The Rayleigh Criterion for Limit of Resolution and Its Application to Vision

Definition

The Rayleigh criterion states that two point sources are said to be just resolved when the principal maximum of the diffraction pattern of one source coincides with the first minimum of the diffraction pattern of the other source.

For a circular aperture, the angular limit of resolution is given by:

$$\theta_{\min} = 1.22 \frac{\lambda}{D}$$

where:

• $\theta_{\min}$ = minimum angular separation that can be resolved
• $\lambda$ = wavelength of light
• $D$ = diameter of the aperture

This criterion gives the minimum separation needed for two objects to be seen as distinct.

Main Content

1. Diffraction and Formation of the Image

• When light passes through an aperture, it does not travel in a perfectly straight line only; instead, it spreads out due to diffraction.
• This spreading causes the image of a point source to become a blurred spot rather than a perfect point. In a circular aperture, this blurred pattern is called the Airy disk, with a bright central spot surrounded by concentric dark and bright rings.

The important consequence of diffraction is that every point object produces its own diffraction pattern. If two points are close enough, their Airy disks overlap. The more the overlap, the harder it becomes to distinguish them separately. Therefore, the resolving power of an optical system depends not only on magnification but also on diffraction effects.

In practical terms:

• A larger aperture produces less diffraction spread and therefore better resolution.
• A smaller aperture produces more spreading and poorer resolution.

This is why a camera lens with a larger diameter can capture finer details than a small hole, and why a telescope with a bigger objective lens or mirror can separate distant stars more clearly.

2. Rayleigh Criterion and Resolving Power

• The Rayleigh criterion provides a standard for deciding whether two closely spaced images are just distinguishable.
• According to this criterion, two images are just resolved when the central maximum of one diffraction pattern falls on the first minimum of the other.

For a circular aperture, the first minimum of the Airy pattern occurs at:

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

This relation shows that:

• Smaller wavelength light gives better resolution.
• Larger aperture size gives better resolution.

Resolving power is the reciprocal of the limit of resolution. If the limit of resolution is small, the resolving power is high. That means the optical system can distinguish finer details.

Example:

• If two distant stars are very close together in the sky, a telescope with a small aperture may show them as one bright point.
• With a larger telescope aperture, the diffraction disks are smaller, so the two stars can be resolved as separate stars.

This criterion is not an absolute physical separation rule but a practical optical standard. In reality, the exact visibility also depends on brightness, contrast, detector sensitivity, and observer judgment.

3. Application to Human Vision

• The human eye behaves like an optical instrument with a circular aperture formed by the pupil.
• Because of diffraction, the eye also has a limit of resolution. Even if the retina is capable of detecting very fine detail, the pupil size and wavelength of visible light set an optical limit.

For the eye:

• The pupil diameter $D$ changes with light conditions.
• In bright light, the pupil is smaller, which increases diffraction and may reduce resolution.
• In moderate conditions, the pupil is of optimum size and the eye can resolve fine details well.
• In very dim light, although the pupil dilates, other factors such as reduced cone activity and low contrast limit vision.

The smallest angular separation the average human eye can resolve is about 1 minute of arc, or approximately:

$$\theta \approx \frac{1}{60^\circ}$$

This corresponds to an object separation of about 0.1 mm at a viewing distance of 25 cm, which is why normal text can be read at that distance if the print is sufficiently large.

Examples in vision:

• Two stars close together may appear as one if their angular separation is below the eye’s resolving limit.
• Fine patterns on a distant building may not be visible because the angular size is too small.
• Tiny letters in a book become unreadable when held too far away because their images on the retina overlap.

The Rayleigh criterion helps explain why the eye has a limit in distinguishing separate points and why visual acuity depends on pupil size, lighting, and wavelength.

Working / Process

1. Light from two closely spaced point objects enters the aperture of an optical system such as a lens, telescope, or the eye.
2. Due to diffraction, each point object forms an Airy diffraction pattern rather than a sharp point image.
3. The patterns overlap, and the system is said to just resolve the two points when the central bright maximum of one pattern coincides with the first minimum of the other, giving the Rayleigh limit.

Advantages / Applications

• Helps determine the maximum resolving power of optical instruments such as microscopes, telescopes, cameras, and the human eye.
• Explains why larger apertures improve image sharpness and detail detection.
• Is widely used in astronomy, microscopy, photography, and vision science to predict and compare resolution limits.

Summary

• The Rayleigh criterion gives the condition for just resolving two closely spaced objects.
• It shows that diffraction limits the sharpness of images formed by optical systems.
• In vision, it explains why the human eye cannot distinguish objects that are too close together angularly.
• Important terms to remember: diffraction, Airy disk, resolving power, limit of resolution, circular aperture, angular separation, pupil.