Numerical aperture (NA)

Numerical aperture is $\text{NA}=n\sin \theta$, where $\theta$ is the half-angle of the widest cone of light that a fiber, lens or waveguide accepts or emits, and $n$ is the refractive index of the medium the cone sits in (1 for air). For a step-index fiber it follows from the core and cladding indices, $\text{NA}=\sqrt{n_{\text{core}}^2-n_{\text{clad}}^2}$: standard single-mode fiber is specified at 0.14, which is an acceptance half-angle of 8.0° in air, and multimode fibers at 0.20 (50 µm core) and 0.275 (62.5 µm core).

The same number describes a lens or microscope objective, where it sets the smallest spot the lens can focus to and the finest detail it can resolve. Because it is a sine multiplied by an index, NA stays the same when light passes through a flat interface between media (Snell's law conserves $n\sin \theta$), which is why it is quoted instead of an angle.

Formulas

Step-index fiber. A ray entering the end face at angle $\theta$ is guided if it meets the core-cladding boundary beyond the critical angle. Applying Snell's law at the end face and at the boundary gives

$$\text{NA}=n_0\sin {\theta }_{\text{max}}=\sqrt{n_{\text{core}}^2-n_{\text{clad}}^2},$$

where $n_0$ is the index outside the fiber. For the small index steps of telecom fiber, with $\mathrm{\Delta }=(n_{\text{core}}-n_{\text{clad}})\mathrm{/}{n}_{\text{core}}$,

$$\text{NA}\approx n_{\text{core}}\sqrt{2\mathrm{\Delta }}\mathrm{.}$$

A core of index 1.4500 in a cladding of 1.4440 ($\mathrm{\Delta }=0.41\mathrm{\%}$) gives NA = 0.132 and an acceptance half-angle of 7.6° in air. The acceptance angle entry covers the ray picture in more detail.

Lens. For a lens that focuses a collimated beam of diameter $D$ to a point at focal length $f$,

$$\text{NA}=n\sin \theta ,\tan \theta =\frac{D}{2f}\mathrm{.}$$

For small angles this is $\text{NA}\approx D\mathrm{/}2f$; for a well-corrected lens that obeys the Abbe sine condition, $D\mathrm{/}2f$ is exact. A 5 mm beam on an 8 mm focal-length asphere gives $D\mathrm{/}2f=0.31$.

NA and f-number. The working f-number $N$ of a lens used with a distant object is

$$N\approx \frac{1}{2\text{NA}},$$

so f/2 corresponds to NA 0.25 and NA 0.5 to f/1. Photographers use $N$; microscopists and fiber engineers use NA.

Acceptance angle from NA. ${\theta }_{\text{max}}=\arcsin (\text{NA}\mathrm{/}{n}_0)$. The full cone is twice this.

Typical values

An NA above 1 is possible only in a medium of index above 1. An oil-immersion objective of NA 1.40 collects light out to a half-angle of 67.5° in the oil.

For high-contrast waveguides the fiber formula returns numbers above 1: silicon ($n=3.48$) in oxide ($n=1.44$) gives $\sqrt{{3.48}^2-{1.44}^2}=3.2$. That means every ray arriving from air is guided (it refracts to within 16.7° of the axis inside the silicon, well inside the guided range), so a silicon waveguide's coupling is set by the size and divergence of its mode, not by an acceptance cone.

The NA of a single-mode fiber and the NA of its beam

For a fully filled step-index multimode fiber the NA is close to a true edge: the far field falls off sharply at the acceptance angle. For single-mode fiber the guided light is a near-Gaussian mode, so the far field has no edge at all and the quoted NA depends on where the edge is drawn. Corning specifies SMF-28 Ultra at 0.14, measured at the 1% power level of a far-field scan at 1310 nm. The Gaussian beam leaving the same fiber has a $1\mathrm{/}{e}^2$ half-angle of

$${\theta }_{1\mathrm{/}{e}^2}=\frac{\lambda }{\pi w_0}=\frac{1.31\mu \text{m}}{\pi \times 4.6\mu \text{m}}=0.091\text{ rad}$$

for a mode field diameter of 9.2 µm ($w_0=4.6$ µm), and 0.095 rad at 1550 nm, where the mode field diameter is 10.4 µm. The two numbers agree: a Gaussian far field falls to 1% of its peak at 1.52 times its $1\mathrm{/}{e}^2$ angle, which gives 0.137.

The difference matters when choosing optics. To couple a collimated beam into single-mode fiber, the lens must reproduce the fiber's mode, so it is chosen to focus to $w_0$, not to fill a cone of NA 0.14. The focused waist of a collimated Gaussian beam of $1\mathrm{/}{e}^2$ radius $w_{\text{in}}$ is $\lambda f\mathrm{/}(\pi w_{\text{in}})$, so

$$f=\frac{\pi w_0{w}_{\text{in}}}{\lambda }\mathrm{.}$$

A 2 mm diameter beam ($w_{\text{in}}=1$ mm) at 1550 nm needs $f=\pi \times 5.2\mu \text{m}\times 1\text{ mm}\mathrm{/}1.55\mu \text{m}=10.5$ mm to match SMF-28's 10.4 µm mode; the full procedure, with the alignment tolerances, is in Coupling a Free-Space Beam into Single-Mode Fiber. Any mismatch in waist size appears as mode mismatch loss. The same reasoning sets the spot size of a lensed fiber or the taper of an edge coupler on a photonic chip.

