The numerical aperture in 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. In Dimensional analysis, a dimensionless quantity (or more precisely a quantity with the dimensions of 1) is a Quantity without any Physical units The exact definition of the term varies slightly between different areas of optics.

## 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. Microscopy is the technical field of using microscopes to view samples or objects An objective in Optics is the lens or Mirror in a Microscope, Telescope, camera or other optical instrument The refractive index (or index of Refraction) of a medium is a measure for how much the speed of light (or other waves such as sound waves is reduced inside the medium 0 for air, 1. Temperature and layers The temperature of the Earth's atmosphere varies with altitude the mathematical relationship between temperature and altitude varies among five 33 for pure water, and up to 1. Water is a common Chemical substance that is essential for the survival of all known forms of Life. 56 for oils), and θ is the half-angle of the maximum cone of light that can enter or exit the lens. An oil is a substance that is in a viscous Liquid state ( "oily") at ambient temperatures or slightly warmer and is In general, this is the angle of the real marginal ray in the system. In Optics, a ray is an idealized narrow Beam of light. Rays are used to model the propagation of Light through an optical system by dividing the real light The angular aperture of the lens is approximately twice this value (within the paraxial approximation). The angular aperture of a lens is the apparent Angle of the lens Aperture as seen from the focal point: a = 2 \arctan In Geometric optics, the paraxial approximation is an Approximation used in ray tracing of light through an optical system (such as a lens) 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 is important because it indicates the resolving power of a lens. Resolving power may refer to Angular resolution Spectral resolution The size of the finest detail that can be resolved is proportional to λ/NA, where λ is the wavelength of the light. In Physics wavelength is the distance between repeating units of a propagating Wave of a given Frequency. A lens with a larger numerical aperture will be able to visualize finer details than a lens with a smaller numerical aperture. Lenses with larger numerical apertures also collect more light and will generally provide a brighter image.

Numerical aperture is used to define the "pit size" in optical disc formats [1]

### Numerical aperture versus f-number

Numerical aperture of a thin lens. In Optics, a thin lens is a lens with a thickness (distance along the Optical axis between the two surfaces of the lens that is negligible compared

Numerical aperture is not typically used in photography. Photography (fә'tɒgrәfi or fә'tɑːgrәfi (from Greek φωτο and γραφία is the process and Art of recording pictures by means of capturing Instead, the angular acceptance of a lens (or an imaging mirror) is expressed by the f-number, written f/# or N, which is defined as the ratio of the focal length to the diameter of the entrance pupil:

N = f / D

This ratio is related to the numerical aperture with respect to the focal point of the lens. A photographic lens (also known as objective lens or photographic objective) is an optical lens or assembly of lenses used in conjunction with The focal length of an optical system is a measure of how strongly it converges (focuses or diverges (diffuses Light. In an optical system the entrance pupil is a virtual aperture that defines the area at the entrance of the system that can accept light Track listing Child – 516 All I Need – 355 Based on the diagram at right, the numerical aperture of the lens in air is:

$\mathrm{NA} = n \sin \theta = n \sin \arctan \frac{D}{2f} \approx \frac {D}{2f}$
thus $N \approx \frac{1}{2\;\mathrm{NA}}.$

This approximation holds when the numerical aperture is small. The f-number describes the light-gathering ability of the lens in the case where the marginal ray before (or after) the lens is collimated. Collimated light is Light whose rays are nearly parallel and therefore will spread slowly as it propagates This case is commonly encountered in photography, where objects being photographed are often far from the camera.

When the object is not distant from the lens the image is no longer formed in the lens's focal plane, and the f-number no longer accurately describes the light-gathering ability of the lens. The cardinal points and the associated cardinal planes are a set of special points and planes in an optical system which help in the analysis In this case, the working f-number is used instead. The working f-number is defined by making the approximate relation above exact:

$N_\mathrm{w} = {1 \over 2 \mathrm{NA}} \approx (1-m)\, N,$

where Nw is the working f-number, and m is the lens's magnification for an object a particular distance away. Magnification is the process of enlarging something only in appearance not in physical size [2] The magnification here is typically negative; in photography, the factor is sometimes written as 1 + m, where m represents the absolute value of the magnification; in either case, the correction factor is 1 or greater. In Mathematics, the absolute value (or modulus) of a Real number is its numerical value without regard to its sign.

