Modules

What each module does and how mature it is. Planned work is listed in WIP.md.

• propagators: mature and tested (Rayleigh-Sommerfeld, Fresnel, Fraunhofer, band-limited and padded propagation; vectorial propagation of [E_x, E_y] with the longitudinal component E_z, any scalar propagator vectorized, Mansuripur’s high NA focusing, checked against the Richards-Wolf integrals; propagation onto free output windows: shifted angular spectrum method with Matsushima’s band limit, scaled Fresnel and Fraunhofer integrals, far field in chosen directions)

• tilted: propagation onto tilted planes, tested: scalar and vectorial fields from z = 0 onto a plane with any rotated normal and back (tilted source to parallel target), sampled in the plane’s own coordinates; exact non-uniform plane wave sum (separable for tilts about x or y, finufft or a direct sum otherwise) and Matsushima’s spectrum rotation with the Jacobian; carrier removal, power flux and ISO 11146 beam widths on inclined planes (detectors); tilted_plane() and vectorial_tilted_plane() are also available from propagators

• fdtd: new, tested (first milestone): finite-difference time-domain solver for Maxwell’s equations in 1D and 2D (polarizations Ez and Hz, with the TE/TM naming explained in the module docstring), NumPy kernels, shapes with sub-cell permittivity averaging, conductive media, convolutional PML, PEC and periodic boundaries, point and line current sources and a total-field/scattered-field plane wave, probes, snapshots (float16, ring buffer, streaming to HDF5), running DFTs and flux monitors, plots and animations (also of the steady state from one DFT); checked against thin_films (Bragg mirror, lossy slab) and the exact cylinder series in both polarizations. Planned: numba kernels, dispersive media, mode sources, Bloch boundaries, 3D (proposals/fdtd.md in the project notes)

• analytic_phase: propagation with analytic handling of the wavefront phase after Wyrowski and coworkers, tested: fields stored as a smooth sampled amplitude times a separable quadratic phase with carrier (QuadraticPhaseField); exact semi-analytical Fourier transforms (SFT) with automatic switch to the FFT; angular spectrum and Fresnel propagation with the same small number of samples at any distance, through the focus; propagation onto tilted planes (spectrum rotation with only the smooth amplitude interpolated); pointwise homeomorphic Fourier transform and geometric propagation with amplitude transport and Gouy phase

• fft, utils, masks, geometry, analytical: mature and tested (fft.czt(): Bluestein’s chirp-z transform; fft.zoom_FT() and zoom_inv_FT() evaluate FT() and inv_FT() on any equidistant grid) (utils.pad()/unpad() for centered zero padding; analytical holds the Airy pattern, its encircled energy, the diffraction limited OTF and the exact series solutions for a dielectric cylinder (TE) and sphere (scalar), also for beams: cylinder_field(), sphere_field())

• bases: mature (Zernike, Legendre, Chebyshev bases, numerical orthogonalization, fits); all bases can be shifted and rotated, Legendre and Chebyshev derive from the abstract TensorProductBasis; tested

• beams: tested (plane, spherical, Hermite-Gaussian and Laguerre-Gaussian beams)

• interfaces, thin_films: mature and tested (Fresnel coefficients, coherent multilayer stacks; stack_transfer_functions() gives r and t on the angular spectrum grid of a field)

• materials: Abbe (n_d, V_d) dispersion model via a two-term Cauchy formula, tested

• polarization.mueller, polarization.jones: Mueller calculus and basic Jones vectors and matrices, tested

• vectorial_imaging: vectorial, partially coherent imaging (Abbe’s method) of 1D and 2D masks with conventional, annular, dipole and quasar sources, TE/TM/unpolarized/partially polarized illumination, contrast and NILS, tested (two-beam contrast of TE and TM waves)