For multimode fiber the rule is different: light outside the fiber's NA cone, or outside its core, is not guided by the core and is lost within a short length of fiber. A lens can trade spot size for angle but cannot shrink both, because the product of beam radius and NA (the one-dimensional étendue) is conserved through any lossless optics. A source whose radius-NA product exceeds the fiber's cannot be coupled into it without loss, whatever lens is used.

Spot size, resolution and depth of focus

A higher NA focuses light to a smaller spot and resolves finer detail, and the depth of focus shrinks faster than the spot does.

• Gaussian beam (fiber outputs, lasers): with NA taken as the $1\mathrm{/}{e}^2$ half-angle of the focused beam, the waist radius is $w_0=\lambda \mathrm{/}(\pi \text{NA})$ while the angles stay small. At 1550 nm, NA 0.10 gives $w_0=4.9$ µm, a 9.9 µm spot.
• Uniformly filled lens (imaging): the radius of the Airy disk, which is also the Rayleigh resolution limit, is $r=0.61\lambda \mathrm{/}\text{NA}$. A 40× objective of NA 0.65 at 532 nm gives $r=0.50$ µm.
• Depth of focus scales as $\lambda \mathrm{/}{\text{NA}}^2$. For the Gaussian beam above, the focus stays within a factor $\sqrt{2}$ of its waist over twice the Rayleigh range, $2z_R=2\lambda \mathrm{/}(\pi {\text{NA}}^2)$, which is 99 µm at NA 0.10 and 1550 nm. Doubling the NA halves the spot and quarters the depth of focus. The depth of focus entry treats imaging systems.

NA also sets the number of modes a fiber guides, through the V number $V=(2\pi a\mathrm{/}\lambda )\text{NA}$, where $a$ is the core radius. A fiber is single-mode for $V<2.405$. That calculation uses the NA from the index profile. The Fiber Numerical Aperture and V Number Calculator works it through for any core, with the acceptance angle, cutoff wavelength and mode field diameter.

How NA is measured

Far-field scan. Light is launched into a short length of fiber, and a detector on a rotation stage, or a camera, records the output intensity against angle at a distance much larger than the core. The NA is the sine of the half-angle at which the intensity falls to a set fraction of its peak. The fraction is part of the definition: the standard test method (IEC 60793-1-43, adopted in the US as TIA-455-177) sets it for multimode fiber, and Corning's single-mode figure uses 1%. A lower threshold gives a larger NA from the same far field, as the single-mode example above shows, so two NA values can be compared only at the same threshold.

Launch conditions. A multimode fiber measured with an underfilled launch (a narrow beam at low angle) reports a smaller NA than one with an overfilled launch. The standard methods overfill the fiber's angular range and use a short sample, so that the high-angle modes have not yet been lost.

Bench estimate. Project the output of a cleaved fiber onto a camera sensor or a card at distance $L$, measure the diameter $d$ of the spot at the chosen intensity level, and take $\text{NA}\approx \sin [\arctan (d\mathrm{/}2L)]$. With a beam-profiling camera and two distances, the difference method removes the uncertainty in where the fiber end sits.

From the index profile. A measured refractive-index profile gives the theoretical NA directly from $n_{\text{core}}$ and $n_{\text{clad}}$. It is the value to use in the V number, and it need not equal the far-field figure on the datasheet.

Common questions

What is the numerical aperture of single-mode fiber?

Standard single-mode fiber (SMF-28 and other ITU-T G.652 fibers) is specified at NA 0.14, an acceptance half-angle of 8° in air. The Gaussian beam it emits has a $1\mathrm{/}{e}^2$ half-angle of about 0.09 rad at 1310 nm and 0.095 rad at 1550 nm, and that is the figure to use when choosing a lens to couple into it.

Can the numerical aperture be greater than 1?

Only when the light is in a medium of index above 1. In air $\sin \theta \le 1$ limits NA to 1; oil-immersion microscope objectives reach NA 1.40 to 1.45 because the cone sits in oil of index 1.515.

How is NA related to f-number?

For a lens imaging a distant object, $N\approx 1\mathrm{/}(2\text{NA})$: f/2 is NA 0.25 and f/1 is NA 0.5.

Is a higher NA always better?

A higher NA collects more light and focuses to a smaller spot, but the depth of focus falls as $1\mathrm{/}{\text{NA}}^2$, aberrations become harder to correct, and a fiber with a higher NA guides more modes for the same core size. High-NA single-mode fibers use smaller cores to stay single-mode, which matches them to small-mode sources such as laser diodes and chip waveguides but raises the splice loss to standard fiber.

References: Saleh & Teich, Fundamentals of Photonics (3rd ed., 2019), chapters on ray optics and fiber optics; Corning, SMF-28 Ultra Optical Fiber product information sheet (NA and mode field diameter); IEC 60793-2-10, sectional specification for category A1 multimode fibres (50 and 62.5 µm NA values); IEC 60793-1-43, Optical fibres, Part 1-43: Measurement methods and test procedures, numerical aperture (TIA-455-177 in the US); D. B. Murphy and M. W. Davidson, Fundamentals of Light Microscopy and Electronic Imaging, 2nd ed. (Wiley-Blackwell, 2012), for objective NAs; Born & Wolf, Principles of Optics (7th ed., 1999), Ch. 8 (diffraction and the Airy pattern).