## Laser physics

In laser physics, the numerical aperture is defined slightly differently. Laser science or laser physics is a branch of Optics that describes the theory and practice of Lasers Laser science is principally concerned with Laser beams spread out as they propagate, but slowly. Far away from the narrowest part of the beam, the spread is roughly linear with distance—the laser beam forms a cone of light in the "far field". The same relation gives the NA,

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

$\mathrm{NA}\simeq \frac{2 \lambda_0}{n \pi D},$

where λ0 is the vacuum wavelength of the light, and D is the diameter of the beam at its narrowest spot, measured between the 1/e2 irradiance points ("Full width at e−2 maximum"). In Physics wavelength is the distance between repeating units of a propagating Wave of a given Frequency. Note that this means that a laser beam that is focused to a small spot will spread out quickly as it moves away from the focus, while a large-diameter laser beam can stay roughly the same size over a very long distance.

## Fiber optics

Multimode optical fiber will only propagate light that enters the fiber within a certain cone, known as the acceptance cone of the fiber. Multi-mode optical fiber ( multimode fiber or MM fiber or fibre) is a type of Optical fiber mostly used for communication over shorter distances A guided ray (also bound ray or trapped ray) is a ray of light in a Multimode optical fiber, which is confined by the core. The half-angle of this cone is called the acceptance angle, θmax. For step-index multimode fiber, the acceptance angle is determined only by the indices of refraction:

$n \sin \theta_\max = \sqrt{n_1^2 - n_2^2},$

where n1 is the refractive index of the fiber core, and n2 is the refractive index of the cladding. For an Optical fiber, a step-index profile is a Refractive index profile characterized by a uniform refractive index within the core and a sharp decrease

When a light ray is incident from a medium of refractive index n to the core of index n1, Snell's law at medium-core interface gives

$n\sin\theta_i = n_1\sin\theta_r.\$

From the above figure and using trigonometry, we get :

$\sin\theta_{r} = \sin\theta_{2} = \sin\left({90^\circ} - \theta_{c} \right) = \cos\theta_{c}\$

where $\theta_{c} = \sin^{-1} \frac{n_{2}}{n_{1}}$is the critical angle for total internal reflection, since

Substituting for sin θr in Snell's law we get:

$\frac{n}{n_{1}}\sin\theta_{i} = \cos\theta_{c}.$

By squaring both sides

$\frac{n^{2}}{n_{1}^{2}}\sin^{2}\theta_{i} = \cos ^{2}\theta_{c} = 1 - \sin^{2}\theta_{c} = 1 - \frac{n_{2}^{2}}{n_{1}^{2}}.$

Thus,

$n \sin \theta_{i} = \sqrt{n_1^2 - n_2^2},$

from where the formula given above follows. The refractive index (or index of Refraction) of a medium is a measure for how much the speed of light (or other waves such as sound waves is reduced inside the medium In Optics and Physics, Snell's law (also known as Descartes' law or the law of refraction) is a formula used to describe the relationship

This has the same form as the numerical aperture in other optical systems, so it has become common to define the NA of any type of fiber to be

$\mathrm{NA} = \sqrt{n_1^2 - n_2^2},$

where n1 is the refractive index along the central axis of the fiber. Note that when this definition is used, the connection between the NA and the acceptance angle of the fiber becomes only an approximation. In particular, manufacturers often quote "NA" for single-mode fiber based on this formula, even though the acceptance angle for single-mode fiber is quite different and cannot be determined from the indices of refraction alone. In Fiber-optic communication, a single-mode optical fiber ( SMF) is an Optical fiber designed to carry only a single ray of light (mode

The number of bound modes, the mode volume, is related to the normalized frequency and thus to the NA. In Fiber optics, mode volume is the number of bound modes that an Optical fiber is capable of supporting Normalized frequency is the Ratio of an actual frequency and a reference value

In multimode fibers, the term equilibrium numerical aperture is sometimes used. This refers to the numerical aperture with respect to the extreme exit angle of a ray emerging from a fiber in which equilibrium mode distribution has been established. The equilibrium mode distribution of light travelling in an optical Waveguide or fiber, is the distribution of light that is no longer changing with fibre length