KAHIBARO
Table of Contents
Resolving Two Nearby Sources
In wave optics, images are limited not only by the quality of lenses and mirrors, but also by diffraction. Even a perfect optical instrument cannot form an infinitely sharp image of a point source. Instead, the image spreads into a diffraction pattern. The Rayleigh criterion gives a practical rule for deciding when two nearby point sources are just distinguishable.
If two objects are too close together, their diffraction patterns overlap so strongly that they appear as one blurred object. If they are far enough apart, the eye or detector can recognize them as separate. The Rayleigh criterion defines the boundary between these two cases.
The Basic Idea
For a circular aperture, such as a telescope lens or a microscope objective, the image of a point source is not a point. It is a central bright spot surrounded by weaker rings. This pattern is called the Airy pattern, and its central bright region is called the Airy disk.
According to the Rayleigh criterion, two point sources are just resolved when the central maximum of one diffraction pattern falls at the first minimum of the other.
This is a convention, not an absolute law of nature. It provides a useful and widely used standard for optical resolution.
For a circular aperture, the Rayleigh criterion gives the minimum angular separation
$$
\theta_{\min} = 1.22 \frac{\lambda}{D}
$$
where $\lambda$ is the wavelength of light and $D$ is the aperture diameter.
Here, $\theta_{\min}$ is measured in radians, and it tells us the smallest angular distance between two distant point sources that can still be distinguished.
Why Diffraction Sets a Limit
When light passes through a finite aperture, it spreads out. This spreading is diffraction. Because of diffraction, the instrument cannot reproduce a perfect point image.
A larger aperture causes less spreading, so it gives better resolution. A shorter wavelength also causes less spreading, so it also improves resolution. This explains why large telescopes can distinguish finer detail, and why shorter wavelength radiation can reveal smaller structures.
Resolution improves when $\lambda$ decreases or $D$ increases.
Visual Meaning of the Criterion
Imagine two stars very far away. Each star forms an Airy pattern in the telescope image. If the stars are extremely close together, the two central bright spots overlap almost completely. The image looks like one star. As the separation increases, the combined image begins to show a dip between the two peaks. At the Rayleigh limit, this dip is just enough that the sources are considered distinguishable.
The real Airy pattern is more complicated than a simple bell-shaped curve, but this sketch shows the essential idea of overlap and partial separation.
Circular Apertures and the Factor 1.22
The number $1.22$ comes from the diffraction pattern produced by a circular aperture. The first dark ring in the Airy pattern occurs at the angle
$$
\theta = 1.22 \frac{\lambda}{D}
$$
This is why the Rayleigh criterion for circular openings includes the factor $1.22$.
For other aperture shapes, the formula changes. For example, for a single slit, the diffraction minima follow a different condition. The Rayleigh criterion discussed here is specifically the standard form for circular apertures.
Linear Resolution
Sometimes we want the minimum distance between two objects, not their angular separation. If the objects are at distance $L$ from the instrument, and the angle is small, then
$$
s_{\min} \approx L \theta_{\min}
$$
Substituting the Rayleigh result gives
$$
s_{\min} \approx 1.22 \frac{\lambda L}{D}
$$
This is the minimum linear separation that can be resolved at distance $L$.
For distant objects and small angles,
$$
s_{\min} \approx 1.22 \frac{\lambda L}{D}
$$
This connects angular resolution to actual separation.
Example
Suppose a telescope has aperture diameter $D = 0.10 \, \text{m}$ and uses light of wavelength $\lambda = 550 \times 10^{-9} \, \text{m}$.
Then the minimum angular separation is
$$
\theta_{\min} = 1.22 \frac{\lambda}{D}
= 1.22 \frac{550 \times 10^{-9}}{0.10}
\approx 6.7 \times 10^{-6} \, \text{rad}
$$
This is a very small angle. It shows that even a modest optical instrument can resolve very fine angular details.
Applications
The Rayleigh criterion is important in astronomy, microscopy, photography, and optical engineering. In astronomy, it helps determine whether two nearby stars can be seen separately. In microscopy, it helps determine the smallest detail that can be distinguished in a specimen. In camera systems, it helps describe how aperture size affects sharpness.
The same physical idea appears whenever waves form images. Resolution is never unlimited when diffraction is present.
What the Criterion Does and Does Not Mean
The Rayleigh criterion is a practical standard, not a strict yes or no boundary of perception. Real resolution also depends on brightness, noise, detector quality, contrast, and image processing. Under some conditions, two sources slightly closer than the Rayleigh limit may still be detected as separate. Under worse conditions, even larger separations may not be easy to resolve.
Still, the criterion remains extremely useful because it gives a simple estimate of the fundamental diffraction limit of an optical system.
The Rayleigh criterion is a diffraction-based estimate of the best possible resolution for an ideal optical system.
Summary Formula Table
| Quantity | Meaning | Formula |
|---|---|---|
| $\theta_{\min}$ | Minimum angular separation | $\theta_{\min} = 1.22 \dfrac{\lambda}{D}$ |
| $s_{\min}$ | Minimum linear separation at distance $L$ | $s_{\min} \approx L\theta_{\min}$ |
| Combined form | Linear resolution for small angles | $s_{\min} \approx 1.22 \dfrac{\lambda L}{D}$ |
Simple Aperture Sketch
The circular aperture limits how sharply each source can be imaged, and this leads directly to the Rayleigh criterion.