Note: this box searches only for keywords in the titles of articles, and for acronyms. For full-text searches on the whole website, use our search page.

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 <$M^2$>.

There would be little to do in a microbiology laboratory without a microscope, because the objects of our attention (bacteria, fungi, and other single celled creatures) are otherwise too small to see. Microscopes are optical instruments that permit us to view the microbial world. Lenses produce the magnified images that allow us to visualize the form and structure of these tiniest of living beings.

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.

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.

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?

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?

Working f-number

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.

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.

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:

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.

For a compound microscope, the optical path leading to a detectable image involves two lenses – the objective lens and the ocular lens. The objective lens magnifies the object and creates a real image, which will appear to be 4, 10, 40, or 100 times larger than the object actually is, depending on the lens used. The ocular lens further magnifies the real image by an additional factor of 10, to produce a vastly larger virtual image of the object when viewed by you.

Once you’ve made the observations using the prepared slide, obtain a glass slide and a sterile swab. Collect a sample from the container of commercially prepared yogurt by swirling the swab in the yogurt, then scraping of the excess on the edge of the container. Smear this over the surface of the slide, making sure that you leave only a thin film of yogurt on the surface. Make a second smear from the container of freshly prepared homemade yogurt, if available. Allow both smears to air dry, and then heat fix them.

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 ability of a lens to resolve detail is ultimately limited by diffraction of light waves, and therefore, the practical limit of resolution for most microscopes is about 0.2 µm. Therefore, it would not be practical to try to observe objects smaller than 0.2 µm with a standard optical microscope. In addition, cells of all types of organisms lack contrast because many cellular components refract light to a similar extent. This is especially true of bacteria. To overcome this problem and increase contrast, biological specimens may be stained with selective dyes.

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.

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.

Enter input values with units, where appropriate. After you have modified some inputs, click the “calc” button to recalculate the output.

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.

Light from an illuminator (light source) below the stage is focused on the object by the condenser lens, which is located just below the stage and adjustable with the condenser adjustment knob. The condenser focuses light through the specimen to match the aperture of the objective lens above, as illustrated in Figure 2.

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).

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.

Numerical aperture

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

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.

Microbiology: A Laboratory Experience Copyright © by Holly Ahern is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.

Once the sample is heat fixed, stain it with safranin. This is a pinkish-red colored stain, and all cells (both bacterial and your mouth cells) will take up the stain and increase the contrast in the image.

Obtain a prepared slide labeled “mouth smear.” On this slide you will see large cells with a nucleus, clearly visible with both the low power and high power objective lenses. These are squamous epithelial cells that form the outermost layer of the oral mucosa. At high power, you should start to see small cells on the surface of the larger epithelial cells. With the oil immersion objective lens, you will be able to tell the smaller cells are bacteria.

Here is a final consideration related to objective lenses and magnification. Look at the lenses on your microscope, and note that as the magnification increases, the length of the lens increases and the lens aperture decreases in size. As a result, you will need to adjust your illumination to compensate for a darkening image. There are essentially three ways to vary the brightness; by increasing or decreasing the light intensity (using the on/off knob), by moving the condenser lens closer to or farther from the object using the condenser adjustment knob, and/or by opening/closing the iris diaphragm. Don’t be afraid to experiment to create the best image possible.

Obtain a prepared slide labeled “yogurt smear” and view it with the microscope. The milk proteins in the yogurt will be visible as lightly stained amorphous blobs. By now you should have a pretty good idea of what bacteria look like, so locate and focus on areas where you see bacterial cells.

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.

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.

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):

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.

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

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.

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.

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.

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 lens with highest magnifying power is the oil immersion lens, which achieves a total magnification of 1000X with a resolution of 0.2 µm. This lens deserves special attention, because without it our time in lab would be frustrating.

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.

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.

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.

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?

So the microscope makes small cells look big. But why can’t we just use more or different lenses with greater magnifying power until the images we see are really, really big and easier to see?

Appropriate use of the condenser, which on most microscopes includes an iris diaphragm, is essential in the quest for a perfect image. Raising the condenser to a position just below the stage creates a spotlight effect on the specimen, which is critical when higher magnification lenses with small apertures are in use. On the other hand, the condenser should be lowered when using the scanning and low power lenses because the apertures are much larger, and too much light can be blinding. For creating the best possible contrast in the image, the iris diaphragm can be opened to make the image brighter or closed to dim the light. These adjustments are subjective and should suit the preferences of the person viewing the image.

Here you can submit questions and comments. As far as they get accepted by the author, they will appear above this paragraph together with the author’s answer. The author will decide on acceptance based on certain criteria. Essentially, the issue must be of sufficiently broad interest.

The answer is resolution. Consider what happens when you try to magnify the fine print from a book with a magnifying glass. As you move the lens away from the print, it gets larger, right? But as you keep moving the lens, you notice that while the letters are still getting larger, they are becoming blurry and hard to read. This is referred to as “empty magnification” because the image is larger, but not clear enough to read. Empty magnification occurs when you exceed the resolving power of the lens.

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.

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.