• litho: lithography imaging toolbox built on vectorial_imaging, tested: illumination sources (conventional, annular, dipole, quadrupole, quasar, multipole with sector or circular poles, freeform maps) for any sigma and any polarization (x, y, TE/azimuthal, TM/radial, unpolarized, Jones, Stokes, degree of polarization); mask layouts with analytic diffraction orders (lines and spaces, isolated lines, trenches, dense, isolated and staggered contact holes, any rectangles, clear or dark field); mask stacks in the thin mask approximation (binary, attenuated PSM), with absorber transmissions from thin_films (angle and TE/TM dependent) and an optional boundary layer model for thick mask edges; projector with immersion (or dry), vectorial or scalar model, Fringe Zernike aberrations and pupil apodization (amplitude transmission 1 + sum of Fringe Zernikes, optionally as log amplitude, or any transmission function); images in the resist on a thin film wafer stack (BARC, top coat, substrate; TE/TM Fresnel transmission, standing waves); metrics (contrast, ILS/NILS, threshold CD, dose to size, MEEF, focus-exposure matrix, Bossung curves, process window, exposure latitude vs. depth of focus, Gaussian resist blur) and plots (source, mask, pupil fill, images, Bossung curves, process windows); litho.lens hands a ray traced lens (raytrace) to the projector (Fringe Zernikes on the projector’s pupil, distortion, telecentricity, placement of a small feature), computes CD, placement and best focus across a scanner slit with scan averaging and the linear scanner correction, and Zernike sensitivities with a quick aberration budget; litho.hopkins computes transmission cross coefficients (TCC) and their SOCS decomposition, equal to Abbe’s method for thin masks and fast for many masks with the same optics; swing curves and BARC optimization (litho.wafer, absorbed energy exact against thin_films) and litho.resist, a chemically amplified resist in an x-z cross section (exposure, PEB with acid diffusion, quencher and amplification, Mack development with a fast marching front; profile, CD at top/middle/bottom, sidewall angle, top loss, dose to size, E0); litho.rcwa: rigorous mask diffraction of 1D layouts by RCWA (Li’s factorization, conical incidence, enhanced transmittance matrices; exact against thin_films, energy conserving), RigorousMask for Abbe and Hopkins imaging and a calibration of the boundary layer model; litho.jones_pupil: polarization of the lens as Jones pupils in the representations of the literature (Pauli-Zernike, polar decomposition, retardance and diattenuation in orientation Zernike polynomials) with fits, used by Projector(jones_pupil=...) in Abbe and Hopkins imaging, and a polarization ray trace of coated or uncoated lenses with optional CaF2 elements and their intrinsic birefringence (litho.lens.lens_jones_pupil on top of raytrace.System.trace_polarization, materials.CubicCrystal); litho.source_opt: source optimization of a pixelated source with symmetry, pupil fill and common dose constraints, maximizing the worst NILS over clips and foci by linear programming on the Abbe responses; litho.opc: optical proximity correction of Manhattan polygons (MaskLayout.from_polygons, exact unions of rectangles): bias tables through pitch, SRAF rules with a printability check, fragmentation, edge placement errors and model-based OPC with a regularized Jacobian update (Abbe, Hopkins or SOCS imaging); litho.illuminator: a ray traced KrF illuminator for the Caldwell lens (zoom-axicon of US 5,675,401 adapted to fused silica, diffuser, glass rod, REMA objective of US 5,982,558): sigma 0.32 to 0.83 and annular settings, pupil fill per field point as an Illumination, field uniformity, telecentricity against the lens

• polarization: TE/TM bases, Stokes to Jones decomposition, DOP and IPS

• imaging: scalar imaging, tested: coherent imaging through a circular pupil, PSF with pupil masks (obscuration, apodization) and phases (also zoomed onto any output window via the chirp-z transform), OTF and MTF, Strehl ratio and Maréchal approximations

