# Numerical Aperture

Author: the photonics expert Dr. Rüdiger Paschotta

Acronym: NA

Definition: the sine of the acceptance angle of an optical system or a waveguide

Categories: general optics, fiber optics and waveguides

Units: (dimensionless)

Formula symbol: NA

## Key questions:

The numerical aperture (NA) of an optical system (e.g. an imaging system) is a measure for its angular acceptance for incoming light. It is defined based on geometrical considerations and is thus a theoretical parameter which is calculated from the optical design. It cannot be directly measured, except in limiting cases with rather large apertures and negligible diffraction effects.

## Numerical Aperture of an Optical System

The numerical aperture of an optical system is defined as the product of the refractive index of the beam from which the light input is received and the sine of the maximum ray angle against the axis, for which light can be transmitted through the system based on purely geometric considerations (*ray optics*):

For the maximum incidence angle, it is demanded that the light can get through the whole system and not only through an entrance aperture.

### NA of a Lens

A simple case is that of a collimating lens:

The extreme rays are limited by the size of the lens, or in some cases somewhat less if there is a non-transparent facet.

It is often not recommended to operate a lens or its full area, since there could be substantial spherical aberrations. The numerical aperture, however, is a completely geometrical measure, which is not considering such aspects.

In the example case above, the numerical aperture of the lens is determined by its diameter and its focal length. Note, however, that a lens may not be designed for collimating light, but for example for imaging objects in a larger distance. In that case, one will consider rays coming from that object distance, and the obtained numerical aperture will be correspondingly smaller – sometimes even much smaller. This shows that the numerical aperture depends on the location of some object plane determined by the designer according to the intended use.

Some lenses are used for focusing collimated laser beams to small spots. The numerical aperture of such a lens depends on its aperture and focal length, just as for the collimation lens discussed above. The beam radius <$w_\textrm{lens}$> at the lens must be small enough to avoid truncation or excessive spherical aberrations. Typically, it will be of the order of half the aperture radius of the lens (or perhaps slight larger), and in that case (<$w_\textrm{lens} = D / 4 = {\rm NA} \cdot f / 2$>, with the beam divergence angle being only half the NA) the achievable beam radius in the focus is

$${w_{\rm{f}}} = \frac{{\lambda \;f}}{{\pi {w_0}}} = \frac{{4 \; \lambda \;f}}{{\pi D}} \approx \frac{{2\;\lambda }}{{\pi \;{\rm{NA}}}}$$where <$D$> is the aperture diameter, <$f$> the focal length and <$\lambda$> the optical wavelength. Note that the calculation is based on the paraxial approximation and therefore not accurate for cases with very high NA.

A somewhat smaller spot size may be possible with correspondingly larger input beam radius, if the performance is not spoiled by aberrations. In case of doubt, one should ask the manufacturer what maximum input beam radius is appropriate for a certain lens.

High-NA lenses (e.g. with NA above 0.6 or even 0.8) are required for various applications:

- In players and recorders of optical data storage media such as CDs, DVDs and Blu-ray Discs it is important to focus laser light to a small spot (pit) and receive light from such a spot.
- Lenses with high NA are also required for collimating laser beams which originate from small apertures. For example, this is the case for low-power single-mode laser diodes. When a lens with too low NA is used, the resulting collimated beam can be distorted (aberrated) or even truncated.

There is a weak dependence of numerical aperture on the optical wavelength due to the wavelength dependence of the focal length, which also causes chromatic aberrations.

### NA of a Microscope Objective

The same kind of considerations apply to microscope objectives. Such an objective is designed for operation with a certain working distance, and depending on the type of microscope with which it is supposed to be used, it may be designed for producing an image at a finite distance or at infinity. In any case, the opening angle on which the numerical aperture definition is based is taken from the center of the intended object plane. It is usually limited by the optical aperture on the object side, i.e., at the light entrance.

