The Rayleigh criterion is a principle used in optics to determine the minimum angular separation between two objects that allows them to be seen as distinct entities when viewed through an optical instrument, such as a telescope or microscope.
Understanding the Rayleigh Criterion
At its core, the Rayleigh criterion addresses the limit imposed on resolution by the wave nature of light, specifically the phenomenon of diffraction. When light from a point source passes through an aperture (like the lens of a telescope), it doesn't produce a perfect point image but rather a diffraction pattern, typically a bright central spot (the Airy disk) surrounded by fainter rings.
According to the provided definition, the Rayleigh criterion states that:
"two images are just resolvable when the centre of the diffraction pattern of one is directly over the first minimum of the diffraction pattern of the other."
This specific alignment is considered the point where the dip between the two central maxima is about 26% of the peak intensity, which is a commonly accepted threshold for distinguishing them as separate.
Why is Diffraction Important for Resolution?
Diffraction causes light from two closely spaced point sources to spread out and overlap. If the sources are too close together, their diffraction patterns merge to such an extent that the observer cannot tell if they are looking at two separate objects or just one blurred object. The Rayleigh criterion provides a quantitative measure of the minimum separation required for resolution.
Quantifying Resolution: The Rayleigh Limit
The angular separation ($\Theta$) at which two objects are just resolvable according to the Rayleigh criterion is given by the formula:
$\Theta = \frac{1.22 \lambda}{D}$
Where:
- $\Theta$ is the minimum resolvable angular separation (in radians).
- $\lambda$ (lambda) is the wavelength of the light being observed.
- $D$ is the diameter of the aperture (e.g., the diameter of the telescope mirror or lens, or the diameter of the pupil of your eye).
- The factor 1.22 arises from the mathematical analysis of the diffraction pattern from a circular aperture.
This formula tells us that to improve angular resolution (make $\Theta$ smaller), you can either use light with a shorter wavelength ($\lambda$) or increase the diameter of the aperture ($D$).
Practical Implications
The Rayleigh criterion is fundamental in designing and evaluating optical instruments:
- Telescopes: Larger diameter objectives ($D$) lead to better resolution, allowing astronomers to distinguish between stars that are very close together.
- Microscopes: Using shorter wavelengths of light (like blue light or even UV light) or techniques like electron microscopy (which uses much shorter wavelengths than visible light) improves resolution.
- Cameras: While often limited by pixel size, diffraction still plays a role, especially with smaller apertures (larger f-numbers), which can reduce sharpness.
In essence, the Rayleigh criterion sets a theoretical upper limit on the resolution of an optical system based purely on the physics of diffraction. Real-world resolution can also be affected by other factors like lens aberrations or atmospheric turbulence, but diffraction remains a fundamental constraint.
