19 Jun 2025

Optical Fibres: Numerical Aperture, Index Profiles, and Modes

Acceptance cone, numerical aperture, step and graded indices, and single- and multimode fibres.

bsc semester-v electromagnetic-theory mj-8 unit-ii optical-fibre numerical-aperture

An optical fibre is a cylindrical dielectric waveguide. Its core index $n_1$ exceeds its cladding index $n_2$, so a suitable field is confined by total internal reflection and by the corresponding evanescent cladding field.

Acceptance angle and numerical aperture

Let the external medium have index $n_0$, and let the largest accepted meridional ray enter at half-angle $\theta_a$ to the fibre axis. Refraction at the flat input face gives

\[n_0\sin\theta_a=n_1\sin r,\]

where $r$ is the ray angle inside the core relative to the axis. At the core-cladding boundary the incidence angle to the normal is $90^\circ-r$. At the limiting accepted ray this equals the critical angle $\theta_c$:

\[\sin\theta_c=\frac{n_2}{n_1}.\]

Therefore

\[\sin r_{\max}=\cos\theta_c =\sqrt{1-\frac{n_2^2}{n_1^2}}.\]

Substitution at the input face gives

\[n_0\sin\theta_a =n_1\sqrt{1-\frac{n_2^2}{n_1^2}} =\sqrt{n_1^2-n_2^2}.\]

The numerical aperture is

\[\boxed{\mathrm{NA}\equiv n_0\sin\theta_a =\sqrt{n_1^2-n_2^2}}.\]

NA is dimensionless. In air, $n_0\simeq1$, so $\theta_a=\sin^{-1}(\mathrm{NA})$.

Define the exact relative index parameter

\[\boxed{\Delta=\frac{n_1^2-n_2^2}{2n_1^2}} \simeq\frac{n_1-n_2}{n_1}\qquad(\Delta\ll1).\]

It follows exactly from this definition that

\[\boxed{\mathrm{NA}=n_1\sqrt{2\Delta}}.\]

The acceptance cone follows from Snell's law and the core-cladding critical-angle condition. Editable TikZ source.

Step-index fibre

For core radius $a$, an ideal step-index profile is

\[\boxed{n(r)=\begin{cases} n_1,&0\le r<a,\\ n_2,&r\ge a. \end{cases}}\]

The abrupt boundary produces total internal reflection in the ray picture. In the wave picture, core solutions are oscillatory Bessel functions and cladding solutions decay exponentially; continuity of tangential $\mathbf E$ and $\mathbf H$ selects the allowed propagation constants.

Graded-index fibre

In a graded-index fibre the core index decreases continuously away from the axis. A common model is

\[\boxed{n^2(r)=n_1^2\left[1-2\Delta\left(\frac ra\right)^g\right], \quad 0\le r<a},\]

with $n(r)=n_2$ in the cladding. The exact definition above ensures $n^2(a)=n_1^2(1-2\Delta)=n_2^2$, so the ideal profile joins continuously at the core boundary. The exponent $g$ fixes the profile; $g=2$ is approximately parabolic. Rays bend continuously toward the high-index axis. The parabolic profile reduces intermodal transit-time differences because rays travelling farther from the axis also travel through lower-index, higher-speed regions.

Normalized frequency and guided modes

The fibre normalized frequency is

\[\boxed{V=\frac{2\pi a}{\lambda_0}\mathrm{NA} =\frac{2\pi a}{\lambda_0}\sqrt{n_1^2-n_2^2}}.\]

$V$ is dimensionless. It combines core size, vacuum wavelength, and index contrast.

A weakly guiding step-index fibre is single mode when

\[\boxed{V<2.405}.\]

The fundamental $\mathrm{LP}_{01}$ mode then propagates, while the next mode is below cutoff. A fibre with $V>2.405$ can support several modes and is called multimode. For large $V$, the approximate number of guided modes including polarization degeneracy is

\[\boxed{M\simeq\frac{V^2}{2}\quad\text{(step index)}}.\]

For an approximately parabolic graded-index fibre,

\[\boxed{M\simeq\frac{V^2}{4}\quad\text{(graded index)}}.\]

Single-mode fibres avoid intermodal dispersion; multimode fibres accept a larger family of spatial field patterns. These classifications concern transverse guided modes, not optical frequency components.

The exact numerical-aperture identity and its weak-guidance expansion are checked in the Unit II Maxima worksheet; every printed residual is zero.