Numerical Aperture

The numerical aperture (NA) is a dimensionless quantity that measures the range of angles over which an optical system can accept or emit light. It is defined as NA = n · sin(θ_max), where n is the refractive index of the medium from which light is received and θ_max is the maximum half-angle of the cone of light that can pass through the system.

For a lens focusing a collimated laser beam, a higher numerical aperture allows for a smaller focused spot size. The achievable beam radius in the focus is approximately w_f ≈ 2λ / (π · NA), where λ is the optical wavelength.

The NA of a microscope objective sets a fundamental limit on the spatial resolution. The finest resolvable details have a diameter of approximately λ / (2 · NA), meaning a higher NA allows for resolving smaller features.

A microscope objective can achieve an NA greater than 1 by using an immersion oil with a refractive index significantly higher than 1 (air) between the objective and the object. Since NA is proportional to the refractive index of this medium, using oil (e.g., with n ≈ 1.5) allows the NA to exceed 1.

In photography, the f-number is used because photographic objectives are not designed for a fixed working distance. The f-number, defined as the focal length divided by the entrance pupil diameter, is a more practical measure of the aperture size in such variable-focus systems.

For a step-index fiber, the numerical aperture is determined by the refractive indices of its core (n_core) and cladding (n_cladding). The formula is NA = sqrt(n_core² - n_cladding²), derived from the maximum angle for which total internal reflection occurs at the core–cladding interface.

A higher NA in a fiber leads to stronger light guidance. This results in a smaller required core diameter for single-mode operation, reduced bend losses, and an increased number of supported modes in multimode fibers. However, it can also increase nonlinearity and birefringence.

The concept of numerical aperture is based on geometrical ray optics. This approach is not suitable for describing the propagation of laser beams, where wave optics and diffraction are dominant. For laser beams, the term beam divergence is the appropriate measure of the beam's angular spread.

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2020年03月31日

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.

2020年04月14日

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 ($\arcsin(1 / n_\textrm{YAG})$), but note that this is measured against the surface normal. The angle to the rod axis is ($\pi / 2 - \arcsin(1 / n_\textrm{YAG})$), and the sine of that gives you the NA.

2020年05月20日

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 to avoid substantial beam clipping and aberrations.

2020年05月20日

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年07月14日

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年07月17日

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年07月21日

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年08月20日

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年10月21日

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年11月12日

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年12月08日

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年01月10日

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年01月28日

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年02月04日

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年03月18日

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年04月03日

How does the fiber core diameter influence the numerical aperture?

The author's answer:

Not at all.

2021年04月22日

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年04月25日

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年06月10日

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

The author's answer:

At 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 tend to 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年10月29日

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 M²?

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年11月07日

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年12月02日

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年01月06日

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年09月30日

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年10月07日

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年12月06日

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年12月11日

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年01月18日

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年05月03日

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年07月07日

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 have the same NA). The angular limitations of the bundle are the same as those for each fiber.

2023年11月03日

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年11月14日

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年04月17日

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.

2024年09月20日

Does the NA correspond to 1/e² (13.5%) beam width or 1% beam width?

The author's answer:

Neither of those: It is a purely geometric quantity which applies to light rays.

2025年01月10日

If we have a focusing lens with low NA, for example 0.1, and the fiber has high NA, for example 0.7, as we want to collimate the diverging beam after passing the fiber, how should we determine the NA of the collimation lens? How output divergence angle is influenced by the difference of the NA of the focusing lens and the fiber? Is there any equations regarding this?

The author's answer:

For collimating the output of a fiber, the lens NA should be at least that of the fiber.

By the way, an NA of 0.7 appears to be unrealistically high for a fiber.

2025年08月18日

If a lens has a numerical aperture (NA) of 0.26 and the multimode fiber has an NA of 0.22, what is the efficiency of coupling light between them?

The author's answer:

That is not clear from these data, but at least the lens is good enough for fairly efficient launching if the beam parameters are right.

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