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   Copyright 2026 FieldAlgo

   Licensed under the Apache License, Version 2.0 (the "License");
   you may not use this file except in compliance with the License.
   You may obtain a copy of the License at

       http://www.apache.org/licenses/LICENSE-2.0

   Unless required by applicable law or agreed to in writing, software
   distributed under the License is distributed on an "AS IS" BASIS,
   WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
   See the License for the specific language governing permissions and
   limitations under the License.

   ---
   This software is derived from 'meent' (https://github.com/kc-ml2/meent),
   Copyright (c) 2022 ML2, originally licensed under the MIT License.
   The MIT License requires that its copyright notice and permission notice
   be retained; both are reproduced in full in the file LICENSE.meent.

   ---
   This software also contains code ported from 'fmmax'
   (https://github.com/facebookresearch/fmmax), Copyright (c) 2023 Meta Research,
   licensed under the MIT License: the eigendecomposition gradient in
   src/wavebit/_common/backend.py is a port of `_eig_bwd` from that project's
   src/fmmax/utils.py. The MIT License requires that its copyright notice and
   permission notice be retained; both are reproduced in full in the file
   LICENSE.fmmax.

   fmmax's maintained fork (https://github.com/invrs-io/fmmax) carries the same
   function in src/fmmax/eig.py and re-states its copyright as 'Copyright: 2025
   invrs.io LLC', under the same MIT terms. Which of the two the port was taken
   from is not recorded, so both notices are stated here. LICENSE.fmmax itself is
   kept as the verbatim upstream text and nothing else, so that licence scanners
   identify it as MIT -- a note appended there is 29% of MIT's short body and
   drops the file below the match threshold GitHub's `licensee` uses.

   ---
   The Li-rule Fourier factorization of rasters in
   src/wavebit/rcwa/_engine/factorization.py was rewritten on 2026-09-25 from
   L. Li's published factorization rules (J. Opt. Soc. Am. A 13, 1870 (1996);
   14, 2758 (1997)): li_conv_matrices, li_conv_matrices_integrated and the
   helpers written with them, the bodies of fourier2d_piecewise_constant,
   fourier2d_pointwise and fourier2d_integrated, and the Li parts of
   to_conv_mat_raster_sampled and to_conv_mat_raster_integrated. The implementer
   of that rewrite did not consult RETICOLO's or meent's source or the bodies of
   the functions it replaced. compression_indices and compress_cell were then
   reimplemented by an implementer working from a written specification of what
   they return, in a copy of this repository without its history or the earlier
   bodies; the specification's author had seen the earlier functions and
   RETICOLO's ret2li. The other Li factorizations run on the rewrite as well:
   the Cartesian shape path (src/wavebit/rcwa/_engine/rectilinear.py) through
   fourier2d_integrated, and the tensor path's row transform
   (src/wavebit/rcwa/_engine/tensor_li.py, added 2026-09-24), which called the
   earlier helpers directly until it was moved onto the rewritten series. That
   move, and _row_dft, which restates the documented rule of _normalized_row_fft
   (a JAX workaround added beside _fourier2d_grid on 2026-09-24 and deleted with
   it), were written by an implementer working from written specifications whose
   authors had seen the earlier bodies. The rewritten code still calls helpers
   kept from before it, written in this repository in August 2026:
   _pixel_correction, _half_pixel_phase and _inverse_axes. It also calls
   _circulant_index, which came with the 2026-08-03 single-source port of
   meent's dfs2d and was rewritten from its definition on 2026-09-26 by an
   implementer who had not read the earlier body. The commit messages of those
   dates, and of later changes, record how.

   The rewritten builders' default nesting ('fixed') corresponds to the
   default, parm.res1.li = 1, of 'RETICOLO', the grating software of J.P.
   Hugonin and P. Lalanne (Institut d'Optique/CNRS): J.P. Hugonin and P.
   Lalanne, "RETICOLO software for grating analysis", Institut d'Optique,
   Orsay, France (2005), arXiv:2101.00901. Their 'symmetric' nesting
   corresponds to RETICOLO's parm.res1.li = 0.

