A step-index optical fiber guides light because the refractive index of the core is slightly larger than the refractive index of the cladding.
If the core index is \(n_1\) and the cladding index is \(n_2\), with \(n_1>n_2\), then light entering within a certain range of angles can be trapped by total internal reflection and propagate along the fiber.
Three of the most important quantities used to describe this behavior are the numerical aperture, the normalized frequency or V-number, and the approximate number of guided modes.
The numerical aperture, abbreviated \(NA\), measures the light-gathering ability of the fiber.
For a step-index fiber in air, it is
\[
NA=\sqrt{n_1^2-n_2^2}.
\]
This comes from the total-internal-reflection condition together with Snell’s law.
If the surrounding medium is air, with refractive index approximately 1, then the acceptance half-angle \(\theta_a\) satisfies
\[
\sin\theta_a = NA,
\qquad
\theta_a=\sin^{-1}(NA).
\]
This acceptance angle determines the cone of input rays that can be coupled into the guided modes of the fiber.
A larger numerical aperture means a wider acceptance cone and easier light coupling.
The second key quantity is the normalized frequency or V-number,
\[
V=\frac{2\pi a}{\lambda}NA,
\]
where \(a\) is the core radius and \(\lambda\) is the free-space wavelength.
The V-number controls how many guided modes the fiber can support.
In a step-index fiber, the single-mode cutoff occurs at
\[
V < 2.405.
\]
If the V-number is below this cutoff, the fiber supports only the fundamental guided mode.
If the V-number is larger, the fiber becomes multimode.
This is one of the most important design rules in fiber optics because it links geometry, wavelength, and refractive indices in a single compact formula.
For a multimode step-index fiber, the approximate number of guided modes is
\[
M \approx \frac{V^2}{2}.
\]
This is an approximation, but it is widely used for quick estimates.
It shows immediately that the number of modes grows rapidly with core size and numerical aperture, and decreases as the wavelength increases.
Short wavelengths and large cores therefore tend to produce strongly multimode propagation.
Another useful derived quantity is the cutoff wavelength, found by setting \(V=2.405\):
\[
\lambda_c=\frac{2\pi a\,NA}{2.405}.
\]
For wavelengths longer than \(\lambda_c\), a step-index fiber of fixed geometry operates in the single-mode regime.
For wavelengths shorter than \(\lambda_c\), the same fiber becomes multimode.
This is why a fiber can be single-mode at one wavelength and multimode at another.
The sample input in this calculator is
\(n_1=1.46\),
\(n_2=1.45\),
\(a=25\,\mu\text{m}\),
and
\(\lambda=1550\,\text{nm}\).
The numerical aperture is
\[
NA=\sqrt{1.46^2-1.45^2}\approx 0.171.
\]
Then the V-number is
\[
V=\frac{2\pi(25\times10^{-6})}{1550\times10^{-9}}(0.171)\approx 21.9.
\]
Since this is much larger than \(2.405\), the fiber is multimode.
The approximate mode count is
\[
M\approx \frac{V^2}{2}\approx \frac{(21.9)^2}{2}\approx 240.
\]
So the sample fiber supports on the order of a few hundred guided modes.
This is typical of a large-core multimode fiber.
It is also common to define the fractional index difference
\[
\Delta=\frac{n_1-n_2}{n_1},
\]
which measures how weakly or strongly guiding the fiber is.
In weakly guiding fibers, \(\Delta\) is small, and the usual LP-mode approximations become very accurate.
That is the regime most often discussed in introductory optical-fiber theory.
At a more advanced university level, one studies graded-index fibers, vector modes, polarization effects, material dispersion, waveguide dispersion, and bend losses.
In graded-index fibers, the simple multimode estimate \(V^2/2\) is replaced by different approximations, and the ray picture becomes less direct.
But for a basic step-index fiber, the three central quantities remain
the numerical aperture,
the V-number,
and the cutoff condition.
\[
NA=\sqrt{n_1^2-n_2^2},
\qquad
V=\frac{2\pi a}{\lambda}NA,
\qquad
V < 2.405 \text{ for single-mode operation}.
\]
These formulas let you estimate the acceptance cone, decide whether the fiber is single-mode or multimode, and predict roughly how many guided modes it can support.
