API Reference

module VectorPSFs

The primary module for computing vector PSFs (point spread functions) with or without aberrations caused by plane-parallel plates. Also supports NV-center weighted PSFs, Strehl ratio calculations, and lens objectives.

NVCenter

A struct for storing an NV emission spectrum, specifying:

• λs::Vector{Float64}: the sampling wavelengths in micrometers (µm)
• weight::Vector{Float64}: weighting (normalized spectral intensity)

Example usage:

# Construct from an existing array of wavelengths & weights
center = NVCenter([0.632, 0.64, 0.65], [0.3, 0.5, 0.2])

# Or use the fitted NV default (pulling from `nv_spectrum_spline`):
center2 = NVCenter(0.5:0.01:0.7)

NVCenter(λs::Real, weight::Real)

Constructor that takes arrays of wavelengths λs and weights weight (must be same length). Converts them to Float64 and returns a new NVCenter.

NVCenter(λs::Vector{Float64})

Constructor that uses the global NV spectrum spline (nv_spectrum_spline) to obtain weights for the given array of wavelengths λs. Weights are normalized (so their sum is 1).

struct NoAberration <: PSFMode end

No plate, no tilt scenario.

struct NormalIncidence <: PSFMode end

Normal incidence with a plate scenario.

Objective

A microscope objective characterized by:

• f::Float64: focal length (µm)
• NA::Float64: numerical aperture
• n::Float64: refractive index of immersion medium (e.g., 1.0 for air)

abstract type PSFMode

Abstract type for dispatching different PSF modes (e.g. NoAberration, NormalIncidence, TiltedIncidence).

mutable struct PSFParams{M<:PSFMode}

Holds parameters for a certain PSF mode M.

• λ::Float64: wavelength in µm
• obj::Objective: the objective
• plate: either a PlaneParallelPlate or nothing
• α::Float64: tilt angle (radians)

PSFParams(λ::Float64, obj::Objective, plate::PlaneParallelPlate, α::Float64)

Construct a PSFParams{TiltedIncidence}, i.e. tilted incidence with plate. α in radians.

PSFParams(λ::Float64, obj::Objective, plate::PlaneParallelPlate)

Construct a PSFParams{NormalIncidence}, i.e. normal incidence with plate.

PSFParams(λ::Float64, obj::Objective)

Construct a PSFParams{NoAberration}, i.e. no plate, no tilt.

PlaneParallelPlate

A PlaneParallelPlate has thickness t (in micrometers) and a refractive index function n_λ(λ), where λ is in micrometers. The field n_λ returns the refractive index at that wavelength.

struct TiltedIncidence <: PSFMode end

Tilted incidence with a plate scenario.

BorosilicateCrown(t)

Creates a PlaneParallelPlate of thickness t (µm) for BK7 (borosilicate crown glass). Uses the Schott Sellmeier coefficients:

B = [1.040, 0.230, 1.010]
C = [0.006, 0.020, 103.560]

Source: refractiveindex.info BK7 page.

CustomPlate(t, B, C)

Create a PlaneParallelPlate of thickness t (µm), with a refractive index defined by the Sellmeier coefficients B and C, where

n^2 = 1 + ∑( Bᵢ * λ² / (λ² - Cᵢ) )

(λ in micrometers). B and C must be the same length.

Diamond(t)

Approximate dispersion for diamond with thickness t (in µm). λ is assumed in micrometers (µm), but the formula internally converts λ to nm by multiplying by 1e3 for the Sellmeier-like expression.

FusedSilica(t)

Creates a PlaneParallelPlate of thickness t (µm) for Fused Silica (SiO₂), using the Malitson Sellmeier coefficients:

B = [0.6961663, 0.4079426, 0.8974794]
C = [0.0684043^2, 0.1162414^2, 9.896161^2]

Source: Malitson data.

LMPLFLN100XBD()

Returns an Objective resembling the Olympus LMPLFLN 100x BD:

• f=1800 µm
• NA=0.8
• n=1.0 (air)

M10x()

Returns an Objective resembling the Newport M10x:

• f=16500 µm
• NA=0.25
• n=1.0 (air)

MPlanApo100x()

Returns an Objective resembling the Mitsutoyo MPlanApo 100x:

• f=2000 µm
• NA=0.7
• n=1.0 (air)

MPlanApo50x()

Returns an Objective resembling the Mitsutoyo MPlanApo 50x:

• f=4000 µm
• NA=0.55
• n=1.0 (air)

MagnesiumFluoride(t)

Creates a PlaneParallelPlate of thickness t (µm) for MgF₂ using the ordinary-ray Sellmeier coefficients (Li):

B = [0.48755108, 0.39875031, 2.3120353]
C = [0.04338408^2, 0.09461442^2, 23.793604^2]

Source: MgF₂ (Li) data.

PSF(x, y, z, λ, NA; rtol=1e-3, atol=1e-4)

Aberration-free PSF with no objective structure, using air (n=1).

• λ in µm
• NA numeric aperture
• (x, y, z) in µm
• rtol, atol tolerances

PSF(x, y, z, λ, NA::Float64, plate::PlaneParallelPlate, α; ...)

