Rayleigh Criterion
A fundamental principle in optics that defines the minimum angular separation needed to distinguish two point sources of light as separate objects.
Definition
The Rayleigh Criterion, formulated by Lord John William Strutt, 3rd Baron Rayleigh, establishes the fundamental limit for the resolution of optical instruments. It states that two point sources are just resolvable when the central maximum of one diffraction pattern coincides with the first minimum of the other.
Mathematical Expression
The angular resolution θ is given by:
θ = 1.22 λ/D
where:
- λ is the wavelength of light
- D is the diameter of the aperture
- θ is measured in radians
Applications
Astronomy
The Rayleigh Criterion is crucial in:
- Determining telescope resolving power
- Binary Star separation measurements
- Planetary observation
- Design of optical telescopes
Microscopy
Applications include:
- Setting resolution limits for optical microscopes
- Cellular imaging capabilities
- Super-resolution microscopy beyond classical limits
Physical Basis
The criterion emerges from the wave nature of light and the resulting Airy Disk pattern formed when light passes through a circular aperture. This pattern consists of:
- A bright central maximum
- Surrounding rings of decreasing intensity
- Dark rings at specific angular distances
Historical Context
Rayleigh developed this criterion while studying optical instruments in the late 19th century. It has since become a cornerstone of:
- Optical design
- Image processing
- Resolution limits physical limitations
Modern Extensions
Contemporary developments include:
- Quantum optics approaches to overcome the limit
- Digital image processing techniques
- Adaptive optics systems
Limitations
The criterion assumes:
- Incoherent light illumination
- Circular apertures
- Perfect optical conditions
- Diffraction-limited systems
These assumptions may not hold in all practical applications, leading to various modifications and alternatives in modern optical theory.