2. Optical Fiber as a Waveguide

Now to the fiber itself. We will derive how light gets trapped, what the maximum acceptance angle is, what numerical aperture really means, why fibers come in single-mode and multimode flavors, and what graded-index profiles are good for.

2.1 Geometry and materials

A telecom fiber is a thin glass strand with three concentric layers:

       buffer / coating  (250 µm OD, polymer)
    ┌──────────────────────────────────────┐
    │   cladding (125 µm OD, low-index SiO2)│
    │   ┌──────────────────────────────┐    │
    │   │                              │    │
    │   │  core (8-9 µm SMF, 50 µm MMF)│    │
    │   │  high-index (Ge-doped SiO2)  │    │
    │   │                              │    │
    │   └──────────────────────────────┘    │
    │                                        │
    └────────────────────────────────────────┘

• Core: high-refractive-index silica, doped with a few percent germanium oxide (GeO$_2$) to bump the index up by about 0.3% relative to the cladding. This is where light propagates.
• Cladding: pure or fluorine-doped silica with a slightly lower refractive index. Crucial: the cladding is not just a coating. It has to be optically continuous with the core, because the wave actually penetrates a fraction of a wavelength into it. A fiber with a perfect core but a scratched cladding is not a working fiber.
• Buffer / coating: an acrylate polymer (typically 250 µm outer diameter) that protects the glass from microcracks, scratches, and bending stress. The buffer is what your fingers touch when you handle a fiber. Inside the buffer, the cladding is what guides the light.

Standard telecom dimensions: 9 µm core for single-mode, 50 or 62.5 µm core for multimode, 125 µm cladding outer diameter, 250 µm coated. The 125 µm number has been the international standard since the 1980s; every connector and splicer in the world is built around it.

2.2 Total internal reflection: the trapping mechanism

We met the critical angle in Chapter 9. Recall: when light goes from a higher-index medium to a lower-index medium, Snell's law

$n_1 \sin\theta_1 = n_2 \sin\theta_2$

forces $\sin\theta_2 = (n_1/n_2)\sin\theta_1$. If $n_1 > n_2$, that ratio is greater than one, so for large enough $\theta_1$ the right side exceeds one and there is no real $\theta_2$. Physically, the wave cannot escape into medium 2. It is totally reflected back into medium 1. The threshold angle, measured from the normal, is

$\theta_c = \arcsin(n_2/n_1).$

For typical fiber, $n_1 = 1.4675$, $n_2 = 1.4625$, giving $\theta_c \approx 85.3°$ from the normal. That is, the ray has to hit the core-cladding boundary at glancing incidence (nearly parallel to the boundary) to be trapped. Almost-parallel rays are exactly the rays that propagate down the fiber's length, which is why the geometry works out.

Whispering-gallery analogy. Stand at one focus of an elliptical dome (St. Paul's Cathedral has a famous one). A whisper at one focus reflects off the curved wall and arrives clearly at the other focus, even tens of meters away, while you cannot hear someone talking five meters away in any other direction. The sound is trapped by the geometry of the wall. Total internal reflection in fiber is the optical equivalent: the cladding boundary is a wall the light cannot cross, so the light hops along the boundary forever, even around gentle bends. The whispering gallery only works for small enough angles between the wall and the sound's path; the fiber only works for small enough angles between the boundary and the ray's path. Both are bound by a critical-angle condition.

The wave does not stop at the boundary; it leaks an evanescent tail a fraction of a wavelength into the cladding. This is why the cladding has to be transparent (any absorption in the cladding shows up as fiber loss) and clean (any scratch in the cladding scatters the evanescent tail).

2.3 Acceptance angle and numerical aperture

When light enters the fiber from air, what is the maximum angle it can come in at and still be trapped?

