Chapters 2 and 5. A good beam deflector at one wavelength can lose efficiency or send light in the wrong direction nearby. Predict which requirement limits its useful band, then test the fixed Chapter 5 geometry.
Executed: 161 wavelengths, two harmonic resolutions, three spectral grids, target efficiency, angular acceptance and modal group delay. Constant real indices isolate structural dispersion; measured material dispersion, absorption and finite apertures are outside this model.
What are we simulating?¶
The six-ridge sequence repeats every 3.9 µm. Height is 1.8 µm, pitch 0.65 µm and ridge index 2.4; y is invariant. Gaps and the incident region contain air; substrate index is 1.5. This exact geometry stays fixed throughout the sweep.
Use normal incidence, electric field along y and time dependence exp(−iωt). Reference reflection to z=0 and transmission to z=h. Sweep vacuum wavelength, keeping widths, pitch, height and indices fixed. Redesigning a different cell for each wavelength does not demonstrate a single broadband device.
Explore, then run¶
Open the browser lab in a new tab · Executed notebook · Complete source bundle
The browser selects saved calculations. Extract the complete ZIP, preserve folders, install requirements-labs.txt in Python 3.12, launch JupyterLab and restart/run notebooks/09-dispersion.ipynb. Physical inputs, performance requirements and numerical controls are separate parameters. One BLAS thread is efficient for these small matrices.
Define the metric first¶
For incident-normalized TE power and the target angle at normal incidence:
The efficiency band requires T₊₁≥0.8. The useful band also requires |θ₊₁(λ)−θ₊₁(1.55 µm)|≤0.5°. Choose the connected passing interval containing 1.55 µm. Interpolate its threshold crossings and flag edges clipped by the scan. Report Δλ, Δλ/λ₀ and frequency width:
Total transmission and T₊₁/ΣTₘ answer different questions; neither replaces incident-normalized efficiency. A threshold band is distinct from resonance FWHM. The substrate’s m=±4 orders reach grazing at 1.4625 µm. The grid avoids this exact singular point and records the changing open-channel count.
Investigation sequence¶
Predict: explain why angle and complex response vary even with constant refractive indices.
Sweep: compare target efficiency, total transmission and angle over 1.36–1.74 µm. Identify which requirement sets each useful-band edge.
Resolve: compare 41, 81 and 161 samples at the same harmonic resolution. Report edge changes rather than unsupported decimal places. Do not merge disconnected passing windows.
Refine independently: compare M=96/144 at every wavelength. Small power-balance error does not replace refinement. Narrow features can require further spectral and harmonic resolution.
Differentiate: for exp(−iωt), compute τ=d arg(t₊₁)/dω. Sort frequency, unwrap phase, mask weak transmission and neighboring derivative stencils, and compare 81/161-point derivatives. This is modal delay at fixed faces, not detector arrival time; power convergence does not guarantee derivative convergence.
Change the requirement: repeat with another threshold or angular tolerance while preserving the device.
Deliverable and extensions¶
Submit the geometry, editable notebook, original plots, both bandwidth definitions, edge/clipping reports, spectral/harmonic sensitivity and model limits. Run all three new notebooks, then python scripts/validate_design_studies.py for the fixed published-reference audit. It does not certify arbitrary edited designs.
Further outline: add a documented causal dispersive material model and check its range, passivity and absorption. Finite-aperture spectra, multiwavelength optimization, independent FEM/FDTD and measured-device bandwidth remain extensions.
Sources and next lab¶
Use Chapter 2 for flux and phase conventions and Chapter 5 for the design. The Fourier method is discussed by Moharam and Gaylord. Our calculations and figures are original. Follow the source and reuse guide.