In many cases, the light input comes from air, where the refractive index is close to 1. The numerical aperture is then necessarily smaller than 1, but for some microscope objectives it is at least not much lower, for example 0.9. Other microscope objectives for particularly high image resolution are designed for the use of some immersion oil between the object and the entrance aperture. Due to its higher refractive index (often somewhat above 1.5), the numerical aperture can then be significantly larger than 1 (for example, 1.3).

The NA of a microscope objective is important particularly concerning the following aspects:

- It determines how bright the observed image can be for a given illumination intensity. Obviously, a high-NA objective can collect more light than one with a low numerical aperture.
- More importantly, the NA sets a limit to the obtained spatial resolution: the finest resolvable details have a diameter of approximately <$\lambda / (2 \; {\rm NA})$>, assuming that the objective does not produce excessive image aberrations.
- A high NA leads to a small depth of field: only objects within a small range of distances from the objective can be seen with a sharp image.

### Photographic Objectives

In photography, it is *not* common to specify the numerical aperture of an objective because such objectives are not thought to be used with a fixed working distance. Instead, one often specifies the aperture size with the so-called *f-number*, which is the focal length divided by the diameter of the entrance pupil. Usually, such an objective allows the adjustment of the f-number in a certain range.

## Numerical Aperture of an Optical Fiber or Waveguide

Although an optical fiber or other kind of waveguide can be seen as a special kind of optical system, there are some special aspects of the term *numerical aperture* in such cases.

In the case of a step-index fiber, one can define the numerical aperture based on the input ray with the maximum angle for which total internal reflection is possible at the core–cladding interface:

The numerical aperture (NA) of the fiber is the sine of that maximum angle of an incident ray with respect to the fiber axis. It can be calculated from the refractive index difference between core and cladding, more precisely with the following relation:

$${\rm{NA}} = \sqrt {{n_{{\rm{core}}}}^2 - {n_{{\rm{cladding}}}}^{\rm{2}}} $$Note that the NA is independent of the refractive index of the medium around the fiber. For an input medium with higher refractive index, for example, the maximum input angle will be smaller, but the numerical aperture remains unchanged.

For a homogeneous fiber *without* a core region, the surrounding medium (e.g. air) is effectively the cladding, and the NA is then typically rather high. In that case, of course, the refractive medium of the surrounding medium matters.

The equation given above holds only for straight fibers. For bent fibers, some modified equations have been suggested, delivering a reduced NA value, called an *effective numerical aperture* of the bent fiber. Proper references for such equations are missing at the moment.

For fibers or other waveguides not having a step-index profile, the concept of the numerical aperture becomes questionable. The maximum input ray angle then generally depends on the position of the input surface. Some authors calculate the numerical aperture of a graded-index fiber based on the maximum refractive index difference between core and cladding, using the equation derived for step-index fibers. However, some common formula in fiber optics involving the NA can then not be applied.

Light propagation in most optical fibers, and particularly in single-mode fibers, cannot be properly described based on a purely geometrical picture (with geometrical optics) because the wave nature of light is very important; diffraction becomes strong for tightly confined light. Therefore, there is no close relation between properties of fiber modes and the numerical aperture. Only, high-NA fibers tend to have modes with larger divergence of the light exiting the fiber. However, that beam divergence also depends on the core diameter. As an example, Figure 3 shows how the mode radius and mode divergence of a fiber depend on the core radius for fixed value of the numerical aperture. The mode divergence stays well below the numerical aperture.

In Figure 4 one can see that the angular intensity distribution somewhat extends beyond the value corresponding to the numerical aperture. This demonstrates that the angular limit from the purely geometrical consideration is not a strict limit for waves.

## Case Study: The Numerical Aperture of a Fiber: a Strict Limit for the Acceptance Angle?

The requirement of total internal reflection would seem to set a strict limit for the angular distributions of fiber modes. However, some modes are found to exceed that limit significantly. We investigate that in detail for single-mode, few-mode and multimode fibers.

