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# Numerical aperture

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 Title: Numerical aperture Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### Numerical aperture

The numerical aperture with respect to a point P depends on the half-angle θ of the maximum cone of light that can enter or exit the lens.

In optics, the numerical aperture (NA) of an optical system is a dimensionless number that characterizes the range of angles over which the system can accept or emit light. By incorporating index of refraction in its definition, NA has the property that it is constant for a beam as it goes from one material to another provided there is no optical power at the interface. The exact definition of the term varies slightly between different areas of optics. Numerical aperture is commonly used in microscopy to describe the acceptance cone of an objective (and hence its light-gathering ability and resolution), and in fiber optics, in which it describes the range of angles within which light that is incident on the fiber will be transmitted along it.

## Contents

• General optics 1
• Numerical aperture versus f-number 1.1
• Working (effective) f-number 1.2
• Laser physics 2
• Fiber optics 3
• References 5

## General optics

In most areas of optics, and especially in microscopy, the numerical aperture of an optical system such as an objective lens is defined by

\mathrm{NA} = n \sin \theta\;

where n is the index of refraction of the medium in which the lens is working (1.00 for air, 1.33 for pure water, and typically 1.52 for immersion oil;[1] see also list of refractive indices), and θ is the half-angle of the maximum cone of light that can enter or exit the lens. In general, this is the angle of the real marginal ray in the system. Because the index of refraction is included, the NA of a pencil of rays is an invariant as a pencil of rays passes from one material to another through a flat surface. This is easily shown by rearranging Snell's law to find that n \sin \theta is constant across an interface.

In air, the angular aperture of the lens is approximately twice this value (within the paraxial approximation). The NA is generally measured with respect to a particular object or image point and will vary as that point is moved. In microscopy, NA generally refers to object-space NA unless otherwise noted.

In microscopy, NA is important because it indicates the resolving power of a lens. The size of the finest detail that can be resolved is proportional to λ/2NA, where λ is the wavelength of the light. A lens with a larger numerical aperture will be able to visualize finer details than a lens with a smaller numerical aperture. Assuming quality (diffraction limited) optics, lenses with larger numerical apertures collect more light and will generally provide a brighter image, but will provide shallower depth of field.

Numerical aperture is used to define the "pit size" in optical disc formats.[2]

### Numerical aperture versus f-number

Numerical aperture of a thin lens.

Numerical aperture is not typically used in

• "Microscope Objectives: Numerical Aperture and Resolution" by Mortimer Abramowitz and Michael W. Davidson, Molecular Expressions: Optical Microscopy Primer (website), Florida State University, April 22, 2004.
• "Basic Concepts and Formulas in Microscopy: Numerical Aperture" by Michael W. Davidson, Nikon MicroscopyU (website).
• "Numerical aperture", Encyclopedia of Laser Physics and Technology (website).
• "Numerical Aperture and Resolution", UCLA Brain Research Institute Microscopy Core Facilities (website), 2007.