Picture a ray coming in from air ($n_0 = 1$) at angle $\theta_0$ to the fiber axis. It refracts into the core at angle $\theta_1$ to the axis, where

$\sin\theta_0 = n_1 \sin\theta_1$

(Snell at the entrance face, which we assume is perpendicular to the axis). To be trapped by TIR at the core-cladding boundary, the ray needs to hit that boundary at an angle to the boundary normal greater than $\theta_c = \arcsin(n_2/n_1)$. The angle between the ray and the boundary normal equals $90° - \theta_1$, so the trapping condition is

$90° - \theta_1 \geq \theta_c \quad\Rightarrow\quad \theta_1 \leq 90° - \theta_c.$

Now use $\sin(90° - \theta_c) = \cos\theta_c = \sqrt{1 - \sin^2\theta_c} = \sqrt{1 - (n_2/n_1)^2}$.

Plug into the entrance Snell relation:

$\sin\theta_0 = n_1 \sin\theta_1 \leq n_1 \sqrt{1 - (n_2/n_1)^2} = \sqrt{n_1^2 - n_2^2}.$

The largest $\theta_0$ that still gets trapped is the acceptance angle, and its sine is the numerical aperture:

$\boxed{\text{NA} = \sin\theta_a = \sqrt{n_1^2 - n_2^2}}$

For $n_1 = 1.4675, n_2 = 1.4625$, NA $\approx 0.121$, an acceptance half-angle of about $7°$. For a multimode fiber with a larger core-cladding index difference, NA can go up to 0.3, opening a much wider entry cone. Higher NA gathers more light from a divergent source like an LED, but, as we are about to see, more NA also means more modes and more dispersion.

The square-root form is occasionally written using the relative index difference $\Delta = (n_1 - n_2)/n_1$ as $\text{NA} \approx n_1\sqrt{2\Delta}$. The two forms are equivalent for small $\Delta$.

2.4 Modes and the V-number

The ray-optic picture (drawing zigzag rays in the core) is intuitive but incomplete. Light is a wave, not a particle, and a finite-width waveguide can only support certain self-consistent field patterns called modes. Each mode is a solution of Maxwell's equations that satisfies the boundary conditions at the core-cladding interface and propagates without changing shape down the fiber. Modes are the equivalent of standing-wave patterns on a string, but extending in two transverse dimensions (radial and azimuthal) plus one propagation dimension.

How many modes a fiber supports depends on a single dimensionless number, the V-number (or normalized frequency):

$\boxed{V = \frac{2\pi a}{\lambda}\sqrt{n_1^2 - n_2^2} = \frac{2\pi a}{\lambda}\,\text{NA}}$

where $a$ is the core radius and $\lambda$ is the operating wavelength in vacuum.

The cutoff for the second-lowest mode (the $\text{LP}_{11}$ mode) is at $V = 2.405$, the first zero of the Bessel function $J_0$. So:

• $V < 2.405$: the fiber supports only the lowest mode, called $\text{LP}_{01}$. Single-mode operation.
• $V > 2.405$: multiple modes propagate. Multimode. The total number of modes is approximately $V^2/2$ for step-index fiber.

A standard 9 µm core SMF at 1550 nm with NA = 0.13 gives $V = 2\pi \cdot 4.5 \cdot 10^{-6} \cdot 0.13 / (1.55 \cdot 10^{-6}) \approx 2.37$, just barely below 2.405. Single-mode. The same fiber at 850 nm would give $V \approx 4.32$, and would support multiple modes. This is why a fiber labeled "single-mode" is single-mode only at certain operating wavelengths; below the cutoff wavelength (typically 1260 nm for telecom SMF), it goes multimode.

A standard 50 µm core multimode fiber at 850 nm with NA = 0.2 gives $V \approx 36.9$, supporting about 680 modes.

