# Huygens Slit

Surface · z = f(x, y, t)

`z = ∫₋₂² cos(2 · hypot(x − u, y) − 2t) du`

[Open in the app](https://www.wavelace.com/app#p=98) · [This page](https://www.wavelace.com/presets/huygens-slit)

### What it draws

The height is an integral, taken numerically at every mesh point: `z = ∫₋₂² cos(2·hypot(x − u, y) − 2·t) du`. `u` runs along `y = 0` from −2 to 2, a slit of width 4. `hypot(x − u, y)` is the distance to the point `(u, 0)` of the slit, so the integrand is a circular wave from that point and the integral sums a continuum of them.

The 2 in front of the distance is the wavenumber: crests are `π ≈ 3.14` apart. The 2 on `t` is the angular frequency: a period of `π ≈ 3.14` seconds at `Speed` 1, and crests travelling at speed 1. Nothing fades and the sources radiate both ways, so the pattern repeats below the slit.

### Why the pattern

This is Huygens' principle as a definition: a wave reaching an opening goes on as if every point of it were a source. On the axis the sources are at nearly the same distance, their wavelets arrive in step, and the height climbs towards 4, the slit's width. Off the axis the distances to the slit's two ends differ, and once that difference is a whole wavelength the wavelets cancel in pairs.

The one null is where `sin φ = λ/a ≈ 0.79`, about 52° from the axis, with `φ` measured from the `y` axis. Beyond it a side lobe climbs to ≈ 0.19 of the central height at 90°, its crest out of reach. Far from the slit the amplitude along an arc follows `2·sin(4·sin φ)/(2·sin φ)`, the sinc envelope of Fraunhofer diffraction. Near the slit the sum is Fresnel's near field: on the axis at `y = 1` the height is only ≈ 2.9, still climbing.

### History

Christiaan Huygens proposed in 1678 that every point reached by a wave is the source of a new one, and published it in his Traité de la Lumière in 1690. It gave reflection and refraction, but not the fringes. Augustin-Jean Fresnel added the interference of the wavelets in 1818, the sum this formula computes, and so explained diffraction. The far-field limit bears Joseph von Fraunhofer's name, though the theory was not his.

### Try

- `Top`: the central beam leaves the slit between two dark rays near 52°, a lobe beyond each, mirrored below `y = 0`.
- Narrow the slit below a wavelength, `integral(cos(2·hypot(x − u, y) − 2·t), u, −1, 1)`: `λ/a ≈ 1.57` exceeds 1, so no null: the beam fills the half-plane.
- Halve the wavelength, `integral(cos(4·hypot(x − u, y) − 2·t), u, −2, 2)`: the beam narrows, nulls near 23° and 52°.
- Raise `Span of x, y` to 20: the near field shrinks to a patch at the slit and the null straightens into a ray, the Fraunhofer picture. At span 8 the sheet ends ≈ 11 units out and only approaches it.

### Read more

- [Huygens–Fresnel principle](https://en.wikipedia.org/wiki/Huygens%E2%80%93Fresnel_principle)
- [Diffraction](https://en.wikipedia.org/wiki/Diffraction)
- [Fraunhofer diffraction](https://en.wikipedia.org/wiki/Fraunhofer_diffraction)
- [Fresnel diffraction](https://en.wikipedia.org/wiki/Fresnel_diffraction)
- [Christiaan Huygens](https://en.wikipedia.org/wiki/Christiaan_Huygens)