Point a telescope at a close double star and the pair arrives as one fuzzy spot. Open the aperture, shorten the wavelength, or switch to a larger instrument, and the spot splits into two. The Rayleigh criterion is the standard rule for when that split happens: two equally bright point sources are just resolved when the center of one Airy disk lands on the first dark ring of the other.

For a circular aperture the rule is short:

$$\theta_R = 1.22 \frac{\lambda}{D}$$

Here $\lambda$ is the wavelength of the light, $D$ is the aperture diameter, and $\theta_R$ is the angular separation in radians. This page derives the 1.22, shows what the image looks like at the limit, and compares the criterion with the other resolution rules in common use. The interactive Airy disk and diffraction overview covers where the pattern itself comes from.

What “Just Resolved” Means

Resolution measures whether two sources can be told apart. Magnification only makes the recorded image larger, so magnifying a blurred pair keeps the blur and adds no new detail. What limits the pair is how wide each source’s diffraction pattern is.

The Rayleigh criterion turns that limit into a test. Two ideal point sources sit some angle apart, and each forms its own Airy pattern. Far apart, the image shows two distinct bright cores. As the sources move closer, the patterns overlap and their intensities add. The test calls the pair “just resolved” at the separation where the center of one Airy pattern sits on the first dark ring of the other. At that spacing the summed profile still has two peaks with a dip between them, which is the minimum structure needed to read it as two sources.

Choosing the first dark ring is a convention, and a convenient one. The first dark ring is a sharply defined landmark, so the criterion gives a single number that anyone can compute from wavelength and aperture alone. Other conventions exist, and the next sections place them side by side.

Where the 1.22 Factor Comes From

The Airy radius from the overview is $r_1 = 1.22\lambda f/D$, the distance from the center of the pattern to the first dark ring on a sensor placed at the focal length $f$. Dividing by the focal length turns that linear distance into an angle:

$$\theta_1 = \frac{r_1}{f} = 1.22\frac{\lambda}{D}$$

A distant source arrives as a nearly flat wavefront, so the angle is small and the sine, tangent, and angle are interchangeable for this purpose.

The 1.22 is not a measured constant. It falls out of the Fraunhofer diffraction pattern of a uniformly lit circular aperture, whose intensity at angle $\theta$ is:

$$I(\theta) = I_0\left[\frac{2J_1(x)}{x}\right]^2, \qquad x = \frac{\pi D \sin\theta}{\lambda}$$

$J_1$ is the first-order Bessel function. It begins at zero, rises to a peak, and crosses zero again at $x_1 = 3.8317$. Setting $x$ equal to that zero locates the first dark ring:

$$\sin\theta_1 = \frac{3.8317}{\pi}\frac{\lambda}{D} = 1.2197\frac{\lambda}{D} \approx 1.22\frac{\lambda}{D}$$

For small angles the sine and the angle agree, so the first dark ring sits at $1.22\lambda/D$. The Rayleigh criterion sets the source separation equal to that angle.

The 1.22 belongs to the circular aperture specifically. A long rectangular slit produces a sinc pattern, and its first zero sits at $\sin\theta = \lambda/D$, so the same style of criterion would read $1.0\lambda/D$ for a slit. A round aperture spreads light over a slightly wider angle because its rim runs in every direction, while a slit limits the wavefront along only one axis.

The Rayleigh Criterion Formula in Three Forms

The angular form, $\theta_R = 1.22\lambda/D$, is what astronomy uses. Stars are effectively infinitely far away, so only the angle between them matters, and the formula reads directly in radians or arcseconds.

On a camera sensor the useful quantity is the linear radius of the first dark ring. Substituting the f-number $N = f/D$ gives:

$$r_1 = 1.22\frac{\lambda f}{D} = 1.22\lambda N$$

Two point sources at the Rayleigh angle land their patterns $r_1$ apart on the sensor. That is why stopping down a lens, which raises $N$, widens the diffraction footprint even when every other aberration improves.

A microscope uses the numerical aperture $NA = n\sin\alpha$, where $n$ is the refractive index between the objective and the sample and $\alpha$ is the half-angle of the cone of light the objective accepts. The aperture and focal length are tied together by that cone, so $f/D = 1/(2NA)$ in air, and the Airy radius becomes an object-space distance:

$$d = \frac{0.61\lambda}{NA}$$

This is the same Rayleigh condition written for a microscope. It says two points in the sample are just resolved when they sit that far apart. Immersion oil raises $n$ above 1, which raises $NA$ and pushes the limit down.

Read across all three forms and the recipe is the same: shorter wavelength, larger aperture, and higher numerical aperture each shrink the limit. Red light and blue light through the same aperture resolve different detail for exactly that reason.

What a Just-Resolved Pair Looks Like

The explorer places two point sources in the left image and plots their summed brightness on the right. The faint curves are the two individual Airy patterns, and the bold curve is the sum. Dragging either source marker toward the other lowers the separation. The aperture rim sets the diameter, and the spectrum marker sets the wavelength, so the reader can move the ruler and the scene independently.

The dip between the peaks is the whole story, and it has a clean value at the Rayleigh limit. The midpoint between the two sources sits half an Airy radius from each. Each pattern contributes about 0.368 of its peak there, so the two add to roughly 0.735 of a single-source peak. The saddle is about 26.5 percent below the peaks, which is shallow but visible. Closer than the limit the saddle fills in, and the pair collapses into one broad hump.

