Huygens' Principle: How Every Point Becomes a Source

Drop a pebble in a pond and a ring spreads out. Now imagine that every point on that ring were itself a tiny pebble, launching its own ripple. Add up all those little ripples and you get the next ring. That is Huygens' principle, one of the oldest and most useful ideas in wave physics, and the meaning of the line on our Christiaan Huygens sticker: every point a source.

What did Huygens actually say?

Christiaan Huygens drafted his Treatise on Light in 1678 and published it in Leiden in 1690 Huygens, Treatise on Light^[1] Huygens–Fresnel principle, Wikipedia^[2]. He pictured light as a disturbance passing through a medium of tiny particles, and argued that each particle reached by the disturbance passes it on in all directions. In Silvanus Thompson's 1912 translation:

“So it arises that around each particle there is made a wave of which that particle is the centre.” Huygens, Treatise on Light^[1]

He then made the key step. These secondary waves are individually weak, and only where they all touch a common surface do they add up to a visible front. Each, he wrote, “can be infinitely feeble only as compared with the wave DCF, to the composition of which all the others contribute” Huygens, Treatise on Light^[1]. In modern language, the new wavefront is the envelope of all the secondary wavelets.

What can the principle explain?

Huygens used it to derive the laws of reflection and refraction Huygens–Fresnel principle, Wikipedia^[2]. Refraction falls out neatly. If light travels more slowly in glass, the wavelets on the part of a slanted front that has already entered the glass make less forward progress than those still in air, so the whole front swings round to a new direction Huygens–Fresnel principle, Wikipedia^[2]. The result is Snell's law, with the ratio of the sines of the angles equal to the ratio of the wave speeds. The same reasoning explains how a gradient in air temperature bends light into a mirage.

What Huygens could not explain was diffraction, the spreading of light around edges and through small openings Huygens–Fresnel principle, Wikipedia^[2]. The effect had been observed carefully and named by Francesco Maria Grimaldi, whose results appeared in 1665 Diffraction, Wikipedia^[3]. Huygens' construction had no wavelength and no phase, so it could not say why light leaks into a shadow in bright and dark bands.

How did Fresnel turn it into a theory of diffraction?

In 1818 Augustin-Jean Fresnel combined Huygens' wavelets with the principle of interference Huygens–Fresnel principle, Wikipedia^[2]. Now each wavelet has a phase as well as an amplitude, and where wavelets from different parts of the front arrive out of step they cancel. To match experiments, Fresnel had to multiply each secondary wave by a constant −i/λ, a quarter-cycle phase shift, and by an inclination factor K(χ) that weakens wavelets sent off at large angles Huygens–Fresnel principle, Wikipedia^[2]. In modern form, the field at a point P is

U(P) = (−i/λ)·∬ U(Q)·(e^ikr/r)·K(χ) dS

where U(Q) is the complex field at a point Q on the wavefront, λ is the wavelength in metres, k = 2π/λ is the wavenumber in m⁻¹, r is the distance from Q to P in metres, χ is the angle between the wavefront's normal and the direction to P, and the integral runs over the surface S of the front Huygens–Fresnel principle, Wikipedia^[2]. These additions had no obvious physical basis at the time, but they predicted experiments very well Arago spot, Wikipedia^[4].

The spot that should not exist

Fresnel's theory was entered in an 1818 competition of the French Academy of Sciences. One of the judges, Siméon Denis Poisson, a supporter of the particle theory of light, worked out a consequence that seemed absurd: in the shadow of a round disk, all the wavelets from the rim travel the same distance to the centre, so the very centre of the shadow should be bright Arago spot, Wikipedia^[4]. The committee's head, François Arago, did the experiment with a 2 mm metal disk fixed to glass with wax and saw the spot, confirming the prediction in 1819 Arago spot, Wikipedia^[4]. Arago later noted that Joseph-Nicolas Delisle and Giacomo Maraldi had already seen the effect a century earlier Arago spot, Wikipedia^[4]. The spot is drawn on our Arago: The Spot of Arago tee.

In 1882 Gustav Kirchhoff put Fresnel's theory on a rigorous footing, showing that it is an approximate form of an integral theorem for solutions of the wave equation, and derived the inclination factor rather than assuming it Huygens–Fresnel principle, Wikipedia^[2] Kirchhoff's diffraction formula, Wikipedia^[5].

Worked example: single-slit diffraction

Send light of wavelength λ through a slit of width a. Split the slit into two halves. At an angle where wavelets from the top and the middle differ in path by half a wavelength, every wavelet in the top half is cancelled by its partner in the bottom half and the screen goes dark Diffraction, Wikipedia^[3]. The dark fringes sit at

a·sin θ_min = m·λ (m = ±1, ±2, …)

and the full far-field intensity, the Fraunhofer pattern, is Diffraction, Wikipedia^[3]

I(θ) = I₀·(sin β / β)², β = πa·sin θ / λ

where I₀ is the intensity straight ahead and θ is measured from the forward direction. For red light of 633 nm and a 50 µm slit, our arithmetic puts the first dark fringe at θ ≈ 0.73°, which on a screen 2 m away is about 2.5 cm from the centre. A narrower slit spreads the light more. The same physics, for a round hole, gives the Airy pattern that limits every telescope, covered in our post on the Airy disk. Our diagram below does the Huygens sum directly: 200 point wavelets across the slit, added with their phases, land on the analytic curve.

Is Huygens' principle exactly true? (partly open)

Not in its original form. Taken literally, point wavelets radiate backwards as well as forwards, which would create a reverse wave that is not observed; Fresnel's inclination factor was a patch for this Huygens–Fresnel principle, Wikipedia^[2]. In 1991 David Miller suggested that treating each secondary source as a dipole, rather than the simple monopole Huygens assumed, cancels the backward wave and makes the construction quantitatively correct Miller 1991^[6] Huygens–Fresnel principle, Wikipedia^[2]. Kirchhoff's formula itself rests on approximations that work when the wavelength is much smaller than the aperture Huygens–Fresnel principle, Wikipedia^[2], and how best to interpret the secondary sources physically is still discussed.

Quick FAQ

Is it Huygens' principle or the Huygens–Fresnel principle?

Huygens' principle is the 1678 wavelet construction; the Huygens–Fresnel principle adds interference and phase, which is what predicts diffraction Huygens–Fresnel principle, Wikipedia^[2].

Does it apply to sound and water waves?

Yes, it is a property of waves in general, not only light Diffraction, Wikipedia^[3].

Where can I find more optics?

Browse the Waves & Optics collection.

References

1. Christiaan Huygens, Treatise on Light (Traité de la Lumière, drafted 1678, published Leiden 1690), translated by Silvanus P. Thompson (London: Macmillan, 1912); Project Gutenberg eBook #14725. https://www.gutenberg.org/ebooks/14725
2. Wikipedia, “Huygens–Fresnel principle”. https://en.wikipedia.org/wiki/Huygens%E2%80%93Fresnel_principle
3. Wikipedia, “Diffraction”. https://en.wikipedia.org/wiki/Diffraction
4. Wikipedia, “Arago spot”. https://en.wikipedia.org/wiki/Arago_spot
5. Wikipedia, “Kirchhoff's diffraction formula”. https://en.wikipedia.org/wiki/Kirchhoff%27s_diffraction_formula
6. David A. B. Miller (1991), “Huygens's wave propagation principle corrected,” Optics Letters 16, 1370 (DOI 10.1364/OL.16.001370; metadata verified via Crossref). https://doi.org/10.1364/OL.16.001370

Written by Pixelated Physics. Every factual claim is linked to the source we checked; points that are still debated are labelled open.