Lab 5: Synthesize, Simulate, and Assess a Metasurface

Topic IDs: ms-05-beam-steering (full-wave supercell), ms-08-metalenses (scalar line focus), and ms-06-polarization (analytical Jones model).

Companions: Chapter 5 and seven worked examples.

Question: how much of a target wavefront survives sampling, response constraints, and interactions between different elements?

Before reading the results, use the model and geometry guide. Identify the object, dimensions, incident wave, invariant/repeated directions, and reference planes in the drawing.

A. Predict the effect of a geometry sequence¶

Switch between the expanded library, original limited library, identical ridges, and reversed sequence. Compare incident-normalized desired-order power, the fraction conditional on transmission, local-mask estimates, and the full-wave order table. The reversed sequence’s desired order is $m=-1$.

Open the browser lab in a new tab

Prediction: which directions are allowed, which carry power for identical cells, and what changes after reversal? Explain why a local-mask power imbalance cannot be called physical absorption here.

Deliverable: a comparison of the four cases, with denominator definitions, model limits, and one symmetry prediction checked against the data. The explorer displays the selected design with its evidence and metadata.

B. Execute beam synthesis and full-wave verification¶

Status: executed reference available in the beam-steering notebook.

Download the complete runnable bundle: chapter-05-synthesis.zip. Extract it before running. It preserves both notebooks, both Python modules, requirements, the original Chapter 3 library, Chapter 5 data, and the audit script. Use the existing environment instructions.

Individual downloads: Chapter 5 notebook: synthesize and verify a beam deflector, metasurface_synthesis.py, beam-steering.json, extended-library.csv, and validation.json. A notebook alone requires its companion modules/data; the ZIP is the easiest complete starting point.

Inputs¶

Procedure¶

1. Derive the target gradient, desired angle, and complete exterior open-order sets.

2. Restart the kernel and run the notebook. Inspect the declared height screen and distinguish exploratory curves from accepted library entries.

3. Recompute the 161-width library at $M=24$ and check each row at $M=32$. Retain complex response, power, phase, and validation metadata.

4. Search the fixed global-offset grid using unit-cell phase error, then select widths subject to $T\geq0.8$. Preserve the limited-library result as a comparison.

5. Compute exact staircase coefficients, including the sinc factor, for local reflected/transmitted estimates.

6. Solve the complete coplanar-ridge supercells for all four sequences. Account for every open order separately on each side.

7. Refine the expanded geometry over the reported $M$ sequence. Compare every final design at $M=144$ versus 192, separately from power residuals.

8. Reconstruct the repeated field, while using modal coefficients for far-field power fractions.

9. Export geometry, selection, source hashes, conventions, full-wave powers, local estimates, and numerical changes.

Reference acceptance¶

Run the numerical audit from the extracted folder or repository root:

python -m pip install -r requirements-validation.txt
python scripts/validate_chapter05.py --grcwa

The optional comparison refines grcwa’s raster from 512 to 4096 cells; the course solver uses analytic rectangle Fourier coefficients. Both use RCWA. Without --grcwa, the script records that this optional comparison was not run. The audit includes reference-specific configurations; adapt and revalidate them for a changed physical experiment.

Deliverables: executed notebook, declared objective/selection grid, both libraries, four geometry sequences, all-channel power tables, field/response figures, convergence, audit, and an interpretation of local-versus-collective differences. These are ideal infinite-periodic TE results, not finite-device or experimental efficiencies.

C. Execute focusing and polarization models¶

Status: executed reference available in the wavefront-model notebook. Run the beam notebook first because it writes the expanded library. Downloads: Chapter 5 notebook: sampled focusing and geometric phase and wavefront-models.json.

Scalar line-focus activity¶

Use a 48-cell, $31.2\,\mu\mathrm m$ aperture, design distance $30\,\mu\mathrm m$, and output index 1.5. Derive the exact path-length phase, compare continuous/sampled/local-library fields, propagate with the outgoing angular spectrum, and inspect FWHM, a consistently normalized peak, and line-intensity concentration within $\pm1\,\mu\mathrm m$.

