README

This dataset includes the results of a symmetry-based analysis of two-dimensional photonic crystals, spanning 11 distinct symmetry settings, two field polarizations, and five dielectric contrasts. For each of these settings, the dataset includes results for 10 000 randomly generated photonic crystal unit cells. These results assume time-reversal symmetry. In addition, results for time-reversal broken settings are included for 4 of the 11 symmetry settings, at a single dielectric contrast of 16 and across 4 time-reversal breaking amplitudes.

Usage and loading data

File structure

The dataset consists of several configs, which each containing several .jld2 files. Below, we briefly summarize the contents of each:

• lattices → lattices-planegroup$num.jld2: The unit cell geometry for photonic crystal samples in the plane group with number $num.
• bands-planegroup$num-tr → bands-planegroup$num-epsid$epsid-$mode-tr.jld2: Results for photonic crystals with time-reversal symmetry in plane group $num (corresponding to those in lattices-planegroup$num, in the same access-order), with dielectric contrast ID $epsid (running from 1 to 5, corresponding to a dielectric contrast of 8, 12, 16, 24, and 32), and mode polarization $mode (te or tm, corresponding to transverse electric and magnetic (TE and TM) polarization).
• bands-planegroup$num-notr → bands-planegroup$num-gidx$gidx-notr.jld2: Results for photonic crystal with broken time-reversal symmetry in plane group $num (for TE modes and at a dielectric contrast of 16). The time-reversal breaking amplitude is set by $gidx, which runs from 1 to 4, corresponding to normalized magnetic field amplitudes of 0.1, 0.4, 0.7, and 0.01, respectively.

Each JLD2 file contains three 10 000 element vectors: flatv, isovalv and Rsv, corresponding to vectors of Fourier lattices (stored as ModulatedFourierLattice objects---see Crystalline.jl), isovalues determining the filling fraction, and real space lattice vectors.

Required packages

The data was processed and generated using the Julia packages Crystalline.jl (v0.4.21), SymmetryBases.jl (0.4.0), and MPBUtils.jl (v0.1.10) (which are compatible, at least, with Julia v1.6-v1.10), and are represented using data structures defined in these packages. Accordingly, they must be installed in order to load the data:

julia> using Pkg
julia> Pkg.add(name="Crystalline", version="0.4.21")
julia> Pkg.add(url="https://github.com/thchr/SymmetryBases.jl", rev="b349a4")
julia> Pkg.add(url="https://github.com/thchr/MPBUtils.jl", rev="f9e728")

Note that the above adds the packages at the specified versions; later versions may be incompatible.

The data is stored as .jld2 files, which additionally requires the JDL2.jl package (v0.4.46) to load:

julia> Pkg.add(name="JLD2", version="0.4.46")

Usage

Photonic crystal unit cells

Each photonic crystal unit cell is stored as a level-set function and a lattice basis. The dataset provides access to these quantities through three vectors, indexed by sample id (running from 1 to 10 000). Specifically, we include a listing of primitive direct basis vectors (Rs) and parameterizations of the level-set functions in terms of the reciprocal lattice vectors and coefficients (flatv) and isovalues (isovalv). Each element of these vectors corresponds to an individual photonic crystal sample.

These quantities can be used to construct the associated real-space unit cells via Crystalline.jl (and visualized via either GLMakie.jl or PyPlot.jl). E.g., to load and visualize the unit cell corresponding to the first photonic crystla sample in plane group 2:

julia> using JLD2, Crystalline, PyPlot

julia> num = 2 # plane group number
julia> lattice_data = load("lattices-planegroup$num.jld2")

julia> id     = 1
julia> flat   = lattice_data["flatv"][id]
julia> isoval = lattice_data["isovalv"][id]
julia> Rs     = lattice_data["Rsv"][id]

julia> plot(flat, Rs; isoval)

The value of the level-set function at a given point r₁R₁+r₂R₂ in real-space can be obtained from real(flat(r₁, r₂)); if it is greater than isoval, ε(r₁, r₂) is equal to the high-index dielectric value, otherwise 1.

Symmetry-analysis results

As an example, to load the file corresponding to results for TE-polarized modes in photonic crystals with plane group 2 symmetry, at a dielectric contrast of 8 (with time reversal):

julia> using JLD2, MPBUtils, Crystalline

julia> num = 2     # plane group number
julia> epsid = 1   # (ε = 8, 12, 16, 24, 32 at ids = 1, 2, 3, 4, 5, respectively)
julia> mode = "te" # "te" or "tm"

julia> data       = load("bands-planegroup$num-epsid$epsid-$mode-tr.jld2")
julia> lgirsd     = data["lgirsd"]     # small irreps, indexed by high-julia> symmetry k-points
julia> brs        = data["brs"]        # elementary band representations
julia> summariesv = data["summariesv"] # per-multiplet band topology, indexed by sample

The primary quanties of interest are summariesv (lgirsd and brs contain auxiliary general-purpose information that could also be obtained directly from Crystalline.jl). They are distinguished by whether they give band information multiplet-by-multiplet (summariesv). Each is a 10 000 element vector, whose elements correspond to distinct photonic crystal samples. The contents of each such element are best illustrated by example. E.g., taking the first sample:

julia> id = 1
julia> summaries = summariesv[id]
40-element Vector{BandSummary}:
 1-band (trivial): [Y₁, B₂, A₂, Γ₁]
 1-band (nontrivial): [Y₂, B₁, A₁, Γ₁]
 1-band (trivial): [Y₁, B₂, A₁, Γ₂]
 1-band (trivial): [Y₂, B₁, A₂, Γ₁]
 ⋮

The listings are per multiplet, counting from zero frequency and up, such that summaries[n] corresponds to the nth multiplet. We can also inspect individual multiplets in greater detail:

julia> summaries[1]
1-band BandSummary:
 bands:      1:1
 n:          Y₁, B₂, A₂, Γ₁
 topology:   trivial

I.e., the first multiplet (summaries[1]) is a single, trivial band with symmetry vector [Y₁, B₂, A₂, Γ₁].

BandSummarys can be added, which corresponds to stacking bands:

julia> summaries[1]+summaries[2]
2-band BandSummary:
 bands:      1:2
 n:          Y₁+Y₂, B₁+B₂, A₁+A₂, 2Γ₁
 topology:   nontrivial
 indicators: 1 ∈ Z₂

Cumulative band topology, i.e., associated with the consecutive stacking of multiplets from zero frequency and upwards, can consequently be obtained via:

julia> cumsum(summaries)

Individual properties can be extracted by field-access:

julia> summary  = summaries[id]
julia> topology = summary.topology
julia> bands    = summary.band     # band indices in multiplet
julia> n        = summary.n        # symmetry vector

The order of elements in the symmetry vector n is the same as those of the irreps in brs.

Finally, data["symeigsdv"] contains the unprocessed symmetry eigenvalues obtained from MPB.

Citation

If you use this dataset in your research, please cite:

@dataset{ghorashi_2024_10836524,
  author       = {Ghorashi, Ali and
                  Soljacic, Marin and
                  Vaidya, Sachin and
                  Benalcazar, Wladimir and
                  Christensen, Thomas and
                  Rechtsman, Mikael},
  title        = {Prevalence of two-dimensional photonic topology},
  month        = mar,
  year         = 2024,
  publisher    = {Zenodo},
  doi          = {10.5281/zenodo.10836524},
  url          = {https://doi.org/10.5281/zenodo.10836524},
}