• microscopy: microscope imaging, tested: partially coherent bright field images of thin objects after Abbe, with Köhler illumination sampled on the frequency grid (disk, annulus, oblique or any source points, source_points()), pupils with exact defocus, aberrations and masks (pupil()), weak object amplitude and phase transfer functions (weak_object_transfer()) and the grating cutoff λ/(NA + NA_c); Zernike phase rings (phase_ring()) and the Nomarski DIC effective object with shear and bias (dic_object()); scalar 3D amplitude PSFs from the pupil (uniform or aplanatic, psf_3d()), fluorescence confocal PSFs for any pinhole (confocal_psf(), Airy units), the thin plane response of optical sectioning and the reflection mirror response; Fourier ptychography: LED array illumination, the low resolution forward model and the reconstruction by alternating projections with embedded pupil recovery (quasi-Newton or EPRY step), pupil overlap in k-space

• abcd: ABCD (ray transfer) matrices, tested: thin and thick lenses, spherical interfaces, tilted mirrors (tangential/sagittal), GRIN media; OpticalSystem builds systems element by element and traces rays and Gaussian beams (beam radius, wavefront radius, Gouy phase) anywhere, also inside elements; cardinal points and images; complex beam parameter q and the ABCD law (also in media), mode matching efficiency; resonator stability, eigenmodes and round trip Gouy phase; collins() propagates any field through an ABCD system (Collins integral, separate x and y matrices for astigmatic systems)

• unwrap: 2D phase unwrapping, tested: least squares unwrapping with a DCT Poisson solver (Ghiglia-Romero), weighted least squares with modulation or amplitude weights and masks (conjugate gradients preconditioned by the DCT solver), quality guided unwrapping by reliability sorting (Herráez), unwrapping with a reference estimate (e.g. a TIE phase) and two-wavelength (synthetic wavelength) unwrapping, residue maps, wrap() and a 1D helper

• interferometry: fringe analysis, tested: simulated interferograms (Zernike test wavefronts, tilt carrier, visibility, camera noise via processing.CameraNoiser, miscalibrated phase shifters); phase-shifting algorithms (N-step least squares for arbitrary known shifts, 4-step, Carré with unknown step, Hariharan’s 5-step), each returning the wrapped phase and the modulation; Takeda’s Fourier-transform method for single interferograms with a carrier (sideband window, automatic carrier search, masks, apodization, overlap warning); phase error vs. shifter miscalibration

• holography: digital holography, tested: off-axis and in-line holograms (tilted reference, phase shifts; camera noise via processing.CameraNoiser), sampling condition for the reference angle; reconstruction by sideband filtering (circular, Tukey, Hann, Gaussian windows, automatic carrier search), phase-shifting holography (via interferometry.psi_n_step()) and Gabor holography with iterative twin image suppression; zero order suppression; numerical refocusing (angular spectrum or Fresnel, padded); autofocus (amplitude variance, Tamura coefficient, gradient, Laplacian); quantitative phase with background compensation, tilt removal and unwrapping (unwrap)

• raywave: conversion between ray ensembles and wave fields, tested: ray field summation from a structured ray grid (ray tubes with the amplitude from the ray density, overlapping sheets added with their Gouy (Maslov) phases), Gaussian beamlets launched from rays or from sampled fields (also from an analytic_phase.QuadraticPhaseField) and propagated in closed form through foci and caustics, or traced through the surfaces of a raytrace.System (base ray and parabasal rays, complex ABCD law, Gouy phase on the continuous branch), rays along the local phase gradient (traceable with raytrace.System.trace()), rays from local spectra (sampled Husimi function) and the Wigner function of a 1D field; structured rays from fields for the round trip through ray-traced optics

• fresnel: Fresnel lenses, tested: a parent surface (sphere, conic, even asphere) collapsed into zones of constant pitch, constant depth or given radii, with curved or flat (conical) facets and drafted risers (FresnelSurface); ray tracing on the base plane with the parent’s normals (ideal model) or through the real sawtooth (riser hits, shadowing and TIR lost or traced on as stray light, optical path with the steps); classification of the rays of a pupil (facet_losses()), exact facet slopes for a focus (focusing_slopes()) and the fit of a parent to slopes (surface_from_slopes()); kinoform depth, scalar efficiency and the diffractive Abbe number; PMMA dispersion