Refine window size independently from sample spacing for each of the continuous, sampled-phase, and local-library apertures. The displayed reference uses $\Delta x=0.10\,\mu\mathrm m$ and a $204.8\,\mu\mathrm m$ FFT window. The window study fixes $\Delta x=0.05\,\mu\mathrm m$ and uses $102.4/204.8/409.6\,\mu\mathrm m$ windows; the grid study fixes the $409.6\,\mu\mathrm m$ window and uses $\Delta x=0.10/0.05/0.025/0.0125\,\mu\mathrm m$. Every peak uses one frozen continuous-reference denominator. Integrate between the same physical focal-window endpoints and report both the integral divided by aperture width and its fraction of each model’s total line intensity. The final independent pairs check profile, FWHM, peak, fixed-window integral, and concentration changes against the limits saved in the notebook/JSON; a failed check must remain visible. The notebook also checks plane-wave phase and propagating-bin norm. These concern prescribed scalar fields; no finite-lens Maxwell validation or physical focusing-efficiency claim follows.

Jones-matrix activity¶

Define the laboratory circular basis explicitly. Verify converted phase $2\sigma\theta$ for ideal half-wave elements, unconverted amplitude, and conversion fraction $\sin^2(\delta/2)$ for ideal retarders. Check analytical identities and unitarity to 10^-12. Explain why this model needs a physical anisotropic vector library before device claims.

Deliverables: phase/field profiles, scalar metrics with denominators, window/grid changes, axial response, and Jones conversion/phase plots. State which figures come from each model and which physical effects are absent.

D. Finite-device and independent-method outline¶

Status: outline; a finite full-wave beam deflector/metalens and an FEM/FDTD cross-check have not been executed.

Select a small aperture and preserve its full geometry, illumination beam, material model, and reference planes. Begin with matched interface/film benchmarks, then reproduce a periodic reference, then simulate finite aperture edges with appropriate absorbing boundaries. For FDTD refine mesh, PML, time decay, and source/monitor placement; for FEM refine mesh/order, exterior domain, and port/near-to-far settings.

Compare a precisely defined angular or focal observable with the local/scalar prediction. For a lens define incident power, transmitted power, focal integration region, polarization channel, and collection geometry. Do not substitute a scalar line-intensity fraction for a full-wave power measurement. Relevant methods appear in Meep’s near-to-far examples.

Deliverables: solver/model files, independent refinements, geometry-to-model mapping, order/angular or focal metrics, numerical cost, and unresolved differences.

E. Amplitude, bandwidth, and robust-design outline¶

Now available: ms-09 executes fixed-geometry spectral sweeps, ms-10 studies local/supercell discrepancies, and ms-11 executes nominal bounded width optimization with independent checks. The extensions below go beyond those restricted demonstrations.

Status: outline; hologram reconstruction, a vector full-wave geometric-phase device, material-dispersive broadband optimization, and fabrication-tolerance optimization have not been run.

Choose one extension with a declared target and constraints. Examples are an amplitude-apodized scalar aperture followed by physical realization, a vector anisotropic library with desired/unwanted polarization channels, a multiwavelength design using justified dispersive materials, or a bounded geometry optimization with independent verification. Specify dimensions and fabrication limits before fitting or optimizing. Hold out wavelengths, incidence conditions, or tolerance cases where the research claim requires generalization.

Deliverables: objective and feasible domain, data/model/solver version, baseline, proposed design, independent verification, and a cost/robustness report. These stages connect naturally to Chapter 6 and Chapter 10.

Contribution handoff¶

Preserve the topic IDs and original evidence labels. Submit code, editable notebooks, data, environment, references, plots, and an execution-status note through the contribution guide. Routine publication displays saved results; it does not rerun heavy simulations.