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Obstacle
Pick what the plane wave hits.
Light source
Wavelength
Geometry and exposure
Move the screen and zoom the view.
Zone plate
Design focal length is quoted at 550 nm.
Camera
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The science
Scalar Fresnel diffraction
A plane wave of wavelength lambda passes a circularly symmetric screen with transmission T(rho). In the paraxial (Fresnel) approximation the field at radius r on a screen a distance z behind it is
U(r) = -i (k/z) ∫ T(rho) exp(i k rho² / 2z) J0(k rho r / z) rho d rho, with k = 2 pi / lambda. Unobstructed, U = 1. The intensity is I/I0 = |U|². This page evaluates the integral numerically (Simpson's rule, many points per Fresnel zone).
Arago-Poisson spot
An opaque disk is the complement of a hole (Babinet's principle): Udisk = 1 - Uhole. On the axis the hole gives Uhole = 1 - exp(i pi N) with Fresnel number N = a²/(lambda z), so Udisk = exp(i pi N) and I = 1 at the centre of the shadow, whatever the disk radius. Poisson derived this as an objection to wave theory; Arago measured it in 1818. Off axis the spot follows J0²(k a r / z) for large N, whose first zero is at r = 0.383 lambda z / a.
Circular hole
Axis intensity I = 4 sin²(pi N / 2): it flickers between 0 and 4 as the hole opens one Fresnel zone at a time (N = number of half-period zones exposed).
Fresnel zone plate
Zone n lies between sqrt((n-1) lambdad f) and sqrt(n lambdad f). At z = f every zone contributes the same amplitude 2 (in units of the unobstructed field), so blocking the even zones leaves M open zones with U = 2M and I = 4M². A phase plate shifts the even zones by pi instead, so all 2M zones add: I = 16M². The focal length is chromatic, f(lambda) = fd lambdad/lambda, and there are weaker foci at f/3, f/5, ....
Honest simplifications
- Illumination is an ideal plane wave of uniform amplitude; real beams are Gaussian and need a collimator.
- Scalar, paraxial theory: no obliquity factor, no polarisation, perfectly sharp edges. Real ball bearings add edge roughness that fills in the spot.
- The 3D scene is drawn at an exaggerated scale (the screen slides on the rail, the obstacle radius is scaled to fit); the numbers come from the integral, not from the drawing.
- White light is eight spectral lines added in intensity, not a continuous spectrum.