Fresnel Zone

Fresnel zones are concentric annular regions on a wavefront, defined such that the optical path length from the edge of each zone to an observation point differs by half a wavelength ($\lambda/2$) from the previous zone.

Definition

For a circular aperture of radius $R$ at distance $z$ from an on-axis observation point, the $m$th Fresnel zone is the annular region between radii $r_{m-1}$ and $r_m$, where:

$$ r_m = \sqrt{m \lambda z} $$

The area of each zone is approximately equal: $\pi \lambda z$, independent of $m$.

On-Axis Interference

Successive Fresnel zones contribute to the on-axis field with alternating sign:

• Odd zones ($m = 1, 3, 5, \ldots$): contribute constructively
• Even zones ($m = 2, 4, 6, \ldots$): contribute destructively

When a circular aperture exposes an odd number of half-zones, the on-axis intensity can reach four times the unobstructed intensity. When an even number is exposed, the contributions cancel and the on-axis intensity drops to zero.

Connection to Fresnel Diffraction

The oscillatory on-axis intensity behind a circular aperture:

$$ I(0, 0, z) = 4\sin^2\left(\frac{\pi}{2\lambda z}\right) $$

is a direct consequence of Fresnel zone interference. As the observation distance $z$ decreases, more zones fit within the aperture, and the intensity oscillates with increasing rapidity.

Applications

• Fresnel zone plates: Binary optics that block alternate zones to create a focusing effect, acting as diffractive lenses
• Radio propagation: Fresnel zones determine the clearance required between a transmitter and receiver for unobstructed signal paths