Diffraction and the Airy disk diameter

The Airy disk is the central bright region of the diffraction pattern produced when a point source of light passes through a circular aperture, such as a camera lens or a telescope objective. It represents the fundamental limit of resolution imposed by diffraction in an optical system with a circular pupil.

When light passes through a circular aperture, it does not form a perfect point on the image plane. Instead, due to diffraction, it spreads into a pattern of concentric rings. The Airy disk refers specifically to:

• The central bright spot of this pattern

• Surrounded by alternating dark and bright rings

• Formed by constructive and destructive interference of diffracted light

This pattern is described mathematically by the square of the first-order Bessel function of the first kind, and its intensity drops off non monotonically with distance from the center.

The radius of the first dark ring (the first null) occurs at an angle where:

sin(theta) = 1.22 * lambda / D

Where:

1. • theta is the angular radius of the Airy disk

   • lambda is the wavelength of light

   • D is the diameter of the aperture

• The Airy disk sets the diffraction limit for optical resolution

• Two point sources are just resolvable (by the Rayleigh criterion) when the center of one Airy disk coincides with the first minimum of the other

• Smaller f-numbers (larger apertures) produce smaller Airy disks, allowing finer resolution

In practical photography, the Airy disk is rarely visible directly, but it governs the transition between lens-limited and diffraction-limited performance. It becomes increasingly important at small apertures (e.g., f/11 and smaller on full-frame sensors) or in precision optical systems like microscopes and telescopes.

There are lots of ways to measure the size of the Airy disk.

First Null Diameter (or First Minimum)
This is the most widely cited definition. It refers to the distance between the first zero of the Airy pattern intensity function.

Formula:
d_Airy = 2.44 * lambda * N
or, more generally:
d_Airy = 1.22 * lambda / NA

Where:

• lambda is the wavelength of light
• f is the focal length
• N is the f-number
• NA is the numerical aperture. In air, for distant subjects, NA ≈ 1 / (2 * N)

Use case: resolution limits, Rayleigh criterion, point spread function bounds.

Full Width at Half Maximum (FWHM)
This is the width of the central Airy disk at half its peak intensity. It is narrower than the first-null diameter.

Approximate value:
FWHM ≈ 1.03 * lambda * N

Use case: practical resolution assessment, especially in astronomical imaging or Gaussian PSF fitting.

Encircled Energy Diameter
This is the diameter of a circle around the center of the Airy pattern that contains a specific fraction of the total energy.

Common values:

• 50% energy: d_50 ≈ 1.03 * lambda * N
• 70% energy: d_70 ≈ 1.60 * lambda * N
• 80% energy: d_80 ≈ 1.75 * lambda * N
• 90% energy: d_90 ≈ 2.00 * lambda * N

Use case: optical system performance, spot size characterization, fiber coupling efficiency, depth of field analysis. EE70 is a useful compromise between tight energy containment and sensitivity to aberrations.

RMS Spot Radius
This is the root-mean-square radius of the intensity-weighted distribution of the Airy pattern. There is no simple closed-form expression. It is usually evaluated numerically.

Use case: statistical blur modeling, system-level PSF assessment.

MTF Cutoff Frequency
Rather than a spatial diameter, sometimes the diffraction-limited cutoff frequency in the modulation transfer function (MTF) is used:

f_cutoff = 1 / (lambda * N)

The reciprocal of this frequency gives an effective resolution limit in Fourier space.

Use case: resolution assessment in frequency domain, image quality modeling.

For most of my work, I prefer the 70% encircled energy criterion, which is best for me to combine with other sources of blur.

While I’m at it, this is probably a good place to collect some of the common resolution limits for diffraction-limited optical systems.