Vectorial Diffraction Model (Debye-Wolf Integral)
This PSF generator implements the vectorial diffraction theory based on the
Debye-Wolf integral, which provides an exact solution for high-NA optical systems. Unlike scalar
(Airy/Gaussian) approximations, this model correctly accounts for:
- Polarization effects at high numerical apertures
- Apodization from the Abbe sine condition
- Non-paraxial propagation (valid for all NA values)
- Asymmetric axial PSF structure
Mathematical Formulation
For circularly polarized or unpolarized illumination, the total intensity PSF is computed from three
diffraction integrals:
I₀(ρ,z) = ∫₀^θmax √cos(θ) · sin(θ) · (1+cos(θ)) · J₀(kρ·sin(θ)) · exp(ikΦ) dθ
I₁(ρ,z) = ∫₀^θmax √cos(θ) · sin²(θ) · J₁(kρ·sin(θ)) · exp(ikΦ) dθ
I₂(ρ,z) = ∫₀^θmax √cos(θ) · sin(θ) · (1-cos(θ)) · J₂(kρ·sin(θ)) · exp(ikΦ) dθ
PSF(ρ,z) = |I₀|² + 2|I₁|² + |I₂|²
Where θmax = arcsin(NA/nᵢ), k = 2πnᵢ/λ, and Jₙ are Bessel functions of the first kind.
Gibson-Lanni Aberration Model
When RI mismatch is enabled, the optical path difference (OPD) includes contributions from:
Φ(θ) = z·cos(θ) + OPD_mismatch(θ)
OPD_mismatch = Δt_g·(n_g·cos(θ_g) - n_i·cos(θ_i)) [cover glass term]
+ d·(n_s·cos(θ_s) - n_i·cos(θ_i)) [sample depth term]
where Snell's law relates angles in each medium: nᵢ·sin(θ) = n_g·sin(θ_g) = n_s·sin(θ_s)
Physical Assumptions
| Assumption |
Value/Condition |
Notes |
| Illumination |
Circularly polarized / unpolarized |
Standard fluorescence microscopy |
| Objective |
Airy pupil, no aberrations |
Perfect lens (aberration-free) |
| Detection |
Incoherent intensity |
Sum of polarization components |
| Numerical integration |
128-point Gauss-Legendre quadrature |
High accuracy (~10⁻¹⁰ relative error) |
Comparison to Established Tools
| Feature |
This Tool |
PSFGenerator (MATLAB) |
Python-microscPSF |
| Diffraction model |
Vectorial Debye-Wolf |
Vectorial Debye-Wolf |
Scalar/Vectorial |
| RI mismatch |
Gibson-Lanni |
Gibson-Lanni |
Gibson-Lanni |
| Aberrations |
Spherical only (from RI) |
Zernike polynomials |
Varies |
| Precision |
64-bit float (JS) |
64-bit float |
64-bit float |
Browser Limitations
| Limitation |
Impact |
Mitigation |
| JavaScript precision |
IEEE 754 64-bit (~15-17 digits) |
Sufficient for PSF accuracy |
| Memory constraints |
Large 3D stacks may fail |
Grid size limited to 512×512 |
| Single-threaded |
UI may freeze during computation |
Web Worker (async computation) |
| Bessel functions |
Custom implementation |
Polynomial approximation (error < 10⁻⁸) |
References
- Wolf, E. (1959). Electromagnetic diffraction in optical systems. Proc. Roy. Soc. A 253, 349-357.
- Richards, B. & Wolf, E. (1959). Electromagnetic diffraction in optical systems II. Proc. Roy.
Soc. A 253, 358-379.
- Gibson, S.F. & Lanni, F. (1992). Experimental test of an analytical model of aberration in an
oil-immersion objective. J. Opt. Soc. Am. A 9, 154-166.
- Haeberle, O. et al. (2003). The point spread function of optical microscopes imaging through
stratified media. Opt. Express 11, 2249-2259.