Diffraction Integral Calculator
Fresnel diffraction pattern of a circular aperture or slit at any distance, with the Fresnel number, on-axis intensity and the far-field Fraunhofer limit.
Fresnel number N_F = a²/(λz)
0.3161
Fresnel, near the far field
On-axis intensity I(0)/I₀
0.9075
exact Rayleigh–Sommerfeld; paraxial 4 sin²(πN_F/2) = 0.9075 (bright for odd N_F, dark for even)
Far-field first dark ring 1.22λz/(2a)
193.0 µm
Only once N_F ≪ 1; not the pattern at this distance
Far field starts beyond
z ≫ a²/λ = 15.80 mm
The distance where N_F = 1
I/I₀ = (k/z)² |∫₀ᵃ eikr²/(2z) J₀(krρ/z) r dr|², the Fresnel–Hankel integral of a plane wave cut by the aperture.
Far field (N_F ≪ 1): (πN_F)² [2J₁(v)/v]², v = kaρ/z (the Airy pattern). Intensities are relative to the incident I₀.
Intensity on the screen
- Fresnel integral
- Fraunhofer (far field)
- Aperture edge
Model and references
Textbook approximation. A standard approximation from the literature (for example paraxial, plane-wave or Gaussian-beam), valid within the range the page states. About the tiers
Scalar Fresnel (paraxial) diffraction of a plane wave, intensities relative to the incident one. Valid for z ≫ a and small angles; the circular aperture is computed up to N_F = 20. The circular on-axis value is the exact Rayleigh–Sommerfeld one (Heurtley 1973); the page flags when the dropped Fresnel phase πa⁴/(4λz³) exceeds 0.1 rad.
- Goodman J. W. (2005). Introduction to Fourier Optics, 3rd ed. Roberts & Company, ch. 4.
- Born M., Wolf E. (1999). Principles of Optics, 7th ed. Cambridge University Press, ch. 8. doi:10.1017/cbo9781139644181
- Heurtley J. C. (1973). Scalar Rayleigh–Sommerfeld and Kirchhoff diffraction integrals: a comparison of exact evaluations for axial points. JOSA 63, 1003–1008. doi:10.1364/josa.63.001003
- Osterberg H., Smith L. W. (1961). Closed solutions of Rayleigh's diffraction integral for axial points. JOSA 51, 1050–1054. doi:10.1364/josa.51.001050
- Mielenz K. D. (2002). Optical diffraction in close proximity to plane apertures. I. Boundary-value solutions for circular apertures and slits. J. Res. NIST 107, 355–362, eq. (9b). doi:10.6028/jres.107.028