Fresnel Diffraction

What Fresnel diffraction is

When light passes an edge, a slit, a small hole, or a small obstacle, it bends into the geometric shadow and forms a pattern of bright and dark fringes. Fresnel diffraction is the description of that pattern in the near field — close enough to the aperture that you cannot pretend the incoming and outgoing waves are flat planes. The wavefronts are curved, and the extra path length introduced by that curvature (a term that grows as the square of transverse distance) controls everything.

The distinction from its far-field cousin, Fraunhofer diffraction, is not a different physics — it is a different approximation of the same Huygens–Fresnel principle. Every point on a wavefront acts as a source of secondary spherical wavelets; the field at any downstream point is the coherent sum (interference) of all those wavelets. Keep the quadratic phase and you have Fresnel; drop it and you have Fraunhofer.

The Fresnel diffraction integral

For a monochromatic scalar wave of wavelength λ passing through an aperture in the plane z = 0 and observed on a screen at distance L, the field is (in the paraxial approximation):

U(x, y) = (e^{ikL} / iλL) ∬ U₀(x', y') · exp[ ik/(2L) · ((x−x')² + (y−y')²) ] dx' dy'

Symbol by symbol, with SI units:

• U(x, y) — complex field amplitude on the observation screen (V/m for the E-field, arbitrary units for a scalar wave).
• U₀(x', y') — the field in the aperture plane (the "aperture function"; 1 inside an open aperture, 0 where opaque).
• k = 2π/λ — the wavenumber (rad/m).
• λ — wavelength (m); e.g. 500 nm = 5 × 10⁻⁷ m for green light.
• L — aperture-to-screen distance (m).
• (x, y) and (x', y') — transverse coordinates on the screen and in the aperture (m).
• i — imaginary unit; the 1/(iλL) prefactor carries the amplitude scaling and a 90° phase (the "obliquity"/Huygens factor).

The key term is the quadratic phase exp[ik/(2L)·((x−x')²+(y−y')²)]. Expanding it produces a cross term (the Fourier kernel) plus a residual quadratic term ∝ (x'² + y'²)/L. When that residual phase over the aperture is small compared to π — i.e. when a²/(λL) ≪ 1 — it is negligible and the integral collapses to a plain Fourier transform: the Fraunhofer limit. When it is not small, you must keep it, and you are doing Fresnel diffraction. Observed intensity is I = |U|².

The Fresnel number: the single most useful ruler

Everything about "which regime am I in?" is captured by one dimensionless number:

N_F = a² / (λ · L)

where a is the characteristic aperture radius (or half-width), λ the wavelength, and L the screen distance. Physically, N_F is the number of Fresnel zones the aperture reveals as seen from the observation point.

Fresnel number N_F	Regime	Behavior
N_F ≫ 1	Deep near field	Sharp-edged geometric projection with fine edge fringes
N_F ≈ 1–10	Fresnel (near field)	On-axis intensity oscillates bright/dark as L changes; pattern reshapes
N_F ≪ 1	Fraunhofer (far field)	Fixed-shape pattern (Fourier transform of aperture) that only scales

Worked value: a circular hole of radius a = 1 mm, green light λ = 500 nm, screen at L = 1 m gives N_F = (10⁻³)² / (5 × 10⁻⁷ × 1) = 2. That is squarely in the Fresnel regime — move the screen to L = 20 m and N_F drops to 0.1, and you have entered Fraunhofer. This is why the "same" aperture produces a totally different-looking pattern depending on how far away you look.

Fresnel zones and the half-amplitude surprise

Fresnel's brilliant trick was to divide the wavefront into concentric annular zones, drawn so that the path from the outer edge of each successive zone to the observation point is longer by exactly λ/2. The zone radii (for a plane wave reaching a screen at distance L) are approximately:

r_m ≈ √(m · λ · L)     (m = 1, 2, 3, …)

Because each zone is λ/2 further away than the last, adjacent zones arrive π out of phase — their contributions alternate in sign. Summing the slowly-shrinking alternating series A₁ − A₂ + A₃ − A₄ + … gives a remarkable result: the total amplitude of the entire unobstructed wave is only about half that of the first zone alone. Equivalently, if you could block every other zone, the survivors would all add in phase and the on-axis intensity would increase far above the fully-open value.

Zone plates: focusing without refraction

A Fresnel zone plate does exactly that: alternating opaque and transparent rings whose boundaries follow r_m = √(m·λ·f). It removes (or, in a phase zone plate, inverts) the destructive zones so that the remaining zones interfere constructively at a focal point. The primary focal length is:

f = r₁² / λ

where r₁ is the radius of the innermost (first) zone. Note the strong wavelength dependence, f ∝ 1/λ — zone plates are extremely chromatic, and they also have higher-order foci at f/3, f/5, f/7, … Because they focus by diffraction rather than refraction, zone plates are the workhorse imaging elements for X-rays and extreme-ultraviolet light, where no ordinary refractive lens exists. The outermost zone width sets the resolution: the smallest resolvable feature is δ ≈ 1.22 · Δr_N, where Δr_N is the width of the outermost zone, so finer zones mean sharper focus.

