Rayleigh Criterion for Limit of Resolution

The Rayleigh criterion, formulated by Lord Rayleigh, is a guideline used to determine the minimum resolvable detail in an optical system.

The Rayleigh criterion is expressed mathematically as:

\[ \theta = 1.22 \times \frac{\lambda}{D} \]

where:

• \( \theta \) is the angular resolution (in radians),
• \( \lambda \) is the wavelength of light used (in the same units as \( D \)),
• \( D \) is the diameter of the aperture (e.g., the objective lens or mirror) through which light passes.

Here's a more detailed explanation of each term:

1. Angular Resolution (\( \theta \)):
   • Angular resolution refers to the smallest angle between two point sources or details that can still be resolved as distinct entities. It is measured in radians.
   • The smaller the angular resolution, the better the resolving power of the optical system.
2. Wavelength of Light (\( \lambda \)):
   • \( \lambda \) represents the wavelength of the light being used in the optical system. Different colors of light have different wavelengths.
   • Smaller wavelengths allow for higher resolution, meaning that shorter wavelengths can resolve smaller details.
3. Aperture Diameter (\( D \)):
   • \( D \) is the diameter of the aperture, which could be the diameter of the objective lens in a microscope or the diameter of the primary mirror in a telescope.
   • Larger apertures result in better resolution, as they allow more light to enter the system and reduce diffraction effects.

The constant factor 1.22 is an empirical factor that accounts for the Airy disk pattern produced by diffraction. The Airy disk is a circular diffraction pattern formed when light passes through a circular aperture, and it determines the practical limit of resolution.

In summary, the Rayleigh criterion provides a relationship between the angular resolution, wavelength of light, and aperture diameter. It helps optical designers and users understand the limitations of their systems and make informed decisions about the achievable resolution. Keep in mind that the Rayleigh criterion is a guideline, and in practice, the actual performance of an optical system may be influenced by factors beyond the basic theoretical considerations.