Fresnel Phasor
(∫₀^(u − π) cos(s²) ds, (u − π)/2, ∫₀^(u − π) sin(s²) ds)
Open in the app The dials and keys named below are the app's.
What it draws
The first and third expressions are the Fresnel integrals, computed numerically at every point: x = C(ℓ) = ∫₀^ℓ cos(s²) ds and z = S(ℓ) = ∫₀^ℓ sin(s²) ds. The upper limit ℓ = u − π runs from −π to π, and the pair traces the Euler spiral in the ground plane. The integrand is a unit vector, so ℓ is arc length and the tangent points at the angle ℓ². The curvature 2ℓ grows with distance, and each half coils into an eye at (√(π/8), √(π/8)) ≈ (0.63, 0.63) or its negative.
The second expression, ℓ/2, is the height. It lifts each point by half its arc length, so the ends sit at ±π/2 ≈ ±1.57. The flat spiral becomes a coil, loose in the middle and tight at the ends, whose shadow is the Euler spiral itself. No expression mentions t, so nothing moves.
The phasor reading
Light passing a straight edge reaches the screen as the sum of waves from every strip of the open wavefront. Each strip is a small arrow, a phasor, of one length, turned by a phase growing as the square of its distance from the line of sight. Head to tail, the arrows are this curve, arc length for distance and tangent angle ℓ² for phase; the height keeps the strips in order.
The amplitude at the screen is the chord from the strip at the edge to the far eye. With no edge the chord runs from eye to eye, a length of √π ≈ 1.77. At the edge of the geometric shadow the strips start at the origin, and the chord to one eye is half that. Intensity is amplitude squared: the shadow's edge is lit at a quarter of the open field. Deeper in the shadow the chord shrinks steadily; out in the light its start winds round the near eye and it swings about 1.77, the fringes beside every sharp edge.
History
Augustin-Jean Fresnel gave this account of diffraction in his 1818 memoir, which won the Académie des Sciences prize the next year. Alfred Cornu drew the curve in 1874 so amplitudes could be read off with a ruler.
Try
- Top: the shadow alone, the Euler spiral flat in the ground plane with its eyes on the diagonal.
- Front: x against height is the graph of C(ℓ), rising through the origin at slope 1 and settling in shrinking swings to ±0.63.
- Double the pitch. Write the height as u − π: the coil stretches to ±π ≈ ±3.14 and its tight ends stand twice as tall.
- Spin it. Write integral(cos(s^2 + t), s, 0, u − π) for x and integral(sin(s^2 + t), s, 0, u − π) for z: the coil turns about the vertical axis once every 2π ≈ 6.28 seconds at Speed 1.