Optical fibers guide light by total internal reflection in a higher-index core surrounded by a lower-index cladding. A simple and very useful parameter for step-index fibers is the normalized frequency, usually called the V-number:
\[
V=\frac{2\pi a}{\lambda}\sqrt{n_1^2-n_2^2},
\]
where \(a\) is the core radius, \(\lambda\) is the vacuum wavelength, \(n_1\) is the core refractive index, and \(n_2\) is the cladding refractive index. The factor \(\sqrt{n_1^2-n_2^2}\) is closely related to the numerical aperture of the fiber.
The V-number determines whether a step-index fiber behaves as single-mode or multimode. A standard cutoff value is approximately
\[
V_c \approx 2.405.
\]
If \(V < 2.405\), only the fundamental guided mode is supported in the idealized step-index model, so the fiber is treated as single-mode. If \(V > 2.405\), higher-order modes can also propagate, so the fiber is multimode.
For the sample values \(a=5\ \mu\text{m}\), \(\lambda=1550\ \text{nm}\), \(n_1=1.46\), and \(n_2=1.45\), the index factor is
\[
\sqrt{1.46^2-1.45^2}\approx 0.171.
\]
Then
\[
V=\frac{2\pi(5\times10^{-6})}{1550\times10^{-9}}(0.171)\approx 3.47.
\]
So for those exact values, the fiber is actually above the usual single-mode cutoff. This means the quoted sample result \(V\approx 2.0\) is not consistent with \(a=5\ \mu\text{m}\), \(\lambda=1550\ \text{nm}\), and \(\Delta n = 0.01\) interpreted as \(n_1-n_2=0.01\). The calculator above follows the standard formula directly, so it will report the mathematically consistent value.
Another useful preview quantity is the guided-wave speed:
\[
v=\frac{c}{n_{\text{eff}}},
\]
where \(n_{\text{eff}}\) is the effective index of the guided mode and \(c\) is the speed of light in vacuum. Since \(n_{\text{eff}}\) usually lies between \(n_2\) and \(n_1\), the guided-wave speed lies below \(c\) and depends on modal confinement. Different modes can have different effective indices, which is part of why multimode fibers can suffer from modal dispersion.
For a step-index multimode fiber, a rough estimate for the number of guided modes is
\[
M\approx \frac{V^2}{2}.
\]
This is only an approximation, but it gives a quick sense of how modal complexity grows as the core gets larger, the wavelength gets shorter, or the refractive-index contrast gets bigger.
In practice, single-mode fibers are widely used for long-distance communications because they greatly reduce modal dispersion. Multimode fibers can be easier to couple into and are common in shorter-distance systems. More advanced university treatments go beyond the step-index approximation and include graded-index fibers, exact vector modes, polarization effects, and chromatic dispersion, but the V-number remains one of the most important first tools for understanding fiber modal behavior.
