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. The exact definition of the term varies slightly between different areas of 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
NA=n\sin\theta
where n is the index of refraction of the medium in which the lens is working (1.0 for air, 1.33 for pure water, and up to 1.56 for oils), and θ is the halfangle 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. 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 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/NA, 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. 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 is not typically used in photography. Instead, the angular aperture of a lens (or an imaging mirror) is expressed by the fnumber, written or
N
N=f/D
This ratio is related to the imagespace numerical aperture when the lens is focused at infinity. Based on the diagram at right, the imagespace numerical aperture of the lens is:
NA_{i} 
=n\sin\theta=n\sin\arctan
D  
2f 
≈ n
D  
2f 
thus
N ≈
1  

n=1
The approximation holds when the numerical aperture is small, and it is nearly exact even at large numerical apertures for wellcorrected camera lenses. As Rudolf Kingslake explains, "It is a common error to suppose that the ratio [<math>D/2f</math> ] is actually equal to
\tan\theta
\sin\theta
The fnumber describes the lightgathering ability of the lens in the case where the marginal rays on the object side are parallel to the axis of the lens. 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, however, the image is no longer formed in the lens's focal plane, and the fnumber no longer accurately describes the lightgathering ability of the lens or the imageside numerical aperture. In this case, the numerical aperture is related to what is sometimes called the "working fnumber" or "effective fnumber." The working fnumber is defined by modifying the relation above, taking into account the magnification from object to image:
1  

=N_{w}=(1m)N,
where
N_{w}
m
The two equalities in the equation above are each taken by various authors as the definition of working fnumber, as the cited sources illustrate. They are not necessarily both exact, but are often treated as if they are. The actual situation is more complicated — as Allen R. Greenleaf explains, "Illuminance varies inversely as the square of the distance between the exit pupil of the lens and the position of the plate or film. Because the position of the exit pupil usually is unknown to the user of a lens, the rear conjugate focal distance is used instead; the resultant theoretical error so introduced is insignificant with most types of photographic lenses."^{[5]}
Conversely, the objectside numerical aperture is related to the fnumber by way of the magnification (tending to zero for a distant object):
1  

=
m1  
m 
N.
In laser physics, the numerical aperture is defined slightly differently. 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,
NA=n\sin\theta,
but θ is defined differently. Laser beams typically do not have sharp edges like the cone of light that passes through the aperture of a lens does. Instead, the irradiance falls off gradually away from the center of the beam. It is very common for the beam to have a Gaussian profile. Laser physicists typically choose to make θ the divergence of the beam: the farfield angle between the propagation direction and the distance from the beam axis for which the irradiance drops to 1/e^{2} times the wavefront total irradiance. The NA of a Gaussian laser beam is then related to its minimum spot size by
NA\simeq
2λ_{0}  
\piD 
,
where λ_{0} is the vacuum wavelength of the light, and D is the diameter of the beam at its narrowest spot, measured between the 1/e^{2} irradiance points ("Full width at e^{2} maximum"). 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 largediameter laser beam can stay roughly the same size over a very long distance.
Multimode optical fiber will only propagate light that enters the fiber within a certain cone, known as the acceptance cone of the fiber. The halfangle of this cone is called the acceptance angle, θ_{max}. For stepindex multimode fiber, the acceptance angle is determined only by the indices of refraction:
n\sin\theta_{max}=
2  
\sqrt{n  
1 

2},  
n  
2 
where n_{1} is the refractive index of the fiber core, and n_{2} is the refractive index of the cladding.
When a light ray is incident from a medium of refractive index n to the core of index n_{1}, Snell's law at mediumcore interface gives
n\sin\theta_{i}=n_{1\sin\theta}_{r. }
\sin\theta_{r}=\sin\left({90^{\circ}}\theta_{c}\right)=\cos\theta_{c}
where
\theta_{c}=\sin^{1}
n_{2}  
n_{1} 
Substituting for sin θ_{r} in Snell's law we get:
n  
n_{1} 
\sin\theta_{i}=\cos\theta_{c}.
By squaring both sides
n^{2}  

\sin^{2}\theta_{i}=\cos^{2}\theta_{c}=1\sin^{2}\theta_{c}=1
 

.
Thus,
n\sin\theta_{i}=
2  
\sqrt{n  
1 

2},  
n  
2 
from where the formula given above follows.
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
NA=
2  
\sqrt{n  
1 

2},  
n  
2 
where n_{1} 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 singlemode fiber based on this formula, even though the acceptance angle for singlemode fiber is quite different and cannot be determined from the indices of refraction alone.
The number of bound modes, the mode volume, is related to the normalized frequency and thus to the NA.
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 Practical Guide to Optics for Photographers
. Rudolf Kingslake. CaseHoyt Corp. for Garden City Books. 1951. 97–98.