For a single-mode fiber, the NA is typically of the order of 0.1, but can vary roughly between 0.05 and 0.4. (Higher values lead to smaller effective mode areas, smaller bend losses but to tentatively higher propagation losses in the straight form due to scattering.) Multimode fibers typically have a higher numerical aperture of e.g. 0.3. Very high values are possible for photonic crystal fibers.

A higher NA has the following consequences:

- For a given mode area, a fiber with higher NA is more strongly guiding, i.e. it will generally support a larger number of modes.
- Single-mode guidance requires a smaller core diameter. The corresponding mode area is smaller, and the divergence of the beam exiting the fiber is larger. The fiber nonlinearity is correspondingly increased. Conversely, a large mode area single-mode fiber must have a low NA.
- A high NA decreases the influence of random refractive index variations. (Fibers with very low NA may exhibit increased propagation losses.) Bend losses are also reduced with a higher NA; the fiber can be more strongly bent before bend losses become significant.
- If the fiber core becomes somewhat elliptical e.g. due to asymmetries in the fabrication, this leads to birefringence. That effect is stronger for fibers with high NA.
- The higher doping concentration of the fiber core (e.g. with germanium), as required for increasing the refractive index difference, may increase scattering losses due to lower optical homogeneity of the fiber core. The same can be caused by irregularities of the core/cladding interface, which are more important for a larger index difference.

There is only a weak dependence of numerical aperture on the optical wavelength due to chromatic dispersion. For example, the NA of a telecom fiber for the 1.5-μm region is not significantly different from that for the 1.3-μm region.

The relation between the numerical aperture and the beam divergence angle of an *output* beam emerging from a fiber end is generally not trivial:

- For a single-mode fiber, a larger NA will generally lead to stronger divergence, but that also somewhat depends on the mode shape.
- For a highly multimode fiber, the NA determines approximately the maximum possible angle of light emerging from the fiber end. However, the actual beam divergence angle also depends on the distribution of optical power on the fiber modes. The latter depends substantially on the launch conditions and may within the fiber change e.g. due to bending and inhomogeneities. When launching a beam with high beam quality mostly into the fundamental mode while avoiding bending and having no significant fiber imperfections, one get an output beam divergence far below the limit set by the NA.

## Numerical Aperture of a Laser Beam

Occasionally, the literature contains statements on the numerical aperture of a laser beam. This use of the term is actually discouraged because the numerical aperture should be considered to be based on ray optics, which cannot be applied here. Still, it can be relevant to understand what is meant with such a statement. Here, the numerical aperture is taken to be the tangent of the half-angle beam divergence. Within the paraxial approximation, the tangent can be omitted, and the result is <$\lambda / (\pi w_0)$> where <$w_0$> is the beam waist radius.

## More to Learn

Encyclopedia articles:

## Questions and Comments from Users

2020-04-14

What is the NA of a Nd:YAG laser rod with a certain diameter for total internal reflection?

The author's answer:

The critical angle for total internal reflection is <$\arcsin(1 / n_\rm{YAG})$>, but note that this is measured against the surface normal. The angle against the rod axis is <$\pi / 2 - \arcsin(1 / n_\rm{YAG})$>, and the sine of that gives you the NA.

2020-05-20

In your discussion on the NA of a lens above you provide the equation <$w_\textrm{lens} = D / 4 = NA \cdot f / 2$>. Why is there a 1/2 factor in the NA definition? Does this mean NA is normally full angle for lenses?

The author's answer:

No, that factor results from the assumption that the beam radius is chosen to be only half the NA in order to avoid substantial beam clipping and aberrations.

2020-05-20

Could you comment on the effect of the clear aperture of the lens (provided by manufacturers), and the impact of spherical aberrations for w = D/4 vs. D/3 (99.9% vs. 99% transmitted of a Gaussian beam)?

The author's answer:

The clear aperture defines the area to which the light should be restricted. That does not directly translate into a limit for Gaussian beams, which do not have a clear boundary. One will usually limit the Gaussian beam radius to be significantly below that aperture radius – e.g. 2/3 of it.