• raytrace: sequential ray tracing, tested: spherical, conic and even aspheric surfaces, lenses and mirrors, ideal (paraxial) lenses, apertures and vignetting, indices from constants, materials.abbe_dispersion or any function (fused silica after Malitson); decentred and tilted surfaces, coordinate breaks (fold mirrors), tilted image planes, fields in any direction, object space telecentric systems; exact intersections and vector Snell’s law, total internal reflection; ray aiming at the real stop; paraxial marginal and chief rays, cardinal points, entrance and exit pupils (system matrix identical to abcd, to_abcd()), Seidel sums; OPD maps over the pupil relative to the reference sphere on the chief ray, Fringe Zernike fits, RMS OPD and Strehl ratio; PSF and MTF via imaging; spot diagrams, ray fans and layout plots; example: Caldwell’s all fused silica KrF lithography projection lens (NA 0.56, 4x) with decentre and axial shift sensitivities of its elements (with_axial_shift()), a water immersion ArF lens and a KrF REMA objective and zoom-axicon from patents; cones (Axicon), surfaces with a Fringe Zernike departure (ZernikeSurface: figure errors, mount deformations, freeforms), x-z sections in draw(); polarization ray trace with energy (System.trace_polarization(), Chipman’s 3x3 matrices): Fresnel refraction and reflection through thin film coatings on any surface (coatings.Coating, coatings.Layer) with layer thickness variations across the surface (radial, xy or Zernike polynomials), coated metal mirrors, absorbing (materials.Isotropic, n + ik) and birefringent or dichroic media (materials.Uniaxial, materials.CubicCrystal); power along every ray, transmittance, output fields and Jones matrices; vectorial focus of the traced rays (System.vectorial_focus(), Debye-Wolf integral with the coatings and the OPD)

• delano: Delano’s y-ybar diagram, tested: the diagram of any centred raytrace.System (powers, spacings, magnification, NA, pupils and cardinal points agree with the paraxial trace and abcd), thin lens systems from a drawn polygon, thick singlets with the same first order properties (principal planes, thicken()), refine() for hand drawn layouts (Petzval sum, spaces, beam heights, power signs)

• lens_optimization: damped least squares lens optimization, tested (singlet bending for minimum spherical aberration, aspheres, solves): curvatures, thicknesses, object distance and even aspheric terms as variables, solves for total length and object side NA, operands for transverse ray errors, distortion and telecentricity (also of the real chief ray through the stop centre), paraxial targets and clearances (centre and edge thicknesses at the rim of the full beam), stages of growing aperture and field

• beam_quality: beam widths, azimuth and M² after ISO 11146 (second moments, iterated integration window, caustic fit), tested

• modes: mode analysis and fibres, tested: decomposition of any field into Hermite-Gaussian or Laguerre-Gaussian modes (given or fitted waist, curvature and centre; power fractions, reconstruction error, propagation of the expansion), M² from the mode content (exact second moments of coherent superpositions); LP modes of weakly guiding step-index fibres (characteristic equation, cutoffs, b(V), n_eff, mode fields, Marcuse’s Gaussian approximation); coupling efficiency of any field into any mode (generalizes abcd.coupling_efficiency()), sweeps over offset, tilt and defocus, power per fibre mode

• TIE: basic transport of intensity solver, partly tested

• ifta: basic Gerchberg-Saxton and variants for beam shaping, tested (convergence of GS, HIO-HOO and over-compensation on a reachable target, phase quantization)

