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)

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

You should provide exactly one of the optional parameters. If none are provided, an unaberrated pupil is created.

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 waves RMS of defocus (Noll index 4)
zc.phase[5] = 0.3   # 0.3 waves RMS of astigmatism (Noll index 5)
zc.phase[7] = 0.2   # 0.2 waves RMS of coma (Noll index 7)
zc.phase[11] = 0.1  # 0.1 waves 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 waves 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.