The Poisson–Arago spot: a prediction that backfired

In 1818 the French Academy held a prize competition on the nature of light. Fresnel submitted his wave theory. The judge Siméon Denis Poisson, a partisan of Newton's corpuscular theory, used Fresnel's own integral to derive what he considered an obvious absurdity: a wave theory would require a bright spot at the very center of the shadow of a circular disk. Surely no such thing could exist.

The committee chair, François Arago, performed the experiment — and the spot was there. It is sometimes called the spot of Arago, the Poisson spot, or the Fresnel bright spot, and it clinched the prize for Fresnel.

The reason is pure symmetry: every point on the circular rim of the disk is exactly the same distance from the axial point directly behind the disk. So all the edge-diffracted wavelets arrive there in phase and interfere constructively, filling the geometric shadow's center with light. The effect requires a smooth, round edge and good coherence, but it appears even behind everyday round objects. See the dedicated Poisson spot page for the full derivation.

Edge diffraction and the Cornu spiral

For a straight edge or a slit, the two-dimensional Fresnel integral separates into one-dimensional Fresnel integrals:

C(u) = ∫₀ᵘ cos(π t²/2) dt        S(u) = ∫₀ᵘ sin(π t²/2) dt

Plotting S(u) against C(u) traces the elegant double-spiral Cornu spiral (a clothoid), winding onto the fixed points (½, ½) and (−½, −½) as u → ±∞. The complex amplitude reaching any screen point is simply the straight chord connecting the two spiral points that correspond to the edges of the open region, and the intensity is the square of that chord length.

This construction immediately explains the signature of edge diffraction: just inside the illuminated side of a straight edge's shadow, the intensity does not jump cleanly from bright to dark. It overshoots to about 1.37× the free-field value, then oscillates with steadily shrinking fringes as you move into the light, while it decays smoothly and monotonically into the shadow. Those wiggles are the tell-tale fingerprint of Fresnel edge diffraction.

How the pattern morphs with distance

The near field's defining property is that the pattern reshapes as L changes, not merely scales. Consider a small circular aperture and watch the on-axis intensity as you pull the screen back:

• Very close (large N_F): the bright disk is a near-perfect geometric projection of the hole, ringed by fine fringes hugging the edge.
• Intermediate (N_F sweeping through integers): the on-axis point flickers bright, dark, bright, dark… Each time the aperture exposes one more full Fresnel zone (N_F passes through an odd/even integer), the axial amplitude swings between a maximum (odd number of zones) and a near-zero minimum (even number of zones).
• Far (N_F ≪ 1): the oscillations stop and the pattern settles into the smooth central-maximum-plus-rings Airy pattern of Fraunhofer diffraction, which thereafter only grows in size with L.

That on-axis flicker is the most vivid signature of the near field, and it is impossible in the Fraunhofer regime, where the axial intensity varies smoothly and monotonically.

Fresnel vs Fraunhofer at a glance

Feature	Fresnel (near field)	Fraunhofer (far field)
Fresnel number	N_F ≳ 1	N_F ≪ 1
Wavefront model	Curved (spherical)	Planar
Quadratic phase term	Kept	Dropped
Math	Fresnel integral (convolution with chirp)	Fourier transform of aperture
Pattern vs distance	Reshapes; on-axis intensity oscillates	Fixed shape, only scales
Circular obstacle	Bright Poisson–Arago spot on axis	(same axial spot persists; edge rings)
Reaching it with a lens	—	A lens brings it to the focal plane
Typical setup	Shadow of an object, no lens, screen nearby	Laser through a slit onto a distant wall

Common misconceptions

• "Fresnel and Fraunhofer are different phenomena." No — they are two approximations of the same Huygens–Fresnel integral. The only difference is whether you keep the quadratic (near-field) phase term, which is set by the Fresnel number.
• "Near field means very close in absolute terms." No; near field is defined by N_F = a²/(λL) ≳ 1, not by an absolute distance. A large aperture can stay in the Fresnel regime out to enormous L, while a tiny slit reaches Fraunhofer within centimeters.
• "The Poisson spot proves particles are wrong / is a trick of a perfect disk." It is a robust wave-interference effect that appears behind ordinary round objects with a reasonably smooth rim; a mild edge roughness only dims it, it does not abolish it.
• "Geometric shadows are always dark inside." Fresnel diffraction fills shadows with fringes, and a round obstacle even puts a bright point at the shadow's dead center.
• "A zone plate focuses like a lens." Only loosely — it focuses by diffraction, so it is intensely chromatic (f ∝ 1/λ) and produces multiple higher-order foci, unlike a refractive lens.
• "The Fresnel integral is exact." It is a paraxial (small-angle) scalar approximation. Very close to the aperture or at wide angles you need the full Rayleigh–Sommerfeld or vector diffraction theory.

History in one paragraph

Christiaan Huygens proposed secondary wavelets in 1678. Thomas Young demonstrated interference in the early 1800s but lacked a quantitative diffraction theory. Augustin-Jean Fresnel (1788–1827) fused Huygens' wavelets with Young's interference and added the crucial idea of coherent summation with phase, submitting his memoir to the 1818 Académie des Sciences prize. Poisson's attempted reductio ad absurdum — the bright spot behind a disk — was confirmed by Arago and became the theory's triumph. Gustav Kirchhoff later placed the integral on rigorous footing (the Fresnel–Kirchhoff diffraction formula, 1882), and Marie Alfred Cornu introduced the spiral that bears his name to visualize edge diffraction.