F number to naformula

With a bright-field microscope, images are formed as a result of the interplay between light waves, the object, and lenses. How images of biological objects are formed is actually more physics than biology. Since this isn’t a physics course, it’s more important to know how to create exceptional images of the object than it is to know precisely how those images are formed.

f-number calculator

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.

Within the past few years, positive health benefits have been correlated with eating fermented foods containing “live” cultures. Although several types of bacteria are known to ferment milk and produce yogurt, two genera in particular, Lactobacillus and Bifidobacterium, have been singled out as promoting good digestive health and a well-balanced immune response. Both of these are bacilli arranged in pairs or short chains. Streptococcus spp., which are cocci arranged in chains, are also usually involved in the process of making the milk into yogurt, but these are not directly associated with positive health benefits to the person who eats the yogurt.

Let’s say you wanted to look at cells of Bacillus cereus, which are rod-shaped cells that are about 4 µm long. If you were observing B. cereus with a microscope using the high power lens, how big would the cells appear to be when you look at them? ___________________________

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.

Exail (formerly iXblue) offers a wide range of specialty optical fibers for lasers and amplifiers. We master erbium, erbium/ytterbium, ytterbium, thulium, holmium, thulium/holmium, neodymium, dysprosium, and phosphorous gain media. PM version are available, and Large Mode Area (LMA) or Very Large Mode Area (VLMA) versions as well. Depending of the requirement, single clad fibers are available for core pumping, double clad fibers for clad pumping. Triple clad and all glass structures are also available.

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.

This brings us to two additional concepts related to microscopy—working distance and parfocality. Working distance is how much space exists between the objective lens and the specimen on the slide. As you increase the magnification by changing to a higher power lens, the working distance decreases and you will see a much smaller slice of the specimen. Also, once you’ve focused on an object, you should not have to make any major adjustments when you switch lenses, because the lenses on your microscope are designed to be parfocal. This means that something you saw in focus with the low power objective should be nearly in focus when you switch to a high power objective, or vice versa. Thus, for viewing any object and regardless of what lens you will ultimately use to view it, the best practice is to first set the working distance with a lower power lens and adjust it to good focus using the coarse focus knob. From that point on, when you switch objectives, only a small amount of adjustment with the fine focus knob should be necessary.

Resolution is often thought of as how clearly the details in the image can be seen. By definition, resolution is the minimum distance between objects needed to be able to see them as two separate entities. It can also be thought of as the size of the smallest object that we can clearly see.

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?

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?

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.

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?

numerical aperturetof-number calculator

Once the sample(s) are heat fixed, stain them with crystal violet. This is a purple colored stain, and although both the milk proteins and cells will stain this color, the milk stains faintly and the bacteria will appear dark purple. Keep in mind that the probiotic bacteria are bacilli. Below, sketch a representative field as seen with the oil immersion objective for each of the yogurt samples.

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

It is important to remember that you must use a drop of oil whenever you use the oil immersion objective or you will not achieve maximum resolution with that lens. However, you should never use oil with any of the other objectives, and you should be diligent about wiping off the oil and cleaning all of your lenses each time you use your microscope, because the oil will damage the lenses and gum up other parts of the instrument if it is left in place.

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.

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?

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.

Numerical aperture of optical fiber

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?

Note: the article keyword search field and some other of the site's functionality would require Javascript, which however is turned off in your browser.

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?

Together we will review how to effectively achieve an exceptional image using a standard optical microscope. This will include not only locating and focusing on the object, but also using the condenser lens and iris diaphragm to achieve a high degree of contrast and clarity.

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

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 same thing happens as the light passes through the glass slide into the air space between the slide and the lens. The light will be refracted away from the lens aperture. To remedy this, we add a drop of oil to the slide and slip the oil immersion objective into it. Oil and glass have a similar refractive index, and therefore the light bends to a lesser degree and most of it enters the lens aperture to form the image.

Move your stage so you observe 10 different microscope fields. Keep track of the number of different types of bacterial cells you encounter during your survey, and record that information below:

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.

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.

Using our advertising package, you can display your logo, further below your product description, and these will been seen by many photonics professionals.

Add to your illustration the bacterial cells which you should see on or near the larger larger cheek cells. Try to keep the size of the bacterial cells to scale with the size of the cheek cell.

If you are new to microscopy, you may initially feel challenged as you try to achieve high quality images of your specimens, particularly in the category of “Which lens should I use?” A simple rule is: the smaller the specimen, the higher the magnification. The smallest creatures we observe are bacteria, for which the average size is a few micrometers (μm). Other microscopic organisms such as fungi, algae, and protozoa are larger, and you may only need to use the high power objective to get a good view of these cells; in fact, using the oil immersion objective may provide you with less information because you will only be seeing a part of a cell.

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.

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.

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.

Yogurt is produced when lactic acid (homolactic) bacteria that naturally occur in milk ferment the milk sugar lactose and turn it into lactic acid. The lactic acid accumulates and causes the milk proteins to denature (“curdle”) and the liquid milk becomes viscous and semi-solid.

