Why does the dashed curve not match the solid curve?

The dashed line is the ideal Fraunhofer sinc² for the same slit; the solid line is the full Fresnel slit integral. They match best when N is small (L large).

Is N exactly a²/(λL)?

Definitions vary by geometry (slit vs disk; width vs radius). This page uses N = a²/(λL) with a the slit half-width as a simple scale for teaching.

Fresnel vs Fraunhofer

For a 1D slit, the Fresnel–Kirchhoff integral includes the quadratic phase exp(i π (y−y′)²/(λL)) across the aperture. The dimensionless Fresnel number N ≈ a²/(λL) (with a the half-width) separates regimes: small N corresponds to far-field propagation where the pattern approaches the Fraunhofer (Fourier) limit, here sin²(πWy/(λL))/(πWy/(λL))² for width W = 2a. Large N means the near-field Fresnel curvature matters. The Cornu spiral traces C(u) and S(u) from the Fresnel integrals; knife-edge and slit amplitudes are phasor steps on this spiral.

Who it's for: After slit diffraction; pairs with Airy disk for circular apertures.

Key terms

Fresnel number
Fresnel integral
Cornu spiral
Fraunhofer diffraction

How it works

Near field (Fresnel): the quadratic phase exp(i π (y−y′)²/(λL)) across the aperture cannot be dropped — the observed pattern is a Fresnel diffraction integral. Far field (Fraunhofer): for large L (small N = a²/(λL)), the same integral reduces to the Fourier (here sinc) pattern of the aperture. The Cornu spiral plots C(u) and S(u) from the Fresnel integrals; knife-edge and slit amplitudes are vector steps on this spiral. This page uses a direct numerical slit integral (solid) and overlays the Fraunhofer sinc² shape (dashed) for comparison.

Key equations

N ≈ a²/(λL) · small N → Fraunhofer; large N → Fresnel

C(u), S(u) = ∫₀^u cos/sin(π t²/2) dt · Cornu spiral

Frequently asked questions

Why does the dashed curve not match the solid curve?: The dashed line is the ideal Fraunhofer sinc² for the same slit; the solid line is the full Fresnel slit integral. They match best when N is small (L large).
Is N exactly a²/(λL)?: Definitions vary by geometry (slit vs disk; width vs radius). This page uses N = a²/(λL) with a the slit half-width as a simple scale for teaching.