API · Zernike.jl

Zernike.plotconfig — Constant

Zernike plot settings.

Fields / Options:

size::Tuple{Float64, Float64}: window size (DPI scaled resolution);
fontsize::Float64: text size;
colormap::Symbol: Default: :oslo;
theme::Makie.Attributes: Default: theme_black();
focus_on_show::Bool: whether the window is focused on generation (default: true).

There are two methods which can be used to trigger a settings refresh: resize! & reset!.

Tip

Setting the theme to Zernike.Attributes() will use the default Makie theme.

Note

The focus_on_show attribute of plotconfig only controls whether Zernike plots will set that option of the screen configuration, but if using Makie to create other types of plots this property needs to be set using activate! or set_theme!. The Zernike.reset!(plotconfig) function will appropriately reset these along with the rest of the plotconfig.

See also: transform, S.

source

Zernike.sieve — Function

sieve(v::Vector{Float64}, threshold::Float64)

Zero out any elements lower than the threshold.

sieve(a::Vector)

Return Zernike non-sequential indices and the corresponding wavefront expansion coefficients for any non-zero coefficients in the input vector.

source

Zernike.standardize — Function

standardize(noll::Noll)
standardize(fringe::Fringe)

Format a Noll or Fringe specified Zernike expansion coefficient vector according to the ANSI standard.

Floating-point coefficient vectors need to be wrapped in the index types (e.g. standardize(Fringe(v))).

The Fringe method expects unnormalized coefficients; the input coefficients will be re-ordered and normalized in line with the orthonormal standard. As Fringe is a 37 polynomial subset of the full set of Zernike polynomials any coefficients in the standard order missing a counterpart in the input vector will be set to zero.

standardize(v_sub::FloatVec, [j::AbstractVector{Int}])
standardize(v_sub::Vector, orders::Vector{Tuple{Int, Int}})
standardize(W::Wavefront)

Pad a subset Zernike expansion coefficient vector to the full standard length up to n_max (1:j_max+1).

The tuples in orders must be of the form (m, n) associated with the respective coefficients at each index in v_sub.

j is a vector of single-mode ordering indices associated with the coefficients; if this is not supplied the coefficients will be assumed to be in order (0:j).

The Wavefront method pads the W.a coefficient vector.

source

Zernike.transform — Function

transform(v::Vector{T}, ε::T, [δ::Complex{T}], [ϕ::T], [ω::Tuple{T,T}]) where T <: Float64

Compute a new set of Zernike wavefront error expansion coefficients under a given set of transformation factors and plot the result.

Available transformations are scaling, translation, & rotation for circular and elliptical exit pupils. These are essentially coordinate transformations in the pupil plane over the wavefront map.

Main arguments

v: vector of full Zernike expansion coefficients ordered in accordance with the ANSI / OSA single index standard. This is the v vector returned by wavefront(ρ, θ, OPD, n_max);
ε: scaling factor {0 ≤ ε ≤ 1};
δ: translational complex coordinates (displacement of the pupil center in the complex plane);
ϕ: rotation of the pupil in radians (mod 2π), defined positive counter-clockwise from the horizontal x-axis;
ω: elliptical pupil transform parameters; 2-tuple where ω[1] is the ratio of the semi-minor axis length to the semi-major axis length of the ellipse and ω[2] is the angle defined positive counter-clockwise from the horizontal coordinate axis of the exit pupil to the minor axis of the ellipse.

The order the transformations are applied is:
scaling –> translation –> rotation –> elliptical transform.

Keyword argument options:

transform(v, ε, [δ], [ϕ], [ω]; [precision = 3], [finesse::Int])

precision: number of digits to use after the decimal point in computing the expansion coefficients. Results will be rounded according to this precision and any polynomials with zero-valued coefficients will be ignored when pulling in the Zernike functions while constructing the composite wavefront error; this means lower precision values yield faster results.
finesse: determines the size of the plot matrix; sets both dimensions to the given value.

Extended help

ε = r₂/r₁ where r₂ is the new smaller radius, r₁ the original

In particular the radial variable corresponding to the rescaled exit pupil is normalized such that:
ρ = r/r₂; {0 ≤ ρ ≤ 1}
r: radial pupil position, r₂: max. radius
ΔW₂(ρ₂, θ) = ΔW₁(ερ₂, θ)

For translation the shift must be within the bounds of the scaling applied such that:
0.0 ≤ ε + |δ| ≤ 1.0.

For elliptical pupils (usually the result of measuring the wavefront off-axis), the semi-major axis is defined such that it equals the radius of the circle and so ω[1] is the fraction of the circular pupil covered by the semi-minor axis (this is approximated well by a cosine projection factor for angles up to 40 degrees); ω[2] is then the direction of the stretching applied under transformation in converting the ellipse to a circle before fitting the expansion coefficients.

The transformed expansion coefficients are computed using a fast and accurate algorithm suitable for high orders; it is based on a formulation presented by Lundström & Unsbo (2007) doi:10.1364/JOSAA.24.000569.

source

Zernike.transform_coefficients — Function

transform_coefficients(v, ε, δ, ϕ, ω)

Directly compute Zernike wavefront error expansion coefficients under pupil transformations. The argument types are the same as in transform.

Returns a 2-tuple with the new coefficient vector and order n_max.

source

Zernike.wavefront — Function

wavefront(ρ, θ, OPD, n_max)

Fit wavefront errors up to order n_max.

Estimates wavefront error by expressing optical aberrations as a linear combination of weighted Zernike polynomials using a linear least squares method. The accuracy of this type of wavefront reconstruction represented as an expanded series depends upon a sufficiently sampled phase field and a suitable choice of the fitting order n_max.

Main arguments

ρ, θ, and OPD must be floating-point vectors of equal length; at each specific index the values are elements of an ordered triple over the exit pupil.

ρ: normalized radial exit pupil position variable {0 ≤ ρ ≤ 1};
θ: angular exit pupil variable in radians (mod 2π), defined positive counter-clockwise from the horizontal x-axis;
OPD: measured optical path difference in waves;
n_max: maximum radial degree to fit to.

Return values

Returns six values contained within a WavefrontOutput type, with fields:

recap: vector of named tuples containing the Zernike polynomial indices and the corresponding expansion coefficients rounded according to precision;
v: full vector of Zernike wavefront error expansion coefficients;
ssr: the sum of the squared residuals from the fit;
metrics: named 3-tuple with the peak-to-valley error, RMS wavefront error, and Strehl ratio;
W: the Wavefront function ΔW(ρ, θ);
fig: the Makie FigureAxisPlot.