The impact of spherical aberrations depends both on the lens design and the whole optical setup.

2020-07-14

If I have a light beam entering an optical fiber at a slight angle to the optic axis, is all of this beam collected by the SM fiber (ignoring the 4% reflection)? The light path is still well within the numerical aperture of the fiber, but is nevertheless some of the light lost to the coating because the light is not absolutely on axis?

The author's answer:

Indeed, some of the light will then be lost, i.e., not get into the guided mode. You can fully launch into the fundamental mode only if you have the perfect amplitude profile, including flat phase fronts perpendicular to the core axis. Particularly for single-mode fibers, the numerical aperture is *not* providing an accurate criterion.

2020-07-17

Would it be possible to relate the mode diameter of a single mode photonic crystal fiber to its numerical aperture? In other terms, can we use the divergence angle of a Gaussian beam focused to be the same size of the fiber mode to infer the numerical aperture of the fiber?

The author's answer:

According to my knowledge, the numerical aperture of a photonic crystal fiber is not even clearly defined. Some people take it to be the sine of the half divergence angled of a mode, but I don't consider that as appropriate, particularly because the results for a simple step-index fiber do not agree. In science, we should not use conflicting definitions of the same term.

2020-07-21

Can the NA be 1.49?

The author's answer:

I suppose you mean the NA of an optical system. The answer: theoretically yes, practically no, it probably can't be that high.

2020-08-20

Can you comment on the output NA of a multimode fiber depending on the divergence angle of the input beam?

The author's answer:

The NA is a property of the fiber, i.e., it does not depend on launch conditions of the input beam.

However, your question is probably how the divergence of the light emerging from the fiber output depends on the divergence of the launched input beam. That divergence is limited by the fiber's NA, but can also depend on the launch conditions if it is a multimode fiber. Tentatively, you get a lower divergence out if you launch a low-divergence input beam, but this is not always strictly so. For example, higher-order fiber modes, which lead to larger divergence, may be obtained if you launch a low-divergence input beam at some anger against the fiber axis, or if mode mixing arises e.g. from bending of the fiber.

2020-10-21

Could you please comment on how the NA of a gradient index MM-fiber can be determined based on the GI profile?

The author's answer:

The numerical aperture of such a fiber is simply not defined. At least, I am not aware of a reasonable way of defining it.

2020-11-12

There exist microresonators based on silicon nitride as the core (n 2.0) and silica as the cladding (n 1.5). This leads to a NA around 1.32. What does it mean? No light could be sent in/out from/to air?

The author's answer:

If you had an end face, you could couple in light even with large incidence angles. However, you probably mean ring resonators, where you do not have an end face to couple in. The coupling is then usually done via evanescent waves, and that may be hard because the evanescent field decays so fast.

2020-12-08

Let us assume that I have a certain waveguide, where we know that the optimal coupling between laser and the waveguide can be reached when the beam waist at the focal point has a diameter of 38.4 μm. Can we somehow predict the divergence of the output ray at the exit of the waveguide?

The author's answer:

Particularly, if it is a single-mode waveguide, the optimum coupling tells us that there you have best matching to the guided mode of the waveguide. Then you will have a similar mode profile at the output of the waveguide. The divergence in air should then be roughly the same as that of the laser beam. This is not exact, however, since the shape of the waveguide mode may somewhat differ from that of the laser beam.

2021-01-10

Assuming a Gaussian laser beam with e.g. an initial NA of 0.1 passes through a lens with e.g. NA = 0.7, what is the resulting NA after the lens to describe the waist radius?

The author's answer:

Using the term *numerical aperture* for laser beams is actually discouraged. I assume, however, that you mean the divergence angle of a beam which is convergent on the way to the lens.

One might expect to obtain a tighter focus after the lens if the input light is already converging. However, the convergence angle of the light after the lens is limited by the NA of the lens. Trying to operate a lens in that way would mean that you violate its specifications, and the result would probably be substantial beam distortions, which may well prevent you from getting a tighter focus.

