Single-mode fibers are those fibers that allow only the propagation of a single value of incident angle through it.

Any ray of light which comes within the acceptance angle can propagate inside an optical fiber. But all the rays which enter the fiber cannot sustain inside it. This happens because the rays which propagate via total internal reflection through the fiber can interfere among themselves. So only certain allowed angles of incidence which can constructively interfere inside the fiber can sustain inside it. These specific angles correspond to the Modes of the system. 

The allowed angles inside a fiber or the modes of a fiber depend on various factors, like, the wavelength of the incident light inside the fiber, the angle at which the ray is incident at the core-cladding interface, and also the diameter of the core of the fiber.

Fiber is categorized as a single-mode or multimode based on a factor known as the V number or normalized frequency. It is a dimensionless quantity that determines the number of modes in an optical fiber. It is defined as,

We know that the numerical aperture of fiber is given by,

where n₁ and n₂ correspond to the refractive indices of the core and cladding of the fiber respectively and a is the radius of the core of the fiber.

The relative refractive index of any medium is the refractive index of that medium with respect to any other medium. Here we define the relative refractive index as the refractive index of the core with respect to the cladding refractive index. It is given by,

So rewriting NA in terms of the relative refractive index, we get,

So we can write, 

On generalizing,  

Here we take n as the core refractive index.

In a single-mode optical fiber, we transmit only one mode through the fiber. From the above equation, it is clear that the V number varies with the core diameter and the wavelength of the incident light. 

If the V number is less than 2.405, it is seen that optical fiber can support only single-mode and if the V number is greater, then the fiber can support multiple modes. So a single-mode fiber is achieved by keeping the V number less than 2.405. This is achieved by adjusting the core radius of the fiber. Core radius can be reduced such that the V number falls between 0 and 2.405 and thus we can obtain a single-mode fiber.

The total number of modes of an optical fiber is given by,

Single-mode operation reduces dispersion losses in the signal.