Poisson's Spot (Arago Spot)

The intuition — why a shadow has a bright heart

Block a beam of light with a small ball bearing or a coin and your gut says the center of the shadow should be the darkest place of all — the spot most thoroughly hidden from the source. Wave optics says the opposite. There is a bright pinpoint sitting exactly on the axis, in the geometric center of the shadow, and in the perfect case it is just as bright as if the disk weren't there.

The reason is a piece of pure geometry. Light is a wave, and by the Huygens–Fresnel principle every point a wavefront reaches becomes a source of secondary spherical wavelets. The disk blocks the light that hits its face, but the rim — the circular edge — is still bathed in the wavefront. Each point on that rim re-radiates a wavelet that bends into the shadow.

Now pick the point on the screen directly behind the center of the disk. Because the edge is a perfect circle and that point is on the symmetry axis, every point on the rim is exactly the same distance away from it. Equal distance means equal travel time means equal phase. All those edge wavelets arrive crest-on-crest and add up. The result is a bright spot — and only on the axis, because move off-axis and the rim points are no longer equidistant: once the path differences among them grow to about half a wavelength the phases scramble and the brightness collapses. That happens over a transverse distance of order λ·b/R, so the bright core is narrow but still many wavelengths wide.

How the diffraction builds the spot

The rigorous version uses the Fresnel–Kirchhoff diffraction integral. For a point source at distance g in front of the disk and a screen at distance b behind it, the field at a point on the screen is the sum (integral) of contributions from every unobstructed part of the wavefront — i.e., everything outside the disk's circular boundary.

The cleanest way to see the spot is Fresnel zone construction. Divide the open wavefront into concentric annular zones, defined so that the path length from successive zone edges to the on-axis point increases by exactly half a wavelength:

r_m = sqrt(m · λ · (g·b)/(g+b))      (radius of the m-th Fresnel zone)

Adjacent zones are roughly out of phase, so their contributions nearly cancel in pairs. The crucial fact: the disk simply blocks the first few central zones and leaves all the outer ones. The surviving sum is dominated by the innermost surviving zone — the ring of wavefront just grazing the disk's edge. Its net contribution is close to half of a single zone's amplitude, which (for an idealized disk) reconstructs essentially the full on-axis field.

Equivalently, the on-axis amplitude is proportional to the value of the obstructing function at the disk boundary. Because the boundary is a clean circle, the boundary-wave (Rubinowicz) picture gives the same answer: a coherent edge wave radiating inward, refocusing on the axis.

The governing equations

Treat the disk of radius R as blocking the central region of the wavefront. The on-axis field on the screen, relative to the unobstructed field, reduces to a clean boundary term. For a plane wave (source at infinity, g → ∞) hitting a disk of radius R, with screen distance b and wavelength λ:

On-axis intensity (idealized, opaque circular disk):
    I_axis = I_0          ← same as no disk at all

Off-axis profile near the center (point source on axis):
    I(ρ) ≈ I_0 · [ J_0( k · R · ρ / b ) ]²

    k = 2π/λ           (wavenumber)
    ρ                  (radial distance on screen from the axis)
    J_0                (zeroth-order Bessel function of the first kind)

The Bessel-function profile is the signature of a circular aperture/obstacle. The spot is brightest at ρ = 0 and falls to its first dark ring where the argument hits the first zero of J₀ (≈ 2.405). So the size of the spot scales as λ·b / R: shrink the disk or use longer wavelengths and the spot grows.

The Fresnel number sets the regime:

N = R² / (λ · b)          (plane wave)
N = R² · (g + b) / (λ · g · b)   (point source at finite distance g)

The spot is a near-field (Fresnel) phenomenon: it is sharp and high-contrast when N is of order one or less. As N → 0 you are deep in the far field; as N grows very large you approach the geometric-optics limit and need an extremely round, clean edge to keep the spot intact.

Worked example — seeing it on a lab bench

Take a steel ball bearing of radius R = 1.0 mm, a red helium–neon laser (λ = 633 nm) spatially filtered to act as a point source, and a screen b = 1.0 m behind the ball.

Fresnel number (plane-wave approx):
    N = R² / (λ·b)
      = (1.0e-3 m)² / (633e-9 m · 1.0 m)
      = 1e-6 / 6.33e-7
      ≈ 1.6

Radius to the first dark ring (first zero of J_0, arg ≈ 2.405):
    ρ_1 ≈ 2.405 · b / (k·R) = 2.405 · λ·b / (2π·R)
        = 2.405 · 633e-9 · 1.0 / (2π · 1.0e-3)
        ≈ 2.4e-4 m  ≈ 0.24 mm

So the bright spot sits at the exact center of a ~2 mm shadow, with a bright core a few tenths of a millimeter across before the first dark ring. Increase b to 4 m and N drops to ≈ 0.4, the spot broadens (ρ₁ grows to ~1 mm) and becomes easier to resolve by eye — a classic undergraduate demonstration.

