ScalarPSF

The ScalarPSF model implements a three-dimensional point spread function based on scalar diffraction theory. This model accounts for defocus and optical aberrations using a complex pupil function approach, providing a good balance between physical accuracy and computational efficiency.

Mathematical Model

The ScalarPSF uses the Fourier optics approach to calculate the complex field distribution:

\[U(\mathbf{r}) = \int_{pupil} P(\boldsymbol{\rho}) e^{i k \boldsymbol{\rho} \cdot \mathbf{r}} d\boldsymbol{\rho}\]

where:

• $U(\mathbf{r})$ is the complex field amplitude at position $\mathbf{r} = (x, y, z)$
• $P(\boldsymbol{\rho})$ is the complex pupil function at pupil coordinates $\boldsymbol{\rho}$
• $k = 2\pi / \lambda$ is the wave number
• The intensity is calculated as $I(\mathbf{r}) = |U(\mathbf{r})|^2$

The pupil function can incorporate various aberrations, typically represented using Zernike polynomials.

Constructor and Parameters

ScalarPSF(nₐ::Real, λ::Real, n::Real; 
         pupil::Union{Nothing, PupilFunction}=nothing,
         pupil_data::Union{Nothing, AbstractMatrix}=nothing,
         zernike_coeffs::Union{Nothing, ZernikeCoefficients}=nothing,
         grid_size::Integer=128)

Required Parameters

• nₐ: Numerical aperture of the objective
• λ: Wavelength of light in microns
• n: Refractive index of the medium

Optional Parameters

• pupil: Pre-created PupilFunction instance
• pupil_data: Complex matrix to initialize the pupil function
• zernike_coeffs: ZernikeCoefficients instance for representing aberrations
• grid_size: Generated pupil grid size (default: 128). A supplied pupil or pupil_data retains its own grid.

Supply one of pupil, pupil_data, or zernike_coeffs. If none is provided, an unaberrated pupil is created using grid_size. Stored Zernike coefficients are converted to the PSF element type. Evaluation returns Float32 for a ScalarPSF{Float32} with Float32 x, y, and z. Optics and the supplied pupil, pupil data, or coefficients determine the PSF element type by promotion. With Float32 optics, Float64 coefficients or pupil components produce a Float64 PSF. If no pupil, pupil data, or coefficients are supplied, default coefficients are Float32 only when all three optical arguments are Float32; otherwise they are Float64. A supplied pupil must have finite positive energy inside the circular aperture.

Pupil sampling and field of view

The computed field is zero at and beyond L/2 in either lateral coordinate relative to the emitter, where L = (grid_size - 1) * λ / (2nₐ) in microns uses the actual pupil grid size. A larger grid_size widens this field. For NA 1.4, wavelength 0.6 microns, and grid 128, L ≈ 27.21 microns and L/2 ≈ 13.61 microns. support=Inf selects the whole camera; explicit support and spline sampling ranges may extend beyond the field. Increasing grid_size also improves defocused intensity accuracy.

For NA 1.4, wavelength 0.6 microns, index 1.518, and 0.5 rad RMS astigmatism (zc.phase[6] = 0.5), the following maximum absolute intensity errors were measured over a 31×31 grid on [-1.5, 1.5] microns in x and y. Errors are divided by the peak intensity of a 512-sample pupil reference at each z. These measured values depend on the optics, aberrations, and sampled positions.

Evaluation cost and a defocused-image example

The default grid of 128 takes about four times the evaluation time of grid 64. With NA 1.2, wavelength 0.520 microns, index 1.3, and an unaberrated pupil, a 21×21-pixel frame on [-1, 1] microns per axis, support=3, sampling=10, and 10,000 photons gives the following maximum absolute pixel errors divided by the peak of a 1024-sample pupil reference. These finite-reference measurements use the same pixel quadrature at each grid size.

Over z = -1, -0.5, 0, 0.5, and 1 microns, warmed single-image medians from three repetitions per z are 2.449–2.613 s at grid 64 and 9.811–10.269 s at grid 128. An 81×81×21 PSF tabulation with NA 1.2, wavelength 0.6 microns, index 1.33, and zc.phase[6]=0.5 rad takes 27.80 s at grid 64 and 93.05 s at grid 128, including its first tabulation and compilation. These are local measurements on a shared host with four Julia threads; they are not portable latency guarantees.

Key Features

• 3D Imaging: Models PSF behavior throughout 3D space, not just in focus
• Aberration Modeling: Supports arbitrary optical aberrations via Zernike polynomials
• Complex Field: Provides access to both amplitude and phase information
• Physical Realism: Based on physical principles of scalar diffraction theory

Aberration Modeling

A key feature of the ScalarPSF is its ability to incorporate optical aberrations using Zernike polynomials with Noll indexing and RMS normalization:

# Create Zernike coefficients object
zc = ZernikeCoefficients(15)  # Up to 15th Zernike polynomial (Noll indexed)

# Add common aberrations by directly setting coefficients
# Note: The package uses Noll indexing and RMS normalization
zc.phase[4] = 0.5   # 0.5 rad RMS of defocus (Noll index 4)
zc.phase[5] = 0.3   # 0.3 rad RMS of astigmatism (Noll index 5)
zc.phase[7] = 0.2   # 0.2 rad RMS of coma (Noll index 7)
zc.phase[11] = 0.1  # 0.1 rad RMS of spherical aberration (Noll index 11)

# Create PSF with these aberrations
psf = ScalarPSF(1.4, 0.532, 1.518, zernike_coeffs=zc)

Examples

Creating a ScalarPSF:

# Create a basic unaberrated 3D PSF
psf = ScalarPSF(1.4, 0.532, 1.518)  # NA=1.4, λ=532nm, n=1.518

# Create a PSF with spherical aberration
zc = ZernikeCoefficients(15)  # Up to 15th Zernike polynomial (Noll indexed)
zc.phase[11] = 0.5            # Add 0.5 rad RMS of spherical aberration (Noll index 11)
psf_aberrated = ScalarPSF(1.4, 0.532, 1.518, zernike_coeffs=zc)

# Create a PSF with a pre-computed pupil function
pupil = PupilFunction(1.4, 0.532, 1.518, zc)
psf_from_pupil = ScalarPSF(1.4, 0.532, 1.518, pupil=pupil)

Limitations

1. No Polarization Effects: Doesn't account for polarization, which becomes significant at high NA (>1.2)
2. Simplified Refractive Index Interfaces: Simplified treatment of refractive index interfaces
3. Scalar Approximation: Uses scalar diffraction theory instead of full vector theory
4. Moderate NA Assumption: Most accurate for moderate NA values and regions near the focal plane

For standard usage patterns, camera integration, and comparison with other PSF types, see the PSF Overview.