# What is a V-number?

#### Editorial Team - GoPhotonics

Jul 28, 2023

The V-number, represented by V, is a dimensionless parameter used in optical fiber and waveguide technology. It is calculated as the ratio between the core radius of an optical fiber and its numerical aperture (NA). The V-number provides valuable information about the fiber's ability to capture and transmit light efficiently.

The V-number is also called a "normalized frequency" because it represents the frequency of light propagating through an optical fiber or waveguide in a normalized or dimensionless form. The V-number, or a normalized frequency, can be calculated using the formula:

where NA represents the numerical aperture and a denotes the radius of the fiber core.

For a step-index fiber,

Take,

Multiplying and dividing by 2 n1,

Substituting in the equation for V-number,

where a is the radius of the fiber core, λ is the free-space wavelength, n is the average refractive index of the core and cladding, i.e., n=(n1+n2)/2, and ∆ is the normalized index difference, that is

The V-number is directly proportional to the core radius, meaning that as the core radius increases, the V-number increases, and when the core radius decreases, the V-number decreases. Therefore, as the V-number is reduced by decreasing the core size, it eventually reaches a critical value of approximately 2.405.

When the V-number is less than 2.405, there is only one mode, the fundamental mode (LP01) that can propagate through the fiber core. As V is reduced further by reducing the core size, the fiber can still support the LP01 mode, but the mode extends increasingly into the cladding. The finite cladding size may then result in some of the power in the wave being lost. A fiber that is designed to allow only the fundamental mode to propagate at the required wavelength is called a single-mode fiber (SMF). Typically, single-mode fibers have a much smaller core radius and a smaller ∆ than multi-mode fibers (MMFs). If the wavelength λ of the source is reduced sufficiently, a single-mode fiber will become multimode as V will exceed 2.405; higher modes will also contribute to propagation. The cutoff wavelength λc above which the fiber becomes single mode is given by,

When the V-parameter increases above 2.405, the number of modes rises sharply. A good approximation to the number of modes M in a step-index multimode fiber is given by,

Significance of V-Number

• Light Gathering Efficiency: A higher V-number corresponds to a larger numerical aperture, indicating a greater ability to collect light from different angles. This is particularly crucial in applications such as fiber optic sensors and low-power optical systems where efficient light collection is essential.
• Modal Dispersion: The V-number influences the propagation characteristics of light within the fiber, affecting modal dispersion. Higher V-numbers can lead to increased modal dispersion, limiting the data-carrying capacity and transmission distances of the fiber.
• Fiber Coupling: The V-number determines the matching criteria between fibers and optical components. Proper alignment of V-numbers is essential for efficient coupling and minimizing losses at connection points.
• Fiber Type and Applications: Different V-numbers correspond to various fiber types, such as single-mode fibers (SMF) or multi-mode fibers (MMF). Each fiber type has unique characteristics and is suitable for specific applications, including telecommunications, data transmission, and fiber optic sensing.

Optimizing V-Number for Performance

• Single-Mode Fibers: For long-distance, high-bandwidth applications, single-mode fibers with lower V-numbers are preferred, as they minimize modal dispersion and provide better transmission characteristics.
• Multi-Mode Fibers: Multi-mode fibers with higher V-numbers are suitable for short-distance communication, such as local area networks (LANs) or data centers, where modal dispersion can be effectively managed.
• Specialty Fibers: Specialty fibers, designed for specific applications such as sensing or high-power transmission, may have customized V-number requirements to meet the unique demands of those applications.