   Earlier versions of those functions remain in this repository's history, with
   the helpers deleted alongside them (_fourier2d_grid, _fourier1d_integrated,
   _wrap and _circulant_gather); that history includes meent's own, from 2022.
   The Fourier builders descend from meent's dfs2d and cfs2d, written in July
   2024 with reference to the MATLAB source of RETICOLO V9, following its
   ret2li (in retcouche.m), retf.m and rettoeplitz.m; compression_indices and
   compress_cell were split from meent's cell_compression (2022). meent's code
   names RETICOLO elsewhere in that history too: fft_piecewise_constant was
   marked "reference: reticolo" from 2023-02 to 2024-06, and a commented-out
   dfs2d_debug in meent/on_numpy/emsolver/fourier_analysis.py, added on
   2024-07-19, is described as "algorithm from reticolo", as it still is in
   meent's own repository (commit bae2802). Checked on 2026-09-25 against
   RETICOLO's Zenodo records and the packages themselves: retcouche.m, retf.m
   and rettoeplitz.m are byte-identical in V9
   (https://zenodo.org/records/4419063, 2021), in its revision V9a
   (https://zenodo.org/records/5905381, 2022) and in V10
   (https://doi.org/10.5281/zenodo.14631951, January 2025). The V9 and V9a
   records declare no licence; the V10 record states CC BY 4.0. No package
   carries a licence file or licence text in its source. Each package's
   documentation states "copyright (c) 2005 Institut d'Optique/CNRS" and
   requires that publications and reports cite the reference above; V10's also
   calls RETICOLO a freeware. The V10 record also names itself a new version
   of V9 (isNewVersionOf 10.5281/zenodo.4419063).

   Those earlier versions, and the meent code named above, are used under the
   Creative Commons Attribution 4.0 International licence (CC BY 4.0) that the
   V10 record states. They were modified: written in Python, first in meent
   (https://github.com/kc-ml2/meent) and then in this repository, with
   reference to RETICOLO's MATLAB source, and the earlier versions have since
   been replaced by the rewrite described above. Attribution:

       RETICOLO, by J.P. Hugonin and P. Lalanne
       copyright (c) 2005 Institut d'Optique/CNRS
       Material: https://doi.org/10.5281/zenodo.14631951
       Citation: arXiv:2101.00901
       Licence: CC BY 4.0, https://creativecommons.org/licenses/by/4.0/

   Section 5 of that licence (Disclaimer of Warranties and Limitation of
   Liability) applies to that material. RETICOLO itself is not part of this
   software; the comparisons in tests/references/ call a separately installed
   copy, and one bridge script, tests/references/interface/reti_aniso.m, lays
   out its calls after RETICOLO's example exemple_V9_0D_anisotrope.m.
   tests/references/README.md records the check.

   ---
   From 2022-09-20 to 2026-08-03 this repository's history carried scattering-
   matrix helpers whose file header states that "many codes for scattering
   matrix method are from" the 'Rigorous-Coupled-Wave-Analysis' package of
   Nathan Zhao (Stanford University),
   https://github.com/zhaonat/Rigorous-Coupled-Wave-Analysis, reached through
   the fork https://github.com/yonghakim/zhaonat-rcwa: meent/integ/smm_util.py
   and meent/integ/scattering_method.py, later fieldwave/on_numpy/,
   on_jax/ and on_torch/emsolver/smm_util.py and scattering_method.py. That
   package is licensed under the MIT License; its LICENSE file carries the MIT
   text under a template copyright line ("Copyright (c) 2018 The Python
   Packaging Authority"), checked on 2026-10-14. The MIT License requires that
   its copyright notice and permission notice be retained, so this notice
   records the origin for the history that still contains those files. They
   were deleted on 2026-08-03 (commit 244cb819, "delete the triplicated
   transfer/convolution/scattering modules") and nothing in the current tree
   derives from them.