2021-01-28

Can the numerical aperture of a fiber change along its length?

The author's answer:

Normally, that should not be the case. It would require that the refractive index contrast between core and cladding changes, and that is unusual. It may happen that the core diameter somewhat varies, but that does not influence the numerical aperture.

2021-02-04

Is the NA expression still applicable to structures without a cladding, e.g. a homogeneous glass rod surrounded by air?

The author's answer:

Sure. You just get relatively high values of the numerical aperture in such cases.

2021-03-18

Does higher numerical aperture of an optical fiber mean that it can carry more data?

The author's answer:

In principle yes, if you apply the technique of mode division multiplexing: an increased numerical aperture gives you more modes and therefore in principle a potential for higher data rates.

If you don't use that technology, however, it is usually better to have a single-mode fiber. Therefore, the numerical aperture should not be too large.

2021-04-03

How does the fiber core diameter influence the numerical aperture?

The author's answer:

Not at all.

2021-04-22

If an aspherical lens with high NA (> 0.55) is used as focusing optics, how is the beam divergence angle defined after beam focus? Would it be possible to determine the angle directly via the NA?

The author's answer:

The NA is a property of the lens, while the beam divergence depends on other factors such as the beam radius before the lens. So you can generally not do that calculation. At most, you can calculate the maximum beam divergence angle with is possible without excessive aberrations.

2021-04-25

If a beam from a laser diode gets launched into a fiber with a NA (which fits to the fiber), will the output have the same NA?

The author's answer:

The output is a beam, and that cannot have a numerical aperture, but only a beam divergence. So your question should be whether the beam divergence is determined by the NA of the fiber.

The answer to that question is no – it generally the beam divergence also depends on the launch conditions, unless you have a single-mode fiber, where the output beam divergence is determined only by the fiber properties, but not specifically by the NA.

2021-06-10

Would it be possible that the NA is modified if a fiber is tapered?

The author's answer:

At a first glance, you may think it is not possible, since the refractive index contrast stays the same – but it can actually change into different ways:

- The refractive index profile may change by diffusion. That would tentatively reduce the NA.
- If a fiber is strongly tapered, the original small index contrast in the class may no longer be effective to confine the light, and the light field will extend out to the glass–air interface, which will then take over the waveguiding function. In that case, the NA becomes very large due to the low refractive index of air.

2021-10-29

Let's assume that we have a diode laser with fiber output, which has a certain NA and core diameter. How to determine its BPP or M^{2}?

The author's answer:

If it is a single-mode fiber, the <$M^2$> value will be close to 1, somewhat dependent on the mode shape, which can be calculated from the index profile.

In the case of a multimode fiber, the problem is that the beam quality will depend on the unknown power distribution over the fiber modes. The article on multimode fibers contains a formula with which we estimate <$M^2$>.

2021-11-07

How to control the divergence angle from a multimode optical fiber, by changing light launched condition or selecting fibers with different NA.

The author's answer:

A smaller NA can reduce the divergence angle (set a limit to it), but may make it more difficult to get enough light into the fiber. Alternatively, you try to inject light with smaller divergence, while also avoiding tight bending.

2021-12-02

I have a fiber with the following specs: FC/APC, SMF4/125/250/900um. What its NA would be?

The author's answer:

That information is not contained in the given specification.

2022-01-06

How does the NA change as one moves out of the nominal operating wavelength range? I have got a telecom fibre for 1300–1600 nm (NA = 0.14) and launch visible light into it.

The author's answer:

For shorter-wavelength light, the refractive indices of core and cladding will generally increase, and the NA will presumably also increase somewhat. For example, for germanosilicate fibers that would be the case.

2022-09-30

How is the output beam divergence influenced when you have two fibers of different NAs coupled to one another? Is the beam divergence limited by the smallest NA in a fiber coupled system?

The author's answer:

In principle yes, unless the second fiber has a larger NA and some mode mixing occurs, e.g. due to bending.