• phase_retrieval: phase retrieval, tested: Fienup’s error reduction and hybrid input-output with support, reality and positivity constraints, shrink-wrap support and the autocorrelation support, error histories, removal of the trivial ambiguities (shift, constant phase, twin image); pupil phase retrieval from through-focus PSFs (phase diversity) by a multi-plane Gerchberg-Saxton iteration (with unwrapping via unwrap) or a least squares fit of Fringe Zernikes with the analytic Jacobian, with paraxial (Z4) or exact (angular spectrum) defocus; near-field (Fresnel) coherent diffractive imaging with any pyoptics propagator in place of the Fourier transform

• shack_hartmann: Shack-Hartmann wavefront sensor, tested: spot images (Fraunhofer pattern of each lenslet on its camera window) from a pupil phase, e.g. a ray-traced raytrace.Pupil, thresholded centroids with optional reference image, intensity weighted model slopes, modal reconstruction (least squares fit of Fringe Zernike gradients) and zonal reconstruction (Southwell); sensitivity and dynamic range of the design

• wpm: wave propagation method (Cython extension) for inhomogeneous, also absorbing media, 2D and y-invariant 1D fields, multi-step driver; direct O(N^2) sum (OpenMP) or the fast “levels” variant (one FFT pair per distinct index), chosen automatically; tested against Rayleigh-Sommerfeld, plane waves (tilted, evanescent, absorbed), slabs and a GRIN mode

• processing: image processors for single images and image stacks, tested: Scaler (peak, mean or total of field strength or intensity), CameraNoiser (Gaussian, Poisson shot noise, linear camera model after EMVA 1288 with dark current, read noise, full well, gain, ADC and fixed pattern noise), Shifter (subpixel Fourier shift) and Centerer (centering on the intensity centroid, utils.center_of_mass())

• sketch: schematic drawings of optical layouts (thin lenses, stops, prisms, mirrors, detectors) with paraxially traced rays, tested: thin lens ray tracing in both directions; ready-made sketches of a Köhler microscope, Abbe’s 4f picture with filtered orders, phase contrast, DIC, confocal and LED array (Fourier ptychography) microscopes; primitives for folded layouts (beam splitters, rotated lenses and detectors, spherical surfaces, lenslet arrays, beams as bands between edge rays) and sketches of a Twyman-Green and a Fizeau interferometer, a Shack-Hartmann and a phase diversity lens test, off-axis holography (lensless and microscope), a coated fold mirror with s and p, the KrF illuminator with its pupil and field planes, and a spatial filter

• Unfinished stubs: interpolator.TrigInterpolator cannot be evaluated yet, the functions in polarization._polarization are empty

Tests

The core is tested: a suite of about 900 pytest tests checks the numerics against analytical results (Fresnel and Fraunhofer diffraction, Gaussian beam propagation, Mahajan’s on-axis intensity, Airy pattern and OTF, Fresnel equations, thin film stacks, Goos-Hänchen shift, beam widths and M² of Hermite-Gaussian beams, Mueller and Jones calculus, FFT conventions, Richards-Wolf integrals for high NA foci, two-beam interference of TE and TM waves, in air and in a resist, standing waves in a resist on silicon, threshold CDs and process windows of analytic images, apodized two-beam images, ABCD matrices against lens and resonator formulas and the Collins integral against Gaussian beams, fields on tilted planes against plane waves and Gaussian beams at the plane’s 3D points, power flux through tilted planes, phase unwrapping of wrapped Zernike wavefronts, vortices and noisy wavefronts, phase-shifting and Fourier fringe analysis of known Zernike phases, holograms of known objects reconstructed, refocused and autofocused, exact ray traces against Snell’s law, perfect conics, Seidel sums and the Airy pattern, tilted plates and fold mirrors, Fienup’s HIO and phase diversity recovering known objects and Zernike pupil phases, HG/LG mode decompositions and M² from the mode content against ISO caustic fits, LP fibre modes against cutoffs and the Marcuse mode field radius, fibre coupling against the Gaussian formulas) and runs in GitHub Actions on Python 3.9, 3.11 and 3.13 for every push and pull request. A consistent phase convention is used throughout (see Conventions).