Widening the aperture lowers $\theta_R$, so a separation that looked merged becomes resolved. Shifting to a longer wavelength raises $\theta_R$, so a pair that was resolved starts to merge. The aperture changes the ruler; the source markers change what is being measured. Keeping those two ideas separate is what makes the criterion useful for comparing instruments.

Rayleigh Versus Sparrow, Abbe, and Dawes

The Rayleigh criterion is one answer to “how close is too close”. Three other rules are common, and each marks a slightly different event.

The Sparrow criterion removes the requirement of a visible dip and asks when the saddle disappears entirely. For two equal Airy patterns that happens at about $0.95\lambda/D$, noticeably closer than Rayleigh. The threshold is where the midpoint of the summed profile stops curving downward, so it is a property of the profile rather than a rule of thumb.

The Dawes limit is an empirical rule from double-star observers. It reads $\theta = 116/D$ arcseconds with $D$ in millimetres, or $4.56/D$ arcseconds with $D$ in inches. At 550 nanometres it corresponds to about $1.02\lambda/D$, a little more optimistic than Rayleigh. It depends on the observer, the atmospheric seeing, and the brightness ratio of the two stars, so it is a practical benchmark rather than a hard physical constant.

Abbe’s limit asks a different question. Instead of two isolated points, it asks for the finest repeating pattern a microscope objective can transmit. The answer is $d = \lambda/(2NA)$. Two-point resolution and grating resolution are related but not identical, which is why the Abbe and Rayleigh numbers differ.

Criterion	Separation	What it marks
Rayleigh	$1.22\lambda/D$	One Airy center on the other’s first dark ring
Sparrow	about $0.95\lambda/D$	The dip between the peaks disappears
Dawes	about $116/D$ arcseconds, $D$ in millimetres	Empirical limit for equal double stars
Abbe	$\lambda/(2NA)$	Finest grating period an objective transmits

The numbers sit within about 30 percent of each other, so for choosing an instrument they usually lead to the same decision. For a careful measurement, the differences matter, and naming the criterion a number comes from prevents confusion.

Real Systems and Edge Cases

The criterion assumes an ideal setup, and several real effects change the practical answer.

The criterion assumes incoherent sources, meaning their intensities add. Stars and fluorescent emitters behave this way. Coherent light from a laser adds amplitudes instead of intensities, which can create fringes and shift where the pattern looks resolved.

The rule also assumes two equally bright sources. A faint companion next to a bright star produces a much shallower dip, so it needs a wider separation to stand out. Detection then depends on the contrast ratio, the number of collected photons, and the noise.

Aberrations enlarge or distort the pattern. Defocus, coma, astigmatism, and spherical aberration all push the achievable separation above the ideal criterion. A well-corrected system is the one that actually reaches it.

The atmosphere often dominates for ground-based telescopes. At visible wavelengths, turbulence limits typical seeing to around an arcsecond, which is well above the diffraction limit of a large aperture. Adaptive optics corrects the wavefront in real time and recovers the diffraction limit over a small field.

The detector must sample the pattern finely enough. As a rule of thumb, two pixels across the full width at half maximum, $1.03\lambda N$, preserves a resolved pair; coarser pixels can hide structure the optics delivered.

A central obstruction from a secondary mirror turns the aperture into a ring. The core narrows slightly, but more energy moves into the surrounding rings. That can help split a close double star while hurting contrast on faint extended detail, which is why obstruction size is a real design tradeoff.

Finally, techniques such as deconvolution, speckle interferometry, and super-resolution resolve below the classical limit by using prior knowledge or many frames. They reconstruct detail from information the instrument did collect, rather than defeating the diffraction of a single exposure.

Computing the Rayleigh Limit

The formulas convert directly into a few utility functions. The first zero of $J_1$ is the only constant that needs care, and it sets the shape of every derived limit.

const ARCSEC_PER_RAD = 206264.806;

// Angular Rayleigh limit in radians.
const rayleighAngle = (wavelength, aperture) => (1.22 * wavelength) / aperture;

// The same limit in arcseconds, the unit telescope makers use.
const rayleighArcsec = (wavelength, aperture) =>
  rayleighAngle(wavelength, aperture) * ARCSEC_PER_RAD;

// Radius of the first dark ring on a sensor behind a lens of focal length f.
const airyRadius = (wavelength, focalLength, aperture) =>
  (1.22 * wavelength * focalLength) / aperture;

// Smallest object-space spacing for a microscope, with NA = n * sin(alpha).
const microscopeLimit = (wavelength, numericalAperture) =>
  (0.61 * wavelength) / numericalAperture;

// Is a measured separation at or above the limit?
const isResolved = (separation, wavelength, aperture) =>
  separation >= rayleighAngle(wavelength, aperture);

Given a wavelength in metres and an aperture in metres, the angular limit comes back in radians, ready to compare with a measured separation. The microscope form expects the same wavelength unit and returns the spacing in that unit.

Recap

The Rayleigh criterion says two equally bright point sources are just resolved when one Airy center lands on the other’s first dark ring, which happens at $\theta_R = 1.22\lambda/D$. The 1.22 traces to the first zero of the Bessel function $J_1$, and the same condition appears as $r_1 = 1.22\lambda N$ on a sensor and $d = 0.61\lambda/NA$ in a microscope.

At the limit the summed profile leaves a dip of about 26.5 percent, small but real. Sparrow, Dawes, and Abbe mark related thresholds for slightly different questions, and real systems add contrast, noise, aberrations, atmosphere, sampling, and post-processing on top of the ideal rule. Used as a baseline for comparing instruments, the criterion is a compact way to predict what an aperture and a wavelength can separate.