V-number
Author: the photonics expert Dr. Rüdiger Paschotta (RP)
Definition: a normalized frequency parameter, which determines the number of modes of a step-index fiber
Category: 
Related: Number of Modes of a Highly Multimode Fiberfibersstep-index fibersfiber corenumerical aperturesingle-mode fibersmultimode fibersAbbe number
Units: (dimensionless number)
Formula symbol: ($V$)
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DOI: 10.61835/8ty Cite the article: BibTex BibLaTex plain textHTML Link to this page! LinkedIn
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What is the V-Number of a Fiber?
The V-number is a dimensionless parameter which is often used in the context of step-index fibers (but normally not usable for other kinds of refractive index profiles). It is defined as
$$V = \frac{{2\pi }}{\lambda } r_\textrm{co} \;{\textrm{NA}} = \frac{{2\pi }}{\lambda } r_\textrm{co} \sqrt{n_{\textrm{co}}^2 - n_{\textrm{cl}}^2} $$where ($\lambda$) is the vacuum wavelength, ($r_\textrm{co}$) is the radius of the fiber core, and NA is the numerical aperture. Of course, the V-number should not be confused with some velocity ($v$), e.g. the phase velocity of light, and also not with the Abbe number, which is also sometimes called V-number.
The V-number can be interpreted as a kind of normalized optical frequency. (It is proportional to the optical frequency, but rescaled depending on waveguide properties.) It is relevant for various essential properties of a fiber:
- For ($V$) values below ≈ 2.405, a fiber supports only one mode per polarization direction (→ single-mode fibers).
- Multimode fibers can have much higher V-numbers. For large values, the number of supported modes of a step-index fiber (including polarization multiplicity) can be calculated approximately as
- The V-number determines the fraction of the optical power in a certain mode which is confined to the fiber core. For single-mode fibers, that fraction is low for low ($V$) values (e.g. below 1), and reaches ≈ 90% near the single-mode cut-off at ($V$) ≈ 2.405.
- There is also the so-called Marcuse equation for estimating the mode radius of a step-index fiber from the V-number; see the article on mode radius.
- A low V-number makes a fiber sensitive to micro-bend losses and to absorption losses in the cladding. However, a high V-number may increase scattering losses in the core or at the core–cladding interface.
The formula for the estimated number of guided modes can be generalized for arbitrary index profiles, where effectively we use an average value of ($V^2$) in the fiber core:
$$M \approx \frac{2\pi}{\lambda^2} \int {\left(n^2 - n_{\textrm{cl}}^2\right) \: \textrm{d}A}$$(again including polarization multiplicity). This is shown in a case study:
Number of Modes of a Highly Multimode Fiber
We seek a simple equation for estimating for the number of modes of a highly multimode fiber, which can be applied to fiber designs with arbitrary shapes of the refractive index profile. Instead of applying complicated mathematics, we build a hypothesis and subject that to multiple numerical tests.
Effective V-numbers
For certain types of photonic crystal fibers, an effective V-number can be defined, where ($n_\textrm{cladding}$) is replaced by an effective cladding index. The same equations as for step-index fibers can then be used for calculating quantities such as the single-mode cut-off, mode radius and splice losses.
Frequently Asked Questions
This FAQ section was generated with AI based on the article content and has been reviewed by the article’s author (RP).
What is the V-number of an optical fiber?
The V-number is a dimensionless quantity that characterizes an optical fiber's properties. It is calculated from the vacuum wavelength ($\lambda$), the fiber core radius ($r_\textrm{co}$), and the numerical aperture (NA) using the formula ($V = (2\pi / \lambda) r_\textrm{co} \cdot \textrm{NA}$).
What is the significance of the V-number?
The V-number determines key properties of a fiber, such as the number of supported guided modes. For example, a standard step-index fiber is a single-mode fiber if its V-number is below approximately 2.405; it also influences the mode radius and bend losses.
How many modes does a multimode fiber have?
For a multimode fiber with a large V-number, the total number of supported modes (including two polarization directions) can be estimated with the formula ($M \approx V^2 / 2$).
Can the V-number be used for any type of optical fiber?
The standard V-number definition is for step-index fibers. For other fiber types, such as photonic crystal fibers, a generalized or effective V-number is sometimes used to apply similar concepts.
Suppliers
Sponsored content: The RP Photonics Buyer's Guide contains 38 suppliers for multimode fibers. Among them:
Edmund Optics offers a variety of fiber optics, including jacketed and unjacketed optical grade or communications grade optical fibers. Optical grade optical fiber is ideal for general industrial lighting or short distance data transmission. It is designed for optimal visible light transmission for digital or analog links. Jacketed fiber has increased durability while decreasing stray light. Edmund Optics also offers optical fiber components, including patchcords, collimators, faceplates and image conduits, fiber connectors, and the tools needed for cutting or stripping fibers.
Exail (formerly iXblue) offers a wide range of multimode specialty optical fibers, either for lasers and amplifiers or for sensing applications. Hundreds of fiber designs are available from stock on our dedicated e-store. Custom versions are also available. More specifically, multimode would consist of step index or graded index fibers with control of temporal dispersion. Most of the references are also available in radiation resistant versions, either for nuclear environment or for space missions.
Guiding Photonics produces low-loss multi-mode fibers with a range of internal diameters that are ideally suited for mid-IR and/or high-power laser beams. In addition to mid-IR, various option are available for UV through far-IR wavelengths.
LVF offers a large range of fluoride multimode optical fibers, for mid infrared applications:
- ZrF4 (fluorozirconate) multimode fibers with transparency ranging from 0.3 to 4.5 µm (typical optical loss at 2.5 µm: < 10 dB/km)
- InF3 (fluoroindate) multimode fibers with transparency ranging from 0.3 to 5.5 µm (typical optical loss at 3.5 µm: < 10 dB/km)
- GeO2 (germanate) multimode fibers are qualified for high power handling around 2.7–3.0 µm (Er:YAG and Er:YSSG medical lasers).
LVF fluoride multimode fibers are the most transparent fibers on the market in the mid-infrared 2–5 µm band.
Fibercore's multimode (MM) fibers are available in graded index (GRIN) variants with 50 μm and 62.5 μm germanium doped cores, along with bend insensitive options. In addition to these germanium doped cores, Fibercore offers pure silica core GRIN fibers specifically designed for long term use in downhole hydrogen environments as experienced in the oil & gas industry, and applications where resistance to radiation effects is required.
Bibliography
| [1] | A. W. Snyder and J. D. Love, Optical Waveguide Theory, Chapman and Hall, London (1983) |
(Suggest additional literature!)
Questions and Comments from Users
2020-10-21
In the case of Photonic crystal fibers (PCFs) do we use the same concept of V-number for figuring out the number of modes?
The author's answer:
Strictly speaking, the numerical aperture and V-number are not defined for a photonic crystal fiber. Further, the formation of modes is influenced by other physical aspects, e.g. photonic bandgap effects. At most, you can for some of those fibers get a rough estimate of the number of modes, based on an intelligent guess how to assign a reasonable V-number to such a fiber.
2022-12-01
What if the fiber is not cylindrical? How to calculate the V-number of a fiber with irregular core shape, for example, a rectangular shape?
The author's answer:
The V-number is not defined for such cases — only for ordinary step-index fibers.
2020-06-14
What is the relation between V-number and power flow in cladding?
The author's answer:
Tentatively, for fibers with low V-number a larger fraction of the total optical power propagates in the fiber cladding. The numerical value, however, depends on the details, not just the V-number.