Examples

The example notebooks illustrate the modules with examples whose physics is easy to follow, and compare the numerics with analytic results wherever an analytic solution exists. They are shown here with their stored outputs; to run them yourself, see Running the notebooks.

Propagation and diffraction

• Free-space propagation: Fresnel, Rayleigh-Sommerfeld, Fraunhofer

• Vectorial diffraction: longitudinal fields, vector beams and high NA foci

• Propagation onto tilted planes

• Chirp-z transform: zoomed spectra, shifted and scaled propagation

• Propagation in inhomogeneous media: the wave propagation method (WPM)

• Propagation with analytic phase: semi-analytical and homeomorphic Fourier transforms

• FDTD basics: pulses, absorbing boundaries, a Bragg mirror and a cylinder

Beams, modes and paraxial optics

• Gaussian beams, modes and the Gouy phase

• Beam widths and M² after ISO 11146

• ABCD matrices: paraxial rays, Gaussian beams and resonators

• Mode analysis and fibre modes

Interfaces, thin films and polarization

• Interfaces and thin films

• Polarization: Jones and Mueller calculus

• Interactive layer stacks

• Layer stacks as transfer functions: Goos-Hänchen shift, frustrated TIR and etalons

• A coated laser mirror: thin film design, polarization ray trace and vectorial focus

Imaging, aberrations and ray tracing

• Imaging, resolution and aberrations

• Basis functions: Zernike, Legendre, Chebyshev and orthogonalized sets

• PSF, MTF and Strehl ratio

• Sequential ray tracing to OPD maps

• Polarization ray tracing with coatings

• From rays to waves: the focus of a lithography projection lens

• Rays and waves: which way round does a plano-convex lens go?

• Rays and waves in a laser processing head: a diffractive 5 × 5 spot splitter

• Rays and waves in an eye aberrometer: a Shack-Hartmann sensor behind a schematic eye

• Fresnel lenses: from a thick lens to a solar concentrator

• Gaussian beam decomposition through a lens system: a spatial filter

• Zernike aberration gallery: Z2 to Z16 from real lenses

Measurement, phase retrieval and beam shaping

• Image processing: scaling, camera noise and centering

• Phase unwrapping

• Interferometry: phase shifting and Fourier fringe analysis

• Measuring the radius of curvature of lenses with a Fizeau interferometer

• Measuring a lens two ways: Shack-Hartmann versus phase diversity

• Digital holography

• Digital holographic microscopy: holography, TIE and phase unwrapping

• Phase retrieval: Fienup’s algorithms and phase diversity

• Phase retrieval with the transport-of-intensity equation (TIE)

• Beam shaping with the iterative Fourier transform algorithm (IFTA)

Microscopy

• Abbe image formation: diffraction orders, pupil cutoff and partial coherence

• Phase contrast and DIC: making transparent objects visible

• Confocal microscopy: pinhole, PSF product and optical sectioning

• Fourier ptychography: a synthetic aperture from an LED array

• Digital holographic microscopy: holography, TIE and phase unwrapping (in Measurement, phase retrieval and beam shaping)

Lithography

• Vectorial imaging: lines and spaces with s, p and unpolarized light

• The lithography toolbox: sources, masks, mask and wafer stacks, process windows

• From lens design to the wafer: aberrations across the scanner slit

• Hopkins imaging: transmission cross coefficients and SOCS

• Resist model and swing curves

• Rigorous mask diffraction (RCWA) and mask 3D effects

• Mask models compared: Kirchhoff, film, boundary layer and RCWA

• Jones pupils: polarization of the projection lens

• Source optimization (SO lite)

• Optical proximity correction (OPC)

• Telecentricity across the slit: chief ray versus energetic

• A ray traced KrF illuminator for the Caldwell lens

• Delano diagrams: first order layout of a lithography lens

• Seidel aberrations surface by surface: correction and tension in Caldwell’s KrF lens

• Magnification, distortion and telecentricity as Zernikes

Running the notebooks

You need an installation of pyoptics as described in the main README, plus matplotlib, Jupyter and (for notebook 12) ipywidgets, all contained in the notebooks extra:

pip install -e ".[notebooks]"
jupyter lab notebooks/

The widgets of notebook 12 only work with a running kernel; static viewers such as GitHub show the static figures stored below each interactive cell.

Run times on an ordinary laptop range from a few seconds to about a minute per notebook (longest: 01 and 05), with these exceptions:

• about 5 minutes: 32 (lithography toolbox, many focus-exposure matrices and through-pitch series), 41 (telecentricity, some 200 polarization ray traces over the slit)

• about 3 minutes: 21 (rays to waves, angular spectrum propagation to many planes), 28 (phase retrieval, many HIO runs), 47 (rays and waves, propagation to many planes and sums of 7 800 Gaussian beamlets)

• about 2 minutes: 20 (polarization ray trace), 33 (lens aberrations, ray traces over the slit and Zernike sensitivities)

• about 2.5 minutes: 52 (Abbe image formation, partially coherent images summed over thousands of source points)

• about 3 minutes: 56 (Gaussian beam decomposition, sums of 9 000 beamlets over the pinhole plane)

• about 1.5 minutes: 09 (mode analysis, basis fits and large LG expansions), 25 (Shack-Hartmann vs. phase diversity), 26 (holography, many refocusing runs on 1024² padded grids), 31 (vectorial imaging lithography), 37 (mask models, RCWA over a grid of pitches and duty cycles), 45 (Zernike gallery, fifteen lenses tuned, traced and propagated through focus), 53 (phase contrast and DIC), 55 (Fourier ptychography, many reconstructions)

Notebooks 14 (coated laser mirror), 24 (radius interferometer), 27 (holographic microscopy), 34 (Hopkins/SOCS), 35 (resist), 36 (rigorous mask), 38 (Jones pupils) and 54 (confocal microscopy) take 20 to 30 seconds, 39 (source optimization) and 40 (OPC) about 10 seconds; 57 (FDTD basics) takes about a minute.

The FFTs (pyoptics.fft) use all CPU cores; FFT-heavy notebooks such as 04 (chirp-z transform) and 26 (holography) run about 30 to 45 % faster than with single threaded FFTs. Set pyoptics.fft.workers = 1 to use a single core.

To re-execute all notebooks and store their outputs:

jupyter nbconvert --to notebook --execute --inplace notebooks/*.ipynb