AiryPSF

The AiryPSF model represents the diffraction-limited point spread function for a circular aperture under the paraxial approximation. Unlike the simpler Gaussian approximation, this model accurately captures the characteristic diffraction rings that appear in real microscope images.

Mathematical Model

The Airy pattern is defined in terms of its field amplitude:

\[A(r) = \frac{\nu}{\sqrt{4\pi}} \cdot \frac{2J_1(\nu r)}{\nu r}\]

where:

  • $r = \sqrt{x^2 + y^2}$ is the radial distance from the optical axis in microns
  • $\nu = \frac{2\pi \cdot \text{NA}}{\lambda}$ is the optical parameter
  • $J_1$ is the Bessel function of the first kind, order 1
  • $\text{NA}$ is the numerical aperture
  • $\lambda$ is the wavelength in microns

The intensity is given by the squared magnitude of the amplitude:

\[I(r) = |A(r)|^2\]

Constructor and Parameters

AiryPSF(na::Real, wavelength::Real)
  • na: Numerical aperture of the objective
  • wavelength: Wavelength of light in microns

Alternative Constructor

AiryPSF(psf::GaussianPSF; λ::Real=0.532)

Creates an AiryPSF that approximates the provided GaussianPSF, using the specified wavelength.

Key Features

  • Physical Accuracy: Correctly models the diffraction pattern from a circular aperture
  • Diffraction Rings: Captures the characteristic rings in real microscope images
  • Rayleigh Criterion: Naturally demonstrates the Rayleigh resolution criterion
  • Computational Efficiency: More physically accurate than Gaussian while still being computationally efficient

Properties of the Airy Pattern

The Airy pattern has several notable physical properties:

  1. Central Maximum: Contains 83.8% of the total intensity
  2. First Minimum: Occurs at a radius of 1.22λ/NA (the Rayleigh criterion)
  3. First Ring: Contains 7.2% of the total intensity
  4. Subsequent Rings: Contain decreasing fractions of the intensity

Examples

Creating and using an Airy PSF:

# Create an Airy PSF for a high-NA objective with green light
psf = AiryPSF(1.4, 0.532)  # NA=1.4, wavelength=532nm

# Create from a GaussianPSF
gaussian_psf = GaussianPSF(0.15)
airy_equivalent = AiryPSF(gaussian_psf, λ=0.532)

Limitations

  1. 2D Only: Only valid for in-focus imaging (no defocus modeling)
  2. Paraxial Approximation: Less accurate for very high-NA objectives (> 1.4)
  3. No Aberrations: Doesn't account for optical aberrations
  4. No Polarization: Doesn't model polarization effects
  5. No Refractive Index Mismatches: Assumes uniform media

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