Tilted incidence with no explicit Objective.

PSF(x, y, z, λ, NA::Float64, plate::PlaneParallelPlate; ...)

Normal incidence with a plane-parallel plate, no explicit Objective.

PSF(x, y, z, λ, obj::Objective, plate::PlaneParallelPlate, α; ...)

Tilted incidence with plane + objective.

PSF(x, y, z, λ, obj::Objective, plate::PlaneParallelPlate; ...)

Normal incidence with a plane-parallel plate and an explicit Objective.

PSF(x, y, z, λ, obj::Objective; rtol=1e-3, atol=1e-4)

Aberration-free PSF with an Objective specifying immersion index, NA, etc. No plate, normal incidence in air or immersion.

PSF(x, y, z, obj::Objective, nv::NVCenter, plate::PlaneParallelPlate, α::Float64; rtol=1e-3, atol=1e-4)

Compute a polychromatic PSF for NV emission with a tilted-incidence plate. Again, integrates over nv.λs and sums with nv.weight.

PSF(x, y, z, obj::Objective, nv::NVCenter, plate::PlaneParallelPlate; rtol=1e-3, atol=1e-4)

Compute a polychromatic PSF for NV emission with a normal-incidence plate. Broadcasts over nv.λs, calling single-wavelength PSF(...) for each λ, then weights with nv.weight.

PSF(x, y, z, obj::Objective, nv::NVCenter; rtol=1e-3, atol=1e-4)

Compute a polychromatic PSF for an NV center (no plate scenario). Integrates over all wavelengths in nv.λs, calling the single-wavelength PSF(...) for each λ and weighting by nv.weight.

• (x, y, z) in µm
• obj is an Objective
• nv::NVCenter is the NV spectrum data
• rtol, atol are integration tolerances

PSF(x, y, z, param::PSFParams{NoAberration}; rtol=1e-3, atol=1e-4)

Dispatch for the no-aberration case (NoAberration). Calls PSF(x, y, z, λ, obj; rtol, atol).

PSF(x, y, z, param::PSFParams{NormalIncidence}; rtol=1e-3, atol=1e-4)

Dispatch for normal incidence with a plate. Calls PSF(x, y, z, λ, obj, plate; rtol, atol).

PSF(x, y, z, param::PSFParams{TiltedIncidence}; rtol=1e-3, atol=1e-4)

Dispatch for tilted incidence with a plate. Calls PSF(x, y, z, λ, obj, plate, α; rtol, atol).

Sapphire(t)

Creates a PlaneParallelPlate of thickness t (µm) for sapphire (Al₂O₃), using the ordinary-ray Sellmeier coefficients (Li):

B = [1.4313493, 0.65054713, 5.3414021]
C = [0.0726631^2, 0.1193242^2, 18.028251^2]

Source: Sapphire (Li) data.

UPLXAPO100X()

Returns an Objective resembling the Olympus UPLXAPO100X:

• f=1800 µm
• NA=1.45
• n=1.515 (oil immersion)

maxtol_thick(λ::Float64, obj::Objective; plate=Diamond, t_range=(0.0, 500.0), zrange=[-2.0,4.0], ...)

Find thickness t in t_range that yields a Strehl ratio near 0.8. No tilt is assumed. We define:

cost(t) = (strehl(t, λ, obj, plate; zrange=zrange) - 0.8)^2

and minimize with a 1D search. Returns the thickness that best satisfies S ≈ 0.8.

maxtol_thick(λ::Float64, obj::Objective, plate::Function, α::Float64; t_range=(0.0, 500.0), zrange=[-2.0,4.0], ...)

Same as above, but for tilted incidence angle α.

cost(t) = (strehl(t, λ, obj, plate, α; zrange=zrange) - 0.8)^2

psf_inner(ψ, u, v, s, n; rtol=1e-3, atol=1e-4)

Computes the aberration-free PSF intensity by numerically integrating the x-, y-, and z-polarized integrands via hcubature.

• ψ, u, v are phase parameters relating to off-axis, defocus, etc.
• s is the normalized aperture.
• n is the refractive index (1.0 or immersion).
• rtol, atol are integration tolerances.

psf_inner(ψ, u, v, k, s, n, n_a, t; rtol=1e-3, atol=1e-4)

Compute the PSF with normal incidence on a plane-parallel plate via hcubature.

• k is the wave number (2πn_obj / λ).
• t is plate thickness in µm.
• n_a is the plate’s refractive index (possibly scaled by immersion factor).

psf_inner(ψ, u, v, k, s, α, n, n_a, t; rtol=1e-3, atol=1e-4)

Compute the PSF with tilted incidence on a plane-parallel plate via hcubature.

• α is the tilt angle (radians).
• k is wave number (2π n_imm / λ).
• n_a is plate index scaled if needed.
• Other parameters as before.

sellmeier(λ, B, C)

Generic Sellmeier equation:

n^2 = 1 + ∑( Bᵢ * λ² / (λ² - Cᵢ) )

• λ in micrometers (µm).
• B/C are matching-length vectors of Sellmeier coefficients.