2022-10-07

How to calculate the NA of a step index profile fiber if the core and cladding refractive index is not flat along radial direction?

The author's answer:

You cannot calculate the NA for that case. It is only defined for a step index fiber, meaning the core and cladding index are constant. But you may of course take the average values over some range to get an approximation.

2022-12-06

The comment that higher NA decreases optical losses seems misleading as for a multimode fiber higher NA leads to optical path increase inside the core material.

The author's answer:

I see, the idea is that light may do more of a zig-zag path, which is longer than a straight path through the fiber. However, that argument is questionable; for example, consider that light in any guided fiber mode does *not* perform a zig-zag path; it has wavefronts perpendicular to the fiber axis and propagates strictly in that direction. Only for light rays (which are anyway a problematic concept for fibers), you can imagine an increased path length, and that would be a quite weak effect. Anyway, other aspects as explained in the article are definitely more important.

2022-12-11

If I need to collimate the beam from a multimode fiber, should I use an aspheric lens with a NA matched that of the MM fiber?

Does the choice of collimation lens depend on the launch condition?

The author's answer:

That's a good approach. With that NA, it should work for any launch conditions. For favorable launch conditions, it can work even with a lower lens NA.

2023-01-18

Is there any reason an extrusion process for a plastic optical fiber would expect to change index of refraction of either core or cladding materials? It seems that if core and cladding IORs are both known, that calculating/predicting NA for a plastic multi-mode fiber should be highly consistent. When measuring finished samples, however, it seems that the numerical aperture fluctuates about 20%. What would be the reasons for fluctuations in measured NA? Could it be that the test system isn’t gauged properly, or have you seen in your experience that NA values will vary even though the theoretical value based on IORs should be consistent?

The author's answer:

I suspect that the reason is some mixing of the materials, which effectively reduces the refractive index contrast.

2023-05-03

If I want to transmit both 1050 nm and 1550 nm through the same fiber, how do NA, MFD and collimation change for the two wavelengths?

The author's answer:

Let's assume that it is a multimode fiber:

- The NA will be not too different for the two wavelengths.
- The modes will be smaller for the shorter wavelength, but you don't need to focus your input beam more tightly.
- Collimation will also work similarly.

2023-07-07

How to calculate a fiber bundle's NA?

The author's answer:

You can just take the NA of one fiber (assuming that all fibers are having the same NA). The angular limitations of the bundle are the same as those for each fiber.

2023-11-03

Why do we define the numerical aperture of a fiber in this way? Is there any reason that we call it numerical aperture? Does it have a certain relation with the numerical aperture of a lens?

The author's answer:

Obviously, the definition leads to a useful quantity.

I am not sure about the origin of the wording. “Numerical” may just relate to “quantitative”, and “aperture” is a kind of limiting device – in this context, limiting not concerning spatial position, but concerning propagation angles. These aspects also apply to the numerical aperture of a lens.

2023-11-14

Does the numerical aperture of the fiber need to match the numerical aperture of a collimator and objective lens? Does the mismatch of numerical aperture result in higher divergence of the output beam?

The author's answer:

For efficient launching, the NA of the collimator should be at least as large as that of the fiber. A larger value won't hurt. A too small value leads to imperfect collimation, including an increased beam divergence.

2024-04-17

Is there an equation relating the numerical aperture and far-field intensity distribution for a single-mode fiber, like figure 4, or does it need to be modelled in detail for each case?

The author's answer:

Presumably, you wonder whether the NA determines the beam divergence in free space. The answer is no for single-mode fibers. See also our case study on the numerical aperture.

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2020-03-31

What is the equation that relates the divergence of the laser beam to the fiber core (105 μm) and the numerical aperture of the optical fiber?

The author's answer:

As an rule of thumb, the half-angle beam divergence in radians should not exceed the NA of the fiber, regardless of the core diameter. Then you should be able to get most of the light launched, assuming that is also all hits the fiber core at the interface.