Observe your mouth smear with the microscope. When you get to the oil immersion objective, locate and focus on a single cheek cell. As you did with the prepared slide, sketch the larger cheek cell in the circle provided and label the membrane and nucleus . Add the bacterial cells to your sketch, and try to keep the size scale accurate.

When the light waves that have interacted with the specimen are collected by the lenses and eventually get to your eye, the information is processed into dark and light and color, and the object becomes an image that you can see and think more about.

Light waves that pass through and interact with the object may speed up, slow down, or change direction as they travel through “media” (such as air, water, oil, cytoplasm, etc.) of different densities. For example, light passing through a thicker or denser part of a specimen (such as the nucleus of a cell) may be reflected or refracted (“bend” by changing speed or direction) more than those waves passing through a thinner part. This makes the thicker part appear darker in the image, while the thinner parts are lighter.

The magnifying power of each lens is engraved on its surface, followed by an “X.” In the table below, find the magnification, and then calculate the total magnification for each of the four lenses on your microscope.

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.

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?

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.

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?

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?

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 resolving power of this lens is dependent on “immersing” it in a drop of oil, which prevents the loss of at least some of the image-forming light waves because of refraction. Refraction is a change in the direction of light waves due to an increase or decrease in the wave velocity, which typically occurs at the intersection between substances through which the light waves pass. This is a phenomenon you can see when you put a pencil in a glass of water. The pencil appears to “bend” at an angle where the air and water meet (see Figure 3). These two substances have different refractive indices, which means that light passing through the air reaches your eye before the light passing through the water. This makes the pencil appear “broken.”

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.

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:

Locate and focus on a single squamous epithelial cell with obvious bacteria on its surface. Create a sketch of the “cheek” cell (as squamous epithelial cells are sometimes called) in the circle provided. Then label the cell membrane, cytoplasm, and nucleus of the “cheek” cell, which should be easily observed.

Our large mode area photonic crystal fibers are designed for diffraction-limited high-power delivery. The large mode area prevents nonlinear effects and material damage. With standard fibers, you trade large mode areas for single-mode operation. With our large mode area fibers, you get single-mode operation in a wide range of wavelengths. Also available in a polarization-maintaining version.

3. When you are finished with the microscope, check the stage to make sure that you don’t leave a slide clipped in the stage. Make sure to switch the microscope OFF before you unplug it. Gently wrap the cord around the base and cover your microscope with its plastic cover.

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

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.

f-number formula

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.

By submitting the information, you give your consent to the potential publication of your inputs on our website according to our rules. (If you later retract your consent, we will delete those inputs.) As your inputs are first reviewed by the author, they may be published with some delay.

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.

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.

To use this important piece of equipment properly, it is helpful to know how a microscope works. A good place to begin is to learn the name and function of all of the various parts, because when we talk about the ways to improve microscopic images, terms like “ocular lenses” and “condenser” always come up.

We’ll start by looking at a prepared slide of a “rectal smear,” which is quite literally a smear of feces on a slide stained with a common method called the Gram stain. You will observe several different types of bacterial cells in this smear that will appear either pink or purple. While the main purpose of this is to develop proficiency in use of the oil immersion objective lens, it also provides the opportunity to look at bacteria, observe the differences in cell shapes and sizes, and note that when Gram stained they turn out to be either purple or pink.

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?

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)?

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.

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 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?

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.

Numerical aperture formula

The encyclopedia covers this topic well: with a long overview article on laser material processing and specialized articles on

Please do not enter personal data here. (See also our privacy declaration.) If you wish to receive personal feedback or consultancy from the author, please contact him, e.g. via e-mail.

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.

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

2. Clean all of the lenses with either lens paper or Kimwipes (NOT paper towels or nose tissues) BEFORE you use your microscope, AFTER you are done, and before you put it away.

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?

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.

Once you’ve looked at the prepared slide, obtain a glass slide and a sterile swab. Collect a sample of your oral mucosa by gently rubbing the swab over the inside of your cheek. Smear the swab over the surface of the slide (this is known as making a “smear” in microbiology). Allow the smear to dry, and then heat fix by passing the slide through the flame of a Bunsen burner, as demonstrated. Discard the swab in the biohazard waste.

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

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.

1. Carry the microscope to your lab table using two hands, and set it down gently on the bench. Once placed on the bench, do not try to slide it around on its base, because this is extremely jarring to the optical system.

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 microscope you’ll be using in lab has a compound system of lenses. The objective lens magnifies the object “X” number of times to create the real image, which is then magnified by the ocular lens an additional 10X in the virtual image. Therefore, the total magnification, or how much bigger the object will actually appear to you when you view it, can be determined by multiplying the magnification of the objective lens by 10.

Based on the picture of the binocular, compound light microscope in Figure 1, match the name of the major part (listed below) with its location on the microscope, and give a very brief description of what each is used for:

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.

The human mouth is home to numerous microbes, which persist no matter how many times you brush your teeth and use mouthwash. Since these microbes generally inhabit the surface layers of the oral mucosa, we humans have evolved ways to keep their numbers under control, including producing antibacterial chemicals in saliva and constantly turning over the outer layer of epithelial cells that line the inside of the mouth.

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.