, along with a cross section at z=10 μm—corresponding to the average focal length across all wavelengths—are also shown.
Important model settings
Description of important objects and settings used in this model
Unit-cell simulations
Variables in “model”
The object has design, \(w_1\), \(w_2\), \(l_1\), \(l_2\), height and period as its variables. The position and span of the RCWA simulation region are automatically set up using these parameters.
Full lens simulations
PEC aperture
To suppress unintended field excitation beyond the lens region, a perfectly electrically conducting (PEC) aperture was placed immediately in front of the metalens. The aperture radius is automatically set by the model script.
"metalens" structure group
To visualize the target phase slope vs. position, set the "make_plot" to "1" in the "metalens" structure group and click on the “Test” button in the “Script” tab.
Far-field projections
The far-field projections are performed using the farfieldexact3d - Script command . The runtime of the far-field calculation can be significant, especially in the case of 2D projections, depending on the number of points used. Using fewer points will reduce the computation time, but at the expense of far-field resolution.
Furthermore, the full_lens_plot.lsf script performs far-field projections for all simulated wavelengths by default, unless explicitly modified. The wavelengths used in the full metalens simulation are automatically matched to those defined in the unit-cell simulations.
Updating the Model With Your Parameters
Instructions for updating the model based on your device parameters
Geometry
The geometry of the unit-cells can be modified through the RCWA solver “model” parameters. If different parameters are used for the unit-cells, the script needs to be modified to ensure correct computation of the phase slopes with respect to the updated parameters. In addition, the exported .json file must be updated to include the corresponding parameter information.
When different unit-cell geometries are used, the script of the “metalens” structure group needs to be updated accordingly.
Wavelength range
Different wavelength ranges can be set in RCWA solver. The number of wavelength/frequency points must be sufficiently large to enable accurate extraction of phase slopes using the least-squares method.
Period
The unit-cell period used in the simulations is significantly smaller than the minimum wavelength considered. While the period can be adjusted, it is recommended to keep it in the subwavelength regime in order to suppress the excitation of higher-order diffraction modes.
Focal length and metalens radius
When changing the focal length and radius of the metalens, it must be ensured that the phase slopes retrieved from the unit-cell calculations are sufficient to cover the phase slope range required for the full lens.
Taking the Model Further
Information and tips for users that want to further customize the model
Propagation of beam in OpticStudio
After running the full metalens simulation, the near-field distribution can be exported as a .ZBF file. This file can then be imported into the Physical Optics Propagation (POP) module in OpticStudio to propagate the beam through the complete optical system, including any bulk optical components. More information can be found in Step 4 of Small-Scale Metalens – Field Propagation .
Multi-parametric unit-cells
In this example, only a single geometric parameter (width) is swept for each unit-cell design. Multi-parameter sweeps of a single unit-cell configuration are also supported; however, this requires modifications to the script that selects and maps the unit-cell designs that will be used in the full metalens simulation.
Large-aperture achromatic metalenses
Additional Resources
Additional documentation, examples, and training material
Related Publications
- Yechuan Zhu, Siyuan Liu, Ying Chang, et al. "Broadband polarization-insensitive metalens with excellent achromaticity and high efficiency for the entire visible spectrum", Appl. Phys. Lett. 122, 201702 (2023), https://doi.org/10.1063/5.0152474
See Also
Related Ansys Innovation Courses