2.5 Step-index vs graded-index profiles

The simplest fiber has a step in refractive index between core and cladding:

   n(r)
    │
n_1 ├──────────┐
    │          │
n_2 │          └──────────────
    │          a              r

This is a step-index fiber. In multimode operation, different modes correspond to rays bouncing at different angles, which means different path lengths down the fiber. A high-angle mode zigzags more, traveling a longer total path than a low-angle axial mode, and so arrives later. A pulse launched into the fiber spreads out in time at the receiver: modal dispersion.

For step-index multimode fiber, the spread is roughly $L \cdot (n_1 - n_2)/c$ per kilometer, which works out to several nanoseconds per kilometer. That limits the bit rate to maybe 20 MHz·km. Hopeless for 1 Gbps over 1 km.

The graded-index fix is gorgeous. Vary the index continuously from $n_1$ at the center to $n_2$ at the cladding boundary, with a near-parabolic profile:

   n(r)
    │
n_1 ├─╮
    │  ╲
    │   ╲
    │    ╲___
n_2 │        ╲___________
    │         a           r

The trick: high-angle rays now sweep through outer regions where the refractive index is lower, which means they travel faster there (phase velocity is $c/n$). The longer geometric path is compensated by the higher speed in the low-index periphery. With the right profile (close to parabolic, actually a slightly modified power law), all modes arrive nearly simultaneously, and modal dispersion drops by roughly $(n_1 - n_2)/(8 n_1)$, two orders of magnitude better than the step-index case.

Modern graded-index multimode fibers (OM3, OM4, OM5) routinely run 10 Gbps over 300 m and 100 Gbps over 100 m. Datacenter racks live on graded-index fiber.

The naming convention of OM grades:

2.6 Single-mode fiber: the long-haul standard

When $V < 2.405$, only the $\text{LP}_{01}$ mode propagates. There is no modal dispersion (only one mode). The intensity profile in the core is approximately Gaussian, with a characteristic width called the mode field diameter (MFD), slightly larger than the physical core diameter because the field tail extends into the cladding. For standard SMF at 1550 nm, MFD is about 10.4 µm even though the core is 9 µm.

Standard single-mode fiber types:

• G.652 (standard SMF). Zero dispersion at 1310 nm, low loss at 1550 nm. The dominant fiber type in the world.
• G.652.D (low-water-peak SMF). Same as G.652 but with eliminated OH absorption peak around 1383 nm, giving a continuous low-loss band from 1260 to 1625 nm.
• G.653 (dispersion-shifted fiber, DSF). Zero-dispersion shifted to 1550 nm. Less common because it has trouble with WDM nonlinearities.
• G.655 (non-zero dispersion-shifted fiber, NZDSF). A small finite dispersion at 1550 nm, just enough to suppress the WDM nonlinearities that plague DSF.
• G.657 (bend-insensitive fiber). Tighter mode confinement, used in FTTH and inside data centers where the fiber gets bent around tight corners.

Loss specs: about 0.35 dB/km at 1310 nm, 0.20 dB/km at 1550 nm, with the lowest commercial fibers (e.g., Corning SMF-28 ULL) hitting 0.15 dB/km. Trans-Atlantic submarine cables use ULL fiber to stretch the EDFA spacing.

2.7 LP mode notation

The "LP" stands for linearly polarized modes. They are an approximation of the exact vector modes of a fiber when the index difference $n_1 - n_2$ is small (which is almost always the case in telecom fiber, where $\Delta < 1\%$). The exact modes are HE/EH/TE/TM modes — full Maxwell solutions with polarization mixed in — but for small $\Delta$ they pair up into nearly degenerate groups that look like linearly polarized x or y waves with a particular spatial pattern.

The two most important modes:

• $\text{LP}_{01}$: a Gaussian-like single bright spot in the core, the only propagating mode in SMF.
• $\text{LP}_{11}$: a two-lobe pattern (one positive, one negative lobe across the core diameter).

In the few-mode fibers used for mode-division multiplexing experiments, multiple LP modes are launched independently and recovered with multi-input multi-output (MIMO) DSP at the receiver, similar to MIMO in Wi-Fi.