Key conditions and regimes

Requirement	Why it matters	What happens if violated
Coherent, point-like source	Edge wavelets must keep a fixed phase relationship	Extended source smears many shifted spots together → washed out
Smooth circular edge	All rim points equidistant from the axis	Roughness ≳ a Fresnel-zone width scrambles edge phases → spot dims
Opaque obstacle	Defines a sharp boundary wave	Partial transmission adds a background that lowers contrast
Fresnel number N = R²/(λb) ≲ 1	Puts you in the near-field diffraction regime	Very large N → geometric limit, spot needs near-perfect edge
Good on-axis alignment	Spot lives only on the symmetry axis	Off-axis the rim points stop being equidistant → no spot
Monochromatic (or narrowband) light	Different λ place dark rings at different radii	Broadband light still gives a central spot but colored, blurred rings

Disk size, wavelength, and what you actually see

Setup	Disk radius R	Wavelength λ	Screen b	Fresnel number N	Result
HeNe laser + ball bearing	1 mm	633 nm	1 m	≈ 1.6	Sharp central spot, clear ring
Same, longer throw	1 mm	633 nm	4 m	≈ 0.4	Broad, easy-to-see spot
Coin in sunlight (filtered)	10 mm	550 nm	2 m	≈ 91	Faint spot; needs very round edge
Microwave + metal disk	50 mm	30 mm (10 GHz)	1 m	≈ 0.08	Strong, detector-measurable spot
Deuterium molecule beam (2009)	0.5 mm	~50 pm (de Broglie)	0.5 m	~10⁴	Matter-wave Poisson spot observed

Notice that you do not need a small Fresnel number for the effect to exist — only for it to be easy to see by eye. The deuterium experiment ran at enormous N and still recorded a spot, because matter waves are coherent and the disk edge was clean. Smaller N just makes the spot wider and more forgiving of alignment.

The 1818 prize that backfired

In 1818 the French Academy of Sciences ran a prize competition on the nature of light, expecting submissions to support the dominant Newtonian corpuscular theory. Augustin-Jean Fresnel instead submitted a thorough wave theory of diffraction. On the judging committee sat Siméon Denis Poisson, a committed corpuscularist and a formidable mathematician.

Poisson worked through Fresnel's integrals and triumphantly produced what he thought was a fatal absurdity: the theory demanded a bright spot at the very center of a circular object's shadow. Surely no such ridiculous thing could exist — so the wave theory must be wrong.

The committee chair, François Arago, decided to actually check. He set up a 2 mm metal disk and looked. The spot was there, exactly as Fresnel's math required. Fresnel won the prize, the wave theory of light was effectively confirmed, and the effect carries the name of the man who tried to use it as a weapon against it. (It is also called the Arago spot, and sometimes the Fresnel bright spot; Delisle and Maraldi had unknowingly observed it a century earlier.)

Where it shows up today

• Optical metrology and alignment. The spot's position marks the exact symmetry axis of a circular object, so it is used as a precise, no-contact alignment reference in optical systems.
• Telescope baffle and stray-light design. Engineers building space telescopes (e.g., starshade and coronagraph studies) must account for the bright spot a circular occulter throws onto the detector — the Arago spot is a real stray-light term to suppress.
• Lithography and proximity printing. Circular features in a mask diffract; the central bright spot behind opaque dots is a known artifact that limits contrast at small feature sizes.
• Matter-wave and electron interferometry. Poisson-spot diffraction of electrons, neutrons, and molecules is a clean demonstration of de Broglie wave behavior and a tool for measuring coherence of particle beams.
• X-ray and EUV optics. At very short wavelengths the spot from circular obstructions (and dust on optics) is a measurable, sometimes troublesome, diffraction signature.
• Teaching wave optics. It remains the single most dramatic bench demonstration that light is a wave — the prediction so counter-intuitive that even its discoverer expected it to fail.

Common misconceptions and edge cases

• "The spot needs a special disk." Any reasonably round opaque object works — a ball bearing, a coin, a dot on glass. The disk doesn't focus light like a lens; the geometry of its edge does the work.
• "It's only as bright as a tiny bit of leaked light." In the ideal case the on-axis intensity equals the full unobstructed beam. It can be genuinely bright, not a faint glimmer.
• "A bigger disk gives a brighter spot." The opposite. A larger disk needs a rounder, smoother edge to maintain the equidistant ring, and pushes you to large Fresnel number where the spot is fragile and dim relative to the wider, darker shadow.
• "Roughness destroys it completely." The spot is famously robust to small irregularities — that's part of why Arago saw it so readily. It only fades when edge bumps approach a Fresnel-zone width.
• "It's a light-only effect." It is a wave effect, demonstrated with sound, water, microwaves, electrons, neutrons, and molecules. Anything that diffracts produces it.
• "The spot is at the center of the disk." It's at the center of the shadow, on the screen behind the disk — on the symmetry axis, where every edge point is equidistant. Off-axis it fades quickly — over a transverse scale of order λ·b/R (the spot radius), the rim points lose their equal path and the phases scramble.