Numerical aperture
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.
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.
This is easily shown by rearranging Snell's law to find that n sin θ is constant across an interface.
In microscopy, NA generally refers to object-space numerical aperture 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 (the resolution) is proportional to λ/2NA, where λ is the wavelength of the light.
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.
This ratio is related to the image-space numerical aperture when the lens is focused at infinity.
The approximation holds when the numerical aperture is small, but it turns out that for well-corrected optical systems such as camera lenses, a more detailed analysis shows that N is almost exactly equal to 1/(2NAi) even at large numerical apertures.
However, the complete theory of the Abbe sine condition shows that if a lens is corrected for coma and spherical aberration, as all good photographic objectives must be, the second principal plane becomes a portion of a sphere of radius f centered about the focal point".
[4] In this sense, the traditional thin-lens definition and illustration of f-number is misleading, and defining it in terms of numerical aperture may be more meaningful.
This case is commonly encountered in photography, where objects being photographed are often far from the camera.
The working f-number is defined by modifying the relation above, taking into account the magnification from object to image:
[3][5] The magnification here is typically negative, and the pupil magnification is most often assumed to be 1 — 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.
The two equalities in the equation above are each taken by various authors as the definition of working f-number, as the cited sources illustrate.
Conversely, the object-side numerical aperture is related to the f-number by way of the magnification (tending to zero for a distant object):
In laser physics, numerical aperture is defined slightly differently.
The relation used to define the NA of the laser beam is the same as that used for an optical system,
Laser beams typically do not have sharp edges like the cone of light that passes through the aperture of a lens does.
Laser physicists typically choose to make θ the divergence of the beam: the far-field angle between the beam axis and the distance from the axis at which the irradiance drops to e−2 times the on-axis irradiance.
where λ0 is the vacuum wavelength of the light, and 2w0 is the diameter of the beam at its narrowest spot, measured between the e−2 irradiance points ("Full width at e−2 maximum of the intensity").
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.
While the core will accept light at higher angles, those rays will not totally reflect off the core–cladding interface, and so will not be transmitted to the other end of the fiber.
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
Note that when this definition is used, the connection between the numerical aperture and the acceptance angle of the fiber becomes only an approximation.
[7][8] One cannot define an acceptance angle for single-mode fiber based on the indices of refraction alone.
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.
