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.