Numerical aperture

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.

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 (1.0 for air, 1.33 for pure water, and up to 1.56 for oils), 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. 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 versus f-number
Numerical aperture is not typically used in photography. Instead, the angular acceptance of a lens is expressed by the f-number,, which for a thin lens imaging an object at infinity is given by


 * $$f/\# \approx \frac {1}{2\,{N\!A}}.$$

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. This case is commonly encountered in photography, where objects being photographed are often far from the camera.

In optical design, the finite distance between the object and the lens must often be considered. In these cases, the working f-number is used instead. The working f-number is defined by making the approximate relation above exact:


 * $$f/\mathit{\#}_\mathrm{w} = {1 \over 2 {N\!A}} \approx (1-m)\, f/\mathit{\#}, $$

where $$f/\mathit{\#}_\mathrm{w}$$ is the working f-number, and $$m$$ is the lens's magnification for an object a particular distance away.

Laser physics
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&mdash;the laser beam forms a cone of light in the "far field". The same relation gives the NA,


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

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 far-field angle between the propagation direction and the distance from the beam axis for which the irradiance drops to 1/e2 times the wavefront total irradiance. The NA of a Gaussian laser beam is then related to its minimum spot size by


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

where &lambda;0 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&minus;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 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. 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.

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.

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.

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.