The split-step Fourier scheme built from scratch and checked on a case whose answer is known, then pointed at an experiment that has none.

• Build the (x, y) grid and the conjugate (k_x, k_y) grid required by the Fourier implementation
• Generate the initial Gaussian field, waist w₀ = 50 µm at λ = 1.03 µm
• Apply the thin lens phase mask, f = 10 mm, and verify energy conservation before and after propagation
• Propagate with the Fresnel transfer function up to Z_max = 30 mm; transverse intensity at the lens and at Z_max
• Track the beam width along z, where the amplitude drops to 1/e of its maximum
• Extension: index map n(x, y, z) of a single silicon cone, 5 µm base tapering to 1 µm over a height of 10 µm, standing in a 10 µm pool
• Propagate the transmitted field in air up to 200 µm above the tip, plot a longitudinal (x, z) cut, and explain the ring seen in the measurements

The update equations of the lecture turned into a running code, then confronted with an interface.

• Domain of 5 µm sampled with N = 1001 points, time step set by the Courant number, 2000 iterations
• E_x polarized along x, H_y along y, propagation along z; coefficients a_e, a_m and a_s for a medium of conductivity σ
• Point source, Gaussian in time, λ = 800 nm, half width at 1/e of 15 fs
• Propagation in vacuum, then reflection on sharp and on smooth index transitions

The open part of the FDTD block: a 300 by 300 grid, a real material, and a number at the end rather than a pretty map.

• Pulse Gaussian in time (10 fs) and in space (5 µm) at λ = 800 nm, perfectly matched layers at the boundaries
• Propagation in vacuum: decide whether an E_x component appears, and say why
• Tungsten layer 100 nm thick, n = 3, σ = 5 × 10⁵ S/m at 800 nm, then a Drude model for a realistic dispersion
• Surface morphologies on the tungsten (bump, spike, rod, hole, interacting or periodic) and the local field enhancement they produce
• Field enhancement factor against the geometrical parameters, and absorption cross-section from the Poynting flux

The stationary problem, where propagation is replaced by an eigenvalue calculation.

• Unit cell of two layers: l₁ = 200 nm with ε₁ = 1, l₂ = 800 nm with ε₂ = 9
• Fourier expansion of 1/ε(x) on N = 2 numG + 1 plane waves, with numG = 50
• Analytical expansion coefficients, matrix differential operator, band structure across the Brillouin zone

The same formalism in two dimensions, on a lattice whose gaps have to be opened by hand.

• Square lattice first, then the band structure of a hexagonal lattice with r = 20 nm and a = 75 nm
• Arrays of voids: effect of the permittivity contrast on the gaps
• Projection of the complete photonic band gap as a function of r/a
• TE and TM polarizations compared, and the role of the dielectric contrast for each

Note

The course was previously listed under the title Nanophotonics. Slides are in English.