Editing Numerical aperture (section)

==Fiber optics==
[[Image:Optic fibre-numerical aperture diagram.svg|400px|thumb|right|A multi-mode fiber of index {{math|''n''<sub>1</sub>}} with cladding of index {{math|''n''<sub>2</sub>}}.]]

A [[multi-mode optical fiber]] will only propagate light that enters the fiber within a certain range of angles, known as the [[acceptance cone]] of the fiber. The half-angle of this cone is called the [[Acceptance angle (optical fiber)|acceptance angle]], {{math|''θ''<sub>max</sub>}}. For [[Step-index profile|step-index]] multimode fiber in a given medium, the acceptance angle is determined only by the indices of refraction of the core, the cladding, and the medium:
<math display="block">n \sin \theta_\max = \sqrt{n_\text{core}^2 - n_\text{clad}^2},</math>
where {{math|''n''}} is the [[refractive index]] of the medium around the fiber, {{math|''n''<sub>core</sub>}} is the refractive index of the fiber core, and {{math|''n''<sub>clad</sub>}} is the refractive index of the [[Cladding (fiber optics)|cladding]]. While the core will accept light at higher angles, those rays will not [[total internal reflection|totally reflect]] off the core–cladding interface, and so will not be transmitted to the other end of the fiber. The derivation of this formula is given below.

When a light ray is incident from a medium of [[refractive index]] {{mvar|n}} to the core of index {{math|''n''<sub>core</sub>}} at the maximum acceptance angle, [[Snell's law]] at the medium–core interface gives
<math display="block">n\sin\theta_\max = n_\text{core}\sin\theta_r.\ </math>

From the geometry of the above figure we have:
<math display="block">\sin\theta_{r} = \sin\left({90^\circ} - \theta_{c}\right) = \cos\theta_{c}</math>

where
<math display="block"> \theta_{c} = \arcsin \frac{n_\text{clad}}{n_\text{core}}</math>

is the [[critical angle (optics)|critical angle]] for [[total internal reflection]].

Substituting {{math|cos ''θ''<sub>''c''</sub>}} for {{math|sin ''θ''<sub>''r''</sub>}} in Snell's law we get:
<math display="block">\frac{n}{n_\text{core}}\sin\theta_\max = \cos\theta_{c}.</math>

By squaring both sides
<math display="block">\frac{n^{2}}{n_\text{core}^{2}}\sin^{2}\theta_\max = \cos^{2}\theta_{c} = 1 - \sin^{2}\theta_{c} = 1 - \frac{n_\text{clad}^{2}}{n_\text{core}^{2}}.</math>

Solving, we find the formula stated above:

<math display="block">n \sin \theta_\max = \sqrt{n_\text{core}^2 - n_\text{clad}^2},</math>

This has the same form as the numerical aperture in other optical systems, so it has become common to ''define'' the {{abbr|NA|numerical aperture}} of any type of fiber to be
<math display="block">\mathrm{NA} = \sqrt{n_\text{core}^2 - n_\text{clad}^2},</math>

where {{math|''n''<sub>core</sub>}} is the refractive index along the central axis of the fiber. Note that when this definition is used, the connection between the numerical aperture and the acceptance angle of the fiber becomes only an approximation. In particular, "{{abbr|NA|numerical aperture}}" defined this way is not relevant for [[single-mode fiber]].<ref>{{cite web |first=R. |last=Paschotta |title=Numerical Aperture |work=RP Photonics Encyclopedia |access-date=2024-08-25 |doi=10.61835/fov |url=https://www.rp-photonics.com/numerical_aperture.html#:~:text=For%20a%20single%2Dmode%20fiber,straight%20form%20due%20to%20scattering.)}}</ref><ref>{{cite report |last1=Kowalevicz, Jr. |first1=Andrew M. |last2=Bucholtz |first2=Frank |date=Oct 6, 2006 |title=Beam Divergence from an SMF-28 Optical Fiber |url=https://apps.dtic.mil/sti/citations/ADA456331 |publisher=Naval Research Lab |id=NRL/MR/5650--06-8996}}</ref> One cannot define an acceptance angle for single-mode fiber based on the indices of refraction alone.

The number of bound [[transverse mode|modes]], the [[mode volume]], is related to the [[Normalized frequency (fiber optics)|normalized frequency]] and thus to the numerical aperture.

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 [[Line (mathematics)|ray]] emerging from a fiber in which [[equilibrium mode distribution]] has been established.