Returns the refractive index n(λ).

strehl(t, λ, obj::Objective, plate::Function=Diamond; 
       zrange=[-2.0, 4.0], rtol=1e-10, atol=1e-10)

Compute the Strehl ratio when a plate of thickness t (µm) is placed in the optical path.

• λ is wavelength in µm
• obj is the objective (e.g. MPlanApo50x())
• plate is a constructor function for a PlaneParallelPlate (defaults to Diamond).
• zrange in µm is the defocus search range: the function does a 1D optimization to find the best focus.
• rtol, atol are integration tolerances for the PSF.

Returns the maximum on-axis intensity (aberrated) normalized by the unaberrated case. Example:

S = strehl(50.0, 0.7, MPlanApo100x())  # diamond thickness=50 µm, λ=0.7 µm

strehl(t, λ, obj::Objective, z_est::Float64, plate::Function=Diamond; rtol=1e-10, atol=1e-10)

Compute the on-axis Strehl ratio for a plate thickness t (µm) at a specified defocus z_est (µm). No tilt assumed.

strehl(t, λ, obj::Objective, z_est::Float64, plate::Function, α::Float64; rtol=1e-10, atol=1e-10)

Same as above, but with incidence angle α (tilt).

strehl(t, λ, obj::Objective, plate::Function, α::Float64; 
       zrange=[-2.0, 4.0], rtol=1e-10, atol=1e-10)

Compute the Strehl ratio for a plate of thickness t (µm), wavelength λ (µm), Objective obj, and tilt angle α (radians). The search for best defocus is done in zrange (µm).

Ξ(ρ, ϕ, ψ, u, v, k, s, n, n_a, t)

Phase term for normal incidence with a plane-parallel plate. Adds the plate-induced phase k * t * Φ(...) to the aberration-free phase.

Ξ(ρ, ϕ, ψ, u, v, k, s, α, n, n_a, t)

Phase term for tilted incidence with a plane-parallel plate. Adds the tilt-based phase Φ(..., α) plus the base aberration-free term.

Ξ(ρ, ϕ, ψ, u, v, s, n)

Computes the phase term for the aberration-free case.

• ρ, ϕ are polar coordinates in the pupil.
• ψ is an azimuthal shift (e.g. relating to off-axis propagation).
• u, v are defocus/piston terms in the exponent.
• s is the normalized aperture (NA).
• n is the refractive index (often 1.0 for air or obj.n for immersion).

Φ(sρ, ϕ, n, α)

Computes the aberration phase for tilted incidence.

• sρ: again, s * ρ.
• ϕ: azimuthal angle in the pupil plane.
• n: refractive index.
• α: tilt angle (radians).

Returns a phase shift accounting for tilt plus the normal-incidence phase.

Φ(sρ, n)

Computes the aberration phase for normal incidence.

• sρ: the product of normalized aperture s and radial coordinate ρ in the pupil plane.
• n: refractive index (e.g., 1.0 for air or the immersion index).

Returns a dimensionless phase shift due to the optical path difference.

ξx(ρ, ϕ, ψ, u, v, k, s, n, n_a, t)

x-polarized integrand with normal-incidence plate.

ξx(ρ, ϕ, ψ, u, v, k, s, α, n, n_a, t)

x-polarized integrand with tilted-incidence plate.

ξx(ρ, ϕ, Ξ, s)

x-polarized integrand for the aberration-free PSF integral.

• ρ, ϕ are pupil-plane coordinates.
• Ξ is the accumulated phase from Ξ(...).
• s is the normalized aperture.

Returns the complex field contribution in the x-direction.

ξy(ρ, ϕ, ψ, u, v, k, s, n, n_a, t)

y-polarized integrand with normal-incidence plate.

ξy(ρ, ϕ, ψ, u, v, k, s, α, n, n_a, t)

y-polarized integrand with tilted-incidence plate.

ξy(ρ, ϕ, Ξ, s)

y-polarized integrand for the aberration-free PSF integral. Similar arguments as ξx(...).

ξz(ρ, ϕ, ψ, u, v, k, s, n, n_a, t)

z-polarized integrand with normal-incidence plate.

ξz(ρ, ϕ, ψ, u, v, k, s, α, n, n_a, t)

z-polarized integrand with tilted-incidence plate.

ξz(ρ, ϕ, Ξ, s)

z-polarized integrand for the aberration-free PSF integral. Similar arguments as ξx(...).

module NVspectrum

NV center spectrum.

• The wavelength array is given in micrometers (µm).
• The spectrum array holds corresponding intensities (arbitrary units).
• nv_spectrum_spline is a smoothing spline fit of spectrum vs. wavelength.

You can access the raw data via NVspectrum.wavelength and NVspectrum.spectrum, or use nv_spectrum_spline for continuous interpolation.

nv_spectrum_spline

A smoothing spline fit (SmoothingSplines.SmoothingSpline) constructed from wavelength and spectrum, using a smoothing parameter of 250.0. This allows continuous interpolation/extrapolation of the NV emission data.

spectrum

A constant array of raw intensities (arbitrary units) for each entry in wavelength.

wavelength

A constant array of NV emission wavelengths (in µm).