Numerical aperture –Derivation

Numerical Aperture (NA) Derivation

Introduction to Numerical Aperture

Numerical Aperture (NA) is a dimensionless number that characterizes the range of angles over which a lens or optical system can accept or emit light. It is a crucial parameter in optics and microscopy, providing information about the light-gathering ability and resolution of an optical system.

Derivation of Numerical Aperture

The Numerical Aperture is defined as the sine of the half-angle of the maximum cone of light that can enter or exit an optical system:

NA = n * sin(θ)

Where:
n is the refractive index of the medium between the lens and the external medium.
θ is the half-angle of the maximum cone of light.

The half-angle (θ) can be derived from the geometry of the optical system using trigonometry. Consider a lens with a circular entrance or exit pupil. The maximum angle at which light can enter or exit is half of the angle subtended by the diameter of the lens:

sin(θ) = (r / f)

Where:
r is the radius of the lens.
f is the focal length of the lens.

Substituting this expression for sin(θ) into the definition of NA gives the final formula:

NA = n * (r / f)

Significance and Applications

The Numerical Aperture is a critical parameter in microscopy, as it determines the resolving power and light-gathering ability of an optical system. Higher NA values result in better resolution and increased sensitivity to low-intensity signals.

Applications include biological and materials microscopy, where detailed imaging and analysis are crucial.

This provides a basic derivation of Numerical Aperture. Explore further to understand its implications in various optical systems and its role in advanced microscopy techniques.